LMFDB - Embedded newform 402.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [402,4,Mod(1,402)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(402, 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("402.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 402 = 2 \cdot 3 \cdot 67 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	402.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:	$23.7187678223$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	402.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -10.4142 q^{5} +6.00000 q^{6} -18.6569 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -10.4142 q^{5} +6.00000 q^{6} -18.6569 q^{7} +8.00000 q^{8} +9.00000 q^{9} -20.8284 q^{10} -26.6863 q^{11} +12.0000 q^{12} +49.8823 q^{13} -37.3137 q^{14} -31.2426 q^{15} +16.0000 q^{16} -86.2254 q^{17} +18.0000 q^{18} -36.4853 q^{19} -41.6569 q^{20} -55.9706 q^{21} -53.3726 q^{22} -11.5025 q^{23} +24.0000 q^{24} -16.5442 q^{25} +99.7645 q^{26} +27.0000 q^{27} -74.6274 q^{28} -240.250 q^{29} -62.4853 q^{30} -303.357 q^{31} +32.0000 q^{32} -80.0589 q^{33} -172.451 q^{34} +194.296 q^{35} +36.0000 q^{36} -33.2843 q^{37} -72.9706 q^{38} +149.647 q^{39} -83.3137 q^{40} -336.238 q^{41} -111.941 q^{42} +222.137 q^{43} -106.745 q^{44} -93.7279 q^{45} -23.0051 q^{46} +294.098 q^{47} +48.0000 q^{48} +5.07821 q^{49} -33.0883 q^{50} -258.676 q^{51} +199.529 q^{52} +249.291 q^{53} +54.0000 q^{54} +277.917 q^{55} -149.255 q^{56} -109.456 q^{57} -480.500 q^{58} +56.0955 q^{59} -124.971 q^{60} +591.196 q^{61} -606.715 q^{62} -167.912 q^{63} +64.0000 q^{64} -519.484 q^{65} -160.118 q^{66} +67.0000 q^{67} -344.902 q^{68} -34.5076 q^{69} +388.593 q^{70} -343.637 q^{71} +72.0000 q^{72} +776.872 q^{73} -66.5685 q^{74} -49.6325 q^{75} -145.941 q^{76} +497.882 q^{77} +299.294 q^{78} +146.981 q^{79} -166.627 q^{80} +81.0000 q^{81} -672.475 q^{82} +1293.72 q^{83} -223.882 q^{84} +897.970 q^{85} +444.274 q^{86} -720.749 q^{87} -213.490 q^{88} -153.314 q^{89} -187.456 q^{90} -930.646 q^{91} -46.0101 q^{92} -910.072 q^{93} +588.195 q^{94} +379.966 q^{95} +96.0000 q^{96} -747.955 q^{97} +10.1564 q^{98} -240.177 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 18 q^{5} + 12 q^{6} - 26 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 18 * q^5 + 12 * q^6 - 26 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 18 q^{5} + 12 q^{6} - 26 q^{7} + 16 q^{8} + 18 q^{9} - 36 q^{10} - 76 q^{11} + 24 q^{12} - 36 q^{13} - 52 q^{14} - 54 q^{15} + 32 q^{16} - 48 q^{17} + 36 q^{18} - 56 q^{19} - 72 q^{20} - 78 q^{21} - 152 q^{22} - 122 q^{23} + 48 q^{24} - 84 q^{25} - 72 q^{26} + 54 q^{27} - 104 q^{28} - 192 q^{29} - 108 q^{30} - 58 q^{31} + 64 q^{32} - 228 q^{33} - 96 q^{34} + 250 q^{35} + 72 q^{36} - 10 q^{37} - 112 q^{38} - 108 q^{39} - 144 q^{40} - 466 q^{41} - 156 q^{42} + 218 q^{43} - 304 q^{44} - 162 q^{45} - 244 q^{46} - 68 q^{47} + 96 q^{48} - 284 q^{49} - 168 q^{50} - 144 q^{51} - 144 q^{52} + 162 q^{53} + 108 q^{54} + 652 q^{55} - 208 q^{56} - 168 q^{57} - 384 q^{58} - 66 q^{59} - 216 q^{60} + 1024 q^{61} - 116 q^{62} - 234 q^{63} + 128 q^{64} + 132 q^{65} - 456 q^{66} + 134 q^{67} - 192 q^{68} - 366 q^{69} + 500 q^{70} + 116 q^{71} + 144 q^{72} + 1022 q^{73} - 20 q^{74} - 252 q^{75} - 224 q^{76} + 860 q^{77} - 216 q^{78} + 656 q^{79} - 288 q^{80} + 162 q^{81} - 932 q^{82} - 142 q^{83} - 312 q^{84} + 608 q^{85} + 436 q^{86} - 576 q^{87} - 608 q^{88} - 284 q^{89} - 324 q^{90} - 300 q^{91} - 488 q^{92} - 174 q^{93} - 136 q^{94} + 528 q^{95} + 192 q^{96} - 868 q^{97} - 568 q^{98} - 684 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 18 * q^5 + 12 * q^6 - 26 * q^7 + 16 * q^8 + 18 * q^9 - 36 * q^10 - 76 * q^11 + 24 * q^12 - 36 * q^13 - 52 * q^14 - 54 * q^15 + 32 * q^16 - 48 * q^17 + 36 * q^18 - 56 * q^19 - 72 * q^20 - 78 * q^21 - 152 * q^22 - 122 * q^23 + 48 * q^24 - 84 * q^25 - 72 * q^26 + 54 * q^27 - 104 * q^28 - 192 * q^29 - 108 * q^30 - 58 * q^31 + 64 * q^32 - 228 * q^33 - 96 * q^34 + 250 * q^35 + 72 * q^36 - 10 * q^37 - 112 * q^38 - 108 * q^39 - 144 * q^40 - 466 * q^41 - 156 * q^42 + 218 * q^43 - 304 * q^44 - 162 * q^45 - 244 * q^46 - 68 * q^47 + 96 * q^48 - 284 * q^49 - 168 * q^50 - 144 * q^51 - 144 * q^52 + 162 * q^53 + 108 * q^54 + 652 * q^55 - 208 * q^56 - 168 * q^57 - 384 * q^58 - 66 * q^59 - 216 * q^60 + 1024 * q^61 - 116 * q^62 - 234 * q^63 + 128 * q^64 + 132 * q^65 - 456 * q^66 + 134 * q^67 - 192 * q^68 - 366 * q^69 + 500 * q^70 + 116 * q^71 + 144 * q^72 + 1022 * q^73 - 20 * q^74 - 252 * q^75 - 224 * q^76 + 860 * q^77 - 216 * q^78 + 656 * q^79 - 288 * q^80 + 162 * q^81 - 932 * q^82 - 142 * q^83 - 312 * q^84 + 608 * q^85 + 436 * q^86 - 576 * q^87 - 608 * q^88 - 284 * q^89 - 324 * q^90 - 300 * q^91 - 488 * q^92 - 174 * q^93 - 136 * q^94 + 528 * q^95 + 192 * q^96 - 868 * q^97 - 568 * q^98 - 684 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−10.4142	−0.931476	−0.465738	−	0.884923i	$-0.654211\pi$
$5$	−10.4142	−0.931476	−0.465738	+	0.884923i	$0.654211\pi$
$6$	6.00000	0.408248
$7$	−18.6569	−1.00738	−0.503688	−	0.863886i	$-0.668024\pi$
$7$	−18.6569	−1.00738	−0.503688	+	0.863886i	$0.668024\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−20.8284	−0.658653
$11$	−26.6863	−0.731475	−0.365737	−	0.930718i	$-0.619183\pi$
$11$	−26.6863	−0.731475	−0.365737	+	0.930718i	$0.619183\pi$
$12$	12.0000	0.288675
$13$	49.8823	1.06422	0.532110	−	0.846676i	$-0.321399\pi$
$13$	49.8823	1.06422	0.532110	+	0.846676i	$0.321399\pi$
$14$	−37.3137	−0.712322
$15$	−31.2426	−0.537788
$16$	16.0000	0.250000
$17$	−86.2254	−1.23016	−0.615080	−	0.788464i	$-0.710876\pi$
$17$	−86.2254	−1.23016	−0.615080	+	0.788464i	$0.710876\pi$
$18$	18.0000	0.235702
$19$	−36.4853	−0.440542	−0.220271	−	0.975439i	$-0.570694\pi$
$19$	−36.4853	−0.440542	−0.220271	+	0.975439i	$0.570694\pi$
$20$	−41.6569	−0.465738
$21$	−55.9706	−0.581608
$22$	−53.3726	−0.517231
$23$	−11.5025	−0.104280	−0.0521401	−	0.998640i	$-0.516604\pi$
$23$	−11.5025	−0.104280	−0.0521401	+	0.998640i	$0.516604\pi$
$24$	24.0000	0.204124
$25$	−16.5442	−0.132353
$26$	99.7645	0.752516
$27$	27.0000	0.192450
$28$	−74.6274	−0.503688
$29$	−240.250	−1.53839	−0.769194	−	0.639015i	$-0.779342\pi$
$29$	−240.250	−1.53839	−0.769194	+	0.639015i	$0.779342\pi$
$30$	−62.4853	−0.380273
$31$	−303.357	−1.75757	−0.878784	−	0.477220i	$-0.841645\pi$
$31$	−303.357	−1.75757	−0.878784	+	0.477220i	$0.841645\pi$
$32$	32.0000	0.176777
$33$	−80.0589	−0.422317
$34$	−172.451	−0.869855
$35$	194.296	0.938346
$36$	36.0000	0.166667
$37$	−33.2843	−0.147889	−0.0739446	−	0.997262i	$-0.523559\pi$
$37$	−33.2843	−0.147889	−0.0739446	+	0.997262i	$0.523559\pi$
$38$	−72.9706	−0.311510
$39$	149.647	0.614427
$40$	−83.3137	−0.329326
$41$	−336.238	−1.28077	−0.640384	−	0.768055i	$-0.721225\pi$
$41$	−336.238	−1.28077	−0.640384	+	0.768055i	$0.721225\pi$
$42$	−111.941	−0.411259
$43$	222.137	0.787804	0.393902	−	0.919152i	$-0.371125\pi$
$43$	222.137	0.787804	0.393902	+	0.919152i	$0.371125\pi$
$44$	−106.745	−0.365737
$45$	−93.7279	−0.310492
$46$	−23.0051	−0.0737372
$47$	294.098	0.912735	0.456367	−	0.889791i	$-0.349150\pi$
$47$	294.098	0.912735	0.456367	+	0.889791i	$0.349150\pi$
$48$	48.0000	0.144338
$49$	5.07821	0.0148053
$50$	−33.0883	−0.0935879
$51$	−258.676	−0.710234
$52$	199.529	0.532110
$53$	249.291	0.646091	0.323045	−	0.946383i	$-0.395293\pi$
$53$	249.291	0.646091	0.323045	+	0.946383i	$0.395293\pi$
$54$	54.0000	0.136083
$55$	277.917	0.681351
$56$	−149.255	−0.356161
$57$	−109.456	−0.254347
$58$	−480.500	−1.08780
$59$	56.0955	0.123780	0.0618899	−	0.998083i	$-0.480287\pi$
$59$	56.0955	0.123780	0.0618899	+	0.998083i	$0.480287\pi$
$60$	−124.971	−0.268894
$61$	591.196	1.24090	0.620450	−	0.784246i	$-0.286950\pi$
$61$	591.196	1.24090	0.620450	+	0.784246i	$0.286950\pi$
$62$	−606.715	−1.24279
$63$	−167.912	−0.335792
$64$	64.0000	0.125000
$65$	−519.484	−0.991294
$66$	−160.118	−0.298623
$67$	67.0000	0.122169
$67$	67.0000	0.122169
$68$	−344.902	−0.615080
$69$	−34.5076	−0.0602061
$70$	388.593	0.663511
$71$	−343.637	−0.574397	−0.287198	−	0.957871i	$-0.592724\pi$
$71$	−343.637	−0.574397	−0.287198	+	0.957871i	$0.592724\pi$
$72$	72.0000	0.117851
$73$	776.872	1.24556	0.622781	−	0.782396i	$-0.286003\pi$
$73$	776.872	1.24556	0.622781	+	0.782396i	$0.286003\pi$
$74$	−66.5685	−0.104573
$75$	−49.6325	−0.0764142
$76$	−145.941	−0.220271
$77$	497.882	0.736869
$78$	299.294	0.434466
$79$	146.981	0.209324	0.104662	−	0.994508i	$-0.466624\pi$
$79$	146.981	0.209324	0.104662	+	0.994508i	$0.466624\pi$
$80$	−166.627	−0.232869
$81$	81.0000	0.111111
$82$	−672.475	−0.905640
$83$	1293.72	1.71089	0.855445	−	0.517894i	$-0.173284\pi$
$83$	1293.72	1.71089	0.855445	+	0.517894i	$0.173284\pi$
$84$	−223.882	−0.290804
$85$	897.970	1.14586
$86$	444.274	0.557062
$87$	−720.749	−0.888189
$88$	−213.490	−0.258615
$89$	−153.314	−0.182598	−0.0912990	−	0.995824i	$-0.529102\pi$
$89$	−153.314	−0.182598	−0.0912990	+	0.995824i	$0.529102\pi$
$90$	−187.456	−0.219551
$91$	−930.646	−1.07207
$92$	−46.0101	−0.0521401
$93$	−910.072	−1.01473
$94$	588.195	0.645401
$95$	379.966	0.410354
$96$	96.0000	0.102062
$97$	−747.955	−0.782921	−0.391461	−	0.920195i	$-0.628030\pi$
$97$	−747.955	−0.782921	−0.391461	+	0.920195i	$0.628030\pi$
$98$	10.1564	0.0104689
$99$	−240.177	−0.243825
$100$	−66.1766	−0.0661766
$101$	−1508.83	−1.48647	−0.743237	−	0.669028i	$-0.766711\pi$
$101$	−1508.83	−1.48647	−0.743237	+	0.669028i	$0.766711\pi$
$102$	−517.352	−0.502211
$103$	270.014	0.258304	0.129152	−	0.991625i	$-0.458775\pi$
$103$	270.014	0.258304	0.129152	+	0.991625i	$0.458775\pi$
$104$	399.058	0.376258
$105$	582.889	0.541754
$106$	498.583	0.456855
$107$	−1636.52	−1.47858	−0.739291	−	0.673387i	$-0.764839\pi$
$107$	−1636.52	−1.47858	−0.739291	+	0.673387i	$0.764839\pi$
$108$	108.000	0.0962250
$109$	250.926	0.220499	0.110249	−	0.993904i	$-0.464835\pi$
$109$	250.926	0.220499	0.110249	+	0.993904i	$0.464835\pi$
$110$	555.833	0.481788
$111$	−99.8528	−0.0853839
$112$	−298.510	−0.251844
$113$	1459.86	1.21532	0.607662	−	0.794195i	$-0.292107\pi$
$113$	1459.86	1.21532	0.607662	+	0.794195i	$0.292107\pi$
$114$	−218.912	−0.179851
$115$	119.790	0.0971344
$116$	−960.999	−0.769194
$117$	448.940	0.354740
$118$	112.191	0.0875255
$119$	1608.69	1.23923
$120$	−249.941	−0.190137
$121$	−618.842	−0.464945
$122$	1182.39	0.877449
$123$	−1008.71	−0.739452
$124$	−1213.43	−0.878784
$125$	1474.07	1.05476
$126$	−335.823	−0.237441
$127$	−386.235	−0.269865	−0.134933	−	0.990855i	$-0.543082\pi$
$127$	−386.235	−0.269865	−0.134933	+	0.990855i	$0.543082\pi$
$128$	128.000	0.0883883
$129$	666.411	0.454839
$130$	−1038.97	−0.700951
$131$	9.41753	0.00628102	0.00314051	−	0.999995i	$-0.499000\pi$
$131$	9.41753	0.00628102	0.00314051	+	0.999995i	$0.499000\pi$
$132$	−320.235	−0.211159
$133$	680.701	0.443791
$134$	134.000	0.0863868
$135$	−281.184	−0.179263
$136$	−689.803	−0.434927
$137$	−1298.47	−0.809751	−0.404876	−	0.914372i	$-0.632685\pi$
$137$	−1298.47	−0.809751	−0.404876	+	0.914372i	$0.632685\pi$
$138$	−69.0152	−0.0425722
$139$	−382.417	−0.233354	−0.116677	−	0.993170i	$-0.537224\pi$
$139$	−382.417	−0.233354	−0.116677	+	0.993170i	$0.537224\pi$
$140$	777.186	0.469173
$141$	882.293	0.526968
$142$	−687.273	−0.406160
$143$	−1331.17	−0.778449
$144$	144.000	0.0833333
$145$	2502.01	1.43297
$146$	1553.74	0.880745
$147$	15.2346	0.00854783
$148$	−133.137	−0.0739446
$149$	1355.84	0.745466	0.372733	−	0.927939i	$-0.378421\pi$
$149$	1355.84	0.745466	0.372733	+	0.927939i	$0.378421\pi$
$150$	−99.2649	−0.0540330
$151$	−169.645	−0.0914273	−0.0457136	−	0.998955i	$-0.514556\pi$
$151$	−169.645	−0.0914273	−0.0457136	+	0.998955i	$0.514556\pi$
$152$	−291.882	−0.155755
$153$	−776.029	−0.410054
$154$	995.765	0.521045
$155$	3159.23	1.63713
$156$	598.587	0.307214
$157$	2023.10	1.02841	0.514207	−	0.857666i	$-0.328086\pi$
$157$	2023.10	1.02841	0.514207	+	0.857666i	$0.328086\pi$
$158$	293.961	0.148015
$159$	747.874	0.373021
$160$	−333.255	−0.164663
$161$	214.601	0.105049
$162$	162.000	0.0785674
$163$	2398.99	1.15278	0.576392	−	0.817173i	$-0.304460\pi$
$163$	2398.99	1.15278	0.576392	+	0.817173i	$0.304460\pi$
$164$	−1344.95	−0.640384
$165$	833.750	0.393378
$166$	2587.43	1.20978
$167$	−4029.45	−1.86712	−0.933558	−	0.358426i	$-0.883314\pi$
$167$	−4029.45	−1.86712	−0.933558	+	0.358426i	$0.883314\pi$
$168$	−447.765	−0.205630
$169$	291.239	0.132562
$170$	1795.94	0.810249
$171$	−328.368	−0.146847
$172$	888.548	0.393902
$173$	−911.092	−0.400399	−0.200200	−	0.979755i	$-0.564159\pi$
$173$	−911.092	−0.400399	−0.200200	+	0.979755i	$0.564159\pi$
$174$	−1441.50	−0.628044
$175$	308.662	0.133329
$176$	−426.981	−0.182869
$177$	168.286	0.0714643
$178$	−306.627	−0.129116
$179$	−3190.11	−1.33207	−0.666033	−	0.745922i	$-0.732009\pi$
$179$	−3190.11	−1.33207	−0.666033	+	0.745922i	$0.732009\pi$
$180$	−374.912	−0.155246
$181$	348.679	0.143188	0.0715942	−	0.997434i	$-0.477191\pi$
$181$	348.679	0.143188	0.0715942	+	0.997434i	$0.477191\pi$
$182$	−1861.29	−0.758067
$183$	1773.59	0.716434
$184$	−92.0202	−0.0368686
$185$	346.630	0.137755
$186$	−1820.14	−0.717524
$187$	2301.04	0.899831
$188$	1176.39	0.456367
$189$	−503.735	−0.193869
$190$	759.931	0.290164
$191$	−1126.00	−0.426570	−0.213285	−	0.976990i	$-0.568416\pi$
$191$	−1126.00	−0.426570	−0.213285	+	0.976990i	$0.568416\pi$
$192$	192.000	0.0721688
$193$	406.462	0.151595	0.0757973	−	0.997123i	$-0.475850\pi$
$193$	406.462	0.151595	0.0757973	+	0.997123i	$0.475850\pi$
$194$	−1495.91	−0.553609
$195$	−1558.45	−0.572324
$196$	20.3128	0.00740264
$197$	−2052.94	−0.742466	−0.371233	−	0.928540i	$-0.621065\pi$
$197$	−2052.94	−0.742466	−0.371233	+	0.928540i	$0.621065\pi$
$198$	−480.353	−0.172410
$199$	2143.99	0.763734	0.381867	−	0.924217i	$-0.375281\pi$
$199$	2143.99	0.763734	0.381867	+	0.924217i	$0.375281\pi$
$200$	−132.353	−0.0467939
$201$	201.000	0.0705346
$202$	−3017.65	−1.05110
$203$	4482.31	1.54973
$204$	−1034.70	−0.355117
$205$	3501.65	1.19300
$206$	540.029	0.182648
$207$	−103.523	−0.0347600
$208$	798.116	0.266055
$209$	973.657	0.322245
$210$	1165.78	0.383078
$211$	2396.63	0.781947	0.390973	−	0.920402i	$-0.372138\pi$
$211$	2396.63	0.781947	0.390973	+	0.920402i	$0.372138\pi$
$212$	997.166	0.323045
$213$	−1030.91	−0.331628
$214$	−3273.04	−1.04551
$215$	−2313.38	−0.733821
$216$	216.000	0.0680414
$217$	5659.70	1.77053
$218$	501.852	0.155916
$219$	2330.62	0.719125
$220$	1111.67	0.340675
$221$	−4301.12	−1.30916
$222$	−199.706	−0.0603755
$223$	4344.91	1.30474	0.652370	−	0.757901i	$-0.273775\pi$
$223$	4344.91	1.30474	0.652370	+	0.757901i	$0.273775\pi$
$224$	−597.019	−0.178081
$225$	−148.897	−0.0441177
$226$	2919.71	0.859364
$227$	2923.83	0.854895	0.427448	−	0.904040i	$-0.359413\pi$
$227$	2923.83	0.854895	0.427448	+	0.904040i	$0.359413\pi$
$228$	−437.823	−0.127174
$229$	−2965.79	−0.855830	−0.427915	−	0.903819i	$-0.640752\pi$
$229$	−2965.79	−0.855830	−0.427915	+	0.903819i	$0.640752\pi$
$230$	239.580	0.0686844
$231$	1493.65	0.425432
$232$	−1922.00	−0.543902
$233$	−6503.57	−1.82860	−0.914298	−	0.405041i	$-0.867257\pi$
$233$	−6503.57	−1.82860	−0.914298	+	0.405041i	$0.867257\pi$
$234$	897.881	0.250839
$235$	−3062.79	−0.850190
$236$	224.382	0.0618899
$237$	440.942	0.120853
$238$	3217.39	0.876270
$239$	4818.42	1.30409	0.652045	−	0.758181i	$-0.273911\pi$
$239$	4818.42	1.30409	0.652045	+	0.758181i	$0.273911\pi$
$240$	−499.882	−0.134447
$241$	1811.80	0.484267	0.242133	−	0.970243i	$-0.422153\pi$
$241$	1811.80	0.484267	0.242133	+	0.970243i	$0.422153\pi$
$242$	−1237.68	−0.328766
$243$	243.000	0.0641500
$244$	2364.78	0.620450
$245$	−52.8856	−0.0137908
$246$	−2017.43	−0.522871
$247$	−1819.97	−0.468833
$248$	−2426.86	−0.621394
$249$	3881.15	0.987782
$250$	2948.14	0.745828
$251$	−1780.12	−0.447651	−0.223826	−	0.974629i	$-0.571855\pi$
$251$	−1780.12	−0.447651	−0.223826	+	0.974629i	$0.571855\pi$
$252$	−671.647	−0.167896
$253$	306.960	0.0762782
$254$	−772.471	−0.190823
$255$	2693.91	0.661565
$256$	256.000	0.0625000
$257$	−6429.78	−1.56062	−0.780308	−	0.625395i	$-0.784938\pi$
$257$	−6429.78	−1.56062	−0.780308	+	0.625395i	$0.784938\pi$
$258$	1332.82	0.321620
$259$	620.980	0.148980
$260$	−2077.94	−0.495647
$261$	−2162.25	−0.512796
$262$	18.8351	0.00444135
$263$	−1524.96	−0.357540	−0.178770	−	0.983891i	$-0.557212\pi$
$263$	−1524.96	−0.357540	−0.178770	+	0.983891i	$0.557212\pi$
$264$	−640.471	−0.149312
$265$	−2596.17	−0.601818
$266$	1361.40	0.313808
$267$	−459.941	−0.105423
$268$	268.000	0.0610847
$269$	−7615.26	−1.72606	−0.863031	−	0.505151i	$-0.831437\pi$
$269$	−7615.26	−1.72606	−0.863031	+	0.505151i	$0.831437\pi$
$270$	−562.368	−0.126758
$271$	6608.48	1.48131	0.740657	−	0.671883i	$-0.234514\pi$
$271$	6608.48	1.48131	0.740657	+	0.671883i	$0.234514\pi$
$272$	−1379.61	−0.307540
$273$	−2791.94	−0.618959
$274$	−2596.94	−0.572580
$275$	441.502	0.0968130
$276$	−138.030	−0.0301031
$277$	2568.89	0.557219	0.278609	−	0.960404i	$-0.410127\pi$
$277$	2568.89	0.557219	0.278609	+	0.960404i	$0.410127\pi$
$278$	−764.834	−0.165006
$279$	−2730.22	−0.585856
$280$	1554.37	0.331755
$281$	−604.136	−0.128255	−0.0641276	−	0.997942i	$-0.520426\pi$
$281$	−604.136	−0.128255	−0.0641276	+	0.997942i	$0.520426\pi$
$282$	1764.59	0.372622
$283$	−8489.19	−1.78314	−0.891572	−	0.452878i	$-0.850397\pi$
$283$	−8489.19	−1.78314	−0.891572	+	0.452878i	$0.850397\pi$
$284$	−1374.55	−0.287198
$285$	1139.90	0.236918
$286$	−2662.34	−0.550447
$287$	6273.14	1.29021
$288$	288.000	0.0589256
$289$	2521.82	0.513295
$290$	5004.03	1.01326
$291$	−2243.87	−0.452020
$292$	3107.49	0.622781
$293$	2550.21	0.508482	0.254241	−	0.967141i	$-0.418174\pi$
$293$	2550.21	0.508482	0.254241	+	0.967141i	$0.418174\pi$
$294$	30.4693	0.00604423
$295$	−584.190	−0.115298
$296$	−266.274	−0.0522867
$297$	−720.530	−0.140772
$298$	2711.67	0.527124
$299$	−573.772	−0.110977
$300$	−198.530	−0.0382071
$301$	−4144.38	−0.793615
$302$	−339.290	−0.0646488
$303$	−4526.48	−0.858216
$304$	−583.765	−0.110135
$305$	−6156.84	−1.15587
$306$	−1552.06	−0.289952
$307$	−7673.11	−1.42647	−0.713237	−	0.700923i	$-0.752772\pi$
$307$	−7673.11	−1.42647	−0.713237	+	0.700923i	$0.752772\pi$
$308$	1991.53	0.368435
$309$	810.043	0.149132
$310$	6318.46	1.15763
$311$	−1656.21	−0.301978	−0.150989	−	0.988535i	$-0.548246\pi$
$311$	−1656.21	−0.301978	−0.150989	+	0.988535i	$0.548246\pi$
$312$	1197.17	0.217233
$313$	−5047.09	−0.911433	−0.455716	−	0.890125i	$-0.650617\pi$
$313$	−5047.09	−0.911433	−0.455716	+	0.890125i	$0.650617\pi$
$314$	4046.20	0.727198
$315$	1748.67	0.312782
$316$	587.923	0.104662
$317$	8526.36	1.51069	0.755344	−	0.655329i	$-0.227470\pi$
$317$	8526.36	1.51069	0.755344	+	0.655329i	$0.227470\pi$
$318$	1495.75	0.263765
$319$	6411.38	1.12529
$320$	−666.510	−0.116434
$321$	−4909.55	−0.853659
$322$	429.202	0.0742810
$323$	3145.96	0.541937
$324$	324.000	0.0555556
$325$	−825.260	−0.140853
$326$	4797.99	0.815141
$327$	752.778	0.127305
$328$	−2689.90	−0.452820
$329$	−5486.94	−0.919467
$330$	1667.50	0.278160
$331$	7277.55	1.20849	0.604245	−	0.796799i	$-0.293475\pi$
$331$	7277.55	1.20849	0.604245	+	0.796799i	$0.293475\pi$
$332$	5174.86	0.855445
$333$	−299.558	−0.0492964
$334$	−8058.91	−1.32025
$335$	−697.752	−0.113798
$336$	−895.529	−0.145402
$337$	−5468.35	−0.883917	−0.441958	−	0.897036i	$-0.645716\pi$
$337$	−5468.35	−0.883917	−0.441958	+	0.897036i	$0.645716\pi$
$338$	582.478	0.0937356
$339$	4379.57	0.701668
$340$	3591.88	0.572932
$341$	8095.48	1.28562
$342$	−656.735	−0.103837
$343$	6304.56	0.992461
$344$	1777.10	0.278531
$345$	359.369	0.0560806
$346$	−1822.18	−0.283125
$347$	−11931.2	−1.84582	−0.922908	−	0.385021i	$-0.874194\pi$
$347$	−11931.2	−1.84582	−0.922908	+	0.385021i	$0.874194\pi$
$348$	−2883.00	−0.444094
$349$	10399.2	1.59500	0.797499	−	0.603321i	$-0.206156\pi$
$349$	10399.2	1.59500	0.797499	+	0.603321i	$0.206156\pi$
$350$	617.324	0.0942781
$351$	1346.82	0.204809
$352$	−853.961	−0.129308
$353$	−10551.9	−1.59100	−0.795501	−	0.605953i	$-0.792792\pi$
$353$	−10551.9	−1.59100	−0.795501	+	0.605953i	$0.792792\pi$
$354$	336.573	0.0505329
$355$	3578.71	0.535037
$356$	−613.255	−0.0912990
$357$	4826.08	0.715472
$358$	−6380.22	−0.941913
$359$	8709.37	1.28040	0.640199	−	0.768209i	$-0.278852\pi$
$359$	8709.37	1.28040	0.640199	+	0.768209i	$0.278852\pi$
$360$	−749.823	−0.109775
$361$	−5527.82	−0.805923
$362$	697.358	0.101249
$363$	−1856.53	−0.268436
$364$	−3722.58	−0.536034
$365$	−8090.51	−1.16021
$366$	3547.18	0.506595
$367$	−3791.82	−0.539323	−0.269661	−	0.962955i	$-0.586912\pi$
$367$	−3791.82	−0.539323	−0.269661	+	0.962955i	$0.586912\pi$
$368$	−184.040	−0.0260700
$369$	−3026.14	−0.426923
$370$	693.259	0.0974076
$371$	−4650.99	−0.650856
$372$	−3640.29	−0.507366
$373$	−12343.3	−1.71343	−0.856717	−	0.515787i	$-0.827500\pi$
$373$	−12343.3	−1.71343	−0.856717	+	0.515787i	$0.827500\pi$
$374$	4602.07	0.636277
$375$	4422.21	0.608966
$376$	2352.78	0.322700
$377$	−11984.2	−1.63718
$378$	−1007.47	−0.137086
$379$	−7539.33	−1.02182	−0.510909	−	0.859635i	$-0.670691\pi$
$379$	−7539.33	−1.02182	−0.510909	+	0.859635i	$0.670691\pi$
$380$	1519.86	0.205177
$381$	−1158.71	−0.155807
$382$	−2252.01	−0.301630
$383$	3183.73	0.424755	0.212377	−	0.977188i	$-0.431879\pi$
$383$	3183.73	0.424755	0.212377	+	0.977188i	$0.431879\pi$
$384$	384.000	0.0510310
$385$	−5185.05	−0.686376
$386$	812.924	0.107194
$387$	1999.23	0.262601
$388$	−2991.82	−0.391461
$389$	−3632.80	−0.473497	−0.236748	−	0.971571i	$-0.576082\pi$
$389$	−3632.80	−0.473497	−0.236748	+	0.971571i	$0.576082\pi$
$390$	−3116.91	−0.404694
$391$	991.810	0.128281
$392$	40.6257	0.00523446
$393$	28.2526	0.00362635
$394$	−4105.88	−0.525003
$395$	−1530.69	−0.194980
$396$	−960.706	−0.121912
$397$	1313.02	0.165991	0.0829956	−	0.996550i	$-0.473551\pi$
$397$	1313.02	0.165991	0.0829956	+	0.996550i	$0.473551\pi$
$398$	4287.97	0.540042
$399$	2042.10	0.256223
$400$	−264.706	−0.0330883
$401$	−6889.39	−0.857955	−0.428977	−	0.903315i	$-0.641126\pi$
$401$	−6889.39	−0.857955	−0.428977	+	0.903315i	$0.641126\pi$
$402$	402.000	0.0498755
$403$	−15132.2	−1.87044
$404$	−6035.31	−0.743237
$405$	−843.551	−0.103497
$406$	8964.61	1.09583
$407$	888.234	0.108177
$408$	−2069.41	−0.251105
$409$	−1548.99	−0.187268	−0.0936339	−	0.995607i	$-0.529848\pi$
$409$	−1548.99	−0.187268	−0.0936339	+	0.995607i	$0.529848\pi$
$410$	7003.30	0.843581
$411$	−3895.41	−0.467510
$412$	1080.06	0.129152
$413$	−1046.56	−0.124693
$414$	−207.045	−0.0245791
$415$	−13473.0	−1.59365
$416$	1596.23	0.188129
$417$	−1147.25	−0.134727
$418$	1947.31	0.227862
$419$	−4132.09	−0.481779	−0.240890	−	0.970552i	$-0.577439\pi$
$419$	−4132.09	−0.481779	−0.240890	+	0.970552i	$0.577439\pi$
$420$	2331.56	0.270877
$421$	−15398.6	−1.78262	−0.891308	−	0.453398i	$-0.850212\pi$
$421$	−15398.6	−1.78262	−0.891308	+	0.453398i	$0.850212\pi$
$422$	4793.26	0.552920
$423$	2646.88	0.304245
$424$	1994.33	0.228428
$425$	1426.53	0.162816
$426$	−2061.82	−0.234496
$427$	−11029.9	−1.25005
$428$	−6546.07	−0.739291
$429$	−3993.52	−0.449438
$430$	−4626.77	−0.518889
$431$	−4201.01	−0.469502	−0.234751	−	0.972055i	$-0.575428\pi$
$431$	−4201.01	−0.469502	−0.234751	+	0.972055i	$0.575428\pi$
$432$	432.000	0.0481125
$433$	6484.61	0.719700	0.359850	−	0.933010i	$-0.382828\pi$
$433$	6484.61	0.719700	0.359850	+	0.933010i	$0.382828\pi$
$434$	11319.4	1.25195
$435$	7506.04	0.827326
$436$	1003.70	0.110249
$437$	419.673	0.0459398
$438$	4661.23	0.508498
$439$	9830.96	1.06881	0.534403	−	0.845230i	$-0.320536\pi$
$439$	9830.96	1.06881	0.534403	+	0.845230i	$0.320536\pi$
$440$	2223.33	0.240894
$441$	45.7039	0.00493509
$442$	−8602.23	−0.925716
$443$	−3182.57	−0.341328	−0.170664	−	0.985329i	$-0.554591\pi$
$443$	−3182.57	−0.341328	−0.170664	+	0.985329i	$0.554591\pi$
$444$	−399.411	−0.0426919
$445$	1596.64	0.170086
$446$	8689.83	0.922590
$447$	4067.51	0.430395
$448$	−1194.04	−0.125922
$449$	−1281.28	−0.134671	−0.0673353	−	0.997730i	$-0.521450\pi$
$449$	−1281.28	−0.134671	−0.0673353	+	0.997730i	$0.521450\pi$
$450$	−297.795	−0.0311960
$451$	8972.93	0.936849
$452$	5839.42	0.607662
$453$	−508.935	−0.0527856
$454$	5847.66	0.604502
$455$	9691.95	0.998605
$456$	−875.647	−0.0899253
$457$	−14609.6	−1.49543	−0.747713	−	0.664022i	$-0.768848\pi$
$457$	−14609.6	−1.49543	−0.747713	+	0.664022i	$0.768848\pi$
$458$	−5931.58	−0.605163
$459$	−2328.09	−0.236745
$460$	479.159	0.0485672
$461$	−3976.54	−0.401748	−0.200874	−	0.979617i	$-0.564378\pi$
$461$	−3976.54	−0.401748	−0.200874	+	0.979617i	$0.564378\pi$
$462$	2987.29	0.300826
$463$	7479.26	0.750735	0.375368	−	0.926876i	$-0.377516\pi$
$463$	7479.26	0.750735	0.375368	+	0.926876i	$0.377516\pi$
$464$	−3844.00	−0.384597
$465$	9477.69	0.945198
$466$	−13007.1	−1.29301
$467$	3317.47	0.328724	0.164362	−	0.986400i	$-0.447443\pi$
$467$	3317.47	0.328724	0.164362	+	0.986400i	$0.447443\pi$
$468$	1795.76	0.177370
$469$	−1250.01	−0.123070
$470$	−6125.59	−0.601175
$471$	6069.30	0.593755
$472$	448.764	0.0437628
$473$	−5928.02	−0.576259
$474$	881.884	0.0854563
$475$	603.618	0.0583072
$476$	6434.78	0.619617
$477$	2243.62	0.215364
$478$	9636.83	0.922130
$479$	−1364.26	−0.130135	−0.0650676	−	0.997881i	$-0.520726\pi$
$479$	−1364.26	−0.130135	−0.0650676	+	0.997881i	$0.520726\pi$
$480$	−999.765	−0.0950683
$481$	−1660.29	−0.157387
$482$	3623.60	0.342428
$483$	643.803	0.0606502
$484$	−2475.37	−0.232473
$485$	7789.37	0.729272
$486$	486.000	0.0453609
$487$	8222.49	0.765085	0.382543	−	0.923938i	$-0.375048\pi$
$487$	8222.49	0.765085	0.382543	+	0.923938i	$0.375048\pi$
$488$	4729.57	0.438724
$489$	7196.98	0.665560
$490$	−105.771	−0.00975154
$491$	−6699.26	−0.615750	−0.307875	−	0.951427i	$-0.599618\pi$
$491$	−6699.26	−0.615750	−0.307875	+	0.951427i	$0.599618\pi$
$492$	−4034.85	−0.369726
$493$	20715.6	1.89246
$494$	−3639.94	−0.331515
$495$	2501.25	0.227117
$496$	−4853.72	−0.439392
$497$	6411.18	0.578633
$498$	7762.30	0.698468
$499$	2582.06	0.231641	0.115820	−	0.993270i	$-0.463050\pi$
$499$	2582.06	0.231641	0.115820	+	0.993270i	$0.463050\pi$
$500$	5896.28	0.527380
$501$	−12088.4	−1.07798
$502$	−3560.25	−0.316537
$503$	9872.78	0.875160	0.437580	−	0.899179i	$-0.355836\pi$
$503$	9872.78	0.875160	0.437580	+	0.899179i	$0.355836\pi$
$504$	−1343.29	−0.118720
$505$	15713.2	1.38461
$506$	613.919	0.0539369
$507$	873.717	0.0765348
$508$	−1544.94	−0.134933
$509$	2858.39	0.248911	0.124456	−	0.992225i	$-0.460282\pi$
$509$	2858.39	0.248911	0.124456	+	0.992225i	$0.460282\pi$
$510$	5387.82	0.467797
$511$	−14494.0	−1.25475
$512$	512.000	0.0441942
$513$	−985.103	−0.0847823
$514$	−12859.6	−1.10352
$515$	−2811.99	−0.240604
$516$	2665.65	0.227420
$517$	−7848.37	−0.667642
$518$	1241.96	0.105345
$519$	−2733.28	−0.231171
$520$	−4155.88	−0.350475
$521$	8312.54	0.699000	0.349500	−	0.936936i	$-0.386352\pi$
$521$	8312.54	0.699000	0.349500	+	0.936936i	$0.386352\pi$
$522$	−4324.50	−0.362602
$523$	15736.7	1.31571	0.657854	−	0.753145i	$-0.271464\pi$
$523$	15736.7	1.31571	0.657854	+	0.753145i	$0.271464\pi$
$524$	37.6701	0.00314051
$525$	925.986	0.0769778
$526$	−3049.92	−0.252819
$527$	26157.1	2.16209
$528$	−1280.94	−0.105579
$529$	−12034.7	−0.989126
$530$	−5192.35	−0.425549
$531$	504.859	0.0412599
$532$	2722.80	0.221896
$533$	−16772.3	−1.36302
$534$	−919.882	−0.0745453
$535$	17043.0	1.37726
$536$	536.000	0.0431934
$537$	−9570.32	−0.769069
$538$	−15230.5	−1.22051
$539$	−135.519	−0.0108297
$540$	−1124.74	−0.0896313
$541$	19377.4	1.53992	0.769961	−	0.638091i	$-0.220276\pi$
$541$	19377.4	1.53992	0.769961	+	0.638091i	$0.220276\pi$
$542$	13217.0	1.04745
$543$	1046.04	0.0826698
$544$	−2759.21	−0.217464
$545$	−2613.20	−0.205389
$546$	−5583.88	−0.437670
$547$	−8646.62	−0.675873	−0.337937	−	0.941169i	$-0.609729\pi$
$547$	−8646.62	−0.675873	−0.337937	+	0.941169i	$0.609729\pi$
$548$	−5193.89	−0.404876
$549$	5320.76	0.413633
$550$	883.004	0.0684571
$551$	8765.58	0.677725
$552$	−276.061	−0.0212861
$553$	−2742.20	−0.210868
$554$	5137.78	0.394013
$555$	1039.89	0.0795330
$556$	−1529.67	−0.116677
$557$	12372.1	0.941154	0.470577	−	0.882359i	$-0.344046\pi$
$557$	12372.1	0.941154	0.470577	+	0.882359i	$0.344046\pi$
$558$	−5460.43	−0.414263
$559$	11080.7	0.838396
$560$	3108.74	0.234586
$561$	6903.11	0.519518
$562$	−1208.27	−0.0906901
$563$	21917.1	1.64067	0.820335	−	0.571883i	$-0.193787\pi$
$563$	21917.1	1.64067	0.820335	+	0.571883i	$0.193787\pi$
$564$	3529.17	0.263484
$565$	−15203.3	−1.13205
$566$	−16978.4	−1.26087
$567$	−1511.21	−0.111931
$568$	−2749.09	−0.203080
$569$	3149.17	0.232021	0.116011	−	0.993248i	$-0.462989\pi$
$569$	3149.17	0.232021	0.116011	+	0.993248i	$0.462989\pi$
$570$	2279.79	0.167526
$571$	17090.5	1.25256	0.626281	−	0.779597i	$-0.284576\pi$
$571$	17090.5	1.25256	0.626281	+	0.779597i	$0.284576\pi$
$572$	−5324.69	−0.389225
$573$	−3378.01	−0.246280
$574$	12546.3	0.912319
$575$	190.300	0.0138018
$576$	576.000	0.0416667
$577$	9049.23	0.652902	0.326451	−	0.945214i	$-0.394147\pi$
$577$	9049.23	0.652902	0.326451	+	0.945214i	$0.394147\pi$
$578$	5043.64	0.362954
$579$	1219.39	0.0875232
$580$	10008.1	0.716486
$581$	−24136.7	−1.72351
$582$	−4487.73	−0.319626
$583$	−6652.66	−0.472599
$584$	6214.98	0.440373
$585$	−4675.36	−0.330431
$586$	5100.43	0.359551
$587$	21139.6	1.48641	0.743206	−	0.669062i	$-0.233304\pi$
$587$	21139.6	1.48641	0.743206	+	0.669062i	$0.233304\pi$
$588$	60.9385	0.00427392
$589$	11068.1	0.774282
$590$	−1168.38	−0.0815279
$591$	−6158.81	−0.428663
$592$	−532.548	−0.0369723
$593$	6329.91	0.438345	0.219172	−	0.975686i	$-0.429664\pi$
$593$	6329.91	0.438345	0.219172	+	0.975686i	$0.429664\pi$
$594$	−1441.06	−0.0995411
$595$	−16753.3	−1.15432
$596$	5423.34	0.372733
$597$	6431.96	0.440942
$598$	−1147.54	−0.0784725
$599$	−14675.8	−1.00106	−0.500531	−	0.865719i	$-0.666862\pi$
$599$	−14675.8	−1.00106	−0.500531	+	0.865719i	$0.666862\pi$
$600$	−397.060	−0.0270165
$601$	−6715.91	−0.455820	−0.227910	−	0.973682i	$-0.573189\pi$
$601$	−6715.91	−0.455820	−0.227910	+	0.973682i	$0.573189\pi$
$602$	−8288.76	−0.561170
$603$	603.000	0.0407231
$604$	−678.580	−0.0457136
$605$	6444.75	0.433085
$606$	−9052.96	−0.606850
$607$	−11636.9	−0.778137	−0.389068	−	0.921209i	$-0.627203\pi$
$607$	−11636.9	−0.778137	−0.389068	+	0.921209i	$0.627203\pi$
$608$	−1167.53	−0.0778776
$609$	13446.9	0.894740
$610$	−12313.7	−0.817322
$611$	14670.2	0.971350
$612$	−3104.11	−0.205027
$613$	−8952.36	−0.589857	−0.294929	−	0.955519i	$-0.595296\pi$
$613$	−8952.36	−0.589857	−0.294929	+	0.955519i	$0.595296\pi$
$614$	−15346.2	−1.00867
$615$	10505.0	0.688781
$616$	3983.06	0.260523
$617$	2599.63	0.169623	0.0848114	−	0.996397i	$-0.472971\pi$
$617$	2599.63	0.169623	0.0848114	+	0.996397i	$0.472971\pi$
$618$	1620.09	0.105452
$619$	−13133.3	−0.852784	−0.426392	−	0.904539i	$-0.640215\pi$
$619$	−13133.3	−0.852784	−0.426392	+	0.904539i	$0.640215\pi$
$620$	12636.9	0.818566
$621$	−310.568	−0.0200687
$622$	−3312.43	−0.213531
$623$	2860.35	0.183945
$624$	2394.35	0.153607
$625$	−13283.3	−0.850129
$626$	−10094.2	−0.644480
$627$	2920.97	0.186048
$628$	8092.40	0.514207
$629$	2869.95	0.181927
$630$	3497.34	0.221170
$631$	4428.92	0.279418	0.139709	−	0.990193i	$-0.455383\pi$
$631$	4428.92	0.279418	0.139709	+	0.990193i	$0.455383\pi$
$632$	1175.85	0.0740073
$633$	7189.89	0.451457
$634$	17052.7	1.06822
$635$	4022.34	0.251373
$636$	2991.50	0.186510
$637$	253.313	0.0157561
$638$	12822.8	0.795702
$639$	−3092.73	−0.191466
$640$	−1333.02	−0.0823316
$641$	3354.33	0.206690	0.103345	−	0.994646i	$-0.467045\pi$
$641$	3354.33	0.206690	0.103345	+	0.994646i	$0.467045\pi$
$642$	−9819.11	−0.603628
$643$	−31988.4	−1.96190	−0.980948	−	0.194271i	$-0.937766\pi$
$643$	−31988.4	−1.96190	−0.980948	+	0.194271i	$0.937766\pi$
$644$	858.404	0.0525246
$645$	−6940.15	−0.423671
$646$	6291.92	0.383208
$647$	8943.52	0.543440	0.271720	−	0.962376i	$-0.412407\pi$
$647$	8943.52	0.543440	0.271720	+	0.962376i	$0.412407\pi$
$648$	648.000	0.0392837
$649$	−1496.98	−0.0905417
$650$	−1650.52	−0.0995980
$651$	16979.1	1.02222
$652$	9595.98	0.576392
$653$	−6152.14	−0.368686	−0.184343	−	0.982862i	$-0.559016\pi$
$653$	−6152.14	−0.368686	−0.184343	+	0.982862i	$0.559016\pi$
$654$	1505.56	0.0900182
$655$	−98.0762	−0.00585062
$656$	−5379.80	−0.320192
$657$	6991.85	0.415187
$658$	−10973.9	−0.650161
$659$	−691.931	−0.0409011	−0.0204505	−	0.999791i	$-0.506510\pi$
$659$	−691.931	−0.0409011	−0.0204505	+	0.999791i	$0.506510\pi$
$660$	3335.00	0.196689
$661$	8792.19	0.517362	0.258681	−	0.965963i	$-0.416712\pi$
$661$	8792.19	0.517362	0.258681	+	0.965963i	$0.416712\pi$
$662$	14555.1	0.854532
$663$	−12903.4	−0.755844
$664$	10349.7	0.604891
$665$	−7088.96	−0.413381
$666$	−599.117	−0.0348578
$667$	2763.48	0.160423
$668$	−16117.8	−0.933558
$669$	13034.7	0.753292
$670$	−1395.50	−0.0804672
$671$	−15776.8	−0.907687
$672$	−1791.06	−0.102815
$673$	−5846.51	−0.334868	−0.167434	−	0.985883i	$-0.553548\pi$
$673$	−5846.51	−0.334868	−0.167434	+	0.985883i	$0.553548\pi$
$674$	−10936.7	−0.625024
$675$	−446.692	−0.0254714
$676$	1164.96	0.0662811
$677$	24795.3	1.40762	0.703812	−	0.710386i	$-0.251480\pi$
$677$	24795.3	1.40762	0.703812	+	0.710386i	$0.251480\pi$
$678$	8759.14	0.496154
$679$	13954.5	0.788696
$680$	7183.76	0.405124
$681$	8771.48	0.493574
$682$	16191.0	0.909068
$683$	21391.7	1.19843	0.599217	−	0.800587i	$-0.295479\pi$
$683$	21391.7	1.19843	0.599217	+	0.800587i	$0.295479\pi$
$684$	−1313.47	−0.0734237
$685$	13522.6	0.754263
$686$	12609.1	0.701776
$687$	−8897.38	−0.494114
$688$	3554.19	0.196951
$689$	12435.2	0.687582
$690$	718.739	0.0396549
$691$	−2094.63	−0.115316	−0.0576582	−	0.998336i	$-0.518363\pi$
$691$	−2094.63	−0.115316	−0.0576582	+	0.998336i	$0.518363\pi$
$692$	−3644.37	−0.200200
$693$	4480.94	0.245623
$694$	−23862.3	−1.30519
$695$	3982.57	0.217364
$696$	−5765.99	−0.314022
$697$	28992.2	1.57555
$698$	20798.3	1.12783
$699$	−19510.7	−1.05574
$700$	1234.65	0.0666647
$701$	6672.97	0.359536	0.179768	−	0.983709i	$-0.442465\pi$
$701$	6672.97	0.359536	0.179768	+	0.983709i	$0.442465\pi$
$702$	2693.64	0.144822
$703$	1214.39	0.0651514
$704$	−1707.92	−0.0914343
$705$	−9188.38	−0.490858
$706$	−21103.9	−1.12501
$707$	28150.0	1.49744
$708$	673.145	0.0357321
$709$	−16252.2	−0.860878	−0.430439	−	0.902620i	$-0.641641\pi$
$709$	−16252.2	−0.860878	−0.430439	+	0.902620i	$0.641641\pi$
$710$	7157.41	0.378328
$711$	1322.83	0.0697748
$712$	−1226.51	−0.0645581
$713$	3489.38	0.183279
$714$	9652.17	0.505915
$715$	13863.1	0.725106
$716$	−12760.4	−0.666033
$717$	14455.2	0.752916
$718$	17418.7	0.905378
$719$	7474.83	0.387711	0.193856	−	0.981030i	$-0.437901\pi$
$719$	7474.83	0.387711	0.193856	+	0.981030i	$0.437901\pi$
$720$	−1499.65	−0.0776230
$721$	−5037.62	−0.260209
$722$	−11055.6	−0.569873
$723$	5435.40	0.279592
$724$	1394.72	0.0715942
$725$	3974.73	0.203611
$726$	−3713.05	−0.189813
$727$	36720.0	1.87327	0.936637	−	0.350301i	$-0.113921\pi$
$727$	36720.0	1.87327	0.936637	+	0.350301i	$0.113921\pi$
$728$	−7445.17	−0.379033
$729$	729.000	0.0370370
$730$	−16181.0	−0.820393
$731$	−19153.9	−0.969126
$732$	7094.35	0.358217
$733$	−34754.2	−1.75126	−0.875632	−	0.482979i	$-0.839555\pi$
$733$	−34754.2	−1.75126	−0.875632	+	0.482979i	$0.839555\pi$
$734$	−7583.64	−0.381359
$735$	−158.657	−0.00796210
$736$	−368.081	−0.0184343
$737$	−1787.98	−0.0893638
$738$	−6052.28	−0.301880
$739$	−29768.0	−1.48178	−0.740890	−	0.671627i	$-0.765596\pi$
$739$	−29768.0	−1.48178	−0.740890	+	0.671627i	$0.765596\pi$
$740$	1386.52	0.0688776
$741$	−5459.90	−0.270681
$742$	−9301.99	−0.460225
$743$	−22905.7	−1.13100	−0.565498	−	0.824750i	$-0.691316\pi$
$743$	−22905.7	−1.13100	−0.565498	+	0.824750i	$0.691316\pi$
$744$	−7280.58	−0.358762
$745$	−14120.0	−0.694383
$746$	−24686.6	−1.21158
$747$	11643.4	0.570296
$748$	9204.14	0.449916
$749$	30532.3	1.48949
$750$	8844.43	0.430604
$751$	−21967.1	−1.06737	−0.533683	−	0.845685i	$-0.679192\pi$
$751$	−21967.1	−1.06737	−0.533683	+	0.845685i	$0.679192\pi$
$752$	4705.56	0.228184
$753$	−5340.37	−0.258451
$754$	−23968.4	−1.15766
$755$	1766.72	0.0851623
$756$	−2014.94	−0.0969347
$757$	−5578.12	−0.267821	−0.133910	−	0.990993i	$-0.542753\pi$
$757$	−5578.12	−0.267821	−0.133910	+	0.990993i	$0.542753\pi$
$758$	−15078.7	−0.722535
$759$	920.879	0.0440393
$760$	3039.72	0.145082
$761$	−11212.5	−0.534105	−0.267052	−	0.963682i	$-0.586050\pi$
$761$	−11212.5	−0.534105	−0.267052	+	0.963682i	$0.586050\pi$
$762$	−2317.41	−0.110172
$763$	−4681.49	−0.222125
$764$	−4504.02	−0.213285
$765$	8081.73	0.381955
$766$	6367.46	0.300347
$767$	2798.17	0.131729
$768$	768.000	0.0360844
$769$	−30925.5	−1.45020	−0.725100	−	0.688644i	$-0.758206\pi$
$769$	−30925.5	−1.45020	−0.725100	+	0.688644i	$0.758206\pi$
$770$	−10370.1	−0.485341
$771$	−19289.3	−0.901022
$772$	1625.85	0.0757973
$773$	23144.2	1.07689	0.538446	−	0.842660i	$-0.319012\pi$
$773$	23144.2	1.07689	0.538446	+	0.842660i	$0.319012\pi$
$774$	3998.47	0.185687
$775$	5018.79	0.232620
$776$	−5983.64	−0.276804
$777$	1862.94	0.0860136
$778$	−7265.60	−0.334813
$779$	12267.7	0.564232
$780$	−6233.81	−0.286162
$781$	9170.39	0.420157
$782$	1983.62	0.0907086
$783$	−6486.74	−0.296063
$784$	81.2514	0.00370132
$785$	−21069.0	−0.957942
$786$	56.5052	0.00256422
$787$	−6041.21	−0.273629	−0.136814	−	0.990597i	$-0.543686\pi$
$787$	−6041.21	−0.273629	−0.136814	+	0.990597i	$0.543686\pi$
$788$	−8211.75	−0.371233
$789$	−4574.88	−0.206426
$790$	−3061.38	−0.137872
$791$	−27236.3	−1.22429
$792$	−1921.41	−0.0862051
$793$	29490.2	1.32059
$794$	2626.04	0.117374
$795$	−7788.52	−0.347460
$796$	8575.94	0.381867
$797$	−23620.8	−1.04980	−0.524901	−	0.851163i	$-0.675898\pi$
$797$	−23620.8	−1.04980	−0.524901	+	0.851163i	$0.675898\pi$
$798$	4084.20	0.181177
$799$	−25358.7	−1.12281
$800$	−529.413	−0.0233970
$801$	−1379.82	−0.0608660
$802$	−13778.8	−0.606666
$803$	−20731.8	−0.911097
$804$	804.000	0.0352673
$805$	−2234.90	−0.0978508
$806$	−30264.3	−1.32260
$807$	−22845.8	−0.996542
$808$	−12070.6	−0.525548
$809$	−35045.0	−1.52301	−0.761507	−	0.648157i	$-0.775540\pi$
$809$	−35045.0	−1.52301	−0.761507	+	0.648157i	$0.775540\pi$
$810$	−1687.10	−0.0731836
$811$	29575.4	1.28056	0.640279	−	0.768142i	$-0.278819\pi$
$811$	29575.4	1.28056	0.640279	+	0.768142i	$0.278819\pi$
$812$	17929.2	0.774867
$813$	19825.4	0.855238
$814$	1776.47	0.0764928
$815$	−24983.6	−1.07379
$816$	−4138.82	−0.177558
$817$	−8104.73	−0.347061
$818$	−3097.98	−0.132418
$819$	−8375.81	−0.357356
$820$	14006.6	0.596502
$821$	−26948.0	−1.14555	−0.572773	−	0.819714i	$-0.694132\pi$
$821$	−26948.0	−1.14555	−0.572773	+	0.819714i	$0.694132\pi$
$822$	−7790.83	−0.330579
$823$	19119.3	0.809791	0.404895	−	0.914363i	$-0.367308\pi$
$823$	19119.3	0.809791	0.404895	+	0.914363i	$0.367308\pi$
$824$	2160.11	0.0913242
$825$	1324.51	0.0558950
$826$	−2093.13	−0.0881711
$827$	−5617.30	−0.236194	−0.118097	−	0.993002i	$-0.537679\pi$
$827$	−5617.30	−0.236194	−0.118097	+	0.993002i	$0.537679\pi$
$828$	−414.091	−0.0173800
$829$	−37902.9	−1.58796	−0.793981	−	0.607942i	$-0.791995\pi$
$829$	−37902.9	−1.58796	−0.793981	+	0.607942i	$0.791995\pi$
$830$	−26946.1	−1.12688
$831$	7706.67	0.321710
$832$	3192.46	0.133027
$833$	−437.871	−0.0182129
$834$	−2294.50	−0.0952664
$835$	41963.6	1.73917
$836$	3894.63	0.161123
$837$	−8190.65	−0.338244
$838$	−8264.17	−0.340669
$839$	−433.615	−0.0178427	−0.00892137	−	0.999960i	$-0.502840\pi$
$839$	−433.615	−0.0178427	−0.00892137	+	0.999960i	$0.502840\pi$
$840$	4663.12	0.191539
$841$	33331.0	1.36664
$842$	−30797.2	−1.26050
$843$	−1812.41	−0.0740482
$844$	9586.52	0.390973
$845$	−3033.02	−0.123478
$846$	5293.76	0.215134
$847$	11545.6	0.468374
$848$	3988.66	0.161523
$849$	−25467.6	−1.02950
$850$	2853.05	0.115128
$851$	382.853	0.0154219
$852$	−4123.64	−0.165814
$853$	−39341.4	−1.57916	−0.789581	−	0.613646i	$-0.789702\pi$
$853$	−39341.4	−1.57916	−0.789581	+	0.613646i	$0.789702\pi$
$854$	−22059.7	−0.883920
$855$	3419.69	0.136785
$856$	−13092.1	−0.522757
$857$	31202.3	1.24370	0.621849	−	0.783137i	$-0.286382\pi$
$857$	31202.3	1.24370	0.621849	+	0.783137i	$0.286382\pi$
$858$	−7987.03	−0.317801
$859$	−3206.29	−0.127354	−0.0636771	−	0.997971i	$-0.520283\pi$
$859$	−3206.29	−0.127354	−0.0636771	+	0.997971i	$0.520283\pi$
$860$	−9253.53	−0.366910
$861$	18819.4	0.744906
$862$	−8402.02	−0.331988
$863$	−14568.9	−0.574659	−0.287329	−	0.957832i	$-0.592767\pi$
$863$	−14568.9	−0.574659	−0.287329	+	0.957832i	$0.592767\pi$
$864$	864.000	0.0340207
$865$	9488.31	0.372962
$866$	12969.2	0.508905
$867$	7565.46	0.296351
$868$	22638.8	0.885265
$869$	−3922.37	−0.153115
$870$	15012.1	0.585008
$871$	3342.11	0.130015
$872$	2007.41	0.0779580
$873$	−6731.60	−0.260974
$874$	839.346	0.0324843
$875$	−27501.5	−1.06254
$876$	9322.47	0.359563
$877$	12411.4	0.477881	0.238941	−	0.971034i	$-0.423200\pi$
$877$	12411.4	0.477881	0.238941	+	0.971034i	$0.423200\pi$
$878$	19661.9	0.755760
$879$	7650.64	0.293572
$880$	4446.67	0.170338
$881$	−35078.2	−1.34145	−0.670723	−	0.741708i	$-0.734016\pi$
$881$	−35078.2	−1.34145	−0.670723	+	0.741708i	$0.734016\pi$
$882$	91.4078	0.00348964
$883$	−31328.2	−1.19397	−0.596986	−	0.802251i	$-0.703635\pi$
$883$	−31328.2	−1.19397	−0.596986	+	0.802251i	$0.703635\pi$
$884$	−17204.5	−0.654580
$885$	−1752.57	−0.0665672
$886$	−6365.14	−0.241355
$887$	6841.72	0.258988	0.129494	−	0.991580i	$-0.458665\pi$
$887$	6841.72	0.258988	0.129494	+	0.991580i	$0.458665\pi$
$888$	−798.823	−0.0301878
$889$	7205.94	0.271855
$890$	3193.28	0.120269
$891$	−2161.59	−0.0812749
$892$	17379.7	0.652370
$893$	−10730.2	−0.402098
$894$	8135.02	0.304335
$895$	33222.5	1.24079
$896$	−2388.08	−0.0890403
$897$	−1721.32	−0.0640725
$898$	−2562.55	−0.0952265
$899$	72881.6	2.70382
$900$	−595.590	−0.0220589
$901$	−21495.3	−0.794795
$902$	17945.9	0.662452
$903$	−12433.1	−0.458194
$904$	11678.8	0.429682
$905$	−3631.22	−0.133376
$906$	−1017.87	−0.0373250
$907$	17521.2	0.641435	0.320718	−	0.947175i	$-0.396076\pi$
$907$	17521.2	0.641435	0.320718	+	0.947175i	$0.396076\pi$
$908$	11695.3	0.427448
$909$	−13579.4	−0.495491
$910$	19383.9	0.706121
$911$	−17770.1	−0.646267	−0.323134	−	0.946353i	$-0.604736\pi$
$911$	−17770.1	−0.646267	−0.323134	+	0.946353i	$0.604736\pi$
$912$	−1751.29	−0.0635868
$913$	−34524.5	−1.25147
$914$	−29219.3	−1.05743
$915$	−18470.5	−0.667341
$916$	−11863.2	−0.427915
$917$	−175.702	−0.00632734
$918$	−4656.17	−0.167404
$919$	41708.8	1.49711	0.748555	−	0.663072i	$-0.230748\pi$
$919$	41708.8	1.49711	0.748555	+	0.663072i	$0.230748\pi$
$920$	958.318	0.0343422
$921$	−23019.3	−0.823575
$922$	−7953.07	−0.284079
$923$	−17141.4	−0.611284
$924$	5974.59	0.212716
$925$	550.660	0.0195736
$926$	14958.5	0.530850
$927$	2430.13	0.0861013
$928$	−7687.99	−0.271951
$929$	−23756.9	−0.839008	−0.419504	−	0.907754i	$-0.637796\pi$
$929$	−23756.9	−0.839008	−0.419504	+	0.907754i	$0.637796\pi$
$930$	18955.4	0.668356
$931$	−185.280	−0.00652235
$932$	−26014.3	−0.914298
$933$	−4968.64	−0.174347
$934$	6634.94	0.232443
$935$	−23963.5	−0.838171
$936$	3591.52	0.125419
$937$	−19712.0	−0.687259	−0.343630	−	0.939105i	$-0.611657\pi$
$937$	−19712.0	−0.687259	−0.343630	+	0.939105i	$0.611657\pi$
$938$	−2500.02	−0.0870240
$939$	−15141.3	−0.526216
$940$	−12251.2	−0.425095
$941$	15492.5	0.536706	0.268353	−	0.963321i	$-0.413521\pi$
$941$	15492.5	0.536706	0.268353	+	0.963321i	$0.413521\pi$
$942$	12138.6	0.419848
$943$	3867.58	0.133559
$944$	897.527	0.0309449
$945$	5246.00	0.180585
$946$	−11856.0	−0.407477
$947$	21668.8	0.743548	0.371774	−	0.928323i	$-0.378750\pi$
$947$	21668.8	0.743548	0.371774	+	0.928323i	$0.378750\pi$
$948$	1763.77	0.0604267
$949$	38752.1	1.32555
$950$	1207.24	0.0412294
$951$	25579.1	0.872196
$952$	12869.6	0.438135
$953$	20481.2	0.696172	0.348086	−	0.937463i	$-0.386832\pi$
$953$	20481.2	0.696172	0.348086	+	0.937463i	$0.386832\pi$
$954$	4487.25	0.152285
$955$	11726.4	0.397339
$956$	19273.7	0.652045
$957$	19234.1	0.649688
$958$	−2728.52	−0.0920195
$959$	24225.4	0.815723
$960$	−1999.53	−0.0672235
$961$	62234.7	2.08904
$962$	−3320.59	−0.111289
$963$	−14728.7	−0.492860
$964$	7247.20	0.242133
$965$	−4232.98	−0.141207
$966$	1287.61	0.0428862
$967$	−51439.1	−1.71062	−0.855309	−	0.518118i	$-0.826633\pi$
$967$	−51439.1	−1.71062	−0.855309	+	0.518118i	$0.826633\pi$
$968$	−4950.73	−0.164383
$969$	9437.87	0.312888
$970$	15578.7	0.515673
$971$	10083.4	0.333257	0.166629	−	0.986020i	$-0.446712\pi$
$971$	10083.4	0.333257	0.166629	+	0.986020i	$0.446712\pi$
$972$	972.000	0.0320750
$973$	7134.70	0.235075
$974$	16445.0	0.540997
$975$	−2475.78	−0.0813214
$976$	9459.14	0.310225
$977$	−36898.0	−1.20826	−0.604131	−	0.796885i	$-0.706480\pi$
$977$	−36898.0	−1.20826	−0.604131	+	0.796885i	$0.706480\pi$
$978$	14394.0	0.470622
$979$	4091.37	0.133566
$980$	−211.542	−0.00689538
$981$	2258.33	0.0734995
$982$	−13398.5	−0.435401
$983$	23417.1	0.759807	0.379903	−	0.925026i	$-0.375957\pi$
$983$	23417.1	0.759807	0.379903	+	0.925026i	$0.375957\pi$
$984$	−8069.70	−0.261436
$985$	21379.7	0.691589
$986$	41431.3	1.33817
$987$	−16460.8	−0.530854
$988$	−7279.87	−0.234417
$989$	−2555.14	−0.0821523
$990$	5002.50	0.160596
$991$	18996.8	0.608932	0.304466	−	0.952523i	$-0.401522\pi$
$991$	18996.8	0.608932	0.304466	+	0.952523i	$0.401522\pi$
$992$	−9707.44	−0.310697
$993$	21832.6	0.697722
$994$	12822.4	0.409155
$995$	−22327.9	−0.711400
$996$	15524.6	0.493891
$997$	41383.0	1.31456	0.657278	−	0.753648i	$-0.271708\pi$
$997$	41383.0	1.31456	0.657278	+	0.753648i	$0.271708\pi$
$998$	5164.11	0.163795
$999$	−898.675	−0.0284613

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	402.4.a.b.1.1	✓	2
3.2	odd	2		1206.4.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
402.4.a.b.1.1	✓	2	1.1	even	1	trivial
1206.4.a.d.1.2		2	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:	\( N \)	\(=\)	\( 402 = 2 \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	402.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -10.4142 q^{5} +6.00000 q^{6} -18.6569 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -10.4142 q^{5} +6.00000 q^{6} -18.6569 q^{7} +8.00000 q^{8} +9.00000 q^{9} -20.8284 q^{10} -26.6863 q^{11} +12.0000 q^{12} +49.8823 q^{13} -37.3137 q^{14} -31.2426 q^{15} +16.0000 q^{16} -86.2254 q^{17} +18.0000 q^{18} -36.4853 q^{19} -41.6569 q^{20} -55.9706 q^{21} -53.3726 q^{22} -11.5025 q^{23} +24.0000 q^{24} -16.5442 q^{25} +99.7645 q^{26} +27.0000 q^{27} -74.6274 q^{28} -240.250 q^{29} -62.4853 q^{30} -303.357 q^{31} +32.0000 q^{32} -80.0589 q^{33} -172.451 q^{34} +194.296 q^{35} +36.0000 q^{36} -33.2843 q^{37} -72.9706 q^{38} +149.647 q^{39} -83.3137 q^{40} -336.238 q^{41} -111.941 q^{42} +222.137 q^{43} -106.745 q^{44} -93.7279 q^{45} -23.0051 q^{46} +294.098 q^{47} +48.0000 q^{48} +5.07821 q^{49} -33.0883 q^{50} -258.676 q^{51} +199.529 q^{52} +249.291 q^{53} +54.0000 q^{54} +277.917 q^{55} -149.255 q^{56} -109.456 q^{57} -480.500 q^{58} +56.0955 q^{59} -124.971 q^{60} +591.196 q^{61} -606.715 q^{62} -167.912 q^{63} +64.0000 q^{64} -519.484 q^{65} -160.118 q^{66} +67.0000 q^{67} -344.902 q^{68} -34.5076 q^{69} +388.593 q^{70} -343.637 q^{71} +72.0000 q^{72} +776.872 q^{73} -66.5685 q^{74} -49.6325 q^{75} -145.941 q^{76} +497.882 q^{77} +299.294 q^{78} +146.981 q^{79} -166.627 q^{80} +81.0000 q^{81} -672.475 q^{82} +1293.72 q^{83} -223.882 q^{84} +897.970 q^{85} +444.274 q^{86} -720.749 q^{87} -213.490 q^{88} -153.314 q^{89} -187.456 q^{90} -930.646 q^{91} -46.0101 q^{92} -910.072 q^{93} +588.195 q^{94} +379.966 q^{95} +96.0000 q^{96} -747.955 q^{97} +10.1564 q^{98} -240.177 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 18 q^{5} + 12 q^{6} - 26 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 18 * q^5 + 12 * q^6 - 26 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 18 q^{5} + 12 q^{6} - 26 q^{7} + 16 q^{8} + 18 q^{9} - 36 q^{10} - 76 q^{11} + 24 q^{12} - 36 q^{13} - 52 q^{14} - 54 q^{15} + 32 q^{16} - 48 q^{17} + 36 q^{18} - 56 q^{19} - 72 q^{20} - 78 q^{21} - 152 q^{22} - 122 q^{23} + 48 q^{24} - 84 q^{25} - 72 q^{26} + 54 q^{27} - 104 q^{28} - 192 q^{29} - 108 q^{30} - 58 q^{31} + 64 q^{32} - 228 q^{33} - 96 q^{34} + 250 q^{35} + 72 q^{36} - 10 q^{37} - 112 q^{38} - 108 q^{39} - 144 q^{40} - 466 q^{41} - 156 q^{42} + 218 q^{43} - 304 q^{44} - 162 q^{45} - 244 q^{46} - 68 q^{47} + 96 q^{48} - 284 q^{49} - 168 q^{50} - 144 q^{51} - 144 q^{52} + 162 q^{53} + 108 q^{54} + 652 q^{55} - 208 q^{56} - 168 q^{57} - 384 q^{58} - 66 q^{59} - 216 q^{60} + 1024 q^{61} - 116 q^{62} - 234 q^{63} + 128 q^{64} + 132 q^{65} - 456 q^{66} + 134 q^{67} - 192 q^{68} - 366 q^{69} + 500 q^{70} + 116 q^{71} + 144 q^{72} + 1022 q^{73} - 20 q^{74} - 252 q^{75} - 224 q^{76} + 860 q^{77} - 216 q^{78} + 656 q^{79} - 288 q^{80} + 162 q^{81} - 932 q^{82} - 142 q^{83} - 312 q^{84} + 608 q^{85} + 436 q^{86} - 576 q^{87} - 608 q^{88} - 284 q^{89} - 324 q^{90} - 300 q^{91} - 488 q^{92} - 174 q^{93} - 136 q^{94} + 528 q^{95} + 192 q^{96} - 868 q^{97} - 568 q^{98} - 684 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 18 * q^5 + 12 * q^6 - 26 * q^7 + 16 * q^8 + 18 * q^9 - 36 * q^10 - 76 * q^11 + 24 * q^12 - 36 * q^13 - 52 * q^14 - 54 * q^15 + 32 * q^16 - 48 * q^17 + 36 * q^18 - 56 * q^19 - 72 * q^20 - 78 * q^21 - 152 * q^22 - 122 * q^23 + 48 * q^24 - 84 * q^25 - 72 * q^26 + 54 * q^27 - 104 * q^28 - 192 * q^29 - 108 * q^30 - 58 * q^31 + 64 * q^32 - 228 * q^33 - 96 * q^34 + 250 * q^35 + 72 * q^36 - 10 * q^37 - 112 * q^38 - 108 * q^39 - 144 * q^40 - 466 * q^41 - 156 * q^42 + 218 * q^43 - 304 * q^44 - 162 * q^45 - 244 * q^46 - 68 * q^47 + 96 * q^48 - 284 * q^49 - 168 * q^50 - 144 * q^51 - 144 * q^52 + 162 * q^53 + 108 * q^54 + 652 * q^55 - 208 * q^56 - 168 * q^57 - 384 * q^58 - 66 * q^59 - 216 * q^60 + 1024 * q^61 - 116 * q^62 - 234 * q^63 + 128 * q^64 + 132 * q^65 - 456 * q^66 + 134 * q^67 - 192 * q^68 - 366 * q^69 + 500 * q^70 + 116 * q^71 + 144 * q^72 + 1022 * q^73 - 20 * q^74 - 252 * q^75 - 224 * q^76 + 860 * q^77 - 216 * q^78 + 656 * q^79 - 288 * q^80 + 162 * q^81 - 932 * q^82 - 142 * q^83 - 312 * q^84 + 608 * q^85 + 436 * q^86 - 576 * q^87 - 608 * q^88 - 284 * q^89 - 324 * q^90 - 300 * q^91 - 488 * q^92 - 174 * q^93 - 136 * q^94 + 528 * q^95 + 192 * q^96 - 868 * q^97 - 568 * q^98 - 684 * q^99

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