LMFDB - Embedded newform 177.4.a.c.1.8

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			1.8
Root			$5.26363$ 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)

$f(q)$	$=$	$q+5.26363 q^{2} -3.00000 q^{3} +19.7058 q^{4} +11.8799 q^{5} -15.7909 q^{6} -3.36662 q^{7} +61.6149 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+5.26363 q^{2} -3.00000 q^{3} +19.7058 q^{4} +11.8799 q^{5} -15.7909 q^{6} -3.36662 q^{7} +61.6149 q^{8} +9.00000 q^{9} +62.5311 q^{10} -3.40311 q^{11} -59.1173 q^{12} -54.8553 q^{13} -17.7206 q^{14} -35.6396 q^{15} +166.672 q^{16} +68.7547 q^{17} +47.3727 q^{18} +8.88097 q^{19} +234.102 q^{20} +10.0999 q^{21} -17.9127 q^{22} +80.8007 q^{23} -184.845 q^{24} +16.1310 q^{25} -288.738 q^{26} -27.0000 q^{27} -66.3419 q^{28} -235.255 q^{29} -187.593 q^{30} +145.469 q^{31} +384.378 q^{32} +10.2093 q^{33} +361.899 q^{34} -39.9950 q^{35} +177.352 q^{36} -309.554 q^{37} +46.7461 q^{38} +164.566 q^{39} +731.976 q^{40} +37.6647 q^{41} +53.1619 q^{42} -465.320 q^{43} -67.0609 q^{44} +106.919 q^{45} +425.305 q^{46} +271.410 q^{47} -500.015 q^{48} -331.666 q^{49} +84.9076 q^{50} -206.264 q^{51} -1080.97 q^{52} -82.0797 q^{53} -142.118 q^{54} -40.4284 q^{55} -207.434 q^{56} -26.6429 q^{57} -1238.29 q^{58} -59.0000 q^{59} -702.306 q^{60} -736.743 q^{61} +765.695 q^{62} -30.2996 q^{63} +689.850 q^{64} -651.673 q^{65} +53.7381 q^{66} +768.441 q^{67} +1354.86 q^{68} -242.402 q^{69} -210.519 q^{70} -164.728 q^{71} +554.534 q^{72} +445.018 q^{73} -1629.38 q^{74} -48.3930 q^{75} +175.006 q^{76} +11.4570 q^{77} +866.213 q^{78} +602.572 q^{79} +1980.03 q^{80} +81.0000 q^{81} +198.253 q^{82} -779.626 q^{83} +199.026 q^{84} +816.795 q^{85} -2449.27 q^{86} +705.764 q^{87} -209.682 q^{88} +1393.87 q^{89} +562.780 q^{90} +184.677 q^{91} +1592.24 q^{92} -436.407 q^{93} +1428.60 q^{94} +105.505 q^{95} -1153.13 q^{96} -269.591 q^{97} -1745.77 q^{98} -30.6280 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.26363	1.86097	0.930487	−	0.366325i	$-0.119384\pi$
$2$	5.26363	1.86097	0.930487	+	0.366325i	$0.119384\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	19.7058	2.46322
$5$	11.8799	1.06257	0.531283	−	0.847194i	$-0.321710\pi$
$5$	11.8799	1.06257	0.531283	+	0.847194i	$0.321710\pi$
$6$	−15.7909	−1.07443
$7$	−3.36662	−0.181781	−0.0908903	−	0.995861i	$-0.528971\pi$
$7$	−3.36662	−0.181781	−0.0908903	+	0.995861i	$0.528971\pi$
$8$	61.6149	2.72302
$9$	9.00000	0.333333
$10$	62.5311	1.97741
$11$	−3.40311	−0.0932796	−0.0466398	−	0.998912i	$-0.514851\pi$
$11$	−3.40311	−0.0932796	−0.0466398	+	0.998912i	$0.514851\pi$
$12$	−59.1173	−1.42214
$13$	−54.8553	−1.17032	−0.585158	−	0.810919i	$-0.698968\pi$
$13$	−54.8553	−1.17032	−0.585158	+	0.810919i	$0.698968\pi$
$14$	−17.7206	−0.338289
$15$	−35.6396	−0.613473
$16$	166.672	2.60424
$17$	68.7547	0.980909	0.490455	−	0.871467i	$-0.336831\pi$
$17$	68.7547	0.980909	0.490455	+	0.871467i	$0.336831\pi$
$18$	47.3727	0.620325
$19$	8.88097	0.107233	0.0536167	−	0.998562i	$-0.482925\pi$
$19$	8.88097	0.107233	0.0536167	+	0.998562i	$0.482925\pi$
$20$	234.102	2.61734
$21$	10.0999	0.104951
$22$	−17.9127	−0.173591
$23$	80.8007	0.732526	0.366263	−	0.930511i	$-0.380637\pi$
$23$	80.8007	0.732526	0.366263	+	0.930511i	$0.380637\pi$
$24$	−184.845	−1.57214
$25$	16.1310	0.129048
$26$	−288.738	−2.17793
$27$	−27.0000	−0.192450
$28$	−66.3419	−0.447766
$29$	−235.255	−1.50640	−0.753202	−	0.657789i	$-0.771492\pi$
$29$	−235.255	−1.50640	−0.753202	+	0.657789i	$0.771492\pi$
$30$	−187.593	−1.14166
$31$	145.469	0.842807	0.421404	−	0.906873i	$-0.361538\pi$
$31$	145.469	0.842807	0.421404	+	0.906873i	$0.361538\pi$
$32$	384.378	2.12341
$33$	10.2093	0.0538550
$34$	361.899	1.82545
$35$	−39.9950	−0.193154
$36$	177.352	0.821074
$37$	−309.554	−1.37542	−0.687709	−	0.725987i	$-0.741383\pi$
$37$	−309.554	−1.37542	−0.687709	+	0.725987i	$0.741383\pi$
$38$	46.7461	0.199558
$39$	164.566	0.675683
$40$	731.976	2.89339
$41$	37.6647	0.143469	0.0717346	−	0.997424i	$-0.477147\pi$
$41$	37.6647	0.143469	0.0717346	+	0.997424i	$0.477147\pi$
$42$	53.1619	0.195311
$43$	−465.320	−1.65025	−0.825124	−	0.564952i	$-0.808895\pi$
$43$	−465.320	−1.65025	−0.825124	+	0.564952i	$0.808895\pi$
$44$	−67.0609	−0.229768
$45$	106.919	0.354189
$46$	425.305	1.36321
$47$	271.410	0.842324	0.421162	−	0.906986i	$-0.361622\pi$
$47$	271.410	0.842324	0.421162	+	0.906986i	$0.361622\pi$
$48$	−500.015	−1.50356
$49$	−331.666	−0.966956
$50$	84.9076	0.240155
$51$	−206.264	−0.566328
$52$	−1080.97	−2.88275
$53$	−82.0797	−0.212727	−0.106363	−	0.994327i	$-0.533921\pi$
$53$	−82.0797	−0.212727	−0.106363	+	0.994327i	$0.533921\pi$
$54$	−142.118	−0.358145
$55$	−40.4284	−0.0991158
$56$	−207.434	−0.494992
$57$	−26.6429	−0.0619112
$58$	−1238.29	−2.80338
$59$	−59.0000	−0.130189
$59$	−59.0000	−0.130189
$60$	−702.306	−1.51112
$61$	−736.743	−1.54640	−0.773199	−	0.634164i	$-0.781345\pi$
$61$	−736.743	−1.54640	−0.773199	+	0.634164i	$0.781345\pi$
$62$	765.695	1.56844
$63$	−30.2996	−0.0605935
$64$	689.850	1.34736
$65$	−651.673	−1.24354
$66$	53.7381	0.100223
$67$	768.441	1.40119	0.700597	−	0.713557i	$-0.252917\pi$
$67$	768.441	1.40119	0.700597	+	0.713557i	$0.252917\pi$
$68$	1354.86	2.41620
$69$	−242.402	−0.422924
$70$	−210.519	−0.359454
$71$	−164.728	−0.275347	−0.137674	−	0.990478i	$-0.543963\pi$
$71$	−164.728	−0.275347	−0.137674	+	0.990478i	$0.543963\pi$
$72$	554.534	0.907673
$73$	445.018	0.713499	0.356749	−	0.934200i	$-0.383885\pi$
$73$	445.018	0.713499	0.356749	+	0.934200i	$0.383885\pi$
$74$	−1629.38	−2.55961
$75$	−48.3930	−0.0745059
$76$	175.006	0.264140
$77$	11.4570	0.0169564
$78$	866.213	1.25743
$79$	602.572	0.858160	0.429080	−	0.903266i	$-0.358838\pi$
$79$	602.572	0.858160	0.429080	+	0.903266i	$0.358838\pi$
$80$	1980.03	2.76718
$81$	81.0000	0.111111
$82$	198.253	0.266992
$83$	−779.626	−1.03102	−0.515512	−	0.856882i	$-0.672398\pi$
$83$	−779.626	−1.03102	−0.515512	+	0.856882i	$0.672398\pi$
$84$	199.026	0.258518
$85$	816.795	1.04228
$86$	−2449.27	−3.07107
$87$	705.764	0.869723
$88$	−209.682	−0.254002
$89$	1393.87	1.66011	0.830056	−	0.557681i	$-0.188309\pi$
$89$	1393.87	1.66011	0.830056	+	0.557681i	$0.188309\pi$
$90$	562.780	0.659136
$91$	184.677	0.212741
$92$	1592.24	1.80437
$93$	−436.407	−0.486595
$94$	1428.60	1.56754
$95$	105.505	0.113943
$96$	−1153.13	−1.22595
$97$	−269.591	−0.282194	−0.141097	−	0.989996i	$-0.545063\pi$
$97$	−269.591	−0.282194	−0.141097	+	0.989996i	$0.545063\pi$
$98$	−1745.77	−1.79948
$99$	−30.6280	−0.0310932
$100$	317.874	0.317874
$101$	936.949	0.923069	0.461534	−	0.887122i	$-0.347299\pi$
$101$	936.949	0.923069	0.461534	+	0.887122i	$0.347299\pi$
$102$	−1085.70	−1.05392
$103$	1461.76	1.39836	0.699182	−	0.714943i	$-0.253547\pi$
$103$	1461.76	1.39836	0.699182	+	0.714943i	$0.253547\pi$
$104$	−3379.90	−3.18679
$105$	119.985	0.111517
$106$	−432.037	−0.395879
$107$	830.314	0.750182	0.375091	−	0.926988i	$-0.377611\pi$
$107$	830.314	0.750182	0.375091	+	0.926988i	$0.377611\pi$
$108$	−532.056	−0.474047
$109$	1476.02	1.29703	0.648517	−	0.761200i	$-0.275390\pi$
$109$	1476.02	1.29703	0.648517	+	0.761200i	$0.275390\pi$
$110$	−212.800	−0.184452
$111$	928.663	0.794097
$112$	−561.120	−0.473401
$113$	−2355.58	−1.96101	−0.980507	−	0.196482i	$-0.937048\pi$
$113$	−2355.58	−1.96101	−0.980507	+	0.196482i	$0.937048\pi$
$114$	−140.238	−0.115215
$115$	959.900	0.778358
$116$	−4635.88	−3.71061
$117$	−493.698	−0.390106
$118$	−310.554	−0.242278
$119$	−231.471	−0.178310
$120$	−2195.93	−1.67050
$121$	−1319.42	−0.991299
$122$	−3877.94	−2.87781
$123$	−112.994	−0.0828320
$124$	2866.58	2.07602
$125$	−1293.35	−0.925445
$126$	−159.486	−0.112763
$127$	2512.25	1.75533	0.877663	−	0.479279i	$-0.159102\pi$
$127$	2512.25	1.75533	0.877663	+	0.479279i	$0.159102\pi$
$128$	556.092	0.384001
$129$	1395.96	0.952771
$130$	−3430.16	−2.31419
$131$	−907.845	−0.605487	−0.302744	−	0.953072i	$-0.597903\pi$
$131$	−907.845	−0.605487	−0.302744	+	0.953072i	$0.597903\pi$
$132$	201.183	0.132657
$133$	−29.8989	−0.0194929
$134$	4044.79	2.60758
$135$	−320.756	−0.204491
$136$	4236.31	2.67103
$137$	326.883	0.203850	0.101925	−	0.994792i	$-0.467500\pi$
$137$	326.883	0.203850	0.101925	+	0.994792i	$0.467500\pi$
$138$	−1275.91	−0.787051
$139$	3008.75	1.83597	0.917983	−	0.396620i	$-0.129817\pi$
$139$	3008.75	1.83597	0.917983	+	0.396620i	$0.129817\pi$
$140$	−788.132	−0.475781
$141$	−814.230	−0.486316
$142$	−867.069	−0.512414
$143$	186.678	0.109167
$144$	1500.04	0.868081
$145$	−2794.79	−1.60065
$146$	2342.41	1.32780
$147$	994.998	0.558272
$148$	−6100.01	−3.38796
$149$	−144.752	−0.0795873	−0.0397937	−	0.999208i	$-0.512670\pi$
$149$	−144.752	−0.0795873	−0.0397937	+	0.999208i	$0.512670\pi$
$150$	−254.723	−0.138654
$151$	−767.404	−0.413579	−0.206789	−	0.978385i	$-0.566302\pi$
$151$	−767.404	−0.413579	−0.206789	+	0.978385i	$0.566302\pi$
$152$	547.200	0.291998
$153$	618.792	0.326970
$154$	60.3052	0.0315554
$155$	1728.15	0.895539
$156$	3242.90	1.66436
$157$	3523.01	1.79087	0.895436	−	0.445190i	$-0.146864\pi$
$157$	3523.01	1.79087	0.895436	+	0.445190i	$0.146864\pi$
$158$	3171.71	1.59701
$159$	246.239	0.122818
$160$	4566.36	2.25626
$161$	−272.025	−0.133159
$162$	426.354	0.206775
$163$	−1448.84	−0.696209	−0.348104	−	0.937456i	$-0.613175\pi$
$163$	−1448.84	−0.696209	−0.348104	+	0.937456i	$0.613175\pi$
$164$	742.212	0.353397
$165$	121.285	0.0572245
$166$	−4103.66	−1.91871
$167$	−584.113	−0.270659	−0.135329	−	0.990801i	$-0.543209\pi$
$167$	−584.113	−0.270659	−0.135329	+	0.990801i	$0.543209\pi$
$168$	622.302	0.285784
$169$	812.102	0.369641
$170$	4299.31	1.93966
$171$	79.9287	0.0357445
$172$	−9169.49	−4.06493
$173$	2756.29	1.21131	0.605656	−	0.795727i	$-0.292911\pi$
$173$	2756.29	1.21131	0.605656	+	0.795727i	$0.292911\pi$
$174$	3714.88	1.61853
$175$	−54.3070	−0.0234584
$176$	−567.201	−0.242923
$177$	177.000	0.0751646
$178$	7336.81	3.08942
$179$	−3014.07	−1.25856	−0.629280	−	0.777179i	$-0.716650\pi$
$179$	−3014.07	−1.25856	−0.629280	+	0.777179i	$0.716650\pi$
$180$	2106.92	0.872446
$181$	−1361.03	−0.558921	−0.279461	−	0.960157i	$-0.590156\pi$
$181$	−1361.03	−0.558921	−0.279461	+	0.960157i	$0.590156\pi$
$182$	972.071	0.395905
$183$	2210.23	0.892813
$184$	4978.52	1.99468
$185$	−3677.46	−1.46147
$186$	−2297.09	−0.905540
$187$	−233.979	−0.0914988
$188$	5348.34	2.07483
$189$	90.8988	0.0349837
$190$	555.337	0.212044
$191$	182.879	0.0692810	0.0346405	−	0.999400i	$-0.488971\pi$
$191$	182.879	0.0692810	0.0346405	+	0.999400i	$0.488971\pi$
$192$	−2069.55	−0.777901
$193$	4052.56	1.51145	0.755725	−	0.654889i	$-0.227285\pi$
$193$	4052.56	1.51145	0.755725	+	0.654889i	$0.227285\pi$
$194$	−1419.03	−0.525155
$195$	1955.02	0.717958
$196$	−6535.73	−2.38183
$197$	1243.86	0.449856	0.224928	−	0.974375i	$-0.427785\pi$
$197$	1243.86	0.449856	0.224928	+	0.974375i	$0.427785\pi$
$198$	−161.214	−0.0578636
$199$	−191.567	−0.0682404	−0.0341202	−	0.999418i	$-0.510863\pi$
$199$	−191.567	−0.0682404	−0.0341202	+	0.999418i	$0.510863\pi$
$200$	993.910	0.351400
$201$	−2305.32	−0.808980
$202$	4931.75	1.71781
$203$	792.014	0.273835
$204$	−4064.59	−1.39499
$205$	447.451	0.152446
$206$	7694.17	2.60232
$207$	727.206	0.244175
$208$	−9142.81	−3.04779
$209$	−30.2229	−0.0100027
$210$	631.556	0.207531
$211$	3016.73	0.984268	0.492134	−	0.870520i	$-0.336217\pi$
$211$	3016.73	0.984268	0.492134	+	0.870520i	$0.336217\pi$
$212$	−1617.44	−0.523993
$213$	494.185	0.158972
$214$	4370.46	1.39607
$215$	−5527.94	−1.75350
$216$	−1663.60	−0.524045
$217$	−489.740	−0.153206
$218$	7769.20	2.41375
$219$	−1335.05	−0.411939
$220$	−796.674	−0.244144
$221$	−3771.56	−1.14797
$222$	4888.14	1.47779
$223$	−3416.65	−1.02599	−0.512995	−	0.858392i	$-0.671464\pi$
$223$	−3416.65	−1.02599	−0.512995	+	0.858392i	$0.671464\pi$
$224$	−1294.06	−0.385994
$225$	145.179	0.0430160
$226$	−12398.9	−3.64940
$227$	442.483	0.129377	0.0646886	−	0.997906i	$-0.479395\pi$
$227$	442.483	0.129377	0.0646886	+	0.997906i	$0.479395\pi$
$228$	−525.019	−0.152501
$229$	−5469.58	−1.57834	−0.789170	−	0.614175i	$-0.789489\pi$
$229$	−5469.58	−1.57834	−0.789170	+	0.614175i	$0.789489\pi$
$230$	5052.56	1.44850
$231$	−34.3709	−0.00978979
$232$	−14495.2	−4.10197
$233$	3053.40	0.858519	0.429259	−	0.903181i	$-0.358775\pi$
$233$	3053.40	0.858519	0.429259	+	0.903181i	$0.358775\pi$
$234$	−2598.64	−0.725976
$235$	3224.31	0.895025
$236$	−1162.64	−0.320684
$237$	−1807.72	−0.495459
$238$	−1218.38	−0.331830
$239$	−4582.19	−1.24015	−0.620077	−	0.784541i	$-0.712899\pi$
$239$	−4582.19	−1.24015	−0.620077	+	0.784541i	$0.712899\pi$
$240$	−5940.10	−1.59763
$241$	6410.62	1.71346	0.856731	−	0.515764i	$-0.172492\pi$
$241$	6410.62	1.71346	0.856731	+	0.515764i	$0.172492\pi$
$242$	−6944.93	−1.84478
$243$	−243.000	−0.0641500
$244$	−14518.1	−3.80912
$245$	−3940.14	−1.02746
$246$	−594.759	−0.154148
$247$	−487.168	−0.125497
$248$	8963.06	2.29498
$249$	2338.88	0.595262
$250$	−6807.70	−1.72223
$251$	1936.29	0.486922	0.243461	−	0.969911i	$-0.421717\pi$
$251$	1936.29	0.486922	0.243461	+	0.969911i	$0.421717\pi$
$252$	−597.077	−0.149255
$253$	−274.973	−0.0683297
$254$	13223.6	3.26661
$255$	−2450.39	−0.601761
$256$	−2591.74	−0.632749
$257$	−341.077	−0.0827852	−0.0413926	−	0.999143i	$-0.513179\pi$
$257$	−341.077	−0.0827852	−0.0413926	+	0.999143i	$0.513179\pi$
$258$	7347.81	1.77308
$259$	1042.15	0.250024
$260$	−12841.7	−3.06312
$261$	−2117.29	−0.502135
$262$	−4778.56	−1.12680
$263$	5497.72	1.28899	0.644494	−	0.764609i	$-0.277068\pi$
$263$	5497.72	1.28899	0.644494	+	0.764609i	$0.277068\pi$
$264$	629.046	0.146648
$265$	−975.095	−0.226036
$266$	−157.377	−0.0362758
$267$	−4181.61	−0.958466
$268$	15142.7	3.45145
$269$	104.441	0.0236725	0.0118363	−	0.999930i	$-0.496232\pi$
$269$	104.441	0.0236725	0.0118363	+	0.999930i	$0.496232\pi$
$270$	−1688.34	−0.380552
$271$	4199.52	0.941339	0.470669	−	0.882310i	$-0.344012\pi$
$271$	4199.52	0.941339	0.470669	+	0.882310i	$0.344012\pi$
$272$	11459.4	2.55453
$273$	−554.031	−0.122826
$274$	1720.59	0.379360
$275$	−54.8955	−0.0120375
$276$	−4776.72	−1.04176
$277$	1330.67	0.288636	0.144318	−	0.989531i	$-0.453901\pi$
$277$	1330.67	0.288636	0.144318	+	0.989531i	$0.453901\pi$
$278$	15837.0	3.41668
$279$	1309.22	0.280936
$280$	−2464.29	−0.525962
$281$	−4164.03	−0.884004	−0.442002	−	0.897014i	$-0.645732\pi$
$281$	−4164.03	−0.884004	−0.442002	+	0.897014i	$0.645732\pi$
$282$	−4285.80	−0.905021
$283$	−985.954	−0.207099	−0.103549	−	0.994624i	$-0.533020\pi$
$283$	−985.954	−0.207099	−0.103549	+	0.994624i	$0.533020\pi$
$284$	−3246.10	−0.678242
$285$	−316.514	−0.0657848
$286$	982.605	0.203156
$287$	−126.803	−0.0260799
$288$	3459.40	0.707803
$289$	−185.798	−0.0378175
$290$	−14710.8	−2.97878
$291$	808.773	0.162925
$292$	8769.43	1.75751
$293$	−1679.74	−0.334920	−0.167460	−	0.985879i	$-0.553557\pi$
$293$	−1679.74	−0.334920	−0.167460	+	0.985879i	$0.553557\pi$
$294$	5237.30	1.03893
$295$	−700.912	−0.138334
$296$	−19073.2	−3.74529
$297$	91.8839	0.0179517
$298$	−761.918	−0.148110
$299$	−4432.34	−0.857288
$300$	−953.622	−0.183525
$301$	1566.56	0.299983
$302$	−4039.33	−0.769660
$303$	−2810.85	−0.532934
$304$	1480.20	0.279262
$305$	−8752.40	−1.64315
$306$	3257.09	0.608482
$307$	−2907.22	−0.540469	−0.270234	−	0.962795i	$-0.587101\pi$
$307$	−2907.22	−0.540469	−0.270234	+	0.962795i	$0.587101\pi$
$308$	225.769	0.0417674
$309$	−4385.28	−0.807346
$310$	9096.35	1.66657
$311$	−1187.35	−0.216490	−0.108245	−	0.994124i	$-0.534523\pi$
$311$	−1187.35	−0.216490	−0.108245	+	0.994124i	$0.534523\pi$
$312$	10139.7	1.83990
$313$	3620.47	0.653806	0.326903	−	0.945058i	$-0.393995\pi$
$313$	3620.47	0.653806	0.326903	+	0.945058i	$0.393995\pi$
$314$	18543.8	3.33277
$315$	−359.955	−0.0643846
$316$	11874.1	2.11384
$317$	−3700.69	−0.655683	−0.327842	−	0.944733i	$-0.606321\pi$
$317$	−3700.69	−0.655683	−0.327842	+	0.944733i	$0.606321\pi$
$318$	1296.11	0.228561
$319$	800.597	0.140517
$320$	8195.32	1.43166
$321$	−2490.94	−0.433118
$322$	−1431.84	−0.247805
$323$	610.608	0.105186
$324$	1596.17	0.273691
$325$	−884.871	−0.151027
$326$	−7626.16	−1.29563
$327$	−4428.05	−0.748843
$328$	2320.71	0.390669
$329$	−913.735	−0.153118
$330$	638.400	0.106493
$331$	−1595.64	−0.264967	−0.132484	−	0.991185i	$-0.542295\pi$
$331$	−1595.64	−0.264967	−0.132484	+	0.991185i	$0.542295\pi$
$332$	−15363.1	−2.53964
$333$	−2785.99	−0.458472
$334$	−3074.55	−0.503689
$335$	9128.97	1.48886
$336$	1683.36	0.273318
$337$	−5677.39	−0.917707	−0.458853	−	0.888512i	$-0.651740\pi$
$337$	−5677.39	−0.917707	−0.458853	+	0.888512i	$0.651740\pi$
$338$	4274.60	0.687893
$339$	7066.75	1.13219
$340$	16095.6	2.56737
$341$	−495.047	−0.0786167
$342$	420.715	0.0665195
$343$	2271.34	0.357554
$344$	−28670.6	−4.49365
$345$	−2879.70	−0.449385
$346$	14508.1	2.25422
$347$	4193.13	0.648700	0.324350	−	0.945937i	$-0.394854\pi$
$347$	4193.13	0.648700	0.324350	+	0.945937i	$0.394854\pi$
$348$	13907.6	2.14232
$349$	3131.79	0.480346	0.240173	−	0.970730i	$-0.422796\pi$
$349$	3131.79	0.480346	0.240173	+	0.970730i	$0.422796\pi$
$350$	−285.852	−0.0436555
$351$	1481.09	0.225228
$352$	−1308.08	−0.198071
$353$	3984.82	0.600823	0.300412	−	0.953810i	$-0.402876\pi$
$353$	3984.82	0.600823	0.300412	+	0.953810i	$0.402876\pi$
$354$	931.662	0.139879
$355$	−1956.95	−0.292575
$356$	27467.3	4.08922
$357$	694.413	0.102947
$358$	−15864.9	−2.34215
$359$	−4378.11	−0.643643	−0.321822	−	0.946800i	$-0.604295\pi$
$359$	−4378.11	−0.643643	−0.321822	+	0.946800i	$0.604295\pi$
$360$	6587.78	0.964463
$361$	−6780.13	−0.988501
$362$	−7163.97	−1.04014
$363$	3958.26	0.572327
$364$	3639.20	0.524028
$365$	5286.75	0.758140
$366$	11633.8	1.66150
$367$	−8615.60	−1.22542	−0.612712	−	0.790306i	$-0.709922\pi$
$367$	−8615.60	−1.22542	−0.612712	+	0.790306i	$0.709922\pi$
$368$	13467.2	1.90768
$369$	338.982	0.0478231
$370$	−19356.8	−2.71976
$371$	276.331	0.0386695
$372$	−8599.75	−1.19859
$373$	−8463.20	−1.17482	−0.587410	−	0.809290i	$-0.699852\pi$
$373$	−8463.20	−1.17482	−0.587410	+	0.809290i	$0.699852\pi$
$374$	−1231.58	−0.170277
$375$	3880.04	0.534306
$376$	16722.9	2.29366
$377$	12905.0	1.76297
$378$	478.457	0.0651037
$379$	5731.49	0.776800	0.388400	−	0.921491i	$-0.373028\pi$
$379$	5731.49	0.776800	0.388400	+	0.921491i	$0.373028\pi$
$380$	2079.05	0.280666
$381$	−7536.75	−1.01344
$382$	962.608	0.128930
$383$	−12082.4	−1.61197	−0.805983	−	0.591939i	$-0.798363\pi$
$383$	−12082.4	−1.61197	−0.805983	+	0.591939i	$0.798363\pi$
$384$	−1668.28	−0.221703
$385$	136.107	0.0180173
$386$	21331.2	2.81277
$387$	−4187.88	−0.550082
$388$	−5312.50	−0.695106
$389$	−14859.0	−1.93671	−0.968357	−	0.249571i	$-0.919711\pi$
$389$	−14859.0	−1.93671	−0.968357	+	0.249571i	$0.919711\pi$
$390$	10290.5	1.33610
$391$	5555.42	0.718541
$392$	−20435.5	−2.63304
$393$	2723.54	0.349578
$394$	6547.24	0.837171
$395$	7158.47	0.911852
$396$	−603.548	−0.0765894
$397$	11650.0	1.47279	0.736396	−	0.676551i	$-0.236526\pi$
$397$	11650.0	1.47279	0.736396	+	0.676551i	$0.236526\pi$
$398$	−1008.34	−0.126994
$399$	89.6966	0.0112543
$400$	2688.58	0.336072
$401$	−5665.33	−0.705518	−0.352759	−	0.935714i	$-0.614757\pi$
$401$	−5665.33	−0.705518	−0.352759	+	0.935714i	$0.614757\pi$
$402$	−12134.4	−1.50549
$403$	−7979.75	−0.986352
$404$	18463.3	2.27372
$405$	962.268	0.118063
$406$	4168.87	0.509599
$407$	1053.45	0.128298
$408$	−12708.9	−1.54212
$409$	−7654.36	−0.925388	−0.462694	−	0.886518i	$-0.653117\pi$
$409$	−7654.36	−0.925388	−0.462694	+	0.886518i	$0.653117\pi$
$410$	2355.22	0.283697
$411$	−980.649	−0.117693
$412$	28805.1	3.44448
$413$	198.631	0.0236658
$414$	3827.74	0.454404
$415$	−9261.84	−1.09553
$416$	−21085.2	−2.48506
$417$	−9026.26	−1.06000
$418$	−159.082	−0.0186147
$419$	8887.25	1.03621	0.518103	−	0.855318i	$-0.326638\pi$
$419$	8887.25	1.03621	0.518103	+	0.855318i	$0.326638\pi$
$420$	2364.40	0.274692
$421$	−8283.80	−0.958974	−0.479487	−	0.877549i	$-0.659177\pi$
$421$	−8283.80	−0.958974	−0.479487	+	0.877549i	$0.659177\pi$
$422$	15879.0	1.83170
$423$	2442.69	0.280775
$424$	−5057.33	−0.579258
$425$	1109.08	0.126584
$426$	2601.21	0.295842
$427$	2480.33	0.281105
$428$	16362.0	1.84787
$429$	−560.035	−0.0630274
$430$	−29097.0	−3.26321
$431$	−5882.75	−0.657453	−0.328726	−	0.944425i	$-0.606619\pi$
$431$	−5882.75	−0.657453	−0.328726	+	0.944425i	$0.606619\pi$
$432$	−4500.13	−0.501187
$433$	−14513.8	−1.61083	−0.805415	−	0.592711i	$-0.798058\pi$
$433$	−14513.8	−1.61083	−0.805415	+	0.592711i	$0.798058\pi$
$434$	−2577.81	−0.285112
$435$	8384.38	0.924139
$436$	29086.1	3.19488
$437$	717.588	0.0785512
$438$	−7027.23	−0.766607
$439$	3287.31	0.357392	0.178696	−	0.983904i	$-0.442812\pi$
$439$	3287.31	0.357392	0.178696	+	0.983904i	$0.442812\pi$
$440$	−2490.99	−0.269894
$441$	−2984.99	−0.322319
$442$	−19852.1	−2.13635
$443$	12974.8	1.39153	0.695767	−	0.718267i	$-0.255064\pi$
$443$	12974.8	1.39153	0.695767	+	0.718267i	$0.255064\pi$
$444$	18300.0	1.95604
$445$	16559.0	1.76398
$446$	−17984.0	−1.90934
$447$	434.255	0.0459498
$448$	−2322.47	−0.244925
$449$	15485.8	1.62766	0.813831	−	0.581101i	$-0.197378\pi$
$449$	15485.8	1.62766	0.813831	+	0.581101i	$0.197378\pi$
$450$	764.169	0.0800517
$451$	−128.177	−0.0133827
$452$	−46418.6	−4.83042
$453$	2302.21	0.238780
$454$	2329.06	0.240767
$455$	2193.94	0.226051
$456$	−1641.60	−0.168585
$457$	4088.09	0.418452	0.209226	−	0.977867i	$-0.432906\pi$
$457$	4088.09	0.418452	0.209226	+	0.977867i	$0.432906\pi$
$458$	−28789.8	−2.93725
$459$	−1856.38	−0.188776
$460$	18915.6	1.91727
$461$	−7983.57	−0.806577	−0.403289	−	0.915073i	$-0.632133\pi$
$461$	−7983.57	−0.806577	−0.403289	+	0.915073i	$0.632133\pi$
$462$	−180.916	−0.0182185
$463$	9829.43	0.986635	0.493318	−	0.869849i	$-0.335784\pi$
$463$	9829.43	0.986635	0.493318	+	0.869849i	$0.335784\pi$
$464$	−39210.3	−3.92304
$465$	−5184.46	−0.517040
$466$	16072.0	1.59768
$467$	−7096.19	−0.703153	−0.351577	−	0.936159i	$-0.614354\pi$
$467$	−7096.19	−0.703153	−0.351577	+	0.936159i	$0.614354\pi$
$468$	−9728.70	−0.960917
$469$	−2587.05	−0.254710
$470$	16971.6	1.66562
$471$	−10569.0	−1.03396
$472$	−3635.28	−0.354507
$473$	1583.53	0.153934
$474$	−9515.14	−0.922036
$475$	143.259	0.0138383
$476$	−4561.32	−0.439218
$477$	−738.717	−0.0709089
$478$	−24118.9	−2.30790
$479$	−8405.79	−0.801817	−0.400909	−	0.916118i	$-0.631305\pi$
$479$	−8405.79	−0.801817	−0.400909	+	0.916118i	$0.631305\pi$
$480$	−13699.1	−1.30265
$481$	16980.7	1.60967
$482$	33743.1	3.18871
$483$	816.076	0.0768794
$484$	−26000.2	−2.44179
$485$	−3202.70	−0.299850
$486$	−1279.06	−0.119382
$487$	−33.3955	−0.00310738	−0.00155369	−	0.999999i	$-0.500495\pi$
$487$	−33.3955	−0.00310738	−0.00155369	+	0.999999i	$0.500495\pi$
$488$	−45394.3	−4.21087
$489$	4346.53	0.401956
$490$	−20739.4	−1.91207
$491$	−3539.38	−0.325316	−0.162658	−	0.986683i	$-0.552007\pi$
$491$	−3539.38	−0.325316	−0.162658	+	0.986683i	$0.552007\pi$
$492$	−2226.64	−0.204034
$493$	−16174.9	−1.47765
$494$	−2564.27	−0.233547
$495$	−363.856	−0.0330386
$496$	24245.6	2.19487
$497$	554.578	0.0500528
$498$	12311.0	1.10777
$499$	3739.04	0.335435	0.167718	−	0.985835i	$-0.446360\pi$
$499$	3739.04	0.335435	0.167718	+	0.985835i	$0.446360\pi$
$500$	−25486.4	−2.27958
$501$	1752.34	0.156265
$502$	10191.9	0.906148
$503$	18545.4	1.64393	0.821965	−	0.569539i	$-0.192878\pi$
$503$	18545.4	1.64393	0.821965	+	0.569539i	$0.192878\pi$
$504$	−1866.91	−0.164997
$505$	11130.8	0.980822
$506$	−1447.36	−0.127160
$507$	−2436.31	−0.213413
$508$	49505.9	4.32376
$509$	18221.3	1.58673	0.793363	−	0.608749i	$-0.208328\pi$
$509$	18221.3	1.58673	0.793363	+	0.608749i	$0.208328\pi$
$510$	−12897.9	−1.11986
$511$	−1498.21	−0.129700
$512$	−18090.7	−1.56153
$513$	−239.786	−0.0206371
$514$	−1795.30	−0.154061
$515$	17365.5	1.48586
$516$	27508.5	2.34689
$517$	−923.637	−0.0785716
$518$	5485.50	0.465288
$519$	−8268.88	−0.699351
$520$	−40152.7	−3.38618
$521$	15891.9	1.33634	0.668172	−	0.744007i	$-0.267077\pi$
$521$	15891.9	1.33634	0.668172	+	0.744007i	$0.267077\pi$
$522$	−11144.6	−0.934459
$523$	15241.9	1.27434	0.637170	−	0.770723i	$-0.280105\pi$
$523$	15241.9	1.27434	0.637170	+	0.770723i	$0.280105\pi$
$524$	−17889.8	−1.49145
$525$	162.921	0.0135437
$526$	28938.0	2.39877
$527$	10001.7	0.826717
$528$	1701.60	0.140251
$529$	−5638.25	−0.463405
$530$	−5132.54	−0.420647
$531$	−531.000	−0.0433963
$532$	−589.180	−0.0480154
$533$	−2066.11	−0.167904
$534$	−22010.4	−1.78368
$535$	9864.01	0.797118
$536$	47347.4	3.81548
$537$	9042.21	0.726629
$538$	549.741	0.0440539
$539$	1128.69	0.0901972
$540$	−6320.75	−0.503707
$541$	14342.6	1.13981	0.569904	−	0.821711i	$-0.306980\pi$
$541$	14342.6	1.13981	0.569904	+	0.821711i	$0.306980\pi$
$542$	22104.7	1.75181
$543$	4083.10	0.322693
$544$	26427.8	2.08287
$545$	17534.9	1.37819
$546$	−2916.21	−0.228576
$547$	4343.76	0.339536	0.169768	−	0.985484i	$-0.445698\pi$
$547$	4343.76	0.339536	0.169768	+	0.985484i	$0.445698\pi$
$548$	6441.48	0.502129
$549$	−6630.69	−0.515466
$550$	−288.950	−0.0224016
$551$	−2089.29	−0.161537
$552$	−14935.6	−1.15163
$553$	−2028.63	−0.155997
$554$	7004.15	0.537144
$555$	11032.4	0.843782
$556$	59289.9	4.52239
$557$	−7199.47	−0.547669	−0.273834	−	0.961777i	$-0.588292\pi$
$557$	−7199.47	−0.547669	−0.273834	+	0.961777i	$0.588292\pi$
$558$	6891.26	0.522814
$559$	25525.3	1.93131
$560$	−6666.03	−0.503020
$561$	701.938	0.0528268
$562$	−21917.9	−1.64511
$563$	9629.26	0.720826	0.360413	−	0.932793i	$-0.382636\pi$
$563$	9629.26	0.720826	0.360413	+	0.932793i	$0.382636\pi$
$564$	−16045.0	−1.19790
$565$	−27984.0	−2.08371
$566$	−5189.69	−0.385405
$567$	−272.696	−0.0201978
$568$	−10149.7	−0.749776
$569$	17896.5	1.31856	0.659279	−	0.751899i	$-0.270862\pi$
$569$	17896.5	1.31856	0.659279	+	0.751899i	$0.270862\pi$
$570$	−1666.01	−0.122424
$571$	−14247.6	−1.04421	−0.522105	−	0.852881i	$-0.674853\pi$
$571$	−14247.6	−1.04421	−0.522105	+	0.852881i	$0.674853\pi$
$572$	3678.64	0.268902
$573$	−548.638	−0.0399994
$574$	−667.443	−0.0485340
$575$	1303.40	0.0945311
$576$	6208.65	0.449121
$577$	−14178.5	−1.02298	−0.511490	−	0.859289i	$-0.670906\pi$
$577$	−14178.5	−1.02298	−0.511490	+	0.859289i	$0.670906\pi$
$578$	−977.969	−0.0703774
$579$	−12157.7	−0.872636
$580$	−55073.6	−3.94277
$581$	2624.71	0.187420
$582$	4257.08	0.303199
$583$	279.326	0.0198430
$584$	27419.7	1.94287
$585$	−5865.06	−0.414513
$586$	−8841.54	−0.623278
$587$	−17769.4	−1.24944	−0.624720	−	0.780848i	$-0.714787\pi$
$587$	−17769.4	−1.24944	−0.624720	+	0.780848i	$0.714787\pi$
$588$	19607.2	1.37515
$589$	1291.91	0.0903771
$590$	−3689.34	−0.257437
$591$	−3731.59	−0.259725
$592$	−51593.9	−3.58192
$593$	−12282.0	−0.850527	−0.425263	−	0.905070i	$-0.639819\pi$
$593$	−12282.0	−0.850527	−0.425263	+	0.905070i	$0.639819\pi$
$594$	483.643	0.0334076
$595$	−2749.84	−0.189466
$596$	−2852.44	−0.196041
$597$	574.702	0.0393986
$598$	−23330.2	−1.59539
$599$	22571.6	1.53965	0.769827	−	0.638253i	$-0.220343\pi$
$599$	22571.6	1.53965	0.769827	+	0.638253i	$0.220343\pi$
$600$	−2981.73	−0.202881
$601$	−5500.85	−0.373352	−0.186676	−	0.982422i	$-0.559771\pi$
$601$	−5500.85	−0.373352	−0.186676	+	0.982422i	$0.559771\pi$
$602$	8245.77	0.558260
$603$	6915.97	0.467065
$604$	−15122.3	−1.01874
$605$	−15674.5	−1.05332
$606$	−14795.3	−0.991776
$607$	377.433	0.0252381	0.0126190	−	0.999920i	$-0.495983\pi$
$607$	377.433	0.0252381	0.0126190	+	0.999920i	$0.495983\pi$
$608$	3413.65	0.227700
$609$	−2376.04	−0.158099
$610$	−46069.4	−3.05786
$611$	−14888.3	−0.985786
$612$	12193.8	0.805399
$613$	12016.8	0.791766	0.395883	−	0.918301i	$-0.370439\pi$
$613$	12016.8	0.791766	0.395883	+	0.918301i	$0.370439\pi$
$614$	−15302.5	−1.00580
$615$	−1342.35	−0.0880145
$616$	705.920	0.0461726
$617$	−15545.9	−1.01435	−0.507176	−	0.861843i	$-0.669311\pi$
$617$	−15545.9	−1.01435	−0.507176	+	0.861843i	$0.669311\pi$
$618$	−23082.5	−1.50245
$619$	2904.00	0.188565	0.0942825	−	0.995545i	$-0.469944\pi$
$619$	2904.00	0.188565	0.0942825	+	0.995545i	$0.469944\pi$
$620$	34054.6	2.20591
$621$	−2181.62	−0.140975
$622$	−6249.76	−0.402882
$623$	−4692.63	−0.301776
$624$	27428.4	1.75964
$625$	−17381.2	−1.11239
$626$	19056.8	1.21672
$627$	90.6686	0.00577505
$628$	69423.7	4.41132
$629$	−21283.3	−1.34916
$630$	−1894.67	−0.119818
$631$	−16236.4	−1.02434	−0.512172	−	0.858883i	$-0.671159\pi$
$631$	−16236.4	−1.02434	−0.512172	+	0.858883i	$0.671159\pi$
$632$	37127.4	2.33679
$633$	−9050.20	−0.568267
$634$	−19479.1	−1.22021
$635$	29845.2	1.86515
$636$	4852.33	0.302527
$637$	18193.6	1.13164
$638$	4214.05	0.261498
$639$	−1482.56	−0.0917825
$640$	6606.30	0.408026
$641$	−28390.2	−1.74937	−0.874684	−	0.484694i	$-0.838931\pi$
$641$	−28390.2	−1.74937	−0.874684	+	0.484694i	$0.838931\pi$
$642$	−13111.4	−0.806021
$643$	23941.9	1.46839	0.734196	−	0.678937i	$-0.237559\pi$
$643$	23941.9	1.46839	0.734196	+	0.678937i	$0.237559\pi$
$644$	−5360.47	−0.328000
$645$	16583.8	1.01238
$646$	3214.01	0.195749
$647$	3874.64	0.235437	0.117718	−	0.993047i	$-0.462442\pi$
$647$	3874.64	0.235437	0.117718	+	0.993047i	$0.462442\pi$
$648$	4990.80	0.302558
$649$	200.783	0.0121440
$650$	−4657.63	−0.281057
$651$	1469.22	0.0884535
$652$	−28550.6	−1.71492
$653$	−23410.1	−1.40292	−0.701460	−	0.712709i	$-0.747468\pi$
$653$	−23410.1	−1.40292	−0.701460	+	0.712709i	$0.747468\pi$
$654$	−23307.6	−1.39358
$655$	−10785.1	−0.643371
$656$	6277.63	0.373629
$657$	4005.16	0.237833
$658$	−4809.56	−0.284949
$659$	22240.7	1.31468	0.657342	−	0.753593i	$-0.271681\pi$
$659$	22240.7	1.31468	0.657342	+	0.753593i	$0.271681\pi$
$660$	2390.02	0.140957
$661$	30096.9	1.77100	0.885502	−	0.464635i	$-0.153815\pi$
$661$	30096.9	1.77100	0.885502	+	0.464635i	$0.153815\pi$
$662$	−8398.84	−0.493097
$663$	11314.7	0.662783
$664$	−48036.5	−2.80750
$665$	−355.194	−0.0207125
$666$	−14664.4	−0.853205
$667$	−19008.7	−1.10348
$668$	−11510.4	−0.666693
$669$	10249.9	0.592355
$670$	48051.5	2.77073
$671$	2507.21	0.144247
$672$	3882.17	0.222854
$673$	17650.2	1.01094	0.505471	−	0.862844i	$-0.331319\pi$
$673$	17650.2	1.01094	0.505471	+	0.862844i	$0.331319\pi$
$674$	−29883.7	−1.70783
$675$	−435.537	−0.0248353
$676$	16003.1	0.910509
$677$	−8151.07	−0.462734	−0.231367	−	0.972866i	$-0.574320\pi$
$677$	−8151.07	−0.462734	−0.231367	+	0.972866i	$0.574320\pi$
$678$	37196.7	2.10698
$679$	907.611	0.0512974
$680$	50326.7	2.83815
$681$	−1327.45	−0.0746959
$682$	−2605.74	−0.146304
$683$	−23902.8	−1.33912	−0.669558	−	0.742760i	$-0.733516\pi$
$683$	−23902.8	−1.33912	−0.669558	+	0.742760i	$0.733516\pi$
$684$	1575.06	0.0880465
$685$	3883.32	0.216605
$686$	11955.5	0.665399
$687$	16408.7	0.911255
$688$	−77555.6	−4.29764
$689$	4502.50	0.248958
$690$	−15157.7	−0.836294
$691$	−25203.7	−1.38754	−0.693772	−	0.720194i	$-0.744053\pi$
$691$	−25203.7	−1.38754	−0.693772	+	0.720194i	$0.744053\pi$
$692$	54314.9	2.98373
$693$	103.113	0.00565214
$694$	22071.1	1.20721
$695$	35743.6	1.95084
$696$	43485.6	2.36827
$697$	2589.62	0.140730
$698$	16484.6	0.893912
$699$	−9160.20	−0.495666
$700$	−1070.16	−0.0577833
$701$	−6102.41	−0.328794	−0.164397	−	0.986394i	$-0.552568\pi$
$701$	−6102.41	−0.328794	−0.164397	+	0.986394i	$0.552568\pi$
$702$	7795.92	0.419143
$703$	−2749.14	−0.147491
$704$	−2347.63	−0.125682
$705$	−9672.93	−0.516743
$706$	20974.6	1.11812
$707$	−3154.35	−0.167796
$708$	3487.92	0.185147
$709$	−483.971	−0.0256360	−0.0128180	−	0.999918i	$-0.504080\pi$
$709$	−483.971	−0.0256360	−0.0128180	+	0.999918i	$0.504080\pi$
$710$	−10300.7	−0.544474
$711$	5423.15	0.286053
$712$	85883.1	4.52051
$713$	11754.0	0.617378
$714$	3655.13	0.191582
$715$	2217.71	0.115997
$716$	−59394.6	−3.10011
$717$	13746.6	0.716004
$718$	−23044.8	−1.19780
$719$	21253.9	1.10242	0.551208	−	0.834368i	$-0.314167\pi$
$719$	21253.9	1.10242	0.551208	+	0.834368i	$0.314167\pi$
$720$	17820.3	0.922394
$721$	−4921.20	−0.254195
$722$	−35688.1	−1.83957
$723$	−19231.9	−0.989267
$724$	−26820.2	−1.37675
$725$	−3794.90	−0.194399
$726$	20834.8	1.06508
$727$	−26369.1	−1.34522	−0.672610	−	0.739997i	$-0.734827\pi$
$727$	−26369.1	−1.34522	−0.672610	+	0.739997i	$0.734827\pi$
$728$	11378.9	0.579297
$729$	729.000	0.0370370
$730$	27827.5	1.41088
$731$	−31992.9	−1.61874
$732$	43554.3	2.19920
$733$	−6909.57	−0.348173	−0.174086	−	0.984730i	$-0.555697\pi$
$733$	−6909.57	−0.348173	−0.174086	+	0.984730i	$0.555697\pi$
$734$	−45349.3	−2.28048
$735$	11820.4	0.593201
$736$	31058.0	1.55545
$737$	−2615.09	−0.130703
$738$	1784.28	0.0889975
$739$	−34884.6	−1.73647	−0.868235	−	0.496153i	$-0.834746\pi$
$739$	−34884.6	−1.73647	−0.868235	+	0.496153i	$0.834746\pi$
$740$	−72467.3	−3.59993
$741$	1461.50	0.0724557
$742$	1454.50	0.0719630
$743$	34839.2	1.72022	0.860112	−	0.510106i	$-0.170394\pi$
$743$	34839.2	1.72022	0.860112	+	0.510106i	$0.170394\pi$
$744$	−26889.2	−1.32501
$745$	−1719.63	−0.0845668
$746$	−44547.1	−2.18631
$747$	−7016.63	−0.343675
$748$	−4610.75	−0.225382
$749$	−2795.35	−0.136368
$750$	20423.1	0.994329
$751$	32677.5	1.58777	0.793887	−	0.608065i	$-0.208054\pi$
$751$	32677.5	1.58777	0.793887	+	0.608065i	$0.208054\pi$
$752$	45236.3	2.19362
$753$	−5808.86	−0.281124
$754$	67927.0	3.28084
$755$	−9116.65	−0.439455
$756$	1791.23	0.0861726
$757$	−263.405	−0.0126468	−0.00632339	−	0.999980i	$-0.502013\pi$
$757$	−263.405	−0.0126468	−0.00632339	+	0.999980i	$0.502013\pi$
$758$	30168.5	1.44560
$759$	824.920	0.0394502
$760$	6500.66	0.310268
$761$	−23504.7	−1.11964	−0.559819	−	0.828615i	$-0.689130\pi$
$761$	−23504.7	−1.11964	−0.559819	+	0.828615i	$0.689130\pi$
$762$	−39670.7	−1.88598
$763$	−4969.19	−0.235776
$764$	3603.78	0.170655
$765$	7351.16	0.347427
$766$	−63597.4	−2.99983
$767$	3236.46	0.152362
$768$	7775.22	0.365318
$769$	−14106.7	−0.661508	−0.330754	−	0.943717i	$-0.607303\pi$
$769$	−14106.7	−0.661508	−0.330754	+	0.943717i	$0.607303\pi$
$770$	716.418	0.0335297
$771$	1023.23	0.0477961
$772$	79858.9	3.72304
$773$	−8023.79	−0.373345	−0.186672	−	0.982422i	$-0.559770\pi$
$773$	−8023.79	−0.373345	−0.186672	+	0.982422i	$0.559770\pi$
$774$	−22043.4	−1.02369
$775$	2346.56	0.108763
$776$	−16610.8	−0.768419
$777$	−3126.46	−0.144351
$778$	−78212.3	−3.60417
$779$	334.499	0.0153847
$780$	38525.2	1.76849
$781$	560.588	0.0256843
$782$	29241.7	1.33719
$783$	6351.88	0.289908
$784$	−55279.3	−2.51819
$785$	41852.9	1.90292
$786$	14335.7	0.650556
$787$	22881.4	1.03638	0.518192	−	0.855264i	$-0.326605\pi$
$787$	22881.4	1.03638	0.518192	+	0.855264i	$0.326605\pi$
$788$	24511.3	1.10810
$789$	−16493.2	−0.744198
$790$	37679.5	1.69693
$791$	7930.36	0.356474
$792$	−1887.14	−0.0846673
$793$	40414.2	1.80978
$794$	61321.4	2.74083
$795$	2925.28	0.130502
$796$	−3774.98	−0.168091
$797$	−3384.94	−0.150440	−0.0752200	−	0.997167i	$-0.523966\pi$
$797$	−3384.94	−0.150440	−0.0752200	+	0.997167i	$0.523966\pi$
$798$	472.130	0.0209439
$799$	18660.7	0.826243
$800$	6200.40	0.274022
$801$	12544.8	0.553370
$802$	−29820.2	−1.31295
$803$	−1514.44	−0.0665549
$804$	−45428.2	−1.99270
$805$	−3231.62	−0.141490
$806$	−42002.4	−1.83557
$807$	−313.324	−0.0136673
$808$	57730.0	2.51353
$809$	−19080.0	−0.829193	−0.414597	−	0.910005i	$-0.636077\pi$
$809$	−19080.0	−0.829193	−0.414597	+	0.910005i	$0.636077\pi$
$810$	5065.02	0.219712
$811$	−38157.7	−1.65216	−0.826078	−	0.563556i	$-0.809433\pi$
$811$	−38157.7	−1.65216	−0.826078	+	0.563556i	$0.809433\pi$
$812$	15607.3	0.674516
$813$	−12598.6	−0.543482
$814$	5544.95	0.238760
$815$	−17212.0	−0.739768
$816$	−34378.3	−1.47486
$817$	−4132.49	−0.176962
$818$	−40289.7	−1.72212
$819$	1662.09	0.0709136
$820$	8817.38	0.375507
$821$	−24776.7	−1.05324	−0.526621	−	0.850100i	$-0.676541\pi$
$821$	−24776.7	−1.05324	−0.526621	+	0.850100i	$0.676541\pi$
$822$	−5161.77	−0.219024
$823$	−29084.0	−1.23184	−0.615921	−	0.787808i	$-0.711216\pi$
$823$	−29084.0	−1.23184	−0.615921	+	0.787808i	$0.711216\pi$
$824$	90066.2	3.80777
$825$	164.687	0.00694988
$826$	1045.52	0.0440414
$827$	−7505.76	−0.315600	−0.157800	−	0.987471i	$-0.550440\pi$
$827$	−7505.76	−0.315600	−0.157800	+	0.987471i	$0.550440\pi$
$828$	14330.2	0.601458
$829$	−4101.70	−0.171843	−0.0859216	−	0.996302i	$-0.527383\pi$
$829$	−4101.70	−0.171843	−0.0859216	+	0.996302i	$0.527383\pi$
$830$	−48750.9	−2.03876
$831$	−3992.01	−0.166644
$832$	−37841.9	−1.57684
$833$	−22803.6	−0.948496
$834$	−47510.9	−1.97262
$835$	−6939.18	−0.287593
$836$	−595.565	−0.0246388
$837$	−3927.67	−0.162198
$838$	46779.2	1.92835
$839$	−33368.3	−1.37307	−0.686533	−	0.727099i	$-0.740868\pi$
$839$	−33368.3	−1.37307	−0.686533	+	0.727099i	$0.740868\pi$
$840$	7392.86	0.303664
$841$	30955.8	1.26925
$842$	−43602.9	−1.78462
$843$	12492.1	0.510380
$844$	59447.1	2.42447
$845$	9647.66	0.392769
$846$	12857.4	0.522514
$847$	4441.98	0.180199
$848$	−13680.3	−0.553992
$849$	2957.86	0.119568
$850$	5837.79	0.235570
$851$	−25012.2	−1.00753
$852$	9738.30	0.391583
$853$	9085.16	0.364678	0.182339	−	0.983236i	$-0.441633\pi$
$853$	9085.16	0.364678	0.182339	+	0.983236i	$0.441633\pi$
$854$	13055.6	0.523129
$855$	949.542	0.0379809
$856$	51159.7	2.04276
$857$	40753.2	1.62439	0.812196	−	0.583385i	$-0.198272\pi$
$857$	40753.2	1.62439	0.812196	+	0.583385i	$0.198272\pi$
$858$	−2947.82	−0.117292
$859$	−20281.5	−0.805585	−0.402792	−	0.915291i	$-0.631960\pi$
$859$	−20281.5	−0.805585	−0.402792	+	0.915291i	$0.631960\pi$
$860$	−108932.	−4.31926
$861$	380.408	0.0150572
$862$	−30964.6	−1.22350
$863$	20235.1	0.798157	0.399079	−	0.916917i	$-0.369330\pi$
$863$	20235.1	0.798157	0.399079	+	0.916917i	$0.369330\pi$
$864$	−10378.2	−0.408650
$865$	32744.4	1.28710
$866$	−76395.3	−2.99771
$867$	557.393	0.0218340
$868$	−9650.70	−0.377380
$869$	−2050.62	−0.0800488
$870$	44132.3	1.71980
$871$	−42153.0	−1.63984
$872$	90944.6	3.53185
$873$	−2426.32	−0.0940646
$874$	3777.12	0.146182
$875$	4354.21	0.168228
$876$	−26308.3	−1.01470
$877$	32647.0	1.25703	0.628513	−	0.777799i	$-0.283664\pi$
$877$	32647.0	1.25703	0.628513	+	0.777799i	$0.283664\pi$
$878$	17303.2	0.665096
$879$	5039.23	0.193366
$880$	−6738.27	−0.258121
$881$	−30042.8	−1.14888	−0.574442	−	0.818545i	$-0.694781\pi$
$881$	−30042.8	−1.14888	−0.574442	+	0.818545i	$0.694781\pi$
$882$	−15711.9	−0.599826
$883$	−21269.9	−0.810634	−0.405317	−	0.914176i	$-0.632839\pi$
$883$	−21269.9	−0.810634	−0.405317	+	0.914176i	$0.632839\pi$
$884$	−74321.5	−2.82772
$885$	2102.73	0.0798674
$886$	68294.4	2.58961
$887$	−1012.07	−0.0383112	−0.0191556	−	0.999817i	$-0.506098\pi$
$887$	−1012.07	−0.0383112	−0.0191556	+	0.999817i	$0.506098\pi$
$888$	57219.5	2.16234
$889$	−8457.80	−0.319084
$890$	87160.3	3.28272
$891$	−275.652	−0.0103644
$892$	−67327.7	−2.52724
$893$	2410.38	0.0903252
$894$	2285.76	0.0855113
$895$	−35806.7	−1.33730
$896$	−1872.15	−0.0698038
$897$	13297.0	0.494955
$898$	81511.5	3.02904
$899$	−34222.3	−1.26961
$900$	2860.87	0.105958
$901$	−5643.36	−0.208665
$902$	−674.676	−0.0249049
$903$	−4699.67	−0.173195
$904$	−145139.	−5.33988
$905$	−16168.9	−0.593891
$906$	12118.0	0.444363
$907$	−9343.76	−0.342067	−0.171033	−	0.985265i	$-0.554711\pi$
$907$	−9343.76	−0.342067	−0.171033	+	0.985265i	$0.554711\pi$
$908$	8719.47	0.318685
$909$	8432.54	0.307690
$910$	11548.1	0.420675
$911$	−27773.6	−1.01008	−0.505039	−	0.863096i	$-0.668522\pi$
$911$	−27773.6	−1.01008	−0.505039	+	0.863096i	$0.668522\pi$
$912$	−4440.61	−0.161232
$913$	2653.15	0.0961735
$914$	21518.2	0.778729
$915$	26257.2	0.948673
$916$	−107782.	−3.88780
$917$	3056.37	0.110066
$918$	−9771.27	−0.351307
$919$	16876.6	0.605776	0.302888	−	0.953026i	$-0.402049\pi$
$919$	16876.6	0.605776	0.302888	+	0.953026i	$0.402049\pi$
$920$	59144.1	2.11948
$921$	8721.67	0.312040
$922$	−42022.6	−1.50102
$923$	9036.22	0.322244
$924$	−677.306	−0.0241144
$925$	−4993.42	−0.177495
$926$	51738.4	1.83610
$927$	13155.8	0.466122
$928$	−90426.8	−3.19871
$929$	10461.8	0.369474	0.184737	−	0.982788i	$-0.440857\pi$
$929$	10461.8	0.369474	0.184737	+	0.982788i	$0.440857\pi$
$930$	−27289.1	−0.962197
$931$	−2945.51	−0.103690
$932$	60169.6	2.11472
$933$	3562.04	0.124990
$934$	−37351.7	−1.30855
$935$	−2779.64	−0.0972235
$936$	−30419.1	−1.06226
$937$	−13284.7	−0.463173	−0.231586	−	0.972814i	$-0.574392\pi$
$937$	−13284.7	−0.463173	−0.231586	+	0.972814i	$0.574392\pi$
$938$	−13617.3	−0.474008
$939$	−10861.4	−0.377475
$940$	63537.6	2.20465
$941$	2290.28	0.0793422	0.0396711	−	0.999213i	$-0.487369\pi$
$941$	2290.28	0.0793422	0.0396711	+	0.999213i	$0.487369\pi$
$942$	−55631.5	−1.92417
$943$	3043.33	0.105095
$944$	−9833.62	−0.339044
$945$	1079.86	0.0371725
$946$	8335.13	0.286468
$947$	18255.7	0.626431	0.313216	−	0.949682i	$-0.398594\pi$
$947$	18255.7	0.626431	0.313216	+	0.949682i	$0.398594\pi$
$948$	−35622.4	−1.22043
$949$	−24411.6	−0.835020
$950$	754.062	0.0257526
$951$	11102.1	0.378559
$952$	−14262.1	−0.485542
$953$	−7649.51	−0.260013	−0.130006	−	0.991513i	$-0.541500\pi$
$953$	−7649.51	−0.260013	−0.130006	+	0.991513i	$0.541500\pi$
$954$	−3888.33	−0.131960
$955$	2172.58	0.0736157
$956$	−90295.6	−3.05478
$957$	−2401.79	−0.0811274
$958$	−44245.0	−1.49216
$959$	−1100.49	−0.0370560
$960$	−24586.0	−0.826572
$961$	−8629.73	−0.289676
$962$	89380.1	2.99556
$963$	7472.83	0.250061
$964$	126326.	4.22064
$965$	48143.9	1.60602
$966$	4295.52	0.143070
$967$	22498.5	0.748194	0.374097	−	0.927390i	$-0.377953\pi$
$967$	22498.5	0.748194	0.374097	+	0.927390i	$0.377953\pi$
$968$	−81295.8	−2.69933
$969$	−1831.82	−0.0607293
$970$	−16857.8	−0.558013
$971$	16368.8	0.540988	0.270494	−	0.962722i	$-0.412813\pi$
$971$	16368.8	0.540988	0.270494	+	0.962722i	$0.412813\pi$
$972$	−4788.50	−0.158016
$973$	−10129.3	−0.333743
$974$	−175.782	−0.00578276
$975$	2654.61	0.0871955
$976$	−122794.	−4.02719
$977$	45793.1	1.49954	0.749770	−	0.661699i	$-0.230164\pi$
$977$	45793.1	1.49954	0.749770	+	0.661699i	$0.230164\pi$
$978$	22878.5	0.748030
$979$	−4743.49	−0.154854
$980$	−77643.6	−2.53085
$981$	13284.1	0.432345
$982$	−18630.0	−0.605404
$983$	29019.5	0.941587	0.470793	−	0.882244i	$-0.343968\pi$
$983$	29019.5	0.941587	0.470793	+	0.882244i	$0.343968\pi$
$984$	−6962.12	−0.225553
$985$	14776.9	0.478002
$986$	−85138.5	−2.74986
$987$	2741.20	0.0884027
$988$	−9600.03	−0.309127
$989$	−37598.2	−1.20885
$990$	−1915.20	−0.0614839
$991$	24513.2	0.785760	0.392880	−	0.919590i	$-0.371479\pi$
$991$	24513.2	0.785760	0.392880	+	0.919590i	$0.371479\pi$
$992$	55915.1	1.78962
$993$	4786.91	0.152979
$994$	2919.09	0.0931469
$995$	−2275.79	−0.0725100
$996$	46089.4	1.46626
$997$	29685.7	0.942983	0.471491	−	0.881871i	$-0.343716\pi$
$997$	29685.7	0.942983	0.471491	+	0.881871i	$0.343716\pi$
$998$	19680.9	0.624236
$999$	8357.97	0.264699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.4.a.c.1.8	✓	8
3.2	odd	2		531.4.a.f.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.8	✓	8	1.1	even	1	trivial
531.4.a.f.1.1		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)

\(f(q)\)	\(=\)	\(q+5.26363 q^{2} -3.00000 q^{3} +19.7058 q^{4} +11.8799 q^{5} -15.7909 q^{6} -3.36662 q^{7} +61.6149 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+5.26363 q^{2} -3.00000 q^{3} +19.7058 q^{4} +11.8799 q^{5} -15.7909 q^{6} -3.36662 q^{7} +61.6149 q^{8} +9.00000 q^{9} +62.5311 q^{10} -3.40311 q^{11} -59.1173 q^{12} -54.8553 q^{13} -17.7206 q^{14} -35.6396 q^{15} +166.672 q^{16} +68.7547 q^{17} +47.3727 q^{18} +8.88097 q^{19} +234.102 q^{20} +10.0999 q^{21} -17.9127 q^{22} +80.8007 q^{23} -184.845 q^{24} +16.1310 q^{25} -288.738 q^{26} -27.0000 q^{27} -66.3419 q^{28} -235.255 q^{29} -187.593 q^{30} +145.469 q^{31} +384.378 q^{32} +10.2093 q^{33} +361.899 q^{34} -39.9950 q^{35} +177.352 q^{36} -309.554 q^{37} +46.7461 q^{38} +164.566 q^{39} +731.976 q^{40} +37.6647 q^{41} +53.1619 q^{42} -465.320 q^{43} -67.0609 q^{44} +106.919 q^{45} +425.305 q^{46} +271.410 q^{47} -500.015 q^{48} -331.666 q^{49} +84.9076 q^{50} -206.264 q^{51} -1080.97 q^{52} -82.0797 q^{53} -142.118 q^{54} -40.4284 q^{55} -207.434 q^{56} -26.6429 q^{57} -1238.29 q^{58} -59.0000 q^{59} -702.306 q^{60} -736.743 q^{61} +765.695 q^{62} -30.2996 q^{63} +689.850 q^{64} -651.673 q^{65} +53.7381 q^{66} +768.441 q^{67} +1354.86 q^{68} -242.402 q^{69} -210.519 q^{70} -164.728 q^{71} +554.534 q^{72} +445.018 q^{73} -1629.38 q^{74} -48.3930 q^{75} +175.006 q^{76} +11.4570 q^{77} +866.213 q^{78} +602.572 q^{79} +1980.03 q^{80} +81.0000 q^{81} +198.253 q^{82} -779.626 q^{83} +199.026 q^{84} +816.795 q^{85} -2449.27 q^{86} +705.764 q^{87} -209.682 q^{88} +1393.87 q^{89} +562.780 q^{90} +184.677 q^{91} +1592.24 q^{92} -436.407 q^{93} +1428.60 q^{94} +105.505 q^{95} -1153.13 q^{96} -269.591 q^{97} -1745.77 q^{98} -30.6280 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} - 24 q^{3} + 38 q^{4} - 12 q^{5} - 6 q^{6} + 53 q^{7} + 3 q^{8} + 72 q^{9}+O(q^{10}) \) 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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