LMFDB - Embedded newform 37.4.a.b.1.3

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			1.3
Root			$-0.960270$ 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)

$f(q)$	$=$	$q+1.96027 q^{2} +3.37210 q^{3} -4.15734 q^{4} +14.4391 q^{5} +6.61023 q^{6} +7.73331 q^{7} -23.8317 q^{8} -15.6289 q^{9} +O(q^{10})$	\(q+1.96027 q^{2} +3.37210 q^{3} -4.15734 q^{4} +14.4391 q^{5} +6.61023 q^{6} +7.73331 q^{7} -23.8317 q^{8} -15.6289 q^{9} +28.3045 q^{10} -22.3879 q^{11} -14.0190 q^{12} +16.3503 q^{13} +15.1594 q^{14} +48.6901 q^{15} -13.4578 q^{16} -116.372 q^{17} -30.6369 q^{18} -25.6535 q^{19} -60.0282 q^{20} +26.0775 q^{21} -43.8863 q^{22} +161.379 q^{23} -80.3628 q^{24} +83.4873 q^{25} +32.0510 q^{26} -143.749 q^{27} -32.1500 q^{28} +191.599 q^{29} +95.4457 q^{30} +100.698 q^{31} +164.272 q^{32} -75.4942 q^{33} -228.120 q^{34} +111.662 q^{35} +64.9748 q^{36} -37.0000 q^{37} -50.2877 q^{38} +55.1348 q^{39} -344.108 q^{40} +318.600 q^{41} +51.1190 q^{42} +205.491 q^{43} +93.0741 q^{44} -225.668 q^{45} +316.346 q^{46} +60.6783 q^{47} -45.3811 q^{48} -283.196 q^{49} +163.658 q^{50} -392.417 q^{51} -67.9736 q^{52} -726.017 q^{53} -281.787 q^{54} -323.261 q^{55} -184.298 q^{56} -86.5061 q^{57} +375.586 q^{58} +77.6735 q^{59} -202.421 q^{60} +48.7787 q^{61} +197.395 q^{62} -120.863 q^{63} +429.681 q^{64} +236.083 q^{65} -147.989 q^{66} -586.321 q^{67} +483.796 q^{68} +544.186 q^{69} +218.888 q^{70} +880.768 q^{71} +372.464 q^{72} -287.542 q^{73} -72.5300 q^{74} +281.528 q^{75} +106.650 q^{76} -173.132 q^{77} +108.079 q^{78} -277.922 q^{79} -194.318 q^{80} -62.7554 q^{81} +624.541 q^{82} +672.991 q^{83} -108.413 q^{84} -1680.30 q^{85} +402.818 q^{86} +646.091 q^{87} +533.541 q^{88} -280.731 q^{89} -442.369 q^{90} +126.442 q^{91} -670.907 q^{92} +339.563 q^{93} +118.946 q^{94} -370.412 q^{95} +553.943 q^{96} -1727.56 q^{97} -555.141 q^{98} +349.899 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.96027	0.693060	0.346530	−	0.938039i	$-0.387360\pi$
$2$	1.96027	0.693060	0.346530	+	0.938039i	$0.387360\pi$
$3$	3.37210	0.648961	0.324481	−	0.945892i	$-0.394810\pi$
$3$	3.37210	0.648961	0.324481	+	0.945892i	$0.394810\pi$
$4$	−4.15734	−0.519667
$5$	14.4391	1.29147	0.645736	−	0.763561i	$-0.276551\pi$
$5$	14.4391	1.29147	0.645736	+	0.763561i	$0.276551\pi$
$6$	6.61023	0.449769
$7$	7.73331	0.417559	0.208780	−	0.977963i	$-0.433051\pi$
$7$	7.73331	0.417559	0.208780	+	0.977963i	$0.433051\pi$
$8$	−23.8317	−1.05322
$9$	−15.6289	−0.578849
$10$	28.3045	0.895067
$11$	−22.3879	−0.613655	−0.306827	−	0.951765i	$-0.599267\pi$
$11$	−22.3879	−0.613655	−0.306827	+	0.951765i	$0.599267\pi$
$12$	−14.0190	−0.337244
$13$	16.3503	0.348827	0.174413	−	0.984673i	$-0.444197\pi$
$13$	16.3503	0.348827	0.174413	+	0.984673i	$0.444197\pi$
$14$	15.1594	0.289394
$15$	48.6901	0.838115
$16$	−13.4578	−0.210278
$17$	−116.372	−1.66025	−0.830125	−	0.557577i	$-0.811731\pi$
$17$	−116.372	−1.66025	−0.830125	+	0.557577i	$0.811731\pi$
$18$	−30.6369	−0.401177
$19$	−25.6535	−0.309753	−0.154876	−	0.987934i	$-0.549498\pi$
$19$	−25.6535	−0.309753	−0.154876	+	0.987934i	$0.549498\pi$
$20$	−60.0282	−0.671136
$21$	26.0775	0.270980
$22$	−43.8863	−0.425300
$23$	161.379	1.46304	0.731518	−	0.681822i	$-0.238812\pi$
$23$	161.379	1.46304	0.731518	+	0.681822i	$0.238812\pi$
$24$	−80.3628	−0.683500
$25$	83.4873	0.667898
$26$	32.0510	0.241758
$27$	−143.749	−1.02461
$28$	−32.1500	−0.216992
$29$	191.599	1.22686	0.613432	−	0.789748i	$-0.289789\pi$
$29$	191.599	1.22686	0.613432	+	0.789748i	$0.289789\pi$
$30$	95.4457	0.580864
$31$	100.698	0.583415	0.291708	−	0.956508i	$-0.405777\pi$
$31$	100.698	0.583415	0.291708	+	0.956508i	$0.405777\pi$
$32$	164.272	0.907486
$33$	−75.4942	−0.398238
$34$	−228.120	−1.15065
$35$	111.662	0.539266
$36$	64.9748	0.300809
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	−50.2877	−0.214677
$39$	55.1348	0.226375
$40$	−344.108	−1.36020
$41$	318.600	1.21358	0.606791	−	0.794861i	$-0.292456\pi$
$41$	318.600	1.21358	0.606791	+	0.794861i	$0.292456\pi$
$42$	51.1190	0.187805
$43$	205.491	0.728769	0.364385	−	0.931249i	$-0.381279\pi$
$43$	205.491	0.728769	0.364385	+	0.931249i	$0.381279\pi$
$44$	93.0741	0.318896
$45$	−225.668	−0.747567
$46$	316.346	1.01397
$47$	60.6783	0.188316	0.0941579	−	0.995557i	$-0.469984\pi$
$47$	60.6783	0.188316	0.0941579	+	0.995557i	$0.469984\pi$
$48$	−45.3811	−0.136462
$49$	−283.196	−0.825644
$50$	163.658	0.462894
$51$	−392.417	−1.07744
$52$	−67.9736	−0.181274
$53$	−726.017	−1.88162	−0.940812	−	0.338928i	$-0.889936\pi$
$53$	−726.017	−1.88162	−0.940812	+	0.338928i	$0.889936\pi$
$54$	−281.787	−0.710118
$55$	−323.261	−0.792518
$56$	−184.298	−0.439782
$57$	−86.5061	−0.201018
$58$	375.586	0.850290
$59$	77.6735	0.171394	0.0856969	−	0.996321i	$-0.472688\pi$
$59$	77.6735	0.171394	0.0856969	+	0.996321i	$0.472688\pi$
$60$	−202.421	−0.435541
$61$	48.7787	0.102385	0.0511924	−	0.998689i	$-0.483698\pi$
$61$	48.7787	0.102385	0.0511924	+	0.998689i	$0.483698\pi$
$62$	197.395	0.404342
$63$	−120.863	−0.241704
$64$	429.681	0.839220
$65$	236.083	0.450500
$66$	−147.989	−0.276003
$67$	−586.321	−1.06911	−0.534556	−	0.845133i	$-0.679521\pi$
$67$	−586.321	−1.06911	−0.534556	+	0.845133i	$0.679521\pi$
$68$	483.796	0.862778
$69$	544.186	0.949454
$70$	218.888	0.373744
$71$	880.768	1.47222	0.736112	−	0.676859i	$-0.236659\pi$
$71$	880.768	1.47222	0.736112	+	0.676859i	$0.236659\pi$
$72$	372.464	0.609656
$73$	−287.542	−0.461018	−0.230509	−	0.973070i	$-0.574039\pi$
$73$	−287.542	−0.461018	−0.230509	+	0.973070i	$0.574039\pi$
$74$	−72.5300	−0.113938
$75$	281.528	0.433440
$76$	106.650	0.160969
$77$	−173.132	−0.256237
$78$	108.079	0.156892
$79$	−277.922	−0.395805	−0.197903	−	0.980222i	$-0.563413\pi$
$79$	−277.922	−0.395805	−0.197903	+	0.980222i	$0.563413\pi$
$80$	−194.318	−0.271568
$81$	−62.7554	−0.0860842
$82$	624.541	0.841086
$83$	672.991	0.890004	0.445002	−	0.895529i	$-0.353203\pi$
$83$	672.991	0.890004	0.445002	+	0.895529i	$0.353203\pi$
$84$	−108.413	−0.140819
$85$	−1680.30	−2.14417
$86$	402.818	0.505081
$87$	646.091	0.796187
$88$	533.541	0.646314
$89$	−280.731	−0.334354	−0.167177	−	0.985927i	$-0.553465\pi$
$89$	−280.731	−0.334354	−0.167177	+	0.985927i	$0.553465\pi$
$90$	−442.369	−0.518109
$91$	126.442	0.145656
$92$	−670.907	−0.760293
$93$	339.563	0.378614
$94$	118.946	0.130514
$95$	−370.412	−0.400037
$96$	553.943	0.588923
$97$	−1727.56	−1.80832	−0.904161	−	0.427191i	$-0.859503\pi$
$97$	−1727.56	−1.80832	−0.904161	+	0.427191i	$0.859503\pi$
$98$	−555.141	−0.572221
$99$	349.899	0.355214
$100$	−347.085	−0.347085
$101$	−23.5572	−0.0232082	−0.0116041	−	0.999933i	$-0.503694\pi$
$101$	−23.5572	−0.0232082	−0.0116041	+	0.999933i	$0.503694\pi$
$102$	−769.243	−0.746729
$103$	1810.67	1.73214	0.866072	−	0.499920i	$-0.166637\pi$
$103$	1810.67	1.73214	0.866072	+	0.499920i	$0.166637\pi$
$104$	−389.654	−0.367392
$105$	376.535	0.349963
$106$	−1423.19	−1.30408
$107$	555.758	0.502123	0.251061	−	0.967971i	$-0.419220\pi$
$107$	555.758	0.502123	0.251061	+	0.967971i	$0.419220\pi$
$108$	597.614	0.532458
$109$	−1655.26	−1.45454	−0.727272	−	0.686349i	$-0.759212\pi$
$109$	−1655.26	−1.45454	−0.727272	+	0.686349i	$0.759212\pi$
$110$	−633.678	−0.549262
$111$	−124.768	−0.106689
$112$	−104.073	−0.0878037
$113$	1993.35	1.65945	0.829727	−	0.558169i	$-0.188496\pi$
$113$	1993.35	1.65945	0.829727	+	0.558169i	$0.188496\pi$
$114$	−169.575	−0.139317
$115$	2330.17	1.88947
$116$	−796.542	−0.637561
$117$	−255.537	−0.201918
$118$	152.261	0.118786
$119$	−899.937	−0.693253
$120$	−1160.37	−0.882720
$121$	−829.782	−0.623428
$122$	95.6194	0.0709588
$123$	1074.35	0.787568
$124$	−418.635	−0.303182
$125$	−599.406	−0.428900
$126$	−236.925	−0.167515
$127$	2795.57	1.95328	0.976641	−	0.214876i	$-0.0689346\pi$
$127$	2795.57	1.95328	0.976641	+	0.214876i	$0.0689346\pi$
$128$	−471.889	−0.325855
$129$	692.936	0.472943
$130$	462.787	0.312224
$131$	−842.231	−0.561726	−0.280863	−	0.959748i	$-0.590621\pi$
$131$	−842.231	−0.561726	−0.280863	+	0.959748i	$0.590621\pi$
$132$	313.855	0.206951
$133$	−198.386	−0.129340
$134$	−1149.35	−0.740958
$135$	−2075.61	−1.32326
$136$	2773.33	1.74861
$137$	−1261.02	−0.786396	−0.393198	−	0.919454i	$-0.628631\pi$
$137$	−1261.02	−0.786396	−0.393198	+	0.919454i	$0.628631\pi$
$138$	1066.75	0.658029
$139$	−1673.89	−1.02142	−0.510711	−	0.859752i	$-0.670618\pi$
$139$	−1673.89	−1.02142	−0.510711	+	0.859752i	$0.670618\pi$
$140$	−464.217	−0.280239
$141$	204.614	0.122210
$142$	1726.54	1.02034
$143$	−366.048	−0.214059
$144$	210.331	0.121719
$145$	2766.51	1.58446
$146$	−563.661	−0.319513
$147$	−954.965	−0.535811
$148$	153.822	0.0854328
$149$	1384.89	0.761443	0.380722	−	0.924690i	$-0.375676\pi$
$149$	1384.89	0.761443	0.380722	+	0.924690i	$0.375676\pi$
$150$	551.870	0.300400
$151$	853.417	0.459934	0.229967	−	0.973198i	$-0.426138\pi$
$151$	853.417	0.459934	0.229967	+	0.973198i	$0.426138\pi$
$152$	611.365	0.326238
$153$	1818.76	0.961035
$154$	−339.387	−0.177588
$155$	1453.99	0.753464
$156$	−229.214	−0.117640
$157$	1334.13	0.678185	0.339093	−	0.940753i	$-0.389880\pi$
$157$	1334.13	0.678185	0.339093	+	0.940753i	$0.389880\pi$
$158$	−544.802	−0.274317
$159$	−2448.20	−1.22110
$160$	2371.94	1.17199
$161$	1247.99	0.610905
$162$	−123.018	−0.0596616
$163$	−2122.93	−1.02013	−0.510064	−	0.860136i	$-0.670378\pi$
$163$	−2122.93	−1.02013	−0.510064	+	0.860136i	$0.670378\pi$
$164$	−1324.53	−0.630660
$165$	−1090.07	−0.514313
$166$	1319.24	0.616827
$167$	−3903.94	−1.80896	−0.904478	−	0.426521i	$-0.859739\pi$
$167$	−3903.94	−1.80896	−0.904478	+	0.426521i	$0.859739\pi$
$168$	−621.471	−0.285402
$169$	−1929.67	−0.878320
$170$	−3293.84	−1.48604
$171$	400.936	0.179300
$172$	−854.296	−0.378718
$173$	742.025	0.326099	0.163050	−	0.986618i	$-0.447867\pi$
$173$	742.025	0.326099	0.163050	+	0.986618i	$0.447867\pi$
$174$	1266.51	0.551805
$175$	645.633	0.278887
$176$	301.292	0.129038
$177$	261.923	0.111228
$178$	−550.310	−0.231727
$179$	3163.04	1.32076	0.660381	−	0.750931i	$-0.270395\pi$
$179$	3163.04	1.32076	0.660381	+	0.750931i	$0.270395\pi$
$180$	938.176	0.388486
$181$	−2119.39	−0.870347	−0.435173	−	0.900347i	$-0.643313\pi$
$181$	−2119.39	−0.870347	−0.435173	+	0.900347i	$0.643313\pi$
$182$	247.860	0.100948
$183$	164.487	0.0664438
$184$	−3845.93	−1.54090
$185$	−534.246	−0.212317
$186$	665.636	0.262402
$187$	2605.31	1.01882
$188$	−252.260	−0.0978616
$189$	−1111.66	−0.427836
$190$	−726.109	−0.277250
$191$	−3236.85	−1.22623	−0.613117	−	0.789992i	$-0.710084\pi$
$191$	−3236.85	−1.22623	−0.613117	+	0.789992i	$0.710084\pi$
$192$	1448.93	0.544622
$193$	−431.146	−0.160801	−0.0804005	−	0.996763i	$-0.525620\pi$
$193$	−431.146	−0.160801	−0.0804005	+	0.996763i	$0.525620\pi$
$194$	−3386.49	−1.25328
$195$	796.096	0.292357
$196$	1177.34	0.429060
$197$	1340.04	0.484640	0.242320	−	0.970196i	$-0.422092\pi$
$197$	1340.04	0.484640	0.242320	+	0.970196i	$0.422092\pi$
$198$	685.896	0.246184
$199$	−4541.30	−1.61771	−0.808855	−	0.588008i	$-0.799912\pi$
$199$	−4541.30	−1.61771	−0.808855	+	0.588008i	$0.799912\pi$
$200$	−1989.64	−0.703444
$201$	−1977.13	−0.693812
$202$	−46.1785	−0.0160847
$203$	1481.69	0.512288
$204$	1631.41	0.559909
$205$	4600.29	1.56731
$206$	3549.41	1.20048
$207$	−2522.18	−0.846878
$208$	−220.039	−0.0733507
$209$	574.327	0.190081
$210$	738.111	0.242545
$211$	−3191.65	−1.04134	−0.520669	−	0.853758i	$-0.674318\pi$
$211$	−3191.65	−1.04134	−0.520669	+	0.853758i	$0.674318\pi$
$212$	3018.30	0.977819
$213$	2970.04	0.955417
$214$	1089.44	0.348001
$215$	2967.10	0.941185
$216$	3425.78	1.07914
$217$	778.728	0.243610
$218$	−3244.76	−1.00809
$219$	−969.622	−0.299183
$220$	1343.90	0.411846
$221$	−1902.71	−0.579140
$222$	−244.579	−0.0739416
$223$	−2551.80	−0.766285	−0.383142	−	0.923689i	$-0.625158\pi$
$223$	−2551.80	−0.766285	−0.383142	+	0.923689i	$0.625158\pi$
$224$	1270.37	0.378929
$225$	−1304.82	−0.386612
$226$	3907.50	1.15010
$227$	4500.20	1.31581	0.657904	−	0.753101i	$-0.271443\pi$
$227$	4500.20	1.31581	0.657904	+	0.753101i	$0.271443\pi$
$228$	359.635	0.104462
$229$	1269.97	0.366472	0.183236	−	0.983069i	$-0.441343\pi$
$229$	1269.97	0.366472	0.183236	+	0.983069i	$0.441343\pi$
$230$	4567.75	1.30952
$231$	−583.820	−0.166288
$232$	−4566.12	−1.29216
$233$	2020.64	0.568138	0.284069	−	0.958804i	$-0.408315\pi$
$233$	2020.64	0.568138	0.284069	+	0.958804i	$0.408315\pi$
$234$	−500.922	−0.139941
$235$	876.140	0.243205
$236$	−322.915	−0.0890678
$237$	−937.180	−0.256862
$238$	−1764.12	−0.480466
$239$	−6496.71	−1.75831	−0.879157	−	0.476533i	$-0.841893\pi$
$239$	−6496.71	−1.75831	−0.879157	+	0.476533i	$0.841893\pi$
$240$	−655.262	−0.176237
$241$	4475.29	1.19618	0.598088	−	0.801430i	$-0.295927\pi$
$241$	4475.29	1.19618	0.598088	+	0.801430i	$0.295927\pi$
$242$	−1626.60	−0.432073
$243$	3669.61	0.968747
$244$	−202.790	−0.0532060
$245$	−4089.09	−1.06630
$246$	2106.02	0.545832
$247$	−419.441	−0.108050
$248$	−2399.80	−0.614465
$249$	2269.39	0.577578
$250$	−1175.00	−0.297254
$251$	5375.18	1.35171	0.675853	−	0.737037i	$-0.263775\pi$
$251$	5375.18	1.35171	0.675853	+	0.737037i	$0.263775\pi$
$252$	502.470	0.125606
$253$	−3612.94	−0.897800
$254$	5480.08	1.35374
$255$	−5666.14	−1.39148
$256$	−4362.48	−1.06506
$257$	1207.77	0.293146	0.146573	−	0.989200i	$-0.453176\pi$
$257$	1207.77	0.293146	0.146573	+	0.989200i	$0.453176\pi$
$258$	1358.34	0.327778
$259$	−286.132	−0.0686464
$260$	−981.477	−0.234110
$261$	−2994.49	−0.710169
$262$	−1651.00	−0.389310
$263$	2799.10	0.656273	0.328136	−	0.944630i	$-0.393579\pi$
$263$	2799.10	0.656273	0.328136	+	0.944630i	$0.393579\pi$
$264$	1799.15	0.419433
$265$	−10483.0	−2.43006
$266$	−388.890	−0.0896406
$267$	−946.655	−0.216983
$268$	2437.53	0.555582
$269$	3885.15	0.880601	0.440300	−	0.897851i	$-0.354872\pi$
$269$	3885.15	0.880601	0.440300	+	0.897851i	$0.354872\pi$
$270$	−4068.75	−0.917097
$271$	3959.23	0.887476	0.443738	−	0.896157i	$-0.353652\pi$
$271$	3959.23	0.887476	0.443738	+	0.896157i	$0.353652\pi$
$272$	1566.11	0.349114
$273$	426.374	0.0945251
$274$	−2471.94	−0.545020
$275$	−1869.10	−0.409859
$276$	−2262.37	−0.493400
$277$	3207.90	0.695826	0.347913	−	0.937527i	$-0.386890\pi$
$277$	3207.90	0.695826	0.347913	+	0.937527i	$0.386890\pi$
$278$	−3281.28	−0.707908
$279$	−1573.80	−0.337709
$280$	−2661.09	−0.567966
$281$	−7188.49	−1.52608	−0.763042	−	0.646349i	$-0.776295\pi$
$281$	−7188.49	−1.52608	−0.763042	+	0.646349i	$0.776295\pi$
$282$	401.098	0.0846987
$283$	−2652.16	−0.557082	−0.278541	−	0.960424i	$-0.589851\pi$
$283$	−2652.16	−0.557082	−0.278541	+	0.960424i	$0.589851\pi$
$284$	−3661.65	−0.765067
$285$	−1249.07	−0.259609
$286$	−717.553	−0.148356
$287$	2463.83	0.506743
$288$	−2567.40	−0.525297
$289$	8629.34	1.75643
$290$	5423.12	1.09813
$291$	−5825.51	−1.17353
$292$	1195.41	0.239576
$293$	4830.74	0.963191	0.481596	−	0.876394i	$-0.340057\pi$
$293$	4830.74	0.963191	0.481596	+	0.876394i	$0.340057\pi$
$294$	−1871.99	−0.371349
$295$	1121.53	0.221350
$296$	881.772	0.173148
$297$	3218.24	0.628758
$298$	2714.77	0.527726
$299$	2638.59	0.510347
$300$	−1170.41	−0.225245
$301$	1589.13	0.304304
$302$	1672.93	0.318762
$303$	−79.4373	−0.0150612
$304$	345.239	0.0651343
$305$	704.320	0.132227
$306$	3565.27	0.666055
$307$	−1963.81	−0.365083	−0.182542	−	0.983198i	$-0.558432\pi$
$307$	−1963.81	−0.365083	−0.182542	+	0.983198i	$0.558432\pi$
$308$	719.771	0.133158
$309$	6105.77	1.12409
$310$	2850.20	0.522196
$311$	5857.60	1.06802	0.534010	−	0.845478i	$-0.320684\pi$
$311$	5857.60	1.06802	0.534010	+	0.845478i	$0.320684\pi$
$312$	−1313.95	−0.238423
$313$	−4824.15	−0.871172	−0.435586	−	0.900147i	$-0.643459\pi$
$313$	−4824.15	−0.871172	−0.435586	+	0.900147i	$0.643459\pi$
$314$	2615.25	0.470023
$315$	−1745.16	−0.312154
$316$	1155.41	0.205687
$317$	3009.25	0.533175	0.266587	−	0.963811i	$-0.414104\pi$
$317$	3009.25	0.533175	0.266587	+	0.963811i	$0.414104\pi$
$318$	−4799.14	−0.846297
$319$	−4289.50	−0.752871
$320$	6204.20	1.08383
$321$	1874.07	0.325858
$322$	2446.41	0.423394
$323$	2985.33	0.514267
$324$	260.896	0.0447352
$325$	1365.04	0.232981
$326$	−4161.52	−0.707011
$327$	−5581.71	−0.943943
$328$	−7592.76	−1.27817
$329$	469.244	0.0786331
$330$	−2136.83	−0.356450
$331$	2383.99	0.395879	0.197939	−	0.980214i	$-0.436575\pi$
$331$	2383.99	0.395879	0.197939	+	0.980214i	$0.436575\pi$
$332$	−2797.85	−0.462506
$333$	578.270	0.0951622
$334$	−7652.77	−1.25372
$335$	−8465.93	−1.38073
$336$	−350.946	−0.0569812
$337$	3955.22	0.639330	0.319665	−	0.947531i	$-0.396430\pi$
$337$	3955.22	0.639330	0.319665	+	0.947531i	$0.396430\pi$
$338$	−3782.67	−0.608729
$339$	6721.77	1.07692
$340$	6985.58	1.11425
$341$	−2254.41	−0.358015
$342$	785.943	0.124266
$343$	−4842.57	−0.762315
$344$	−4897.19	−0.767555
$345$	7857.55	1.22619
$346$	1454.57	0.226006
$347$	2179.16	0.337128	0.168564	−	0.985691i	$-0.446087\pi$
$347$	2179.16	0.337128	0.168564	+	0.985691i	$0.446087\pi$
$348$	−2686.02	−0.413752
$349$	9470.52	1.45257	0.726283	−	0.687396i	$-0.241246\pi$
$349$	9470.52	1.45257	0.726283	+	0.687396i	$0.241246\pi$
$350$	1265.62	0.193286
$351$	−2350.34	−0.357412
$352$	−3677.71	−0.556883
$353$	−1722.01	−0.259641	−0.129821	−	0.991537i	$-0.541440\pi$
$353$	−1722.01	−0.259641	−0.129821	+	0.991537i	$0.541440\pi$
$354$	513.440	0.0770876
$355$	12717.5	1.90134
$356$	1167.10	0.173753
$357$	−3034.68	−0.449894
$358$	6200.41	0.915368
$359$	12223.9	1.79708	0.898539	−	0.438895i	$-0.144630\pi$
$359$	12223.9	1.79708	0.898539	+	0.438895i	$0.144630\pi$
$360$	5378.03	0.787354
$361$	−6200.90	−0.904053
$362$	−4154.57	−0.603203
$363$	−2798.11	−0.404580
$364$	−525.661	−0.0756927
$365$	−4151.85	−0.595391
$366$	322.438	0.0460495
$367$	−4398.95	−0.625676	−0.312838	−	0.949806i	$-0.601280\pi$
$367$	−4398.95	−0.625676	−0.312838	+	0.949806i	$0.601280\pi$
$368$	−2171.81	−0.307645
$369$	−4979.37	−0.702482
$370$	−1047.27	−0.147148
$371$	−5614.51	−0.785690
$372$	−1411.68	−0.196753
$373$	−35.6048	−0.00494248	−0.00247124	−	0.999997i	$-0.500787\pi$
$373$	−35.6048	−0.00494248	−0.00247124	+	0.999997i	$0.500787\pi$
$374$	5107.12	0.706104
$375$	−2021.26	−0.278339
$376$	−1446.07	−0.198338
$377$	3132.69	0.427963
$378$	−2179.15	−0.296516
$379$	725.870	0.0983785	0.0491893	−	0.998789i	$-0.484336\pi$
$379$	725.870	0.0983785	0.0491893	+	0.998789i	$0.484336\pi$
$380$	1539.93	0.207886
$381$	9426.95	1.26760
$382$	−6345.11	−0.849853
$383$	−4713.91	−0.628902	−0.314451	−	0.949274i	$-0.601820\pi$
$383$	−4713.91	−0.628902	−0.314451	+	0.949274i	$0.601820\pi$
$384$	−1591.26	−0.211467
$385$	−2499.88	−0.330923
$386$	−845.163	−0.111445
$387$	−3211.60	−0.421848
$388$	7182.06	0.939726
$389$	−2116.34	−0.275842	−0.137921	−	0.990443i	$-0.544042\pi$
$389$	−2116.34	−0.275842	−0.137921	+	0.990443i	$0.544042\pi$
$390$	1560.56	0.202621
$391$	−18779.9	−2.42901
$392$	6749.03	0.869586
$393$	−2840.09	−0.364538
$394$	2626.84	0.335884
$395$	−4012.94	−0.511171
$396$	−1454.65	−0.184593
$397$	−820.559	−0.103735	−0.0518673	−	0.998654i	$-0.516517\pi$
$397$	−820.559	−0.103735	−0.0518673	+	0.998654i	$0.516517\pi$
$398$	−8902.18	−1.12117
$399$	−668.978	−0.0839368
$400$	−1123.56	−0.140444
$401$	−5116.11	−0.637124	−0.318562	−	0.947902i	$-0.603200\pi$
$401$	−5116.11	−0.637124	−0.318562	+	0.947902i	$0.603200\pi$
$402$	−3875.71	−0.480853
$403$	1646.44	0.203511
$404$	97.9353	0.0120606
$405$	−906.131	−0.111175
$406$	2904.52	0.355047
$407$	828.352	0.100884
$408$	9351.95	1.13478
$409$	3075.44	0.371811	0.185906	−	0.982568i	$-0.440478\pi$
$409$	3075.44	0.371811	0.185906	+	0.982568i	$0.440478\pi$
$410$	9017.81	1.08624
$411$	−4252.29	−0.510341
$412$	−7527.58	−0.900139
$413$	600.673	0.0715671
$414$	−4944.16	−0.586937
$415$	9717.38	1.14942
$416$	2685.90	0.316555
$417$	−5644.54	−0.662864
$418$	1125.84	0.131738
$419$	1591.60	0.185573	0.0927863	−	0.995686i	$-0.470423\pi$
$419$	1591.60	0.185573	0.0927863	+	0.995686i	$0.470423\pi$
$420$	−1565.39	−0.181864
$421$	−613.147	−0.0709809	−0.0354904	−	0.999370i	$-0.511299\pi$
$421$	−613.147	−0.0709809	−0.0354904	+	0.999370i	$0.511299\pi$
$422$	−6256.50	−0.721711
$423$	−948.338	−0.109006
$424$	17302.2	1.98177
$425$	−9715.54	−1.10888
$426$	5822.08	0.662161
$427$	377.221	0.0427517
$428$	−2310.47	−0.260937
$429$	−1234.35	−0.138916
$430$	5816.32	0.652298
$431$	−4846.06	−0.541593	−0.270797	−	0.962637i	$-0.587287\pi$
$431$	−4846.06	−0.541593	−0.270797	+	0.962637i	$0.587287\pi$
$432$	1934.55	0.215454
$433$	5819.69	0.645904	0.322952	−	0.946415i	$-0.395325\pi$
$433$	5819.69	0.645904	0.322952	+	0.946415i	$0.395325\pi$
$434$	1526.52	0.168837
$435$	9328.97	1.02825
$436$	6881.49	0.755879
$437$	−4139.93	−0.453180
$438$	−1900.72	−0.207352
$439$	5851.11	0.636124	0.318062	−	0.948070i	$-0.396968\pi$
$439$	5851.11	0.636124	0.318062	+	0.948070i	$0.396968\pi$
$440$	7703.84	0.834696
$441$	4426.05	0.477923
$442$	−3729.82	−0.401379
$443$	11983.4	1.28521	0.642605	−	0.766198i	$-0.277854\pi$
$443$	11983.4	1.28521	0.642605	+	0.766198i	$0.277854\pi$
$444$	518.702	0.0554426
$445$	−4053.51	−0.431808
$446$	−5002.23	−0.531082
$447$	4670.01	0.494147
$448$	3322.86	0.350424
$449$	−7338.30	−0.771305	−0.385653	−	0.922644i	$-0.626024\pi$
$449$	−7338.30	−0.771305	−0.385653	+	0.922644i	$0.626024\pi$
$450$	−2557.79	−0.267946
$451$	−7132.77	−0.744721
$452$	−8287.02	−0.862365
$453$	2877.81	0.298480
$454$	8821.60	0.911935
$455$	1825.70	0.188110
$456$	2061.58	0.211716
$457$	−117.438	−0.0120208	−0.00601040	−	0.999982i	$-0.501913\pi$
$457$	−117.438	−0.0120208	−0.00601040	+	0.999982i	$0.501913\pi$
$458$	2489.49	0.253987
$459$	16728.3	1.70111
$460$	−9687.29	−0.981896
$461$	500.319	0.0505470	0.0252735	−	0.999681i	$-0.491954\pi$
$461$	500.319	0.0505470	0.0252735	+	0.999681i	$0.491954\pi$
$462$	−1144.45	−0.115248
$463$	−527.163	−0.0529143	−0.0264572	−	0.999650i	$-0.508423\pi$
$463$	−527.163	−0.0529143	−0.0264572	+	0.999650i	$0.508423\pi$
$464$	−2578.50	−0.257983
$465$	4902.99	0.488969
$466$	3960.99	0.393754
$467$	−6035.95	−0.598095	−0.299048	−	0.954238i	$-0.596669\pi$
$467$	−6035.95	−0.598095	−0.299048	+	0.954238i	$0.596669\pi$
$468$	1062.36	0.104930
$469$	−4534.20	−0.446417
$470$	1717.47	0.168555
$471$	4498.82	0.440116
$472$	−1851.09	−0.180516
$473$	−4600.51	−0.447213
$474$	−1837.13	−0.178021
$475$	−2141.74	−0.206883
$476$	3741.35	0.360261
$477$	11346.9	1.08918
$478$	−12735.3	−1.21862
$479$	17756.3	1.69375	0.846876	−	0.531790i	$-0.178480\pi$
$479$	17756.3	1.69375	0.846876	+	0.531790i	$0.178480\pi$
$480$	7998.44	0.760577
$481$	−604.960	−0.0573468
$482$	8772.77	0.829022
$483$	4208.36	0.396454
$484$	3449.69	0.323975
$485$	−24944.4	−2.33540
$486$	7193.42	0.671400
$487$	2591.61	0.241143	0.120572	−	0.992705i	$-0.461527\pi$
$487$	2591.61	0.241143	0.120572	+	0.992705i	$0.461527\pi$
$488$	−1162.48	−0.107834
$489$	−7158.75	−0.662024
$490$	−8015.72	−0.739007
$491$	−14630.4	−1.34473	−0.672365	−	0.740220i	$-0.734721\pi$
$491$	−14630.4	−1.34473	−0.672365	+	0.740220i	$0.734721\pi$
$492$	−4466.44	−0.409274
$493$	−22296.7	−2.03690
$494$	−822.218	−0.0748853
$495$	5052.22	0.458748
$496$	−1355.17	−0.122680
$497$	6811.25	0.614741
$498$	4448.63	0.400297
$499$	−18450.8	−1.65525	−0.827627	−	0.561278i	$-0.810310\pi$
$499$	−18450.8	−1.65525	−0.827627	+	0.561278i	$0.810310\pi$
$500$	2491.93	0.222885
$501$	−13164.5	−1.17394
$502$	10536.8	0.936813
$503$	−9272.29	−0.821930	−0.410965	−	0.911651i	$-0.634808\pi$
$503$	−9272.29	−0.821930	−0.410965	+	0.911651i	$0.634808\pi$
$504$	2880.38	0.254568
$505$	−340.145	−0.0299727
$506$	−7082.33	−0.622229
$507$	−6507.04	−0.569996
$508$	−11622.1	−1.01506
$509$	5350.96	0.465967	0.232983	−	0.972481i	$-0.425151\pi$
$509$	5350.96	0.465967	0.232983	+	0.972481i	$0.425151\pi$
$510$	−11107.2	−0.964380
$511$	−2223.65	−0.192502
$512$	−4776.52	−0.412294
$513$	3687.66	0.317377
$514$	2367.55	0.203168
$515$	26144.4	2.23701
$516$	−2880.77	−0.245773
$517$	−1358.46	−0.115561
$518$	−560.897	−0.0475761
$519$	2502.18	0.211626
$520$	−5626.25	−0.474476
$521$	−17782.4	−1.49532	−0.747660	−	0.664082i	$-0.768823\pi$
$521$	−17782.4	−1.49532	−0.747660	+	0.664082i	$0.768823\pi$
$522$	−5870.00	−0.492190
$523$	11943.5	0.998574	0.499287	−	0.866437i	$-0.333595\pi$
$523$	11943.5	0.998574	0.499287	+	0.866437i	$0.333595\pi$
$524$	3501.44	0.291911
$525$	2177.14	0.180987
$526$	5486.99	0.454837
$527$	−11718.4	−0.968615
$528$	1015.99	0.0837408
$529$	13876.2	1.14048
$530$	−20549.6	−1.68418
$531$	−1213.95	−0.0992111
$532$	824.758	0.0672139
$533$	5209.19	0.423330
$534$	−1855.70	−0.150382
$535$	8024.63	0.648477
$536$	13973.0	1.12601
$537$	10666.1	0.857124
$538$	7615.94	0.610309
$539$	6340.16	0.506661
$540$	8629.00	0.687654
$541$	21997.1	1.74811	0.874056	−	0.485825i	$-0.161481\pi$
$541$	21997.1	1.74811	0.874056	+	0.485825i	$0.161481\pi$
$542$	7761.16	0.615074
$543$	−7146.79	−0.564821
$544$	−19116.6	−1.50665
$545$	−23900.5	−1.87850
$546$	835.809	0.0655116
$547$	4140.03	0.323611	0.161805	−	0.986823i	$-0.448268\pi$
$547$	4140.03	0.323611	0.161805	+	0.986823i	$0.448268\pi$
$548$	5242.49	0.408664
$549$	−762.359	−0.0592654
$550$	−3663.95	−0.284057
$551$	−4915.17	−0.380024
$552$	−12968.9	−0.999985
$553$	−2149.25	−0.165272
$554$	6288.35	0.482249
$555$	−1801.53	−0.137785
$556$	6958.94	0.530800
$557$	1535.50	0.116806	0.0584031	−	0.998293i	$-0.481399\pi$
$557$	1535.50	0.116806	0.0584031	+	0.998293i	$0.481399\pi$
$558$	−3085.07	−0.234053
$559$	3359.83	0.254214
$560$	−1502.72	−0.113396
$561$	8785.38	0.661175
$562$	−14091.4	−1.05767
$563$	2193.88	0.164229	0.0821146	−	0.996623i	$-0.473833\pi$
$563$	2193.88	0.164229	0.0821146	+	0.996623i	$0.473833\pi$
$564$	−850.648	−0.0635084
$565$	28782.1	2.14314
$566$	−5198.94	−0.386092
$567$	−485.307	−0.0359453
$568$	−20990.2	−1.55058
$569$	110.891	0.00817014	0.00408507	−	0.999992i	$-0.498700\pi$
$569$	110.891	0.00817014	0.00408507	+	0.999992i	$0.498700\pi$
$570$	−2448.51	−0.179924
$571$	11986.8	0.878513	0.439256	−	0.898362i	$-0.355242\pi$
$571$	11986.8	0.878513	0.439256	+	0.898362i	$0.355242\pi$
$572$	1521.79	0.111240
$573$	−10915.0	−0.795778
$574$	4829.77	0.351203
$575$	13473.1	0.977160
$576$	−6715.45	−0.485782
$577$	15131.6	1.09175	0.545873	−	0.837868i	$-0.316198\pi$
$577$	15131.6	1.09175	0.545873	+	0.837868i	$0.316198\pi$
$578$	16915.8	1.21731
$579$	−1453.87	−0.104354
$580$	−11501.3	−0.823392
$581$	5204.45	0.371630
$582$	−11419.6	−0.813328
$583$	16254.0	1.15467
$584$	6852.61	0.485553
$585$	−3689.73	−0.260772
$586$	9469.56	0.667549
$587$	23727.8	1.66840	0.834201	−	0.551461i	$-0.185929\pi$
$587$	23727.8	1.66840	0.834201	+	0.551461i	$0.185929\pi$
$588$	3970.12	0.278444
$589$	−2583.25	−0.180715
$590$	2198.51	0.153409
$591$	4518.76	0.314512
$592$	497.939	0.0345695
$593$	−12468.5	−0.863437	−0.431719	−	0.902008i	$-0.642093\pi$
$593$	−12468.5	−0.863437	−0.431719	+	0.902008i	$0.642093\pi$
$594$	6308.62	0.435767
$595$	−12994.3	−0.895316
$596$	−5757.48	−0.395697
$597$	−15313.7	−1.04983
$598$	5172.35	0.353701
$599$	−22733.4	−1.55069	−0.775345	−	0.631538i	$-0.782424\pi$
$599$	−22733.4	−1.55069	−0.775345	+	0.631538i	$0.782424\pi$
$600$	−6709.27	−0.456508
$601$	26975.1	1.83084	0.915421	−	0.402499i	$-0.131858\pi$
$601$	26975.1	1.83084	0.915421	+	0.402499i	$0.131858\pi$
$602$	3115.11	0.210901
$603$	9163.56	0.618854
$604$	−3547.94	−0.239013
$605$	−11981.3	−0.805139
$606$	−155.719	−0.0104383
$607$	8067.77	0.539474	0.269737	−	0.962934i	$-0.413063\pi$
$607$	8067.77	0.539474	0.269737	+	0.962934i	$0.413063\pi$
$608$	−4214.16	−0.281096
$609$	4996.42	0.332455
$610$	1380.66	0.0916413
$611$	992.107	0.0656896
$612$	−7561.22	−0.499418
$613$	3346.20	0.220476	0.110238	−	0.993905i	$-0.464839\pi$
$613$	3346.20	0.220476	0.110238	+	0.993905i	$0.464839\pi$
$614$	−3849.60	−0.253025
$615$	15512.6	1.01712
$616$	4126.04	0.269875
$617$	18062.7	1.17857	0.589284	−	0.807926i	$-0.299410\pi$
$617$	18062.7	1.17857	0.589284	+	0.807926i	$0.299410\pi$
$618$	11969.0	0.779065
$619$	−25410.9	−1.65000	−0.825001	−	0.565131i	$-0.808826\pi$
$619$	−25410.9	−1.65000	−0.825001	+	0.565131i	$0.808826\pi$
$620$	−6044.71	−0.391551
$621$	−23198.1	−1.49905
$622$	11482.5	0.740202
$623$	−2170.98	−0.139613
$624$	−741.993	−0.0476018
$625$	−19090.8	−1.22181
$626$	−9456.63	−0.603775
$627$	1936.69	0.123355
$628$	−5546.43	−0.352431
$629$	4305.75	0.272943
$630$	−3420.98	−0.216341
$631$	−16027.5	−1.01117	−0.505583	−	0.862778i	$-0.668723\pi$
$631$	−16027.5	−1.01117	−0.505583	+	0.862778i	$0.668723\pi$
$632$	6623.34	0.416871
$633$	−10762.6	−0.675789
$634$	5898.95	0.369522
$635$	40365.5	2.52261
$636$	10178.0	0.634567
$637$	−4630.33	−0.288007
$638$	−8408.57	−0.521785
$639$	−13765.5	−0.852196
$640$	−6813.64	−0.420833
$641$	−4776.64	−0.294330	−0.147165	−	0.989112i	$-0.547015\pi$
$641$	−4776.64	−0.294330	−0.147165	+	0.989112i	$0.547015\pi$
$642$	3673.69	0.225839
$643$	−22418.3	−1.37495	−0.687474	−	0.726209i	$-0.741280\pi$
$643$	−22418.3	−1.37495	−0.687474	+	0.726209i	$0.741280\pi$
$644$	−5188.33	−0.317467
$645$	10005.4	0.610792
$646$	5852.06	0.356418
$647$	−3006.49	−0.182685	−0.0913425	−	0.995820i	$-0.529116\pi$
$647$	−3006.49	−0.182685	−0.0913425	+	0.995820i	$0.529116\pi$
$648$	1495.57	0.0906657
$649$	−1738.95	−0.105177
$650$	2675.85	0.161470
$651$	2625.95	0.158094
$652$	8825.76	0.530128
$653$	−28105.8	−1.68433	−0.842163	−	0.539223i	$-0.818718\pi$
$653$	−28105.8	−1.68433	−0.842163	+	0.539223i	$0.818718\pi$
$654$	−10941.7	−0.654209
$655$	−12161.1	−0.725453
$656$	−4287.65	−0.255190
$657$	4493.98	0.266860
$658$	919.846	0.0544975
$659$	−17059.3	−1.00840	−0.504199	−	0.863587i	$-0.668212\pi$
$659$	−17059.3	−1.00840	−0.504199	+	0.863587i	$0.668212\pi$
$660$	4531.78	0.267272
$661$	11415.0	0.671699	0.335849	−	0.941916i	$-0.390977\pi$
$661$	11415.0	0.671699	0.335849	+	0.941916i	$0.390977\pi$
$662$	4673.26	0.274368
$663$	−6416.12	−0.375839
$664$	−16038.5	−0.937371
$665$	−2864.51	−0.167039
$666$	1133.57	0.0659532
$667$	30920.0	1.79495
$668$	16230.0	0.940055
$669$	−8604.95	−0.497289
$670$	−16595.5	−0.956927
$671$	−1092.05	−0.0628289
$672$	4283.82	0.245910
$673$	−19082.8	−1.09300	−0.546499	−	0.837459i	$-0.684040\pi$
$673$	−19082.8	−1.09300	−0.546499	+	0.837459i	$0.684040\pi$
$674$	7753.29	0.443094
$675$	−12001.2	−0.684336
$676$	8022.29	0.456434
$677$	30995.2	1.75959	0.879794	−	0.475355i	$-0.157681\pi$
$677$	30995.2	1.75959	0.879794	+	0.475355i	$0.157681\pi$
$678$	13176.5	0.746372
$679$	−13359.8	−0.755082
$680$	40044.3	2.25828
$681$	15175.1	0.853909
$682$	−4419.26	−0.248126
$683$	18270.3	1.02356	0.511781	−	0.859116i	$-0.328986\pi$
$683$	18270.3	1.02356	0.511781	+	0.859116i	$0.328986\pi$
$684$	−1666.83	−0.0931765
$685$	−18208.0	−1.01561
$686$	−9492.74	−0.528330
$687$	4282.48	0.237826
$688$	−2765.46	−0.153244
$689$	−11870.6	−0.656361
$690$	15402.9	0.849826
$691$	−7972.30	−0.438901	−0.219450	−	0.975624i	$-0.570426\pi$
$691$	−7972.30	−0.438901	−0.219450	+	0.975624i	$0.570426\pi$
$692$	−3084.85	−0.169463
$693$	2705.88	0.148323
$694$	4271.74	0.233650
$695$	−24169.5	−1.31914
$696$	−15397.4	−0.838561
$697$	−37075.9	−2.01485
$698$	18564.8	1.00672
$699$	6813.79	0.368700
$700$	−2684.12	−0.144929
$701$	−15824.1	−0.852596	−0.426298	−	0.904583i	$-0.640182\pi$
$701$	−15824.1	−0.852596	−0.426298	+	0.904583i	$0.640182\pi$
$702$	−4607.30	−0.247708
$703$	949.178	0.0509231
$704$	−9619.65	−0.514992
$705$	2954.43	0.157830
$706$	−3375.61	−0.179947
$707$	−182.175	−0.00969081
$708$	−1088.90	−0.0578015
$709$	9545.61	0.505632	0.252816	−	0.967514i	$-0.418643\pi$
$709$	9545.61	0.505632	0.252816	+	0.967514i	$0.418643\pi$
$710$	24929.7	1.31774
$711$	4343.62	0.229112
$712$	6690.30	0.352148
$713$	16250.5	0.853558
$714$	−5948.79	−0.311804
$715$	−5285.40	−0.276451
$716$	−13149.8	−0.686357
$717$	−21907.6	−1.14108
$718$	23962.1	1.24548
$719$	−1483.32	−0.0769379	−0.0384689	−	0.999260i	$-0.512248\pi$
$719$	−1483.32	−0.0769379	−0.0384689	+	0.999260i	$0.512248\pi$
$720$	3036.99	0.157197
$721$	14002.5	0.723273
$722$	−12155.4	−0.626563
$723$	15091.1	0.776272
$724$	8811.01	0.452291
$725$	15996.1	0.819420
$726$	−5485.05	−0.280399
$727$	14988.3	0.764629	0.382315	−	0.924032i	$-0.375127\pi$
$727$	14988.3	0.764629	0.382315	+	0.924032i	$0.375127\pi$
$728$	−3013.32	−0.153408
$729$	14068.7	0.714763
$730$	−8138.75	−0.412642
$731$	−23913.3	−1.20994
$732$	−683.827	−0.0345287
$733$	−9390.15	−0.473169	−0.236585	−	0.971611i	$-0.576028\pi$
$733$	−9390.15	−0.473169	−0.236585	+	0.971611i	$0.576028\pi$
$734$	−8623.13	−0.433631
$735$	−13788.8	−0.691985
$736$	26510.1	1.32768
$737$	13126.5	0.656065
$738$	−9760.91	−0.486862
$739$	2397.97	0.119365	0.0596824	−	0.998217i	$-0.480991\pi$
$739$	2397.97	0.119365	0.0596824	+	0.998217i	$0.480991\pi$
$740$	2221.04	0.110334
$741$	−1414.40	−0.0701204
$742$	−11006.0	−0.544531
$743$	30217.0	1.49200	0.745999	−	0.665947i	$-0.231972\pi$
$743$	30217.0	1.49200	0.745999	+	0.665947i	$0.231972\pi$
$744$	−8092.36	−0.398764
$745$	19996.6	0.983382
$746$	−69.7950	−0.00342544
$747$	−10518.1	−0.515178
$748$	−10831.2	−0.529448
$749$	4297.85	0.209666
$750$	−3962.21	−0.192906
$751$	−4654.03	−0.226136	−0.113068	−	0.993587i	$-0.536068\pi$
$751$	−4654.03	−0.226136	−0.113068	+	0.993587i	$0.536068\pi$
$752$	−816.597	−0.0395987
$753$	18125.6	0.877205
$754$	6140.93	0.296604
$755$	12322.6	0.593992
$756$	4621.53	0.222333
$757$	6369.47	0.305815	0.152908	−	0.988240i	$-0.451136\pi$
$757$	6369.47	0.305815	0.152908	+	0.988240i	$0.451136\pi$
$758$	1422.90	0.0681822
$759$	−12183.2	−0.582637
$760$	8827.55	0.421327
$761$	−20110.5	−0.957959	−0.478979	−	0.877826i	$-0.658993\pi$
$761$	−20110.5	−0.957959	−0.478979	+	0.877826i	$0.658993\pi$
$762$	18479.4	0.878527
$763$	−12800.7	−0.607359
$764$	13456.7	0.637233
$765$	26261.3	1.24115
$766$	−9240.54	−0.435867
$767$	1269.98	0.0597867
$768$	−14710.7	−0.691181
$769$	16199.5	0.759647	0.379824	−	0.925059i	$-0.375985\pi$
$769$	16199.5	0.759647	0.379824	+	0.925059i	$0.375985\pi$
$770$	−4900.43	−0.229350
$771$	4072.72	0.190241
$772$	1792.42	0.0835630
$773$	6794.33	0.316139	0.158069	−	0.987428i	$-0.449473\pi$
$773$	6794.33	0.316139	0.158069	+	0.987428i	$0.449473\pi$
$774$	−6295.61	−0.292366
$775$	8406.99	0.389662
$776$	41170.7	1.90456
$777$	−964.868	−0.0445488
$778$	−4148.60	−0.191175
$779$	−8173.18	−0.375911
$780$	−3309.64	−0.151928
$781$	−19718.5	−0.903438
$782$	−36813.7	−1.68345
$783$	−27542.2	−1.25706
$784$	3811.20	0.173615
$785$	19263.6	0.875857
$786$	−5567.34	−0.252647
$787$	−10068.1	−0.456021	−0.228010	−	0.973659i	$-0.573222\pi$
$787$	−10068.1	−0.456021	−0.228010	+	0.973659i	$0.573222\pi$
$788$	−5571.01	−0.251851
$789$	9438.84	0.425896
$790$	−7866.44	−0.354273
$791$	15415.2	0.692921
$792$	−8338.67	−0.374119
$793$	797.545	0.0357146
$794$	−1608.52	−0.0718944
$795$	−35349.8	−1.57702
$796$	18879.7	0.840672
$797$	33737.9	1.49944	0.749722	−	0.661753i	$-0.230187\pi$
$797$	33737.9	1.49944	0.749722	+	0.661753i	$0.230187\pi$
$798$	−1311.38	−0.0581733
$799$	−7061.23	−0.312651
$800$	13714.7	0.606108
$801$	4387.53	0.193540
$802$	−10029.0	−0.441565
$803$	6437.47	0.282906
$804$	8219.61	0.360551
$805$	18019.9	0.788966
$806$	3227.46	0.141045
$807$	13101.1	0.571476
$808$	561.408	0.0244434
$809$	−28133.1	−1.22263	−0.611314	−	0.791388i	$-0.709359\pi$
$809$	−28133.1	−1.22263	−0.611314	+	0.791388i	$0.709359\pi$
$810$	−1776.26	−0.0770512
$811$	−2475.18	−0.107170	−0.0535852	−	0.998563i	$-0.517065\pi$
$811$	−2475.18	−0.107170	−0.0535852	+	0.998563i	$0.517065\pi$
$812$	−6159.91	−0.266220
$813$	13350.9	0.575937
$814$	1623.79	0.0699189
$815$	−30653.2	−1.31747
$816$	5281.07	0.226562
$817$	−5271.55	−0.225738
$818$	6028.70	0.257688
$819$	−1976.15	−0.0843128
$820$	−19125.0	−0.814479
$821$	−20732.4	−0.881324	−0.440662	−	0.897673i	$-0.645256\pi$
$821$	−20732.4	−0.881324	−0.440662	+	0.897673i	$0.645256\pi$
$822$	−8335.64	−0.353697
$823$	5569.58	0.235897	0.117949	−	0.993020i	$-0.462368\pi$
$823$	5569.58	0.235897	0.117949	+	0.993020i	$0.462368\pi$
$824$	−43151.3	−1.82433
$825$	−6302.81	−0.265983
$826$	1177.48	0.0496003
$827$	701.437	0.0294938	0.0147469	−	0.999891i	$-0.495306\pi$
$827$	701.437	0.0294938	0.0147469	+	0.999891i	$0.495306\pi$
$828$	10485.6	0.440095
$829$	−28285.7	−1.18504	−0.592522	−	0.805554i	$-0.701868\pi$
$829$	−28285.7	−1.18504	−0.592522	+	0.805554i	$0.701868\pi$
$830$	19048.7	0.796614
$831$	10817.4	0.451564
$832$	7025.40	0.292743
$833$	32956.0	1.37078
$834$	−11064.8	−0.459405
$835$	−56369.3	−2.33621
$836$	−2387.67	−0.0987791
$837$	−14475.2	−0.597774
$838$	3119.97	0.128613
$839$	−12510.5	−0.514792	−0.257396	−	0.966306i	$-0.582865\pi$
$839$	−12510.5	−0.514792	−0.257396	+	0.966306i	$0.582865\pi$
$840$	−8973.47	−0.368588
$841$	12321.2	0.505193
$842$	−1201.93	−0.0491940
$843$	−24240.3	−0.990369
$844$	13268.8	0.541150
$845$	−27862.7	−1.13432
$846$	−1859.00	−0.0755481
$847$	−6416.96	−0.260318
$848$	9770.60	0.395665
$849$	−8943.34	−0.361525
$850$	−19045.1	−0.768519
$851$	−5971.02	−0.240522
$852$	−12347.5	−0.496499
$853$	6059.93	0.243245	0.121622	−	0.992576i	$-0.461190\pi$
$853$	6059.93	0.243245	0.121622	+	0.992576i	$0.461190\pi$
$854$	739.455	0.0296295
$855$	5789.15	0.231561
$856$	−13244.6	−0.528846
$857$	−18567.4	−0.740080	−0.370040	−	0.929016i	$-0.620656\pi$
$857$	−18567.4	−0.740080	−0.370040	+	0.929016i	$0.620656\pi$
$858$	−2419.66	−0.0962773
$859$	−432.463	−0.0171775	−0.00858874	−	0.999963i	$-0.502734\pi$
$859$	−432.463	−0.0171775	−0.00858874	+	0.999963i	$0.502734\pi$
$860$	−12335.3	−0.489103
$861$	8308.28	0.328857
$862$	−9499.59	−0.375357
$863$	−489.635	−0.0193133	−0.00965664	−	0.999953i	$-0.503074\pi$
$863$	−489.635	−0.0193133	−0.00965664	+	0.999953i	$0.503074\pi$
$864$	−23614.0	−0.929821
$865$	10714.2	0.421148
$866$	11408.2	0.447650
$867$	29099.0	1.13986
$868$	−3237.44	−0.126596
$869$	6222.08	0.242888
$870$	18287.3	0.712641
$871$	−9586.50	−0.372935
$872$	39447.7	1.53196
$873$	26999.9	1.04675
$874$	−8115.38	−0.314081
$875$	−4635.39	−0.179091
$876$	4031.05	0.155475
$877$	24890.1	0.958357	0.479179	−	0.877717i	$-0.340935\pi$
$877$	24890.1	0.958357	0.479179	+	0.877717i	$0.340935\pi$
$878$	11469.8	0.440872
$879$	16289.8	0.625074
$880$	4350.38	0.166649
$881$	14021.1	0.536188	0.268094	−	0.963393i	$-0.413606\pi$
$881$	14021.1	0.536188	0.268094	+	0.963393i	$0.413606\pi$
$882$	8676.25	0.331230
$883$	22122.2	0.843117	0.421558	−	0.906801i	$-0.361483\pi$
$883$	22122.2	0.843117	0.421558	+	0.906801i	$0.361483\pi$
$884$	7910.20	0.300960
$885$	3781.93	0.143648
$886$	23490.7	0.890728
$887$	17688.9	0.669600	0.334800	−	0.942289i	$-0.391331\pi$
$887$	17688.9	0.669600	0.334800	+	0.942289i	$0.391331\pi$
$888$	2973.42	0.112367
$889$	21619.0	0.815612
$890$	−7945.97	−0.299269
$891$	1404.96	0.0528260
$892$	10608.7	0.398213
$893$	−1556.61	−0.0583314
$894$	9154.48	0.342474
$895$	45671.4	1.70573
$896$	−3649.26	−0.136064
$897$	8897.60	0.331195
$898$	−14385.1	−0.534561
$899$	19293.6	0.715770
$900$	5424.57	0.200910
$901$	84487.8	3.12397
$902$	−13982.2	−0.516137
$903$	5358.69	0.197482
$904$	−47504.8	−1.74777
$905$	−30602.0	−1.12403
$906$	5641.28	0.206864
$907$	−48272.9	−1.76723	−0.883613	−	0.468218i	$-0.844896\pi$
$907$	−48272.9	−1.76723	−0.883613	+	0.468218i	$0.844896\pi$
$908$	−18708.9	−0.683783
$909$	368.174	0.0134341
$910$	3578.87	0.130372
$911$	−8091.91	−0.294289	−0.147144	−	0.989115i	$-0.547008\pi$
$911$	−8091.91	−0.294289	−0.147144	+	0.989115i	$0.547008\pi$
$912$	1164.18	0.0422696
$913$	−15066.9	−0.546156
$914$	−230.210	−0.00833114
$915$	2375.04	0.0858102
$916$	−5279.71	−0.190444
$917$	−6513.24	−0.234554
$918$	32792.0	1.17897
$919$	−18742.6	−0.672753	−0.336377	−	0.941728i	$-0.609201\pi$
$919$	−18742.6	−0.672753	−0.336377	+	0.941728i	$0.609201\pi$
$920$	−55531.7	−1.99003
$921$	−6622.17	−0.236925
$922$	980.760	0.0350321
$923$	14400.8	0.513552
$924$	2427.14	0.0864145
$925$	−3089.03	−0.109802
$926$	−1033.38	−0.0366728
$927$	−28298.9	−1.00265
$928$	31474.4	1.11336
$929$	42784.3	1.51099	0.755494	−	0.655155i	$-0.227397\pi$
$929$	42784.3	1.51099	0.755494	+	0.655155i	$0.227397\pi$
$930$	9611.18	0.338885
$931$	7264.95	0.255746
$932$	−8400.47	−0.295243
$933$	19752.4	0.693104
$934$	−11832.1	−0.414516
$935$	37618.4	1.31578
$936$	6089.88	0.212665
$937$	−34117.2	−1.18950	−0.594749	−	0.803912i	$-0.702749\pi$
$937$	−34117.2	−1.18950	−0.594749	+	0.803912i	$0.702749\pi$
$938$	−8888.25	−0.309394
$939$	−16267.5	−0.565357
$940$	−3642.41	−0.126385
$941$	−29051.0	−1.00642	−0.503208	−	0.864166i	$-0.667847\pi$
$941$	−29051.0	−1.00642	−0.503208	+	0.864166i	$0.667847\pi$
$942$	8818.90	0.305027
$943$	51415.3	1.77552
$944$	−1045.32	−0.0360404
$945$	−16051.3	−0.552538
$946$	−9018.24	−0.309945
$947$	−45329.4	−1.55545	−0.777723	−	0.628607i	$-0.783625\pi$
$947$	−45329.4	−1.55545	−0.777723	+	0.628607i	$0.783625\pi$
$948$	3896.18	0.133483
$949$	−4701.39	−0.160815
$950$	−4198.38	−0.143383
$951$	10147.5	0.346010
$952$	21447.0	0.730149
$953$	−6964.01	−0.236712	−0.118356	−	0.992971i	$-0.537762\pi$
$953$	−6964.01	−0.236712	−0.118356	+	0.992971i	$0.537762\pi$
$954$	22242.9	0.754865
$955$	−46737.2	−1.58364
$956$	27009.0	0.913738
$957$	−14464.6	−0.488584
$958$	34807.2	1.17387
$959$	−9751.86	−0.328367
$960$	20921.2	0.703363
$961$	−19650.9	−0.659627
$962$	−1185.89	−0.0397448
$963$	−8685.90	−0.290653
$964$	−18605.3	−0.621614
$965$	−6225.36	−0.207670
$966$	8249.53	0.274766
$967$	56280.2	1.87161	0.935807	−	0.352513i	$-0.114673\pi$
$967$	56280.2	1.87161	0.935807	+	0.352513i	$0.114673\pi$
$968$	19775.1	0.656607
$969$	10066.8	0.333740
$970$	−48897.8	−1.61857
$971$	−2932.28	−0.0969118	−0.0484559	−	0.998825i	$-0.515430\pi$
$971$	−2932.28	−0.0969118	−0.0484559	+	0.998825i	$0.515430\pi$
$972$	−15255.8	−0.503426
$973$	−12944.7	−0.426505
$974$	5080.25	0.167127
$975$	4603.05	0.151196
$976$	−656.454	−0.0215293
$977$	−44214.4	−1.44784	−0.723922	−	0.689882i	$-0.757663\pi$
$977$	−44214.4	−1.44784	−0.723922	+	0.689882i	$0.757663\pi$
$978$	−14033.1	−0.458823
$979$	6284.98	0.205178
$980$	16999.7	0.554119
$981$	25870.0	0.841962
$982$	−28679.6	−0.931979
$983$	−3801.07	−0.123332	−0.0616659	−	0.998097i	$-0.519641\pi$
$983$	−3801.07	−0.123332	−0.0616659	+	0.998097i	$0.519641\pi$
$984$	−25603.6	−0.829484
$985$	19349.0	0.625898
$986$	−43707.5	−1.41169
$987$	1582.34	0.0510298
$988$	1743.76	0.0561502
$989$	33161.9	1.06622
$990$	9903.72	0.317940
$991$	23211.1	0.744022	0.372011	−	0.928228i	$-0.378668\pi$
$991$	23211.1	0.744022	0.372011	+	0.928228i	$0.378668\pi$
$992$	16541.9	0.529441
$993$	8039.06	0.256910
$994$	13351.9	0.426053
$995$	−65572.3	−2.08923
$996$	−9434.64	−0.300149
$997$	44560.6	1.41549	0.707747	−	0.706466i	$-0.249712\pi$
$997$	44560.6	1.41549	0.707747	+	0.706466i	$0.249712\pi$
$998$	−36168.6	−1.14719
$999$	5318.72	0.168445

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.a.b.1.3	✓	5
3.2	odd	2		333.4.a.f.1.3		5
4.3	odd	2		592.4.a.g.1.3		5
5.4	even	2		925.4.a.b.1.3		5
7.6	odd	2		1813.4.a.c.1.3		5
8.3	odd	2		2368.4.a.r.1.3		5
8.5	even	2		2368.4.a.m.1.3		5
37.36	even	2		1369.4.a.d.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.a.b.1.3	✓	5	1.1	even	1	trivial
333.4.a.f.1.3		5	3.2	odd	2
592.4.a.g.1.3		5	4.3	odd	2
925.4.a.b.1.3		5	5.4	even	2
1369.4.a.d.1.3		5	37.36	even	2
1813.4.a.c.1.3		5	7.6	odd	2
2368.4.a.m.1.3		5	8.5	even	2
2368.4.a.r.1.3		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)

\(f(q)\)	\(=\)	\(q+1.96027 q^{2} +3.37210 q^{3} -4.15734 q^{4} +14.4391 q^{5} +6.61023 q^{6} +7.73331 q^{7} -23.8317 q^{8} -15.6289 q^{9} +O(q^{10})\)	\(q+1.96027 q^{2} +3.37210 q^{3} -4.15734 q^{4} +14.4391 q^{5} +6.61023 q^{6} +7.73331 q^{7} -23.8317 q^{8} -15.6289 q^{9} +28.3045 q^{10} -22.3879 q^{11} -14.0190 q^{12} +16.3503 q^{13} +15.1594 q^{14} +48.6901 q^{15} -13.4578 q^{16} -116.372 q^{17} -30.6369 q^{18} -25.6535 q^{19} -60.0282 q^{20} +26.0775 q^{21} -43.8863 q^{22} +161.379 q^{23} -80.3628 q^{24} +83.4873 q^{25} +32.0510 q^{26} -143.749 q^{27} -32.1500 q^{28} +191.599 q^{29} +95.4457 q^{30} +100.698 q^{31} +164.272 q^{32} -75.4942 q^{33} -228.120 q^{34} +111.662 q^{35} +64.9748 q^{36} -37.0000 q^{37} -50.2877 q^{38} +55.1348 q^{39} -344.108 q^{40} +318.600 q^{41} +51.1190 q^{42} +205.491 q^{43} +93.0741 q^{44} -225.668 q^{45} +316.346 q^{46} +60.6783 q^{47} -45.3811 q^{48} -283.196 q^{49} +163.658 q^{50} -392.417 q^{51} -67.9736 q^{52} -726.017 q^{53} -281.787 q^{54} -323.261 q^{55} -184.298 q^{56} -86.5061 q^{57} +375.586 q^{58} +77.6735 q^{59} -202.421 q^{60} +48.7787 q^{61} +197.395 q^{62} -120.863 q^{63} +429.681 q^{64} +236.083 q^{65} -147.989 q^{66} -586.321 q^{67} +483.796 q^{68} +544.186 q^{69} +218.888 q^{70} +880.768 q^{71} +372.464 q^{72} -287.542 q^{73} -72.5300 q^{74} +281.528 q^{75} +106.650 q^{76} -173.132 q^{77} +108.079 q^{78} -277.922 q^{79} -194.318 q^{80} -62.7554 q^{81} +624.541 q^{82} +672.991 q^{83} -108.413 q^{84} -1680.30 q^{85} +402.818 q^{86} +646.091 q^{87} +533.541 q^{88} -280.731 q^{89} -442.369 q^{90} +126.442 q^{91} -670.907 q^{92} +339.563 q^{93} +118.946 q^{94} -370.412 q^{95} +553.943 q^{96} -1727.56 q^{97} -555.141 q^{98} +349.899 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 13 q^{3} + 18 q^{4} + 11 q^{5} + 9 q^{6} + 24 q^{7} + 30 q^{8} + 46 q^{9}+O(q^{10}) \) 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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