LMFDB - Embedded newform 4033.2.a.f.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4033,2,Mod(1,4033)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4033, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4033 = 37 \cdot 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4033.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:	$32.2036671352$
Analytic rank:	$0$
Dimension:	$85$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.03156 q^{2} +3.25300 q^{3} +2.12723 q^{4} +1.52354 q^{5} -6.60865 q^{6} -4.26037 q^{7} -0.258468 q^{8} +7.58200 q^{9} +O(q^{10})$	\(q-2.03156 q^{2} +3.25300 q^{3} +2.12723 q^{4} +1.52354 q^{5} -6.60865 q^{6} -4.26037 q^{7} -0.258468 q^{8} +7.58200 q^{9} -3.09515 q^{10} -0.546050 q^{11} +6.91986 q^{12} +0.570945 q^{13} +8.65518 q^{14} +4.95606 q^{15} -3.72936 q^{16} -1.08709 q^{17} -15.4033 q^{18} +2.45317 q^{19} +3.24091 q^{20} -13.8590 q^{21} +1.10933 q^{22} -8.77079 q^{23} -0.840796 q^{24} -2.67884 q^{25} -1.15991 q^{26} +14.9052 q^{27} -9.06277 q^{28} +4.87255 q^{29} -10.0685 q^{30} +4.77597 q^{31} +8.09335 q^{32} -1.77630 q^{33} +2.20849 q^{34} -6.49083 q^{35} +16.1286 q^{36} +1.00000 q^{37} -4.98375 q^{38} +1.85728 q^{39} -0.393786 q^{40} -4.73451 q^{41} +28.1553 q^{42} +5.48913 q^{43} -1.16157 q^{44} +11.5514 q^{45} +17.8184 q^{46} +8.15516 q^{47} -12.1316 q^{48} +11.1507 q^{49} +5.44221 q^{50} -3.53631 q^{51} +1.21453 q^{52} +13.8731 q^{53} -30.2808 q^{54} -0.831927 q^{55} +1.10117 q^{56} +7.98015 q^{57} -9.89887 q^{58} +14.4576 q^{59} +10.5427 q^{60} -1.32357 q^{61} -9.70265 q^{62} -32.3021 q^{63} -8.98338 q^{64} +0.869855 q^{65} +3.60865 q^{66} +2.92586 q^{67} -2.31249 q^{68} -28.5314 q^{69} +13.1865 q^{70} +8.48759 q^{71} -1.95970 q^{72} +7.16098 q^{73} -2.03156 q^{74} -8.71425 q^{75} +5.21844 q^{76} +2.32637 q^{77} -3.77318 q^{78} +17.5571 q^{79} -5.68182 q^{80} +25.7407 q^{81} +9.61844 q^{82} -15.5410 q^{83} -29.4812 q^{84} -1.65623 q^{85} -11.1515 q^{86} +15.8504 q^{87} +0.141136 q^{88} -15.4639 q^{89} -23.4674 q^{90} -2.43244 q^{91} -18.6575 q^{92} +15.5362 q^{93} -16.5677 q^{94} +3.73749 q^{95} +26.3276 q^{96} +4.97454 q^{97} -22.6534 q^{98} -4.14015 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9}+O(q^{10}) $ 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9	\( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9} + 9 q^{10} + 37 q^{11} + 44 q^{12} + 14 q^{13} + 26 q^{14} + 27 q^{15} + 85 q^{16} + 34 q^{17} + 3 q^{18} + 15 q^{19} + 15 q^{20} + 17 q^{21} + q^{22} + 72 q^{23} + 15 q^{24} + 85 q^{25} + 33 q^{26} + 69 q^{27} + 7 q^{28} + 19 q^{29} - 9 q^{30} + 23 q^{31} + 51 q^{32} + 32 q^{33} + 49 q^{34} + 40 q^{35} + 121 q^{36} + 85 q^{37} + 84 q^{38} + 39 q^{39} + 22 q^{40} + 55 q^{41} - 28 q^{42} + 78 q^{44} + 28 q^{45} + 17 q^{46} + 184 q^{47} + 97 q^{48} + 88 q^{49} + 26 q^{50} + 27 q^{51} + 73 q^{52} + 64 q^{53} + 31 q^{54} + 39 q^{55} + 68 q^{56} - 33 q^{57} + 28 q^{58} + 60 q^{59} - 22 q^{60} + 7 q^{61} + 70 q^{62} + 28 q^{63} + 102 q^{64} + 17 q^{65} - 15 q^{66} + 82 q^{67} + 92 q^{68} + 22 q^{69} - 41 q^{70} + 113 q^{71} - 19 q^{73} + 11 q^{74} + 45 q^{75} + 34 q^{76} + 64 q^{77} + 29 q^{78} + 23 q^{79} + 54 q^{80} + 149 q^{81} + 4 q^{82} + 100 q^{83} - 49 q^{84} - 5 q^{85} - 24 q^{86} + 65 q^{87} + 14 q^{88} + 84 q^{89} - 21 q^{90} + 32 q^{91} + 95 q^{92} + 19 q^{93} - 47 q^{94} + 102 q^{95} + 29 q^{96} + 7 q^{97} + 26 q^{98} + 107 q^{99}+O(q^{100}) \) 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9 + 9 * q^10 + 37 * q^11 + 44 * q^12 + 14 * q^13 + 26 * q^14 + 27 * q^15 + 85 * q^16 + 34 * q^17 + 3 * q^18 + 15 * q^19 + 15 * q^20 + 17 * q^21 + q^22 + 72 * q^23 + 15 * q^24 + 85 * q^25 + 33 * q^26 + 69 * q^27 + 7 * q^28 + 19 * q^29 - 9 * q^30 + 23 * q^31 + 51 * q^32 + 32 * q^33 + 49 * q^34 + 40 * q^35 + 121 * q^36 + 85 * q^37 + 84 * q^38 + 39 * q^39 + 22 * q^40 + 55 * q^41 - 28 * q^42 + 78 * q^44 + 28 * q^45 + 17 * q^46 + 184 * q^47 + 97 * q^48 + 88 * q^49 + 26 * q^50 + 27 * q^51 + 73 * q^52 + 64 * q^53 + 31 * q^54 + 39 * q^55 + 68 * q^56 - 33 * q^57 + 28 * q^58 + 60 * q^59 - 22 * q^60 + 7 * q^61 + 70 * q^62 + 28 * q^63 + 102 * q^64 + 17 * q^65 - 15 * q^66 + 82 * q^67 + 92 * q^68 + 22 * q^69 - 41 * q^70 + 113 * q^71 - 19 * q^73 + 11 * q^74 + 45 * q^75 + 34 * q^76 + 64 * q^77 + 29 * q^78 + 23 * q^79 + 54 * q^80 + 149 * q^81 + 4 * q^82 + 100 * q^83 - 49 * q^84 - 5 * q^85 - 24 * q^86 + 65 * q^87 + 14 * q^88 + 84 * q^89 - 21 * q^90 + 32 * q^91 + 95 * q^92 + 19 * q^93 - 47 * q^94 + 102 * q^95 + 29 * q^96 + 7 * q^97 + 26 * q^98 + 107 * 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 $

$2$	−2.03156	−1.43653	−0.718264	−	0.695771i	$-0.755063\pi$
$2$	−2.03156	−1.43653	−0.718264	+	0.695771i	$0.755063\pi$
$3$	3.25300	1.87812	0.939060	−	0.343754i	$-0.111699\pi$
$3$	3.25300	1.87812	0.939060	+	0.343754i	$0.111699\pi$
$4$	2.12723	1.06361
$5$	1.52354	0.681346	0.340673	−	0.940182i	$-0.389345\pi$
$5$	1.52354	0.681346	0.340673	+	0.940182i	$0.389345\pi$
$6$	−6.60865	−2.69797
$7$	−4.26037	−1.61027	−0.805134	−	0.593093i	$-0.797907\pi$
$7$	−4.26037	−1.61027	−0.805134	+	0.593093i	$0.797907\pi$
$8$	−0.258468	−0.0913823
$9$	7.58200	2.52733
$10$	−3.09515	−0.978773
$11$	−0.546050	−0.164640	−0.0823201	−	0.996606i	$-0.526233\pi$
$11$	−0.546050	−0.164640	−0.0823201	+	0.996606i	$0.526233\pi$
$12$	6.91986	1.99759
$13$	0.570945	0.158352	0.0791758	−	0.996861i	$-0.474771\pi$
$13$	0.570945	0.158352	0.0791758	+	0.996861i	$0.474771\pi$
$14$	8.65518	2.31320
$15$	4.95606	1.27965
$16$	−3.72936	−0.932340
$17$	−1.08709	−0.263659	−0.131829	−	0.991272i	$-0.542085\pi$
$17$	−1.08709	−0.263659	−0.131829	+	0.991272i	$0.542085\pi$
$18$	−15.4033	−3.63058
$19$	2.45317	0.562795	0.281398	−	0.959591i	$-0.409202\pi$
$19$	2.45317	0.562795	0.281398	+	0.959591i	$0.409202\pi$
$20$	3.24091	0.724689
$21$	−13.8590	−3.02428
$22$	1.10933	0.236510
$23$	−8.77079	−1.82884	−0.914418	−	0.404771i	$-0.867351\pi$
$23$	−8.77079	−1.82884	−0.914418	+	0.404771i	$0.867351\pi$
$24$	−0.840796	−0.171627
$25$	−2.67884	−0.535767
$26$	−1.15991	−0.227477
$27$	14.9052	2.86851
$28$	−9.06277	−1.71270
$29$	4.87255	0.904811	0.452405	−	0.891812i	$-0.350566\pi$
$29$	4.87255	0.904811	0.452405	+	0.891812i	$0.350566\pi$
$30$	−10.0685	−1.83825
$31$	4.77597	0.857789	0.428895	−	0.903355i	$-0.358903\pi$
$31$	4.77597	0.857789	0.428895	+	0.903355i	$0.358903\pi$
$32$	8.09335	1.43072
$33$	−1.77630	−0.309214
$34$	2.20849	0.378753
$35$	−6.49083	−1.09715
$36$	16.1286	2.68810
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−4.98375	−0.808471
$39$	1.85728	0.297403
$40$	−0.393786	−0.0622630
$41$	−4.73451	−0.739407	−0.369703	−	0.929150i	$-0.620541\pi$
$41$	−4.73451	−0.739407	−0.369703	+	0.929150i	$0.620541\pi$
$42$	28.1553	4.34446
$43$	5.48913	0.837084	0.418542	−	0.908197i	$-0.362541\pi$
$43$	5.48913	0.837084	0.418542	+	0.908197i	$0.362541\pi$
$44$	−1.16157	−0.175114
$45$	11.5514	1.72199
$46$	17.8184	2.62717
$47$	8.15516	1.18955	0.594776	−	0.803891i	$-0.297241\pi$
$47$	8.15516	1.18955	0.594776	+	0.803891i	$0.297241\pi$
$48$	−12.1316	−1.75105
$49$	11.1507	1.59296
$50$	5.44221	0.769645
$51$	−3.53631	−0.495183
$52$	1.21453	0.168425
$53$	13.8731	1.90561	0.952806	−	0.303581i	$-0.0981824\pi$
$53$	13.8731	1.90561	0.952806	+	0.303581i	$0.0981824\pi$
$54$	−30.2808	−4.12070
$55$	−0.831927	−0.112177
$56$	1.10117	0.147150
$57$	7.98015	1.05700
$58$	−9.89887	−1.29979
$59$	14.4576	1.88222	0.941111	−	0.338097i	$-0.109783\pi$
$59$	14.4576	1.88222	0.941111	+	0.338097i	$0.109783\pi$
$60$	10.5427	1.36105
$61$	−1.32357	−0.169466	−0.0847328	−	0.996404i	$-0.527004\pi$
$61$	−1.32357	−0.169466	−0.0847328	+	0.996404i	$0.527004\pi$
$62$	−9.70265	−1.23224
$63$	−32.3021	−4.06968
$64$	−8.98338	−1.12292
$65$	0.869855	0.107892
$66$	3.60865	0.444195
$67$	2.92586	0.357451	0.178725	−	0.983899i	$-0.442803\pi$
$67$	2.92586	0.357451	0.178725	+	0.983899i	$0.442803\pi$
$68$	−2.31249	−0.280431
$69$	−28.5314	−3.43477
$70$	13.1865	1.57609
$71$	8.48759	1.00729	0.503646	−	0.863910i	$-0.331992\pi$
$71$	8.48759	1.00729	0.503646	+	0.863910i	$0.331992\pi$
$72$	−1.95970	−0.230953
$73$	7.16098	0.838129	0.419065	−	0.907956i	$-0.362358\pi$
$73$	7.16098	0.838129	0.419065	+	0.907956i	$0.362358\pi$
$74$	−2.03156	−0.236164
$75$	−8.71425	−1.00623
$76$	5.21844	0.598597
$77$	2.32637	0.265115
$78$	−3.77318	−0.427228
$79$	17.5571	1.97533	0.987664	−	0.156590i	$-0.0500500\pi$
$79$	17.5571	1.97533	0.987664	+	0.156590i	$0.0500500\pi$
$80$	−5.68182	−0.635246
$81$	25.7407	2.86007
$82$	9.61844	1.06218
$83$	−15.5410	−1.70585	−0.852926	−	0.522032i	$-0.825174\pi$
$83$	−15.5410	−1.70585	−0.852926	+	0.522032i	$0.825174\pi$
$84$	−29.4812	−3.21666
$85$	−1.65623	−0.179643
$86$	−11.1515	−1.20249
$87$	15.8504	1.69934
$88$	0.141136	0.0150452
$89$	−15.4639	−1.63917	−0.819587	−	0.572954i	$-0.805797\pi$
$89$	−15.4639	−1.63917	−0.819587	+	0.572954i	$0.805797\pi$
$90$	−23.4674	−2.47368
$91$	−2.43244	−0.254989
$92$	−18.6575	−1.94517
$93$	15.5362	1.61103
$94$	−16.5677	−1.70883
$95$	3.73749	0.383459
$96$	26.3276	2.68705
$97$	4.97454	0.505088	0.252544	−	0.967585i	$-0.418733\pi$
$97$	4.97454	0.505088	0.252544	+	0.967585i	$0.418733\pi$
$98$	−22.6534	−2.28834
$99$	−4.14015	−0.416100
$100$	−5.69849	−0.569849
$101$	−9.88072	−0.983169	−0.491584	−	0.870830i	$-0.663582\pi$
$101$	−9.88072	−0.983169	−0.491584	+	0.870830i	$0.663582\pi$
$102$	7.18422	0.711344
$103$	12.1767	1.19981	0.599905	−	0.800071i	$-0.295205\pi$
$103$	12.1767	1.19981	0.599905	+	0.800071i	$0.295205\pi$
$104$	−0.147571	−0.0144705
$105$	−21.1146	−2.06058
$106$	−28.1839	−2.73746
$107$	−2.08068	−0.201147	−0.100573	−	0.994930i	$-0.532068\pi$
$107$	−2.08068	−0.201147	−0.100573	+	0.994930i	$0.532068\pi$
$108$	31.7068	3.05099
$109$	−1.00000	−0.0957826
$109$	−1.00000	−0.0957826
$110$	1.69011	0.161145
$111$	3.25300	0.308761
$112$	15.8884	1.50132
$113$	15.7320	1.47994	0.739971	−	0.672639i	$-0.234839\pi$
$113$	15.7320	1.47994	0.739971	+	0.672639i	$0.234839\pi$
$114$	−16.2121	−1.51841
$115$	−13.3626	−1.24607
$116$	10.3650	0.962369
$117$	4.32890	0.400207
$118$	−29.3715	−2.70387
$119$	4.63142	0.424561
$120$	−1.28098	−0.116937
$121$	−10.7018	−0.972894
$122$	2.68891	0.243442
$123$	−15.4014	−1.38869
$124$	10.1596	0.912356
$125$	−11.6990	−1.04639
$126$	65.6236	5.84621
$127$	19.0714	1.69232	0.846158	−	0.532932i	$-0.178910\pi$
$127$	19.0714	1.69232	0.846158	+	0.532932i	$0.178910\pi$
$128$	2.06356	0.182395
$129$	17.8561	1.57214
$130$	−1.76716	−0.154990
$131$	−9.11248	−0.796161	−0.398081	−	0.917350i	$-0.630324\pi$
$131$	−9.11248	−0.796161	−0.398081	+	0.917350i	$0.630324\pi$
$132$	−3.77859	−0.328884
$133$	−10.4514	−0.906251
$134$	−5.94406	−0.513488
$135$	22.7087	1.95445
$136$	0.280979	0.0240937
$137$	19.2948	1.64846	0.824232	−	0.566252i	$-0.191607\pi$
$137$	19.2948	1.64846	0.824232	+	0.566252i	$0.191607\pi$
$138$	57.9631	4.93415
$139$	−2.46802	−0.209335	−0.104667	−	0.994507i	$-0.533378\pi$
$139$	−2.46802	−0.209335	−0.104667	+	0.994507i	$0.533378\pi$
$140$	−13.8075	−1.16694
$141$	26.5287	2.23412
$142$	−17.2430	−1.44700
$143$	−0.311764	−0.0260710
$144$	−28.2760	−2.35633
$145$	7.42352	0.616489
$146$	−14.5479	−1.20400
$147$	36.2733	2.99177
$148$	2.12723	0.174857
$149$	12.6094	1.03300	0.516502	−	0.856286i	$-0.327234\pi$
$149$	12.6094	1.03300	0.516502	+	0.856286i	$0.327234\pi$
$150$	17.7035	1.44548
$151$	6.76150	0.550243	0.275121	−	0.961410i	$-0.411282\pi$
$151$	6.76150	0.550243	0.275121	+	0.961410i	$0.411282\pi$
$152$	−0.634066	−0.0514295
$153$	−8.24234	−0.666353
$154$	−4.72616	−0.380845
$155$	7.27636	0.584451
$156$	3.95086	0.316322
$157$	14.6673	1.17058	0.585289	−	0.810825i	$-0.300981\pi$
$157$	14.6673	1.17058	0.585289	+	0.810825i	$0.300981\pi$
$158$	−35.6683	−2.83761
$159$	45.1290	3.57896
$160$	12.3305	0.974812
$161$	37.3668	2.94492
$162$	−52.2937	−4.10858
$163$	−19.6288	−1.53744	−0.768722	−	0.639583i	$-0.779107\pi$
$163$	−19.6288	−1.53744	−0.768722	+	0.639583i	$0.779107\pi$
$164$	−10.0714	−0.786443
$165$	−2.70626	−0.210682
$166$	31.5725	2.45050
$167$	8.94289	0.692021	0.346011	−	0.938231i	$-0.387536\pi$
$167$	8.94289	0.692021	0.346011	+	0.938231i	$0.387536\pi$
$168$	3.58210	0.276365
$169$	−12.6740	−0.974925
$170$	3.36472	0.258062
$171$	18.5999	1.42237
$172$	11.6766	0.890334
$173$	18.6601	1.41870	0.709350	−	0.704856i	$-0.248988\pi$
$173$	18.6601	1.41870	0.709350	+	0.704856i	$0.248988\pi$
$174$	−32.2010	−2.44115
$175$	11.4128	0.862729
$176$	2.03642	0.153501
$177$	47.0306	3.53504
$178$	31.4159	2.35472
$179$	−8.37869	−0.626252	−0.313126	−	0.949712i	$-0.601376\pi$
$179$	−8.37869	−0.626252	−0.313126	+	0.949712i	$0.601376\pi$
$180$	24.5725	1.83153
$181$	10.2413	0.761229	0.380615	−	0.924734i	$-0.375712\pi$
$181$	10.2413	0.761229	0.380615	+	0.924734i	$0.375712\pi$
$182$	4.94163	0.366298
$183$	−4.30557	−0.318277
$184$	2.26697	0.167123
$185$	1.52354	0.112013
$186$	−31.5627	−2.31429
$187$	0.593607	0.0434088
$188$	17.3479	1.26522
$189$	−63.5017	−4.61907
$190$	−7.59293	−0.550849
$191$	−5.97566	−0.432384	−0.216192	−	0.976351i	$-0.569364\pi$
$191$	−5.97566	−0.432384	−0.216192	+	0.976351i	$0.569364\pi$
$192$	−29.2229	−2.10898
$193$	−3.50362	−0.252196	−0.126098	−	0.992018i	$-0.540245\pi$
$193$	−3.50362	−0.252196	−0.126098	+	0.992018i	$0.540245\pi$
$194$	−10.1061	−0.725572
$195$	2.82964	0.202635
$196$	23.7201	1.69430
$197$	−21.2101	−1.51116	−0.755578	−	0.655059i	$-0.772644\pi$
$197$	−21.2101	−1.51116	−0.755578	+	0.655059i	$0.772644\pi$
$198$	8.41095	0.597740
$199$	−11.4000	−0.808126	−0.404063	−	0.914731i	$-0.632402\pi$
$199$	−11.4000	−0.808126	−0.404063	+	0.914731i	$0.632402\pi$
$200$	0.692394	0.0489596
$201$	9.51782	0.671336
$202$	20.0733	1.41235
$203$	−20.7589	−1.45699
$204$	−7.52254	−0.526683
$205$	−7.21320	−0.503792
$206$	−24.7378	−1.72356
$207$	−66.5001	−4.62208
$208$	−2.12926	−0.147638
$209$	−1.33955	−0.0926587
$210$	42.8956	2.96008
$211$	−19.7674	−1.36085	−0.680423	−	0.732819i	$-0.738204\pi$
$211$	−19.7674	−1.36085	−0.680423	+	0.732819i	$0.738204\pi$
$212$	29.5111	2.02683
$213$	27.6101	1.89181
$214$	4.22702	0.288953
$215$	8.36289	0.570344
$216$	−3.85252	−0.262131
$217$	−20.3474	−1.38127
$218$	2.03156	0.137594
$219$	23.2947	1.57411
$220$	−1.76970	−0.119313
$221$	−0.620670	−0.0417508
$222$	−6.60865	−0.443544
$223$	−0.578478	−0.0387378	−0.0193689	−	0.999812i	$-0.506166\pi$
$223$	−0.578478	−0.0387378	−0.0193689	+	0.999812i	$0.506166\pi$
$224$	−34.4806	−2.30383
$225$	−20.3109	−1.35406
$226$	−31.9605	−2.12598
$227$	−15.2214	−1.01028	−0.505140	−	0.863037i	$-0.668559\pi$
$227$	−15.2214	−1.01028	−0.505140	+	0.863037i	$0.668559\pi$
$228$	16.9756	1.12424
$229$	−21.4627	−1.41830	−0.709149	−	0.705059i	$-0.750921\pi$
$229$	−21.4627	−1.41830	−0.709149	+	0.705059i	$0.750921\pi$
$230$	27.1469	1.79002
$231$	7.56769	0.497917
$232$	−1.25940	−0.0826837
$233$	3.46749	0.227163	0.113581	−	0.993529i	$-0.463768\pi$
$233$	3.46749	0.227163	0.113581	+	0.993529i	$0.463768\pi$
$234$	−8.79441	−0.574909
$235$	12.4247	0.810497
$236$	30.7546	2.00196
$237$	57.1132	3.70990
$238$	−9.40899	−0.609894
$239$	12.3627	0.799674	0.399837	−	0.916586i	$-0.369067\pi$
$239$	12.3627	0.799674	0.399837	+	0.916586i	$0.369067\pi$
$240$	−18.4829	−1.19307
$241$	−29.1046	−1.87479	−0.937395	−	0.348269i	$-0.886770\pi$
$241$	−29.1046	−1.87479	−0.937395	+	0.348269i	$0.886770\pi$
$242$	21.7414	1.39759
$243$	39.0187	2.50305
$244$	−2.81553	−0.180246
$245$	16.9886	1.08536
$246$	31.2888	1.99490
$247$	1.40062	0.0891196
$248$	−1.23444	−0.0783867
$249$	−50.5550	−3.20379
$250$	23.7672	1.50317
$251$	5.58811	0.352718	0.176359	−	0.984326i	$-0.443568\pi$
$251$	5.58811	0.352718	0.176359	+	0.984326i	$0.443568\pi$
$252$	−68.7139	−4.32857
$253$	4.78929	0.301100
$254$	−38.7447	−2.43106
$255$	−5.38770	−0.337391
$256$	13.7745	0.860907
$257$	−0.999058	−0.0623195	−0.0311598	−	0.999514i	$-0.509920\pi$
$257$	−0.999058	−0.0623195	−0.0311598	+	0.999514i	$0.509920\pi$
$258$	−36.2757	−2.25843
$259$	−4.26037	−0.264726
$260$	1.85038	0.114756
$261$	36.9437	2.28676
$262$	18.5125	1.14371
$263$	−0.159863	−0.00985760	−0.00492880	−	0.999988i	$-0.501569\pi$
$263$	−0.159863	−0.00985760	−0.00492880	+	0.999988i	$0.501569\pi$
$264$	0.459117	0.0282567
$265$	21.1361	1.29838
$266$	21.2326	1.30186
$267$	−50.3042	−3.07857
$268$	6.22397	0.380190
$269$	3.46219	0.211093	0.105547	−	0.994414i	$-0.466341\pi$
$269$	3.46219	0.211093	0.105547	+	0.994414i	$0.466341\pi$
$270$	−46.1339	−2.80762
$271$	1.09887	0.0667517	0.0333758	−	0.999443i	$-0.489374\pi$
$271$	1.09887	0.0667517	0.0333758	+	0.999443i	$0.489374\pi$
$272$	4.05416	0.245820
$273$	−7.91271	−0.478899
$274$	−39.1985	−2.36807
$275$	1.46278	0.0882088
$276$	−60.6927	−3.65327
$277$	−14.8721	−0.893580	−0.446790	−	0.894639i	$-0.647433\pi$
$277$	−14.8721	−0.893580	−0.446790	+	0.894639i	$0.647433\pi$
$278$	5.01392	0.300715
$279$	36.2114	2.16792
$280$	1.67767	0.100260
$281$	−22.5014	−1.34232	−0.671160	−	0.741313i	$-0.734204\pi$
$281$	−22.5014	−1.34232	−0.671160	+	0.741313i	$0.734204\pi$
$282$	−53.8946	−3.20938
$283$	1.75989	0.104615	0.0523073	−	0.998631i	$-0.483342\pi$
$283$	1.75989	0.104615	0.0523073	+	0.998631i	$0.483342\pi$
$284$	18.0550	1.07137
$285$	12.1581	0.720181
$286$	0.633367	0.0374518
$287$	20.1708	1.19064
$288$	61.3637	3.61589
$289$	−15.8182	−0.930484
$290$	−15.0813	−0.885604
$291$	16.1822	0.948615
$292$	15.2330	0.891446
$293$	16.4577	0.961472	0.480736	−	0.876866i	$-0.340370\pi$
$293$	16.4577	0.961472	0.480736	+	0.876866i	$0.340370\pi$
$294$	−73.6914	−4.29777
$295$	22.0267	1.28245
$296$	−0.258468	−0.0150232
$297$	−8.13899	−0.472272
$298$	−25.6168	−1.48394
$299$	−5.00764	−0.289599
$300$	−18.5372	−1.07024
$301$	−23.3857	−1.34793
$302$	−13.7364	−0.790439
$303$	−32.1420	−1.84651
$304$	−9.14875	−0.524717
$305$	−2.01651	−0.115465
$306$	16.7448	0.957235
$307$	−17.5076	−0.999211	−0.499605	−	0.866253i	$-0.666522\pi$
$307$	−17.5076	−0.999211	−0.499605	+	0.866253i	$0.666522\pi$
$308$	4.94872	0.281980
$309$	39.6109	2.25339
$310$	−14.7823	−0.839581
$311$	11.8326	0.670966	0.335483	−	0.942046i	$-0.391101\pi$
$311$	11.8326	0.670966	0.335483	+	0.942046i	$0.391101\pi$
$312$	−0.480048	−0.0271774
$313$	24.9255	1.40887	0.704437	−	0.709766i	$-0.251199\pi$
$313$	24.9255	1.40887	0.704437	+	0.709766i	$0.251199\pi$
$314$	−29.7974	−1.68157
$315$	−49.2134	−2.77286
$316$	37.3479	2.10098
$317$	12.4199	0.697568	0.348784	−	0.937203i	$-0.386595\pi$
$317$	12.4199	0.697568	0.348784	+	0.937203i	$0.386595\pi$
$318$	−91.6822	−5.14128
$319$	−2.66066	−0.148968
$320$	−13.6865	−0.765099
$321$	−6.76844	−0.377778
$322$	−75.9128	−4.23045
$323$	−2.66682	−0.148386
$324$	54.7562	3.04201
$325$	−1.52947	−0.0848396
$326$	39.8770	2.20858
$327$	−3.25300	−0.179891
$328$	1.22372	0.0675687
$329$	−34.7440	−1.91550
$330$	5.49792	0.302650
$331$	−4.59258	−0.252431	−0.126215	−	0.992003i	$-0.540283\pi$
$331$	−4.59258	−0.252431	−0.126215	+	0.992003i	$0.540283\pi$
$332$	−33.0593	−1.81437
$333$	7.58200	0.415491
$334$	−18.1680	−0.994108
$335$	4.45766	0.243548
$336$	51.6851	2.81965
$337$	−5.21831	−0.284259	−0.142130	−	0.989848i	$-0.545395\pi$
$337$	−5.21831	−0.284259	−0.142130	+	0.989848i	$0.545395\pi$
$338$	25.7480	1.40051
$339$	51.1762	2.77951
$340$	−3.52317	−0.191071
$341$	−2.60792	−0.141227
$342$	−37.7868	−2.04328
$343$	−17.6837	−0.954829
$344$	−1.41876	−0.0764946
$345$	−43.4686	−2.34027
$346$	−37.9091	−2.03800
$347$	23.1616	1.24338	0.621690	−	0.783264i	$-0.286446\pi$
$347$	23.1616	1.24338	0.621690	+	0.783264i	$0.286446\pi$
$348$	33.7174	1.80744
$349$	−13.4990	−0.722587	−0.361294	−	0.932452i	$-0.617665\pi$
$349$	−13.4990	−0.722587	−0.361294	+	0.932452i	$0.617665\pi$
$350$	−23.1858	−1.23933
$351$	8.51006	0.454233
$352$	−4.41937	−0.235553
$353$	4.26309	0.226902	0.113451	−	0.993544i	$-0.463810\pi$
$353$	4.26309	0.226902	0.113451	+	0.993544i	$0.463810\pi$
$354$	−95.5454	−5.07818
$355$	12.9312	0.686314
$356$	−32.8953	−1.74345
$357$	15.0660	0.797377
$358$	17.0218	0.899629
$359$	−5.33744	−0.281699	−0.140850	−	0.990031i	$-0.544983\pi$
$359$	−5.33744	−0.281699	−0.140850	+	0.990031i	$0.544983\pi$
$360$	−2.98568	−0.157359
$361$	−12.9820	−0.683261
$362$	−20.8058	−1.09353
$363$	−34.8130	−1.82721
$364$	−5.17434	−0.271209
$365$	10.9100	0.571056
$366$	8.74701	0.457214
$367$	17.0429	0.889631	0.444816	−	0.895622i	$-0.353269\pi$
$367$	17.0429	0.889631	0.444816	+	0.895622i	$0.353269\pi$
$368$	32.7094	1.70510
$369$	−35.8971	−1.86873
$370$	−3.09515	−0.160909
$371$	−59.1043	−3.06854
$372$	33.0490	1.71351
$373$	−30.5449	−1.58156	−0.790778	−	0.612102i	$-0.790324\pi$
$373$	−30.5449	−1.58156	−0.790778	+	0.612102i	$0.790324\pi$
$374$	−1.20595	−0.0623580
$375$	−38.0568	−1.96524
$376$	−2.10785	−0.108704
$377$	2.78196	0.143278
$378$	129.007	6.63543
$379$	17.2349	0.885299	0.442650	−	0.896695i	$-0.354039\pi$
$379$	17.2349	0.885299	0.442650	+	0.896695i	$0.354039\pi$
$380$	7.95049	0.407852
$381$	62.0393	3.17837
$382$	12.1399	0.621132
$383$	−20.2777	−1.03614	−0.518072	−	0.855337i	$-0.673350\pi$
$383$	−20.2777	−1.03614	−0.518072	+	0.855337i	$0.673350\pi$
$384$	6.71276	0.342559
$385$	3.54431	0.180635
$386$	7.11780	0.362286
$387$	41.6185	2.11559
$388$	10.5820	0.537218
$389$	−14.5265	−0.736522	−0.368261	−	0.929722i	$-0.620047\pi$
$389$	−14.5265	−0.736522	−0.368261	+	0.929722i	$0.620047\pi$
$390$	−5.74857	−0.291090
$391$	9.53467	0.482189
$392$	−2.88211	−0.145569
$393$	−29.6429	−1.49529
$394$	43.0895	2.17082
$395$	26.7489	1.34588
$396$	−8.80703	−0.442570
$397$	7.68874	0.385887	0.192943	−	0.981210i	$-0.438197\pi$
$397$	7.68874	0.385887	0.192943	+	0.981210i	$0.438197\pi$
$398$	23.1598	1.16090
$399$	−33.9984	−1.70205
$400$	9.99034	0.499517
$401$	28.1492	1.40571	0.702853	−	0.711336i	$-0.251909\pi$
$401$	28.1492	1.40571	0.702853	+	0.711336i	$0.251909\pi$
$402$	−19.3360	−0.964392
$403$	2.72681	0.135832
$404$	−21.0185	−1.04571
$405$	39.2169	1.94870
$406$	42.1729	2.09300
$407$	−0.546050	−0.0270667
$408$	0.914024	0.0452509
$409$	−30.3569	−1.50105	−0.750527	−	0.660840i	$-0.770200\pi$
$409$	−30.3569	−1.50105	−0.750527	+	0.660840i	$0.770200\pi$
$410$	14.6540	0.723711
$411$	62.7659	3.09601
$412$	25.9027	1.27613
$413$	−61.5948	−3.03088
$414$	135.099	6.63974
$415$	−23.6774	−1.16228
$416$	4.62085	0.226556
$417$	−8.02846	−0.393155
$418$	2.72138	0.133107
$419$	24.4611	1.19500	0.597501	−	0.801868i	$-0.296160\pi$
$419$	24.4611	1.19500	0.597501	+	0.801868i	$0.296160\pi$
$420$	−44.9156	−2.19166
$421$	4.33624	0.211335	0.105668	−	0.994401i	$-0.466302\pi$
$421$	4.33624	0.211335	0.105668	+	0.994401i	$0.466302\pi$
$422$	40.1587	1.95489
$423$	61.8324	3.00639
$424$	−3.58574	−0.174139
$425$	2.91214	0.141260
$426$	−56.0915	−2.71764
$427$	5.63889	0.272885
$428$	−4.42607	−0.213942
$429$	−1.01417	−0.0489645
$430$	−16.9897	−0.819315
$431$	−16.8228	−0.810328	−0.405164	−	0.914244i	$-0.632786\pi$
$431$	−16.8228	−0.810328	−0.405164	+	0.914244i	$0.632786\pi$
$432$	−55.5869	−2.67443
$433$	37.5028	1.80227	0.901136	−	0.433537i	$-0.142735\pi$
$433$	37.5028	1.80227	0.901136	+	0.433537i	$0.142735\pi$
$434$	41.3369	1.98423
$435$	24.1487	1.15784
$436$	−2.12723	−0.101876
$437$	−21.5162	−1.02926
$438$	−47.3244	−2.26125
$439$	−32.3433	−1.54366	−0.771830	−	0.635829i	$-0.780658\pi$
$439$	−32.3433	−1.54366	−0.771830	+	0.635829i	$0.780658\pi$
$440$	0.215027	0.0102510
$441$	84.5449	4.02595
$442$	1.26093	0.0599762
$443$	6.14982	0.292187	0.146093	−	0.989271i	$-0.453330\pi$
$443$	6.14982	0.292187	0.146093	+	0.989271i	$0.453330\pi$
$444$	6.91986	0.328402
$445$	−23.5599	−1.11685
$446$	1.17521	0.0556479
$447$	41.0184	1.94011
$448$	38.2725	1.80821
$449$	−37.5136	−1.77038	−0.885189	−	0.465231i	$-0.845971\pi$
$449$	−37.5136	−1.77038	−0.885189	+	0.465231i	$0.845971\pi$
$450$	41.2628	1.94515
$451$	2.58528	0.121736
$452$	33.4655	1.57409
$453$	21.9951	1.03342
$454$	30.9232	1.45130
$455$	−3.70590	−0.173735
$456$	−2.06261	−0.0965908
$457$	−22.8020	−1.06663	−0.533315	−	0.845917i	$-0.679054\pi$
$457$	−22.8020	−1.06663	−0.533315	+	0.845917i	$0.679054\pi$
$458$	43.6028	2.03742
$459$	−16.2034	−0.756308
$460$	−28.4253	−1.32534
$461$	−13.6269	−0.634669	−0.317335	−	0.948314i	$-0.602788\pi$
$461$	−13.6269	−0.634669	−0.317335	+	0.948314i	$0.602788\pi$
$462$	−15.3742	−0.715272
$463$	13.1806	0.612556	0.306278	−	0.951942i	$-0.400916\pi$
$463$	13.1806	0.612556	0.306278	+	0.951942i	$0.400916\pi$
$464$	−18.1715	−0.843591
$465$	23.6700	1.09767
$466$	−7.04440	−0.326326
$467$	−16.2683	−0.752806	−0.376403	−	0.926456i	$-0.622839\pi$
$467$	−16.2683	−0.752806	−0.376403	+	0.926456i	$0.622839\pi$
$468$	9.20855	0.425666
$469$	−12.4653	−0.575592
$470$	−25.2415	−1.16430
$471$	47.7127	2.19848
$472$	−3.73684	−0.172002
$473$	−2.99734	−0.137818
$474$	−116.029	−5.32938
$475$	−6.57163	−0.301527
$476$	9.85208	0.451569
$477$	105.185	4.81611
$478$	−25.1155	−1.14875
$479$	25.5435	1.16711	0.583557	−	0.812072i	$-0.301661\pi$
$479$	25.5435	1.16711	0.583557	+	0.812072i	$0.301661\pi$
$480$	40.1111	1.83081
$481$	0.570945	0.0260328
$482$	59.1276	2.69319
$483$	121.554	5.53090
$484$	−22.7652	−1.03478
$485$	7.57889	0.344140
$486$	−79.2687	−3.59570
$487$	17.2990	0.783891	0.391945	−	0.919988i	$-0.371802\pi$
$487$	17.2990	0.783891	0.391945	+	0.919988i	$0.371802\pi$
$488$	0.342100	0.0154862
$489$	−63.8523	−2.88750
$490$	−34.5132	−1.55915
$491$	11.9903	0.541113	0.270557	−	0.962704i	$-0.412792\pi$
$491$	11.9903	0.541113	0.270557	+	0.962704i	$0.412792\pi$
$492$	−32.7622	−1.47703
$493$	−5.29692	−0.238561
$494$	−2.84545	−0.128023
$495$	−6.30767	−0.283508
$496$	−17.8113	−0.799751
$497$	−36.1603	−1.62201
$498$	102.705	4.60234
$499$	−30.7510	−1.37660	−0.688302	−	0.725424i	$-0.741644\pi$
$499$	−30.7510	−1.37660	−0.688302	+	0.725424i	$0.741644\pi$
$500$	−24.8864	−1.11295
$501$	29.0912	1.29970
$502$	−11.3526	−0.506690
$503$	44.2161	1.97150	0.985750	−	0.168219i	$-0.0538015\pi$
$503$	44.2161	1.97150	0.985750	+	0.168219i	$0.0538015\pi$
$504$	8.34906	0.371897
$505$	−15.0536	−0.669878
$506$	−9.72971	−0.432539
$507$	−41.2286	−1.83102
$508$	40.5693	1.79997
$509$	14.1213	0.625917	0.312958	−	0.949767i	$-0.398680\pi$
$509$	14.1213	0.625917	0.312958	+	0.949767i	$0.398680\pi$
$510$	10.9454	0.484672
$511$	−30.5084	−1.34961
$512$	−32.1108	−1.41911
$513$	36.5650	1.61438
$514$	2.02964	0.0895237
$515$	18.5517	0.817486
$516$	37.9840	1.67215
$517$	−4.45312	−0.195848
$518$	8.65518	0.380287
$519$	60.7012	2.66449
$520$	−0.224830	−0.00985944
$521$	17.8809	0.783376	0.391688	−	0.920098i	$-0.371891\pi$
$521$	17.8809	0.783376	0.391688	+	0.920098i	$0.371891\pi$
$522$	−75.0532	−3.28499
$523$	7.52606	0.329092	0.164546	−	0.986369i	$-0.447384\pi$
$523$	7.52606	0.329092	0.164546	+	0.986369i	$0.447384\pi$
$524$	−19.3843	−0.846808
$525$	37.1259	1.62031
$526$	0.324772	0.0141607
$527$	−5.19192	−0.226164
$528$	6.62446	0.288293
$529$	53.9267	2.34464
$530$	−42.9392	−1.86516
$531$	109.618	4.75700
$532$	−22.2325	−0.963901
$533$	−2.70315	−0.117086
$534$	102.196	4.42245
$535$	−3.16999	−0.137051
$536$	−0.756242	−0.0326647
$537$	−27.2558	−1.17618
$538$	−7.03364	−0.303242
$539$	−6.08886	−0.262266
$540$	48.3064	2.07878
$541$	−12.2104	−0.524968	−0.262484	−	0.964936i	$-0.584542\pi$
$541$	−12.2104	−0.524968	−0.262484	+	0.964936i	$0.584542\pi$
$542$	−2.23242	−0.0958907
$543$	33.3149	1.42968
$544$	−8.79822	−0.377221
$545$	−1.52354	−0.0652611
$546$	16.0751	0.687952
$547$	7.19022	0.307432	0.153716	−	0.988115i	$-0.450876\pi$
$547$	7.19022	0.307432	0.153716	+	0.988115i	$0.450876\pi$
$548$	41.0444	1.75333
$549$	−10.0353	−0.428296
$550$	−2.97172	−0.126714
$551$	11.9532	0.509223
$552$	7.37445	0.313877
$553$	−74.7997	−3.18081
$554$	30.2136	1.28365
$555$	4.95606	0.210373
$556$	−5.25004	−0.222651
$557$	−7.47673	−0.316799	−0.158400	−	0.987375i	$-0.550633\pi$
$557$	−7.47673	−0.316799	−0.158400	+	0.987375i	$0.550633\pi$
$558$	−73.5655	−3.11427
$559$	3.13399	0.132554
$560$	24.2066	1.02292
$561$	1.93100	0.0815270
$562$	45.7128	1.92828
$563$	12.7736	0.538343	0.269171	−	0.963092i	$-0.413250\pi$
$563$	12.7736	0.538343	0.269171	+	0.963092i	$0.413250\pi$
$564$	56.4326	2.37624
$565$	23.9683	1.00835
$566$	−3.57532	−0.150282
$567$	−109.665	−4.60549
$568$	−2.19377	−0.0920486
$569$	−12.6510	−0.530359	−0.265180	−	0.964199i	$-0.585431\pi$
$569$	−12.6510	−0.530359	−0.265180	+	0.964199i	$0.585431\pi$
$570$	−24.6998	−1.03456
$571$	−7.32976	−0.306741	−0.153371	−	0.988169i	$-0.549013\pi$
$571$	−7.32976	−0.306741	−0.153371	+	0.988169i	$0.549013\pi$
$572$	−0.663193	−0.0277295
$573$	−19.4388	−0.812068
$574$	−40.9781	−1.71039
$575$	23.4955	0.979830
$576$	−68.1119	−2.83800
$577$	−24.1092	−1.00368	−0.501839	−	0.864961i	$-0.667343\pi$
$577$	−24.1092	−1.00368	−0.501839	+	0.864961i	$0.667343\pi$
$578$	32.1356	1.33667
$579$	−11.3973	−0.473654
$580$	15.7915	0.655706
$581$	66.2106	2.74688
$582$	−32.8750	−1.36271
$583$	−7.57538	−0.313740
$584$	−1.85089	−0.0765902
$585$	6.59524	0.272680
$586$	−33.4349	−1.38118
$587$	−9.43115	−0.389265	−0.194633	−	0.980876i	$-0.562351\pi$
$587$	−9.43115	−0.389265	−0.194633	+	0.980876i	$0.562351\pi$
$588$	77.1616	3.18209
$589$	11.7163	0.482760
$590$	−44.7486	−1.84227
$591$	−68.9963	−2.83813
$592$	−3.72936	−0.153276
$593$	8.68112	0.356491	0.178245	−	0.983986i	$-0.442958\pi$
$593$	8.68112	0.356491	0.178245	+	0.983986i	$0.442958\pi$
$594$	16.5348	0.678432
$595$	7.05614	0.289273
$596$	26.8231	1.09872
$597$	−37.0843	−1.51776
$598$	10.1733	0.416017
$599$	33.4414	1.36638	0.683189	−	0.730242i	$-0.260592\pi$
$599$	33.4414	1.36638	0.683189	+	0.730242i	$0.260592\pi$
$600$	2.25236	0.0919520
$601$	−47.5882	−1.94116	−0.970581	−	0.240774i	$-0.922599\pi$
$601$	−47.5882	−1.94116	−0.970581	+	0.240774i	$0.922599\pi$
$602$	47.5094	1.93634
$603$	22.1839	0.903397
$604$	14.3832	0.585245
$605$	−16.3046	−0.662877
$606$	65.2983	2.65256
$607$	2.42123	0.0982746	0.0491373	−	0.998792i	$-0.484353\pi$
$607$	2.42123	0.0982746	0.0491373	+	0.998792i	$0.484353\pi$
$608$	19.8543	0.805200
$609$	−67.5286	−2.73640
$610$	4.09665	0.165868
$611$	4.65615	0.188368
$612$	−17.5333	−0.708742
$613$	−15.6552	−0.632307	−0.316153	−	0.948708i	$-0.602391\pi$
$613$	−15.6552	−0.632307	−0.316153	+	0.948708i	$0.602391\pi$
$614$	35.5677	1.43539
$615$	−23.4645	−0.946181
$616$	−0.601293	−0.0242268
$617$	6.59231	0.265397	0.132698	−	0.991156i	$-0.457636\pi$
$617$	6.59231	0.265397	0.132698	+	0.991156i	$0.457636\pi$
$618$	−80.4719	−3.23705
$619$	2.34081	0.0940852	0.0470426	−	0.998893i	$-0.485020\pi$
$619$	2.34081	0.0940852	0.0470426	+	0.998893i	$0.485020\pi$
$620$	15.4785	0.621630
$621$	−130.731	−5.24604
$622$	−24.0386	−0.963861
$623$	65.8821	2.63951
$624$	−6.92648	−0.277281
$625$	−4.42966	−0.177186
$626$	−50.6377	−2.02389
$627$	−4.35756	−0.174024
$628$	31.2007	1.24504
$629$	−1.08709	−0.0433452
$630$	99.9799	3.98330
$631$	1.00802	0.0401286	0.0200643	−	0.999799i	$-0.493613\pi$
$631$	1.00802	0.0401286	0.0200643	+	0.999799i	$0.493613\pi$
$632$	−4.53795	−0.180510
$633$	−64.3034	−2.55583
$634$	−25.2316	−1.00208
$635$	29.0560	1.15305
$636$	95.9997	3.80663
$637$	6.36646	0.252248
$638$	5.40528	0.213997
$639$	64.3529	2.54576
$640$	3.14391	0.124274
$641$	3.56920	0.140975	0.0704874	−	0.997513i	$-0.477545\pi$
$641$	3.56920	0.140975	0.0704874	+	0.997513i	$0.477545\pi$
$642$	13.7505	0.542688
$643$	−7.60317	−0.299840	−0.149920	−	0.988698i	$-0.547902\pi$
$643$	−7.60317	−0.299840	−0.149920	+	0.988698i	$0.547902\pi$
$644$	79.4876	3.13225
$645$	27.2044	1.07117
$646$	5.41780	0.213161
$647$	5.53566	0.217629	0.108815	−	0.994062i	$-0.465294\pi$
$647$	5.53566	0.217629	0.108815	+	0.994062i	$0.465294\pi$
$648$	−6.65314	−0.261360
$649$	−7.89458	−0.309890
$650$	3.10720	0.121874
$651$	−66.1900	−2.59419
$652$	−41.7548	−1.63525
$653$	39.9341	1.56274	0.781371	−	0.624067i	$-0.214521\pi$
$653$	39.9341	1.56274	0.781371	+	0.624067i	$0.214521\pi$
$654$	6.60865	0.258419
$655$	−13.8832	−0.542462
$656$	17.6567	0.689378
$657$	54.2945	2.11823
$658$	70.5844	2.75167
$659$	22.2365	0.866210	0.433105	−	0.901343i	$-0.357418\pi$
$659$	22.2365	0.866210	0.433105	+	0.901343i	$0.357418\pi$
$660$	−5.75682	−0.224084
$661$	3.76253	0.146346	0.0731728	−	0.997319i	$-0.476688\pi$
$661$	3.76253	0.146346	0.0731728	+	0.997319i	$0.476688\pi$
$662$	9.33008	0.362624
$663$	−2.01904	−0.0784130
$664$	4.01687	0.155885
$665$	−15.9231	−0.617471
$666$	−15.4033	−0.596864
$667$	−42.7361	−1.65475
$668$	19.0235	0.736043
$669$	−1.88179	−0.0727541
$670$	−9.05599	−0.349863
$671$	0.722735	0.0279009
$672$	−112.165	−4.32688
$673$	−17.8885	−0.689551	−0.344776	−	0.938685i	$-0.612045\pi$
$673$	−17.8885	−0.689551	−0.344776	+	0.938685i	$0.612045\pi$
$674$	10.6013	0.408347
$675$	−39.9286	−1.53685
$676$	−26.9605	−1.03694
$677$	−24.2267	−0.931108	−0.465554	−	0.885019i	$-0.654145\pi$
$677$	−24.2267	−0.931108	−0.465554	+	0.885019i	$0.654145\pi$
$678$	−103.967	−3.99284
$679$	−21.1934	−0.813326
$680$	0.428082	0.0164162
$681$	−49.5152	−1.89743
$682$	5.29813	0.202876
$683$	−24.8015	−0.949002	−0.474501	−	0.880255i	$-0.657372\pi$
$683$	−24.8015	−0.949002	−0.474501	+	0.880255i	$0.657372\pi$
$684$	39.5662	1.51285
$685$	29.3963	1.12318
$686$	35.9254	1.37164
$687$	−69.8182	−2.66373
$688$	−20.4709	−0.780447
$689$	7.92075	0.301757
$690$	88.3089	3.36186
$691$	28.4367	1.08178	0.540892	−	0.841092i	$-0.318087\pi$
$691$	28.4367	1.08178	0.540892	+	0.841092i	$0.318087\pi$
$692$	39.6942	1.50895
$693$	17.6386	0.670033
$694$	−47.0541	−1.78615
$695$	−3.76012	−0.142629
$696$	−4.09683	−0.155290
$697$	5.14686	0.194951
$698$	27.4241	1.03802
$699$	11.2797	0.426639
$700$	24.2777	0.917610
$701$	15.4812	0.584717	0.292358	−	0.956309i	$-0.405560\pi$
$701$	15.4812	0.584717	0.292358	+	0.956309i	$0.405560\pi$
$702$	−17.2887	−0.652519
$703$	2.45317	0.0925230
$704$	4.90537	0.184878
$705$	40.4175	1.52221
$706$	−8.66072	−0.325951
$707$	42.0955	1.58316
$708$	100.045	3.75991
$709$	−25.5895	−0.961033	−0.480516	−	0.876986i	$-0.659551\pi$
$709$	−25.5895	−0.961033	−0.480516	+	0.876986i	$0.659551\pi$
$710$	−26.2704	−0.985910
$711$	133.118	4.99231
$712$	3.99694	0.149791
$713$	−41.8890	−1.56876
$714$	−30.6074	−1.14545
$715$	−0.474984	−0.0177634
$716$	−17.8234	−0.666090
$717$	40.2157	1.50188
$718$	10.8433	0.404669
$719$	19.7569	0.736809	0.368405	−	0.929666i	$-0.379904\pi$
$719$	19.7569	0.736809	0.368405	+	0.929666i	$0.379904\pi$
$720$	−43.0795	−1.60548
$721$	−51.8774	−1.93202
$722$	26.3736	0.981524
$723$	−94.6771	−3.52108
$724$	21.7856	0.809654
$725$	−13.0528	−0.484768
$726$	70.7247	2.62484
$727$	3.69059	0.136876	0.0684382	−	0.997655i	$-0.478198\pi$
$727$	3.69059	0.136876	0.0684382	+	0.997655i	$0.478198\pi$
$728$	0.628707	0.0233014
$729$	49.7057	1.84095
$730$	−22.1643	−0.820339
$731$	−5.96719	−0.220705
$732$	−9.15892	−0.338523
$733$	−40.1289	−1.48219	−0.741097	−	0.671398i	$-0.765694\pi$
$733$	−40.1289	−1.48219	−0.741097	+	0.671398i	$0.765694\pi$
$734$	−34.6236	−1.27798
$735$	55.2637	2.03843
$736$	−70.9850	−2.61654
$737$	−1.59767	−0.0588508
$738$	72.9269	2.68448
$739$	19.4003	0.713650	0.356825	−	0.934171i	$-0.383859\pi$
$739$	19.4003	0.713650	0.356825	+	0.934171i	$0.383859\pi$
$740$	3.24091	0.119138
$741$	4.55623	0.167377
$742$	120.074	4.40805
$743$	46.6873	1.71279	0.856395	−	0.516321i	$-0.172699\pi$
$743$	46.6873	1.71279	0.856395	+	0.516321i	$0.172699\pi$
$744$	−4.01562	−0.147220
$745$	19.2109	0.703834
$746$	62.0538	2.27195
$747$	−117.832	−4.31125
$748$	1.26274	0.0461702
$749$	8.86445	0.323900
$750$	77.3146	2.82313
$751$	−36.3853	−1.32772	−0.663859	−	0.747858i	$-0.731082\pi$
$751$	−36.3853	−1.32772	−0.663859	+	0.747858i	$0.731082\pi$
$752$	−30.4135	−1.10907
$753$	18.1781	0.662447
$754$	−5.65171	−0.205823
$755$	10.3014	0.374906
$756$	−135.083	−4.91291
$757$	−17.9072	−0.650849	−0.325424	−	0.945568i	$-0.605507\pi$
$757$	−17.9072	−0.650849	−0.325424	+	0.945568i	$0.605507\pi$
$758$	−35.0138	−1.27176
$759$	15.5795	0.565502
$760$	−0.966022	−0.0350413
$761$	29.3886	1.06534	0.532668	−	0.846324i	$-0.321189\pi$
$761$	29.3886	1.06534	0.532668	+	0.846324i	$0.321189\pi$
$762$	−126.036	−4.56582
$763$	4.26037	0.154236
$764$	−12.7116	−0.459889
$765$	−12.5575	−0.454017
$766$	41.1954	1.48845
$767$	8.25451	0.298053
$768$	44.8085	1.61689
$769$	38.7058	1.39577	0.697883	−	0.716212i	$-0.254125\pi$
$769$	38.7058	1.39577	0.697883	+	0.716212i	$0.254125\pi$
$770$	−7.20048	−0.259487
$771$	−3.24993	−0.117043
$772$	−7.45298	−0.268239
$773$	−4.78935	−0.172261	−0.0861304	−	0.996284i	$-0.527450\pi$
$773$	−4.78935	−0.172261	−0.0861304	+	0.996284i	$0.527450\pi$
$774$	−84.5504	−3.03910
$775$	−12.7940	−0.459575
$776$	−1.28576	−0.0461561
$777$	−13.8590	−0.497188
$778$	29.5114	1.05804
$779$	−11.6146	−0.416135
$780$	6.01928	0.215525
$781$	−4.63465	−0.165841
$782$	−19.3702	−0.692678
$783$	72.6265	2.59546
$784$	−41.5851	−1.48518
$785$	22.3462	0.797569
$786$	60.2212	2.14802
$787$	45.8941	1.63595	0.817974	−	0.575255i	$-0.195097\pi$
$787$	45.8941	1.63595	0.817974	+	0.575255i	$0.195097\pi$
$788$	−45.1186	−1.60729
$789$	−0.520036	−0.0185138
$790$	−54.3419	−1.93340
$791$	−67.0241	−2.38310
$792$	1.07010	0.0380242
$793$	−0.755685	−0.0268352
$794$	−15.6201	−0.554337
$795$	68.7557	2.43851
$796$	−24.2504	−0.859534
$797$	−36.8119	−1.30394	−0.651972	−	0.758243i	$-0.726058\pi$
$797$	−36.8119	−1.30394	−0.651972	+	0.758243i	$0.726058\pi$
$798$	69.0697	2.44504
$799$	−8.86542	−0.313636
$800$	−21.6807	−0.766530
$801$	−117.248	−4.14274
$802$	−57.1868	−2.01934
$803$	−3.91025	−0.137990
$804$	20.2466	0.714041
$805$	56.9297	2.00651
$806$	−5.53968	−0.195127
$807$	11.2625	0.396459
$808$	2.55385	0.0898442
$809$	−23.1517	−0.813970	−0.406985	−	0.913435i	$-0.633420\pi$
$809$	−23.1517	−0.813970	−0.406985	+	0.913435i	$0.633420\pi$
$810$	−79.6713	−2.79936
$811$	−15.7483	−0.552999	−0.276500	−	0.961014i	$-0.589174\pi$
$811$	−15.7483	−0.552999	−0.276500	+	0.961014i	$0.589174\pi$
$812$	−44.1588	−1.54967
$813$	3.57463	0.125368
$814$	1.10933	0.0388821
$815$	−29.9051	−1.04753
$816$	13.1882	0.461679
$817$	13.4657	0.471107
$818$	61.6718	2.15631
$819$	−18.4427	−0.644441
$820$	−15.3441	−0.535840
$821$	11.3572	0.396370	0.198185	−	0.980165i	$-0.436495\pi$
$821$	11.3572	0.396370	0.198185	+	0.980165i	$0.436495\pi$
$822$	−127.513	−4.44751
$823$	3.01683	0.105160	0.0525800	−	0.998617i	$-0.483256\pi$
$823$	3.01683	0.105160	0.0525800	+	0.998617i	$0.483256\pi$
$824$	−3.14730	−0.109641
$825$	4.75841	0.165667
$826$	125.133	4.35395
$827$	7.88004	0.274016	0.137008	−	0.990570i	$-0.456251\pi$
$827$	7.88004	0.274016	0.137008	+	0.990570i	$0.456251\pi$
$828$	−141.461	−4.91610
$829$	1.54781	0.0537575	0.0268788	−	0.999639i	$-0.491443\pi$
$829$	1.54781	0.0537575	0.0268788	+	0.999639i	$0.491443\pi$
$830$	48.1019	1.66964
$831$	−48.3790	−1.67825
$832$	−5.12901	−0.177817
$833$	−12.1219	−0.419999
$834$	16.3103	0.564779
$835$	13.6248	0.471506
$836$	−2.84953	−0.0985531
$837$	71.1869	2.46058
$838$	−49.6941	−1.71665
$839$	19.8629	0.685744	0.342872	−	0.939382i	$-0.388600\pi$
$839$	19.8629	0.685744	0.342872	+	0.939382i	$0.388600\pi$
$840$	5.45746	0.188300
$841$	−5.25821	−0.181318
$842$	−8.80932	−0.303589
$843$	−73.1969	−2.52104
$844$	−42.0498	−1.44741
$845$	−19.3093	−0.664261
$846$	−125.616	−4.31877
$847$	45.5937	1.56662
$848$	−51.7376	−1.77668
$849$	5.72492	0.196479
$850$	−5.91619	−0.202924
$851$	−8.77079	−0.300659
$852$	58.7330	2.01216
$853$	−24.2198	−0.829271	−0.414636	−	0.909988i	$-0.636091\pi$
$853$	−24.2198	−0.829271	−0.414636	+	0.909988i	$0.636091\pi$
$854$	−11.4557	−0.392007
$855$	28.3376	0.969127
$856$	0.537789	0.0183812
$857$	9.26327	0.316427	0.158214	−	0.987405i	$-0.449427\pi$
$857$	9.26327	0.316427	0.158214	+	0.987405i	$0.449427\pi$
$858$	2.06034	0.0703389
$859$	36.8697	1.25798	0.628989	−	0.777414i	$-0.283469\pi$
$859$	36.8697	1.25798	0.628989	+	0.777414i	$0.283469\pi$
$860$	17.7898	0.606626
$861$	65.6155	2.23617
$862$	34.1766	1.16406
$863$	−44.8377	−1.52629	−0.763147	−	0.646225i	$-0.776347\pi$
$863$	−44.8377	−1.52629	−0.763147	+	0.646225i	$0.776347\pi$
$864$	120.633	4.10402
$865$	28.4293	0.966626
$866$	−76.1892	−2.58901
$867$	−51.4567	−1.74756
$868$	−43.2835	−1.46914
$869$	−9.58705	−0.325218
$870$	−49.0594	−1.66327
$871$	1.67051	0.0566029
$872$	0.258468	0.00875284
$873$	37.7169	1.27652
$874$	43.7114	1.47856
$875$	49.8420	1.68497
$876$	49.5530	1.67424
$877$	46.2569	1.56199	0.780993	−	0.624540i	$-0.214713\pi$
$877$	46.2569	1.56199	0.780993	+	0.624540i	$0.214713\pi$
$878$	65.7072	2.21751
$879$	53.5370	1.80576
$880$	3.10256	0.104587
$881$	3.17860	0.107090	0.0535449	−	0.998565i	$-0.482948\pi$
$881$	3.17860	0.107090	0.0535449	+	0.998565i	$0.482948\pi$
$882$	−171.758	−5.78338
$883$	−42.9231	−1.44448	−0.722238	−	0.691645i	$-0.756886\pi$
$883$	−42.9231	−1.44448	−0.722238	+	0.691645i	$0.756886\pi$
$884$	−1.32031	−0.0444067
$885$	71.6529	2.40859
$886$	−12.4937	−0.419734
$887$	−3.07092	−0.103111	−0.0515556	−	0.998670i	$-0.516418\pi$
$887$	−3.07092	−0.103111	−0.0515556	+	0.998670i	$0.516418\pi$
$888$	−0.840796	−0.0282153
$889$	−81.2513	−2.72508
$890$	47.8633	1.60438
$891$	−14.0557	−0.470883
$892$	−1.23055	−0.0412020
$893$	20.0060	0.669475
$894$	−83.3313	−2.78702
$895$	−12.7652	−0.426695
$896$	−8.79153	−0.293704
$897$	−16.2898	−0.543902
$898$	76.2111	2.54320
$899$	23.2712	0.776137
$900$	−43.2059	−1.44020
$901$	−15.0813	−0.502431
$902$	−5.25214	−0.174877
$903$	−76.0736	−2.53157
$904$	−4.06622	−0.135240
$905$	15.6030	0.518661
$906$	−44.6844	−1.48454
$907$	34.8612	1.15755	0.578773	−	0.815489i	$-0.303532\pi$
$907$	34.8612	1.15755	0.578773	+	0.815489i	$0.303532\pi$
$908$	−32.3794	−1.07455
$909$	−74.9156	−2.48479
$910$	7.52876	0.249576
$911$	11.6735	0.386759	0.193379	−	0.981124i	$-0.438055\pi$
$911$	11.6735	0.386759	0.193379	+	0.981124i	$0.438055\pi$
$912$	−29.7609	−0.985480
$913$	8.48619	0.280852
$914$	46.3235	1.53224
$915$	−6.55969	−0.216857
$916$	−45.6561	−1.50852
$917$	38.8225	1.28203
$918$	32.9181	1.08646
$919$	−21.3173	−0.703193	−0.351596	−	0.936152i	$-0.614361\pi$
$919$	−21.3173	−0.703193	−0.351596	+	0.936152i	$0.614361\pi$
$920$	3.45381	0.113869
$921$	−56.9521	−1.87664
$922$	27.6839	0.911721
$923$	4.84595	0.159506
$924$	16.0982	0.529591
$925$	−2.67884	−0.0880796
$926$	−26.7772	−0.879954
$927$	92.3240	3.03232
$928$	39.4353	1.29453
$929$	19.1324	0.627715	0.313858	−	0.949470i	$-0.398379\pi$
$929$	19.1324	0.627715	0.313858	+	0.949470i	$0.398379\pi$
$930$	−48.0870	−1.57683
$931$	27.3546	0.896512
$932$	7.37613	0.241613
$933$	38.4915	1.26015
$934$	33.0499	1.08143
$935$	0.904382	0.0295765
$936$	−1.11888	−0.0365718
$937$	−2.54020	−0.0829847	−0.0414924	−	0.999139i	$-0.513211\pi$
$937$	−2.54020	−0.0829847	−0.0414924	+	0.999139i	$0.513211\pi$
$938$	25.3239	0.826854
$939$	81.0827	2.64603
$940$	26.4301	0.862056
$941$	−32.3291	−1.05390	−0.526950	−	0.849897i	$-0.676664\pi$
$941$	−32.3291	−1.05390	−0.526950	+	0.849897i	$0.676664\pi$
$942$	−96.9310	−3.15818
$943$	41.5254	1.35225
$944$	−53.9177	−1.75487
$945$	−96.7472	−3.14719
$946$	6.08926	0.197979
$947$	−50.2228	−1.63202	−0.816011	−	0.578037i	$-0.803819\pi$
$947$	−50.2228	−1.63202	−0.816011	+	0.578037i	$0.803819\pi$
$948$	121.493	3.94590
$949$	4.08853	0.132719
$950$	13.3507	0.433152
$951$	40.4017	1.31012
$952$	−1.19707	−0.0387974
$953$	45.9492	1.48844	0.744221	−	0.667934i	$-0.232821\pi$
$953$	45.9492	1.48844	0.744221	+	0.667934i	$0.232821\pi$
$954$	−213.690	−6.91848
$955$	−9.10414	−0.294603
$956$	26.2982	0.850544
$957$	−8.65511	−0.279780
$958$	−51.8931	−1.67659
$959$	−82.2029	−2.65447
$960$	−44.5222	−1.43695
$961$	−8.19013	−0.264198
$962$	−1.15991	−0.0373969
$963$	−15.7757	−0.508364
$964$	−61.9120	−1.99405
$965$	−5.33789	−0.171833
$966$	−246.944	−7.94530
$967$	−26.8671	−0.863988	−0.431994	−	0.901877i	$-0.642190\pi$
$967$	−26.8671	−0.863988	−0.431994	+	0.901877i	$0.642190\pi$
$968$	2.76608	0.0889052
$969$	−8.67517	−0.278687
$970$	−15.3969	−0.494366
$971$	9.99159	0.320645	0.160323	−	0.987065i	$-0.448747\pi$
$971$	9.99159	0.320645	0.160323	+	0.987065i	$0.448747\pi$
$972$	83.0016	2.66228
$973$	10.5147	0.337085
$974$	−35.1438	−1.12608
$975$	−4.97536	−0.159339
$976$	4.93607	0.158000
$977$	17.5745	0.562258	0.281129	−	0.959670i	$-0.409291\pi$
$977$	17.5745	0.562258	0.281129	+	0.959670i	$0.409291\pi$
$978$	129.720	4.14798
$979$	8.44408	0.269874
$980$	36.1385	1.15440
$981$	−7.58200	−0.242074
$982$	−24.3589	−0.777325
$983$	35.4648	1.13115	0.565576	−	0.824696i	$-0.308654\pi$
$983$	35.4648	1.13115	0.565576	+	0.824696i	$0.308654\pi$
$984$	3.98076	0.126902
$985$	−32.3143	−1.02962
$986$	10.7610	0.342700
$987$	−113.022	−3.59753
$988$	2.97944	0.0947887
$989$	−48.1440	−1.53089
$990$	12.8144	0.407268
$991$	4.33294	0.137640	0.0688202	−	0.997629i	$-0.478077\pi$
$991$	4.33294	0.137640	0.0688202	+	0.997629i	$0.478077\pi$
$992$	38.6536	1.22725
$993$	−14.9396	−0.474095
$994$	73.4617	2.33006
$995$	−17.3684	−0.550614
$996$	−107.542	−3.40760
$997$	4.82838	0.152916	0.0764581	−	0.997073i	$-0.475639\pi$
$997$	4.82838	0.152916	0.0764581	+	0.997073i	$0.475639\pi$
$998$	62.4725	1.97753
$999$	14.9052	0.471580

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4033.2.a.f.1.13	✓	85

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4033.2.a.f.1.13	✓	85	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:	\( N \)	\(=\)	\( 4033 = 37 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4033.a (trivial)

\(f(q)\)	\(=\)	\(q-2.03156 q^{2} +3.25300 q^{3} +2.12723 q^{4} +1.52354 q^{5} -6.60865 q^{6} -4.26037 q^{7} -0.258468 q^{8} +7.58200 q^{9} +O(q^{10})\)	\(q-2.03156 q^{2} +3.25300 q^{3} +2.12723 q^{4} +1.52354 q^{5} -6.60865 q^{6} -4.26037 q^{7} -0.258468 q^{8} +7.58200 q^{9} -3.09515 q^{10} -0.546050 q^{11} +6.91986 q^{12} +0.570945 q^{13} +8.65518 q^{14} +4.95606 q^{15} -3.72936 q^{16} -1.08709 q^{17} -15.4033 q^{18} +2.45317 q^{19} +3.24091 q^{20} -13.8590 q^{21} +1.10933 q^{22} -8.77079 q^{23} -0.840796 q^{24} -2.67884 q^{25} -1.15991 q^{26} +14.9052 q^{27} -9.06277 q^{28} +4.87255 q^{29} -10.0685 q^{30} +4.77597 q^{31} +8.09335 q^{32} -1.77630 q^{33} +2.20849 q^{34} -6.49083 q^{35} +16.1286 q^{36} +1.00000 q^{37} -4.98375 q^{38} +1.85728 q^{39} -0.393786 q^{40} -4.73451 q^{41} +28.1553 q^{42} +5.48913 q^{43} -1.16157 q^{44} +11.5514 q^{45} +17.8184 q^{46} +8.15516 q^{47} -12.1316 q^{48} +11.1507 q^{49} +5.44221 q^{50} -3.53631 q^{51} +1.21453 q^{52} +13.8731 q^{53} -30.2808 q^{54} -0.831927 q^{55} +1.10117 q^{56} +7.98015 q^{57} -9.89887 q^{58} +14.4576 q^{59} +10.5427 q^{60} -1.32357 q^{61} -9.70265 q^{62} -32.3021 q^{63} -8.98338 q^{64} +0.869855 q^{65} +3.60865 q^{66} +2.92586 q^{67} -2.31249 q^{68} -28.5314 q^{69} +13.1865 q^{70} +8.48759 q^{71} -1.95970 q^{72} +7.16098 q^{73} -2.03156 q^{74} -8.71425 q^{75} +5.21844 q^{76} +2.32637 q^{77} -3.77318 q^{78} +17.5571 q^{79} -5.68182 q^{80} +25.7407 q^{81} +9.61844 q^{82} -15.5410 q^{83} -29.4812 q^{84} -1.65623 q^{85} -11.1515 q^{86} +15.8504 q^{87} +0.141136 q^{88} -15.4639 q^{89} -23.4674 q^{90} -2.43244 q^{91} -18.6575 q^{92} +15.5362 q^{93} -16.5677 q^{94} +3.73749 q^{95} +26.3276 q^{96} +4.97454 q^{97} -22.6534 q^{98} -4.14015 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9}+O(q^{10}) \) 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9	\( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9} + 9 q^{10} + 37 q^{11} + 44 q^{12} + 14 q^{13} + 26 q^{14} + 27 q^{15} + 85 q^{16} + 34 q^{17} + 3 q^{18} + 15 q^{19} + 15 q^{20} + 17 q^{21} + q^{22} + 72 q^{23} + 15 q^{24} + 85 q^{25} + 33 q^{26} + 69 q^{27} + 7 q^{28} + 19 q^{29} - 9 q^{30} + 23 q^{31} + 51 q^{32} + 32 q^{33} + 49 q^{34} + 40 q^{35} + 121 q^{36} + 85 q^{37} + 84 q^{38} + 39 q^{39} + 22 q^{40} + 55 q^{41} - 28 q^{42} + 78 q^{44} + 28 q^{45} + 17 q^{46} + 184 q^{47} + 97 q^{48} + 88 q^{49} + 26 q^{50} + 27 q^{51} + 73 q^{52} + 64 q^{53} + 31 q^{54} + 39 q^{55} + 68 q^{56} - 33 q^{57} + 28 q^{58} + 60 q^{59} - 22 q^{60} + 7 q^{61} + 70 q^{62} + 28 q^{63} + 102 q^{64} + 17 q^{65} - 15 q^{66} + 82 q^{67} + 92 q^{68} + 22 q^{69} - 41 q^{70} + 113 q^{71} - 19 q^{73} + 11 q^{74} + 45 q^{75} + 34 q^{76} + 64 q^{77} + 29 q^{78} + 23 q^{79} + 54 q^{80} + 149 q^{81} + 4 q^{82} + 100 q^{83} - 49 q^{84} - 5 q^{85} - 24 q^{86} + 65 q^{87} + 14 q^{88} + 84 q^{89} - 21 q^{90} + 32 q^{91} + 95 q^{92} + 19 q^{93} - 47 q^{94} + 102 q^{95} + 29 q^{96} + 7 q^{97} + 26 q^{98} + 107 q^{99}+O(q^{100}) \) 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9 + 9 * q^10 + 37 * q^11 + 44 * q^12 + 14 * q^13 + 26 * q^14 + 27 * q^15 + 85 * q^16 + 34 * q^17 + 3 * q^18 + 15 * q^19 + 15 * q^20 + 17 * q^21 + q^22 + 72 * q^23 + 15 * q^24 + 85 * q^25 + 33 * q^26 + 69 * q^27 + 7 * q^28 + 19 * q^29 - 9 * q^30 + 23 * q^31 + 51 * q^32 + 32 * q^33 + 49 * q^34 + 40 * q^35 + 121 * q^36 + 85 * q^37 + 84 * q^38 + 39 * q^39 + 22 * q^40 + 55 * q^41 - 28 * q^42 + 78 * q^44 + 28 * q^45 + 17 * q^46 + 184 * q^47 + 97 * q^48 + 88 * q^49 + 26 * q^50 + 27 * q^51 + 73 * q^52 + 64 * q^53 + 31 * q^54 + 39 * q^55 + 68 * q^56 - 33 * q^57 + 28 * q^58 + 60 * q^59 - 22 * q^60 + 7 * q^61 + 70 * q^62 + 28 * q^63 + 102 * q^64 + 17 * q^65 - 15 * q^66 + 82 * q^67 + 92 * q^68 + 22 * q^69 - 41 * q^70 + 113 * q^71 - 19 * q^73 + 11 * q^74 + 45 * q^75 + 34 * q^76 + 64 * q^77 + 29 * q^78 + 23 * q^79 + 54 * q^80 + 149 * q^81 + 4 * q^82 + 100 * q^83 - 49 * q^84 - 5 * q^85 - 24 * q^86 + 65 * q^87 + 14 * q^88 + 84 * q^89 - 21 * q^90 + 32 * q^91 + 95 * q^92 + 19 * q^93 - 47 * q^94 + 102 * q^95 + 29 * q^96 + 7 * q^97 + 26 * q^98 + 107 * q^99

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