LMFDB - Embedded newform 37.4.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [37,4,Mod(1,37)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(37, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("37.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	37.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:	$2.18307067021$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 3x^{2} + 176x + 144 $ x^5 - x^4 - 27x^3 + 3x^2 + 176*x + 144
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.40805$ of defining polynomial
Character	$\chi$	$=$	37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.40805 q^{2} +7.32702 q^{3} +3.61479 q^{4} -17.2892 q^{5} +24.9708 q^{6} -15.1635 q^{7} -14.9450 q^{8} +26.6852 q^{9} +O(q^{10})$	\(q+3.40805 q^{2} +7.32702 q^{3} +3.61479 q^{4} -17.2892 q^{5} +24.9708 q^{6} -15.1635 q^{7} -14.9450 q^{8} +26.6852 q^{9} -58.9226 q^{10} +61.3864 q^{11} +26.4857 q^{12} +59.3647 q^{13} -51.6780 q^{14} -126.679 q^{15} -79.8516 q^{16} +62.2512 q^{17} +90.9446 q^{18} -71.9018 q^{19} -62.4971 q^{20} -111.103 q^{21} +209.208 q^{22} -39.7229 q^{23} -109.502 q^{24} +173.918 q^{25} +202.318 q^{26} -2.30624 q^{27} -54.8130 q^{28} -132.354 q^{29} -431.727 q^{30} +59.1047 q^{31} -152.578 q^{32} +449.779 q^{33} +212.155 q^{34} +262.166 q^{35} +96.4617 q^{36} -37.0000 q^{37} -245.045 q^{38} +434.966 q^{39} +258.388 q^{40} +146.066 q^{41} -378.646 q^{42} -369.236 q^{43} +221.899 q^{44} -461.368 q^{45} -135.377 q^{46} -101.221 q^{47} -585.074 q^{48} -113.068 q^{49} +592.721 q^{50} +456.116 q^{51} +214.591 q^{52} +244.643 q^{53} -7.85979 q^{54} -1061.32 q^{55} +226.619 q^{56} -526.826 q^{57} -451.070 q^{58} +522.117 q^{59} -457.917 q^{60} -682.007 q^{61} +201.432 q^{62} -404.642 q^{63} +118.819 q^{64} -1026.37 q^{65} +1532.87 q^{66} +235.648 q^{67} +225.025 q^{68} -291.050 q^{69} +893.474 q^{70} +965.213 q^{71} -398.811 q^{72} +665.592 q^{73} -126.098 q^{74} +1274.30 q^{75} -259.910 q^{76} -930.834 q^{77} +1482.39 q^{78} -82.1421 q^{79} +1380.57 q^{80} -737.399 q^{81} +497.800 q^{82} +1294.65 q^{83} -401.616 q^{84} -1076.28 q^{85} -1258.37 q^{86} -969.763 q^{87} -917.419 q^{88} -24.4755 q^{89} -1572.36 q^{90} -900.178 q^{91} -143.590 q^{92} +433.061 q^{93} -344.965 q^{94} +1243.13 q^{95} -1117.94 q^{96} -676.067 q^{97} -385.340 q^{98} +1638.11 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 13 q^{3} + 18 q^{4} + 11 q^{5} + 9 q^{6} + 24 q^{7} + 30 q^{8} + 46 q^{9}+O(q^{10}) $ 5 * q + 4 * q^2 + 13 * q^3 + 18 * q^4 + 11 * q^5 + 9 * q^6 + 24 * q^7 + 30 * q^8 + 46 * q^9	\( 5 q + 4 q^{2} + 13 q^{3} + 18 q^{4} + 11 q^{5} + 9 q^{6} + 24 q^{7} + 30 q^{8} + 46 q^{9} - 75 q^{10} + 61 q^{11} - 65 q^{12} - 37 q^{13} - 128 q^{14} - 116 q^{15} - 182 q^{16} + 130 q^{17} - 159 q^{18} - 22 q^{19} - 59 q^{20} - 44 q^{21} - 95 q^{22} + 73 q^{23} - 105 q^{24} + 26 q^{25} + 197 q^{26} + 472 q^{27} - 2 q^{28} + 271 q^{29} - 196 q^{30} + 363 q^{31} + 74 q^{32} + 198 q^{33} + 272 q^{34} + 604 q^{35} - 251 q^{36} - 185 q^{37} + 576 q^{38} - 65 q^{39} + 97 q^{40} + 381 q^{41} - 376 q^{42} - 408 q^{43} + 235 q^{44} - 704 q^{45} - 325 q^{46} + 276 q^{47} - 889 q^{48} - 949 q^{49} + 415 q^{50} - 38 q^{51} - 403 q^{52} + 156 q^{53} - 1068 q^{54} - 843 q^{55} + 578 q^{56} - 1618 q^{57} - 31 q^{58} + 100 q^{59} - 952 q^{60} - 1711 q^{61} + 1305 q^{62} + 94 q^{63} + 370 q^{64} - 890 q^{65} + 2490 q^{66} + 787 q^{67} + 2464 q^{68} - 2335 q^{69} + 308 q^{70} + 1578 q^{71} + 753 q^{72} - 313 q^{73} - 148 q^{74} + 684 q^{75} + 2080 q^{76} - 342 q^{77} + 3955 q^{78} + 569 q^{79} + 189 q^{80} + 385 q^{81} + 119 q^{82} + 2422 q^{83} - 1098 q^{84} - 2210 q^{85} + 1566 q^{86} + 2371 q^{87} - 669 q^{88} - 2466 q^{89} - 1930 q^{90} - 1678 q^{91} - 373 q^{92} - 1142 q^{93} - 1660 q^{94} + 794 q^{95} - 1797 q^{96} - 2406 q^{97} - 2010 q^{98} + 1746 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + 13 * q^3 + 18 * q^4 + 11 * q^5 + 9 * q^6 + 24 * q^7 + 30 * q^8 + 46 * q^9 - 75 * q^10 + 61 * q^11 - 65 * q^12 - 37 * q^13 - 128 * q^14 - 116 * q^15 - 182 * q^16 + 130 * q^17 - 159 * q^18 - 22 * q^19 - 59 * q^20 - 44 * q^21 - 95 * q^22 + 73 * q^23 - 105 * q^24 + 26 * q^25 + 197 * q^26 + 472 * q^27 - 2 * q^28 + 271 * q^29 - 196 * q^30 + 363 * q^31 + 74 * q^32 + 198 * q^33 + 272 * q^34 + 604 * q^35 - 251 * q^36 - 185 * q^37 + 576 * q^38 - 65 * q^39 + 97 * q^40 + 381 * q^41 - 376 * q^42 - 408 * q^43 + 235 * q^44 - 704 * q^45 - 325 * q^46 + 276 * q^47 - 889 * q^48 - 949 * q^49 + 415 * q^50 - 38 * q^51 - 403 * q^52 + 156 * q^53 - 1068 * q^54 - 843 * q^55 + 578 * q^56 - 1618 * q^57 - 31 * q^58 + 100 * q^59 - 952 * q^60 - 1711 * q^61 + 1305 * q^62 + 94 * q^63 + 370 * q^64 - 890 * q^65 + 2490 * q^66 + 787 * q^67 + 2464 * q^68 - 2335 * q^69 + 308 * q^70 + 1578 * q^71 + 753 * q^72 - 313 * q^73 - 148 * q^74 + 684 * q^75 + 2080 * q^76 - 342 * q^77 + 3955 * q^78 + 569 * q^79 + 189 * q^80 + 385 * q^81 + 119 * q^82 + 2422 * q^83 - 1098 * q^84 - 2210 * q^85 + 1566 * q^86 + 2371 * q^87 - 669 * q^88 - 2466 * q^89 - 1930 * q^90 - 1678 * q^91 - 373 * q^92 - 1142 * q^93 - 1660 * q^94 + 794 * q^95 - 1797 * q^96 - 2406 * q^97 - 2010 * q^98 + 1746 * 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 $

$2$	3.40805	1.20493	0.602464	−	0.798146i	$-0.294186\pi$
$2$	3.40805	1.20493	0.602464	+	0.798146i	$0.294186\pi$
$3$	7.32702	1.41009	0.705043	−	0.709165i	$-0.250928\pi$
$3$	7.32702	1.41009	0.705043	+	0.709165i	$0.250928\pi$
$4$	3.61479	0.451849
$5$	−17.2892	−1.54640	−0.773198	−	0.634164i	$-0.781344\pi$
$5$	−17.2892	−1.54640	−0.773198	+	0.634164i	$0.781344\pi$
$6$	24.9708	1.69905
$7$	−15.1635	−0.818753	−0.409377	−	0.912365i	$-0.634254\pi$
$7$	−15.1635	−0.818753	−0.409377	+	0.912365i	$0.634254\pi$
$8$	−14.9450	−0.660482
$9$	26.6852	0.988342
$10$	−58.9226	−1.86330
$11$	61.3864	1.68261	0.841304	−	0.540562i	$-0.181788\pi$
$11$	61.3864	1.68261	0.841304	+	0.540562i	$0.181788\pi$
$12$	26.4857	0.637146
$13$	59.3647	1.26652	0.633262	−	0.773938i	$-0.281716\pi$
$13$	59.3647	1.26652	0.633262	+	0.773938i	$0.281716\pi$
$14$	−51.6780	−0.986538
$15$	−126.679	−2.18055
$16$	−79.8516	−1.24768
$17$	62.2512	0.888125	0.444062	−	0.895996i	$-0.353537\pi$
$17$	62.2512	0.888125	0.444062	+	0.895996i	$0.353537\pi$
$18$	90.9446	1.19088
$19$	−71.9018	−0.868179	−0.434090	−	0.900870i	$-0.642930\pi$
$19$	−71.9018	−0.868179	−0.434090	+	0.900870i	$0.642930\pi$
$20$	−62.4971	−0.698738
$21$	−111.103	−1.15451
$22$	209.208	2.02742
$23$	−39.7229	−0.360121	−0.180061	−	0.983656i	$-0.557629\pi$
$23$	−39.7229	−0.360121	−0.180061	+	0.983656i	$0.557629\pi$
$24$	−109.502	−0.931336
$25$	173.918	1.39134
$26$	202.318	1.52607
$27$	−2.30624	−0.0164384
$28$	−54.8130	−0.369953
$29$	−132.354	−0.847503	−0.423751	−	0.905779i	$-0.639287\pi$
$29$	−132.354	−0.847503	−0.423751	+	0.905779i	$0.639287\pi$
$30$	−431.727	−2.62741
$31$	59.1047	0.342436	0.171218	−	0.985233i	$-0.445230\pi$
$31$	59.1047	0.342436	0.171218	+	0.985233i	$0.445230\pi$
$32$	−152.578	−0.842884
$33$	449.779	2.37262
$34$	212.155	1.07013
$35$	262.166	1.26612
$36$	96.4617	0.446582
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	−245.045	−1.04609
$39$	434.966	1.78591
$40$	258.388	1.02137
$41$	146.066	0.556383	0.278191	−	0.960526i	$-0.410265\pi$
$41$	146.066	0.556383	0.278191	+	0.960526i	$0.410265\pi$
$42$	−378.646	−1.39110
$43$	−369.236	−1.30949	−0.654743	−	0.755851i	$-0.727223\pi$
$43$	−369.236	−1.30949	−0.654743	+	0.755851i	$0.727223\pi$
$44$	221.899	0.760285
$45$	−461.368	−1.52837
$46$	−135.377	−0.433920
$47$	−101.221	−0.314140	−0.157070	−	0.987588i	$-0.550205\pi$
$47$	−101.221	−0.314140	−0.157070	+	0.987588i	$0.550205\pi$
$48$	−585.074	−1.75934
$49$	−113.068	−0.329643
$50$	592.721	1.67647
$51$	456.116	1.25233
$52$	214.591	0.572278
$53$	244.643	0.634042	0.317021	−	0.948418i	$-0.397317\pi$
$53$	244.643	0.634042	0.317021	+	0.948418i	$0.397317\pi$
$54$	−7.85979	−0.0198071
$55$	−1061.32	−2.60198
$56$	226.619	0.540772
$57$	−526.826	−1.22421
$58$	−451.070	−1.02118
$59$	522.117	1.15210	0.576050	−	0.817415i	$-0.304593\pi$
$59$	522.117	1.15210	0.576050	+	0.817415i	$0.304593\pi$
$60$	−457.917	−0.985281
$61$	−682.007	−1.43151	−0.715754	−	0.698352i	$-0.753917\pi$
$61$	−682.007	−1.43151	−0.715754	+	0.698352i	$0.753917\pi$
$62$	201.432	0.412610
$63$	−404.642	−0.809209
$64$	118.819	0.232068
$65$	−1026.37	−1.95855
$66$	1532.87	2.85884
$67$	235.648	0.429687	0.214843	−	0.976649i	$-0.431076\pi$
$67$	235.648	0.429687	0.214843	+	0.976649i	$0.431076\pi$
$68$	225.025	0.401299
$69$	−291.050	−0.507802
$70$	893.474	1.52558
$71$	965.213	1.61338	0.806688	−	0.590978i	$-0.201258\pi$
$71$	965.213	1.61338	0.806688	+	0.590978i	$0.201258\pi$
$72$	−398.811	−0.652782
$73$	665.592	1.06715	0.533573	−	0.845754i	$-0.320849\pi$
$73$	665.592	1.06715	0.533573	+	0.845754i	$0.320849\pi$
$74$	−126.098	−0.198089
$75$	1274.30	1.96191
$76$	−259.910	−0.392286
$77$	−930.834	−1.37764
$78$	1482.39	2.15189
$79$	−82.1421	−0.116984	−0.0584918	−	0.998288i	$-0.518629\pi$
$79$	−82.1421	−0.116984	−0.0584918	+	0.998288i	$0.518629\pi$
$80$	1380.57	1.92941
$81$	−737.399	−1.01152
$82$	497.800	0.670401
$83$	1294.65	1.71213	0.856064	−	0.516870i	$-0.172903\pi$
$83$	1294.65	1.71213	0.856064	+	0.516870i	$0.172903\pi$
$84$	−401.616	−0.521666
$85$	−1076.28	−1.37339
$86$	−1258.37	−1.57784
$87$	−969.763	−1.19505
$88$	−917.419	−1.11133
$89$	−24.4755	−0.0291505	−0.0145753	−	0.999894i	$-0.504640\pi$
$89$	−24.4755	−0.0291505	−0.0145753	+	0.999894i	$0.504640\pi$
$90$	−1572.36	−1.84157
$91$	−900.178	−1.03697
$92$	−143.590	−0.162721
$93$	433.061	0.482864
$94$	−344.965	−0.378515
$95$	1243.13	1.34255
$96$	−1117.94	−1.18854
$97$	−676.067	−0.707673	−0.353836	−	0.935307i	$-0.615123\pi$
$97$	−676.067	−0.707673	−0.353836	+	0.935307i	$0.615123\pi$
$98$	−385.340	−0.397196
$99$	1638.11	1.66299
$100$	628.678	0.628678
$101$	1111.48	1.09501	0.547505	−	0.836803i	$-0.315578\pi$
$101$	1111.48	1.09501	0.547505	+	0.836803i	$0.315578\pi$
$102$	1554.46	1.50897
$103$	−348.978	−0.333843	−0.166922	−	0.985970i	$-0.553383\pi$
$103$	−348.978	−0.333843	−0.166922	+	0.985970i	$0.553383\pi$
$104$	−887.205	−0.836516
$105$	1920.90	1.78533
$106$	833.754	0.763975
$107$	−1674.35	−1.51276	−0.756382	−	0.654131i	$-0.773035\pi$
$107$	−1674.35	−1.51276	−0.756382	+	0.654131i	$0.773035\pi$
$108$	−8.33660	−0.00742768
$109$	−442.404	−0.388758	−0.194379	−	0.980927i	$-0.562269\pi$
$109$	−442.404	−0.388758	−0.194379	+	0.980927i	$0.562269\pi$
$110$	−3617.04	−3.13520
$111$	−271.100	−0.231817
$112$	1210.83	1.02154
$113$	−1.77808	−0.00148025	−0.000740124	−	1.00000i	$-0.500236\pi$
$113$	−1.77808	−0.00148025	−0.000740124		1.00000i	$0.500236\pi$
$114$	−1795.45	−1.47508
$115$	686.778	0.556890
$116$	−478.434	−0.382944
$117$	1584.16	1.25176
$118$	1779.40	1.38820
$119$	−943.947	−0.727155
$120$	1893.21	1.44021
$121$	2437.29	1.83117
$122$	−2324.31	−1.72486
$123$	1070.23	0.784548
$124$	213.651	0.154729
$125$	−845.754	−0.605173
$126$	−1379.04	−0.975037
$127$	−723.414	−0.505454	−0.252727	−	0.967538i	$-0.581327\pi$
$127$	−723.414	−0.505454	−0.252727	+	0.967538i	$0.581327\pi$
$128$	1625.57	1.12251
$129$	−2705.40	−1.84649
$130$	−3497.92	−2.35991
$131$	237.650	0.158501	0.0792504	−	0.996855i	$-0.474747\pi$
$131$	237.650	0.158501	0.0792504	+	0.996855i	$0.474747\pi$
$132$	1625.86	1.07207
$133$	1090.29	0.710825
$134$	803.100	0.517741
$135$	39.8732	0.0254203
$136$	−930.343	−0.586590
$137$	−1884.46	−1.17519	−0.587593	−	0.809156i	$-0.699924\pi$
$137$	−1884.46	−1.17519	−0.587593	+	0.809156i	$0.699924\pi$
$138$	−991.913	−0.611864
$139$	−5.60755	−0.00342177	−0.00171089	−	0.999999i	$-0.500545\pi$
$139$	−5.60755	−0.00342177	−0.00171089	+	0.999999i	$0.500545\pi$
$140$	947.676	0.572094
$141$	−741.647	−0.442964
$142$	3289.49	1.94400
$143$	3644.18	2.13106
$144$	−2130.86	−1.23314
$145$	2288.31	1.31058
$146$	2268.37	1.28583
$147$	−828.448	−0.464825
$148$	−133.747	−0.0742836
$149$	−2515.01	−1.38280	−0.691402	−	0.722470i	$-0.743007\pi$
$149$	−2515.01	−1.38280	−0.691402	+	0.722470i	$0.743007\pi$
$150$	4342.88	2.36396
$151$	−166.517	−0.0897417	−0.0448708	−	0.998993i	$-0.514288\pi$
$151$	−166.517	−0.0897417	−0.0448708	+	0.998993i	$0.514288\pi$
$152$	1074.57	0.573417
$153$	1661.19	0.877771
$154$	−3172.33	−1.65996
$155$	−1021.87	−0.529542
$156$	1572.31	0.806961
$157$	−2314.06	−1.17632	−0.588159	−	0.808745i	$-0.700147\pi$
$157$	−2314.06	−1.17632	−0.588159	+	0.808745i	$0.700147\pi$
$158$	−279.944	−0.140957
$159$	1792.50	0.894054
$160$	2637.96	1.30343
$161$	602.339	0.294851
$162$	−2513.09	−1.21881
$163$	461.333	0.221683	0.110842	−	0.993838i	$-0.464645\pi$
$163$	461.333	0.221683	0.110842	+	0.993838i	$0.464645\pi$
$164$	527.999	0.251401
$165$	−7776.34	−3.66902
$166$	4412.24	2.06299
$167$	3281.70	1.52063	0.760316	−	0.649553i	$-0.225044\pi$
$167$	3281.70	1.52063	0.760316	+	0.649553i	$0.225044\pi$
$168$	1660.44	0.762534
$169$	1327.17	0.604082
$170$	−3668.00	−1.65484
$171$	−1918.72	−0.858058
$172$	−1334.71	−0.591690
$173$	1993.94	0.876281	0.438141	−	0.898906i	$-0.355637\pi$
$173$	1993.94	0.876281	0.438141	+	0.898906i	$0.355637\pi$
$174$	−3305.00	−1.43995
$175$	−2637.21	−1.13917
$176$	−4901.80	−2.09936
$177$	3825.56	1.62456
$178$	−83.4136	−0.0351242
$179$	−42.1135	−0.0175850	−0.00879249	−	0.999961i	$-0.502799\pi$
$179$	−42.1135	−0.0175850	−0.00879249	+	0.999961i	$0.502799\pi$
$180$	−1667.75	−0.690593
$181$	2809.68	1.15382	0.576911	−	0.816807i	$-0.304258\pi$
$181$	2809.68	1.15382	0.576911	+	0.816807i	$0.304258\pi$
$182$	−3067.85	−1.24947
$183$	−4997.08	−2.01855
$184$	593.658	0.237853
$185$	639.702	0.254226
$186$	1475.89	0.581816
$187$	3821.37	1.49437
$188$	−365.892	−0.141944
$189$	34.9708	0.0134590
$190$	4236.64	1.61768
$191$	−4690.79	−1.77704	−0.888518	−	0.458841i	$-0.848265\pi$
$191$	−4690.79	−1.77704	−0.888518	+	0.458841i	$0.848265\pi$
$192$	870.589	0.327236
$193$	−1984.44	−0.740120	−0.370060	−	0.929008i	$-0.620663\pi$
$193$	−1984.44	−0.740120	−0.370060	+	0.929008i	$0.620663\pi$
$194$	−2304.07	−0.852694
$195$	−7520.24	−2.76172
$196$	−408.716	−0.148949
$197$	3500.72	1.26607	0.633035	−	0.774123i	$-0.281809\pi$
$197$	3500.72	1.26607	0.633035	+	0.774123i	$0.281809\pi$
$198$	5582.76	2.00378
$199$	2472.02	0.880586	0.440293	−	0.897854i	$-0.354874\pi$
$199$	2472.02	0.880586	0.440293	+	0.897854i	$0.354874\pi$
$200$	−2599.20	−0.918957
$201$	1726.60	0.605895
$202$	3787.96	1.31941
$203$	2006.96	0.693896
$204$	1648.76	0.565865
$205$	−2525.37	−0.860389
$206$	−1189.33	−0.402257
$207$	−1060.01	−0.355923
$208$	−4740.37	−1.58022
$209$	−4413.79	−1.46081
$210$	6546.50	2.15120
$211$	−2725.47	−0.889238	−0.444619	−	0.895720i	$-0.646661\pi$
$211$	−2725.47	−0.889238	−0.444619	+	0.895720i	$0.646661\pi$
$212$	884.333	0.286492
$213$	7072.13	2.27500
$214$	−5706.27	−1.82277
$215$	6383.81	2.02499
$216$	34.4668	0.0108573
$217$	−896.235	−0.280370
$218$	−1507.73	−0.468425
$219$	4876.81	1.50477
$220$	−3836.47	−1.17570
$221$	3695.52	1.12483
$222$	−923.921	−0.279322
$223$	−2126.12	−0.638455	−0.319227	−	0.947678i	$-0.603423\pi$
$223$	−2126.12	−0.638455	−0.319227	+	0.947678i	$0.603423\pi$
$224$	2313.62	0.690114
$225$	4641.04	1.37512
$226$	−6.05980	−0.00178359
$227$	92.8360	0.0271442	0.0135721	−	0.999908i	$-0.495680\pi$
$227$	92.8360	0.0271442	0.0135721	+	0.999908i	$0.495680\pi$
$228$	−1904.37	−0.553157
$229$	1810.52	0.522455	0.261228	−	0.965277i	$-0.415873\pi$
$229$	1810.52	0.522455	0.261228	+	0.965277i	$0.415873\pi$
$230$	2340.57	0.671012
$231$	−6820.24	−1.94259
$232$	1978.03	0.559760
$233$	−3015.67	−0.847910	−0.423955	−	0.905683i	$-0.639359\pi$
$233$	−3015.67	−0.847910	−0.423955	+	0.905683i	$0.639359\pi$
$234$	5398.90	1.50828
$235$	1750.03	0.485785
$236$	1887.35	0.520575
$237$	−601.857	−0.164957
$238$	−3217.02	−0.876169
$239$	−905.734	−0.245134	−0.122567	−	0.992460i	$-0.539113\pi$
$239$	−905.734	−0.245134	−0.122567	+	0.992460i	$0.539113\pi$
$240$	10115.5	2.72064
$241$	−6486.04	−1.73362	−0.866810	−	0.498639i	$-0.833833\pi$
$241$	−6486.04	−1.73362	−0.866810	+	0.498639i	$0.833833\pi$
$242$	8306.39	2.20643
$243$	−5340.67	−1.40989
$244$	−2465.31	−0.646826
$245$	1954.85	0.509759
$246$	3647.39	0.945323
$247$	−4268.43	−1.09957
$248$	−883.319	−0.226173
$249$	9485.95	2.41425
$250$	−2882.37	−0.729189
$251$	−498.941	−0.125470	−0.0627348	−	0.998030i	$-0.519982\pi$
$251$	−498.941	−0.125470	−0.0627348	+	0.998030i	$0.519982\pi$
$252$	−1462.70	−0.365640
$253$	−2438.44	−0.605943
$254$	−2465.43	−0.609035
$255$	−7885.89	−1.93660
$256$	4589.46	1.12047
$257$	4569.66	1.10914	0.554568	−	0.832139i	$-0.312884\pi$
$257$	4569.66	1.10914	0.554568	+	0.832139i	$0.312884\pi$
$258$	−9220.13	−2.22488
$259$	561.050	0.134602
$260$	−3710.12	−0.884969
$261$	−3531.91	−0.837623
$262$	809.924	0.190982
$263$	949.544	0.222629	0.111314	−	0.993785i	$-0.464494\pi$
$263$	949.544	0.222629	0.111314	+	0.993785i	$0.464494\pi$
$264$	−6721.95	−1.56707
$265$	−4229.69	−0.980481
$266$	3715.74	0.856492
$267$	−179.332	−0.0411047
$268$	851.820	0.194154
$269$	−5858.86	−1.32796	−0.663979	−	0.747751i	$-0.731134\pi$
$269$	−5858.86	−1.32796	−0.663979	+	0.747751i	$0.731134\pi$
$270$	135.890	0.0306296
$271$	4542.25	1.01816	0.509081	−	0.860719i	$-0.329985\pi$
$271$	4542.25	1.01816	0.509081	+	0.860719i	$0.329985\pi$
$272$	−4970.86	−1.10810
$273$	−6595.62	−1.46222
$274$	−6422.34	−1.41601
$275$	10676.2	2.34109
$276$	−1052.09	−0.229450
$277$	−1446.58	−0.313778	−0.156889	−	0.987616i	$-0.550147\pi$
$277$	−1446.58	−0.313778	−0.156889	+	0.987616i	$0.550147\pi$
$278$	−19.1108	−0.00412298
$279$	1577.22	0.338444
$280$	−3918.07	−0.836248
$281$	−5196.89	−1.10328	−0.551638	−	0.834084i	$-0.685997\pi$
$281$	−5196.89	−1.10328	−0.551638	+	0.834084i	$0.685997\pi$
$282$	−2527.57	−0.533739
$283$	7650.69	1.60702	0.803509	−	0.595292i	$-0.202964\pi$
$283$	7650.69	1.60702	0.803509	+	0.595292i	$0.202964\pi$
$284$	3489.05	0.729003
$285$	9108.43	1.89311
$286$	12419.6	2.56778
$287$	−2214.88	−0.455540
$288$	−4071.59	−0.833057
$289$	−1037.79	−0.211234
$290$	7798.66	1.57915
$291$	−4953.56	−0.997879
$292$	2405.98	0.482189
$293$	1588.21	0.316669	0.158335	−	0.987386i	$-0.449388\pi$
$293$	1588.21	0.316669	0.158335	+	0.987386i	$0.449388\pi$
$294$	−2823.39	−0.560080
$295$	−9027.01	−1.78160
$296$	552.965	0.108583
$297$	−141.572	−0.0276594
$298$	−8571.28	−1.66618
$299$	−2358.14	−0.456102
$300$	4606.33	0.886489
$301$	5598.91	1.07215
$302$	−567.499	−0.108132
$303$	8143.81	1.54406
$304$	5741.48	1.08321
$305$	11791.4	2.21368
$306$	5661.41	1.05765
$307$	4345.90	0.807928	0.403964	−	0.914775i	$-0.367632\pi$
$307$	4345.90	0.807928	0.403964	+	0.914775i	$0.367632\pi$
$308$	−3364.77	−0.622486
$309$	−2556.97	−0.470747
$310$	−3482.60	−0.638059
$311$	1457.37	0.265724	0.132862	−	0.991135i	$-0.457583\pi$
$311$	1457.37	0.265724	0.132862	+	0.991135i	$0.457583\pi$
$312$	−6500.57	−1.17956
$313$	2407.17	0.434700	0.217350	−	0.976094i	$-0.430259\pi$
$313$	2407.17	0.434700	0.217350	+	0.976094i	$0.430259\pi$
$314$	−7886.43	−1.41738
$315$	6995.96	1.25136
$316$	−296.927	−0.0528589
$317$	4714.42	0.835294	0.417647	−	0.908609i	$-0.362855\pi$
$317$	4714.42	0.835294	0.417647	+	0.908609i	$0.362855\pi$
$318$	6108.93	1.07727
$319$	−8124.75	−1.42602
$320$	−2054.29	−0.358870
$321$	−12268.0	−2.13313
$322$	2052.80	0.355273
$323$	−4475.97	−0.771052
$324$	−2665.55	−0.457055
$325$	10324.6	1.76217
$326$	1572.24	0.267112
$327$	−3241.50	−0.548182
$328$	−2182.96	−0.367481
$329$	1534.86	0.257203
$330$	−26502.2	−4.42090
$331$	5525.25	0.917509	0.458754	−	0.888563i	$-0.348296\pi$
$331$	5525.25	0.917509	0.458754	+	0.888563i	$0.348296\pi$
$332$	4679.90	0.773624
$333$	−987.354	−0.162482
$334$	11184.2	1.83225
$335$	−4074.18	−0.664466
$336$	8871.79	1.44046
$337$	531.342	0.0858873	0.0429437	−	0.999077i	$-0.486326\pi$
$337$	531.342	0.0858873	0.0429437	+	0.999077i	$0.486326\pi$
$338$	4523.05	0.727874
$339$	−13.0281	−0.00208728
$340$	−3890.51	−0.620567
$341$	3628.22	0.576185
$342$	−6539.08	−1.03390
$343$	6915.59	1.08865
$344$	5518.22	0.864892
$345$	5032.04	0.785263
$346$	6795.45	1.05585
$347$	3277.22	0.507005	0.253502	−	0.967335i	$-0.418417\pi$
$347$	3277.22	0.507005	0.253502	+	0.967335i	$0.418417\pi$
$348$	−3505.49	−0.539983
$349$	−177.861	−0.0272800	−0.0136400	−	0.999907i	$-0.504342\pi$
$349$	−177.861	−0.0272800	−0.0136400	+	0.999907i	$0.504342\pi$
$350$	−8987.74	−1.37261
$351$	−136.909	−0.0208196
$352$	−9366.22	−1.41824
$353$	−9855.08	−1.48593	−0.742964	−	0.669331i	$-0.766581\pi$
$353$	−9855.08	−1.48593	−0.742964	+	0.669331i	$0.766581\pi$
$354$	13037.7	1.95748
$355$	−16687.8	−2.49492
$356$	−88.4738	−0.0131716
$357$	−6916.32	−1.02535
$358$	−143.525	−0.0211886
$359$	2375.50	0.349231	0.174615	−	0.984637i	$-0.444132\pi$
$359$	2375.50	0.349231	0.174615	+	0.984637i	$0.444132\pi$
$360$	6895.14	1.00946
$361$	−1689.13	−0.246264
$362$	9575.53	1.39027
$363$	17858.1	2.58211
$364$	−3253.96	−0.468554
$365$	−11507.6	−1.65023
$366$	−17030.3	−2.43221
$367$	6177.27	0.878613	0.439307	−	0.898337i	$-0.355224\pi$
$367$	6177.27	0.878613	0.439307	+	0.898337i	$0.355224\pi$
$368$	3171.93	0.449317
$369$	3897.81	0.549897
$370$	2180.14	0.306324
$371$	−3709.64	−0.519124
$372$	1565.43	0.218182
$373$	5863.33	0.813919	0.406959	−	0.913446i	$-0.366589\pi$
$373$	5863.33	0.813919	0.406959	+	0.913446i	$0.366589\pi$
$374$	13023.4	1.80060
$375$	−6196.86	−0.853345
$376$	1512.74	0.207483
$377$	−7857.18	−1.07338
$378$	119.182	0.0162171
$379$	12485.0	1.69212	0.846058	−	0.533090i	$-0.178969\pi$
$379$	12485.0	1.69212	0.846058	+	0.533090i	$0.178969\pi$
$380$	4493.65	0.606630
$381$	−5300.47	−0.712733
$382$	−15986.5	−2.14120
$383$	−3450.06	−0.460287	−0.230144	−	0.973157i	$-0.573920\pi$
$383$	−3450.06	−0.460287	−0.230144	+	0.973157i	$0.573920\pi$
$384$	11910.6	1.58283
$385$	16093.4	2.13038
$386$	−6763.07	−0.891790
$387$	−9853.14	−1.29422
$388$	−2443.84	−0.319761
$389$	−850.997	−0.110918	−0.0554592	−	0.998461i	$-0.517662\pi$
$389$	−850.997	−0.110918	−0.0554592	+	0.998461i	$0.517662\pi$
$390$	−25629.3	−3.32767
$391$	−2472.79	−0.319833
$392$	1689.79	0.217723
$393$	1741.27	0.223500
$394$	11930.6	1.52552
$395$	1420.17	0.180903
$396$	5921.43	0.751422
$397$	−5602.71	−0.708292	−0.354146	−	0.935190i	$-0.615228\pi$
$397$	−5602.71	−0.708292	−0.354146	+	0.935190i	$0.615228\pi$
$398$	8424.76	1.06104
$399$	7988.54	1.00232
$400$	−13887.6	−1.73595
$401$	−11956.7	−1.48900	−0.744502	−	0.667620i	$-0.767313\pi$
$401$	−11956.7	−1.48900	−0.744502	+	0.667620i	$0.767313\pi$
$402$	5884.33	0.730059
$403$	3508.73	0.433703
$404$	4017.76	0.494779
$405$	12749.1	1.56421
$406$	6839.81	0.836094
$407$	−2271.30	−0.276619
$408$	−6816.64	−0.827143
$409$	−10776.3	−1.30282	−0.651410	−	0.758726i	$-0.725822\pi$
$409$	−10776.3	−1.30282	−0.651410	+	0.758726i	$0.725822\pi$
$410$	−8606.59	−1.03671
$411$	−13807.5	−1.65711
$412$	−1261.48	−0.150847
$413$	−7917.14	−0.943285
$414$	−3612.58	−0.428861
$415$	−22383.6	−2.64763
$416$	−9057.76	−1.06753
$417$	−41.0866	−0.00482499
$418$	−15042.4	−1.76016
$419$	−11387.0	−1.32766	−0.663831	−	0.747882i	$-0.731071\pi$
$419$	−11387.0	−1.32766	−0.663831	+	0.747882i	$0.731071\pi$
$420$	6943.64	0.806702
$421$	2276.64	0.263556	0.131778	−	0.991279i	$-0.457931\pi$
$421$	2276.64	0.263556	0.131778	+	0.991279i	$0.457931\pi$
$422$	−9288.54	−1.07147
$423$	−2701.10	−0.310477
$424$	−3656.18	−0.418773
$425$	10826.6	1.23569
$426$	24102.2	2.74121
$427$	10341.6	1.17205
$428$	−6052.44	−0.683541
$429$	26701.0	3.00498
$430$	21756.3	2.43996
$431$	−4140.24	−0.462710	−0.231355	−	0.972869i	$-0.574316\pi$
$431$	−4140.24	−0.462710	−0.231355	+	0.972869i	$0.574316\pi$
$432$	184.157	0.0205099
$433$	−5067.65	−0.562438	−0.281219	−	0.959644i	$-0.590739\pi$
$433$	−5067.65	−0.562438	−0.281219	+	0.959644i	$0.590739\pi$
$434$	−3054.41	−0.337826
$435$	16766.5	1.84802
$436$	−1599.20	−0.175660
$437$	2856.15	0.312650
$438$	16620.4	1.81313
$439$	5284.70	0.574545	0.287272	−	0.957849i	$-0.407252\pi$
$439$	5284.70	0.574545	0.287272	+	0.957849i	$0.407252\pi$
$440$	15861.5	1.71856
$441$	−3017.23	−0.325800
$442$	12594.5	1.35534
$443$	2692.24	0.288741	0.144370	−	0.989524i	$-0.453884\pi$
$443$	2692.24	0.288741	0.144370	+	0.989524i	$0.453884\pi$
$444$	−979.970	−0.104746
$445$	423.162	0.0450783
$446$	−7245.91	−0.769291
$447$	−18427.5	−1.94987
$448$	−1801.71	−0.190007
$449$	11463.8	1.20492	0.602462	−	0.798148i	$-0.294187\pi$
$449$	11463.8	1.20492	0.602462	+	0.798148i	$0.294187\pi$
$450$	15816.9	1.65692
$451$	8966.47	0.936174
$452$	−6.42741	−0.000668849 0
$453$	−1220.08	−0.126543
$454$	316.390	0.0327068
$455$	15563.4	1.60357
$456$	7873.41	0.808567
$457$	14221.3	1.45567	0.727836	−	0.685751i	$-0.240526\pi$
$457$	14221.3	1.45567	0.727836	+	0.685751i	$0.240526\pi$
$458$	6170.33	0.629521
$459$	−143.566	−0.0145994
$460$	2482.56	0.251631
$461$	−11584.3	−1.17036	−0.585179	−	0.810904i	$-0.698976\pi$
$461$	−11584.3	−1.17036	−0.585179	+	0.810904i	$0.698976\pi$
$462$	−23243.7	−2.34068
$463$	−14141.1	−1.41943	−0.709713	−	0.704491i	$-0.751176\pi$
$463$	−14141.1	−1.41943	−0.709713	+	0.704491i	$0.751176\pi$
$464$	10568.7	1.05741
$465$	−7487.30	−0.746699
$466$	−10277.6	−1.02167
$467$	1546.70	0.153260	0.0766301	−	0.997060i	$-0.475584\pi$
$467$	1546.70	0.153260	0.0766301	+	0.997060i	$0.475584\pi$
$468$	5726.42	0.565606
$469$	−3573.26	−0.351807
$470$	5964.19	0.585335
$471$	−16955.2	−1.65871
$472$	−7803.04	−0.760940
$473$	−22666.0	−2.20335
$474$	−2051.16	−0.198761
$475$	−12505.0	−1.20794
$476$	−3412.17	−0.328565
$477$	6528.35	0.626651
$478$	−3086.79	−0.295369
$479$	−6041.36	−0.576277	−0.288139	−	0.957589i	$-0.593036\pi$
$479$	−6041.36	−0.576277	−0.288139	+	0.957589i	$0.593036\pi$
$480$	19328.4	1.83795
$481$	−2196.49	−0.208215
$482$	−22104.7	−2.08889
$483$	4413.35	0.415765
$484$	8810.29	0.827413
$485$	11688.7	1.09434
$486$	−18201.3	−1.69882
$487$	1453.48	0.135243	0.0676215	−	0.997711i	$-0.478459\pi$
$487$	1453.48	0.135243	0.0676215	+	0.997711i	$0.478459\pi$
$488$	10192.6	0.945485
$489$	3380.20	0.312592
$490$	6662.23	0.614222
$491$	3583.72	0.329391	0.164696	−	0.986344i	$-0.447336\pi$
$491$	3583.72	0.329391	0.164696	+	0.986344i	$0.447336\pi$
$492$	3868.66	0.354497
$493$	−8239.21	−0.752688
$494$	−14547.0	−1.32490
$495$	−28321.7	−2.57165
$496$	−4719.60	−0.427251
$497$	−14636.0	−1.32096
$498$	32328.6	2.90899
$499$	13744.7	1.23306	0.616531	−	0.787330i	$-0.288537\pi$
$499$	13744.7	1.23306	0.616531	+	0.787330i	$0.288537\pi$
$500$	−3057.23	−0.273447
$501$	24045.1	2.14422
$502$	−1700.41	−0.151182
$503$	18258.7	1.61852	0.809261	−	0.587449i	$-0.199868\pi$
$503$	18258.7	1.61852	0.809261	+	0.587449i	$0.199868\pi$
$504$	6047.38	0.534467
$505$	−19216.6	−1.69332
$506$	−8310.33	−0.730117
$507$	9724.18	0.851807
$508$	−2614.99	−0.228389
$509$	−11784.5	−1.02621	−0.513105	−	0.858326i	$-0.671505\pi$
$509$	−11784.5	−1.02621	−0.513105	+	0.858326i	$0.671505\pi$
$510$	−26875.5	−2.33347
$511$	−10092.7	−0.873729
$512$	2636.56	0.227580
$513$	165.823	0.0142715
$514$	15573.6	1.33643
$515$	6033.57	0.516254
$516$	−9779.46	−0.834334
$517$	−6213.57	−0.528574
$518$	1912.09	0.162186
$519$	14609.7	1.23563
$520$	15339.1	1.29359
$521$	−6162.73	−0.518223	−0.259112	−	0.965847i	$-0.583430\pi$
$521$	−6162.73	−0.518223	−0.259112	+	0.965847i	$0.583430\pi$
$522$	−12036.9	−1.00927
$523$	−2010.65	−0.168106	−0.0840532	−	0.996461i	$-0.526787\pi$
$523$	−2010.65	−0.168106	−0.0840532	+	0.996461i	$0.526787\pi$
$524$	859.057	0.0716185
$525$	−19322.9	−1.60632
$526$	3236.09	0.268252
$527$	3679.33	0.304126
$528$	−35915.6	−2.96028
$529$	−10589.1	−0.870313
$530$	−14415.0	−1.18141
$531$	13932.8	1.13867
$532$	3941.16	0.321186
$533$	8671.17	0.704672
$534$	−611.173	−0.0495282
$535$	28948.3	2.33933
$536$	−3521.76	−0.283800
$537$	−308.567	−0.0247963
$538$	−19967.3	−1.60009
$539$	−6940.80	−0.554660
$540$	144.133	0.0114861
$541$	−897.373	−0.0713144	−0.0356572	−	0.999364i	$-0.511352\pi$
$541$	−897.373	−0.0713144	−0.0356572	+	0.999364i	$0.511352\pi$
$542$	15480.2	1.22681
$543$	20586.6	1.62699
$544$	−9498.17	−0.748586
$545$	7648.83	0.601174
$546$	−22478.2	−1.76187
$547$	20436.2	1.59742	0.798708	−	0.601719i	$-0.205517\pi$
$547$	20436.2	1.59742	0.798708	+	0.601719i	$0.205517\pi$
$548$	−6811.95	−0.531007
$549$	−18199.5	−1.41482
$550$	36385.0	2.82084
$551$	9516.52	0.735785
$552$	4349.74	0.335394
$553$	1245.56	0.0957807
$554$	−4930.02	−0.378080
$555$	4687.11	0.358481
$556$	−20.2701	−0.00154612
$557$	20236.7	1.53942	0.769708	−	0.638396i	$-0.220402\pi$
$557$	20236.7	1.53942	0.769708	+	0.638396i	$0.220402\pi$
$558$	5375.25	0.407800
$559$	−21919.6	−1.65850
$560$	−20934.4	−1.57971
$561$	27999.3	2.10718
$562$	−17711.3	−1.32937
$563$	20916.3	1.56575	0.782874	−	0.622180i	$-0.213753\pi$
$563$	20916.3	1.56575	0.782874	+	0.622180i	$0.213753\pi$
$564$	−2680.90	−0.200153
$565$	30.7417	0.00228905
$566$	26073.9	1.93634
$567$	11181.6	0.828187
$568$	−14425.1	−1.06561
$569$	4024.22	0.296493	0.148246	−	0.988950i	$-0.452637\pi$
$569$	4024.22	0.296493	0.148246	+	0.988950i	$0.452637\pi$
$570$	31042.0	2.28106
$571$	7615.45	0.558138	0.279069	−	0.960271i	$-0.409974\pi$
$571$	7615.45	0.558138	0.279069	+	0.960271i	$0.409974\pi$
$572$	13173.0	0.962919
$573$	−34369.5	−2.50577
$574$	−7548.41	−0.548893
$575$	−6908.52	−0.501052
$576$	3170.71	0.229363
$577$	6950.86	0.501504	0.250752	−	0.968051i	$-0.419322\pi$
$577$	6950.86	0.501504	0.250752	+	0.968051i	$0.419322\pi$
$578$	−3536.85	−0.254522
$579$	−14540.0	−1.04363
$580$	8271.76	0.592183
$581$	−19631.5	−1.40181
$582$	−16882.0	−1.20237
$583$	15017.7	1.06684
$584$	−9947.27	−0.704830
$585$	−27389.0	−1.93572
$586$	5412.69	0.381563
$587$	24268.2	1.70640	0.853198	−	0.521587i	$-0.174660\pi$
$587$	24268.2	1.70640	0.853198	+	0.521587i	$0.174660\pi$
$588$	−2994.67	−0.210031
$589$	−4249.73	−0.297296
$590$	−30764.5	−2.14670
$591$	25649.9	1.78527
$592$	2954.51	0.205118
$593$	17703.8	1.22599	0.612993	−	0.790089i	$-0.289965\pi$
$593$	17703.8	1.22599	0.612993	+	0.790089i	$0.289965\pi$
$594$	−482.484	−0.0333275
$595$	16320.1	1.12447
$596$	−9091.25	−0.624819
$597$	18112.5	1.24170
$598$	−8036.64	−0.549570
$599$	3262.54	0.222544	0.111272	−	0.993790i	$-0.464508\pi$
$599$	3262.54	0.222544	0.111272	+	0.993790i	$0.464508\pi$
$600$	−19044.4	−1.29581
$601$	2210.60	0.150037	0.0750185	−	0.997182i	$-0.476098\pi$
$601$	2210.60	0.150037	0.0750185	+	0.997182i	$0.476098\pi$
$602$	19081.4	1.29186
$603$	6288.33	0.424678
$604$	−601.926	−0.0405497
$605$	−42138.9	−2.83172
$606$	27754.5	1.86048
$607$	−603.474	−0.0403529	−0.0201765	−	0.999796i	$-0.506423\pi$
$607$	−603.474	−0.0403529	−0.0201765	+	0.999796i	$0.506423\pi$
$608$	10970.7	0.731774
$609$	14705.0	0.978453
$610$	40185.6	2.66732
$611$	−6008.94	−0.397865
$612$	6004.85	0.396620
$613$	−19160.1	−1.26243	−0.631213	−	0.775609i	$-0.717443\pi$
$613$	−19160.1	−1.26243	−0.631213	+	0.775609i	$0.717443\pi$
$614$	14811.1	0.973494
$615$	−18503.5	−1.21322
$616$	13911.3	0.909907
$617$	−5337.77	−0.348283	−0.174142	−	0.984721i	$-0.555715\pi$
$617$	−5337.77	−0.348283	−0.174142	+	0.984721i	$0.555715\pi$
$618$	−8714.28	−0.567216
$619$	−20398.0	−1.32450	−0.662251	−	0.749282i	$-0.730399\pi$
$619$	−20398.0	−1.32450	−0.662251	+	0.749282i	$0.730399\pi$
$620$	−3693.87	−0.239273
$621$	91.6106	0.00591982
$622$	4966.80	0.320178
$623$	371.134	0.0238671
$624$	−34732.8	−2.22824
$625$	−7117.29	−0.455507
$626$	8203.74	0.523782
$627$	−32340.0	−2.05986
$628$	−8364.85	−0.531519
$629$	−2303.29	−0.146007
$630$	23842.6	1.50779
$631$	−17137.4	−1.08119	−0.540595	−	0.841283i	$-0.681801\pi$
$631$	−17137.4	−1.08119	−0.540595	+	0.841283i	$0.681801\pi$
$632$	1227.61	0.0772655
$633$	−19969.6	−1.25390
$634$	16067.0	1.00647
$635$	12507.3	0.781632
$636$	6479.52	0.403978
$637$	−6712.22	−0.417500
$638$	−27689.6	−1.71824
$639$	25756.9	1.59457
$640$	−28104.8	−1.73584
$641$	24230.4	1.49305	0.746523	−	0.665360i	$-0.231722\pi$
$641$	24230.4	1.49305	0.746523	+	0.665360i	$0.231722\pi$
$642$	−41810.0	−2.57026
$643$	−4364.81	−0.267701	−0.133850	−	0.991002i	$-0.542734\pi$
$643$	−4364.81	−0.267701	−0.133850	+	0.991002i	$0.542734\pi$
$644$	2177.33	0.133228
$645$	46774.3	2.85540
$646$	−15254.3	−0.929061
$647$	−28488.9	−1.73109	−0.865543	−	0.500834i	$-0.833027\pi$
$647$	−28488.9	−1.73109	−0.865543	+	0.500834i	$0.833027\pi$
$648$	11020.4	0.668092
$649$	32050.9	1.93853
$650$	35186.7	2.12329
$651$	−6566.73	−0.395346
$652$	1667.62	0.100167
$653$	−15055.3	−0.902236	−0.451118	−	0.892464i	$-0.648975\pi$
$653$	−15055.3	−0.902236	−0.451118	+	0.892464i	$0.648975\pi$
$654$	−11047.2	−0.660519
$655$	−4108.79	−0.245105
$656$	−11663.6	−0.694189
$657$	17761.5	1.05471
$658$	5230.89	0.309911
$659$	−16004.1	−0.946027	−0.473014	−	0.881055i	$-0.656834\pi$
$659$	−16004.1	−0.946027	−0.473014	+	0.881055i	$0.656834\pi$
$660$	−28109.9	−1.65784
$661$	−1963.67	−0.115549	−0.0577745	−	0.998330i	$-0.518400\pi$
$661$	−1963.67	−0.115549	−0.0577745	+	0.998330i	$0.518400\pi$
$662$	18830.3	1.10553
$663$	27077.2	1.58611
$664$	−19348.6	−1.13083
$665$	−18850.2	−1.09922
$666$	−3364.95	−0.195780
$667$	5257.49	0.305204
$668$	11862.7	0.687096
$669$	−15578.1	−0.900276
$670$	−13885.0	−0.800633
$671$	−41865.9	−2.40867
$672$	16952.0	0.973120
$673$	−20465.2	−1.17218	−0.586088	−	0.810248i	$-0.699333\pi$
$673$	−20465.2	−1.17218	−0.586088	+	0.810248i	$0.699333\pi$
$674$	1810.84	0.103488
$675$	−401.097	−0.0228715
$676$	4797.44	0.272954
$677$	10550.4	0.598944	0.299472	−	0.954105i	$-0.403189\pi$
$677$	10550.4	0.598944	0.299472	+	0.954105i	$0.403189\pi$
$678$	−44.4003	−0.00251502
$679$	10251.6	0.579409
$680$	16084.9	0.907101
$681$	680.211	0.0382757
$682$	12365.2	0.694261
$683$	−10914.8	−0.611485	−0.305742	−	0.952114i	$-0.598905\pi$
$683$	−10914.8	−0.611485	−0.305742	+	0.952114i	$0.598905\pi$
$684$	−6935.77	−0.387713
$685$	32580.9	1.81730
$686$	23568.7	1.31174
$687$	13265.7	0.736707
$688$	29484.1	1.63382
$689$	14523.1	0.803030
$690$	17149.4	0.946185
$691$	−15266.3	−0.840460	−0.420230	−	0.907418i	$-0.638051\pi$
$691$	−15266.3	−0.840460	−0.420230	+	0.907418i	$0.638051\pi$
$692$	7207.69	0.395947
$693$	−24839.5	−1.36158
$694$	11168.9	0.610904
$695$	96.9503	0.00529142
$696$	14493.1	0.789310
$697$	9092.79	0.494137
$698$	−606.160	−0.0328704
$699$	−22095.9	−1.19563
$700$	−9532.97	−0.514732
$701$	22944.6	1.23624	0.618121	−	0.786083i	$-0.287894\pi$
$701$	22944.6	1.23624	0.618121	+	0.786083i	$0.287894\pi$
$702$	−466.594	−0.0250861
$703$	2660.37	0.142728
$704$	7293.86	0.390480
$705$	12822.5	0.684998
$706$	−33586.6	−1.79044
$707$	−16853.9	−0.896543
$708$	13828.6	0.734056
$709$	−1061.67	−0.0562367	−0.0281183	−	0.999605i	$-0.508952\pi$
$709$	−1061.67	−0.0562367	−0.0281183	+	0.999605i	$0.508952\pi$
$710$	−56872.8	−3.00620
$711$	−2191.98	−0.115620
$712$	365.786	0.0192534
$713$	−2347.81	−0.123318
$714$	−23571.2	−1.23547
$715$	−63005.2	−3.29547
$716$	−152.232	−0.00794576
$717$	−6636.33	−0.345660
$718$	8095.81	0.420798
$719$	17882.2	0.927532	0.463766	−	0.885958i	$-0.346498\pi$
$719$	17882.2	0.927532	0.463766	+	0.885958i	$0.346498\pi$
$720$	36841.0	1.90692
$721$	5291.74	0.273335
$722$	−5756.63	−0.296731
$723$	−47523.3	−2.44455
$724$	10156.4	0.521354
$725$	−23018.8	−1.17917
$726$	60861.1	3.11125
$727$	16646.5	0.849224	0.424612	−	0.905375i	$-0.360410\pi$
$727$	16646.5	0.849224	0.424612	+	0.905375i	$0.360410\pi$
$728$	13453.2	0.684900
$729$	−19221.4	−0.976550
$730$	−39218.4	−1.98841
$731$	−22985.3	−1.16299
$732$	−18063.4	−0.912080
$733$	21358.0	1.07623	0.538113	−	0.842873i	$-0.319137\pi$
$733$	21358.0	1.07623	0.538113	+	0.842873i	$0.319137\pi$
$734$	21052.4	1.05866
$735$	14323.2	0.718804
$736$	6060.84	0.303540
$737$	14465.6	0.722994
$738$	13283.9	0.662585
$739$	−22156.5	−1.10290	−0.551448	−	0.834210i	$-0.685924\pi$
$739$	−22156.5	−1.10290	−0.551448	+	0.834210i	$0.685924\pi$
$740$	2312.39	0.114872
$741$	−31274.9	−1.55049
$742$	−12642.6	−0.625507
$743$	18574.8	0.917153	0.458576	−	0.888655i	$-0.348360\pi$
$743$	18574.8	0.917153	0.458576	+	0.888655i	$0.348360\pi$
$744$	−6472.09	−0.318923
$745$	43482.7	2.13836
$746$	19982.5	0.980712
$747$	34548.1	1.69217
$748$	13813.5	0.675228
$749$	25389.1	1.23858
$750$	−21119.2	−1.02822
$751$	17123.4	0.832015	0.416008	−	0.909361i	$-0.363429\pi$
$751$	17123.4	0.832015	0.416008	+	0.909361i	$0.363429\pi$
$752$	8082.64	0.391946
$753$	−3655.75	−0.176923
$754$	−26777.6	−1.29335
$755$	2878.96	0.138776
$756$	126.412	0.00608144
$757$	32394.0	1.55532	0.777662	−	0.628682i	$-0.216405\pi$
$757$	32394.0	1.55532	0.777662	+	0.628682i	$0.216405\pi$
$758$	42549.5	2.03888
$759$	−17866.5	−0.854432
$760$	−18578.5	−0.886730
$761$	−14769.8	−0.703552	−0.351776	−	0.936084i	$-0.614422\pi$
$761$	−14769.8	−0.703552	−0.351776	+	0.936084i	$0.614422\pi$
$762$	−18064.3	−0.858792
$763$	6708.40	0.318297
$764$	−16956.3	−0.802953
$765$	−28720.7	−1.35738
$766$	−11758.0	−0.554613
$767$	30995.3	1.45916
$768$	33627.1	1.57996
$769$	−34871.2	−1.63523	−0.817613	−	0.575768i	$-0.804703\pi$
$769$	−34871.2	−1.63523	−0.817613	+	0.575768i	$0.804703\pi$
$770$	54847.1	2.56695
$771$	33482.0	1.56398
$772$	−7173.34	−0.334423
$773$	−32967.8	−1.53399	−0.766993	−	0.641656i	$-0.778248\pi$
$773$	−32967.8	−1.53399	−0.766993	+	0.641656i	$0.778248\pi$
$774$	−33580.0	−1.55944
$775$	10279.4	0.476446
$776$	10103.8	0.467405
$777$	4110.83	0.189801
$778$	−2900.24	−0.133649
$779$	−10502.4	−0.483040
$780$	−27184.1	−1.24788
$781$	59250.9	2.71468
$782$	−8427.40	−0.385375
$783$	305.241	0.0139316
$784$	9028.62	0.411289
$785$	40008.3	1.81906
$786$	5934.33	0.269301
$787$	18353.5	0.831298	0.415649	−	0.909525i	$-0.363554\pi$
$787$	18353.5	0.831298	0.415649	+	0.909525i	$0.363554\pi$
$788$	12654.4	0.572073
$789$	6957.33	0.313926
$790$	4840.02	0.217975
$791$	26.9620	0.00121196
$792$	−24481.5	−1.09838
$793$	−40487.1	−1.81304
$794$	−19094.3	−0.853440
$795$	−30991.0	−1.38256
$796$	8935.84	0.397892
$797$	4624.20	0.205518	0.102759	−	0.994706i	$-0.467233\pi$
$797$	4624.20	0.205518	0.102759	+	0.994706i	$0.467233\pi$
$798$	27225.3	1.20773
$799$	−6301.11	−0.278995
$800$	−26536.1	−1.17274
$801$	−653.134	−0.0288107
$802$	−40749.1	−1.79414
$803$	40858.3	1.79559
$804$	6241.30	0.273773
$805$	−10414.0	−0.455956
$806$	11957.9	0.522580
$807$	−42928.0	−1.87254
$808$	−16611.0	−0.723234
$809$	37224.5	1.61773	0.808865	−	0.587994i	$-0.200082\pi$
$809$	37224.5	1.61773	0.808865	+	0.587994i	$0.200082\pi$
$810$	43449.5	1.88476
$811$	15627.0	0.676618	0.338309	−	0.941035i	$-0.390145\pi$
$811$	15627.0	0.676618	0.338309	+	0.941035i	$0.390145\pi$
$812$	7254.74	0.313536
$813$	33281.1	1.43570
$814$	−7740.69	−0.333306
$815$	−7976.10	−0.342810
$816$	−36421.6	−1.56251
$817$	26548.7	1.13687
$818$	−36726.1	−1.56980
$819$	−24021.5	−1.02488
$820$	−9128.70	−0.388766
$821$	−40736.8	−1.73170	−0.865849	−	0.500305i	$-0.833221\pi$
$821$	−40736.8	−1.73170	−0.865849	+	0.500305i	$0.833221\pi$
$822$	−47056.6	−1.99670
$823$	−36184.5	−1.53258	−0.766290	−	0.642495i	$-0.777899\pi$
$823$	−36184.5	−1.53258	−0.766290	+	0.642495i	$0.777899\pi$
$824$	5215.48	0.220497
$825$	78224.7	3.30113
$826$	−26982.0	−1.13659
$827$	40562.7	1.70557	0.852783	−	0.522266i	$-0.174913\pi$
$827$	40562.7	1.70557	0.852783	+	0.522266i	$0.174913\pi$
$828$	−3831.73	−0.160824
$829$	−15647.9	−0.655580	−0.327790	−	0.944751i	$-0.606304\pi$
$829$	−15647.9	−0.655580	−0.327790	+	0.944751i	$0.606304\pi$
$830$	−76284.3	−3.19020
$831$	−10599.1	−0.442454
$832$	7053.65	0.293920
$833$	−7038.58	−0.292764
$834$	−140.025	−0.00581376
$835$	−56738.1	−2.35150
$836$	−15955.0	−0.660064
$837$	−136.310	−0.00562910
$838$	−38807.4	−1.59974
$839$	3611.33	0.148602	0.0743010	−	0.997236i	$-0.476327\pi$
$839$	3611.33	0.148602	0.0743010	+	0.997236i	$0.476327\pi$
$840$	−28707.8	−1.17918
$841$	−6871.33	−0.281739
$842$	7758.92	0.317565
$843$	−38077.7	−1.55571
$844$	−9852.02	−0.401801
$845$	−22945.7	−0.934150
$846$	−9205.48	−0.374103
$847$	−36957.9	−1.49928
$848$	−19535.1	−0.791083
$849$	56056.8	2.26603
$850$	36897.6	1.48891
$851$	1469.75	0.0592036
$852$	25564.3	1.02796
$853$	24661.1	0.989893	0.494946	−	0.868923i	$-0.335188\pi$
$853$	24661.1	0.989893	0.494946	+	0.868923i	$0.335188\pi$
$854$	35244.8	1.41224
$855$	33173.2	1.32690
$856$	25023.2	0.999152
$857$	−2534.30	−0.101015	−0.0505076	−	0.998724i	$-0.516084\pi$
$857$	−2534.30	−0.101015	−0.0505076	+	0.998724i	$0.516084\pi$
$858$	90998.3	3.62078
$859$	−6967.97	−0.276768	−0.138384	−	0.990379i	$-0.544191\pi$
$859$	−6967.97	−0.276768	−0.138384	+	0.990379i	$0.544191\pi$
$860$	23076.1	0.914988
$861$	−16228.5	−0.642351
$862$	−14110.1	−0.557532
$863$	−20109.9	−0.793222	−0.396611	−	0.917987i	$-0.629814\pi$
$863$	−20109.9	−0.793222	−0.396611	+	0.917987i	$0.629814\pi$
$864$	351.883	0.0138557
$865$	−34473.8	−1.35508
$866$	−17270.8	−0.677697
$867$	−7603.94	−0.297858
$868$	−3239.70	−0.126685
$869$	−5042.40	−0.196838
$870$	57140.9	2.22673
$871$	13989.2	0.544208
$872$	6611.72	0.256767
$873$	−18041.0	−0.699423
$874$	9733.88	0.376720
$875$	12824.6	0.495487
$876$	17628.7	0.679928
$877$	19107.7	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$877$	19107.7	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$878$	18010.5	0.692284
$879$	11636.8	0.446531
$880$	84748.4	3.24644
$881$	−49065.2	−1.87633	−0.938167	−	0.346183i	$-0.887478\pi$
$881$	−49065.2	−1.87633	−0.938167	+	0.346183i	$0.887478\pi$
$882$	−10282.9	−0.392565
$883$	12342.2	0.470381	0.235191	−	0.971949i	$-0.424429\pi$
$883$	12342.2	0.470381	0.235191	+	0.971949i	$0.424429\pi$
$884$	13358.5	0.508254
$885$	−66141.1	−2.51221
$886$	9175.28	0.347911
$887$	1885.82	0.0713863	0.0356932	−	0.999363i	$-0.488636\pi$
$887$	1885.82	0.0713863	0.0356932	+	0.999363i	$0.488636\pi$
$888$	4051.58	0.153111
$889$	10969.5	0.413842
$890$	1442.16	0.0543160
$891$	−45266.3	−1.70199
$892$	−7685.48	−0.288485
$893$	7277.96	0.272730
$894$	−62802.0	−2.34945
$895$	728.111	0.0271934
$896$	−24649.3	−0.919058
$897$	−17278.1	−0.643143
$898$	39069.2	1.45185
$899$	−7822.76	−0.290215
$900$	16776.4	0.621349
$901$	15229.3	0.563109
$902$	30558.2	1.12802
$903$	41023.4	1.51182
$904$	26.5735	0.000977677 0
$905$	−48577.2	−1.78427
$906$	−4158.08	−0.152476
$907$	−32004.6	−1.17166	−0.585831	−	0.810433i	$-0.699232\pi$
$907$	−32004.6	−1.17166	−0.585831	+	0.810433i	$0.699232\pi$
$908$	335.583	0.0122651
$909$	29660.0	1.08224
$910$	53040.8	1.93218
$911$	26451.4	0.961991	0.480995	−	0.876723i	$-0.340275\pi$
$911$	26451.4	0.961991	0.480995	+	0.876723i	$0.340275\pi$
$912$	42067.9	1.52742
$913$	79474.0	2.88084
$914$	48466.7	1.75398
$915$	86395.7	3.12148
$916$	6544.65	0.236071
$917$	−3603.62	−0.129773
$918$	−489.281	−0.0175912
$919$	−54432.9	−1.95384	−0.976918	−	0.213617i	$-0.931476\pi$
$919$	−54432.9	−1.95384	−0.976918	+	0.213617i	$0.931476\pi$
$920$	−10263.9	−0.367816
$921$	31842.5	1.13925
$922$	−39479.9	−1.41020
$923$	57299.6	2.04338
$924$	−24653.8	−0.877759
$925$	−6434.96	−0.228735
$926$	−48193.7	−1.71031
$927$	−9312.57	−0.329951
$928$	20194.4	0.714346
$929$	−50860.2	−1.79620	−0.898099	−	0.439793i	$-0.855052\pi$
$929$	−50860.2	−1.79620	−0.898099	+	0.439793i	$0.855052\pi$
$930$	−25517.1	−0.899718
$931$	8129.76	0.286189
$932$	−10901.0	−0.383128
$933$	10678.2	0.374694
$934$	5271.21	0.184667
$935$	−66068.6	−2.31088
$936$	−23675.3	−0.826764
$937$	42688.1	1.48832	0.744162	−	0.667999i	$-0.232849\pi$
$937$	42688.1	1.48832	0.744162	+	0.667999i	$0.232849\pi$
$938$	−12177.8	−0.423902
$939$	17637.4	0.612964
$940$	6326.00	0.219501
$941$	32789.1	1.13591	0.567956	−	0.823059i	$-0.307734\pi$
$941$	32789.1	1.13591	0.567956	+	0.823059i	$0.307734\pi$
$942$	−57784.0	−1.99863
$943$	−5802.16	−0.200365
$944$	−41691.9	−1.43745
$945$	−604.618	−0.0208130
$946$	−77247.0	−2.65488
$947$	14996.6	0.514599	0.257299	−	0.966332i	$-0.417167\pi$
$947$	14996.6	0.514599	0.257299	+	0.966332i	$0.417167\pi$
$948$	−2175.59	−0.0745357
$949$	39512.7	1.35157
$950$	−42617.7	−1.45547
$951$	34542.6	1.17784
$952$	14107.3	0.480273
$953$	7373.11	0.250618	0.125309	−	0.992118i	$-0.460008\pi$
$953$	7373.11	0.250618	0.125309	+	0.992118i	$0.460008\pi$
$954$	22248.9	0.755069
$955$	81100.3	2.74800
$956$	−3274.04	−0.110764
$957$	−59530.2	−2.01080
$958$	−20589.2	−0.694372
$959$	28575.1	0.962188
$960$	−15051.8	−0.506037
$961$	−26297.6	−0.882738
$962$	−7485.76	−0.250884
$963$	−44680.5	−1.49513
$964$	−23445.7	−0.783335
$965$	34309.5	1.14452
$966$	15040.9	0.500966
$967$	−35306.7	−1.17413	−0.587067	−	0.809538i	$-0.699718\pi$
$967$	−35306.7	−1.17413	−0.587067	+	0.809538i	$0.699718\pi$
$968$	−36425.2	−1.20945
$969$	−32795.5	−1.08725
$970$	39835.6	1.31860
$971$	−42629.0	−1.40889	−0.704443	−	0.709761i	$-0.748803\pi$
$971$	−42629.0	−1.40889	−0.704443	+	0.709761i	$0.748803\pi$
$972$	−19305.4	−0.637060
$973$	85.0302	0.00280159
$974$	4953.52	0.162958
$975$	75648.5	2.48481
$976$	54459.3	1.78607
$977$	−22751.7	−0.745028	−0.372514	−	0.928027i	$-0.621504\pi$
$977$	−22751.7	−0.745028	−0.372514	+	0.928027i	$0.621504\pi$
$978$	11519.9	0.376651
$979$	−1502.46	−0.0490489
$980$	7066.39	0.230334
$981$	−11805.7	−0.384226
$982$	12213.5	0.396892
$983$	27626.4	0.896385	0.448193	−	0.893937i	$-0.352068\pi$
$983$	27626.4	0.896385	0.448193	+	0.893937i	$0.352068\pi$
$984$	−15994.6	−0.518179
$985$	−60524.8	−1.95785
$986$	−28079.6	−0.906935
$987$	11246.0	0.362678
$988$	−15429.5	−0.496840
$989$	14667.1	0.471574
$990$	−96521.7	−3.09865
$991$	48652.8	1.55954	0.779772	−	0.626064i	$-0.215335\pi$
$991$	48652.8	1.55954	0.779772	+	0.626064i	$0.215335\pi$
$992$	−9018.08	−0.288633
$993$	40483.7	1.29377
$994$	−49880.3	−1.59166
$995$	−42739.3	−1.36174
$996$	34289.7	1.09088
$997$	−61036.9	−1.93887	−0.969437	−	0.245342i	$-0.921100\pi$
$997$	−61036.9	−1.93887	−0.969437	+	0.245342i	$0.921100\pi$
$998$	46842.7	1.48575
$999$	85.3310	0.00270246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.a.b.1.4	✓	5
3.2	odd	2		333.4.a.f.1.2		5
4.3	odd	2		592.4.a.g.1.2		5
5.4	even	2		925.4.a.b.1.2		5
7.6	odd	2		1813.4.a.c.1.4		5
8.3	odd	2		2368.4.a.r.1.4		5
8.5	even	2		2368.4.a.m.1.2		5
37.36	even	2		1369.4.a.d.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.a.b.1.4	✓	5	1.1	even	1	trivial
333.4.a.f.1.2		5	3.2	odd	2
592.4.a.g.1.2		5	4.3	odd	2
925.4.a.b.1.2		5	5.4	even	2
1369.4.a.d.1.2		5	37.36	even	2
1813.4.a.c.1.4		5	7.6	odd	2
2368.4.a.m.1.2		5	8.5	even	2
2368.4.a.r.1.4		5	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	37.a (trivial)

\(f(q)\)	\(=\)	\(q+3.40805 q^{2} +7.32702 q^{3} +3.61479 q^{4} -17.2892 q^{5} +24.9708 q^{6} -15.1635 q^{7} -14.9450 q^{8} +26.6852 q^{9} +O(q^{10})\)	\(q+3.40805 q^{2} +7.32702 q^{3} +3.61479 q^{4} -17.2892 q^{5} +24.9708 q^{6} -15.1635 q^{7} -14.9450 q^{8} +26.6852 q^{9} -58.9226 q^{10} +61.3864 q^{11} +26.4857 q^{12} +59.3647 q^{13} -51.6780 q^{14} -126.679 q^{15} -79.8516 q^{16} +62.2512 q^{17} +90.9446 q^{18} -71.9018 q^{19} -62.4971 q^{20} -111.103 q^{21} +209.208 q^{22} -39.7229 q^{23} -109.502 q^{24} +173.918 q^{25} +202.318 q^{26} -2.30624 q^{27} -54.8130 q^{28} -132.354 q^{29} -431.727 q^{30} +59.1047 q^{31} -152.578 q^{32} +449.779 q^{33} +212.155 q^{34} +262.166 q^{35} +96.4617 q^{36} -37.0000 q^{37} -245.045 q^{38} +434.966 q^{39} +258.388 q^{40} +146.066 q^{41} -378.646 q^{42} -369.236 q^{43} +221.899 q^{44} -461.368 q^{45} -135.377 q^{46} -101.221 q^{47} -585.074 q^{48} -113.068 q^{49} +592.721 q^{50} +456.116 q^{51} +214.591 q^{52} +244.643 q^{53} -7.85979 q^{54} -1061.32 q^{55} +226.619 q^{56} -526.826 q^{57} -451.070 q^{58} +522.117 q^{59} -457.917 q^{60} -682.007 q^{61} +201.432 q^{62} -404.642 q^{63} +118.819 q^{64} -1026.37 q^{65} +1532.87 q^{66} +235.648 q^{67} +225.025 q^{68} -291.050 q^{69} +893.474 q^{70} +965.213 q^{71} -398.811 q^{72} +665.592 q^{73} -126.098 q^{74} +1274.30 q^{75} -259.910 q^{76} -930.834 q^{77} +1482.39 q^{78} -82.1421 q^{79} +1380.57 q^{80} -737.399 q^{81} +497.800 q^{82} +1294.65 q^{83} -401.616 q^{84} -1076.28 q^{85} -1258.37 q^{86} -969.763 q^{87} -917.419 q^{88} -24.4755 q^{89} -1572.36 q^{90} -900.178 q^{91} -143.590 q^{92} +433.061 q^{93} -344.965 q^{94} +1243.13 q^{95} -1117.94 q^{96} -676.067 q^{97} -385.340 q^{98} +1638.11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 13 q^{3} + 18 q^{4} + 11 q^{5} + 9 q^{6} + 24 q^{7} + 30 q^{8} + 46 q^{9}+O(q^{10}) \) 5 * q + 4 * q^2 + 13 * q^3 + 18 * q^4 + 11 * q^5 + 9 * q^6 + 24 * q^7 + 30 * q^8 + 46 * q^9	\( 5 q + 4 q^{2} + 13 q^{3} + 18 q^{4} + 11 q^{5} + 9 q^{6} + 24 q^{7} + 30 q^{8} + 46 q^{9} - 75 q^{10} + 61 q^{11} - 65 q^{12} - 37 q^{13} - 128 q^{14} - 116 q^{15} - 182 q^{16} + 130 q^{17} - 159 q^{18} - 22 q^{19} - 59 q^{20} - 44 q^{21} - 95 q^{22} + 73 q^{23} - 105 q^{24} + 26 q^{25} + 197 q^{26} + 472 q^{27} - 2 q^{28} + 271 q^{29} - 196 q^{30} + 363 q^{31} + 74 q^{32} + 198 q^{33} + 272 q^{34} + 604 q^{35} - 251 q^{36} - 185 q^{37} + 576 q^{38} - 65 q^{39} + 97 q^{40} + 381 q^{41} - 376 q^{42} - 408 q^{43} + 235 q^{44} - 704 q^{45} - 325 q^{46} + 276 q^{47} - 889 q^{48} - 949 q^{49} + 415 q^{50} - 38 q^{51} - 403 q^{52} + 156 q^{53} - 1068 q^{54} - 843 q^{55} + 578 q^{56} - 1618 q^{57} - 31 q^{58} + 100 q^{59} - 952 q^{60} - 1711 q^{61} + 1305 q^{62} + 94 q^{63} + 370 q^{64} - 890 q^{65} + 2490 q^{66} + 787 q^{67} + 2464 q^{68} - 2335 q^{69} + 308 q^{70} + 1578 q^{71} + 753 q^{72} - 313 q^{73} - 148 q^{74} + 684 q^{75} + 2080 q^{76} - 342 q^{77} + 3955 q^{78} + 569 q^{79} + 189 q^{80} + 385 q^{81} + 119 q^{82} + 2422 q^{83} - 1098 q^{84} - 2210 q^{85} + 1566 q^{86} + 2371 q^{87} - 669 q^{88} - 2466 q^{89} - 1930 q^{90} - 1678 q^{91} - 373 q^{92} - 1142 q^{93} - 1660 q^{94} + 794 q^{95} - 1797 q^{96} - 2406 q^{97} - 2010 q^{98} + 1746 q^{99}+O(q^{100}) \) 5 * q + 4 * q^2 + 13 * q^3 + 18 * q^4 + 11 * q^5 + 9 * q^6 + 24 * q^7 + 30 * q^8 + 46 * q^9 - 75 * q^10 + 61 * q^11 - 65 * q^12 - 37 * q^13 - 128 * q^14 - 116 * q^15 - 182 * q^16 + 130 * q^17 - 159 * q^18 - 22 * q^19 - 59 * q^20 - 44 * q^21 - 95 * q^22 + 73 * q^23 - 105 * q^24 + 26 * q^25 + 197 * q^26 + 472 * q^27 - 2 * q^28 + 271 * q^29 - 196 * q^30 + 363 * q^31 + 74 * q^32 + 198 * q^33 + 272 * q^34 + 604 * q^35 - 251 * q^36 - 185 * q^37 + 576 * q^38 - 65 * q^39 + 97 * q^40 + 381 * q^41 - 376 * q^42 - 408 * q^43 + 235 * q^44 - 704 * q^45 - 325 * q^46 + 276 * q^47 - 889 * q^48 - 949 * q^49 + 415 * q^50 - 38 * q^51 - 403 * q^52 + 156 * q^53 - 1068 * q^54 - 843 * q^55 + 578 * q^56 - 1618 * q^57 - 31 * q^58 + 100 * q^59 - 952 * q^60 - 1711 * q^61 + 1305 * q^62 + 94 * q^63 + 370 * q^64 - 890 * q^65 + 2490 * q^66 + 787 * q^67 + 2464 * q^68 - 2335 * q^69 + 308 * q^70 + 1578 * q^71 + 753 * q^72 - 313 * q^73 - 148 * q^74 + 684 * q^75 + 2080 * q^76 - 342 * q^77 + 3955 * q^78 + 569 * q^79 + 189 * q^80 + 385 * q^81 + 119 * q^82 + 2422 * q^83 - 1098 * q^84 - 2210 * q^85 + 1566 * q^86 + 2371 * q^87 - 669 * q^88 - 2466 * q^89 - 1930 * q^90 - 1678 * q^91 - 373 * q^92 - 1142 * q^93 - 1660 * q^94 + 794 * q^95 - 1797 * q^96 - 2406 * q^97 - 2010 * q^98 + 1746 * q^99

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