LMFDB - Embedded newform 8038.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1837531447$
Analytic rank:	$1$
Dimension:	$84$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.36115 q^{3} +1.00000 q^{4} -3.51543 q^{5} +3.36115 q^{6} -3.19269 q^{7} -1.00000 q^{8} +8.29735 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.36115 q^{3} +1.00000 q^{4} -3.51543 q^{5} +3.36115 q^{6} -3.19269 q^{7} -1.00000 q^{8} +8.29735 q^{9} +3.51543 q^{10} -5.56306 q^{11} -3.36115 q^{12} -6.75995 q^{13} +3.19269 q^{14} +11.8159 q^{15} +1.00000 q^{16} -5.12738 q^{17} -8.29735 q^{18} +0.938201 q^{19} -3.51543 q^{20} +10.7311 q^{21} +5.56306 q^{22} -3.14603 q^{23} +3.36115 q^{24} +7.35824 q^{25} +6.75995 q^{26} -17.8052 q^{27} -3.19269 q^{28} -7.79363 q^{29} -11.8159 q^{30} -2.41100 q^{31} -1.00000 q^{32} +18.6983 q^{33} +5.12738 q^{34} +11.2237 q^{35} +8.29735 q^{36} -4.92716 q^{37} -0.938201 q^{38} +22.7212 q^{39} +3.51543 q^{40} -5.53848 q^{41} -10.7311 q^{42} +10.3294 q^{43} -5.56306 q^{44} -29.1687 q^{45} +3.14603 q^{46} -5.75395 q^{47} -3.36115 q^{48} +3.19326 q^{49} -7.35824 q^{50} +17.2339 q^{51} -6.75995 q^{52} +2.10884 q^{53} +17.8052 q^{54} +19.5565 q^{55} +3.19269 q^{56} -3.15344 q^{57} +7.79363 q^{58} +2.44408 q^{59} +11.8159 q^{60} -13.3910 q^{61} +2.41100 q^{62} -26.4909 q^{63} +1.00000 q^{64} +23.7641 q^{65} -18.6983 q^{66} +0.517910 q^{67} -5.12738 q^{68} +10.5743 q^{69} -11.2237 q^{70} -4.29915 q^{71} -8.29735 q^{72} -7.78255 q^{73} +4.92716 q^{74} -24.7322 q^{75} +0.938201 q^{76} +17.7611 q^{77} -22.7212 q^{78} -4.65989 q^{79} -3.51543 q^{80} +34.9540 q^{81} +5.53848 q^{82} -16.6736 q^{83} +10.7311 q^{84} +18.0249 q^{85} -10.3294 q^{86} +26.1956 q^{87} +5.56306 q^{88} +13.0346 q^{89} +29.1687 q^{90} +21.5824 q^{91} -3.14603 q^{92} +8.10373 q^{93} +5.75395 q^{94} -3.29818 q^{95} +3.36115 q^{96} +17.1790 q^{97} -3.19326 q^{98} -46.1587 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) $ 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.36115	−1.94056	−0.970281	−	0.241980i	$-0.922203\pi$
$3$	−3.36115	−1.94056	−0.970281	+	0.241980i	$0.922203\pi$
$4$	1.00000	0.500000
$5$	−3.51543	−1.57215	−0.786074	−	0.618133i	$-0.787889\pi$
$5$	−3.51543	−1.57215	−0.786074	+	0.618133i	$0.787889\pi$
$6$	3.36115	1.37219
$7$	−3.19269	−1.20672	−0.603361	−	0.797468i	$-0.706172\pi$
$7$	−3.19269	−1.20672	−0.603361	+	0.797468i	$0.706172\pi$
$8$	−1.00000	−0.353553
$9$	8.29735	2.76578
$10$	3.51543	1.11168
$11$	−5.56306	−1.67733	−0.838663	−	0.544651i	$-0.816662\pi$
$11$	−5.56306	−1.67733	−0.838663	+	0.544651i	$0.816662\pi$
$12$	−3.36115	−0.970281
$13$	−6.75995	−1.87487	−0.937436	−	0.348158i	$-0.886807\pi$
$13$	−6.75995	−1.87487	−0.937436	+	0.348158i	$0.886807\pi$
$14$	3.19269	0.853282
$15$	11.8159	3.05085
$16$	1.00000	0.250000
$17$	−5.12738	−1.24357	−0.621786	−	0.783187i	$-0.713593\pi$
$17$	−5.12738	−1.24357	−0.621786	+	0.783187i	$0.713593\pi$
$18$	−8.29735	−1.95570
$19$	0.938201	0.215238	0.107619	−	0.994192i	$-0.465677\pi$
$19$	0.938201	0.215238	0.107619	+	0.994192i	$0.465677\pi$
$20$	−3.51543	−0.786074
$21$	10.7311	2.34172
$22$	5.56306	1.18605
$23$	−3.14603	−0.655993	−0.327996	−	0.944679i	$-0.606373\pi$
$23$	−3.14603	−0.655993	−0.327996	+	0.944679i	$0.606373\pi$
$24$	3.36115	0.686093
$25$	7.35824	1.47165
$26$	6.75995	1.32573
$27$	−17.8052	−3.42661
$28$	−3.19269	−0.603361
$29$	−7.79363	−1.44724	−0.723621	−	0.690198i	$-0.757523\pi$
$29$	−7.79363	−1.44724	−0.723621	+	0.690198i	$0.757523\pi$
$30$	−11.8159	−2.15728
$31$	−2.41100	−0.433028	−0.216514	−	0.976280i	$-0.569469\pi$
$31$	−2.41100	−0.433028	−0.216514	+	0.976280i	$0.569469\pi$
$32$	−1.00000	−0.176777
$33$	18.6983	3.25496
$34$	5.12738	0.879338
$35$	11.2237	1.89715
$36$	8.29735	1.38289
$37$	−4.92716	−0.810020	−0.405010	−	0.914312i	$-0.632732\pi$
$37$	−4.92716	−0.810020	−0.405010	+	0.914312i	$0.632732\pi$
$38$	−0.938201	−0.152196
$39$	22.7212	3.63831
$40$	3.51543	0.555838
$41$	−5.53848	−0.864965	−0.432483	−	0.901642i	$-0.642362\pi$
$41$	−5.53848	−0.864965	−0.432483	+	0.901642i	$0.642362\pi$
$42$	−10.7311	−1.65585
$43$	10.3294	1.57522	0.787609	−	0.616175i	$-0.211319\pi$
$43$	10.3294	1.57522	0.787609	+	0.616175i	$0.211319\pi$
$44$	−5.56306	−0.838663
$45$	−29.1687	−4.34822
$46$	3.14603	0.463857
$47$	−5.75395	−0.839300	−0.419650	−	0.907686i	$-0.637847\pi$
$47$	−5.75395	−0.839300	−0.419650	+	0.907686i	$0.637847\pi$
$48$	−3.36115	−0.485141
$49$	3.19326	0.456180
$50$	−7.35824	−1.04061
$51$	17.2339	2.41323
$52$	−6.75995	−0.937436
$53$	2.10884	0.289671	0.144836	−	0.989456i	$-0.453735\pi$
$53$	2.10884	0.289671	0.144836	+	0.989456i	$0.453735\pi$
$54$	17.8052	2.42298
$55$	19.5565	2.63700
$56$	3.19269	0.426641
$57$	−3.15344	−0.417683
$58$	7.79363	1.02335
$59$	2.44408	0.318193	0.159096	−	0.987263i	$-0.449142\pi$
$59$	2.44408	0.318193	0.159096	+	0.987263i	$0.449142\pi$
$60$	11.8159	1.52543
$61$	−13.3910	−1.71454	−0.857269	−	0.514869i	$-0.827841\pi$
$61$	−13.3910	−1.71454	−0.857269	+	0.514869i	$0.827841\pi$
$62$	2.41100	0.306197
$63$	−26.4909	−3.33753
$64$	1.00000	0.125000
$65$	23.7641	2.94757
$66$	−18.6983	−2.30160
$67$	0.517910	0.0632728	0.0316364	−	0.999499i	$-0.489928\pi$
$67$	0.517910	0.0632728	0.0316364	+	0.999499i	$0.489928\pi$
$68$	−5.12738	−0.621786
$69$	10.5743	1.27300
$70$	−11.2237	−1.34148
$71$	−4.29915	−0.510216	−0.255108	−	0.966913i	$-0.582111\pi$
$71$	−4.29915	−0.510216	−0.255108	+	0.966913i	$0.582111\pi$
$72$	−8.29735	−0.977852
$73$	−7.78255	−0.910879	−0.455439	−	0.890267i	$-0.650518\pi$
$73$	−7.78255	−0.910879	−0.455439	+	0.890267i	$0.650518\pi$
$74$	4.92716	0.572771
$75$	−24.7322	−2.85582
$76$	0.938201	0.107619
$77$	17.7611	2.02407
$78$	−22.7212	−2.57267
$79$	−4.65989	−0.524278	−0.262139	−	0.965030i	$-0.584428\pi$
$79$	−4.65989	−0.524278	−0.262139	+	0.965030i	$0.584428\pi$
$80$	−3.51543	−0.393037
$81$	34.9540	3.88377
$82$	5.53848	0.611623
$83$	−16.6736	−1.83016	−0.915081	−	0.403271i	$-0.867873\pi$
$83$	−16.6736	−1.83016	−0.915081	+	0.403271i	$0.867873\pi$
$84$	10.7311	1.17086
$85$	18.0249	1.95508
$86$	−10.3294	−1.11385
$87$	26.1956	2.80846
$88$	5.56306	0.593024
$89$	13.0346	1.38167	0.690834	−	0.723014i	$-0.257244\pi$
$89$	13.0346	1.38167	0.690834	+	0.723014i	$0.257244\pi$
$90$	29.1687	3.07466
$91$	21.5824	2.26245
$92$	−3.14603	−0.327996
$93$	8.10373	0.840318
$94$	5.75395	0.593474
$95$	−3.29818	−0.338386
$96$	3.36115	0.343046
$97$	17.1790	1.74426	0.872132	−	0.489271i	$-0.162737\pi$
$97$	17.1790	1.74426	0.872132	+	0.489271i	$0.162737\pi$
$98$	−3.19326	−0.322568
$99$	−46.1587	−4.63912
$100$	7.35824	0.735824
$101$	−10.4275	−1.03758	−0.518789	−	0.854902i	$-0.673617\pi$
$101$	−10.4275	−1.03758	−0.518789	+	0.854902i	$0.673617\pi$
$102$	−17.2339	−1.70641
$103$	−0.237817	−0.0234328	−0.0117164	−	0.999931i	$-0.503730\pi$
$103$	−0.237817	−0.0234328	−0.0117164	+	0.999931i	$0.503730\pi$
$104$	6.75995	0.662867
$105$	−37.7245	−3.68153
$106$	−2.10884	−0.204829
$107$	9.98003	0.964806	0.482403	−	0.875949i	$-0.339764\pi$
$107$	9.98003	0.964806	0.482403	+	0.875949i	$0.339764\pi$
$108$	−17.8052	−1.71331
$109$	12.6088	1.20770	0.603850	−	0.797098i	$-0.293633\pi$
$109$	12.6088	1.20770	0.603850	+	0.797098i	$0.293633\pi$
$110$	−19.5565	−1.86464
$111$	16.5609	1.57189
$112$	−3.19269	−0.301681
$113$	−8.94606	−0.841575	−0.420787	−	0.907159i	$-0.638246\pi$
$113$	−8.94606	−0.841575	−0.420787	+	0.907159i	$0.638246\pi$
$114$	3.15344	0.295346
$115$	11.0596	1.03132
$116$	−7.79363	−0.723621
$117$	−56.0896	−5.18549
$118$	−2.44408	−0.224996
$119$	16.3701	1.50065
$120$	−11.8159	−1.07864
$121$	19.9476	1.81342
$122$	13.3910	1.21236
$123$	18.6157	1.67852
$124$	−2.41100	−0.216514
$125$	−8.29021	−0.741499
$126$	26.4909	2.35999
$127$	−7.64491	−0.678376	−0.339188	−	0.940719i	$-0.610152\pi$
$127$	−7.64491	−0.678376	−0.339188	+	0.940719i	$0.610152\pi$
$128$	−1.00000	−0.0883883
$129$	−34.7187	−3.05681
$130$	−23.7641	−2.08425
$131$	7.35255	0.642396	0.321198	−	0.947012i	$-0.395915\pi$
$131$	7.35255	0.642396	0.321198	+	0.947012i	$0.395915\pi$
$132$	18.6983	1.62748
$133$	−2.99538	−0.259733
$134$	−0.517910	−0.0447406
$135$	62.5929	5.38714
$136$	5.12738	0.439669
$137$	−10.2231	−0.873418	−0.436709	−	0.899603i	$-0.643856\pi$
$137$	−10.2231	−0.873418	−0.436709	+	0.899603i	$0.643856\pi$
$138$	−10.5743	−0.900144
$139$	−14.6346	−1.24129	−0.620645	−	0.784092i	$-0.713129\pi$
$139$	−14.6346	−1.24129	−0.620645	+	0.784092i	$0.713129\pi$
$140$	11.2237	0.948573
$141$	19.3399	1.62871
$142$	4.29915	0.360777
$143$	37.6060	3.14477
$144$	8.29735	0.691446
$145$	27.3980	2.27528
$146$	7.78255	0.644089
$147$	−10.7330	−0.885245
$148$	−4.92716	−0.405010
$149$	8.75129	0.716933	0.358467	−	0.933543i	$-0.383300\pi$
$149$	8.75129	0.716933	0.358467	+	0.933543i	$0.383300\pi$
$150$	24.7322	2.01937
$151$	13.9998	1.13929	0.569645	−	0.821891i	$-0.307081\pi$
$151$	13.9998	1.13929	0.569645	+	0.821891i	$0.307081\pi$
$152$	−0.938201	−0.0760981
$153$	−42.5436	−3.43945
$154$	−17.7611	−1.43123
$155$	8.47569	0.680784
$156$	22.7212	1.81915
$157$	3.90708	0.311819	0.155909	−	0.987771i	$-0.450169\pi$
$157$	3.90708	0.311819	0.155909	+	0.987771i	$0.450169\pi$
$158$	4.65989	0.370721
$159$	−7.08813	−0.562125
$160$	3.51543	0.277919
$161$	10.0443	0.791602
$162$	−34.9540	−2.74624
$163$	−19.9600	−1.56339	−0.781695	−	0.623661i	$-0.785645\pi$
$163$	−19.9600	−1.56339	−0.781695	+	0.623661i	$0.785645\pi$
$164$	−5.53848	−0.432483
$165$	−65.7325	−5.11727
$166$	16.6736	1.29412
$167$	20.3486	1.57462	0.787311	−	0.616556i	$-0.211473\pi$
$167$	20.3486	1.57462	0.787311	+	0.616556i	$0.211473\pi$
$168$	−10.7311	−0.827923
$169$	32.6969	2.51514
$170$	−18.0249	−1.38245
$171$	7.78458	0.595302
$172$	10.3294	0.787609
$173$	−22.4203	−1.70459	−0.852293	−	0.523064i	$-0.824789\pi$
$173$	−22.4203	−1.70459	−0.852293	+	0.523064i	$0.824789\pi$
$174$	−26.1956	−1.98588
$175$	−23.4926	−1.77587
$176$	−5.56306	−0.419331
$177$	−8.21494	−0.617473
$178$	−13.0346	−0.976986
$179$	8.41226	0.628762	0.314381	−	0.949297i	$-0.398203\pi$
$179$	8.41226	0.628762	0.314381	+	0.949297i	$0.398203\pi$
$180$	−29.1687	−2.17411
$181$	−21.2864	−1.58221	−0.791104	−	0.611682i	$-0.790493\pi$
$181$	−21.2864	−1.58221	−0.791104	+	0.611682i	$0.790493\pi$
$182$	−21.5824	−1.59979
$183$	45.0091	3.32717
$184$	3.14603	0.231929
$185$	17.3211	1.27347
$186$	−8.10373	−0.594194
$187$	28.5239	2.08588
$188$	−5.75395	−0.419650
$189$	56.8465	4.13497
$190$	3.29818	0.239275
$191$	−11.7768	−0.852141	−0.426070	−	0.904690i	$-0.640102\pi$
$191$	−11.7768	−0.852141	−0.426070	+	0.904690i	$0.640102\pi$
$192$	−3.36115	−0.242570
$193$	0.725352	0.0522120	0.0261060	−	0.999659i	$-0.491689\pi$
$193$	0.725352	0.0522120	0.0261060	+	0.999659i	$0.491689\pi$
$194$	−17.1790	−1.23338
$195$	−79.8748	−5.71995
$196$	3.19326	0.228090
$197$	7.20195	0.513118	0.256559	−	0.966529i	$-0.417411\pi$
$197$	7.20195	0.513118	0.256559	+	0.966529i	$0.417411\pi$
$198$	46.1587	3.28035
$199$	14.7177	1.04331	0.521655	−	0.853157i	$-0.325315\pi$
$199$	14.7177	1.04331	0.521655	+	0.853157i	$0.325315\pi$
$200$	−7.35824	−0.520306
$201$	−1.74077	−0.122785
$202$	10.4275	0.733679
$203$	24.8826	1.74642
$204$	17.2339	1.20661
$205$	19.4701	1.35985
$206$	0.237817	0.0165695
$207$	−26.1037	−1.81433
$208$	−6.75995	−0.468718
$209$	−5.21927	−0.361024
$210$	37.7245	2.60324
$211$	28.1086	1.93507	0.967537	−	0.252731i	$-0.0813288\pi$
$211$	28.1086	1.93507	0.967537	+	0.252731i	$0.0813288\pi$
$212$	2.10884	0.144836
$213$	14.4501	0.990106
$214$	−9.98003	−0.682221
$215$	−36.3122	−2.47647
$216$	17.8052	1.21149
$217$	7.69756	0.522545
$218$	−12.6088	−0.853972
$219$	26.1583	1.76762
$220$	19.5565	1.31850
$221$	34.6608	2.33154
$222$	−16.5609	−1.11150
$223$	8.10523	0.542766	0.271383	−	0.962471i	$-0.412519\pi$
$223$	8.10523	0.542766	0.271383	+	0.962471i	$0.412519\pi$
$224$	3.19269	0.213320
$225$	61.0539	4.07026
$226$	8.94606	0.595083
$227$	9.58448	0.636144	0.318072	−	0.948066i	$-0.396965\pi$
$227$	9.58448	0.636144	0.318072	+	0.948066i	$0.396965\pi$
$228$	−3.15344	−0.208841
$229$	8.19651	0.541641	0.270820	−	0.962630i	$-0.412705\pi$
$229$	8.19651	0.541641	0.270820	+	0.962630i	$0.412705\pi$
$230$	−11.0596	−0.729252
$231$	−59.6978	−3.92783
$232$	7.79363	0.511677
$233$	2.09878	0.137495	0.0687477	−	0.997634i	$-0.478100\pi$
$233$	2.09878	0.137495	0.0687477	+	0.997634i	$0.478100\pi$
$234$	56.0896	3.66669
$235$	20.2276	1.31950
$236$	2.44408	0.159096
$237$	15.6626	1.01740
$238$	−16.3701	−1.06112
$239$	18.2173	1.17838	0.589189	−	0.807995i	$-0.299447\pi$
$239$	18.2173	1.17838	0.589189	+	0.807995i	$0.299447\pi$
$240$	11.8159	0.762713
$241$	10.3910	0.669345	0.334673	−	0.942334i	$-0.391374\pi$
$241$	10.3910	0.669345	0.334673	+	0.942334i	$0.391374\pi$
$242$	−19.9476	−1.28228
$243$	−64.0700	−4.11009
$244$	−13.3910	−0.857269
$245$	−11.2257	−0.717182
$246$	−18.6157	−1.18689
$247$	−6.34218	−0.403544
$248$	2.41100	0.153099
$249$	56.0424	3.55154
$250$	8.29021	0.524319
$251$	−12.4584	−0.786369	−0.393185	−	0.919459i	$-0.628627\pi$
$251$	−12.4584	−0.786369	−0.393185	+	0.919459i	$0.628627\pi$
$252$	−26.4909	−1.66877
$253$	17.5016	1.10031
$254$	7.64491	0.479684
$255$	−60.5845	−3.79395
$256$	1.00000	0.0625000
$257$	−24.8192	−1.54818	−0.774091	−	0.633075i	$-0.781793\pi$
$257$	−24.8192	−1.54818	−0.774091	+	0.633075i	$0.781793\pi$
$258$	34.7187	2.16149
$259$	15.7309	0.977469
$260$	23.7641	1.47379
$261$	−64.6665	−4.00276
$262$	−7.35255	−0.454242
$263$	7.71818	0.475923	0.237962	−	0.971275i	$-0.423521\pi$
$263$	7.71818	0.475923	0.237962	+	0.971275i	$0.423521\pi$
$264$	−18.6983	−1.15080
$265$	−7.41347	−0.455406
$266$	2.99538	0.183659
$267$	−43.8114	−2.68121
$268$	0.517910	0.0316364
$269$	−6.51266	−0.397084	−0.198542	−	0.980092i	$-0.563621\pi$
$269$	−6.51266	−0.397084	−0.198542	+	0.980092i	$0.563621\pi$
$270$	−62.5929	−3.80928
$271$	−2.15068	−0.130645	−0.0653223	−	0.997864i	$-0.520808\pi$
$271$	−2.15068	−0.130645	−0.0653223	+	0.997864i	$0.520808\pi$
$272$	−5.12738	−0.310893
$273$	−72.5417	−4.39043
$274$	10.2231	0.617600
$275$	−40.9343	−2.46843
$276$	10.5743	0.636498
$277$	−22.9616	−1.37963	−0.689813	−	0.723988i	$-0.742307\pi$
$277$	−22.9616	−1.37963	−0.689813	+	0.723988i	$0.742307\pi$
$278$	14.6346	0.877724
$279$	−20.0049	−1.19766
$280$	−11.2237	−0.670742
$281$	−12.1402	−0.724226	−0.362113	−	0.932134i	$-0.617944\pi$
$281$	−12.1402	−0.724226	−0.362113	+	0.932134i	$0.617944\pi$
$282$	−19.3399	−1.15167
$283$	4.33092	0.257446	0.128723	−	0.991681i	$-0.458912\pi$
$283$	4.33092	0.257446	0.128723	+	0.991681i	$0.458912\pi$
$284$	−4.29915	−0.255108
$285$	11.0857	0.656659
$286$	−37.6060	−2.22369
$287$	17.6826	1.04377
$288$	−8.29735	−0.488926
$289$	9.29000	0.546471
$290$	−27.3980	−1.60886
$291$	−57.7412	−3.38485
$292$	−7.78255	−0.455439
$293$	−17.7496	−1.03694	−0.518471	−	0.855095i	$-0.673498\pi$
$293$	−17.7496	−1.03694	−0.518471	+	0.855095i	$0.673498\pi$
$294$	10.7330	0.625963
$295$	−8.59200	−0.500246
$296$	4.92716	0.286385
$297$	99.0514	5.74755
$298$	−8.75129	−0.506948
$299$	21.2670	1.22990
$300$	−24.7322	−1.42791
$301$	−32.9785	−1.90085
$302$	−13.9998	−0.805600
$303$	35.0485	2.01349
$304$	0.938201	0.0538095
$305$	47.0750	2.69551
$306$	42.5436	2.43206
$307$	21.5661	1.23084	0.615420	−	0.788199i	$-0.288986\pi$
$307$	21.5661	1.23084	0.615420	+	0.788199i	$0.288986\pi$
$308$	17.7611	1.01203
$309$	0.799339	0.0454728
$310$	−8.47569	−0.481387
$311$	28.4933	1.61571	0.807853	−	0.589384i	$-0.200629\pi$
$311$	28.4933	1.61571	0.807853	+	0.589384i	$0.200629\pi$
$312$	−22.7212	−1.28634
$313$	−19.5523	−1.10516	−0.552581	−	0.833460i	$-0.686357\pi$
$313$	−19.5523	−1.10516	−0.552581	+	0.833460i	$0.686357\pi$
$314$	−3.90708	−0.220489
$315$	93.1267	5.24709
$316$	−4.65989	−0.262139
$317$	−6.21692	−0.349177	−0.174589	−	0.984641i	$-0.555860\pi$
$317$	−6.21692	−0.349177	−0.174589	+	0.984641i	$0.555860\pi$
$318$	7.08813	0.397483
$319$	43.3565	2.42750
$320$	−3.51543	−0.196518
$321$	−33.5444	−1.87227
$322$	−10.0443	−0.559747
$323$	−4.81051	−0.267664
$324$	34.9540	1.94189
$325$	−49.7413	−2.75915
$326$	19.9600	1.10548
$327$	−42.3799	−2.34362
$328$	5.53848	0.305811
$329$	18.3706	1.01280
$330$	65.7325	3.61846
$331$	9.59994	0.527661	0.263830	−	0.964569i	$-0.415014\pi$
$331$	9.59994	0.527661	0.263830	+	0.964569i	$0.415014\pi$
$332$	−16.6736	−0.915081
$333$	−40.8824	−2.24034
$334$	−20.3486	−1.11343
$335$	−1.82067	−0.0994741
$336$	10.7311	0.585430
$337$	26.6124	1.44967	0.724834	−	0.688923i	$-0.241916\pi$
$337$	26.6124	1.44967	0.724834	+	0.688923i	$0.241916\pi$
$338$	−32.6969	−1.77847
$339$	30.0691	1.63313
$340$	18.0249	0.977539
$341$	13.4125	0.726329
$342$	−7.78458	−0.420942
$343$	12.1537	0.656240
$344$	−10.3294	−0.556924
$345$	−37.1732	−2.00134
$346$	22.4203	1.20532
$347$	5.09385	0.273452	0.136726	−	0.990609i	$-0.456342\pi$
$347$	5.09385	0.273452	0.136726	+	0.990609i	$0.456342\pi$
$348$	26.1956	1.40423
$349$	−15.3287	−0.820524	−0.410262	−	0.911968i	$-0.634563\pi$
$349$	−15.3287	−0.820524	−0.410262	+	0.911968i	$0.634563\pi$
$350$	23.4926	1.25573
$351$	120.362	6.42446
$352$	5.56306	0.296512
$353$	−18.1472	−0.965876	−0.482938	−	0.875655i	$-0.660430\pi$
$353$	−18.1472	−0.965876	−0.482938	+	0.875655i	$0.660430\pi$
$354$	8.21494	0.436619
$355$	15.1134	0.802134
$356$	13.0346	0.690834
$357$	−55.0225	−2.91210
$358$	−8.41226	−0.444602
$359$	−14.4643	−0.763395	−0.381697	−	0.924287i	$-0.624660\pi$
$359$	−14.4643	−0.763395	−0.381697	+	0.924287i	$0.624660\pi$
$360$	29.1687	1.53733
$361$	−18.1198	−0.953673
$362$	21.2864	1.11879
$363$	−67.0471	−3.51906
$364$	21.5824	1.13123
$365$	27.3590	1.43204
$366$	−45.0091	−2.35266
$367$	−18.1624	−0.948071	−0.474035	−	0.880506i	$-0.657203\pi$
$367$	−18.1624	−0.948071	−0.474035	+	0.880506i	$0.657203\pi$
$368$	−3.14603	−0.163998
$369$	−45.9547	−2.39231
$370$	−17.3211	−0.900480
$371$	−6.73287	−0.349553
$372$	8.10373	0.420159
$373$	−33.1677	−1.71736	−0.858680	−	0.512512i	$-0.828715\pi$
$373$	−33.1677	−1.71736	−0.858680	+	0.512512i	$0.828715\pi$
$374$	−28.5239	−1.47494
$375$	27.8647	1.43892
$376$	5.75395	0.296737
$377$	52.6845	2.71339
$378$	−56.8465	−2.92387
$379$	−8.21890	−0.422177	−0.211088	−	0.977467i	$-0.567701\pi$
$379$	−8.21890	−0.422177	−0.211088	+	0.977467i	$0.567701\pi$
$380$	−3.29818	−0.169193
$381$	25.6957	1.31643
$382$	11.7768	0.602555
$383$	−22.6373	−1.15671	−0.578356	−	0.815784i	$-0.696306\pi$
$383$	−22.6373	−1.15671	−0.578356	+	0.815784i	$0.696306\pi$
$384$	3.36115	0.171523
$385$	−62.4379	−3.18213
$386$	−0.725352	−0.0369195
$387$	85.7066	4.35671
$388$	17.1790	0.872132
$389$	−33.8132	−1.71440	−0.857199	−	0.514985i	$-0.827798\pi$
$389$	−33.8132	−1.71440	−0.857199	+	0.514985i	$0.827798\pi$
$390$	79.8748	4.04462
$391$	16.1309	0.815774
$392$	−3.19326	−0.161284
$393$	−24.7131	−1.24661
$394$	−7.20195	−0.362829
$395$	16.3815	0.824243
$396$	−46.1587	−2.31956
$397$	27.3384	1.37207	0.686037	−	0.727566i	$-0.259349\pi$
$397$	27.3384	1.37207	0.686037	+	0.727566i	$0.259349\pi$
$398$	−14.7177	−0.737731
$399$	10.0679	0.504027
$400$	7.35824	0.367912
$401$	−20.6073	−1.02908	−0.514539	−	0.857467i	$-0.672037\pi$
$401$	−20.6073	−1.02908	−0.514539	+	0.857467i	$0.672037\pi$
$402$	1.74077	0.0868219
$403$	16.2982	0.811872
$404$	−10.4275	−0.518789
$405$	−122.878	−6.10587
$406$	−24.8826	−1.23490
$407$	27.4101	1.35867
$408$	−17.2339	−0.853205
$409$	18.1594	0.897924	0.448962	−	0.893551i	$-0.351794\pi$
$409$	18.1594	0.897924	0.448962	+	0.893551i	$0.351794\pi$
$410$	−19.4701	−0.961561
$411$	34.3614	1.69492
$412$	−0.237817	−0.0117164
$413$	−7.80320	−0.383970
$414$	26.1037	1.28293
$415$	58.6147	2.87728
$416$	6.75995	0.331434
$417$	49.1891	2.40880
$418$	5.21927	0.255283
$419$	9.46526	0.462408	0.231204	−	0.972905i	$-0.425733\pi$
$419$	9.46526	0.462408	0.231204	+	0.972905i	$0.425733\pi$
$420$	−37.7245	−1.84077
$421$	−27.8159	−1.35566	−0.677831	−	0.735218i	$-0.737080\pi$
$421$	−27.8159	−1.35566	−0.677831	+	0.735218i	$0.737080\pi$
$422$	−28.1086	−1.36830
$423$	−47.7425	−2.32132
$424$	−2.10884	−0.102414
$425$	−37.7285	−1.83010
$426$	−14.4501	−0.700110
$427$	42.7532	2.06897
$428$	9.98003	0.482403
$429$	−126.399	−6.10262
$430$	36.3122	1.75113
$431$	−6.72579	−0.323970	−0.161985	−	0.986793i	$-0.551790\pi$
$431$	−6.72579	−0.323970	−0.161985	+	0.986793i	$0.551790\pi$
$432$	−17.8052	−0.856653
$433$	−6.52903	−0.313765	−0.156883	−	0.987617i	$-0.550144\pi$
$433$	−6.52903	−0.313765	−0.156883	+	0.987617i	$0.550144\pi$
$434$	−7.69756	−0.369495
$435$	−92.0887	−4.41532
$436$	12.6088	0.603850
$437$	−2.95161	−0.141195
$438$	−26.1583	−1.24989
$439$	−0.449577	−0.0214571	−0.0107286	−	0.999942i	$-0.503415\pi$
$439$	−0.449577	−0.0214571	−0.0107286	+	0.999942i	$0.503415\pi$
$440$	−19.5565	−0.932322
$441$	26.4956	1.26169
$442$	−34.6608	−1.64865
$443$	5.42560	0.257778	0.128889	−	0.991659i	$-0.458859\pi$
$443$	5.42560	0.257778	0.128889	+	0.991659i	$0.458859\pi$
$444$	16.5609	0.785947
$445$	−45.8223	−2.17218
$446$	−8.10523	−0.383794
$447$	−29.4144	−1.39125
$448$	−3.19269	−0.150840
$449$	36.3201	1.71405	0.857027	−	0.515272i	$-0.172309\pi$
$449$	36.3201	1.71405	0.857027	+	0.515272i	$0.172309\pi$
$450$	−61.0539	−2.87811
$451$	30.8109	1.45083
$452$	−8.94606	−0.420787
$453$	−47.0556	−2.21086
$454$	−9.58448	−0.449822
$455$	−75.8714	−3.55690
$456$	3.15344	0.147673
$457$	−14.9263	−0.698221	−0.349111	−	0.937081i	$-0.613516\pi$
$457$	−14.9263	−0.698221	−0.349111	+	0.937081i	$0.613516\pi$
$458$	−8.19651	−0.382998
$459$	91.2940	4.26124
$460$	11.0596	0.515659
$461$	−3.55523	−0.165583	−0.0827917	−	0.996567i	$-0.526384\pi$
$461$	−3.55523	−0.165583	−0.0827917	+	0.996567i	$0.526384\pi$
$462$	59.6978	2.77739
$463$	4.93536	0.229366	0.114683	−	0.993402i	$-0.463415\pi$
$463$	4.93536	0.229366	0.114683	+	0.993402i	$0.463415\pi$
$464$	−7.79363	−0.361810
$465$	−28.4881	−1.32110
$466$	−2.09878	−0.0972239
$467$	25.4192	1.17626	0.588130	−	0.808766i	$-0.299864\pi$
$467$	25.4192	1.17626	0.588130	+	0.808766i	$0.299864\pi$
$468$	−56.0896	−2.59274
$469$	−1.65352	−0.0763527
$470$	−20.2276	−0.933029
$471$	−13.1323	−0.605104
$472$	−2.44408	−0.112498
$473$	−57.4630	−2.64215
$474$	−15.6626	−0.719407
$475$	6.90350	0.316754
$476$	16.3701	0.750323
$477$	17.4978	0.801168
$478$	−18.2173	−0.833239
$479$	16.1703	0.738840	0.369420	−	0.929263i	$-0.379556\pi$
$479$	16.1703	0.738840	0.369420	+	0.929263i	$0.379556\pi$
$480$	−11.8159	−0.539319
$481$	33.3073	1.51868
$482$	−10.3910	−0.473299
$483$	−33.7604	−1.53615
$484$	19.9476	0.906711
$485$	−60.3915	−2.74224
$486$	64.0700	2.90628
$487$	−35.0084	−1.58638	−0.793192	−	0.608972i	$-0.791582\pi$
$487$	−35.0084	−1.58638	−0.793192	+	0.608972i	$0.791582\pi$
$488$	13.3910	0.606181
$489$	67.0887	3.03386
$490$	11.2257	0.507124
$491$	−13.4989	−0.609197	−0.304599	−	0.952481i	$-0.598522\pi$
$491$	−13.4989	−0.609197	−0.304599	+	0.952481i	$0.598522\pi$
$492$	18.6157	0.839259
$493$	39.9609	1.79975
$494$	6.34218	0.285348
$495$	162.267	7.29338
$496$	−2.41100	−0.108257
$497$	13.7259	0.615689
$498$	−56.0424	−2.51132
$499$	−9.70218	−0.434329	−0.217165	−	0.976135i	$-0.569681\pi$
$499$	−9.70218	−0.434329	−0.217165	+	0.976135i	$0.569681\pi$
$500$	−8.29021	−0.370749
$501$	−68.3948	−3.05565
$502$	12.4584	0.556047
$503$	−24.9357	−1.11183	−0.555914	−	0.831240i	$-0.687632\pi$
$503$	−24.9357	−1.11183	−0.555914	+	0.831240i	$0.687632\pi$
$504$	26.4909	1.18000
$505$	36.6572	1.63123
$506$	−17.5016	−0.778039
$507$	−109.899	−4.88079
$508$	−7.64491	−0.339188
$509$	7.87360	0.348991	0.174496	−	0.984658i	$-0.444171\pi$
$509$	7.87360	0.348991	0.174496	+	0.984658i	$0.444171\pi$
$510$	60.5845	2.68273
$511$	24.8473	1.09918
$512$	−1.00000	−0.0441942
$513$	−16.7049	−0.737537
$514$	24.8192	1.09473
$515$	0.836028	0.0368398
$516$	−34.7187	−1.52840
$517$	32.0096	1.40778
$518$	−15.7309	−0.691175
$519$	75.3582	3.30786
$520$	−23.7641	−1.04212
$521$	−16.8480	−0.738125	−0.369062	−	0.929405i	$-0.620321\pi$
$521$	−16.8480	−0.738125	−0.369062	+	0.929405i	$0.620321\pi$
$522$	64.6665	2.83038
$523$	−8.21139	−0.359059	−0.179530	−	0.983753i	$-0.557458\pi$
$523$	−8.21139	−0.359059	−0.179530	+	0.983753i	$0.557458\pi$
$524$	7.35255	0.321198
$525$	78.9621	3.44619
$526$	−7.71818	−0.336529
$527$	12.3621	0.538501
$528$	18.6983	0.813739
$529$	−13.1025	−0.569673
$530$	7.41347	0.322021
$531$	20.2794	0.880052
$532$	−2.99538	−0.129866
$533$	37.4398	1.62170
$534$	43.8114	1.89590
$535$	−35.0841	−1.51682
$536$	−0.517910	−0.0223703
$537$	−28.2749	−1.22015
$538$	6.51266	0.280781
$539$	−17.7643	−0.765162
$540$	62.5929	2.69357
$541$	−30.8242	−1.32524	−0.662618	−	0.748958i	$-0.730555\pi$
$541$	−30.8242	−1.32524	−0.662618	+	0.748958i	$0.730555\pi$
$542$	2.15068	0.0923796
$543$	71.5470	3.07037
$544$	5.12738	0.219835
$545$	−44.3252	−1.89868
$546$	72.5417	3.10450
$547$	−20.9337	−0.895059	−0.447529	−	0.894269i	$-0.647696\pi$
$547$	−20.9337	−0.895059	−0.447529	+	0.894269i	$0.647696\pi$
$548$	−10.2231	−0.436709
$549$	−111.110	−4.74204
$550$	40.9343	1.74544
$551$	−7.31199	−0.311501
$552$	−10.5743	−0.450072
$553$	14.8776	0.632659
$554$	22.9616	0.975543
$555$	−58.2188	−2.47125
$556$	−14.6346	−0.620645
$557$	−4.39422	−0.186189	−0.0930946	−	0.995657i	$-0.529676\pi$
$557$	−4.39422	−0.186189	−0.0930946	+	0.995657i	$0.529676\pi$
$558$	20.0049	0.846875
$559$	−69.8261	−2.95333
$560$	11.2237	0.474286
$561$	−95.8732	−4.04777
$562$	12.1402	0.512105
$563$	7.41435	0.312478	0.156239	−	0.987719i	$-0.450063\pi$
$563$	7.41435	0.312478	0.156239	+	0.987719i	$0.450063\pi$
$564$	19.3399	0.814357
$565$	31.4492	1.32308
$566$	−4.33092	−0.182042
$567$	−111.597	−4.68664
$568$	4.29915	0.180389
$569$	12.7586	0.534869	0.267435	−	0.963576i	$-0.413824\pi$
$569$	12.7586	0.534869	0.267435	+	0.963576i	$0.413824\pi$
$570$	−11.0857	−0.464328
$571$	−5.86097	−0.245274	−0.122637	−	0.992452i	$-0.539135\pi$
$571$	−5.86097	−0.245274	−0.122637	+	0.992452i	$0.539135\pi$
$572$	37.6060	1.57239
$573$	39.5837	1.65363
$574$	−17.6826	−0.738059
$575$	−23.1492	−0.965390
$576$	8.29735	0.345723
$577$	0.718531	0.0299128	0.0149564	−	0.999888i	$-0.495239\pi$
$577$	0.718531	0.0299128	0.0149564	+	0.999888i	$0.495239\pi$
$578$	−9.29000	−0.386413
$579$	−2.43802	−0.101321
$580$	27.3980	1.13764
$581$	53.2335	2.20850
$582$	57.7412	2.39345
$583$	−11.7316	−0.485873
$584$	7.78255	0.322044
$585$	197.179	8.15235
$586$	17.7496	0.733228
$587$	−42.7236	−1.76339	−0.881697	−	0.471817i	$-0.843598\pi$
$587$	−42.7236	−1.76339	−0.881697	+	0.471817i	$0.843598\pi$
$588$	−10.7330	−0.442623
$589$	−2.26200	−0.0932041
$590$	8.59200	0.353727
$591$	−24.2069	−0.995737
$592$	−4.92716	−0.202505
$593$	−18.8802	−0.775316	−0.387658	−	0.921803i	$-0.626716\pi$
$593$	−18.8802	−0.775316	−0.387658	+	0.921803i	$0.626716\pi$
$594$	−99.0514	−4.06413
$595$	−57.5480	−2.35924
$596$	8.75129	0.358467
$597$	−49.4684	−2.02461
$598$	−21.2670	−0.869672
$599$	31.0047	1.26682	0.633408	−	0.773818i	$-0.281656\pi$
$599$	31.0047	1.26682	0.633408	+	0.773818i	$0.281656\pi$
$600$	24.7322	1.00969
$601$	29.0695	1.18577	0.592886	−	0.805287i	$-0.297989\pi$
$601$	29.0695	1.18577	0.592886	+	0.805287i	$0.297989\pi$
$602$	32.9785	1.34410
$603$	4.29728	0.174999
$604$	13.9998	0.569645
$605$	−70.1245	−2.85097
$606$	−35.0485	−1.42375
$607$	−45.1196	−1.83135	−0.915674	−	0.401922i	$-0.868342\pi$
$607$	−45.1196	−1.83135	−0.915674	+	0.401922i	$0.868342\pi$
$608$	−0.938201	−0.0380491
$609$	−83.6344	−3.38904
$610$	−47.0750	−1.90601
$611$	38.8964	1.57358
$612$	−42.5436	−1.71973
$613$	21.1465	0.854101	0.427050	−	0.904228i	$-0.359553\pi$
$613$	21.1465	0.854101	0.427050	+	0.904228i	$0.359553\pi$
$614$	−21.5661	−0.870336
$615$	−65.4421	−2.63888
$616$	−17.7611	−0.715616
$617$	−17.7044	−0.712751	−0.356375	−	0.934343i	$-0.615988\pi$
$617$	−17.7044	−0.712751	−0.356375	+	0.934343i	$0.615988\pi$
$618$	−0.799339	−0.0321541
$619$	35.0810	1.41002	0.705012	−	0.709195i	$-0.250941\pi$
$619$	35.0810	1.41002	0.705012	+	0.709195i	$0.250941\pi$
$620$	8.47569	0.340392
$621$	56.0157	2.24783
$622$	−28.4933	−1.14248
$623$	−41.6155	−1.66729
$624$	22.7212	0.909576
$625$	−7.64755	−0.305902
$626$	19.5523	0.781467
$627$	17.5428	0.700590
$628$	3.90708	0.155909
$629$	25.2634	1.00732
$630$	−93.1267	−3.71026
$631$	−42.9763	−1.71086	−0.855430	−	0.517919i	$-0.826707\pi$
$631$	−42.9763	−1.71086	−0.855430	+	0.517919i	$0.826707\pi$
$632$	4.65989	0.185360
$633$	−94.4772	−3.75513
$634$	6.21692	0.246906
$635$	26.8751	1.06651
$636$	−7.08813	−0.281063
$637$	−21.5863	−0.855278
$638$	−43.3565	−1.71650
$639$	−35.6716	−1.41115
$640$	3.51543	0.138960
$641$	−0.801304	−0.0316496	−0.0158248	−	0.999875i	$-0.505037\pi$
$641$	−0.801304	−0.0316496	−0.0158248	+	0.999875i	$0.505037\pi$
$642$	33.5444	1.32389
$643$	29.6166	1.16796	0.583982	−	0.811766i	$-0.301494\pi$
$643$	29.6166	1.16796	0.583982	+	0.811766i	$0.301494\pi$
$644$	10.0443	0.395801
$645$	122.051	4.80575
$646$	4.81051	0.189267
$647$	−12.1967	−0.479501	−0.239750	−	0.970835i	$-0.577066\pi$
$647$	−12.1967	−0.479501	−0.239750	+	0.970835i	$0.577066\pi$
$648$	−34.9540	−1.37312
$649$	−13.5966	−0.533713
$650$	49.7413	1.95101
$651$	−25.8727	−1.01403
$652$	−19.9600	−0.781695
$653$	26.3718	1.03201	0.516005	−	0.856586i	$-0.327419\pi$
$653$	26.3718	1.03201	0.516005	+	0.856586i	$0.327419\pi$
$654$	42.3799	1.65719
$655$	−25.8474	−1.00994
$656$	−5.53848	−0.216241
$657$	−64.5746	−2.51929
$658$	−18.3706	−0.716159
$659$	−7.98701	−0.311130	−0.155565	−	0.987826i	$-0.549720\pi$
$659$	−7.98701	−0.311130	−0.155565	+	0.987826i	$0.549720\pi$
$660$	−65.7325	−2.55864
$661$	−18.9091	−0.735477	−0.367739	−	0.929929i	$-0.619868\pi$
$661$	−18.9091	−0.735477	−0.367739	+	0.929929i	$0.619868\pi$
$662$	−9.59994	−0.373112
$663$	−116.500	−4.52449
$664$	16.6736	0.647060
$665$	10.5300	0.408338
$666$	40.8824	1.58416
$667$	24.5190	0.949380
$668$	20.3486	0.787311
$669$	−27.2429	−1.05327
$670$	1.82067	0.0703388
$671$	74.4948	2.87584
$672$	−10.7311	−0.413962
$673$	−8.90945	−0.343434	−0.171717	−	0.985146i	$-0.554932\pi$
$673$	−8.90945	−0.343434	−0.171717	+	0.985146i	$0.554932\pi$
$674$	−26.6124	−1.02507
$675$	−131.015	−5.04277
$676$	32.6969	1.25757
$677$	23.6797	0.910086	0.455043	−	0.890470i	$-0.349624\pi$
$677$	23.6797	0.910086	0.455043	+	0.890470i	$0.349624\pi$
$678$	−30.0691	−1.15480
$679$	−54.8472	−2.10484
$680$	−18.0249	−0.691224
$681$	−32.2149	−1.23448
$682$	−13.4125	−0.513592
$683$	−42.3017	−1.61863	−0.809314	−	0.587376i	$-0.800161\pi$
$683$	−42.3017	−1.61863	−0.809314	+	0.587376i	$0.800161\pi$
$684$	7.78458	0.297651
$685$	35.9386	1.37314
$686$	−12.1537	−0.464032
$687$	−27.5497	−1.05109
$688$	10.3294	0.393804
$689$	−14.2556	−0.543097
$690$	37.1732	1.41516
$691$	−16.2167	−0.616913	−0.308457	−	0.951238i	$-0.599812\pi$
$691$	−16.2167	−0.616913	−0.308457	+	0.951238i	$0.599812\pi$
$692$	−22.4203	−0.852293
$693$	147.370	5.59813
$694$	−5.09385	−0.193360
$695$	51.4468	1.95149
$696$	−26.1956	−0.992942
$697$	28.3979	1.07565
$698$	15.3287	0.580198
$699$	−7.05431	−0.266818
$700$	−23.4926	−0.887935
$701$	−22.4449	−0.847733	−0.423866	−	0.905725i	$-0.639327\pi$
$701$	−22.4449	−0.847733	−0.423866	+	0.905725i	$0.639327\pi$
$702$	−120.362	−4.54278
$703$	−4.62266	−0.174347
$704$	−5.56306	−0.209666
$705$	−67.9880	−2.56058
$706$	18.1472	0.682978
$707$	33.2919	1.25207
$708$	−8.21494	−0.308736
$709$	−21.2150	−0.796745	−0.398373	−	0.917224i	$-0.630425\pi$
$709$	−21.2150	−0.796745	−0.398373	+	0.917224i	$0.630425\pi$
$710$	−15.1134	−0.567195
$711$	−38.6647	−1.45004
$712$	−13.0346	−0.488493
$713$	7.58507	0.284063
$714$	55.0225	2.05916
$715$	−132.201	−4.94404
$716$	8.41226	0.314381
$717$	−61.2310	−2.28672
$718$	14.4643	0.539802
$719$	−8.22798	−0.306852	−0.153426	−	0.988160i	$-0.549031\pi$
$719$	−8.22798	−0.306852	−0.153426	+	0.988160i	$0.549031\pi$
$720$	−29.1687	−1.08705
$721$	0.759275	0.0282769
$722$	18.1198	0.674348
$723$	−34.9259	−1.29891
$724$	−21.2864	−0.791104
$725$	−57.3474	−2.12983
$726$	67.0471	2.48835
$727$	51.2073	1.89917	0.949587	−	0.313503i	$-0.101503\pi$
$727$	51.2073	1.89917	0.949587	+	0.313503i	$0.101503\pi$
$728$	−21.5824	−0.799897
$729$	110.487	4.09212
$730$	−27.3590	−1.01260
$731$	−52.9627	−1.95890
$732$	45.0091	1.66358
$733$	−11.3587	−0.419544	−0.209772	−	0.977750i	$-0.567272\pi$
$733$	−11.3587	−0.419544	−0.209772	+	0.977750i	$0.567272\pi$
$734$	18.1624	0.670387
$735$	37.7312	1.39174
$736$	3.14603	0.115964
$737$	−2.88116	−0.106129
$738$	45.9547	1.69162
$739$	22.2437	0.818249	0.409124	−	0.912479i	$-0.365834\pi$
$739$	22.2437	0.818249	0.409124	+	0.912479i	$0.365834\pi$
$740$	17.3211	0.636735
$741$	21.3171	0.783102
$742$	6.73287	0.247171
$743$	5.97361	0.219150	0.109575	−	0.993979i	$-0.465051\pi$
$743$	5.97361	0.219150	0.109575	+	0.993979i	$0.465051\pi$
$744$	−8.10373	−0.297097
$745$	−30.7645	−1.12712
$746$	33.1677	1.21436
$747$	−138.346	−5.06183
$748$	28.5239	1.04294
$749$	−31.8631	−1.16425
$750$	−27.8647	−1.01747
$751$	21.9580	0.801258	0.400629	−	0.916240i	$-0.368792\pi$
$751$	21.9580	0.801258	0.400629	+	0.916240i	$0.368792\pi$
$752$	−5.75395	−0.209825
$753$	41.8747	1.52600
$754$	−52.6845	−1.91866
$755$	−49.2154	−1.79113
$756$	56.8465	2.06749
$757$	−20.2814	−0.737139	−0.368569	−	0.929600i	$-0.620152\pi$
$757$	−20.2814	−0.737139	−0.368569	+	0.929600i	$0.620152\pi$
$758$	8.21890	0.298524
$759$	−58.8254	−2.13523
$760$	3.29818	0.119637
$761$	11.7581	0.426231	0.213115	−	0.977027i	$-0.431639\pi$
$761$	11.7581	0.426231	0.213115	+	0.977027i	$0.431639\pi$
$762$	−25.6957	−0.930858
$763$	−40.2558	−1.45736
$764$	−11.7768	−0.426070
$765$	149.559	5.40732
$766$	22.6373	0.817919
$767$	−16.5219	−0.596570
$768$	−3.36115	−0.121285
$769$	15.1272	0.545499	0.272750	−	0.962085i	$-0.412067\pi$
$769$	15.1272	0.545499	0.272750	+	0.962085i	$0.412067\pi$
$770$	62.4379	2.25011
$771$	83.4212	3.00434
$772$	0.725352	0.0261060
$773$	−10.4504	−0.375876	−0.187938	−	0.982181i	$-0.560180\pi$
$773$	−10.4504	−0.375876	−0.187938	+	0.982181i	$0.560180\pi$
$774$	−85.7066	−3.08066
$775$	−17.7407	−0.637264
$776$	−17.1790	−0.616690
$777$	−52.8739	−1.89684
$778$	33.8132	1.21226
$779$	−5.19620	−0.186173
$780$	−79.8748	−2.85998
$781$	23.9165	0.855798
$782$	−16.1309	−0.576840
$783$	138.767	4.95914
$784$	3.19326	0.114045
$785$	−13.7351	−0.490225
$786$	24.7131	0.881486
$787$	−16.2123	−0.577904	−0.288952	−	0.957344i	$-0.593307\pi$
$787$	−16.2123	−0.577904	−0.288952	+	0.957344i	$0.593307\pi$
$788$	7.20195	0.256559
$789$	−25.9420	−0.923559
$790$	−16.3815	−0.582828
$791$	28.5620	1.01555
$792$	46.1587	1.64018
$793$	90.5222	3.21454
$794$	−27.3384	−0.970203
$795$	24.9178	0.883744
$796$	14.7177	0.521655
$797$	1.22885	0.0435282	0.0217641	−	0.999763i	$-0.493072\pi$
$797$	1.22885	0.0435282	0.0217641	+	0.999763i	$0.493072\pi$
$798$	−10.0679	−0.356401
$799$	29.5027	1.04373
$800$	−7.35824	−0.260153
$801$	108.153	3.82139
$802$	20.6073	0.727668
$803$	43.2948	1.52784
$804$	−1.74077	−0.0613924
$805$	−35.3100	−1.24451
$806$	−16.2982	−0.574080
$807$	21.8900	0.770566
$808$	10.4275	0.366839
$809$	7.77481	0.273348	0.136674	−	0.990616i	$-0.456359\pi$
$809$	7.77481	0.273348	0.136674	+	0.990616i	$0.456359\pi$
$810$	122.878	4.31750
$811$	6.73422	0.236470	0.118235	−	0.992986i	$-0.462276\pi$
$811$	6.73422	0.236470	0.118235	+	0.992986i	$0.462276\pi$
$812$	24.8826	0.873210
$813$	7.22876	0.253524
$814$	−27.4101	−0.960723
$815$	70.1680	2.45788
$816$	17.2339	0.603307
$817$	9.69104	0.339047
$818$	−18.1594	−0.634928
$819$	179.077	6.25745
$820$	19.4701	0.679926
$821$	−7.10368	−0.247920	−0.123960	−	0.992287i	$-0.539559\pi$
$821$	−7.10368	−0.247920	−0.123960	+	0.992287i	$0.539559\pi$
$822$	−34.3614	−1.19849
$823$	−20.7159	−0.722110	−0.361055	−	0.932544i	$-0.617583\pi$
$823$	−20.7159	−0.722110	−0.361055	+	0.932544i	$0.617583\pi$
$824$	0.237817	0.00828474
$825$	137.586	4.79015
$826$	7.80320	0.271508
$827$	24.0262	0.835471	0.417736	−	0.908569i	$-0.362824\pi$
$827$	24.0262	0.835471	0.417736	+	0.908569i	$0.362824\pi$
$828$	−26.1037	−0.907167
$829$	−25.3339	−0.879884	−0.439942	−	0.898026i	$-0.645001\pi$
$829$	−25.3339	−0.879884	−0.439942	+	0.898026i	$0.645001\pi$
$830$	−58.6147	−2.03455
$831$	77.1773	2.67725
$832$	−6.75995	−0.234359
$833$	−16.3730	−0.567292
$834$	−49.1891	−1.70328
$835$	−71.5341	−2.47554
$836$	−5.21927	−0.180512
$837$	42.9283	1.48382
$838$	−9.46526	−0.326972
$839$	19.4783	0.672466	0.336233	−	0.941779i	$-0.390847\pi$
$839$	19.4783	0.672466	0.336233	+	0.941779i	$0.390847\pi$
$840$	37.7245	1.30162
$841$	31.7407	1.09451
$842$	27.8159	0.958598
$843$	40.8052	1.40541
$844$	28.1086	0.967537
$845$	−114.943	−3.95418
$846$	47.7425	1.64142
$847$	−63.6866	−2.18830
$848$	2.10884	0.0724178
$849$	−14.5569	−0.499590
$850$	37.7285	1.29408
$851$	15.5010	0.531367
$852$	14.4501	0.495053
$853$	49.6486	1.69994	0.849968	−	0.526835i	$-0.176621\pi$
$853$	49.6486	1.69994	0.849968	+	0.526835i	$0.176621\pi$
$854$	−42.7532	−1.46298
$855$	−27.3661	−0.935902
$856$	−9.98003	−0.341110
$857$	−1.46730	−0.0501220	−0.0250610	−	0.999686i	$-0.507978\pi$
$857$	−1.46730	−0.0501220	−0.0250610	+	0.999686i	$0.507978\pi$
$858$	126.399	4.31521
$859$	−16.4413	−0.560971	−0.280486	−	0.959858i	$-0.590495\pi$
$859$	−16.4413	−0.560971	−0.280486	+	0.959858i	$0.590495\pi$
$860$	−36.3122	−1.23824
$861$	−59.4341	−2.02551
$862$	6.72579	0.229081
$863$	−1.89649	−0.0645573	−0.0322787	−	0.999479i	$-0.510276\pi$
$863$	−1.89649	−0.0645573	−0.0322787	+	0.999479i	$0.510276\pi$
$864$	17.8052	0.605745
$865$	78.8171	2.67986
$866$	6.52903	0.221865
$867$	−31.2251	−1.06046
$868$	7.69756	0.261272
$869$	25.9232	0.879386
$870$	92.0887	3.12210
$871$	−3.50104	−0.118628
$872$	−12.6088	−0.426986
$873$	142.540	4.82425
$874$	2.95161	0.0998396
$875$	26.4680	0.894783
$876$	26.1583	0.883809
$877$	−18.7425	−0.632888	−0.316444	−	0.948611i	$-0.602489\pi$
$877$	−18.7425	−0.632888	−0.316444	+	0.948611i	$0.602489\pi$
$878$	0.449577	0.0151725
$879$	59.6590	2.01225
$880$	19.5565	0.659251
$881$	26.3624	0.888173	0.444086	−	0.895984i	$-0.353528\pi$
$881$	26.3624	0.888173	0.444086	+	0.895984i	$0.353528\pi$
$882$	−26.4956	−0.892153
$883$	−24.6596	−0.829861	−0.414931	−	0.909853i	$-0.636194\pi$
$883$	−24.6596	−0.829861	−0.414931	+	0.909853i	$0.636194\pi$
$884$	34.6608	1.16577
$885$	28.8790	0.970758
$886$	−5.42560	−0.182277
$887$	9.83733	0.330305	0.165153	−	0.986268i	$-0.447188\pi$
$887$	9.83733	0.330305	0.165153	+	0.986268i	$0.447188\pi$
$888$	−16.5609	−0.555749
$889$	24.4078	0.818612
$890$	45.8223	1.53597
$891$	−194.451	−6.51436
$892$	8.10523	0.271383
$893$	−5.39836	−0.180649
$894$	29.4144	0.983765
$895$	−29.5727	−0.988506
$896$	3.19269	0.106660
$897$	−71.4817	−2.38670
$898$	−36.3201	−1.21202
$899$	18.7904	0.626696
$900$	61.0539	2.03513
$901$	−10.8128	−0.360227
$902$	−30.8109	−1.02589
$903$	110.846	3.68872
$904$	8.94606	0.297542
$905$	74.8309	2.48746
$906$	47.0556	1.56332
$907$	48.4521	1.60883	0.804413	−	0.594070i	$-0.202480\pi$
$907$	48.4521	1.60883	0.804413	+	0.594070i	$0.202480\pi$
$908$	9.58448	0.318072
$909$	−86.5209	−2.86972
$910$	75.8714	2.51511
$911$	−35.2504	−1.16790	−0.583949	−	0.811790i	$-0.698493\pi$
$911$	−35.2504	−1.16790	−0.583949	+	0.811790i	$0.698493\pi$
$912$	−3.15344	−0.104421
$913$	92.7561	3.06978
$914$	14.9263	0.493717
$915$	−158.226	−5.23080
$916$	8.19651	0.270820
$917$	−23.4744	−0.775193
$918$	−91.2940	−3.01315
$919$	10.7620	0.355005	0.177502	−	0.984120i	$-0.443198\pi$
$919$	10.7620	0.355005	0.177502	+	0.984120i	$0.443198\pi$
$920$	−11.0596	−0.364626
$921$	−72.4869	−2.38852
$922$	3.55523	0.117085
$923$	29.0621	0.956589
$924$	−59.6978	−1.96391
$925$	−36.2552	−1.19206
$926$	−4.93536	−0.162186
$927$	−1.97325	−0.0648100
$928$	7.79363	0.255839
$929$	48.0414	1.57619	0.788094	−	0.615554i	$-0.211068\pi$
$929$	48.0414	1.57619	0.788094	+	0.615554i	$0.211068\pi$
$930$	28.4881	0.934161
$931$	2.99592	0.0981872
$932$	2.09878	0.0687477
$933$	−95.7703	−3.13538
$934$	−25.4192	−0.831742
$935$	−100.274	−3.27930
$936$	56.0896	1.83335
$937$	−1.54791	−0.0505679	−0.0252839	−	0.999680i	$-0.508049\pi$
$937$	−1.54791	−0.0505679	−0.0252839	+	0.999680i	$0.508049\pi$
$938$	1.65352	0.0539895
$939$	65.7183	2.14463
$940$	20.2276	0.659751
$941$	36.7554	1.19819	0.599097	−	0.800677i	$-0.295527\pi$
$941$	36.7554	1.19819	0.599097	+	0.800677i	$0.295527\pi$
$942$	13.1323	0.427873
$943$	17.4242	0.567411
$944$	2.44408	0.0795481
$945$	−199.840	−6.50079
$946$	57.4630	1.86828
$947$	−2.80309	−0.0910883	−0.0455442	−	0.998962i	$-0.514502\pi$
$947$	−2.80309	−0.0910883	−0.0455442	+	0.998962i	$0.514502\pi$
$948$	15.6626	0.508698
$949$	52.6096	1.70778
$950$	−6.90350	−0.223979
$951$	20.8960	0.677600
$952$	−16.3701	−0.530559
$953$	−0.546575	−0.0177053	−0.00885265	−	0.999961i	$-0.502818\pi$
$953$	−0.546575	−0.0177053	−0.00885265	+	0.999961i	$0.502818\pi$
$954$	−17.4978	−0.566511
$955$	41.4006	1.33969
$956$	18.2173	0.589189
$957$	−145.728	−4.71071
$958$	−16.1703	−0.522439
$959$	32.6392	1.05397
$960$	11.8159	0.381356
$961$	−25.1871	−0.812487
$962$	−33.3073	−1.07387
$963$	82.8078	2.66844
$964$	10.3910	0.334673
$965$	−2.54992	−0.0820850
$966$	33.7604	1.08622
$967$	46.1677	1.48465	0.742327	−	0.670038i	$-0.233722\pi$
$967$	46.1677	1.48465	0.742327	+	0.670038i	$0.233722\pi$
$968$	−19.9476	−0.641142
$969$	16.1689	0.519418
$970$	60.3915	1.93906
$971$	−32.1557	−1.03193	−0.515964	−	0.856611i	$-0.672566\pi$
$971$	−32.1557	−1.03193	−0.515964	+	0.856611i	$0.672566\pi$
$972$	−64.0700	−2.05505
$973$	46.7237	1.49789
$974$	35.0084	1.12174
$975$	167.188	5.35430
$976$	−13.3910	−0.428635
$977$	−10.8392	−0.346776	−0.173388	−	0.984854i	$-0.555471\pi$
$977$	−10.8392	−0.346776	−0.173388	+	0.984854i	$0.555471\pi$
$978$	−67.0887	−2.14526
$979$	−72.5124	−2.31751
$980$	−11.2257	−0.358591
$981$	104.619	3.34023
$982$	13.4989	0.430767
$983$	−55.4410	−1.76829	−0.884147	−	0.467209i	$-0.845260\pi$
$983$	−55.4410	−1.76829	−0.884147	+	0.467209i	$0.845260\pi$
$984$	−18.6157	−0.593446
$985$	−25.3179	−0.806697
$986$	−39.9609	−1.27261
$987$	−61.7463	−1.96541
$988$	−6.34218	−0.201772
$989$	−32.4966	−1.03333
$990$	−162.267	−5.15720
$991$	45.2644	1.43787	0.718936	−	0.695076i	$-0.244630\pi$
$991$	45.2644	1.43787	0.718936	+	0.695076i	$0.244630\pi$
$992$	2.41100	0.0765493
$993$	−32.2669	−1.02396
$994$	−13.7259	−0.435358
$995$	−51.7390	−1.64024
$996$	56.0424	1.77577
$997$	26.3484	0.834463	0.417232	−	0.908800i	$-0.363000\pi$
$997$	26.3484	0.834463	0.417232	+	0.908800i	$0.363000\pi$
$998$	9.70218	0.307117
$999$	87.7291	2.77562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.c.1.1	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.c.1.1	✓	84	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.36115 q^{3} +1.00000 q^{4} -3.51543 q^{5} +3.36115 q^{6} -3.19269 q^{7} -1.00000 q^{8} +8.29735 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.36115 q^{3} +1.00000 q^{4} -3.51543 q^{5} +3.36115 q^{6} -3.19269 q^{7} -1.00000 q^{8} +8.29735 q^{9} +3.51543 q^{10} -5.56306 q^{11} -3.36115 q^{12} -6.75995 q^{13} +3.19269 q^{14} +11.8159 q^{15} +1.00000 q^{16} -5.12738 q^{17} -8.29735 q^{18} +0.938201 q^{19} -3.51543 q^{20} +10.7311 q^{21} +5.56306 q^{22} -3.14603 q^{23} +3.36115 q^{24} +7.35824 q^{25} +6.75995 q^{26} -17.8052 q^{27} -3.19269 q^{28} -7.79363 q^{29} -11.8159 q^{30} -2.41100 q^{31} -1.00000 q^{32} +18.6983 q^{33} +5.12738 q^{34} +11.2237 q^{35} +8.29735 q^{36} -4.92716 q^{37} -0.938201 q^{38} +22.7212 q^{39} +3.51543 q^{40} -5.53848 q^{41} -10.7311 q^{42} +10.3294 q^{43} -5.56306 q^{44} -29.1687 q^{45} +3.14603 q^{46} -5.75395 q^{47} -3.36115 q^{48} +3.19326 q^{49} -7.35824 q^{50} +17.2339 q^{51} -6.75995 q^{52} +2.10884 q^{53} +17.8052 q^{54} +19.5565 q^{55} +3.19269 q^{56} -3.15344 q^{57} +7.79363 q^{58} +2.44408 q^{59} +11.8159 q^{60} -13.3910 q^{61} +2.41100 q^{62} -26.4909 q^{63} +1.00000 q^{64} +23.7641 q^{65} -18.6983 q^{66} +0.517910 q^{67} -5.12738 q^{68} +10.5743 q^{69} -11.2237 q^{70} -4.29915 q^{71} -8.29735 q^{72} -7.78255 q^{73} +4.92716 q^{74} -24.7322 q^{75} +0.938201 q^{76} +17.7611 q^{77} -22.7212 q^{78} -4.65989 q^{79} -3.51543 q^{80} +34.9540 q^{81} +5.53848 q^{82} -16.6736 q^{83} +10.7311 q^{84} +18.0249 q^{85} -10.3294 q^{86} +26.1956 q^{87} +5.56306 q^{88} +13.0346 q^{89} +29.1687 q^{90} +21.5824 q^{91} -3.14603 q^{92} +8.10373 q^{93} +5.75395 q^{94} -3.29818 q^{95} +3.36115 q^{96} +17.1790 q^{97} -3.19326 q^{98} -46.1587 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9}+O(q^{10}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9	\( 84 q - 84 q^{2} - 19 q^{3} + 84 q^{4} - 32 q^{5} + 19 q^{6} + q^{7} - 84 q^{8} + 77 q^{9} + 32 q^{10} - 6 q^{11} - 19 q^{12} - 29 q^{13} - q^{14} - 4 q^{15} + 84 q^{16} - 36 q^{17} - 77 q^{18} + 33 q^{19} - 32 q^{20} - 21 q^{21} + 6 q^{22} - 62 q^{23} + 19 q^{24} + 82 q^{25} + 29 q^{26} - 82 q^{27} + q^{28} - 51 q^{29} + 4 q^{30} + 39 q^{31} - 84 q^{32} - 32 q^{33} + 36 q^{34} - 34 q^{35} + 77 q^{36} - 32 q^{37} - 33 q^{38} + 29 q^{39} + 32 q^{40} - 38 q^{41} + 21 q^{42} - 6 q^{44} - 91 q^{45} + 62 q^{46} - 58 q^{47} - 19 q^{48} + 83 q^{49} - 82 q^{50} - q^{51} - 29 q^{52} - 106 q^{53} + 82 q^{54} + 32 q^{55} - q^{56} - 44 q^{57} + 51 q^{58} - 42 q^{59} - 4 q^{60} - 41 q^{61} - 39 q^{62} - 9 q^{63} + 84 q^{64} - 49 q^{65} + 32 q^{66} - 16 q^{67} - 36 q^{68} - 45 q^{69} + 34 q^{70} - 62 q^{71} - 77 q^{72} - 16 q^{73} + 32 q^{74} - 80 q^{75} + 33 q^{76} - 134 q^{77} - 29 q^{78} + 53 q^{79} - 32 q^{80} + 56 q^{81} + 38 q^{82} - 90 q^{83} - 21 q^{84} - 60 q^{85} - 3 q^{87} + 6 q^{88} - 54 q^{89} + 91 q^{90} + 33 q^{91} - 62 q^{92} - 69 q^{93} + 58 q^{94} - 47 q^{95} + 19 q^{96} - 31 q^{97} - 83 q^{98} + 23 q^{99}+O(q^{100}) \) 84 * q - 84 * q^2 - 19 * q^3 + 84 * q^4 - 32 * q^5 + 19 * q^6 + q^7 - 84 * q^8 + 77 * q^9 + 32 * q^10 - 6 * q^11 - 19 * q^12 - 29 * q^13 - q^14 - 4 * q^15 + 84 * q^16 - 36 * q^17 - 77 * q^18 + 33 * q^19 - 32 * q^20 - 21 * q^21 + 6 * q^22 - 62 * q^23 + 19 * q^24 + 82 * q^25 + 29 * q^26 - 82 * q^27 + q^28 - 51 * q^29 + 4 * q^30 + 39 * q^31 - 84 * q^32 - 32 * q^33 + 36 * q^34 - 34 * q^35 + 77 * q^36 - 32 * q^37 - 33 * q^38 + 29 * q^39 + 32 * q^40 - 38 * q^41 + 21 * q^42 - 6 * q^44 - 91 * q^45 + 62 * q^46 - 58 * q^47 - 19 * q^48 + 83 * q^49 - 82 * q^50 - q^51 - 29 * q^52 - 106 * q^53 + 82 * q^54 + 32 * q^55 - q^56 - 44 * q^57 + 51 * q^58 - 42 * q^59 - 4 * q^60 - 41 * q^61 - 39 * q^62 - 9 * q^63 + 84 * q^64 - 49 * q^65 + 32 * q^66 - 16 * q^67 - 36 * q^68 - 45 * q^69 + 34 * q^70 - 62 * q^71 - 77 * q^72 - 16 * q^73 + 32 * q^74 - 80 * q^75 + 33 * q^76 - 134 * q^77 - 29 * q^78 + 53 * q^79 - 32 * q^80 + 56 * q^81 + 38 * q^82 - 90 * q^83 - 21 * q^84 - 60 * q^85 - 3 * q^87 + 6 * q^88 - 54 * q^89 + 91 * q^90 + 33 * q^91 - 62 * q^92 - 69 * q^93 + 58 * q^94 - 47 * q^95 + 19 * q^96 - 31 * q^97 - 83 * q^98 + 23 * q^99

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