LMFDB - Embedded newform 177.4.a.a.1.4

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 34x^{5} + 25x^{4} + 315x^{3} - 146x^{2} - 736x + 512 $ x^7 - x^6 - 34x^5 + 25x^4 + 315x^3 - 146x^2 - 736*x + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.775001$ 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)

$f(q)$	$=$	$q-1.77500 q^{2} +3.00000 q^{3} -4.84937 q^{4} +6.23028 q^{5} -5.32500 q^{6} -18.0779 q^{7} +22.8076 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.77500 q^{2} +3.00000 q^{3} -4.84937 q^{4} +6.23028 q^{5} -5.32500 q^{6} -18.0779 q^{7} +22.8076 q^{8} +9.00000 q^{9} -11.0588 q^{10} +13.3400 q^{11} -14.5481 q^{12} -66.3734 q^{13} +32.0883 q^{14} +18.6908 q^{15} -1.68865 q^{16} -97.6720 q^{17} -15.9750 q^{18} +109.769 q^{19} -30.2129 q^{20} -54.2336 q^{21} -23.6785 q^{22} -147.736 q^{23} +68.4229 q^{24} -86.1836 q^{25} +117.813 q^{26} +27.0000 q^{27} +87.6663 q^{28} -173.639 q^{29} -33.1763 q^{30} +148.613 q^{31} -179.464 q^{32} +40.0199 q^{33} +173.368 q^{34} -112.630 q^{35} -43.6443 q^{36} -446.332 q^{37} -194.840 q^{38} -199.120 q^{39} +142.098 q^{40} +182.883 q^{41} +96.2648 q^{42} -223.613 q^{43} -64.6904 q^{44} +56.0725 q^{45} +262.231 q^{46} -529.113 q^{47} -5.06595 q^{48} -16.1904 q^{49} +152.976 q^{50} -293.016 q^{51} +321.869 q^{52} +398.029 q^{53} -47.9250 q^{54} +83.1117 q^{55} -412.314 q^{56} +329.306 q^{57} +308.210 q^{58} -59.0000 q^{59} -90.6388 q^{60} +788.427 q^{61} -263.789 q^{62} -162.701 q^{63} +332.058 q^{64} -413.525 q^{65} -71.0354 q^{66} +288.375 q^{67} +473.648 q^{68} -443.207 q^{69} +199.919 q^{70} -139.803 q^{71} +205.269 q^{72} +549.077 q^{73} +792.240 q^{74} -258.551 q^{75} -532.310 q^{76} -241.158 q^{77} +353.439 q^{78} -190.617 q^{79} -10.5208 q^{80} +81.0000 q^{81} -324.617 q^{82} +410.618 q^{83} +262.999 q^{84} -608.524 q^{85} +396.914 q^{86} -520.918 q^{87} +304.253 q^{88} +1291.81 q^{89} -99.5288 q^{90} +1199.89 q^{91} +716.425 q^{92} +445.840 q^{93} +939.176 q^{94} +683.891 q^{95} -538.392 q^{96} +1488.11 q^{97} +28.7380 q^{98} +120.060 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9}+O(q^{10}) $ 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9} - 79 q^{10} - 131 q^{11} + 66 q^{12} - 123 q^{13} - 117 q^{14} - 84 q^{15} + 202 q^{16} - 235 q^{17} - 72 q^{18} - 80 q^{19} + 61 q^{20} - 177 q^{21} + 688 q^{22} - 274 q^{23} - 351 q^{24} + 193 q^{25} - 180 q^{26} + 189 q^{27} - 118 q^{28} - 406 q^{29} - 237 q^{30} - 346 q^{31} - 854 q^{32} - 393 q^{33} + 178 q^{34} - 424 q^{35} + 198 q^{36} - 157 q^{37} - 129 q^{38} - 369 q^{39} - 590 q^{40} - 825 q^{41} - 351 q^{42} - 815 q^{43} - 1690 q^{44} - 252 q^{45} + 1457 q^{46} - 1196 q^{47} + 606 q^{48} + 914 q^{49} + 713 q^{50} - 705 q^{51} + 1030 q^{52} - 900 q^{53} - 216 q^{54} - 1044 q^{55} + 2172 q^{56} - 240 q^{57} + 1242 q^{58} - 413 q^{59} + 183 q^{60} + 420 q^{61} + 646 q^{62} - 531 q^{63} + 3541 q^{64} + 190 q^{65} + 2064 q^{66} + 1316 q^{67} - 611 q^{68} - 822 q^{69} + 4658 q^{70} - 173 q^{71} - 1053 q^{72} - 418 q^{73} + 660 q^{74} + 579 q^{75} + 1540 q^{76} - 753 q^{77} - 540 q^{78} + 2635 q^{79} + 6155 q^{80} + 567 q^{81} - 125 q^{82} + 457 q^{83} - 354 q^{84} + 1270 q^{85} + 3482 q^{86} - 1218 q^{87} + 7685 q^{88} + 592 q^{89} - 711 q^{90} + 3179 q^{91} - 3500 q^{92} - 1038 q^{93} + 2064 q^{94} - 2250 q^{95} - 2562 q^{96} - 1906 q^{97} + 2994 q^{98} - 1179 q^{99}+O(q^{100}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9 - 79 * q^10 - 131 * q^11 + 66 * q^12 - 123 * q^13 - 117 * q^14 - 84 * q^15 + 202 * q^16 - 235 * q^17 - 72 * q^18 - 80 * q^19 + 61 * q^20 - 177 * q^21 + 688 * q^22 - 274 * q^23 - 351 * q^24 + 193 * q^25 - 180 * q^26 + 189 * q^27 - 118 * q^28 - 406 * q^29 - 237 * q^30 - 346 * q^31 - 854 * q^32 - 393 * q^33 + 178 * q^34 - 424 * q^35 + 198 * q^36 - 157 * q^37 - 129 * q^38 - 369 * q^39 - 590 * q^40 - 825 * q^41 - 351 * q^42 - 815 * q^43 - 1690 * q^44 - 252 * q^45 + 1457 * q^46 - 1196 * q^47 + 606 * q^48 + 914 * q^49 + 713 * q^50 - 705 * q^51 + 1030 * q^52 - 900 * q^53 - 216 * q^54 - 1044 * q^55 + 2172 * q^56 - 240 * q^57 + 1242 * q^58 - 413 * q^59 + 183 * q^60 + 420 * q^61 + 646 * q^62 - 531 * q^63 + 3541 * q^64 + 190 * q^65 + 2064 * q^66 + 1316 * q^67 - 611 * q^68 - 822 * q^69 + 4658 * q^70 - 173 * q^71 - 1053 * q^72 - 418 * q^73 + 660 * q^74 + 579 * q^75 + 1540 * q^76 - 753 * q^77 - 540 * q^78 + 2635 * q^79 + 6155 * q^80 + 567 * q^81 - 125 * q^82 + 457 * q^83 - 354 * q^84 + 1270 * q^85 + 3482 * q^86 - 1218 * q^87 + 7685 * q^88 + 592 * q^89 - 711 * q^90 + 3179 * q^91 - 3500 * q^92 - 1038 * q^93 + 2064 * q^94 - 2250 * q^95 - 2562 * q^96 - 1906 * q^97 + 2994 * q^98 - 1179 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.77500	−0.627558	−0.313779	−	0.949496i	$-0.601595\pi$
$2$	−1.77500	−0.627558	−0.313779	+	0.949496i	$0.601595\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−4.84937	−0.606171
$5$	6.23028	0.557253	0.278627	−	0.960399i	$-0.410121\pi$
$5$	6.23028	0.557253	0.278627	+	0.960399i	$0.410121\pi$
$6$	−5.32500	−0.362321
$7$	−18.0779	−0.976114	−0.488057	−	0.872812i	$-0.662294\pi$
$7$	−18.0779	−0.976114	−0.488057	+	0.872812i	$0.662294\pi$
$8$	22.8076	1.00797
$9$	9.00000	0.333333
$10$	−11.0588	−0.349709
$11$	13.3400	0.365650	0.182825	−	0.983145i	$-0.441476\pi$
$11$	13.3400	0.365650	0.182825	+	0.983145i	$0.441476\pi$
$12$	−14.5481	−0.349973
$13$	−66.3734	−1.41605	−0.708026	−	0.706187i	$-0.750414\pi$
$13$	−66.3734	−1.41605	−0.708026	+	0.706187i	$0.750414\pi$
$14$	32.0883	0.612568
$15$	18.6908	0.321730
$16$	−1.68865	−0.0263851
$17$	−97.6720	−1.39347	−0.696733	−	0.717330i	$-0.745364\pi$
$17$	−97.6720	−1.39347	−0.696733	+	0.717330i	$0.745364\pi$
$18$	−15.9750	−0.209186
$19$	109.769	1.32540	0.662702	−	0.748883i	$-0.269409\pi$
$19$	109.769	1.32540	0.662702	+	0.748883i	$0.269409\pi$
$20$	−30.2129	−0.337791
$21$	−54.2336	−0.563559
$22$	−23.6785	−0.229466
$23$	−147.736	−1.33935	−0.669674	−	0.742655i	$-0.733566\pi$
$23$	−147.736	−1.33935	−0.669674	+	0.742655i	$0.733566\pi$
$24$	68.4229	0.581949
$25$	−86.1836	−0.689469
$26$	117.813	0.888654
$27$	27.0000	0.192450
$28$	87.6663	0.591692
$29$	−173.639	−1.11186	−0.555931	−	0.831228i	$-0.687638\pi$
$29$	−173.639	−1.11186	−0.555931	+	0.831228i	$0.687638\pi$
$30$	−33.1763	−0.201904
$31$	148.613	0.861024	0.430512	−	0.902585i	$-0.358333\pi$
$31$	148.613	0.861024	0.430512	+	0.902585i	$0.358333\pi$
$32$	−179.464	−0.991407
$33$	40.0199	0.211108
$34$	173.368	0.874481
$35$	−112.630	−0.543943
$36$	−43.6443	−0.202057
$37$	−446.332	−1.98315	−0.991574	−	0.129539i	$-0.958650\pi$
$37$	−446.332	−1.98315	−0.991574	+	0.129539i	$0.958650\pi$
$38$	−194.840	−0.831768
$39$	−199.120	−0.817557
$40$	142.098	0.561692
$41$	182.883	0.696622	0.348311	−	0.937379i	$-0.386755\pi$
$41$	182.883	0.696622	0.348311	+	0.937379i	$0.386755\pi$
$42$	96.2648	0.353666
$43$	−223.613	−0.793040	−0.396520	−	0.918026i	$-0.629782\pi$
$43$	−223.613	−0.793040	−0.396520	+	0.918026i	$0.629782\pi$
$44$	−64.6904	−0.221647
$45$	56.0725	0.185751
$46$	262.231	0.840519
$47$	−529.113	−1.64211	−0.821053	−	0.570851i	$-0.806613\pi$
$47$	−529.113	−1.64211	−0.821053	+	0.570851i	$0.806613\pi$
$48$	−5.06595	−0.0152335
$49$	−16.1904	−0.0472023
$50$	152.976	0.432681
$51$	−293.016	−0.804518
$52$	321.869	0.858369
$53$	398.029	1.03157	0.515787	−	0.856717i	$-0.327500\pi$
$53$	398.029	1.03157	0.515787	+	0.856717i	$0.327500\pi$
$54$	−47.9250	−0.120774
$55$	83.1117	0.203760
$56$	−412.314	−0.983889
$57$	329.306	0.765223
$58$	308.210	0.697758
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	−90.6388	−0.195024
$61$	788.427	1.65488	0.827440	−	0.561554i	$-0.189796\pi$
$61$	788.427	1.65488	0.827440	+	0.561554i	$0.189796\pi$
$62$	−263.789	−0.540342
$63$	−162.701	−0.325371
$64$	332.058	0.648550
$65$	−413.525	−0.789099
$66$	−71.0354	−0.132483
$67$	288.375	0.525829	0.262915	−	0.964819i	$-0.415316\pi$
$67$	288.375	0.525829	0.262915	+	0.964819i	$0.415316\pi$
$68$	473.648	0.844680
$69$	−443.207	−0.773273
$70$	199.919	0.341355
$71$	−139.803	−0.233684	−0.116842	−	0.993150i	$-0.537277\pi$
$71$	−139.803	−0.233684	−0.116842	+	0.993150i	$0.537277\pi$
$72$	205.269	0.335988
$73$	549.077	0.880336	0.440168	−	0.897915i	$-0.354919\pi$
$73$	549.077	0.880336	0.440168	+	0.897915i	$0.354919\pi$
$74$	792.240	1.24454
$75$	−258.551	−0.398065
$76$	−532.310	−0.803422
$77$	−241.158	−0.356916
$78$	353.439	0.513065
$79$	−190.617	−0.271469	−0.135735	−	0.990745i	$-0.543339\pi$
$79$	−190.617	−0.271469	−0.135735	+	0.990745i	$0.543339\pi$
$80$	−10.5208	−0.0147032
$81$	81.0000	0.111111
$82$	−324.617	−0.437171
$83$	410.618	0.543026	0.271513	−	0.962435i	$-0.412476\pi$
$83$	410.618	0.543026	0.271513	+	0.962435i	$0.412476\pi$
$84$	262.999	0.341614
$85$	−608.524	−0.776514
$86$	396.914	0.497679
$87$	−520.918	−0.641934
$88$	304.253	0.368562
$89$	1291.81	1.53856	0.769281	−	0.638911i	$-0.220615\pi$
$89$	1291.81	1.53856	0.769281	+	0.638911i	$0.220615\pi$
$90$	−99.5288	−0.116570
$91$	1199.89	1.38223
$92$	716.425	0.811875
$93$	445.840	0.497112
$94$	939.176	1.03052
$95$	683.891	0.738586
$96$	−538.392	−0.572389
$97$	1488.11	1.55768	0.778839	−	0.627225i	$-0.215809\pi$
$97$	1488.11	1.55768	0.778839	+	0.627225i	$0.215809\pi$
$98$	28.7380	0.0296222
$99$	120.060	0.121883
$100$	417.936	0.417936
$101$	−1287.68	−1.26861	−0.634303	−	0.773085i	$-0.718713\pi$
$101$	−1287.68	−1.26861	−0.634303	+	0.773085i	$0.718713\pi$
$102$	520.104	0.504882
$103$	−469.022	−0.448680	−0.224340	−	0.974511i	$-0.572023\pi$
$103$	−469.022	−0.448680	−0.224340	+	0.974511i	$0.572023\pi$
$104$	−1513.82	−1.42733
$105$	−337.891	−0.314045
$106$	−706.501	−0.647372
$107$	−1237.22	−1.11782	−0.558908	−	0.829230i	$-0.688779\pi$
$107$	−1237.22	−1.11782	−0.558908	+	0.829230i	$0.688779\pi$
$108$	−130.933	−0.116658
$109$	2145.12	1.88500	0.942501	−	0.334204i	$-0.108468\pi$
$109$	2145.12	1.88500	0.942501	+	0.334204i	$0.108468\pi$
$110$	−147.523	−0.127871
$111$	−1339.00	−1.14497
$112$	30.5272	0.0257549
$113$	1114.62	0.927919	0.463960	−	0.885856i	$-0.346428\pi$
$113$	1114.62	0.927919	0.463960	+	0.885856i	$0.346428\pi$
$114$	−584.519	−0.480222
$115$	−920.435	−0.746357
$116$	842.041	0.673979
$117$	−597.360	−0.472017
$118$	104.725	0.0817011
$119$	1765.70	1.36018
$120$	426.294	0.324293
$121$	−1153.05	−0.866300
$122$	−1399.46	−1.03853
$123$	548.649	0.402195
$124$	−720.681	−0.521928
$125$	−1315.73	−0.941462
$126$	288.794	0.204189
$127$	−1267.75	−0.885782	−0.442891	−	0.896576i	$-0.646047\pi$
$127$	−1267.75	−0.885782	−0.442891	+	0.896576i	$0.646047\pi$
$128$	846.308	0.584404
$129$	−670.840	−0.457862
$130$	734.007	0.495205
$131$	−2119.49	−1.41359	−0.706796	−	0.707418i	$-0.749860\pi$
$131$	−2119.49	−1.41359	−0.706796	+	0.707418i	$0.749860\pi$
$132$	−194.071	−0.127968
$133$	−1984.39	−1.29375
$134$	−511.865	−0.329988
$135$	168.218	0.107243
$136$	−2227.67	−1.40457
$137$	−845.583	−0.527322	−0.263661	−	0.964615i	$-0.584930\pi$
$137$	−845.583	−0.527322	−0.263661	+	0.964615i	$0.584930\pi$
$138$	786.693	0.485274
$139$	−1390.93	−0.848754	−0.424377	−	0.905486i	$-0.639507\pi$
$139$	−1390.93	−0.848754	−0.424377	+	0.905486i	$0.639507\pi$
$140$	546.186	0.329722
$141$	−1587.34	−0.948071
$142$	248.151	0.146650
$143$	−885.418	−0.517779
$144$	−15.1978	−0.00879504
$145$	−1081.82	−0.619589
$146$	−974.612	−0.552462
$147$	−48.5712	−0.0272523
$148$	2164.43	1.20213
$149$	−1007.80	−0.554110	−0.277055	−	0.960854i	$-0.589358\pi$
$149$	−1007.80	−0.554110	−0.277055	+	0.960854i	$0.589358\pi$
$150$	458.928	0.249809
$151$	−1926.18	−1.03808	−0.519041	−	0.854750i	$-0.673711\pi$
$151$	−1926.18	−1.03808	−0.519041	+	0.854750i	$0.673711\pi$
$152$	2503.57	1.33596
$153$	−879.048	−0.464489
$154$	428.056	0.223985
$155$	925.903	0.479808
$156$	965.607	0.495580
$157$	−1235.58	−0.628092	−0.314046	−	0.949408i	$-0.601685\pi$
$157$	−1235.58	−0.628092	−0.314046	+	0.949408i	$0.601685\pi$
$158$	338.345	0.170363
$159$	1194.09	0.595580
$160$	−1118.11	−0.552465
$161$	2670.75	1.30736
$162$	−143.775	−0.0697286
$163$	3664.53	1.76091	0.880454	−	0.474131i	$-0.157238\pi$
$163$	3664.53	1.76091	0.880454	+	0.474131i	$0.157238\pi$
$164$	−886.867	−0.422272
$165$	249.335	0.117641
$166$	−728.847	−0.340780
$167$	3953.29	1.83183	0.915913	−	0.401377i	$-0.131468\pi$
$167$	3953.29	1.83183	0.915913	+	0.401377i	$0.131468\pi$
$168$	−1236.94	−0.568048
$169$	2208.43	1.00520
$170$	1080.13	0.487307
$171$	987.919	0.441802
$172$	1084.38	0.480718
$173$	−2225.27	−0.977945	−0.488973	−	0.872299i	$-0.662628\pi$
$173$	−2225.27	−0.977945	−0.488973	+	0.872299i	$0.662628\pi$
$174$	924.630	0.402851
$175$	1558.02	0.673000
$176$	−22.5265	−0.00964772
$177$	−177.000	−0.0751646
$178$	−2292.97	−0.965536
$179$	−2130.61	−0.889662	−0.444831	−	0.895615i	$-0.646736\pi$
$179$	−2130.61	−0.889662	−0.444831	+	0.895615i	$0.646736\pi$
$180$	−271.917	−0.112597
$181$	1429.58	0.587072	0.293536	−	0.955948i	$-0.405168\pi$
$181$	1429.58	0.587072	0.293536	+	0.955948i	$0.405168\pi$
$182$	−2129.81	−0.867427
$183$	2365.28	0.955446
$184$	−3369.50	−1.35002
$185$	−2780.77	−1.10512
$186$	−791.366	−0.311967
$187$	−1302.94	−0.509521
$188$	2565.86	0.995398
$189$	−488.103	−0.187853
$190$	−1213.91	−0.463506
$191$	−920.502	−0.348718	−0.174359	−	0.984682i	$-0.555785\pi$
$191$	−920.502	−0.348718	−0.174359	+	0.984682i	$0.555785\pi$
$192$	996.173	0.374441
$193$	−1870.05	−0.697458	−0.348729	−	0.937224i	$-0.613387\pi$
$193$	−1870.05	−0.697458	−0.348729	+	0.937224i	$0.613387\pi$
$194$	−2641.40	−0.977532
$195$	−1240.57	−0.455587
$196$	78.5132	0.0286127
$197$	−476.512	−0.172335	−0.0861676	−	0.996281i	$-0.527462\pi$
$197$	−476.512	−0.172335	−0.0861676	+	0.996281i	$0.527462\pi$
$198$	−213.106	−0.0764888
$199$	4164.08	1.48334	0.741668	−	0.670767i	$-0.234035\pi$
$199$	4164.08	1.48334	0.741668	+	0.670767i	$0.234035\pi$
$200$	−1965.64	−0.694960
$201$	865.124	0.303588
$202$	2285.64	0.796123
$203$	3139.03	1.08530
$204$	1420.94	0.487676
$205$	1139.41	0.388195
$206$	832.514	0.281573
$207$	−1329.62	−0.446450
$208$	112.081	0.0373627
$209$	1464.31	0.484634
$210$	599.757	0.197082
$211$	−2499.21	−0.815415	−0.407707	−	0.913113i	$-0.633672\pi$
$211$	−2499.21	−0.815415	−0.407707	+	0.913113i	$0.633672\pi$
$212$	−1930.19	−0.625311
$213$	−419.410	−0.134918
$214$	2196.06	0.701494
$215$	−1393.18	−0.441924
$216$	615.807	0.193983
$217$	−2686.61	−0.840457
$218$	−3807.59	−1.18295
$219$	1647.23	0.508262
$220$	−403.040	−0.123513
$221$	6482.82	1.97322
$222$	2376.72	0.718536
$223$	2567.55	0.771013	0.385506	−	0.922705i	$-0.374027\pi$
$223$	2567.55	0.771013	0.385506	+	0.922705i	$0.374027\pi$
$224$	3244.33	0.967726
$225$	−775.652	−0.229823
$226$	−1978.46	−0.582323
$227$	−969.018	−0.283330	−0.141665	−	0.989915i	$-0.545246\pi$
$227$	−969.018	−0.283330	−0.141665	+	0.989915i	$0.545246\pi$
$228$	−1596.93	−0.463856
$229$	4217.16	1.21693	0.608467	−	0.793579i	$-0.291785\pi$
$229$	4217.16	1.21693	0.608467	+	0.793579i	$0.291785\pi$
$230$	1633.77	0.468382
$231$	−723.475	−0.206065
$232$	−3960.30	−1.12072
$233$	382.087	0.107431	0.0537154	−	0.998556i	$-0.482894\pi$
$233$	382.087	0.107431	0.0537154	+	0.998556i	$0.482894\pi$
$234$	1060.32	0.296218
$235$	−3296.52	−0.915070
$236$	286.113	0.0789168
$237$	−571.851	−0.156733
$238$	−3134.12	−0.853593
$239$	−4069.70	−1.10145	−0.550726	−	0.834686i	$-0.685649\pi$
$239$	−4069.70	−1.10145	−0.550726	+	0.834686i	$0.685649\pi$
$240$	−31.5623	−0.00848890
$241$	4573.11	1.22232	0.611162	−	0.791506i	$-0.290703\pi$
$241$	4573.11	1.22232	0.611162	+	0.791506i	$0.290703\pi$
$242$	2046.66	0.543653
$243$	243.000	0.0641500
$244$	−3823.37	−1.00314
$245$	−100.871	−0.0263036
$246$	−973.852	−0.252401
$247$	−7285.73	−1.87684
$248$	3389.52	0.867882
$249$	1231.85	0.313516
$250$	2335.43	0.590822
$251$	4263.66	1.07219	0.536095	−	0.844158i	$-0.319899\pi$
$251$	4263.66	1.07219	0.536095	+	0.844158i	$0.319899\pi$
$252$	788.997	0.197231
$253$	−1970.79	−0.489733
$254$	2250.25	0.555879
$255$	−1825.57	−0.448321
$256$	−4158.66	−1.01530
$257$	3758.60	0.912277	0.456138	−	0.889909i	$-0.349232\pi$
$257$	3758.60	0.912277	0.456138	+	0.889909i	$0.349232\pi$
$258$	1190.74	0.287335
$259$	8068.73	1.93578
$260$	2005.34	0.478329
$261$	−1562.75	−0.370621
$262$	3762.09	0.887110
$263$	−2832.72	−0.664157	−0.332078	−	0.943252i	$-0.607750\pi$
$263$	−2832.72	−0.664157	−0.332078	+	0.943252i	$0.607750\pi$
$264$	912.760	0.212790
$265$	2479.83	0.574848
$266$	3522.29	0.811900
$267$	3875.44	0.888289
$268$	−1398.44	−0.318743
$269$	−7351.53	−1.66629	−0.833143	−	0.553058i	$-0.813461\pi$
$269$	−7351.53	−1.66629	−0.833143	+	0.553058i	$0.813461\pi$
$270$	−298.587	−0.0673015
$271$	−4053.05	−0.908507	−0.454253	−	0.890872i	$-0.650094\pi$
$271$	−4053.05	−0.908507	−0.454253	+	0.890872i	$0.650094\pi$
$272$	164.934	0.0367668
$273$	3599.67	0.798029
$274$	1500.91	0.330925
$275$	−1149.69	−0.252104
$276$	2149.28	0.468736
$277$	3054.14	0.662475	0.331238	−	0.943547i	$-0.392534\pi$
$277$	3054.14	0.662475	0.331238	+	0.943547i	$0.392534\pi$
$278$	2468.90	0.532642
$279$	1337.52	0.287008
$280$	−2568.83	−0.548275
$281$	−3512.24	−0.745632	−0.372816	−	0.927905i	$-0.621608\pi$
$281$	−3512.24	−0.745632	−0.372816	+	0.927905i	$0.621608\pi$
$282$	2817.53	0.594969
$283$	−7955.47	−1.67104	−0.835518	−	0.549463i	$-0.814832\pi$
$283$	−7955.47	−1.67104	−0.835518	+	0.549463i	$0.814832\pi$
$284$	677.958	0.141653
$285$	2051.67	0.426423
$286$	1571.62	0.324936
$287$	−3306.13	−0.679982
$288$	−1615.17	−0.330469
$289$	4626.82	0.941750
$290$	1920.24	0.388828
$291$	4464.33	0.899325
$292$	−2662.68	−0.533635
$293$	4933.26	0.983632	0.491816	−	0.870699i	$-0.336333\pi$
$293$	4933.26	0.983632	0.491816	+	0.870699i	$0.336333\pi$
$294$	86.2139	0.0171024
$295$	−367.587	−0.0725482
$296$	−10179.8	−1.99895
$297$	360.179	0.0703694
$298$	1788.85	0.347736
$299$	9805.72	1.89659
$300$	1253.81	0.241296
$301$	4042.46	0.774098
$302$	3418.97	0.651456
$303$	−3863.05	−0.732430
$304$	−185.361	−0.0349710
$305$	4912.12	0.922188
$306$	1560.31	0.291494
$307$	−4362.63	−0.811037	−0.405518	−	0.914087i	$-0.632909\pi$
$307$	−4362.63	−0.811037	−0.405518	+	0.914087i	$0.632909\pi$
$308$	1169.47	0.216352
$309$	−1407.07	−0.259046
$310$	−1643.48	−0.301107
$311$	−3542.33	−0.645875	−0.322937	−	0.946420i	$-0.604670\pi$
$311$	−3542.33	−0.645875	−0.322937	+	0.946420i	$0.604670\pi$
$312$	−4541.46	−0.824069
$313$	−1808.65	−0.326616	−0.163308	−	0.986575i	$-0.552216\pi$
$313$	−1808.65	−0.326616	−0.163308	+	0.986575i	$0.552216\pi$
$314$	2193.16	0.394164
$315$	−1013.67	−0.181314
$316$	924.372	0.164557
$317$	302.078	0.0535217	0.0267609	−	0.999642i	$-0.491481\pi$
$317$	302.078	0.0535217	0.0267609	+	0.999642i	$0.491481\pi$
$318$	−2119.50	−0.373761
$319$	−2316.34	−0.406552
$320$	2068.81	0.361407
$321$	−3711.65	−0.645371
$322$	−4740.58	−0.820442
$323$	−10721.3	−1.84691
$324$	−392.799	−0.0673524
$325$	5720.30	0.976323
$326$	−6504.54	−1.10507
$327$	6435.36	1.08831
$328$	4171.13	0.702171
$329$	9565.23	1.60288
$330$	−442.570	−0.0738263
$331$	−4133.17	−0.686343	−0.343172	−	0.939273i	$-0.611501\pi$
$331$	−4133.17	−0.686343	−0.343172	+	0.939273i	$0.611501\pi$
$332$	−1991.24	−0.329167
$333$	−4016.99	−0.661050
$334$	−7017.10	−1.14958
$335$	1796.66	0.293020
$336$	91.5815	0.0148696
$337$	−3246.37	−0.524750	−0.262375	−	0.964966i	$-0.584506\pi$
$337$	−3246.37	−0.524750	−0.262375	+	0.964966i	$0.584506\pi$
$338$	−3919.96	−0.630821
$339$	3343.87	0.535734
$340$	2950.96	0.470701
$341$	1982.50	0.314833
$342$	−1753.56	−0.277256
$343$	6493.40	1.02219
$344$	−5100.10	−0.799357
$345$	−2761.31	−0.430909
$346$	3949.87	0.613717
$347$	463.911	0.0717696	0.0358848	−	0.999356i	$-0.488575\pi$
$347$	463.911	0.0717696	0.0358848	+	0.999356i	$0.488575\pi$
$348$	2526.12	0.389122
$349$	−6606.06	−1.01322	−0.506611	−	0.862175i	$-0.669102\pi$
$349$	−6606.06	−1.01322	−0.506611	+	0.862175i	$0.669102\pi$
$350$	−2765.48	−0.422346
$351$	−1792.08	−0.272519
$352$	−2394.04	−0.362508
$353$	−6284.55	−0.947572	−0.473786	−	0.880640i	$-0.657113\pi$
$353$	−6284.55	−0.947572	−0.473786	+	0.880640i	$0.657113\pi$
$354$	314.175	0.0471701
$355$	−871.014	−0.130221
$356$	−6264.48	−0.932632
$357$	5297.11	0.785301
$358$	3781.84	0.558314
$359$	5773.93	0.848848	0.424424	−	0.905464i	$-0.360477\pi$
$359$	5773.93	0.848848	0.424424	+	0.905464i	$0.360477\pi$
$360$	1278.88	0.187231
$361$	5190.19	0.756698
$362$	−2537.51	−0.368421
$363$	−3459.14	−0.500159
$364$	−5818.71	−0.837866
$365$	3420.90	0.490570
$366$	−4198.38	−0.599597
$367$	−669.964	−0.0952911	−0.0476456	−	0.998864i	$-0.515172\pi$
$367$	−669.964	−0.0952911	−0.0476456	+	0.998864i	$0.515172\pi$
$368$	249.474	0.0353389
$369$	1645.95	0.232207
$370$	4935.88	0.693524
$371$	−7195.51	−1.00693
$372$	−2162.04	−0.301335
$373$	3345.79	0.464446	0.232223	−	0.972663i	$-0.425400\pi$
$373$	3345.79	0.464446	0.232223	+	0.972663i	$0.425400\pi$
$374$	2312.72	0.319754
$375$	−3947.20	−0.543553
$376$	−12067.8	−1.65519
$377$	11525.0	1.57445
$378$	866.383	0.117889
$379$	6881.01	0.932595	0.466298	−	0.884628i	$-0.345588\pi$
$379$	6881.01	0.932595	0.466298	+	0.884628i	$0.345588\pi$
$380$	−3316.44	−0.447710
$381$	−3803.24	−0.511406
$382$	1633.89	0.218841
$383$	−4080.85	−0.544444	−0.272222	−	0.962235i	$-0.587758\pi$
$383$	−4080.85	−0.544444	−0.272222	+	0.962235i	$0.587758\pi$
$384$	2538.92	0.337406
$385$	−1502.48	−0.198893
$386$	3319.35	0.437695
$387$	−2012.52	−0.264347
$388$	−7216.40	−0.944219
$389$	−4698.80	−0.612439	−0.306220	−	0.951961i	$-0.599064\pi$
$389$	−4698.80	−0.612439	−0.306220	+	0.951961i	$0.599064\pi$
$390$	2202.02	0.285907
$391$	14429.6	1.86634
$392$	−369.265	−0.0475783
$393$	−6358.46	−0.816138
$394$	845.809	0.108150
$395$	−1187.60	−0.151277
$396$	−582.214	−0.0738822
$397$	10293.0	1.30124	0.650620	−	0.759404i	$-0.274509\pi$
$397$	10293.0	1.30124	0.650620	+	0.759404i	$0.274509\pi$
$398$	−7391.25	−0.930879
$399$	−5953.16	−0.746944
$400$	145.534	0.0181917
$401$	7673.35	0.955583	0.477792	−	0.878473i	$-0.341437\pi$
$401$	7673.35	0.955583	0.477792	+	0.878473i	$0.341437\pi$
$402$	−1535.60	−0.190519
$403$	−9863.97	−1.21925
$404$	6244.45	0.768992
$405$	504.653	0.0619170
$406$	−5571.78	−0.681091
$407$	−5954.05	−0.725138
$408$	−6683.01	−0.810927
$409$	3707.61	0.448239	0.224119	−	0.974562i	$-0.428049\pi$
$409$	3707.61	0.448239	0.224119	+	0.974562i	$0.428049\pi$
$410$	−2022.46	−0.243615
$411$	−2536.75	−0.304449
$412$	2274.46	0.271977
$413$	1066.59	0.127079
$414$	2360.08	0.280173
$415$	2558.27	0.302603
$416$	11911.6	1.40388
$417$	−4172.78	−0.490029
$418$	−2599.16	−0.304136
$419$	−11357.0	−1.32416	−0.662082	−	0.749431i	$-0.730327\pi$
$419$	−11357.0	−1.32416	−0.662082	+	0.749431i	$0.730327\pi$
$420$	1638.56	0.190365
$421$	3814.03	0.441531	0.220766	−	0.975327i	$-0.429144\pi$
$421$	3814.03	0.441531	0.220766	+	0.975327i	$0.429144\pi$
$422$	4436.10	0.511720
$423$	−4762.01	−0.547369
$424$	9078.10	1.03979
$425$	8417.72	0.960752
$426$	744.453	0.0846687
$427$	−14253.1	−1.61535
$428$	5999.72	0.677588
$429$	−2656.26	−0.298940
$430$	2472.89	0.277333
$431$	−1307.09	−0.146079	−0.0730396	−	0.997329i	$-0.523270\pi$
$431$	−1307.09	−0.146079	−0.0730396	+	0.997329i	$0.523270\pi$
$432$	−45.5935	−0.00507782
$433$	7915.74	0.878536	0.439268	−	0.898356i	$-0.355238\pi$
$433$	7915.74	0.878536	0.439268	+	0.898356i	$0.355238\pi$
$434$	4768.74	0.527435
$435$	−3245.47	−0.357720
$436$	−10402.5	−1.14263
$437$	−16216.8	−1.77518
$438$	−2923.83	−0.318964
$439$	2570.45	0.279456	0.139728	−	0.990190i	$-0.455377\pi$
$439$	2570.45	0.279456	0.139728	+	0.990190i	$0.455377\pi$
$440$	1895.58	0.205383
$441$	−145.713	−0.0157341
$442$	−11507.0	−1.23831
$443$	−633.121	−0.0679017	−0.0339509	−	0.999424i	$-0.510809\pi$
$443$	−633.121	−0.0679017	−0.0339509	+	0.999424i	$0.510809\pi$
$444$	6493.29	0.694049
$445$	8048.36	0.857369
$446$	−4557.40	−0.483855
$447$	−3023.41	−0.319915
$448$	−6002.90	−0.633059
$449$	10146.6	1.06648	0.533238	−	0.845965i	$-0.320975\pi$
$449$	10146.6	1.06648	0.533238	+	0.845965i	$0.320975\pi$
$450$	1376.78	0.144227
$451$	2439.65	0.254720
$452$	−5405.22	−0.562478
$453$	−5778.54	−0.599336
$454$	1720.01	0.177806
$455$	7475.65	0.770250
$456$	7510.71	0.771318
$457$	−7173.92	−0.734315	−0.367157	−	0.930159i	$-0.619669\pi$
$457$	−7173.92	−0.734315	−0.367157	+	0.930159i	$0.619669\pi$
$458$	−7485.47	−0.763697
$459$	−2637.14	−0.268173
$460$	4463.53	0.452420
$461$	−15959.8	−1.61242	−0.806208	−	0.591632i	$-0.798484\pi$
$461$	−15959.8	−1.61242	−0.806208	+	0.591632i	$0.798484\pi$
$462$	1284.17	0.129318
$463$	10486.8	1.05262	0.526308	−	0.850294i	$-0.323576\pi$
$463$	10486.8	1.05262	0.526308	+	0.850294i	$0.323576\pi$
$464$	293.216	0.0293366
$465$	2777.71	0.277018
$466$	−678.205	−0.0674190
$467$	−11871.2	−1.17630	−0.588150	−	0.808752i	$-0.700144\pi$
$467$	−11871.2	−1.17630	−0.588150	+	0.808752i	$0.700144\pi$
$468$	2896.82	0.286123
$469$	−5213.20	−0.513269
$470$	5851.33	0.574259
$471$	−3706.75	−0.362629
$472$	−1345.65	−0.131226
$473$	−2983.00	−0.289975
$474$	1015.04	0.0983590
$475$	−9460.27	−0.913825
$476$	−8562.54	−0.824503
$477$	3582.26	0.343858
$478$	7223.72	0.691225
$479$	−9570.85	−0.912950	−0.456475	−	0.889736i	$-0.650888\pi$
$479$	−9570.85	−0.912950	−0.456475	+	0.889736i	$0.650888\pi$
$480$	−3354.33	−0.318966
$481$	29624.6	2.80824
$482$	−8117.28	−0.767078
$483$	8012.24	0.754803
$484$	5591.54	0.525126
$485$	9271.34	0.868021
$486$	−431.325	−0.0402578
$487$	11208.2	1.04290	0.521451	−	0.853281i	$-0.325391\pi$
$487$	11208.2	1.04290	0.521451	+	0.853281i	$0.325391\pi$
$488$	17982.2	1.66806
$489$	10993.6	1.01666
$490$	179.046	0.0165071
$491$	7720.19	0.709587	0.354793	−	0.934945i	$-0.384551\pi$
$491$	7720.19	0.709587	0.354793	+	0.934945i	$0.384551\pi$
$492$	−2660.60	−0.243799
$493$	16959.7	1.54934
$494$	12932.2	1.17783
$495$	748.006	0.0679199
$496$	−250.956	−0.0227182
$497$	2527.35	0.228103
$498$	−2186.54	−0.196750
$499$	−3679.73	−0.330115	−0.165057	−	0.986284i	$-0.552781\pi$
$499$	−3679.73	−0.330115	−0.165057	+	0.986284i	$0.552781\pi$
$500$	6380.48	0.570687
$501$	11859.9	1.05761
$502$	−7568.00	−0.672861
$503$	15306.9	1.35686	0.678431	−	0.734664i	$-0.262660\pi$
$503$	15306.9	1.35686	0.678431	+	0.734664i	$0.262660\pi$
$504$	−3710.82	−0.327963
$505$	−8022.62	−0.706935
$506$	3498.15	0.307336
$507$	6625.28	0.580353
$508$	6147.77	0.536935
$509$	−9361.34	−0.815195	−0.407597	−	0.913162i	$-0.633633\pi$
$509$	−9361.34	−0.815195	−0.407597	+	0.913162i	$0.633633\pi$
$510$	3240.39	0.281347
$511$	−9926.14	−0.859308
$512$	611.164	0.0527537
$513$	2963.76	0.255074
$514$	−6671.52	−0.572506
$515$	−2922.14	−0.250029
$516$	3253.15	0.277543
$517$	−7058.34	−0.600436
$518$	−14322.0	−1.21481
$519$	−6675.82	−0.564617
$520$	−9431.53	−0.795385
$521$	−5933.23	−0.498924	−0.249462	−	0.968385i	$-0.580254\pi$
$521$	−5933.23	−0.498924	−0.249462	+	0.968385i	$0.580254\pi$
$522$	2773.89	0.232586
$523$	17269.3	1.44385	0.721925	−	0.691972i	$-0.243258\pi$
$523$	17269.3	1.44385	0.721925	+	0.691972i	$0.243258\pi$
$524$	10278.2	0.856879
$525$	4674.05	0.388557
$526$	5028.09	0.416797
$527$	−14515.4	−1.19981
$528$	−67.5795	−0.00557012
$529$	9658.83	0.793855
$530$	−4401.70	−0.360750
$531$	−531.000	−0.0433963
$532$	9623.03	0.784231
$533$	−12138.6	−0.986453
$534$	−6878.91	−0.557453
$535$	−7708.21	−0.622907
$536$	6577.15	0.530018
$537$	−6391.84	−0.513647
$538$	13049.0	1.04569
$539$	−215.979	−0.0172595
$540$	−815.750	−0.0650079
$541$	−1132.58	−0.0900067	−0.0450034	−	0.998987i	$-0.514330\pi$
$541$	−1132.58	−0.0900067	−0.0450034	+	0.998987i	$0.514330\pi$
$542$	7194.17	0.570141
$543$	4288.75	0.338946
$544$	17528.6	1.38149
$545$	13364.7	1.05042
$546$	−6389.42	−0.500809
$547$	8546.77	0.668069	0.334034	−	0.942561i	$-0.391590\pi$
$547$	8546.77	0.668069	0.334034	+	0.942561i	$0.391590\pi$
$548$	4100.55	0.319647
$549$	7095.84	0.551627
$550$	2040.69	0.158210
$551$	−19060.2	−1.47367
$552$	−10108.5	−0.779433
$553$	3445.95	0.264985
$554$	−5421.11	−0.415741
$555$	−8342.32	−0.638039
$556$	6745.12	0.514490
$557$	−16689.2	−1.26956	−0.634778	−	0.772695i	$-0.718908\pi$
$557$	−16689.2	−1.26956	−0.634778	+	0.772695i	$0.718908\pi$
$558$	−2374.10	−0.180114
$559$	14842.0	1.12299
$560$	190.193	0.0143520
$561$	−3908.82	−0.294172
$562$	6234.23	0.467927
$563$	−6844.25	−0.512346	−0.256173	−	0.966631i	$-0.582462\pi$
$563$	−6844.25	−0.512346	−0.256173	+	0.966631i	$0.582462\pi$
$564$	7697.59	0.574693
$565$	6944.41	0.517086
$566$	14121.0	1.04867
$567$	−1464.31	−0.108457
$568$	−3188.58	−0.235546
$569$	−12047.3	−0.887611	−0.443806	−	0.896123i	$-0.646372\pi$
$569$	−12047.3	−0.887611	−0.443806	+	0.896123i	$0.646372\pi$
$570$	−3641.72	−0.267605
$571$	−14990.8	−1.09868	−0.549338	−	0.835600i	$-0.685120\pi$
$571$	−14990.8	−1.09868	−0.549338	+	0.835600i	$0.685120\pi$
$572$	4293.72	0.313863
$573$	−2761.51	−0.201333
$574$	5868.39	0.426728
$575$	12732.4	0.923439
$576$	2988.52	0.216183
$577$	−5773.70	−0.416572	−0.208286	−	0.978068i	$-0.566788\pi$
$577$	−5773.70	−0.416572	−0.208286	+	0.978068i	$0.566788\pi$
$578$	−8212.61	−0.591002
$579$	−5610.16	−0.402677
$580$	5246.16	0.375577
$581$	−7423.10	−0.530055
$582$	−7924.19	−0.564379
$583$	5309.69	0.377195
$584$	12523.1	0.887348
$585$	−3721.72	−0.263033
$586$	−8756.54	−0.617286
$587$	−1595.79	−0.112207	−0.0561034	−	0.998425i	$-0.517868\pi$
$587$	−1595.79	−0.112207	−0.0561034	+	0.998425i	$0.517868\pi$
$588$	235.540	0.0165195
$589$	16313.1	1.14121
$590$	652.467	0.0455282
$591$	−1429.53	−0.0994978
$592$	753.698	0.0523256
$593$	5301.19	0.367106	0.183553	−	0.983010i	$-0.441240\pi$
$593$	5301.19	0.367106	0.183553	+	0.983010i	$0.441240\pi$
$594$	−639.318	−0.0441608
$595$	11000.8	0.757966
$596$	4887.20	0.335885
$597$	12492.2	0.856405
$598$	−17405.2	−1.19022
$599$	12610.9	0.860211	0.430105	−	0.902779i	$-0.358476\pi$
$599$	12610.9	0.860211	0.430105	+	0.902779i	$0.358476\pi$
$600$	−5896.93	−0.401236
$601$	−27751.9	−1.88357	−0.941783	−	0.336221i	$-0.890851\pi$
$601$	−27751.9	−1.88357	−0.941783	+	0.336221i	$0.890851\pi$
$602$	−7175.37	−0.485791
$603$	2595.37	0.175276
$604$	9340.76	0.629255
$605$	−7183.80	−0.482749
$606$	6856.91	0.459642
$607$	−20229.0	−1.35267	−0.676334	−	0.736595i	$-0.736433\pi$
$607$	−20229.0	−1.35267	−0.676334	+	0.736595i	$0.736433\pi$
$608$	−19699.5	−1.31402
$609$	9417.09	0.626601
$610$	−8719.02	−0.578726
$611$	35119.0	2.32531
$612$	4262.83	0.281560
$613$	−21903.1	−1.44316	−0.721580	−	0.692331i	$-0.756584\pi$
$613$	−21903.1	−1.44316	−0.721580	+	0.692331i	$0.756584\pi$
$614$	7743.67	0.508972
$615$	3418.24	0.224125
$616$	−5500.25	−0.359759
$617$	8023.42	0.523519	0.261759	−	0.965133i	$-0.415697\pi$
$617$	8023.42	0.523519	0.261759	+	0.965133i	$0.415697\pi$
$618$	2497.54	0.162566
$619$	−6262.61	−0.406649	−0.203324	−	0.979111i	$-0.565175\pi$
$619$	−6262.61	−0.406649	−0.203324	+	0.979111i	$0.565175\pi$
$620$	−4490.05	−0.290846
$621$	−3988.86	−0.257758
$622$	6287.64	0.405324
$623$	−23353.2	−1.50181
$624$	336.244	0.0215714
$625$	2575.56	0.164836
$626$	3210.35	0.204970
$627$	4392.93	0.279804
$628$	5991.81	0.380731
$629$	43594.1	2.76345
$630$	1799.27	0.113785
$631$	−6018.53	−0.379705	−0.189852	−	0.981813i	$-0.560801\pi$
$631$	−6018.53	−0.379705	−0.189852	+	0.981813i	$0.560801\pi$
$632$	−4347.53	−0.273632
$633$	−7497.62	−0.470780
$634$	−536.189	−0.0335880
$635$	−7898.42	−0.493605
$636$	−5790.56	−0.361023
$637$	1074.61	0.0668409
$638$	4111.51	0.255135
$639$	−1258.23	−0.0778948
$640$	5272.74	0.325661
$641$	3669.06	0.226083	0.113041	−	0.993590i	$-0.463941\pi$
$641$	3669.06	0.226083	0.113041	+	0.993590i	$0.463941\pi$
$642$	6588.19	0.405008
$643$	−9099.99	−0.558116	−0.279058	−	0.960274i	$-0.590022\pi$
$643$	−9099.99	−0.558116	−0.279058	+	0.960274i	$0.590022\pi$
$644$	−12951.4	−0.792482
$645$	−4179.53	−0.255145
$646$	19030.4	1.15904
$647$	−1873.56	−0.113844	−0.0569222	−	0.998379i	$-0.518129\pi$
$647$	−1873.56	−0.113844	−0.0569222	+	0.998379i	$0.518129\pi$
$648$	1847.42	0.111996
$649$	−787.058	−0.0476036
$650$	−10153.5	−0.612699
$651$	−8059.84	−0.485238
$652$	−17770.7	−1.06741
$653$	−23009.5	−1.37892	−0.689458	−	0.724325i	$-0.742151\pi$
$653$	−23009.5	−1.37892	−0.689458	+	0.724325i	$0.742151\pi$
$654$	−11422.8	−0.682975
$655$	−13205.0	−0.787729
$656$	−308.825	−0.0183805
$657$	4941.69	0.293445
$658$	−16978.3	−1.00590
$659$	−10038.1	−0.593368	−0.296684	−	0.954976i	$-0.595881\pi$
$659$	−10038.1	−0.593368	−0.296684	+	0.954976i	$0.595881\pi$
$660$	−1209.12	−0.0713104
$661$	−21406.5	−1.25963	−0.629816	−	0.776744i	$-0.716870\pi$
$661$	−21406.5	−1.25963	−0.629816	+	0.776744i	$0.716870\pi$
$662$	7336.38	0.430720
$663$	19448.5	1.13924
$664$	9365.23	0.547351
$665$	−12363.3	−0.720944
$666$	7130.16	0.414847
$667$	25652.7	1.48917
$668$	−19171.0	−1.11040
$669$	7702.65	0.445144
$670$	−3189.07	−0.183887
$671$	10517.6	0.605107
$672$	9732.98	0.558717
$673$	−934.831	−0.0535439	−0.0267720	−	0.999642i	$-0.508523\pi$
$673$	−934.831	−0.0535439	−0.0267720	+	0.999642i	$0.508523\pi$
$674$	5762.30	0.329311
$675$	−2326.96	−0.132688
$676$	−10709.5	−0.609324
$677$	−3937.51	−0.223531	−0.111766	−	0.993735i	$-0.535651\pi$
$677$	−3937.51	−0.223531	−0.111766	+	0.993735i	$0.535651\pi$
$678$	−5935.37	−0.336204
$679$	−26901.9	−1.52047
$680$	−13879.0	−0.782699
$681$	−2907.05	−0.163581
$682$	−3518.93	−0.197576
$683$	−12374.6	−0.693268	−0.346634	−	0.938001i	$-0.612675\pi$
$683$	−12374.6	−0.693268	−0.346634	+	0.938001i	$0.612675\pi$
$684$	−4790.79	−0.267807
$685$	−5268.22	−0.293852
$686$	−11525.8	−0.641482
$687$	12651.5	0.702598
$688$	377.605	0.0209245
$689$	−26418.5	−1.46076
$690$	4901.32	0.270420
$691$	−5887.06	−0.324102	−0.162051	−	0.986782i	$-0.551811\pi$
$691$	−5887.06	−0.324102	−0.162051	+	0.986782i	$0.551811\pi$
$692$	10791.2	0.592802
$693$	−2170.42	−0.118972
$694$	−823.443	−0.0450396
$695$	−8665.87	−0.472971
$696$	−11880.9	−0.647047
$697$	−17862.5	−0.970720
$698$	11725.8	0.635855
$699$	1146.26	0.0620252
$700$	−7555.40	−0.407953
$701$	7809.40	0.420766	0.210383	−	0.977619i	$-0.432529\pi$
$701$	7809.40	0.420766	0.210383	+	0.977619i	$0.432529\pi$
$702$	3180.95	0.171022
$703$	−48993.3	−2.62847
$704$	4429.64	0.237142
$705$	−9889.56	−0.528316
$706$	11155.1	0.594656
$707$	23278.6	1.23830
$708$	858.339	0.0455626
$709$	13649.5	0.723018	0.361509	−	0.932369i	$-0.382262\pi$
$709$	13649.5	0.723018	0.361509	+	0.932369i	$0.382262\pi$
$710$	1546.05	0.0817215
$711$	−1715.55	−0.0904898
$712$	29463.2	1.55082
$713$	−21955.5	−1.15321
$714$	−9402.37	−0.492822
$715$	−5516.41	−0.288534
$716$	10332.1	0.539288
$717$	−12209.1	−0.635923
$718$	−10248.7	−0.532701
$719$	475.869	0.0246828	0.0123414	−	0.999924i	$-0.496072\pi$
$719$	475.869	0.0246828	0.0123414	+	0.999924i	$0.496072\pi$
$720$	−94.6868	−0.00490107
$721$	8478.92	0.437963
$722$	−9212.60	−0.474872
$723$	13719.3	0.705709
$724$	−6932.57	−0.355866
$725$	14964.9	0.766594
$726$	6139.97	0.313878
$727$	−8057.43	−0.411050	−0.205525	−	0.978652i	$-0.565890\pi$
$727$	−8057.43	−0.411050	−0.205525	+	0.978652i	$0.565890\pi$
$728$	27366.7	1.39324
$729$	729.000	0.0370370
$730$	−6072.11	−0.307861
$731$	21840.8	1.10508
$732$	−11470.1	−0.579164
$733$	28918.2	1.45719	0.728594	−	0.684946i	$-0.240174\pi$
$733$	28918.2	1.45719	0.728594	+	0.684946i	$0.240174\pi$
$734$	1189.19	0.0598007
$735$	−302.612	−0.0151864
$736$	26513.2	1.32784
$737$	3846.91	0.192270
$738$	−2921.56	−0.145724
$739$	−25403.3	−1.26451	−0.632256	−	0.774760i	$-0.717871\pi$
$739$	−25403.3	−1.26451	−0.632256	+	0.774760i	$0.717871\pi$
$740$	13485.0	0.669890
$741$	−21857.2	−1.08359
$742$	12772.0	0.631909
$743$	−30468.3	−1.50441	−0.752204	−	0.658930i	$-0.771009\pi$
$743$	−30468.3	−1.50441	−0.752204	+	0.658930i	$0.771009\pi$
$744$	10168.6	0.501072
$745$	−6278.89	−0.308779
$746$	−5938.77	−0.291466
$747$	3695.56	0.181009
$748$	6318.44	0.308857
$749$	22366.3	1.09112
$750$	7006.29	0.341111
$751$	−1951.23	−0.0948088	−0.0474044	−	0.998876i	$-0.515095\pi$
$751$	−1951.23	−0.0948088	−0.0474044	+	0.998876i	$0.515095\pi$
$752$	893.485	0.0433272
$753$	12791.0	0.619029
$754$	−20456.9	−0.988061
$755$	−12000.6	−0.578474
$756$	2366.99	0.113871
$757$	13529.5	0.649588	0.324794	−	0.945785i	$-0.394705\pi$
$757$	13529.5	0.649588	0.324794	+	0.945785i	$0.394705\pi$
$758$	−12213.8	−0.585257
$759$	−5912.37	−0.282747
$760$	15597.9	0.744469
$761$	31379.5	1.49475	0.747376	−	0.664402i	$-0.231314\pi$
$761$	31379.5	1.49475	0.747376	+	0.664402i	$0.231314\pi$
$762$	6750.75	0.320937
$763$	−38779.2	−1.83998
$764$	4463.86	0.211383
$765$	−5476.72	−0.258838
$766$	7243.52	0.341670
$767$	3916.03	0.184354
$768$	−12476.0	−0.586182
$769$	−1805.38	−0.0846604	−0.0423302	−	0.999104i	$-0.513478\pi$
$769$	−1805.38	−0.0846604	−0.0423302	+	0.999104i	$0.513478\pi$
$770$	2666.91	0.124817
$771$	11275.8	0.526703
$772$	9068.58	0.422779
$773$	27743.2	1.29089	0.645443	−	0.763808i	$-0.276673\pi$
$773$	27743.2	1.29089	0.645443	+	0.763808i	$0.276673\pi$
$774$	3572.23	0.165893
$775$	−12808.0	−0.593649
$776$	33940.3	1.57008
$777$	24206.2	1.11762
$778$	8340.38	0.384341
$779$	20074.8	0.923306
$780$	6016.01	0.276164
$781$	−1864.97	−0.0854467
$782$	−25612.6	−1.17123
$783$	−4688.26	−0.213978
$784$	27.3399	0.00124544
$785$	−7698.04	−0.350006
$786$	11286.3	0.512173
$787$	−2980.67	−0.135005	−0.0675027	−	0.997719i	$-0.521503\pi$
$787$	−2980.67	−0.135005	−0.0675027	+	0.997719i	$0.521503\pi$
$788$	2310.78	0.104465
$789$	−8498.17	−0.383451
$790$	2107.99	0.0949352
$791$	−20150.0	−0.905755
$792$	2738.28	0.122854
$793$	−52330.6	−2.34340
$794$	−18270.1	−0.816603
$795$	7439.49	0.331889
$796$	−20193.2	−0.899156
$797$	26151.2	1.16226	0.581131	−	0.813810i	$-0.302610\pi$
$797$	26151.2	1.16226	0.581131	+	0.813810i	$0.302610\pi$
$798$	10566.9	0.468751
$799$	51679.5	2.28822
$800$	15466.8	0.683544
$801$	11626.3	0.512854
$802$	−13620.2	−0.599684
$803$	7324.66	0.321895
$804$	−4195.31	−0.184026
$805$	16639.5	0.728529
$806$	17508.6	0.765152
$807$	−22054.6	−0.962030
$808$	−29369.0	−1.27871
$809$	−30368.8	−1.31979	−0.659895	−	0.751358i	$-0.729399\pi$
$809$	−30368.8	−1.31979	−0.659895	+	0.751358i	$0.729399\pi$
$810$	−895.760	−0.0388565
$811$	−5497.72	−0.238041	−0.119020	−	0.992892i	$-0.537975\pi$
$811$	−5497.72	−0.238041	−0.119020	+	0.992892i	$0.537975\pi$
$812$	−15222.3	−0.657880
$813$	−12159.2	−0.524527
$814$	10568.4	0.455066
$815$	22831.0	0.981272
$816$	494.801	0.0212273
$817$	−24545.8	−1.05110
$818$	−6581.02	−0.281296
$819$	10799.0	0.460742
$820$	−5525.43	−0.235313
$821$	−33287.7	−1.41504	−0.707521	−	0.706692i	$-0.750186\pi$
$821$	−33287.7	−1.41504	−0.707521	+	0.706692i	$0.750186\pi$
$822$	4502.74	0.191060
$823$	−13109.6	−0.555253	−0.277626	−	0.960689i	$-0.589548\pi$
$823$	−13109.6	−0.555253	−0.277626	+	0.960689i	$0.589548\pi$
$824$	−10697.3	−0.452254
$825$	−3449.06	−0.145552
$826$	−1893.21	−0.0797495
$827$	31629.7	1.32995	0.664977	−	0.746864i	$-0.268441\pi$
$827$	31629.7	1.32995	0.664977	+	0.746864i	$0.268441\pi$
$828$	6447.83	0.270625
$829$	23175.2	0.970938	0.485469	−	0.874254i	$-0.338649\pi$
$829$	23175.2	0.970938	0.485469	+	0.874254i	$0.338649\pi$
$830$	−4540.92	−0.189901
$831$	9162.42	0.382480
$832$	−22039.8	−0.918380
$833$	1581.35	0.0657748
$834$	7406.69	0.307521
$835$	24630.1	1.02079
$836$	−7100.99	−0.293771
$837$	4012.56	0.165704
$838$	20158.7	0.830990
$839$	24731.6	1.01767	0.508837	−	0.860863i	$-0.330076\pi$
$839$	24731.6	1.01767	0.508837	+	0.860863i	$0.330076\pi$
$840$	−7706.50	−0.316547
$841$	5761.61	0.236238
$842$	−6769.91	−0.277086
$843$	−10536.7	−0.430491
$844$	12119.6	0.494281
$845$	13759.1	0.560151
$846$	8452.58	0.343506
$847$	20844.6	0.845607
$848$	−672.130	−0.0272182
$849$	−23866.4	−0.964773
$850$	−14941.5	−0.602927
$851$	65939.1	2.65613
$852$	2033.87	0.0817833
$853$	26779.6	1.07493	0.537466	−	0.843285i	$-0.319382\pi$
$853$	26779.6	1.07493	0.537466	+	0.843285i	$0.319382\pi$
$854$	25299.2	1.01373
$855$	6155.02	0.246195
$856$	−28218.0	−1.12672
$857$	32339.7	1.28904	0.644518	−	0.764589i	$-0.277058\pi$
$857$	32339.7	1.28904	0.644518	+	0.764589i	$0.277058\pi$
$858$	4714.86	0.187602
$859$	41682.2	1.65562	0.827810	−	0.561009i	$-0.189587\pi$
$859$	41682.2	1.65562	0.827810	+	0.561009i	$0.189587\pi$
$860$	6756.02	0.267882
$861$	−9918.40	−0.392588
$862$	2320.08	0.0916732
$863$	−15307.7	−0.603802	−0.301901	−	0.953339i	$-0.597621\pi$
$863$	−15307.7	−0.603802	−0.301901	+	0.953339i	$0.597621\pi$
$864$	−4845.52	−0.190796
$865$	−13864.1	−0.544963
$866$	−14050.4	−0.551332
$867$	13880.5	0.543720
$868$	13028.4	0.509461
$869$	−2542.82	−0.0992628
$870$	5760.71	0.224490
$871$	−19140.4	−0.744601
$872$	48925.1	1.90002
$873$	13393.0	0.519226
$874$	28784.8	1.11403
$875$	23785.7	0.918974
$876$	−7988.03	−0.308094
$877$	−42064.0	−1.61961	−0.809807	−	0.586696i	$-0.800428\pi$
$877$	−42064.0	−1.61961	−0.809807	+	0.586696i	$0.800428\pi$
$878$	−4562.56	−0.175374
$879$	14799.8	0.567900
$880$	−140.346	−0.00537623
$881$	−983.391	−0.0376065	−0.0188032	−	0.999823i	$-0.505986\pi$
$881$	−983.391	−0.0376065	−0.0188032	+	0.999823i	$0.505986\pi$
$882$	258.642	0.00987405
$883$	3067.46	0.116906	0.0584531	−	0.998290i	$-0.481383\pi$
$883$	3067.46	0.116906	0.0584531	+	0.998290i	$0.481383\pi$
$884$	−31437.6	−1.19611
$885$	−1102.76	−0.0418857
$886$	1123.79	0.0426123
$887$	15178.8	0.574583	0.287291	−	0.957843i	$-0.407245\pi$
$887$	15178.8	0.574583	0.287291	+	0.957843i	$0.407245\pi$
$888$	−30539.3	−1.15409
$889$	22918.2	0.864624
$890$	−14285.9	−0.538048
$891$	1080.54	0.0406278
$892$	−12451.0	−0.467366
$893$	−58080.1	−2.17646
$894$	5366.55	0.200765
$895$	−13274.3	−0.495767
$896$	−15299.4	−0.570445
$897$	29417.1	1.09499
$898$	−18010.2	−0.669275
$899$	−25805.1	−0.957340
$900$	3761.42	0.139312
$901$	−38876.2	−1.43746
$902$	−4330.38	−0.159851
$903$	12127.4	0.446925
$904$	25421.9	0.935310
$905$	8906.70	0.327148
$906$	10256.9	0.376118
$907$	−2893.09	−0.105914	−0.0529568	−	0.998597i	$-0.516865\pi$
$907$	−2893.09	−0.105914	−0.0529568	+	0.998597i	$0.516865\pi$
$908$	4699.13	0.171747
$909$	−11589.1	−0.422869
$910$	−13269.3	−0.483377
$911$	−2299.05	−0.0836124	−0.0418062	−	0.999126i	$-0.513311\pi$
$911$	−2299.05	−0.0836124	−0.0418062	+	0.999126i	$0.513311\pi$
$912$	−556.083	−0.0201905
$913$	5477.63	0.198557
$914$	12733.7	0.460825
$915$	14736.4	0.532425
$916$	−20450.6	−0.737671
$917$	38315.8	1.37983
$918$	4680.93	0.168294
$919$	−45794.9	−1.64378	−0.821891	−	0.569645i	$-0.807081\pi$
$919$	−45794.9	−1.64378	−0.821891	+	0.569645i	$0.807081\pi$
$920$	−20993.0	−0.752301
$921$	−13087.9	−0.468252
$922$	28328.7	1.01188
$923$	9279.22	0.330909
$924$	3508.40	0.124911
$925$	38466.5	1.36732
$926$	−18614.0	−0.660578
$927$	−4221.20	−0.149560
$928$	31162.0	1.10231
$929$	−43137.0	−1.52344	−0.761721	−	0.647905i	$-0.775646\pi$
$929$	−43137.0	−1.52344	−0.761721	+	0.647905i	$0.775646\pi$
$930$	−4930.44	−0.173844
$931$	−1777.20	−0.0625621
$932$	−1852.88	−0.0651214
$933$	−10627.0	−0.372896
$934$	21071.4	0.738197
$935$	−8117.69	−0.283932
$936$	−13624.4	−0.475777
$937$	20420.6	0.711966	0.355983	−	0.934492i	$-0.384146\pi$
$937$	20420.6	0.711966	0.355983	+	0.934492i	$0.384146\pi$
$938$	9253.44	0.322106
$939$	−5425.94	−0.188572
$940$	15986.1	0.554689
$941$	−21235.7	−0.735668	−0.367834	−	0.929892i	$-0.619900\pi$
$941$	−21235.7	−0.735668	−0.367834	+	0.929892i	$0.619900\pi$
$942$	6579.49	0.227571
$943$	−27018.3	−0.933020
$944$	99.6303	0.00343505
$945$	−3041.02	−0.104682
$946$	5294.82	0.181976
$947$	−21631.9	−0.742283	−0.371141	−	0.928576i	$-0.621034\pi$
$947$	−21631.9	−0.742283	−0.371141	+	0.928576i	$0.621034\pi$
$948$	2773.12	0.0950070
$949$	−36444.1	−1.24660
$950$	16792.0	0.573478
$951$	906.234	0.0309008
$952$	40271.5	1.37102
$953$	869.752	0.0295635	0.0147818	−	0.999891i	$-0.495295\pi$
$953$	869.752	0.0295635	0.0147818	+	0.999891i	$0.495295\pi$
$954$	−6358.51	−0.215791
$955$	−5734.99	−0.194324
$956$	19735.5	0.667668
$957$	−6949.03	−0.234723
$958$	16988.3	0.572929
$959$	15286.4	0.514726
$960$	6206.44	0.208658
$961$	−7705.09	−0.258638
$962$	−52583.6	−1.76233
$963$	−11135.0	−0.372605
$964$	−22176.7	−0.740937
$965$	−11651.0	−0.388661
$966$	−14221.7	−0.473682
$967$	45869.5	1.52540	0.762700	−	0.646752i	$-0.223873\pi$
$967$	45869.5	1.52540	0.762700	+	0.646752i	$0.223873\pi$
$968$	−26298.3	−0.873200
$969$	−32164.0	−1.06631
$970$	−16456.6	−0.544733
$971$	4456.05	0.147272	0.0736361	−	0.997285i	$-0.476540\pi$
$971$	4456.05	0.147272	0.0736361	+	0.997285i	$0.476540\pi$
$972$	−1178.40	−0.0388859
$973$	25145.0	0.828481
$974$	−19894.6	−0.654481
$975$	17160.9	0.563680
$976$	−1331.38	−0.0436642
$977$	−34528.0	−1.13065	−0.565326	−	0.824867i	$-0.691250\pi$
$977$	−34528.0	−1.13065	−0.565326	+	0.824867i	$0.691250\pi$
$978$	−19513.6	−0.638013
$979$	17232.7	0.562575
$980$	489.159	0.0159445
$981$	19306.1	0.628334
$982$	−13703.3	−0.445307
$983$	17284.8	0.560834	0.280417	−	0.959878i	$-0.409527\pi$
$983$	17284.8	0.560834	0.280417	+	0.959878i	$0.409527\pi$
$984$	12513.4	0.405399
$985$	−2968.80	−0.0960344
$986$	−30103.5	−0.972303
$987$	28695.7	0.925425
$988$	35331.2	1.13769
$989$	33035.7	1.06216
$990$	−1327.71	−0.0426237
$991$	−49628.1	−1.59081	−0.795403	−	0.606081i	$-0.792741\pi$
$991$	−49628.1	−1.59081	−0.795403	+	0.606081i	$0.792741\pi$
$992$	−26670.7	−0.853625
$993$	−12399.5	−0.396260
$994$	−4486.04	−0.143148
$995$	25943.4	0.826594
$996$	−5973.71	−0.190045
$997$	−1153.92	−0.0366549	−0.0183274	−	0.999832i	$-0.505834\pi$
$997$	−1153.92	−0.0366549	−0.0183274	+	0.999832i	$0.505834\pi$
$998$	6531.53	0.207166
$999$	−12051.0	−0.381657

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.a.1.4	✓	7
3.2	odd	2		531.4.a.d.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.a.1.4	✓	7	1.1	even	1	trivial
531.4.a.d.1.4		7	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)

\(f(q)\)	\(=\)	\(q-1.77500 q^{2} +3.00000 q^{3} -4.84937 q^{4} +6.23028 q^{5} -5.32500 q^{6} -18.0779 q^{7} +22.8076 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.77500 q^{2} +3.00000 q^{3} -4.84937 q^{4} +6.23028 q^{5} -5.32500 q^{6} -18.0779 q^{7} +22.8076 q^{8} +9.00000 q^{9} -11.0588 q^{10} +13.3400 q^{11} -14.5481 q^{12} -66.3734 q^{13} +32.0883 q^{14} +18.6908 q^{15} -1.68865 q^{16} -97.6720 q^{17} -15.9750 q^{18} +109.769 q^{19} -30.2129 q^{20} -54.2336 q^{21} -23.6785 q^{22} -147.736 q^{23} +68.4229 q^{24} -86.1836 q^{25} +117.813 q^{26} +27.0000 q^{27} +87.6663 q^{28} -173.639 q^{29} -33.1763 q^{30} +148.613 q^{31} -179.464 q^{32} +40.0199 q^{33} +173.368 q^{34} -112.630 q^{35} -43.6443 q^{36} -446.332 q^{37} -194.840 q^{38} -199.120 q^{39} +142.098 q^{40} +182.883 q^{41} +96.2648 q^{42} -223.613 q^{43} -64.6904 q^{44} +56.0725 q^{45} +262.231 q^{46} -529.113 q^{47} -5.06595 q^{48} -16.1904 q^{49} +152.976 q^{50} -293.016 q^{51} +321.869 q^{52} +398.029 q^{53} -47.9250 q^{54} +83.1117 q^{55} -412.314 q^{56} +329.306 q^{57} +308.210 q^{58} -59.0000 q^{59} -90.6388 q^{60} +788.427 q^{61} -263.789 q^{62} -162.701 q^{63} +332.058 q^{64} -413.525 q^{65} -71.0354 q^{66} +288.375 q^{67} +473.648 q^{68} -443.207 q^{69} +199.919 q^{70} -139.803 q^{71} +205.269 q^{72} +549.077 q^{73} +792.240 q^{74} -258.551 q^{75} -532.310 q^{76} -241.158 q^{77} +353.439 q^{78} -190.617 q^{79} -10.5208 q^{80} +81.0000 q^{81} -324.617 q^{82} +410.618 q^{83} +262.999 q^{84} -608.524 q^{85} +396.914 q^{86} -520.918 q^{87} +304.253 q^{88} +1291.81 q^{89} -99.5288 q^{90} +1199.89 q^{91} +716.425 q^{92} +445.840 q^{93} +939.176 q^{94} +683.891 q^{95} -538.392 q^{96} +1488.11 q^{97} +28.7380 q^{98} +120.060 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9}+O(q^{10}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9	\( 7 q - 8 q^{2} + 21 q^{3} + 22 q^{4} - 28 q^{5} - 24 q^{6} - 59 q^{7} - 117 q^{8} + 63 q^{9} - 79 q^{10} - 131 q^{11} + 66 q^{12} - 123 q^{13} - 117 q^{14} - 84 q^{15} + 202 q^{16} - 235 q^{17} - 72 q^{18} - 80 q^{19} + 61 q^{20} - 177 q^{21} + 688 q^{22} - 274 q^{23} - 351 q^{24} + 193 q^{25} - 180 q^{26} + 189 q^{27} - 118 q^{28} - 406 q^{29} - 237 q^{30} - 346 q^{31} - 854 q^{32} - 393 q^{33} + 178 q^{34} - 424 q^{35} + 198 q^{36} - 157 q^{37} - 129 q^{38} - 369 q^{39} - 590 q^{40} - 825 q^{41} - 351 q^{42} - 815 q^{43} - 1690 q^{44} - 252 q^{45} + 1457 q^{46} - 1196 q^{47} + 606 q^{48} + 914 q^{49} + 713 q^{50} - 705 q^{51} + 1030 q^{52} - 900 q^{53} - 216 q^{54} - 1044 q^{55} + 2172 q^{56} - 240 q^{57} + 1242 q^{58} - 413 q^{59} + 183 q^{60} + 420 q^{61} + 646 q^{62} - 531 q^{63} + 3541 q^{64} + 190 q^{65} + 2064 q^{66} + 1316 q^{67} - 611 q^{68} - 822 q^{69} + 4658 q^{70} - 173 q^{71} - 1053 q^{72} - 418 q^{73} + 660 q^{74} + 579 q^{75} + 1540 q^{76} - 753 q^{77} - 540 q^{78} + 2635 q^{79} + 6155 q^{80} + 567 q^{81} - 125 q^{82} + 457 q^{83} - 354 q^{84} + 1270 q^{85} + 3482 q^{86} - 1218 q^{87} + 7685 q^{88} + 592 q^{89} - 711 q^{90} + 3179 q^{91} - 3500 q^{92} - 1038 q^{93} + 2064 q^{94} - 2250 q^{95} - 2562 q^{96} - 1906 q^{97} + 2994 q^{98} - 1179 q^{99}+O(q^{100}) \) 7 * q - 8 * q^2 + 21 * q^3 + 22 * q^4 - 28 * q^5 - 24 * q^6 - 59 * q^7 - 117 * q^8 + 63 * q^9 - 79 * q^10 - 131 * q^11 + 66 * q^12 - 123 * q^13 - 117 * q^14 - 84 * q^15 + 202 * q^16 - 235 * q^17 - 72 * q^18 - 80 * q^19 + 61 * q^20 - 177 * q^21 + 688 * q^22 - 274 * q^23 - 351 * q^24 + 193 * q^25 - 180 * q^26 + 189 * q^27 - 118 * q^28 - 406 * q^29 - 237 * q^30 - 346 * q^31 - 854 * q^32 - 393 * q^33 + 178 * q^34 - 424 * q^35 + 198 * q^36 - 157 * q^37 - 129 * q^38 - 369 * q^39 - 590 * q^40 - 825 * q^41 - 351 * q^42 - 815 * q^43 - 1690 * q^44 - 252 * q^45 + 1457 * q^46 - 1196 * q^47 + 606 * q^48 + 914 * q^49 + 713 * q^50 - 705 * q^51 + 1030 * q^52 - 900 * q^53 - 216 * q^54 - 1044 * q^55 + 2172 * q^56 - 240 * q^57 + 1242 * q^58 - 413 * q^59 + 183 * q^60 + 420 * q^61 + 646 * q^62 - 531 * q^63 + 3541 * q^64 + 190 * q^65 + 2064 * q^66 + 1316 * q^67 - 611 * q^68 - 822 * q^69 + 4658 * q^70 - 173 * q^71 - 1053 * q^72 - 418 * q^73 + 660 * q^74 + 579 * q^75 + 1540 * q^76 - 753 * q^77 - 540 * q^78 + 2635 * q^79 + 6155 * q^80 + 567 * q^81 - 125 * q^82 + 457 * q^83 - 354 * q^84 + 1270 * q^85 + 3482 * q^86 - 1218 * q^87 + 7685 * q^88 + 592 * q^89 - 711 * q^90 + 3179 * q^91 - 3500 * q^92 - 1038 * q^93 + 2064 * q^94 - 2250 * q^95 - 2562 * q^96 - 1906 * q^97 + 2994 * q^98 - 1179 * q^99

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