LMFDB - Embedded newform 1045.4.a.b.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + \cdots + 17756160 $ x^20 - 8x^19 - 82x^18 + 700x^17 + 2826x^16 - 25467x^15 - 53768x^14 + 498499x^13 + 623613x^12 - 5673747x^11 - 4539454x^10 + 37893109x^9 + 19879768x^8 - 143049638x^7 - 45064360x^6 + 280461480x^5 + 43097920x^4 - 240447168x^3 - 36068096x^2 + 74703872*x + 17756160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.38435$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.384350 q^{2} -2.95996 q^{3} -7.85228 q^{4} +5.00000 q^{5} -1.13766 q^{6} +3.33433 q^{7} -6.09282 q^{8} -18.2387 q^{9} +O(q^{10})$	\(q+0.384350 q^{2} -2.95996 q^{3} -7.85228 q^{4} +5.00000 q^{5} -1.13766 q^{6} +3.33433 q^{7} -6.09282 q^{8} -18.2387 q^{9} +1.92175 q^{10} +11.0000 q^{11} +23.2424 q^{12} -16.3239 q^{13} +1.28155 q^{14} -14.7998 q^{15} +60.4764 q^{16} +0.290513 q^{17} -7.01002 q^{18} +19.0000 q^{19} -39.2614 q^{20} -9.86947 q^{21} +4.22785 q^{22} +149.375 q^{23} +18.0345 q^{24} +25.0000 q^{25} -6.27410 q^{26} +133.904 q^{27} -26.1821 q^{28} +93.9932 q^{29} -5.68829 q^{30} -293.593 q^{31} +71.9866 q^{32} -32.5595 q^{33} +0.111659 q^{34} +16.6717 q^{35} +143.215 q^{36} -299.079 q^{37} +7.30264 q^{38} +48.3181 q^{39} -30.4641 q^{40} +371.735 q^{41} -3.79333 q^{42} +204.117 q^{43} -86.3750 q^{44} -91.1933 q^{45} +57.4122 q^{46} -4.54042 q^{47} -179.008 q^{48} -331.882 q^{49} +9.60874 q^{50} -0.859905 q^{51} +128.180 q^{52} -54.3698 q^{53} +51.4661 q^{54} +55.0000 q^{55} -20.3155 q^{56} -56.2391 q^{57} +36.1262 q^{58} +16.9282 q^{59} +116.212 q^{60} -38.2212 q^{61} -112.842 q^{62} -60.8137 q^{63} -456.143 q^{64} -81.6197 q^{65} -12.5142 q^{66} +1037.76 q^{67} -2.28119 q^{68} -442.143 q^{69} +6.40774 q^{70} +649.596 q^{71} +111.125 q^{72} -360.910 q^{73} -114.951 q^{74} -73.9989 q^{75} -149.193 q^{76} +36.6776 q^{77} +18.5711 q^{78} -460.987 q^{79} +302.382 q^{80} +96.0929 q^{81} +142.876 q^{82} -1034.45 q^{83} +77.4978 q^{84} +1.45256 q^{85} +78.4523 q^{86} -278.216 q^{87} -67.0210 q^{88} -919.712 q^{89} -35.0501 q^{90} -54.4294 q^{91} -1172.93 q^{92} +869.022 q^{93} -1.74511 q^{94} +95.0000 q^{95} -213.077 q^{96} -113.688 q^{97} -127.559 q^{98} -200.625 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) $ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.384350	0.135888	0.0679441	−	0.997689i	$-0.478356\pi$
$2$	0.384350	0.135888	0.0679441	+	0.997689i	$0.478356\pi$
$3$	−2.95996	−0.569644	−0.284822	−	0.958580i	$-0.591934\pi$
$3$	−2.95996	−0.569644	−0.284822	+	0.958580i	$0.591934\pi$
$4$	−7.85228	−0.981534
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−1.13766	−0.0774078
$7$	3.33433	0.180037	0.0900185	−	0.995940i	$-0.471307\pi$
$7$	3.33433	0.180037	0.0900185	+	0.995940i	$0.471307\pi$
$8$	−6.09282	−0.269267
$9$	−18.2387	−0.675506
$10$	1.92175	0.0607710
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	23.2424	0.559125
$13$	−16.3239	−0.348265	−0.174133	−	0.984722i	$-0.555712\pi$
$13$	−16.3239	−0.348265	−0.174133	+	0.984722i	$0.555712\pi$
$14$	1.28155	0.0244649
$15$	−14.7998	−0.254752
$16$	60.4764	0.944944
$17$	0.290513	0.00414469	0.00207235	−	0.999998i	$-0.499340\pi$
$17$	0.290513	0.00414469	0.00207235	+	0.999998i	$0.499340\pi$
$18$	−7.01002	−0.0917933
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−39.2614	−0.438956
$21$	−9.86947	−0.102557
$22$	4.22785	0.0409718
$23$	149.375	1.35421	0.677105	−	0.735887i	$-0.263234\pi$
$23$	149.375	1.35421	0.677105	+	0.735887i	$0.263234\pi$
$24$	18.0345	0.153386
$25$	25.0000	0.200000
$26$	−6.27410	−0.0473251
$27$	133.904	0.954441
$28$	−26.1821	−0.176712
$29$	93.9932	0.601865	0.300933	−	0.953645i	$-0.402702\pi$
$29$	93.9932	0.601865	0.300933	+	0.953645i	$0.402702\pi$
$30$	−5.68829	−0.0346178
$31$	−293.593	−1.70100	−0.850498	−	0.525978i	$-0.823699\pi$
$31$	−293.593	−1.70100	−0.850498	+	0.525978i	$0.823699\pi$
$32$	71.9866	0.397674
$33$	−32.5595	−0.171754
$34$	0.111659	0.000563214 0
$35$	16.6717	0.0805149
$36$	143.215	0.663033
$37$	−299.079	−1.32887	−0.664435	−	0.747346i	$-0.731328\pi$
$37$	−299.079	−1.32887	−0.664435	+	0.747346i	$0.731328\pi$
$38$	7.30264	0.0311749
$39$	48.3181	0.198387
$40$	−30.4641	−0.120420
$41$	371.735	1.41598	0.707991	−	0.706222i	$-0.249602\pi$
$41$	371.735	1.41598	0.707991	+	0.706222i	$0.249602\pi$
$42$	−3.79333	−0.0139363
$43$	204.117	0.723897	0.361949	−	0.932198i	$-0.382112\pi$
$43$	204.117	0.723897	0.361949	+	0.932198i	$0.382112\pi$
$44$	−86.3750	−0.295944
$45$	−91.1933	−0.302096
$46$	57.4122	0.184021
$47$	−4.54042	−0.0140913	−0.00704563	−	0.999975i	$-0.502243\pi$
$47$	−4.54042	−0.0140913	−0.00704563	+	0.999975i	$0.502243\pi$
$48$	−179.008	−0.538281
$49$	−331.882	−0.967587
$50$	9.60874	0.0271776
$51$	−0.859905	−0.00236100
$52$	128.180	0.341834
$53$	−54.3698	−0.140911	−0.0704553	−	0.997515i	$-0.522445\pi$
$53$	−54.3698	−0.140911	−0.0704553	+	0.997515i	$0.522445\pi$
$54$	51.4661	0.129697
$55$	55.0000	0.134840
$56$	−20.3155	−0.0484780
$57$	−56.2391	−0.130685
$58$	36.1262	0.0817863
$59$	16.9282	0.0373536	0.0186768	−	0.999826i	$-0.494055\pi$
$59$	16.9282	0.0373536	0.0186768	+	0.999826i	$0.494055\pi$
$60$	116.212	0.250048
$61$	−38.2212	−0.0802250	−0.0401125	−	0.999195i	$-0.512772\pi$
$61$	−38.2212	−0.0802250	−0.0401125	+	0.999195i	$0.512772\pi$
$62$	−112.842	−0.231145
$63$	−60.8137	−0.121616
$64$	−456.143	−0.890905
$65$	−81.6197	−0.155749
$66$	−12.5142	−0.0233393
$67$	1037.76	1.89228	0.946140	−	0.323758i	$-0.104946\pi$
$67$	1037.76	1.89228	0.946140	+	0.323758i	$0.104946\pi$
$68$	−2.28119	−0.00406816
$69$	−442.143	−0.771417
$70$	6.40774	0.0109410
$71$	649.596	1.08581	0.542907	−	0.839793i	$-0.317323\pi$
$71$	649.596	1.08581	0.542907	+	0.839793i	$0.317323\pi$
$72$	111.125	0.181892
$73$	−360.910	−0.578647	−0.289324	−	0.957231i	$-0.593430\pi$
$73$	−360.910	−0.578647	−0.289324	+	0.957231i	$0.593430\pi$
$74$	−114.951	−0.180578
$75$	−73.9989	−0.113929
$76$	−149.193	−0.225179
$77$	36.6776	0.0542832
$78$	18.5711	0.0269584
$79$	−460.987	−0.656521	−0.328260	−	0.944587i	$-0.606462\pi$
$79$	−460.987	−0.656521	−0.328260	+	0.944587i	$0.606462\pi$
$80$	302.382	0.422592
$81$	96.0929	0.131815
$82$	142.876	0.192415
$83$	−1034.45	−1.36801	−0.684007	−	0.729476i	$-0.739764\pi$
$83$	−1034.45	−1.36801	−0.684007	+	0.729476i	$0.739764\pi$
$84$	77.4978	0.100663
$85$	1.45256	0.00185356
$86$	78.4523	0.0983690
$87$	−278.216	−0.342849
$88$	−67.0210	−0.0811871
$89$	−919.712	−1.09539	−0.547693	−	0.836680i	$-0.684494\pi$
$89$	−919.712	−1.09539	−0.547693	+	0.836680i	$0.684494\pi$
$90$	−35.0501	−0.0410512
$91$	−54.4294	−0.0627006
$92$	−1172.93	−1.32920
$93$	869.022	0.968962
$94$	−1.74511	−0.00191483
$95$	95.0000	0.102598
$96$	−213.077	−0.226532
$97$	−113.688	−0.119003	−0.0595014	−	0.998228i	$-0.518951\pi$
$97$	−113.688	−0.119003	−0.0595014	+	0.998228i	$0.518951\pi$
$98$	−127.559	−0.131484
$99$	−200.625	−0.203673
$100$	−196.307	−0.196307
$101$	642.611	0.633091	0.316545	−	0.948577i	$-0.397477\pi$
$101$	642.611	0.633091	0.316545	+	0.948577i	$0.397477\pi$
$102$	−0.330504	−0.000320831 0
$103$	−1624.36	−1.55391	−0.776956	−	0.629555i	$-0.783237\pi$
$103$	−1624.36	−1.55391	−0.776956	+	0.629555i	$0.783237\pi$
$104$	99.4588	0.0937763
$105$	−49.3473	−0.0458648
$106$	−20.8970	−0.0191481
$107$	425.497	0.384433	0.192216	−	0.981353i	$-0.438432\pi$
$107$	425.497	0.384433	0.192216	+	0.981353i	$0.438432\pi$
$108$	−1051.45	−0.936817
$109$	−1056.40	−0.928304	−0.464152	−	0.885756i	$-0.653641\pi$
$109$	−1056.40	−0.928304	−0.464152	+	0.885756i	$0.653641\pi$
$110$	21.1392	0.0183232
$111$	885.259	0.756983
$112$	201.648	0.170125
$113$	−8.55897	−0.00712531	−0.00356266	−	0.999994i	$-0.501134\pi$
$113$	−8.55897	−0.00712531	−0.00356266	+	0.999994i	$0.501134\pi$
$114$	−21.6155	−0.0177586
$115$	746.874	0.605621
$116$	−738.060	−0.590751
$117$	297.727	0.235255
$118$	6.50635	0.00507592
$119$	0.968666	0.000746197 0
$120$	90.1723	0.0685964
$121$	121.000	0.0909091
$122$	−14.6903	−0.0109016
$123$	−1100.32	−0.806605
$124$	2305.37	1.66959
$125$	125.000	0.0894427
$126$	−23.3737	−0.0165262
$127$	594.031	0.415053	0.207527	−	0.978229i	$-0.433459\pi$
$127$	594.031	0.415053	0.207527	+	0.978229i	$0.433459\pi$
$128$	−751.212	−0.518737
$129$	−604.178	−0.412363
$130$	−31.3705	−0.0211644
$131$	−2295.57	−1.53103	−0.765513	−	0.643420i	$-0.777515\pi$
$131$	−2295.57	−1.53103	−0.765513	+	0.643420i	$0.777515\pi$
$132$	255.666	0.168582
$133$	63.3523	0.0413033
$134$	398.863	0.257138
$135$	669.522	0.426839
$136$	−1.77004	−0.00111603
$137$	−2501.83	−1.56019	−0.780093	−	0.625664i	$-0.784828\pi$
$137$	−2501.83	−1.56019	−0.780093	+	0.625664i	$0.784828\pi$
$138$	−169.937	−0.104826
$139$	−3226.99	−1.96913	−0.984566	−	0.175012i	$-0.944004\pi$
$139$	−3226.99	−1.96913	−0.984566	+	0.175012i	$0.944004\pi$
$140$	−130.910	−0.0790282
$141$	13.4395	0.00802699
$142$	249.672	0.147549
$143$	−179.563	−0.105006
$144$	−1103.01	−0.638316
$145$	469.966	0.269162
$146$	−138.715	−0.0786313
$147$	982.357	0.551180
$148$	2348.45	1.30433
$149$	−2458.16	−1.35154	−0.675772	−	0.737111i	$-0.736190\pi$
$149$	−2458.16	−1.35154	−0.675772	+	0.737111i	$0.736190\pi$
$150$	−28.4414	−0.0154816
$151$	−1364.61	−0.735434	−0.367717	−	0.929938i	$-0.619860\pi$
$151$	−1364.61	−0.735434	−0.367717	+	0.929938i	$0.619860\pi$
$152$	−115.764	−0.0617741
$153$	−5.29857	−0.00279976
$154$	14.0970	0.00737644
$155$	−1467.97	−0.760709
$156$	−379.407	−0.194724
$157$	−2176.21	−1.10624	−0.553122	−	0.833100i	$-0.686564\pi$
$157$	−2176.21	−1.10624	−0.553122	+	0.833100i	$0.686564\pi$
$158$	−177.180	−0.0892134
$159$	160.932	0.0802688
$160$	359.933	0.177845
$161$	498.065	0.243808
$162$	36.9333	0.0179121
$163$	−1068.27	−0.513333	−0.256667	−	0.966500i	$-0.582624\pi$
$163$	−1068.27	−0.513333	−0.256667	+	0.966500i	$0.582624\pi$
$164$	−2918.96	−1.38983
$165$	−162.798	−0.0768107
$166$	−397.589	−0.185897
$167$	−2142.22	−0.992633	−0.496317	−	0.868142i	$-0.665315\pi$
$167$	−2142.22	−0.992633	−0.496317	+	0.868142i	$0.665315\pi$
$168$	60.1329	0.0276152
$169$	−1930.53	−0.878711
$170$	0.558293	0.000251877 0
$171$	−346.535	−0.154972
$172$	−1602.78	−0.710530
$173$	1875.74	0.824334	0.412167	−	0.911108i	$-0.364772\pi$
$173$	1875.74	0.824334	0.412167	+	0.911108i	$0.364772\pi$
$174$	−106.932	−0.0465891
$175$	83.3583	0.0360074
$176$	665.241	0.284911
$177$	−50.1067	−0.0212783
$178$	−353.491	−0.148850
$179$	2721.59	1.13643	0.568217	−	0.822879i	$-0.307634\pi$
$179$	2721.59	1.13643	0.568217	+	0.822879i	$0.307634\pi$
$180$	716.075	0.296517
$181$	−3919.27	−1.60949	−0.804743	−	0.593624i	$-0.797697\pi$
$181$	−3919.27	−1.60949	−0.804743	+	0.593624i	$0.797697\pi$
$182$	−20.9199	−0.00852026
$183$	113.133	0.0456997
$184$	−910.114	−0.364644
$185$	−1495.39	−0.594289
$186$	334.008	0.131670
$187$	3.19564	0.00124967
$188$	35.6527	0.0138311
$189$	446.482	0.171835
$190$	36.5132	0.0139418
$191$	1612.09	0.610716	0.305358	−	0.952238i	$-0.401224\pi$
$191$	1612.09	0.610716	0.305358	+	0.952238i	$0.401224\pi$
$192$	1350.16	0.507498
$193$	1034.22	0.385725	0.192862	−	0.981226i	$-0.438223\pi$
$193$	1034.22	0.385725	0.192862	+	0.981226i	$0.438223\pi$
$194$	−43.6959	−0.0161711
$195$	241.591	0.0887214
$196$	2606.03	0.949720
$197$	4965.97	1.79599	0.897997	−	0.440002i	$-0.145022\pi$
$197$	4965.97	1.79599	0.897997	+	0.440002i	$0.145022\pi$
$198$	−77.1103	−0.0276767
$199$	−1585.50	−0.564790	−0.282395	−	0.959298i	$-0.591129\pi$
$199$	−1585.50	−0.564790	−0.282395	+	0.959298i	$0.591129\pi$
$200$	−152.320	−0.0538534
$201$	−3071.73	−1.07793
$202$	246.987	0.0860295
$203$	313.404	0.108358
$204$	6.75221	0.00231740
$205$	1858.67	0.633246
$206$	−624.322	−0.211158
$207$	−2724.40	−0.914777
$208$	−987.214	−0.329091
$209$	209.000	0.0691714
$210$	−18.9666	−0.00623249
$211$	572.339	0.186737	0.0933684	−	0.995632i	$-0.470237\pi$
$211$	572.339	0.186737	0.0933684	+	0.995632i	$0.470237\pi$
$212$	426.926	0.138309
$213$	−1922.77	−0.618528
$214$	163.539	0.0522399
$215$	1020.59	0.323737
$216$	−815.855	−0.257000
$217$	−978.936	−0.306242
$218$	−406.028	−0.126145
$219$	1068.28	0.329623
$220$	−431.875	−0.132350
$221$	−4.74232	−0.00144345
$222$	340.249	0.102865
$223$	−2877.66	−0.864138	−0.432069	−	0.901841i	$-0.642216\pi$
$223$	−2877.66	−0.864138	−0.432069	+	0.901841i	$0.642216\pi$
$224$	240.027	0.0715959
$225$	−455.967	−0.135101
$226$	−3.28964	−0.000968245 0
$227$	5890.32	1.72227	0.861133	−	0.508379i	$-0.169755\pi$
$227$	5890.32	1.72227	0.861133	+	0.508379i	$0.169755\pi$
$228$	441.605	0.128272
$229$	−2420.52	−0.698483	−0.349242	−	0.937033i	$-0.613561\pi$
$229$	−2420.52	−0.698483	−0.349242	+	0.937033i	$0.613561\pi$
$230$	287.061	0.0822967
$231$	−108.564	−0.0309221
$232$	−572.683	−0.162062
$233$	−2448.85	−0.688538	−0.344269	−	0.938871i	$-0.611873\pi$
$233$	−2448.85	−0.688538	−0.344269	+	0.938871i	$0.611873\pi$
$234$	114.431	0.0319684
$235$	−22.7021	−0.00630180
$236$	−132.925	−0.0366639
$237$	1364.50	0.373983
$238$	0.372306	0.000101399 0
$239$	3498.41	0.946834	0.473417	−	0.880838i	$-0.343020\pi$
$239$	3498.41	0.946834	0.473417	+	0.880838i	$0.343020\pi$
$240$	−895.038	−0.240727
$241$	1255.25	0.335511	0.167755	−	0.985829i	$-0.446348\pi$
$241$	1255.25	0.335511	0.167755	+	0.985829i	$0.446348\pi$
$242$	46.5063	0.0123535
$243$	−3899.85	−1.02953
$244$	300.123	0.0787436
$245$	−1659.41	−0.432718
$246$	−422.907	−0.109608
$247$	−310.155	−0.0798975
$248$	1788.81	0.458022
$249$	3061.91	0.779280
$250$	48.0437	0.0121542
$251$	3541.76	0.890653	0.445327	−	0.895368i	$-0.353088\pi$
$251$	3541.76	0.890653	0.445327	+	0.895368i	$0.353088\pi$
$252$	477.526	0.119370
$253$	1643.12	0.408309
$254$	228.316	0.0564008
$255$	−4.29953	−0.00105587
$256$	3360.42	0.820415
$257$	−5455.79	−1.32421	−0.662107	−	0.749409i	$-0.730338\pi$
$257$	−5455.79	−1.32421	−0.662107	+	0.749409i	$0.730338\pi$
$258$	−232.215	−0.0560353
$259$	−997.227	−0.239246
$260$	640.900	0.152873
$261$	−1714.31	−0.406564
$262$	−882.300	−0.208048
$263$	3629.09	0.850872	0.425436	−	0.904989i	$-0.360121\pi$
$263$	3629.09	0.850872	0.425436	+	0.904989i	$0.360121\pi$
$264$	198.379	0.0462477
$265$	−271.849	−0.0630171
$266$	24.3494	0.00561263
$267$	2722.31	0.623979
$268$	−8148.79	−1.85734
$269$	3750.68	0.850122	0.425061	−	0.905165i	$-0.360253\pi$
$269$	3750.68	0.850122	0.425061	+	0.905165i	$0.360253\pi$
$270$	257.331	0.0580024
$271$	7843.37	1.75812	0.879060	−	0.476711i	$-0.158171\pi$
$271$	7843.37	1.75812	0.879060	+	0.476711i	$0.158171\pi$
$272$	17.5692	0.00391650
$273$	161.109	0.0357170
$274$	−961.576	−0.212011
$275$	275.000	0.0603023
$276$	3471.83	0.757172
$277$	5860.53	1.27121	0.635605	−	0.772014i	$-0.280751\pi$
$277$	5860.53	1.27121	0.635605	+	0.772014i	$0.280751\pi$
$278$	−1240.29	−0.267582
$279$	5354.75	1.14903
$280$	−101.577	−0.0216800
$281$	4691.34	0.995949	0.497975	−	0.867192i	$-0.334077\pi$
$281$	4691.34	0.995949	0.497975	+	0.867192i	$0.334077\pi$
$282$	5.16545	0.00109077
$283$	−3911.65	−0.821637	−0.410819	−	0.911717i	$-0.634757\pi$
$283$	−3911.65	−0.821637	−0.410819	+	0.911717i	$0.634757\pi$
$284$	−5100.81	−1.06576
$285$	−281.196	−0.0584442
$286$	−69.0151	−0.0142691
$287$	1239.49	0.254929
$288$	−1312.94	−0.268631
$289$	−4912.92	−0.999983
$290$	180.631	0.0365760
$291$	336.511	0.0677892
$292$	2833.96	0.567962
$293$	465.752	0.0928652	0.0464326	−	0.998921i	$-0.485215\pi$
$293$	465.752	0.0928652	0.0464326	+	0.998921i	$0.485215\pi$
$294$	377.568	0.0748988
$295$	84.6410	0.0167051
$296$	1822.23	0.357821
$297$	1472.95	0.287775
$298$	−944.792	−0.183659
$299$	−2438.39	−0.471624
$300$	581.060	0.111825
$301$	680.594	0.130328
$302$	−524.488	−0.0999368
$303$	−1902.10	−0.360636
$304$	1149.05	0.216785
$305$	−191.106	−0.0358777
$306$	−2.03650	−0.000380455 0
$307$	−9248.79	−1.71940	−0.859701	−	0.510798i	$-0.829350\pi$
$307$	−9248.79	−1.71940	−0.859701	+	0.510798i	$0.829350\pi$
$308$	−288.003	−0.0532808
$309$	4808.03	0.885176
$310$	−564.212	−0.103371
$311$	4566.86	0.832678	0.416339	−	0.909209i	$-0.363313\pi$
$311$	4566.86	0.832678	0.416339	+	0.909209i	$0.363313\pi$
$312$	−294.394	−0.0534191
$313$	948.110	0.171215	0.0856075	−	0.996329i	$-0.472717\pi$
$313$	948.110	0.171215	0.0856075	+	0.996329i	$0.472717\pi$
$314$	−836.425	−0.150325
$315$	−304.069	−0.0543883
$316$	3619.80	0.644398
$317$	6577.25	1.16535	0.582674	−	0.812706i	$-0.302006\pi$
$317$	6577.25	1.16535	0.582674	+	0.812706i	$0.302006\pi$
$318$	61.8542	0.0109076
$319$	1033.92	0.181469
$320$	−2280.72	−0.398425
$321$	−1259.45	−0.218990
$322$	191.431	0.0331306
$323$	5.51975	0.000950857 0
$324$	−754.548	−0.129381
$325$	−408.099	−0.0696530
$326$	−410.589	−0.0697559
$327$	3126.91	0.528802
$328$	−2264.91	−0.381277
$329$	−15.1393	−0.00253695
$330$	−62.5712	−0.0104377
$331$	−2530.01	−0.420127	−0.210063	−	0.977688i	$-0.567367\pi$
$331$	−2530.01	−0.420127	−0.210063	+	0.977688i	$0.567367\pi$
$332$	8122.75	1.34275
$333$	5454.79	0.897660
$334$	−823.361	−0.134887
$335$	5188.81	0.846253
$336$	−596.870	−0.0969105
$337$	−9669.47	−1.56300	−0.781498	−	0.623907i	$-0.785544\pi$
$337$	−9669.47	−1.56300	−0.781498	+	0.623907i	$0.785544\pi$
$338$	−741.998	−0.119406
$339$	25.3342	0.00405889
$340$	−11.4059	−0.00181933
$341$	−3229.52	−0.512870
$342$	−133.190	−0.0210588
$343$	−2250.28	−0.354238
$344$	−1243.65	−0.194922
$345$	−2210.71	−0.344988
$346$	720.940	0.112017
$347$	−8842.00	−1.36791	−0.683953	−	0.729526i	$-0.739741\pi$
$347$	−8842.00	−1.36791	−0.683953	+	0.729526i	$0.739741\pi$
$348$	2184.62	0.336518
$349$	10968.6	1.68233	0.841167	−	0.540776i	$-0.181869\pi$
$349$	10968.6	1.68233	0.841167	+	0.540776i	$0.181869\pi$
$350$	32.0387	0.00489298
$351$	−2185.85	−0.332399
$352$	791.853	0.119903
$353$	958.478	0.144517	0.0722587	−	0.997386i	$-0.476979\pi$
$353$	958.478	0.144517	0.0722587	+	0.997386i	$0.476979\pi$
$354$	−19.2585	−0.00289146
$355$	3247.98	0.485591
$356$	7221.83	1.07516
$357$	−2.86721	−0.000425066 0
$358$	1046.04	0.154428
$359$	−4362.92	−0.641409	−0.320705	−	0.947179i	$-0.603920\pi$
$359$	−4362.92	−0.641409	−0.320705	+	0.947179i	$0.603920\pi$
$360$	555.624	0.0813444
$361$	361.000	0.0526316
$362$	−1506.37	−0.218710
$363$	−358.155	−0.0517858
$364$	427.395	0.0615428
$365$	−1804.55	−0.258779
$366$	43.4827	0.00621004
$367$	−7069.28	−1.00549	−0.502743	−	0.864436i	$-0.667676\pi$
$367$	−7069.28	−1.00549	−0.502743	+	0.864436i	$0.667676\pi$
$368$	9033.66	1.27965
$369$	−6779.95	−0.956504
$370$	−574.754	−0.0807568
$371$	−181.287	−0.0253691
$372$	−6823.80	−0.951069
$373$	−2819.30	−0.391361	−0.195681	−	0.980668i	$-0.562692\pi$
$373$	−2819.30	−0.391361	−0.195681	+	0.980668i	$0.562692\pi$
$374$	1.22824	0.000169815 0
$375$	−369.994	−0.0509505
$376$	27.6640	0.00379431
$377$	−1534.34	−0.209609
$378$	171.605	0.0233503
$379$	−4880.95	−0.661524	−0.330762	−	0.943714i	$-0.607306\pi$
$379$	−4880.95	−0.661524	−0.330762	+	0.943714i	$0.607306\pi$
$380$	−745.966	−0.100703
$381$	−1758.31	−0.236432
$382$	619.606	0.0829890
$383$	−13189.9	−1.75972	−0.879859	−	0.475235i	$-0.842363\pi$
$383$	−13189.9	−1.75972	−0.879859	+	0.475235i	$0.842363\pi$
$384$	2223.55	0.295495
$385$	183.388	0.0242762
$386$	397.503	0.0524154
$387$	−3722.82	−0.488997
$388$	892.710	0.116805
$389$	−9426.16	−1.22860	−0.614300	−	0.789073i	$-0.710561\pi$
$389$	−9426.16	−1.22860	−0.614300	+	0.789073i	$0.710561\pi$
$390$	92.8553	0.0120562
$391$	43.3953	0.00561278
$392$	2022.10	0.260539
$393$	6794.77	0.872139
$394$	1908.67	0.244054
$395$	−2304.94	−0.293605
$396$	1575.37	0.199912
$397$	−2872.63	−0.363156	−0.181578	−	0.983377i	$-0.558120\pi$
$397$	−2872.63	−0.363156	−0.181578	+	0.983377i	$0.558120\pi$
$398$	−609.387	−0.0767483
$399$	−187.520	−0.0235282
$400$	1511.91	0.188989
$401$	3502.90	0.436226	0.218113	−	0.975924i	$-0.430010\pi$
$401$	3502.90	0.436226	0.218113	+	0.975924i	$0.430010\pi$
$402$	−1180.62	−0.146477
$403$	4792.60	0.592398
$404$	−5045.96	−0.621401
$405$	480.465	0.0589493
$406$	120.457	0.0147246
$407$	−3289.86	−0.400670
$408$	5.23924	0.000635738 0
$409$	8650.57	1.04583	0.522913	−	0.852386i	$-0.324845\pi$
$409$	8650.57	1.04583	0.522913	+	0.852386i	$0.324845\pi$
$410$	714.381	0.0860506
$411$	7405.29	0.888750
$412$	12754.9	1.52522
$413$	56.4442	0.00672503
$414$	−1047.12	−0.124307
$415$	−5172.23	−0.611794
$416$	−1175.11	−0.138496
$417$	9551.74	1.12170
$418$	80.3291	0.00939958
$419$	−7838.87	−0.913971	−0.456985	−	0.889474i	$-0.651071\pi$
$419$	−7838.87	−0.913971	−0.456985	+	0.889474i	$0.651071\pi$
$420$	387.489	0.0450179
$421$	137.348	0.0159001	0.00795006	−	0.999968i	$-0.497469\pi$
$421$	137.348	0.0159001	0.00795006	+	0.999968i	$0.497469\pi$
$422$	219.978	0.0253753
$423$	82.8113	0.00951873
$424$	331.265	0.0379426
$425$	7.26282	0.000828938 0
$426$	−739.018	−0.0840505
$427$	−127.442	−0.0144435
$428$	−3341.12	−0.377334
$429$	531.499	0.0598159
$430$	392.262	0.0439920
$431$	10087.5	1.12738	0.563688	−	0.825988i	$-0.309382\pi$
$431$	10087.5	1.12738	0.563688	+	0.825988i	$0.309382\pi$
$432$	8098.06	0.901894
$433$	3964.08	0.439957	0.219979	−	0.975505i	$-0.429401\pi$
$433$	3964.08	0.439957	0.219979	+	0.975505i	$0.429401\pi$
$434$	−376.254	−0.0416147
$435$	−1391.08	−0.153327
$436$	8295.17	0.911162
$437$	2838.12	0.310677
$438$	410.591	0.0447918
$439$	−1663.86	−0.180892	−0.0904460	−	0.995901i	$-0.528829\pi$
$439$	−1663.86	−0.180892	−0.0904460	+	0.995901i	$0.528829\pi$
$440$	−335.105	−0.0363080
$441$	6053.09	0.653611
$442$	−1.82271	−0.000196148 0
$443$	2980.69	0.319677	0.159838	−	0.987143i	$-0.448903\pi$
$443$	2980.69	0.319677	0.159838	+	0.987143i	$0.448903\pi$
$444$	−6951.30	−0.743004
$445$	−4598.56	−0.489871
$446$	−1106.03	−0.117426
$447$	7276.03	0.769898
$448$	−1520.93	−0.160396
$449$	−3102.61	−0.326105	−0.163052	−	0.986617i	$-0.552134\pi$
$449$	−3102.61	−0.326105	−0.163052	+	0.986617i	$0.552134\pi$
$450$	−175.251	−0.0183587
$451$	4089.08	0.426934
$452$	67.2074	0.00699374
$453$	4039.19	0.418935
$454$	2263.94	0.234036
$455$	−272.147	−0.0280406
$456$	342.655	0.0351892
$457$	9400.25	0.962199	0.481100	−	0.876666i	$-0.340238\pi$
$457$	9400.25	0.962199	0.481100	+	0.876666i	$0.340238\pi$
$458$	−930.327	−0.0949156
$459$	38.9010	0.00395586
$460$	−5864.66	−0.594438
$461$	−11396.2	−1.15136	−0.575678	−	0.817676i	$-0.695262\pi$
$461$	−11396.2	−1.15136	−0.575678	+	0.817676i	$0.695262\pi$
$462$	−41.7266	−0.00420194
$463$	−10459.4	−1.04987	−0.524936	−	0.851142i	$-0.675911\pi$
$463$	−10459.4	−1.04987	−0.524936	+	0.851142i	$0.675911\pi$
$464$	5684.37	0.568729
$465$	4345.11	0.433333
$466$	−941.213	−0.0935641
$467$	−25.2633	−0.00250331	−0.00125166	−	0.999999i	$-0.500398\pi$
$467$	−25.2633	−0.00250331	−0.00125166	+	0.999999i	$0.500398\pi$
$468$	−2337.83	−0.230911
$469$	3460.24	0.340680
$470$	−8.72555	−0.000856340 0
$471$	6441.48	0.630165
$472$	−103.140	−0.0100581
$473$	2245.29	0.218263
$474$	524.446	0.0508198
$475$	475.000	0.0458831
$476$	−7.60623	−0.000732418 0
$477$	991.632	0.0951860
$478$	1344.61	0.128664
$479$	13211.0	1.26018	0.630091	−	0.776521i	$-0.283018\pi$
$479$	13211.0	1.26018	0.630091	+	0.776521i	$0.283018\pi$
$480$	−1065.39	−0.101308
$481$	4882.14	0.462799
$482$	482.457	0.0455919
$483$	−1474.25	−0.138883
$484$	−950.125	−0.0892304
$485$	−568.440	−0.0532197
$486$	−1498.91	−0.139901
$487$	−8207.59	−0.763699	−0.381850	−	0.924224i	$-0.624713\pi$
$487$	−8207.59	−0.763699	−0.381850	+	0.924224i	$0.624713\pi$
$488$	232.875	0.0216019
$489$	3162.03	0.292417
$490$	−637.794	−0.0588012
$491$	5552.20	0.510321	0.255160	−	0.966899i	$-0.417872\pi$
$491$	5552.20	0.510321	0.255160	+	0.966899i	$0.417872\pi$
$492$	8640.00	0.791710
$493$	27.3062	0.00249455
$494$	−119.208	−0.0108571
$495$	−1003.13	−0.0910852
$496$	−17755.5	−1.60735
$497$	2165.97	0.195487
$498$	1176.84	0.105895
$499$	20226.8	1.81458	0.907292	−	0.420500i	$-0.138145\pi$
$499$	20226.8	1.81458	0.907292	+	0.420500i	$0.138145\pi$
$500$	−981.534	−0.0877911
$501$	6340.87	0.565447
$502$	1361.27	0.121029
$503$	−18085.7	−1.60319	−0.801594	−	0.597869i	$-0.796014\pi$
$503$	−18085.7	−1.60319	−0.801594	+	0.597869i	$0.796014\pi$
$504$	370.527	0.0327472
$505$	3213.05	0.283127
$506$	631.534	0.0554844
$507$	5714.28	0.500552
$508$	−4664.50	−0.407389
$509$	8111.00	0.706314	0.353157	−	0.935564i	$-0.385108\pi$
$509$	8111.00	0.706314	0.353157	+	0.935564i	$0.385108\pi$
$510$	−1.65252	−0.000143480 0
$511$	−1203.39	−0.104178
$512$	7301.27	0.630222
$513$	2544.18	0.218964
$514$	−2096.93	−0.179945
$515$	−8121.80	−0.694930
$516$	4744.17	0.404749
$517$	−49.9447	−0.00424867
$518$	−383.284	−0.0325107
$519$	−5552.11	−0.469577
$520$	497.294	0.0419380
$521$	5744.67	0.483068	0.241534	−	0.970392i	$-0.422349\pi$
$521$	5744.67	0.483068	0.241534	+	0.970392i	$0.422349\pi$
$522$	−658.894	−0.0552472
$523$	−17835.3	−1.49118	−0.745588	−	0.666407i	$-0.767831\pi$
$523$	−17835.3	−1.49118	−0.745588	+	0.666407i	$0.767831\pi$
$524$	18025.4	1.50276
$525$	−246.737	−0.0205114
$526$	1394.84	0.115623
$527$	−85.2926	−0.00705010
$528$	−1969.08	−0.162298
$529$	10145.8	0.833882
$530$	−104.485	−0.00856328
$531$	−308.748	−0.0252326
$532$	−497.459	−0.0405406
$533$	−6068.18	−0.493137
$534$	1046.32	0.0847914
$535$	2127.48	0.171924
$536$	−6322.89	−0.509528
$537$	−8055.80	−0.647362
$538$	1441.57	0.115522
$539$	−3650.70	−0.291738
$540$	−5257.27	−0.418957
$541$	−22649.9	−1.79999	−0.899995	−	0.435900i	$-0.856430\pi$
$541$	−22649.9	−1.79999	−0.899995	+	0.435900i	$0.856430\pi$
$542$	3014.59	0.238908
$543$	11600.9	0.916833
$544$	20.9130	0.00164823
$545$	−5282.02	−0.415150
$546$	61.9220	0.00485351
$547$	−3738.05	−0.292189	−0.146094	−	0.989271i	$-0.546670\pi$
$547$	−3738.05	−0.292189	−0.146094	+	0.989271i	$0.546670\pi$
$548$	19645.0	1.53138
$549$	697.104	0.0541925
$550$	105.696	0.00819436
$551$	1785.87	0.138077
$552$	2693.90	0.207717
$553$	−1537.08	−0.118198
$554$	2252.49	0.172742
$555$	4426.30	0.338533
$556$	25339.2	1.93277
$557$	164.394	0.0125056	0.00625279	−	0.999980i	$-0.498010\pi$
$557$	164.394	0.0125056	0.00625279	+	0.999980i	$0.498010\pi$
$558$	2058.09	0.156140
$559$	−3332.00	−0.252108
$560$	1008.24	0.0760821
$561$	−9.45896	−0.000711867 0
$562$	1803.11	0.135338
$563$	−10248.1	−0.767148	−0.383574	−	0.923510i	$-0.625307\pi$
$563$	−10248.1	−0.767148	−0.383574	+	0.923510i	$0.625307\pi$
$564$	−105.530	−0.00787877
$565$	−42.7948	−0.00318654
$566$	−1503.44	−0.111651
$567$	320.406	0.0237315
$568$	−3957.87	−0.292374
$569$	−12760.0	−0.940117	−0.470059	−	0.882635i	$-0.655767\pi$
$569$	−12760.0	−0.940117	−0.470059	+	0.882635i	$0.655767\pi$
$570$	−108.077	−0.00794187
$571$	24384.4	1.78714	0.893570	−	0.448924i	$-0.148193\pi$
$571$	24384.4	1.78714	0.893570	+	0.448924i	$0.148193\pi$
$572$	1409.98	0.103067
$573$	−4771.71	−0.347890
$574$	476.396	0.0346418
$575$	3734.37	0.270842
$576$	8319.45	0.601812
$577$	−268.407	−0.0193656	−0.00968278	−	0.999953i	$-0.503082\pi$
$577$	−268.407	−0.0193656	−0.00968278	+	0.999953i	$0.503082\pi$
$578$	−1888.28	−0.135886
$579$	−3061.25	−0.219726
$580$	−3690.30	−0.264192
$581$	−3449.18	−0.246293
$582$	129.338	0.00921174
$583$	−598.067	−0.0424861
$584$	2198.96	0.155811
$585$	1488.63	0.105209
$586$	179.012	0.0126193
$587$	−5716.83	−0.401974	−0.200987	−	0.979594i	$-0.564415\pi$
$587$	−5716.83	−0.401974	−0.200987	+	0.979594i	$0.564415\pi$
$588$	−7713.73	−0.541002
$589$	−5578.27	−0.390235
$590$	32.5318	0.00227002
$591$	−14699.0	−1.02308
$592$	−18087.2	−1.25571
$593$	−7424.36	−0.514135	−0.257068	−	0.966393i	$-0.582756\pi$
$593$	−7424.36	−0.514135	−0.257068	+	0.966393i	$0.582756\pi$
$594$	566.127	0.0391052
$595$	4.84333	0.000333710 0
$596$	19302.1	1.32659
$597$	4693.01	0.321729
$598$	−937.193	−0.0640881
$599$	23079.3	1.57428	0.787141	−	0.616773i	$-0.211560\pi$
$599$	23079.3	1.57428	0.787141	+	0.616773i	$0.211560\pi$
$600$	450.862	0.0306772
$601$	−20960.9	−1.42265	−0.711326	−	0.702862i	$-0.751905\pi$
$601$	−20960.9	−1.42265	−0.711326	+	0.702862i	$0.751905\pi$
$602$	261.586	0.0177101
$603$	−18927.4	−1.27825
$604$	10715.3	0.721854
$605$	605.000	0.0406558
$606$	−731.071	−0.0490062
$607$	11949.8	0.799055	0.399527	−	0.916721i	$-0.369174\pi$
$607$	11949.8	0.799055	0.399527	+	0.916721i	$0.369174\pi$
$608$	1367.75	0.0912326
$609$	−927.662	−0.0617254
$610$	−73.4515	−0.00487535
$611$	74.1176	0.00490749
$612$	41.6058	0.00274806
$613$	17917.8	1.18058	0.590288	−	0.807193i	$-0.299014\pi$
$613$	17917.8	1.18058	0.590288	+	0.807193i	$0.299014\pi$
$614$	−3554.77	−0.233646
$615$	−5501.59	−0.360725
$616$	−223.470	−0.0146167
$617$	−23896.6	−1.55923	−0.779613	−	0.626261i	$-0.784584\pi$
$617$	−23896.6	−1.55923	−0.779613	+	0.626261i	$0.784584\pi$
$618$	1847.96	0.120285
$619$	15353.8	0.996965	0.498483	−	0.866900i	$-0.333891\pi$
$619$	15353.8	0.996965	0.498483	+	0.866900i	$0.333891\pi$
$620$	11526.9	0.746662
$621$	20002.0	1.29251
$622$	1755.27	0.113151
$623$	−3066.62	−0.197210
$624$	2922.11	0.187465
$625$	625.000	0.0400000
$626$	364.406	0.0232661
$627$	−618.631	−0.0394031
$628$	17088.2	1.08582
$629$	−86.8862	−0.00550776
$630$	−116.869	−0.00739073
$631$	−22966.3	−1.44893	−0.724466	−	0.689311i	$-0.757913\pi$
$631$	−22966.3	−1.44893	−0.724466	+	0.689311i	$0.757913\pi$
$632$	2808.71	0.176779
$633$	−1694.10	−0.106373
$634$	2527.96	0.158357
$635$	2970.16	0.185617
$636$	−1263.68	−0.0787866
$637$	5417.63	0.336977
$638$	397.389	0.0246595
$639$	−11847.8	−0.733475
$640$	−3756.06	−0.231986
$641$	30136.4	1.85697	0.928483	−	0.371375i	$-0.121114\pi$
$641$	30136.4	1.85697	0.928483	+	0.371375i	$0.121114\pi$
$642$	−484.070	−0.0297581
$643$	−1429.72	−0.0876866	−0.0438433	−	0.999038i	$-0.513960\pi$
$643$	−1429.72	−0.0876866	−0.0438433	+	0.999038i	$0.513960\pi$
$644$	−3910.94	−0.239306
$645$	−3020.89	−0.184414
$646$	2.12151	0.000129210 0
$647$	−24147.1	−1.46727	−0.733633	−	0.679546i	$-0.762177\pi$
$647$	−24147.1	−1.46727	−0.733633	+	0.679546i	$0.762177\pi$
$648$	−585.477	−0.0354934
$649$	186.210	0.0112625
$650$	−156.853	−0.00946502
$651$	2897.61	0.174449
$652$	8388.35	0.503854
$653$	−8610.79	−0.516028	−0.258014	−	0.966141i	$-0.583068\pi$
$653$	−8610.79	−0.516028	−0.258014	+	0.966141i	$0.583068\pi$
$654$	1201.83	0.0718580
$655$	−11477.8	−0.684696
$656$	22481.2	1.33802
$657$	6582.51	0.390880
$658$	−5.81877	−0.000344741 0
$659$	−19242.7	−1.13747	−0.568734	−	0.822522i	$-0.692567\pi$
$659$	−19242.7	−1.13747	−0.568734	+	0.822522i	$0.692567\pi$
$660$	1278.33	0.0753924
$661$	14899.8	0.876758	0.438379	−	0.898790i	$-0.355553\pi$
$661$	14899.8	0.876758	0.438379	+	0.898790i	$0.355553\pi$
$662$	−972.409	−0.0570903
$663$	14.0370	0.000822253 0
$664$	6302.69	0.368361
$665$	316.761	0.0184714
$666$	2096.55	0.121981
$667$	14040.2	0.815051
$668$	16821.3	0.974304
$669$	8517.76	0.492250
$670$	1994.32	0.114996
$671$	−420.433	−0.0241887
$672$	−710.470	−0.0407842
$673$	24035.7	1.37668	0.688341	−	0.725388i	$-0.258340\pi$
$673$	24035.7	1.37668	0.688341	+	0.725388i	$0.258340\pi$
$674$	−3716.46	−0.212393
$675$	3347.61	0.190888
$676$	15159.0	0.862485
$677$	30961.3	1.75767	0.878833	−	0.477129i	$-0.158323\pi$
$677$	30961.3	1.75767	0.878833	+	0.477129i	$0.158323\pi$
$678$	9.73718	0.000551555 0
$679$	−379.073	−0.0214249
$680$	−8.85021	−0.000499103 0
$681$	−17435.1	−0.981078
$682$	−1241.27	−0.0696929
$683$	8961.07	0.502029	0.251015	−	0.967983i	$-0.419236\pi$
$683$	8961.07	0.502029	0.251015	+	0.967983i	$0.419236\pi$
$684$	2721.09	0.152110
$685$	−12509.1	−0.697736
$686$	−864.894	−0.0481368
$687$	7164.64	0.397887
$688$	12344.3	0.684042
$689$	887.529	0.0490742
$690$	−849.687	−0.0468798
$691$	−7506.36	−0.413249	−0.206625	−	0.978420i	$-0.566248\pi$
$691$	−7506.36	−0.413249	−0.206625	+	0.978420i	$0.566248\pi$
$692$	−14728.8	−0.809113
$693$	−668.951	−0.0366686
$694$	−3398.42	−0.185882
$695$	−16134.9	−0.880623
$696$	1695.12	0.0923178
$697$	107.994	0.00586880
$698$	4215.77	0.228609
$699$	7248.48	0.392221
$700$	−654.552	−0.0353425
$701$	9806.45	0.528366	0.264183	−	0.964473i	$-0.414898\pi$
$701$	9806.45	0.528366	0.264183	+	0.964473i	$0.414898\pi$
$702$	−840.130	−0.0451690
$703$	−5682.49	−0.304864
$704$	−5017.58	−0.268618
$705$	67.1973	0.00358978
$706$	368.391	0.0196382
$707$	2142.68	0.113980
$708$	393.452	0.0208853
$709$	−4159.70	−0.220340	−0.110170	−	0.993913i	$-0.535139\pi$
$709$	−4159.70	−0.220340	−0.110170	+	0.993913i	$0.535139\pi$
$710$	1248.36	0.0659861
$711$	8407.80	0.443484
$712$	5603.64	0.294951
$713$	−43855.4	−2.30350
$714$	−1.10201	−5.77615e−5 0
$715$	−897.817	−0.0469601
$716$	−21370.7	−1.11545
$717$	−10355.1	−0.539358
$718$	−1676.89	−0.0871599
$719$	12.8241	0.000665171 0	0.000332585	−	1.00000i	$-0.499894\pi$
$719$	12.8241	0.000665171 0	0.000332585		1.00000i	$0.499894\pi$
$720$	−5515.05	−0.285463
$721$	−5416.15	−0.279761
$722$	138.750	0.00715201
$723$	−3715.50	−0.191122
$724$	30775.2	1.57977
$725$	2349.83	0.120373
$726$	−137.657	−0.00703707
$727$	−18632.9	−0.950559	−0.475279	−	0.879835i	$-0.657653\pi$
$727$	−18632.9	−0.950559	−0.475279	+	0.879835i	$0.657653\pi$
$728$	331.628	0.0168832
$729$	8948.87	0.454650
$730$	−693.577	−0.0351650
$731$	59.2987	0.00300033
$732$	−888.352	−0.0448558
$733$	−7663.22	−0.386149	−0.193075	−	0.981184i	$-0.561846\pi$
$733$	−7663.22	−0.386149	−0.193075	+	0.981184i	$0.561846\pi$
$734$	−2717.07	−0.136634
$735$	4911.78	0.246495
$736$	10753.0	0.538533
$737$	11415.4	0.570544
$738$	−2605.87	−0.129978
$739$	−34280.6	−1.70640	−0.853201	−	0.521582i	$-0.825342\pi$
$739$	−34280.6	−1.70640	−0.853201	+	0.521582i	$0.825342\pi$
$740$	11742.2	0.583315
$741$	918.045	0.0455131
$742$	−69.6775	−0.00344736
$743$	−31121.5	−1.53666	−0.768328	−	0.640056i	$-0.778911\pi$
$743$	−31121.5	−1.53666	−0.768328	+	0.640056i	$0.778911\pi$
$744$	−5294.79	−0.260909
$745$	−12290.8	−0.604429
$746$	−1083.60	−0.0531814
$747$	18866.9	0.924102
$748$	−25.0931	−0.00122660
$749$	1418.75	0.0692121
$750$	−142.207	−0.00692356
$751$	−35994.2	−1.74893	−0.874464	−	0.485090i	$-0.838787\pi$
$751$	−35994.2	−1.74893	−0.874464	+	0.485090i	$0.838787\pi$
$752$	−274.589	−0.0133154
$753$	−10483.5	−0.507355
$754$	−589.723	−0.0284833
$755$	−6823.06	−0.328896
$756$	−3505.90	−0.168662
$757$	−2262.01	−0.108605	−0.0543026	−	0.998525i	$-0.517294\pi$
$757$	−2262.01	−0.108605	−0.0543026	+	0.998525i	$0.517294\pi$
$758$	−1875.99	−0.0898932
$759$	−4863.57	−0.232591
$760$	−578.818	−0.0276262
$761$	5716.95	0.272325	0.136163	−	0.990687i	$-0.456523\pi$
$761$	5716.95	0.272325	0.136163	+	0.990687i	$0.456523\pi$
$762$	−675.804	−0.0321284
$763$	−3522.40	−0.167129
$764$	−12658.6	−0.599439
$765$	−26.4928	−0.00125209
$766$	−5069.53	−0.239125
$767$	−276.335	−0.0130090
$768$	−9946.69	−0.467344
$769$	19589.0	0.918592	0.459296	−	0.888283i	$-0.348102\pi$
$769$	19589.0	0.918592	0.459296	+	0.888283i	$0.348102\pi$
$770$	70.4852	0.00329884
$771$	16148.9	0.754330
$772$	−8120.99	−0.378602
$773$	1508.47	0.0701888	0.0350944	−	0.999384i	$-0.488827\pi$
$773$	1508.47	0.0701888	0.0350944	+	0.999384i	$0.488827\pi$
$774$	−1430.87	−0.0664489
$775$	−7339.83	−0.340199
$776$	692.680	0.0320435
$777$	2951.75	0.136285
$778$	−3622.94	−0.166952
$779$	7062.96	0.324848
$780$	−1897.04	−0.0870831
$781$	7145.56	0.327386
$782$	16.6790	0.000762710 0
$783$	12586.1	0.574445
$784$	−20071.1	−0.914315
$785$	−10881.0	−0.494728
$786$	2611.57	0.118513
$787$	−40051.1	−1.81406	−0.907031	−	0.421064i	$-0.861657\pi$
$787$	−40051.1	−1.81406	−0.907031	+	0.421064i	$0.861657\pi$
$788$	−38994.2	−1.76283
$789$	−10741.9	−0.484694
$790$	−885.902	−0.0398974
$791$	−28.5384	−0.00128282
$792$	1222.37	0.0548424
$793$	623.921	0.0279396
$794$	−1104.09	−0.0493486
$795$	804.660	0.0358973
$796$	12449.8	0.554361
$797$	14925.3	0.663340	0.331670	−	0.943396i	$-0.392388\pi$
$797$	14925.3	0.663340	0.331670	+	0.943396i	$0.392388\pi$
$798$	−72.0732	−0.00319720
$799$	−1.31905	−5.84039e−5 0
$800$	1799.67	0.0795347
$801$	16774.3	0.739940
$802$	1346.34	0.0592779
$803$	−3970.00	−0.174469
$804$	24120.0	1.05802
$805$	2490.33	0.109034
$806$	1842.03	0.0804998
$807$	−11101.8	−0.484267
$808$	−3915.31	−0.170470
$809$	15283.1	0.664185	0.332092	−	0.943247i	$-0.392245\pi$
$809$	15283.1	0.664185	0.332092	+	0.943247i	$0.392245\pi$
$810$	184.666	0.00801051
$811$	37852.7	1.63895	0.819475	−	0.573115i	$-0.194265\pi$
$811$	37852.7	1.63895	0.819475	+	0.573115i	$0.194265\pi$
$812$	−2460.94	−0.106357
$813$	−23216.0	−1.00150
$814$	−1264.46	−0.0544462
$815$	−5341.35	−0.229570
$816$	−52.0040	−0.00223101
$817$	3878.23	0.166073
$818$	3324.84	0.142115
$819$	992.720	0.0423546
$820$	−14594.8	−0.621553
$821$	−3931.63	−0.167131	−0.0835657	−	0.996502i	$-0.526631\pi$
$821$	−3931.63	−0.167131	−0.0835657	+	0.996502i	$0.526631\pi$
$822$	2846.22	0.120771
$823$	5469.87	0.231674	0.115837	−	0.993268i	$-0.463045\pi$
$823$	5469.87	0.231674	0.115837	+	0.993268i	$0.463045\pi$
$824$	9896.92	0.418417
$825$	−813.988	−0.0343508
$826$	21.6943	0.000913852 0
$827$	14313.3	0.601842	0.300921	−	0.953649i	$-0.402706\pi$
$827$	14313.3	0.601842	0.300921	+	0.953649i	$0.402706\pi$
$828$	21392.7	0.897885
$829$	15352.1	0.643184	0.321592	−	0.946878i	$-0.395782\pi$
$829$	15352.1	0.643184	0.321592	+	0.946878i	$0.395782\pi$
$830$	−1987.94	−0.0831356
$831$	−17346.9	−0.724137
$832$	7446.06	0.310271
$833$	−96.4161	−0.00401035
$834$	3671.21	0.152426
$835$	−10711.1	−0.443919
$836$	−1641.13	−0.0678942
$837$	−39313.4	−1.62350
$838$	−3012.87	−0.124198
$839$	7925.08	0.326107	0.163054	−	0.986617i	$-0.447866\pi$
$839$	7925.08	0.326107	0.163054	+	0.986617i	$0.447866\pi$
$840$	300.664	0.0123499
$841$	−15554.3	−0.637758
$842$	52.7898	0.00216064
$843$	−13886.1	−0.567336
$844$	−4494.17	−0.183289
$845$	−9652.64	−0.392972
$846$	31.8285	0.00129348
$847$	403.454	0.0163670
$848$	−3288.09	−0.133153
$849$	11578.3	0.468040
$850$	2.79146	0.000112643 0
$851$	−44674.8	−1.79957
$852$	15098.2	0.607106
$853$	−29245.5	−1.17391	−0.586957	−	0.809618i	$-0.699674\pi$
$853$	−29245.5	−1.17391	−0.586957	+	0.809618i	$0.699674\pi$
$854$	−48.9823	−0.00196269
$855$	−1732.67	−0.0693055
$856$	−2592.47	−0.103515
$857$	−14921.1	−0.594742	−0.297371	−	0.954762i	$-0.596110\pi$
$857$	−14921.1	−0.594742	−0.297371	+	0.954762i	$0.596110\pi$
$858$	204.282	0.00812828
$859$	5613.11	0.222953	0.111477	−	0.993767i	$-0.464442\pi$
$859$	5613.11	0.222953	0.111477	+	0.993767i	$0.464442\pi$
$860$	−8013.92	−0.317759
$861$	−3668.83	−0.145219
$862$	3877.13	0.153197
$863$	−26523.4	−1.04620	−0.523098	−	0.852273i	$-0.675224\pi$
$863$	−26523.4	−1.04620	−0.523098	+	0.852273i	$0.675224\pi$
$864$	9639.33	0.379556
$865$	9378.70	0.368654
$866$	1523.59	0.0597850
$867$	14542.0	0.569634
$868$	7686.88	0.300587
$869$	−5070.86	−0.197948
$870$	−534.660	−0.0208353
$871$	−16940.4	−0.659015
$872$	6436.47	0.249962
$873$	2073.52	0.0803871
$874$	1090.83	0.0422173
$875$	416.791	0.0161030
$876$	−8388.40	−0.323536
$877$	−17072.9	−0.657367	−0.328684	−	0.944440i	$-0.606605\pi$
$877$	−17072.9	−0.657367	−0.328684	+	0.944440i	$0.606605\pi$
$878$	−639.503	−0.0245811
$879$	−1378.60	−0.0529001
$880$	3326.20	0.127416
$881$	−4445.99	−0.170022	−0.0850108	−	0.996380i	$-0.527092\pi$
$881$	−4445.99	−0.170022	−0.0850108	+	0.996380i	$0.527092\pi$
$882$	2326.50	0.0888179
$883$	12560.6	0.478706	0.239353	−	0.970933i	$-0.423065\pi$
$883$	12560.6	0.478706	0.239353	+	0.970933i	$0.423065\pi$
$884$	37.2380	0.00141680
$885$	−250.534	−0.00951593
$886$	1145.63	0.0434403
$887$	11148.3	0.422012	0.211006	−	0.977485i	$-0.432326\pi$
$887$	11148.3	0.422012	0.211006	+	0.977485i	$0.432326\pi$
$888$	−5393.72	−0.203830
$889$	1980.70	0.0747249
$890$	−1767.46	−0.0665677
$891$	1057.02	0.0397436
$892$	22596.2	0.848181
$893$	−86.2681	−0.00323276
$894$	2796.54	0.104620
$895$	13608.0	0.508228
$896$	−2504.79	−0.0933918
$897$	7217.51	0.268658
$898$	−1192.49	−0.0443138
$899$	−27595.7	−1.02377
$900$	3580.38	0.132607
$901$	−15.7951	−0.000584031 0
$902$	1571.64	0.0580153
$903$	−2014.53	−0.0742406
$904$	52.1482	0.00191861
$905$	−19596.3	−0.719784
$906$	1552.46	0.0569283
$907$	7923.96	0.290089	0.145044	−	0.989425i	$-0.453667\pi$
$907$	7923.96	0.290089	0.145044	+	0.989425i	$0.453667\pi$
$908$	−46252.4	−1.69046
$909$	−11720.4	−0.427657
$910$	−104.600	−0.00381038
$911$	−51321.4	−1.86647	−0.933235	−	0.359266i	$-0.883027\pi$
$911$	−51321.4	−1.86647	−0.933235	+	0.359266i	$0.883027\pi$
$912$	−3401.14	−0.123490
$913$	−11378.9	−0.412472
$914$	3612.98	0.130751
$915$	565.665	0.0204375
$916$	19006.6	0.685586
$917$	−7654.17	−0.275641
$918$	14.9516	0.000537555 0
$919$	−36933.2	−1.32570	−0.662848	−	0.748754i	$-0.730652\pi$
$919$	−36933.2	−1.32570	−0.662848	+	0.748754i	$0.730652\pi$
$920$	−4550.57	−0.163074
$921$	27376.0	0.979446
$922$	−4380.14	−0.156456
$923$	−10604.0	−0.378152
$924$	852.475	0.0303511
$925$	−7476.96	−0.265774
$926$	−4020.08	−0.142665
$927$	29626.1	1.04968
$928$	6766.25	0.239346
$929$	−35185.7	−1.24263	−0.621316	−	0.783560i	$-0.713402\pi$
$929$	−35185.7	−1.24263	−0.621316	+	0.783560i	$0.713402\pi$
$930$	1670.04	0.0588848
$931$	−6305.76	−0.221980
$932$	19229.0	0.675824
$933$	−13517.7	−0.474330
$934$	−9.70995	−0.000340171 0
$935$	15.9782	0.000558870 0
$936$	−1814.00	−0.0633465
$937$	50097.1	1.74664	0.873320	−	0.487146i	$-0.161962\pi$
$937$	50097.1	1.74664	0.873320	+	0.487146i	$0.161962\pi$
$938$	1329.94	0.0462944
$939$	−2806.36	−0.0975316
$940$	178.263	0.00618543
$941$	−523.895	−0.0181493	−0.00907464	−	0.999959i	$-0.502889\pi$
$941$	−523.895	−0.0181493	−0.00907464	+	0.999959i	$0.502889\pi$
$942$	2475.78	0.0856320
$943$	55527.8	1.91753
$944$	1023.76	0.0352971
$945$	2232.41	0.0768468
$946$	862.976	0.0296594
$947$	−49063.2	−1.68357	−0.841785	−	0.539813i	$-0.818495\pi$
$947$	−49063.2	−1.68357	−0.841785	+	0.539813i	$0.818495\pi$
$948$	−10714.4	−0.367077
$949$	5891.47	0.201523
$950$	182.566	0.00623497
$951$	−19468.4	−0.663833
$952$	−5.90190	−0.000200926 0
$953$	−55812.2	−1.89710	−0.948549	−	0.316631i	$-0.897448\pi$
$953$	−55812.2	−1.89710	−0.948549	+	0.316631i	$0.897448\pi$
$954$	381.133	0.0129346
$955$	8060.45	0.273120
$956$	−27470.5	−0.929350
$957$	−3060.37	−0.103373
$958$	5077.65	0.171244
$959$	−8341.91	−0.280891
$960$	6750.82	0.226960
$961$	56405.9	1.89339
$962$	1876.45	0.0628889
$963$	−7760.49	−0.259687
$964$	−9856.61	−0.329315
$965$	5171.11	0.172501
$966$	−566.628	−0.0188726
$967$	−19207.0	−0.638734	−0.319367	−	0.947631i	$-0.603470\pi$
$967$	−19207.0	−0.638734	−0.319367	+	0.947631i	$0.603470\pi$
$968$	−737.231	−0.0244788
$969$	−16.3382	−0.000541650 0
$970$	−218.480	−0.00723192
$971$	2225.68	0.0735587	0.0367794	−	0.999323i	$-0.488290\pi$
$971$	2225.68	0.0735587	0.0367794	+	0.999323i	$0.488290\pi$
$972$	30622.7	1.01052
$973$	−10759.8	−0.354517
$974$	−3154.59	−0.103778
$975$	1207.95	0.0396774
$976$	−2311.48	−0.0758081
$977$	8448.14	0.276643	0.138321	−	0.990387i	$-0.455829\pi$
$977$	8448.14	0.276643	0.138321	+	0.990387i	$0.455829\pi$
$978$	1215.33	0.0397360
$979$	−10116.8	−0.330271
$980$	13030.2	0.424728
$981$	19267.4	0.627075
$982$	2133.99	0.0693465
$983$	54327.1	1.76273	0.881367	−	0.472433i	$-0.156624\pi$
$983$	54327.1	1.76273	0.881367	+	0.472433i	$0.156624\pi$
$984$	6704.04	0.217192
$985$	24829.9	0.803193
$986$	10.4951	0.000338979 0
$987$	44.8116	0.00144515
$988$	2435.42	0.0784222
$989$	30490.0	0.980308
$990$	−385.551	−0.0123774
$991$	28913.7	0.926816	0.463408	−	0.886145i	$-0.346626\pi$
$991$	28913.7	0.926816	0.463408	+	0.886145i	$0.346626\pi$
$992$	−21134.8	−0.676441
$993$	7488.72	0.239323
$994$	832.489	0.0265643
$995$	−7927.51	−0.252582
$996$	−24043.0	−0.764890
$997$	1125.68	0.0357580	0.0178790	−	0.999840i	$-0.494309\pi$
$997$	1125.68	0.0357580	0.0178790	+	0.999840i	$0.494309\pi$
$998$	7774.18	0.246581
$999$	−40047.9	−1.26833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.13	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.13	✓	20	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+0.384350 q^{2} -2.95996 q^{3} -7.85228 q^{4} +5.00000 q^{5} -1.13766 q^{6} +3.33433 q^{7} -6.09282 q^{8} -18.2387 q^{9} +O(q^{10})\)	\(q+0.384350 q^{2} -2.95996 q^{3} -7.85228 q^{4} +5.00000 q^{5} -1.13766 q^{6} +3.33433 q^{7} -6.09282 q^{8} -18.2387 q^{9} +1.92175 q^{10} +11.0000 q^{11} +23.2424 q^{12} -16.3239 q^{13} +1.28155 q^{14} -14.7998 q^{15} +60.4764 q^{16} +0.290513 q^{17} -7.01002 q^{18} +19.0000 q^{19} -39.2614 q^{20} -9.86947 q^{21} +4.22785 q^{22} +149.375 q^{23} +18.0345 q^{24} +25.0000 q^{25} -6.27410 q^{26} +133.904 q^{27} -26.1821 q^{28} +93.9932 q^{29} -5.68829 q^{30} -293.593 q^{31} +71.9866 q^{32} -32.5595 q^{33} +0.111659 q^{34} +16.6717 q^{35} +143.215 q^{36} -299.079 q^{37} +7.30264 q^{38} +48.3181 q^{39} -30.4641 q^{40} +371.735 q^{41} -3.79333 q^{42} +204.117 q^{43} -86.3750 q^{44} -91.1933 q^{45} +57.4122 q^{46} -4.54042 q^{47} -179.008 q^{48} -331.882 q^{49} +9.60874 q^{50} -0.859905 q^{51} +128.180 q^{52} -54.3698 q^{53} +51.4661 q^{54} +55.0000 q^{55} -20.3155 q^{56} -56.2391 q^{57} +36.1262 q^{58} +16.9282 q^{59} +116.212 q^{60} -38.2212 q^{61} -112.842 q^{62} -60.8137 q^{63} -456.143 q^{64} -81.6197 q^{65} -12.5142 q^{66} +1037.76 q^{67} -2.28119 q^{68} -442.143 q^{69} +6.40774 q^{70} +649.596 q^{71} +111.125 q^{72} -360.910 q^{73} -114.951 q^{74} -73.9989 q^{75} -149.193 q^{76} +36.6776 q^{77} +18.5711 q^{78} -460.987 q^{79} +302.382 q^{80} +96.0929 q^{81} +142.876 q^{82} -1034.45 q^{83} +77.4978 q^{84} +1.45256 q^{85} +78.4523 q^{86} -278.216 q^{87} -67.0210 q^{88} -919.712 q^{89} -35.0501 q^{90} -54.4294 q^{91} -1172.93 q^{92} +869.022 q^{93} -1.74511 q^{94} +95.0000 q^{95} -213.077 q^{96} -113.688 q^{97} -127.559 q^{98} -200.625 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

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