LMFDB - Embedded newform 8042.2.a.d.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$0$
Dimension:	$101$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.31266 q^{3} +1.00000 q^{4} -2.37478 q^{5} -2.31266 q^{6} -3.49882 q^{7} +1.00000 q^{8} +2.34838 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.31266 q^{3} +1.00000 q^{4} -2.37478 q^{5} -2.31266 q^{6} -3.49882 q^{7} +1.00000 q^{8} +2.34838 q^{9} -2.37478 q^{10} -5.67957 q^{11} -2.31266 q^{12} +6.77003 q^{13} -3.49882 q^{14} +5.49205 q^{15} +1.00000 q^{16} +0.816767 q^{17} +2.34838 q^{18} -2.50124 q^{19} -2.37478 q^{20} +8.09156 q^{21} -5.67957 q^{22} -6.71098 q^{23} -2.31266 q^{24} +0.639581 q^{25} +6.77003 q^{26} +1.50698 q^{27} -3.49882 q^{28} -8.05370 q^{29} +5.49205 q^{30} -4.06757 q^{31} +1.00000 q^{32} +13.1349 q^{33} +0.816767 q^{34} +8.30892 q^{35} +2.34838 q^{36} -7.21360 q^{37} -2.50124 q^{38} -15.6567 q^{39} -2.37478 q^{40} -3.15523 q^{41} +8.09156 q^{42} -2.38123 q^{43} -5.67957 q^{44} -5.57688 q^{45} -6.71098 q^{46} -10.0772 q^{47} -2.31266 q^{48} +5.24171 q^{49} +0.639581 q^{50} -1.88890 q^{51} +6.77003 q^{52} -4.00381 q^{53} +1.50698 q^{54} +13.4877 q^{55} -3.49882 q^{56} +5.78450 q^{57} -8.05370 q^{58} +3.01909 q^{59} +5.49205 q^{60} +2.83303 q^{61} -4.06757 q^{62} -8.21654 q^{63} +1.00000 q^{64} -16.0773 q^{65} +13.1349 q^{66} -10.3806 q^{67} +0.816767 q^{68} +15.5202 q^{69} +8.30892 q^{70} -3.78963 q^{71} +2.34838 q^{72} -1.78878 q^{73} -7.21360 q^{74} -1.47913 q^{75} -2.50124 q^{76} +19.8718 q^{77} -15.6567 q^{78} +4.13696 q^{79} -2.37478 q^{80} -10.5303 q^{81} -3.15523 q^{82} +12.1249 q^{83} +8.09156 q^{84} -1.93964 q^{85} -2.38123 q^{86} +18.6254 q^{87} -5.67957 q^{88} -4.92462 q^{89} -5.57688 q^{90} -23.6871 q^{91} -6.71098 q^{92} +9.40690 q^{93} -10.0772 q^{94} +5.93989 q^{95} -2.31266 q^{96} -4.43915 q^{97} +5.24171 q^{98} -13.3378 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9}+O(q^{10}) $ 101 * q + 101 * q^2 + 10 * q^3 + 101 * q^4 + 19 * q^5 + 10 * q^6 + 42 * q^7 + 101 * q^8 + 147 * q^9	\( 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9} + 19 q^{10} + 4 q^{11} + 10 q^{12} + 58 q^{13} + 42 q^{14} + 27 q^{15} + 101 q^{16} + 34 q^{17} + 147 q^{18} + 36 q^{19} + 19 q^{20} + 45 q^{21} + 4 q^{22} + 47 q^{23} + 10 q^{24} + 174 q^{25} + 58 q^{26} + 31 q^{27} + 42 q^{28} + 62 q^{29} + 27 q^{30} + 47 q^{31} + 101 q^{32} + 55 q^{33} + 34 q^{34} + 16 q^{35} + 147 q^{36} + 90 q^{37} + 36 q^{38} + 50 q^{39} + 19 q^{40} + 54 q^{41} + 45 q^{42} + 65 q^{43} + 4 q^{44} + 47 q^{45} + 47 q^{46} + 54 q^{47} + 10 q^{48} + 189 q^{49} + 174 q^{50} + 36 q^{51} + 58 q^{52} + 94 q^{53} + 31 q^{54} + 68 q^{55} + 42 q^{56} + 79 q^{57} + 62 q^{58} - 6 q^{59} + 27 q^{60} + 58 q^{61} + 47 q^{62} + 117 q^{63} + 101 q^{64} + 89 q^{65} + 55 q^{66} + 127 q^{67} + 34 q^{68} + 45 q^{69} + 16 q^{70} + 87 q^{71} + 147 q^{72} + 83 q^{73} + 90 q^{74} - 4 q^{75} + 36 q^{76} + 53 q^{77} + 50 q^{78} + 74 q^{79} + 19 q^{80} + 241 q^{81} + 54 q^{82} + 11 q^{83} + 45 q^{84} + 120 q^{85} + 65 q^{86} + 37 q^{87} + 4 q^{88} + 89 q^{89} + 47 q^{90} + 31 q^{91} + 47 q^{92} + 123 q^{93} + 54 q^{94} + 61 q^{95} + 10 q^{96} + 85 q^{97} + 189 q^{98} - 55 q^{99}+O(q^{100}) \) 101 * q + 101 * q^2 + 10 * q^3 + 101 * q^4 + 19 * q^5 + 10 * q^6 + 42 * q^7 + 101 * q^8 + 147 * q^9 + 19 * q^10 + 4 * q^11 + 10 * q^12 + 58 * q^13 + 42 * q^14 + 27 * q^15 + 101 * q^16 + 34 * q^17 + 147 * q^18 + 36 * q^19 + 19 * q^20 + 45 * q^21 + 4 * q^22 + 47 * q^23 + 10 * q^24 + 174 * q^25 + 58 * q^26 + 31 * q^27 + 42 * q^28 + 62 * q^29 + 27 * q^30 + 47 * q^31 + 101 * q^32 + 55 * q^33 + 34 * q^34 + 16 * q^35 + 147 * q^36 + 90 * q^37 + 36 * q^38 + 50 * q^39 + 19 * q^40 + 54 * q^41 + 45 * q^42 + 65 * q^43 + 4 * q^44 + 47 * q^45 + 47 * q^46 + 54 * q^47 + 10 * q^48 + 189 * q^49 + 174 * q^50 + 36 * q^51 + 58 * q^52 + 94 * q^53 + 31 * q^54 + 68 * q^55 + 42 * q^56 + 79 * q^57 + 62 * q^58 - 6 * q^59 + 27 * q^60 + 58 * q^61 + 47 * q^62 + 117 * q^63 + 101 * q^64 + 89 * q^65 + 55 * q^66 + 127 * q^67 + 34 * q^68 + 45 * q^69 + 16 * q^70 + 87 * q^71 + 147 * q^72 + 83 * q^73 + 90 * q^74 - 4 * q^75 + 36 * q^76 + 53 * q^77 + 50 * q^78 + 74 * q^79 + 19 * q^80 + 241 * q^81 + 54 * q^82 + 11 * q^83 + 45 * q^84 + 120 * q^85 + 65 * q^86 + 37 * q^87 + 4 * q^88 + 89 * q^89 + 47 * q^90 + 31 * q^91 + 47 * q^92 + 123 * q^93 + 54 * q^94 + 61 * q^95 + 10 * q^96 + 85 * q^97 + 189 * q^98 - 55 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−2.31266	−1.33521	−0.667606	−	0.744515i	$-0.732681\pi$
$3$	−2.31266	−1.33521	−0.667606	+	0.744515i	$0.732681\pi$
$4$	1.00000	0.500000
$5$	−2.37478	−1.06203	−0.531017	−	0.847361i	$-0.678190\pi$
$5$	−2.37478	−1.06203	−0.531017	+	0.847361i	$0.678190\pi$
$6$	−2.31266	−0.944138
$7$	−3.49882	−1.32243	−0.661214	−	0.750197i	$-0.729958\pi$
$7$	−3.49882	−1.32243	−0.661214	+	0.750197i	$0.729958\pi$
$8$	1.00000	0.353553
$9$	2.34838	0.782792
$10$	−2.37478	−0.750971
$11$	−5.67957	−1.71246	−0.856228	−	0.516599i	$-0.827198\pi$
$11$	−5.67957	−1.71246	−0.856228	+	0.516599i	$0.827198\pi$
$12$	−2.31266	−0.667606
$13$	6.77003	1.87767	0.938834	−	0.344369i	$-0.111907\pi$
$13$	6.77003	1.87767	0.938834	+	0.344369i	$0.111907\pi$
$14$	−3.49882	−0.935098
$15$	5.49205	1.41804
$16$	1.00000	0.250000
$17$	0.816767	0.198095	0.0990476	−	0.995083i	$-0.468420\pi$
$17$	0.816767	0.198095	0.0990476	+	0.995083i	$0.468420\pi$
$18$	2.34838	0.553518
$19$	−2.50124	−0.573823	−0.286911	−	0.957957i	$-0.592629\pi$
$19$	−2.50124	−0.573823	−0.286911	+	0.957957i	$0.592629\pi$
$20$	−2.37478	−0.531017
$21$	8.09156	1.76572
$22$	−5.67957	−1.21089
$23$	−6.71098	−1.39934	−0.699668	−	0.714468i	$-0.746669\pi$
$23$	−6.71098	−1.39934	−0.699668	+	0.714468i	$0.746669\pi$
$24$	−2.31266	−0.472069
$25$	0.639581	0.127916
$26$	6.77003	1.32771
$27$	1.50698	0.290018
$28$	−3.49882	−0.661214
$29$	−8.05370	−1.49553	−0.747767	−	0.663961i	$-0.768874\pi$
$29$	−8.05370	−1.49553	−0.747767	+	0.663961i	$0.768874\pi$
$30$	5.49205	1.00271
$31$	−4.06757	−0.730558	−0.365279	−	0.930898i	$-0.619026\pi$
$31$	−4.06757	−0.730558	−0.365279	+	0.930898i	$0.619026\pi$
$32$	1.00000	0.176777
$33$	13.1349	2.28649
$34$	0.816767	0.140074
$35$	8.30892	1.40446
$36$	2.34838	0.391396
$37$	−7.21360	−1.18591	−0.592955	−	0.805236i	$-0.702039\pi$
$37$	−7.21360	−1.18591	−0.592955	+	0.805236i	$0.702039\pi$
$38$	−2.50124	−0.405754
$39$	−15.6567	−2.50709
$40$	−2.37478	−0.375486
$41$	−3.15523	−0.492765	−0.246382	−	0.969173i	$-0.579242\pi$
$41$	−3.15523	−0.492765	−0.246382	+	0.969173i	$0.579242\pi$
$42$	8.09156	1.24855
$43$	−2.38123	−0.363134	−0.181567	−	0.983379i	$-0.558117\pi$
$43$	−2.38123	−0.363134	−0.181567	+	0.983379i	$0.558117\pi$
$44$	−5.67957	−0.856228
$45$	−5.57688	−0.831352
$46$	−6.71098	−0.989480
$47$	−10.0772	−1.46992	−0.734958	−	0.678112i	$-0.762798\pi$
$47$	−10.0772	−1.46992	−0.734958	+	0.678112i	$0.762798\pi$
$48$	−2.31266	−0.333803
$49$	5.24171	0.748816
$50$	0.639581	0.0904504
$51$	−1.88890	−0.264499
$52$	6.77003	0.938834
$53$	−4.00381	−0.549966	−0.274983	−	0.961449i	$-0.588672\pi$
$53$	−4.00381	−0.549966	−0.274983	+	0.961449i	$0.588672\pi$
$54$	1.50698	0.205074
$55$	13.4877	1.81869
$56$	−3.49882	−0.467549
$57$	5.78450	0.766175
$58$	−8.05370	−1.05750
$59$	3.01909	0.393052	0.196526	−	0.980499i	$-0.437034\pi$
$59$	3.01909	0.393052	0.196526	+	0.980499i	$0.437034\pi$
$60$	5.49205	0.709020
$61$	2.83303	0.362732	0.181366	−	0.983416i	$-0.441948\pi$
$61$	2.83303	0.362732	0.181366	+	0.983416i	$0.441948\pi$
$62$	−4.06757	−0.516582
$63$	−8.21654	−1.03519
$64$	1.00000	0.125000
$65$	−16.0773	−1.99415
$66$	13.1349	1.61679
$67$	−10.3806	−1.26819	−0.634095	−	0.773255i	$-0.718627\pi$
$67$	−10.3806	−1.26819	−0.634095	+	0.773255i	$0.718627\pi$
$68$	0.816767	0.0990476
$69$	15.5202	1.86841
$70$	8.30892	0.993106
$71$	−3.78963	−0.449746	−0.224873	−	0.974388i	$-0.572197\pi$
$71$	−3.78963	−0.449746	−0.224873	+	0.974388i	$0.572197\pi$
$72$	2.34838	0.276759
$73$	−1.78878	−0.209361	−0.104680	−	0.994506i	$-0.533382\pi$
$73$	−1.78878	−0.209361	−0.104680	+	0.994506i	$0.533382\pi$
$74$	−7.21360	−0.838564
$75$	−1.47913	−0.170795
$76$	−2.50124	−0.286911
$77$	19.8718	2.26460
$78$	−15.6567	−1.77278
$79$	4.13696	0.465444	0.232722	−	0.972543i	$-0.425237\pi$
$79$	4.13696	0.465444	0.232722	+	0.972543i	$0.425237\pi$
$80$	−2.37478	−0.265508
$81$	−10.5303	−1.17003
$82$	−3.15523	−0.348437
$83$	12.1249	1.33088	0.665441	−	0.746450i	$-0.268243\pi$
$83$	12.1249	1.33088	0.665441	+	0.746450i	$0.268243\pi$
$84$	8.09156	0.882861
$85$	−1.93964	−0.210384
$86$	−2.38123	−0.256774
$87$	18.6254	1.99686
$88$	−5.67957	−0.605444
$89$	−4.92462	−0.522008	−0.261004	−	0.965338i	$-0.584054\pi$
$89$	−4.92462	−0.522008	−0.261004	+	0.965338i	$0.584054\pi$
$90$	−5.57688	−0.587855
$91$	−23.6871	−2.48308
$92$	−6.71098	−0.699668
$93$	9.40690	0.975450
$94$	−10.0772	−1.03939
$95$	5.93989	0.609419
$96$	−2.31266	−0.236034
$97$	−4.43915	−0.450727	−0.225364	−	0.974275i	$-0.572357\pi$
$97$	−4.43915	−0.450727	−0.225364	+	0.974275i	$0.572357\pi$
$98$	5.24171	0.529493
$99$	−13.3378	−1.34050
$100$	0.639581	0.0639581
$101$	8.41454	0.837278	0.418639	−	0.908153i	$-0.362507\pi$
$101$	8.41454	0.837278	0.418639	+	0.908153i	$0.362507\pi$
$102$	−1.88890	−0.187029
$103$	−13.4067	−1.32101	−0.660503	−	0.750824i	$-0.729657\pi$
$103$	−13.4067	−1.32101	−0.660503	+	0.750824i	$0.729657\pi$
$104$	6.77003	0.663856
$105$	−19.2157	−1.87526
$106$	−4.00381	−0.388885
$107$	−18.5138	−1.78980	−0.894901	−	0.446266i	$-0.852754\pi$
$107$	−18.5138	−1.78980	−0.894901	+	0.446266i	$0.852754\pi$
$108$	1.50698	0.145009
$109$	−18.5522	−1.77698	−0.888491	−	0.458893i	$-0.848246\pi$
$109$	−18.5522	−1.77698	−0.888491	+	0.458893i	$0.848246\pi$
$110$	13.4877	1.28601
$111$	16.6826	1.58344
$112$	−3.49882	−0.330607
$113$	−8.25832	−0.776877	−0.388439	−	0.921475i	$-0.626985\pi$
$113$	−8.25832	−0.776877	−0.388439	+	0.921475i	$0.626985\pi$
$114$	5.78450	0.541768
$115$	15.9371	1.48614
$116$	−8.05370	−0.747767
$117$	15.8986	1.46982
$118$	3.01909	0.277930
$119$	−2.85772	−0.261967
$120$	5.49205	0.501353
$121$	21.2575	1.93250
$122$	2.83303	0.256490
$123$	7.29697	0.657945
$124$	−4.06757	−0.365279
$125$	10.3550	0.926183
$126$	−8.21654	−0.731987
$127$	3.93473	0.349151	0.174575	−	0.984644i	$-0.444145\pi$
$127$	3.93473	0.349151	0.174575	+	0.984644i	$0.444145\pi$
$128$	1.00000	0.0883883
$129$	5.50696	0.484861
$130$	−16.0773	−1.41008
$131$	−14.2604	−1.24594	−0.622969	−	0.782246i	$-0.714074\pi$
$131$	−14.2604	−1.24594	−0.622969	+	0.782246i	$0.714074\pi$
$132$	13.1349	1.14325
$133$	8.75136	0.758839
$134$	−10.3806	−0.896746
$135$	−3.57875	−0.308009
$136$	0.816767	0.0700372
$137$	−13.5350	−1.15637	−0.578187	−	0.815904i	$-0.696240\pi$
$137$	−13.5350	−1.15637	−0.578187	+	0.815904i	$0.696240\pi$
$138$	15.5202	1.32117
$139$	−17.4450	−1.47966	−0.739831	−	0.672793i	$-0.765095\pi$
$139$	−17.4450	−1.47966	−0.739831	+	0.672793i	$0.765095\pi$
$140$	8.30892	0.702232
$141$	23.3052	1.96265
$142$	−3.78963	−0.318018
$143$	−38.4509	−3.21542
$144$	2.34838	0.195698
$145$	19.1258	1.58831
$146$	−1.78878	−0.148040
$147$	−12.1223	−0.999828
$148$	−7.21360	−0.592955
$149$	−7.06705	−0.578955	−0.289478	−	0.957185i	$-0.593482\pi$
$149$	−7.06705	−0.578955	−0.289478	+	0.957185i	$0.593482\pi$
$150$	−1.47913	−0.120770
$151$	18.1369	1.47596	0.737981	−	0.674821i	$-0.235779\pi$
$151$	18.1369	1.47596	0.737981	+	0.674821i	$0.235779\pi$
$152$	−2.50124	−0.202877
$153$	1.91808	0.155067
$154$	19.8718	1.60131
$155$	9.65959	0.775877
$156$	−15.6567	−1.25354
$157$	−0.144835	−0.0115591	−0.00577953	−	0.999983i	$-0.501840\pi$
$157$	−0.144835	−0.0115591	−0.00577953	+	0.999983i	$0.501840\pi$
$158$	4.13696	0.329119
$159$	9.25944	0.734322
$160$	−2.37478	−0.187743
$161$	23.4805	1.85052
$162$	−10.5303	−0.827335
$163$	18.9784	1.48651	0.743253	−	0.669010i	$-0.233282\pi$
$163$	18.9784	1.48651	0.743253	+	0.669010i	$0.233282\pi$
$164$	−3.15523	−0.246382
$165$	−31.1925	−2.42833
$166$	12.1249	0.941076
$167$	0.637257	0.0493124	0.0246562	−	0.999696i	$-0.492151\pi$
$167$	0.637257	0.0493124	0.0246562	+	0.999696i	$0.492151\pi$
$168$	8.09156	0.624277
$169$	32.8333	2.52564
$170$	−1.93964	−0.148764
$171$	−5.87384	−0.449184
$172$	−2.38123	−0.181567
$173$	7.70272	0.585627	0.292814	−	0.956170i	$-0.405408\pi$
$173$	7.70272	0.585627	0.292814	+	0.956170i	$0.405408\pi$
$174$	18.6254	1.41199
$175$	−2.23777	−0.169160
$176$	−5.67957	−0.428114
$177$	−6.98212	−0.524808
$178$	−4.92462	−0.369116
$179$	22.6778	1.69502	0.847509	−	0.530781i	$-0.178101\pi$
$179$	22.6778	1.69502	0.847509	+	0.530781i	$0.178101\pi$
$180$	−5.57688	−0.415676
$181$	24.5295	1.82326	0.911630	−	0.411011i	$-0.134824\pi$
$181$	24.5295	1.82326	0.911630	+	0.411011i	$0.134824\pi$
$182$	−23.6871	−1.75580
$183$	−6.55182	−0.484324
$184$	−6.71098	−0.494740
$185$	17.1307	1.25948
$186$	9.40690	0.689747
$187$	−4.63889	−0.339229
$188$	−10.0772	−0.734958
$189$	−5.27265	−0.383529
$190$	5.93989	0.430925
$191$	−6.36075	−0.460248	−0.230124	−	0.973161i	$-0.573913\pi$
$191$	−6.36075	−0.460248	−0.230124	+	0.973161i	$0.573913\pi$
$192$	−2.31266	−0.166902
$193$	24.5392	1.76637	0.883186	−	0.469023i	$-0.155394\pi$
$193$	24.5392	1.76637	0.883186	+	0.469023i	$0.155394\pi$
$194$	−4.43915	−0.318712
$195$	37.1813	2.66261
$196$	5.24171	0.374408
$197$	22.5752	1.60842	0.804208	−	0.594347i	$-0.202590\pi$
$197$	22.5752	1.60842	0.804208	+	0.594347i	$0.202590\pi$
$198$	−13.3378	−0.947874
$199$	7.80557	0.553322	0.276661	−	0.960968i	$-0.410772\pi$
$199$	7.80557	0.553322	0.276661	+	0.960968i	$0.410772\pi$
$200$	0.639581	0.0452252
$201$	24.0067	1.69330
$202$	8.41454	0.592045
$203$	28.1784	1.97774
$204$	−1.88890	−0.132250
$205$	7.49299	0.523333
$206$	−13.4067	−0.934092
$207$	−15.7599	−1.09539
$208$	6.77003	0.469417
$209$	14.2059	0.982646
$210$	−19.2157	−1.32601
$211$	−2.12462	−0.146265	−0.0731325	−	0.997322i	$-0.523300\pi$
$211$	−2.12462	−0.146265	−0.0731325	+	0.997322i	$0.523300\pi$
$212$	−4.00381	−0.274983
$213$	8.76410	0.600506
$214$	−18.5138	−1.26558
$215$	5.65489	0.385660
$216$	1.50698	0.102537
$217$	14.2317	0.966110
$218$	−18.5522	−1.25652
$219$	4.13683	0.279541
$220$	13.4877	0.909343
$221$	5.52954	0.371957
$222$	16.6826	1.11966
$223$	−10.7798	−0.721870	−0.360935	−	0.932591i	$-0.617542\pi$
$223$	−10.7798	−0.721870	−0.360935	+	0.932591i	$0.617542\pi$
$224$	−3.49882	−0.233774
$225$	1.50198	0.100132
$226$	−8.25832	−0.549335
$227$	−19.9218	−1.32226	−0.661128	−	0.750273i	$-0.729922\pi$
$227$	−19.9218	−1.32226	−0.661128	+	0.750273i	$0.729922\pi$
$228$	5.78450	0.383088
$229$	−26.5699	−1.75579	−0.877894	−	0.478855i	$-0.841052\pi$
$229$	−26.5699	−1.75579	−0.877894	+	0.478855i	$0.841052\pi$
$230$	15.9371	1.05086
$231$	−45.9566	−3.02372
$232$	−8.05370	−0.528751
$233$	16.6924	1.09356	0.546778	−	0.837277i	$-0.315854\pi$
$233$	16.6924	1.09356	0.546778	+	0.837277i	$0.315854\pi$
$234$	15.8986	1.03932
$235$	23.9312	1.56110
$236$	3.01909	0.196526
$237$	−9.56736	−0.621467
$238$	−2.85772	−0.185238
$239$	12.3884	0.801339	0.400670	−	0.916223i	$-0.368778\pi$
$239$	12.3884	0.801339	0.400670	+	0.916223i	$0.368778\pi$
$240$	5.49205	0.354510
$241$	3.33049	0.214536	0.107268	−	0.994230i	$-0.465790\pi$
$241$	3.33049	0.214536	0.107268	+	0.994230i	$0.465790\pi$
$242$	21.2575	1.36649
$243$	19.8319	1.27222
$244$	2.83303	0.181366
$245$	−12.4479	−0.795268
$246$	7.29697	0.465238
$247$	−16.9334	−1.07745
$248$	−4.06757	−0.258291
$249$	−28.0408	−1.77701
$250$	10.3550	0.654910
$251$	−30.8430	−1.94679	−0.973395	−	0.229133i	$-0.926411\pi$
$251$	−30.8430	−1.94679	−0.973395	+	0.229133i	$0.926411\pi$
$252$	−8.21654	−0.517593
$253$	38.1155	2.39630
$254$	3.93473	0.246887
$255$	4.48573	0.280907
$256$	1.00000	0.0625000
$257$	−9.22497	−0.575438	−0.287719	−	0.957715i	$-0.592897\pi$
$257$	−9.22497	−0.575438	−0.287719	+	0.957715i	$0.592897\pi$
$258$	5.50696	0.342848
$259$	25.2391	1.56828
$260$	−16.0773	−0.997074
$261$	−18.9131	−1.17069
$262$	−14.2604	−0.881011
$263$	−29.8728	−1.84204	−0.921019	−	0.389517i	$-0.872642\pi$
$263$	−29.8728	−1.84204	−0.921019	+	0.389517i	$0.872642\pi$
$264$	13.1349	0.808397
$265$	9.50818	0.584083
$266$	8.75136	0.536581
$267$	11.3889	0.696992
$268$	−10.3806	−0.634095
$269$	21.5868	1.31617	0.658085	−	0.752943i	$-0.271367\pi$
$269$	21.5868	1.31617	0.658085	+	0.752943i	$0.271367\pi$
$270$	−3.57875	−0.217796
$271$	25.7079	1.56164	0.780821	−	0.624755i	$-0.214801\pi$
$271$	25.7079	1.56164	0.780821	+	0.624755i	$0.214801\pi$
$272$	0.816767	0.0495238
$273$	54.7801	3.31544
$274$	−13.5350	−0.817680
$275$	−3.63254	−0.219051
$276$	15.5202	0.934205
$277$	−21.1787	−1.27251	−0.636253	−	0.771481i	$-0.719516\pi$
$277$	−21.1787	−1.27251	−0.636253	+	0.771481i	$0.719516\pi$
$278$	−17.4450	−1.04628
$279$	−9.55219	−0.571875
$280$	8.30892	0.496553
$281$	32.3338	1.92887	0.964435	−	0.264320i	$-0.0851475\pi$
$281$	32.3338	1.92887	0.964435	+	0.264320i	$0.0851475\pi$
$282$	23.3052	1.38780
$283$	−18.6228	−1.10701	−0.553506	−	0.832845i	$-0.686711\pi$
$283$	−18.6228	−1.10701	−0.553506	+	0.832845i	$0.686711\pi$
$284$	−3.78963	−0.224873
$285$	−13.7369	−0.813704
$286$	−38.4509	−2.27365
$287$	11.0396	0.651646
$288$	2.34838	0.138379
$289$	−16.3329	−0.960758
$290$	19.1258	1.12310
$291$	10.2662	0.601817
$292$	−1.78878	−0.104680
$293$	−7.37662	−0.430947	−0.215473	−	0.976510i	$-0.569129\pi$
$293$	−7.37662	−0.430947	−0.215473	+	0.976510i	$0.569129\pi$
$294$	−12.1223	−0.706985
$295$	−7.16968	−0.417435
$296$	−7.21360	−0.419282
$297$	−8.55900	−0.496644
$298$	−7.06705	−0.409383
$299$	−45.4335	−2.62749
$300$	−1.47913	−0.0853976
$301$	8.33148	0.480218
$302$	18.1369	1.04366
$303$	−19.4599	−1.11794
$304$	−2.50124	−0.143456
$305$	−6.72782	−0.385234
$306$	1.91808	0.109649
$307$	12.9056	0.736559	0.368279	−	0.929715i	$-0.379947\pi$
$307$	12.9056	0.736559	0.368279	+	0.929715i	$0.379947\pi$
$308$	19.8718	1.13230
$309$	31.0052	1.76382
$310$	9.65959	0.548628
$311$	−7.92889	−0.449606	−0.224803	−	0.974404i	$-0.572174\pi$
$311$	−7.92889	−0.449606	−0.224803	+	0.974404i	$0.572174\pi$
$312$	−15.6567	−0.886389
$313$	−9.33641	−0.527725	−0.263863	−	0.964560i	$-0.584997\pi$
$313$	−9.33641	−0.527725	−0.263863	+	0.964560i	$0.584997\pi$
$314$	−0.144835	−0.00817350
$315$	19.5125	1.09940
$316$	4.13696	0.232722
$317$	−18.3394	−1.03005	−0.515023	−	0.857176i	$-0.672217\pi$
$317$	−18.3394	−1.03005	−0.515023	+	0.857176i	$0.672217\pi$
$318$	9.25944	0.519244
$319$	45.7415	2.56103
$320$	−2.37478	−0.132754
$321$	42.8162	2.38976
$322$	23.4805	1.30852
$323$	−2.04293	−0.113672
$324$	−10.5303	−0.585014
$325$	4.32998	0.240184
$326$	18.9784	1.05112
$327$	42.9050	2.37265
$328$	−3.15523	−0.174219
$329$	35.2584	1.94386
$330$	−31.1925	−1.71709
$331$	21.0373	1.15632	0.578158	−	0.815925i	$-0.303772\pi$
$331$	21.0373	1.15632	0.578158	+	0.815925i	$0.303772\pi$
$332$	12.1249	0.665441
$333$	−16.9403	−0.928321
$334$	0.637257	0.0348692
$335$	24.6516	1.34686
$336$	8.09156	0.441431
$337$	1.97437	0.107551	0.0537753	−	0.998553i	$-0.482875\pi$
$337$	1.97437	0.107551	0.0537753	+	0.998553i	$0.482875\pi$
$338$	32.8333	1.78590
$339$	19.0986	1.03730
$340$	−1.93964	−0.105192
$341$	23.1021	1.25105
$342$	−5.87384	−0.317621
$343$	6.15193	0.332173
$344$	−2.38123	−0.128387
$345$	−36.8570	−1.98432
$346$	7.70272	0.414101
$347$	0.302801	0.0162552	0.00812761	−	0.999967i	$-0.497413\pi$
$347$	0.302801	0.0162552	0.00812761	+	0.999967i	$0.497413\pi$
$348$	18.6254	0.998428
$349$	12.6274	0.675931	0.337965	−	0.941159i	$-0.390261\pi$
$349$	12.6274	0.675931	0.337965	+	0.941159i	$0.390261\pi$
$350$	−2.23777	−0.119614
$351$	10.2023	0.544558
$352$	−5.67957	−0.302722
$353$	24.7975	1.31984	0.659920	−	0.751336i	$-0.270590\pi$
$353$	24.7975	1.31984	0.659920	+	0.751336i	$0.270590\pi$
$354$	−6.98212	−0.371096
$355$	8.99953	0.477645
$356$	−4.92462	−0.261004
$357$	6.60892	0.349781
$358$	22.6778	1.19856
$359$	−32.2276	−1.70091	−0.850454	−	0.526049i	$-0.823673\pi$
$359$	−32.2276	−1.70091	−0.850454	+	0.526049i	$0.823673\pi$
$360$	−5.57688	−0.293927
$361$	−12.7438	−0.670727
$362$	24.5295	1.28924
$363$	−49.1614	−2.58030
$364$	−23.6871	−1.24154
$365$	4.24795	0.222348
$366$	−6.55182	−0.342469
$367$	29.0525	1.51653	0.758265	−	0.651947i	$-0.226048\pi$
$367$	29.0525	1.51653	0.758265	+	0.651947i	$0.226048\pi$
$368$	−6.71098	−0.349834
$369$	−7.40968	−0.385732
$370$	17.1307	0.890584
$371$	14.0086	0.727291
$372$	9.40690	0.487725
$373$	24.6738	1.27756	0.638780	−	0.769389i	$-0.279439\pi$
$373$	24.6738	1.27756	0.638780	+	0.769389i	$0.279439\pi$
$374$	−4.63889	−0.239871
$375$	−23.9476	−1.23665
$376$	−10.0772	−0.519694
$377$	−54.5238	−2.80812
$378$	−5.27265	−0.271196
$379$	−19.1914	−0.985797	−0.492898	−	0.870087i	$-0.664063\pi$
$379$	−19.1914	−0.985797	−0.492898	+	0.870087i	$0.664063\pi$
$380$	5.93989	0.304710
$381$	−9.09967	−0.466190
$382$	−6.36075	−0.325444
$383$	−24.5399	−1.25393	−0.626964	−	0.779048i	$-0.715703\pi$
$383$	−24.5399	−1.25393	−0.626964	+	0.779048i	$0.715703\pi$
$384$	−2.31266	−0.118017
$385$	−47.1911	−2.40508
$386$	24.5392	1.24901
$387$	−5.59202	−0.284258
$388$	−4.43915	−0.225364
$389$	−26.1698	−1.32686	−0.663432	−	0.748237i	$-0.730901\pi$
$389$	−26.1698	−1.32686	−0.663432	+	0.748237i	$0.730901\pi$
$390$	37.1813	1.88275
$391$	−5.48131	−0.277202
$392$	5.24171	0.264746
$393$	32.9794	1.66359
$394$	22.5752	1.13732
$395$	−9.82436	−0.494317
$396$	−13.3378	−0.670248
$397$	23.5952	1.18421	0.592104	−	0.805861i	$-0.298297\pi$
$397$	23.5952	1.18421	0.592104	+	0.805861i	$0.298297\pi$
$398$	7.80557	0.391258
$399$	−20.2389	−1.01321
$400$	0.639581	0.0319790
$401$	0.199599	0.00996747	0.00498374	−	0.999988i	$-0.498414\pi$
$401$	0.199599	0.00996747	0.00498374	+	0.999988i	$0.498414\pi$
$402$	24.0067	1.19735
$403$	−27.5376	−1.37175
$404$	8.41454	0.418639
$405$	25.0070	1.24261
$406$	28.1784	1.39847
$407$	40.9702	2.03082
$408$	−1.88890	−0.0935146
$409$	−13.3572	−0.660473	−0.330236	−	0.943898i	$-0.607128\pi$
$409$	−13.3572	−0.660473	−0.330236	+	0.943898i	$0.607128\pi$
$410$	7.49299	0.370052
$411$	31.3018	1.54400
$412$	−13.4067	−0.660503
$413$	−10.5632	−0.519783
$414$	−15.7599	−0.774557
$415$	−28.7940	−1.41344
$416$	6.77003	0.331928
$417$	40.3442	1.97566
$418$	14.2059	0.694836
$419$	9.36972	0.457741	0.228870	−	0.973457i	$-0.426497\pi$
$419$	9.36972	0.457741	0.228870	+	0.973457i	$0.426497\pi$
$420$	−19.2157	−0.937629
$421$	−22.1679	−1.08040	−0.540199	−	0.841537i	$-0.681651\pi$
$421$	−22.1679	−1.08040	−0.540199	+	0.841537i	$0.681651\pi$
$422$	−2.12462	−0.103425
$423$	−23.6652	−1.15064
$424$	−4.00381	−0.194442
$425$	0.522389	0.0253396
$426$	8.76410	0.424622
$427$	−9.91224	−0.479687
$428$	−18.5138	−0.894901
$429$	88.9236	4.29327
$430$	5.65489	0.272703
$431$	31.6691	1.52545	0.762724	−	0.646724i	$-0.223861\pi$
$431$	31.6691	1.52545	0.762724	+	0.646724i	$0.223861\pi$
$432$	1.50698	0.0725046
$433$	−26.9676	−1.29598	−0.647989	−	0.761650i	$-0.724390\pi$
$433$	−26.9676	−1.29598	−0.647989	+	0.761650i	$0.724390\pi$
$434$	14.2317	0.683143
$435$	−44.2313	−2.12073
$436$	−18.5522	−0.888491
$437$	16.7857	0.802971
$438$	4.13683	0.197665
$439$	−12.8523	−0.613406	−0.306703	−	0.951805i	$-0.599226\pi$
$439$	−12.8523	−0.613406	−0.306703	+	0.951805i	$0.599226\pi$
$440$	13.4877	0.643003
$441$	12.3095	0.586167
$442$	5.52954	0.263013
$443$	−15.7113	−0.746466	−0.373233	−	0.927738i	$-0.621751\pi$
$443$	−15.7113	−0.746466	−0.373233	+	0.927738i	$0.621751\pi$
$444$	16.6826	0.791720
$445$	11.6949	0.554391
$446$	−10.7798	−0.510439
$447$	16.3437	0.773028
$448$	−3.49882	−0.165304
$449$	−17.6898	−0.834833	−0.417417	−	0.908715i	$-0.637064\pi$
$449$	−17.6898	−0.834833	−0.417417	+	0.908715i	$0.637064\pi$
$450$	1.50198	0.0708038
$451$	17.9204	0.843838
$452$	−8.25832	−0.388439
$453$	−41.9445	−1.97072
$454$	−19.9218	−0.934976
$455$	56.2516	2.63712
$456$	5.78450	0.270884
$457$	6.58922	0.308231	0.154115	−	0.988053i	$-0.450747\pi$
$457$	6.58922	0.308231	0.154115	+	0.988053i	$0.450747\pi$
$458$	−26.5699	−1.24153
$459$	1.23085	0.0574512
$460$	15.9371	0.743071
$461$	−26.9441	−1.25491	−0.627456	−	0.778652i	$-0.715904\pi$
$461$	−26.9441	−1.25491	−0.627456	+	0.778652i	$0.715904\pi$
$462$	−45.9566	−2.13809
$463$	4.59023	0.213326	0.106663	−	0.994295i	$-0.465983\pi$
$463$	4.59023	0.213326	0.106663	+	0.994295i	$0.465983\pi$
$464$	−8.05370	−0.373883
$465$	−22.3393	−1.03596
$466$	16.6924	0.773261
$467$	−15.4406	−0.714504	−0.357252	−	0.934008i	$-0.616286\pi$
$467$	−15.4406	−0.714504	−0.357252	+	0.934008i	$0.616286\pi$
$468$	15.8986	0.734912
$469$	36.3198	1.67709
$470$	23.9312	1.10387
$471$	0.334953	0.0154338
$472$	3.01909	0.138965
$473$	13.5244	0.621851
$474$	−9.56736	−0.439443
$475$	−1.59974	−0.0734012
$476$	−2.85772	−0.130983
$477$	−9.40246	−0.430509
$478$	12.3884	0.566632
$479$	18.8547	0.861494	0.430747	−	0.902473i	$-0.358250\pi$
$479$	18.8547	0.861494	0.430747	+	0.902473i	$0.358250\pi$
$480$	5.49205	0.250677
$481$	−48.8363	−2.22674
$482$	3.33049	0.151700
$483$	−54.3023	−2.47084
$484$	21.2575	0.966252
$485$	10.5420	0.478688
$486$	19.8319	0.899594
$487$	25.1738	1.14073	0.570366	−	0.821391i	$-0.306801\pi$
$487$	25.1738	1.14073	0.570366	+	0.821391i	$0.306801\pi$
$488$	2.83303	0.128245
$489$	−43.8906	−1.98480
$490$	−12.4479	−0.562339
$491$	−19.3742	−0.874346	−0.437173	−	0.899377i	$-0.644020\pi$
$491$	−19.3742	−0.874346	−0.437173	+	0.899377i	$0.644020\pi$
$492$	7.29697	0.328973
$493$	−6.57799	−0.296258
$494$	−16.9334	−0.761872
$495$	31.6743	1.42365
$496$	−4.06757	−0.182639
$497$	13.2592	0.594757
$498$	−28.0408	−1.25654
$499$	1.42716	0.0638886	0.0319443	−	0.999490i	$-0.489830\pi$
$499$	1.42716	0.0638886	0.0319443	+	0.999490i	$0.489830\pi$
$500$	10.3550	0.463091
$501$	−1.47376	−0.0658426
$502$	−30.8430	−1.37659
$503$	−30.9559	−1.38025	−0.690127	−	0.723688i	$-0.742445\pi$
$503$	−30.9559	−1.38025	−0.690127	+	0.723688i	$0.742445\pi$
$504$	−8.21654	−0.365994
$505$	−19.9827	−0.889218
$506$	38.1155	1.69444
$507$	−75.9321	−3.37226
$508$	3.93473	0.174575
$509$	−4.77683	−0.211729	−0.105865	−	0.994381i	$-0.533761\pi$
$509$	−4.77683	−0.211729	−0.105865	+	0.994381i	$0.533761\pi$
$510$	4.48573	0.198631
$511$	6.25860	0.276864
$512$	1.00000	0.0441942
$513$	−3.76931	−0.166419
$514$	−9.22497	−0.406896
$515$	31.8381	1.40295
$516$	5.50696	0.242430
$517$	57.2344	2.51717
$518$	25.2391	1.10894
$519$	−17.8137	−0.781937
$520$	−16.0773	−0.705038
$521$	41.2832	1.80865	0.904324	−	0.426847i	$-0.140376\pi$
$521$	41.2832	1.80865	0.904324	+	0.426847i	$0.140376\pi$
$522$	−18.9131	−0.827804
$523$	36.5100	1.59647	0.798235	−	0.602346i	$-0.205767\pi$
$523$	36.5100	1.59647	0.798235	+	0.602346i	$0.205767\pi$
$524$	−14.2604	−0.622969
$525$	5.17520	0.225864
$526$	−29.8728	−1.30252
$527$	−3.32226	−0.144720
$528$	13.1349	0.571623
$529$	22.0372	0.958141
$530$	9.50818	0.413009
$531$	7.08997	0.307678
$532$	8.75136	0.379420
$533$	−21.3610	−0.925249
$534$	11.3889	0.492848
$535$	43.9663	1.90083
$536$	−10.3806	−0.448373
$537$	−52.4459	−2.26321
$538$	21.5868	0.930673
$539$	−29.7707	−1.28231
$540$	−3.57875	−0.154005
$541$	−45.0498	−1.93684	−0.968421	−	0.249322i	$-0.919792\pi$
$541$	−45.0498	−1.93684	−0.968421	+	0.249322i	$0.919792\pi$
$542$	25.7079	1.10425
$543$	−56.7282	−2.43444
$544$	0.816767	0.0350186
$545$	44.0575	1.88722
$546$	54.7801	2.34437
$547$	−4.75899	−0.203480	−0.101740	−	0.994811i	$-0.532441\pi$
$547$	−4.75899	−0.203480	−0.101740	+	0.994811i	$0.532441\pi$
$548$	−13.5350	−0.578187
$549$	6.65302	0.283944
$550$	−3.63254	−0.154892
$551$	20.1442	0.858171
$552$	15.5202	0.660583
$553$	−14.4745	−0.615516
$554$	−21.1787	−0.899797
$555$	−39.6175	−1.68167
$556$	−17.4450	−0.739831
$557$	2.26627	0.0960248	0.0480124	−	0.998847i	$-0.484711\pi$
$557$	2.26627	0.0960248	0.0480124	+	0.998847i	$0.484711\pi$
$558$	−9.55219	−0.404377
$559$	−16.1210	−0.681845
$560$	8.30892	0.351116
$561$	10.7282	0.452943
$562$	32.3338	1.36392
$563$	−6.55207	−0.276137	−0.138068	−	0.990423i	$-0.544089\pi$
$563$	−6.55207	−0.276137	−0.138068	+	0.990423i	$0.544089\pi$
$564$	23.3052	0.981325
$565$	19.6117	0.825070
$566$	−18.6228	−0.782776
$567$	36.8434	1.54728
$568$	−3.78963	−0.159009
$569$	−21.4854	−0.900713	−0.450356	−	0.892849i	$-0.648703\pi$
$569$	−21.4854	−0.900713	−0.450356	+	0.892849i	$0.648703\pi$
$570$	−13.7369	−0.575376
$571$	7.76834	0.325095	0.162547	−	0.986701i	$-0.448029\pi$
$571$	7.76834	0.325095	0.162547	+	0.986701i	$0.448029\pi$
$572$	−38.4509	−1.60771
$573$	14.7102	0.614528
$574$	11.0396	0.460783
$575$	−4.29221	−0.178998
$576$	2.34838	0.0978490
$577$	−2.48378	−0.103401	−0.0517005	−	0.998663i	$-0.516464\pi$
$577$	−2.48378	−0.103401	−0.0517005	+	0.998663i	$0.516464\pi$
$578$	−16.3329	−0.679359
$579$	−56.7508	−2.35848
$580$	19.1258	0.794154
$581$	−42.4229	−1.76000
$582$	10.2662	0.425549
$583$	22.7400	0.941793
$584$	−1.78878	−0.0740201
$585$	−37.7556	−1.56100
$586$	−7.37662	−0.304726
$587$	27.8876	1.15105	0.575523	−	0.817786i	$-0.304798\pi$
$587$	27.8876	1.15105	0.575523	+	0.817786i	$0.304798\pi$
$588$	−12.1223	−0.499914
$589$	10.1740	0.419211
$590$	−7.16968	−0.295171
$591$	−52.2087	−2.14758
$592$	−7.21360	−0.296477
$593$	−34.6149	−1.42146	−0.710732	−	0.703463i	$-0.751636\pi$
$593$	−34.6149	−1.42146	−0.710732	+	0.703463i	$0.751636\pi$
$594$	−8.55900	−0.351180
$595$	6.78645	0.278217
$596$	−7.06705	−0.289478
$597$	−18.0516	−0.738802
$598$	−45.4335	−1.85792
$599$	−4.92643	−0.201288	−0.100644	−	0.994922i	$-0.532090\pi$
$599$	−4.92643	−0.201288	−0.100644	+	0.994922i	$0.532090\pi$
$600$	−1.47913	−0.0603852
$601$	35.0057	1.42791	0.713956	−	0.700191i	$-0.246902\pi$
$601$	35.0057	1.42791	0.713956	+	0.700191i	$0.246902\pi$
$602$	8.33148	0.339566
$603$	−24.3775	−0.992730
$604$	18.1369	0.737981
$605$	−50.4820	−2.05238
$606$	−19.4599	−0.790506
$607$	−14.1480	−0.574248	−0.287124	−	0.957893i	$-0.592699\pi$
$607$	−14.1480	−0.574248	−0.287124	+	0.957893i	$0.592699\pi$
$608$	−2.50124	−0.101439
$609$	−65.1669	−2.64070
$610$	−6.72782	−0.272401
$611$	−68.2232	−2.76002
$612$	1.91808	0.0775337
$613$	36.5062	1.47447	0.737237	−	0.675635i	$-0.236130\pi$
$613$	36.5062	1.47447	0.737237	+	0.675635i	$0.236130\pi$
$614$	12.9056	0.520826
$615$	−17.3287	−0.698760
$616$	19.8718	0.800657
$617$	28.9401	1.16508	0.582542	−	0.812800i	$-0.302058\pi$
$617$	28.9401	1.16508	0.582542	+	0.812800i	$0.302058\pi$
$618$	31.0052	1.24721
$619$	−4.14869	−0.166750	−0.0833750	−	0.996518i	$-0.526570\pi$
$619$	−4.14869	−0.166750	−0.0833750	+	0.996518i	$0.526570\pi$
$620$	9.65959	0.387939
$621$	−10.1133	−0.405833
$622$	−7.92889	−0.317920
$623$	17.2303	0.690318
$624$	−15.6567	−0.626772
$625$	−27.7888	−1.11155
$626$	−9.33641	−0.373158
$627$	−32.8535	−1.31204
$628$	−0.144835	−0.00577953
$629$	−5.89183	−0.234923
$630$	19.5125	0.777395
$631$	36.7584	1.46333	0.731665	−	0.681664i	$-0.238744\pi$
$631$	36.7584	1.46333	0.731665	+	0.681664i	$0.238744\pi$
$632$	4.13696	0.164559
$633$	4.91352	0.195295
$634$	−18.3394	−0.728353
$635$	−9.34411	−0.370810
$636$	9.25944	0.367161
$637$	35.4865	1.40603
$638$	45.7415	1.81093
$639$	−8.89947	−0.352058
$640$	−2.37478	−0.0938714
$641$	−31.3834	−1.23957	−0.619785	−	0.784772i	$-0.712780\pi$
$641$	−31.3834	−1.23957	−0.619785	+	0.784772i	$0.712780\pi$
$642$	42.8162	1.68982
$643$	26.1422	1.03095	0.515475	−	0.856905i	$-0.327616\pi$
$643$	26.1422	1.03095	0.515475	+	0.856905i	$0.327616\pi$
$644$	23.4805	0.925261
$645$	−13.0778	−0.514939
$646$	−2.04293	−0.0803779
$647$	13.5123	0.531224	0.265612	−	0.964080i	$-0.414426\pi$
$647$	13.5123	0.531224	0.265612	+	0.964080i	$0.414426\pi$
$648$	−10.5303	−0.413668
$649$	−17.1472	−0.673085
$650$	4.32998	0.169836
$651$	−32.9130	−1.28996
$652$	18.9784	0.743253
$653$	17.2267	0.674133	0.337067	−	0.941481i	$-0.390565\pi$
$653$	17.2267	0.674133	0.337067	+	0.941481i	$0.390565\pi$
$654$	42.9050	1.67772
$655$	33.8654	1.32323
$656$	−3.15523	−0.123191
$657$	−4.20072	−0.163886
$658$	35.2584	1.37452
$659$	−32.9031	−1.28172	−0.640862	−	0.767656i	$-0.721423\pi$
$659$	−32.9031	−1.28172	−0.640862	+	0.767656i	$0.721423\pi$
$660$	−31.1925	−1.21417
$661$	−1.62207	−0.0630910	−0.0315455	−	0.999502i	$-0.510043\pi$
$661$	−1.62207	−0.0630910	−0.0315455	+	0.999502i	$0.510043\pi$
$662$	21.0373	0.817639
$663$	−12.7879	−0.496642
$664$	12.1249	0.470538
$665$	−20.7826	−0.805913
$666$	−16.9403	−0.656422
$667$	54.0482	2.09275
$668$	0.637257	0.0246562
$669$	24.9300	0.963850
$670$	24.6516	0.952375
$671$	−16.0904	−0.621162
$672$	8.09156	0.312139
$673$	−11.5105	−0.443698	−0.221849	−	0.975081i	$-0.571209\pi$
$673$	−11.5105	−0.443698	−0.221849	+	0.975081i	$0.571209\pi$
$674$	1.97437	0.0760497
$675$	0.963835	0.0370980
$676$	32.8333	1.26282
$677$	32.5164	1.24971	0.624854	−	0.780742i	$-0.285159\pi$
$677$	32.5164	1.24971	0.624854	+	0.780742i	$0.285159\pi$
$678$	19.0986	0.733479
$679$	15.5318	0.596054
$680$	−1.93964	−0.0743819
$681$	46.0722	1.76549
$682$	23.1021	0.884624
$683$	−17.0968	−0.654193	−0.327096	−	0.944991i	$-0.606070\pi$
$683$	−17.0968	−0.654193	−0.327096	+	0.944991i	$0.606070\pi$
$684$	−5.87384	−0.224592
$685$	32.1427	1.22811
$686$	6.15193	0.234882
$687$	61.4470	2.34435
$688$	−2.38123	−0.0907835
$689$	−27.1059	−1.03265
$690$	−36.8570	−1.40312
$691$	−46.0170	−1.75057	−0.875285	−	0.483607i	$-0.839326\pi$
$691$	−46.0170	−1.75057	−0.875285	+	0.483607i	$0.839326\pi$
$692$	7.70272	0.292814
$693$	46.6664	1.77271
$694$	0.302801	0.0114942
$695$	41.4279	1.57145
$696$	18.6254	0.705995
$697$	−2.57709	−0.0976143
$698$	12.6274	0.477955
$699$	−38.6038	−1.46013
$700$	−2.23777	−0.0845799
$701$	19.2774	0.728096	0.364048	−	0.931380i	$-0.381394\pi$
$701$	19.2774	0.728096	0.364048	+	0.931380i	$0.381394\pi$
$702$	10.2023	0.385061
$703$	18.0429	0.680502
$704$	−5.67957	−0.214057
$705$	−55.3447	−2.08440
$706$	24.7975	0.933268
$707$	−29.4409	−1.10724
$708$	−6.98212	−0.262404
$709$	9.84193	0.369621	0.184811	−	0.982774i	$-0.440833\pi$
$709$	9.84193	0.369621	0.184811	+	0.982774i	$0.440833\pi$
$710$	8.99953	0.337746
$711$	9.71514	0.364346
$712$	−4.92462	−0.184558
$713$	27.2974	1.02230
$714$	6.60892	0.247333
$715$	91.3124	3.41489
$716$	22.6778	0.847509
$717$	−28.6501	−1.06996
$718$	−32.2276	−1.20272
$719$	−17.8163	−0.664436	−0.332218	−	0.943203i	$-0.607797\pi$
$719$	−17.8163	−0.664436	−0.332218	+	0.943203i	$0.607797\pi$
$720$	−5.57688	−0.207838
$721$	46.9077	1.74693
$722$	−12.7438	−0.474276
$723$	−7.70228	−0.286451
$724$	24.5295	0.911630
$725$	−5.15099	−0.191303
$726$	−49.1614	−1.82455
$727$	−14.5359	−0.539106	−0.269553	−	0.962985i	$-0.586876\pi$
$727$	−14.5359	−0.539106	−0.269553	+	0.962985i	$0.586876\pi$
$728$	−23.6871	−0.877902
$729$	−14.2736	−0.528653
$730$	4.24795	0.157224
$731$	−1.94491	−0.0719350
$732$	−6.55182	−0.242162
$733$	−29.5015	−1.08966	−0.544831	−	0.838546i	$-0.683406\pi$
$733$	−29.5015	−1.08966	−0.544831	+	0.838546i	$0.683406\pi$
$734$	29.0525	1.07235
$735$	28.7877	1.06185
$736$	−6.71098	−0.247370
$737$	58.9573	2.17172
$738$	−7.40968	−0.272754
$739$	−21.8350	−0.803215	−0.401608	−	0.915812i	$-0.631548\pi$
$739$	−21.8350	−0.803215	−0.401608	+	0.915812i	$0.631548\pi$
$740$	17.1307	0.629738
$741$	39.1612	1.43862
$742$	14.0086	0.514272
$743$	11.0550	0.405570	0.202785	−	0.979223i	$-0.435001\pi$
$743$	11.0550	0.405570	0.202785	+	0.979223i	$0.435001\pi$
$744$	9.40690	0.344874
$745$	16.7827	0.614870
$746$	24.6738	0.903372
$747$	28.4739	1.04180
$748$	−4.63889	−0.169615
$749$	64.7765	2.36688
$750$	−23.9476	−0.874444
$751$	32.5908	1.18926	0.594628	−	0.804001i	$-0.297299\pi$
$751$	32.5908	1.18926	0.594628	+	0.804001i	$0.297299\pi$
$752$	−10.0772	−0.367479
$753$	71.3292	2.59938
$754$	−54.5238	−1.98564
$755$	−43.0712	−1.56752
$756$	−5.27265	−0.191764
$757$	−30.6344	−1.11343	−0.556714	−	0.830704i	$-0.687938\pi$
$757$	−30.6344	−1.11343	−0.556714	+	0.830704i	$0.687938\pi$
$758$	−19.1914	−0.697064
$759$	−88.1480	−3.19957
$760$	5.93989	0.215462
$761$	38.2466	1.38644	0.693218	−	0.720728i	$-0.256192\pi$
$761$	38.2466	1.38644	0.693218	+	0.720728i	$0.256192\pi$
$762$	−9.09967	−0.329646
$763$	64.9109	2.34993
$764$	−6.36075	−0.230124
$765$	−4.55501	−0.164687
$766$	−24.5399	−0.886661
$767$	20.4393	0.738022
$768$	−2.31266	−0.0834508
$769$	−27.1192	−0.977945	−0.488972	−	0.872299i	$-0.662628\pi$
$769$	−27.1192	−0.977945	−0.488972	+	0.872299i	$0.662628\pi$
$770$	−47.1911	−1.70065
$771$	21.3342	0.768331
$772$	24.5392	0.883186
$773$	−21.7495	−0.782276	−0.391138	−	0.920332i	$-0.627919\pi$
$773$	−21.7495	−0.782276	−0.391138	+	0.920332i	$0.627919\pi$
$774$	−5.59202	−0.201001
$775$	−2.60154	−0.0934501
$776$	−4.43915	−0.159356
$777$	−58.3693	−2.09399
$778$	−26.1698	−0.938234
$779$	7.89198	0.282760
$780$	37.1813	1.33131
$781$	21.5235	0.770170
$782$	−5.48131	−0.196011
$783$	−12.1368	−0.433732
$784$	5.24171	0.187204
$785$	0.343951	0.0122761
$786$	32.9794	1.17634
$787$	38.4761	1.37152	0.685762	−	0.727825i	$-0.259469\pi$
$787$	38.4761	1.37152	0.685762	+	0.727825i	$0.259469\pi$
$788$	22.5752	0.804208
$789$	69.0856	2.45951
$790$	−9.82436	−0.349535
$791$	28.8943	1.02736
$792$	−13.3378	−0.473937
$793$	19.1797	0.681090
$794$	23.5952	0.837362
$795$	−21.9891	−0.779875
$796$	7.80557	0.276661
$797$	−21.7817	−0.771547	−0.385773	−	0.922594i	$-0.626065\pi$
$797$	−21.7817	−0.771547	−0.385773	+	0.922594i	$0.626065\pi$
$798$	−20.2389	−0.716449
$799$	−8.23076	−0.291183
$800$	0.639581	0.0226126
$801$	−11.5649	−0.408624
$802$	0.199599	0.00704807
$803$	10.1595	0.358521
$804$	24.0067	0.846652
$805$	−55.7610	−1.96532
$806$	−27.5376	−0.969970
$807$	−49.9229	−1.75737
$808$	8.41454	0.296023
$809$	0.755164	0.0265502	0.0132751	−	0.999912i	$-0.495774\pi$
$809$	0.755164	0.0265502	0.0132751	+	0.999912i	$0.495774\pi$
$810$	25.0070	0.878658
$811$	0.849057	0.0298144	0.0149072	−	0.999889i	$-0.495255\pi$
$811$	0.849057	0.0298144	0.0149072	+	0.999889i	$0.495255\pi$
$812$	28.1784	0.988868
$813$	−59.4534	−2.08512
$814$	40.9702	1.43600
$815$	−45.0696	−1.57872
$816$	−1.88890	−0.0661248
$817$	5.95601	0.208374
$818$	−13.3572	−0.467025
$819$	−55.6262	−1.94374
$820$	7.49299	0.261666
$821$	−32.4218	−1.13153	−0.565764	−	0.824567i	$-0.691419\pi$
$821$	−32.4218	−1.13153	−0.565764	+	0.824567i	$0.691419\pi$
$822$	31.3018	1.09178
$823$	36.0983	1.25831	0.629154	−	0.777281i	$-0.283402\pi$
$823$	36.0983	1.25831	0.629154	+	0.777281i	$0.283402\pi$
$824$	−13.4067	−0.467046
$825$	8.40083	0.292479
$826$	−10.5632	−0.367542
$827$	−50.7297	−1.76404	−0.882022	−	0.471207i	$-0.843818\pi$
$827$	−50.7297	−1.76404	−0.882022	+	0.471207i	$0.843818\pi$
$828$	−15.7599	−0.547695
$829$	−27.9507	−0.970766	−0.485383	−	0.874302i	$-0.661320\pi$
$829$	−27.9507	−0.970766	−0.485383	+	0.874302i	$0.661320\pi$
$830$	−28.7940	−0.999455
$831$	48.9791	1.69906
$832$	6.77003	0.234709
$833$	4.28126	0.148337
$834$	40.3442	1.39700
$835$	−1.51335	−0.0523715
$836$	14.2059	0.491323
$837$	−6.12975	−0.211875
$838$	9.36972	0.323672
$839$	−23.1805	−0.800281	−0.400140	−	0.916454i	$-0.631039\pi$
$839$	−23.1805	−0.800281	−0.400140	+	0.916454i	$0.631039\pi$
$840$	−19.2157	−0.663004
$841$	35.8620	1.23662
$842$	−22.1679	−0.763957
$843$	−74.7768	−2.57545
$844$	−2.12462	−0.0731325
$845$	−77.9719	−2.68231
$846$	−23.6652	−0.813625
$847$	−74.3762	−2.55560
$848$	−4.00381	−0.137492
$849$	43.0682	1.47810
$850$	0.522389	0.0179178
$851$	48.4103	1.65949
$852$	8.76410	0.300253
$853$	0.844806	0.0289256	0.0144628	−	0.999895i	$-0.495396\pi$
$853$	0.844806	0.0289256	0.0144628	+	0.999895i	$0.495396\pi$
$854$	−9.91224	−0.339190
$855$	13.9491	0.477049
$856$	−18.5138	−0.632790
$857$	−36.1677	−1.23547	−0.617733	−	0.786388i	$-0.711949\pi$
$857$	−36.1677	−1.23547	−0.617733	+	0.786388i	$0.711949\pi$
$858$	88.9236	3.03580
$859$	14.4930	0.494494	0.247247	−	0.968953i	$-0.420474\pi$
$859$	14.4930	0.494494	0.247247	+	0.968953i	$0.420474\pi$
$860$	5.65489	0.192830
$861$	−25.5307	−0.870086
$862$	31.6691	1.07866
$863$	37.1108	1.26327	0.631634	−	0.775267i	$-0.282385\pi$
$863$	37.1108	1.26327	0.631634	+	0.775267i	$0.282385\pi$
$864$	1.50698	0.0512685
$865$	−18.2923	−0.621956
$866$	−26.9676	−0.916395
$867$	37.7724	1.28282
$868$	14.2317	0.483055
$869$	−23.4961	−0.797052
$870$	−44.2313	−1.49958
$871$	−70.2769	−2.38124
$872$	−18.5522	−0.628258
$873$	−10.4248	−0.352826
$874$	16.7857	0.567786
$875$	−36.2304	−1.22481
$876$	4.13683	0.139770
$877$	−25.6474	−0.866050	−0.433025	−	0.901382i	$-0.642554\pi$
$877$	−25.6474	−0.866050	−0.433025	+	0.901382i	$0.642554\pi$
$878$	−12.8523	−0.433744
$879$	17.0596	0.575406
$880$	13.4877	0.454671
$881$	−19.1318	−0.644567	−0.322283	−	0.946643i	$-0.604450\pi$
$881$	−19.1318	−0.644567	−0.322283	+	0.946643i	$0.604450\pi$
$882$	12.3095	0.414483
$883$	−10.0621	−0.338616	−0.169308	−	0.985563i	$-0.554153\pi$
$883$	−10.0621	−0.338616	−0.169308	+	0.985563i	$0.554153\pi$
$884$	5.52954	0.185979
$885$	16.5810	0.557364
$886$	−15.7113	−0.527831
$887$	19.8489	0.666459	0.333230	−	0.942846i	$-0.391862\pi$
$887$	19.8489	0.666459	0.333230	+	0.942846i	$0.391862\pi$
$888$	16.6826	0.559831
$889$	−13.7669	−0.461727
$890$	11.6949	0.392013
$891$	59.8074	2.00362
$892$	−10.7798	−0.360935
$893$	25.2055	0.843472
$894$	16.3437	0.546614
$895$	−53.8548	−1.80017
$896$	−3.49882	−0.116887
$897$	105.072	3.50826
$898$	−17.6898	−0.590316
$899$	32.7590	1.09257
$900$	1.50198	0.0500659
$901$	−3.27018	−0.108946
$902$	17.9204	0.596683
$903$	−19.2678	−0.641194
$904$	−8.25832	−0.274668
$905$	−58.2521	−1.93636
$906$	−41.9445	−1.39351
$907$	−53.0501	−1.76150	−0.880750	−	0.473582i	$-0.842961\pi$
$907$	−53.0501	−1.76150	−0.880750	+	0.473582i	$0.842961\pi$
$908$	−19.9218	−0.661128
$909$	19.7605	0.655415
$910$	56.2516	1.86472
$911$	23.1909	0.768348	0.384174	−	0.923261i	$-0.374486\pi$
$911$	23.1909	0.768348	0.384174	+	0.923261i	$0.374486\pi$
$912$	5.78450	0.191544
$913$	−68.8644	−2.27908
$914$	6.58922	0.217952
$915$	15.5591	0.514369
$916$	−26.5699	−0.877894
$917$	49.8946	1.64766
$918$	1.23085	0.0406242
$919$	−7.08701	−0.233779	−0.116889	−	0.993145i	$-0.537292\pi$
$919$	−7.08701	−0.233779	−0.116889	+	0.993145i	$0.537292\pi$
$920$	15.9371	0.525431
$921$	−29.8461	−0.983462
$922$	−26.9441	−0.887357
$923$	−25.6559	−0.844474
$924$	−45.9566	−1.51186
$925$	−4.61368	−0.151697
$926$	4.59023	0.150844
$927$	−31.4841	−1.03407
$928$	−8.05370	−0.264375
$929$	−9.44484	−0.309875	−0.154938	−	0.987924i	$-0.549518\pi$
$929$	−9.44484	−0.309875	−0.154938	+	0.987924i	$0.549518\pi$
$930$	−22.3393	−0.732535
$931$	−13.1108	−0.429688
$932$	16.6924	0.546778
$933$	18.3368	0.600320
$934$	−15.4406	−0.505231
$935$	11.0163	0.360273
$936$	15.8986	0.519661
$937$	4.38701	0.143317	0.0716586	−	0.997429i	$-0.477171\pi$
$937$	4.38701	0.143317	0.0716586	+	0.997429i	$0.477171\pi$
$938$	36.3198	1.18588
$939$	21.5919	0.704625
$940$	23.9312	0.780551
$941$	22.5666	0.735649	0.367824	−	0.929895i	$-0.380103\pi$
$941$	22.5666	0.735649	0.367824	+	0.929895i	$0.380103\pi$
$942$	0.334953	0.0109134
$943$	21.1747	0.689543
$944$	3.01909	0.0982631
$945$	12.5214	0.407320
$946$	13.5244	0.439715
$947$	−41.7645	−1.35716	−0.678582	−	0.734525i	$-0.737405\pi$
$947$	−41.7645	−1.35716	−0.678582	+	0.734525i	$0.737405\pi$
$948$	−9.56736	−0.310733
$949$	−12.1101	−0.393110
$950$	−1.59974	−0.0519025
$951$	42.4128	1.37533
$952$	−2.85772	−0.0926192
$953$	34.2045	1.10799	0.553996	−	0.832520i	$-0.313102\pi$
$953$	34.2045	1.10799	0.553996	+	0.832520i	$0.313102\pi$
$954$	−9.40246	−0.304416
$955$	15.1054	0.488798
$956$	12.3884	0.400670
$957$	−105.784	−3.41953
$958$	18.8547	0.609168
$959$	47.3565	1.52922
$960$	5.49205	0.177255
$961$	−14.4548	−0.466285
$962$	−48.8363	−1.57455
$963$	−43.4775	−1.40104
$964$	3.33049	0.107268
$965$	−58.2753	−1.87595
$966$	−54.3023	−1.74715
$967$	44.1467	1.41966	0.709831	−	0.704372i	$-0.248771\pi$
$967$	44.1467	1.41966	0.709831	+	0.704372i	$0.248771\pi$
$968$	21.2575	0.683243
$969$	4.72459	0.151776
$970$	10.5420	0.338483
$971$	29.8442	0.957748	0.478874	−	0.877884i	$-0.341045\pi$
$971$	29.8442	0.957748	0.478874	+	0.877884i	$0.341045\pi$
$972$	19.8319	0.636109
$973$	61.0367	1.95675
$974$	25.1738	0.806619
$975$	−10.0138	−0.320697
$976$	2.83303	0.0906830
$977$	−33.0161	−1.05628	−0.528139	−	0.849158i	$-0.677110\pi$
$977$	−33.0161	−1.05628	−0.528139	+	0.849158i	$0.677110\pi$
$978$	−43.8906	−1.40347
$979$	27.9697	0.893916
$980$	−12.4479	−0.397634
$981$	−43.5677	−1.39101
$982$	−19.3742	−0.618256
$983$	−2.06747	−0.0659422	−0.0329711	−	0.999456i	$-0.510497\pi$
$983$	−2.06747	−0.0659422	−0.0329711	+	0.999456i	$0.510497\pi$
$984$	7.29697	0.232619
$985$	−53.6112	−1.70819
$986$	−6.57799	−0.209486
$987$	−81.5405	−2.59546
$988$	−16.9334	−0.538725
$989$	15.9804	0.508146
$990$	31.6743	1.00667
$991$	−2.49595	−0.0792864	−0.0396432	−	0.999214i	$-0.512622\pi$
$991$	−2.49595	−0.0792864	−0.0396432	+	0.999214i	$0.512622\pi$
$992$	−4.06757	−0.129146
$993$	−48.6521	−1.54393
$994$	13.2592	0.420556
$995$	−18.5365	−0.587647
$996$	−28.0408	−0.888506
$997$	−23.4605	−0.743002	−0.371501	−	0.928433i	$-0.621157\pi$
$997$	−23.4605	−0.743002	−0.371501	+	0.928433i	$0.621157\pi$
$998$	1.42716	0.0451761
$999$	−10.8708	−0.343936

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.d.1.19	✓	101

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.d.1.19	✓	101	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.31266 q^{3} +1.00000 q^{4} -2.37478 q^{5} -2.31266 q^{6} -3.49882 q^{7} +1.00000 q^{8} +2.34838 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.31266 q^{3} +1.00000 q^{4} -2.37478 q^{5} -2.31266 q^{6} -3.49882 q^{7} +1.00000 q^{8} +2.34838 q^{9} -2.37478 q^{10} -5.67957 q^{11} -2.31266 q^{12} +6.77003 q^{13} -3.49882 q^{14} +5.49205 q^{15} +1.00000 q^{16} +0.816767 q^{17} +2.34838 q^{18} -2.50124 q^{19} -2.37478 q^{20} +8.09156 q^{21} -5.67957 q^{22} -6.71098 q^{23} -2.31266 q^{24} +0.639581 q^{25} +6.77003 q^{26} +1.50698 q^{27} -3.49882 q^{28} -8.05370 q^{29} +5.49205 q^{30} -4.06757 q^{31} +1.00000 q^{32} +13.1349 q^{33} +0.816767 q^{34} +8.30892 q^{35} +2.34838 q^{36} -7.21360 q^{37} -2.50124 q^{38} -15.6567 q^{39} -2.37478 q^{40} -3.15523 q^{41} +8.09156 q^{42} -2.38123 q^{43} -5.67957 q^{44} -5.57688 q^{45} -6.71098 q^{46} -10.0772 q^{47} -2.31266 q^{48} +5.24171 q^{49} +0.639581 q^{50} -1.88890 q^{51} +6.77003 q^{52} -4.00381 q^{53} +1.50698 q^{54} +13.4877 q^{55} -3.49882 q^{56} +5.78450 q^{57} -8.05370 q^{58} +3.01909 q^{59} +5.49205 q^{60} +2.83303 q^{61} -4.06757 q^{62} -8.21654 q^{63} +1.00000 q^{64} -16.0773 q^{65} +13.1349 q^{66} -10.3806 q^{67} +0.816767 q^{68} +15.5202 q^{69} +8.30892 q^{70} -3.78963 q^{71} +2.34838 q^{72} -1.78878 q^{73} -7.21360 q^{74} -1.47913 q^{75} -2.50124 q^{76} +19.8718 q^{77} -15.6567 q^{78} +4.13696 q^{79} -2.37478 q^{80} -10.5303 q^{81} -3.15523 q^{82} +12.1249 q^{83} +8.09156 q^{84} -1.93964 q^{85} -2.38123 q^{86} +18.6254 q^{87} -5.67957 q^{88} -4.92462 q^{89} -5.57688 q^{90} -23.6871 q^{91} -6.71098 q^{92} +9.40690 q^{93} -10.0772 q^{94} +5.93989 q^{95} -2.31266 q^{96} -4.43915 q^{97} +5.24171 q^{98} -13.3378 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9}+O(q^{10}) \) 101 * q + 101 * q^2 + 10 * q^3 + 101 * q^4 + 19 * q^5 + 10 * q^6 + 42 * q^7 + 101 * q^8 + 147 * q^9	\( 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9} + 19 q^{10} + 4 q^{11} + 10 q^{12} + 58 q^{13} + 42 q^{14} + 27 q^{15} + 101 q^{16} + 34 q^{17} + 147 q^{18} + 36 q^{19} + 19 q^{20} + 45 q^{21} + 4 q^{22} + 47 q^{23} + 10 q^{24} + 174 q^{25} + 58 q^{26} + 31 q^{27} + 42 q^{28} + 62 q^{29} + 27 q^{30} + 47 q^{31} + 101 q^{32} + 55 q^{33} + 34 q^{34} + 16 q^{35} + 147 q^{36} + 90 q^{37} + 36 q^{38} + 50 q^{39} + 19 q^{40} + 54 q^{41} + 45 q^{42} + 65 q^{43} + 4 q^{44} + 47 q^{45} + 47 q^{46} + 54 q^{47} + 10 q^{48} + 189 q^{49} + 174 q^{50} + 36 q^{51} + 58 q^{52} + 94 q^{53} + 31 q^{54} + 68 q^{55} + 42 q^{56} + 79 q^{57} + 62 q^{58} - 6 q^{59} + 27 q^{60} + 58 q^{61} + 47 q^{62} + 117 q^{63} + 101 q^{64} + 89 q^{65} + 55 q^{66} + 127 q^{67} + 34 q^{68} + 45 q^{69} + 16 q^{70} + 87 q^{71} + 147 q^{72} + 83 q^{73} + 90 q^{74} - 4 q^{75} + 36 q^{76} + 53 q^{77} + 50 q^{78} + 74 q^{79} + 19 q^{80} + 241 q^{81} + 54 q^{82} + 11 q^{83} + 45 q^{84} + 120 q^{85} + 65 q^{86} + 37 q^{87} + 4 q^{88} + 89 q^{89} + 47 q^{90} + 31 q^{91} + 47 q^{92} + 123 q^{93} + 54 q^{94} + 61 q^{95} + 10 q^{96} + 85 q^{97} + 189 q^{98} - 55 q^{99}+O(q^{100}) \) 101 * q + 101 * q^2 + 10 * q^3 + 101 * q^4 + 19 * q^5 + 10 * q^6 + 42 * q^7 + 101 * q^8 + 147 * q^9 + 19 * q^10 + 4 * q^11 + 10 * q^12 + 58 * q^13 + 42 * q^14 + 27 * q^15 + 101 * q^16 + 34 * q^17 + 147 * q^18 + 36 * q^19 + 19 * q^20 + 45 * q^21 + 4 * q^22 + 47 * q^23 + 10 * q^24 + 174 * q^25 + 58 * q^26 + 31 * q^27 + 42 * q^28 + 62 * q^29 + 27 * q^30 + 47 * q^31 + 101 * q^32 + 55 * q^33 + 34 * q^34 + 16 * q^35 + 147 * q^36 + 90 * q^37 + 36 * q^38 + 50 * q^39 + 19 * q^40 + 54 * q^41 + 45 * q^42 + 65 * q^43 + 4 * q^44 + 47 * q^45 + 47 * q^46 + 54 * q^47 + 10 * q^48 + 189 * q^49 + 174 * q^50 + 36 * q^51 + 58 * q^52 + 94 * q^53 + 31 * q^54 + 68 * q^55 + 42 * q^56 + 79 * q^57 + 62 * q^58 - 6 * q^59 + 27 * q^60 + 58 * q^61 + 47 * q^62 + 117 * q^63 + 101 * q^64 + 89 * q^65 + 55 * q^66 + 127 * q^67 + 34 * q^68 + 45 * q^69 + 16 * q^70 + 87 * q^71 + 147 * q^72 + 83 * q^73 + 90 * q^74 - 4 * q^75 + 36 * q^76 + 53 * q^77 + 50 * q^78 + 74 * q^79 + 19 * q^80 + 241 * q^81 + 54 * q^82 + 11 * q^83 + 45 * q^84 + 120 * q^85 + 65 * q^86 + 37 * q^87 + 4 * q^88 + 89 * q^89 + 47 * q^90 + 31 * q^91 + 47 * q^92 + 123 * q^93 + 54 * q^94 + 61 * q^95 + 10 * q^96 + 85 * q^97 + 189 * q^98 - 55 * q^99

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