LMFDB - Embedded newform 4031.2.a.c.1.61

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.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:	$32.1876970548$
Analytic rank:	$1$
Dimension:	$61$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.61
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.66551 q^{2} -2.32686 q^{3} +5.10492 q^{4} +0.610111 q^{5} -6.20227 q^{6} -0.943250 q^{7} +8.27618 q^{8} +2.41430 q^{9} +O(q^{10})$	\(q+2.66551 q^{2} -2.32686 q^{3} +5.10492 q^{4} +0.610111 q^{5} -6.20227 q^{6} -0.943250 q^{7} +8.27618 q^{8} +2.41430 q^{9} +1.62626 q^{10} -3.25430 q^{11} -11.8785 q^{12} -0.245980 q^{13} -2.51424 q^{14} -1.41965 q^{15} +11.8504 q^{16} -4.59580 q^{17} +6.43532 q^{18} -4.18839 q^{19} +3.11457 q^{20} +2.19481 q^{21} -8.67435 q^{22} -0.591566 q^{23} -19.2575 q^{24} -4.62776 q^{25} -0.655662 q^{26} +1.36285 q^{27} -4.81521 q^{28} -1.00000 q^{29} -3.78407 q^{30} +1.97744 q^{31} +15.0348 q^{32} +7.57231 q^{33} -12.2501 q^{34} -0.575487 q^{35} +12.3248 q^{36} -11.1135 q^{37} -11.1642 q^{38} +0.572363 q^{39} +5.04939 q^{40} -1.50454 q^{41} +5.85029 q^{42} +7.20157 q^{43} -16.6129 q^{44} +1.47299 q^{45} -1.57682 q^{46} +0.483429 q^{47} -27.5742 q^{48} -6.11028 q^{49} -12.3353 q^{50} +10.6938 q^{51} -1.25571 q^{52} -5.53768 q^{53} +3.63269 q^{54} -1.98548 q^{55} -7.80650 q^{56} +9.74582 q^{57} -2.66551 q^{58} -1.37587 q^{59} -7.24718 q^{60} +10.0674 q^{61} +5.27088 q^{62} -2.27728 q^{63} +16.3747 q^{64} -0.150075 q^{65} +20.1840 q^{66} -15.9307 q^{67} -23.4612 q^{68} +1.37649 q^{69} -1.53396 q^{70} +7.08536 q^{71} +19.9811 q^{72} +9.30661 q^{73} -29.6231 q^{74} +10.7682 q^{75} -21.3814 q^{76} +3.06962 q^{77} +1.52564 q^{78} -16.3662 q^{79} +7.23004 q^{80} -10.4141 q^{81} -4.01036 q^{82} +4.17322 q^{83} +11.2043 q^{84} -2.80395 q^{85} +19.1958 q^{86} +2.32686 q^{87} -26.9331 q^{88} -3.17530 q^{89} +3.92626 q^{90} +0.232021 q^{91} -3.01990 q^{92} -4.60124 q^{93} +1.28858 q^{94} -2.55539 q^{95} -34.9840 q^{96} +13.7722 q^{97} -16.2870 q^{98} -7.85684 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) $ 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * 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 $

$2$	2.66551	1.88480	0.942398	−	0.334492i	$-0.108565\pi$
$2$	2.66551	1.88480	0.942398	+	0.334492i	$0.108565\pi$
$3$	−2.32686	−1.34342	−0.671708	−	0.740816i	$-0.734439\pi$
$3$	−2.32686	−1.34342	−0.671708	+	0.740816i	$0.734439\pi$
$4$	5.10492	2.55246
$5$	0.610111	0.272850	0.136425	−	0.990650i	$-0.456439\pi$
$5$	0.610111	0.272850	0.136425	+	0.990650i	$0.456439\pi$
$6$	−6.20227	−2.53207
$7$	−0.943250	−0.356515	−0.178257	−	0.983984i	$-0.557046\pi$
$7$	−0.943250	−0.356515	−0.178257	+	0.983984i	$0.557046\pi$
$8$	8.27618	2.92607
$9$	2.41430	0.804766
$10$	1.62626	0.514267
$11$	−3.25430	−0.981208	−0.490604	−	0.871383i	$-0.663224\pi$
$11$	−3.25430	−0.981208	−0.490604	+	0.871383i	$0.663224\pi$
$12$	−11.8785	−3.42901
$13$	−0.245980	−0.0682227	−0.0341114	−	0.999418i	$-0.510860\pi$
$13$	−0.245980	−0.0682227	−0.0341114	+	0.999418i	$0.510860\pi$
$14$	−2.51424	−0.671958
$15$	−1.41965	−0.366551
$16$	11.8504	2.96259
$17$	−4.59580	−1.11465	−0.557323	−	0.830296i	$-0.688171\pi$
$17$	−4.59580	−1.11465	−0.557323	+	0.830296i	$0.688171\pi$
$18$	6.43532	1.51682
$19$	−4.18839	−0.960883	−0.480442	−	0.877027i	$-0.659524\pi$
$19$	−4.18839	−0.960883	−0.480442	+	0.877027i	$0.659524\pi$
$20$	3.11457	0.696439
$21$	2.19481	0.478948
$22$	−8.67435	−1.84938
$23$	−0.591566	−0.123350	−0.0616750	−	0.998096i	$-0.519644\pi$
$23$	−0.591566	−0.123350	−0.0616750	+	0.998096i	$0.519644\pi$
$24$	−19.2575	−3.93093
$25$	−4.62776	−0.925553
$26$	−0.655662	−0.128586
$27$	1.36285	0.262281
$28$	−4.81521	−0.909989
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−3.78407	−0.690874
$31$	1.97744	0.355159	0.177580	−	0.984106i	$-0.443173\pi$
$31$	1.97744	0.355159	0.177580	+	0.984106i	$0.443173\pi$
$32$	15.0348	2.65781
$33$	7.57231	1.31817
$34$	−12.2501	−2.10088
$35$	−0.575487	−0.0972751
$36$	12.3248	2.05413
$37$	−11.1135	−1.82705	−0.913524	−	0.406784i	$-0.866650\pi$
$37$	−11.1135	−1.82705	−0.913524	+	0.406784i	$0.866650\pi$
$38$	−11.1642	−1.81107
$39$	0.572363	0.0916514
$40$	5.04939	0.798379
$41$	−1.50454	−0.234970	−0.117485	−	0.993075i	$-0.537483\pi$
$41$	−1.50454	−0.234970	−0.117485	+	0.993075i	$0.537483\pi$
$42$	5.85029	0.902719
$43$	7.20157	1.09823	0.549114	−	0.835747i	$-0.314965\pi$
$43$	7.20157	1.09823	0.549114	+	0.835747i	$0.314965\pi$
$44$	−16.6129	−2.50449
$45$	1.47299	0.219580
$46$	−1.57682	−0.232490
$47$	0.483429	0.0705154	0.0352577	−	0.999378i	$-0.488775\pi$
$47$	0.483429	0.0705154	0.0352577	+	0.999378i	$0.488775\pi$
$48$	−27.5742	−3.97999
$49$	−6.11028	−0.872897
$50$	−12.3353	−1.74448
$51$	10.6938	1.49743
$52$	−1.25571	−0.174136
$53$	−5.53768	−0.760659	−0.380329	−	0.924851i	$-0.624189\pi$
$53$	−5.53768	−0.760659	−0.380329	+	0.924851i	$0.624189\pi$
$54$	3.63269	0.494346
$55$	−1.98548	−0.267723
$56$	−7.80650	−1.04319
$57$	9.74582	1.29087
$58$	−2.66551	−0.349998
$59$	−1.37587	−0.179123	−0.0895616	−	0.995981i	$-0.528547\pi$
$59$	−1.37587	−0.179123	−0.0895616	+	0.995981i	$0.528547\pi$
$60$	−7.24718	−0.935607
$61$	10.0674	1.28900	0.644500	−	0.764604i	$-0.277065\pi$
$61$	10.0674	1.28900	0.644500	+	0.764604i	$0.277065\pi$
$62$	5.27088	0.669403
$63$	−2.27728	−0.286911
$64$	16.3747	2.04684
$65$	−0.150075	−0.0186146
$66$	20.1840	2.48448
$67$	−15.9307	−1.94625	−0.973124	−	0.230283i	$-0.926035\pi$
$67$	−15.9307	−1.94625	−0.973124	+	0.230283i	$0.926035\pi$
$68$	−23.4612	−2.84509
$69$	1.37649	0.165710
$70$	−1.53396	−0.183344
$71$	7.08536	0.840878	0.420439	−	0.907321i	$-0.361876\pi$
$71$	7.08536	0.840878	0.420439	+	0.907321i	$0.361876\pi$
$72$	19.9811	2.35480
$73$	9.30661	1.08926	0.544628	−	0.838678i	$-0.316671\pi$
$73$	9.30661	1.08926	0.544628	+	0.838678i	$0.316671\pi$
$74$	−29.6231	−3.44362
$75$	10.7682	1.24340
$76$	−21.3814	−2.45262
$77$	3.06962	0.349815
$78$	1.52564	0.172744
$79$	−16.3662	−1.84134	−0.920669	−	0.390345i	$-0.872356\pi$
$79$	−16.3662	−1.84134	−0.920669	+	0.390345i	$0.872356\pi$
$80$	7.23004	0.808343
$81$	−10.4141	−1.15712
$82$	−4.01036	−0.442870
$83$	4.17322	0.458070	0.229035	−	0.973418i	$-0.426443\pi$
$83$	4.17322	0.458070	0.229035	+	0.973418i	$0.426443\pi$
$84$	11.2043	1.22249
$85$	−2.80395	−0.304131
$86$	19.1958	2.06994
$87$	2.32686	0.249466
$88$	−26.9331	−2.87108
$89$	−3.17530	−0.336581	−0.168290	−	0.985737i	$-0.553825\pi$
$89$	−3.17530	−0.336581	−0.168290	+	0.985737i	$0.553825\pi$
$90$	3.92626	0.413864
$91$	0.232021	0.0243224
$92$	−3.01990	−0.314846
$93$	−4.60124	−0.477126
$94$	1.28858	0.132907
$95$	−2.55539	−0.262177
$96$	−34.9840	−3.57054
$97$	13.7722	1.39836	0.699179	−	0.714947i	$-0.253549\pi$
$97$	13.7722	1.39836	0.699179	+	0.714947i	$0.253549\pi$
$98$	−16.2870	−1.64523
$99$	−7.85684	−0.789642
$100$	−23.6244	−2.36244
$101$	2.08548	0.207513	0.103757	−	0.994603i	$-0.466914\pi$
$101$	2.08548	0.207513	0.103757	+	0.994603i	$0.466914\pi$
$102$	28.5044	2.82236
$103$	10.4087	1.02560	0.512801	−	0.858507i	$-0.328608\pi$
$103$	10.4087	1.02560	0.512801	+	0.858507i	$0.328608\pi$
$104$	−2.03578	−0.199624
$105$	1.33908	0.130681
$106$	−14.7607	−1.43369
$107$	−2.18388	−0.211124	−0.105562	−	0.994413i	$-0.533664\pi$
$107$	−2.18388	−0.211124	−0.105562	+	0.994413i	$0.533664\pi$
$108$	6.95724	0.669461
$109$	8.38212	0.802861	0.401431	−	0.915889i	$-0.368513\pi$
$109$	8.38212	0.802861	0.401431	+	0.915889i	$0.368513\pi$
$110$	−5.29232	−0.504603
$111$	25.8596	2.45449
$112$	−11.1778	−1.05621
$113$	−13.7602	−1.29445	−0.647224	−	0.762300i	$-0.724070\pi$
$113$	−13.7602	−1.29445	−0.647224	+	0.762300i	$0.724070\pi$
$114$	25.9775	2.43302
$115$	−0.360921	−0.0336561
$116$	−5.10492	−0.473980
$117$	−0.593870	−0.0549033
$118$	−3.66739	−0.337611
$119$	4.33499	0.397388
$120$	−11.7492	−1.07255
$121$	−0.409541	−0.0372310
$122$	26.8347	2.42950
$123$	3.50086	0.315662
$124$	10.0947	0.906529
$125$	−5.87401	−0.525387
$126$	−6.07011	−0.540769
$127$	2.95668	0.262363	0.131181	−	0.991358i	$-0.458123\pi$
$127$	2.95668	0.262363	0.131181	+	0.991358i	$0.458123\pi$
$128$	13.5772	1.20007
$129$	−16.7571	−1.47538
$130$	−0.400027	−0.0350847
$131$	14.7869	1.29194	0.645968	−	0.763365i	$-0.276454\pi$
$131$	14.7869	1.29194	0.645968	+	0.763365i	$0.276454\pi$
$132$	38.6560	3.36458
$133$	3.95070	0.342569
$134$	−42.4634	−3.66828
$135$	0.831491	0.0715634
$136$	−38.0357	−3.26153
$137$	−7.40461	−0.632619	−0.316309	−	0.948656i	$-0.602444\pi$
$137$	−7.40461	−0.632619	−0.316309	+	0.948656i	$0.602444\pi$
$138$	3.66905	0.312330
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	−2.93782	−0.248291
$141$	−1.12487	−0.0947315
$142$	18.8861	1.58488
$143$	0.800494	0.0669407
$144$	28.6103	2.38419
$145$	−0.610111	−0.0506670
$146$	24.8068	2.05303
$147$	14.2178	1.17266
$148$	−56.7335	−4.66347
$149$	−16.6805	−1.36652	−0.683261	−	0.730174i	$-0.739439\pi$
$149$	−16.6805	−1.36652	−0.683261	+	0.730174i	$0.739439\pi$
$150$	28.7026	2.34356
$151$	9.84737	0.801367	0.400684	−	0.916216i	$-0.368773\pi$
$151$	9.84737	0.801367	0.400684	+	0.916216i	$0.368773\pi$
$152$	−34.6639	−2.81161
$153$	−11.0956	−0.897029
$154$	8.18208	0.659331
$155$	1.20646	0.0969052
$156$	2.92187	0.233937
$157$	−17.0266	−1.35887	−0.679434	−	0.733736i	$-0.737775\pi$
$157$	−17.0266	−1.35887	−0.679434	+	0.733736i	$0.737775\pi$
$158$	−43.6241	−3.47055
$159$	12.8854	1.02188
$160$	9.17292	0.725183
$161$	0.557994	0.0439761
$162$	−27.7587	−2.18093
$163$	−2.64951	−0.207525	−0.103763	−	0.994602i	$-0.533088\pi$
$163$	−2.64951	−0.207525	−0.103763	+	0.994602i	$0.533088\pi$
$164$	−7.68055	−0.599750
$165$	4.61995	0.359663
$166$	11.1237	0.863370
$167$	−22.5385	−1.74408	−0.872042	−	0.489431i	$-0.837204\pi$
$167$	−22.5385	−1.74408	−0.872042	+	0.489431i	$0.837204\pi$
$168$	18.1647	1.40143
$169$	−12.9395	−0.995346
$170$	−7.47395	−0.573225
$171$	−10.1120	−0.773286
$172$	36.7634	2.80318
$173$	25.2423	1.91914	0.959569	−	0.281472i	$-0.0908227\pi$
$173$	25.2423	1.91914	0.959569	+	0.281472i	$0.0908227\pi$
$174$	6.20227	0.470193
$175$	4.36514	0.329973
$176$	−38.5646	−2.90692
$177$	3.20147	0.240637
$178$	−8.46377	−0.634387
$179$	0.0947056	0.00707863	0.00353931	−	0.999994i	$-0.498873\pi$
$179$	0.0947056	0.00707863	0.00353931	+	0.999994i	$0.498873\pi$
$180$	7.51949	0.560470
$181$	−17.2170	−1.27973	−0.639864	−	0.768488i	$-0.721009\pi$
$181$	−17.2170	−1.27973	−0.639864	+	0.768488i	$0.721009\pi$
$182$	0.618453	0.0458428
$183$	−23.4255	−1.73166
$184$	−4.89591	−0.360931
$185$	−6.78048	−0.498510
$186$	−12.2646	−0.899286
$187$	14.9561	1.09370
$188$	2.46787	0.179988
$189$	−1.28551	−0.0935070
$190$	−6.81140	−0.494151
$191$	−18.1952	−1.31656	−0.658279	−	0.752774i	$-0.728715\pi$
$191$	−18.1952	−1.31656	−0.658279	+	0.752774i	$0.728715\pi$
$192$	−38.1017	−2.74975
$193$	10.7479	0.773653	0.386827	−	0.922152i	$-0.373571\pi$
$193$	10.7479	0.773653	0.386827	+	0.922152i	$0.373571\pi$
$194$	36.7100	2.63562
$195$	0.349205	0.0250071
$196$	−31.1925	−2.22803
$197$	6.83923	0.487275	0.243638	−	0.969866i	$-0.421659\pi$
$197$	6.83923	0.487275	0.243638	+	0.969866i	$0.421659\pi$
$198$	−20.9425	−1.48832
$199$	5.20257	0.368800	0.184400	−	0.982851i	$-0.440966\pi$
$199$	5.20257	0.368800	0.184400	+	0.982851i	$0.440966\pi$
$200$	−38.3002	−2.70823
$201$	37.0686	2.61462
$202$	5.55886	0.391120
$203$	0.943250	0.0662031
$204$	54.5910	3.82213
$205$	−0.917937	−0.0641115
$206$	27.7445	1.93305
$207$	−1.42822	−0.0992679
$208$	−2.91496	−0.202116
$209$	13.6303	0.942827
$210$	3.56933	0.246307
$211$	22.3560	1.53905	0.769525	−	0.638617i	$-0.220493\pi$
$211$	22.3560	1.53905	0.769525	+	0.638617i	$0.220493\pi$
$212$	−28.2694	−1.94155
$213$	−16.4867	−1.12965
$214$	−5.82115	−0.397926
$215$	4.39376	0.299652
$216$	11.2792	0.767452
$217$	−1.86522	−0.126619
$218$	22.3426	1.51323
$219$	−21.6552	−1.46332
$220$	−10.1357	−0.683351
$221$	1.13048	0.0760441
$222$	68.9289	4.62621
$223$	0.936202	0.0626927	0.0313463	−	0.999509i	$-0.490021\pi$
$223$	0.936202	0.0626927	0.0313463	+	0.999509i	$0.490021\pi$
$224$	−14.1816	−0.947548
$225$	−11.1728	−0.744853
$226$	−36.6778	−2.43977
$227$	1.25322	0.0831789	0.0415895	−	0.999135i	$-0.486758\pi$
$227$	1.25322	0.0831789	0.0415895	+	0.999135i	$0.486758\pi$
$228$	49.7516	3.29488
$229$	−6.09073	−0.402487	−0.201243	−	0.979541i	$-0.564498\pi$
$229$	−6.09073	−0.402487	−0.201243	+	0.979541i	$0.564498\pi$
$230$	−0.962037	−0.0634349
$231$	−7.14258	−0.469947
$232$	−8.27618	−0.543358
$233$	10.4210	0.682704	0.341352	−	0.939936i	$-0.389115\pi$
$233$	10.4210	0.682704	0.341352	+	0.939936i	$0.389115\pi$
$234$	−1.58296	−0.103482
$235$	0.294946	0.0192401
$236$	−7.02371	−0.457205
$237$	38.0818	2.47368
$238$	11.5549	0.748995
$239$	16.7869	1.08585	0.542927	−	0.839780i	$-0.317316\pi$
$239$	16.7869	1.08585	0.542927	+	0.839780i	$0.317316\pi$
$240$	−16.8233	−1.08594
$241$	−23.4061	−1.50772	−0.753858	−	0.657037i	$-0.771810\pi$
$241$	−23.4061	−1.50772	−0.753858	+	0.657037i	$0.771810\pi$
$242$	−1.09163	−0.0701728
$243$	20.1436	1.29221
$244$	51.3933	3.29012
$245$	−3.72795	−0.238170
$246$	9.33156	0.594958
$247$	1.03026	0.0655541
$248$	16.3657	1.03922
$249$	−9.71051	−0.615379
$250$	−15.6572	−0.990248
$251$	−20.1219	−1.27008	−0.635041	−	0.772478i	$-0.719017\pi$
$251$	−20.1219	−1.27008	−0.635041	+	0.772478i	$0.719017\pi$
$252$	−11.6254	−0.732328
$253$	1.92513	0.121032
$254$	7.88104	0.494501
$255$	6.52441	0.408575
$256$	3.44071	0.215044
$257$	−5.77422	−0.360186	−0.180093	−	0.983650i	$-0.557640\pi$
$257$	−5.77422	−0.360186	−0.180093	+	0.983650i	$0.557640\pi$
$258$	−44.6660	−2.78079
$259$	10.4828	0.651370
$260$	−0.766123	−0.0475129
$261$	−2.41430	−0.149441
$262$	39.4145	2.43504
$263$	11.1297	0.686287	0.343144	−	0.939283i	$-0.388508\pi$
$263$	11.1297	0.686287	0.343144	+	0.939283i	$0.388508\pi$
$264$	62.6698	3.85706
$265$	−3.37860	−0.207546
$266$	10.5306	0.645673
$267$	7.38849	0.452168
$268$	−81.3250	−4.96772
$269$	−2.10916	−0.128598	−0.0642989	−	0.997931i	$-0.520481\pi$
$269$	−2.10916	−0.128598	−0.0642989	+	0.997931i	$0.520481\pi$
$270$	2.21634	0.134882
$271$	0.902505	0.0548233	0.0274116	−	0.999624i	$-0.491274\pi$
$271$	0.902505	0.0548233	0.0274116	+	0.999624i	$0.491274\pi$
$272$	−54.4619	−3.30224
$273$	−0.539881	−0.0326751
$274$	−19.7370	−1.19236
$275$	15.0601	0.908160
$276$	7.02689	0.422969
$277$	17.5108	1.05212	0.526062	−	0.850446i	$-0.323668\pi$
$277$	17.5108	1.05212	0.526062	+	0.850446i	$0.323668\pi$
$278$	−2.66551	−0.159866
$279$	4.77413	0.285820
$280$	−4.76283	−0.284634
$281$	5.36251	0.319900	0.159950	−	0.987125i	$-0.448867\pi$
$281$	5.36251	0.319900	0.159950	+	0.987125i	$0.448867\pi$
$282$	−2.99836	−0.178550
$283$	−26.3047	−1.56365	−0.781826	−	0.623497i	$-0.785711\pi$
$283$	−26.3047	−1.56365	−0.781826	+	0.623497i	$0.785711\pi$
$284$	36.1702	2.14631
$285$	5.94604	0.352213
$286$	2.13372	0.126170
$287$	1.41916	0.0837701
$288$	36.2985	2.13891
$289$	4.12139	0.242435
$290$	−1.62626	−0.0954970
$291$	−32.0461	−1.87858
$292$	47.5095	2.78028
$293$	29.1843	1.70496	0.852482	−	0.522756i	$-0.175096\pi$
$293$	29.1843	1.70496	0.852482	+	0.522756i	$0.175096\pi$
$294$	37.8976	2.21023
$295$	−0.839435	−0.0488738
$296$	−91.9773	−5.34607
$297$	−4.43512	−0.257352
$298$	−44.4620	−2.57562
$299$	0.145514	0.00841527
$300$	54.9707	3.17373
$301$	−6.79287	−0.391535
$302$	26.2482	1.51041
$303$	−4.85263	−0.278776
$304$	−49.6339	−2.84670
$305$	6.14224	0.351704
$306$	−29.5755	−1.69072
$307$	27.7702	1.58493	0.792464	−	0.609919i	$-0.208798\pi$
$307$	27.7702	1.58493	0.792464	+	0.609919i	$0.208798\pi$
$308$	15.6701	0.892889
$309$	−24.2197	−1.37781
$310$	3.21583	0.182647
$311$	9.96965	0.565327	0.282664	−	0.959219i	$-0.408782\pi$
$311$	9.96965	0.565327	0.282664	+	0.959219i	$0.408782\pi$
$312$	4.73698	0.268179
$313$	−8.95119	−0.505951	−0.252975	−	0.967473i	$-0.581409\pi$
$313$	−8.95119	−0.505951	−0.252975	+	0.967473i	$0.581409\pi$
$314$	−45.3844	−2.56119
$315$	−1.38940	−0.0782837
$316$	−83.5479	−4.69994
$317$	−19.1432	−1.07519	−0.537593	−	0.843204i	$-0.680666\pi$
$317$	−19.1432	−1.07519	−0.537593	+	0.843204i	$0.680666\pi$
$318$	34.3462	1.92604
$319$	3.25430	0.182206
$320$	9.99039	0.558480
$321$	5.08160	0.283627
$322$	1.48734	0.0828861
$323$	19.2490	1.07104
$324$	−53.1629	−2.95350
$325$	1.13834	0.0631437
$326$	−7.06227	−0.391143
$327$	−19.5040	−1.07858
$328$	−12.4518	−0.687537
$329$	−0.455994	−0.0251398
$330$	12.3145	0.677891
$331$	−8.23189	−0.452465	−0.226233	−	0.974073i	$-0.572641\pi$
$331$	−8.23189	−0.452465	−0.226233	+	0.974073i	$0.572641\pi$
$332$	21.3039	1.16921
$333$	−26.8313	−1.47035
$334$	−60.0766	−3.28724
$335$	−9.71952	−0.531034
$336$	26.0093	1.41892
$337$	−23.7600	−1.29429	−0.647145	−	0.762367i	$-0.724037\pi$
$337$	−23.7600	−1.29429	−0.647145	+	0.762367i	$0.724037\pi$
$338$	−34.4903	−1.87602
$339$	32.0181	1.73898
$340$	−14.3139	−0.776282
$341$	−6.43519	−0.348485
$342$	−26.9537	−1.45749
$343$	12.3663	0.667716
$344$	59.6014	3.21349
$345$	0.839815	0.0452141
$346$	67.2836	3.61719
$347$	−13.2565	−0.711647	−0.355824	−	0.934553i	$-0.615800\pi$
$347$	−13.2565	−0.711647	−0.355824	+	0.934553i	$0.615800\pi$
$348$	11.8785	0.636752
$349$	−4.22611	−0.226219	−0.113109	−	0.993583i	$-0.536081\pi$
$349$	−4.22611	−0.226219	−0.113109	+	0.993583i	$0.536081\pi$
$350$	11.6353	0.621933
$351$	−0.335235	−0.0178935
$352$	−48.9278	−2.60786
$353$	−1.34637	−0.0716602	−0.0358301	−	0.999358i	$-0.511408\pi$
$353$	−1.34637	−0.0716602	−0.0358301	+	0.999358i	$0.511408\pi$
$354$	8.53353	0.453552
$355$	4.32286	0.229434
$356$	−16.2096	−0.859109
$357$	−10.0869	−0.533857
$358$	0.252438	0.0133418
$359$	33.6119	1.77397	0.886984	−	0.461800i	$-0.152796\pi$
$359$	33.6119	1.77397	0.886984	+	0.461800i	$0.152796\pi$
$360$	12.1907	0.642508
$361$	−1.45736	−0.0767029
$362$	−45.8920	−2.41203
$363$	0.952946	0.0500167
$364$	1.18445	0.0620819
$365$	5.67807	0.297204
$366$	−62.4408	−3.26383
$367$	3.85438	0.201197	0.100599	−	0.994927i	$-0.467924\pi$
$367$	3.85438	0.201197	0.100599	+	0.994927i	$0.467924\pi$
$368$	−7.01027	−0.365435
$369$	−3.63241	−0.189095
$370$	−18.0734	−0.939591
$371$	5.22341	0.271186
$372$	−23.4890	−1.21785
$373$	−6.82178	−0.353218	−0.176609	−	0.984281i	$-0.556513\pi$
$373$	−6.82178	−0.353218	−0.176609	+	0.984281i	$0.556513\pi$
$374$	39.8656	2.06140
$375$	13.6680	0.705814
$376$	4.00095	0.206333
$377$	0.245980	0.0126686
$378$	−3.42653	−0.176242
$379$	27.4464	1.40983	0.704913	−	0.709294i	$-0.250986\pi$
$379$	27.4464	1.40983	0.704913	+	0.709294i	$0.250986\pi$
$380$	−13.0450	−0.669197
$381$	−6.87979	−0.352462
$382$	−48.4994	−2.48145
$383$	−3.18237	−0.162612	−0.0813058	−	0.996689i	$-0.525909\pi$
$383$	−3.18237	−0.162612	−0.0813058	+	0.996689i	$0.525909\pi$
$384$	−31.5923	−1.61219
$385$	1.87281	0.0954471
$386$	28.6487	1.45818
$387$	17.3867	0.883817
$388$	70.3061	3.56925
$389$	25.8003	1.30813	0.654063	−	0.756440i	$-0.273063\pi$
$389$	25.8003	1.30813	0.654063	+	0.756440i	$0.273063\pi$
$390$	0.930808	0.0471333
$391$	2.71872	0.137492
$392$	−50.5698	−2.55416
$393$	−34.4071	−1.73561
$394$	18.2300	0.918415
$395$	−9.98518	−0.502409
$396$	−40.1085	−2.01553
$397$	−0.580692	−0.0291441	−0.0145720	−	0.999894i	$-0.504639\pi$
$397$	−0.580692	−0.0291441	−0.0145720	+	0.999894i	$0.504639\pi$
$398$	13.8675	0.695113
$399$	−9.19274	−0.460213
$400$	−54.8406	−2.74203
$401$	12.9560	0.646991	0.323495	−	0.946230i	$-0.395142\pi$
$401$	12.9560	0.646991	0.323495	+	0.946230i	$0.395142\pi$
$402$	98.8066	4.92803
$403$	−0.486412	−0.0242299
$404$	10.6462	0.529669
$405$	−6.35374	−0.315720
$406$	2.51424	0.124779
$407$	36.1667	1.79271
$408$	88.5038	4.38159
$409$	7.20619	0.356323	0.178161	−	0.984001i	$-0.442985\pi$
$409$	7.20619	0.356323	0.178161	+	0.984001i	$0.442985\pi$
$410$	−2.44677	−0.120837
$411$	17.2295	0.849870
$412$	53.1357	2.61781
$413$	1.29779	0.0638601
$414$	−3.80692	−0.187100
$415$	2.54613	0.124985
$416$	−3.69827	−0.181323
$417$	2.32686	0.113947
$418$	36.3316	1.77704
$419$	29.6459	1.44830	0.724149	−	0.689643i	$-0.242233\pi$
$419$	29.6459	1.44830	0.724149	+	0.689643i	$0.242233\pi$
$420$	6.83590	0.333558
$421$	−1.04207	−0.0507874	−0.0253937	−	0.999678i	$-0.508084\pi$
$421$	−1.04207	−0.0507874	−0.0253937	+	0.999678i	$0.508084\pi$
$422$	59.5900	2.90079
$423$	1.16714	0.0567484
$424$	−45.8308	−2.22574
$425$	21.2683	1.03166
$426$	−43.9453	−2.12916
$427$	−9.49608	−0.459548
$428$	−11.1485	−0.538885
$429$	−1.86264	−0.0899291
$430$	11.7116	0.564783
$431$	−20.5127	−0.988063	−0.494031	−	0.869444i	$-0.664477\pi$
$431$	−20.5127	−0.988063	−0.494031	+	0.869444i	$0.664477\pi$
$432$	16.1503	0.777030
$433$	−35.6604	−1.71373	−0.856864	−	0.515542i	$-0.827591\pi$
$433$	−35.6604	−1.71373	−0.856864	+	0.515542i	$0.827591\pi$
$434$	−4.97176	−0.238652
$435$	1.41965	0.0680668
$436$	42.7900	2.04927
$437$	2.47771	0.118525
$438$	−57.7221	−2.75807
$439$	19.9751	0.953360	0.476680	−	0.879077i	$-0.341840\pi$
$439$	19.9751	0.953360	0.476680	+	0.879077i	$0.341840\pi$
$440$	−16.4322	−0.783375
$441$	−14.7520	−0.702478
$442$	3.01329	0.143328
$443$	22.5317	1.07051	0.535256	−	0.844690i	$-0.320215\pi$
$443$	22.5317	1.07051	0.535256	+	0.844690i	$0.320215\pi$
$444$	132.011	6.26498
$445$	−1.93729	−0.0918361
$446$	2.49545	0.118163
$447$	38.8133	1.83581
$448$	−15.4454	−0.729728
$449$	25.9544	1.22486	0.612432	−	0.790523i	$-0.290191\pi$
$449$	25.9544	1.22486	0.612432	+	0.790523i	$0.290191\pi$
$450$	−29.7811	−1.40390
$451$	4.89622	0.230554
$452$	−70.2446	−3.30403
$453$	−22.9135	−1.07657
$454$	3.34046	0.156775
$455$	0.141559	0.00663637
$456$	80.6582	3.77716
$457$	−2.80526	−0.131225	−0.0656123	−	0.997845i	$-0.520900\pi$
$457$	−2.80526	−0.131225	−0.0656123	+	0.997845i	$0.520900\pi$
$458$	−16.2349	−0.758606
$459$	−6.26339	−0.292350
$460$	−1.84247	−0.0859058
$461$	−17.3159	−0.806483	−0.403241	−	0.915094i	$-0.632117\pi$
$461$	−17.3159	−0.806483	−0.403241	+	0.915094i	$0.632117\pi$
$462$	−19.0386	−0.885755
$463$	17.9620	0.834767	0.417383	−	0.908731i	$-0.362947\pi$
$463$	17.9620	0.834767	0.417383	+	0.908731i	$0.362947\pi$
$464$	−11.8504	−0.550139
$465$	−2.80727	−0.130184
$466$	27.7773	1.28676
$467$	25.2200	1.16704	0.583522	−	0.812097i	$-0.301674\pi$
$467$	25.2200	1.16704	0.583522	+	0.812097i	$0.301674\pi$
$468$	−3.03166	−0.140138
$469$	15.0266	0.693866
$470$	0.786179	0.0362637
$471$	39.6185	1.82553
$472$	−11.3870	−0.524127
$473$	−23.4360	−1.07759
$474$	101.507	4.66239
$475$	19.3829	0.889348
$476$	22.1298	1.01432
$477$	−13.3696	−0.612152
$478$	44.7455	2.04661
$479$	−2.87397	−0.131315	−0.0656576	−	0.997842i	$-0.520914\pi$
$479$	−2.87397	−0.131315	−0.0656576	+	0.997842i	$0.520914\pi$
$480$	−21.3441	−0.974222
$481$	2.73370	0.124646
$482$	−62.3890	−2.84174
$483$	−1.29838	−0.0590782
$484$	−2.09067	−0.0950306
$485$	8.40260	0.381542
$486$	53.6927	2.43555
$487$	27.8521	1.26210	0.631049	−	0.775743i	$-0.282625\pi$
$487$	27.8521	1.26210	0.631049	+	0.775743i	$0.282625\pi$
$488$	83.3197	3.77171
$489$	6.16504	0.278793
$490$	−9.93688	−0.448902
$491$	−37.0788	−1.67334	−0.836672	−	0.547704i	$-0.815502\pi$
$491$	−37.0788	−1.67334	−0.836672	+	0.547704i	$0.815502\pi$
$492$	17.8716	0.805714
$493$	4.59580	0.206984
$494$	2.74617	0.123556
$495$	−4.79355	−0.215454
$496$	23.4334	1.05219
$497$	−6.68326	−0.299785
$498$	−25.8834	−1.15986
$499$	18.0797	0.809357	0.404679	−	0.914459i	$-0.367383\pi$
$499$	18.0797	0.809357	0.404679	+	0.914459i	$0.367383\pi$
$500$	−29.9863	−1.34103
$501$	52.4441	2.34303
$502$	−53.6350	−2.39385
$503$	14.6346	0.652525	0.326262	−	0.945279i	$-0.394211\pi$
$503$	14.6346	0.652525	0.326262	+	0.945279i	$0.394211\pi$
$504$	−18.8472	−0.839521
$505$	1.27238	0.0566200
$506$	5.13145	0.228121
$507$	30.1084	1.33716
$508$	15.0936	0.669670
$509$	−33.8266	−1.49934	−0.749669	−	0.661813i	$-0.769787\pi$
$509$	−33.8266	−1.49934	−0.749669	+	0.661813i	$0.769787\pi$
$510$	17.3909	0.770080
$511$	−8.77845	−0.388336
$512$	−17.9832	−0.794752
$513$	−5.70816	−0.252021
$514$	−15.3912	−0.678877
$515$	6.35048	0.279836
$516$	−85.5434	−3.76584
$517$	−1.57322	−0.0691903
$518$	27.9420	1.22770
$519$	−58.7355	−2.57820
$520$	−1.24205	−0.0544675
$521$	−7.25453	−0.317827	−0.158913	−	0.987293i	$-0.550799\pi$
$521$	−7.25453	−0.317827	−0.158913	+	0.987293i	$0.550799\pi$
$522$	−6.43532	−0.281666
$523$	34.9514	1.52832	0.764158	−	0.645029i	$-0.223155\pi$
$523$	34.9514	1.52832	0.764158	+	0.645029i	$0.223155\pi$
$524$	75.4858	3.29761
$525$	−10.1571	−0.443291
$526$	29.6663	1.29351
$527$	−9.08793	−0.395877
$528$	89.7346	3.90520
$529$	−22.6500	−0.984785
$530$	−9.00568	−0.391182
$531$	−3.32176	−0.144152
$532$	20.1680	0.874394
$533$	0.370087	0.0160303
$534$	19.6941	0.852245
$535$	−1.33241	−0.0576052
$536$	−131.845	−5.69486
$537$	−0.220367	−0.00950954
$538$	−5.62198	−0.242381
$539$	19.8847	0.856494
$540$	4.24469	0.182663
$541$	18.4622	0.793751	0.396876	−	0.917872i	$-0.370094\pi$
$541$	18.4622	0.793751	0.396876	+	0.917872i	$0.370094\pi$
$542$	2.40563	0.103331
$543$	40.0616	1.71921
$544$	−69.0971	−2.96251
$545$	5.11402	0.219061
$546$	−1.43906	−0.0615859
$547$	35.6296	1.52341	0.761706	−	0.647923i	$-0.224362\pi$
$547$	35.6296	1.52341	0.761706	+	0.647923i	$0.224362\pi$
$548$	−37.7999	−1.61473
$549$	24.3057	1.03734
$550$	40.1428	1.71170
$551$	4.18839	0.178432
$552$	11.3921	0.484880
$553$	15.4374	0.656464
$554$	46.6753	1.98304
$555$	15.7772	0.669707
$556$	−5.10492	−0.216497
$557$	−33.5075	−1.41976	−0.709880	−	0.704323i	$-0.751251\pi$
$557$	−33.5075	−1.41976	−0.709880	+	0.704323i	$0.751251\pi$
$558$	12.7255	0.538712
$559$	−1.77144	−0.0749241
$560$	−6.81973	−0.288186
$561$	−34.8008	−1.46929
$562$	14.2938	0.602947
$563$	30.0323	1.26571	0.632856	−	0.774270i	$-0.281883\pi$
$563$	30.0323	1.26571	0.632856	+	0.774270i	$0.281883\pi$
$564$	−5.74239	−0.241798
$565$	−8.39524	−0.353190
$566$	−70.1153	−2.94717
$567$	9.82306	0.412530
$568$	58.6397	2.46047
$569$	7.14106	0.299369	0.149684	−	0.988734i	$-0.452174\pi$
$569$	7.14106	0.299369	0.149684	+	0.988734i	$0.452174\pi$
$570$	15.8492	0.663850
$571$	31.2195	1.30650	0.653248	−	0.757144i	$-0.273406\pi$
$571$	31.2195	1.30650	0.653248	+	0.757144i	$0.273406\pi$
$572$	4.08646	0.170863
$573$	42.3378	1.76869
$574$	3.78277	0.157890
$575$	2.73763	0.114167
$576$	39.5334	1.64723
$577$	12.5739	0.523460	0.261730	−	0.965141i	$-0.415707\pi$
$577$	12.5739	0.523460	0.261730	+	0.965141i	$0.415707\pi$
$578$	10.9856	0.456940
$579$	−25.0090	−1.03934
$580$	−3.11457	−0.129325
$581$	−3.93639	−0.163309
$582$	−85.4191	−3.54073
$583$	18.0213	0.746364
$584$	77.0231	3.18724
$585$	−0.362327	−0.0149804
$586$	77.7909	3.21351
$587$	−6.94900	−0.286816	−0.143408	−	0.989664i	$-0.545806\pi$
$587$	−6.94900	−0.286816	−0.143408	+	0.989664i	$0.545806\pi$
$588$	72.5807	2.99318
$589$	−8.28231	−0.341267
$590$	−2.23752	−0.0921172
$591$	−15.9140	−0.654613
$592$	−131.699	−5.41279
$593$	8.10447	0.332811	0.166405	−	0.986057i	$-0.446784\pi$
$593$	8.10447	0.332811	0.166405	+	0.986057i	$0.446784\pi$
$594$	−11.8218	−0.485056
$595$	2.64483	0.108427
$596$	−85.1527	−3.48799
$597$	−12.1057	−0.495452
$598$	0.387868	0.0158611
$599$	24.5947	1.00491	0.502457	−	0.864602i	$-0.332430\pi$
$599$	24.5947	1.00491	0.502457	+	0.864602i	$0.332430\pi$
$600$	89.1193	3.63828
$601$	−29.8625	−1.21812	−0.609059	−	0.793125i	$-0.708453\pi$
$601$	−29.8625	−1.21812	−0.609059	+	0.793125i	$0.708453\pi$
$602$	−18.1064	−0.737963
$603$	−38.4615	−1.56627
$604$	50.2700	2.04546
$605$	−0.249866	−0.0101585
$606$	−12.9347	−0.525437
$607$	−39.7956	−1.61525	−0.807626	−	0.589694i	$-0.799248\pi$
$607$	−39.7956	−1.61525	−0.807626	+	0.589694i	$0.799248\pi$
$608$	−62.9718	−2.55384
$609$	−2.19481	−0.0889383
$610$	16.3722	0.662890
$611$	−0.118914	−0.00481075
$612$	−56.6423	−2.28963
$613$	1.54207	0.0622835	0.0311418	−	0.999515i	$-0.490086\pi$
$613$	1.54207	0.0622835	0.0311418	+	0.999515i	$0.490086\pi$
$614$	74.0215	2.98727
$615$	2.13591	0.0861284
$616$	25.4047	1.02358
$617$	−25.2326	−1.01582	−0.507912	−	0.861409i	$-0.669583\pi$
$617$	−25.2326	−1.01582	−0.507912	+	0.861409i	$0.669583\pi$
$618$	−64.5577	−2.59689
$619$	−20.0780	−0.807003	−0.403502	−	0.914979i	$-0.632207\pi$
$619$	−20.0780	−0.807003	−0.403502	+	0.914979i	$0.632207\pi$
$620$	6.15888	0.247347
$621$	−0.806217	−0.0323524
$622$	26.5742	1.06553
$623$	2.99510	0.119996
$624$	6.78271	0.271526
$625$	19.5550	0.782201
$626$	−23.8594	−0.953615
$627$	−31.7158	−1.26661
$628$	−86.9193	−3.46846
$629$	51.0755	2.03651
$630$	−3.70345	−0.147549
$631$	−43.2842	−1.72312	−0.861558	−	0.507659i	$-0.830511\pi$
$631$	−43.2842	−1.72312	−0.861558	+	0.507659i	$0.830511\pi$
$632$	−135.449	−5.38788
$633$	−52.0193	−2.06758
$634$	−51.0262	−2.02651
$635$	1.80390	0.0715857
$636$	65.7790	2.60831
$637$	1.50301	0.0595514
$638$	8.67435	0.343421
$639$	17.1062	0.676710
$640$	8.28361	0.327438
$641$	−6.41945	−0.253553	−0.126777	−	0.991931i	$-0.540463\pi$
$641$	−6.41945	−0.253553	−0.126777	+	0.991931i	$0.540463\pi$
$642$	13.5450	0.534580
$643$	−16.0624	−0.633440	−0.316720	−	0.948519i	$-0.602582\pi$
$643$	−16.0624	−0.633440	−0.316720	+	0.948519i	$0.602582\pi$
$644$	2.84852	0.112247
$645$	−10.2237	−0.402557
$646$	51.3084	2.01870
$647$	−20.4202	−0.802801	−0.401401	−	0.915903i	$-0.631477\pi$
$647$	−20.4202	−0.802801	−0.401401	+	0.915903i	$0.631477\pi$
$648$	−86.1886	−3.38581
$649$	4.47750	0.175757
$650$	3.03425	0.119013
$651$	4.34012	0.170103
$652$	−13.5255	−0.529700
$653$	0.650164	0.0254429	0.0127214	−	0.999919i	$-0.495951\pi$
$653$	0.650164	0.0254429	0.0127214	+	0.999919i	$0.495951\pi$
$654$	−51.9881	−2.03290
$655$	9.02165	0.352505
$656$	−17.8293	−0.696118
$657$	22.4689	0.876596
$658$	−1.21546	−0.0473834
$659$	−25.3458	−0.987333	−0.493667	−	0.869651i	$-0.664344\pi$
$659$	−25.3458	−0.987333	−0.493667	+	0.869651i	$0.664344\pi$
$660$	23.5845	0.918025
$661$	−18.4109	−0.716101	−0.358051	−	0.933702i	$-0.616558\pi$
$661$	−18.4109	−0.716101	−0.358051	+	0.933702i	$0.616558\pi$
$662$	−21.9421	−0.852805
$663$	−2.63047	−0.102159
$664$	34.5383	1.34035
$665$	2.41037	0.0934700
$666$	−71.5190	−2.77130
$667$	0.591566	0.0229055
$668$	−115.057	−4.45170
$669$	−2.17841	−0.0842224
$670$	−25.9074	−1.00089
$671$	−32.7624	−1.26478
$672$	32.9986	1.27295
$673$	7.83360	0.301963	0.150982	−	0.988537i	$-0.451757\pi$
$673$	7.83360	0.301963	0.150982	+	0.988537i	$0.451757\pi$
$674$	−63.3324	−2.43947
$675$	−6.30695	−0.242755
$676$	−66.0551	−2.54058
$677$	17.4599	0.671037	0.335519	−	0.942034i	$-0.391088\pi$
$677$	17.4599	0.671037	0.335519	+	0.942034i	$0.391088\pi$
$678$	85.3443	3.27763
$679$	−12.9907	−0.498535
$680$	−23.2060	−0.889909
$681$	−2.91607	−0.111744
$682$	−17.1530	−0.656823
$683$	−16.9794	−0.649698	−0.324849	−	0.945766i	$-0.605314\pi$
$683$	−16.9794	−0.649698	−0.324849	+	0.945766i	$0.605314\pi$
$684$	−51.6211	−1.97378
$685$	−4.51764	−0.172610
$686$	32.9623	1.25851
$687$	14.1723	0.540707
$688$	85.3411	3.25360
$689$	1.36216	0.0518942
$690$	2.23853	0.0852194
$691$	−30.9505	−1.17741	−0.588707	−	0.808347i	$-0.700363\pi$
$691$	−30.9505	−1.17741	−0.588707	+	0.808347i	$0.700363\pi$
$692$	128.860	4.89852
$693$	7.41096	0.281519
$694$	−35.3353	−1.34131
$695$	−0.610111	−0.0231428
$696$	19.2575	0.729955
$697$	6.91456	0.261908
$698$	−11.2647	−0.426376
$699$	−24.2483	−0.917155
$700$	22.2837	0.842243
$701$	−14.0133	−0.529276	−0.264638	−	0.964348i	$-0.585253\pi$
$701$	−14.0133	−0.529276	−0.264638	+	0.964348i	$0.585253\pi$
$702$	−0.893570	−0.0337256
$703$	46.5477	1.75558
$704$	−53.2882	−2.00837
$705$	−0.686299	−0.0258475
$706$	−3.58876	−0.135065
$707$	−1.96713	−0.0739815
$708$	16.3432	0.614216
$709$	−24.3553	−0.914681	−0.457341	−	0.889292i	$-0.651198\pi$
$709$	−24.3553	−0.914681	−0.457341	+	0.889292i	$0.651198\pi$
$710$	11.5226	0.432436
$711$	−39.5128	−1.48184
$712$	−26.2793	−0.984859
$713$	−1.16979	−0.0438089
$714$	−26.8868	−1.00621
$715$	0.488390	0.0182648
$716$	0.483464	0.0180679
$717$	−39.0608	−1.45875
$718$	89.5927	3.34357
$719$	−11.9962	−0.447384	−0.223692	−	0.974660i	$-0.571811\pi$
$719$	−11.9962	−0.447384	−0.223692	+	0.974660i	$0.571811\pi$
$720$	17.4555	0.650526
$721$	−9.81803	−0.365643
$722$	−3.88459	−0.144569
$723$	54.4627	2.02549
$724$	−87.8913	−3.26645
$725$	4.62776	0.171871
$726$	2.54008	0.0942713
$727$	21.5254	0.798331	0.399166	−	0.916879i	$-0.369300\pi$
$727$	21.5254	0.798331	0.399166	+	0.916879i	$0.369300\pi$
$728$	1.92025	0.0711691
$729$	−15.6291	−0.578857
$730$	15.1349	0.560169
$731$	−33.0970	−1.22414
$732$	−119.585	−4.42000
$733$	−19.0756	−0.704574	−0.352287	−	0.935892i	$-0.614596\pi$
$733$	−19.0756	−0.704574	−0.352287	+	0.935892i	$0.614596\pi$
$734$	10.2739	0.379216
$735$	8.67444	0.319961
$736$	−8.89409	−0.327841
$737$	51.8433	1.90967
$738$	−9.68219	−0.356407
$739$	−34.0978	−1.25431	−0.627153	−	0.778896i	$-0.715780\pi$
$739$	−34.0978	−1.25431	−0.627153	+	0.778896i	$0.715780\pi$
$740$	−34.6138	−1.27243
$741$	−2.39728	−0.0880664
$742$	13.9230	0.511131
$743$	−20.4920	−0.751779	−0.375889	−	0.926665i	$-0.622663\pi$
$743$	−20.4920	−0.751779	−0.375889	+	0.926665i	$0.622663\pi$
$744$	−38.0807	−1.39611
$745$	−10.1770	−0.372856
$746$	−18.1835	−0.665745
$747$	10.0754	0.368639
$748$	76.3497	2.79162
$749$	2.05995	0.0752688
$750$	36.4322	1.33032
$751$	−8.67701	−0.316629	−0.158314	−	0.987389i	$-0.550606\pi$
$751$	−8.67701	−0.316629	−0.158314	+	0.987389i	$0.550606\pi$
$752$	5.72881	0.208908
$753$	46.8209	1.70625
$754$	0.655662	0.0238778
$755$	6.00799	0.218653
$756$	−6.56242	−0.238673
$757$	−18.5983	−0.675968	−0.337984	−	0.941152i	$-0.609745\pi$
$757$	−18.5983	−0.675968	−0.337984	+	0.941152i	$0.609745\pi$
$758$	73.1585	2.65724
$759$	−4.47952	−0.162596
$760$	−21.1488	−0.767149
$761$	21.3654	0.774495	0.387247	−	0.921976i	$-0.373426\pi$
$761$	21.3654	0.774495	0.387247	+	0.921976i	$0.373426\pi$
$762$	−18.3381	−0.664320
$763$	−7.90643	−0.286232
$764$	−92.8850	−3.36046
$765$	−6.76957	−0.244754
$766$	−8.48263	−0.306490
$767$	0.338438	0.0122203
$768$	−8.00605	−0.288894
$769$	37.6207	1.35664	0.678318	−	0.734769i	$-0.262709\pi$
$769$	37.6207	1.35664	0.678318	+	0.734769i	$0.262709\pi$
$770$	4.99198	0.179898
$771$	13.4358	0.483879
$772$	54.8673	1.97472
$773$	−10.0222	−0.360474	−0.180237	−	0.983623i	$-0.557687\pi$
$773$	−10.0222	−0.360474	−0.180237	+	0.983623i	$0.557687\pi$
$774$	46.3444	1.66581
$775$	−9.15114	−0.328719
$776$	113.981	4.09169
$777$	−24.3921	−0.875061
$778$	68.7708	2.46555
$779$	6.30160	0.225778
$780$	1.78266	0.0638296
$781$	−23.0579	−0.825076
$782$	7.24676	0.259144
$783$	−1.36285	−0.0487043
$784$	−72.4090	−2.58604
$785$	−10.3881	−0.370767
$786$	−91.7122	−3.27127
$787$	37.2653	1.32836	0.664181	−	0.747571i	$-0.268780\pi$
$787$	37.2653	1.32836	0.664181	+	0.747571i	$0.268780\pi$
$788$	34.9137	1.24375
$789$	−25.8973	−0.921969
$790$	−26.6156	−0.946939
$791$	12.9793	0.461490
$792$	−65.0246	−2.31055
$793$	−2.47639	−0.0879391
$794$	−1.54784	−0.0549307
$795$	7.86155	0.278820
$796$	26.5587	0.941347
$797$	−45.7511	−1.62059	−0.810293	−	0.586025i	$-0.800692\pi$
$797$	−45.7511	−1.62059	−0.810293	+	0.586025i	$0.800692\pi$
$798$	−24.5033	−0.867408
$799$	−2.22174	−0.0785997
$800$	−69.5776	−2.45994
$801$	−7.66611	−0.270869
$802$	34.5342	1.21945
$803$	−30.2865	−1.06879
$804$	189.232	6.67371
$805$	0.340439	0.0119989
$806$	−1.29653	−0.0456685
$807$	4.90773	0.172760
$808$	17.2598	0.607198
$809$	46.5655	1.63716	0.818579	−	0.574394i	$-0.194762\pi$
$809$	46.5655	1.63716	0.818579	+	0.574394i	$0.194762\pi$
$810$	−16.9359	−0.595068
$811$	−34.4472	−1.20960	−0.604802	−	0.796376i	$-0.706748\pi$
$811$	−34.4472	−1.20960	−0.604802	+	0.796376i	$0.706748\pi$
$812$	4.81521	0.168981
$813$	−2.10001	−0.0736505
$814$	96.4024	3.37890
$815$	−1.61649	−0.0566233
$816$	126.725	4.43628
$817$	−30.1630	−1.05527
$818$	19.2081	0.671596
$819$	0.560167	0.0195738
$820$	−4.68599	−0.163642
$821$	−4.01003	−0.139951	−0.0699755	−	0.997549i	$-0.522292\pi$
$821$	−4.01003	−0.139951	−0.0699755	+	0.997549i	$0.522292\pi$
$822$	45.9254	1.60183
$823$	−32.0647	−1.11771	−0.558853	−	0.829267i	$-0.688758\pi$
$823$	−32.0647	−1.11771	−0.558853	+	0.829267i	$0.688758\pi$
$824$	86.1445	3.00099
$825$	−35.0429	−1.22004
$826$	3.45927	0.120363
$827$	13.3821	0.465343	0.232671	−	0.972555i	$-0.425253\pi$
$827$	13.3821	0.465343	0.232671	+	0.972555i	$0.425253\pi$
$828$	−7.29093	−0.253377
$829$	3.56851	0.123940	0.0619698	−	0.998078i	$-0.480262\pi$
$829$	3.56851	0.123940	0.0619698	+	0.998078i	$0.480262\pi$
$830$	6.78672	0.235570
$831$	−40.7454	−1.41344
$832$	−4.02786	−0.139641
$833$	28.0816	0.972971
$834$	6.20227	0.214767
$835$	−13.7510	−0.475873
$836$	69.5815	2.40653
$837$	2.69496	0.0931514
$838$	79.0214	2.72975
$839$	2.03561	0.0702772	0.0351386	−	0.999382i	$-0.488813\pi$
$839$	2.03561	0.0702772	0.0351386	+	0.999382i	$0.488813\pi$
$840$	11.0825	0.382381
$841$	1.00000	0.0344828
$842$	−2.77765	−0.0957240
$843$	−12.4778	−0.429759
$844$	114.125	3.92836
$845$	−7.89453	−0.271580
$846$	3.11102	0.106959
$847$	0.386299	0.0132734
$848$	−65.6234	−2.25352
$849$	61.2074	2.10063
$850$	56.6907	1.94448
$851$	6.57437	0.225367
$852$	−84.1631	−2.88338
$853$	29.3567	1.00515	0.502577	−	0.864533i	$-0.332385\pi$
$853$	29.3567	1.00515	0.502577	+	0.864533i	$0.332385\pi$
$854$	−25.3119	−0.866154
$855$	−6.16946	−0.210991
$856$	−18.0742	−0.617764
$857$	−3.98594	−0.136157	−0.0680786	−	0.997680i	$-0.521687\pi$
$857$	−3.98594	−0.136157	−0.0680786	+	0.997680i	$0.521687\pi$
$858$	−4.96488	−0.169498
$859$	1.87313	0.0639103	0.0319552	−	0.999489i	$-0.489827\pi$
$859$	1.87313	0.0639103	0.0319552	+	0.999489i	$0.489827\pi$
$860$	22.4298	0.764849
$861$	−3.30218	−0.112538
$862$	−54.6767	−1.86230
$863$	−42.6658	−1.45236	−0.726180	−	0.687505i	$-0.758706\pi$
$863$	−42.6658	−1.45236	−0.726180	+	0.687505i	$0.758706\pi$
$864$	20.4902	0.697092
$865$	15.4006	0.523637
$866$	−95.0529	−3.23003
$867$	−9.58992	−0.325691
$868$	−9.52180	−0.323191
$869$	53.2604	1.80673
$870$	3.78407	0.128292
$871$	3.91865	0.132778
$872$	69.3719	2.34923
$873$	33.2503	1.12535
$874$	6.60435	0.223396
$875$	5.54066	0.187308
$876$	−110.548	−3.73507
$877$	24.6023	0.830762	0.415381	−	0.909648i	$-0.363648\pi$
$877$	24.6023	0.830762	0.415381	+	0.909648i	$0.363648\pi$
$878$	53.2438	1.79689
$879$	−67.9079	−2.29048
$880$	−23.5287	−0.793152
$881$	−7.91861	−0.266785	−0.133392	−	0.991063i	$-0.542587\pi$
$881$	−7.91861	−0.266785	−0.133392	+	0.991063i	$0.542587\pi$
$882$	−39.3216	−1.32403
$883$	−15.2813	−0.514256	−0.257128	−	0.966377i	$-0.582776\pi$
$883$	−15.2813	−0.514256	−0.257128	+	0.966377i	$0.582776\pi$
$884$	5.77099	0.194100
$885$	1.95325	0.0656578
$886$	60.0583	2.01770
$887$	25.7441	0.864402	0.432201	−	0.901777i	$-0.357737\pi$
$887$	25.7441	0.864402	0.432201	+	0.901777i	$0.357737\pi$
$888$	214.019	7.18200
$889$	−2.78889	−0.0935362
$890$	−5.16384	−0.173092
$891$	33.8905	1.13537
$892$	4.77923	0.160021
$893$	−2.02479	−0.0677571
$894$	103.457	3.46012
$895$	0.0577810	0.00193140
$896$	−12.8067	−0.427842
$897$	−0.338591	−0.0113052
$898$	69.1816	2.30862
$899$	−1.97744	−0.0659514
$900$	−57.0362	−1.90121
$901$	25.4501	0.847865
$902$	13.0509	0.434547
$903$	15.8061	0.525994
$904$	−113.882	−3.78765
$905$	−10.5043	−0.349174
$906$	−61.0760	−2.02912
$907$	4.32009	0.143446	0.0717232	−	0.997425i	$-0.477150\pi$
$907$	4.32009	0.143446	0.0717232	+	0.997425i	$0.477150\pi$
$908$	6.39757	0.212311
$909$	5.03497	0.166999
$910$	0.377325	0.0125082
$911$	47.7605	1.58237	0.791187	−	0.611574i	$-0.209463\pi$
$911$	47.7605	1.58237	0.791187	+	0.611574i	$0.209463\pi$
$912$	115.491	3.82430
$913$	−13.5809	−0.449462
$914$	−7.47744	−0.247332
$915$	−14.2922	−0.472485
$916$	−31.0927	−1.02733
$917$	−13.9477	−0.460594
$918$	−16.6951	−0.551021
$919$	−44.9917	−1.48414	−0.742070	−	0.670322i	$-0.766156\pi$
$919$	−44.9917	−1.48414	−0.742070	+	0.670322i	$0.766156\pi$
$920$	−2.98705	−0.0984800
$921$	−64.6174	−2.12922
$922$	−46.1557	−1.52006
$923$	−1.74286	−0.0573669
$924$	−36.4623	−1.19952
$925$	51.4307	1.69103
$926$	47.8779	1.57337
$927$	25.1298	0.825370
$928$	−15.0348	−0.493542
$929$	−46.4255	−1.52317	−0.761586	−	0.648063i	$-0.775579\pi$
$929$	−46.4255	−1.52317	−0.761586	+	0.648063i	$0.775579\pi$
$930$	−7.48279	−0.245370
$931$	25.5923	0.838753
$932$	53.1984	1.74257
$933$	−23.1980	−0.759469
$934$	67.2241	2.19964
$935$	9.12489	0.298416
$936$	−4.91497	−0.160651
$937$	−38.4858	−1.25728	−0.628638	−	0.777698i	$-0.716387\pi$
$937$	−38.4858	−1.25728	−0.628638	+	0.777698i	$0.716387\pi$
$938$	40.0536	1.30780
$939$	20.8282	0.679702
$940$	1.50567	0.0491097
$941$	−8.39448	−0.273652	−0.136826	−	0.990595i	$-0.543690\pi$
$941$	−8.39448	−0.273652	−0.136826	+	0.990595i	$0.543690\pi$
$942$	105.603	3.44074
$943$	0.890035	0.0289835
$944$	−16.3046	−0.530669
$945$	−0.784304	−0.0255134
$946$	−62.4689	−2.03104
$947$	−27.2129	−0.884301	−0.442151	−	0.896941i	$-0.645784\pi$
$947$	−27.2129	−0.884301	−0.442151	+	0.896941i	$0.645784\pi$
$948$	194.405	6.31397
$949$	−2.28924	−0.0743120
$950$	51.6652	1.67624
$951$	44.5435	1.44442
$952$	35.8771	1.16278
$953$	39.5443	1.28097	0.640483	−	0.767972i	$-0.278734\pi$
$953$	39.5443	1.28097	0.640483	+	0.767972i	$0.278734\pi$
$954$	−35.6367	−1.15378
$955$	−11.1011	−0.359223
$956$	85.6957	2.77160
$957$	−7.57231	−0.244778
$958$	−7.66059	−0.247502
$959$	6.98439	0.225538
$960$	−23.2463	−0.750271
$961$	−27.0897	−0.873862
$962$	7.28670	0.234933
$963$	−5.27254	−0.169905
$964$	−119.486	−3.84839
$965$	6.55744	0.211091
$966$	−3.46083	−0.111350
$967$	−34.9770	−1.12478	−0.562392	−	0.826870i	$-0.690119\pi$
$967$	−34.9770	−1.12478	−0.562392	+	0.826870i	$0.690119\pi$
$968$	−3.38943	−0.108940
$969$	−44.7899	−1.43886
$970$	22.3972	0.719130
$971$	45.9988	1.47617	0.738086	−	0.674707i	$-0.235730\pi$
$971$	45.9988	1.47617	0.738086	+	0.674707i	$0.235730\pi$
$972$	102.831	3.29831
$973$	0.943250	0.0302392
$974$	74.2399	2.37880
$975$	−2.64876	−0.0848283
$976$	119.302	3.81878
$977$	−14.8522	−0.475164	−0.237582	−	0.971367i	$-0.576355\pi$
$977$	−14.8522	−0.475164	−0.237582	+	0.971367i	$0.576355\pi$
$978$	16.4329	0.525468
$979$	10.3334	0.330256
$980$	−19.0309	−0.607919
$981$	20.2369	0.646115
$982$	−98.8338	−3.15391
$983$	−7.37177	−0.235123	−0.117562	−	0.993066i	$-0.537508\pi$
$983$	−7.37177	−0.235123	−0.117562	+	0.993066i	$0.537508\pi$
$984$	28.9737	0.923649
$985$	4.17269	0.132953
$986$	12.2501	0.390124
$987$	1.06104	0.0337732
$988$	5.25941	0.167324
$989$	−4.26020	−0.135467
$990$	−12.7772	−0.406087
$991$	−56.2304	−1.78622	−0.893109	−	0.449840i	$-0.851481\pi$
$991$	−56.2304	−1.78622	−0.893109	+	0.449840i	$0.851481\pi$
$992$	29.7305	0.943944
$993$	19.1545	0.607849
$994$	−17.8143	−0.565034
$995$	3.17415	0.100627
$996$	−49.5714	−1.57073
$997$	30.8281	0.976335	0.488167	−	0.872750i	$-0.337666\pi$
$997$	30.8281	0.976335	0.488167	+	0.872750i	$0.337666\pi$
$998$	48.1915	1.52547
$999$	−15.1461	−0.479200

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.c.1.61	✓	61

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.c.1.61	✓	61	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q+2.66551 q^{2} -2.32686 q^{3} +5.10492 q^{4} +0.610111 q^{5} -6.20227 q^{6} -0.943250 q^{7} +8.27618 q^{8} +2.41430 q^{9} +O(q^{10})\)	\(q+2.66551 q^{2} -2.32686 q^{3} +5.10492 q^{4} +0.610111 q^{5} -6.20227 q^{6} -0.943250 q^{7} +8.27618 q^{8} +2.41430 q^{9} +1.62626 q^{10} -3.25430 q^{11} -11.8785 q^{12} -0.245980 q^{13} -2.51424 q^{14} -1.41965 q^{15} +11.8504 q^{16} -4.59580 q^{17} +6.43532 q^{18} -4.18839 q^{19} +3.11457 q^{20} +2.19481 q^{21} -8.67435 q^{22} -0.591566 q^{23} -19.2575 q^{24} -4.62776 q^{25} -0.655662 q^{26} +1.36285 q^{27} -4.81521 q^{28} -1.00000 q^{29} -3.78407 q^{30} +1.97744 q^{31} +15.0348 q^{32} +7.57231 q^{33} -12.2501 q^{34} -0.575487 q^{35} +12.3248 q^{36} -11.1135 q^{37} -11.1642 q^{38} +0.572363 q^{39} +5.04939 q^{40} -1.50454 q^{41} +5.85029 q^{42} +7.20157 q^{43} -16.6129 q^{44} +1.47299 q^{45} -1.57682 q^{46} +0.483429 q^{47} -27.5742 q^{48} -6.11028 q^{49} -12.3353 q^{50} +10.6938 q^{51} -1.25571 q^{52} -5.53768 q^{53} +3.63269 q^{54} -1.98548 q^{55} -7.80650 q^{56} +9.74582 q^{57} -2.66551 q^{58} -1.37587 q^{59} -7.24718 q^{60} +10.0674 q^{61} +5.27088 q^{62} -2.27728 q^{63} +16.3747 q^{64} -0.150075 q^{65} +20.1840 q^{66} -15.9307 q^{67} -23.4612 q^{68} +1.37649 q^{69} -1.53396 q^{70} +7.08536 q^{71} +19.9811 q^{72} +9.30661 q^{73} -29.6231 q^{74} +10.7682 q^{75} -21.3814 q^{76} +3.06962 q^{77} +1.52564 q^{78} -16.3662 q^{79} +7.23004 q^{80} -10.4141 q^{81} -4.01036 q^{82} +4.17322 q^{83} +11.2043 q^{84} -2.80395 q^{85} +19.1958 q^{86} +2.32686 q^{87} -26.9331 q^{88} -3.17530 q^{89} +3.92626 q^{90} +0.232021 q^{91} -3.01990 q^{92} -4.60124 q^{93} +1.28858 q^{94} -2.55539 q^{95} -34.9840 q^{96} +13.7722 q^{97} -16.2870 q^{98} -7.85684 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9	\( 61 q - q^{2} - 4 q^{3} + 43 q^{4} - 7 q^{5} - 13 q^{6} - 10 q^{7} - 6 q^{8} + 23 q^{9} - 16 q^{10} - 3 q^{11} - 18 q^{12} - 28 q^{13} - 14 q^{14} - 12 q^{15} + 11 q^{16} - 21 q^{17} - 17 q^{18} - 36 q^{19} - 16 q^{20} - 12 q^{21} - 42 q^{22} - 15 q^{23} - 28 q^{24} - 16 q^{25} - 13 q^{26} - 10 q^{27} - 25 q^{28} - 61 q^{29} - 12 q^{30} - 18 q^{31} - 3 q^{32} - 42 q^{33} - 22 q^{34} - 29 q^{35} - 38 q^{36} - 30 q^{37} - 27 q^{38} - 31 q^{39} - 22 q^{40} - 28 q^{41} - 9 q^{42} - 58 q^{43} - 2 q^{44} - 31 q^{45} - 40 q^{46} - 6 q^{47} - 37 q^{48} - 37 q^{49} - 15 q^{50} - 44 q^{51} - 43 q^{52} - 27 q^{53} - 18 q^{54} - 38 q^{55} - 22 q^{56} - 50 q^{57} + q^{58} - 24 q^{59} + 6 q^{60} - 76 q^{61} - 17 q^{62} - 6 q^{63} - 60 q^{64} - 65 q^{65} - 7 q^{66} - 45 q^{67} - 31 q^{68} - 16 q^{69} - 48 q^{70} - 28 q^{71} - 40 q^{72} - 50 q^{73} - 17 q^{74} - 35 q^{75} - 100 q^{76} + q^{77} - 6 q^{78} - 66 q^{79} - 10 q^{80} - 63 q^{81} - 5 q^{82} - 9 q^{83} - 24 q^{84} - 77 q^{85} + 29 q^{86} + 4 q^{87} - 62 q^{88} - 30 q^{89} + 50 q^{90} - 52 q^{91} - 53 q^{92} - 42 q^{93} - 92 q^{94} - 20 q^{95} - 47 q^{96} - 34 q^{97} + 36 q^{98} - 29 q^{99}+O(q^{100}) \) 61 * q - q^2 - 4 * q^3 + 43 * q^4 - 7 * q^5 - 13 * q^6 - 10 * q^7 - 6 * q^8 + 23 * q^9 - 16 * q^10 - 3 * q^11 - 18 * q^12 - 28 * q^13 - 14 * q^14 - 12 * q^15 + 11 * q^16 - 21 * q^17 - 17 * q^18 - 36 * q^19 - 16 * q^20 - 12 * q^21 - 42 * q^22 - 15 * q^23 - 28 * q^24 - 16 * q^25 - 13 * q^26 - 10 * q^27 - 25 * q^28 - 61 * q^29 - 12 * q^30 - 18 * q^31 - 3 * q^32 - 42 * q^33 - 22 * q^34 - 29 * q^35 - 38 * q^36 - 30 * q^37 - 27 * q^38 - 31 * q^39 - 22 * q^40 - 28 * q^41 - 9 * q^42 - 58 * q^43 - 2 * q^44 - 31 * q^45 - 40 * q^46 - 6 * q^47 - 37 * q^48 - 37 * q^49 - 15 * q^50 - 44 * q^51 - 43 * q^52 - 27 * q^53 - 18 * q^54 - 38 * q^55 - 22 * q^56 - 50 * q^57 + q^58 - 24 * q^59 + 6 * q^60 - 76 * q^61 - 17 * q^62 - 6 * q^63 - 60 * q^64 - 65 * q^65 - 7 * q^66 - 45 * q^67 - 31 * q^68 - 16 * q^69 - 48 * q^70 - 28 * q^71 - 40 * q^72 - 50 * q^73 - 17 * q^74 - 35 * q^75 - 100 * q^76 + q^77 - 6 * q^78 - 66 * q^79 - 10 * q^80 - 63 * q^81 - 5 * q^82 - 9 * q^83 - 24 * q^84 - 77 * q^85 + 29 * q^86 + 4 * q^87 - 62 * q^88 - 30 * q^89 + 50 * q^90 - 52 * q^91 - 53 * q^92 - 42 * q^93 - 92 * q^94 - 20 * q^95 - 47 * q^96 - 34 * q^97 + 36 * q^98 - 29 * q^99

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