LMFDB - Embedded newform 538.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,2,Mod(1,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, 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("538.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	538.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:	$4.29595162874$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 $ x^7 - x^6 - 15x^5 + 16x^4 + 49x^3 - 53x^2 - 44*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.22542$ of defining polynomial
Character	$\chi$	$=$	538.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.383827 q^{3} +1.00000 q^{4} +4.15424 q^{5} -0.383827 q^{6} +4.87818 q^{7} +1.00000 q^{8} -2.85268 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.383827 q^{3} +1.00000 q^{4} +4.15424 q^{5} -0.383827 q^{6} +4.87818 q^{7} +1.00000 q^{8} -2.85268 q^{9} +4.15424 q^{10} -1.62910 q^{11} -0.383827 q^{12} -2.16682 q^{13} +4.87818 q^{14} -1.59451 q^{15} +1.00000 q^{16} -6.78448 q^{17} -2.85268 q^{18} -8.61766 q^{19} +4.15424 q^{20} -1.87238 q^{21} -1.62910 q^{22} +2.76765 q^{23} -0.383827 q^{24} +12.2577 q^{25} -2.16682 q^{26} +2.24641 q^{27} +4.87818 q^{28} +2.10510 q^{29} -1.59451 q^{30} -3.62414 q^{31} +1.00000 q^{32} +0.625290 q^{33} -6.78448 q^{34} +20.2651 q^{35} -2.85268 q^{36} +1.19760 q^{37} -8.61766 q^{38} +0.831683 q^{39} +4.15424 q^{40} +5.43715 q^{41} -1.87238 q^{42} -9.63402 q^{43} -1.62910 q^{44} -11.8507 q^{45} +2.76765 q^{46} -7.07613 q^{47} -0.383827 q^{48} +16.7967 q^{49} +12.2577 q^{50} +2.60406 q^{51} -2.16682 q^{52} -3.03890 q^{53} +2.24641 q^{54} -6.76765 q^{55} +4.87818 q^{56} +3.30769 q^{57} +2.10510 q^{58} +4.21827 q^{59} -1.59451 q^{60} +8.74744 q^{61} -3.62414 q^{62} -13.9159 q^{63} +1.00000 q^{64} -9.00148 q^{65} +0.625290 q^{66} -1.50761 q^{67} -6.78448 q^{68} -1.06230 q^{69} +20.2651 q^{70} +11.0270 q^{71} -2.85268 q^{72} +1.37039 q^{73} +1.19760 q^{74} -4.70482 q^{75} -8.61766 q^{76} -7.94703 q^{77} +0.831683 q^{78} -14.9749 q^{79} +4.15424 q^{80} +7.69580 q^{81} +5.43715 q^{82} -8.08276 q^{83} -1.87238 q^{84} -28.1843 q^{85} -9.63402 q^{86} -0.807993 q^{87} -1.62910 q^{88} +3.31299 q^{89} -11.8507 q^{90} -10.5701 q^{91} +2.76765 q^{92} +1.39104 q^{93} -7.07613 q^{94} -35.7998 q^{95} -0.383827 q^{96} +16.8474 q^{97} +16.7967 q^{98} +4.64729 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9}+O(q^{10}) $ 7 * q + 7 * q^2 + q^3 + 7 * q^4 + 7 * q^5 + q^6 + 6 * q^7 + 7 * q^8 + 12 * q^9	\( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9} + 7 q^{10} - 3 q^{11} + q^{12} - 9 q^{13} + 6 q^{14} + 8 q^{15} + 7 q^{16} + 8 q^{17} + 12 q^{18} - 11 q^{19} + 7 q^{20} - 6 q^{21} - 3 q^{22} + 12 q^{23} + q^{24} + 22 q^{25} - 9 q^{26} - 14 q^{27} + 6 q^{28} - 5 q^{29} + 8 q^{30} + 14 q^{31} + 7 q^{32} - 4 q^{33} + 8 q^{34} - 4 q^{35} + 12 q^{36} + 13 q^{37} - 11 q^{38} - 18 q^{39} + 7 q^{40} + 12 q^{41} - 6 q^{42} - 11 q^{43} - 3 q^{44} + 3 q^{45} + 12 q^{46} + 2 q^{47} + q^{48} + 15 q^{49} + 22 q^{50} - 26 q^{51} - 9 q^{52} + 19 q^{53} - 14 q^{54} - 40 q^{55} + 6 q^{56} - 12 q^{57} - 5 q^{58} - 9 q^{59} + 8 q^{60} - 3 q^{61} + 14 q^{62} - 26 q^{63} + 7 q^{64} - 10 q^{65} - 4 q^{66} - 33 q^{67} + 8 q^{68} - 64 q^{69} - 4 q^{70} + 28 q^{71} + 12 q^{72} - 14 q^{73} + 13 q^{74} - 45 q^{75} - 11 q^{76} + 10 q^{77} - 18 q^{78} + 2 q^{79} + 7 q^{80} + 15 q^{81} + 12 q^{82} - 7 q^{83} - 6 q^{84} - 16 q^{85} - 11 q^{86} + 16 q^{87} - 3 q^{88} + 18 q^{89} + 3 q^{90} - 26 q^{91} + 12 q^{92} + 6 q^{93} + 2 q^{94} - 34 q^{95} + q^{96} - 4 q^{97} + 15 q^{98} - 37 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 + q^3 + 7 * q^4 + 7 * q^5 + q^6 + 6 * q^7 + 7 * q^8 + 12 * q^9 + 7 * q^10 - 3 * q^11 + q^12 - 9 * q^13 + 6 * q^14 + 8 * q^15 + 7 * q^16 + 8 * q^17 + 12 * q^18 - 11 * q^19 + 7 * q^20 - 6 * q^21 - 3 * q^22 + 12 * q^23 + q^24 + 22 * q^25 - 9 * q^26 - 14 * q^27 + 6 * q^28 - 5 * q^29 + 8 * q^30 + 14 * q^31 + 7 * q^32 - 4 * q^33 + 8 * q^34 - 4 * q^35 + 12 * q^36 + 13 * q^37 - 11 * q^38 - 18 * q^39 + 7 * q^40 + 12 * q^41 - 6 * q^42 - 11 * q^43 - 3 * q^44 + 3 * q^45 + 12 * q^46 + 2 * q^47 + q^48 + 15 * q^49 + 22 * q^50 - 26 * q^51 - 9 * q^52 + 19 * q^53 - 14 * q^54 - 40 * q^55 + 6 * q^56 - 12 * q^57 - 5 * q^58 - 9 * q^59 + 8 * q^60 - 3 * q^61 + 14 * q^62 - 26 * q^63 + 7 * q^64 - 10 * q^65 - 4 * q^66 - 33 * q^67 + 8 * q^68 - 64 * q^69 - 4 * q^70 + 28 * q^71 + 12 * q^72 - 14 * q^73 + 13 * q^74 - 45 * q^75 - 11 * q^76 + 10 * q^77 - 18 * q^78 + 2 * q^79 + 7 * q^80 + 15 * q^81 + 12 * q^82 - 7 * q^83 - 6 * q^84 - 16 * q^85 - 11 * q^86 + 16 * q^87 - 3 * q^88 + 18 * q^89 + 3 * q^90 - 26 * q^91 + 12 * q^92 + 6 * q^93 + 2 * q^94 - 34 * q^95 + q^96 - 4 * q^97 + 15 * q^98 - 37 * 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
$3$	−0.383827	−0.221602	−0.110801	−	0.993843i	$-0.535342\pi$
$3$	−0.383827	−0.221602	−0.110801	+	0.993843i	$0.535342\pi$
$4$	1.00000	0.500000
$5$	4.15424	1.85783	0.928916	−	0.370291i	$-0.120742\pi$
$5$	4.15424	1.85783	0.928916	+	0.370291i	$0.120742\pi$
$6$	−0.383827	−0.156697
$7$	4.87818	1.84378	0.921890	−	0.387452i	$-0.126645\pi$
$7$	4.87818	1.84378	0.921890	+	0.387452i	$0.126645\pi$
$8$	1.00000	0.353553
$9$	−2.85268	−0.950892
$10$	4.15424	1.31369
$11$	−1.62910	−0.491191	−0.245596	−	0.969372i	$-0.578984\pi$
$11$	−1.62910	−0.491191	−0.245596	+	0.969372i	$0.578984\pi$
$12$	−0.383827	−0.110801
$13$	−2.16682	−0.600968	−0.300484	−	0.953787i	$-0.597148\pi$
$13$	−2.16682	−0.600968	−0.300484	+	0.953787i	$0.597148\pi$
$14$	4.87818	1.30375
$15$	−1.59451	−0.411700
$16$	1.00000	0.250000
$17$	−6.78448	−1.64548	−0.822739	−	0.568420i	$-0.807555\pi$
$17$	−6.78448	−1.64548	−0.822739	+	0.568420i	$0.807555\pi$
$18$	−2.85268	−0.672382
$19$	−8.61766	−1.97703	−0.988513	−	0.151136i	$-0.951707\pi$
$19$	−8.61766	−1.97703	−0.988513	+	0.151136i	$0.951707\pi$
$20$	4.15424	0.928916
$21$	−1.87238	−0.408586
$22$	−1.62910	−0.347325
$23$	2.76765	0.577096	0.288548	−	0.957465i	$-0.406828\pi$
$23$	2.76765	0.577096	0.288548	+	0.957465i	$0.406828\pi$
$24$	−0.383827	−0.0783483
$25$	12.2577	2.45154
$26$	−2.16682	−0.424948
$27$	2.24641	0.432322
$28$	4.87818	0.921890
$29$	2.10510	0.390907	0.195454	−	0.980713i	$-0.437382\pi$
$29$	2.10510	0.390907	0.195454	+	0.980713i	$0.437382\pi$
$30$	−1.59451	−0.291116
$31$	−3.62414	−0.650914	−0.325457	−	0.945557i	$-0.605518\pi$
$31$	−3.62414	−0.650914	−0.325457	+	0.945557i	$0.605518\pi$
$32$	1.00000	0.176777
$33$	0.625290	0.108849
$34$	−6.78448	−1.16353
$35$	20.2651	3.42543
$36$	−2.85268	−0.475446
$37$	1.19760	0.196885	0.0984424	−	0.995143i	$-0.468614\pi$
$37$	1.19760	0.196885	0.0984424	+	0.995143i	$0.468614\pi$
$38$	−8.61766	−1.39797
$39$	0.831683	0.133176
$40$	4.15424	0.656843
$41$	5.43715	0.849140	0.424570	−	0.905395i	$-0.360425\pi$
$41$	5.43715	0.849140	0.424570	+	0.905395i	$0.360425\pi$
$42$	−1.87238	−0.288914
$43$	−9.63402	−1.46917	−0.734587	−	0.678514i	$-0.762624\pi$
$43$	−9.63402	−1.46917	−0.734587	+	0.678514i	$0.762624\pi$
$44$	−1.62910	−0.245596
$45$	−11.8507	−1.76660
$46$	2.76765	0.408068
$47$	−7.07613	−1.03216	−0.516080	−	0.856541i	$-0.672609\pi$
$47$	−7.07613	−1.03216	−0.516080	+	0.856541i	$0.672609\pi$
$48$	−0.383827	−0.0554006
$49$	16.7967	2.39952
$50$	12.2577	1.73350
$51$	2.60406	0.364642
$52$	−2.16682	−0.300484
$53$	−3.03890	−0.417425	−0.208712	−	0.977977i	$-0.566927\pi$
$53$	−3.03890	−0.417425	−0.208712	+	0.977977i	$0.566927\pi$
$54$	2.24641	0.305698
$55$	−6.76765	−0.912550
$56$	4.87818	0.651875
$57$	3.30769	0.438114
$58$	2.10510	0.276413
$59$	4.21827	0.549172	0.274586	−	0.961563i	$-0.411459\pi$
$59$	4.21827	0.549172	0.274586	+	0.961563i	$0.411459\pi$
$60$	−1.59451	−0.205850
$61$	8.74744	1.11999	0.559997	−	0.828494i	$-0.310802\pi$
$61$	8.74744	1.11999	0.559997	+	0.828494i	$0.310802\pi$
$62$	−3.62414	−0.460266
$63$	−13.9159	−1.75324
$64$	1.00000	0.125000
$65$	−9.00148	−1.11650
$66$	0.625290	0.0769679
$67$	−1.50761	−0.184184	−0.0920921	−	0.995750i	$-0.529355\pi$
$67$	−1.50761	−0.184184	−0.0920921	+	0.995750i	$0.529355\pi$
$68$	−6.78448	−0.822739
$69$	−1.06230	−0.127886
$70$	20.2651	2.42215
$71$	11.0270	1.30866	0.654331	−	0.756208i	$-0.272950\pi$
$71$	11.0270	1.30866	0.654331	+	0.756208i	$0.272950\pi$
$72$	−2.85268	−0.336191
$73$	1.37039	0.160391	0.0801957	−	0.996779i	$-0.474445\pi$
$73$	1.37039	0.160391	0.0801957	+	0.996779i	$0.474445\pi$
$74$	1.19760	0.139219
$75$	−4.70482	−0.543266
$76$	−8.61766	−0.988513
$77$	−7.94703	−0.905648
$78$	0.831683	0.0941695
$79$	−14.9749	−1.68480	−0.842401	−	0.538850i	$-0.818859\pi$
$79$	−14.9749	−1.68480	−0.842401	+	0.538850i	$0.818859\pi$
$80$	4.15424	0.464458
$81$	7.69580	0.855089
$82$	5.43715	0.600432
$83$	−8.08276	−0.887198	−0.443599	−	0.896225i	$-0.646299\pi$
$83$	−8.08276	−0.887198	−0.443599	+	0.896225i	$0.646299\pi$
$84$	−1.87238	−0.204293
$85$	−28.1843	−3.05702
$86$	−9.63402	−1.03886
$87$	−0.807993	−0.0866259
$88$	−1.62910	−0.173662
$89$	3.31299	0.351176	0.175588	−	0.984464i	$-0.443817\pi$
$89$	3.31299	0.351176	0.175588	+	0.984464i	$0.443817\pi$
$90$	−11.8507	−1.24917
$91$	−10.5701	−1.10805
$92$	2.76765	0.288548
$93$	1.39104	0.144244
$94$	−7.07613	−0.729847
$95$	−35.7998	−3.67298
$96$	−0.383827	−0.0391741
$97$	16.8474	1.71059	0.855296	−	0.518139i	$-0.173375\pi$
$97$	16.8474	1.71059	0.855296	+	0.518139i	$0.173375\pi$
$98$	16.7967	1.69672
$99$	4.64729	0.467070
$100$	12.2577	1.22577
$101$	2.93034	0.291580	0.145790	−	0.989316i	$-0.453428\pi$
$101$	2.93034	0.291580	0.145790	+	0.989316i	$0.453428\pi$
$102$	2.60406	0.257841
$103$	1.25819	0.123973	0.0619867	−	0.998077i	$-0.480256\pi$
$103$	1.25819	0.123973	0.0619867	+	0.998077i	$0.480256\pi$
$104$	−2.16682	−0.212474
$105$	−7.77829	−0.759084
$106$	−3.03890	−0.295164
$107$	9.35946	0.904813	0.452407	−	0.891812i	$-0.350566\pi$
$107$	9.35946	0.904813	0.452407	+	0.891812i	$0.350566\pi$
$108$	2.24641	0.216161
$109$	11.4192	1.09376	0.546882	−	0.837210i	$-0.315815\pi$
$109$	11.4192	1.09376	0.546882	+	0.837210i	$0.315815\pi$
$110$	−6.76765	−0.645270
$111$	−0.459672	−0.0436301
$112$	4.87818	0.460945
$113$	0.316816	0.0298036	0.0149018	−	0.999889i	$-0.495256\pi$
$113$	0.316816	0.0298036	0.0149018	+	0.999889i	$0.495256\pi$
$114$	3.30769	0.309793
$115$	11.4975	1.07215
$116$	2.10510	0.195454
$117$	6.18124	0.571456
$118$	4.21827	0.388323
$119$	−33.0959	−3.03390
$120$	−1.59451	−0.145558
$121$	−8.34604	−0.758731
$122$	8.74744	0.791956
$123$	−2.08692	−0.188171
$124$	−3.62414	−0.325457
$125$	30.1501	2.69671
$126$	−13.9159	−1.23973
$127$	−2.08772	−0.185255	−0.0926276	−	0.995701i	$-0.529527\pi$
$127$	−2.08772	−0.185255	−0.0926276	+	0.995701i	$0.529527\pi$
$128$	1.00000	0.0883883
$129$	3.69779	0.325572
$130$	−9.00148	−0.789482
$131$	−5.17397	−0.452052	−0.226026	−	0.974121i	$-0.572573\pi$
$131$	−5.17397	−0.452052	−0.226026	+	0.974121i	$0.572573\pi$
$132$	0.625290	0.0544245
$133$	−42.0385	−3.64520
$134$	−1.50761	−0.130238
$135$	9.33213	0.803182
$136$	−6.78448	−0.581764
$137$	6.35949	0.543328	0.271664	−	0.962392i	$-0.412426\pi$
$137$	6.35949	0.543328	0.271664	+	0.962392i	$0.412426\pi$
$138$	−1.06230	−0.0904289
$139$	−10.6786	−0.905746	−0.452873	−	0.891575i	$-0.649601\pi$
$139$	−10.6786	−0.905746	−0.452873	+	0.891575i	$0.649601\pi$
$140$	20.2651	1.71272
$141$	2.71601	0.228729
$142$	11.0270	0.925364
$143$	3.52996	0.295190
$144$	−2.85268	−0.237723
$145$	8.74508	0.726239
$146$	1.37039	0.113414
$147$	−6.44701	−0.531740
$148$	1.19760	0.0984424
$149$	−3.47426	−0.284622	−0.142311	−	0.989822i	$-0.545453\pi$
$149$	−3.47426	−0.284622	−0.142311	+	0.989822i	$0.545453\pi$
$150$	−4.70482	−0.384147
$151$	11.2407	0.914752	0.457376	−	0.889273i	$-0.348789\pi$
$151$	11.2407	0.914752	0.457376	+	0.889273i	$0.348789\pi$
$152$	−8.61766	−0.698984
$153$	19.3539	1.56467
$154$	−7.94703	−0.640390
$155$	−15.0555	−1.20929
$156$	0.831683	0.0665879
$157$	8.59533	0.685982	0.342991	−	0.939339i	$-0.388560\pi$
$157$	8.59533	0.685982	0.342991	+	0.939339i	$0.388560\pi$
$158$	−14.9749	−1.19134
$159$	1.16641	0.0925024
$160$	4.15424	0.328421
$161$	13.5011	1.06404
$162$	7.69580	0.604639
$163$	10.1530	0.795241	0.397621	−	0.917550i	$-0.369836\pi$
$163$	10.1530	0.795241	0.397621	+	0.917550i	$0.369836\pi$
$164$	5.43715	0.424570
$165$	2.59760	0.202223
$166$	−8.08276	−0.627344
$167$	4.37572	0.338604	0.169302	−	0.985564i	$-0.445849\pi$
$167$	4.37572	0.338604	0.169302	+	0.985564i	$0.445849\pi$
$168$	−1.87238	−0.144457
$169$	−8.30489	−0.638838
$170$	−28.1843	−2.16164
$171$	24.5834	1.87994
$172$	−9.63402	−0.734587
$173$	−22.7651	−1.73080	−0.865398	−	0.501085i	$-0.832934\pi$
$173$	−22.7651	−1.73080	−0.865398	+	0.501085i	$0.832934\pi$
$174$	−0.807993	−0.0612538
$175$	59.7952	4.52009
$176$	−1.62910	−0.122798
$177$	−1.61908	−0.121698
$178$	3.31299	0.248319
$179$	12.1634	0.909139	0.454569	−	0.890711i	$-0.349793\pi$
$179$	12.1634	0.909139	0.454569	+	0.890711i	$0.349793\pi$
$180$	−11.8507	−0.883299
$181$	−4.61048	−0.342694	−0.171347	−	0.985211i	$-0.554812\pi$
$181$	−4.61048	−0.342694	−0.171347	+	0.985211i	$0.554812\pi$
$182$	−10.5701	−0.783511
$183$	−3.35750	−0.248193
$184$	2.76765	0.204034
$185$	4.97513	0.365779
$186$	1.39104	0.101996
$187$	11.0526	0.808244
$188$	−7.07613	−0.516080
$189$	10.9584	0.797107
$190$	−35.7998	−2.59719
$191$	3.30553	0.239180	0.119590	−	0.992823i	$-0.461842\pi$
$191$	3.30553	0.239180	0.119590	+	0.992823i	$0.461842\pi$
$192$	−0.383827	−0.0277003
$193$	−18.5157	−1.33279	−0.666394	−	0.745600i	$-0.732163\pi$
$193$	−18.5157	−1.33279	−0.666394	+	0.745600i	$0.732163\pi$
$194$	16.8474	1.20957
$195$	3.45501	0.247418
$196$	16.7967	1.19976
$197$	8.78109	0.625627	0.312814	−	0.949815i	$-0.398729\pi$
$197$	8.78109	0.625627	0.312814	+	0.949815i	$0.398729\pi$
$198$	4.64729	0.330268
$199$	−12.4370	−0.881634	−0.440817	−	0.897597i	$-0.645311\pi$
$199$	−12.4370	−0.881634	−0.440817	+	0.897597i	$0.645311\pi$
$200$	12.2577	0.866749
$201$	0.578661	0.0408156
$202$	2.93034	0.206178
$203$	10.2691	0.720746
$204$	2.60406	0.182321
$205$	22.5872	1.57756
$206$	1.25819	0.0876625
$207$	−7.89522	−0.548756
$208$	−2.16682	−0.150242
$209$	14.0390	0.971098
$210$	−7.77829	−0.536753
$211$	−10.9290	−0.752381	−0.376191	−	0.926542i	$-0.622766\pi$
$211$	−10.9290	−0.752381	−0.376191	+	0.926542i	$0.622766\pi$
$212$	−3.03890	−0.208712
$213$	−4.23245	−0.290003
$214$	9.35946	0.639800
$215$	−40.0220	−2.72948
$216$	2.24641	0.152849
$217$	−17.6792	−1.20014
$218$	11.4192	0.773408
$219$	−0.525990	−0.0355431
$220$	−6.76765	−0.456275
$221$	14.7007	0.988879
$222$	−0.459672	−0.0308512
$223$	8.79372	0.588871	0.294436	−	0.955671i	$-0.404868\pi$
$223$	8.79372	0.588871	0.294436	+	0.955671i	$0.404868\pi$
$224$	4.87818	0.325937
$225$	−34.9672	−2.33115
$226$	0.316816	0.0210743
$227$	6.74871	0.447928	0.223964	−	0.974597i	$-0.428100\pi$
$227$	6.74871	0.447928	0.223964	+	0.974597i	$0.428100\pi$
$228$	3.30769	0.219057
$229$	−23.2606	−1.53711	−0.768553	−	0.639787i	$-0.779023\pi$
$229$	−23.2606	−1.53711	−0.768553	+	0.639787i	$0.779023\pi$
$230$	11.4975	0.758122
$231$	3.05028	0.200694
$232$	2.10510	0.138206
$233$	11.0721	0.725357	0.362679	−	0.931914i	$-0.381862\pi$
$233$	11.0721	0.725357	0.362679	+	0.931914i	$0.381862\pi$
$234$	6.18124	0.404080
$235$	−29.3959	−1.91758
$236$	4.21827	0.274586
$237$	5.74775	0.373356
$238$	−33.0959	−2.14529
$239$	−3.99072	−0.258138	−0.129069	−	0.991636i	$-0.541199\pi$
$239$	−3.99072	−0.258138	−0.129069	+	0.991636i	$0.541199\pi$
$240$	−1.59451	−0.102925
$241$	−16.5408	−1.06549	−0.532745	−	0.846276i	$-0.678839\pi$
$241$	−16.5408	−1.06549	−0.532745	+	0.846276i	$0.678839\pi$
$242$	−8.34604	−0.536504
$243$	−9.69309	−0.621812
$244$	8.74744	0.559997
$245$	69.7773	4.45791
$246$	−2.08692	−0.133057
$247$	18.6729	1.18813
$248$	−3.62414	−0.230133
$249$	3.10238	0.196605
$250$	30.1501	1.90686
$251$	24.0533	1.51823	0.759117	−	0.650955i	$-0.225631\pi$
$251$	24.0533	1.51823	0.759117	+	0.650955i	$0.225631\pi$
$252$	−13.9159	−0.876618
$253$	−4.50877	−0.283464
$254$	−2.08772	−0.130995
$255$	10.8179	0.677443
$256$	1.00000	0.0625000
$257$	−13.6822	−0.853470	−0.426735	−	0.904377i	$-0.640336\pi$
$257$	−13.6822	−0.853470	−0.426735	+	0.904377i	$0.640336\pi$
$258$	3.69779	0.230214
$259$	5.84213	0.363012
$260$	−9.00148	−0.558248
$261$	−6.00517	−0.371711
$262$	−5.17397	−0.319649
$263$	11.3942	0.702596	0.351298	−	0.936264i	$-0.385740\pi$
$263$	11.3942	0.702596	0.351298	+	0.936264i	$0.385740\pi$
$264$	0.625290	0.0384840
$265$	−12.6243	−0.775505
$266$	−42.0385	−2.57755
$267$	−1.27161	−0.0778214
$268$	−1.50761	−0.0920921
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	9.33213	0.567935
$271$	−24.6167	−1.49536	−0.747679	−	0.664061i	$-0.768832\pi$
$271$	−24.6167	−1.49536	−0.747679	+	0.664061i	$0.768832\pi$
$272$	−6.78448	−0.411369
$273$	4.05710	0.245547
$274$	6.35949	0.384191
$275$	−19.9690	−1.20417
$276$	−1.06230	−0.0639429
$277$	4.70762	0.282853	0.141427	−	0.989949i	$-0.454831\pi$
$277$	4.70762	0.282853	0.141427	+	0.989949i	$0.454831\pi$
$278$	−10.6786	−0.640459
$279$	10.3385	0.618949
$280$	20.2651	1.21107
$281$	6.51540	0.388676	0.194338	−	0.980935i	$-0.437744\pi$
$281$	6.51540	0.388676	0.194338	+	0.980935i	$0.437744\pi$
$282$	2.71601	0.161736
$283$	−8.67417	−0.515626	−0.257813	−	0.966195i	$-0.583002\pi$
$283$	−8.67417	−0.515626	−0.257813	+	0.966195i	$0.583002\pi$
$284$	11.0270	0.654331
$285$	13.7409	0.813941
$286$	3.52996	0.208731
$287$	26.5234	1.56563
$288$	−2.85268	−0.168096
$289$	29.0291	1.70760
$290$	8.74508	0.513529
$291$	−6.46647	−0.379071
$292$	1.37039	0.0801957
$293$	20.3798	1.19060	0.595300	−	0.803504i	$-0.297033\pi$
$293$	20.3798	1.19060	0.595300	+	0.803504i	$0.297033\pi$
$294$	−6.44701	−0.375997
$295$	17.5237	1.02027
$296$	1.19760	0.0696093
$297$	−3.65962	−0.212353
$298$	−3.47426	−0.201258
$299$	−5.99701	−0.346816
$300$	−4.70482	−0.271633
$301$	−46.9965	−2.70883
$302$	11.2407	0.646827
$303$	−1.12474	−0.0646147
$304$	−8.61766	−0.494256
$305$	36.3389	2.08076
$306$	19.3539	1.10639
$307$	−17.5028	−0.998939	−0.499470	−	0.866331i	$-0.666472\pi$
$307$	−17.5028	−0.998939	−0.499470	+	0.866331i	$0.666472\pi$
$308$	−7.94703	−0.452824
$309$	−0.482928	−0.0274728
$310$	−15.0555	−0.855096
$311$	22.4715	1.27424	0.637120	−	0.770764i	$-0.280125\pi$
$311$	22.4715	1.27424	0.637120	+	0.770764i	$0.280125\pi$
$312$	0.831683	0.0470848
$313$	−32.6321	−1.84447	−0.922236	−	0.386626i	$-0.873640\pi$
$313$	−32.6321	−1.84447	−0.922236	+	0.386626i	$0.873640\pi$
$314$	8.59533	0.485063
$315$	−57.8099	−3.25722
$316$	−14.9749	−0.842401
$317$	−19.5532	−1.09822	−0.549110	−	0.835750i	$-0.685033\pi$
$317$	−19.5532	−1.09822	−0.549110	+	0.835750i	$0.685033\pi$
$318$	1.16641	0.0654090
$319$	−3.42941	−0.192010
$320$	4.15424	0.232229
$321$	−3.59241	−0.200509
$322$	13.5011	0.752388
$323$	58.4663	3.25315
$324$	7.69580	0.427544
$325$	−26.5602	−1.47329
$326$	10.1530	0.562321
$327$	−4.38300	−0.242381
$328$	5.43715	0.300216
$329$	−34.5186	−1.90307
$330$	2.59760	0.142993
$331$	−20.1680	−1.10853	−0.554266	−	0.832340i	$-0.687001\pi$
$331$	−20.1680	−1.10853	−0.554266	+	0.832340i	$0.687001\pi$
$332$	−8.08276	−0.443599
$333$	−3.41638	−0.187216
$334$	4.37572	0.239429
$335$	−6.26298	−0.342183
$336$	−1.87238	−0.102146
$337$	29.1409	1.58741	0.793704	−	0.608305i	$-0.208150\pi$
$337$	29.1409	1.58741	0.793704	+	0.608305i	$0.208150\pi$
$338$	−8.30489	−0.451727
$339$	−0.121603	−0.00660454
$340$	−28.1843	−1.52851
$341$	5.90407	0.319723
$342$	24.5834	1.32932
$343$	47.7899	2.58041
$344$	−9.63402	−0.519432
$345$	−4.41304	−0.237590
$346$	−22.7651	−1.22386
$347$	12.2141	0.655686	0.327843	−	0.944732i	$-0.393678\pi$
$347$	12.2141	0.655686	0.327843	+	0.944732i	$0.393678\pi$
$348$	−0.807993	−0.0433130
$349$	15.3346	0.820843	0.410421	−	0.911896i	$-0.365382\pi$
$349$	15.3346	0.820843	0.410421	+	0.911896i	$0.365382\pi$
$350$	59.7952	3.19619
$351$	−4.86757	−0.259812
$352$	−1.62910	−0.0868311
$353$	30.6859	1.63324	0.816622	−	0.577173i	$-0.195844\pi$
$353$	30.6859	1.63324	0.816622	+	0.577173i	$0.195844\pi$
$354$	−1.61908	−0.0860533
$355$	45.8087	2.43127
$356$	3.31299	0.175588
$357$	12.7031	0.672319
$358$	12.1634	0.642858
$359$	0.344824	0.0181991	0.00909956	−	0.999959i	$-0.497103\pi$
$359$	0.344824	0.0181991	0.00909956	+	0.999959i	$0.497103\pi$
$360$	−11.8507	−0.624587
$361$	55.2640	2.90863
$362$	−4.61048	−0.242322
$363$	3.20343	0.168137
$364$	−10.5701	−0.554026
$365$	5.69291	0.297980
$366$	−3.35750	−0.175499
$367$	22.3072	1.16443	0.582214	−	0.813035i	$-0.302186\pi$
$367$	22.3072	1.16443	0.582214	+	0.813035i	$0.302186\pi$
$368$	2.76765	0.144274
$369$	−15.5104	−0.807440
$370$	4.97513	0.258645
$371$	−14.8243	−0.769640
$372$	1.39104	0.0721220
$373$	−5.92843	−0.306962	−0.153481	−	0.988152i	$-0.549048\pi$
$373$	−5.92843	−0.306962	−0.153481	+	0.988152i	$0.549048\pi$
$374$	11.0526	0.571515
$375$	−11.5724	−0.597597
$376$	−7.07613	−0.364923
$377$	−4.56137	−0.234922
$378$	10.9584	0.563640
$379$	−2.01119	−0.103308	−0.0516540	−	0.998665i	$-0.516449\pi$
$379$	−2.01119	−0.103308	−0.0516540	+	0.998665i	$0.516449\pi$
$380$	−35.7998	−1.83649
$381$	0.801322	0.0410530
$382$	3.30553	0.169126
$383$	4.21696	0.215477	0.107738	−	0.994179i	$-0.465639\pi$
$383$	4.21696	0.215477	0.107738	+	0.994179i	$0.465639\pi$
$384$	−0.383827	−0.0195871
$385$	−33.0138	−1.68254
$386$	−18.5157	−0.942423
$387$	27.4827	1.39703
$388$	16.8474	0.855296
$389$	4.54939	0.230663	0.115332	−	0.993327i	$-0.463207\pi$
$389$	4.54939	0.230663	0.115332	+	0.993327i	$0.463207\pi$
$390$	3.45501	0.174951
$391$	−18.7771	−0.949598
$392$	16.7967	0.848360
$393$	1.98591	0.100176
$394$	8.78109	0.442385
$395$	−62.2091	−3.13008
$396$	4.64729	0.233535
$397$	−15.2162	−0.763677	−0.381839	−	0.924229i	$-0.624709\pi$
$397$	−15.2162	−0.763677	−0.381839	+	0.924229i	$0.624709\pi$
$398$	−12.4370	−0.623409
$399$	16.1355	0.807785
$400$	12.2577	0.612884
$401$	18.4280	0.920250	0.460125	−	0.887854i	$-0.347805\pi$
$401$	18.4280	0.920250	0.460125	+	0.887854i	$0.347805\pi$
$402$	0.578661	0.0288610
$403$	7.85285	0.391178
$404$	2.93034	0.145790
$405$	31.9702	1.58861
$406$	10.2691	0.509645
$407$	−1.95101	−0.0967081
$408$	2.60406	0.128920
$409$	36.3293	1.79637	0.898184	−	0.439619i	$-0.144887\pi$
$409$	36.3293	1.79637	0.898184	+	0.439619i	$0.144887\pi$
$410$	22.5872	1.11550
$411$	−2.44094	−0.120403
$412$	1.25819	0.0619867
$413$	20.5775	1.01255
$414$	−7.89522	−0.388029
$415$	−33.5777	−1.64826
$416$	−2.16682	−0.106237
$417$	4.09872	0.200715
$418$	14.0390	0.686670
$419$	−6.63242	−0.324015	−0.162007	−	0.986790i	$-0.551797\pi$
$419$	−6.63242	−0.324015	−0.162007	+	0.986790i	$0.551797\pi$
$420$	−7.77829	−0.379542
$421$	−33.1090	−1.61363	−0.806817	−	0.590801i	$-0.798812\pi$
$421$	−33.1090	−1.61363	−0.806817	+	0.590801i	$0.798812\pi$
$422$	−10.9290	−0.532014
$423$	20.1859	0.981472
$424$	−3.03890	−0.147582
$425$	−83.1620	−4.03395
$426$	−4.23245	−0.205063
$427$	42.6716	2.06502
$428$	9.35946	0.452407
$429$	−1.35489	−0.0654148
$430$	−40.0220	−1.93003
$431$	−16.6498	−0.801995	−0.400997	−	0.916079i	$-0.631336\pi$
$431$	−16.6498	−0.801995	−0.400997	+	0.916079i	$0.631336\pi$
$432$	2.24641	0.108081
$433$	5.36131	0.257648	0.128824	−	0.991667i	$-0.458880\pi$
$433$	5.36131	0.257648	0.128824	+	0.991667i	$0.458880\pi$
$434$	−17.6792	−0.848629
$435$	−3.35659	−0.160936
$436$	11.4192	0.546882
$437$	−23.8507	−1.14093
$438$	−0.525990	−0.0251328
$439$	−5.42961	−0.259141	−0.129571	−	0.991570i	$-0.541360\pi$
$439$	−5.42961	−0.259141	−0.129571	+	0.991570i	$0.541360\pi$
$440$	−6.76765	−0.322635
$441$	−47.9155	−2.28169
$442$	14.7007	0.699243
$443$	3.66946	0.174341	0.0871706	−	0.996193i	$-0.472217\pi$
$443$	3.66946	0.174341	0.0871706	+	0.996193i	$0.472217\pi$
$444$	−0.459672	−0.0218151
$445$	13.7629	0.652426
$446$	8.79372	0.416395
$447$	1.33351	0.0630729
$448$	4.87818	0.230472
$449$	6.43513	0.303693	0.151846	−	0.988404i	$-0.451478\pi$
$449$	6.43513	0.303693	0.151846	+	0.988404i	$0.451478\pi$
$450$	−34.9672	−1.64837
$451$	−8.85764	−0.417090
$452$	0.316816	0.0149018
$453$	−4.31446	−0.202711
$454$	6.74871	0.316733
$455$	−43.9109	−2.05857
$456$	3.30769	0.154897
$457$	2.97606	0.139214	0.0696071	−	0.997574i	$-0.477825\pi$
$457$	2.97606	0.139214	0.0696071	+	0.997574i	$0.477825\pi$
$458$	−23.2606	−1.08690
$459$	−15.2407	−0.711377
$460$	11.4975	0.536073
$461$	9.73958	0.453617	0.226809	−	0.973939i	$-0.427171\pi$
$461$	9.73958	0.453617	0.226809	+	0.973939i	$0.427171\pi$
$462$	3.05028	0.141912
$463$	−19.9810	−0.928598	−0.464299	−	0.885679i	$-0.653694\pi$
$463$	−19.9810	−0.928598	−0.464299	+	0.885679i	$0.653694\pi$
$464$	2.10510	0.0977268
$465$	5.77871	0.267981
$466$	11.0721	0.512905
$467$	−14.3949	−0.666115	−0.333057	−	0.942907i	$-0.608080\pi$
$467$	−14.3949	−0.666115	−0.333057	+	0.942907i	$0.608080\pi$
$468$	6.18124	0.285728
$469$	−7.35441	−0.339595
$470$	−29.3959	−1.35593
$471$	−3.29912	−0.152015
$472$	4.21827	0.194161
$473$	15.6947	0.721645
$474$	5.74775	0.264003
$475$	−105.633	−4.84675
$476$	−33.0959	−1.51695
$477$	8.66900	0.396926
$478$	−3.99072	−0.182531
$479$	−22.4679	−1.02659	−0.513293	−	0.858213i	$-0.671575\pi$
$479$	−22.4679	−1.02659	−0.513293	+	0.858213i	$0.671575\pi$
$480$	−1.59451	−0.0727789
$481$	−2.59499	−0.118321
$482$	−16.5408	−0.753415
$483$	−5.18209	−0.235793
$484$	−8.34604	−0.379366
$485$	69.9880	3.17799
$486$	−9.69309	−0.439687
$487$	−15.3289	−0.694618	−0.347309	−	0.937751i	$-0.612904\pi$
$487$	−15.3289	−0.694618	−0.347309	+	0.937751i	$0.612904\pi$
$488$	8.74744	0.395978
$489$	−3.89698	−0.176227
$490$	69.7773	3.15222
$491$	−23.8945	−1.07834	−0.539172	−	0.842196i	$-0.681263\pi$
$491$	−23.8945	−1.07834	−0.539172	+	0.842196i	$0.681263\pi$
$492$	−2.08692	−0.0940857
$493$	−14.2820	−0.643229
$494$	18.6729	0.840134
$495$	19.3059	0.867737
$496$	−3.62414	−0.162729
$497$	53.7917	2.41289
$498$	3.10238	0.139021
$499$	−15.7627	−0.705635	−0.352818	−	0.935692i	$-0.614776\pi$
$499$	−15.7627	−0.705635	−0.352818	+	0.935692i	$0.614776\pi$
$500$	30.1501	1.34836
$501$	−1.67952	−0.0750353
$502$	24.0533	1.07355
$503$	0.635388	0.0283305	0.0141653	−	0.999900i	$-0.495491\pi$
$503$	0.635388	0.0283305	0.0141653	+	0.999900i	$0.495491\pi$
$504$	−13.9159	−0.619863
$505$	12.1733	0.541706
$506$	−4.50877	−0.200439
$507$	3.18764	0.141568
$508$	−2.08772	−0.0926276
$509$	1.74035	0.0771399	0.0385699	−	0.999256i	$-0.487720\pi$
$509$	1.74035	0.0771399	0.0385699	+	0.999256i	$0.487720\pi$
$510$	10.8179	0.479024
$511$	6.68499	0.295727
$512$	1.00000	0.0441942
$513$	−19.3588	−0.854712
$514$	−13.6822	−0.603495
$515$	5.22683	0.230322
$516$	3.69779	0.162786
$517$	11.5277	0.506987
$518$	5.84213	0.256688
$519$	8.73784	0.383548
$520$	−9.00148	−0.394741
$521$	10.5683	0.463004	0.231502	−	0.972834i	$-0.425636\pi$
$521$	10.5683	0.463004	0.231502	+	0.972834i	$0.425636\pi$
$522$	−6.00517	−0.262839
$523$	10.5329	0.460572	0.230286	−	0.973123i	$-0.426034\pi$
$523$	10.5329	0.460572	0.230286	+	0.973123i	$0.426034\pi$
$524$	−5.17397	−0.226026
$525$	−22.9510	−1.00166
$526$	11.3942	0.496810
$527$	24.5879	1.07106
$528$	0.625290	0.0272123
$529$	−15.3401	−0.666961
$530$	−12.6243	−0.548365
$531$	−12.0334	−0.522203
$532$	−42.0385	−1.82260
$533$	−11.7813	−0.510305
$534$	−1.27161	−0.0550281
$535$	38.8814	1.68099
$536$	−1.50761	−0.0651189
$537$	−4.66865	−0.201467
$538$	−1.00000	−0.0431131
$539$	−27.3634	−1.17862
$540$	9.33213	0.401591
$541$	−0.0839881	−0.00361093	−0.00180546	−	0.999998i	$-0.500575\pi$
$541$	−0.0839881	−0.00361093	−0.00180546	+	0.999998i	$0.500575\pi$
$542$	−24.6167	−1.05738
$543$	1.76963	0.0759419
$544$	−6.78448	−0.290882
$545$	47.4382	2.03203
$546$	4.05710	0.173628
$547$	1.38614	0.0592669	0.0296334	−	0.999561i	$-0.490566\pi$
$547$	1.38614	0.0592669	0.0296334	+	0.999561i	$0.490566\pi$
$548$	6.35949	0.271664
$549$	−24.9536	−1.06499
$550$	−19.9690	−0.851479
$551$	−18.1410	−0.772833
$552$	−1.06230	−0.0452144
$553$	−73.0501	−3.10641
$554$	4.70762	0.200008
$555$	−1.90959	−0.0810575
$556$	−10.6786	−0.452873
$557$	17.1840	0.728111	0.364056	−	0.931377i	$-0.381392\pi$
$557$	17.1840	0.728111	0.364056	+	0.931377i	$0.381392\pi$
$558$	10.3385	0.437663
$559$	20.8752	0.882926
$560$	20.2651	0.856358
$561$	−4.24227	−0.179109
$562$	6.51540	0.274836
$563$	−12.1968	−0.514033	−0.257017	−	0.966407i	$-0.582740\pi$
$563$	−12.1968	−0.514033	−0.257017	+	0.966407i	$0.582740\pi$
$564$	2.71601	0.114364
$565$	1.31613	0.0553700
$566$	−8.67417	−0.364602
$567$	37.5415	1.57660
$568$	11.0270	0.462682
$569$	23.4816	0.984401	0.492201	−	0.870482i	$-0.336193\pi$
$569$	23.4816	0.984401	0.492201	+	0.870482i	$0.336193\pi$
$570$	13.7409	0.575543
$571$	−2.61940	−0.109619	−0.0548093	−	0.998497i	$-0.517455\pi$
$571$	−2.61940	−0.109619	−0.0548093	+	0.998497i	$0.517455\pi$
$572$	3.52996	0.147595
$573$	−1.26875	−0.0530028
$574$	26.5234	1.10707
$575$	33.9250	1.41477
$576$	−2.85268	−0.118862
$577$	12.1035	0.503874	0.251937	−	0.967744i	$-0.418932\pi$
$577$	12.1035	0.503874	0.251937	+	0.967744i	$0.418932\pi$
$578$	29.0291	1.20745
$579$	7.10681	0.295349
$580$	8.74508	0.363120
$581$	−39.4292	−1.63580
$582$	−6.46647	−0.268044
$583$	4.95066	0.205035
$584$	1.37039	0.0567070
$585$	25.6783	1.06167
$586$	20.3798	0.841881
$587$	−3.09578	−0.127776	−0.0638882	−	0.997957i	$-0.520350\pi$
$587$	−3.09578	−0.127776	−0.0638882	+	0.997957i	$0.520350\pi$
$588$	−6.44701	−0.265870
$589$	31.2316	1.28687
$590$	17.5237	0.721439
$591$	−3.37042	−0.138640
$592$	1.19760	0.0492212
$593$	−24.1730	−0.992666	−0.496333	−	0.868132i	$-0.665321\pi$
$593$	−24.1730	−0.992666	−0.496333	+	0.868132i	$0.665321\pi$
$594$	−3.65962	−0.150156
$595$	−137.488	−5.63647
$596$	−3.47426	−0.142311
$597$	4.77364	0.195372
$598$	−5.99701	−0.245236
$599$	39.0984	1.59752	0.798759	−	0.601651i	$-0.205490\pi$
$599$	39.0984	1.59752	0.798759	+	0.601651i	$0.205490\pi$
$600$	−4.70482	−0.192074
$601$	−29.5069	−1.20361	−0.601807	−	0.798642i	$-0.705552\pi$
$601$	−29.5069	−1.20361	−0.601807	+	0.798642i	$0.705552\pi$
$602$	−46.9965	−1.91543
$603$	4.30073	0.175139
$604$	11.2407	0.457376
$605$	−34.6714	−1.40959
$606$	−1.12474	−0.0456895
$607$	17.2674	0.700861	0.350431	−	0.936589i	$-0.386035\pi$
$607$	17.2674	0.700861	0.350431	+	0.936589i	$0.386035\pi$
$608$	−8.61766	−0.349492
$609$	−3.94154	−0.159719
$610$	36.3389	1.47132
$611$	15.3327	0.620294
$612$	19.3539	0.782336
$613$	−7.94254	−0.320796	−0.160398	−	0.987052i	$-0.551278\pi$
$613$	−7.94254	−0.320796	−0.160398	+	0.987052i	$0.551278\pi$
$614$	−17.5028	−0.706357
$615$	−8.66956	−0.349591
$616$	−7.94703	−0.320195
$617$	18.0886	0.728218	0.364109	−	0.931356i	$-0.381374\pi$
$617$	18.0886	0.728218	0.364109	+	0.931356i	$0.381374\pi$
$618$	−0.482928	−0.0194262
$619$	−33.4949	−1.34627	−0.673136	−	0.739518i	$-0.735053\pi$
$619$	−33.4949	−1.34627	−0.673136	+	0.739518i	$0.735053\pi$
$620$	−15.0555	−0.604644
$621$	6.21729	0.249491
$622$	22.4715	0.901024
$623$	16.1614	0.647491
$624$	0.831683	0.0332940
$625$	63.9624	2.55850
$626$	−32.6321	−1.30424
$627$	−5.38854	−0.215197
$628$	8.59533	0.342991
$629$	−8.12512	−0.323970
$630$	−57.8099	−2.30320
$631$	37.3162	1.48554	0.742768	−	0.669549i	$-0.233512\pi$
$631$	37.3162	1.48554	0.742768	+	0.669549i	$0.233512\pi$
$632$	−14.9749	−0.595668
$633$	4.19483	0.166729
$634$	−19.5532	−0.776559
$635$	−8.67288	−0.344173
$636$	1.16641	0.0462512
$637$	−36.3954	−1.44204
$638$	−3.42941	−0.135772
$639$	−31.4564	−1.24440
$640$	4.15424	0.164211
$641$	−18.0393	−0.712509	−0.356254	−	0.934389i	$-0.615946\pi$
$641$	−18.0393	−0.712509	−0.356254	+	0.934389i	$0.615946\pi$
$642$	−3.59241	−0.141781
$643$	−33.0496	−1.30335	−0.651675	−	0.758498i	$-0.725933\pi$
$643$	−33.0496	−1.30335	−0.651675	+	0.758498i	$0.725933\pi$
$644$	13.5011	0.532019
$645$	15.3615	0.604859
$646$	58.4663	2.30033
$647$	−40.3067	−1.58462	−0.792310	−	0.610118i	$-0.791122\pi$
$647$	−40.3067	−1.58462	−0.792310	+	0.610118i	$0.791122\pi$
$648$	7.69580	0.302320
$649$	−6.87196	−0.269748
$650$	−26.5602	−1.04178
$651$	6.78575	0.265954
$652$	10.1530	0.397621
$653$	−21.3784	−0.836601	−0.418300	−	0.908309i	$-0.637374\pi$
$653$	−21.3784	−0.836601	−0.418300	+	0.908309i	$0.637374\pi$
$654$	−4.38300	−0.171389
$655$	−21.4939	−0.839836
$656$	5.43715	0.212285
$657$	−3.90927	−0.152515
$658$	−34.5186	−1.34568
$659$	23.1777	0.902875	0.451438	−	0.892303i	$-0.350911\pi$
$659$	23.1777	0.902875	0.451438	+	0.892303i	$0.350911\pi$
$660$	2.59760	0.101112
$661$	−3.96791	−0.154334	−0.0771669	−	0.997018i	$-0.524587\pi$
$661$	−3.96791	−0.154334	−0.0771669	+	0.997018i	$0.524587\pi$
$662$	−20.1680	−0.783850
$663$	−5.64253	−0.219138
$664$	−8.08276	−0.313672
$665$	−174.638	−6.77217
$666$	−3.41638	−0.132382
$667$	5.82618	0.225591
$668$	4.37572	0.169302
$669$	−3.37526	−0.130495
$670$	−6.26298	−0.241960
$671$	−14.2504	−0.550131
$672$	−1.87238	−0.0722285
$673$	32.9589	1.27047	0.635235	−	0.772319i	$-0.280903\pi$
$673$	32.9589	1.27047	0.635235	+	0.772319i	$0.280903\pi$
$674$	29.1409	1.12247
$675$	27.5358	1.05985
$676$	−8.30489	−0.319419
$677$	17.1159	0.657816	0.328908	−	0.944362i	$-0.393319\pi$
$677$	17.1159	0.657816	0.328908	+	0.944362i	$0.393319\pi$
$678$	−0.121603	−0.00467012
$679$	82.1846	3.15396
$680$	−28.1843	−1.08082
$681$	−2.59033	−0.0992618
$682$	5.90407	0.226078
$683$	33.2813	1.27347	0.636736	−	0.771082i	$-0.280284\pi$
$683$	33.2813	1.27347	0.636736	+	0.771082i	$0.280284\pi$
$684$	24.5834	0.939969
$685$	26.4188	1.00941
$686$	47.7899	1.82463
$687$	8.92804	0.340626
$688$	−9.63402	−0.367294
$689$	6.58475	0.250859
$690$	−4.41304	−0.168002
$691$	−21.3236	−0.811189	−0.405594	−	0.914053i	$-0.632935\pi$
$691$	−21.3236	−0.811189	−0.405594	+	0.914053i	$0.632935\pi$
$692$	−22.7651	−0.865398
$693$	22.6703	0.861174
$694$	12.2141	0.463640
$695$	−44.3614	−1.68272
$696$	−0.807993	−0.0306269
$697$	−36.8882	−1.39724
$698$	15.3346	0.580423
$699$	−4.24976	−0.160741
$700$	59.7952	2.26005
$701$	−28.5333	−1.07769	−0.538844	−	0.842406i	$-0.681139\pi$
$701$	−28.5333	−1.07769	−0.538844	+	0.842406i	$0.681139\pi$
$702$	−4.86757	−0.183715
$703$	−10.3205	−0.389246
$704$	−1.62910	−0.0613989
$705$	11.2829	0.424940
$706$	30.6859	1.15488
$707$	14.2947	0.537609
$708$	−1.61908	−0.0608489
$709$	4.38294	0.164605	0.0823024	−	0.996607i	$-0.473773\pi$
$709$	4.38294	0.164605	0.0823024	+	0.996607i	$0.473773\pi$
$710$	45.8087	1.71917
$711$	42.7184	1.60207
$712$	3.31299	0.124159
$713$	−10.0304	−0.375640
$714$	12.7031	0.475401
$715$	14.6643	0.548413
$716$	12.1634	0.454569
$717$	1.53174	0.0572040
$718$	0.344824	0.0128687
$719$	−36.1805	−1.34930	−0.674652	−	0.738136i	$-0.735706\pi$
$719$	−36.1805	−1.34930	−0.674652	+	0.738136i	$0.735706\pi$
$720$	−11.8507	−0.441649
$721$	6.13770	0.228580
$722$	55.2640	2.05671
$723$	6.34881	0.236115
$724$	−4.61048	−0.171347
$725$	25.8036	0.958323
$726$	3.20343	0.118891
$727$	−23.0348	−0.854312	−0.427156	−	0.904178i	$-0.640485\pi$
$727$	−23.0348	−0.854312	−0.427156	+	0.904178i	$0.640485\pi$
$728$	−10.5701	−0.391756
$729$	−19.3669	−0.717294
$730$	5.69291	0.210704
$731$	65.3618	2.41749
$732$	−3.35750	−0.124097
$733$	−41.3032	−1.52557	−0.762785	−	0.646652i	$-0.776169\pi$
$733$	−41.3032	−1.52557	−0.762785	+	0.646652i	$0.776169\pi$
$734$	22.3072	0.823375
$735$	−26.7824	−0.987884
$736$	2.76765	0.102017
$737$	2.45605	0.0904696
$738$	−15.5104	−0.570947
$739$	−15.2014	−0.559191	−0.279596	−	0.960118i	$-0.590200\pi$
$739$	−15.2014	−0.559191	−0.279596	+	0.960118i	$0.590200\pi$
$740$	4.97513	0.182889
$741$	−7.16716	−0.263292
$742$	−14.8243	−0.544218
$743$	51.4130	1.88616	0.943079	−	0.332568i	$-0.107915\pi$
$743$	51.4130	1.88616	0.943079	+	0.332568i	$0.107915\pi$
$744$	1.39104	0.0509980
$745$	−14.4329	−0.528780
$746$	−5.92843	−0.217055
$747$	23.0575	0.843630
$748$	11.0526	0.404122
$749$	45.6572	1.66828
$750$	−11.5724	−0.422565
$751$	37.2197	1.35816	0.679082	−	0.734062i	$-0.262378\pi$
$751$	37.2197	1.35816	0.679082	+	0.734062i	$0.262378\pi$
$752$	−7.07613	−0.258040
$753$	−9.23231	−0.336444
$754$	−4.56137	−0.166115
$755$	46.6964	1.69945
$756$	10.9584	0.398554
$757$	51.7191	1.87976	0.939882	−	0.341499i	$-0.110935\pi$
$757$	51.7191	1.87976	0.939882	+	0.341499i	$0.110935\pi$
$758$	−2.01119	−0.0730499
$759$	1.73059	0.0628163
$760$	−35.7998	−1.29859
$761$	16.8448	0.610623	0.305312	−	0.952253i	$-0.401239\pi$
$761$	16.8448	0.610623	0.305312	+	0.952253i	$0.401239\pi$
$762$	0.801322	0.0290288
$763$	55.7051	2.01666
$764$	3.30553	0.119590
$765$	80.4008	2.90690
$766$	4.21696	0.152365
$767$	−9.14022	−0.330034
$768$	−0.383827	−0.0138501
$769$	10.2220	0.368614	0.184307	−	0.982869i	$-0.440996\pi$
$769$	10.2220	0.368614	0.184307	+	0.982869i	$0.440996\pi$
$770$	−33.0138	−1.18974
$771$	5.25158	0.189131
$772$	−18.5157	−0.666394
$773$	−31.6879	−1.13973	−0.569867	−	0.821737i	$-0.693005\pi$
$773$	−31.6879	−1.13973	−0.569867	+	0.821737i	$0.693005\pi$
$774$	27.4827	0.987847
$775$	−44.4235	−1.59574
$776$	16.8474	0.604786
$777$	−2.24236	−0.0804444
$778$	4.54939	0.163103
$779$	−46.8555	−1.67877
$780$	3.45501	0.123709
$781$	−17.9640	−0.642803
$782$	−18.7771	−0.671467
$783$	4.72892	0.168998
$784$	16.7967	0.599881
$785$	35.7070	1.27444
$786$	1.98591	0.0708350
$787$	21.1252	0.753033	0.376516	−	0.926410i	$-0.377122\pi$
$787$	21.1252	0.753033	0.376516	+	0.926410i	$0.377122\pi$
$788$	8.78109	0.312814
$789$	−4.37339	−0.155697
$790$	−62.2091	−2.21330
$791$	1.54549	0.0549512
$792$	4.64729	0.165134
$793$	−18.9541	−0.673080
$794$	−15.2162	−0.540001
$795$	4.84554	0.171854
$796$	−12.4370	−0.440817
$797$	−4.27630	−0.151474	−0.0757371	−	0.997128i	$-0.524131\pi$
$797$	−4.27630	−0.151474	−0.0757371	+	0.997128i	$0.524131\pi$
$798$	16.1355	0.571190
$799$	48.0078	1.69839
$800$	12.2577	0.433375
$801$	−9.45089	−0.333931
$802$	18.4280	0.650715
$803$	−2.23249	−0.0787829
$804$	0.578661	0.0204078
$805$	56.0868	1.97680
$806$	7.85285	0.276605
$807$	0.383827	0.0135113
$808$	2.93034	0.103089
$809$	4.78386	0.168192	0.0840958	−	0.996458i	$-0.473200\pi$
$809$	4.78386	0.168192	0.0840958	+	0.996458i	$0.473200\pi$
$810$	31.9702	1.12332
$811$	−34.4127	−1.20839	−0.604197	−	0.796835i	$-0.706506\pi$
$811$	−34.4127	−1.20839	−0.604197	+	0.796835i	$0.706506\pi$
$812$	10.2691	0.360373
$813$	9.44854	0.331375
$814$	−1.95101	−0.0683829
$815$	42.1778	1.47742
$816$	2.60406	0.0911604
$817$	83.0227	2.90460
$818$	36.3293	1.27022
$819$	30.1532	1.05364
$820$	22.5872	0.788779
$821$	−38.7973	−1.35404	−0.677018	−	0.735967i	$-0.736728\pi$
$821$	−38.7973	−1.35404	−0.677018	+	0.735967i	$0.736728\pi$
$822$	−2.44094	−0.0851375
$823$	−8.52567	−0.297186	−0.148593	−	0.988898i	$-0.547474\pi$
$823$	−8.52567	−0.297186	−0.148593	+	0.988898i	$0.547474\pi$
$824$	1.25819	0.0438312
$825$	7.66461	0.266848
$826$	20.5775	0.715982
$827$	3.84932	0.133854	0.0669271	−	0.997758i	$-0.478681\pi$
$827$	3.84932	0.133854	0.0669271	+	0.997758i	$0.478681\pi$
$828$	−7.89522	−0.274378
$829$	1.76719	0.0613771	0.0306886	−	0.999529i	$-0.490230\pi$
$829$	1.76719	0.0613771	0.0306886	+	0.999529i	$0.490230\pi$
$830$	−33.5777	−1.16550
$831$	−1.80691	−0.0626810
$832$	−2.16682	−0.0751210
$833$	−113.957	−3.94836
$834$	4.09872	0.141927
$835$	18.1778	0.629068
$836$	14.0390	0.485549
$837$	−8.14131	−0.281405
$838$	−6.63242	−0.229113
$839$	48.4527	1.67277	0.836386	−	0.548141i	$-0.184664\pi$
$839$	48.4527	1.67277	0.836386	+	0.548141i	$0.184664\pi$
$840$	−7.77829	−0.268377
$841$	−24.5686	−0.847192
$842$	−33.1090	−1.14101
$843$	−2.50078	−0.0861315
$844$	−10.9290	−0.376191
$845$	−34.5005	−1.18685
$846$	20.1859	0.694006
$847$	−40.7135	−1.39893
$848$	−3.03890	−0.104356
$849$	3.32938	0.114264
$850$	−83.1620	−2.85243
$851$	3.31455	0.113621
$852$	−4.23245	−0.145001
$853$	35.2024	1.20531	0.602654	−	0.798002i	$-0.294110\pi$
$853$	35.2024	1.20531	0.602654	+	0.798002i	$0.294110\pi$
$854$	42.6716	1.46019
$855$	102.125	3.49261
$856$	9.35946	0.319900
$857$	8.30685	0.283757	0.141878	−	0.989884i	$-0.454686\pi$
$857$	8.30685	0.283757	0.141878	+	0.989884i	$0.454686\pi$
$858$	−1.35489	−0.0462552
$859$	48.5886	1.65782	0.828911	−	0.559381i	$-0.188961\pi$
$859$	48.5886	1.65782	0.828911	+	0.559381i	$0.188961\pi$
$860$	−40.0220	−1.36474
$861$	−10.1804	−0.346947
$862$	−16.6498	−0.567096
$863$	50.0516	1.70378	0.851888	−	0.523723i	$-0.175457\pi$
$863$	50.0516	1.70378	0.851888	+	0.523723i	$0.175457\pi$
$864$	2.24641	0.0764245
$865$	−94.5715	−3.21553
$866$	5.36131	0.182185
$867$	−11.1421	−0.378407
$868$	−17.6792	−0.600071
$869$	24.3955	0.827560
$870$	−3.35659	−0.113799
$871$	3.26672	0.110689
$872$	11.4192	0.386704
$873$	−48.0601	−1.62659
$874$	−23.8507	−0.806761
$875$	147.078	4.97214
$876$	−0.525990	−0.0177716
$877$	−35.1921	−1.18835	−0.594177	−	0.804334i	$-0.702522\pi$
$877$	−35.1921	−1.18835	−0.594177	+	0.804334i	$0.702522\pi$
$878$	−5.42961	−0.183240
$879$	−7.82230	−0.263840
$880$	−6.76765	−0.228138
$881$	−29.4895	−0.993527	−0.496764	−	0.867886i	$-0.665478\pi$
$881$	−29.4895	−0.993527	−0.496764	+	0.867886i	$0.665478\pi$
$882$	−47.9155	−1.61340
$883$	−28.1353	−0.946829	−0.473415	−	0.880840i	$-0.656979\pi$
$883$	−28.1353	−0.946829	−0.473415	+	0.880840i	$0.656979\pi$
$884$	14.7007	0.494439
$885$	−6.72605	−0.226094
$886$	3.66946	0.123278
$887$	−12.3917	−0.416072	−0.208036	−	0.978121i	$-0.566707\pi$
$887$	−12.3917	−0.416072	−0.208036	+	0.978121i	$0.566707\pi$
$888$	−0.459672	−0.0154256
$889$	−10.1843	−0.341570
$890$	13.7629	0.461335
$891$	−12.5372	−0.420012
$892$	8.79372	0.294436
$893$	60.9796	2.04061
$894$	1.33351	0.0445993
$895$	50.5298	1.68903
$896$	4.87818	0.162969
$897$	2.30181	0.0768552
$898$	6.43513	0.214743
$899$	−7.62916	−0.254447
$900$	−34.9672	−1.16557
$901$	20.6173	0.686863
$902$	−8.85764	−0.294927
$903$	18.0385	0.600284
$904$	0.316816	0.0105372
$905$	−19.1530	−0.636669
$906$	−4.31446	−0.143338
$907$	−14.9432	−0.496181	−0.248090	−	0.968737i	$-0.579803\pi$
$907$	−14.9432	−0.496181	−0.248090	+	0.968737i	$0.579803\pi$
$908$	6.74871	0.223964
$909$	−8.35931	−0.277261
$910$	−43.9109	−1.45563
$911$	29.3458	0.972269	0.486134	−	0.873884i	$-0.338407\pi$
$911$	29.3458	0.972269	0.486134	+	0.873884i	$0.338407\pi$
$912$	3.30769	0.109528
$913$	13.1676	0.435784
$914$	2.97606	0.0984393
$915$	−13.9478	−0.461101
$916$	−23.2606	−0.768553
$917$	−25.2396	−0.833485
$918$	−15.2407	−0.503019
$919$	24.7551	0.816595	0.408297	−	0.912849i	$-0.366123\pi$
$919$	24.7551	0.816595	0.408297	+	0.912849i	$0.366123\pi$
$920$	11.4975	0.379061
$921$	6.71805	0.221367
$922$	9.73958	0.320756
$923$	−23.8935	−0.786464
$924$	3.05028	0.100347
$925$	14.6799	0.482671
$926$	−19.9810	−0.656618
$927$	−3.58922	−0.117885
$928$	2.10510	0.0691032
$929$	−27.1303	−0.890117	−0.445059	−	0.895501i	$-0.646817\pi$
$929$	−27.1303	−0.890117	−0.445059	+	0.895501i	$0.646817\pi$
$930$	5.77871	0.189491
$931$	−144.748	−4.74392
$932$	11.0721	0.362679
$933$	−8.62515	−0.282375
$934$	−14.3949	−0.471014
$935$	45.9150	1.50158
$936$	6.18124	0.202040
$937$	44.2931	1.44699	0.723496	−	0.690328i	$-0.242534\pi$
$937$	44.2931	1.44699	0.723496	+	0.690328i	$0.242534\pi$
$938$	−7.35441	−0.240130
$939$	12.5250	0.408740
$940$	−29.3959	−0.958789
$941$	−27.0894	−0.883090	−0.441545	−	0.897239i	$-0.645569\pi$
$941$	−27.0894	−0.883090	−0.441545	+	0.897239i	$0.645569\pi$
$942$	−3.29912	−0.107491
$943$	15.0481	0.490035
$944$	4.21827	0.137293
$945$	45.5238	1.48089
$946$	15.6947	0.510280
$947$	44.9196	1.45969	0.729845	−	0.683613i	$-0.239592\pi$
$947$	44.9196	1.45969	0.729845	+	0.683613i	$0.239592\pi$
$948$	5.74775	0.186678
$949$	−2.96938	−0.0963901
$950$	−105.633	−3.42717
$951$	7.50505	0.243368
$952$	−33.0959	−1.07264
$953$	−21.0858	−0.683036	−0.341518	−	0.939875i	$-0.610941\pi$
$953$	−21.0858	−0.683036	−0.341518	+	0.939875i	$0.610941\pi$
$954$	8.66900	0.280669
$955$	13.7320	0.444356
$956$	−3.99072	−0.129069
$957$	1.31630	0.0425499
$958$	−22.4679	−0.725906
$959$	31.0227	1.00178
$960$	−1.59451	−0.0514625
$961$	−17.8656	−0.576311
$962$	−2.59499	−0.0836659
$963$	−26.6995	−0.860380
$964$	−16.5408	−0.532745
$965$	−76.9185	−2.47609
$966$	−5.18209	−0.166731
$967$	25.2415	0.811711	0.405855	−	0.913937i	$-0.366974\pi$
$967$	25.2415	0.811711	0.405855	+	0.913937i	$0.366974\pi$
$968$	−8.34604	−0.268252
$969$	−22.4409	−0.720906
$970$	69.9880	2.24718
$971$	−34.5382	−1.10838	−0.554192	−	0.832389i	$-0.686973\pi$
$971$	−34.5382	−1.10838	−0.554192	+	0.832389i	$0.686973\pi$
$972$	−9.69309	−0.310906
$973$	−52.0921	−1.67000
$974$	−15.3289	−0.491169
$975$	10.1945	0.326486
$976$	8.74744	0.279999
$977$	61.0213	1.95224	0.976122	−	0.217225i	$-0.0697004\pi$
$977$	61.0213	1.95224	0.976122	+	0.217225i	$0.0697004\pi$
$978$	−3.89698	−0.124612
$979$	−5.39718	−0.172495
$980$	69.7773	2.22896
$981$	−32.5754	−1.04005
$982$	−23.8945	−0.762504
$983$	−47.5260	−1.51585	−0.757923	−	0.652345i	$-0.773786\pi$
$983$	−47.5260	−1.51585	−0.757923	+	0.652345i	$0.773786\pi$
$984$	−2.08692	−0.0665286
$985$	36.4787	1.16231
$986$	−14.2820	−0.454831
$987$	13.2492	0.421726
$988$	18.6729	0.594064
$989$	−26.6636	−0.847854
$990$	19.3059	0.613583
$991$	−35.6855	−1.13359	−0.566794	−	0.823860i	$-0.691816\pi$
$991$	−35.6855	−1.13359	−0.566794	+	0.823860i	$0.691816\pi$
$992$	−3.62414	−0.115066
$993$	7.74100	0.245653
$994$	53.7917	1.70617
$995$	−51.6662	−1.63793
$996$	3.10238	0.0983026
$997$	12.2019	0.386438	0.193219	−	0.981156i	$-0.438107\pi$
$997$	12.2019	0.386438	0.193219	+	0.981156i	$0.438107\pi$
$998$	−15.7627	−0.498960
$999$	2.69031	0.0851177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.2.a.e.1.3	✓	7
3.2	odd	2		4842.2.a.n.1.1		7
4.3	odd	2		4304.2.a.h.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.e.1.3	✓	7	1.1	even	1	trivial
4304.2.a.h.1.5		7	4.3	odd	2
4842.2.a.n.1.1		7	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.383827 q^{3} +1.00000 q^{4} +4.15424 q^{5} -0.383827 q^{6} +4.87818 q^{7} +1.00000 q^{8} -2.85268 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.383827 q^{3} +1.00000 q^{4} +4.15424 q^{5} -0.383827 q^{6} +4.87818 q^{7} +1.00000 q^{8} -2.85268 q^{9} +4.15424 q^{10} -1.62910 q^{11} -0.383827 q^{12} -2.16682 q^{13} +4.87818 q^{14} -1.59451 q^{15} +1.00000 q^{16} -6.78448 q^{17} -2.85268 q^{18} -8.61766 q^{19} +4.15424 q^{20} -1.87238 q^{21} -1.62910 q^{22} +2.76765 q^{23} -0.383827 q^{24} +12.2577 q^{25} -2.16682 q^{26} +2.24641 q^{27} +4.87818 q^{28} +2.10510 q^{29} -1.59451 q^{30} -3.62414 q^{31} +1.00000 q^{32} +0.625290 q^{33} -6.78448 q^{34} +20.2651 q^{35} -2.85268 q^{36} +1.19760 q^{37} -8.61766 q^{38} +0.831683 q^{39} +4.15424 q^{40} +5.43715 q^{41} -1.87238 q^{42} -9.63402 q^{43} -1.62910 q^{44} -11.8507 q^{45} +2.76765 q^{46} -7.07613 q^{47} -0.383827 q^{48} +16.7967 q^{49} +12.2577 q^{50} +2.60406 q^{51} -2.16682 q^{52} -3.03890 q^{53} +2.24641 q^{54} -6.76765 q^{55} +4.87818 q^{56} +3.30769 q^{57} +2.10510 q^{58} +4.21827 q^{59} -1.59451 q^{60} +8.74744 q^{61} -3.62414 q^{62} -13.9159 q^{63} +1.00000 q^{64} -9.00148 q^{65} +0.625290 q^{66} -1.50761 q^{67} -6.78448 q^{68} -1.06230 q^{69} +20.2651 q^{70} +11.0270 q^{71} -2.85268 q^{72} +1.37039 q^{73} +1.19760 q^{74} -4.70482 q^{75} -8.61766 q^{76} -7.94703 q^{77} +0.831683 q^{78} -14.9749 q^{79} +4.15424 q^{80} +7.69580 q^{81} +5.43715 q^{82} -8.08276 q^{83} -1.87238 q^{84} -28.1843 q^{85} -9.63402 q^{86} -0.807993 q^{87} -1.62910 q^{88} +3.31299 q^{89} -11.8507 q^{90} -10.5701 q^{91} +2.76765 q^{92} +1.39104 q^{93} -7.07613 q^{94} -35.7998 q^{95} -0.383827 q^{96} +16.8474 q^{97} +16.7967 q^{98} +4.64729 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9}+O(q^{10}) \) 7 * q + 7 * q^2 + q^3 + 7 * q^4 + 7 * q^5 + q^6 + 6 * q^7 + 7 * q^8 + 12 * q^9	\( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9} + 7 q^{10} - 3 q^{11} + q^{12} - 9 q^{13} + 6 q^{14} + 8 q^{15} + 7 q^{16} + 8 q^{17} + 12 q^{18} - 11 q^{19} + 7 q^{20} - 6 q^{21} - 3 q^{22} + 12 q^{23} + q^{24} + 22 q^{25} - 9 q^{26} - 14 q^{27} + 6 q^{28} - 5 q^{29} + 8 q^{30} + 14 q^{31} + 7 q^{32} - 4 q^{33} + 8 q^{34} - 4 q^{35} + 12 q^{36} + 13 q^{37} - 11 q^{38} - 18 q^{39} + 7 q^{40} + 12 q^{41} - 6 q^{42} - 11 q^{43} - 3 q^{44} + 3 q^{45} + 12 q^{46} + 2 q^{47} + q^{48} + 15 q^{49} + 22 q^{50} - 26 q^{51} - 9 q^{52} + 19 q^{53} - 14 q^{54} - 40 q^{55} + 6 q^{56} - 12 q^{57} - 5 q^{58} - 9 q^{59} + 8 q^{60} - 3 q^{61} + 14 q^{62} - 26 q^{63} + 7 q^{64} - 10 q^{65} - 4 q^{66} - 33 q^{67} + 8 q^{68} - 64 q^{69} - 4 q^{70} + 28 q^{71} + 12 q^{72} - 14 q^{73} + 13 q^{74} - 45 q^{75} - 11 q^{76} + 10 q^{77} - 18 q^{78} + 2 q^{79} + 7 q^{80} + 15 q^{81} + 12 q^{82} - 7 q^{83} - 6 q^{84} - 16 q^{85} - 11 q^{86} + 16 q^{87} - 3 q^{88} + 18 q^{89} + 3 q^{90} - 26 q^{91} + 12 q^{92} + 6 q^{93} + 2 q^{94} - 34 q^{95} + q^{96} - 4 q^{97} + 15 q^{98} - 37 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 + q^3 + 7 * q^4 + 7 * q^5 + q^6 + 6 * q^7 + 7 * q^8 + 12 * q^9 + 7 * q^10 - 3 * q^11 + q^12 - 9 * q^13 + 6 * q^14 + 8 * q^15 + 7 * q^16 + 8 * q^17 + 12 * q^18 - 11 * q^19 + 7 * q^20 - 6 * q^21 - 3 * q^22 + 12 * q^23 + q^24 + 22 * q^25 - 9 * q^26 - 14 * q^27 + 6 * q^28 - 5 * q^29 + 8 * q^30 + 14 * q^31 + 7 * q^32 - 4 * q^33 + 8 * q^34 - 4 * q^35 + 12 * q^36 + 13 * q^37 - 11 * q^38 - 18 * q^39 + 7 * q^40 + 12 * q^41 - 6 * q^42 - 11 * q^43 - 3 * q^44 + 3 * q^45 + 12 * q^46 + 2 * q^47 + q^48 + 15 * q^49 + 22 * q^50 - 26 * q^51 - 9 * q^52 + 19 * q^53 - 14 * q^54 - 40 * q^55 + 6 * q^56 - 12 * q^57 - 5 * q^58 - 9 * q^59 + 8 * q^60 - 3 * q^61 + 14 * q^62 - 26 * q^63 + 7 * q^64 - 10 * q^65 - 4 * q^66 - 33 * q^67 + 8 * q^68 - 64 * q^69 - 4 * q^70 + 28 * q^71 + 12 * q^72 - 14 * q^73 + 13 * q^74 - 45 * q^75 - 11 * q^76 + 10 * q^77 - 18 * q^78 + 2 * q^79 + 7 * q^80 + 15 * q^81 + 12 * q^82 - 7 * q^83 - 6 * q^84 - 16 * q^85 - 11 * q^86 + 16 * q^87 - 3 * q^88 + 18 * q^89 + 3 * q^90 - 26 * q^91 + 12 * q^92 + 6 * q^93 + 2 * q^94 - 34 * q^95 + q^96 - 4 * q^97 + 15 * q^98 - 37 * q^99

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