LMFDB - Embedded newform 177.4.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,4,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("177.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	177.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:	$10.4433380710$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 $ x^8 - 2x^7 - 49x^6 + 89x^5 + 648x^4 - 1023x^3 - 1476x^2 + 1940*x - 384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.780043$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.780043 q^{2} -3.00000 q^{3} -7.39153 q^{4} -21.9196 q^{5} -2.34013 q^{6} +32.1153 q^{7} -12.0061 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+0.780043 q^{2} -3.00000 q^{3} -7.39153 q^{4} -21.9196 q^{5} -2.34013 q^{6} +32.1153 q^{7} -12.0061 q^{8} +9.00000 q^{9} -17.0983 q^{10} -35.3089 q^{11} +22.1746 q^{12} +34.5155 q^{13} +25.0513 q^{14} +65.7589 q^{15} +49.7670 q^{16} -27.9810 q^{17} +7.02038 q^{18} +74.3473 q^{19} +162.020 q^{20} -96.3458 q^{21} -27.5424 q^{22} +128.081 q^{23} +36.0182 q^{24} +355.471 q^{25} +26.9235 q^{26} -27.0000 q^{27} -237.381 q^{28} +42.7099 q^{29} +51.2948 q^{30} -211.536 q^{31} +134.869 q^{32} +105.927 q^{33} -21.8263 q^{34} -703.955 q^{35} -66.5238 q^{36} -165.794 q^{37} +57.9940 q^{38} -103.546 q^{39} +263.168 q^{40} +377.795 q^{41} -75.1538 q^{42} -393.343 q^{43} +260.987 q^{44} -197.277 q^{45} +99.9085 q^{46} +261.503 q^{47} -149.301 q^{48} +688.391 q^{49} +277.282 q^{50} +83.9429 q^{51} -255.122 q^{52} -113.158 q^{53} -21.0611 q^{54} +773.957 q^{55} -385.578 q^{56} -223.042 q^{57} +33.3155 q^{58} -59.0000 q^{59} -486.059 q^{60} +337.123 q^{61} -165.007 q^{62} +289.037 q^{63} -292.933 q^{64} -756.567 q^{65} +82.6272 q^{66} +183.638 q^{67} +206.822 q^{68} -384.242 q^{69} -549.115 q^{70} -168.306 q^{71} -108.054 q^{72} +805.659 q^{73} -129.326 q^{74} -1066.41 q^{75} -549.540 q^{76} -1133.95 q^{77} -80.7706 q^{78} +797.644 q^{79} -1090.88 q^{80} +81.0000 q^{81} +294.696 q^{82} +251.649 q^{83} +712.143 q^{84} +613.333 q^{85} -306.824 q^{86} -128.130 q^{87} +423.920 q^{88} +653.582 q^{89} -153.884 q^{90} +1108.47 q^{91} -946.713 q^{92} +634.607 q^{93} +203.983 q^{94} -1629.67 q^{95} -404.606 q^{96} -1574.87 q^{97} +536.974 q^{98} -317.780 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) $ 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9} + 29 q^{10} - 27 q^{11} - 114 q^{12} + 89 q^{13} - 37 q^{14} + 36 q^{15} + 362 q^{16} + 79 q^{17} + 18 q^{18} + 288 q^{19} + 457 q^{20} - 159 q^{21} + 596 q^{22} + 202 q^{23} - 9 q^{24} + 264 q^{25} + 270 q^{26} - 216 q^{27} + 702 q^{28} - 114 q^{29} - 87 q^{30} + 538 q^{31} + 316 q^{32} + 81 q^{33} + 498 q^{34} - 196 q^{35} + 342 q^{36} + 395 q^{37} + 397 q^{38} - 267 q^{39} + 918 q^{40} - 39 q^{41} + 111 q^{42} + 527 q^{43} + 64 q^{44} - 108 q^{45} - 539 q^{46} + 860 q^{47} - 1086 q^{48} + 347 q^{49} - 591 q^{50} - 237 q^{51} - 644 q^{52} - 812 q^{53} - 54 q^{54} + 536 q^{55} - 2218 q^{56} - 864 q^{57} - 1154 q^{58} - 472 q^{59} - 1371 q^{60} - 460 q^{61} - 2014 q^{62} + 477 q^{63} - 451 q^{64} - 986 q^{65} - 1788 q^{66} + 1934 q^{67} - 69 q^{68} - 606 q^{69} - 1028 q^{70} - 1687 q^{71} + 27 q^{72} + 1980 q^{73} - 2400 q^{74} - 792 q^{75} - 940 q^{76} - 821 q^{77} - 810 q^{78} + 3319 q^{79} - 2119 q^{80} + 648 q^{81} + 429 q^{82} + 2057 q^{83} - 2106 q^{84} + 566 q^{85} - 6690 q^{86} + 342 q^{87} + 1189 q^{88} + 1668 q^{89} + 261 q^{90} + 2427 q^{91} - 980 q^{92} - 1614 q^{93} + 332 q^{94} + 2146 q^{95} - 948 q^{96} + 1956 q^{97} - 2026 q^{98} - 243 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9 + 29 * q^10 - 27 * q^11 - 114 * q^12 + 89 * q^13 - 37 * q^14 + 36 * q^15 + 362 * q^16 + 79 * q^17 + 18 * q^18 + 288 * q^19 + 457 * q^20 - 159 * q^21 + 596 * q^22 + 202 * q^23 - 9 * q^24 + 264 * q^25 + 270 * q^26 - 216 * q^27 + 702 * q^28 - 114 * q^29 - 87 * q^30 + 538 * q^31 + 316 * q^32 + 81 * q^33 + 498 * q^34 - 196 * q^35 + 342 * q^36 + 395 * q^37 + 397 * q^38 - 267 * q^39 + 918 * q^40 - 39 * q^41 + 111 * q^42 + 527 * q^43 + 64 * q^44 - 108 * q^45 - 539 * q^46 + 860 * q^47 - 1086 * q^48 + 347 * q^49 - 591 * q^50 - 237 * q^51 - 644 * q^52 - 812 * q^53 - 54 * q^54 + 536 * q^55 - 2218 * q^56 - 864 * q^57 - 1154 * q^58 - 472 * q^59 - 1371 * q^60 - 460 * q^61 - 2014 * q^62 + 477 * q^63 - 451 * q^64 - 986 * q^65 - 1788 * q^66 + 1934 * q^67 - 69 * q^68 - 606 * q^69 - 1028 * q^70 - 1687 * q^71 + 27 * q^72 + 1980 * q^73 - 2400 * q^74 - 792 * q^75 - 940 * q^76 - 821 * q^77 - 810 * q^78 + 3319 * q^79 - 2119 * q^80 + 648 * q^81 + 429 * q^82 + 2057 * q^83 - 2106 * q^84 + 566 * q^85 - 6690 * q^86 + 342 * q^87 + 1189 * q^88 + 1668 * q^89 + 261 * q^90 + 2427 * q^91 - 980 * q^92 - 1614 * q^93 + 332 * q^94 + 2146 * q^95 - 948 * q^96 + 1956 * q^97 - 2026 * q^98 - 243 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.780043	0.275787	0.137893	−	0.990447i	$-0.455967\pi$
$2$	0.780043	0.275787	0.137893	+	0.990447i	$0.455967\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.39153	−0.923942
$5$	−21.9196	−1.96055	−0.980276	−	0.197633i	$-0.936675\pi$
$5$	−21.9196	−1.96055	−0.980276	+	0.197633i	$0.936675\pi$
$6$	−2.34013	−0.159226
$7$	32.1153	1.73406	0.867031	−	0.498255i	$-0.166026\pi$
$7$	32.1153	1.73406	0.867031	+	0.498255i	$0.166026\pi$
$8$	−12.0061	−0.530598
$9$	9.00000	0.333333
$10$	−17.0983	−0.540694
$11$	−35.3089	−0.967820	−0.483910	−	0.875118i	$-0.660784\pi$
$11$	−35.3089	−0.967820	−0.483910	+	0.875118i	$0.660784\pi$
$12$	22.1746	0.533438
$13$	34.5155	0.736375	0.368187	−	0.929752i	$-0.379978\pi$
$13$	34.5155	0.736375	0.368187	+	0.929752i	$0.379978\pi$
$14$	25.0513	0.478231
$15$	65.7589	1.13193
$16$	49.7670	0.777610
$17$	−27.9810	−0.399199	−0.199599	−	0.979878i	$-0.563964\pi$
$17$	−27.9810	−0.399199	−0.199599	+	0.979878i	$0.563964\pi$
$18$	7.02038	0.0919289
$19$	74.3473	0.897707	0.448853	−	0.893605i	$-0.351833\pi$
$19$	74.3473	0.897707	0.448853	+	0.893605i	$0.351833\pi$
$20$	162.020	1.81144
$21$	−96.3458	−1.00116
$22$	−27.5424	−0.266912
$23$	128.081	1.16116	0.580580	−	0.814203i	$-0.302826\pi$
$23$	128.081	1.16116	0.580580	+	0.814203i	$0.302826\pi$
$24$	36.0182	0.306341
$25$	355.471	2.84377
$26$	26.9235	0.203082
$27$	−27.0000	−0.192450
$28$	−237.381	−1.60217
$29$	42.7099	0.273484	0.136742	−	0.990607i	$-0.456337\pi$
$29$	42.7099	0.273484	0.136742	+	0.990607i	$0.456337\pi$
$30$	51.2948	0.312170
$31$	−211.536	−1.22558	−0.612790	−	0.790246i	$-0.709953\pi$
$31$	−211.536	−1.22558	−0.612790	+	0.790246i	$0.709953\pi$
$32$	134.869	0.745052
$33$	105.927	0.558771
$34$	−21.8263	−0.110094
$35$	−703.955	−3.39972
$36$	−66.5238	−0.307981
$37$	−165.794	−0.736658	−0.368329	−	0.929695i	$-0.620070\pi$
$37$	−165.794	−0.736658	−0.368329	+	0.929695i	$0.620070\pi$
$38$	57.9940	0.247576
$39$	−103.546	−0.425146
$40$	263.168	1.04026
$41$	377.795	1.43906	0.719532	−	0.694459i	$-0.244356\pi$
$41$	377.795	1.43906	0.719532	+	0.694459i	$0.244356\pi$
$42$	−75.1538	−0.276107
$43$	−393.343	−1.39498	−0.697491	−	0.716594i	$-0.745700\pi$
$43$	−393.343	−1.39498	−0.697491	+	0.716594i	$0.745700\pi$
$44$	260.987	0.894209
$45$	−197.277	−0.653517
$46$	99.9085	0.320233
$47$	261.503	0.811577	0.405789	−	0.913967i	$-0.366997\pi$
$47$	261.503	0.811577	0.405789	+	0.913967i	$0.366997\pi$
$48$	−149.301	−0.448953
$49$	688.391	2.00697
$50$	277.282	0.784273
$51$	83.9429	0.230478
$52$	−255.122	−0.680367
$53$	−113.158	−0.293273	−0.146636	−	0.989190i	$-0.546845\pi$
$53$	−113.158	−0.293273	−0.146636	+	0.989190i	$0.546845\pi$
$54$	−21.0611	−0.0530752
$55$	773.957	1.89746
$56$	−385.578	−0.920089
$57$	−223.042	−0.518291
$58$	33.3155	0.0754232
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	−486.059	−1.04583
$61$	337.123	0.707610	0.353805	−	0.935319i	$-0.384888\pi$
$61$	337.123	0.707610	0.353805	+	0.935319i	$0.384888\pi$
$62$	−165.007	−0.337998
$63$	289.037	0.578021
$64$	−292.933	−0.572135
$65$	−756.567	−1.44370
$66$	82.6272	0.154102
$67$	183.638	0.334850	0.167425	−	0.985885i	$-0.446455\pi$
$67$	183.638	0.334850	0.167425	+	0.985885i	$0.446455\pi$
$68$	206.822	0.368837
$69$	−384.242	−0.670396
$70$	−549.115	−0.937597
$71$	−168.306	−0.281327	−0.140664	−	0.990057i	$-0.544924\pi$
$71$	−168.306	−0.281327	−0.140664	+	0.990057i	$0.544924\pi$
$72$	−108.054	−0.176866
$73$	805.659	1.29172	0.645858	−	0.763457i	$-0.276500\pi$
$73$	805.659	1.29172	0.645858	+	0.763457i	$0.276500\pi$
$74$	−129.326	−0.203161
$75$	−1066.41	−1.64185
$76$	−549.540	−0.829429
$77$	−1133.95	−1.67826
$78$	−80.7706	−0.117250
$79$	797.644	1.13597	0.567987	−	0.823038i	$-0.307722\pi$
$79$	797.644	1.13597	0.567987	+	0.823038i	$0.307722\pi$
$80$	−1090.88	−1.52454
$81$	81.0000	0.111111
$82$	294.696	0.396875
$83$	251.649	0.332796	0.166398	−	0.986059i	$-0.446786\pi$
$83$	251.649	0.332796	0.166398	+	0.986059i	$0.446786\pi$
$84$	712.143	0.925014
$85$	613.333	0.782650
$86$	−306.824	−0.384717
$87$	−128.130	−0.157896
$88$	423.920	0.513523
$89$	653.582	0.778422	0.389211	−	0.921149i	$-0.372748\pi$
$89$	653.582	0.778422	0.389211	+	0.921149i	$0.372748\pi$
$90$	−153.884	−0.180231
$91$	1108.47	1.27692
$92$	−946.713	−1.07284
$93$	634.607	0.707589
$94$	203.983	0.223822
$95$	−1629.67	−1.76000
$96$	−404.606	−0.430156
$97$	−1574.87	−1.64849	−0.824246	−	0.566231i	$-0.808401\pi$
$97$	−1574.87	−1.64849	−0.824246	+	0.566231i	$0.808401\pi$
$98$	536.974	0.553495
$99$	−317.780	−0.322607
$100$	−2627.47	−2.62747
$101$	1309.13	1.28973	0.644867	−	0.764295i	$-0.276913\pi$
$101$	1309.13	1.28973	0.644867	+	0.764295i	$0.276913\pi$
$102$	65.4790	0.0635627
$103$	855.586	0.818479	0.409240	−	0.912427i	$-0.365794\pi$
$103$	855.586	0.818479	0.409240	+	0.912427i	$0.365794\pi$
$104$	−414.395	−0.390719
$105$	2111.87	1.96283
$106$	−88.2681	−0.0808807
$107$	661.333	0.597509	0.298755	−	0.954330i	$-0.403429\pi$
$107$	661.333	0.597509	0.298755	+	0.954330i	$0.403429\pi$
$108$	199.571	0.177813
$109$	979.622	0.860832	0.430416	−	0.902631i	$-0.358367\pi$
$109$	979.622	0.860832	0.430416	+	0.902631i	$0.358367\pi$
$110$	603.720	0.523295
$111$	497.382	0.425310
$112$	1598.28	1.34842
$113$	1814.46	1.51053	0.755267	−	0.655417i	$-0.227507\pi$
$113$	1814.46	1.51053	0.755267	+	0.655417i	$0.227507\pi$
$114$	−173.982	−0.142938
$115$	−2807.48	−2.27652
$116$	−315.692	−0.252683
$117$	310.639	0.245458
$118$	−46.0225	−0.0359044
$119$	−898.616	−0.692235
$120$	−789.505	−0.600597
$121$	−84.2847	−0.0633244
$122$	262.970	0.195149
$123$	−1133.38	−0.830844
$124$	1563.57	1.13236
$125$	−5051.83	−3.61480
$126$	225.461	0.159410
$127$	−2057.03	−1.43726	−0.718629	−	0.695393i	$-0.755230\pi$
$127$	−2057.03	−1.43726	−0.718629	+	0.695393i	$0.755230\pi$
$128$	−1307.45	−0.902839
$129$	1180.03	0.805393
$130$	−590.154	−0.398154
$131$	2007.37	1.33881	0.669407	−	0.742896i	$-0.266548\pi$
$131$	2007.37	1.33881	0.669407	+	0.742896i	$0.266548\pi$
$132$	−782.960	−0.516272
$133$	2387.68	1.55668
$134$	143.245	0.0923471
$135$	591.830	0.377308
$136$	335.941	0.211814
$137$	−1424.64	−0.888434	−0.444217	−	0.895919i	$-0.646518\pi$
$137$	−1424.64	−0.888434	−0.444217	+	0.895919i	$0.646518\pi$
$138$	−299.725	−0.184886
$139$	455.829	0.278150	0.139075	−	0.990282i	$-0.455587\pi$
$139$	455.829	0.278150	0.139075	+	0.990282i	$0.455587\pi$
$140$	5203.31	3.14114
$141$	−784.509	−0.468564
$142$	−131.286	−0.0775863
$143$	−1218.70	−0.712678
$144$	447.903	0.259203
$145$	−936.185	−0.536179
$146$	628.449	0.356238
$147$	−2065.17	−1.15872
$148$	1225.47	0.680629
$149$	2370.80	1.30351	0.651756	−	0.758429i	$-0.274033\pi$
$149$	2370.80	1.30351	0.651756	+	0.758429i	$0.274033\pi$
$150$	−831.847	−0.452800
$151$	413.879	0.223053	0.111526	−	0.993761i	$-0.464426\pi$
$151$	413.879	0.223053	0.111526	+	0.993761i	$0.464426\pi$
$152$	−892.617	−0.476321
$153$	−251.829	−0.133066
$154$	−884.532	−0.462842
$155$	4636.79	2.40281
$156$	765.367	0.392810
$157$	−833.858	−0.423880	−0.211940	−	0.977283i	$-0.567978\pi$
$157$	−833.858	−0.423880	−0.211940	+	0.977283i	$0.567978\pi$
$158$	622.196	0.313286
$159$	339.474	0.169321
$160$	−2956.28	−1.46071
$161$	4113.35	2.01352
$162$	63.1834	0.0306430
$163$	3278.36	1.57534	0.787672	−	0.616095i	$-0.211286\pi$
$163$	3278.36	1.57534	0.787672	+	0.616095i	$0.211286\pi$
$164$	−2792.48	−1.32961
$165$	−2321.87	−1.09550
$166$	196.297	0.0917808
$167$	938.229	0.434745	0.217372	−	0.976089i	$-0.430251\pi$
$167$	938.229	0.434745	0.217372	+	0.976089i	$0.430251\pi$
$168$	1156.73	0.531213
$169$	−1005.68	−0.457752
$170$	478.426	0.215845
$171$	669.125	0.299236
$172$	2907.41	1.28888
$173$	−1635.88	−0.718924	−0.359462	−	0.933160i	$-0.617040\pi$
$173$	−1635.88	−0.718924	−0.359462	+	0.933160i	$0.617040\pi$
$174$	−99.9466	−0.0435456
$175$	11416.0	4.93126
$176$	−1757.22	−0.752587
$177$	177.000	0.0751646
$178$	509.822	0.214678
$179$	−1356.18	−0.566287	−0.283144	−	0.959078i	$-0.591377\pi$
$179$	−1356.18	−0.566287	−0.283144	+	0.959078i	$0.591377\pi$
$180$	1458.18	0.603812
$181$	−2778.89	−1.14118	−0.570590	−	0.821235i	$-0.693285\pi$
$181$	−2778.89	−1.14118	−0.570590	+	0.821235i	$0.693285\pi$
$182$	864.657	0.352157
$183$	−1011.37	−0.408539
$184$	−1537.74	−0.616109
$185$	3634.14	1.44426
$186$	495.021	0.195143
$187$	987.976	0.386353
$188$	−1932.91	−0.749850
$189$	−867.112	−0.333720
$190$	−1271.21	−0.485385
$191$	−3200.32	−1.21239	−0.606197	−	0.795315i	$-0.707306\pi$
$191$	−3200.32	−1.21239	−0.606197	+	0.795315i	$0.707306\pi$
$192$	878.799	0.330322
$193$	945.056	0.352470	0.176235	−	0.984348i	$-0.443608\pi$
$193$	945.056	0.352470	0.176235	+	0.984348i	$0.443608\pi$
$194$	−1228.47	−0.454632
$195$	2269.70	0.833521
$196$	−5088.26	−1.85432
$197$	−135.114	−0.0488653	−0.0244327	−	0.999701i	$-0.507778\pi$
$197$	−135.114	−0.0488653	−0.0244327	+	0.999701i	$0.507778\pi$
$198$	−247.882	−0.0889706
$199$	−1993.07	−0.709976	−0.354988	−	0.934871i	$-0.615515\pi$
$199$	−1993.07	−0.709976	−0.354988	+	0.934871i	$0.615515\pi$
$200$	−4267.80	−1.50889
$201$	−550.913	−0.193326
$202$	1021.18	0.355691
$203$	1371.64	0.474238
$204$	−620.467	−0.212948
$205$	−8281.12	−2.82136
$206$	667.393	0.225726
$207$	1152.73	0.387053
$208$	1717.73	0.572612
$209$	−2625.12	−0.868819
$210$	1647.34	0.541322
$211$	1231.14	0.401684	0.200842	−	0.979624i	$-0.435632\pi$
$211$	1231.14	0.401684	0.200842	+	0.979624i	$0.435632\pi$
$212$	836.412	0.270967
$213$	504.917	0.162424
$214$	515.868	0.164785
$215$	8621.93	2.73493
$216$	324.163	0.102114
$217$	−6793.53	−2.12523
$218$	764.147	0.237406
$219$	−2416.98	−0.745773
$220$	−5720.73	−1.75314
$221$	−965.777	−0.293960
$222$	387.979	0.117295
$223$	4085.96	1.22698	0.613489	−	0.789703i	$-0.289765\pi$
$223$	4085.96	1.22698	0.613489	+	0.789703i	$0.289765\pi$
$224$	4331.35	1.29197
$225$	3199.24	0.947922
$226$	1415.36	0.416585
$227$	5198.03	1.51985	0.759924	−	0.650012i	$-0.225236\pi$
$227$	5198.03	1.51985	0.759924	+	0.650012i	$0.225236\pi$
$228$	1648.62	0.478871
$229$	−1237.19	−0.357014	−0.178507	−	0.983939i	$-0.557127\pi$
$229$	−1237.19	−0.357014	−0.178507	+	0.983939i	$0.557127\pi$
$230$	−2189.96	−0.627833
$231$	3401.86	0.968944
$232$	−512.777	−0.145110
$233$	−5514.40	−1.55047	−0.775236	−	0.631671i	$-0.782369\pi$
$233$	−5514.40	−1.55047	−0.775236	+	0.631671i	$0.782369\pi$
$234$	242.312	0.0676941
$235$	−5732.05	−1.59114
$236$	436.100	0.120287
$237$	−2392.93	−0.655855
$238$	−700.959	−0.190909
$239$	−3600.71	−0.974522	−0.487261	−	0.873256i	$-0.662004\pi$
$239$	−3600.71	−0.974522	−0.487261	+	0.873256i	$0.662004\pi$
$240$	3272.63	0.880196
$241$	6157.76	1.64588	0.822938	−	0.568131i	$-0.192333\pi$
$241$	6157.76	1.64588	0.822938	+	0.568131i	$0.192333\pi$
$242$	−65.7457	−0.0174640
$243$	−243.000	−0.0641500
$244$	−2491.86	−0.653790
$245$	−15089.3	−3.93477
$246$	−884.088	−0.229136
$247$	2566.13	0.661049
$248$	2539.71	0.650289
$249$	−754.948	−0.192140
$250$	−3940.65	−0.996913
$251$	6014.95	1.51259	0.756296	−	0.654230i	$-0.227007\pi$
$251$	6014.95	1.51259	0.756296	+	0.654230i	$0.227007\pi$
$252$	−2136.43	−0.534057
$253$	−4522.39	−1.12379
$254$	−1604.57	−0.396377
$255$	−1840.00	−0.451863
$256$	1323.60	0.323144
$257$	−3933.41	−0.954706	−0.477353	−	0.878712i	$-0.658404\pi$
$257$	−3933.41	−0.954706	−0.477353	+	0.878712i	$0.658404\pi$
$258$	920.472	0.222117
$259$	−5324.52	−1.27741
$260$	5592.19	1.33390
$261$	384.389	0.0911612
$262$	1565.83	0.369227
$263$	1974.57	0.462955	0.231477	−	0.972840i	$-0.425644\pi$
$263$	1974.57	0.462955	0.231477	+	0.972840i	$0.425644\pi$
$264$	−1271.76	−0.296483
$265$	2480.38	0.574977
$266$	1862.49	0.429311
$267$	−1960.75	−0.449422
$268$	−1357.36	−0.309381
$269$	334.493	0.0758155	0.0379078	−	0.999281i	$-0.487931\pi$
$269$	334.493	0.0758155	0.0379078	+	0.999281i	$0.487931\pi$
$270$	461.653	0.104057
$271$	−4193.71	−0.940036	−0.470018	−	0.882657i	$-0.655753\pi$
$271$	−4193.71	−0.940036	−0.470018	+	0.882657i	$0.655753\pi$
$272$	−1392.53	−0.310421
$273$	−3325.42	−0.737230
$274$	−1111.28	−0.245018
$275$	−12551.3	−2.75225
$276$	2840.14	0.619407
$277$	2515.63	0.545666	0.272833	−	0.962061i	$-0.412039\pi$
$277$	2515.63	0.545666	0.272833	+	0.962061i	$0.412039\pi$
$278$	355.566	0.0767101
$279$	−1903.82	−0.408526
$280$	8451.72	1.80388
$281$	−616.348	−0.130848	−0.0654239	−	0.997858i	$-0.520840\pi$
$281$	−616.348	−0.130848	−0.0654239	+	0.997858i	$0.520840\pi$
$282$	−611.950	−0.129224
$283$	−3782.80	−0.794573	−0.397286	−	0.917695i	$-0.630048\pi$
$283$	−3782.80	−0.794573	−0.397286	+	0.917695i	$0.630048\pi$
$284$	1244.04	0.259930
$285$	4889.00	1.01614
$286$	−950.640	−0.196547
$287$	12133.0	2.49543
$288$	1213.82	0.248351
$289$	−4130.07	−0.840640
$290$	−730.264	−0.147871
$291$	4724.61	0.951758
$292$	−5955.06	−1.19347
$293$	1298.66	0.258938	0.129469	−	0.991583i	$-0.458673\pi$
$293$	1298.66	0.258938	0.129469	+	0.991583i	$0.458673\pi$
$294$	−1610.92	−0.319561
$295$	1293.26	0.255242
$296$	1990.53	0.390869
$297$	953.339	0.186257
$298$	1849.32	0.359491
$299$	4420.77	0.855049
$300$	7882.42	1.51697
$301$	−12632.3	−2.41898
$302$	322.843	0.0615150
$303$	−3927.38	−0.744628
$304$	3700.04	0.698066
$305$	−7389.62	−1.38731
$306$	−196.437	−0.0366979
$307$	−2669.82	−0.496335	−0.248167	−	0.968717i	$-0.579828\pi$
$307$	−2669.82	−0.496335	−0.248167	+	0.968717i	$0.579828\pi$
$308$	8381.65	1.55061
$309$	−2566.76	−0.472549
$310$	3616.89	0.662664
$311$	3834.46	0.699139	0.349570	−	0.936910i	$-0.386328\pi$
$311$	3834.46	0.699139	0.349570	+	0.936910i	$0.386328\pi$
$312$	1243.18	0.225582
$313$	6620.03	1.19548	0.597741	−	0.801689i	$-0.296065\pi$
$313$	6620.03	1.19548	0.597741	+	0.801689i	$0.296065\pi$
$314$	−650.445	−0.116900
$315$	−6335.60	−1.13324
$316$	−5895.81	−1.04957
$317$	−2662.49	−0.471736	−0.235868	−	0.971785i	$-0.575793\pi$
$317$	−2662.49	−0.471736	−0.235868	+	0.971785i	$0.575793\pi$
$318$	264.804	0.0466965
$319$	−1508.04	−0.264683
$320$	6420.98	1.12170
$321$	−1984.00	−0.344972
$322$	3208.59	0.555303
$323$	−2080.31	−0.358364
$324$	−598.714	−0.102660
$325$	12269.2	2.09408
$326$	2557.26	0.434459
$327$	−2938.86	−0.497002
$328$	−4535.82	−0.763564
$329$	8398.24	1.40732
$330$	−1811.16	−0.302124
$331$	−6571.43	−1.09123	−0.545617	−	0.838035i	$-0.683705\pi$
$331$	−6571.43	−1.09123	−0.545617	+	0.838035i	$0.683705\pi$
$332$	−1860.07	−0.307484
$333$	−1492.15	−0.245553
$334$	731.859	0.119897
$335$	−4025.27	−0.656490
$336$	−4794.85	−0.778513
$337$	5603.74	0.905802	0.452901	−	0.891561i	$-0.350389\pi$
$337$	5603.74	0.905802	0.452901	+	0.891561i	$0.350389\pi$
$338$	−784.474	−0.126242
$339$	−5443.39	−0.872107
$340$	−4533.47	−0.723123
$341$	7469.09	1.18614
$342$	521.946	0.0825252
$343$	11092.3	1.74615
$344$	4722.49	0.740174
$345$	8422.45	1.31435
$346$	−1276.06	−0.198270
$347$	653.011	0.101024	0.0505122	−	0.998723i	$-0.483915\pi$
$347$	653.011	0.101024	0.0505122	+	0.998723i	$0.483915\pi$
$348$	947.075	0.145887
$349$	1891.17	0.290063	0.145031	−	0.989427i	$-0.453672\pi$
$349$	1891.17	0.290063	0.145031	+	0.989427i	$0.453672\pi$
$350$	8904.99	1.35998
$351$	−931.918	−0.141715
$352$	−4762.06	−0.721076
$353$	−10774.6	−1.62457	−0.812285	−	0.583261i	$-0.801776\pi$
$353$	−10774.6	−1.62457	−0.812285	+	0.583261i	$0.801776\pi$
$354$	138.068	0.0207294
$355$	3689.20	0.551557
$356$	−4830.97	−0.719216
$357$	2695.85	0.399662
$358$	−1057.88	−0.156175
$359$	2797.37	0.411253	0.205626	−	0.978631i	$-0.434077\pi$
$359$	2797.37	0.411253	0.205626	+	0.978631i	$0.434077\pi$
$360$	2368.52	0.346755
$361$	−1331.48	−0.194122
$362$	−2167.66	−0.314722
$363$	252.854	0.0365603
$364$	−8193.32	−1.17980
$365$	−17659.8	−2.53248
$366$	−788.911	−0.112670
$367$	3198.93	0.454994	0.227497	−	0.973779i	$-0.426946\pi$
$367$	3198.93	0.454994	0.227497	+	0.973779i	$0.426946\pi$
$368$	6374.20	0.902930
$369$	3400.15	0.479688
$370$	2834.79	0.398307
$371$	−3634.10	−0.508553
$372$	−4690.72	−0.653771
$373$	263.678	0.0366025	0.0183013	−	0.999833i	$-0.494174\pi$
$373$	263.678	0.0366025	0.0183013	+	0.999833i	$0.494174\pi$
$374$	770.663	0.106551
$375$	15155.5	2.08700
$376$	−3139.62	−0.430621
$377$	1474.15	0.201387
$378$	−676.384	−0.0920356
$379$	1657.03	0.224580	0.112290	−	0.993675i	$-0.464181\pi$
$379$	1657.03	0.224580	0.112290	+	0.993675i	$0.464181\pi$
$380$	12045.7	1.62614
$381$	6171.09	0.829802
$382$	−2496.39	−0.334362
$383$	12320.1	1.64367	0.821837	−	0.569723i	$-0.192949\pi$
$383$	12320.1	1.64367	0.821837	+	0.569723i	$0.192949\pi$
$384$	3922.35	0.521254
$385$	24855.9	3.29032
$386$	737.184	0.0972064
$387$	−3540.08	−0.464994
$388$	11640.7	1.52311
$389$	−4911.58	−0.640172	−0.320086	−	0.947389i	$-0.603712\pi$
$389$	−4911.58	−0.640172	−0.320086	+	0.947389i	$0.603712\pi$
$390$	1770.46	0.229874
$391$	−3583.82	−0.463534
$392$	−8264.85	−1.06489
$393$	−6022.11	−0.772965
$394$	−105.395	−0.0134764
$395$	−17484.1	−2.22714
$396$	2348.88	0.298070
$397$	15177.7	1.91875	0.959377	−	0.282126i	$-0.0910397\pi$
$397$	15177.7	1.91875	0.959377	+	0.282126i	$0.0910397\pi$
$398$	−1554.68	−0.195802
$399$	−7163.05	−0.898749
$400$	17690.7	2.21134
$401$	−6165.32	−0.767785	−0.383892	−	0.923378i	$-0.625417\pi$
$401$	−6165.32	−0.767785	−0.383892	+	0.923378i	$0.625417\pi$
$402$	−429.736	−0.0533166
$403$	−7301.26	−0.902486
$404$	−9676.47	−1.19164
$405$	−1775.49	−0.217839
$406$	1069.94	0.130788
$407$	5853.99	0.712953
$408$	−1007.82	−0.122291
$409$	−7801.41	−0.943166	−0.471583	−	0.881822i	$-0.656317\pi$
$409$	−7801.41	−0.943166	−0.471583	+	0.881822i	$0.656317\pi$
$410$	−6459.63	−0.778093
$411$	4273.93	0.512938
$412$	−6324.09	−0.756227
$413$	−1894.80	−0.225756
$414$	899.176	0.106744
$415$	−5516.06	−0.652465
$416$	4655.06	0.548638
$417$	−1367.49	−0.160590
$418$	−2047.70	−0.239609
$419$	−8108.97	−0.945463	−0.472732	−	0.881207i	$-0.656732\pi$
$419$	−8108.97	−0.945463	−0.472732	+	0.881207i	$0.656732\pi$
$420$	−15609.9	−1.81354
$421$	9487.25	1.09829	0.549145	−	0.835727i	$-0.314953\pi$
$421$	9487.25	1.09829	0.549145	+	0.835727i	$0.314953\pi$
$422$	960.343	0.110779
$423$	2353.53	0.270526
$424$	1358.58	0.155610
$425$	−9946.41	−1.13523
$426$	393.857	0.0447945
$427$	10826.8	1.22704
$428$	−4888.27	−0.552064
$429$	3656.11	0.411465
$430$	6725.47	0.754258
$431$	11388.1	1.27273	0.636366	−	0.771387i	$-0.280437\pi$
$431$	11388.1	1.27273	0.636366	+	0.771387i	$0.280437\pi$
$432$	−1343.71	−0.149651
$433$	15467.8	1.71671	0.858353	−	0.513060i	$-0.171488\pi$
$433$	15467.8	1.71671	0.858353	+	0.513060i	$0.171488\pi$
$434$	−5299.24	−0.586110
$435$	2808.56	0.309563
$436$	−7240.91	−0.795359
$437$	9522.46	1.04238
$438$	−1885.35	−0.205674
$439$	8989.92	0.977370	0.488685	−	0.872460i	$-0.337477\pi$
$439$	8989.92	0.977370	0.488685	+	0.872460i	$0.337477\pi$
$440$	−9292.17	−1.00679
$441$	6195.51	0.668990
$442$	−753.347	−0.0810703
$443$	12756.2	1.36809	0.684045	−	0.729439i	$-0.260219\pi$
$443$	12756.2	1.36809	0.684045	+	0.729439i	$0.260219\pi$
$444$	−3676.41	−0.392961
$445$	−14326.3	−1.52614
$446$	3187.22	0.338384
$447$	−7112.39	−0.752583
$448$	−9407.62	−0.992116
$449$	−7974.32	−0.838154	−0.419077	−	0.907951i	$-0.637646\pi$
$449$	−7974.32	−0.838154	−0.419077	+	0.907951i	$0.637646\pi$
$450$	2495.54	0.261424
$451$	−13339.5	−1.39275
$452$	−13411.7	−1.39565
$453$	−1241.64	−0.128780
$454$	4054.69	0.419154
$455$	−24297.4	−2.50347
$456$	2677.85	0.275004
$457$	−2012.86	−0.206034	−0.103017	−	0.994680i	$-0.532850\pi$
$457$	−2012.86	−0.206034	−0.103017	+	0.994680i	$0.532850\pi$
$458$	−965.065	−0.0984596
$459$	755.486	0.0768259
$460$	20751.6	2.10337
$461$	−12665.8	−1.27962	−0.639810	−	0.768533i	$-0.720987\pi$
$461$	−12665.8	−1.27962	−0.639810	+	0.768533i	$0.720987\pi$
$462$	2653.60	0.267222
$463$	−3474.03	−0.348709	−0.174354	−	0.984683i	$-0.555784\pi$
$463$	−3474.03	−0.348709	−0.174354	+	0.984683i	$0.555784\pi$
$464$	2125.54	0.212664
$465$	−13910.4	−1.38726
$466$	−4301.46	−0.427600
$467$	317.104	0.0314214	0.0157107	−	0.999877i	$-0.494999\pi$
$467$	317.104	0.0314214	0.0157107	+	0.999877i	$0.494999\pi$
$468$	−2296.10	−0.226789
$469$	5897.58	0.580650
$470$	−4471.24	−0.438815
$471$	2501.57	0.244727
$472$	708.357	0.0690779
$473$	13888.5	1.35009
$474$	−1866.59	−0.180876
$475$	26428.3	2.55287
$476$	6642.15	0.639585
$477$	−1018.42	−0.0977576
$478$	−2808.71	−0.268760
$479$	19601.8	1.86979	0.934895	−	0.354923i	$-0.115493\pi$
$479$	19601.8	1.86979	0.934895	+	0.354923i	$0.115493\pi$
$480$	8868.83	0.843343
$481$	−5722.46	−0.542457
$482$	4803.31	0.453911
$483$	−12340.0	−1.16251
$484$	622.993	0.0585080
$485$	34520.6	3.23196
$486$	−189.550	−0.0176917
$487$	−13669.1	−1.27189	−0.635943	−	0.771736i	$-0.719389\pi$
$487$	−13669.1	−1.27189	−0.635943	+	0.771736i	$0.719389\pi$
$488$	−4047.52	−0.375456
$489$	−9835.09	−0.909525
$490$	−11770.3	−1.08516
$491$	−13294.5	−1.22194	−0.610970	−	0.791653i	$-0.709221\pi$
$491$	−13294.5	−1.22194	−0.610970	+	0.791653i	$0.709221\pi$
$492$	8377.45	0.767651
$493$	−1195.06	−0.109174
$494$	2001.69	0.182308
$495$	6965.62	0.632487
$496$	−10527.5	−0.953023
$497$	−5405.19	−0.487839
$498$	−588.892	−0.0529897
$499$	4779.40	0.428768	0.214384	−	0.976749i	$-0.431226\pi$
$499$	4779.40	0.428768	0.214384	+	0.976749i	$0.431226\pi$
$500$	37340.8	3.33986
$501$	−2814.69	−0.251000
$502$	4691.92	0.417152
$503$	−6846.81	−0.606927	−0.303463	−	0.952843i	$-0.598143\pi$
$503$	−6846.81	−0.606927	−0.303463	+	0.952843i	$0.598143\pi$
$504$	−3470.20	−0.306696
$505$	−28695.6	−2.52859
$506$	−3527.65	−0.309928
$507$	3017.04	0.264283
$508$	15204.6	1.32794
$509$	2306.50	0.200852	0.100426	−	0.994945i	$-0.467979\pi$
$509$	2306.50	0.200852	0.100426	+	0.994945i	$0.467979\pi$
$510$	−1435.28	−0.124618
$511$	25874.0	2.23992
$512$	11492.1	0.991958
$513$	−2007.38	−0.172764
$514$	−3068.23	−0.263295
$515$	−18754.1	−1.60467
$516$	−8722.22	−0.744136
$517$	−9233.37	−0.785461
$518$	−4153.35	−0.352293
$519$	4907.65	0.415071
$520$	9083.38	0.766024
$521$	−2031.49	−0.170828	−0.0854139	−	0.996346i	$-0.527221\pi$
$521$	−2031.49	−0.170828	−0.0854139	+	0.996346i	$0.527221\pi$
$522$	299.840	0.0251411
$523$	−16135.1	−1.34902	−0.674512	−	0.738264i	$-0.735646\pi$
$523$	−16135.1	−1.34902	−0.674512	+	0.738264i	$0.735646\pi$
$524$	−14837.5	−1.23699
$525$	−34248.1	−2.84707
$526$	1540.25	0.127677
$527$	5918.98	0.489250
$528$	5271.65	0.434506
$529$	4237.69	0.348293
$530$	1934.81	0.158571
$531$	−531.000	−0.0433963
$532$	−17648.6	−1.43828
$533$	13039.8	1.05969
$534$	−1529.47	−0.123945
$535$	−14496.2	−1.17145
$536$	−2204.76	−0.177670
$537$	4068.53	0.326946
$538$	260.919	0.0209089
$539$	−24306.3	−1.94239
$540$	−4374.53	−0.348611
$541$	4833.44	0.384114	0.192057	−	0.981384i	$-0.438484\pi$
$541$	4833.44	0.384114	0.192057	+	0.981384i	$0.438484\pi$
$542$	−3271.27	−0.259249
$543$	8336.68	0.658860
$544$	−3773.76	−0.297424
$545$	−21473.0	−1.68771
$546$	−2593.97	−0.203318
$547$	−6411.62	−0.501172	−0.250586	−	0.968094i	$-0.580623\pi$
$547$	−6411.62	−0.501172	−0.250586	+	0.968094i	$0.580623\pi$
$548$	10530.3	0.820861
$549$	3034.11	0.235870
$550$	−9790.52	−0.759035
$551$	3175.36	0.245508
$552$	4613.23	0.355711
$553$	25616.5	1.96985
$554$	1962.30	0.150488
$555$	−10902.4	−0.833842
$556$	−3369.27	−0.256995
$557$	−4099.19	−0.311828	−0.155914	−	0.987771i	$-0.549832\pi$
$557$	−4099.19	−0.311828	−0.155914	+	0.987771i	$0.549832\pi$
$558$	−1485.06	−0.112666
$559$	−13576.4	−1.02723
$560$	−35033.8	−2.64365
$561$	−2963.93	−0.223061
$562$	−480.777	−0.0360861
$563$	−4795.83	−0.359006	−0.179503	−	0.983757i	$-0.557449\pi$
$563$	−4795.83	−0.359006	−0.179503	+	0.983757i	$0.557449\pi$
$564$	5798.72	0.432926
$565$	−39772.4	−2.96148
$566$	−2950.74	−0.219133
$567$	2601.34	0.192674
$568$	2020.69	0.149271
$569$	14582.8	1.07441	0.537207	−	0.843451i	$-0.319480\pi$
$569$	14582.8	1.07441	0.537207	+	0.843451i	$0.319480\pi$
$570$	3813.63	0.280237
$571$	−17268.6	−1.26562	−0.632809	−	0.774308i	$-0.718098\pi$
$571$	−17268.6	−1.26562	−0.632809	+	0.774308i	$0.718098\pi$
$572$	9008.08	0.658473
$573$	9600.96	0.699976
$574$	9464.24	0.688205
$575$	45529.0	3.30207
$576$	−2636.40	−0.190712
$577$	−10005.8	−0.721918	−0.360959	−	0.932582i	$-0.617551\pi$
$577$	−10005.8	−0.721918	−0.360959	+	0.932582i	$0.617551\pi$
$578$	−3221.63	−0.231837
$579$	−2835.17	−0.203498
$580$	6919.85	0.495398
$581$	8081.78	0.577089
$582$	3685.40	0.262482
$583$	3995.48	0.283835
$584$	−9672.79	−0.685382
$585$	−6809.10	−0.481234
$586$	1013.01	0.0714116
$587$	−1313.62	−0.0923662	−0.0461831	−	0.998933i	$-0.514706\pi$
$587$	−1313.62	−0.0923662	−0.0461831	+	0.998933i	$0.514706\pi$
$588$	15264.8	1.07059
$589$	−15727.1	−1.10021
$590$	1008.80	0.0703924
$591$	405.342	0.0282124
$592$	−8251.07	−0.572833
$593$	18128.5	1.25539	0.627697	−	0.778458i	$-0.283998\pi$
$593$	18128.5	1.25539	0.627697	+	0.778458i	$0.283998\pi$
$594$	743.645	0.0513672
$595$	19697.3	1.35716
$596$	−17523.8	−1.20437
$597$	5979.22	0.409905
$598$	3448.39	0.235811
$599$	636.072	0.0433876	0.0216938	−	0.999765i	$-0.493094\pi$
$599$	636.072	0.0433876	0.0216938	+	0.999765i	$0.493094\pi$
$600$	12803.4	0.871161
$601$	12022.0	0.815950	0.407975	−	0.912993i	$-0.366235\pi$
$601$	12022.0	0.815950	0.407975	+	0.912993i	$0.366235\pi$
$602$	−9853.74	−0.667123
$603$	1652.74	0.111617
$604$	−3059.20	−0.206088
$605$	1847.49	0.124151
$606$	−3063.53	−0.205359
$607$	4317.86	0.288726	0.144363	−	0.989525i	$-0.453887\pi$
$607$	4317.86	0.288726	0.144363	+	0.989525i	$0.453887\pi$
$608$	10027.1	0.668838
$609$	−4114.92	−0.273801
$610$	−5764.22	−0.382600
$611$	9025.90	0.597625
$612$	1861.40	0.122946
$613$	7724.56	0.508959	0.254480	−	0.967078i	$-0.418096\pi$
$613$	7724.56	0.508959	0.254480	+	0.967078i	$0.418096\pi$
$614$	−2082.57	−0.136883
$615$	24843.4	1.62891
$616$	13614.3	0.890480
$617$	−13543.2	−0.883676	−0.441838	−	0.897095i	$-0.645673\pi$
$617$	−13543.2	−0.883676	−0.441838	+	0.897095i	$0.645673\pi$
$618$	−2002.18	−0.130323
$619$	24983.1	1.62222	0.811112	−	0.584891i	$-0.198863\pi$
$619$	24983.1	1.62222	0.811112	+	0.584891i	$0.198863\pi$
$620$	−34273.0	−2.22006
$621$	−3458.18	−0.223465
$622$	2991.04	0.192813
$623$	20990.0	1.34983
$624$	−5153.20	−0.330598
$625$	66300.6	4.24324
$626$	5163.90	0.329698
$627$	7875.35	0.501613
$628$	6163.49	0.391640
$629$	4639.07	0.294073
$630$	−4942.03	−0.312532
$631$	2921.67	0.184327	0.0921633	−	0.995744i	$-0.470622\pi$
$631$	2921.67	0.184327	0.0921633	+	0.995744i	$0.470622\pi$
$632$	−9576.55	−0.602745
$633$	−3693.43	−0.231912
$634$	−2076.85	−0.130098
$635$	45089.3	2.81782
$636$	−2509.24	−0.156443
$637$	23760.1	1.47788
$638$	−1176.33	−0.0729960
$639$	−1514.75	−0.0937757
$640$	28658.8	1.77006
$641$	7252.88	0.446914	0.223457	−	0.974714i	$-0.428266\pi$
$641$	7252.88	0.446914	0.223457	+	0.974714i	$0.428266\pi$
$642$	−1547.60	−0.0951388
$643$	21945.7	1.34596	0.672981	−	0.739659i	$-0.265013\pi$
$643$	21945.7	1.34596	0.672981	+	0.739659i	$0.265013\pi$
$644$	−30404.0	−1.86038
$645$	−25865.8	−1.57901
$646$	−1622.73	−0.0988319
$647$	−4657.90	−0.283031	−0.141515	−	0.989936i	$-0.545197\pi$
$647$	−4657.90	−0.283031	−0.141515	+	0.989936i	$0.545197\pi$
$648$	−972.490	−0.0589553
$649$	2083.22	0.125999
$650$	9570.53	0.577519
$651$	20380.6	1.22700
$652$	−24232.1	−1.45553
$653$	−6157.88	−0.369030	−0.184515	−	0.982830i	$-0.559071\pi$
$653$	−6157.88	−0.369030	−0.184515	+	0.982830i	$0.559071\pi$
$654$	−2292.44	−0.137066
$655$	−44000.8	−2.62482
$656$	18801.7	1.11903
$657$	7250.94	0.430572
$658$	6550.98	0.388121
$659$	−5471.34	−0.323419	−0.161709	−	0.986838i	$-0.551701\pi$
$659$	−5471.34	−0.323419	−0.161709	+	0.986838i	$0.551701\pi$
$660$	17162.2	1.01218
$661$	14162.9	0.833396	0.416698	−	0.909045i	$-0.363187\pi$
$661$	14162.9	0.833396	0.416698	+	0.909045i	$0.363187\pi$
$662$	−5126.00	−0.300948
$663$	2897.33	0.169718
$664$	−3021.31	−0.176581
$665$	−52337.1	−3.05195
$666$	−1163.94	−0.0677202
$667$	5470.32	0.317558
$668$	−6934.95	−0.401679
$669$	−12257.9	−0.708397
$670$	−3139.88	−0.181051
$671$	−11903.4	−0.684839
$672$	−12994.0	−0.745917
$673$	−9280.39	−0.531550	−0.265775	−	0.964035i	$-0.585628\pi$
$673$	−9280.39	−0.531550	−0.265775	+	0.964035i	$0.585628\pi$
$674$	4371.16	0.249808
$675$	−9597.71	−0.547283
$676$	7433.53	0.422936
$677$	17040.6	0.967391	0.483695	−	0.875236i	$-0.339294\pi$
$677$	17040.6	0.967391	0.483695	+	0.875236i	$0.339294\pi$
$678$	−4246.08	−0.240516
$679$	−50577.4	−2.85859
$680$	−7363.70	−0.415272
$681$	−15594.1	−0.877485
$682$	5826.21	0.327122
$683$	20830.7	1.16700	0.583502	−	0.812112i	$-0.301682\pi$
$683$	20830.7	1.16700	0.583502	+	0.812112i	$0.301682\pi$
$684$	−4945.86	−0.276476
$685$	31227.7	1.74182
$686$	8652.47	0.481564
$687$	3711.58	0.206122
$688$	−19575.5	−1.08475
$689$	−3905.71	−0.215959
$690$	6569.87	0.362479
$691$	13181.3	0.725671	0.362835	−	0.931853i	$-0.381809\pi$
$691$	13181.3	0.725671	0.362835	+	0.931853i	$0.381809\pi$
$692$	12091.7	0.664244
$693$	−10205.6	−0.559420
$694$	509.376	0.0278612
$695$	−9991.60	−0.545328
$696$	1538.33	0.0837792
$697$	−10571.1	−0.574473
$698$	1475.19	0.0799955
$699$	16543.2	0.895166
$700$	−84382.0	−4.55620
$701$	−2796.51	−0.150675	−0.0753373	−	0.997158i	$-0.524003\pi$
$701$	−2796.51	−0.150675	−0.0753373	+	0.997158i	$0.524003\pi$
$702$	−726.936	−0.0390832
$703$	−12326.3	−0.661303
$704$	10343.1	0.553723
$705$	17196.2	0.918645
$706$	−8404.63	−0.448035
$707$	42043.0	2.23648
$708$	−1308.30	−0.0694477
$709$	−8280.22	−0.438604	−0.219302	−	0.975657i	$-0.570378\pi$
$709$	−8280.22	−0.438604	−0.219302	+	0.975657i	$0.570378\pi$
$710$	2877.74	0.152112
$711$	7178.79	0.378658
$712$	−7846.94	−0.413029
$713$	−27093.7	−1.42309
$714$	2102.88	0.110222
$715$	26713.5	1.39724
$716$	10024.2	0.523216
$717$	10802.1	0.562641
$718$	2182.07	0.113418
$719$	−30936.1	−1.60462	−0.802310	−	0.596907i	$-0.796396\pi$
$719$	−30936.1	−1.60462	−0.802310	+	0.596907i	$0.796396\pi$
$720$	−9817.88	−0.508182
$721$	27477.4	1.41929
$722$	−1038.61	−0.0535363
$723$	−18473.3	−0.950247
$724$	20540.3	1.05438
$725$	15182.1	0.777723
$726$	197.237	0.0100829
$727$	24053.2	1.22708	0.613538	−	0.789665i	$-0.289746\pi$
$727$	24053.2	1.22708	0.613538	+	0.789665i	$0.289746\pi$
$728$	−13308.4	−0.677530
$729$	729.000	0.0370370
$730$	−13775.4	−0.698424
$731$	11006.1	0.556875
$732$	7475.57	0.377466
$733$	−4234.34	−0.213368	−0.106684	−	0.994293i	$-0.534023\pi$
$733$	−4234.34	−0.213368	−0.106684	+	0.994293i	$0.534023\pi$
$734$	2495.30	0.125481
$735$	45267.8	2.27174
$736$	17274.1	0.865125
$737$	−6484.04	−0.324074
$738$	2652.26	0.132292
$739$	−29248.6	−1.45592	−0.727961	−	0.685619i	$-0.759532\pi$
$739$	−29248.6	−1.45592	−0.727961	+	0.685619i	$0.759532\pi$
$740$	−26861.9	−1.33441
$741$	−7698.40	−0.381657
$742$	−2834.75	−0.140252
$743$	−5051.56	−0.249426	−0.124713	−	0.992193i	$-0.539801\pi$
$743$	−5051.56	−0.249426	−0.124713	+	0.992193i	$0.539801\pi$
$744$	−7619.13	−0.375445
$745$	−51967.0	−2.55560
$746$	205.680	0.0100945
$747$	2264.84	0.110932
$748$	−7302.66	−0.356967
$749$	21238.9	1.03612
$750$	11821.9	0.575568
$751$	12132.8	0.589526	0.294763	−	0.955570i	$-0.404759\pi$
$751$	12132.8	0.589526	0.294763	+	0.955570i	$0.404759\pi$
$752$	13014.2	0.631090
$753$	−18044.9	−0.873295
$754$	1149.90	0.0555397
$755$	−9072.07	−0.437307
$756$	6409.29	0.308338
$757$	−14667.2	−0.704214	−0.352107	−	0.935960i	$-0.614535\pi$
$757$	−14667.2	−0.704214	−0.352107	+	0.935960i	$0.614535\pi$
$758$	1292.55	0.0619362
$759$	13567.2	0.648823
$760$	19565.8	0.933852
$761$	−2328.04	−0.110895	−0.0554477	−	0.998462i	$-0.517659\pi$
$761$	−2328.04	−0.110895	−0.0554477	+	0.998462i	$0.517659\pi$
$762$	4813.71	0.228848
$763$	31460.8	1.49274
$764$	23655.3	1.12018
$765$	5519.99	0.260883
$766$	9610.19	0.453303
$767$	−2036.41	−0.0958678
$768$	−3970.79	−0.186567
$769$	25456.4	1.19373	0.596867	−	0.802340i	$-0.296412\pi$
$769$	25456.4	1.19373	0.596867	+	0.802340i	$0.296412\pi$
$770$	19388.6	0.907425
$771$	11800.2	0.551200
$772$	−6985.41	−0.325661
$773$	2331.11	0.108466	0.0542331	−	0.998528i	$-0.482729\pi$
$773$	2331.11	0.108466	0.0542331	+	0.998528i	$0.482729\pi$
$774$	−2761.42	−0.128239
$775$	−75194.8	−3.48526
$776$	18908.0	0.874686
$777$	15973.6	0.737513
$778$	−3831.24	−0.176551
$779$	28088.0	1.29186
$780$	−16776.6	−0.770125
$781$	5942.69	0.272274
$782$	−2795.54	−0.127836
$783$	−1153.17	−0.0526320
$784$	34259.2	1.56064
$785$	18277.9	0.831038
$786$	−4697.50	−0.213173
$787$	−15041.1	−0.681268	−0.340634	−	0.940196i	$-0.610642\pi$
$787$	−15041.1	−0.681268	−0.340634	+	0.940196i	$0.610642\pi$
$788$	998.699	0.0451487
$789$	−5923.70	−0.267287
$790$	−13638.3	−0.614214
$791$	58272.0	2.61936
$792$	3815.28	0.171174
$793$	11636.0	0.521066
$794$	11839.2	0.529167
$795$	−7441.15	−0.331963
$796$	14731.9	0.655976
$797$	38360.4	1.70489	0.852444	−	0.522818i	$-0.175119\pi$
$797$	38360.4	1.70489	0.852444	+	0.522818i	$0.175119\pi$
$798$	−5587.48	−0.247863
$799$	−7317.10	−0.323981
$800$	47941.9	2.11875
$801$	5882.24	0.259474
$802$	−4809.22	−0.211745
$803$	−28446.9	−1.25015
$804$	4072.09	0.178621
$805$	−90163.1	−3.94762
$806$	−5695.29	−0.248894
$807$	−1003.48	−0.0437721
$808$	−15717.5	−0.684330
$809$	20027.1	0.870351	0.435175	−	0.900346i	$-0.356686\pi$
$809$	20027.1	0.870351	0.435175	+	0.900346i	$0.356686\pi$
$810$	−1384.96	−0.0600771
$811$	−27584.7	−1.19436	−0.597182	−	0.802106i	$-0.703713\pi$
$811$	−27584.7	−1.19436	−0.597182	+	0.802106i	$0.703713\pi$
$812$	−10138.5	−0.438168
$813$	12581.1	0.542730
$814$	4566.36	0.196623
$815$	−71860.5	−3.08854
$816$	4177.59	0.179222
$817$	−29244.0	−1.25228
$818$	−6085.43	−0.260113
$819$	9976.27	0.425640
$820$	61210.2	2.60677
$821$	−5945.56	−0.252742	−0.126371	−	0.991983i	$-0.540333\pi$
$821$	−5945.56	−0.252742	−0.126371	+	0.991983i	$0.540333\pi$
$822$	3333.85	0.141461
$823$	−5880.56	−0.249069	−0.124534	−	0.992215i	$-0.539744\pi$
$823$	−5880.56	−0.249069	−0.124534	+	0.992215i	$0.539744\pi$
$824$	−10272.2	−0.434283
$825$	37653.8	1.58901
$826$	−1478.03	−0.0622604
$827$	37847.0	1.59138	0.795689	−	0.605705i	$-0.207109\pi$
$827$	37847.0	1.59138	0.795689	+	0.605705i	$0.207109\pi$
$828$	−8520.42	−0.357615
$829$	−38100.9	−1.59626	−0.798130	−	0.602485i	$-0.794177\pi$
$829$	−38100.9	−1.59626	−0.798130	+	0.602485i	$0.794177\pi$
$830$	−4302.76	−0.179941
$831$	−7546.89	−0.315041
$832$	−10110.7	−0.421305
$833$	−19261.8	−0.801180
$834$	−1066.70	−0.0442886
$835$	−20565.7	−0.852340
$836$	19403.6	0.802738
$837$	5711.47	0.235863
$838$	−6325.34	−0.260746
$839$	−15102.5	−0.621450	−0.310725	−	0.950500i	$-0.600572\pi$
$839$	−15102.5	−0.621450	−0.310725	+	0.950500i	$0.600572\pi$
$840$	−25355.2	−1.04147
$841$	−22564.9	−0.925207
$842$	7400.46	0.302894
$843$	1849.04	0.0755450
$844$	−9100.03	−0.371133
$845$	22044.2	0.897447
$846$	1835.85	0.0746074
$847$	−2706.83	−0.109808
$848$	−5631.54	−0.228052
$849$	11348.4	0.458747
$850$	−7758.63	−0.313081
$851$	−21235.0	−0.855378
$852$	−3732.11	−0.150071
$853$	32185.9	1.29194	0.645970	−	0.763363i	$-0.276453\pi$
$853$	32185.9	1.29194	0.645970	+	0.763363i	$0.276453\pi$
$854$	8445.36	0.338401
$855$	−14667.0	−0.586667
$856$	−7940.00	−0.317037
$857$	−9750.86	−0.388661	−0.194331	−	0.980936i	$-0.562253\pi$
$857$	−9750.86	−0.388661	−0.194331	+	0.980936i	$0.562253\pi$
$858$	2851.92	0.113477
$859$	11337.4	0.450323	0.225162	−	0.974321i	$-0.427709\pi$
$859$	11337.4	0.450323	0.225162	+	0.974321i	$0.427709\pi$
$860$	−63729.3	−2.52692
$861$	−36398.9	−1.44073
$862$	8883.24	0.351003
$863$	25245.6	0.995794	0.497897	−	0.867236i	$-0.334106\pi$
$863$	25245.6	0.995794	0.497897	+	0.867236i	$0.334106\pi$
$864$	−3641.46	−0.143385
$865$	35857.9	1.40949
$866$	12065.5	0.473445
$867$	12390.2	0.485344
$868$	50214.6	1.96359
$869$	−28163.9	−1.09942
$870$	2190.79	0.0853734
$871$	6338.35	0.246575
$872$	−11761.4	−0.456756
$873$	−14173.8	−0.549498
$874$	7427.92	0.287475
$875$	−162241.	−6.26828
$876$	17865.2	0.689051
$877$	−9052.34	−0.348547	−0.174274	−	0.984697i	$-0.555758\pi$
$877$	−9052.34	−0.348547	−0.174274	+	0.984697i	$0.555758\pi$
$878$	7012.52	0.269546
$879$	−3895.99	−0.149498
$880$	38517.6	1.47549
$881$	−32080.2	−1.22680	−0.613400	−	0.789772i	$-0.710199\pi$
$881$	−32080.2	−1.22680	−0.613400	+	0.789772i	$0.710199\pi$
$882$	4832.77	0.184498
$883$	−39328.6	−1.49888	−0.749440	−	0.662072i	$-0.769677\pi$
$883$	−39328.6	−1.49888	−0.749440	+	0.662072i	$0.769677\pi$
$884$	7138.57	0.271602
$885$	−3879.78	−0.147364
$886$	9950.36	0.377301
$887$	50922.9	1.92765	0.963823	−	0.266542i	$-0.0858813\pi$
$887$	50922.9	1.92765	0.963823	+	0.266542i	$0.0858813\pi$
$888$	−5971.59	−0.225668
$889$	−66062.1	−2.49230
$890$	−11175.1	−0.420888
$891$	−2860.02	−0.107536
$892$	−30201.5	−1.13366
$893$	19442.0	0.728558
$894$	−5547.97	−0.207552
$895$	29726.9	1.11024
$896$	−41989.1	−1.56558
$897$	−13262.3	−0.493663
$898$	−6220.31	−0.231152
$899$	−9034.67	−0.335176
$900$	−23647.3	−0.875824
$901$	3166.27	0.117074
$902$	−10405.4	−0.384103
$903$	37896.9	1.39660
$904$	−21784.5	−0.801486
$905$	60912.3	2.23734
$906$	−968.529	−0.0355157
$907$	−43998.4	−1.61074	−0.805370	−	0.592772i	$-0.798034\pi$
$907$	−43998.4	−1.61074	−0.805370	+	0.592772i	$0.798034\pi$
$908$	−38421.4	−1.40425
$909$	11782.2	0.429911
$910$	−18953.0	−0.690423
$911$	−28063.3	−1.02061	−0.510306	−	0.859993i	$-0.670468\pi$
$911$	−28063.3	−1.02061	−0.510306	+	0.859993i	$0.670468\pi$
$912$	−11100.1	−0.403029
$913$	−8885.45	−0.322087
$914$	−1570.11	−0.0568214
$915$	22168.9	0.800961
$916$	9144.77	0.329860
$917$	64467.2	2.32159
$918$	589.311	0.0211876
$919$	29137.9	1.04589	0.522944	−	0.852367i	$-0.324834\pi$
$919$	29137.9	1.04589	0.522944	+	0.852367i	$0.324834\pi$
$920$	33706.8	1.20791
$921$	8009.46	0.286559
$922$	−9879.87	−0.352902
$923$	−5809.16	−0.207162
$924$	−25145.0	−0.895247
$925$	−58934.9	−2.09488
$926$	−2709.89	−0.0961692
$927$	7700.27	0.272826
$928$	5760.23	0.203760
$929$	−9282.01	−0.327807	−0.163904	−	0.986476i	$-0.552409\pi$
$929$	−9282.01	−0.327807	−0.163904	+	0.986476i	$0.552409\pi$
$930$	−10850.7	−0.382589
$931$	51180.0	1.80167
$932$	40759.8	1.43255
$933$	−11503.4	−0.403648
$934$	247.354	0.00866561
$935$	−21656.1	−0.757465
$936$	−3729.55	−0.130240
$937$	−26659.3	−0.929478	−0.464739	−	0.885448i	$-0.653852\pi$
$937$	−26659.3	−0.929478	−0.464739	+	0.885448i	$0.653852\pi$
$938$	4600.36	0.160135
$939$	−19860.1	−0.690212
$940$	42368.6	1.47012
$941$	−56714.2	−1.96475	−0.982375	−	0.186923i	$-0.940149\pi$
$941$	−56714.2	−1.96475	−0.982375	+	0.186923i	$0.940149\pi$
$942$	1951.33	0.0674925
$943$	48388.2	1.67098
$944$	−2936.26	−0.101236
$945$	19006.8	0.654276
$946$	10833.6	0.372337
$947$	−3737.90	−0.128264	−0.0641318	−	0.997941i	$-0.520428\pi$
$947$	−3737.90	−0.128264	−0.0641318	+	0.997941i	$0.520428\pi$
$948$	17687.4	0.605972
$949$	27807.7	0.951188
$950$	20615.2	0.704047
$951$	7987.46	0.272357
$952$	10788.8	0.367298
$953$	−29404.8	−0.999489	−0.499745	−	0.866173i	$-0.666573\pi$
$953$	−29404.8	−0.999489	−0.499745	+	0.866173i	$0.666573\pi$
$954$	−794.413	−0.0269602
$955$	70149.9	2.37696
$956$	26614.8	0.900402
$957$	4524.11	0.152815
$958$	15290.3	0.515663
$959$	−45752.8	−1.54060
$960$	−19263.0	−0.647614
$961$	14956.4	0.502044
$962$	−4463.76	−0.149602
$963$	5952.00	0.199170
$964$	−45515.3	−1.52069
$965$	−20715.3	−0.691035
$966$	−9625.76	−0.320604
$967$	52879.0	1.75851	0.879253	−	0.476356i	$-0.158043\pi$
$967$	52879.0	1.75851	0.879253	+	0.476356i	$0.158043\pi$
$968$	1011.93	0.0335997
$969$	6240.92	0.206901
$970$	26927.5	0.891330
$971$	25027.8	0.827167	0.413584	−	0.910466i	$-0.364277\pi$
$971$	25027.8	0.827167	0.413584	+	0.910466i	$0.364277\pi$
$972$	1796.14	0.0592709
$973$	14639.1	0.482330
$974$	−10662.5	−0.350769
$975$	−36807.7	−1.20902
$976$	16777.6	0.550244
$977$	−29643.3	−0.970701	−0.485350	−	0.874320i	$-0.661308\pi$
$977$	−29643.3	−0.970701	−0.485350	+	0.874320i	$0.661308\pi$
$978$	−7671.79	−0.250835
$979$	−23077.2	−0.753372
$980$	111533.	3.63550
$981$	8816.59	0.286944
$982$	−10370.3	−0.336995
$983$	−14434.3	−0.468345	−0.234173	−	0.972195i	$-0.575238\pi$
$983$	−14434.3	−0.468345	−0.234173	+	0.972195i	$0.575238\pi$
$984$	13607.5	0.440844
$985$	2961.65	0.0958030
$986$	−932.201	−0.0301088
$987$	−25194.7	−0.812519
$988$	−18967.6	−0.610771
$989$	−50379.6	−1.61980
$990$	5433.48	0.174432
$991$	15333.7	0.491516	0.245758	−	0.969331i	$-0.420963\pi$
$991$	15333.7	0.491516	0.245758	+	0.969331i	$0.420963\pi$
$992$	−28529.6	−0.913120
$993$	19714.3	0.630024
$994$	−4216.28	−0.134539
$995$	43687.4	1.39194
$996$	5580.22	0.177526
$997$	3787.12	0.120300	0.0601501	−	0.998189i	$-0.480842\pi$
$997$	3787.12	0.120300	0.0601501	+	0.998189i	$0.480842\pi$
$998$	3728.14	0.118249
$999$	4476.44	0.141770

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.c.1.5	✓	8
3.2	odd	2		531.4.a.f.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.5	✓	8	1.1	even	1	trivial
531.4.a.f.1.4		8	3.2	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q+0.780043 q^{2} -3.00000 q^{3} -7.39153 q^{4} -21.9196 q^{5} -2.34013 q^{6} +32.1153 q^{7} -12.0061 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+0.780043 q^{2} -3.00000 q^{3} -7.39153 q^{4} -21.9196 q^{5} -2.34013 q^{6} +32.1153 q^{7} -12.0061 q^{8} +9.00000 q^{9} -17.0983 q^{10} -35.3089 q^{11} +22.1746 q^{12} +34.5155 q^{13} +25.0513 q^{14} +65.7589 q^{15} +49.7670 q^{16} -27.9810 q^{17} +7.02038 q^{18} +74.3473 q^{19} +162.020 q^{20} -96.3458 q^{21} -27.5424 q^{22} +128.081 q^{23} +36.0182 q^{24} +355.471 q^{25} +26.9235 q^{26} -27.0000 q^{27} -237.381 q^{28} +42.7099 q^{29} +51.2948 q^{30} -211.536 q^{31} +134.869 q^{32} +105.927 q^{33} -21.8263 q^{34} -703.955 q^{35} -66.5238 q^{36} -165.794 q^{37} +57.9940 q^{38} -103.546 q^{39} +263.168 q^{40} +377.795 q^{41} -75.1538 q^{42} -393.343 q^{43} +260.987 q^{44} -197.277 q^{45} +99.9085 q^{46} +261.503 q^{47} -149.301 q^{48} +688.391 q^{49} +277.282 q^{50} +83.9429 q^{51} -255.122 q^{52} -113.158 q^{53} -21.0611 q^{54} +773.957 q^{55} -385.578 q^{56} -223.042 q^{57} +33.3155 q^{58} -59.0000 q^{59} -486.059 q^{60} +337.123 q^{61} -165.007 q^{62} +289.037 q^{63} -292.933 q^{64} -756.567 q^{65} +82.6272 q^{66} +183.638 q^{67} +206.822 q^{68} -384.242 q^{69} -549.115 q^{70} -168.306 q^{71} -108.054 q^{72} +805.659 q^{73} -129.326 q^{74} -1066.41 q^{75} -549.540 q^{76} -1133.95 q^{77} -80.7706 q^{78} +797.644 q^{79} -1090.88 q^{80} +81.0000 q^{81} +294.696 q^{82} +251.649 q^{83} +712.143 q^{84} +613.333 q^{85} -306.824 q^{86} -128.130 q^{87} +423.920 q^{88} +653.582 q^{89} -153.884 q^{90} +1108.47 q^{91} -946.713 q^{92} +634.607 q^{93} +203.983 q^{94} -1629.67 q^{95} -404.606 q^{96} -1574.87 q^{97} +536.974 q^{98} -317.780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9} + 29 q^{10} - 27 q^{11} - 114 q^{12} + 89 q^{13} - 37 q^{14} + 36 q^{15} + 362 q^{16} + 79 q^{17} + 18 q^{18} + 288 q^{19} + 457 q^{20} - 159 q^{21} + 596 q^{22} + 202 q^{23} - 9 q^{24} + 264 q^{25} + 270 q^{26} - 216 q^{27} + 702 q^{28} - 114 q^{29} - 87 q^{30} + 538 q^{31} + 316 q^{32} + 81 q^{33} + 498 q^{34} - 196 q^{35} + 342 q^{36} + 395 q^{37} + 397 q^{38} - 267 q^{39} + 918 q^{40} - 39 q^{41} + 111 q^{42} + 527 q^{43} + 64 q^{44} - 108 q^{45} - 539 q^{46} + 860 q^{47} - 1086 q^{48} + 347 q^{49} - 591 q^{50} - 237 q^{51} - 644 q^{52} - 812 q^{53} - 54 q^{54} + 536 q^{55} - 2218 q^{56} - 864 q^{57} - 1154 q^{58} - 472 q^{59} - 1371 q^{60} - 460 q^{61} - 2014 q^{62} + 477 q^{63} - 451 q^{64} - 986 q^{65} - 1788 q^{66} + 1934 q^{67} - 69 q^{68} - 606 q^{69} - 1028 q^{70} - 1687 q^{71} + 27 q^{72} + 1980 q^{73} - 2400 q^{74} - 792 q^{75} - 940 q^{76} - 821 q^{77} - 810 q^{78} + 3319 q^{79} - 2119 q^{80} + 648 q^{81} + 429 q^{82} + 2057 q^{83} - 2106 q^{84} + 566 q^{85} - 6690 q^{86} + 342 q^{87} + 1189 q^{88} + 1668 q^{89} + 261 q^{90} + 2427 q^{91} - 980 q^{92} - 1614 q^{93} + 332 q^{94} + 2146 q^{95} - 948 q^{96} + 1956 q^{97} - 2026 q^{98} - 243 q^{99}+O(q^{100}) \) 8 * q + 2 * q^2 - 24 * q^3 + 38 * q^4 - 12 * q^5 - 6 * q^6 + 53 * q^7 + 3 * q^8 + 72 * q^9 + 29 * q^10 - 27 * q^11 - 114 * q^12 + 89 * q^13 - 37 * q^14 + 36 * q^15 + 362 * q^16 + 79 * q^17 + 18 * q^18 + 288 * q^19 + 457 * q^20 - 159 * q^21 + 596 * q^22 + 202 * q^23 - 9 * q^24 + 264 * q^25 + 270 * q^26 - 216 * q^27 + 702 * q^28 - 114 * q^29 - 87 * q^30 + 538 * q^31 + 316 * q^32 + 81 * q^33 + 498 * q^34 - 196 * q^35 + 342 * q^36 + 395 * q^37 + 397 * q^38 - 267 * q^39 + 918 * q^40 - 39 * q^41 + 111 * q^42 + 527 * q^43 + 64 * q^44 - 108 * q^45 - 539 * q^46 + 860 * q^47 - 1086 * q^48 + 347 * q^49 - 591 * q^50 - 237 * q^51 - 644 * q^52 - 812 * q^53 - 54 * q^54 + 536 * q^55 - 2218 * q^56 - 864 * q^57 - 1154 * q^58 - 472 * q^59 - 1371 * q^60 - 460 * q^61 - 2014 * q^62 + 477 * q^63 - 451 * q^64 - 986 * q^65 - 1788 * q^66 + 1934 * q^67 - 69 * q^68 - 606 * q^69 - 1028 * q^70 - 1687 * q^71 + 27 * q^72 + 1980 * q^73 - 2400 * q^74 - 792 * q^75 - 940 * q^76 - 821 * q^77 - 810 * q^78 + 3319 * q^79 - 2119 * q^80 + 648 * q^81 + 429 * q^82 + 2057 * q^83 - 2106 * q^84 + 566 * q^85 - 6690 * q^86 + 342 * q^87 + 1189 * q^88 + 1668 * q^89 + 261 * q^90 + 2427 * q^91 - 980 * q^92 - 1614 * q^93 + 332 * q^94 + 2146 * q^95 - 948 * q^96 + 1956 * q^97 - 2026 * q^98 - 243 * q^99

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