LMFDB - Embedded newform 58.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [58,4,Mod(1,58)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(58, 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("58.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 58 = 2 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	58.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:	$3.42211078033$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
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			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	58.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +1.44949 q^{3} +4.00000 q^{4} -19.6969 q^{5} -2.89898 q^{6} +11.5959 q^{7} -8.00000 q^{8} -24.8990 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +1.44949 q^{3} +4.00000 q^{4} -19.6969 q^{5} -2.89898 q^{6} +11.5959 q^{7} -8.00000 q^{8} -24.8990 q^{9} +39.3939 q^{10} -37.6515 q^{11} +5.79796 q^{12} -44.5959 q^{13} -23.1918 q^{14} -28.5505 q^{15} +16.0000 q^{16} -61.1918 q^{17} +49.7980 q^{18} +63.7980 q^{19} -78.7878 q^{20} +16.8082 q^{21} +75.3031 q^{22} +177.060 q^{23} -11.5959 q^{24} +262.969 q^{25} +89.1918 q^{26} -75.2270 q^{27} +46.3837 q^{28} +29.0000 q^{29} +57.1010 q^{30} -233.994 q^{31} -32.0000 q^{32} -54.5755 q^{33} +122.384 q^{34} -228.404 q^{35} -99.5959 q^{36} +10.2020 q^{37} -127.596 q^{38} -64.6413 q^{39} +157.576 q^{40} +347.959 q^{41} -33.6163 q^{42} -194.823 q^{43} -150.606 q^{44} +490.434 q^{45} -354.120 q^{46} -14.5005 q^{47} +23.1918 q^{48} -208.535 q^{49} -525.939 q^{50} -88.6969 q^{51} -178.384 q^{52} -606.373 q^{53} +150.454 q^{54} +741.620 q^{55} -92.7673 q^{56} +92.4745 q^{57} -58.0000 q^{58} -702.372 q^{59} -114.202 q^{60} +543.394 q^{61} +467.989 q^{62} -288.727 q^{63} +64.0000 q^{64} +878.403 q^{65} +109.151 q^{66} -407.010 q^{67} -244.767 q^{68} +256.647 q^{69} +456.808 q^{70} +314.717 q^{71} +199.192 q^{72} -859.110 q^{73} -20.4041 q^{74} +381.171 q^{75} +255.192 q^{76} -436.604 q^{77} +129.283 q^{78} +725.266 q^{79} -315.151 q^{80} +563.232 q^{81} -695.918 q^{82} -820.919 q^{83} +67.2327 q^{84} +1205.29 q^{85} +389.646 q^{86} +42.0352 q^{87} +301.212 q^{88} -648.363 q^{89} -980.867 q^{90} -517.131 q^{91} +708.241 q^{92} -339.172 q^{93} +29.0010 q^{94} -1256.62 q^{95} -46.3837 q^{96} -60.9490 q^{97} +417.069 q^{98} +937.485 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} - 16 q^{8} - 40 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 - 16 * q^8 - 40 * q^9	\( 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} - 16 q^{8} - 40 q^{9} + 20 q^{10} - 90 q^{11} - 8 q^{12} - 50 q^{13} + 32 q^{14} - 62 q^{15} + 32 q^{16} - 44 q^{17} + 80 q^{18} + 108 q^{19} - 40 q^{20} + 112 q^{21} + 180 q^{22} - 28 q^{23} + 16 q^{24} + 232 q^{25} + 100 q^{26} + 70 q^{27} - 64 q^{28} + 58 q^{29} + 124 q^{30} + 66 q^{31} - 64 q^{32} + 126 q^{33} + 88 q^{34} - 496 q^{35} - 160 q^{36} + 40 q^{37} - 216 q^{38} - 46 q^{39} + 80 q^{40} + 304 q^{41} - 224 q^{42} - 130 q^{43} - 360 q^{44} + 344 q^{45} + 56 q^{46} - 514 q^{47} - 32 q^{48} + 210 q^{49} - 464 q^{50} - 148 q^{51} - 200 q^{52} - 958 q^{53} - 140 q^{54} + 234 q^{55} + 128 q^{56} - 60 q^{57} - 116 q^{58} - 180 q^{59} - 248 q^{60} + 1028 q^{61} - 132 q^{62} + 128 q^{63} + 128 q^{64} + 826 q^{65} - 252 q^{66} - 912 q^{67} - 176 q^{68} + 964 q^{69} + 992 q^{70} + 796 q^{71} + 320 q^{72} - 856 q^{73} - 80 q^{74} + 488 q^{75} + 432 q^{76} + 1008 q^{77} + 92 q^{78} - 318 q^{79} - 160 q^{80} + 470 q^{81} - 608 q^{82} - 1828 q^{83} + 448 q^{84} + 1372 q^{85} + 260 q^{86} - 58 q^{87} + 720 q^{88} - 944 q^{89} - 688 q^{90} - 368 q^{91} - 112 q^{92} - 1374 q^{93} + 1028 q^{94} - 828 q^{95} + 64 q^{96} + 368 q^{97} - 420 q^{98} + 1728 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 - 16 * q^8 - 40 * q^9 + 20 * q^10 - 90 * q^11 - 8 * q^12 - 50 * q^13 + 32 * q^14 - 62 * q^15 + 32 * q^16 - 44 * q^17 + 80 * q^18 + 108 * q^19 - 40 * q^20 + 112 * q^21 + 180 * q^22 - 28 * q^23 + 16 * q^24 + 232 * q^25 + 100 * q^26 + 70 * q^27 - 64 * q^28 + 58 * q^29 + 124 * q^30 + 66 * q^31 - 64 * q^32 + 126 * q^33 + 88 * q^34 - 496 * q^35 - 160 * q^36 + 40 * q^37 - 216 * q^38 - 46 * q^39 + 80 * q^40 + 304 * q^41 - 224 * q^42 - 130 * q^43 - 360 * q^44 + 344 * q^45 + 56 * q^46 - 514 * q^47 - 32 * q^48 + 210 * q^49 - 464 * q^50 - 148 * q^51 - 200 * q^52 - 958 * q^53 - 140 * q^54 + 234 * q^55 + 128 * q^56 - 60 * q^57 - 116 * q^58 - 180 * q^59 - 248 * q^60 + 1028 * q^61 - 132 * q^62 + 128 * q^63 + 128 * q^64 + 826 * q^65 - 252 * q^66 - 912 * q^67 - 176 * q^68 + 964 * q^69 + 992 * q^70 + 796 * q^71 + 320 * q^72 - 856 * q^73 - 80 * q^74 + 488 * q^75 + 432 * q^76 + 1008 * q^77 + 92 * q^78 - 318 * q^79 - 160 * q^80 + 470 * q^81 - 608 * q^82 - 1828 * q^83 + 448 * q^84 + 1372 * q^85 + 260 * q^86 - 58 * q^87 + 720 * q^88 - 944 * q^89 - 688 * q^90 - 368 * q^91 - 112 * q^92 - 1374 * q^93 + 1028 * q^94 - 828 * q^95 + 64 * q^96 + 368 * q^97 - 420 * q^98 + 1728 * 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
$3$	1.44949	0.278954	0.139477	−	0.990225i	$-0.455458\pi$
$3$	1.44949	0.278954	0.139477	+	0.990225i	$0.455458\pi$
$4$	4.00000	0.500000
$5$	−19.6969	−1.76175	−0.880874	−	0.473351i	$-0.843044\pi$
$5$	−19.6969	−1.76175	−0.880874	+	0.473351i	$0.843044\pi$
$6$	−2.89898	−0.197251
$7$	11.5959	0.626121	0.313060	−	0.949733i	$-0.398646\pi$
$7$	11.5959	0.626121	0.313060	+	0.949733i	$0.398646\pi$
$8$	−8.00000	−0.353553
$9$	−24.8990	−0.922184
$10$	39.3939	1.24574
$11$	−37.6515	−1.03203	−0.516017	−	0.856579i	$-0.672586\pi$
$11$	−37.6515	−1.03203	−0.516017	+	0.856579i	$0.672586\pi$
$12$	5.79796	0.139477
$13$	−44.5959	−0.951437	−0.475719	−	0.879598i	$-0.657812\pi$
$13$	−44.5959	−0.951437	−0.475719	+	0.879598i	$0.657812\pi$
$14$	−23.1918	−0.442734
$15$	−28.5505	−0.491447
$16$	16.0000	0.250000
$17$	−61.1918	−0.873012	−0.436506	−	0.899701i	$-0.643784\pi$
$17$	−61.1918	−0.873012	−0.436506	+	0.899701i	$0.643784\pi$
$18$	49.7980	0.652083
$19$	63.7980	0.770329	0.385165	−	0.922848i	$-0.374145\pi$
$19$	63.7980	0.770329	0.385165	+	0.922848i	$0.374145\pi$
$20$	−78.7878	−0.880874
$21$	16.8082	0.174659
$22$	75.3031	0.729758
$23$	177.060	1.60520	0.802600	−	0.596517i	$-0.203449\pi$
$23$	177.060	1.60520	0.802600	+	0.596517i	$0.203449\pi$
$24$	−11.5959	−0.0986253
$25$	262.969	2.10376
$26$	89.1918	0.672768
$27$	−75.2270	−0.536202
$28$	46.3837	0.313060
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	57.1010	0.347506
$31$	−233.994	−1.35570	−0.677849	−	0.735201i	$-0.737088\pi$
$31$	−233.994	−1.35570	−0.677849	+	0.735201i	$0.737088\pi$
$32$	−32.0000	−0.176777
$33$	−54.5755	−0.287890
$34$	122.384	0.617313
$35$	−228.404	−1.10307
$36$	−99.5959	−0.461092
$37$	10.2020	0.0453299	0.0226649	−	0.999743i	$-0.492785\pi$
$37$	10.2020	0.0453299	0.0226649	+	0.999743i	$0.492785\pi$
$38$	−127.596	−0.544705
$39$	−64.6413	−0.265408
$40$	157.576	0.622872
$41$	347.959	1.32542	0.662708	−	0.748877i	$-0.269407\pi$
$41$	347.959	1.32542	0.662708	+	0.748877i	$0.269407\pi$
$42$	−33.6163	−0.123503
$43$	−194.823	−0.690935	−0.345468	−	0.938431i	$-0.612280\pi$
$43$	−194.823	−0.690935	−0.345468	+	0.938431i	$0.612280\pi$
$44$	−150.606	−0.516017
$45$	490.434	1.62466
$46$	−354.120	−1.13505
$47$	−14.5005	−0.0450025	−0.0225013	−	0.999747i	$-0.507163\pi$
$47$	−14.5005	−0.0450025	−0.0225013	+	0.999747i	$0.507163\pi$
$48$	23.1918	0.0697386
$49$	−208.535	−0.607973
$50$	−525.939	−1.48758
$51$	−88.6969	−0.243531
$52$	−178.384	−0.475719
$53$	−606.373	−1.57154	−0.785772	−	0.618517i	$-0.787734\pi$
$53$	−606.373	−1.57154	−0.785772	+	0.618517i	$0.787734\pi$
$54$	150.454	0.379152
$55$	741.620	1.81818
$56$	−92.7673	−0.221367
$57$	92.4745	0.214887
$58$	−58.0000	−0.131306
$59$	−702.372	−1.54985	−0.774925	−	0.632054i	$-0.782212\pi$
$59$	−702.372	−1.54985	−0.774925	+	0.632054i	$0.782212\pi$
$60$	−114.202	−0.245724
$61$	543.394	1.14056	0.570282	−	0.821449i	$-0.306834\pi$
$61$	543.394	1.14056	0.570282	+	0.821449i	$0.306834\pi$
$62$	467.989	0.958623
$63$	−288.727	−0.577399
$64$	64.0000	0.125000
$65$	878.403	1.67619
$66$	109.151	0.203569
$67$	−407.010	−0.742152	−0.371076	−	0.928602i	$-0.621011\pi$
$67$	−407.010	−0.742152	−0.371076	+	0.928602i	$0.621011\pi$
$68$	−244.767	−0.436506
$69$	256.647	0.447778
$70$	456.808	0.779986
$71$	314.717	0.526057	0.263029	−	0.964788i	$-0.415279\pi$
$71$	314.717	0.526057	0.263029	+	0.964788i	$0.415279\pi$
$72$	199.192	0.326041
$73$	−859.110	−1.37741	−0.688707	−	0.725040i	$-0.741821\pi$
$73$	−859.110	−1.37741	−0.688707	+	0.725040i	$0.741821\pi$
$74$	−20.4041	−0.0320531
$75$	381.171	0.586852
$76$	255.192	0.385165
$77$	−436.604	−0.646177
$78$	129.283	0.187672
$79$	725.266	1.03290	0.516448	−	0.856319i	$-0.327254\pi$
$79$	725.266	1.03290	0.516448	+	0.856319i	$0.327254\pi$
$80$	−315.151	−0.440437
$81$	563.232	0.772609
$82$	−695.918	−0.937211
$83$	−820.919	−1.08563	−0.542817	−	0.839851i	$-0.682642\pi$
$83$	−820.919	−1.08563	−0.542817	+	0.839851i	$0.682642\pi$
$84$	67.2327	0.0873296
$85$	1205.29	1.53803
$86$	389.646	0.488565
$87$	42.0352	0.0518005
$88$	301.212	0.364879
$89$	−648.363	−0.772206	−0.386103	−	0.922456i	$-0.626179\pi$
$89$	−648.363	−0.772206	−0.386103	+	0.922456i	$0.626179\pi$
$90$	−980.867	−1.14881
$91$	−517.131	−0.595714
$92$	708.241	0.802600
$93$	−339.172	−0.378178
$94$	29.0010	0.0318216
$95$	−1256.62	−1.35713
$96$	−46.3837	−0.0493126
$97$	−60.9490	−0.0637983	−0.0318991	−	0.999491i	$-0.510156\pi$
$97$	−60.9490	−0.0637983	−0.0318991	+	0.999491i	$0.510156\pi$
$98$	417.069	0.429902
$99$	937.485	0.951725
$100$	1051.88	1.05188
$101$	1100.16	1.08386	0.541930	−	0.840423i	$-0.317694\pi$
$101$	1100.16	1.08386	0.541930	+	0.840423i	$0.317694\pi$
$102$	177.394	0.172202
$103$	747.930	0.715492	0.357746	−	0.933819i	$-0.383545\pi$
$103$	747.930	0.715492	0.357746	+	0.933819i	$0.383545\pi$
$104$	356.767	0.336384
$105$	−331.069	−0.307705
$106$	1212.75	1.11125
$107$	176.059	0.159068	0.0795340	−	0.996832i	$-0.474657\pi$
$107$	176.059	0.159068	0.0795340	+	0.996832i	$0.474657\pi$
$108$	−300.908	−0.268101
$109$	173.856	0.152774	0.0763871	−	0.997078i	$-0.475662\pi$
$109$	173.856	0.152774	0.0763871	+	0.997078i	$0.475662\pi$
$110$	−1483.24	−1.28565
$111$	14.7878	0.0126450
$112$	185.535	0.156530
$113$	−1707.65	−1.42161	−0.710807	−	0.703387i	$-0.751670\pi$
$113$	−1707.65	−1.42161	−0.710807	+	0.703387i	$0.751670\pi$
$114$	−184.949	−0.151948
$115$	−3487.54	−2.82796
$116$	116.000	0.0928477
$117$	1110.39	0.877400
$118$	1404.74	1.09591
$119$	−709.576	−0.546611
$120$	228.404	0.173753
$121$	86.6378	0.0650922
$122$	−1086.79	−0.806501
$123$	504.363	0.369731
$124$	−935.978	−0.677849
$125$	−2717.57	−1.94454
$126$	577.453	0.408283
$127$	−881.473	−0.615891	−0.307945	−	0.951404i	$-0.599641\pi$
$127$	−881.473	−0.615891	−0.307945	+	0.951404i	$0.599641\pi$
$128$	−128.000	−0.0883883
$129$	−282.394	−0.192739
$130$	−1756.81	−1.18525
$131$	2025.49	1.35090	0.675451	−	0.737405i	$-0.263949\pi$
$131$	2025.49	1.35090	0.675451	+	0.737405i	$0.263949\pi$
$132$	−218.302	−0.143945
$133$	739.796	0.482319
$134$	814.020	0.524781
$135$	1481.74	0.944652
$136$	489.535	0.308656
$137$	1594.32	0.994248	0.497124	−	0.867679i	$-0.334389\pi$
$137$	1594.32	0.994248	0.497124	+	0.867679i	$0.334389\pi$
$138$	−513.294	−0.316627
$139$	−2855.40	−1.74239	−0.871195	−	0.490938i	$-0.836654\pi$
$139$	−2855.40	−1.74239	−0.871195	+	0.490938i	$0.836654\pi$
$140$	−913.616	−0.551533
$141$	−21.0183	−0.0125536
$142$	−629.435	−0.371979
$143$	1679.10	0.981915
$144$	−398.384	−0.230546
$145$	−571.211	−0.327148
$146$	1718.22	0.973979
$147$	−302.269	−0.169597
$148$	40.8082	0.0226649
$149$	−18.3755	−0.0101032	−0.00505162	−	0.999987i	$-0.501608\pi$
$149$	−18.3755	−0.0101032	−0.00505162	+	0.999987i	$0.501608\pi$
$150$	−762.343	−0.414967
$151$	−778.806	−0.419724	−0.209862	−	0.977731i	$-0.567301\pi$
$151$	−778.806	−0.419724	−0.209862	+	0.977731i	$0.567301\pi$
$152$	−510.384	−0.272353
$153$	1523.61	0.805078
$154$	873.208	0.456916
$155$	4608.97	2.38840
$156$	−258.565	−0.132704
$157$	−511.896	−0.260215	−0.130107	−	0.991500i	$-0.541532\pi$
$157$	−511.896	−0.260215	−0.130107	+	0.991500i	$0.541532\pi$
$158$	−1450.53	−0.730368
$159$	−878.932	−0.438389
$160$	630.302	0.311436
$161$	2053.18	1.00505
$162$	−1126.46	−0.546317
$163$	−1451.59	−0.697529	−0.348764	−	0.937210i	$-0.613399\pi$
$163$	−1451.59	−0.697529	−0.348764	+	0.937210i	$0.613399\pi$
$164$	1391.84	0.662708
$165$	1074.97	0.507190
$166$	1641.84	0.767659
$167$	600.374	0.278194	0.139097	−	0.990279i	$-0.455580\pi$
$167$	600.374	0.278194	0.139097	+	0.990279i	$0.455580\pi$
$168$	−134.465	−0.0617513
$169$	−208.204	−0.0947675
$170$	−2410.58	−1.08755
$171$	−1588.50	−0.710386
$172$	−779.292	−0.345468
$173$	−2574.20	−1.13129	−0.565644	−	0.824650i	$-0.691372\pi$
$173$	−2574.20	−1.13129	−0.565644	+	0.824650i	$0.691372\pi$
$174$	−84.0704	−0.0366285
$175$	3049.37	1.31720
$176$	−602.424	−0.258008
$177$	−1018.08	−0.432337
$178$	1296.73	0.546032
$179$	3828.72	1.59872	0.799362	−	0.600850i	$-0.205171\pi$
$179$	3828.72	1.59872	0.799362	+	0.600850i	$0.205171\pi$
$180$	1961.73	0.812328
$181$	2075.64	0.852381	0.426190	−	0.904633i	$-0.359855\pi$
$181$	2075.64	0.852381	0.426190	+	0.904633i	$0.359855\pi$
$182$	1034.26	0.421234
$183$	787.644	0.318166
$184$	−1416.48	−0.567524
$185$	−200.949	−0.0798598
$186$	678.345	0.267412
$187$	2303.97	0.900977
$188$	−58.0021	−0.0225013
$189$	−872.327	−0.335727
$190$	2513.25	0.959633
$191$	4070.08	1.54189	0.770944	−	0.636902i	$-0.219785\pi$
$191$	4070.08	1.54189	0.770944	+	0.636902i	$0.219785\pi$
$192$	92.7673	0.0348693
$193$	2373.24	0.885129	0.442565	−	0.896737i	$-0.354069\pi$
$193$	2373.24	0.885129	0.442565	+	0.896737i	$0.354069\pi$
$194$	121.898	0.0451122
$195$	1273.24	0.467581
$196$	−834.139	−0.303986
$197$	−3108.58	−1.12425	−0.562125	−	0.827052i	$-0.690016\pi$
$197$	−3108.58	−1.12425	−0.562125	+	0.827052i	$0.690016\pi$
$198$	−1874.97	−0.672971
$199$	−4048.30	−1.44209	−0.721046	−	0.692887i	$-0.756338\pi$
$199$	−4048.30	−1.44209	−0.721046	+	0.692887i	$0.756338\pi$
$200$	−2103.76	−0.743790
$201$	−589.957	−0.207027
$202$	−2200.32	−0.766405
$203$	336.282	0.116268
$204$	−354.788	−0.121765
$205$	−6853.73	−2.33505
$206$	−1495.86	−0.505929
$207$	−4408.62	−1.48029
$208$	−713.535	−0.237859
$209$	−2402.09	−0.795005
$210$	662.139	0.217581
$211$	3591.66	1.17185	0.585924	−	0.810366i	$-0.300732\pi$
$211$	3591.66	1.17185	0.585924	+	0.810366i	$0.300732\pi$
$212$	−2425.49	−0.785772
$213$	456.180	0.146746
$214$	−352.118	−0.112478
$215$	3837.42	1.21725
$216$	601.816	0.189576
$217$	−2713.38	−0.848830
$218$	−347.712	−0.108028
$219$	−1245.27	−0.384236
$220$	2966.48	0.909091
$221$	2728.91	0.830616
$222$	−29.5755	−0.00894134
$223$	−772.085	−0.231850	−0.115925	−	0.993258i	$-0.536983\pi$
$223$	−772.085	−0.231850	−0.115925	+	0.993258i	$0.536983\pi$
$224$	−371.069	−0.110684
$225$	−6547.67	−1.94005
$226$	3415.31	1.00523
$227$	−4435.16	−1.29679	−0.648397	−	0.761303i	$-0.724560\pi$
$227$	−4435.16	−1.29679	−0.648397	+	0.761303i	$0.724560\pi$
$228$	369.898	0.107443
$229$	2213.12	0.638634	0.319317	−	0.947648i	$-0.396546\pi$
$229$	2213.12	0.638634	0.319317	+	0.947648i	$0.396546\pi$
$230$	6975.09	1.99967
$231$	−632.853	−0.180254
$232$	−232.000	−0.0656532
$233$	−2980.41	−0.837998	−0.418999	−	0.907987i	$-0.637619\pi$
$233$	−2980.41	−0.837998	−0.418999	+	0.907987i	$0.637619\pi$
$234$	−2220.79	−0.620416
$235$	285.616	0.0792831
$236$	−2809.49	−0.774925
$237$	1051.27	0.288131
$238$	1419.15	0.386512
$239$	−557.093	−0.150775	−0.0753877	−	0.997154i	$-0.524019\pi$
$239$	−557.093	−0.150775	−0.0753877	+	0.997154i	$0.524019\pi$
$240$	−456.808	−0.122862
$241$	−4168.65	−1.11422	−0.557109	−	0.830439i	$-0.688089\pi$
$241$	−4168.65	−1.11422	−0.557109	+	0.830439i	$0.688089\pi$
$242$	−173.276	−0.0460272
$243$	2847.53	0.751724
$244$	2173.58	0.570282
$245$	4107.49	1.07109
$246$	−1008.73	−0.261439
$247$	−2845.13	−0.732920
$248$	1871.96	0.479312
$249$	−1189.91	−0.302842
$250$	5435.15	1.37500
$251$	−7132.56	−1.79364	−0.896819	−	0.442398i	$-0.854128\pi$
$251$	−7132.56	−1.79364	−0.896819	+	0.442398i	$0.854128\pi$
$252$	−1154.91	−0.288699
$253$	−6666.59	−1.65662
$254$	1762.95	0.435501
$255$	1747.06	0.429039
$256$	256.000	0.0625000
$257$	5564.76	1.35066	0.675331	−	0.737514i	$-0.264001\pi$
$257$	5564.76	1.35066	0.675331	+	0.737514i	$0.264001\pi$
$258$	564.788	0.136287
$259$	118.302	0.0283820
$260$	3513.61	0.838096
$261$	−722.070	−0.171245
$262$	−4050.98	−0.955231
$263$	7188.72	1.68546	0.842729	−	0.538338i	$-0.180947\pi$
$263$	7188.72	1.68546	0.842729	+	0.538338i	$0.180947\pi$
$264$	436.604	0.101785
$265$	11943.7	2.76866
$266$	−1479.59	−0.341051
$267$	−939.796	−0.215410
$268$	−1628.04	−0.371076
$269$	3184.38	0.721765	0.360883	−	0.932611i	$-0.382476\pi$
$269$	3184.38	0.721765	0.360883	+	0.932611i	$0.382476\pi$
$270$	−2963.48	−0.667970
$271$	1732.34	0.388311	0.194155	−	0.980971i	$-0.437803\pi$
$271$	1732.34	0.388311	0.194155	+	0.980971i	$0.437803\pi$
$272$	−979.069	−0.218253
$273$	−749.576	−0.166177
$274$	−3188.64	−0.703040
$275$	−9901.20	−2.17114
$276$	1026.59	0.223889
$277$	6061.31	1.31476	0.657381	−	0.753558i	$-0.271664\pi$
$277$	6061.31	1.31476	0.657381	+	0.753558i	$0.271664\pi$
$278$	5710.81	1.23206
$279$	5826.22	1.25020
$280$	1827.23	0.389993
$281$	6183.75	1.31278	0.656391	−	0.754421i	$-0.272082\pi$
$281$	6183.75	1.31278	0.656391	+	0.754421i	$0.272082\pi$
$282$	42.0367	0.00887677
$283$	−9076.75	−1.90656	−0.953281	−	0.302086i	$-0.902317\pi$
$283$	−9076.75	−1.90656	−0.953281	+	0.302086i	$0.902317\pi$
$284$	1258.87	0.263029
$285$	−1821.46	−0.378576
$286$	−3358.21	−0.694318
$287$	4034.91	0.829871
$288$	796.767	0.163021
$289$	−1168.56	−0.237850
$290$	1142.42	0.231329
$291$	−88.3449	−0.0177968
$292$	−3436.44	−0.688707
$293$	−3636.66	−0.725105	−0.362553	−	0.931963i	$-0.618095\pi$
$293$	−3636.66	−0.725105	−0.362553	+	0.931963i	$0.618095\pi$
$294$	604.538	0.119923
$295$	13834.6	2.73044
$296$	−81.6163	−0.0160265
$297$	2832.41	0.553378
$298$	36.7511	0.00714407
$299$	−7896.16	−1.52725
$300$	1524.69	0.293426
$301$	−2259.15	−0.432609
$302$	1557.61	0.296790
$303$	1594.67	0.302348
$304$	1020.77	0.192582
$305$	−10703.2	−2.00939
$306$	−3047.23	−0.569276
$307$	−4619.06	−0.858708	−0.429354	−	0.903136i	$-0.641259\pi$
$307$	−4619.06	−0.858708	−0.429354	+	0.903136i	$0.641259\pi$
$308$	−1746.42	−0.323089
$309$	1084.12	0.199590
$310$	−9217.95	−1.68885
$311$	8094.07	1.47580	0.737898	−	0.674912i	$-0.235818\pi$
$311$	8094.07	1.47580	0.737898	+	0.674912i	$0.235818\pi$
$312$	517.131	0.0938358
$313$	6901.79	1.24637	0.623183	−	0.782076i	$-0.285839\pi$
$313$	6901.79	1.24637	0.623183	+	0.782076i	$0.285839\pi$
$314$	1023.79	0.184000
$315$	5687.03	1.01723
$316$	2901.06	0.516448
$317$	−6119.75	−1.08429	−0.542144	−	0.840286i	$-0.682387\pi$
$317$	−6119.75	−1.08429	−0.542144	+	0.840286i	$0.682387\pi$
$318$	1757.86	0.309988
$319$	−1091.89	−0.191644
$320$	−1260.60	−0.220218
$321$	255.196	0.0443727
$322$	−4106.35	−0.710677
$323$	−3903.91	−0.672506
$324$	2252.93	0.386304
$325$	−11727.4	−2.00159
$326$	2903.18	0.493227
$327$	252.003	0.0426171
$328$	−2783.67	−0.468606
$329$	−168.147	−0.0281770
$330$	−2149.94	−0.358637
$331$	198.924	0.0330328	0.0165164	−	0.999864i	$-0.494742\pi$
$331$	198.924	0.0330328	0.0165164	+	0.999864i	$0.494742\pi$
$332$	−3283.68	−0.542817
$333$	−254.020	−0.0418025
$334$	−1200.75	−0.196713
$335$	8016.85	1.30749
$336$	268.931	0.0436648
$337$	2102.43	0.339841	0.169921	−	0.985458i	$-0.445649\pi$
$337$	2102.43	0.339841	0.169921	+	0.985458i	$0.445649\pi$
$338$	416.408	0.0670107
$339$	−2475.23	−0.396566
$340$	4821.17	0.769013
$341$	8810.25	1.39912
$342$	3177.01	0.502318
$343$	−6395.55	−1.00679
$344$	1558.58	0.244283
$345$	−5055.16	−0.788871
$346$	5148.40	0.799941
$347$	6543.83	1.01237	0.506183	−	0.862426i	$-0.331056\pi$
$347$	6543.83	1.01237	0.506183	+	0.862426i	$0.331056\pi$
$348$	168.141	0.0259003
$349$	793.427	0.121694	0.0608469	−	0.998147i	$-0.480620\pi$
$349$	793.427	0.121694	0.0608469	+	0.998147i	$0.480620\pi$
$350$	−6098.74	−0.931404
$351$	3354.82	0.510162
$352$	1204.85	0.182439
$353$	7378.84	1.11257	0.556284	−	0.830992i	$-0.312227\pi$
$353$	7378.84	1.11257	0.556284	+	0.830992i	$0.312227\pi$
$354$	2036.16	0.305709
$355$	−6198.97	−0.926780
$356$	−2593.45	−0.386103
$357$	−1028.52	−0.152479
$358$	−7657.43	−1.13047
$359$	−7142.97	−1.05012	−0.525058	−	0.851066i	$-0.675956\pi$
$359$	−7142.97	−1.05012	−0.525058	+	0.851066i	$0.675956\pi$
$360$	−3923.47	−0.574403
$361$	−2788.82	−0.406593
$362$	−4151.28	−0.602724
$363$	125.581	0.0181578
$364$	−2068.52	−0.297857
$365$	16921.8	2.42666
$366$	−1575.29	−0.224977
$367$	−1806.47	−0.256940	−0.128470	−	0.991713i	$-0.541007\pi$
$367$	−1806.47	−0.256940	−0.128470	+	0.991713i	$0.541007\pi$
$368$	2832.96	0.401300
$369$	−8663.83	−1.22228
$370$	401.898	0.0564694
$371$	−7031.46	−0.983976
$372$	−1356.69	−0.189089
$373$	9396.05	1.30431	0.652157	−	0.758084i	$-0.273864\pi$
$373$	9396.05	1.30431	0.652157	+	0.758084i	$0.273864\pi$
$374$	−4607.93	−0.637087
$375$	−3939.10	−0.542437
$376$	116.004	0.0159108
$377$	−1293.28	−0.176677
$378$	1744.65	0.237395
$379$	−421.482	−0.0571242	−0.0285621	−	0.999592i	$-0.509093\pi$
$379$	−421.482	−0.0571242	−0.0285621	+	0.999592i	$0.509093\pi$
$380$	−5026.50	−0.678563
$381$	−1277.69	−0.171805
$382$	−8140.16	−1.09028
$383$	3189.84	0.425570	0.212785	−	0.977099i	$-0.431747\pi$
$383$	3189.84	0.425570	0.212785	+	0.977099i	$0.431747\pi$
$384$	−185.535	−0.0246563
$385$	8599.76	1.13840
$386$	−4746.49	−0.625881
$387$	4850.89	0.637170
$388$	−243.796	−0.0318991
$389$	−4110.96	−0.535820	−0.267910	−	0.963444i	$-0.586333\pi$
$389$	−4110.96	−0.535820	−0.267910	+	0.963444i	$0.586333\pi$
$390$	−2546.47	−0.330630
$391$	−10834.6	−1.40136
$392$	1668.28	0.214951
$393$	2935.93	0.376840
$394$	6217.17	0.794965
$395$	−14285.5	−1.81970
$396$	3749.94	0.475862
$397$	−827.505	−0.104613	−0.0523064	−	0.998631i	$-0.516657\pi$
$397$	−827.505	−0.104613	−0.0523064	+	0.998631i	$0.516657\pi$
$398$	8096.60	1.01971
$399$	1072.33	0.134545
$400$	4207.51	0.525939
$401$	675.145	0.0840776	0.0420388	−	0.999116i	$-0.486615\pi$
$401$	675.145	0.0840776	0.0420388	+	0.999116i	$0.486615\pi$
$402$	1179.91	0.146390
$403$	10435.2	1.28986
$404$	4400.64	0.541930
$405$	−11093.9	−1.36114
$406$	−672.563	−0.0822137
$407$	−384.122	−0.0467819
$408$	709.576	0.0861010
$409$	764.847	0.0924676	0.0462338	−	0.998931i	$-0.485278\pi$
$409$	764.847	0.0924676	0.0462338	+	0.998931i	$0.485278\pi$
$410$	13707.5	1.65113
$411$	2310.95	0.277350
$412$	2991.72	0.357746
$413$	−8144.65	−0.970393
$414$	8817.24	1.04672
$415$	16169.6	1.91261
$416$	1427.07	0.168192
$417$	−4138.88	−0.486047
$418$	4804.18	0.562154
$419$	−2573.33	−0.300037	−0.150019	−	0.988683i	$-0.547933\pi$
$419$	−2573.33	−0.300037	−0.150019	+	0.988683i	$0.547933\pi$
$420$	−1324.28	−0.153853
$421$	14287.4	1.65398	0.826992	−	0.562213i	$-0.190050\pi$
$421$	14287.4	1.65398	0.826992	+	0.562213i	$0.190050\pi$
$422$	−7183.32	−0.828622
$423$	361.048	0.0415006
$424$	4850.99	0.555625
$425$	−16091.6	−1.83660
$426$	−912.359	−0.103765
$427$	6301.15	0.714131
$428$	704.237	0.0795340
$429$	2433.84	0.273909
$430$	−7674.83	−0.860728
$431$	−11283.2	−1.26100	−0.630500	−	0.776189i	$-0.717150\pi$
$431$	−11283.2	−1.26100	−0.630500	+	0.776189i	$0.717150\pi$
$432$	−1203.63	−0.134050
$433$	8069.64	0.895617	0.447809	−	0.894129i	$-0.352205\pi$
$433$	8069.64	0.895617	0.447809	+	0.894129i	$0.352205\pi$
$434$	5426.76	0.600214
$435$	−827.965	−0.0912595
$436$	695.424	0.0763871
$437$	11296.1	1.23653
$438$	2490.54	0.271696
$439$	−7819.42	−0.850116	−0.425058	−	0.905166i	$-0.639746\pi$
$439$	−7819.42	−0.850116	−0.425058	+	0.905166i	$0.639746\pi$
$440$	−5932.96	−0.642824
$441$	5192.30	0.560663
$442$	−5457.81	−0.587334
$443$	−10153.4	−1.08894	−0.544472	−	0.838779i	$-0.683270\pi$
$443$	−10153.4	−1.08894	−0.544472	+	0.838779i	$0.683270\pi$
$444$	59.1510	0.00632248
$445$	12770.8	1.36043
$446$	1544.17	0.163943
$447$	−26.6351	−0.00281834
$448$	742.139	0.0782651
$449$	13608.3	1.43033	0.715163	−	0.698958i	$-0.246353\pi$
$449$	13608.3	1.43033	0.715163	+	0.698958i	$0.246353\pi$
$450$	13095.3	1.37182
$451$	−13101.2	−1.36787
$452$	−6830.61	−0.710807
$453$	−1128.87	−0.117084
$454$	8870.32	0.916971
$455$	10185.9	1.04950
$456$	−739.796	−0.0759739
$457$	−1882.55	−0.192696	−0.0963479	−	0.995348i	$-0.530716\pi$
$457$	−1882.55	−0.192696	−0.0963479	+	0.995348i	$0.530716\pi$
$458$	−4426.24	−0.451583
$459$	4603.28	0.468111
$460$	−13950.2	−1.41398
$461$	−13947.2	−1.40908	−0.704542	−	0.709662i	$-0.748847\pi$
$461$	−13947.2	−1.40908	−0.704542	+	0.709662i	$0.748847\pi$
$462$	1265.71	0.127459
$463$	−12454.8	−1.25016	−0.625079	−	0.780562i	$-0.714933\pi$
$463$	−12454.8	−1.25016	−0.625079	+	0.780562i	$0.714933\pi$
$464$	464.000	0.0464238
$465$	6680.66	0.666254
$466$	5960.83	0.592554
$467$	−6561.59	−0.650180	−0.325090	−	0.945683i	$-0.605395\pi$
$467$	−6561.59	−0.650180	−0.325090	+	0.945683i	$0.605395\pi$
$468$	4441.57	0.438700
$469$	−4719.66	−0.464677
$470$	−571.232	−0.0560616
$471$	−741.988	−0.0725881
$472$	5618.98	0.547954
$473$	7335.38	0.713068
$474$	−2102.53	−0.203739
$475$	16776.9	1.62058
$476$	−2838.30	−0.273305
$477$	15098.1	1.44925
$478$	1114.19	0.106614
$479$	−17407.5	−1.66048	−0.830240	−	0.557406i	$-0.811797\pi$
$479$	−17407.5	−1.66048	−0.830240	+	0.557406i	$0.811797\pi$
$480$	913.616	0.0868764
$481$	−454.969	−0.0431285
$482$	8337.31	0.787871
$483$	2976.06	0.280363
$484$	346.551	0.0325461
$485$	1200.51	0.112396
$486$	−5695.06	−0.531549
$487$	−2544.79	−0.236787	−0.118393	−	0.992967i	$-0.537774\pi$
$487$	−2544.79	−0.236787	−0.118393	+	0.992967i	$0.537774\pi$
$488$	−4347.15	−0.403251
$489$	−2104.06	−0.194579
$490$	−8214.99	−0.757378
$491$	2712.89	0.249351	0.124675	−	0.992198i	$-0.460211\pi$
$491$	2712.89	0.249351	0.124675	+	0.992198i	$0.460211\pi$
$492$	2017.45	0.184865
$493$	−1774.56	−0.162114
$494$	5690.26	0.518253
$495$	−18465.6	−1.67670
$496$	−3743.91	−0.338924
$497$	3649.44	0.329375
$498$	2379.83	0.214142
$499$	4583.79	0.411220	0.205610	−	0.978634i	$-0.434082\pi$
$499$	4583.79	0.411220	0.205610	+	0.978634i	$0.434082\pi$
$500$	−10870.3	−0.972269
$501$	870.237	0.0776034
$502$	14265.1	1.26829
$503$	−1950.19	−0.172872	−0.0864359	−	0.996257i	$-0.527548\pi$
$503$	−1950.19	−0.172872	−0.0864359	+	0.996257i	$0.527548\pi$
$504$	2309.81	0.204141
$505$	−21669.8	−1.90949
$506$	13333.2	1.17141
$507$	−301.790	−0.0264358
$508$	−3525.89	−0.307945
$509$	−8853.80	−0.770997	−0.385499	−	0.922708i	$-0.625971\pi$
$509$	−8853.80	−0.770997	−0.385499	+	0.922708i	$0.625971\pi$
$510$	−3494.12	−0.303377
$511$	−9962.17	−0.862428
$512$	−512.000	−0.0441942
$513$	−4799.33	−0.413052
$514$	−11129.5	−0.955063
$515$	−14731.9	−1.26052
$516$	−1129.58	−0.0963697
$517$	545.967	0.0464441
$518$	−236.604	−0.0200691
$519$	−3731.28	−0.315578
$520$	−7027.22	−0.592623
$521$	−19939.5	−1.67671	−0.838355	−	0.545125i	$-0.816482\pi$
$521$	−19939.5	−1.67671	−0.838355	+	0.545125i	$0.816482\pi$
$522$	1444.14	0.121089
$523$	−12185.8	−1.01883	−0.509416	−	0.860520i	$-0.670138\pi$
$523$	−12185.8	−1.01883	−0.509416	+	0.860520i	$0.670138\pi$
$524$	8101.97	0.675451
$525$	4420.03	0.367440
$526$	−14377.4	−1.19180
$527$	14318.5	1.18354
$528$	−873.208	−0.0719725
$529$	19183.3	1.57667
$530$	−23887.4	−1.95774
$531$	17488.4	1.42925
$532$	2959.18	0.241160
$533$	−15517.6	−1.26105
$534$	1879.59	0.152318
$535$	−3467.83	−0.280238
$536$	3256.08	0.262390
$537$	5549.68	0.445971
$538$	−6368.75	−0.510365
$539$	7851.65	0.627448
$540$	5926.97	0.472326
$541$	700.188	0.0556440	0.0278220	−	0.999613i	$-0.491143\pi$
$541$	700.188	0.0556440	0.0278220	+	0.999613i	$0.491143\pi$
$542$	−3464.68	−0.274577
$543$	3008.62	0.237775
$544$	1958.14	0.154328
$545$	−3424.43	−0.269150
$546$	1499.15	0.117505
$547$	10664.2	0.833576	0.416788	−	0.909004i	$-0.363156\pi$
$547$	10664.2	0.833576	0.416788	+	0.909004i	$0.363156\pi$
$548$	6377.28	0.497124
$549$	−13530.0	−1.05181
$550$	19802.4	1.53523
$551$	1850.14	0.143047
$552$	−2053.18	−0.158313
$553$	8410.12	0.646718
$554$	−12122.6	−0.929677
$555$	−291.273	−0.0222772
$556$	−11421.6	−0.871195
$557$	8410.60	0.639800	0.319900	−	0.947451i	$-0.396351\pi$
$557$	8410.60	0.639800	0.319900	+	0.947451i	$0.396351\pi$
$558$	−11652.4	−0.884027
$559$	8688.31	0.657382
$560$	−3654.47	−0.275767
$561$	3339.58	0.251332
$562$	−12367.5	−0.928277
$563$	8770.15	0.656515	0.328257	−	0.944588i	$-0.393539\pi$
$563$	8770.15	0.656515	0.328257	+	0.944588i	$0.393539\pi$
$564$	−84.0734	−0.00627682
$565$	33635.5	2.50453
$566$	18153.5	1.34814
$567$	6531.19	0.483746
$568$	−2517.74	−0.185989
$569$	−10493.4	−0.773125	−0.386562	−	0.922263i	$-0.626338\pi$
$569$	−10493.4	−0.773125	−0.386562	+	0.922263i	$0.626338\pi$
$570$	3642.93	0.267694
$571$	−2517.50	−0.184508	−0.0922542	−	0.995735i	$-0.529407\pi$
$571$	−2517.50	−0.184508	−0.0922542	+	0.995735i	$0.529407\pi$
$572$	6716.42	0.490957
$573$	5899.54	0.430117
$574$	−8069.81	−0.586807
$575$	46561.4	3.37695
$576$	−1593.53	−0.115273
$577$	1394.88	0.100641	0.0503203	−	0.998733i	$-0.483976\pi$
$577$	1394.88	0.100641	0.0503203	+	0.998733i	$0.483976\pi$
$578$	2337.12	0.168186
$579$	3439.99	0.246911
$580$	−2284.84	−0.163574
$581$	−9519.31	−0.679738
$582$	176.690	0.0125842
$583$	22830.9	1.62188
$584$	6872.88	0.486989
$585$	−21871.3	−1.54576
$586$	7273.31	0.512727
$587$	24391.0	1.71504	0.857518	−	0.514455i	$-0.172006\pi$
$587$	24391.0	1.71504	0.857518	+	0.514455i	$0.172006\pi$
$588$	−1209.08	−0.0847984
$589$	−14928.4	−1.04433
$590$	−27669.2	−1.93071
$591$	−4505.86	−0.313615
$592$	163.233	0.0113325
$593$	−16089.4	−1.11418	−0.557092	−	0.830451i	$-0.688083\pi$
$593$	−16089.4	−1.11418	−0.557092	+	0.830451i	$0.688083\pi$
$594$	−5664.83	−0.391297
$595$	13976.5	0.962990
$596$	−73.5021	−0.00505162
$597$	−5867.97	−0.402278
$598$	15792.3	1.07993
$599$	−19228.2	−1.31159	−0.655796	−	0.754938i	$-0.727667\pi$
$599$	−19228.2	−1.31159	−0.655796	+	0.754938i	$0.727667\pi$
$600$	−3049.37	−0.207483
$601$	−2345.94	−0.159223	−0.0796115	−	0.996826i	$-0.525368\pi$
$601$	−2345.94	−0.159223	−0.0796115	+	0.996826i	$0.525368\pi$
$602$	4518.30	0.305901
$603$	10134.1	0.684401
$604$	−3115.22	−0.209862
$605$	−1706.50	−0.114676
$606$	−3189.34	−0.213792
$607$	−4621.11	−0.309004	−0.154502	−	0.987993i	$-0.549377\pi$
$607$	−4621.11	−0.309004	−0.154502	+	0.987993i	$0.549377\pi$
$608$	−2041.53	−0.136176
$609$	487.437	0.0324334
$610$	21406.4	1.42085
$611$	646.664	0.0428170
$612$	6094.46	0.402539
$613$	−10464.8	−0.689511	−0.344755	−	0.938693i	$-0.612038\pi$
$613$	−10464.8	−0.689511	−0.344755	+	0.938693i	$0.612038\pi$
$614$	9238.11	0.607198
$615$	−9934.41	−0.651373
$616$	3492.83	0.228458
$617$	−11873.3	−0.774718	−0.387359	−	0.921929i	$-0.626613\pi$
$617$	−11873.3	−0.774718	−0.387359	+	0.921929i	$0.626613\pi$
$618$	−2168.23	−0.141131
$619$	13102.2	0.850762	0.425381	−	0.905014i	$-0.360140\pi$
$619$	13102.2	0.850762	0.425381	+	0.905014i	$0.360140\pi$
$620$	18435.9	1.19420
$621$	−13319.7	−0.860711
$622$	−16188.1	−1.04355
$623$	−7518.37	−0.483494
$624$	−1034.26	−0.0663519
$625$	20656.7	1.32203
$626$	−13803.6	−0.881313
$627$	−3481.81	−0.221770
$628$	−2047.58	−0.130107
$629$	−624.282	−0.0395735
$630$	−11374.1	−0.719291
$631$	12279.7	0.774719	0.387359	−	0.921929i	$-0.373387\pi$
$631$	12279.7	0.774719	0.387359	+	0.921929i	$0.373387\pi$
$632$	−5802.13	−0.365184
$633$	5206.07	0.326892
$634$	12239.5	0.766707
$635$	17362.3	1.08504
$636$	−3515.73	−0.219195
$637$	9299.80	0.578448
$638$	2183.79	0.135513
$639$	−7836.14	−0.485122
$640$	2521.21	0.155718
$641$	17054.4	1.05087	0.525435	−	0.850834i	$-0.323903\pi$
$641$	17054.4	1.05087	0.525435	+	0.850834i	$0.323903\pi$
$642$	−510.392	−0.0313763
$643$	−20697.3	−1.26940	−0.634698	−	0.772761i	$-0.718875\pi$
$643$	−20697.3	−1.26940	−0.634698	+	0.772761i	$0.718875\pi$
$644$	8212.70	0.502525
$645$	5562.29	0.339558
$646$	7807.83	0.475534
$647$	−10613.6	−0.644924	−0.322462	−	0.946582i	$-0.604510\pi$
$647$	−10613.6	−0.644924	−0.322462	+	0.946582i	$0.604510\pi$
$648$	−4505.85	−0.273158
$649$	26445.4	1.59950
$650$	23454.7	1.41534
$651$	−3933.02	−0.236785
$652$	−5806.35	−0.348764
$653$	−10019.4	−0.600441	−0.300220	−	0.953870i	$-0.597060\pi$
$653$	−10019.4	−0.600441	−0.300220	+	0.953870i	$0.597060\pi$
$654$	−504.005	−0.0301348
$655$	−39896.0	−2.37995
$656$	5567.35	0.331354
$657$	21391.0	1.27023
$658$	336.294	0.0199241
$659$	2400.70	0.141909	0.0709545	−	0.997480i	$-0.477395\pi$
$659$	2400.70	0.141909	0.0709545	+	0.997480i	$0.477395\pi$
$660$	4299.88	0.253595
$661$	−5037.05	−0.296397	−0.148199	−	0.988958i	$-0.547347\pi$
$661$	−5037.05	−0.296397	−0.148199	+	0.988958i	$0.547347\pi$
$662$	−397.848	−0.0233577
$663$	3955.52	0.231704
$664$	6567.36	0.383830
$665$	−14571.7	−0.849725
$666$	508.041	0.0295588
$667$	5134.75	0.298078
$668$	2401.50	0.139097
$669$	−1119.13	−0.0646757
$670$	−16033.7	−0.924532
$671$	−20459.6	−1.17710
$672$	−537.861	−0.0308757
$673$	−20842.1	−1.19377	−0.596884	−	0.802328i	$-0.703595\pi$
$673$	−20842.1	−1.19377	−0.596884	+	0.802328i	$0.703595\pi$
$674$	−4204.85	−0.240304
$675$	−19782.4	−1.12804
$676$	−832.816	−0.0473837
$677$	−23394.0	−1.32807	−0.664037	−	0.747700i	$-0.731158\pi$
$677$	−23394.0	−1.32807	−0.664037	+	0.747700i	$0.731158\pi$
$678$	4950.45	0.280414
$679$	−706.759	−0.0399454
$680$	−9642.33	−0.543774
$681$	−6428.72	−0.361746
$682$	−17620.5	−0.989331
$683$	−8567.11	−0.479958	−0.239979	−	0.970778i	$-0.577141\pi$
$683$	−8567.11	−0.479958	−0.239979	+	0.970778i	$0.577141\pi$
$684$	−6354.02	−0.355193
$685$	−31403.2	−1.75161
$686$	12791.1	0.711905
$687$	3207.90	0.178150
$688$	−3117.17	−0.172734
$689$	27041.8	1.49522
$690$	10110.3	0.557816
$691$	−11987.6	−0.659959	−0.329979	−	0.943988i	$-0.607042\pi$
$691$	−11987.6	−0.659959	−0.329979	+	0.943988i	$0.607042\pi$
$692$	−10296.8	−0.565644
$693$	10871.0	0.595895
$694$	−13087.7	−0.715851
$695$	56242.7	3.06965
$696$	−336.282	−0.0183143
$697$	−21292.3	−1.15710
$698$	−1586.85	−0.0860505
$699$	−4320.08	−0.233763
$700$	12197.5	0.658602
$701$	19429.6	1.04686	0.523428	−	0.852070i	$-0.324653\pi$
$701$	19429.6	1.04686	0.523428	+	0.852070i	$0.324653\pi$
$702$	−6709.64	−0.360739
$703$	650.869	0.0349189
$704$	−2409.70	−0.129004
$705$	413.997	0.0221164
$706$	−14757.7	−0.786704
$707$	12757.4	0.678628
$708$	−4072.33	−0.216169
$709$	−804.014	−0.0425887	−0.0212944	−	0.999773i	$-0.506779\pi$
$709$	−804.014	−0.0425887	−0.0212944	+	0.999773i	$0.506779\pi$
$710$	12397.9	0.655333
$711$	−18058.4	−0.952521
$712$	5186.91	0.273016
$713$	−41431.1	−2.17617
$714$	2057.04	0.107819
$715$	−33073.2	−1.72989
$716$	15314.9	0.799362
$717$	−807.500	−0.0420595
$718$	14285.9	0.742544
$719$	4975.94	0.258096	0.129048	−	0.991638i	$-0.458808\pi$
$719$	4975.94	0.258096	0.129048	+	0.991638i	$0.458808\pi$
$720$	7846.94	0.406164
$721$	8672.93	0.447984
$722$	5577.64	0.287505
$723$	−6042.42	−0.310816
$724$	8302.55	0.426190
$725$	7626.11	0.390658
$726$	−251.161	−0.0128395
$727$	−3761.32	−0.191884	−0.0959419	−	0.995387i	$-0.530586\pi$
$727$	−3761.32	−0.191884	−0.0959419	+	0.995387i	$0.530586\pi$
$728$	4137.04	0.210617
$729$	−11079.8	−0.562912
$730$	−33843.7	−1.71591
$731$	11921.6	0.603195
$732$	3150.58	0.159083
$733$	21439.2	1.08032	0.540161	−	0.841562i	$-0.318363\pi$
$733$	21439.2	1.08032	0.540161	+	0.841562i	$0.318363\pi$
$734$	3612.94	0.181684
$735$	5953.77	0.298787
$736$	−5665.93	−0.283762
$737$	15324.6	0.765926
$738$	17327.7	0.864282
$739$	22000.3	1.09512	0.547561	−	0.836766i	$-0.315556\pi$
$739$	22000.3	1.09512	0.547561	+	0.836766i	$0.315556\pi$
$740$	−803.796	−0.0399299
$741$	−4123.98	−0.204451
$742$	14062.9	0.695776
$743$	−13371.8	−0.660246	−0.330123	−	0.943938i	$-0.607090\pi$
$743$	−13371.8	−0.660246	−0.330123	+	0.943938i	$0.607090\pi$
$744$	2713.38	0.133706
$745$	361.942	0.0177994
$746$	−18792.1	−0.922289
$747$	20440.1	1.00115
$748$	9215.87	0.450489
$749$	2041.57	0.0995958
$750$	7878.19	0.383561
$751$	−17764.8	−0.863178	−0.431589	−	0.902070i	$-0.642047\pi$
$751$	−17764.8	−0.863178	−0.431589	+	0.902070i	$0.642047\pi$
$752$	−232.008	−0.0112506
$753$	−10338.6	−0.500343
$754$	2586.56	0.124930
$755$	15340.1	0.739448
$756$	−3489.31	−0.167864
$757$	33266.1	1.59719	0.798597	−	0.601867i	$-0.205576\pi$
$757$	33266.1	1.59719	0.798597	+	0.601867i	$0.205576\pi$
$758$	842.963	0.0403929
$759$	−9663.15	−0.462121
$760$	10053.0	0.479816
$761$	−25327.3	−1.20646	−0.603228	−	0.797569i	$-0.706119\pi$
$761$	−25327.3	−1.20646	−0.603228	+	0.797569i	$0.706119\pi$
$762$	2555.37	0.121485
$763$	2016.02	0.0956551
$764$	16280.3	0.770944
$765$	−30010.5	−1.41834
$766$	−6379.69	−0.300924
$767$	31322.9	1.47458
$768$	371.069	0.0174347
$769$	−9587.28	−0.449579	−0.224789	−	0.974407i	$-0.572169\pi$
$769$	−9587.28	−0.449579	−0.224789	+	0.974407i	$0.572169\pi$
$770$	−17199.5	−0.804971
$771$	8066.07	0.376773
$772$	9492.98	0.442565
$773$	2825.54	0.131472	0.0657359	−	0.997837i	$-0.479061\pi$
$773$	2825.54	0.131472	0.0657359	+	0.997837i	$0.479061\pi$
$774$	−9701.79	−0.450547
$775$	−61533.4	−2.85206
$776$	487.592	0.0225561
$777$	171.478	0.00791728
$778$	8221.92	0.378882
$779$	22199.1	1.02101
$780$	5092.94	0.233791
$781$	−11849.6	−0.542909
$782$	21669.3	0.990910
$783$	−2181.58	−0.0995702
$784$	−3336.55	−0.151993
$785$	10082.8	0.458433
$786$	−5871.86	−0.266466
$787$	2132.09	0.0965702	0.0482851	−	0.998834i	$-0.484624\pi$
$787$	2132.09	0.0965702	0.0482851	+	0.998834i	$0.484624\pi$
$788$	−12434.3	−0.562125
$789$	10420.0	0.470166
$790$	28571.0	1.28672
$791$	−19801.8	−0.890103
$792$	−7499.88	−0.336486
$793$	−24233.1	−1.08518
$794$	1655.01	0.0739724
$795$	17312.3	0.772331
$796$	−16193.2	−0.721046
$797$	5443.62	0.241936	0.120968	−	0.992656i	$-0.461400\pi$
$797$	5443.62	0.241936	0.120968	+	0.992656i	$0.461400\pi$
$798$	−2144.65	−0.0951377
$799$	887.313	0.0392877
$800$	−8415.02	−0.371895
$801$	16143.6	0.712117
$802$	−1350.29	−0.0594519
$803$	32346.8	1.42154
$804$	−2359.83	−0.103513
$805$	−40441.3	−1.77064
$806$	−20870.4	−0.912070
$807$	4615.72	0.201340
$808$	−8801.27	−0.383203
$809$	41193.5	1.79022	0.895109	−	0.445847i	$-0.147098\pi$
$809$	41193.5	1.79022	0.895109	+	0.445847i	$0.147098\pi$
$810$	22187.9	0.962472
$811$	−18225.9	−0.789145	−0.394572	−	0.918865i	$-0.629107\pi$
$811$	−18225.9	−0.789145	−0.394572	+	0.918865i	$0.629107\pi$
$812$	1345.13	0.0581338
$813$	2511.01	0.108321
$814$	768.245	0.0330798
$815$	28591.8	1.22887
$816$	−1419.15	−0.0608826
$817$	−12429.3	−0.532248
$818$	−1529.69	−0.0653845
$819$	12876.0	0.549359
$820$	−27414.9	−1.16753
$821$	46061.7	1.95806	0.979029	−	0.203720i	$-0.0653033\pi$
$821$	46061.7	1.95806	0.979029	+	0.203720i	$0.0653033\pi$
$822$	−4621.90	−0.196116
$823$	40166.3	1.70122	0.850612	−	0.525793i	$-0.176231\pi$
$823$	40166.3	1.70122	0.850612	+	0.525793i	$0.176231\pi$
$824$	−5983.44	−0.252965
$825$	−14351.7	−0.605650
$826$	16289.3	0.686171
$827$	−23994.5	−1.00891	−0.504457	−	0.863437i	$-0.668307\pi$
$827$	−23994.5	−1.00891	−0.504457	+	0.863437i	$0.668307\pi$
$828$	−17634.5	−0.740145
$829$	−13549.6	−0.567668	−0.283834	−	0.958873i	$-0.591606\pi$
$829$	−13549.6	−0.567668	−0.283834	+	0.958873i	$0.591606\pi$
$830$	−32339.2	−1.35242
$831$	8785.81	0.366759
$832$	−2854.14	−0.118930
$833$	12760.6	0.530767
$834$	8277.75	0.343687
$835$	−11825.5	−0.490107
$836$	−9608.36	−0.397503
$837$	17602.7	0.726928
$838$	5146.67	0.212158
$839$	34768.5	1.43068	0.715341	−	0.698776i	$-0.246271\pi$
$839$	34768.5	1.43068	0.715341	+	0.698776i	$0.246271\pi$
$840$	2648.55	0.108790
$841$	841.000	0.0344828
$842$	−28574.9	−1.16954
$843$	8963.29	0.366207
$844$	14366.6	0.585924
$845$	4100.98	0.166956
$846$	−722.096	−0.0293454
$847$	1004.64	0.0407556
$848$	−9701.98	−0.392886
$849$	−13156.7	−0.531844
$850$	32183.2	1.29867
$851$	1806.38	0.0727635
$852$	1824.72	0.0733730
$853$	4691.07	0.188299	0.0941496	−	0.995558i	$-0.469987\pi$
$853$	4691.07	0.188299	0.0941496	+	0.995558i	$0.469987\pi$
$854$	−12602.3	−0.504967
$855$	31288.7	1.25152
$856$	−1408.47	−0.0562391
$857$	−14362.7	−0.572485	−0.286242	−	0.958157i	$-0.592406\pi$
$857$	−14362.7	−0.572485	−0.286242	+	0.958157i	$0.592406\pi$
$858$	−4867.69	−0.193683
$859$	1392.43	0.0553073	0.0276537	−	0.999618i	$-0.491196\pi$
$859$	1392.43	0.0553073	0.0276537	+	0.999618i	$0.491196\pi$
$860$	15349.7	0.608627
$861$	5848.55	0.231496
$862$	22566.3	0.891661
$863$	19162.2	0.755840	0.377920	−	0.925838i	$-0.376639\pi$
$863$	19162.2	0.755840	0.377920	+	0.925838i	$0.376639\pi$
$864$	2407.27	0.0947880
$865$	50703.9	1.99304
$866$	−16139.3	−0.633297
$867$	−1693.81	−0.0663494
$868$	−10853.5	−0.424415
$869$	−27307.4	−1.06598
$870$	1655.93	0.0645302
$871$	18151.0	0.706111
$872$	−1390.85	−0.0540139
$873$	1517.57	0.0588338
$874$	−22592.2	−0.874361
$875$	−31512.8	−1.21752
$876$	−4981.09	−0.192118
$877$	−6509.75	−0.250649	−0.125324	−	0.992116i	$-0.539997\pi$
$877$	−6509.75	−0.250649	−0.125324	+	0.992116i	$0.539997\pi$
$878$	15638.8	0.601122
$879$	−5271.30	−0.202271
$880$	11865.9	0.454545
$881$	−589.496	−0.0225433	−0.0112716	−	0.999936i	$-0.503588\pi$
$881$	−589.496	−0.0225433	−0.0112716	+	0.999936i	$0.503588\pi$
$882$	−10384.6	−0.396449
$883$	−3903.57	−0.148772	−0.0743859	−	0.997230i	$-0.523700\pi$
$883$	−3903.57	−0.148772	−0.0743859	+	0.997230i	$0.523700\pi$
$884$	10915.6	0.415308
$885$	20053.1	0.761669
$886$	20306.8	0.770000
$887$	−24136.6	−0.913674	−0.456837	−	0.889550i	$-0.651018\pi$
$887$	−24136.6	−0.913674	−0.456837	+	0.889550i	$0.651018\pi$
$888$	−118.302	−0.00447067
$889$	−10221.5	−0.385622
$890$	−25541.5	−0.961971
$891$	−21206.5	−0.797358
$892$	−3088.34	−0.115925
$893$	−925.103	−0.0346667
$894$	53.2703	0.00199287
$895$	−75414.0	−2.81655
$896$	−1484.28	−0.0553418
$897$	−11445.4	−0.426032
$898$	−27216.6	−1.01139
$899$	−6785.84	−0.251747
$900$	−26190.7	−0.970025
$901$	37105.1	1.37198
$902$	26202.4	0.967233
$903$	−3274.62	−0.120678
$904$	13661.2	0.502617
$905$	−40883.7	−1.50168
$906$	2257.74	0.0827908
$907$	22174.3	0.811781	0.405890	−	0.913922i	$-0.366962\pi$
$907$	22174.3	0.811781	0.405890	+	0.913922i	$0.366962\pi$
$908$	−17740.6	−0.648397
$909$	−27392.8	−0.999519
$910$	−20371.8	−0.742108
$911$	−46956.1	−1.70771	−0.853856	−	0.520509i	$-0.825742\pi$
$911$	−46956.1	−1.70771	−0.853856	+	0.520509i	$0.825742\pi$
$912$	1479.59	0.0537217
$913$	30908.9	1.12041
$914$	3765.10	0.136257
$915$	−15514.2	−0.560528
$916$	8852.49	0.319317
$917$	23487.4	0.845827
$918$	−9206.56	−0.331004
$919$	−37166.4	−1.33407	−0.667033	−	0.745028i	$-0.732436\pi$
$919$	−37166.4	−1.33407	−0.667033	+	0.745028i	$0.732436\pi$
$920$	27900.4	0.999834
$921$	−6695.27	−0.239540
$922$	27894.5	0.996373
$923$	−14035.1	−0.500511
$924$	−2531.41	−0.0901270
$925$	2682.82	0.0953629
$926$	24909.6	0.883995
$927$	−18622.7	−0.659816
$928$	−928.000	−0.0328266
$929$	29571.7	1.04437	0.522183	−	0.852833i	$-0.325118\pi$
$929$	29571.7	1.04437	0.522183	+	0.852833i	$0.325118\pi$
$930$	−13361.3	−0.471113
$931$	−13304.1	−0.468339
$932$	−11921.7	−0.418999
$933$	11732.3	0.411680
$934$	13123.2	0.459747
$935$	−45381.1	−1.58729
$936$	−8883.14	−0.310208
$937$	8691.51	0.303030	0.151515	−	0.988455i	$-0.451585\pi$
$937$	8691.51	0.303030	0.151515	+	0.988455i	$0.451585\pi$
$938$	9439.31	0.328576
$939$	10004.1	0.347679
$940$	1142.46	0.0396415
$941$	−2862.43	−0.0991632	−0.0495816	−	0.998770i	$-0.515789\pi$
$941$	−2862.43	−0.0991632	−0.0495816	+	0.998770i	$0.515789\pi$
$942$	1483.98	0.0513275
$943$	61609.7	2.12756
$944$	−11238.0	−0.387462
$945$	17182.2	0.591466
$946$	−14670.8	−0.504215
$947$	36997.8	1.26955	0.634777	−	0.772695i	$-0.281092\pi$
$947$	36997.8	1.26955	0.634777	+	0.772695i	$0.281092\pi$
$948$	4205.06	0.144065
$949$	38312.8	1.31052
$950$	−33553.8	−1.14593
$951$	−8870.51	−0.302467
$952$	5676.60	0.193256
$953$	20998.8	0.713764	0.356882	−	0.934150i	$-0.383840\pi$
$953$	20998.8	0.713764	0.356882	+	0.934150i	$0.383840\pi$
$954$	−30196.2	−1.02478
$955$	−80168.1	−2.71642
$956$	−2228.37	−0.0753877
$957$	−1582.69	−0.0534599
$958$	34815.0	1.17414
$959$	18487.6	0.622519
$960$	−1827.23	−0.0614309
$961$	24962.4	0.837917
$962$	909.939	0.0304965
$963$	−4383.69	−0.146690
$964$	−16674.6	−0.557109
$965$	−46745.7	−1.55937
$966$	−5952.11	−0.198247
$967$	−30803.0	−1.02436	−0.512181	−	0.858878i	$-0.671162\pi$
$967$	−30803.0	−1.02436	−0.512181	+	0.858878i	$0.671162\pi$
$968$	−693.102	−0.0230136
$969$	−5658.68	−0.187599
$970$	−2401.02	−0.0794763
$971$	−755.015	−0.0249532	−0.0124766	−	0.999922i	$-0.503972\pi$
$971$	−755.015	−0.0249532	−0.0124766	+	0.999922i	$0.503972\pi$
$972$	11390.1	0.375862
$973$	−33111.0	−1.09095
$974$	5089.57	0.167434
$975$	−16998.7	−0.558353
$976$	8694.30	0.285141
$977$	−44011.7	−1.44121	−0.720603	−	0.693348i	$-0.756135\pi$
$977$	−44011.7	−1.44121	−0.720603	+	0.693348i	$0.756135\pi$
$978$	4208.12	0.137588
$979$	24411.9	0.796943
$980$	16430.0	0.535547
$981$	−4328.84	−0.140886
$982$	−5425.79	−0.176318
$983$	−36334.8	−1.17894	−0.589471	−	0.807790i	$-0.700664\pi$
$983$	−36334.8	−1.17894	−0.589471	+	0.807790i	$0.700664\pi$
$984$	−4034.91	−0.130720
$985$	61229.6	1.98065
$986$	3549.13	0.114632
$987$	−243.727	−0.00786010
$988$	−11380.5	−0.366460
$989$	−34495.4	−1.10909
$990$	36931.2	1.18561
$991$	9848.60	0.315692	0.157846	−	0.987464i	$-0.449545\pi$
$991$	9848.60	0.315692	0.157846	+	0.987464i	$0.449545\pi$
$992$	7487.82	0.239656
$993$	288.338	0.00921464
$994$	−7298.87	−0.232904
$995$	79739.1	2.54060
$996$	−4759.66	−0.151421
$997$	3378.02	0.107305	0.0536525	−	0.998560i	$-0.482914\pi$
$997$	3378.02	0.107305	0.0536525	+	0.998560i	$0.482914\pi$
$998$	−9167.59	−0.290777
$999$	−767.469	−0.0243060

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	58.4.a.c.1.2	✓	2
3.2	odd	2		522.4.a.j.1.2		2
4.3	odd	2		464.4.a.e.1.1		2
5.4	even	2		1450.4.a.g.1.1		2
8.3	odd	2		1856.4.a.i.1.2		2
8.5	even	2		1856.4.a.l.1.1		2
29.28	even	2		1682.4.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.c.1.2	✓	2	1.1	even	1	trivial
464.4.a.e.1.1		2	4.3	odd	2
522.4.a.j.1.2		2	3.2	odd	2
1450.4.a.g.1.1		2	5.4	even	2
1682.4.a.c.1.1		2	29.28	even	2
1856.4.a.i.1.2		2	8.3	odd	2
1856.4.a.l.1.1		2	8.5	even	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:	\( N \)	\(=\)	\( 58 = 2 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	58.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +1.44949 q^{3} +4.00000 q^{4} -19.6969 q^{5} -2.89898 q^{6} +11.5959 q^{7} -8.00000 q^{8} -24.8990 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +1.44949 q^{3} +4.00000 q^{4} -19.6969 q^{5} -2.89898 q^{6} +11.5959 q^{7} -8.00000 q^{8} -24.8990 q^{9} +39.3939 q^{10} -37.6515 q^{11} +5.79796 q^{12} -44.5959 q^{13} -23.1918 q^{14} -28.5505 q^{15} +16.0000 q^{16} -61.1918 q^{17} +49.7980 q^{18} +63.7980 q^{19} -78.7878 q^{20} +16.8082 q^{21} +75.3031 q^{22} +177.060 q^{23} -11.5959 q^{24} +262.969 q^{25} +89.1918 q^{26} -75.2270 q^{27} +46.3837 q^{28} +29.0000 q^{29} +57.1010 q^{30} -233.994 q^{31} -32.0000 q^{32} -54.5755 q^{33} +122.384 q^{34} -228.404 q^{35} -99.5959 q^{36} +10.2020 q^{37} -127.596 q^{38} -64.6413 q^{39} +157.576 q^{40} +347.959 q^{41} -33.6163 q^{42} -194.823 q^{43} -150.606 q^{44} +490.434 q^{45} -354.120 q^{46} -14.5005 q^{47} +23.1918 q^{48} -208.535 q^{49} -525.939 q^{50} -88.6969 q^{51} -178.384 q^{52} -606.373 q^{53} +150.454 q^{54} +741.620 q^{55} -92.7673 q^{56} +92.4745 q^{57} -58.0000 q^{58} -702.372 q^{59} -114.202 q^{60} +543.394 q^{61} +467.989 q^{62} -288.727 q^{63} +64.0000 q^{64} +878.403 q^{65} +109.151 q^{66} -407.010 q^{67} -244.767 q^{68} +256.647 q^{69} +456.808 q^{70} +314.717 q^{71} +199.192 q^{72} -859.110 q^{73} -20.4041 q^{74} +381.171 q^{75} +255.192 q^{76} -436.604 q^{77} +129.283 q^{78} +725.266 q^{79} -315.151 q^{80} +563.232 q^{81} -695.918 q^{82} -820.919 q^{83} +67.2327 q^{84} +1205.29 q^{85} +389.646 q^{86} +42.0352 q^{87} +301.212 q^{88} -648.363 q^{89} -980.867 q^{90} -517.131 q^{91} +708.241 q^{92} -339.172 q^{93} +29.0010 q^{94} -1256.62 q^{95} -46.3837 q^{96} -60.9490 q^{97} +417.069 q^{98} +937.485 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} - 16 q^{8} - 40 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 - 16 * q^8 - 40 * q^9	\( 2 q - 4 q^{2} - 2 q^{3} + 8 q^{4} - 10 q^{5} + 4 q^{6} - 16 q^{7} - 16 q^{8} - 40 q^{9} + 20 q^{10} - 90 q^{11} - 8 q^{12} - 50 q^{13} + 32 q^{14} - 62 q^{15} + 32 q^{16} - 44 q^{17} + 80 q^{18} + 108 q^{19} - 40 q^{20} + 112 q^{21} + 180 q^{22} - 28 q^{23} + 16 q^{24} + 232 q^{25} + 100 q^{26} + 70 q^{27} - 64 q^{28} + 58 q^{29} + 124 q^{30} + 66 q^{31} - 64 q^{32} + 126 q^{33} + 88 q^{34} - 496 q^{35} - 160 q^{36} + 40 q^{37} - 216 q^{38} - 46 q^{39} + 80 q^{40} + 304 q^{41} - 224 q^{42} - 130 q^{43} - 360 q^{44} + 344 q^{45} + 56 q^{46} - 514 q^{47} - 32 q^{48} + 210 q^{49} - 464 q^{50} - 148 q^{51} - 200 q^{52} - 958 q^{53} - 140 q^{54} + 234 q^{55} + 128 q^{56} - 60 q^{57} - 116 q^{58} - 180 q^{59} - 248 q^{60} + 1028 q^{61} - 132 q^{62} + 128 q^{63} + 128 q^{64} + 826 q^{65} - 252 q^{66} - 912 q^{67} - 176 q^{68} + 964 q^{69} + 992 q^{70} + 796 q^{71} + 320 q^{72} - 856 q^{73} - 80 q^{74} + 488 q^{75} + 432 q^{76} + 1008 q^{77} + 92 q^{78} - 318 q^{79} - 160 q^{80} + 470 q^{81} - 608 q^{82} - 1828 q^{83} + 448 q^{84} + 1372 q^{85} + 260 q^{86} - 58 q^{87} + 720 q^{88} - 944 q^{89} - 688 q^{90} - 368 q^{91} - 112 q^{92} - 1374 q^{93} + 1028 q^{94} - 828 q^{95} + 64 q^{96} + 368 q^{97} - 420 q^{98} + 1728 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 - 2 * q^3 + 8 * q^4 - 10 * q^5 + 4 * q^6 - 16 * q^7 - 16 * q^8 - 40 * q^9 + 20 * q^10 - 90 * q^11 - 8 * q^12 - 50 * q^13 + 32 * q^14 - 62 * q^15 + 32 * q^16 - 44 * q^17 + 80 * q^18 + 108 * q^19 - 40 * q^20 + 112 * q^21 + 180 * q^22 - 28 * q^23 + 16 * q^24 + 232 * q^25 + 100 * q^26 + 70 * q^27 - 64 * q^28 + 58 * q^29 + 124 * q^30 + 66 * q^31 - 64 * q^32 + 126 * q^33 + 88 * q^34 - 496 * q^35 - 160 * q^36 + 40 * q^37 - 216 * q^38 - 46 * q^39 + 80 * q^40 + 304 * q^41 - 224 * q^42 - 130 * q^43 - 360 * q^44 + 344 * q^45 + 56 * q^46 - 514 * q^47 - 32 * q^48 + 210 * q^49 - 464 * q^50 - 148 * q^51 - 200 * q^52 - 958 * q^53 - 140 * q^54 + 234 * q^55 + 128 * q^56 - 60 * q^57 - 116 * q^58 - 180 * q^59 - 248 * q^60 + 1028 * q^61 - 132 * q^62 + 128 * q^63 + 128 * q^64 + 826 * q^65 - 252 * q^66 - 912 * q^67 - 176 * q^68 + 964 * q^69 + 992 * q^70 + 796 * q^71 + 320 * q^72 - 856 * q^73 - 80 * q^74 + 488 * q^75 + 432 * q^76 + 1008 * q^77 + 92 * q^78 - 318 * q^79 - 160 * q^80 + 470 * q^81 - 608 * q^82 - 1828 * q^83 + 448 * q^84 + 1372 * q^85 + 260 * q^86 - 58 * q^87 + 720 * q^88 - 944 * q^89 - 688 * q^90 - 368 * q^91 - 112 * q^92 - 1374 * q^93 + 1028 * q^94 - 828 * q^95 + 64 * q^96 + 368 * q^97 - 420 * q^98 + 1728 * q^99

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