LMFDB - Embedded newform 4013.2.a.b.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.81982 q^{2} +1.03159 q^{3} +5.95140 q^{4} +1.63220 q^{5} -2.90891 q^{6} +3.00127 q^{7} -11.1422 q^{8} -1.93581 q^{9} +O(q^{10})$	\(q-2.81982 q^{2} +1.03159 q^{3} +5.95140 q^{4} +1.63220 q^{5} -2.90891 q^{6} +3.00127 q^{7} -11.1422 q^{8} -1.93581 q^{9} -4.60252 q^{10} +4.90998 q^{11} +6.13943 q^{12} -5.47917 q^{13} -8.46305 q^{14} +1.68377 q^{15} +19.5163 q^{16} -6.48207 q^{17} +5.45865 q^{18} -4.81887 q^{19} +9.71388 q^{20} +3.09610 q^{21} -13.8453 q^{22} -0.298932 q^{23} -11.4943 q^{24} -2.33592 q^{25} +15.4503 q^{26} -5.09176 q^{27} +17.8618 q^{28} +7.24040 q^{29} -4.74793 q^{30} -6.64697 q^{31} -32.7481 q^{32} +5.06510 q^{33} +18.2783 q^{34} +4.89868 q^{35} -11.5208 q^{36} +4.71805 q^{37} +13.5883 q^{38} -5.65228 q^{39} -18.1864 q^{40} +2.11712 q^{41} -8.73044 q^{42} -3.63811 q^{43} +29.2212 q^{44} -3.15964 q^{45} +0.842934 q^{46} -4.84762 q^{47} +20.1329 q^{48} +2.00764 q^{49} +6.58687 q^{50} -6.68687 q^{51} -32.6087 q^{52} +2.37433 q^{53} +14.3579 q^{54} +8.01407 q^{55} -33.4409 q^{56} -4.97112 q^{57} -20.4166 q^{58} +7.57648 q^{59} +10.0208 q^{60} -6.90355 q^{61} +18.7433 q^{62} -5.80990 q^{63} +53.3112 q^{64} -8.94311 q^{65} -14.2827 q^{66} -0.975192 q^{67} -38.5774 q^{68} -0.308376 q^{69} -13.8134 q^{70} +9.25846 q^{71} +21.5693 q^{72} -11.2782 q^{73} -13.3040 q^{74} -2.40972 q^{75} -28.6790 q^{76} +14.7362 q^{77} +15.9384 q^{78} +11.5118 q^{79} +31.8546 q^{80} +0.554809 q^{81} -5.96991 q^{82} +10.2684 q^{83} +18.4261 q^{84} -10.5801 q^{85} +10.2588 q^{86} +7.46916 q^{87} -54.7081 q^{88} -14.4094 q^{89} +8.90962 q^{90} -16.4445 q^{91} -1.77906 q^{92} -6.85698 q^{93} +13.6694 q^{94} -7.86536 q^{95} -33.7828 q^{96} -17.5407 q^{97} -5.66118 q^{98} -9.50479 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.81982	−1.99392	−0.996958	−	0.0779447i	$-0.975164\pi$
$2$	−2.81982	−1.99392	−0.996958	+	0.0779447i	$0.975164\pi$
$3$	1.03159	0.595591	0.297796	−	0.954630i	$-0.403749\pi$
$3$	1.03159	0.595591	0.297796	+	0.954630i	$0.403749\pi$
$4$	5.95140	2.97570
$5$	1.63220	0.729943	0.364971	−	0.931019i	$-0.381079\pi$
$5$	1.63220	0.729943	0.364971	+	0.931019i	$0.381079\pi$
$6$	−2.90891	−1.18756
$7$	3.00127	1.13437	0.567187	−	0.823589i	$-0.308032\pi$
$7$	3.00127	1.13437	0.567187	+	0.823589i	$0.308032\pi$
$8$	−11.1422	−3.93938
$9$	−1.93581	−0.645271
$10$	−4.60252	−1.45544
$11$	4.90998	1.48041	0.740207	−	0.672379i	$-0.234728\pi$
$11$	4.90998	1.48041	0.740207	+	0.672379i	$0.234728\pi$
$12$	6.13943	1.77230
$13$	−5.47917	−1.51965	−0.759824	−	0.650129i	$-0.774715\pi$
$13$	−5.47917	−1.51965	−0.759824	+	0.650129i	$0.774715\pi$
$14$	−8.46305	−2.26185
$15$	1.68377	0.434748
$16$	19.5163	4.87908
$17$	−6.48207	−1.57213	−0.786067	−	0.618141i	$-0.787886\pi$
$17$	−6.48207	−1.57213	−0.786067	+	0.618141i	$0.787886\pi$
$18$	5.45865	1.28662
$19$	−4.81887	−1.10552	−0.552762	−	0.833339i	$-0.686426\pi$
$19$	−4.81887	−1.10552	−0.552762	+	0.833339i	$0.686426\pi$
$20$	9.71388	2.17209
$21$	3.09610	0.675624
$22$	−13.8453	−2.95182
$23$	−0.298932	−0.0623315	−0.0311658	−	0.999514i	$-0.509922\pi$
$23$	−0.298932	−0.0623315	−0.0311658	+	0.999514i	$0.509922\pi$
$24$	−11.4943	−2.34626
$25$	−2.33592	−0.467183
$26$	15.4503	3.03005
$27$	−5.09176	−0.979909
$28$	17.8618	3.37556
$29$	7.24040	1.34451	0.672254	−	0.740320i	$-0.265326\pi$
$29$	7.24040	1.34451	0.672254	+	0.740320i	$0.265326\pi$
$30$	−4.74793	−0.866850
$31$	−6.64697	−1.19383	−0.596916	−	0.802304i	$-0.703607\pi$
$31$	−6.64697	−1.19383	−0.596916	+	0.802304i	$0.703607\pi$
$32$	−32.7481	−5.78910
$33$	5.06510	0.881721
$34$	18.2783	3.13470
$35$	4.89868	0.828029
$36$	−11.5208	−1.92013
$37$	4.71805	0.775642	0.387821	−	0.921735i	$-0.373228\pi$
$37$	4.71805	0.775642	0.387821	+	0.921735i	$0.373228\pi$
$38$	13.5883	2.20432
$39$	−5.65228	−0.905089
$40$	−18.1864	−2.87552
$41$	2.11712	0.330639	0.165319	−	0.986240i	$-0.447134\pi$
$41$	2.11712	0.330639	0.165319	+	0.986240i	$0.447134\pi$
$42$	−8.73044	−1.34714
$43$	−3.63811	−0.554807	−0.277404	−	0.960753i	$-0.589474\pi$
$43$	−3.63811	−0.554807	−0.277404	+	0.960753i	$0.589474\pi$
$44$	29.2212	4.40526
$45$	−3.15964	−0.471011
$46$	0.842934	0.124284
$47$	−4.84762	−0.707097	−0.353549	−	0.935416i	$-0.615025\pi$
$47$	−4.84762	−0.707097	−0.353549	+	0.935416i	$0.615025\pi$
$48$	20.1329	2.90594
$49$	2.00764	0.286805
$50$	6.58687	0.931524
$51$	−6.68687	−0.936349
$52$	−32.6087	−4.52201
$53$	2.37433	0.326139	0.163070	−	0.986615i	$-0.447860\pi$
$53$	2.37433	0.326139	0.163070	+	0.986615i	$0.447860\pi$
$54$	14.3579	1.95386
$55$	8.01407	1.08062
$56$	−33.4409	−4.46873
$57$	−4.97112	−0.658440
$58$	−20.4166	−2.68084
$59$	7.57648	0.986374	0.493187	−	0.869923i	$-0.335832\pi$
$59$	7.57648	0.986374	0.493187	+	0.869923i	$0.335832\pi$
$60$	10.0208	1.29368
$61$	−6.90355	−0.883909	−0.441955	−	0.897037i	$-0.645715\pi$
$61$	−6.90355	−0.883909	−0.441955	+	0.897037i	$0.645715\pi$
$62$	18.7433	2.38040
$63$	−5.80990	−0.731979
$64$	53.3112	6.66390
$65$	−8.94311	−1.10926
$66$	−14.2827	−1.75808
$67$	−0.975192	−0.119139	−0.0595693	−	0.998224i	$-0.518973\pi$
$67$	−0.975192	−0.119139	−0.0595693	+	0.998224i	$0.518973\pi$
$68$	−38.5774	−4.67820
$69$	−0.308376	−0.0371241
$70$	−13.8134	−1.65102
$71$	9.25846	1.09878	0.549389	−	0.835567i	$-0.314861\pi$
$71$	9.25846	1.09878	0.549389	+	0.835567i	$0.314861\pi$
$72$	21.5693	2.54196
$73$	−11.2782	−1.32001	−0.660004	−	0.751262i	$-0.729445\pi$
$73$	−11.2782	−1.32001	−0.660004	+	0.751262i	$0.729445\pi$
$74$	−13.3040	−1.54656
$75$	−2.40972	−0.278250
$76$	−28.6790	−3.28971
$77$	14.7362	1.67934
$78$	15.9384	1.80467
$79$	11.5118	1.29517	0.647587	−	0.761992i	$-0.275778\pi$
$79$	11.5118	1.29517	0.647587	+	0.761992i	$0.275778\pi$
$80$	31.8546	3.56145
$81$	0.554809	0.0616454
$82$	−5.96991	−0.659266
$83$	10.2684	1.12711	0.563554	−	0.826080i	$-0.309434\pi$
$83$	10.2684	1.12711	0.563554	+	0.826080i	$0.309434\pi$
$84$	18.4261	2.01045
$85$	−10.5801	−1.14757
$86$	10.2588	1.10624
$87$	7.46916	0.800778
$88$	−54.7081	−5.83190
$89$	−14.4094	−1.52739	−0.763697	−	0.645574i	$-0.776618\pi$
$89$	−14.4094	−1.52739	−0.763697	+	0.645574i	$0.776618\pi$
$90$	8.90962	0.939156
$91$	−16.4445	−1.72385
$92$	−1.77906	−0.185480
$93$	−6.85698	−0.711036
$94$	13.6694	1.40989
$95$	−7.86536	−0.806969
$96$	−33.7828	−3.44794
$97$	−17.5407	−1.78098	−0.890492	−	0.454998i	$-0.849640\pi$
$97$	−17.5407	−1.78098	−0.890492	+	0.454998i	$0.849640\pi$
$98$	−5.66118	−0.571865
$99$	−9.50479	−0.955268
$100$	−13.9020	−1.39020
$101$	−13.0898	−1.30248	−0.651242	−	0.758870i	$-0.725752\pi$
$101$	−13.0898	−1.30248	−0.651242	+	0.758870i	$0.725752\pi$
$102$	18.8558	1.86700
$103$	−11.9455	−1.17702	−0.588511	−	0.808489i	$-0.700286\pi$
$103$	−11.9455	−1.17702	−0.588511	+	0.808489i	$0.700286\pi$
$104$	61.0502	5.98646
$105$	5.05345	0.493167
$106$	−6.69519	−0.650294
$107$	3.55055	0.343245	0.171623	−	0.985163i	$-0.445099\pi$
$107$	3.55055	0.343245	0.171623	+	0.985163i	$0.445099\pi$
$108$	−30.3031	−2.91591
$109$	−14.0076	−1.34168	−0.670842	−	0.741600i	$-0.734067\pi$
$109$	−14.0076	−1.34168	−0.670842	+	0.741600i	$0.734067\pi$
$110$	−22.5983	−2.15466
$111$	4.86711	0.461966
$112$	58.5738	5.53471
$113$	−4.46545	−0.420074	−0.210037	−	0.977693i	$-0.567358\pi$
$113$	−4.46545	−0.420074	−0.210037	+	0.977693i	$0.567358\pi$
$114$	14.0177	1.31287
$115$	−0.487917	−0.0454985
$116$	43.0905	4.00085
$117$	10.6066	0.980584
$118$	−21.3643	−1.96675
$119$	−19.4545	−1.78339
$120$	−18.7610	−1.71263
$121$	13.1079	1.19162
$122$	19.4668	1.76244
$123$	2.18401	0.196926
$124$	−39.5588	−3.55248
$125$	−11.9737	−1.07096
$126$	16.3829	1.45950
$127$	−5.69251	−0.505128	−0.252564	−	0.967580i	$-0.581274\pi$
$127$	−5.69251	−0.505128	−0.252564	+	0.967580i	$0.581274\pi$
$128$	−84.8318	−7.49815
$129$	−3.75306	−0.330438
$130$	25.2180	2.21176
$131$	−6.12789	−0.535397	−0.267698	−	0.963503i	$-0.586263\pi$
$131$	−6.12789	−0.535397	−0.267698	+	0.963503i	$0.586263\pi$
$132$	30.1444	2.62374
$133$	−14.4627	−1.25408
$134$	2.74987	0.237552
$135$	−8.31078	−0.715278
$136$	72.2248	6.19323
$137$	4.66038	0.398163	0.199082	−	0.979983i	$-0.436204\pi$
$137$	4.66038	0.398163	0.199082	+	0.979983i	$0.436204\pi$
$138$	0.869566	0.0740224
$139$	10.7876	0.914994	0.457497	−	0.889211i	$-0.348746\pi$
$139$	10.7876	0.914994	0.457497	+	0.889211i	$0.348746\pi$
$140$	29.1540	2.46396
$141$	−5.00077	−0.421141
$142$	−26.1072	−2.19087
$143$	−26.9026	−2.24971
$144$	−37.7800	−3.14833
$145$	11.8178	0.981415
$146$	31.8024	2.63198
$147$	2.07107	0.170819
$148$	28.0790	2.30808
$149$	−11.9737	−0.980926	−0.490463	−	0.871462i	$-0.663172\pi$
$149$	−11.9737	−0.980926	−0.490463	+	0.871462i	$0.663172\pi$
$150$	6.79498	0.554808
$151$	−12.3400	−1.00421	−0.502107	−	0.864805i	$-0.667442\pi$
$151$	−12.3400	−1.00421	−0.502107	+	0.864805i	$0.667442\pi$
$152$	53.6929	4.35507
$153$	12.5481	1.01445
$154$	−41.5534	−3.34847
$155$	−10.8492	−0.871429
$156$	−33.6390	−2.69327
$157$	−8.94198	−0.713648	−0.356824	−	0.934172i	$-0.616140\pi$
$157$	−8.94198	−0.713648	−0.356824	+	0.934172i	$0.616140\pi$
$158$	−32.4611	−2.58247
$159$	2.44934	0.194246
$160$	−53.4515	−4.22572
$161$	−0.897175	−0.0707073
$162$	−1.56446	−0.122916
$163$	10.8844	0.852535	0.426267	−	0.904597i	$-0.359828\pi$
$163$	10.8844	0.852535	0.426267	+	0.904597i	$0.359828\pi$
$164$	12.5998	0.983882
$165$	8.26727	0.643606
$166$	−28.9552	−2.24736
$167$	−7.35515	−0.569159	−0.284579	−	0.958653i	$-0.591854\pi$
$167$	−7.35515	−0.569159	−0.284579	+	0.958653i	$0.591854\pi$
$168$	−34.4974	−2.66154
$169$	17.0213	1.30933
$170$	29.8339	2.28815
$171$	9.32842	0.713362
$172$	−21.6519	−1.65094
$173$	−7.00564	−0.532629	−0.266314	−	0.963886i	$-0.585806\pi$
$173$	−7.00564	−0.532629	−0.266314	+	0.963886i	$0.585806\pi$
$174$	−21.0617	−1.59668
$175$	−7.01072	−0.529961
$176$	95.8247	7.22306
$177$	7.81586	0.587476
$178$	40.6320	3.04550
$179$	−2.90858	−0.217397	−0.108699	−	0.994075i	$-0.534668\pi$
$179$	−2.90858	−0.217397	−0.108699	+	0.994075i	$0.534668\pi$
$180$	−18.8043	−1.40159
$181$	−13.0636	−0.971008	−0.485504	−	0.874235i	$-0.661364\pi$
$181$	−13.0636	−0.971008	−0.485504	+	0.874235i	$0.661364\pi$
$182$	46.3705	3.43721
$183$	−7.12167	−0.526449
$184$	3.33077	0.245547
$185$	7.70080	0.566174
$186$	19.3355	1.41775
$187$	−31.8268	−2.32741
$188$	−28.8501	−2.10411
$189$	−15.2818	−1.11158
$190$	22.1789	1.60903
$191$	14.5844	1.05529	0.527646	−	0.849464i	$-0.323075\pi$
$191$	14.5844	1.05529	0.527646	+	0.849464i	$0.323075\pi$
$192$	54.9955	3.96896
$193$	−3.72783	−0.268335	−0.134167	−	0.990959i	$-0.542836\pi$
$193$	−3.72783	−0.268335	−0.134167	+	0.990959i	$0.542836\pi$
$194$	49.4615	3.55113
$195$	−9.22566	−0.660663
$196$	11.9482	0.853446
$197$	23.8734	1.70091	0.850455	−	0.526048i	$-0.176327\pi$
$197$	23.8734	1.70091	0.850455	+	0.526048i	$0.176327\pi$
$198$	26.8018	1.90472
$199$	19.0289	1.34892	0.674462	−	0.738310i	$-0.264376\pi$
$199$	19.0289	1.34892	0.674462	+	0.738310i	$0.264376\pi$
$200$	26.0273	1.84041
$201$	−1.00600	−0.0709579
$202$	36.9109	2.59704
$203$	21.7304	1.52518
$204$	−39.7962	−2.78629
$205$	3.45557	0.241348
$206$	33.6841	2.34688
$207$	0.578675	0.0402207
$208$	−106.933	−7.41449
$209$	−23.6605	−1.63663
$210$	−14.2498	−0.983333
$211$	5.29850	0.364764	0.182382	−	0.983228i	$-0.441619\pi$
$211$	5.29850	0.364764	0.182382	+	0.983228i	$0.441619\pi$
$212$	14.1306	0.970492
$213$	9.55098	0.654422
$214$	−10.0119	−0.684402
$215$	−5.93814	−0.404978
$216$	56.7336	3.86023
$217$	−19.9494	−1.35425
$218$	39.4989	2.67521
$219$	−11.6345	−0.786185
$220$	47.6949	3.21559
$221$	35.5164	2.38909
$222$	−13.7244	−0.921120
$223$	−2.49768	−0.167257	−0.0836286	−	0.996497i	$-0.526651\pi$
$223$	−2.49768	−0.167257	−0.0836286	+	0.996497i	$0.526651\pi$
$224$	−98.2860	−6.56701
$225$	4.52190	0.301460
$226$	12.5918	0.837592
$227$	16.8104	1.11574	0.557872	−	0.829927i	$-0.311618\pi$
$227$	16.8104	1.11574	0.557872	+	0.829927i	$0.311618\pi$
$228$	−29.5851	−1.95932
$229$	23.0956	1.52620	0.763100	−	0.646280i	$-0.223676\pi$
$229$	23.0956	1.52620	0.763100	+	0.646280i	$0.223676\pi$
$230$	1.37584	0.0907201
$231$	15.2018	1.00020
$232$	−80.6743	−5.29653
$233$	−28.0004	−1.83437	−0.917183	−	0.398467i	$-0.869542\pi$
$233$	−28.0004	−1.83437	−0.917183	+	0.398467i	$0.869542\pi$
$234$	−29.9088	−1.95520
$235$	−7.91229	−0.516141
$236$	45.0907	2.93515
$237$	11.8755	0.771394
$238$	54.8581	3.55593
$239$	−5.86560	−0.379414	−0.189707	−	0.981841i	$-0.560754\pi$
$239$	−5.86560	−0.379414	−0.189707	+	0.981841i	$0.560754\pi$
$240$	32.8610	2.12117
$241$	1.30460	0.0840366	0.0420183	−	0.999117i	$-0.486621\pi$
$241$	1.30460	0.0840366	0.0420183	+	0.999117i	$0.486621\pi$
$242$	−36.9618	−2.37600
$243$	15.8476	1.01662
$244$	−41.0858	−2.63025
$245$	3.27687	0.209351
$246$	−6.15852	−0.392653
$247$	26.4034	1.68001
$248$	74.0622	4.70295
$249$	10.5929	0.671295
$250$	33.7637	2.13540
$251$	28.8995	1.82412	0.912059	−	0.410060i	$-0.134492\pi$
$251$	28.8995	1.82412	0.912059	+	0.410060i	$0.134492\pi$
$252$	−34.5770	−2.17815
$253$	−1.46775	−0.0922764
$254$	16.0519	1.00718
$255$	−10.9143	−0.683482
$256$	132.588	8.28677
$257$	9.90390	0.617788	0.308894	−	0.951096i	$-0.400041\pi$
$257$	9.90390	0.617788	0.308894	+	0.951096i	$0.400041\pi$
$258$	10.5830	0.658866
$259$	14.1601	0.879868
$260$	−53.2240	−3.30081
$261$	−14.0161	−0.867572
$262$	17.2796	1.06754
$263$	10.2382	0.631316	0.315658	−	0.948873i	$-0.397775\pi$
$263$	10.2382	0.631316	0.315658	+	0.948873i	$0.397775\pi$
$264$	−56.4366	−3.47343
$265$	3.87538	0.238063
$266$	40.7823	2.50052
$267$	−14.8647	−0.909703
$268$	−5.80375	−0.354521
$269$	−9.88154	−0.602488	−0.301244	−	0.953547i	$-0.597402\pi$
$269$	−9.88154	−0.602488	−0.301244	+	0.953547i	$0.597402\pi$
$270$	23.4349	1.42620
$271$	−22.9893	−1.39650	−0.698250	−	0.715854i	$-0.746037\pi$
$271$	−22.9893	−1.39650	−0.698250	+	0.715854i	$0.746037\pi$
$272$	−126.506	−7.67057
$273$	−16.9640	−1.02671
$274$	−13.1414	−0.793904
$275$	−11.4693	−0.691624
$276$	−1.83527	−0.110470
$277$	2.97484	0.178741	0.0893705	−	0.995998i	$-0.471514\pi$
$277$	2.97484	0.178741	0.0893705	+	0.995998i	$0.471514\pi$
$278$	−30.4192	−1.82442
$279$	12.8673	0.770345
$280$	−54.5823	−3.26192
$281$	−25.0556	−1.49469	−0.747346	−	0.664435i	$-0.768672\pi$
$281$	−25.0556	−1.49469	−0.747346	+	0.664435i	$0.768672\pi$
$282$	14.1013	0.839720
$283$	22.8372	1.35753	0.678766	−	0.734355i	$-0.262515\pi$
$283$	22.8372	1.35753	0.678766	+	0.734355i	$0.262515\pi$
$284$	55.1008	3.26963
$285$	−8.11387	−0.480624
$286$	75.8605	4.48572
$287$	6.35406	0.375068
$288$	63.3942	3.73554
$289$	25.0173	1.47160
$290$	−33.3241	−1.95686
$291$	−18.0949	−1.06074
$292$	−67.1208	−3.92794
$293$	−28.3112	−1.65396	−0.826978	−	0.562235i	$-0.809942\pi$
$293$	−28.3112	−1.65396	−0.826978	+	0.562235i	$0.809942\pi$
$294$	−5.84004	−0.340598
$295$	12.3664	0.719997
$296$	−52.5696	−3.05554
$297$	−25.0004	−1.45067
$298$	33.7638	1.95588
$299$	1.63790	0.0947220
$300$	−14.3412	−0.827989
$301$	−10.9190	−0.629359
$302$	34.7966	2.00232
$303$	−13.5034	−0.775749
$304$	−94.0466	−5.39394
$305$	−11.2680	−0.645203
$306$	−35.3834	−2.02273
$307$	−12.6484	−0.721882	−0.360941	−	0.932589i	$-0.617544\pi$
$307$	−12.6484	−0.721882	−0.360941	+	0.932589i	$0.617544\pi$
$308$	87.7008	4.99722
$309$	−12.3229	−0.701024
$310$	30.5928	1.73756
$311$	−0.962252	−0.0545643	−0.0272822	−	0.999628i	$-0.508685\pi$
$311$	−0.962252	−0.0545643	−0.0272822	+	0.999628i	$0.508685\pi$
$312$	62.9790	3.56549
$313$	−14.3470	−0.810938	−0.405469	−	0.914109i	$-0.632892\pi$
$313$	−14.3470	−0.810938	−0.405469	+	0.914109i	$0.632892\pi$
$314$	25.2148	1.42295
$315$	−9.48293	−0.534303
$316$	68.5110	3.85405
$317$	21.0823	1.18410	0.592049	−	0.805902i	$-0.298319\pi$
$317$	21.0823	1.18410	0.592049	+	0.805902i	$0.298319\pi$
$318$	−6.90672	−0.387309
$319$	35.5502	1.99043
$320$	87.0146	4.86427
$321$	3.66273	0.204434
$322$	2.52987	0.140984
$323$	31.2362	1.73803
$324$	3.30189	0.183438
$325$	12.7989	0.709954
$326$	−30.6922	−1.69988
$327$	−14.4502	−0.799096
$328$	−23.5895	−1.30251
$329$	−14.5490	−0.802113
$330$	−23.3122	−1.28330
$331$	−3.09575	−0.170158	−0.0850790	−	0.996374i	$-0.527114\pi$
$331$	−3.09575	−0.170158	−0.0850790	+	0.996374i	$0.527114\pi$
$332$	61.1115	3.35393
$333$	−9.13325	−0.500499
$334$	20.7402	1.13485
$335$	−1.59171	−0.0869644
$336$	60.4244	3.29642
$337$	−4.30290	−0.234394	−0.117197	−	0.993109i	$-0.537391\pi$
$337$	−4.30290	−0.234394	−0.117197	+	0.993109i	$0.537391\pi$
$338$	−47.9970	−2.61069
$339$	−4.60653	−0.250192
$340$	−62.9661	−3.41482
$341$	−32.6365	−1.76736
$342$	−26.3045	−1.42238
$343$	−14.9834	−0.809030
$344$	40.5367	2.18559
$345$	−0.503332	−0.0270985
$346$	19.7547	1.06202
$347$	6.23494	0.334709	0.167355	−	0.985897i	$-0.446478\pi$
$347$	6.23494	0.334709	0.167355	+	0.985897i	$0.446478\pi$
$348$	44.4519	2.38287
$349$	−16.8738	−0.903232	−0.451616	−	0.892212i	$-0.649152\pi$
$349$	−16.8738	−0.903232	−0.451616	+	0.892212i	$0.649152\pi$
$350$	19.7690	1.05670
$351$	27.8986	1.48912
$352$	−160.792	−8.57027
$353$	2.28194	0.121455	0.0607276	−	0.998154i	$-0.480658\pi$
$353$	2.28194	0.121455	0.0607276	+	0.998154i	$0.480658\pi$
$354$	−22.0393	−1.17138
$355$	15.1117	0.802045
$356$	−85.7561	−4.54507
$357$	−20.0691	−1.06217
$358$	8.20167	0.433472
$359$	−22.1412	−1.16857	−0.584283	−	0.811550i	$-0.698624\pi$
$359$	−22.1412	−1.16857	−0.584283	+	0.811550i	$0.698624\pi$
$360$	35.2054	1.85549
$361$	4.22147	0.222183
$362$	36.8369	1.93611
$363$	13.5220	0.709721
$364$	−97.8676	−5.12966
$365$	−18.4082	−0.963530
$366$	20.0818	1.04969
$367$	4.05357	0.211595	0.105797	−	0.994388i	$-0.466260\pi$
$367$	4.05357	0.211595	0.105797	+	0.994388i	$0.466260\pi$
$368$	−5.83405	−0.304121
$369$	−4.09835	−0.213352
$370$	−21.7149	−1.12890
$371$	7.12601	0.369964
$372$	−40.8086	−2.11583
$373$	−3.35329	−0.173627	−0.0868133	−	0.996225i	$-0.527668\pi$
$373$	−3.35329	−0.173627	−0.0868133	+	0.996225i	$0.527668\pi$
$374$	89.7460	4.64065
$375$	−12.3520	−0.637855
$376$	54.0133	2.78552
$377$	−39.6714	−2.04318
$378$	43.0918	2.21640
$379$	−5.00597	−0.257139	−0.128570	−	0.991700i	$-0.541039\pi$
$379$	−5.00597	−0.257139	−0.128570	+	0.991700i	$0.541039\pi$
$380$	−46.8099	−2.40130
$381$	−5.87236	−0.300850
$382$	−41.1255	−2.10416
$383$	8.96283	0.457979	0.228990	−	0.973429i	$-0.426458\pi$
$383$	8.96283	0.457979	0.228990	+	0.973429i	$0.426458\pi$
$384$	−87.5121	−4.46583
$385$	24.0524	1.22582
$386$	10.5118	0.535037
$387$	7.04271	0.358001
$388$	−104.391	−5.29967
$389$	−24.7213	−1.25342	−0.626710	−	0.779253i	$-0.715599\pi$
$389$	−24.7213	−1.25342	−0.626710	+	0.779253i	$0.715599\pi$
$390$	26.0147	1.31731
$391$	1.93770	0.0979935
$392$	−22.3696	−1.12983
$393$	−6.32150	−0.318878
$394$	−67.3188	−3.39147
$395$	18.7895	0.945403
$396$	−56.5668	−2.84259
$397$	20.2495	1.01629	0.508147	−	0.861270i	$-0.330331\pi$
$397$	20.2495	1.01629	0.508147	+	0.861270i	$0.330331\pi$
$398$	−53.6582	−2.68964
$399$	−14.9197	−0.746918
$400$	−45.5885	−2.27943
$401$	20.9110	1.04425	0.522123	−	0.852870i	$-0.325140\pi$
$401$	20.9110	1.04425	0.522123	+	0.852870i	$0.325140\pi$
$402$	2.83675	0.141484
$403$	36.4199	1.81420
$404$	−77.9027	−3.87580
$405$	0.905560	0.0449977
$406$	−61.2759	−3.04107
$407$	23.1655	1.14827
$408$	74.5067	3.68863
$409$	−15.1092	−0.747102	−0.373551	−	0.927610i	$-0.621860\pi$
$409$	−15.1092	−0.747102	−0.373551	+	0.927610i	$0.621860\pi$
$410$	−9.74409	−0.481227
$411$	4.80762	0.237143
$412$	−71.0922	−3.50246
$413$	22.7391	1.11892
$414$	−1.63176	−0.0801967
$415$	16.7602	0.822724
$416$	179.432	8.79740
$417$	11.1285	0.544963
$418$	66.7184	3.26331
$419$	22.0687	1.07813	0.539063	−	0.842265i	$-0.318778\pi$
$419$	22.0687	1.07813	0.539063	+	0.842265i	$0.318778\pi$
$420$	30.0751	1.46752
$421$	−40.0155	−1.95024	−0.975118	−	0.221689i	$-0.928843\pi$
$421$	−40.0155	−1.95024	−0.975118	+	0.221689i	$0.928843\pi$
$422$	−14.9408	−0.727308
$423$	9.38408	0.456269
$424$	−26.4553	−1.28478
$425$	15.1416	0.734475
$426$	−26.9321	−1.30486
$427$	−20.7194	−1.00268
$428$	21.1308	1.02139
$429$	−27.7526	−1.33991
$430$	16.7445	0.807491
$431$	−31.4461	−1.51471	−0.757353	−	0.653005i	$-0.773508\pi$
$431$	−31.4461	−1.51471	−0.757353	+	0.653005i	$0.773508\pi$
$432$	−99.3724	−4.78106
$433$	−27.0905	−1.30189	−0.650944	−	0.759126i	$-0.725627\pi$
$433$	−27.0905	−1.30189	−0.650944	+	0.759126i	$0.725627\pi$
$434$	56.2537	2.70026
$435$	12.1912	0.584522
$436$	−83.3648	−3.99245
$437$	1.44051	0.0689090
$438$	32.8072	1.56759
$439$	−7.96189	−0.380000	−0.190000	−	0.981784i	$-0.560849\pi$
$439$	−7.96189	−0.380000	−0.190000	+	0.981784i	$0.560849\pi$
$440$	−89.2947	−4.25696
$441$	−3.88641	−0.185067
$442$	−100.150	−4.76364
$443$	32.3402	1.53653	0.768264	−	0.640133i	$-0.221121\pi$
$443$	32.3402	1.53653	0.768264	+	0.640133i	$0.221121\pi$
$444$	28.9661	1.37467
$445$	−23.5191	−1.11491
$446$	7.04302	0.333497
$447$	−12.3520	−0.584231
$448$	160.001	7.55936
$449$	−8.57427	−0.404645	−0.202322	−	0.979319i	$-0.564849\pi$
$449$	−8.57427	−0.404645	−0.202322	+	0.979319i	$0.564849\pi$
$450$	−12.7509	−0.601085
$451$	10.3950	0.489482
$452$	−26.5756	−1.25001
$453$	−12.7299	−0.598102
$454$	−47.4022	−2.22470
$455$	−26.8407	−1.25831
$456$	55.3894	2.59384
$457$	13.8460	0.647688	0.323844	−	0.946110i	$-0.395025\pi$
$457$	13.8460	0.647688	0.323844	+	0.946110i	$0.395025\pi$
$458$	−65.1255	−3.04312
$459$	33.0051	1.54055
$460$	−2.90379	−0.135390
$461$	25.2053	1.17393	0.586964	−	0.809613i	$-0.300323\pi$
$461$	25.2053	1.17393	0.586964	+	0.809613i	$0.300323\pi$
$462$	−42.8663	−1.99432
$463$	14.9530	0.694925	0.347462	−	0.937694i	$-0.387043\pi$
$463$	14.9530	0.694925	0.347462	+	0.937694i	$0.387043\pi$
$464$	141.306	6.55997
$465$	−11.1920	−0.519016
$466$	78.9561	3.65757
$467$	−11.7295	−0.542777	−0.271389	−	0.962470i	$-0.587483\pi$
$467$	−11.7295	−0.542777	−0.271389	+	0.962470i	$0.587483\pi$
$468$	63.1243	2.91792
$469$	−2.92682	−0.135148
$470$	22.3112	1.02914
$471$	−9.22450	−0.425042
$472$	−84.4190	−3.88570
$473$	−17.8631	−0.821344
$474$	−33.4867	−1.53809
$475$	11.2565	0.516482
$476$	−115.781	−5.30683
$477$	−4.59626	−0.210448
$478$	16.5399	0.756519
$479$	−30.8541	−1.40976	−0.704880	−	0.709326i	$-0.748999\pi$
$479$	−30.8541	−1.40976	−0.704880	+	0.709326i	$0.748999\pi$
$480$	−55.1403	−2.51680
$481$	−25.8510	−1.17870
$482$	−3.67874	−0.167562
$483$	−0.925521	−0.0421127
$484$	78.0101	3.54591
$485$	−28.6299	−1.30002
$486$	−44.6874	−2.02706
$487$	24.9298	1.12968	0.564840	−	0.825201i	$-0.308938\pi$
$487$	24.9298	1.12968	0.564840	+	0.825201i	$0.308938\pi$
$488$	76.9210	3.48205
$489$	11.2283	0.507762
$490$	−9.24018	−0.417429
$491$	2.50375	0.112992	0.0564962	−	0.998403i	$-0.482007\pi$
$491$	2.50375	0.112992	0.0564962	+	0.998403i	$0.482007\pi$
$492$	12.9979	0.585991
$493$	−46.9328	−2.11375
$494$	−74.4528	−3.34979
$495$	−15.5137	−0.697291
$496$	−129.725	−5.82480
$497$	27.7872	1.24642
$498$	−29.8700	−1.33851
$499$	42.0583	1.88279	0.941394	−	0.337310i	$-0.109517\pi$
$499$	42.0583	1.88279	0.941394	+	0.337310i	$0.109517\pi$
$500$	−71.2602	−3.18685
$501$	−7.58753	−0.338986
$502$	−81.4913	−3.63714
$503$	23.5105	1.04828	0.524142	−	0.851631i	$-0.324386\pi$
$503$	23.5105	1.04828	0.524142	+	0.851631i	$0.324386\pi$
$504$	64.7353	2.88354
$505$	−21.3652	−0.950740
$506$	4.13878	0.183991
$507$	17.5591	0.779825
$508$	−33.8784	−1.50311
$509$	33.5711	1.48801	0.744007	−	0.668171i	$-0.232923\pi$
$509$	33.5711	1.48801	0.744007	+	0.668171i	$0.232923\pi$
$510$	30.7765	1.36280
$511$	−33.8488	−1.49738
$512$	−204.212	−9.02497
$513$	24.5365	1.08331
$514$	−27.9272	−1.23182
$515$	−19.4974	−0.859159
$516$	−22.3359	−0.983285
$517$	−23.8017	−1.04680
$518$	−39.9291	−1.75438
$519$	−7.22698	−0.317229
$520$	99.6462	4.36978
$521$	19.6828	0.862319	0.431159	−	0.902276i	$-0.358105\pi$
$521$	19.6828	0.862319	0.431159	+	0.902276i	$0.358105\pi$
$522$	39.5228	1.72987
$523$	−24.5399	−1.07305	−0.536527	−	0.843883i	$-0.680264\pi$
$523$	−24.5399	−1.07305	−0.536527	+	0.843883i	$0.680264\pi$
$524$	−36.4695	−1.59318
$525$	−7.23222	−0.315640
$526$	−28.8700	−1.25879
$527$	43.0862	1.87686
$528$	98.8523	4.30199
$529$	−22.9106	−0.996115
$530$	−10.9279	−0.474677
$531$	−14.6667	−0.636478
$532$	−86.0734	−3.73176
$533$	−11.6001	−0.502455
$534$	41.9157	1.81387
$535$	5.79522	0.250549
$536$	10.8658	0.469332
$537$	−3.00047	−0.129480
$538$	27.8642	1.20131
$539$	9.85744	0.424590
$540$	−49.4607	−2.12845
$541$	−2.70038	−0.116098	−0.0580492	−	0.998314i	$-0.518488\pi$
$541$	−2.70038	−0.116098	−0.0580492	+	0.998314i	$0.518488\pi$
$542$	64.8257	2.78450
$543$	−13.4763	−0.578324
$544$	212.276	9.10124
$545$	−22.8632	−0.979353
$546$	47.8355	2.04717
$547$	−7.84479	−0.335419	−0.167709	−	0.985836i	$-0.553637\pi$
$547$	−7.84479	−0.335419	−0.167709	+	0.985836i	$0.553637\pi$
$548$	27.7358	1.18481
$549$	13.3640	0.570361
$550$	32.3414	1.37904
$551$	−34.8905	−1.48639
$552$	3.43600	0.146246
$553$	34.5499	1.46921
$554$	−8.38853	−0.356394
$555$	7.94411	0.337209
$556$	64.2014	2.72275
$557$	34.0031	1.44076	0.720378	−	0.693582i	$-0.243968\pi$
$557$	34.0031	1.44076	0.720378	+	0.693582i	$0.243968\pi$
$558$	−36.2835	−1.53600
$559$	19.9338	0.843111
$560$	95.6043	4.04002
$561$	−32.8324	−1.38618
$562$	70.6524	2.98029
$563$	−7.67566	−0.323490	−0.161745	−	0.986833i	$-0.551712\pi$
$563$	−7.67566	−0.323490	−0.161745	+	0.986833i	$0.551712\pi$
$564$	−29.7616	−1.25319
$565$	−7.28851	−0.306630
$566$	−64.3969	−2.70680
$567$	1.66513	0.0699290
$568$	−103.160	−4.32850
$569$	7.21892	0.302633	0.151316	−	0.988485i	$-0.451649\pi$
$569$	7.21892	0.302633	0.151316	+	0.988485i	$0.451649\pi$
$570$	22.8797	0.958323
$571$	36.6934	1.53557	0.767786	−	0.640707i	$-0.221359\pi$
$571$	36.6934	1.53557	0.767786	+	0.640707i	$0.221359\pi$
$572$	−160.108	−6.69445
$573$	15.0452	0.628523
$574$	−17.9173	−0.747854
$575$	0.698279	0.0291202
$576$	−103.200	−4.30002
$577$	26.6827	1.11082	0.555408	−	0.831578i	$-0.312562\pi$
$577$	26.6827	1.11082	0.555408	+	0.831578i	$0.312562\pi$
$578$	−70.5443	−2.93426
$579$	−3.84560	−0.159818
$580$	70.3324	2.92039
$581$	30.8184	1.27856
$582$	51.0243	2.11502
$583$	11.6579	0.482821
$584$	125.664	5.20001
$585$	17.3122	0.715771
$586$	79.8324	3.29785
$587$	−14.4442	−0.596175	−0.298088	−	0.954539i	$-0.596349\pi$
$587$	−14.4442	−0.596175	−0.298088	+	0.954539i	$0.596349\pi$
$588$	12.3257	0.508305
$589$	32.0309	1.31981
$590$	−34.8709	−1.43561
$591$	24.6277	1.01305
$592$	92.0789	3.78442
$593$	7.89910	0.324377	0.162189	−	0.986760i	$-0.448145\pi$
$593$	7.89910	0.324377	0.162189	+	0.986760i	$0.448145\pi$
$594$	70.4967	2.89251
$595$	−31.7536	−1.30177
$596$	−71.2604	−2.91894
$597$	19.6301	0.803408
$598$	−4.61857	−0.188868
$599$	−44.8641	−1.83310	−0.916549	−	0.399922i	$-0.869037\pi$
$599$	−44.8641	−1.83310	−0.916549	+	0.399922i	$0.869037\pi$
$600$	26.8497	1.09613
$601$	10.0406	0.409566	0.204783	−	0.978807i	$-0.434351\pi$
$601$	10.0406	0.409566	0.204783	+	0.978807i	$0.434351\pi$
$602$	30.7896	1.25489
$603$	1.88779	0.0768767
$604$	−73.4402	−2.98824
$605$	21.3947	0.869817
$606$	38.0771	1.54678
$607$	−18.6721	−0.757876	−0.378938	−	0.925422i	$-0.623711\pi$
$607$	−18.6721	−0.757876	−0.378938	+	0.925422i	$0.623711\pi$
$608$	157.809	6.39999
$609$	22.4170	0.908382
$610$	31.7737	1.28648
$611$	26.5609	1.07454
$612$	74.6786	3.01870
$613$	10.1753	0.410976	0.205488	−	0.978660i	$-0.434122\pi$
$613$	10.1753	0.410976	0.205488	+	0.978660i	$0.434122\pi$
$614$	35.6662	1.43937
$615$	3.56475	0.143745
$616$	−164.194	−6.61556
$617$	−48.1240	−1.93740	−0.968700	−	0.248233i	$-0.920150\pi$
$617$	−48.1240	−1.93740	−0.968700	+	0.248233i	$0.920150\pi$
$618$	34.7483	1.39778
$619$	42.3661	1.70284	0.851418	−	0.524487i	$-0.175743\pi$
$619$	42.3661	1.70284	0.851418	+	0.524487i	$0.175743\pi$
$620$	−64.5679	−2.59311
$621$	1.52209	0.0610792
$622$	2.71338	0.108797
$623$	−43.2466	−1.73264
$624$	−110.312	−4.41600
$625$	−7.86391	−0.314557
$626$	40.4558	1.61694
$627$	−24.4081	−0.974764
$628$	−53.2173	−2.12360
$629$	−30.5827	−1.21941
$630$	26.7402	1.06535
$631$	16.1789	0.644071	0.322036	−	0.946728i	$-0.395633\pi$
$631$	16.1789	0.644071	0.322036	+	0.946728i	$0.395633\pi$
$632$	−128.267	−5.10217
$633$	5.46591	0.217250
$634$	−59.4483	−2.36099
$635$	−9.29132	−0.368715
$636$	14.5770	0.578017
$637$	−11.0002	−0.435843
$638$	−100.245	−3.96875
$639$	−17.9227	−0.709009
$640$	−138.463	−5.47322
$641$	−26.4679	−1.04542	−0.522710	−	0.852511i	$-0.675079\pi$
$641$	−26.4679	−1.04542	−0.522710	+	0.852511i	$0.675079\pi$
$642$	−10.3283	−0.407624
$643$	−32.7273	−1.29064	−0.645319	−	0.763913i	$-0.723276\pi$
$643$	−32.7273	−1.29064	−0.645319	+	0.763913i	$0.723276\pi$
$644$	−5.33944	−0.210404
$645$	−6.12575	−0.241201
$646$	−88.0807	−3.46549
$647$	21.9733	0.863861	0.431930	−	0.901907i	$-0.357833\pi$
$647$	21.9733	0.863861	0.431930	+	0.901907i	$0.357833\pi$
$648$	−6.18181	−0.242845
$649$	37.2003	1.46024
$650$	−36.0906	−1.41559
$651$	−20.5797	−0.806581
$652$	64.7776	2.53689
$653$	−25.8305	−1.01083	−0.505413	−	0.862878i	$-0.668660\pi$
$653$	−25.8305	−1.01083	−0.505413	+	0.862878i	$0.668660\pi$
$654$	40.7469	1.59333
$655$	−10.0020	−0.390809
$656$	41.3185	1.61321
$657$	21.8324	0.851763
$658$	41.0256	1.59935
$659$	27.4121	1.06782	0.533912	−	0.845540i	$-0.320721\pi$
$659$	27.4121	1.06782	0.533912	+	0.845540i	$0.320721\pi$
$660$	49.2018	1.91518
$661$	4.67037	0.181657	0.0908283	−	0.995867i	$-0.471049\pi$
$661$	4.67037	0.181657	0.0908283	+	0.995867i	$0.471049\pi$
$662$	8.72948	0.339281
$663$	36.6385	1.42292
$664$	−114.413	−4.44010
$665$	−23.6061	−0.915405
$666$	25.7541	0.997953
$667$	−2.16438	−0.0838053
$668$	−43.7734	−1.69364
$669$	−2.57660	−0.0996170
$670$	4.48834	0.173400
$671$	−33.8963	−1.30855
$672$	−101.391	−3.91125
$673$	−34.6556	−1.33587	−0.667937	−	0.744218i	$-0.732823\pi$
$673$	−34.6556	−1.33587	−0.667937	+	0.744218i	$0.732823\pi$
$674$	12.1334	0.467362
$675$	11.8939	0.457797
$676$	101.300	3.89617
$677$	0.880503	0.0338405	0.0169202	−	0.999857i	$-0.494614\pi$
$677$	0.880503	0.0338405	0.0169202	+	0.999857i	$0.494614\pi$
$678$	12.9896	0.498862
$679$	−52.6443	−2.02030
$680$	117.885	4.52070
$681$	17.3415	0.664527
$682$	92.0291	3.52398
$683$	−26.9586	−1.03154	−0.515772	−	0.856726i	$-0.672495\pi$
$683$	−26.9586	−1.03154	−0.515772	+	0.856726i	$0.672495\pi$
$684$	55.5171	2.12275
$685$	7.60668	0.290636
$686$	42.2506	1.61314
$687$	23.8253	0.908992
$688$	−71.0027	−2.70695
$689$	−13.0093	−0.495617
$690$	1.41931	0.0540321
$691$	4.01561	0.152761	0.0763805	−	0.997079i	$-0.475664\pi$
$691$	4.01561	0.152761	0.0763805	+	0.997079i	$0.475664\pi$
$692$	−41.6933	−1.58494
$693$	−28.5265	−1.08363
$694$	−17.5814	−0.667382
$695$	17.6076	0.667894
$696$	−83.2231	−3.15456
$697$	−13.7233	−0.519809
$698$	47.5810	1.80097
$699$	−28.8850	−1.09253
$700$	−41.7236	−1.57700
$701$	−12.9604	−0.489508	−0.244754	−	0.969585i	$-0.578707\pi$
$701$	−12.9604	−0.489508	−0.244754	+	0.969585i	$0.578707\pi$
$702$	−78.6691	−2.96917
$703$	−22.7356	−0.857491
$704$	261.757	9.86532
$705$	−8.16227	−0.307409
$706$	−6.43466	−0.242171
$707$	−39.2861	−1.47751
$708$	46.5153	1.74815
$709$	−22.8221	−0.857103	−0.428552	−	0.903517i	$-0.640976\pi$
$709$	−22.8221	−0.857103	−0.428552	+	0.903517i	$0.640976\pi$
$710$	−42.6123	−1.59921
$711$	−22.2846	−0.835738
$712$	160.553	6.01698
$713$	1.98699	0.0744134
$714$	56.5914	2.11788
$715$	−43.9104	−1.64216
$716$	−17.3101	−0.646909
$717$	−6.05092	−0.225976
$718$	62.4342	2.33002
$719$	15.8102	0.589620	0.294810	−	0.955556i	$-0.404744\pi$
$719$	15.8102	0.589620	0.294810	+	0.955556i	$0.404744\pi$
$720$	−61.6645	−2.29810
$721$	−35.8516	−1.33518
$722$	−11.9038	−0.443013
$723$	1.34582	0.0500515
$724$	−77.7465	−2.88943
$725$	−16.9130	−0.628132
$726$	−38.1296	−1.41512
$727$	−22.1698	−0.822232	−0.411116	−	0.911583i	$-0.634861\pi$
$727$	−22.1698	−0.822232	−0.411116	+	0.911583i	$0.634861\pi$
$728$	183.228	6.79089
$729$	14.6839	0.543847
$730$	51.9079	1.92120
$731$	23.5825	0.872231
$732$	−42.3839	−1.56655
$733$	−20.6038	−0.761019	−0.380510	−	0.924777i	$-0.624251\pi$
$733$	−20.6038	−0.761019	−0.380510	+	0.924777i	$0.624251\pi$
$734$	−11.4304	−0.421902
$735$	3.38040	0.124688
$736$	9.78944	0.360844
$737$	−4.78817	−0.176374
$738$	11.5566	0.425405
$739$	10.6714	0.392554	0.196277	−	0.980549i	$-0.437115\pi$
$739$	10.6714	0.392554	0.196277	+	0.980549i	$0.437115\pi$
$740$	45.8305	1.68476
$741$	27.2376	1.00060
$742$	−20.0941	−0.737677
$743$	6.38222	0.234141	0.117070	−	0.993124i	$-0.462650\pi$
$743$	6.38222	0.234141	0.117070	+	0.993124i	$0.462650\pi$
$744$	76.4021	2.80104
$745$	−19.5435	−0.716020
$746$	9.45567	0.346197
$747$	−19.8778	−0.727289
$748$	−189.414	−6.92566
$749$	10.6562	0.389368
$750$	34.8304	1.27183
$751$	−17.3836	−0.634337	−0.317169	−	0.948369i	$-0.602732\pi$
$751$	−17.3836	−0.634337	−0.317169	+	0.948369i	$0.602732\pi$
$752$	−94.6077	−3.44999
$753$	29.8125	1.08643
$754$	111.866	4.07393
$755$	−20.1414	−0.733020
$756$	−90.9478	−3.30774
$757$	37.9814	1.38046	0.690229	−	0.723591i	$-0.257510\pi$
$757$	37.9814	1.38046	0.690229	+	0.723591i	$0.257510\pi$
$758$	14.1159	0.512714
$759$	−1.51412	−0.0549591
$760$	87.6377	3.17896
$761$	25.7815	0.934577	0.467289	−	0.884105i	$-0.345231\pi$
$761$	25.7815	0.934577	0.467289	+	0.884105i	$0.345231\pi$
$762$	16.5590	0.599869
$763$	−42.0406	−1.52197
$764$	86.7977	3.14023
$765$	20.4810	0.740492
$766$	−25.2736	−0.913171
$767$	−41.5128	−1.49894
$768$	136.777	4.93553
$769$	37.2405	1.34293	0.671463	−	0.741039i	$-0.265666\pi$
$769$	37.2405	1.34293	0.671463	+	0.741039i	$0.265666\pi$
$770$	−67.8235	−2.44419
$771$	10.2168	0.367949
$772$	−22.1858	−0.798483
$773$	27.8431	1.00145	0.500723	−	0.865608i	$-0.333068\pi$
$773$	27.8431	1.00145	0.500723	+	0.865608i	$0.333068\pi$
$774$	−19.8592	−0.713824
$775$	15.5268	0.557738
$776$	195.442	7.01597
$777$	14.6075	0.524042
$778$	69.7097	2.49921
$779$	−10.2021	−0.365529
$780$	−54.9056	−1.96593
$781$	45.4588	1.62664
$782$	−5.46396	−0.195391
$783$	−36.8664	−1.31750
$784$	39.1817	1.39935
$785$	−14.5951	−0.520922
$786$	17.8255	0.635815
$787$	45.1279	1.60864	0.804318	−	0.594199i	$-0.202531\pi$
$787$	45.1279	1.60864	0.804318	+	0.594199i	$0.202531\pi$
$788$	142.080	5.06140
$789$	10.5617	0.376007
$790$	−52.9831	−1.88505
$791$	−13.4020	−0.476521
$792$	105.905	3.76316
$793$	37.8257	1.34323
$794$	−57.1000	−2.02640
$795$	3.99783	0.141788
$796$	113.249	4.01399
$797$	0.519595	0.0184050	0.00920250	−	0.999958i	$-0.497071\pi$
$797$	0.519595	0.0184050	0.00920250	+	0.999958i	$0.497071\pi$
$798$	42.0708	1.48929
$799$	31.4226	1.11165
$800$	76.4968	2.70457
$801$	27.8939	0.985583
$802$	−58.9654	−2.08214
$803$	−55.3754	−1.95416
$804$	−5.98712	−0.211149
$805$	−1.46437	−0.0516123
$806$	−102.698	−3.61737
$807$	−10.1937	−0.358837
$808$	145.850	5.13098
$809$	−11.4395	−0.402192	−0.201096	−	0.979571i	$-0.564450\pi$
$809$	−11.4395	−0.402192	−0.201096	+	0.979571i	$0.564450\pi$
$810$	−2.55352	−0.0897215
$811$	−20.8150	−0.730912	−0.365456	−	0.930829i	$-0.619087\pi$
$811$	−20.8150	−0.730912	−0.365456	+	0.930829i	$0.619087\pi$
$812$	129.326	4.53846
$813$	−23.7156	−0.831743
$814$	−65.3226	−2.28955
$815$	17.7656	0.622302
$816$	−130.503	−4.56853
$817$	17.5316	0.613353
$818$	42.6053	1.48966
$819$	31.8334	1.11235
$820$	20.5655	0.718177
$821$	−10.1073	−0.352748	−0.176374	−	0.984323i	$-0.556437\pi$
$821$	−10.1073	−0.352748	−0.176374	+	0.984323i	$0.556437\pi$
$822$	−13.5566	−0.472842
$823$	21.4947	0.749257	0.374628	−	0.927175i	$-0.377770\pi$
$823$	21.4947	0.749257	0.374628	+	0.927175i	$0.377770\pi$
$824$	133.099	4.63673
$825$	−11.8317	−0.411926
$826$	−64.1202	−2.23103
$827$	30.5221	1.06136	0.530679	−	0.847573i	$-0.321937\pi$
$827$	30.5221	1.06136	0.530679	+	0.847573i	$0.321937\pi$
$828$	3.44393	0.119685
$829$	23.1867	0.805308	0.402654	−	0.915352i	$-0.368088\pi$
$829$	23.1867	0.805308	0.402654	+	0.915352i	$0.368088\pi$
$830$	−47.2607	−1.64044
$831$	3.06883	0.106457
$832$	−292.101	−10.1268
$833$	−13.0136	−0.450896
$834$	−31.3803	−1.08661
$835$	−12.0051	−0.415453
$836$	−140.813	−4.87012
$837$	33.8448	1.16985
$838$	−62.2298	−2.14969
$839$	−40.1019	−1.38447	−0.692235	−	0.721672i	$-0.743374\pi$
$839$	−40.1019	−1.38447	−0.692235	+	0.721672i	$0.743374\pi$
$840$	−56.3068	−1.94277
$841$	23.4234	0.807704
$842$	112.837	3.88860
$843$	−25.8472	−0.890226
$844$	31.5335	1.08543
$845$	27.7822	0.955735
$846$	−26.4614	−0.909763
$847$	39.3403	1.35175
$848$	46.3382	1.59126
$849$	23.5588	0.808535
$850$	−42.6966	−1.46448
$851$	−1.41037	−0.0483469
$852$	56.8417	1.94736
$853$	28.1507	0.963861	0.481930	−	0.876210i	$-0.339936\pi$
$853$	28.1507	0.963861	0.481930	+	0.876210i	$0.339936\pi$
$854$	58.4251	1.99927
$855$	15.2259	0.520714
$856$	−39.5611	−1.35217
$857$	−45.2090	−1.54431	−0.772155	−	0.635434i	$-0.780821\pi$
$857$	−45.2090	−1.54431	−0.772155	+	0.635434i	$0.780821\pi$
$858$	78.2573	2.67166
$859$	−5.37755	−0.183480	−0.0917399	−	0.995783i	$-0.529243\pi$
$859$	−5.37755	−0.183480	−0.0917399	+	0.995783i	$0.529243\pi$
$860$	−35.3402	−1.20509
$861$	6.55481	0.223387
$862$	88.6725	3.02020
$863$	−18.1486	−0.617784	−0.308892	−	0.951097i	$-0.599958\pi$
$863$	−18.1486	−0.617784	−0.308892	+	0.951097i	$0.599958\pi$
$864$	166.745	5.67280
$865$	−11.4346	−0.388789
$866$	76.3904	2.59585
$867$	25.8077	0.876475
$868$	−118.727	−4.02985
$869$	56.5224	1.91739
$870$	−34.3770	−1.16549
$871$	5.34324	0.181049
$872$	156.076	5.28540
$873$	33.9554	1.14922
$874$	−4.06198	−0.137399
$875$	−35.9363	−1.21487
$876$	−69.2414	−2.33945
$877$	4.16524	0.140650	0.0703251	−	0.997524i	$-0.477596\pi$
$877$	4.16524	0.140650	0.0703251	+	0.997524i	$0.477596\pi$
$878$	22.4511	0.757688
$879$	−29.2056	−0.985081
$880$	156.405	5.27242
$881$	−14.7796	−0.497938	−0.248969	−	0.968511i	$-0.580092\pi$
$881$	−14.7796	−0.497938	−0.248969	+	0.968511i	$0.580092\pi$
$882$	10.9590	0.369008
$883$	−17.9986	−0.605702	−0.302851	−	0.953038i	$-0.597938\pi$
$883$	−17.9986	−0.605702	−0.302851	+	0.953038i	$0.597938\pi$
$884$	211.372	7.10921
$885$	12.7571	0.428824
$886$	−91.1935	−3.06371
$887$	45.5760	1.53029	0.765145	−	0.643858i	$-0.222667\pi$
$887$	45.5760	1.53029	0.765145	+	0.643858i	$0.222667\pi$
$888$	−54.2305	−1.81986
$889$	−17.0848	−0.573005
$890$	66.3196	2.22304
$891$	2.72410	0.0912607
$892$	−14.8647	−0.497707
$893$	23.3600	0.781713
$894$	34.8305	1.16491
$895$	−4.74739	−0.158688
$896$	−254.603	−8.50570
$897$	1.68964	0.0564156
$898$	24.1779	0.806827
$899$	−48.1268	−1.60512
$900$	26.9116	0.897053
$901$	−15.3906	−0.512734
$902$	−29.3121	−0.975986
$903$	−11.2640	−0.374841
$904$	49.7551	1.65483
$905$	−21.3224	−0.708780
$906$	35.8960	1.19256
$907$	−13.1327	−0.436063	−0.218032	−	0.975942i	$-0.569964\pi$
$907$	−13.1327	−0.436063	−0.218032	+	0.975942i	$0.569964\pi$
$908$	100.045	3.32012
$909$	25.3394	0.840456
$910$	75.6860	2.50897
$911$	56.3744	1.86777	0.933883	−	0.357577i	$-0.116397\pi$
$911$	56.3744	1.86777	0.933883	+	0.357577i	$0.116397\pi$
$912$	−97.0179	−3.21259
$913$	50.4178	1.66858
$914$	−39.0432	−1.29144
$915$	−11.6240	−0.384278
$916$	137.451	4.54151
$917$	−18.3915	−0.607340
$918$	−93.0686	−3.07172
$919$	−17.0940	−0.563878	−0.281939	−	0.959432i	$-0.590977\pi$
$919$	−17.0940	−0.563878	−0.281939	+	0.959432i	$0.590977\pi$
$920$	5.43648	0.179236
$921$	−13.0480	−0.429947
$922$	−71.0745	−2.34071
$923$	−50.7287	−1.66975
$924$	90.4717	2.97630
$925$	−11.0210	−0.362367
$926$	−42.1648	−1.38562
$927$	23.1242	0.759498
$928$	−237.109	−7.78350
$929$	11.3970	0.373923	0.186961	−	0.982367i	$-0.440136\pi$
$929$	11.3970	0.373923	0.186961	+	0.982367i	$0.440136\pi$
$930$	31.5594	1.03487
$931$	−9.67453	−0.317070
$932$	−166.641	−5.45852
$933$	−0.992654	−0.0324980
$934$	33.0752	1.08225
$935$	−51.9478	−1.69888
$936$	−118.182	−3.86289
$937$	−28.3827	−0.927222	−0.463611	−	0.886039i	$-0.653447\pi$
$937$	−28.3827	−0.927222	−0.463611	+	0.886039i	$0.653447\pi$
$938$	8.25310	0.269473
$939$	−14.8002	−0.482987
$940$	−47.0892	−1.53588
$941$	20.1284	0.656167	0.328084	−	0.944649i	$-0.393597\pi$
$941$	20.1284	0.656167	0.328084	+	0.944649i	$0.393597\pi$
$942$	26.0114	0.847498
$943$	−0.632874	−0.0206092
$944$	147.865	4.81260
$945$	−24.9429	−0.811393
$946$	50.3706	1.63769
$947$	43.4788	1.41287	0.706435	−	0.707778i	$-0.250302\pi$
$947$	43.4788	1.41287	0.706435	+	0.707778i	$0.250302\pi$
$948$	70.6756	2.29544
$949$	61.7949	2.00595
$950$	−31.7412	−1.02982
$951$	21.7484	0.705239
$952$	216.766	7.02544
$953$	−30.4139	−0.985202	−0.492601	−	0.870255i	$-0.663954\pi$
$953$	−30.4139	−0.985202	−0.492601	+	0.870255i	$0.663954\pi$
$954$	12.9606	0.419616
$955$	23.8047	0.770303
$956$	−34.9085	−1.12902
$957$	36.6734	1.18548
$958$	87.0031	2.81094
$959$	13.9871	0.451666
$960$	89.7638	2.89711
$961$	13.1823	0.425234
$962$	72.8951	2.35023
$963$	−6.87321	−0.221486
$964$	7.76419	0.250068
$965$	−6.08456	−0.195869
$966$	2.60980	0.0839691
$967$	7.56190	0.243174	0.121587	−	0.992581i	$-0.461202\pi$
$967$	7.56190	0.243174	0.121587	+	0.992581i	$0.461202\pi$
$968$	−146.051	−4.69425
$969$	32.2231	1.03516
$970$	80.7312	2.59212
$971$	−47.1478	−1.51305	−0.756523	−	0.653968i	$-0.773103\pi$
$971$	−47.1478	−1.51305	−0.756523	+	0.653968i	$0.773103\pi$
$972$	94.3154	3.02517
$973$	32.3766	1.03795
$974$	−70.2977	−2.25248
$975$	13.2032	0.422842
$976$	−134.732	−4.31267
$977$	−48.3929	−1.54823	−0.774114	−	0.633046i	$-0.781804\pi$
$977$	−48.3929	−1.54823	−0.774114	+	0.633046i	$0.781804\pi$
$978$	−31.6619	−1.01244
$979$	−70.7499	−2.26118
$980$	19.5019	0.622967
$981$	27.1161	0.865750
$982$	−7.06012	−0.225297
$983$	42.6912	1.36164	0.680819	−	0.732452i	$-0.261624\pi$
$983$	42.6912	1.36164	0.680819	+	0.732452i	$0.261624\pi$
$984$	−24.3348	−0.775764
$985$	38.9662	1.24157
$986$	132.342	4.21463
$987$	−15.0087	−0.477732
$988$	157.137	4.99919
$989$	1.08755	0.0345820
$990$	43.7460	1.39034
$991$	12.9519	0.411430	0.205715	−	0.978612i	$-0.434048\pi$
$991$	12.9519	0.411430	0.205715	+	0.978612i	$0.434048\pi$
$992$	217.676	6.91122
$993$	−3.19356	−0.101345
$994$	−78.3549	−2.48527
$995$	31.0590	0.984638
$996$	63.0423	1.99757
$997$	−38.3038	−1.21309	−0.606546	−	0.795048i	$-0.707445\pi$
$997$	−38.3038	−1.21309	−0.606546	+	0.795048i	$0.707445\pi$
$998$	−118.597	−3.75412
$999$	−24.0231	−0.760059

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4013.2.a.b.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-2.81982 q^{2} +1.03159 q^{3} +5.95140 q^{4} +1.63220 q^{5} -2.90891 q^{6} +3.00127 q^{7} -11.1422 q^{8} -1.93581 q^{9} +O(q^{10})\)	\(q-2.81982 q^{2} +1.03159 q^{3} +5.95140 q^{4} +1.63220 q^{5} -2.90891 q^{6} +3.00127 q^{7} -11.1422 q^{8} -1.93581 q^{9} -4.60252 q^{10} +4.90998 q^{11} +6.13943 q^{12} -5.47917 q^{13} -8.46305 q^{14} +1.68377 q^{15} +19.5163 q^{16} -6.48207 q^{17} +5.45865 q^{18} -4.81887 q^{19} +9.71388 q^{20} +3.09610 q^{21} -13.8453 q^{22} -0.298932 q^{23} -11.4943 q^{24} -2.33592 q^{25} +15.4503 q^{26} -5.09176 q^{27} +17.8618 q^{28} +7.24040 q^{29} -4.74793 q^{30} -6.64697 q^{31} -32.7481 q^{32} +5.06510 q^{33} +18.2783 q^{34} +4.89868 q^{35} -11.5208 q^{36} +4.71805 q^{37} +13.5883 q^{38} -5.65228 q^{39} -18.1864 q^{40} +2.11712 q^{41} -8.73044 q^{42} -3.63811 q^{43} +29.2212 q^{44} -3.15964 q^{45} +0.842934 q^{46} -4.84762 q^{47} +20.1329 q^{48} +2.00764 q^{49} +6.58687 q^{50} -6.68687 q^{51} -32.6087 q^{52} +2.37433 q^{53} +14.3579 q^{54} +8.01407 q^{55} -33.4409 q^{56} -4.97112 q^{57} -20.4166 q^{58} +7.57648 q^{59} +10.0208 q^{60} -6.90355 q^{61} +18.7433 q^{62} -5.80990 q^{63} +53.3112 q^{64} -8.94311 q^{65} -14.2827 q^{66} -0.975192 q^{67} -38.5774 q^{68} -0.308376 q^{69} -13.8134 q^{70} +9.25846 q^{71} +21.5693 q^{72} -11.2782 q^{73} -13.3040 q^{74} -2.40972 q^{75} -28.6790 q^{76} +14.7362 q^{77} +15.9384 q^{78} +11.5118 q^{79} +31.8546 q^{80} +0.554809 q^{81} -5.96991 q^{82} +10.2684 q^{83} +18.4261 q^{84} -10.5801 q^{85} +10.2588 q^{86} +7.46916 q^{87} -54.7081 q^{88} -14.4094 q^{89} +8.90962 q^{90} -16.4445 q^{91} -1.77906 q^{92} -6.85698 q^{93} +13.6694 q^{94} -7.86536 q^{95} -33.7828 q^{96} -17.5407 q^{97} -5.66118 q^{98} -9.50479 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 157 q - 15 q^{2} - 51 q^{3} + 137 q^{4} - 13 q^{5} - 15 q^{6} - 49 q^{7} - 42 q^{8} + 144 q^{9}+O(q^{10}) \) 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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