LMFDB - Embedded newform 43.4.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [43,4,Mod(1,43)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(43, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("43.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	43.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$2.53708213025$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.45868.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 11x + 1 $ x^4 - x^3 - 10x^2 + 11x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.21390$ of defining polynomial
Character	$\chi$	$=$	43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.32005 q^{2} -9.13635 q^{3} -2.61739 q^{4} -12.9617 q^{5} -21.1967 q^{6} +21.6165 q^{7} -24.6328 q^{8} +56.4728 q^{9} +O(q^{10})$	\(q+2.32005 q^{2} -9.13635 q^{3} -2.61739 q^{4} -12.9617 q^{5} -21.1967 q^{6} +21.6165 q^{7} -24.6328 q^{8} +56.4728 q^{9} -30.0718 q^{10} -29.2253 q^{11} +23.9134 q^{12} +2.67349 q^{13} +50.1512 q^{14} +118.423 q^{15} -36.2102 q^{16} -137.211 q^{17} +131.020 q^{18} -51.9326 q^{19} +33.9259 q^{20} -197.496 q^{21} -67.8040 q^{22} +92.6938 q^{23} +225.054 q^{24} +43.0067 q^{25} +6.20263 q^{26} -269.274 q^{27} -56.5787 q^{28} +148.453 q^{29} +274.747 q^{30} +130.108 q^{31} +113.053 q^{32} +267.012 q^{33} -318.335 q^{34} -280.187 q^{35} -147.811 q^{36} -211.403 q^{37} -120.486 q^{38} -24.4260 q^{39} +319.284 q^{40} -474.175 q^{41} -458.199 q^{42} +43.0000 q^{43} +76.4939 q^{44} -731.986 q^{45} +215.054 q^{46} +174.793 q^{47} +330.829 q^{48} +124.272 q^{49} +99.7776 q^{50} +1253.60 q^{51} -6.99757 q^{52} -543.612 q^{53} -624.727 q^{54} +378.811 q^{55} -532.475 q^{56} +474.475 q^{57} +344.417 q^{58} +85.7755 q^{59} -309.959 q^{60} +238.340 q^{61} +301.857 q^{62} +1220.74 q^{63} +551.970 q^{64} -34.6531 q^{65} +619.481 q^{66} -316.481 q^{67} +359.133 q^{68} -846.883 q^{69} -650.047 q^{70} +51.1352 q^{71} -1391.08 q^{72} -219.515 q^{73} -490.465 q^{74} -392.924 q^{75} +135.928 q^{76} -631.748 q^{77} -56.6694 q^{78} +1172.35 q^{79} +469.347 q^{80} +935.412 q^{81} -1100.11 q^{82} -921.819 q^{83} +516.922 q^{84} +1778.49 q^{85} +99.7620 q^{86} -1356.31 q^{87} +719.901 q^{88} +910.505 q^{89} -1698.24 q^{90} +57.7915 q^{91} -242.616 q^{92} -1188.71 q^{93} +405.527 q^{94} +673.137 q^{95} -1032.89 q^{96} -884.342 q^{97} +288.317 q^{98} -1650.43 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 11 q^{3} + 2 q^{4} - 27 q^{5} - 27 q^{6} - 20 q^{7} - 66 q^{8} - 9 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 11 * q^3 + 2 * q^4 - 27 * q^5 - 27 * q^6 - 20 * q^7 - 66 * q^8 - 9 * q^9	\( 4 q - 4 q^{2} - 11 q^{3} + 2 q^{4} - 27 q^{5} - 27 q^{6} - 20 q^{7} - 66 q^{8} - 9 q^{9} - 3 q^{10} - 62 q^{11} + 61 q^{12} - 2 q^{13} + 112 q^{14} + 92 q^{15} + 202 q^{16} - 207 q^{17} + 299 q^{18} + 99 q^{19} + 81 q^{20} - 90 q^{21} + 202 q^{22} - 103 q^{23} + 209 q^{24} - 101 q^{25} - 50 q^{26} - 218 q^{27} - 80 q^{28} - 99 q^{29} + 300 q^{30} + 131 q^{31} - 342 q^{32} - 32 q^{33} + 53 q^{34} - 374 q^{35} - 379 q^{36} - 449 q^{37} - 609 q^{38} + 98 q^{39} + 133 q^{40} - 491 q^{41} - 394 q^{42} + 172 q^{43} - 764 q^{44} - 338 q^{45} + 1061 q^{46} + 19 q^{47} + 237 q^{48} + 236 q^{49} + 599 q^{50} + 1649 q^{51} + 224 q^{52} - 1220 q^{53} - 322 q^{54} + 1360 q^{55} + 344 q^{56} + 232 q^{57} - 771 q^{58} + 816 q^{59} - 156 q^{60} + 372 q^{61} - 97 q^{62} + 1914 q^{63} + 434 q^{64} - 350 q^{65} + 812 q^{66} + 110 q^{67} - 1697 q^{68} - 1238 q^{69} - 718 q^{70} + 468 q^{71} - 315 q^{72} + 628 q^{73} + 395 q^{74} + 62 q^{75} + 1671 q^{76} - 2044 q^{77} - 90 q^{78} + 1095 q^{79} - 31 q^{80} + 2056 q^{81} - 2287 q^{82} - 980 q^{83} - 610 q^{84} - 152 q^{85} - 172 q^{86} - 507 q^{87} + 1816 q^{88} - 738 q^{89} - 2398 q^{90} + 852 q^{91} - 2517 q^{92} - 35 q^{93} - 2233 q^{94} + 1149 q^{95} - 1551 q^{96} - 1765 q^{97} + 1652 q^{98} - 58 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 11 * q^3 + 2 * q^4 - 27 * q^5 - 27 * q^6 - 20 * q^7 - 66 * q^8 - 9 * q^9 - 3 * q^10 - 62 * q^11 + 61 * q^12 - 2 * q^13 + 112 * q^14 + 92 * q^15 + 202 * q^16 - 207 * q^17 + 299 * q^18 + 99 * q^19 + 81 * q^20 - 90 * q^21 + 202 * q^22 - 103 * q^23 + 209 * q^24 - 101 * q^25 - 50 * q^26 - 218 * q^27 - 80 * q^28 - 99 * q^29 + 300 * q^30 + 131 * q^31 - 342 * q^32 - 32 * q^33 + 53 * q^34 - 374 * q^35 - 379 * q^36 - 449 * q^37 - 609 * q^38 + 98 * q^39 + 133 * q^40 - 491 * q^41 - 394 * q^42 + 172 * q^43 - 764 * q^44 - 338 * q^45 + 1061 * q^46 + 19 * q^47 + 237 * q^48 + 236 * q^49 + 599 * q^50 + 1649 * q^51 + 224 * q^52 - 1220 * q^53 - 322 * q^54 + 1360 * q^55 + 344 * q^56 + 232 * q^57 - 771 * q^58 + 816 * q^59 - 156 * q^60 + 372 * q^61 - 97 * q^62 + 1914 * q^63 + 434 * q^64 - 350 * q^65 + 812 * q^66 + 110 * q^67 - 1697 * q^68 - 1238 * q^69 - 718 * q^70 + 468 * q^71 - 315 * q^72 + 628 * q^73 + 395 * q^74 + 62 * q^75 + 1671 * q^76 - 2044 * q^77 - 90 * q^78 + 1095 * q^79 - 31 * q^80 + 2056 * q^81 - 2287 * q^82 - 980 * q^83 - 610 * q^84 - 152 * q^85 - 172 * q^86 - 507 * q^87 + 1816 * q^88 - 738 * q^89 - 2398 * q^90 + 852 * q^91 - 2517 * q^92 - 35 * q^93 - 2233 * q^94 + 1149 * q^95 - 1551 * q^96 - 1765 * q^97 + 1652 * q^98 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.32005	0.820260	0.410130	−	0.912027i	$-0.365483\pi$
$2$	2.32005	0.820260	0.410130	+	0.912027i	$0.365483\pi$
$3$	−9.13635	−1.75829	−0.879145	−	0.476554i	$-0.841886\pi$
$3$	−9.13635	−1.75829	−0.879145	+	0.476554i	$0.841886\pi$
$4$	−2.61739	−0.327173
$5$	−12.9617	−1.15933	−0.579667	−	0.814854i	$-0.696817\pi$
$5$	−12.9617	−1.15933	−0.579667	+	0.814854i	$0.696817\pi$
$6$	−21.1967	−1.44226
$7$	21.6165	1.16718	0.583590	−	0.812048i	$-0.301648\pi$
$7$	21.6165	1.16718	0.583590	+	0.812048i	$0.301648\pi$
$8$	−24.6328	−1.08863
$9$	56.4728	2.09159
$10$	−30.0718	−0.950955
$11$	−29.2253	−0.801069	−0.400534	−	0.916282i	$-0.631175\pi$
$11$	−29.2253	−0.801069	−0.400534	+	0.916282i	$0.631175\pi$
$12$	23.9134	0.575266
$13$	2.67349	0.0570380	0.0285190	−	0.999593i	$-0.490921\pi$
$13$	2.67349	0.0570380	0.0285190	+	0.999593i	$0.490921\pi$
$14$	50.1512	0.957391
$15$	118.423	2.03844
$16$	−36.2102	−0.565784
$17$	−137.211	−1.95756	−0.978779	−	0.204920i	$-0.934307\pi$
$17$	−137.211	−1.95756	−0.978779	+	0.204920i	$0.934307\pi$
$18$	131.020	1.71564
$19$	−51.9326	−0.627061	−0.313531	−	0.949578i	$-0.601512\pi$
$19$	−51.9326	−0.627061	−0.313531	+	0.949578i	$0.601512\pi$
$20$	33.9259	0.379303
$21$	−197.496	−2.05224
$22$	−67.8040	−0.657085
$23$	92.6938	0.840348	0.420174	−	0.907444i	$-0.361969\pi$
$23$	92.6938	0.840348	0.420174	+	0.907444i	$0.361969\pi$
$24$	225.054	1.91412
$25$	43.0067	0.344054
$26$	6.20263	0.0467860
$27$	−269.274	−1.91932
$28$	−56.5787	−0.381870
$29$	148.453	0.950584	0.475292	−	0.879828i	$-0.342342\pi$
$29$	148.453	0.950584	0.475292	+	0.879828i	$0.342342\pi$
$30$	274.747	1.67205
$31$	130.108	0.753811	0.376906	−	0.926252i	$-0.376988\pi$
$31$	130.108	0.753811	0.376906	+	0.926252i	$0.376988\pi$
$32$	113.053	0.624537
$33$	267.012	1.40851
$34$	−318.335	−1.60571
$35$	−280.187	−1.35315
$36$	−147.811	−0.684311
$37$	−211.403	−0.939311	−0.469655	−	0.882850i	$-0.655622\pi$
$37$	−211.403	−0.939311	−0.469655	+	0.882850i	$0.655622\pi$
$38$	−120.486	−0.514353
$39$	−24.4260	−0.100289
$40$	319.284	1.26208
$41$	−474.175	−1.80619	−0.903094	−	0.429444i	$-0.858710\pi$
$41$	−474.175	−1.80619	−0.903094	+	0.429444i	$0.858710\pi$
$42$	−458.199	−1.68337
$43$	43.0000	0.152499
$43$	43.0000	0.152499
$44$	76.4939	0.262088
$45$	−731.986	−2.42484
$46$	215.054	0.689304
$47$	174.793	0.542471	0.271236	−	0.962513i	$-0.412568\pi$
$47$	174.793	0.542471	0.271236	+	0.962513i	$0.412568\pi$
$48$	330.829	0.994813
$49$	124.272	0.362309
$50$	99.7776	0.282214
$51$	1253.60	3.44195
$52$	−6.99757	−0.0186613
$53$	−543.612	−1.40888	−0.704442	−	0.709761i	$-0.748803\pi$
$53$	−543.612	−1.40888	−0.704442	+	0.709761i	$0.748803\pi$
$54$	−624.727	−1.57435
$55$	378.811	0.928706
$56$	−532.475	−1.27062
$57$	474.475	1.10256
$58$	344.417	0.779726
$59$	85.7755	0.189272	0.0946358	−	0.995512i	$-0.469831\pi$
$59$	85.7755	0.189272	0.0946358	+	0.995512i	$0.469831\pi$
$60$	−309.959	−0.666925
$61$	238.340	0.500267	0.250134	−	0.968211i	$-0.419525\pi$
$61$	238.340	0.500267	0.250134	+	0.968211i	$0.419525\pi$
$62$	301.857	0.618321
$63$	1220.74	2.44126
$64$	551.970	1.07807
$65$	−34.6531	−0.0661261
$66$	619.481	1.15535
$67$	−316.481	−0.577079	−0.288540	−	0.957468i	$-0.593170\pi$
$67$	−316.481	−0.577079	−0.288540	+	0.957468i	$0.593170\pi$
$68$	359.133	0.640461
$69$	−846.883	−1.47758
$70$	−650.047	−1.10994
$71$	51.1352	0.0854738	0.0427369	−	0.999086i	$-0.486392\pi$
$71$	51.1352	0.0854738	0.0427369	+	0.999086i	$0.486392\pi$
$72$	−1391.08	−2.27696
$73$	−219.515	−0.351948	−0.175974	−	0.984395i	$-0.556308\pi$
$73$	−219.515	−0.351948	−0.175974	+	0.984395i	$0.556308\pi$
$74$	−490.465	−0.770479
$75$	−392.924	−0.604946
$76$	135.928	0.205158
$77$	−631.748	−0.934991
$78$	−56.6694	−0.0822634
$79$	1172.35	1.66961	0.834805	−	0.550545i	$-0.185580\pi$
$79$	1172.35	1.66961	0.834805	+	0.550545i	$0.185580\pi$
$80$	469.347	0.655932
$81$	935.412	1.28314
$82$	−1100.11	−1.48154
$83$	−921.819	−1.21907	−0.609535	−	0.792759i	$-0.708644\pi$
$83$	−921.819	−1.21907	−0.609535	+	0.792759i	$0.708644\pi$
$84$	516.922	0.671439
$85$	1778.49	2.26946
$86$	99.7620	0.125088
$87$	−1356.31	−1.67140
$88$	719.901	0.872065
$89$	910.505	1.08442	0.542210	−	0.840243i	$-0.317588\pi$
$89$	910.505	1.08442	0.542210	+	0.840243i	$0.317588\pi$
$90$	−1698.24	−1.98900
$91$	57.7915	0.0665736
$92$	−242.616	−0.274939
$93$	−1188.71	−1.32542
$94$	405.527	0.444968
$95$	673.137	0.726973
$96$	−1032.89	−1.09812
$97$	−884.342	−0.925683	−0.462842	−	0.886441i	$-0.653170\pi$
$97$	−884.342	−0.925683	−0.462842	+	0.886441i	$0.653170\pi$
$98$	288.317	0.297188
$99$	−1650.43	−1.67550
$100$	−112.565	−0.112565
$101$	−1297.02	−1.27781	−0.638904	−	0.769286i	$-0.720612\pi$
$101$	−1297.02	−1.27781	−0.638904	+	0.769286i	$0.720612\pi$
$102$	2908.42	2.82330
$103$	−999.529	−0.956180	−0.478090	−	0.878311i	$-0.658671\pi$
$103$	−999.529	−0.956180	−0.478090	+	0.878311i	$0.658671\pi$
$104$	−65.8557	−0.0620931
$105$	2559.89	2.37923
$106$	−1261.21	−1.15565
$107$	−777.580	−0.702537	−0.351269	−	0.936275i	$-0.614250\pi$
$107$	−777.580	−0.702537	−0.351269	+	0.936275i	$0.614250\pi$
$108$	704.794	0.627952
$109$	1683.85	1.47967	0.739833	−	0.672791i	$-0.234905\pi$
$109$	1683.85	1.47967	0.739833	+	0.672791i	$0.234905\pi$
$110$	878.858	0.761780
$111$	1931.45	1.65158
$112$	−782.737	−0.660372
$113$	−288.278	−0.239990	−0.119995	−	0.992774i	$-0.538288\pi$
$113$	−288.278	−0.239990	−0.119995	+	0.992774i	$0.538288\pi$
$114$	1100.80	0.904383
$115$	−1201.47	−0.974243
$116$	−388.558	−0.311006
$117$	150.980	0.119300
$118$	199.003	0.155252
$119$	−2966.01	−2.28482
$120$	−2917.09	−2.21911
$121$	−476.883	−0.358289
$122$	552.960	0.410349
$123$	4332.23	3.17580
$124$	−340.544	−0.246627
$125$	1062.78	0.760460
$126$	2832.18	2.00247
$127$	85.9145	0.0600290	0.0300145	−	0.999549i	$-0.490445\pi$
$127$	85.9145	0.0600290	0.0300145	+	0.999549i	$0.490445\pi$
$128$	376.170	0.259758
$129$	−392.863	−0.268137
$130$	−80.3969	−0.0542406
$131$	365.279	0.243623	0.121811	−	0.992553i	$-0.461130\pi$
$131$	365.279	0.243623	0.121811	+	0.992553i	$0.461130\pi$
$132$	−698.875	−0.460827
$133$	−1122.60	−0.731893
$134$	−734.251	−0.473355
$135$	3490.26	2.22514
$136$	3379.89	2.13105
$137$	777.397	0.484799	0.242400	−	0.970176i	$-0.422065\pi$
$137$	777.397	0.484799	0.242400	+	0.970176i	$0.422065\pi$
$138$	−1964.81	−1.21200
$139$	−1227.84	−0.749236	−0.374618	−	0.927179i	$-0.622226\pi$
$139$	−1227.84	−0.749236	−0.374618	+	0.927179i	$0.622226\pi$
$140$	733.358	0.442715
$141$	−1596.97	−0.953822
$142$	118.636	0.0701107
$143$	−78.1337	−0.0456914
$144$	−2044.89	−1.18339
$145$	−1924.20	−1.10204
$146$	−509.284	−0.288689
$147$	−1135.39	−0.637045
$148$	553.324	0.307317
$149$	1617.47	0.889317	0.444659	−	0.895700i	$-0.353325\pi$
$149$	1617.47	0.889317	0.444659	+	0.895700i	$0.353325\pi$
$150$	−911.602	−0.496213
$151$	−378.309	−0.203883	−0.101942	−	0.994790i	$-0.532505\pi$
$151$	−378.309	−0.203883	−0.101942	+	0.994790i	$0.532505\pi$
$152$	1279.25	0.682636
$153$	−7748.67	−4.09440
$154$	−1465.68	−0.766936
$155$	−1686.43	−0.873918
$156$	63.9322	0.0328120
$157$	404.282	0.205511	0.102755	−	0.994707i	$-0.467234\pi$
$157$	404.282	0.205511	0.102755	+	0.994707i	$0.467234\pi$
$158$	2719.90	1.36951
$159$	4966.63	2.47723
$160$	−1465.37	−0.724047
$161$	2003.71	0.980837
$162$	2170.20	1.05251
$163$	−392.165	−0.188446	−0.0942231	−	0.995551i	$-0.530037\pi$
$163$	−392.165	−0.188446	−0.0942231	+	0.995551i	$0.530037\pi$
$164$	1241.10	0.590936
$165$	−3460.94	−1.63293
$166$	−2138.66	−0.999954
$167$	254.961	0.118140	0.0590702	−	0.998254i	$-0.481186\pi$
$167$	254.961	0.118140	0.0590702	+	0.998254i	$0.481186\pi$
$168$	4864.87	2.23413
$169$	−2189.85	−0.996747
$170$	4126.18	1.86155
$171$	−2932.78	−1.31155
$172$	−112.548	−0.0498935
$173$	−536.300	−0.235689	−0.117844	−	0.993032i	$-0.537598\pi$
$173$	−536.300	−0.235689	−0.117844	+	0.993032i	$0.537598\pi$
$174$	−3146.71	−1.37099
$175$	929.654	0.401573
$176$	1058.25	0.453232
$177$	−783.675	−0.332794
$178$	2112.41	0.889506
$179$	−1944.34	−0.811882	−0.405941	−	0.913899i	$-0.633056\pi$
$179$	−1944.34	−0.811882	−0.405941	+	0.913899i	$0.633056\pi$
$180$	1915.89	0.793345
$181$	4143.16	1.70143	0.850714	−	0.525630i	$-0.176170\pi$
$181$	4143.16	1.70143	0.850714	+	0.525630i	$0.176170\pi$
$182$	134.079	0.0546077
$183$	−2177.56	−0.879616
$184$	−2283.31	−0.914826
$185$	2740.15	1.08897
$186$	−2757.87	−1.08719
$187$	4010.02	1.56814
$188$	−457.501	−0.177482
$189$	−5820.75	−2.24020
$190$	1561.71	0.596307
$191$	−293.088	−0.111032	−0.0555160	−	0.998458i	$-0.517680\pi$
$191$	−293.088	−0.111032	−0.0555160	+	0.998458i	$0.517680\pi$
$192$	−5042.99	−1.89556
$193$	−2417.36	−0.901584	−0.450792	−	0.892629i	$-0.648858\pi$
$193$	−2417.36	−0.901584	−0.450792	+	0.892629i	$0.648858\pi$
$194$	−2051.71	−0.759301
$195$	316.603	0.116269
$196$	−325.268	−0.118538
$197$	1085.38	0.392540	0.196270	−	0.980550i	$-0.437117\pi$
$197$	1085.38	0.392540	0.196270	+	0.980550i	$0.437117\pi$
$198$	−3829.08	−1.37435
$199$	187.257	0.0667052	0.0333526	−	0.999444i	$-0.489382\pi$
$199$	187.257	0.0667052	0.0333526	+	0.999444i	$0.489382\pi$
$200$	−1059.38	−0.374546
$201$	2891.48	1.01467
$202$	−3009.15	−1.04814
$203$	3209.02	1.10950
$204$	−3281.17	−1.12612
$205$	6146.13	2.09397
$206$	−2318.95	−0.784316
$207$	5234.68	1.75766
$208$	−96.8078	−0.0322712
$209$	1517.75	0.502319
$210$	5939.05	1.95159
$211$	−472.381	−0.154124	−0.0770618	−	0.997026i	$-0.524554\pi$
$211$	−472.381	−0.154124	−0.0770618	+	0.997026i	$0.524554\pi$
$212$	1422.84	0.460950
$213$	−467.189	−0.150288
$214$	−1804.02	−0.576263
$215$	−557.355	−0.176797
$216$	6632.97	2.08943
$217$	2812.48	0.879833
$218$	3906.61	1.21371
$219$	2005.56	0.618828
$220$	−991.494	−0.303848
$221$	−366.832	−0.111655
$222$	4481.06	1.35473
$223$	−2899.24	−0.870616	−0.435308	−	0.900282i	$-0.643361\pi$
$223$	−2899.24	−0.870616	−0.435308	+	0.900282i	$0.643361\pi$
$224$	2443.81	0.728947
$225$	2428.71	0.719618
$226$	−668.818	−0.196854
$227$	3204.94	0.937088	0.468544	−	0.883440i	$-0.344779\pi$
$227$	3204.94	0.937088	0.468544	+	0.883440i	$0.344779\pi$
$228$	−1241.88	−0.360727
$229$	−2555.11	−0.737322	−0.368661	−	0.929564i	$-0.620184\pi$
$229$	−2555.11	−0.737322	−0.368661	+	0.929564i	$0.620184\pi$
$230$	−2787.47	−0.799133
$231$	5771.87	1.64399
$232$	−3656.81	−1.03483
$233$	−4748.72	−1.33519	−0.667594	−	0.744525i	$-0.732676\pi$
$233$	−4748.72	−1.33519	−0.667594	+	0.744525i	$0.732676\pi$
$234$	350.280	0.0978569
$235$	−2265.62	−0.628905
$236$	−224.508	−0.0619246
$237$	−10711.0	−2.93566
$238$	−6881.28	−1.87415
$239$	5452.31	1.47565	0.737826	−	0.674991i	$-0.235852\pi$
$239$	5452.31	1.47565	0.737826	+	0.674991i	$0.235852\pi$
$240$	−4288.12	−1.15332
$241$	6676.52	1.78453	0.892267	−	0.451509i	$-0.149114\pi$
$241$	6676.52	1.78453	0.892267	+	0.451509i	$0.149114\pi$
$242$	−1106.39	−0.293890
$243$	−1275.86	−0.336816
$244$	−623.828	−0.163674
$245$	−1610.78	−0.420037
$246$	10051.0	2.60498
$247$	−138.842	−0.0357663
$248$	−3204.94	−0.820619
$249$	8422.06	2.14348
$250$	2465.69	0.623775
$251$	297.282	0.0747582	0.0373791	−	0.999301i	$-0.488099\pi$
$251$	297.282	0.0747582	0.0373791	+	0.999301i	$0.488099\pi$
$252$	−3195.16	−0.798714
$253$	−2709.00	−0.673176
$254$	199.326	0.0492394
$255$	−16248.9	−3.99037
$256$	−3543.03	−0.864998
$257$	−5752.39	−1.39620	−0.698101	−	0.715999i	$-0.745971\pi$
$257$	−5752.39	−1.39620	−0.698101	+	0.715999i	$0.745971\pi$
$258$	−911.460	−0.219942
$259$	−4569.79	−1.09634
$260$	90.7007	0.0216347
$261$	8383.53	1.98823
$262$	847.464	0.199834
$263$	−3046.57	−0.714294	−0.357147	−	0.934048i	$-0.616251\pi$
$263$	−3046.57	−0.714294	−0.357147	+	0.934048i	$0.616251\pi$
$264$	−6577.27	−1.53334
$265$	7046.16	1.63337
$266$	−2604.49	−0.600343
$267$	−8318.69	−1.90672
$268$	828.354	0.188805
$269$	2040.69	0.462539	0.231269	−	0.972890i	$-0.425712\pi$
$269$	2040.69	0.462539	0.231269	+	0.972890i	$0.425712\pi$
$270$	8097.56	1.82519
$271$	−5824.13	−1.30550	−0.652750	−	0.757573i	$-0.726385\pi$
$271$	−5824.13	−1.30550	−0.652750	+	0.757573i	$0.726385\pi$
$272$	4968.42	1.10756
$273$	−528.003	−0.117056
$274$	1803.60	0.397662
$275$	−1256.88	−0.275611
$276$	2216.62	0.483423
$277$	−1072.21	−0.232573	−0.116287	−	0.993216i	$-0.537099\pi$
$277$	−1072.21	−0.232573	−0.116287	+	0.993216i	$0.537099\pi$
$278$	−2848.64	−0.614568
$279$	7347.58	1.57666
$280$	6901.80	1.47308
$281$	−7783.22	−1.65234	−0.826171	−	0.563420i	$-0.809485\pi$
$281$	−7783.22	−1.65234	−0.826171	+	0.563420i	$0.809485\pi$
$282$	−3705.04	−0.782382
$283$	3934.47	0.826430	0.413215	−	0.910633i	$-0.364406\pi$
$283$	3934.47	0.826430	0.413215	+	0.910633i	$0.364406\pi$
$284$	−133.841	−0.0279647
$285$	−6150.02	−1.27823
$286$	−181.274	−0.0374788
$287$	−10250.0	−2.10815
$288$	6384.44	1.30627
$289$	13913.8	2.83203
$290$	−4464.24	−0.903963
$291$	8079.65	1.62762
$292$	574.555	0.115148
$293$	−947.548	−0.188930	−0.0944648	−	0.995528i	$-0.530114\pi$
$293$	−947.548	−0.188930	−0.0944648	+	0.995528i	$0.530114\pi$
$294$	−2634.16	−0.522542
$295$	−1111.80	−0.219429
$296$	5207.46	1.02256
$297$	7869.60	1.53751
$298$	3752.60	0.729472
$299$	247.817	0.0479318
$300$	1028.43	0.197922
$301$	929.508	0.177993
$302$	−877.695	−0.167237
$303$	11850.1	2.24676
$304$	1880.49	0.354781
$305$	−3089.30	−0.579977
$306$	−17977.3	−3.35847
$307$	6651.57	1.23656	0.618282	−	0.785956i	$-0.287829\pi$
$307$	6651.57	1.23656	0.618282	+	0.785956i	$0.287829\pi$
$308$	1653.53	0.305904
$309$	9132.04	1.68124
$310$	−3912.60	−0.716840
$311$	4994.49	0.910648	0.455324	−	0.890326i	$-0.349523\pi$
$311$	4994.49	0.910648	0.455324	+	0.890326i	$0.349523\pi$
$312$	601.681	0.109178
$313$	1298.36	0.234465	0.117233	−	0.993104i	$-0.462598\pi$
$313$	1298.36	0.234465	0.117233	+	0.993104i	$0.462598\pi$
$314$	937.952	0.168572
$315$	−15823.0	−2.83023
$316$	−3068.48	−0.546252
$317$	−6897.11	−1.22202	−0.611010	−	0.791623i	$-0.709237\pi$
$317$	−6897.11	−1.22202	−0.611010	+	0.791623i	$0.709237\pi$
$318$	11522.8	2.03197
$319$	−4338.57	−0.761483
$320$	−7154.50	−1.24984
$321$	7104.24	1.23526
$322$	4648.71	0.804542
$323$	7125.71	1.22751
$324$	−2448.34	−0.419811
$325$	114.978	0.0196241
$326$	−909.841	−0.154575
$327$	−15384.2	−2.60168
$328$	11680.3	1.96627
$329$	3778.41	0.633162
$330$	−8029.55	−1.33943
$331$	−2650.59	−0.440149	−0.220075	−	0.975483i	$-0.570630\pi$
$331$	−2650.59	−0.440149	−0.220075	+	0.975483i	$0.570630\pi$
$332$	2412.76	0.398847
$333$	−11938.5	−1.96465
$334$	591.520	0.0969058
$335$	4102.15	0.669027
$336$	7151.35	1.16113
$337$	−1004.48	−0.162367	−0.0811835	−	0.996699i	$-0.525870\pi$
$337$	−1004.48	−0.162367	−0.0811835	+	0.996699i	$0.525870\pi$
$338$	−5080.56	−0.817592
$339$	2633.81	0.421972
$340$	−4654.99	−0.742507
$341$	−3802.45	−0.603854
$342$	−6804.19	−1.07581
$343$	−4728.13	−0.744300
$344$	−1059.21	−0.166014
$345$	10977.1	1.71300
$346$	−1244.24	−0.193326
$347$	−1588.96	−0.245820	−0.122910	−	0.992418i	$-0.539223\pi$
$347$	−1588.96	−0.245820	−0.122910	+	0.992418i	$0.539223\pi$
$348$	3550.00	0.546839
$349$	6154.36	0.943941	0.471971	−	0.881614i	$-0.343543\pi$
$349$	6154.36	0.943941	0.471971	+	0.881614i	$0.343543\pi$
$350$	2156.84	0.329394
$351$	−719.902	−0.109474
$352$	−3304.02	−0.500297
$353$	6182.29	0.932153	0.466077	−	0.884744i	$-0.345667\pi$
$353$	6182.29	0.932153	0.466077	+	0.884744i	$0.345667\pi$
$354$	−1818.16	−0.272978
$355$	−662.802	−0.0990926
$356$	−2383.14	−0.354793
$357$	27098.5	4.01738
$358$	−4510.96	−0.665954
$359$	−11058.1	−1.62569	−0.812843	−	0.582482i	$-0.802082\pi$
$359$	−11058.1	−1.62569	−0.812843	+	0.582482i	$0.802082\pi$
$360$	18030.9	2.63975
$361$	−4162.00	−0.606794
$362$	9612.31	1.39561
$363$	4356.96	0.629976
$364$	−151.263	−0.0217811
$365$	2845.29	0.408026
$366$	−5052.03	−0.721514
$367$	−6012.26	−0.855143	−0.427572	−	0.903981i	$-0.640631\pi$
$367$	−6012.26	−0.855143	−0.427572	+	0.903981i	$0.640631\pi$
$368$	−3356.46	−0.475456
$369$	−26778.0	−3.77780
$370$	6357.29	0.893242
$371$	−11751.0	−1.64442
$372$	3111.33	0.433642
$373$	5198.95	0.721693	0.360847	−	0.932625i	$-0.382488\pi$
$373$	5198.95	0.721693	0.360847	+	0.932625i	$0.382488\pi$
$374$	9303.43	1.28628
$375$	−9709.89	−1.33711
$376$	−4305.64	−0.590549
$377$	396.887	0.0542194
$378$	−13504.4	−1.83754
$379$	−8139.82	−1.10320	−0.551602	−	0.834107i	$-0.685983\pi$
$379$	−8139.82	−1.10320	−0.551602	+	0.834107i	$0.685983\pi$
$380$	−1761.86	−0.237846
$381$	−784.945	−0.105548
$382$	−679.978	−0.0910751
$383$	11443.4	1.52672	0.763359	−	0.645975i	$-0.223549\pi$
$383$	11443.4	1.52672	0.763359	+	0.645975i	$0.223549\pi$
$384$	−3436.82	−0.456731
$385$	8188.55	1.08397
$386$	−5608.40	−0.739533
$387$	2428.33	0.318964
$388$	2314.66	0.302859
$389$	2838.64	0.369987	0.184994	−	0.982740i	$-0.440774\pi$
$389$	2838.64	0.369987	0.184994	+	0.982740i	$0.440774\pi$
$390$	734.534	0.0953707
$391$	−12718.6	−1.64503
$392$	−3061.17	−0.394420
$393$	−3337.31	−0.428359
$394$	2518.14	0.321985
$395$	−15195.6	−1.93563
$396$	4319.82	0.548180
$397$	−1907.29	−0.241118	−0.120559	−	0.992706i	$-0.538469\pi$
$397$	−1907.29	−0.241118	−0.120559	+	0.992706i	$0.538469\pi$
$398$	434.446	0.0547156
$399$	10256.5	1.28688
$400$	−1557.28	−0.194660
$401$	7894.11	0.983075	0.491538	−	0.870856i	$-0.336435\pi$
$401$	7894.11	0.983075	0.491538	+	0.870856i	$0.336435\pi$
$402$	6708.37	0.832296
$403$	347.844	0.0429959
$404$	3394.81	0.418065
$405$	−12124.6	−1.48759
$406$	7445.08	0.910081
$407$	6178.32	0.752452
$408$	−30879.8	−3.74701
$409$	−13591.0	−1.64311	−0.821554	−	0.570131i	$-0.806893\pi$
$409$	−13591.0	−1.64311	−0.821554	+	0.570131i	$0.806893\pi$
$410$	14259.3	1.71760
$411$	−7102.57	−0.852418
$412$	2616.15	0.312837
$413$	1854.16	0.220914
$414$	12144.7	1.44174
$415$	11948.4	1.41331
$416$	302.247	0.0356224
$417$	11217.9	1.31737
$418$	3521.24	0.412032
$419$	12331.7	1.43781	0.718904	−	0.695110i	$-0.244644\pi$
$419$	12331.7	1.43781	0.718904	+	0.695110i	$0.244644\pi$
$420$	−6700.21	−0.778421
$421$	−14571.2	−1.68684	−0.843419	−	0.537256i	$-0.819461\pi$
$421$	−14571.2	−1.68684	−0.843419	+	0.537256i	$0.819461\pi$
$422$	−1095.95	−0.126421
$423$	9871.04	1.13463
$424$	13390.7	1.53375
$425$	−5900.98	−0.673505
$426$	−1083.90	−0.123275
$427$	5152.07	0.583902
$428$	2035.23	0.229851
$429$	713.856	0.0803387
$430$	−1293.09	−0.145019
$431$	−6932.25	−0.774744	−0.387372	−	0.921923i	$-0.626617\pi$
$431$	−6932.25	−0.774744	−0.387372	+	0.921923i	$0.626617\pi$
$432$	9750.45	1.08592
$433$	−13384.7	−1.48551	−0.742757	−	0.669561i	$-0.766482\pi$
$433$	−13384.7	−1.48551	−0.742757	+	0.669561i	$0.766482\pi$
$434$	6525.09	0.721692
$435$	17580.2	1.93771
$436$	−4407.29	−0.484107
$437$	−4813.84	−0.526950
$438$	4652.99	0.507600
$439$	−1927.39	−0.209543	−0.104772	−	0.994496i	$-0.533411\pi$
$439$	−1927.39	−0.209543	−0.104772	+	0.994496i	$0.533411\pi$
$440$	−9331.17	−1.01101
$441$	7017.99	0.757800
$442$	−851.067	−0.0915863
$443$	−3297.59	−0.353665	−0.176832	−	0.984241i	$-0.556585\pi$
$443$	−3297.59	−0.353665	−0.176832	+	0.984241i	$0.556585\pi$
$444$	−5055.36	−0.540353
$445$	−11801.7	−1.25720
$446$	−6726.37	−0.714132
$447$	−14777.8	−1.56368
$448$	11931.7	1.25830
$449$	−4121.02	−0.433147	−0.216574	−	0.976266i	$-0.569488\pi$
$449$	−4121.02	−0.433147	−0.216574	+	0.976266i	$0.569488\pi$
$450$	5634.72	0.590274
$451$	13857.9	1.44688
$452$	754.535	0.0785184
$453$	3456.36	0.358486
$454$	7435.60	0.768656
$455$	−749.079	−0.0771810
$456$	−11687.6	−1.20027
$457$	11967.8	1.22501	0.612507	−	0.790465i	$-0.290161\pi$
$457$	11967.8	1.22501	0.612507	+	0.790465i	$0.290161\pi$
$458$	−5927.98	−0.604796
$459$	36947.2	3.75719
$460$	3144.72	0.318746
$461$	−9166.23	−0.926061	−0.463030	−	0.886342i	$-0.653238\pi$
$461$	−9166.23	−0.926061	−0.463030	+	0.886342i	$0.653238\pi$
$462$	13391.0	1.34850
$463$	−5287.39	−0.530725	−0.265362	−	0.964149i	$-0.585492\pi$
$463$	−5287.39	−0.530725	−0.265362	+	0.964149i	$0.585492\pi$
$464$	−5375.50	−0.537826
$465$	15407.8	1.53660
$466$	−11017.2	−1.09520
$467$	7257.05	0.719092	0.359546	−	0.933127i	$-0.382931\pi$
$467$	7257.05	0.719092	0.359546	+	0.933127i	$0.382931\pi$
$468$	−395.172	−0.0390317
$469$	−6841.21	−0.673556
$470$	−5256.34	−0.515866
$471$	−3693.66	−0.361348
$472$	−2112.89	−0.206046
$473$	−1256.69	−0.122162
$474$	−24849.9	−2.40800
$475$	−2233.45	−0.215743
$476$	7763.20	0.747533
$477$	−30699.3	−2.94680
$478$	12649.6	1.21042
$479$	−3513.79	−0.335176	−0.167588	−	0.985857i	$-0.553598\pi$
$479$	−3513.79	−0.335176	−0.167588	+	0.985857i	$0.553598\pi$
$480$	13388.1	1.27308
$481$	−565.186	−0.0535764
$482$	15489.8	1.46378
$483$	−18306.6	−1.72460
$484$	1248.19	0.117223
$485$	11462.6	1.07318
$486$	−2960.05	−0.276277
$487$	−398.977	−0.0371240	−0.0185620	−	0.999828i	$-0.505909\pi$
$487$	−398.977	−0.0371240	−0.0185620	+	0.999828i	$0.505909\pi$
$488$	−5870.99	−0.544605
$489$	3582.95	0.331343
$490$	−3737.09	−0.344540
$491$	11811.6	1.08564	0.542822	−	0.839848i	$-0.317356\pi$
$491$	11811.6	1.08564	0.542822	+	0.839848i	$0.317356\pi$
$492$	−11339.1	−1.03904
$493$	−20369.3	−1.86082
$494$	−322.119	−0.0293377
$495$	21392.5	1.94247
$496$	−4711.25	−0.426494
$497$	1105.36	0.0997633
$498$	19539.6	1.75821
$499$	1194.80	0.107187	0.0535937	−	0.998563i	$-0.482932\pi$
$499$	1194.80	0.107187	0.0535937	+	0.998563i	$0.482932\pi$
$500$	−2781.70	−0.248802
$501$	−2329.41	−0.207725
$502$	689.709	0.0613211
$503$	18815.1	1.66784	0.833920	−	0.551885i	$-0.186091\pi$
$503$	18815.1	1.66784	0.833920	+	0.551885i	$0.186091\pi$
$504$	−30070.3	−2.65762
$505$	16811.7	1.48141
$506$	−6285.01	−0.552180
$507$	20007.2	1.75257
$508$	−224.872	−0.0196399
$509$	−4842.32	−0.421674	−0.210837	−	0.977521i	$-0.567619\pi$
$509$	−4842.32	−0.421674	−0.210837	+	0.977521i	$0.567619\pi$
$510$	−37698.2	−3.27314
$511$	−4745.13	−0.410787
$512$	−11229.4	−0.969281
$513$	13984.1	1.20353
$514$	−13345.8	−1.14525
$515$	12955.6	1.10853
$516$	1028.27	0.0877272
$517$	−5108.37	−0.434557
$518$	−10602.1	−0.899288
$519$	4899.82	0.414409
$520$	853.605	0.0719866
$521$	9595.11	0.806851	0.403426	−	0.915012i	$-0.367819\pi$
$521$	9595.11	0.806851	0.403426	+	0.915012i	$0.367819\pi$
$522$	19450.2	1.63086
$523$	5782.05	0.483425	0.241712	−	0.970348i	$-0.422291\pi$
$523$	5782.05	0.483425	0.241712	+	0.970348i	$0.422291\pi$
$524$	−956.076	−0.0797069
$525$	−8493.64	−0.706081
$526$	−7068.17	−0.585907
$527$	−17852.2	−1.47563
$528$	−9668.57	−0.796914
$529$	−3574.85	−0.293815
$530$	16347.4	1.33979
$531$	4843.99	0.395878
$532$	2938.28	0.239456
$533$	−1267.70	−0.103021
$534$	−19299.7	−1.56401
$535$	10078.8	0.814475
$536$	7795.82	0.628225
$537$	17764.2	1.42752
$538$	4734.49	0.379402
$539$	−3631.88	−0.290234
$540$	−9135.35	−0.728005
$541$	10095.6	0.802302	0.401151	−	0.916012i	$-0.368610\pi$
$541$	10095.6	0.802302	0.401151	+	0.916012i	$0.368610\pi$
$542$	−13512.2	−1.07085
$543$	−37853.3	−2.99160
$544$	−15512.1	−1.22257
$545$	−21825.6	−1.71543
$546$	−1224.99	−0.0960162
$547$	−10688.4	−0.835473	−0.417737	−	0.908568i	$-0.637177\pi$
$547$	−10688.4	−0.835473	−0.417737	+	0.908568i	$0.637177\pi$
$548$	−2034.75	−0.158613
$549$	13459.7	1.04635
$550$	−2916.03	−0.226072
$551$	−7709.53	−0.596075
$552$	20861.1	1.60853
$553$	25342.0	1.94874
$554$	−2487.57	−0.190771
$555$	−25035.0	−1.91473
$556$	3213.72	0.245130
$557$	7257.02	0.552046	0.276023	−	0.961151i	$-0.410983\pi$
$557$	7257.02	0.552046	0.276023	+	0.961151i	$0.410983\pi$
$558$	17046.7	1.29327
$559$	114.960	0.00869821
$560$	10145.6	0.765591
$561$	−36636.9	−2.75724
$562$	−18057.4	−1.35535
$563$	6918.09	0.517873	0.258937	−	0.965894i	$-0.416628\pi$
$563$	6918.09	0.517873	0.258937	+	0.965894i	$0.416628\pi$
$564$	4179.88	0.312065
$565$	3736.58	0.278229
$566$	9128.14	0.677888
$567$	20220.3	1.49766
$568$	−1259.61	−0.0930491
$569$	12407.7	0.914158	0.457079	−	0.889426i	$-0.348896\pi$
$569$	12407.7	0.914158	0.457079	+	0.889426i	$0.348896\pi$
$570$	−14268.3	−1.04848
$571$	−24702.9	−1.81048	−0.905240	−	0.424900i	$-0.860309\pi$
$571$	−24702.9	−1.81048	−0.905240	+	0.424900i	$0.860309\pi$
$572$	204.506	0.0149490
$573$	2677.75	0.195226
$574$	−23780.4	−1.72923
$575$	3986.46	0.289125
$576$	31171.3	2.25487
$577$	−4375.84	−0.315717	−0.157858	−	0.987462i	$-0.550459\pi$
$577$	−4375.84	−0.315717	−0.157858	+	0.987462i	$0.550459\pi$
$578$	32280.6	2.32300
$579$	22085.9	1.58525
$580$	5036.39	0.360559
$581$	−19926.5	−1.42287
$582$	18745.2	1.33507
$583$	15887.2	1.12861
$584$	5407.26	0.383141
$585$	−1956.96	−0.138308
$586$	−2198.36	−0.154971
$587$	10810.6	0.760141	0.380070	−	0.924958i	$-0.375900\pi$
$587$	10810.6	0.760141	0.380070	+	0.924958i	$0.375900\pi$
$588$	2971.76	0.208424
$589$	−6756.87	−0.472686
$590$	−2579.43	−0.179989
$591$	−9916.45	−0.690200
$592$	7654.95	0.531447
$593$	−7.55631	−0.000523272 0	−0.000261636	−	1.00000i	$-0.500083\pi$
$593$	−7.55631	−0.000523272 0	−0.000261636		1.00000i	$0.500083\pi$
$594$	18257.8	1.26116
$595$	38444.7	2.64887
$596$	−4233.54	−0.290961
$597$	−1710.85	−0.117287
$598$	574.946	0.0393165
$599$	4175.20	0.284798	0.142399	−	0.989809i	$-0.454518\pi$
$599$	4175.20	0.284798	0.142399	+	0.989809i	$0.454518\pi$
$600$	9678.83	0.658561
$601$	13011.8	0.883131	0.441566	−	0.897229i	$-0.354423\pi$
$601$	13011.8	0.883131	0.441566	+	0.897229i	$0.354423\pi$
$602$	2156.50	0.146001
$603$	−17872.6	−1.20701
$604$	990.182	0.0667052
$605$	6181.23	0.415376
$606$	27492.7	1.84293
$607$	−9246.84	−0.618316	−0.309158	−	0.951011i	$-0.600047\pi$
$607$	−9246.84	−0.618316	−0.309158	+	0.951011i	$0.600047\pi$
$608$	−5871.16	−0.391623
$609$	−29318.7	−1.95083
$610$	−7167.32	−0.475732
$611$	467.308	0.0309415
$612$	20281.3	1.33958
$613$	140.053	0.00922788	0.00461394	−	0.999989i	$-0.498531\pi$
$613$	140.053	0.00922788	0.00461394	+	0.999989i	$0.498531\pi$
$614$	15432.0	1.01430
$615$	−56153.2	−3.68181
$616$	15561.7	1.01786
$617$	26919.5	1.75647	0.878233	−	0.478234i	$-0.158723\pi$
$617$	26919.5	1.75647	0.878233	+	0.478234i	$0.158723\pi$
$618$	21186.8	1.37906
$619$	−15533.0	−1.00860	−0.504300	−	0.863529i	$-0.668249\pi$
$619$	−15533.0	−1.00860	−0.504300	+	0.863529i	$0.668249\pi$
$620$	4414.04	0.285923
$621$	−24960.0	−1.61290
$622$	11587.4	0.746968
$623$	19681.9	1.26571
$624$	884.469	0.0567421
$625$	−19151.3	−1.22568
$626$	3012.25	0.192322
$627$	−13866.7	−0.883223
$628$	−1058.16	−0.0672376
$629$	29006.8	1.83875
$630$	−36710.0	−2.32152
$631$	18931.2	1.19436	0.597180	−	0.802108i	$-0.296288\pi$
$631$	18931.2	1.19436	0.597180	+	0.802108i	$0.296288\pi$
$632$	−28878.2	−1.81758
$633$	4315.84	0.270994
$634$	−16001.6	−1.00237
$635$	−1113.60	−0.0695936
$636$	−12999.6	−0.810483
$637$	332.241	0.0206654
$638$	−10065.7	−0.624614
$639$	2887.75	0.178776
$640$	−4875.82	−0.301146
$641$	−29592.9	−1.82348	−0.911739	−	0.410769i	$-0.865260\pi$
$641$	−29592.9	−1.82348	−0.911739	+	0.410769i	$0.865260\pi$
$642$	16482.2	1.01324
$643$	−2519.80	−0.154543	−0.0772716	−	0.997010i	$-0.524621\pi$
$643$	−2519.80	−0.154543	−0.0772716	+	0.997010i	$0.524621\pi$
$644$	−5244.50	−0.320904
$645$	5092.19	0.310860
$646$	16532.0	1.00688
$647$	1600.75	0.0972671	0.0486335	−	0.998817i	$-0.484513\pi$
$647$	1600.75	0.0972671	0.0486335	+	0.998817i	$0.484513\pi$
$648$	−23041.8	−1.39687
$649$	−2506.81	−0.151620
$650$	266.755	0.0160969
$651$	−25695.8	−1.54700
$652$	1026.45	0.0616546
$653$	−3189.90	−0.191164	−0.0955822	−	0.995422i	$-0.530471\pi$
$653$	−3189.90	−0.191164	−0.0955822	+	0.995422i	$0.530471\pi$
$654$	−35692.1	−2.13406
$655$	−4734.65	−0.282440
$656$	17170.0	1.02191
$657$	−12396.6	−0.736130
$658$	8766.07	0.519357
$659$	10614.5	0.627439	0.313720	−	0.949516i	$-0.398425\pi$
$659$	10614.5	0.627439	0.313720	+	0.949516i	$0.398425\pi$
$660$	9058.63	0.534253
$661$	−10430.3	−0.613753	−0.306876	−	0.951749i	$-0.599284\pi$
$661$	−10430.3	−0.613753	−0.306876	+	0.951749i	$0.599284\pi$
$662$	−6149.48	−0.361037
$663$	3351.50	0.196322
$664$	22707.0	1.32711
$665$	14550.9	0.848508
$666$	−27698.0	−1.61152
$667$	13760.6	0.798822
$668$	−667.330	−0.0386524
$669$	26488.4	1.53080
$670$	9517.17	0.548777
$671$	−6965.56	−0.400749
$672$	−22327.5	−1.28170
$673$	2497.60	0.143054	0.0715269	−	0.997439i	$-0.477213\pi$
$673$	2497.60	0.143054	0.0715269	+	0.997439i	$0.477213\pi$
$674$	−2330.45	−0.133183
$675$	−11580.6	−0.660351
$676$	5731.69	0.326109
$677$	−17363.2	−0.985705	−0.492853	−	0.870113i	$-0.664046\pi$
$677$	−17363.2	−0.985705	−0.492853	+	0.870113i	$0.664046\pi$
$678$	6110.55	0.346127
$679$	−19116.3	−1.08044
$680$	−43809.2	−2.47060
$681$	−29281.4	−1.64767
$682$	−8821.86	−0.495318
$683$	−8602.77	−0.481956	−0.240978	−	0.970531i	$-0.577468\pi$
$683$	−8602.77	−0.481956	−0.240978	+	0.970531i	$0.577468\pi$
$684$	7676.23	0.429105
$685$	−10076.4	−0.562044
$686$	−10969.5	−0.610520
$687$	23344.4	1.29643
$688$	−1557.04	−0.0862813
$689$	−1453.34	−0.0803600
$690$	25467.3	1.40511
$691$	3155.19	0.173704	0.0868518	−	0.996221i	$-0.472319\pi$
$691$	3155.19	0.173704	0.0868518	+	0.996221i	$0.472319\pi$
$692$	1403.70	0.0771111
$693$	−35676.6	−1.95561
$694$	−3686.45	−0.201636
$695$	15914.9	0.868614
$696$	33409.8	1.81954
$697$	65061.9	3.53572
$698$	14278.4	0.774277
$699$	43385.9	2.34765
$700$	−2433.26	−0.131384
$701$	575.377	0.0310010	0.0155005	−	0.999880i	$-0.495066\pi$
$701$	575.377	0.0310010	0.0155005	+	0.999880i	$0.495066\pi$
$702$	−1670.21	−0.0897975
$703$	10978.7	0.589005
$704$	−16131.5	−0.863606
$705$	20699.5	1.10580
$706$	14343.2	0.764608
$707$	−28037.1	−1.49143
$708$	2051.18	0.108881
$709$	9563.36	0.506572	0.253286	−	0.967391i	$-0.418489\pi$
$709$	9563.36	0.506572	0.253286	+	0.967391i	$0.418489\pi$
$710$	−1537.73	−0.0812817
$711$	66205.7	3.49213
$712$	−22428.3	−1.18053
$713$	12060.2	0.633464
$714$	62869.8	3.29530
$715$	1012.75	0.0529715
$716$	5089.09	0.265626
$717$	−49814.2	−2.59462
$718$	−25655.2	−1.33349
$719$	30679.9	1.59133	0.795667	−	0.605734i	$-0.207121\pi$
$719$	30679.9	1.59133	0.795667	+	0.605734i	$0.207121\pi$
$720$	26505.3	1.37194
$721$	−21606.3	−1.11603
$722$	−9656.03	−0.497729
$723$	−60999.0	−3.13773
$724$	−10844.2	−0.556662
$725$	6384.46	0.327052
$726$	10108.4	0.516744
$727$	7171.95	0.365877	0.182939	−	0.983124i	$-0.441439\pi$
$727$	7171.95	0.365877	0.182939	+	0.983124i	$0.441439\pi$
$728$	−1423.57	−0.0724739
$729$	−13599.5	−0.690924
$730$	6601.20	0.334687
$731$	−5900.06	−0.298525
$732$	5699.51	0.287787
$733$	−14119.7	−0.711491	−0.355745	−	0.934583i	$-0.615773\pi$
$733$	−14119.7	−0.711491	−0.355745	+	0.934583i	$0.615773\pi$
$734$	−13948.7	−0.701440
$735$	14716.7	0.738547
$736$	10479.3	0.524829
$737$	9249.25	0.462280
$738$	−62126.2	−3.09877
$739$	−17508.7	−0.871542	−0.435771	−	0.900058i	$-0.643524\pi$
$739$	−17508.7	−0.871542	−0.435771	+	0.900058i	$0.643524\pi$
$740$	−7172.05	−0.356283
$741$	1268.51	0.0628876
$742$	−27262.8	−1.34885
$743$	6152.28	0.303775	0.151888	−	0.988398i	$-0.451465\pi$
$743$	6152.28	0.303775	0.151888	+	0.988398i	$0.451465\pi$
$744$	29281.4	1.44289
$745$	−20965.2	−1.03102
$746$	12061.8	0.591976
$747$	−52057.7	−2.54979
$748$	−10495.8	−0.513053
$749$	−16808.5	−0.819987
$750$	−22527.4	−1.09678
$751$	−38610.0	−1.87603	−0.938015	−	0.346594i	$-0.887338\pi$
$751$	−38610.0	−1.87603	−0.938015	+	0.346594i	$0.887338\pi$
$752$	−6329.28	−0.306922
$753$	−2716.07	−0.131447
$754$	920.796	0.0444740
$755$	4903.55	0.236369
$756$	15235.2	0.732933
$757$	9462.94	0.454341	0.227171	−	0.973855i	$-0.427052\pi$
$757$	9462.94	0.454341	0.227171	+	0.973855i	$0.427052\pi$
$758$	−18884.8	−0.904915
$759$	24750.4	1.18364
$760$	−16581.3	−0.791403
$761$	−11917.8	−0.567703	−0.283851	−	0.958868i	$-0.591612\pi$
$761$	−11917.8	−0.567703	−0.283851	+	0.958868i	$0.591612\pi$
$762$	−1821.11	−0.0865771
$763$	36398.9	1.72704
$764$	767.125	0.0363267
$765$	100436.	4.74677
$766$	26549.3	1.25231
$767$	229.320	0.0107957
$768$	32370.4	1.52092
$769$	−4964.41	−0.232797	−0.116399	−	0.993203i	$-0.537135\pi$
$769$	−4964.41	−0.232797	−0.116399	+	0.993203i	$0.537135\pi$
$770$	18997.8	0.889135
$771$	52555.8	2.45493
$772$	6327.18	0.294974
$773$	−35514.9	−1.65250	−0.826250	−	0.563304i	$-0.809530\pi$
$773$	−35514.9	−1.65250	−0.826250	+	0.563304i	$0.809530\pi$
$774$	5633.84	0.261633
$775$	5595.53	0.259351
$776$	21783.8	1.00772
$777$	41751.2	1.92769
$778$	6585.78	0.303486
$779$	24625.2	1.13259
$780$	−828.673	−0.0380401
$781$	−1494.44	−0.0684704
$782$	−29507.7	−1.34935
$783$	−39974.4	−1.82448
$784$	−4499.91	−0.204989
$785$	−5240.19	−0.238255
$786$	−7742.72	−0.351366
$787$	20332.4	0.920932	0.460466	−	0.887677i	$-0.347682\pi$
$787$	20332.4	0.920932	0.460466	+	0.887677i	$0.347682\pi$
$788$	−2840.87	−0.128429
$789$	27834.5	1.25594
$790$	−35254.6	−1.58772
$791$	−6231.55	−0.280112
$792$	40654.9	1.82400
$793$	637.201	0.0285343
$794$	−4424.99	−0.197780
$795$	−64376.2	−2.87193
$796$	−490.125	−0.0218242
$797$	−1894.62	−0.0842045	−0.0421022	−	0.999113i	$-0.513406\pi$
$797$	−1894.62	−0.0842045	−0.0421022	+	0.999113i	$0.513406\pi$
$798$	23795.5	1.05558
$799$	−23983.4	−1.06192
$800$	4862.05	0.214874
$801$	51418.8	2.26816
$802$	18314.7	0.806377
$803$	6415.38	0.281935
$804$	−7568.12	−0.331974
$805$	−25971.6	−1.13712
$806$	807.014	0.0352678
$807$	−18644.4	−0.813277
$808$	31949.4	1.39106
$809$	−25986.2	−1.12933	−0.564665	−	0.825320i	$-0.690994\pi$
$809$	−25986.2	−1.12933	−0.564665	+	0.825320i	$0.690994\pi$
$810$	−28129.6	−1.22021
$811$	−39080.9	−1.69213	−0.846064	−	0.533081i	$-0.821034\pi$
$811$	−39080.9	−1.69213	−0.846064	+	0.533081i	$0.821034\pi$
$812$	−8399.25	−0.363000
$813$	53211.2	2.29545
$814$	14334.0	0.617207
$815$	5083.14	0.218472
$816$	−45393.2	−1.94740
$817$	−2233.10	−0.0956260
$818$	−31531.7	−1.34778
$819$	3263.65	0.139244
$820$	−16086.8	−0.685092
$821$	31884.2	1.35538	0.677690	−	0.735348i	$-0.262981\pi$
$821$	31884.2	1.35538	0.677690	+	0.735348i	$0.262981\pi$
$822$	−16478.3	−0.699205
$823$	15560.9	0.659077	0.329538	−	0.944142i	$-0.393107\pi$
$823$	15560.9	0.659077	0.329538	+	0.944142i	$0.393107\pi$
$824$	24621.2	1.04092
$825$	11483.3	0.484604
$826$	4301.75	0.181207
$827$	−33729.4	−1.41824	−0.709121	−	0.705087i	$-0.750908\pi$
$827$	−33729.4	−1.41824	−0.709121	+	0.705087i	$0.750908\pi$
$828$	−13701.2	−0.575059
$829$	−23504.4	−0.984731	−0.492366	−	0.870388i	$-0.663868\pi$
$829$	−23504.4	−0.984731	−0.492366	+	0.870388i	$0.663868\pi$
$830$	27720.8	1.15928
$831$	9796.07	0.408931
$832$	1475.69	0.0614908
$833$	−17051.4	−0.709241
$834$	26026.1	1.08059
$835$	−3304.73	−0.136964
$836$	−3972.53	−0.164345
$837$	−35034.8	−1.44681
$838$	28610.0	1.17938
$839$	30947.0	1.27343	0.636715	−	0.771099i	$-0.280293\pi$
$839$	30947.0	1.27343	0.636715	+	0.771099i	$0.280293\pi$
$840$	−63057.2	−2.59010
$841$	−2350.84	−0.0963893
$842$	−33806.0	−1.38365
$843$	71110.2	2.90530
$844$	1236.40	0.0504251
$845$	28384.3	1.15556
$846$	22901.3	0.930688
$847$	−10308.5	−0.418188
$848$	19684.3	0.797125
$849$	−35946.6	−1.45310
$850$	−13690.5	−0.552449
$851$	−19595.8	−0.789348
$852$	1222.82	0.0491701
$853$	5946.93	0.238709	0.119355	−	0.992852i	$-0.461917\pi$
$853$	5946.93	0.238709	0.119355	+	0.992852i	$0.461917\pi$
$854$	11953.0	0.478952
$855$	38014.0	1.52053
$856$	19154.0	0.764801
$857$	−41841.9	−1.66779	−0.833894	−	0.551925i	$-0.813893\pi$
$857$	−41841.9	−1.66779	−0.833894	+	0.551925i	$0.813893\pi$
$858$	1656.18	0.0658986
$859$	−23426.9	−0.930520	−0.465260	−	0.885174i	$-0.654039\pi$
$859$	−23426.9	−0.930520	−0.465260	+	0.885174i	$0.654039\pi$
$860$	1458.81	0.0578432
$861$	93647.5	3.70673
$862$	−16083.1	−0.635492
$863$	−15054.3	−0.593808	−0.296904	−	0.954907i	$-0.595954\pi$
$863$	−15054.3	−0.593808	−0.296904	+	0.954907i	$0.595954\pi$
$864$	−30442.3	−1.19869
$865$	6951.38	0.273242
$866$	−31053.1	−1.21851
$867$	−127121.	−4.97953
$868$	−7361.36	−0.287858
$869$	−34262.1	−1.33747
$870$	40786.8	1.58943
$871$	−846.111	−0.0329155
$872$	−41478.0	−1.61080
$873$	−49941.2	−1.93615
$874$	−11168.3	−0.432236
$875$	22973.5	0.887594
$876$	−5249.33	−0.202464
$877$	40790.1	1.57056	0.785281	−	0.619139i	$-0.212518\pi$
$877$	40790.1	1.57056	0.785281	+	0.619139i	$0.212518\pi$
$878$	−4471.64	−0.171880
$879$	8657.13	0.332193
$880$	−13716.8	−0.525447
$881$	−14608.4	−0.558647	−0.279323	−	0.960197i	$-0.590110\pi$
$881$	−14608.4	−0.558647	−0.279323	+	0.960197i	$0.590110\pi$
$882$	16282.1	0.621593
$883$	34181.5	1.30272	0.651359	−	0.758770i	$-0.274199\pi$
$883$	34181.5	1.30272	0.651359	+	0.758770i	$0.274199\pi$
$884$	960.141	0.0365306
$885$	10157.8	0.385820
$886$	−7650.57	−0.290097
$887$	39324.3	1.48859	0.744296	−	0.667850i	$-0.232785\pi$
$887$	39324.3	1.48859	0.744296	+	0.667850i	$0.232785\pi$
$888$	−47577.2	−1.79796
$889$	1857.17	0.0700646
$890$	−27380.5	−1.03123
$891$	−27337.7	−1.02789
$892$	7588.43	0.284842
$893$	−9077.45	−0.340163
$894$	−34285.1	−1.28262
$895$	25202.0	0.941242
$896$	8131.47	0.303185
$897$	−2264.14	−0.0842780
$898$	−9560.96	−0.355293
$899$	19314.9	0.716561
$900$	−6356.87	−0.235440
$901$	74589.4	2.75797
$902$	32151.0	1.18682
$903$	−8492.31	−0.312964
$904$	7101.10	0.261260
$905$	−53702.5	−1.97252
$906$	8018.92	0.294052
$907$	38633.2	1.41433	0.707163	−	0.707051i	$-0.249975\pi$
$907$	38633.2	1.41433	0.707163	+	0.707051i	$0.249975\pi$
$908$	−8388.56	−0.306590
$909$	−73246.6	−2.67265
$910$	−1737.90	−0.0633085
$911$	37387.7	1.35973	0.679863	−	0.733339i	$-0.262039\pi$
$911$	37387.7	1.35973	0.679863	+	0.733339i	$0.262039\pi$
$912$	−17180.8	−0.623809
$913$	26940.4	0.976559
$914$	27765.9	1.00483
$915$	28224.9	1.01977
$916$	6687.72	0.241232
$917$	7896.04	0.284352
$918$	85719.3	3.08187
$919$	26028.7	0.934285	0.467142	−	0.884182i	$-0.345284\pi$
$919$	26028.7	0.934285	0.467142	+	0.884182i	$0.345284\pi$
$920$	29595.7	1.06059
$921$	−60771.1	−2.17424
$922$	−21266.1	−0.759611
$923$	136.710	0.00487525
$924$	−15107.2	−0.537869
$925$	−9091.76	−0.323173
$926$	−12267.0	−0.435333
$927$	−56446.2	−1.99993
$928$	16783.1	0.593675
$929$	6657.92	0.235134	0.117567	−	0.993065i	$-0.462491\pi$
$929$	6657.92	0.235134	0.117567	+	0.993065i	$0.462491\pi$
$930$	35746.8	1.26041
$931$	−6453.77	−0.227190
$932$	12429.2	0.436838
$933$	−45631.4	−1.60118
$934$	16836.7	0.589843
$935$	−51976.9	−1.81799
$936$	−3719.06	−0.129873
$937$	42235.7	1.47255	0.736276	−	0.676681i	$-0.236583\pi$
$937$	42235.7	1.47255	0.736276	+	0.676681i	$0.236583\pi$
$938$	−15871.9	−0.552491
$939$	−11862.3	−0.412258
$940$	5930.00	0.205761
$941$	18897.2	0.654656	0.327328	−	0.944911i	$-0.393852\pi$
$941$	18897.2	0.654656	0.327328	+	0.944911i	$0.393852\pi$
$942$	−8569.45	−0.296399
$943$	−43953.1	−1.51783
$944$	−3105.95	−0.107087
$945$	75447.0	2.59713
$946$	−2915.57	−0.100204
$947$	18864.3	0.647314	0.323657	−	0.946175i	$-0.395088\pi$
$947$	18864.3	0.647314	0.323657	+	0.946175i	$0.395088\pi$
$948$	28034.7	0.960470
$949$	−586.871	−0.0200744
$950$	−5181.71	−0.176965
$951$	63014.4	2.14867
$952$	73061.2	2.48732
$953$	19158.3	0.651204	0.325602	−	0.945507i	$-0.394433\pi$
$953$	19158.3	0.651204	0.325602	+	0.945507i	$0.394433\pi$
$954$	−71223.8	−2.41715
$955$	3798.93	0.128723
$956$	−14270.8	−0.482794
$957$	39638.7	1.33891
$958$	−8152.15	−0.274931
$959$	16804.6	0.565848
$960$	65366.0	2.19758
$961$	−12862.8	−0.431769
$962$	−1311.26	−0.0439466
$963$	−43912.1	−1.46942
$964$	−17475.0	−0.583852
$965$	31333.2	1.04524
$966$	−42472.2	−1.41462
$967$	22964.8	0.763699	0.381850	−	0.924224i	$-0.375287\pi$
$967$	22964.8	0.763699	0.381850	+	0.924224i	$0.375287\pi$
$968$	11747.0	0.390043
$969$	−65103.0	−2.15832
$970$	26593.8	0.880283
$971$	39311.5	1.29924	0.649622	−	0.760257i	$-0.274927\pi$
$971$	39311.5	1.29924	0.649622	+	0.760257i	$0.274927\pi$
$972$	3339.41	0.110197
$973$	−26541.5	−0.874493
$974$	−925.646	−0.0304513
$975$	−1050.48	−0.0345049
$976$	−8630.34	−0.283043
$977$	−18444.9	−0.603998	−0.301999	−	0.953308i	$-0.597654\pi$
$977$	−18444.9	−0.603998	−0.301999	+	0.953308i	$0.597654\pi$
$978$	8312.62	0.271788
$979$	−26609.8	−0.868694
$980$	4216.04	0.137425
$981$	95091.7	3.09485
$982$	27403.5	0.890510
$983$	−27707.8	−0.899024	−0.449512	−	0.893274i	$-0.648402\pi$
$983$	−27707.8	−0.899024	−0.449512	+	0.893274i	$0.648402\pi$
$984$	−106715.	−3.45727
$985$	−14068.5	−0.455085
$986$	−47257.7	−1.52636
$987$	−34520.8	−1.11328
$988$	363.402	0.0117018
$989$	3985.84	0.128152
$990$	49631.6	1.59333
$991$	−10277.5	−0.329439	−0.164720	−	0.986340i	$-0.552672\pi$
$991$	−10277.5	−0.329439	−0.164720	+	0.986340i	$0.552672\pi$
$992$	14709.2	0.470783
$993$	24216.7	0.773910
$994$	2564.49	0.0818318
$995$	−2427.18	−0.0773335
$996$	−22043.8	−0.701289
$997$	−32313.7	−1.02646	−0.513232	−	0.858250i	$-0.671552\pi$
$997$	−32313.7	−1.02646	−0.513232	+	0.858250i	$0.671552\pi$
$998$	2771.99	0.0879216
$999$	56925.4	1.80284

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.4.a.a.1.4	✓	4
3.2	odd	2		387.4.a.e.1.1		4
4.3	odd	2		688.4.a.f.1.4		4
5.4	even	2		1075.4.a.a.1.1		4
7.6	odd	2		2107.4.a.b.1.4		4
43.42	odd	2		1849.4.a.b.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.a.1.4	✓	4	1.1	even	1	trivial
387.4.a.e.1.1		4	3.2	odd	2
688.4.a.f.1.4		4	4.3	odd	2
1075.4.a.a.1.1		4	5.4	even	2
1849.4.a.b.1.1		4	43.42	odd	2
2107.4.a.b.1.4		4	7.6	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:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	43.a (trivial)

\(f(q)\)	\(=\)	\(q+2.32005 q^{2} -9.13635 q^{3} -2.61739 q^{4} -12.9617 q^{5} -21.1967 q^{6} +21.6165 q^{7} -24.6328 q^{8} +56.4728 q^{9} +O(q^{10})\)	\(q+2.32005 q^{2} -9.13635 q^{3} -2.61739 q^{4} -12.9617 q^{5} -21.1967 q^{6} +21.6165 q^{7} -24.6328 q^{8} +56.4728 q^{9} -30.0718 q^{10} -29.2253 q^{11} +23.9134 q^{12} +2.67349 q^{13} +50.1512 q^{14} +118.423 q^{15} -36.2102 q^{16} -137.211 q^{17} +131.020 q^{18} -51.9326 q^{19} +33.9259 q^{20} -197.496 q^{21} -67.8040 q^{22} +92.6938 q^{23} +225.054 q^{24} +43.0067 q^{25} +6.20263 q^{26} -269.274 q^{27} -56.5787 q^{28} +148.453 q^{29} +274.747 q^{30} +130.108 q^{31} +113.053 q^{32} +267.012 q^{33} -318.335 q^{34} -280.187 q^{35} -147.811 q^{36} -211.403 q^{37} -120.486 q^{38} -24.4260 q^{39} +319.284 q^{40} -474.175 q^{41} -458.199 q^{42} +43.0000 q^{43} +76.4939 q^{44} -731.986 q^{45} +215.054 q^{46} +174.793 q^{47} +330.829 q^{48} +124.272 q^{49} +99.7776 q^{50} +1253.60 q^{51} -6.99757 q^{52} -543.612 q^{53} -624.727 q^{54} +378.811 q^{55} -532.475 q^{56} +474.475 q^{57} +344.417 q^{58} +85.7755 q^{59} -309.959 q^{60} +238.340 q^{61} +301.857 q^{62} +1220.74 q^{63} +551.970 q^{64} -34.6531 q^{65} +619.481 q^{66} -316.481 q^{67} +359.133 q^{68} -846.883 q^{69} -650.047 q^{70} +51.1352 q^{71} -1391.08 q^{72} -219.515 q^{73} -490.465 q^{74} -392.924 q^{75} +135.928 q^{76} -631.748 q^{77} -56.6694 q^{78} +1172.35 q^{79} +469.347 q^{80} +935.412 q^{81} -1100.11 q^{82} -921.819 q^{83} +516.922 q^{84} +1778.49 q^{85} +99.7620 q^{86} -1356.31 q^{87} +719.901 q^{88} +910.505 q^{89} -1698.24 q^{90} +57.7915 q^{91} -242.616 q^{92} -1188.71 q^{93} +405.527 q^{94} +673.137 q^{95} -1032.89 q^{96} -884.342 q^{97} +288.317 q^{98} -1650.43 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 11 q^{3} + 2 q^{4} - 27 q^{5} - 27 q^{6} - 20 q^{7} - 66 q^{8} - 9 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 11 * q^3 + 2 * q^4 - 27 * q^5 - 27 * q^6 - 20 * q^7 - 66 * q^8 - 9 * q^9	\( 4 q - 4 q^{2} - 11 q^{3} + 2 q^{4} - 27 q^{5} - 27 q^{6} - 20 q^{7} - 66 q^{8} - 9 q^{9} - 3 q^{10} - 62 q^{11} + 61 q^{12} - 2 q^{13} + 112 q^{14} + 92 q^{15} + 202 q^{16} - 207 q^{17} + 299 q^{18} + 99 q^{19} + 81 q^{20} - 90 q^{21} + 202 q^{22} - 103 q^{23} + 209 q^{24} - 101 q^{25} - 50 q^{26} - 218 q^{27} - 80 q^{28} - 99 q^{29} + 300 q^{30} + 131 q^{31} - 342 q^{32} - 32 q^{33} + 53 q^{34} - 374 q^{35} - 379 q^{36} - 449 q^{37} - 609 q^{38} + 98 q^{39} + 133 q^{40} - 491 q^{41} - 394 q^{42} + 172 q^{43} - 764 q^{44} - 338 q^{45} + 1061 q^{46} + 19 q^{47} + 237 q^{48} + 236 q^{49} + 599 q^{50} + 1649 q^{51} + 224 q^{52} - 1220 q^{53} - 322 q^{54} + 1360 q^{55} + 344 q^{56} + 232 q^{57} - 771 q^{58} + 816 q^{59} - 156 q^{60} + 372 q^{61} - 97 q^{62} + 1914 q^{63} + 434 q^{64} - 350 q^{65} + 812 q^{66} + 110 q^{67} - 1697 q^{68} - 1238 q^{69} - 718 q^{70} + 468 q^{71} - 315 q^{72} + 628 q^{73} + 395 q^{74} + 62 q^{75} + 1671 q^{76} - 2044 q^{77} - 90 q^{78} + 1095 q^{79} - 31 q^{80} + 2056 q^{81} - 2287 q^{82} - 980 q^{83} - 610 q^{84} - 152 q^{85} - 172 q^{86} - 507 q^{87} + 1816 q^{88} - 738 q^{89} - 2398 q^{90} + 852 q^{91} - 2517 q^{92} - 35 q^{93} - 2233 q^{94} + 1149 q^{95} - 1551 q^{96} - 1765 q^{97} + 1652 q^{98} - 58 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - 11 * q^3 + 2 * q^4 - 27 * q^5 - 27 * q^6 - 20 * q^7 - 66 * q^8 - 9 * q^9 - 3 * q^10 - 62 * q^11 + 61 * q^12 - 2 * q^13 + 112 * q^14 + 92 * q^15 + 202 * q^16 - 207 * q^17 + 299 * q^18 + 99 * q^19 + 81 * q^20 - 90 * q^21 + 202 * q^22 - 103 * q^23 + 209 * q^24 - 101 * q^25 - 50 * q^26 - 218 * q^27 - 80 * q^28 - 99 * q^29 + 300 * q^30 + 131 * q^31 - 342 * q^32 - 32 * q^33 + 53 * q^34 - 374 * q^35 - 379 * q^36 - 449 * q^37 - 609 * q^38 + 98 * q^39 + 133 * q^40 - 491 * q^41 - 394 * q^42 + 172 * q^43 - 764 * q^44 - 338 * q^45 + 1061 * q^46 + 19 * q^47 + 237 * q^48 + 236 * q^49 + 599 * q^50 + 1649 * q^51 + 224 * q^52 - 1220 * q^53 - 322 * q^54 + 1360 * q^55 + 344 * q^56 + 232 * q^57 - 771 * q^58 + 816 * q^59 - 156 * q^60 + 372 * q^61 - 97 * q^62 + 1914 * q^63 + 434 * q^64 - 350 * q^65 + 812 * q^66 + 110 * q^67 - 1697 * q^68 - 1238 * q^69 - 718 * q^70 + 468 * q^71 - 315 * q^72 + 628 * q^73 + 395 * q^74 + 62 * q^75 + 1671 * q^76 - 2044 * q^77 - 90 * q^78 + 1095 * q^79 - 31 * q^80 + 2056 * q^81 - 2287 * q^82 - 980 * q^83 - 610 * q^84 - 152 * q^85 - 172 * q^86 - 507 * q^87 + 1816 * q^88 - 738 * q^89 - 2398 * q^90 + 852 * q^91 - 2517 * q^92 - 35 * q^93 - 2233 * q^94 + 1149 * q^95 - 1551 * q^96 - 1765 * q^97 + 1652 * q^98 - 58 * q^99

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