LMFDB - Embedded newform 4013.2.a.b.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4013,2,Mod(1,4013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4013, 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("4013.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4013 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4013.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0439663311$
Analytic rank:	$1$
Dimension:	$157$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34257 q^{2} -1.48585 q^{3} +3.48762 q^{4} +0.440014 q^{5} +3.48071 q^{6} +1.63946 q^{7} -3.48485 q^{8} -0.792243 q^{9} +O(q^{10})$	\(q-2.34257 q^{2} -1.48585 q^{3} +3.48762 q^{4} +0.440014 q^{5} +3.48071 q^{6} +1.63946 q^{7} -3.48485 q^{8} -0.792243 q^{9} -1.03076 q^{10} +0.186392 q^{11} -5.18209 q^{12} -2.90368 q^{13} -3.84054 q^{14} -0.653796 q^{15} +1.18826 q^{16} -5.04104 q^{17} +1.85588 q^{18} +4.73345 q^{19} +1.53460 q^{20} -2.43599 q^{21} -0.436636 q^{22} +1.76210 q^{23} +5.17798 q^{24} -4.80639 q^{25} +6.80206 q^{26} +5.63471 q^{27} +5.71781 q^{28} +5.10553 q^{29} +1.53156 q^{30} +10.4307 q^{31} +4.18613 q^{32} -0.276951 q^{33} +11.8090 q^{34} +0.721385 q^{35} -2.76304 q^{36} -10.0480 q^{37} -11.0884 q^{38} +4.31444 q^{39} -1.53339 q^{40} +6.81696 q^{41} +5.70647 q^{42} -8.15336 q^{43} +0.650066 q^{44} -0.348598 q^{45} -4.12783 q^{46} +7.87539 q^{47} -1.76558 q^{48} -4.31218 q^{49} +11.2593 q^{50} +7.49025 q^{51} -10.1269 q^{52} -9.79790 q^{53} -13.1997 q^{54} +0.0820153 q^{55} -5.71327 q^{56} -7.03321 q^{57} -11.9601 q^{58} +4.65600 q^{59} -2.28019 q^{60} -10.1428 q^{61} -24.4345 q^{62} -1.29885 q^{63} -12.1828 q^{64} -1.27766 q^{65} +0.648777 q^{66} -9.21424 q^{67} -17.5813 q^{68} -2.61822 q^{69} -1.68989 q^{70} -5.86442 q^{71} +2.76085 q^{72} +6.59087 q^{73} +23.5382 q^{74} +7.14158 q^{75} +16.5085 q^{76} +0.305582 q^{77} -10.1069 q^{78} +15.0728 q^{79} +0.522852 q^{80} -5.99562 q^{81} -15.9692 q^{82} +14.6554 q^{83} -8.49582 q^{84} -2.21813 q^{85} +19.0998 q^{86} -7.58607 q^{87} -0.649550 q^{88} -9.27492 q^{89} +0.816615 q^{90} -4.76045 q^{91} +6.14553 q^{92} -15.4984 q^{93} -18.4486 q^{94} +2.08279 q^{95} -6.21997 q^{96} +4.50666 q^{97} +10.1016 q^{98} -0.147668 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9}+O(q^{10}) $ 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9} - 61 q^{10} - 27 q^{11} - 93 q^{12} - 97 q^{13} - 12 q^{14} - 36 q^{15} + 105 q^{16} - 45 q^{17} - 68 q^{18} - 128 q^{19} - 30 q^{20} - 26 q^{21} - 68 q^{22} - 41 q^{23} - 40 q^{24} + 102 q^{25} - 5 q^{26} - 189 q^{27} - 115 q^{28} - 26 q^{29} - 12 q^{30} - 88 q^{31} - 89 q^{32} - 52 q^{33} - 61 q^{34} - 87 q^{35} + 110 q^{36} - 62 q^{37} - 37 q^{38} - 20 q^{39} - 161 q^{40} - 34 q^{41} - 53 q^{42} - 254 q^{43} - 19 q^{44} - 46 q^{45} - 52 q^{46} - 76 q^{47} - 162 q^{48} + 96 q^{49} - 54 q^{50} - 76 q^{51} - 259 q^{52} - 48 q^{53} - 12 q^{54} - 194 q^{55} - 10 q^{56} - 30 q^{57} - 52 q^{58} - 64 q^{59} - 31 q^{60} - 107 q^{61} - 51 q^{62} - 106 q^{63} + 54 q^{64} - 17 q^{65} - 13 q^{66} - 193 q^{67} - 118 q^{68} - 55 q^{69} - 86 q^{70} - 11 q^{71} - 172 q^{72} - 173 q^{73} - 11 q^{74} - 209 q^{75} - 213 q^{76} - 84 q^{77} - 30 q^{78} - 111 q^{79} - 6 q^{80} + 157 q^{81} - 117 q^{82} - 154 q^{83} - 6 q^{84} - 91 q^{85} + 28 q^{86} - 165 q^{87} - 165 q^{88} - 32 q^{89} - 103 q^{90} - 200 q^{91} - 86 q^{92} - 39 q^{93} - 118 q^{94} - 22 q^{95} - 28 q^{96} - 151 q^{97} - 38 q^{98} - 91 q^{99}+O(q^{100}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9 - 61 * q^10 - 27 * q^11 - 93 * q^12 - 97 * q^13 - 12 * q^14 - 36 * q^15 + 105 * q^16 - 45 * q^17 - 68 * q^18 - 128 * q^19 - 30 * q^20 - 26 * q^21 - 68 * q^22 - 41 * q^23 - 40 * q^24 + 102 * q^25 - 5 * q^26 - 189 * q^27 - 115 * q^28 - 26 * q^29 - 12 * q^30 - 88 * q^31 - 89 * q^32 - 52 * q^33 - 61 * q^34 - 87 * q^35 + 110 * q^36 - 62 * q^37 - 37 * q^38 - 20 * q^39 - 161 * q^40 - 34 * q^41 - 53 * q^42 - 254 * q^43 - 19 * q^44 - 46 * q^45 - 52 * q^46 - 76 * q^47 - 162 * q^48 + 96 * q^49 - 54 * q^50 - 76 * q^51 - 259 * q^52 - 48 * q^53 - 12 * q^54 - 194 * q^55 - 10 * q^56 - 30 * q^57 - 52 * q^58 - 64 * q^59 - 31 * q^60 - 107 * q^61 - 51 * q^62 - 106 * q^63 + 54 * q^64 - 17 * q^65 - 13 * q^66 - 193 * q^67 - 118 * q^68 - 55 * q^69 - 86 * q^70 - 11 * q^71 - 172 * q^72 - 173 * q^73 - 11 * q^74 - 209 * q^75 - 213 * q^76 - 84 * q^77 - 30 * q^78 - 111 * q^79 - 6 * q^80 + 157 * q^81 - 117 * q^82 - 154 * q^83 - 6 * q^84 - 91 * q^85 + 28 * q^86 - 165 * q^87 - 165 * q^88 - 32 * q^89 - 103 * q^90 - 200 * q^91 - 86 * q^92 - 39 * q^93 - 118 * q^94 - 22 * q^95 - 28 * q^96 - 151 * q^97 - 38 * q^98 - 91 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.34257	−1.65645	−0.828223	−	0.560399i	$-0.810648\pi$
$2$	−2.34257	−1.65645	−0.828223	+	0.560399i	$0.810648\pi$
$3$	−1.48585	−0.857857	−0.428929	−	0.903338i	$-0.641109\pi$
$3$	−1.48585	−0.857857	−0.428929	+	0.903338i	$0.641109\pi$
$4$	3.48762	1.74381
$5$	0.440014	0.196780	0.0983902	−	0.995148i	$-0.468631\pi$
$5$	0.440014	0.196780	0.0983902	+	0.995148i	$0.468631\pi$
$6$	3.48071	1.42099
$7$	1.63946	0.619657	0.309828	−	0.950793i	$-0.399728\pi$
$7$	1.63946	0.619657	0.309828	+	0.950793i	$0.399728\pi$
$8$	−3.48485	−1.23208
$9$	−0.792243	−0.264081
$10$	−1.03076	−0.325956
$11$	0.186392	0.0561994	0.0280997	−	0.999605i	$-0.491054\pi$
$11$	0.186392	0.0561994	0.0280997	+	0.999605i	$0.491054\pi$
$12$	−5.18209	−1.49594
$13$	−2.90368	−0.805335	−0.402668	−	0.915346i	$-0.631917\pi$
$13$	−2.90368	−0.805335	−0.402668	+	0.915346i	$0.631917\pi$
$14$	−3.84054	−1.02643
$15$	−0.653796	−0.168810
$16$	1.18826	0.297065
$17$	−5.04104	−1.22263	−0.611316	−	0.791386i	$-0.709360\pi$
$17$	−5.04104	−1.22263	−0.611316	+	0.791386i	$0.709360\pi$
$18$	1.85588	0.437436
$19$	4.73345	1.08593	0.542964	−	0.839756i	$-0.317302\pi$
$19$	4.73345	1.08593	0.542964	+	0.839756i	$0.317302\pi$
$20$	1.53460	0.343148
$21$	−2.43599	−0.531577
$22$	−0.436636	−0.0930912
$23$	1.76210	0.367423	0.183711	−	0.982980i	$-0.441189\pi$
$23$	1.76210	0.367423	0.183711	+	0.982980i	$0.441189\pi$
$24$	5.17798	1.05695
$25$	−4.80639	−0.961277
$26$	6.80206	1.33399
$27$	5.63471	1.08440
$28$	5.71781	1.08056
$29$	5.10553	0.948073	0.474037	−	0.880505i	$-0.342796\pi$
$29$	5.10553	0.948073	0.474037	+	0.880505i	$0.342796\pi$
$30$	1.53156	0.279624
$31$	10.4307	1.87340	0.936702	−	0.350129i	$-0.113862\pi$
$31$	10.4307	1.87340	0.936702	+	0.350129i	$0.113862\pi$
$32$	4.18613	0.740010
$33$	−0.276951	−0.0482110
$34$	11.8090	2.02522
$35$	0.721385	0.121936
$36$	−2.76304	−0.460507
$37$	−10.0480	−1.65188	−0.825942	−	0.563755i	$-0.809356\pi$
$37$	−10.0480	−1.65188	−0.825942	+	0.563755i	$0.809356\pi$
$38$	−11.0884	−1.79878
$39$	4.31444	0.690863
$40$	−1.53339	−0.242450
$41$	6.81696	1.06463	0.532315	−	0.846546i	$-0.321322\pi$
$41$	6.81696	1.06463	0.532315	+	0.846546i	$0.321322\pi$
$42$	5.70647	0.880528
$43$	−8.15336	−1.24338	−0.621688	−	0.783265i	$-0.713553\pi$
$43$	−8.15336	−1.24338	−0.621688	+	0.783265i	$0.713553\pi$
$44$	0.650066	0.0980011
$45$	−0.348598	−0.0519660
$46$	−4.12783	−0.608616
$47$	7.87539	1.14874	0.574372	−	0.818594i	$-0.305246\pi$
$47$	7.87539	1.14874	0.574372	+	0.818594i	$0.305246\pi$
$48$	−1.76558	−0.254840
$49$	−4.31218	−0.616026
$50$	11.2593	1.59230
$51$	7.49025	1.04884
$52$	−10.1269	−1.40435
$53$	−9.79790	−1.34585	−0.672923	−	0.739713i	$-0.734961\pi$
$53$	−9.79790	−1.34585	−0.672923	+	0.739713i	$0.734961\pi$
$54$	−13.1997	−1.79625
$55$	0.0820153	0.0110589
$56$	−5.71327	−0.763468
$57$	−7.03321	−0.931572
$58$	−11.9601	−1.57043
$59$	4.65600	0.606160	0.303080	−	0.952965i	$-0.401985\pi$
$59$	4.65600	0.606160	0.303080	+	0.952965i	$0.401985\pi$
$60$	−2.28019	−0.294372
$61$	−10.1428	−1.29865	−0.649324	−	0.760512i	$-0.724948\pi$
$61$	−10.1428	−1.29865	−0.649324	+	0.760512i	$0.724948\pi$
$62$	−24.4345	−3.10319
$63$	−1.29885	−0.163640
$64$	−12.1828	−1.52285
$65$	−1.27766	−0.158474
$66$	0.648777	0.0798589
$67$	−9.21424	−1.12570	−0.562850	−	0.826559i	$-0.690295\pi$
$67$	−9.21424	−1.12570	−0.562850	+	0.826559i	$0.690295\pi$
$68$	−17.5813	−2.13204
$69$	−2.61822	−0.315196
$70$	−1.68989	−0.201981
$71$	−5.86442	−0.695979	−0.347990	−	0.937498i	$-0.613136\pi$
$71$	−5.86442	−0.695979	−0.347990	+	0.937498i	$0.613136\pi$
$72$	2.76085	0.325369
$73$	6.59087	0.771403	0.385701	−	0.922624i	$-0.373960\pi$
$73$	6.59087	0.771403	0.385701	+	0.922624i	$0.373960\pi$
$74$	23.5382	2.73626
$75$	7.14158	0.824639
$76$	16.5085	1.89365
$77$	0.305582	0.0348243
$78$	−10.1069	−1.14438
$79$	15.0728	1.69582	0.847911	−	0.530139i	$-0.177860\pi$
$79$	15.0728	1.69582	0.847911	+	0.530139i	$0.177860\pi$
$80$	0.522852	0.0584566
$81$	−5.99562	−0.666180
$82$	−15.9692	−1.76350
$83$	14.6554	1.60864	0.804319	−	0.594198i	$-0.202531\pi$
$83$	14.6554	1.60864	0.804319	+	0.594198i	$0.202531\pi$
$84$	−8.49582	−0.926970
$85$	−2.21813	−0.240590
$86$	19.0998	2.05958
$87$	−7.58607	−0.813312
$88$	−0.649550	−0.0692422
$89$	−9.27492	−0.983139	−0.491570	−	0.870838i	$-0.663577\pi$
$89$	−9.27492	−0.983139	−0.491570	+	0.870838i	$0.663577\pi$
$90$	0.816615	0.0860788
$91$	−4.76045	−0.499031
$92$	6.14553	0.640716
$93$	−15.4984	−1.60711
$94$	−18.4486	−1.90283
$95$	2.08279	0.213690
$96$	−6.21997	−0.634823
$97$	4.50666	0.457582	0.228791	−	0.973476i	$-0.426523\pi$
$97$	4.50666	0.457582	0.228791	+	0.973476i	$0.426523\pi$
$98$	10.1016	1.02041
$99$	−0.147668	−0.0148412
$100$	−16.7629	−1.67629
$101$	5.11405	0.508867	0.254434	−	0.967090i	$-0.418111\pi$
$101$	5.11405	0.508867	0.254434	+	0.967090i	$0.418111\pi$
$102$	−17.5464	−1.73735
$103$	2.17153	0.213967	0.106984	−	0.994261i	$-0.465881\pi$
$103$	2.17153	0.213967	0.106984	+	0.994261i	$0.465881\pi$
$104$	10.1189	0.992239
$105$	−1.07187	−0.104604
$106$	22.9522	2.22932
$107$	14.6241	1.41377	0.706884	−	0.707330i	$-0.250100\pi$
$107$	14.6241	1.41377	0.706884	+	0.707330i	$0.250100\pi$
$108$	19.6517	1.89099
$109$	−12.3420	−1.18215	−0.591075	−	0.806617i	$-0.701296\pi$
$109$	−12.3420	−1.18215	−0.591075	+	0.806617i	$0.701296\pi$
$110$	−0.192126	−0.0183185
$111$	14.9299	1.41708
$112$	1.94810	0.184078
$113$	−0.421195	−0.0396227	−0.0198113	−	0.999804i	$-0.506307\pi$
$113$	−0.421195	−0.0396227	−0.0198113	+	0.999804i	$0.506307\pi$
$114$	16.4758	1.54310
$115$	0.775348	0.0723016
$116$	17.8062	1.65326
$117$	2.30042	0.212674
$118$	−10.9070	−1.00407
$119$	−8.26458	−0.757613
$120$	2.27838	0.207987
$121$	−10.9653	−0.996842
$122$	23.7601	2.15114
$123$	−10.1290	−0.913300
$124$	36.3782	3.26686
$125$	−4.31495	−0.385941
$126$	3.04264	0.271060
$127$	1.43576	0.127403	0.0637015	−	0.997969i	$-0.479709\pi$
$127$	1.43576	0.127403	0.0637015	+	0.997969i	$0.479709\pi$
$128$	20.1668	1.78251
$129$	12.1147	1.06664
$130$	2.99300	0.262504
$131$	−6.54943	−0.572227	−0.286113	−	0.958196i	$-0.592363\pi$
$131$	−6.54943	−0.572227	−0.286113	+	0.958196i	$0.592363\pi$
$132$	−0.965901	−0.0840709
$133$	7.76030	0.672903
$134$	21.5850	1.86466
$135$	2.47936	0.213389
$136$	17.5673	1.50638
$137$	−2.83072	−0.241845	−0.120923	−	0.992662i	$-0.538585\pi$
$137$	−2.83072	−0.241845	−0.120923	+	0.992662i	$0.538585\pi$
$138$	6.13335	0.522105
$139$	1.56989	0.133156	0.0665781	−	0.997781i	$-0.478792\pi$
$139$	1.56989	0.133156	0.0665781	+	0.997781i	$0.478792\pi$
$140$	2.51592	0.212634
$141$	−11.7017	−0.985458
$142$	13.7378	1.15285
$143$	−0.541223	−0.0452593
$144$	−0.941391	−0.0784493
$145$	2.24651	0.186562
$146$	−15.4395	−1.27779
$147$	6.40726	0.528462
$148$	−35.0437	−2.88057
$149$	12.9127	1.05785	0.528923	−	0.848670i	$-0.322596\pi$
$149$	12.9127	1.05785	0.528923	+	0.848670i	$0.322596\pi$
$150$	−16.7296	−1.36597
$151$	−14.0614	−1.14430	−0.572149	−	0.820149i	$-0.693890\pi$
$151$	−14.0614	−1.14430	−0.572149	+	0.820149i	$0.693890\pi$
$152$	−16.4954	−1.33795
$153$	3.99373	0.322874
$154$	−0.715847	−0.0576846
$155$	4.58964	0.368649
$156$	15.0471	1.20473
$157$	−1.19676	−0.0955121	−0.0477560	−	0.998859i	$-0.515207\pi$
$157$	−1.19676	−0.0955121	−0.0477560	+	0.998859i	$0.515207\pi$
$158$	−35.3090	−2.80904
$159$	14.5582	1.15454
$160$	1.84196	0.145619
$161$	2.88888	0.227676
$162$	14.0451	1.10349
$163$	−11.1397	−0.872526	−0.436263	−	0.899819i	$-0.643698\pi$
$163$	−11.1397	−0.872526	−0.436263	+	0.899819i	$0.643698\pi$
$164$	23.7750	1.85651
$165$	−0.121863	−0.00948699
$166$	−34.3312	−2.66462
$167$	−19.2137	−1.48680	−0.743400	−	0.668848i	$-0.766788\pi$
$167$	−19.2137	−1.48680	−0.743400	+	0.668848i	$0.766788\pi$
$168$	8.48907	0.654946
$169$	−4.56866	−0.351435
$170$	5.19612	0.398524
$171$	−3.75004	−0.286773
$172$	−28.4358	−2.16821
$173$	21.4953	1.63425	0.817127	−	0.576457i	$-0.195565\pi$
$173$	21.4953	1.63425	0.817127	+	0.576457i	$0.195565\pi$
$174$	17.7709	1.34721
$175$	−7.87987	−0.595662
$176$	0.221483	0.0166949
$177$	−6.91813	−0.519998
$178$	21.7271	1.62852
$179$	−8.80160	−0.657862	−0.328931	−	0.944354i	$-0.606688\pi$
$179$	−8.80160	−0.657862	−0.328931	+	0.944354i	$0.606688\pi$
$180$	−1.21578	−0.0906188
$181$	24.9196	1.85226	0.926131	−	0.377201i	$-0.123114\pi$
$181$	24.9196	1.85226	0.926131	+	0.377201i	$0.123114\pi$
$182$	11.1517	0.826618
$183$	15.0707	1.11405
$184$	−6.14065	−0.452695
$185$	−4.42127	−0.325059
$186$	36.3061	2.66209
$187$	−0.939611	−0.0687112
$188$	27.4664	2.00319
$189$	9.23787	0.671956
$190$	−4.87907	−0.353965
$191$	17.4429	1.26212	0.631061	−	0.775733i	$-0.282620\pi$
$191$	17.4429	1.26212	0.631061	+	0.775733i	$0.282620\pi$
$192$	18.1018	1.30639
$193$	18.4684	1.32938	0.664692	−	0.747118i	$-0.268563\pi$
$193$	18.4684	1.32938	0.664692	+	0.747118i	$0.268563\pi$
$194$	−10.5572	−0.757960
$195$	1.89841	0.135948
$196$	−15.0392	−1.07423
$197$	−15.5972	−1.11126	−0.555628	−	0.831431i	$-0.687522\pi$
$197$	−15.5972	−1.11126	−0.555628	+	0.831431i	$0.687522\pi$
$198$	0.345922	0.0245836
$199$	0.107169	0.00759698	0.00379849	−	0.999993i	$-0.498791\pi$
$199$	0.107169	0.00759698	0.00379849	+	0.999993i	$0.498791\pi$
$200$	16.7496	1.18437
$201$	13.6910	0.965689
$202$	−11.9800	−0.842911
$203$	8.37030	0.587480
$204$	26.1231	1.82899
$205$	2.99956	0.209498
$206$	−5.08695	−0.354425
$207$	−1.39601	−0.0970293
$208$	−3.45033	−0.239237
$209$	0.882279	0.0610285
$210$	2.51093	0.173271
$211$	2.37391	0.163427	0.0817134	−	0.996656i	$-0.473961\pi$
$211$	2.37391	0.163427	0.0817134	+	0.996656i	$0.473961\pi$
$212$	−34.1714	−2.34690
$213$	8.71367	0.597051
$214$	−34.2580	−2.34183
$215$	−3.58760	−0.244672
$216$	−19.6361	−1.33607
$217$	17.1006	1.16087
$218$	28.9120	1.95817
$219$	−9.79305	−0.661753
$220$	0.286038	0.0192847
$221$	14.6376	0.984629
$222$	−34.9742	−2.34732
$223$	−2.79786	−0.187359	−0.0936793	−	0.995602i	$-0.529863\pi$
$223$	−2.79786	−0.187359	−0.0936793	+	0.995602i	$0.529863\pi$
$224$	6.86298	0.458552
$225$	3.80783	0.253855
$226$	0.986677	0.0656328
$227$	−15.7844	−1.04764	−0.523822	−	0.851828i	$-0.675494\pi$
$227$	−15.7844	−1.04764	−0.523822	+	0.851828i	$0.675494\pi$
$228$	−24.5292	−1.62448
$229$	−0.428956	−0.0283462	−0.0141731	−	0.999900i	$-0.504512\pi$
$229$	−0.428956	−0.0283462	−0.0141731	+	0.999900i	$0.504512\pi$
$230$	−1.81631	−0.119764
$231$	−0.454050	−0.0298743
$232$	−17.7920	−1.16810
$233$	−18.4627	−1.20953	−0.604767	−	0.796402i	$-0.706734\pi$
$233$	−18.4627	−1.20953	−0.604767	+	0.796402i	$0.706734\pi$
$234$	−5.38888	−0.352282
$235$	3.46529	0.226050
$236$	16.2384	1.05703
$237$	−22.3959	−1.45477
$238$	19.3603	1.25494
$239$	−12.7092	−0.822087	−0.411043	−	0.911616i	$-0.634836\pi$
$239$	−12.7092	−0.822087	−0.411043	+	0.911616i	$0.634836\pi$
$240$	−0.776881	−0.0501474
$241$	−4.91411	−0.316546	−0.158273	−	0.987395i	$-0.550593\pi$
$241$	−4.91411	−0.316546	−0.158273	+	0.987395i	$0.550593\pi$
$242$	25.6869	1.65121
$243$	−7.99553	−0.512913
$244$	−35.3741	−2.26460
$245$	−1.89742	−0.121222
$246$	23.7278	1.51283
$247$	−13.7444	−0.874536
$248$	−36.3494	−2.30819
$249$	−21.7757	−1.37998
$250$	10.1081	0.639290
$251$	−1.62177	−0.102365	−0.0511827	−	0.998689i	$-0.516299\pi$
$251$	−1.62177	−0.102365	−0.0511827	+	0.998689i	$0.516299\pi$
$252$	−4.52989	−0.285356
$253$	0.328441	0.0206489
$254$	−3.36336	−0.211036
$255$	3.29582	0.206392
$256$	−22.8764	−1.42978
$257$	11.1772	0.697213	0.348606	−	0.937269i	$-0.386655\pi$
$257$	11.1772	0.697213	0.348606	+	0.937269i	$0.386655\pi$
$258$	−28.3795	−1.76683
$259$	−16.4733	−1.02360
$260$	−4.45599	−0.276349
$261$	−4.04482	−0.250368
$262$	15.3425	0.947862
$263$	−0.285515	−0.0176056	−0.00880281	−	0.999961i	$-0.502802\pi$
$263$	−0.285515	−0.0176056	−0.00880281	+	0.999961i	$0.502802\pi$
$264$	0.965135	0.0593999
$265$	−4.31122	−0.264836
$266$	−18.1790	−1.11463
$267$	13.7812	0.843393
$268$	−32.1358	−1.96301
$269$	5.48568	0.334468	0.167234	−	0.985917i	$-0.446516\pi$
$269$	5.48568	0.334468	0.167234	+	0.985917i	$0.446516\pi$
$270$	−5.80806	−0.353467
$271$	26.2688	1.59571	0.797857	−	0.602847i	$-0.205967\pi$
$271$	26.2688	1.59571	0.797857	+	0.602847i	$0.205967\pi$
$272$	−5.99007	−0.363202
$273$	7.07333	0.428098
$274$	6.63116	0.400603
$275$	−0.895873	−0.0540232
$276$	−9.13135	−0.549643
$277$	−26.4761	−1.59080	−0.795399	−	0.606086i	$-0.792739\pi$
$277$	−26.4761	−1.59080	−0.795399	+	0.606086i	$0.792739\pi$
$278$	−3.67757	−0.220566
$279$	−8.26362	−0.494730
$280$	−2.51392	−0.150236
$281$	8.02940	0.478994	0.239497	−	0.970897i	$-0.423018\pi$
$281$	8.02940	0.478994	0.239497	+	0.970897i	$0.423018\pi$
$282$	27.4119	1.63236
$283$	−16.1944	−0.962660	−0.481330	−	0.876540i	$-0.659846\pi$
$283$	−16.1944	−0.962660	−0.481330	+	0.876540i	$0.659846\pi$
$284$	−20.4529	−1.21366
$285$	−3.09472	−0.183315
$286$	1.26785	0.0749696
$287$	11.1761	0.659705
$288$	−3.31643	−0.195422
$289$	8.41212	0.494831
$290$	−5.26260	−0.309030
$291$	−6.69623	−0.392540
$292$	22.9864	1.34518
$293$	−14.5414	−0.849516	−0.424758	−	0.905307i	$-0.639641\pi$
$293$	−14.5414	−0.849516	−0.424758	+	0.905307i	$0.639641\pi$
$294$	−15.0094	−0.875368
$295$	2.04871	0.119280
$296$	35.0159	2.03526
$297$	1.05027	0.0609427
$298$	−30.2488	−1.75226
$299$	−5.11656	−0.295898
$300$	24.9071	1.43801
$301$	−13.3671	−0.770467
$302$	32.9397	1.89547
$303$	−7.59873	−0.436535
$304$	5.62458	0.322592
$305$	−4.46296	−0.255549
$306$	−9.35558	−0.534823
$307$	13.5684	0.774391	0.387196	−	0.921998i	$-0.373444\pi$
$307$	13.5684	0.774391	0.387196	+	0.921998i	$0.373444\pi$
$308$	1.06575	0.0607270
$309$	−3.22657	−0.183553
$310$	−10.7516	−0.610647
$311$	16.9221	0.959563	0.479782	−	0.877388i	$-0.340716\pi$
$311$	16.9221	0.959563	0.479782	+	0.877388i	$0.340716\pi$
$312$	−15.0352	−0.851199
$313$	−10.9153	−0.616971	−0.308485	−	0.951229i	$-0.599822\pi$
$313$	−10.9153	−0.616971	−0.308485	+	0.951229i	$0.599822\pi$
$314$	2.80350	0.158211
$315$	−0.571512	−0.0322011
$316$	52.5682	2.95719
$317$	−2.31429	−0.129983	−0.0649917	−	0.997886i	$-0.520702\pi$
$317$	−2.31429	−0.129983	−0.0649917	+	0.997886i	$0.520702\pi$
$318$	−34.1036	−1.91244
$319$	0.951631	0.0532811
$320$	−5.36061	−0.299667
$321$	−21.7293	−1.21281
$322$	−6.76740	−0.377133
$323$	−23.8615	−1.32769
$324$	−20.9105	−1.16169
$325$	13.9562	0.774150
$326$	26.0954	1.44529
$327$	18.3384	1.01412
$328$	−23.7561	−1.31171
$329$	12.9114	0.711827
$330$	0.285471	0.0157147
$331$	−15.0614	−0.827849	−0.413924	−	0.910311i	$-0.635842\pi$
$331$	−15.0614	−0.827849	−0.413924	+	0.910311i	$0.635842\pi$
$332$	51.1124	2.80516
$333$	7.96047	0.436231
$334$	45.0093	2.46280
$335$	−4.05440	−0.221516
$336$	−2.89459	−0.157913
$337$	−19.5126	−1.06292	−0.531461	−	0.847083i	$-0.678357\pi$
$337$	−19.5126	−1.06292	−0.531461	+	0.847083i	$0.678357\pi$
$338$	10.7024	0.582134
$339$	0.625833	0.0339906
$340$	−7.73600	−0.419544
$341$	1.94420	0.105284
$342$	8.78473	0.475024
$343$	−18.5458	−1.00138
$344$	28.4133	1.53194
$345$	−1.15205	−0.0620244
$346$	−50.3541	−2.70705
$347$	33.8401	1.81663	0.908317	−	0.418283i	$-0.137368\pi$
$347$	33.8401	1.81663	0.908317	+	0.418283i	$0.137368\pi$
$348$	−26.4573	−1.41826
$349$	27.6042	1.47762	0.738810	−	0.673913i	$-0.235388\pi$
$349$	27.6042	1.47762	0.738810	+	0.673913i	$0.235388\pi$
$350$	18.4591	0.986682
$351$	−16.3614	−0.873306
$352$	0.780261	0.0415881
$353$	−15.0793	−0.802590	−0.401295	−	0.915949i	$-0.631440\pi$
$353$	−15.0793	−0.802590	−0.401295	+	0.915949i	$0.631440\pi$
$354$	16.2062	0.861349
$355$	−2.58043	−0.136955
$356$	−32.3474	−1.71441
$357$	12.2799	0.649923
$358$	20.6183	1.08971
$359$	−5.45716	−0.288018	−0.144009	−	0.989576i	$-0.545999\pi$
$359$	−5.45716	−0.288018	−0.144009	+	0.989576i	$0.545999\pi$
$360$	1.21481	0.0640263
$361$	3.40558	0.179241
$362$	−58.3759	−3.06817
$363$	16.2928	0.855148
$364$	−16.6027	−0.870216
$365$	2.90008	0.151797
$366$	−35.3040	−1.84537
$367$	12.2702	0.640498	0.320249	−	0.947333i	$-0.396233\pi$
$367$	12.2702	0.640498	0.320249	+	0.947333i	$0.396233\pi$
$368$	2.09383	0.109148
$369$	−5.40069	−0.281148
$370$	10.3571	0.538442
$371$	−16.0632	−0.833962
$372$	−54.0527	−2.80250
$373$	−29.2009	−1.51197	−0.755983	−	0.654591i	$-0.772841\pi$
$373$	−29.2009	−1.51197	−0.755983	+	0.654591i	$0.772841\pi$
$374$	2.20110	0.113816
$375$	6.41138	0.331082
$376$	−27.4446	−1.41535
$377$	−14.8248	−0.763517
$378$	−21.6403	−1.11306
$379$	−12.6471	−0.649640	−0.324820	−	0.945776i	$-0.605304\pi$
$379$	−12.6471	−0.649640	−0.324820	+	0.945776i	$0.605304\pi$
$380$	7.26398	0.372634
$381$	−2.13333	−0.109294
$382$	−40.8611	−2.09064
$383$	−35.6009	−1.81912	−0.909560	−	0.415572i	$-0.863581\pi$
$383$	−35.6009	−1.81912	−0.909560	+	0.415572i	$0.863581\pi$
$384$	−29.9649	−1.52914
$385$	0.134461	0.00685274
$386$	−43.2634	−2.20205
$387$	6.45945	0.328352
$388$	15.7175	0.797937
$389$	−29.8662	−1.51428	−0.757139	−	0.653254i	$-0.773403\pi$
$389$	−29.8662	−1.51428	−0.757139	+	0.653254i	$0.773403\pi$
$390$	−4.44716	−0.225191
$391$	−8.88281	−0.449223
$392$	15.0273	0.758994
$393$	9.73149	0.490889
$394$	36.5375	1.84073
$395$	6.63225	0.333705
$396$	−0.515010	−0.0258802
$397$	−35.8949	−1.80151	−0.900756	−	0.434325i	$-0.856987\pi$
$397$	−35.8949	−1.80151	−0.900756	+	0.434325i	$0.856987\pi$
$398$	−0.251050	−0.0125840
$399$	−11.5307	−0.577255
$400$	−5.71124	−0.285562
$401$	10.8117	0.539910	0.269955	−	0.962873i	$-0.412991\pi$
$401$	10.8117	0.539910	0.269955	+	0.962873i	$0.412991\pi$
$402$	−32.0721	−1.59961
$403$	−30.2873	−1.50872
$404$	17.8359	0.887368
$405$	−2.63816	−0.131091
$406$	−19.6080	−0.973129
$407$	−1.87287	−0.0928349
$408$	−26.1024	−1.29226
$409$	−27.3699	−1.35336	−0.676678	−	0.736279i	$-0.736581\pi$
$409$	−27.3699	−1.35336	−0.676678	+	0.736279i	$0.736581\pi$
$410$	−7.02667	−0.347022
$411$	4.20604	0.207469
$412$	7.57347	0.373118
$413$	7.63331	0.375611
$414$	3.27025	0.160724
$415$	6.44858	0.316548
$416$	−12.1552	−0.595956
$417$	−2.33262	−0.114229
$418$	−2.06680	−0.101090
$419$	−17.2185	−0.841177	−0.420589	−	0.907251i	$-0.638176\pi$
$419$	−17.2185	−0.841177	−0.420589	+	0.907251i	$0.638176\pi$
$420$	−3.73828	−0.182410
$421$	−29.2414	−1.42514	−0.712569	−	0.701602i	$-0.752469\pi$
$421$	−29.2414	−1.42514	−0.712569	+	0.701602i	$0.752469\pi$
$422$	−5.56105	−0.270708
$423$	−6.23922	−0.303361
$424$	34.1442	1.65819
$425$	24.2292	1.17529
$426$	−20.4124	−0.988982
$427$	−16.6286	−0.804716
$428$	51.0034	2.46534
$429$	0.804177	0.0388260
$430$	8.40419	0.405286
$431$	−7.44970	−0.358839	−0.179420	−	0.983773i	$-0.557422\pi$
$431$	−7.44970	−0.358839	−0.179420	+	0.983773i	$0.557422\pi$
$432$	6.69551	0.322138
$433$	19.6657	0.945073	0.472536	−	0.881311i	$-0.343339\pi$
$433$	19.6657	0.945073	0.472536	+	0.881311i	$0.343339\pi$
$434$	−40.0594	−1.92291
$435$	−3.33798	−0.160044
$436$	−43.0443	−2.06145
$437$	8.34081	0.398995
$438$	22.9409	1.09616
$439$	7.97384	0.380571	0.190285	−	0.981729i	$-0.439059\pi$
$439$	7.97384	0.380571	0.190285	+	0.981729i	$0.439059\pi$
$440$	−0.285811	−0.0136255
$441$	3.41629	0.162681
$442$	−34.2895	−1.63098
$443$	−6.88390	−0.327064	−0.163532	−	0.986538i	$-0.552289\pi$
$443$	−6.88390	−0.327064	−0.163532	+	0.986538i	$0.552289\pi$
$444$	52.0698	2.47112
$445$	−4.08110	−0.193463
$446$	6.55418	0.310349
$447$	−19.1863	−0.907481
$448$	−19.9732	−0.943644
$449$	18.9799	0.895718	0.447859	−	0.894104i	$-0.352187\pi$
$449$	18.9799	0.895718	0.447859	+	0.894104i	$0.352187\pi$
$450$	−8.92009	−0.420497
$451$	1.27063	0.0598315
$452$	−1.46897	−0.0690944
$453$	20.8931	0.981645
$454$	36.9759	1.73537
$455$	−2.09467	−0.0981996
$456$	24.5097	1.14777
$457$	13.3883	0.626278	0.313139	−	0.949707i	$-0.398619\pi$
$457$	13.3883	0.626278	0.313139	+	0.949707i	$0.398619\pi$
$458$	1.00486	0.0469540
$459$	−28.4048	−1.32582
$460$	2.70412	0.126080
$461$	8.56681	0.398996	0.199498	−	0.979898i	$-0.436069\pi$
$461$	8.56681	0.398996	0.199498	+	0.979898i	$0.436069\pi$
$462$	1.06364	0.0494851
$463$	0.230303	0.0107031	0.00535155	−	0.999986i	$-0.498297\pi$
$463$	0.230303	0.0107031	0.00535155	+	0.999986i	$0.498297\pi$
$464$	6.06670	0.281640
$465$	−6.81953	−0.316248
$466$	43.2502	2.00353
$467$	−15.5168	−0.718033	−0.359016	−	0.933331i	$-0.616888\pi$
$467$	−15.5168	−0.718033	−0.359016	+	0.933331i	$0.616888\pi$
$468$	8.02299	0.370863
$469$	−15.1064	−0.697547
$470$	−8.11767	−0.374440
$471$	1.77821	0.0819357
$472$	−16.2255	−0.746838
$473$	−1.51972	−0.0698770
$474$	52.4640	2.40975
$475$	−22.7508	−1.04388
$476$	−28.8237	−1.32113
$477$	7.76232	0.355412
$478$	29.7720	1.36174
$479$	27.0017	1.23374	0.616870	−	0.787065i	$-0.288400\pi$
$479$	27.0017	1.23374	0.616870	+	0.787065i	$0.288400\pi$
$480$	−2.73687	−0.124921
$481$	29.1762	1.33032
$482$	11.5116	0.524341
$483$	−4.29245	−0.195313
$484$	−38.2427	−1.73830
$485$	1.98300	0.0900432
$486$	18.7301	0.849613
$487$	−33.6104	−1.52303	−0.761516	−	0.648146i	$-0.775545\pi$
$487$	−33.6104	−1.52303	−0.761516	+	0.648146i	$0.775545\pi$
$488$	35.3461	1.60004
$489$	16.5519	0.748503
$490$	4.44484	0.200797
$491$	28.7402	1.29703	0.648514	−	0.761203i	$-0.275391\pi$
$491$	28.7402	1.29703	0.648514	+	0.761203i	$0.275391\pi$
$492$	−35.3261	−1.59262
$493$	−25.7372	−1.15915
$494$	32.1972	1.44862
$495$	−0.0649760	−0.00292045
$496$	12.3944	0.556523
$497$	−9.61447	−0.431268
$498$	51.0111	2.28586
$499$	−0.0952938	−0.00426594	−0.00213297	−	0.999998i	$-0.500679\pi$
$499$	−0.0952938	−0.00426594	−0.00213297	+	0.999998i	$0.500679\pi$
$500$	−15.0489	−0.673008
$501$	28.5487	1.27546
$502$	3.79911	0.169563
$503$	−34.1977	−1.52480	−0.762401	−	0.647105i	$-0.775980\pi$
$503$	−34.1977	−1.52480	−0.762401	+	0.647105i	$0.775980\pi$
$504$	4.52630	0.201617
$505$	2.25026	0.100135
$506$	−0.769396	−0.0342038
$507$	6.78835	0.301481
$508$	5.00739	0.222167
$509$	−22.1636	−0.982386	−0.491193	−	0.871051i	$-0.663439\pi$
$509$	−22.1636	−0.982386	−0.491193	+	0.871051i	$0.663439\pi$
$510$	−7.72067	−0.341877
$511$	10.8054	0.478005
$512$	13.2560	0.585840
$513$	26.6717	1.17758
$514$	−26.1833	−1.15489
$515$	0.955504	0.0421045
$516$	42.2515	1.86002
$517$	1.46791	0.0645587
$518$	38.5898	1.69554
$519$	−31.9388	−1.40196
$520$	4.45246	0.195253
$521$	−13.6724	−0.598999	−0.299500	−	0.954096i	$-0.596820\pi$
$521$	−13.6724	−0.598999	−0.299500	+	0.954096i	$0.596820\pi$
$522$	9.47527	0.414721
$523$	11.9272	0.521539	0.260770	−	0.965401i	$-0.416024\pi$
$523$	11.9272	0.521539	0.260770	+	0.965401i	$0.416024\pi$
$524$	−22.8419	−0.997855
$525$	11.7083	0.510993
$526$	0.668839	0.0291628
$527$	−52.5815	−2.29048
$528$	−0.329090	−0.0143218
$529$	−19.8950	−0.865001
$530$	10.0993	0.438686
$531$	−3.68868	−0.160075
$532$	27.0650	1.17342
$533$	−19.7942	−0.857384
$534$	−32.2833	−1.39703
$535$	6.43482	0.278202
$536$	32.1103	1.38695
$537$	13.0779	0.564352
$538$	−12.8506	−0.554028
$539$	−0.803757	−0.0346202
$540$	8.64705	0.372110
$541$	−39.5711	−1.70129	−0.850647	−	0.525737i	$-0.823790\pi$
$541$	−39.5711	−1.70129	−0.850647	+	0.525737i	$0.823790\pi$
$542$	−61.5363	−2.64321
$543$	−37.0269	−1.58898
$544$	−21.1024	−0.904760
$545$	−5.43066	−0.232624
$546$	−16.5698	−0.709120
$547$	11.7641	0.502999	0.251499	−	0.967857i	$-0.419076\pi$
$547$	11.7641	0.502999	0.251499	+	0.967857i	$0.419076\pi$
$548$	−9.87249	−0.421732
$549$	8.03553	0.342948
$550$	2.09864	0.0894864
$551$	24.1668	1.02954
$552$	9.12410	0.388347
$553$	24.7112	1.05083
$554$	62.0222	2.63507
$555$	6.56936	0.278854
$556$	5.47518	0.232199
$557$	−8.09170	−0.342856	−0.171428	−	0.985197i	$-0.554838\pi$
$557$	−8.09170	−0.342856	−0.171428	+	0.985197i	$0.554838\pi$
$558$	19.3581	0.819493
$559$	23.6747	1.00133
$560$	0.857194	0.0362230
$561$	1.39612	0.0589444
$562$	−18.8094	−0.793427
$563$	10.1223	0.426605	0.213302	−	0.976986i	$-0.431578\pi$
$563$	10.1223	0.426605	0.213302	+	0.976986i	$0.431578\pi$
$564$	−40.8110	−1.71845
$565$	−0.185332	−0.00779696
$566$	37.9366	1.59459
$567$	−9.82957	−0.412803
$568$	20.4367	0.857503
$569$	−31.0835	−1.30309	−0.651544	−	0.758610i	$-0.725879\pi$
$569$	−31.0835	−1.30309	−0.651544	+	0.758610i	$0.725879\pi$
$570$	7.24958	0.303651
$571$	5.84051	0.244418	0.122209	−	0.992504i	$-0.461002\pi$
$571$	5.84051	0.244418	0.122209	+	0.992504i	$0.461002\pi$
$572$	−1.88758	−0.0789237
$573$	−25.9176	−1.08272
$574$	−26.1808	−1.09277
$575$	−8.46932	−0.353195
$576$	9.65174	0.402156
$577$	33.2557	1.38445	0.692227	−	0.721679i	$-0.256629\pi$
$577$	33.2557	1.38445	0.692227	+	0.721679i	$0.256629\pi$
$578$	−19.7060	−0.819660
$579$	−27.4413	−1.14042
$580$	7.83497	0.325329
$581$	24.0269	0.996803
$582$	15.6864	0.650221
$583$	−1.82625	−0.0756356
$584$	−22.9682	−0.950431
$585$	1.01222	0.0418500
$586$	34.0642	1.40718
$587$	−31.0783	−1.28274	−0.641369	−	0.767233i	$-0.721633\pi$
$587$	−31.0783	−1.28274	−0.641369	+	0.767233i	$0.721633\pi$
$588$	22.3461	0.921538
$589$	49.3731	2.03438
$590$	−4.79923	−0.197581
$591$	23.1752	0.953299
$592$	−11.9397	−0.490717
$593$	−6.62861	−0.272204	−0.136102	−	0.990695i	$-0.543458\pi$
$593$	−6.62861	−0.272204	−0.136102	+	0.990695i	$0.543458\pi$
$594$	−2.46032	−0.100948
$595$	−3.63653	−0.149083
$596$	45.0345	1.84468
$597$	−0.159237	−0.00651713
$598$	11.9859	0.490139
$599$	−34.5212	−1.41050	−0.705248	−	0.708960i	$-0.749165\pi$
$599$	−34.5212	−1.41050	−0.705248	+	0.708960i	$0.749165\pi$
$600$	−24.8874	−1.01602
$601$	−41.6834	−1.70030	−0.850151	−	0.526539i	$-0.823490\pi$
$601$	−41.6834	−1.70030	−0.850151	+	0.526539i	$0.823490\pi$
$602$	31.3133	1.27624
$603$	7.29992	0.297276
$604$	−49.0408	−1.99544
$605$	−4.82487	−0.196159
$606$	17.8005	0.723097
$607$	2.67405	0.108536	0.0542682	−	0.998526i	$-0.482717\pi$
$607$	2.67405	0.108536	0.0542682	+	0.998526i	$0.482717\pi$
$608$	19.8148	0.803598
$609$	−12.4370	−0.503974
$610$	10.4548	0.423302
$611$	−22.8676	−0.925124
$612$	13.9286	0.563031
$613$	−29.1796	−1.17855	−0.589277	−	0.807931i	$-0.700587\pi$
$613$	−29.1796	−1.17855	−0.589277	+	0.807931i	$0.700587\pi$
$614$	−31.7850	−1.28274
$615$	−4.45690	−0.179720
$616$	−1.06491	−0.0429064
$617$	13.6167	0.548188	0.274094	−	0.961703i	$-0.411622\pi$
$617$	13.6167	0.548188	0.274094	+	0.961703i	$0.411622\pi$
$618$	7.55846	0.304046
$619$	45.1641	1.81530	0.907650	−	0.419728i	$-0.137874\pi$
$619$	45.1641	1.81530	0.907650	+	0.419728i	$0.137874\pi$
$620$	16.0069	0.642854
$621$	9.92891	0.398434
$622$	−39.6411	−1.58946
$623$	−15.2058	−0.609209
$624$	5.12667	0.205231
$625$	22.1333	0.885332
$626$	25.5699	1.02198
$627$	−1.31094	−0.0523537
$628$	−4.17386	−0.166555
$629$	50.6525	2.01965
$630$	1.33881	0.0533393
$631$	−25.1828	−1.00251	−0.501255	−	0.865300i	$-0.667128\pi$
$631$	−25.1828	−1.00251	−0.501255	+	0.865300i	$0.667128\pi$
$632$	−52.5265	−2.08939
$633$	−3.52728	−0.140197
$634$	5.42137	0.215310
$635$	0.631755	0.0250704
$636$	50.7736	2.01330
$637$	12.5212	0.496107
$638$	−2.22926	−0.0882573
$639$	4.64605	0.183795
$640$	8.87368	0.350763
$641$	24.3222	0.960667	0.480334	−	0.877086i	$-0.340516\pi$
$641$	24.3222	0.960667	0.480334	+	0.877086i	$0.340516\pi$
$642$	50.9023	2.00895
$643$	8.20020	0.323384	0.161692	−	0.986841i	$-0.448305\pi$
$643$	8.20020	0.323384	0.161692	+	0.986841i	$0.448305\pi$
$644$	10.0753	0.397024
$645$	5.33064	0.209894
$646$	55.8973	2.19925
$647$	20.9249	0.822645	0.411322	−	0.911490i	$-0.365067\pi$
$647$	20.9249	0.822645	0.411322	+	0.911490i	$0.365067\pi$
$648$	20.8939	0.820789
$649$	0.867842	0.0340658
$650$	−32.6933	−1.28234
$651$	−25.4090	−0.995858
$652$	−38.8509	−1.52152
$653$	37.5690	1.47019	0.735094	−	0.677965i	$-0.237138\pi$
$653$	37.5690	1.47019	0.735094	+	0.677965i	$0.237138\pi$
$654$	−42.9589	−1.67983
$655$	−2.88185	−0.112603
$656$	8.10032	0.316264
$657$	−5.22157	−0.203713
$658$	−30.2458	−1.17910
$659$	−49.3477	−1.92231	−0.961157	−	0.276002i	$-0.910990\pi$
$659$	−49.3477	−1.92231	−0.961157	+	0.276002i	$0.910990\pi$
$660$	−0.425011	−0.0165435
$661$	−27.9095	−1.08555	−0.542777	−	0.839877i	$-0.682627\pi$
$661$	−27.9095	−1.08555	−0.542777	+	0.839877i	$0.682627\pi$
$662$	35.2823	1.37129
$663$	−21.7493	−0.844671
$664$	−51.0719	−1.98197
$665$	3.41464	0.132414
$666$	−18.6479	−0.722593
$667$	8.99644	0.348344
$668$	−67.0100	−2.59270
$669$	4.15721	0.160727
$670$	9.49771	0.366928
$671$	−1.89053	−0.0729832
$672$	−10.1974	−0.393372
$673$	−45.5613	−1.75626	−0.878130	−	0.478422i	$-0.841209\pi$
$673$	−45.5613	−1.75626	−0.878130	+	0.478422i	$0.841209\pi$
$674$	45.7097	1.76067
$675$	−27.0826	−1.04241
$676$	−15.9338	−0.612837
$677$	23.7480	0.912711	0.456356	−	0.889798i	$-0.349155\pi$
$677$	23.7480	0.912711	0.456356	+	0.889798i	$0.349155\pi$
$678$	−1.46606	−0.0563035
$679$	7.38848	0.283544
$680$	7.72987	0.296427
$681$	23.4532	0.898729
$682$	−4.55441	−0.174397
$683$	15.4519	0.591252	0.295626	−	0.955304i	$-0.404472\pi$
$683$	15.4519	0.591252	0.295626	+	0.955304i	$0.404472\pi$
$684$	−13.0787	−0.500078
$685$	−1.24556	−0.0475904
$686$	43.4449	1.65873
$687$	0.637365	0.0243170
$688$	−9.68832	−0.369364
$689$	28.4499	1.08386
$690$	2.69876	0.102740
$691$	17.5575	0.667920	0.333960	−	0.942587i	$-0.391615\pi$
$691$	17.5575	0.667920	0.333960	+	0.942587i	$0.391615\pi$
$692$	74.9673	2.84983
$693$	−0.242095	−0.00919644
$694$	−79.2728	−3.00915
$695$	0.690774	0.0262025
$696$	26.4363	1.00207
$697$	−34.3646	−1.30165
$698$	−64.6648	−2.44760
$699$	27.4329	1.03761
$700$	−27.4820	−1.03872
$701$	19.9988	0.755346	0.377673	−	0.925939i	$-0.376724\pi$
$701$	19.9988	0.755346	0.377673	+	0.925939i	$0.376724\pi$
$702$	38.3276	1.44658
$703$	−47.5618	−1.79383
$704$	−2.27078	−0.0855832
$705$	−5.14890	−0.193919
$706$	35.3243	1.32945
$707$	8.38427	0.315323
$708$	−24.1278	−0.906779
$709$	−0.571766	−0.0214731	−0.0107366	−	0.999942i	$-0.503418\pi$
$709$	−0.571766	−0.0214731	−0.0107366	+	0.999942i	$0.503418\pi$
$710$	6.04483	0.226859
$711$	−11.9413	−0.447834
$712$	32.3217	1.21131
$713$	18.3799	0.688331
$714$	−28.7666	−1.07656
$715$	−0.238146	−0.00890615
$716$	−30.6966	−1.14719
$717$	18.8839	0.705233
$718$	12.7838	0.477086
$719$	16.9911	0.633663	0.316831	−	0.948482i	$-0.397381\pi$
$719$	16.9911	0.633663	0.316831	+	0.948482i	$0.397381\pi$
$720$	−0.414226	−0.0154373
$721$	3.56013	0.132586
$722$	−7.97780	−0.296903
$723$	7.30164	0.271551
$724$	86.9103	3.23000
$725$	−24.5392	−0.911362
$726$	−38.1669	−1.41651
$727$	−8.83263	−0.327584	−0.163792	−	0.986495i	$-0.552373\pi$
$727$	−8.83263	−0.327584	−0.163792	+	0.986495i	$0.552373\pi$
$728$	16.5895	0.614847
$729$	29.8670	1.10619
$730$	−6.79362	−0.251443
$731$	41.1015	1.52019
$732$	52.5607	1.94270
$733$	−33.1760	−1.22538	−0.612692	−	0.790322i	$-0.709913\pi$
$733$	−33.1760	−1.22538	−0.612692	+	0.790322i	$0.709913\pi$
$734$	−28.7437	−1.06095
$735$	2.81929	0.103991
$736$	7.37636	0.271896
$737$	−1.71746	−0.0632636
$738$	12.6515	0.465707
$739$	−28.1265	−1.03465	−0.517325	−	0.855789i	$-0.673072\pi$
$739$	−28.1265	−1.03465	−0.517325	+	0.855789i	$0.673072\pi$
$740$	−15.4197	−0.566841
$741$	20.4222	0.750227
$742$	37.6292	1.38141
$743$	−19.9810	−0.733033	−0.366517	−	0.930412i	$-0.619450\pi$
$743$	−19.9810	−0.733033	−0.366517	+	0.930412i	$0.619450\pi$
$744$	54.0098	1.98009
$745$	5.68176	0.208164
$746$	68.4051	2.50449
$747$	−11.6106	−0.424810
$748$	−3.27701	−0.119819
$749$	23.9756	0.876050
$750$	−15.0191	−0.548420
$751$	5.65565	0.206378	0.103189	−	0.994662i	$-0.467095\pi$
$751$	5.65565	0.206378	0.103189	+	0.994662i	$0.467095\pi$
$752$	9.35802	0.341252
$753$	2.40972	0.0878149
$754$	34.7281	1.26472
$755$	−6.18721	−0.225176
$756$	32.2182	1.17176
$757$	−51.9114	−1.88675	−0.943377	−	0.331723i	$-0.892370\pi$
$757$	−51.9114	−1.88675	−0.943377	+	0.331723i	$0.892370\pi$
$758$	29.6268	1.07609
$759$	−0.488015	−0.0177138
$760$	−7.25821	−0.263283
$761$	5.52644	0.200333	0.100167	−	0.994971i	$-0.468062\pi$
$761$	5.52644	0.200333	0.100167	+	0.994971i	$0.468062\pi$
$762$	4.99746	0.181039
$763$	−20.2342	−0.732527
$764$	60.8342	2.20090
$765$	1.75730	0.0635353
$766$	83.3975	3.01327
$767$	−13.5195	−0.488162
$768$	33.9910	1.22655
$769$	−40.4653	−1.45922	−0.729608	−	0.683865i	$-0.760298\pi$
$769$	−40.4653	−1.45922	−0.729608	+	0.683865i	$0.760298\pi$
$770$	−0.314983	−0.0113512
$771$	−16.6076	−0.598109
$772$	64.4107	2.31819
$773$	−17.0885	−0.614632	−0.307316	−	0.951608i	$-0.599431\pi$
$773$	−17.0885	−0.614632	−0.307316	+	0.951608i	$0.599431\pi$
$774$	−15.1317	−0.543897
$775$	−50.1338	−1.80086
$776$	−15.7051	−0.563779
$777$	24.4769	0.878104
$778$	69.9636	2.50832
$779$	32.2677	1.15611
$780$	6.62095	0.237068
$781$	−1.09308	−0.0391136
$782$	20.8086	0.744113
$783$	28.7682	1.02809
$784$	−5.12399	−0.183000
$785$	−0.526593	−0.0187949
$786$	−22.7967	−0.813130
$787$	−7.88142	−0.280942	−0.140471	−	0.990085i	$-0.544862\pi$
$787$	−7.88142	−0.280942	−0.140471	+	0.990085i	$0.544862\pi$
$788$	−54.3972	−1.93782
$789$	0.424234	0.0151031
$790$	−15.5365	−0.552763
$791$	−0.690531	−0.0245524
$792$	0.514601	0.0182856
$793$	29.4513	1.04585
$794$	84.0862	2.98411
$795$	6.40583	0.227191
$796$	0.373764	0.0132477
$797$	−31.1406	−1.10306	−0.551528	−	0.834156i	$-0.685955\pi$
$797$	−31.1406	−1.10306	−0.551528	+	0.834156i	$0.685955\pi$
$798$	27.0113	0.956191
$799$	−39.7002	−1.40449
$800$	−20.1201	−0.711354
$801$	7.34799	0.259628
$802$	−25.3271	−0.894331
$803$	1.22849	0.0433523
$804$	47.7490	1.68398
$805$	1.27115	0.0448022
$806$	70.9500	2.49911
$807$	−8.15091	−0.286926
$808$	−17.8217	−0.626966
$809$	−30.7407	−1.08079	−0.540393	−	0.841413i	$-0.681724\pi$
$809$	−30.7407	−1.08079	−0.540393	+	0.841413i	$0.681724\pi$
$810$	6.18007	0.217145
$811$	−26.0169	−0.913578	−0.456789	−	0.889575i	$-0.651001\pi$
$811$	−26.0169	−0.913578	−0.456789	+	0.889575i	$0.651001\pi$
$812$	29.1924	1.02445
$813$	−39.0315	−1.36889
$814$	4.38733	0.153776
$815$	−4.90161	−0.171696
$816$	8.90037	0.311575
$817$	−38.5936	−1.35022
$818$	64.1159	2.24176
$819$	3.77144	0.131785
$820$	10.4613	0.365325
$821$	21.5595	0.752432	0.376216	−	0.926532i	$-0.377225\pi$
$821$	21.5595	0.752432	0.376216	+	0.926532i	$0.377225\pi$
$822$	−9.85293	−0.343660
$823$	−17.3170	−0.603633	−0.301816	−	0.953366i	$-0.597593\pi$
$823$	−17.3170	−0.603633	−0.301816	+	0.953366i	$0.597593\pi$
$824$	−7.56746	−0.263625
$825$	1.33114	0.0463442
$826$	−17.8816	−0.622179
$827$	−14.1681	−0.492673	−0.246337	−	0.969184i	$-0.579227\pi$
$827$	−14.1681	−0.492673	−0.246337	+	0.969184i	$0.579227\pi$
$828$	−4.86875	−0.169201
$829$	40.1616	1.39487	0.697434	−	0.716649i	$-0.254325\pi$
$829$	40.1616	1.39487	0.697434	+	0.716649i	$0.254325\pi$
$830$	−15.1062	−0.524345
$831$	39.3396	1.36468
$832$	35.3749	1.22640
$833$	21.7379	0.753173
$834$	5.46432	0.189214
$835$	−8.45430	−0.292573
$836$	3.07705	0.106422
$837$	58.7738	2.03152
$838$	40.3354	1.39336
$839$	−19.1787	−0.662121	−0.331061	−	0.943610i	$-0.607406\pi$
$839$	−19.1787	−0.662121	−0.331061	+	0.943610i	$0.607406\pi$
$840$	3.73532	0.128881
$841$	−2.93355	−0.101157
$842$	68.4999	2.36066
$843$	−11.9305	−0.410908
$844$	8.27931	0.284985
$845$	−2.01028	−0.0691556
$846$	14.6158	0.502502
$847$	−17.9771	−0.617700
$848$	−11.6425	−0.399804
$849$	24.0625	0.825825
$850$	−56.7585	−1.94680
$851$	−17.7056	−0.606940
$852$	30.3900	1.04114
$853$	−32.6038	−1.11633	−0.558167	−	0.829729i	$-0.688495\pi$
$853$	−32.6038	−1.11633	−0.558167	+	0.829729i	$0.688495\pi$
$854$	38.9537	1.33297
$855$	−1.65007	−0.0564313
$856$	−50.9629	−1.74188
$857$	−19.6471	−0.671133	−0.335566	−	0.942017i	$-0.608928\pi$
$857$	−19.6471	−0.671133	−0.335566	+	0.942017i	$0.608928\pi$
$858$	−1.88384	−0.0643132
$859$	4.71539	0.160887	0.0804435	−	0.996759i	$-0.474366\pi$
$859$	4.71539	0.160887	0.0804435	+	0.996759i	$0.474366\pi$
$860$	−12.5122	−0.426662
$861$	−16.6060	−0.565933
$862$	17.4514	0.594398
$863$	31.6357	1.07689	0.538445	−	0.842661i	$-0.319012\pi$
$863$	31.6357	1.07689	0.538445	+	0.842661i	$0.319012\pi$
$864$	23.5876	0.802467
$865$	9.45822	0.321589
$866$	−46.0682	−1.56546
$867$	−12.4992	−0.424494
$868$	59.6406	2.02433
$869$	2.80945	0.0953041
$870$	7.81944	0.265104
$871$	26.7552	0.906565
$872$	43.0101	1.45651
$873$	−3.57037	−0.120839
$874$	−19.5389	−0.660913
$875$	−7.07418	−0.239151
$876$	−34.1545	−1.15397
$877$	−26.5413	−0.896235	−0.448118	−	0.893975i	$-0.647905\pi$
$877$	−26.5413	−0.896235	−0.448118	+	0.893975i	$0.647905\pi$
$878$	−18.6793	−0.630394
$879$	21.6063	0.728764
$880$	0.0974555	0.00328523
$881$	42.3680	1.42741	0.713706	−	0.700445i	$-0.247015\pi$
$881$	42.3680	1.42741	0.713706	+	0.700445i	$0.247015\pi$
$882$	−8.00290	−0.269472
$883$	34.3853	1.15716	0.578579	−	0.815627i	$-0.303608\pi$
$883$	34.3853	1.15716	0.578579	+	0.815627i	$0.303608\pi$
$884$	51.0503	1.71701
$885$	−3.04408	−0.102326
$886$	16.1260	0.541764
$887$	40.7946	1.36975	0.684874	−	0.728661i	$-0.259857\pi$
$887$	40.7946	1.36975	0.684874	+	0.728661i	$0.259857\pi$
$888$	−52.0284	−1.74596
$889$	2.35387	0.0789462
$890$	9.56025	0.320460
$891$	−1.11754	−0.0374389
$892$	−9.75788	−0.326718
$893$	37.2778	1.24745
$894$	44.9452	1.50319
$895$	−3.87283	−0.129454
$896$	33.0626	1.10454
$897$	7.60245	0.253839
$898$	−44.4617	−1.48371
$899$	53.2541	1.77612
$900$	13.2803	0.442675
$901$	49.3916	1.64547
$902$	−2.97653	−0.0991076
$903$	19.8615	0.660950
$904$	1.46780	0.0488184
$905$	10.9650	0.364489
$906$	−48.9436	−1.62604
$907$	0.766312	0.0254450	0.0127225	−	0.999919i	$-0.495950\pi$
$907$	0.766312	0.0254450	0.0127225	+	0.999919i	$0.495950\pi$
$908$	−55.0499	−1.82689
$909$	−4.05157	−0.134382
$910$	4.90690	0.162662
$911$	6.57518	0.217845	0.108923	−	0.994050i	$-0.465260\pi$
$911$	6.57518	0.217845	0.108923	+	0.994050i	$0.465260\pi$
$912$	−8.35729	−0.276738
$913$	2.73165	0.0904044
$914$	−31.3630	−1.03739
$915$	6.63130	0.219224
$916$	−1.49604	−0.0494304
$917$	−10.7375	−0.354584
$918$	66.5402	2.19615
$919$	22.2207	0.732993	0.366496	−	0.930419i	$-0.380557\pi$
$919$	22.2207	0.732993	0.366496	+	0.930419i	$0.380557\pi$
$920$	−2.70198	−0.0890815
$921$	−20.1607	−0.664317
$922$	−20.0683	−0.660915
$923$	17.0284	0.560496
$924$	−1.58355	−0.0520951
$925$	48.2947	1.58792
$926$	−0.539501	−0.0177291
$927$	−1.72038	−0.0565046
$928$	21.3724	0.701583
$929$	−41.9985	−1.37792	−0.688962	−	0.724797i	$-0.741933\pi$
$929$	−41.9985	−1.37792	−0.688962	+	0.724797i	$0.741933\pi$
$930$	15.9752	0.523848
$931$	−20.4115	−0.668960
$932$	−64.3910	−2.10920
$933$	−25.1437	−0.823168
$934$	36.3492	1.18938
$935$	−0.413443	−0.0135210
$936$	−8.01662	−0.262031
$937$	31.5254	1.02989	0.514944	−	0.857224i	$-0.327813\pi$
$937$	31.5254	1.02989	0.514944	+	0.857224i	$0.327813\pi$
$938$	35.3877	1.15545
$939$	16.2186	0.529273
$940$	12.0856	0.394189
$941$	13.5961	0.443220	0.221610	−	0.975135i	$-0.428869\pi$
$941$	13.5961	0.443220	0.221610	+	0.975135i	$0.428869\pi$
$942$	−4.16558	−0.135722
$943$	12.0121	0.391169
$944$	5.53254	0.180069
$945$	4.06480	0.132228
$946$	3.56005	0.115747
$947$	−34.1267	−1.10897	−0.554485	−	0.832194i	$-0.687085\pi$
$947$	−34.1267	−1.10897	−0.554485	+	0.832194i	$0.687085\pi$
$948$	−78.1086	−2.53685
$949$	−19.1377	−0.621238
$950$	53.2953	1.72913
$951$	3.43869	0.111507
$952$	28.8008	0.933441
$953$	−5.94042	−0.192429	−0.0962146	−	0.995361i	$-0.530674\pi$
$953$	−5.94042	−0.192429	−0.0962146	+	0.995361i	$0.530674\pi$
$954$	−18.1838	−0.588721
$955$	7.67512	0.248361
$956$	−44.3247	−1.43356
$957$	−1.41398	−0.0457076
$958$	−63.2534	−2.04362
$959$	−4.64085	−0.149861
$960$	7.96507	0.257072
$961$	77.7988	2.50964
$962$	−68.3472	−2.20360
$963$	−11.5859	−0.373349
$964$	−17.1386	−0.551996
$965$	8.12636	0.261597
$966$	10.0554	0.323526
$967$	19.5837	0.629770	0.314885	−	0.949130i	$-0.398034\pi$
$967$	19.5837	0.629770	0.314885	+	0.949130i	$0.398034\pi$
$968$	38.2123	1.22819
$969$	35.4547	1.13897
$970$	−4.64530	−0.149152
$971$	−45.5929	−1.46315	−0.731573	−	0.681763i	$-0.761213\pi$
$971$	−45.5929	−1.46315	−0.731573	+	0.681763i	$0.761213\pi$
$972$	−27.8854	−0.894424
$973$	2.57377	0.0825111
$974$	78.7346	2.52282
$975$	−20.7368	−0.664111
$976$	−12.0523	−0.385783
$977$	−52.2905	−1.67292	−0.836460	−	0.548028i	$-0.815379\pi$
$977$	−52.2905	−1.67292	−0.836460	+	0.548028i	$0.815379\pi$
$978$	−38.7739	−1.23985
$979$	−1.72877	−0.0552518
$980$	−6.61749	−0.211388
$981$	9.77787	0.312183
$982$	−67.3259	−2.14846
$983$	−9.23211	−0.294459	−0.147229	−	0.989102i	$-0.547036\pi$
$983$	−9.23211	−0.294459	−0.147229	+	0.989102i	$0.547036\pi$
$984$	35.2980	1.12526
$985$	−6.86300	−0.218673
$986$	60.2911	1.92006
$987$	−19.1844	−0.610646
$988$	−47.9353	−1.52503
$989$	−14.3670	−0.456845
$990$	0.152211	0.00483757
$991$	−8.62448	−0.273966	−0.136983	−	0.990573i	$-0.543741\pi$
$991$	−8.62448	−0.273966	−0.136983	+	0.990573i	$0.543741\pi$
$992$	43.6641	1.38634
$993$	22.3790	0.710176
$994$	22.5226	0.714372
$995$	0.0471557	0.00149494
$996$	−75.9455	−2.40643
$997$	−58.3550	−1.84812	−0.924061	−	0.382245i	$-0.875151\pi$
$997$	−58.3550	−1.84812	−0.924061	+	0.382245i	$0.875151\pi$
$998$	0.223232	0.00706629
$999$	−56.6177	−1.79131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4013.2.a.b.1.18	✓	157

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4013.2.a.b.1.18	✓	157	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4013 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4013.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34257 q^{2} -1.48585 q^{3} +3.48762 q^{4} +0.440014 q^{5} +3.48071 q^{6} +1.63946 q^{7} -3.48485 q^{8} -0.792243 q^{9} +O(q^{10})\)	\(q-2.34257 q^{2} -1.48585 q^{3} +3.48762 q^{4} +0.440014 q^{5} +3.48071 q^{6} +1.63946 q^{7} -3.48485 q^{8} -0.792243 q^{9} -1.03076 q^{10} +0.186392 q^{11} -5.18209 q^{12} -2.90368 q^{13} -3.84054 q^{14} -0.653796 q^{15} +1.18826 q^{16} -5.04104 q^{17} +1.85588 q^{18} +4.73345 q^{19} +1.53460 q^{20} -2.43599 q^{21} -0.436636 q^{22} +1.76210 q^{23} +5.17798 q^{24} -4.80639 q^{25} +6.80206 q^{26} +5.63471 q^{27} +5.71781 q^{28} +5.10553 q^{29} +1.53156 q^{30} +10.4307 q^{31} +4.18613 q^{32} -0.276951 q^{33} +11.8090 q^{34} +0.721385 q^{35} -2.76304 q^{36} -10.0480 q^{37} -11.0884 q^{38} +4.31444 q^{39} -1.53339 q^{40} +6.81696 q^{41} +5.70647 q^{42} -8.15336 q^{43} +0.650066 q^{44} -0.348598 q^{45} -4.12783 q^{46} +7.87539 q^{47} -1.76558 q^{48} -4.31218 q^{49} +11.2593 q^{50} +7.49025 q^{51} -10.1269 q^{52} -9.79790 q^{53} -13.1997 q^{54} +0.0820153 q^{55} -5.71327 q^{56} -7.03321 q^{57} -11.9601 q^{58} +4.65600 q^{59} -2.28019 q^{60} -10.1428 q^{61} -24.4345 q^{62} -1.29885 q^{63} -12.1828 q^{64} -1.27766 q^{65} +0.648777 q^{66} -9.21424 q^{67} -17.5813 q^{68} -2.61822 q^{69} -1.68989 q^{70} -5.86442 q^{71} +2.76085 q^{72} +6.59087 q^{73} +23.5382 q^{74} +7.14158 q^{75} +16.5085 q^{76} +0.305582 q^{77} -10.1069 q^{78} +15.0728 q^{79} +0.522852 q^{80} -5.99562 q^{81} -15.9692 q^{82} +14.6554 q^{83} -8.49582 q^{84} -2.21813 q^{85} +19.0998 q^{86} -7.58607 q^{87} -0.649550 q^{88} -9.27492 q^{89} +0.816615 q^{90} -4.76045 q^{91} +6.14553 q^{92} -15.4984 q^{93} -18.4486 q^{94} +2.08279 q^{95} -6.21997 q^{96} +4.50666 q^{97} +10.1016 q^{98} -0.147668 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9}+O(q^{10}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9} - 61 q^{10} - 27 q^{11} - 93 q^{12} - 97 q^{13} - 12 q^{14} - 36 q^{15} + 105 q^{16} - 45 q^{17} - 68 q^{18} - 128 q^{19} - 30 q^{20} - 26 q^{21} - 68 q^{22} - 41 q^{23} - 40 q^{24} + 102 q^{25} - 5 q^{26} - 189 q^{27} - 115 q^{28} - 26 q^{29} - 12 q^{30} - 88 q^{31} - 89 q^{32} - 52 q^{33} - 61 q^{34} - 87 q^{35} + 110 q^{36} - 62 q^{37} - 37 q^{38} - 20 q^{39} - 161 q^{40} - 34 q^{41} - 53 q^{42} - 254 q^{43} - 19 q^{44} - 46 q^{45} - 52 q^{46} - 76 q^{47} - 162 q^{48} + 96 q^{49} - 54 q^{50} - 76 q^{51} - 259 q^{52} - 48 q^{53} - 12 q^{54} - 194 q^{55} - 10 q^{56} - 30 q^{57} - 52 q^{58} - 64 q^{59} - 31 q^{60} - 107 q^{61} - 51 q^{62} - 106 q^{63} + 54 q^{64} - 17 q^{65} - 13 q^{66} - 193 q^{67} - 118 q^{68} - 55 q^{69} - 86 q^{70} - 11 q^{71} - 172 q^{72} - 173 q^{73} - 11 q^{74} - 209 q^{75} - 213 q^{76} - 84 q^{77} - 30 q^{78} - 111 q^{79} - 6 q^{80} + 157 q^{81} - 117 q^{82} - 154 q^{83} - 6 q^{84} - 91 q^{85} + 28 q^{86} - 165 q^{87} - 165 q^{88} - 32 q^{89} - 103 q^{90} - 200 q^{91} - 86 q^{92} - 39 q^{93} - 118 q^{94} - 22 q^{95} - 28 q^{96} - 151 q^{97} - 38 q^{98} - 91 q^{99}+O(q^{100}) \) 157 * q - 15 * q^2 - 51 * q^3 + 137 * q^4 - 13 * q^5 - 15 * q^6 - 49 * q^7 - 42 * q^8 + 144 * q^9 - 61 * q^10 - 27 * q^11 - 93 * q^12 - 97 * q^13 - 12 * q^14 - 36 * q^15 + 105 * q^16 - 45 * q^17 - 68 * q^18 - 128 * q^19 - 30 * q^20 - 26 * q^21 - 68 * q^22 - 41 * q^23 - 40 * q^24 + 102 * q^25 - 5 * q^26 - 189 * q^27 - 115 * q^28 - 26 * q^29 - 12 * q^30 - 88 * q^31 - 89 * q^32 - 52 * q^33 - 61 * q^34 - 87 * q^35 + 110 * q^36 - 62 * q^37 - 37 * q^38 - 20 * q^39 - 161 * q^40 - 34 * q^41 - 53 * q^42 - 254 * q^43 - 19 * q^44 - 46 * q^45 - 52 * q^46 - 76 * q^47 - 162 * q^48 + 96 * q^49 - 54 * q^50 - 76 * q^51 - 259 * q^52 - 48 * q^53 - 12 * q^54 - 194 * q^55 - 10 * q^56 - 30 * q^57 - 52 * q^58 - 64 * q^59 - 31 * q^60 - 107 * q^61 - 51 * q^62 - 106 * q^63 + 54 * q^64 - 17 * q^65 - 13 * q^66 - 193 * q^67 - 118 * q^68 - 55 * q^69 - 86 * q^70 - 11 * q^71 - 172 * q^72 - 173 * q^73 - 11 * q^74 - 209 * q^75 - 213 * q^76 - 84 * q^77 - 30 * q^78 - 111 * q^79 - 6 * q^80 + 157 * q^81 - 117 * q^82 - 154 * q^83 - 6 * q^84 - 91 * q^85 + 28 * q^86 - 165 * q^87 - 165 * q^88 - 32 * q^89 - 103 * q^90 - 200 * q^91 - 86 * q^92 - 39 * q^93 - 118 * q^94 - 22 * q^95 - 28 * q^96 - 151 * q^97 - 38 * q^98 - 91 * q^99

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