LMFDB - Embedded newform 37.4.a.a.1.2

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.21208.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 13x - 3 $ x^4 - x^3 - 8x^2 + 13x - 3
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			1.2
Root			$1.54469$ 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)

$f(q)$	$=$	$q-2.06923 q^{2} +0.265551 q^{3} -3.71830 q^{4} -5.08678 q^{5} -0.549486 q^{6} -27.2773 q^{7} +24.2478 q^{8} -26.9295 q^{9} +O(q^{10})$	\(q-2.06923 q^{2} +0.265551 q^{3} -3.71830 q^{4} -5.08678 q^{5} -0.549486 q^{6} -27.2773 q^{7} +24.2478 q^{8} -26.9295 q^{9} +10.5257 q^{10} -32.8577 q^{11} -0.987398 q^{12} +77.0620 q^{13} +56.4430 q^{14} -1.35080 q^{15} -20.4279 q^{16} -43.9281 q^{17} +55.7232 q^{18} +141.397 q^{19} +18.9141 q^{20} -7.24353 q^{21} +67.9901 q^{22} -161.240 q^{23} +6.43904 q^{24} -99.1247 q^{25} -159.459 q^{26} -14.3210 q^{27} +101.425 q^{28} -64.9923 q^{29} +2.79511 q^{30} -101.684 q^{31} -151.713 q^{32} -8.72540 q^{33} +90.8973 q^{34} +138.754 q^{35} +100.132 q^{36} +37.0000 q^{37} -292.583 q^{38} +20.4639 q^{39} -123.343 q^{40} +245.393 q^{41} +14.9885 q^{42} -306.268 q^{43} +122.175 q^{44} +136.984 q^{45} +333.643 q^{46} -114.835 q^{47} -5.42466 q^{48} +401.053 q^{49} +205.112 q^{50} -11.6652 q^{51} -286.539 q^{52} -10.1590 q^{53} +29.6335 q^{54} +167.140 q^{55} -661.416 q^{56} +37.5482 q^{57} +134.484 q^{58} -414.054 q^{59} +5.02267 q^{60} +492.558 q^{61} +210.408 q^{62} +734.565 q^{63} +477.351 q^{64} -391.997 q^{65} +18.0548 q^{66} +73.8213 q^{67} +163.338 q^{68} -42.8175 q^{69} -287.113 q^{70} -763.314 q^{71} -652.981 q^{72} -642.749 q^{73} -76.5614 q^{74} -26.3227 q^{75} -525.757 q^{76} +896.270 q^{77} -42.3445 q^{78} +268.849 q^{79} +103.912 q^{80} +723.293 q^{81} -507.775 q^{82} -729.587 q^{83} +26.9336 q^{84} +223.453 q^{85} +633.739 q^{86} -17.2588 q^{87} -796.728 q^{88} +617.016 q^{89} -283.452 q^{90} -2102.05 q^{91} +599.538 q^{92} -27.0024 q^{93} +237.621 q^{94} -719.256 q^{95} -40.2875 q^{96} -459.757 q^{97} -829.871 q^{98} +884.841 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} - 11 q^{3} + 6 q^{4} - 29 q^{5} - 27 q^{6} - 32 q^{7} - 90 q^{8} + q^{9}+O(q^{10}) $ 4 * q - 6 * q^2 - 11 * q^3 + 6 * q^4 - 29 * q^5 - 27 * q^6 - 32 * q^7 - 90 * q^8 + q^9	\( 4 q - 6 q^{2} - 11 q^{3} + 6 q^{4} - 29 q^{5} - 27 q^{6} - 32 q^{7} - 90 q^{8} + q^{9} + 81 q^{10} - 11 q^{11} + 143 q^{12} - 45 q^{13} + 82 q^{14} - 16 q^{15} + 378 q^{16} - 56 q^{17} + 255 q^{18} - 144 q^{19} - 165 q^{20} + 108 q^{21} + 73 q^{22} - 275 q^{23} + 9 q^{24} + 91 q^{25} + 97 q^{26} - 272 q^{27} - 202 q^{28} - 29 q^{29} + 96 q^{30} + 111 q^{31} - 946 q^{32} - 250 q^{33} + 212 q^{34} - 636 q^{35} - 723 q^{36} + 148 q^{37} - 76 q^{38} + 361 q^{39} + 937 q^{40} - 9 q^{41} + 446 q^{42} - 190 q^{43} + 211 q^{44} + 1018 q^{45} - 269 q^{46} - 472 q^{47} + 203 q^{48} + 1586 q^{49} + 83 q^{50} - 94 q^{51} - 1165 q^{52} + 318 q^{53} + 306 q^{54} + 779 q^{55} + 518 q^{56} + 1330 q^{57} - 915 q^{58} - 1530 q^{59} - 840 q^{60} - 31 q^{61} - 747 q^{62} - 1190 q^{63} + 3490 q^{64} + 570 q^{65} - 484 q^{66} - 5 q^{67} - 404 q^{68} + 1535 q^{69} - 1468 q^{70} + 390 q^{71} + 255 q^{72} - 967 q^{73} - 222 q^{74} - 320 q^{75} - 708 q^{76} - 1050 q^{77} + 1163 q^{78} + 17 q^{79} - 4653 q^{80} + 1888 q^{81} - 153 q^{82} - 3138 q^{83} - 1078 q^{84} + 302 q^{85} + 1118 q^{86} + 1025 q^{87} - 1931 q^{88} + 224 q^{89} - 1146 q^{90} - 1470 q^{91} + 3625 q^{92} - 458 q^{93} + 1666 q^{94} + 1382 q^{95} - 4391 q^{96} - 2532 q^{97} + 1722 q^{98} + 3166 q^{99}+O(q^{100}) \) 4 * q - 6 * q^2 - 11 * q^3 + 6 * q^4 - 29 * q^5 - 27 * q^6 - 32 * q^7 - 90 * q^8 + q^9 + 81 * q^10 - 11 * q^11 + 143 * q^12 - 45 * q^13 + 82 * q^14 - 16 * q^15 + 378 * q^16 - 56 * q^17 + 255 * q^18 - 144 * q^19 - 165 * q^20 + 108 * q^21 + 73 * q^22 - 275 * q^23 + 9 * q^24 + 91 * q^25 + 97 * q^26 - 272 * q^27 - 202 * q^28 - 29 * q^29 + 96 * q^30 + 111 * q^31 - 946 * q^32 - 250 * q^33 + 212 * q^34 - 636 * q^35 - 723 * q^36 + 148 * q^37 - 76 * q^38 + 361 * q^39 + 937 * q^40 - 9 * q^41 + 446 * q^42 - 190 * q^43 + 211 * q^44 + 1018 * q^45 - 269 * q^46 - 472 * q^47 + 203 * q^48 + 1586 * q^49 + 83 * q^50 - 94 * q^51 - 1165 * q^52 + 318 * q^53 + 306 * q^54 + 779 * q^55 + 518 * q^56 + 1330 * q^57 - 915 * q^58 - 1530 * q^59 - 840 * q^60 - 31 * q^61 - 747 * q^62 - 1190 * q^63 + 3490 * q^64 + 570 * q^65 - 484 * q^66 - 5 * q^67 - 404 * q^68 + 1535 * q^69 - 1468 * q^70 + 390 * q^71 + 255 * q^72 - 967 * q^73 - 222 * q^74 - 320 * q^75 - 708 * q^76 - 1050 * q^77 + 1163 * q^78 + 17 * q^79 - 4653 * q^80 + 1888 * q^81 - 153 * q^82 - 3138 * q^83 - 1078 * q^84 + 302 * q^85 + 1118 * q^86 + 1025 * q^87 - 1931 * q^88 + 224 * q^89 - 1146 * q^90 - 1470 * q^91 + 3625 * q^92 - 458 * q^93 + 1666 * q^94 + 1382 * q^95 - 4391 * q^96 - 2532 * q^97 + 1722 * q^98 + 3166 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.06923	−0.731583	−0.365791	−	0.930697i	$-0.619202\pi$
$2$	−2.06923	−0.731583	−0.365791	+	0.930697i	$0.619202\pi$
$3$	0.265551	0.0511054	0.0255527	−	0.999673i	$-0.491865\pi$
$3$	0.265551	0.0511054	0.0255527	+	0.999673i	$0.491865\pi$
$4$	−3.71830	−0.464787
$5$	−5.08678	−0.454975	−0.227488	−	0.973781i	$-0.573051\pi$
$5$	−5.08678	−0.454975	−0.227488	+	0.973781i	$0.573051\pi$
$6$	−0.549486	−0.0373878
$7$	−27.2773	−1.47284	−0.736419	−	0.676526i	$-0.763485\pi$
$7$	−27.2773	−1.47284	−0.736419	+	0.676526i	$0.763485\pi$
$8$	24.2478	1.07161
$9$	−26.9295	−0.997388
$10$	10.5257	0.332852
$11$	−32.8577	−0.900633	−0.450317	−	0.892869i	$-0.648689\pi$
$11$	−32.8577	−0.900633	−0.450317	+	0.892869i	$0.648689\pi$
$12$	−0.987398	−0.0237531
$13$	77.0620	1.64409	0.822044	−	0.569423i	$-0.192834\pi$
$13$	77.0620	1.64409	0.822044	+	0.569423i	$0.192834\pi$
$14$	56.4430	1.07750
$15$	−1.35080	−0.0232517
$16$	−20.4279	−0.319186
$17$	−43.9281	−0.626714	−0.313357	−	0.949635i	$-0.601454\pi$
$17$	−43.9281	−0.626714	−0.313357	+	0.949635i	$0.601454\pi$
$18$	55.7232	0.729672
$19$	141.397	1.70730	0.853652	−	0.520844i	$-0.174383\pi$
$19$	141.397	1.70730	0.853652	+	0.520844i	$0.174383\pi$
$20$	18.9141	0.211467
$21$	−7.24353	−0.0752699
$22$	67.9901	0.658888
$23$	−161.240	−1.46178	−0.730889	−	0.682497i	$-0.760894\pi$
$23$	−161.240	−1.46178	−0.730889	+	0.682497i	$0.760894\pi$
$24$	6.43904	0.0547652
$25$	−99.1247	−0.792998
$26$	−159.459	−1.20279
$27$	−14.3210	−0.102077
$28$	101.425	0.684556
$29$	−64.9923	−0.416164	−0.208082	−	0.978111i	$-0.566722\pi$
$29$	−64.9923	−0.416164	−0.208082	+	0.978111i	$0.566722\pi$
$30$	2.79511	0.0170105
$31$	−101.684	−0.589131	−0.294566	−	0.955631i	$-0.595175\pi$
$31$	−101.684	−0.589131	−0.294566	+	0.955631i	$0.595175\pi$
$32$	−151.713	−0.838102
$33$	−8.72540	−0.0460272
$34$	90.8973	0.458493
$35$	138.754	0.670105
$36$	100.132	0.463573
$37$	37.0000	0.164399
$37$	37.0000	0.164399
$38$	−292.583	−1.24903
$39$	20.4639	0.0840218
$40$	−123.343	−0.487557
$41$	245.393	0.934732	0.467366	−	0.884064i	$-0.345203\pi$
$41$	245.393	0.934732	0.467366	+	0.884064i	$0.345203\pi$
$42$	14.9885	0.0550662
$43$	−306.268	−1.08617	−0.543087	−	0.839676i	$-0.682745\pi$
$43$	−306.268	−1.08617	−0.543087	+	0.839676i	$0.682745\pi$
$44$	122.175	0.418603
$45$	136.984	0.453787
$46$	333.643	1.06941
$47$	−114.835	−0.356393	−0.178197	−	0.983995i	$-0.557026\pi$
$47$	−114.835	−0.356393	−0.178197	+	0.983995i	$0.557026\pi$
$48$	−5.42466	−0.0163121
$49$	401.053	1.16925
$50$	205.112	0.580143
$51$	−11.6652	−0.0320284
$52$	−286.539	−0.764151
$53$	−10.1590	−0.0263291	−0.0131646	−	0.999913i	$-0.504191\pi$
$53$	−10.1590	−0.0263291	−0.0131646	+	0.999913i	$0.504191\pi$
$54$	29.6335	0.0746779
$55$	167.140	0.409766
$56$	−661.416	−1.57831
$57$	37.5482	0.0872524
$58$	134.484	0.304459
$59$	−414.054	−0.913648	−0.456824	−	0.889557i	$-0.651013\pi$
$59$	−414.054	−0.913648	−0.456824	+	0.889557i	$0.651013\pi$
$60$	5.02267	0.0108071
$61$	492.558	1.03386	0.516931	−	0.856027i	$-0.327074\pi$
$61$	492.558	1.03386	0.516931	+	0.856027i	$0.327074\pi$
$62$	210.408	0.430998
$63$	734.565	1.46899
$64$	477.351	0.932327
$65$	−391.997	−0.748020
$66$	18.0548	0.0336727
$67$	73.8213	0.134608	0.0673038	−	0.997733i	$-0.478560\pi$
$67$	73.8213	0.134608	0.0673038	+	0.997733i	$0.478560\pi$
$68$	163.338	0.291288
$69$	−42.8175	−0.0747047
$70$	−287.113	−0.490237
$71$	−763.314	−1.27590	−0.637948	−	0.770079i	$-0.720217\pi$
$71$	−763.314	−1.27590	−0.637948	+	0.770079i	$0.720217\pi$
$72$	−652.981	−1.06881
$73$	−642.749	−1.03052	−0.515261	−	0.857033i	$-0.672305\pi$
$73$	−642.749	−1.03052	−0.515261	+	0.857033i	$0.672305\pi$
$74$	−76.5614	−0.120271
$75$	−26.3227	−0.0405264
$76$	−525.757	−0.793532
$77$	896.270	1.32649
$78$	−42.3445	−0.0614689
$79$	268.849	0.382885	0.191442	−	0.981504i	$-0.438683\pi$
$79$	268.849	0.382885	0.191442	+	0.981504i	$0.438683\pi$
$80$	103.912	0.145222
$81$	723.293	0.992172
$82$	−507.775	−0.683834
$83$	−729.587	−0.964850	−0.482425	−	0.875937i	$-0.660244\pi$
$83$	−729.587	−0.964850	−0.482425	+	0.875937i	$0.660244\pi$
$84$	26.9336	0.0349845
$85$	223.453	0.285139
$86$	633.739	0.794626
$87$	−17.2588	−0.0212682
$88$	−796.728	−0.965130
$89$	617.016	0.734872	0.367436	−	0.930049i	$-0.380236\pi$
$89$	617.016	0.734872	0.367436	+	0.930049i	$0.380236\pi$
$90$	−283.452	−0.331983
$91$	−2102.05	−2.42148
$92$	599.538	0.679415
$93$	−27.0024	−0.0301078
$94$	237.621	0.260731
$95$	−719.256	−0.776781
$96$	−40.2875	−0.0428315
$97$	−459.757	−0.481250	−0.240625	−	0.970618i	$-0.577352\pi$
$97$	−459.757	−0.481250	−0.240625	+	0.970618i	$0.577352\pi$
$98$	−829.871	−0.855404
$99$	884.841	0.898281
$100$	368.575	0.368575
$101$	1875.56	1.84778	0.923889	−	0.382660i	$-0.124992\pi$
$101$	1875.56	1.84778	0.923889	+	0.382660i	$0.124992\pi$
$102$	24.1379	0.0234315
$103$	1579.80	1.51128	0.755642	−	0.654985i	$-0.227325\pi$
$103$	1579.80	1.51128	0.755642	+	0.654985i	$0.227325\pi$
$104$	1868.59	1.76183
$105$	36.8462	0.0342459
$106$	21.0213	0.0192619
$107$	−1699.15	−1.53517	−0.767586	−	0.640945i	$-0.778542\pi$
$107$	−1699.15	−1.53517	−0.767586	+	0.640945i	$0.778542\pi$
$108$	53.2499	0.0474442
$109$	889.402	0.781553	0.390776	−	0.920486i	$-0.372207\pi$
$109$	889.402	0.781553	0.390776	+	0.920486i	$0.372207\pi$
$110$	−345.850	−0.299778
$111$	9.82540	0.00840167
$112$	557.219	0.470109
$113$	366.085	0.304764	0.152382	−	0.988322i	$-0.451306\pi$
$113$	366.085	0.304764	0.152382	+	0.988322i	$0.451306\pi$
$114$	−77.6959	−0.0638323
$115$	820.192	0.665073
$116$	241.661	0.193428
$117$	−2075.24	−1.63979
$118$	856.772	0.668409
$119$	1198.24	0.923048
$120$	−32.7540	−0.0249168
$121$	−251.372	−0.188860
$122$	−1019.22	−0.756356
$123$	65.1645	0.0477698
$124$	378.093	0.273821
$125$	1140.07	0.815769
$126$	−1519.98	−1.07469
$127$	222.455	0.155430	0.0777152	−	0.996976i	$-0.475238\pi$
$127$	222.455	0.155430	0.0777152	+	0.996976i	$0.475238\pi$
$128$	225.952	0.156028
$129$	−81.3299	−0.0555093
$130$	811.132	0.547238
$131$	−386.073	−0.257492	−0.128746	−	0.991678i	$-0.541095\pi$
$131$	−386.073	−0.257492	−0.128746	+	0.991678i	$0.541095\pi$
$132$	32.4436	0.0213928
$133$	−3856.94	−2.51458
$134$	−152.753	−0.0984765
$135$	72.8479	0.0464426
$136$	−1065.16	−0.671595
$137$	−2557.81	−1.59510	−0.797550	−	0.603253i	$-0.793871\pi$
$137$	−2557.81	−1.59510	−0.797550	+	0.603253i	$0.793871\pi$
$138$	88.5992	0.0546526
$139$	334.998	0.204418	0.102209	−	0.994763i	$-0.467409\pi$
$139$	334.998	0.204418	0.102209	+	0.994763i	$0.467409\pi$
$140$	−515.927	−0.311456
$141$	−30.4947	−0.0182136
$142$	1579.47	0.933424
$143$	−2532.08	−1.48072
$144$	550.113	0.318353
$145$	330.601	0.189345
$146$	1329.99	0.753912
$147$	106.500	0.0597550
$148$	−137.577	−0.0764105
$149$	−1426.18	−0.784140	−0.392070	−	0.919935i	$-0.628241\pi$
$149$	−1426.18	−0.784140	−0.392070	+	0.919935i	$0.628241\pi$
$150$	54.4676	0.0296484
$151$	−1230.68	−0.663256	−0.331628	−	0.943410i	$-0.607598\pi$
$151$	−1230.68	−0.663256	−0.331628	+	0.943410i	$0.607598\pi$
$152$	3428.58	1.82957
$153$	1182.96	0.625077
$154$	−1854.59	−0.970435
$155$	517.246	0.268040
$156$	−76.0909	−0.0390522
$157$	−236.139	−0.120038	−0.0600189	−	0.998197i	$-0.519116\pi$
$157$	−236.139	−0.120038	−0.0600189	+	0.998197i	$0.519116\pi$
$158$	−556.310	−0.280112
$159$	−2.69773	−0.00134556
$160$	771.728	0.381315
$161$	4398.20	2.15296
$162$	−1496.66	−0.725855
$163$	−2840.41	−1.36490	−0.682449	−	0.730933i	$-0.739085\pi$
$163$	−2840.41	−1.36490	−0.682449	+	0.730933i	$0.739085\pi$
$164$	−912.445	−0.434451
$165$	44.3842	0.0209412
$166$	1509.68	0.705868
$167$	513.319	0.237855	0.118928	−	0.992903i	$-0.462054\pi$
$167$	513.319	0.237855	0.118928	+	0.992903i	$0.462054\pi$
$168$	−175.640	−0.0806602
$169$	3741.55	1.70303
$170$	−462.374	−0.208603
$171$	−3807.76	−1.70284
$172$	1138.80	0.504839
$173$	216.851	0.0952998	0.0476499	−	0.998864i	$-0.484827\pi$
$173$	216.851	0.0952998	0.0476499	+	0.998864i	$0.484827\pi$
$174$	35.7124	0.0155595
$175$	2703.86	1.16796
$176$	671.214	0.287470
$177$	−109.953	−0.0466923
$178$	−1276.75	−0.537619
$179$	3241.45	1.35351	0.676753	−	0.736210i	$-0.263387\pi$
$179$	3241.45	1.35351	0.676753	+	0.736210i	$0.263387\pi$
$180$	−509.348	−0.210914
$181$	−1116.74	−0.458602	−0.229301	−	0.973356i	$-0.573644\pi$
$181$	−1116.74	−0.458602	−0.229301	+	0.973356i	$0.573644\pi$
$182$	4349.61	1.77151
$183$	130.799	0.0528359
$184$	−3909.72	−1.56646
$185$	−188.211	−0.0747975
$186$	55.8742	0.0220263
$187$	1443.38	0.564440
$188$	426.992	0.165647
$189$	390.640	0.150343
$190$	1488.31	0.568279
$191$	−1067.66	−0.404465	−0.202233	−	0.979338i	$-0.564820\pi$
$191$	−1067.66	−0.404465	−0.202233	+	0.979338i	$0.564820\pi$
$192$	126.761	0.0476469
$193$	690.750	0.257623	0.128812	−	0.991669i	$-0.458884\pi$
$193$	690.750	0.257623	0.128812	+	0.991669i	$0.458884\pi$
$194$	951.342	0.352074
$195$	−104.095	−0.0382278
$196$	−1491.23	−0.543453
$197$	3122.53	1.12929	0.564647	−	0.825332i	$-0.309012\pi$
$197$	3122.53	1.12929	0.564647	+	0.825332i	$0.309012\pi$
$198$	−1830.94	−0.657167
$199$	−2816.63	−1.00334	−0.501672	−	0.865058i	$-0.667282\pi$
$199$	−2816.63	−1.00334	−0.501672	+	0.865058i	$0.667282\pi$
$200$	−2403.56	−0.849786
$201$	19.6033	0.00687917
$202$	−3880.97	−1.35180
$203$	1772.82	0.612943
$204$	43.3746	0.0148864
$205$	−1248.26	−0.425280
$206$	−3268.96	−1.10563
$207$	4342.11	1.45796
$208$	−1574.22	−0.524770
$209$	−4645.99	−1.53765
$210$	−76.2433	−0.0250537
$211$	2180.07	0.711290	0.355645	−	0.934621i	$-0.384261\pi$
$211$	2180.07	0.711290	0.355645	+	0.934621i	$0.384261\pi$
$212$	37.7741	0.0122374
$213$	−202.699	−0.0652052
$214$	3515.94	1.12311
$215$	1557.92	0.494182
$216$	−347.254	−0.109387
$217$	2773.68	0.867695
$218$	−1840.37	−0.571770
$219$	−170.683	−0.0526652
$220$	−621.475	−0.190454
$221$	−3385.19	−1.03037
$222$	−20.3310	−0.00614652
$223$	1470.43	0.441558	0.220779	−	0.975324i	$-0.429140\pi$
$223$	1470.43	0.441558	0.220779	+	0.975324i	$0.429140\pi$
$224$	4138.32	1.23439
$225$	2669.38	0.790926
$226$	−757.512	−0.222960
$227$	−3651.40	−1.06763	−0.533814	−	0.845602i	$-0.679242\pi$
$227$	−3651.40	−1.06763	−0.533814	+	0.845602i	$0.679242\pi$
$228$	−139.615	−0.0405538
$229$	−1631.69	−0.470853	−0.235427	−	0.971892i	$-0.575649\pi$
$229$	−1631.69	−0.470853	−0.235427	+	0.971892i	$0.575649\pi$
$230$	−1697.17	−0.486555
$231$	238.006	0.0677906
$232$	−1575.92	−0.445967
$233$	−3477.88	−0.977868	−0.488934	−	0.872321i	$-0.662614\pi$
$233$	−3477.88	−0.977868	−0.488934	+	0.872321i	$0.662614\pi$
$234$	4294.14	1.19965
$235$	584.142	0.162150
$236$	1539.58	0.424652
$237$	71.3933	0.0195675
$238$	−2479.44	−0.675286
$239$	−2070.74	−0.560441	−0.280220	−	0.959936i	$-0.590408\pi$
$239$	−2070.74	−0.560441	−0.280220	+	0.959936i	$0.590408\pi$
$240$	27.5940	0.00742161
$241$	−4764.25	−1.27341	−0.636706	−	0.771107i	$-0.719703\pi$
$241$	−4764.25	−1.27341	−0.636706	+	0.771107i	$0.719703\pi$
$242$	520.146	0.138166
$243$	578.740	0.152783
$244$	−1831.48	−0.480526
$245$	−2040.07	−0.531980
$246$	−134.840	−0.0349476
$247$	10896.4	2.80696
$248$	−2465.63	−0.631321
$249$	−193.743	−0.0493090
$250$	−2359.07	−0.596803
$251$	2561.25	0.644083	0.322041	−	0.946726i	$-0.395631\pi$
$251$	2561.25	0.644083	0.322041	+	0.946726i	$0.395631\pi$
$252$	−2731.33	−0.682768
$253$	5297.98	1.31653
$254$	−460.309	−0.113710
$255$	59.3381	0.0145721
$256$	−4286.36	−1.04647
$257$	−7757.57	−1.88289	−0.941447	−	0.337161i	$-0.890533\pi$
$257$	−7757.57	−1.88289	−0.941447	+	0.337161i	$0.890533\pi$
$258$	168.290	0.0406096
$259$	−1009.26	−0.242133
$260$	1457.56	0.347670
$261$	1750.21	0.415078
$262$	798.874	0.188376
$263$	6944.82	1.62827	0.814137	−	0.580673i	$-0.197211\pi$
$263$	6944.82	1.62827	0.814137	+	0.580673i	$0.197211\pi$
$264$	−211.572	−0.0493233
$265$	51.6765	0.0119791
$266$	7980.89	1.83962
$267$	163.849	0.0375559
$268$	−274.489	−0.0625638
$269$	927.481	0.210221	0.105111	−	0.994461i	$-0.466480\pi$
$269$	927.481	0.210221	0.105111	+	0.994461i	$0.466480\pi$
$270$	−150.739	−0.0339766
$271$	221.600	0.0496724	0.0248362	−	0.999692i	$-0.492094\pi$
$271$	221.600	0.0496724	0.0248362	+	0.999692i	$0.492094\pi$
$272$	897.360	0.200038
$273$	−558.201	−0.123750
$274$	5292.70	1.16695
$275$	3257.01	0.714200
$276$	159.208	0.0347218
$277$	−6556.47	−1.42217	−0.711083	−	0.703108i	$-0.751795\pi$
$277$	−6556.47	−1.42217	−0.711083	+	0.703108i	$0.751795\pi$
$278$	−693.187	−0.149549
$279$	2738.31	0.587593
$280$	3364.48	0.718093
$281$	−7572.29	−1.60756	−0.803781	−	0.594926i	$-0.797181\pi$
$281$	−7572.29	−1.60756	−0.803781	+	0.594926i	$0.797181\pi$
$282$	63.1005	0.0133248
$283$	−625.308	−0.131345	−0.0656726	−	0.997841i	$-0.520919\pi$
$283$	−625.308	−0.131345	−0.0656726	+	0.997841i	$0.520919\pi$
$284$	2838.23	0.593020
$285$	−190.999	−0.0396977
$286$	5239.45	1.08327
$287$	−6693.68	−1.37671
$288$	4085.54	0.835913
$289$	−2983.32	−0.607230
$290$	−684.090	−0.138521
$291$	−122.089	−0.0245945
$292$	2389.93	0.478973
$293$	8599.01	1.71454	0.857269	−	0.514869i	$-0.172159\pi$
$293$	8599.01	1.71454	0.857269	+	0.514869i	$0.172159\pi$
$294$	−220.373	−0.0437157
$295$	2106.20	0.415687
$296$	897.170	0.176172
$297$	470.556	0.0919342
$298$	2951.08	0.573663
$299$	−12425.5	−2.40329
$300$	97.8755	0.0188362
$301$	8354.18	1.59976
$302$	2546.57	0.485227
$303$	498.059	0.0944314
$304$	−2888.45	−0.544948
$305$	−2505.53	−0.470382
$306$	−2447.82	−0.457296
$307$	−4214.17	−0.783438	−0.391719	−	0.920085i	$-0.628119\pi$
$307$	−4214.17	−0.783438	−0.391719	+	0.920085i	$0.628119\pi$
$308$	−3332.60	−0.616534
$309$	419.518	0.0772347
$310$	−1070.30	−0.196093
$311$	−471.209	−0.0859157	−0.0429579	−	0.999077i	$-0.513678\pi$
$311$	−471.209	−0.0859157	−0.0429579	+	0.999077i	$0.513678\pi$
$312$	496.205	0.0900388
$313$	1096.68	0.198045	0.0990224	−	0.995085i	$-0.468428\pi$
$313$	1096.68	0.198045	0.0990224	+	0.995085i	$0.468428\pi$
$314$	488.625	0.0878175
$315$	−3736.57	−0.668355
$316$	−999.661	−0.177960
$317$	−5341.19	−0.946344	−0.473172	−	0.880970i	$-0.656891\pi$
$317$	−5341.19	−0.946344	−0.473172	+	0.880970i	$0.656891\pi$
$318$	5.58223	0.000984389 0
$319$	2135.50	0.374812
$320$	−2428.18	−0.424185
$321$	−451.213	−0.0784556
$322$	−9100.88	−1.57507
$323$	−6211.32	−1.06999
$324$	−2689.42	−0.461148
$325$	−7638.75	−1.30376
$326$	5877.46	0.998536
$327$	236.182	0.0399415
$328$	5950.26	1.00167
$329$	3132.41	0.524909
$330$	−91.8410	−0.0153202
$331$	2324.99	0.386082	0.193041	−	0.981191i	$-0.438165\pi$
$331$	2324.99	0.386082	0.193041	+	0.981191i	$0.438165\pi$
$332$	2712.82	0.448450
$333$	−996.391	−0.163970
$334$	−1062.17	−0.174011
$335$	−375.512	−0.0612431
$336$	147.970	0.0240251
$337$	3786.59	0.612073	0.306037	−	0.952020i	$-0.400997\pi$
$337$	3786.59	0.612073	0.306037	+	0.952020i	$0.400997\pi$
$338$	−7742.12	−1.24591
$339$	97.2142	0.0155751
$340$	−830.863	−0.132529
$341$	3341.12	0.530591
$342$	7879.11	1.24577
$343$	−1583.54	−0.249280
$344$	−7426.34	−1.16396
$345$	217.803	0.0339888
$346$	−448.714	−0.0697197
$347$	−3848.24	−0.595344	−0.297672	−	0.954668i	$-0.596210\pi$
$347$	−3848.24	−0.595344	−0.297672	+	0.954668i	$0.596210\pi$
$348$	64.1733	0.00988520
$349$	−9522.63	−1.46056	−0.730279	−	0.683149i	$-0.760610\pi$
$349$	−9522.63	−1.46056	−0.730279	+	0.683149i	$0.760610\pi$
$350$	−5594.90	−0.854457
$351$	−1103.61	−0.167824
$352$	4984.93	0.754822
$353$	3779.61	0.569882	0.284941	−	0.958545i	$-0.408026\pi$
$353$	3779.61	0.569882	0.284941	+	0.958545i	$0.408026\pi$
$354$	227.517	0.0341593
$355$	3882.81	0.580501
$356$	−2294.25	−0.341559
$357$	318.195	0.0471727
$358$	−6707.30	−0.990201
$359$	3351.07	0.492653	0.246327	−	0.969187i	$-0.420776\pi$
$359$	3351.07	0.492653	0.246327	+	0.969187i	$0.420776\pi$
$360$	3321.57	0.486284
$361$	13134.2	1.91488
$362$	2310.80	0.335505
$363$	−66.7522	−0.00965174
$364$	7816.03	1.12547
$365$	3269.52	0.468862
$366$	−270.654	−0.0386538
$367$	9438.03	1.34240	0.671200	−	0.741276i	$-0.265779\pi$
$367$	9438.03	1.34240	0.671200	+	0.741276i	$0.265779\pi$
$368$	3293.80	0.466579
$369$	−6608.32	−0.932291
$370$	389.451	0.0547205
$371$	277.110	0.0387786
$372$	100.403	0.0139937
$373$	709.309	0.0984628	0.0492314	−	0.998787i	$-0.484323\pi$
$373$	709.309	0.0984628	0.0492314	+	0.998787i	$0.484323\pi$
$374$	−2986.68	−0.412934
$375$	302.748	0.0416902
$376$	−2784.51	−0.381915
$377$	−5008.44	−0.684211
$378$	−808.323	−0.109989
$379$	3932.35	0.532959	0.266480	−	0.963841i	$-0.414140\pi$
$379$	3932.35	0.532959	0.266480	+	0.963841i	$0.414140\pi$
$380$	2674.41	0.361037
$381$	59.0731	0.00794333
$382$	2209.22	0.295900
$383$	4245.51	0.566412	0.283206	−	0.959059i	$-0.408602\pi$
$383$	4245.51	0.566412	0.283206	+	0.959059i	$0.408602\pi$
$384$	60.0019	0.00797385
$385$	−4559.13	−0.603519
$386$	−1429.32	−0.188473
$387$	8247.65	1.08334
$388$	1709.51	0.223679
$389$	12447.9	1.62245	0.811227	−	0.584731i	$-0.198800\pi$
$389$	12447.9	1.62245	0.811227	+	0.584731i	$0.198800\pi$
$390$	215.397	0.0279668
$391$	7082.98	0.916117
$392$	9724.67	1.25298
$393$	−102.522	−0.0131592
$394$	−6461.22	−0.826172
$395$	−1367.58	−0.174203
$396$	−3290.10	−0.417509
$397$	−7042.08	−0.890256	−0.445128	−	0.895467i	$-0.646842\pi$
$397$	−7042.08	−0.890256	−0.445128	+	0.895467i	$0.646842\pi$
$398$	5828.25	0.734029
$399$	−1024.22	−0.128509
$400$	2024.91	0.253114
$401$	−5223.04	−0.650439	−0.325220	−	0.945639i	$-0.605438\pi$
$401$	−5223.04	−0.650439	−0.325220	+	0.945639i	$0.605438\pi$
$402$	−40.5638	−0.00503268
$403$	−7836.01	−0.968584
$404$	−6973.90	−0.858823
$405$	−3679.23	−0.451413
$406$	−3668.36	−0.448418
$407$	−1215.73	−0.148063
$408$	−282.855	−0.0343221
$409$	1483.24	0.179320	0.0896598	−	0.995972i	$-0.471422\pi$
$409$	1483.24	0.179320	0.0896598	+	0.995972i	$0.471422\pi$
$410$	2582.94	0.311127
$411$	−679.230	−0.0815181
$412$	−5874.16	−0.702425
$413$	11294.3	1.34566
$414$	−8984.82	−1.06662
$415$	3711.25	0.438983
$416$	−11691.3	−1.37791
$417$	88.9592	0.0104469
$418$	9613.61	1.12492
$419$	−11099.4	−1.29413	−0.647065	−	0.762435i	$-0.724004\pi$
$419$	−11099.4	−1.29413	−0.647065	+	0.762435i	$0.724004\pi$
$420$	−137.005	−0.0159171
$421$	5497.12	0.636374	0.318187	−	0.948028i	$-0.396926\pi$
$421$	5497.12	0.636374	0.318187	+	0.948028i	$0.396926\pi$
$422$	−4511.06	−0.520368
$423$	3092.46	0.355462
$424$	−246.333	−0.0282146
$425$	4354.36	0.496983
$426$	419.430	0.0477030
$427$	−13435.7	−1.52271
$428$	6317.96	0.713528
$429$	−672.397	−0.0756728
$430$	−3223.69	−0.361535
$431$	11441.4	1.27868	0.639341	−	0.768923i	$-0.279207\pi$
$431$	11441.4	1.27868	0.639341	+	0.768923i	$0.279207\pi$
$432$	292.549	0.0325816
$433$	9236.75	1.02515	0.512575	−	0.858642i	$-0.328692\pi$
$433$	9236.75	1.02515	0.512575	+	0.858642i	$0.328692\pi$
$434$	−5739.38	−0.634791
$435$	87.7916	0.00967652
$436$	−3307.06	−0.363255
$437$	−22798.9	−2.49570
$438$	353.182	0.0385289
$439$	−5713.59	−0.621173	−0.310586	−	0.950545i	$-0.600525\pi$
$439$	−5713.59	−0.621173	−0.310586	+	0.950545i	$0.600525\pi$
$440$	4052.78	0.439110
$441$	−10800.2	−1.16620
$442$	7004.73	0.753803
$443$	10461.4	1.12198	0.560991	−	0.827822i	$-0.310420\pi$
$443$	10461.4	1.12198	0.560991	+	0.827822i	$0.310420\pi$
$444$	−36.5337	−0.00390499
$445$	−3138.62	−0.334348
$446$	−3042.66	−0.323036
$447$	−378.723	−0.0400738
$448$	−13020.9	−1.37317
$449$	13584.1	1.42778	0.713891	−	0.700256i	$-0.246931\pi$
$449$	13584.1	1.42778	0.713891	+	0.700256i	$0.246931\pi$
$450$	−5523.55	−0.578628
$451$	−8063.06	−0.841851
$452$	−1361.21	−0.141650
$453$	−326.810	−0.0338959
$454$	7555.57	0.781059
$455$	10692.6	1.10171
$456$	910.463	0.0935007
$457$	−652.511	−0.0667903	−0.0333952	−	0.999442i	$-0.510632\pi$
$457$	−652.511	−0.0667903	−0.0333952	+	0.999442i	$0.510632\pi$
$458$	3376.35	0.344468
$459$	629.097	0.0639732
$460$	−3049.72	−0.309117
$461$	3717.66	0.375594	0.187797	−	0.982208i	$-0.439865\pi$
$461$	3717.66	0.375594	0.187797	+	0.982208i	$0.439865\pi$
$462$	−492.488	−0.0495944
$463$	−13123.6	−1.31729	−0.658646	−	0.752453i	$-0.728870\pi$
$463$	−13123.6	−1.31729	−0.658646	+	0.752453i	$0.728870\pi$
$464$	1327.66	0.132834
$465$	137.355	0.0136983
$466$	7196.52	0.715391
$467$	2173.59	0.215378	0.107689	−	0.994185i	$-0.465655\pi$
$467$	2173.59	0.215378	0.107689	+	0.994185i	$0.465655\pi$
$468$	7716.35	0.762155
$469$	−2013.65	−0.198255
$470$	−1208.72	−0.118626
$471$	−62.7070	−0.00613457
$472$	−10039.9	−0.979077
$473$	10063.3	0.978244
$474$	−147.729	−0.0143152
$475$	−14016.0	−1.35389
$476$	−4455.42	−0.429021
$477$	273.576	0.0262604
$478$	4284.84	0.410009
$479$	6638.22	0.633211	0.316606	−	0.948557i	$-0.397457\pi$
$479$	6638.22	0.633211	0.316606	+	0.948557i	$0.397457\pi$
$480$	204.933	0.0194873
$481$	2851.29	0.270287
$482$	9858.31	0.931605
$483$	1167.95	0.110028
$484$	934.676	0.0877794
$485$	2338.68	0.218957
$486$	−1197.54	−0.111773
$487$	−4190.94	−0.389958	−0.194979	−	0.980807i	$-0.562464\pi$
$487$	−4190.94	−0.389958	−0.194979	+	0.980807i	$0.562464\pi$
$488$	11943.5	1.10790
$489$	−754.276	−0.0697536
$490$	4221.37	0.389188
$491$	2099.28	0.192952	0.0964758	−	0.995335i	$-0.469243\pi$
$491$	2099.28	0.192952	0.0964758	+	0.995335i	$0.469243\pi$
$492$	−242.301	−0.0222028
$493$	2854.99	0.260816
$494$	−22547.0	−2.05352
$495$	−4500.99	−0.408696
$496$	2077.20	0.188043
$497$	20821.2	1.87919
$498$	400.898	0.0360736
$499$	−10710.1	−0.960823	−0.480412	−	0.877043i	$-0.659513\pi$
$499$	−10710.1	−0.960823	−0.480412	+	0.877043i	$0.659513\pi$
$500$	−4239.13	−0.379159
$501$	136.312	0.0121557
$502$	−5299.81	−0.471200
$503$	−15349.8	−1.36066	−0.680331	−	0.732905i	$-0.738164\pi$
$503$	−15349.8	−1.36066	−0.680331	+	0.732905i	$0.738164\pi$
$504$	17811.6	1.57419
$505$	−9540.58	−0.840693
$506$	−10962.7	−0.963147
$507$	993.574	0.0870339
$508$	−827.152	−0.0722420
$509$	−3404.58	−0.296474	−0.148237	−	0.988952i	$-0.547360\pi$
$509$	−3404.58	−0.296474	−0.148237	+	0.988952i	$0.547360\pi$
$510$	−122.784	−0.0106607
$511$	17532.5	1.51779
$512$	7061.83	0.609554
$513$	−2024.96	−0.174277
$514$	16052.2	1.37749
$515$	−8036.09	−0.687597
$516$	302.409	0.0258000
$517$	3773.23	0.320980
$518$	2088.39	0.177140
$519$	57.5850	0.00487033
$520$	−9505.08	−0.801587
$521$	23349.4	1.96345	0.981724	−	0.190311i	$-0.0609498\pi$
$521$	23349.4	1.96345	0.981724	+	0.190311i	$0.0609498\pi$
$522$	−3621.58	−0.303664
$523$	−12055.1	−1.00790	−0.503951	−	0.863732i	$-0.668121\pi$
$523$	−12055.1	−1.00790	−0.503951	+	0.863732i	$0.668121\pi$
$524$	1435.54	0.119679
$525$	718.013	0.0596889
$526$	−14370.4	−1.19122
$527$	4466.81	0.369217
$528$	178.242	0.0146912
$529$	13831.4	1.13679
$530$	−106.931	−0.00876371
$531$	11150.3	0.911262
$532$	14341.3	1.16874
$533$	18910.5	1.53678
$534$	−339.042	−0.0274752
$535$	8643.22	0.698466
$536$	1790.01	0.144247
$537$	860.772	0.0691714
$538$	−1919.17	−0.153794
$539$	−13177.7	−1.05307
$540$	−270.870	−0.0215859
$541$	8970.96	0.712924	0.356462	−	0.934310i	$-0.383983\pi$
$541$	8970.96	0.712924	0.356462	+	0.934310i	$0.383983\pi$
$542$	−458.540	−0.0363395
$543$	−296.553	−0.0234370
$544$	6664.45	0.525250
$545$	−4524.19	−0.355587
$546$	1155.05	0.0905337
$547$	−21511.8	−1.68149	−0.840747	−	0.541428i	$-0.817884\pi$
$547$	−21511.8	−1.68149	−0.840747	+	0.541428i	$0.817884\pi$
$548$	9510.70	0.741381
$549$	−13264.3	−1.03116
$550$	−6739.49	−0.522496
$551$	−9189.74	−0.710519
$552$	−1038.23	−0.0800545
$553$	−7333.49	−0.563927
$554$	13566.8	1.04043
$555$	−49.9796	−0.00382255
$556$	−1245.62	−0.0950110
$557$	−16422.7	−1.24928	−0.624642	−	0.780912i	$-0.714755\pi$
$557$	−16422.7	−1.24928	−0.624642	+	0.780912i	$0.714755\pi$
$558$	−5666.19	−0.429873
$559$	−23601.6	−1.78577
$560$	−2834.45	−0.213888
$561$	383.291	0.0288459
$562$	15668.8	1.17606
$563$	4634.56	0.346933	0.173467	−	0.984840i	$-0.444503\pi$
$563$	4634.56	0.346933	0.173467	+	0.984840i	$0.444503\pi$
$564$	113.388	0.00846544
$565$	−1862.19	−0.138660
$566$	1293.90	0.0960899
$567$	−19729.5	−1.46131
$568$	−18508.7	−1.36727
$569$	−8644.46	−0.636897	−0.318449	−	0.947940i	$-0.603162\pi$
$569$	−8644.46	−0.636897	−0.318449	+	0.947940i	$0.603162\pi$
$570$	395.221	0.0290421
$571$	22981.8	1.68434	0.842171	−	0.539210i	$-0.181277\pi$
$571$	22981.8	1.68434	0.842171	+	0.539210i	$0.181277\pi$
$572$	9415.02	0.688220
$573$	−283.517	−0.0206703
$574$	13850.7	1.00718
$575$	15982.9	1.15919
$576$	−12854.8	−0.929892
$577$	8204.55	0.591958	0.295979	−	0.955194i	$-0.404354\pi$
$577$	8204.55	0.591958	0.295979	+	0.955194i	$0.404354\pi$
$578$	6173.17	0.444239
$579$	183.430	0.0131659
$580$	−1229.27	−0.0880049
$581$	19901.2	1.42107
$582$	252.630	0.0179929
$583$	333.801	0.0237129
$584$	−15585.3	−1.10432
$585$	10556.3	0.746066
$586$	−17793.3	−1.25433
$587$	6347.95	0.446351	0.223175	−	0.974778i	$-0.428358\pi$
$587$	6347.95	0.446351	0.223175	+	0.974778i	$0.428358\pi$
$588$	−395.999	−0.0277734
$589$	−14377.9	−1.00583
$590$	−4358.21	−0.304109
$591$	829.192	0.0577130
$592$	−755.833	−0.0524739
$593$	−21633.9	−1.49814	−0.749070	−	0.662490i	$-0.769500\pi$
$593$	−21633.9	−1.49814	−0.749070	+	0.662490i	$0.769500\pi$
$594$	−973.689	−0.0672574
$595$	−6095.19	−0.419964
$596$	5302.95	0.364458
$597$	−747.959	−0.0512763
$598$	25711.2	1.75821
$599$	−20138.9	−1.37371	−0.686856	−	0.726793i	$-0.741010\pi$
$599$	−20138.9	−1.37371	−0.686856	+	0.726793i	$0.741010\pi$
$600$	−638.268	−0.0434286
$601$	−27237.3	−1.84864	−0.924321	−	0.381616i	$-0.875368\pi$
$601$	−27237.3	−1.84864	−0.924321	+	0.381616i	$0.875368\pi$
$602$	−17286.7	−1.17036
$603$	−1987.97	−0.134256
$604$	4576.05	0.308273
$605$	1278.67	0.0859264
$606$	−1030.60	−0.0690844
$607$	23918.0	1.59934	0.799672	−	0.600437i	$-0.205007\pi$
$607$	23918.0	1.59934	0.799672	+	0.600437i	$0.205007\pi$
$608$	−21451.7	−1.43089
$609$	470.774	0.0313247
$610$	5184.52	0.344123
$611$	−8849.45	−0.585942
$612$	−4398.60	−0.290528
$613$	−15880.6	−1.04635	−0.523173	−	0.852226i	$-0.675252\pi$
$613$	−15880.6	−1.04635	−0.523173	+	0.852226i	$0.675252\pi$
$614$	8720.08	0.573149
$615$	−331.477	−0.0217341
$616$	21732.6	1.42148
$617$	−14389.8	−0.938919	−0.469459	−	0.882954i	$-0.655551\pi$
$617$	−14389.8	−0.938919	−0.469459	+	0.882954i	$0.655551\pi$
$618$	−868.078	−0.0565036
$619$	24824.3	1.61191	0.805956	−	0.591976i	$-0.201652\pi$
$619$	24824.3	1.61191	0.805956	+	0.591976i	$0.201652\pi$
$620$	−1923.27	−0.124582
$621$	2309.13	0.149214
$622$	975.038	0.0628545
$623$	−16830.6	−1.08235
$624$	−418.035	−0.0268186
$625$	6591.29	0.421843
$626$	−2269.28	−0.144886
$627$	−1233.75	−0.0785824
$628$	878.034	0.0557920
$629$	−1625.34	−0.103031
$630$	7731.81	0.488957
$631$	−30318.1	−1.91275	−0.956375	−	0.292141i	$-0.905632\pi$
$631$	−30318.1	−1.91275	−0.956375	+	0.292141i	$0.905632\pi$
$632$	6519.01	0.410304
$633$	578.921	0.0363508
$634$	11052.1	0.692329
$635$	−1131.58	−0.0707170
$636$	10.0310	0.000625399 0
$637$	30906.0	1.92235
$638$	−4418.83	−0.274206
$639$	20555.6	1.27256
$640$	−1149.37	−0.0709887
$641$	−18889.8	−1.16397	−0.581983	−	0.813201i	$-0.697723\pi$
$641$	−18889.8	−1.16397	−0.581983	+	0.813201i	$0.697723\pi$
$642$	933.662	0.0573967
$643$	18450.5	1.13159	0.565797	−	0.824544i	$-0.308568\pi$
$643$	18450.5	1.13159	0.565797	+	0.824544i	$0.308568\pi$
$644$	−16353.8	−1.00067
$645$	413.707	0.0252554
$646$	12852.6	0.782787
$647$	17829.7	1.08340	0.541698	−	0.840573i	$-0.317782\pi$
$647$	17829.7	1.08340	0.541698	+	0.840573i	$0.317782\pi$
$648$	17538.3	1.06322
$649$	13604.9	0.822862
$650$	15806.3	0.953807
$651$	736.555	0.0443439
$652$	10561.5	0.634387
$653$	14712.1	0.881670	0.440835	−	0.897588i	$-0.354682\pi$
$653$	14712.1	0.881670	0.440835	+	0.897588i	$0.354682\pi$
$654$	−488.714	−0.0292205
$655$	1963.87	0.117152
$656$	−5012.88	−0.298353
$657$	17308.9	1.02783
$658$	−6481.66	−0.384015
$659$	−12656.7	−0.748157	−0.374078	−	0.927397i	$-0.622041\pi$
$659$	−12656.7	−0.748157	−0.374078	+	0.927397i	$0.622041\pi$
$660$	−165.033	−0.00973321
$661$	−10471.2	−0.616161	−0.308081	−	0.951360i	$-0.599687\pi$
$661$	−10471.2	−0.616161	−0.308081	+	0.951360i	$0.599687\pi$
$662$	−4810.94	−0.282451
$663$	−898.942	−0.0526576
$664$	−17690.9	−1.03395
$665$	19619.4	1.14407
$666$	2061.76	0.119957
$667$	10479.4	0.608340
$668$	−1908.67	−0.110552
$669$	390.475	0.0225660
$670$	777.021	0.0448044
$671$	−16184.3	−0.931131
$672$	1098.94	0.0630838
$673$	−20416.8	−1.16941	−0.584703	−	0.811247i	$-0.698789\pi$
$673$	−20416.8	−1.16941	−0.584703	+	0.811247i	$0.698789\pi$
$674$	−7835.32	−0.447782
$675$	1419.57	0.0809470
$676$	−13912.2	−0.791545
$677$	21811.7	1.23824	0.619122	−	0.785295i	$-0.287489\pi$
$677$	21811.7	1.23824	0.619122	+	0.785295i	$0.287489\pi$
$678$	−201.158	−0.0113945
$679$	12541.0	0.708803
$680$	5418.24	0.305559
$681$	−969.633	−0.0545616
$682$	−6913.53	−0.388171
$683$	2811.45	0.157507	0.0787534	−	0.996894i	$-0.474906\pi$
$683$	2811.45	0.157507	0.0787534	+	0.996894i	$0.474906\pi$
$684$	14158.4	0.791460
$685$	13011.0	0.725731
$686$	3276.70	0.182369
$687$	−433.298	−0.0240631
$688$	6256.42	0.346692
$689$	−782.872	−0.0432875
$690$	−450.684	−0.0248656
$691$	9797.31	0.539374	0.269687	−	0.962948i	$-0.413080\pi$
$691$	9797.31	0.539374	0.269687	+	0.962948i	$0.413080\pi$
$692$	−806.316	−0.0442941
$693$	−24136.1	−1.32302
$694$	7962.88	0.435543
$695$	−1704.06	−0.0930053
$696$	−418.488	−0.0227913
$697$	−10779.7	−0.585810
$698$	19704.5	1.06852
$699$	−923.555	−0.0499743
$700$	−10053.7	−0.542851
$701$	34362.7	1.85144	0.925722	−	0.378206i	$-0.123459\pi$
$701$	34362.7	1.85144	0.925722	+	0.378206i	$0.123459\pi$
$702$	2283.62	0.122777
$703$	5231.70	0.280679
$704$	−15684.7	−0.839684
$705$	155.120	0.00828674
$706$	−7820.88	−0.416916
$707$	−51160.4	−2.72148
$708$	408.836	0.0217020
$709$	−15868.1	−0.840532	−0.420266	−	0.907401i	$-0.638063\pi$
$709$	−15868.1	−0.840532	−0.420266	+	0.907401i	$0.638063\pi$
$710$	−8034.41	−0.424685
$711$	−7239.97	−0.381885
$712$	14961.3	0.787498
$713$	16395.6	0.861179
$714$	−658.418	−0.0345107
$715$	12880.1	0.673691
$716$	−12052.7	−0.629092
$717$	−549.889	−0.0286415
$718$	−6934.12	−0.360417
$719$	−13004.8	−0.674545	−0.337273	−	0.941407i	$-0.609504\pi$
$719$	−13004.8	−0.674545	−0.337273	+	0.941407i	$0.609504\pi$
$720$	−2798.30	−0.144842
$721$	−43092.7	−2.22588
$722$	−27177.6	−1.40090
$723$	−1265.15	−0.0650782
$724$	4152.39	0.213152
$725$	6442.34	0.330017
$726$	138.125	0.00706104
$727$	6412.30	0.327124	0.163562	−	0.986533i	$-0.447702\pi$
$727$	6412.30	0.327124	0.163562	+	0.986533i	$0.447702\pi$
$728$	−50970.1	−2.59488
$729$	−19375.2	−0.984364
$730$	−6765.39	−0.343011
$731$	13453.8	0.680720
$732$	−486.351	−0.0245574
$733$	−21597.5	−1.08830	−0.544150	−	0.838988i	$-0.683148\pi$
$733$	−21597.5	−1.08830	−0.544150	+	0.838988i	$0.683148\pi$
$734$	−19529.4	−0.982077
$735$	−541.743	−0.0271871
$736$	24462.2	1.22512
$737$	−2425.60	−0.121232
$738$	13674.1	0.682048
$739$	16788.2	0.835677	0.417838	−	0.908521i	$-0.362788\pi$
$739$	16788.2	0.835677	0.417838	+	0.908521i	$0.362788\pi$
$740$	699.823	0.0347649
$741$	2893.54	0.143451
$742$	−573.404	−0.0283697
$743$	19087.7	0.942477	0.471238	−	0.882006i	$-0.343807\pi$
$743$	19087.7	0.942477	0.471238	+	0.882006i	$0.343807\pi$
$744$	−654.751	−0.0322639
$745$	7254.64	0.356764
$746$	−1467.72	−0.0720336
$747$	19647.4	0.962330
$748$	−5366.90	−0.262344
$749$	46348.4	2.26106
$750$	−626.454	−0.0304998
$751$	28722.3	1.39559	0.697796	−	0.716296i	$-0.254164\pi$
$751$	28722.3	1.39559	0.697796	+	0.716296i	$0.254164\pi$
$752$	2345.85	0.113756
$753$	680.144	0.0329161
$754$	10363.6	0.500557
$755$	6260.22	0.301765
$756$	−1452.51	−0.0698776
$757$	−11677.9	−0.560686	−0.280343	−	0.959900i	$-0.590448\pi$
$757$	−11677.9	−0.560686	−0.280343	+	0.959900i	$0.590448\pi$
$758$	−8136.94	−0.389904
$759$	1406.88	0.0672815
$760$	−17440.4	−0.832408
$761$	31156.4	1.48412	0.742062	−	0.670332i	$-0.233848\pi$
$761$	31156.4	1.48412	0.742062	+	0.670332i	$0.233848\pi$
$762$	−122.236	−0.00581120
$763$	−24260.5	−1.15110
$764$	3969.86	0.187990
$765$	−6017.46	−0.284395
$766$	−8784.94	−0.414377
$767$	−31907.8	−1.50212
$768$	−1138.25	−0.0534804
$769$	−10780.8	−0.505545	−0.252772	−	0.967526i	$-0.581342\pi$
$769$	−10780.8	−0.505545	−0.252772	+	0.967526i	$0.581342\pi$
$770$	9433.87	0.441524
$771$	−2060.03	−0.0962260
$772$	−2568.41	−0.119740
$773$	−31993.3	−1.48864	−0.744321	−	0.667822i	$-0.767227\pi$
$773$	−31993.3	−1.48864	−0.744321	+	0.667822i	$0.767227\pi$
$774$	−17066.3	−0.792551
$775$	10079.4	0.467180
$776$	−11148.1	−0.515714
$777$	−268.011	−0.0123743
$778$	−25757.6	−1.18696
$779$	34698.0	1.59587
$780$	387.057	0.0177678
$781$	25080.7	1.14912
$782$	−14656.3	−0.670215
$783$	930.758	0.0424809
$784$	−8192.68	−0.373209
$785$	1201.19	0.0546142
$786$	212.142	0.00962704
$787$	553.829	0.0250850	0.0125425	−	0.999921i	$-0.496008\pi$
$787$	553.829	0.0250850	0.0125425	+	0.999921i	$0.496008\pi$
$788$	−11610.5	−0.524881
$789$	1844.21	0.0832135
$790$	2829.83	0.127444
$791$	−9985.81	−0.448868
$792$	21455.5	0.962609
$793$	37957.5	1.69976
$794$	14571.7	0.651296
$795$	13.7228	0.000612197 0
$796$	10473.1	0.466341
$797$	−18370.4	−0.816452	−0.408226	−	0.912881i	$-0.633852\pi$
$797$	−18370.4	−0.816452	−0.408226	+	0.912881i	$0.633852\pi$
$798$	2119.34	0.0940146
$799$	5044.51	0.223357
$800$	15038.5	0.664613
$801$	−16615.9	−0.732952
$802$	10807.7	0.475850
$803$	21119.3	0.928123
$804$	−72.8910	−0.00319735
$805$	−22372.7	−0.979544
$806$	16214.5	0.708599
$807$	246.294	0.0107434
$808$	45478.4	1.98010
$809$	−4035.41	−0.175374	−0.0876870	−	0.996148i	$-0.527948\pi$
$809$	−4035.41	−0.175374	−0.0876870	+	0.996148i	$0.527948\pi$
$810$	7613.17	0.330246
$811$	11553.5	0.500244	0.250122	−	0.968214i	$-0.419529\pi$
$811$	11553.5	0.500244	0.250122	+	0.968214i	$0.419529\pi$
$812$	−6591.86	−0.284888
$813$	58.8461	0.00253853
$814$	2515.63	0.108320
$815$	14448.6	0.620995
$816$	238.295	0.0102230
$817$	−43305.5	−1.85443
$818$	−3069.17	−0.131187
$819$	56607.0	2.41515
$820$	4641.40	0.197664
$821$	−6940.51	−0.295037	−0.147519	−	0.989059i	$-0.547129\pi$
$821$	−6940.51	−0.295037	−0.147519	+	0.989059i	$0.547129\pi$
$822$	1405.48	0.0596373
$823$	−41952.7	−1.77689	−0.888445	−	0.458984i	$-0.848214\pi$
$823$	−41952.7	−1.77689	−0.888445	+	0.458984i	$0.848214\pi$
$824$	38306.7	1.61951
$825$	864.903	0.0364995
$826$	−23370.5	−0.984458
$827$	−13167.8	−0.553676	−0.276838	−	0.960917i	$-0.589287\pi$
$827$	−13167.8	−0.553676	−0.276838	+	0.960917i	$0.589287\pi$
$828$	−16145.3	−0.677641
$829$	4454.34	0.186617	0.0933086	−	0.995637i	$-0.470256\pi$
$829$	4454.34	0.186617	0.0933086	+	0.995637i	$0.470256\pi$
$830$	−7679.41	−0.321152
$831$	−1741.08	−0.0726804
$832$	36785.6	1.53283
$833$	−17617.5	−0.732786
$834$	−184.077	−0.00764275
$835$	−2611.14	−0.108218
$836$	17275.2	0.714682
$837$	1456.23	0.0601369
$838$	22967.1	0.946762
$839$	22423.9	0.922717	0.461359	−	0.887214i	$-0.347362\pi$
$839$	22423.9	0.922717	0.461359	+	0.887214i	$0.347362\pi$
$840$	893.441	0.0366984
$841$	−20165.0	−0.826807
$842$	−11374.8	−0.465560
$843$	−2010.83	−0.0821550
$844$	−8106.15	−0.330598
$845$	−19032.4	−0.774835
$846$	−6399.01	−0.260050
$847$	6856.76	0.278159
$848$	207.527	0.00840390
$849$	−166.051	−0.00671244
$850$	−9010.17	−0.363584
$851$	−5965.88	−0.240315
$852$	753.695	0.0303065
$853$	−8503.87	−0.341345	−0.170672	−	0.985328i	$-0.554594\pi$
$853$	−8503.87	−0.341345	−0.170672	+	0.985328i	$0.554594\pi$
$854$	27801.5	1.11399
$855$	19369.2	0.774752
$856$	−41200.8	−1.64511
$857$	16403.6	0.653834	0.326917	−	0.945053i	$-0.393990\pi$
$857$	16403.6	0.653834	0.326917	+	0.945053i	$0.393990\pi$
$858$	1391.34	0.0553609
$859$	−10948.7	−0.434885	−0.217442	−	0.976073i	$-0.569771\pi$
$859$	−10948.7	−0.434885	−0.217442	+	0.976073i	$0.569771\pi$
$860$	−5792.80	−0.229689
$861$	−1777.52	−0.0703572
$862$	−23674.8	−0.935462
$863$	−4652.56	−0.183517	−0.0917585	−	0.995781i	$-0.529249\pi$
$863$	−4652.56	−0.183517	−0.0917585	+	0.995781i	$0.529249\pi$
$864$	2172.68	0.0855511
$865$	−1103.07	−0.0433591
$866$	−19112.9	−0.749982
$867$	−792.224	−0.0310327
$868$	−10313.4	−0.403293
$869$	−8833.77	−0.344839
$870$	−181.661	−0.00707917
$871$	5688.81	0.221307
$872$	21566.1	0.837522
$873$	12381.0	0.479993
$874$	47176.2	1.82581
$875$	−31098.1	−1.20150
$876$	634.649	0.0244781
$877$	13397.1	0.515836	0.257918	−	0.966167i	$-0.416964\pi$
$877$	13397.1	0.515836	0.257918	+	0.966167i	$0.416964\pi$
$878$	11822.7	0.454439
$879$	2283.48	0.0876221
$880$	−3414.32	−0.130792
$881$	5858.92	0.224054	0.112027	−	0.993705i	$-0.464266\pi$
$881$	5858.92	0.224054	0.112027	+	0.993705i	$0.464266\pi$
$882$	22348.0	0.853170
$883$	23797.6	0.906968	0.453484	−	0.891264i	$-0.350181\pi$
$883$	23797.6	0.906968	0.453484	+	0.891264i	$0.350181\pi$
$884$	12587.1	0.478904
$885$	559.304	0.0212438
$886$	−21647.1	−0.820823
$887$	−13556.5	−0.513173	−0.256586	−	0.966521i	$-0.582598\pi$
$887$	−13556.5	−0.513173	−0.256586	+	0.966521i	$0.582598\pi$
$888$	238.245	0.00900334
$889$	−6067.97	−0.228924
$890$	6494.53	0.244603
$891$	−23765.7	−0.893583
$892$	−5467.50	−0.205230
$893$	−16237.4	−0.608471
$894$	783.664	0.0293173
$895$	−16488.5	−0.615812
$896$	−6163.37	−0.229803
$897$	−3299.60	−0.122821
$898$	−28108.7	−1.04454
$899$	6608.71	0.245176
$900$	−9925.53	−0.367612
$901$	446.266	0.0165008
$902$	16684.3	0.615883
$903$	2218.46	0.0817562
$904$	8876.75	0.326589
$905$	5680.63	0.208653
$906$	676.244	0.0247977
$907$	17705.9	0.648197	0.324099	−	0.946023i	$-0.394939\pi$
$907$	17705.9	0.648197	0.324099	+	0.946023i	$0.394939\pi$
$908$	13577.0	0.496220
$909$	−50508.0	−1.84295
$910$	−22125.5	−0.805993
$911$	−6846.11	−0.248981	−0.124490	−	0.992221i	$-0.539730\pi$
$911$	−6846.11	−0.248981	−0.124490	+	0.992221i	$0.539730\pi$
$912$	−767.032	−0.0278497
$913$	23972.5	0.868976
$914$	1350.19	0.0488627
$915$	−665.348	−0.0240390
$916$	6067.12	0.218846
$917$	10531.1	0.379243
$918$	−1301.74	−0.0468017
$919$	5031.25	0.180594	0.0902969	−	0.995915i	$-0.471218\pi$
$919$	5031.25	0.180594	0.0902969	+	0.995915i	$0.471218\pi$
$920$	19887.9	0.712700
$921$	−1119.08	−0.0400379
$922$	−7692.68	−0.274778
$923$	−58822.5	−2.09769
$924$	−884.976	−0.0315082
$925$	−3667.61	−0.130368
$926$	27155.7	0.963707
$927$	−42543.2	−1.50734
$928$	9860.15	0.348788
$929$	3932.03	0.138865	0.0694326	−	0.997587i	$-0.477881\pi$
$929$	3932.03	0.138865	0.0694326	+	0.997587i	$0.477881\pi$
$930$	−284.220	−0.0100214
$931$	56707.8	1.99627
$932$	12931.8	0.454500
$933$	−125.130	−0.00439076
$934$	−4497.64	−0.157567
$935$	−7342.14	−0.256806
$936$	−50320.1	−1.75722
$937$	24197.2	0.843637	0.421818	−	0.906680i	$-0.361392\pi$
$937$	24197.2	0.843637	0.421818	+	0.906680i	$0.361392\pi$
$938$	4166.70	0.145040
$939$	291.225	0.0101212
$940$	−2172.01	−0.0753652
$941$	−23077.8	−0.799485	−0.399742	−	0.916628i	$-0.630900\pi$
$941$	−23077.8	−0.799485	−0.399742	+	0.916628i	$0.630900\pi$
$942$	129.755	0.00448795
$943$	−39567.3	−1.36637
$944$	8458.26	0.291624
$945$	−1987.10	−0.0684024
$946$	−20823.2	−0.715667
$947$	−51851.4	−1.77924	−0.889622	−	0.456697i	$-0.849032\pi$
$947$	−51851.4	−1.77924	−0.889622	+	0.456697i	$0.849032\pi$
$948$	−265.461	−0.00909471
$949$	−49531.5	−1.69427
$950$	29002.2	0.990480
$951$	−1418.36	−0.0483632
$952$	29054.8	0.989150
$953$	−2357.75	−0.0801417	−0.0400708	−	0.999197i	$-0.512758\pi$
$953$	−2357.75	−0.0801417	−0.0400708	+	0.999197i	$0.512758\pi$
$954$	−566.092	−0.0192116
$955$	5430.92	0.184022
$956$	7699.64	0.260485
$957$	567.084	0.0191549
$958$	−13736.0	−0.463246
$959$	69770.3	2.34932
$960$	−644.806	−0.0216782
$961$	−19451.3	−0.652924
$962$	−5899.98	−0.197737
$963$	45757.4	1.53116
$964$	17714.9	0.591865
$965$	−3513.69	−0.117212
$966$	−2416.75	−0.0804945
$967$	10985.1	0.365313	0.182656	−	0.983177i	$-0.441530\pi$
$967$	10985.1	0.365313	0.182656	+	0.983177i	$0.441530\pi$
$968$	−6095.23	−0.202384
$969$	−1649.42	−0.0546823
$970$	−4839.27	−0.160185
$971$	24081.2	0.795882	0.397941	−	0.917411i	$-0.369725\pi$
$971$	24081.2	0.795882	0.397941	+	0.917411i	$0.369725\pi$
$972$	−2151.92	−0.0710113
$973$	−9137.86	−0.301075
$974$	8672.00	0.285286
$975$	−2028.48	−0.0666291
$976$	−10061.9	−0.329995
$977$	−5406.02	−0.177025	−0.0885127	−	0.996075i	$-0.528211\pi$
$977$	−5406.02	−0.177025	−0.0885127	+	0.996075i	$0.528211\pi$
$978$	1560.77	0.0510305
$979$	−20273.7	−0.661850
$980$	7585.58	0.247258
$981$	−23951.1	−0.779511
$982$	−4343.89	−0.141160
$983$	−16592.3	−0.538364	−0.269182	−	0.963089i	$-0.586753\pi$
$983$	−16592.3	−0.538364	−0.269182	+	0.963089i	$0.586753\pi$
$984$	1580.10	0.0511907
$985$	−15883.6	−0.513801
$986$	−5907.63	−0.190809
$987$	831.815	0.0268257
$988$	−40515.9	−1.30464
$989$	49382.7	1.58774
$990$	9313.57	0.298995
$991$	8748.89	0.280442	0.140221	−	0.990120i	$-0.455219\pi$
$991$	8748.89	0.280442	0.140221	+	0.990120i	$0.455219\pi$
$992$	15426.8	0.493752
$993$	617.405	0.0197309
$994$	−43083.7	−1.37478
$995$	14327.6	0.456497
$996$	720.393	0.0229182
$997$	56755.3	1.80287	0.901434	−	0.432917i	$-0.142516\pi$
$997$	56755.3	1.80287	0.901434	+	0.432917i	$0.142516\pi$
$998$	22161.7	0.702921
$999$	−529.879	−0.0167814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.a.a.1.2	✓	4
3.2	odd	2		333.4.a.e.1.3		4
4.3	odd	2		592.4.a.f.1.2		4
5.4	even	2		925.4.a.a.1.3		4
7.6	odd	2		1813.4.a.b.1.2		4
8.3	odd	2		2368.4.a.g.1.3		4
8.5	even	2		2368.4.a.l.1.2		4
37.36	even	2		1369.4.a.c.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.a.a.1.2	✓	4	1.1	even	1	trivial
333.4.a.e.1.3		4	3.2	odd	2
592.4.a.f.1.2		4	4.3	odd	2
925.4.a.a.1.3		4	5.4	even	2
1369.4.a.c.1.3		4	37.36	even	2
1813.4.a.b.1.2		4	7.6	odd	2
2368.4.a.g.1.3		4	8.3	odd	2
2368.4.a.l.1.2		4	8.5	even	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)

\(f(q)\)	\(=\)	\(q-2.06923 q^{2} +0.265551 q^{3} -3.71830 q^{4} -5.08678 q^{5} -0.549486 q^{6} -27.2773 q^{7} +24.2478 q^{8} -26.9295 q^{9} +O(q^{10})\)	\(q-2.06923 q^{2} +0.265551 q^{3} -3.71830 q^{4} -5.08678 q^{5} -0.549486 q^{6} -27.2773 q^{7} +24.2478 q^{8} -26.9295 q^{9} +10.5257 q^{10} -32.8577 q^{11} -0.987398 q^{12} +77.0620 q^{13} +56.4430 q^{14} -1.35080 q^{15} -20.4279 q^{16} -43.9281 q^{17} +55.7232 q^{18} +141.397 q^{19} +18.9141 q^{20} -7.24353 q^{21} +67.9901 q^{22} -161.240 q^{23} +6.43904 q^{24} -99.1247 q^{25} -159.459 q^{26} -14.3210 q^{27} +101.425 q^{28} -64.9923 q^{29} +2.79511 q^{30} -101.684 q^{31} -151.713 q^{32} -8.72540 q^{33} +90.8973 q^{34} +138.754 q^{35} +100.132 q^{36} +37.0000 q^{37} -292.583 q^{38} +20.4639 q^{39} -123.343 q^{40} +245.393 q^{41} +14.9885 q^{42} -306.268 q^{43} +122.175 q^{44} +136.984 q^{45} +333.643 q^{46} -114.835 q^{47} -5.42466 q^{48} +401.053 q^{49} +205.112 q^{50} -11.6652 q^{51} -286.539 q^{52} -10.1590 q^{53} +29.6335 q^{54} +167.140 q^{55} -661.416 q^{56} +37.5482 q^{57} +134.484 q^{58} -414.054 q^{59} +5.02267 q^{60} +492.558 q^{61} +210.408 q^{62} +734.565 q^{63} +477.351 q^{64} -391.997 q^{65} +18.0548 q^{66} +73.8213 q^{67} +163.338 q^{68} -42.8175 q^{69} -287.113 q^{70} -763.314 q^{71} -652.981 q^{72} -642.749 q^{73} -76.5614 q^{74} -26.3227 q^{75} -525.757 q^{76} +896.270 q^{77} -42.3445 q^{78} +268.849 q^{79} +103.912 q^{80} +723.293 q^{81} -507.775 q^{82} -729.587 q^{83} +26.9336 q^{84} +223.453 q^{85} +633.739 q^{86} -17.2588 q^{87} -796.728 q^{88} +617.016 q^{89} -283.452 q^{90} -2102.05 q^{91} +599.538 q^{92} -27.0024 q^{93} +237.621 q^{94} -719.256 q^{95} -40.2875 q^{96} -459.757 q^{97} -829.871 q^{98} +884.841 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} - 11 q^{3} + 6 q^{4} - 29 q^{5} - 27 q^{6} - 32 q^{7} - 90 q^{8} + q^{9}+O(q^{10}) \) 4 * q - 6 * q^2 - 11 * q^3 + 6 * q^4 - 29 * q^5 - 27 * q^6 - 32 * q^7 - 90 * q^8 + q^9	\( 4 q - 6 q^{2} - 11 q^{3} + 6 q^{4} - 29 q^{5} - 27 q^{6} - 32 q^{7} - 90 q^{8} + q^{9} + 81 q^{10} - 11 q^{11} + 143 q^{12} - 45 q^{13} + 82 q^{14} - 16 q^{15} + 378 q^{16} - 56 q^{17} + 255 q^{18} - 144 q^{19} - 165 q^{20} + 108 q^{21} + 73 q^{22} - 275 q^{23} + 9 q^{24} + 91 q^{25} + 97 q^{26} - 272 q^{27} - 202 q^{28} - 29 q^{29} + 96 q^{30} + 111 q^{31} - 946 q^{32} - 250 q^{33} + 212 q^{34} - 636 q^{35} - 723 q^{36} + 148 q^{37} - 76 q^{38} + 361 q^{39} + 937 q^{40} - 9 q^{41} + 446 q^{42} - 190 q^{43} + 211 q^{44} + 1018 q^{45} - 269 q^{46} - 472 q^{47} + 203 q^{48} + 1586 q^{49} + 83 q^{50} - 94 q^{51} - 1165 q^{52} + 318 q^{53} + 306 q^{54} + 779 q^{55} + 518 q^{56} + 1330 q^{57} - 915 q^{58} - 1530 q^{59} - 840 q^{60} - 31 q^{61} - 747 q^{62} - 1190 q^{63} + 3490 q^{64} + 570 q^{65} - 484 q^{66} - 5 q^{67} - 404 q^{68} + 1535 q^{69} - 1468 q^{70} + 390 q^{71} + 255 q^{72} - 967 q^{73} - 222 q^{74} - 320 q^{75} - 708 q^{76} - 1050 q^{77} + 1163 q^{78} + 17 q^{79} - 4653 q^{80} + 1888 q^{81} - 153 q^{82} - 3138 q^{83} - 1078 q^{84} + 302 q^{85} + 1118 q^{86} + 1025 q^{87} - 1931 q^{88} + 224 q^{89} - 1146 q^{90} - 1470 q^{91} + 3625 q^{92} - 458 q^{93} + 1666 q^{94} + 1382 q^{95} - 4391 q^{96} - 2532 q^{97} + 1722 q^{98} + 3166 q^{99}+O(q^{100}) \) 4 * q - 6 * q^2 - 11 * q^3 + 6 * q^4 - 29 * q^5 - 27 * q^6 - 32 * q^7 - 90 * q^8 + q^9 + 81 * q^10 - 11 * q^11 + 143 * q^12 - 45 * q^13 + 82 * q^14 - 16 * q^15 + 378 * q^16 - 56 * q^17 + 255 * q^18 - 144 * q^19 - 165 * q^20 + 108 * q^21 + 73 * q^22 - 275 * q^23 + 9 * q^24 + 91 * q^25 + 97 * q^26 - 272 * q^27 - 202 * q^28 - 29 * q^29 + 96 * q^30 + 111 * q^31 - 946 * q^32 - 250 * q^33 + 212 * q^34 - 636 * q^35 - 723 * q^36 + 148 * q^37 - 76 * q^38 + 361 * q^39 + 937 * q^40 - 9 * q^41 + 446 * q^42 - 190 * q^43 + 211 * q^44 + 1018 * q^45 - 269 * q^46 - 472 * q^47 + 203 * q^48 + 1586 * q^49 + 83 * q^50 - 94 * q^51 - 1165 * q^52 + 318 * q^53 + 306 * q^54 + 779 * q^55 + 518 * q^56 + 1330 * q^57 - 915 * q^58 - 1530 * q^59 - 840 * q^60 - 31 * q^61 - 747 * q^62 - 1190 * q^63 + 3490 * q^64 + 570 * q^65 - 484 * q^66 - 5 * q^67 - 404 * q^68 + 1535 * q^69 - 1468 * q^70 + 390 * q^71 + 255 * q^72 - 967 * q^73 - 222 * q^74 - 320 * q^75 - 708 * q^76 - 1050 * q^77 + 1163 * q^78 + 17 * q^79 - 4653 * q^80 + 1888 * q^81 - 153 * q^82 - 3138 * q^83 - 1078 * q^84 + 302 * q^85 + 1118 * q^86 + 1025 * q^87 - 1931 * q^88 + 224 * q^89 - 1146 * q^90 - 1470 * q^91 + 3625 * q^92 - 458 * q^93 + 1666 * q^94 + 1382 * q^95 - 4391 * q^96 - 2532 * q^97 + 1722 * q^98 + 3166 * q^99

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