LMFDB - Embedded newform 4022.2.a.d.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, 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("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$32.1158316930$
Analytic rank:	$1$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.0611935 q^{3} +1.00000 q^{4} -3.49914 q^{5} +0.0611935 q^{6} -5.08120 q^{7} -1.00000 q^{8} -2.99626 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.0611935 q^{3} +1.00000 q^{4} -3.49914 q^{5} +0.0611935 q^{6} -5.08120 q^{7} -1.00000 q^{8} -2.99626 q^{9} +3.49914 q^{10} -0.435748 q^{11} -0.0611935 q^{12} +0.685830 q^{13} +5.08120 q^{14} +0.214124 q^{15} +1.00000 q^{16} +4.65973 q^{17} +2.99626 q^{18} +8.26410 q^{19} -3.49914 q^{20} +0.310936 q^{21} +0.435748 q^{22} -1.27945 q^{23} +0.0611935 q^{24} +7.24396 q^{25} -0.685830 q^{26} +0.366932 q^{27} -5.08120 q^{28} -0.893012 q^{29} -0.214124 q^{30} +4.10479 q^{31} -1.00000 q^{32} +0.0266650 q^{33} -4.65973 q^{34} +17.7798 q^{35} -2.99626 q^{36} -10.6262 q^{37} -8.26410 q^{38} -0.0419683 q^{39} +3.49914 q^{40} +8.20672 q^{41} -0.310936 q^{42} -3.04884 q^{43} -0.435748 q^{44} +10.4843 q^{45} +1.27945 q^{46} +4.10573 q^{47} -0.0611935 q^{48} +18.8186 q^{49} -7.24396 q^{50} -0.285145 q^{51} +0.685830 q^{52} +9.92221 q^{53} -0.366932 q^{54} +1.52474 q^{55} +5.08120 q^{56} -0.505709 q^{57} +0.893012 q^{58} -10.1672 q^{59} +0.214124 q^{60} -11.7413 q^{61} -4.10479 q^{62} +15.2246 q^{63} +1.00000 q^{64} -2.39981 q^{65} -0.0266650 q^{66} -8.02486 q^{67} +4.65973 q^{68} +0.0782940 q^{69} -17.7798 q^{70} +1.52492 q^{71} +2.99626 q^{72} +3.89753 q^{73} +10.6262 q^{74} -0.443283 q^{75} +8.26410 q^{76} +2.21412 q^{77} +0.0419683 q^{78} +0.879981 q^{79} -3.49914 q^{80} +8.96631 q^{81} -8.20672 q^{82} +16.3188 q^{83} +0.310936 q^{84} -16.3050 q^{85} +3.04884 q^{86} +0.0546465 q^{87} +0.435748 q^{88} -2.11755 q^{89} -10.4843 q^{90} -3.48484 q^{91} -1.27945 q^{92} -0.251186 q^{93} -4.10573 q^{94} -28.9172 q^{95} +0.0611935 q^{96} +1.77788 q^{97} -18.8186 q^{98} +1.30561 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9}+O(q^{10}) $ 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9	\( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9} + 13 q^{10} + 12 q^{11} - 5 q^{12} - 36 q^{13} + 22 q^{14} - 7 q^{15} + 37 q^{16} + 4 q^{17} - 32 q^{18} - 10 q^{19} - 13 q^{20} - 11 q^{21} - 12 q^{22} - q^{23} + 5 q^{24} + 8 q^{25} + 36 q^{26} - 20 q^{27} - 22 q^{28} - 6 q^{29} + 7 q^{30} - 23 q^{31} - 37 q^{32} - 27 q^{33} - 4 q^{34} + 24 q^{35} + 32 q^{36} - 46 q^{37} + 10 q^{38} + 5 q^{39} + 13 q^{40} + 11 q^{41} + 11 q^{42} - 22 q^{43} + 12 q^{44} - 57 q^{45} + q^{46} - 18 q^{47} - 5 q^{48} - q^{49} - 8 q^{50} + 18 q^{51} - 36 q^{52} - 25 q^{53} + 20 q^{54} - 25 q^{55} + 22 q^{56} - 25 q^{57} + 6 q^{58} + 24 q^{59} - 7 q^{60} - 40 q^{61} + 23 q^{62} - 38 q^{63} + 37 q^{64} + 14 q^{65} + 27 q^{66} - 49 q^{67} + 4 q^{68} - 19 q^{69} - 24 q^{70} + 27 q^{71} - 32 q^{72} - 87 q^{73} + 46 q^{74} - 5 q^{75} - 10 q^{76} - 50 q^{77} - 5 q^{78} + 11 q^{79} - 13 q^{80} + 5 q^{81} - 11 q^{82} - 9 q^{83} - 11 q^{84} - 68 q^{85} + 22 q^{86} - 12 q^{87} - 12 q^{88} - 3 q^{89} + 57 q^{90} - 19 q^{91} - q^{92} - 69 q^{93} + 18 q^{94} + 27 q^{95} + 5 q^{96} - 98 q^{97} + q^{98} + 33 q^{99}+O(q^{100}) \) 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9 + 13 * q^10 + 12 * q^11 - 5 * q^12 - 36 * q^13 + 22 * q^14 - 7 * q^15 + 37 * q^16 + 4 * q^17 - 32 * q^18 - 10 * q^19 - 13 * q^20 - 11 * q^21 - 12 * q^22 - q^23 + 5 * q^24 + 8 * q^25 + 36 * q^26 - 20 * q^27 - 22 * q^28 - 6 * q^29 + 7 * q^30 - 23 * q^31 - 37 * q^32 - 27 * q^33 - 4 * q^34 + 24 * q^35 + 32 * q^36 - 46 * q^37 + 10 * q^38 + 5 * q^39 + 13 * q^40 + 11 * q^41 + 11 * q^42 - 22 * q^43 + 12 * q^44 - 57 * q^45 + q^46 - 18 * q^47 - 5 * q^48 - q^49 - 8 * q^50 + 18 * q^51 - 36 * q^52 - 25 * q^53 + 20 * q^54 - 25 * q^55 + 22 * q^56 - 25 * q^57 + 6 * q^58 + 24 * q^59 - 7 * q^60 - 40 * q^61 + 23 * q^62 - 38 * q^63 + 37 * q^64 + 14 * q^65 + 27 * q^66 - 49 * q^67 + 4 * q^68 - 19 * q^69 - 24 * q^70 + 27 * q^71 - 32 * q^72 - 87 * q^73 + 46 * q^74 - 5 * q^75 - 10 * q^76 - 50 * q^77 - 5 * q^78 + 11 * q^79 - 13 * q^80 + 5 * q^81 - 11 * q^82 - 9 * q^83 - 11 * q^84 - 68 * q^85 + 22 * q^86 - 12 * q^87 - 12 * q^88 - 3 * q^89 + 57 * q^90 - 19 * q^91 - q^92 - 69 * q^93 + 18 * q^94 + 27 * q^95 + 5 * q^96 - 98 * q^97 + q^98 + 33 * 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
$3$	−0.0611935	−0.0353301	−0.0176650	−	0.999844i	$-0.505623\pi$
$3$	−0.0611935	−0.0353301	−0.0176650	+	0.999844i	$0.505623\pi$
$4$	1.00000	0.500000
$5$	−3.49914	−1.56486	−0.782431	−	0.622738i	$-0.786020\pi$
$5$	−3.49914	−1.56486	−0.782431	+	0.622738i	$0.786020\pi$
$6$	0.0611935	0.0249821
$7$	−5.08120	−1.92051	−0.960257	−	0.279119i	$-0.909958\pi$
$7$	−5.08120	−1.92051	−0.960257	+	0.279119i	$0.909958\pi$
$8$	−1.00000	−0.353553
$9$	−2.99626	−0.998752
$10$	3.49914	1.10652
$11$	−0.435748	−0.131383	−0.0656915	−	0.997840i	$-0.520925\pi$
$11$	−0.435748	−0.131383	−0.0656915	+	0.997840i	$0.520925\pi$
$12$	−0.0611935	−0.0176650
$13$	0.685830	0.190215	0.0951075	−	0.995467i	$-0.469681\pi$
$13$	0.685830	0.190215	0.0951075	+	0.995467i	$0.469681\pi$
$14$	5.08120	1.35801
$15$	0.214124	0.0552867
$16$	1.00000	0.250000
$17$	4.65973	1.13015	0.565075	−	0.825039i	$-0.308847\pi$
$17$	4.65973	1.13015	0.565075	+	0.825039i	$0.308847\pi$
$18$	2.99626	0.706224
$19$	8.26410	1.89592	0.947958	−	0.318397i	$-0.103144\pi$
$19$	8.26410	1.89592	0.947958	+	0.318397i	$0.103144\pi$
$20$	−3.49914	−0.782431
$21$	0.310936	0.0678518
$22$	0.435748	0.0929019
$23$	−1.27945	−0.266784	−0.133392	−	0.991063i	$-0.542587\pi$
$23$	−1.27945	−0.266784	−0.133392	+	0.991063i	$0.542587\pi$
$24$	0.0611935	0.0124911
$25$	7.24396	1.44879
$26$	−0.685830	−0.134502
$27$	0.366932	0.0706160
$28$	−5.08120	−0.960257
$29$	−0.893012	−0.165828	−0.0829141	−	0.996557i	$-0.526423\pi$
$29$	−0.893012	−0.165828	−0.0829141	+	0.996557i	$0.526423\pi$
$30$	−0.214124	−0.0390936
$31$	4.10479	0.737242	0.368621	−	0.929580i	$-0.379830\pi$
$31$	4.10479	0.737242	0.368621	+	0.929580i	$0.379830\pi$
$32$	−1.00000	−0.176777
$33$	0.0266650	0.00464177
$34$	−4.65973	−0.799137
$35$	17.7798	3.00534
$36$	−2.99626	−0.499376
$37$	−10.6262	−1.74694	−0.873472	−	0.486875i	$-0.838137\pi$
$37$	−10.6262	−1.74694	−0.873472	+	0.486875i	$0.838137\pi$
$38$	−8.26410	−1.34061
$39$	−0.0419683	−0.00672031
$40$	3.49914	0.553262
$41$	8.20672	1.28167	0.640837	−	0.767677i	$-0.278588\pi$
$41$	8.20672	1.28167	0.640837	+	0.767677i	$0.278588\pi$
$42$	−0.310936	−0.0479785
$43$	−3.04884	−0.464944	−0.232472	−	0.972603i	$-0.574681\pi$
$43$	−3.04884	−0.464944	−0.232472	+	0.972603i	$0.574681\pi$
$44$	−0.435748	−0.0656915
$45$	10.4843	1.56291
$46$	1.27945	0.188645
$47$	4.10573	0.598883	0.299441	−	0.954115i	$-0.403200\pi$
$47$	4.10573	0.598883	0.299441	+	0.954115i	$0.403200\pi$
$48$	−0.0611935	−0.00883251
$49$	18.8186	2.68837
$50$	−7.24396	−1.02445
$51$	−0.285145	−0.0399283
$52$	0.685830	0.0951075
$53$	9.92221	1.36292	0.681460	−	0.731855i	$-0.261345\pi$
$53$	9.92221	1.36292	0.681460	+	0.731855i	$0.261345\pi$
$54$	−0.366932	−0.0499331
$55$	1.52474	0.205596
$56$	5.08120	0.679004
$57$	−0.505709	−0.0669828
$58$	0.893012	0.117258
$59$	−10.1672	−1.32366	−0.661831	−	0.749653i	$-0.730220\pi$
$59$	−10.1672	−1.32366	−0.661831	+	0.749653i	$0.730220\pi$
$60$	0.214124	0.0276433
$61$	−11.7413	−1.50332	−0.751659	−	0.659552i	$-0.770746\pi$
$61$	−11.7413	−1.50332	−0.751659	+	0.659552i	$0.770746\pi$
$62$	−4.10479	−0.521309
$63$	15.2246	1.91812
$64$	1.00000	0.125000
$65$	−2.39981	−0.297660
$66$	−0.0266650	−0.00328223
$67$	−8.02486	−0.980392	−0.490196	−	0.871612i	$-0.663075\pi$
$67$	−8.02486	−0.980392	−0.490196	+	0.871612i	$0.663075\pi$
$68$	4.65973	0.565075
$69$	0.0782940	0.00942549
$70$	−17.7798	−2.12509
$71$	1.52492	0.180975	0.0904873	−	0.995898i	$-0.471158\pi$
$71$	1.52492	0.180975	0.0904873	+	0.995898i	$0.471158\pi$
$72$	2.99626	0.353112
$73$	3.89753	0.456171	0.228085	−	0.973641i	$-0.426753\pi$
$73$	3.89753	0.456171	0.228085	+	0.973641i	$0.426753\pi$
$74$	10.6262	1.23528
$75$	−0.443283	−0.0511859
$76$	8.26410	0.947958
$77$	2.21412	0.252323
$78$	0.0419683	0.00475198
$79$	0.879981	0.0990056	0.0495028	−	0.998774i	$-0.484236\pi$
$79$	0.879981	0.0990056	0.0495028	+	0.998774i	$0.484236\pi$
$80$	−3.49914	−0.391215
$81$	8.96631	0.996257
$82$	−8.20672	−0.906280
$83$	16.3188	1.79122	0.895612	−	0.444836i	$-0.146738\pi$
$83$	16.3188	1.79122	0.895612	+	0.444836i	$0.146738\pi$
$84$	0.310936	0.0339259
$85$	−16.3050	−1.76853
$86$	3.04884	0.328765
$87$	0.0546465	0.00585872
$88$	0.435748	0.0464509
$89$	−2.11755	−0.224460	−0.112230	−	0.993682i	$-0.535799\pi$
$89$	−2.11755	−0.224460	−0.112230	+	0.993682i	$0.535799\pi$
$90$	−10.4843	−1.10514
$91$	−3.48484	−0.365311
$92$	−1.27945	−0.133392
$93$	−0.251186	−0.0260468
$94$	−4.10573	−0.423474
$95$	−28.9172	−2.96684
$96$	0.0611935	0.00624553
$97$	1.77788	0.180516	0.0902581	−	0.995918i	$-0.471231\pi$
$97$	1.77788	0.180516	0.0902581	+	0.995918i	$0.471231\pi$
$98$	−18.8186	−1.90097
$99$	1.30561	0.131219
$100$	7.24396	0.724396
$101$	1.43584	0.142872	0.0714358	−	0.997445i	$-0.477242\pi$
$101$	1.43584	0.142872	0.0714358	+	0.997445i	$0.477242\pi$
$102$	0.285145	0.0282336
$103$	−2.93501	−0.289195	−0.144597	−	0.989491i	$-0.546189\pi$
$103$	−2.93501	−0.289195	−0.144597	+	0.989491i	$0.546189\pi$
$104$	−0.685830	−0.0672512
$105$	−1.08801	−0.106179
$106$	−9.92221	−0.963730
$107$	−5.29527	−0.511913	−0.255956	−	0.966688i	$-0.582390\pi$
$107$	−5.29527	−0.511913	−0.255956	+	0.966688i	$0.582390\pi$
$108$	0.366932	0.0353080
$109$	−9.98409	−0.956302	−0.478151	−	0.878278i	$-0.658693\pi$
$109$	−9.98409	−0.956302	−0.478151	+	0.878278i	$0.658693\pi$
$110$	−1.52474	−0.145379
$111$	0.650256	0.0617196
$112$	−5.08120	−0.480128
$113$	2.51508	0.236599	0.118299	−	0.992978i	$-0.462256\pi$
$113$	2.51508	0.236599	0.118299	+	0.992978i	$0.462256\pi$
$114$	0.505709	0.0473640
$115$	4.47697	0.417480
$116$	−0.893012	−0.0829141
$117$	−2.05492	−0.189978
$118$	10.1672	0.935970
$119$	−23.6770	−2.17047
$120$	−0.214124	−0.0195468
$121$	−10.8101	−0.982738
$122$	11.7413	1.06301
$123$	−0.502197	−0.0452816
$124$	4.10479	0.368621
$125$	−7.85192	−0.702297
$126$	−15.2246	−1.35631
$127$	−18.2688	−1.62110	−0.810548	−	0.585672i	$-0.800831\pi$
$127$	−18.2688	−1.62110	−0.810548	+	0.585672i	$0.800831\pi$
$128$	−1.00000	−0.0883883
$129$	0.186569	0.0164265
$130$	2.39981	0.210478
$131$	6.84595	0.598133	0.299067	−	0.954232i	$-0.403325\pi$
$131$	6.84595	0.598133	0.299067	+	0.954232i	$0.403325\pi$
$132$	0.0266650	0.00232089
$133$	−41.9916	−3.64113
$134$	8.02486	0.693242
$135$	−1.28394	−0.110504
$136$	−4.65973	−0.399569
$137$	−18.1594	−1.55146	−0.775732	−	0.631063i	$-0.782619\pi$
$137$	−18.1594	−1.55146	−0.775732	+	0.631063i	$0.782619\pi$
$138$	−0.0782940	−0.00666483
$139$	10.7728	0.913735	0.456868	−	0.889535i	$-0.348971\pi$
$139$	10.7728	0.913735	0.456868	+	0.889535i	$0.348971\pi$
$140$	17.7798	1.50267
$141$	−0.251244	−0.0211586
$142$	−1.52492	−0.127968
$143$	−0.298849	−0.0249910
$144$	−2.99626	−0.249688
$145$	3.12477	0.259498
$146$	−3.89753	−0.322562
$147$	−1.15157	−0.0949803
$148$	−10.6262	−0.873472
$149$	−10.0866	−0.826323	−0.413161	−	0.910658i	$-0.635575\pi$
$149$	−10.0866	−0.826323	−0.413161	+	0.910658i	$0.635575\pi$
$150$	0.443283	0.0361939
$151$	11.2302	0.913903	0.456952	−	0.889492i	$-0.348941\pi$
$151$	11.2302	0.913903	0.456952	+	0.889492i	$0.348941\pi$
$152$	−8.26410	−0.670307
$153$	−13.9617	−1.12874
$154$	−2.21412	−0.178419
$155$	−14.3632	−1.15368
$156$	−0.0419683	−0.00336015
$157$	18.3948	1.46806	0.734031	−	0.679116i	$-0.237637\pi$
$157$	18.3948	1.46806	0.734031	+	0.679116i	$0.237637\pi$
$158$	−0.879981	−0.0700076
$159$	−0.607174	−0.0481520
$160$	3.49914	0.276631
$161$	6.50115	0.512362
$162$	−8.96631	−0.704460
$163$	−18.9079	−1.48098	−0.740490	−	0.672067i	$-0.765407\pi$
$163$	−18.9079	−1.48098	−0.740490	+	0.672067i	$0.765407\pi$
$164$	8.20672	0.640837
$165$	−0.0933043	−0.00726373
$166$	−16.3188	−1.26659
$167$	−8.29004	−0.641502	−0.320751	−	0.947164i	$-0.603935\pi$
$167$	−8.29004	−0.641502	−0.320751	+	0.947164i	$0.603935\pi$
$168$	−0.310936	−0.0239892
$169$	−12.5296	−0.963818
$170$	16.3050	1.25054
$171$	−24.7614	−1.89355
$172$	−3.04884	−0.232472
$173$	3.30808	0.251509	0.125754	−	0.992061i	$-0.459865\pi$
$173$	3.30808	0.251509	0.125754	+	0.992061i	$0.459865\pi$
$174$	−0.0546465	−0.00414274
$175$	−36.8080	−2.78242
$176$	−0.435748	−0.0328458
$177$	0.622168	0.0467650
$178$	2.11755	0.158717
$179$	17.3494	1.29676	0.648378	−	0.761319i	$-0.275448\pi$
$179$	17.3494	1.29676	0.648378	+	0.761319i	$0.275448\pi$
$180$	10.4843	0.781454
$181$	25.4989	1.89532	0.947658	−	0.319288i	$-0.103444\pi$
$181$	25.4989	1.89532	0.947658	+	0.319288i	$0.103444\pi$
$182$	3.48484	0.258314
$183$	0.718490	0.0531123
$184$	1.27945	0.0943224
$185$	37.1827	2.73372
$186$	0.251186	0.0184179
$187$	−2.03047	−0.148483
$188$	4.10573	0.299441
$189$	−1.86445	−0.135619
$190$	28.9172	2.09788
$191$	22.8773	1.65534	0.827671	−	0.561214i	$-0.189665\pi$
$191$	22.8773	1.65534	0.827671	+	0.561214i	$0.189665\pi$
$192$	−0.0611935	−0.00441626
$193$	6.99022	0.503167	0.251584	−	0.967836i	$-0.419049\pi$
$193$	6.99022	0.503167	0.251584	+	0.967836i	$0.419049\pi$
$194$	−1.77788	−0.127644
$195$	0.146853	0.0105164
$196$	18.8186	1.34419
$197$	−2.50115	−0.178200	−0.0890998	−	0.996023i	$-0.528399\pi$
$197$	−2.50115	−0.178200	−0.0890998	+	0.996023i	$0.528399\pi$
$198$	−1.30561	−0.0927859
$199$	6.69601	0.474667	0.237334	−	0.971428i	$-0.423727\pi$
$199$	6.69601	0.474667	0.237334	+	0.971428i	$0.423727\pi$
$200$	−7.24396	−0.512225
$201$	0.491069	0.0346373
$202$	−1.43584	−0.101025
$203$	4.53757	0.318475
$204$	−0.285145	−0.0199641
$205$	−28.7164	−2.00564
$206$	2.93501	0.204492
$207$	3.83356	0.266451
$208$	0.685830	0.0475538
$209$	−3.60107	−0.249091
$210$	1.08801	0.0750797
$211$	−26.0947	−1.79643	−0.898217	−	0.439553i	$-0.855137\pi$
$211$	−26.0947	−1.79643	−0.898217	+	0.439553i	$0.855137\pi$
$212$	9.92221	0.681460
$213$	−0.0933150	−0.00639384
$214$	5.29527	0.361977
$215$	10.6683	0.727573
$216$	−0.366932	−0.0249665
$217$	−20.8572	−1.41588
$218$	9.98409	0.676208
$219$	−0.238503	−0.0161165
$220$	1.52474	0.102798
$221$	3.19578	0.214972
$222$	−0.650256	−0.0436423
$223$	−5.41806	−0.362820	−0.181410	−	0.983408i	$-0.558066\pi$
$223$	−5.41806	−0.362820	−0.181410	+	0.983408i	$0.558066\pi$
$224$	5.08120	0.339502
$225$	−21.7048	−1.44698
$226$	−2.51508	−0.167301
$227$	−4.42350	−0.293598	−0.146799	−	0.989166i	$-0.546897\pi$
$227$	−4.42350	−0.293598	−0.146799	+	0.989166i	$0.546897\pi$
$228$	−0.505709	−0.0334914
$229$	−0.821632	−0.0542950	−0.0271475	−	0.999631i	$-0.508642\pi$
$229$	−0.821632	−0.0542950	−0.0271475	+	0.999631i	$0.508642\pi$
$230$	−4.47697	−0.295203
$231$	−0.135490	−0.00891458
$232$	0.893012	0.0586291
$233$	15.1979	0.995645	0.497822	−	0.867279i	$-0.334133\pi$
$233$	15.1979	0.995645	0.497822	+	0.867279i	$0.334133\pi$
$234$	2.05492	0.134334
$235$	−14.3665	−0.937169
$236$	−10.1672	−0.661831
$237$	−0.0538491	−0.00349788
$238$	23.6770	1.53475
$239$	4.92727	0.318719	0.159359	−	0.987221i	$-0.449057\pi$
$239$	4.92727	0.318719	0.159359	+	0.987221i	$0.449057\pi$
$240$	0.214124	0.0138217
$241$	18.4369	1.18762	0.593811	−	0.804604i	$-0.297623\pi$
$241$	18.4369	1.18762	0.593811	+	0.804604i	$0.297623\pi$
$242$	10.8101	0.694901
$243$	−1.64947	−0.105814
$244$	−11.7413	−0.751659
$245$	−65.8488	−4.20693
$246$	0.502197	0.0320189
$247$	5.66777	0.360632
$248$	−4.10479	−0.260654
$249$	−0.998606	−0.0632841
$250$	7.85192	0.496599
$251$	22.6972	1.43264	0.716318	−	0.697774i	$-0.245826\pi$
$251$	22.6972	1.43264	0.716318	+	0.697774i	$0.245826\pi$
$252$	15.2246	0.959058
$253$	0.557519	0.0350509
$254$	18.2688	1.14629
$255$	0.997762	0.0624823
$256$	1.00000	0.0625000
$257$	−12.2930	−0.766816	−0.383408	−	0.923579i	$-0.625250\pi$
$257$	−12.2930	−0.766816	−0.383408	+	0.923579i	$0.625250\pi$
$258$	−0.186569	−0.0116153
$259$	53.9940	3.35503
$260$	−2.39981	−0.148830
$261$	2.67569	0.165621
$262$	−6.84595	−0.422944
$263$	−28.1035	−1.73293	−0.866467	−	0.499234i	$-0.833615\pi$
$263$	−28.1035	−1.73293	−0.866467	+	0.499234i	$0.833615\pi$
$264$	−0.0266650	−0.00164111
$265$	−34.7192	−2.13278
$266$	41.9916	2.57467
$267$	0.129580	0.00793018
$268$	−8.02486	−0.490196
$269$	−14.3762	−0.876534	−0.438267	−	0.898845i	$-0.644408\pi$
$269$	−14.3762	−0.876534	−0.438267	+	0.898845i	$0.644408\pi$
$270$	1.28394	0.0781383
$271$	−10.6452	−0.646651	−0.323325	−	0.946288i	$-0.604801\pi$
$271$	−10.6452	−0.646651	−0.323325	+	0.946288i	$0.604801\pi$
$272$	4.65973	0.282538
$273$	0.213249	0.0129064
$274$	18.1594	1.09705
$275$	−3.15654	−0.190347
$276$	0.0782940	0.00471275
$277$	−23.5976	−1.41784	−0.708921	−	0.705288i	$-0.750818\pi$
$277$	−23.5976	−1.41784	−0.708921	+	0.705288i	$0.750818\pi$
$278$	−10.7728	−0.646108
$279$	−12.2990	−0.736321
$280$	−17.7798	−1.06255
$281$	−4.46587	−0.266411	−0.133206	−	0.991088i	$-0.542527\pi$
$281$	−4.46587	−0.266411	−0.133206	+	0.991088i	$0.542527\pi$
$282$	0.251244	0.0149614
$283$	30.4143	1.80794	0.903972	−	0.427592i	$-0.140638\pi$
$283$	30.4143	1.80794	0.903972	+	0.427592i	$0.140638\pi$
$284$	1.52492	0.0904873
$285$	1.76954	0.104819
$286$	0.298849	0.0176713
$287$	−41.7000	−2.46147
$288$	2.99626	0.176556
$289$	4.71309	0.277241
$290$	−3.12477	−0.183493
$291$	−0.108795	−0.00637765
$292$	3.89753	0.228085
$293$	−29.8683	−1.74493	−0.872463	−	0.488680i	$-0.837479\pi$
$293$	−29.8683	−1.74493	−0.872463	+	0.488680i	$0.837479\pi$
$294$	1.15157	0.0671612
$295$	35.5766	2.07135
$296$	10.6262	0.617638
$297$	−0.159890	−0.00927775
$298$	10.0866	0.584298
$299$	−0.877486	−0.0507463
$300$	−0.443283	−0.0255929
$301$	15.4918	0.892931
$302$	−11.2302	−0.646227
$303$	−0.0878641	−0.00504766
$304$	8.26410	0.473979
$305$	41.0844	2.35248
$306$	13.9617	0.798140
$307$	21.2830	1.21468	0.607342	−	0.794441i	$-0.292236\pi$
$307$	21.2830	1.21468	0.607342	+	0.794441i	$0.292236\pi$
$308$	2.21412	0.126161
$309$	0.179603	0.0102173
$310$	14.3632	0.815776
$311$	22.1697	1.25713	0.628563	−	0.777758i	$-0.283643\pi$
$311$	22.1697	1.25713	0.628563	+	0.777758i	$0.283643\pi$
$312$	0.0419683	0.00237599
$313$	−25.9576	−1.46721	−0.733604	−	0.679577i	$-0.762163\pi$
$313$	−25.9576	−1.46721	−0.733604	+	0.679577i	$0.762163\pi$
$314$	−18.3948	−1.03808
$315$	−53.2729	−3.00159
$316$	0.879981	0.0495028
$317$	26.5281	1.48997	0.744985	−	0.667082i	$-0.232457\pi$
$317$	26.5281	1.48997	0.744985	+	0.667082i	$0.232457\pi$
$318$	0.607174	0.0340486
$319$	0.389128	0.0217870
$320$	−3.49914	−0.195608
$321$	0.324036	0.0180859
$322$	−6.50115	−0.362295
$323$	38.5085	2.14267
$324$	8.96631	0.498128
$325$	4.96813	0.275582
$326$	18.9079	1.04721
$327$	0.610961	0.0337862
$328$	−8.20672	−0.453140
$329$	−20.8621	−1.15016
$330$	0.0933043	0.00513623
$331$	7.32483	0.402609	0.201304	−	0.979529i	$-0.435482\pi$
$331$	7.32483	0.402609	0.201304	+	0.979529i	$0.435482\pi$
$332$	16.3188	0.895612
$333$	31.8389	1.74476
$334$	8.29004	0.453611
$335$	28.0801	1.53418
$336$	0.310936	0.0169630
$337$	28.2581	1.53932	0.769658	−	0.638456i	$-0.220427\pi$
$337$	28.2581	1.53932	0.769658	+	0.638456i	$0.220427\pi$
$338$	12.5296	0.681522
$339$	−0.153906	−0.00835905
$340$	−16.3050	−0.884265
$341$	−1.78865	−0.0968611
$342$	24.7614	1.33894
$343$	−60.0526	−3.24254
$344$	3.04884	0.164382
$345$	−0.273961	−0.0147496
$346$	−3.30808	−0.177843
$347$	−9.50897	−0.510468	−0.255234	−	0.966879i	$-0.582153\pi$
$347$	−9.50897	−0.510468	−0.255234	+	0.966879i	$0.582153\pi$
$348$	0.0546465	0.00292936
$349$	−17.2692	−0.924396	−0.462198	−	0.886777i	$-0.652939\pi$
$349$	−17.2692	−0.924396	−0.462198	+	0.886777i	$0.652939\pi$
$350$	36.8080	1.96747
$351$	0.251653	0.0134322
$352$	0.435748	0.0232255
$353$	−6.93260	−0.368985	−0.184493	−	0.982834i	$-0.559064\pi$
$353$	−6.93260	−0.368985	−0.184493	+	0.982834i	$0.559064\pi$
$354$	−0.622168	−0.0330679
$355$	−5.33590	−0.283200
$356$	−2.11755	−0.112230
$357$	1.44888	0.0766828
$358$	−17.3494	−0.916944
$359$	1.92339	0.101513	0.0507563	−	0.998711i	$-0.483837\pi$
$359$	1.92339	0.101513	0.0507563	+	0.998711i	$0.483837\pi$
$360$	−10.4843	−0.552572
$361$	49.2954	2.59449
$362$	−25.4989	−1.34019
$363$	0.661509	0.0347202
$364$	−3.48484	−0.182655
$365$	−13.6380	−0.713844
$366$	−0.718490	−0.0375561
$367$	9.66206	0.504356	0.252178	−	0.967681i	$-0.418853\pi$
$367$	9.66206	0.504356	0.252178	+	0.967681i	$0.418853\pi$
$368$	−1.27945	−0.0666960
$369$	−24.5894	−1.28007
$370$	−37.1827	−1.93303
$371$	−50.4167	−2.61751
$372$	−0.251186	−0.0130234
$373$	1.27250	0.0658876	0.0329438	−	0.999457i	$-0.489512\pi$
$373$	1.27250	0.0658876	0.0329438	+	0.999457i	$0.489512\pi$
$374$	2.03047	0.104993
$375$	0.480486	0.0248122
$376$	−4.10573	−0.211737
$377$	−0.612454	−0.0315430
$378$	1.86445	0.0958971
$379$	−25.8845	−1.32960	−0.664798	−	0.747023i	$-0.731483\pi$
$379$	−25.8845	−1.32960	−0.664798	+	0.747023i	$0.731483\pi$
$380$	−28.9172	−1.48342
$381$	1.11793	0.0572734
$382$	−22.8773	−1.17050
$383$	−7.70473	−0.393693	−0.196847	−	0.980434i	$-0.563070\pi$
$383$	−7.70473	−0.393693	−0.196847	+	0.980434i	$0.563070\pi$
$384$	0.0611935	0.00312277
$385$	−7.74753	−0.394850
$386$	−6.99022	−0.355793
$387$	9.13511	0.464364
$388$	1.77788	0.0902581
$389$	−7.85027	−0.398025	−0.199012	−	0.979997i	$-0.563773\pi$
$389$	−7.85027	−0.398025	−0.199012	+	0.979997i	$0.563773\pi$
$390$	−0.146853	−0.00743619
$391$	−5.96190	−0.301506
$392$	−18.8186	−0.950483
$393$	−0.418927	−0.0211321
$394$	2.50115	0.126006
$395$	−3.07918	−0.154930
$396$	1.30561	0.0656095
$397$	6.42948	0.322686	0.161343	−	0.986898i	$-0.448417\pi$
$397$	6.42948	0.322686	0.161343	+	0.986898i	$0.448417\pi$
$398$	−6.69601	−0.335641
$399$	2.56961	0.128641
$400$	7.24396	0.362198
$401$	−18.2141	−0.909566	−0.454783	−	0.890602i	$-0.650283\pi$
$401$	−18.2141	−0.909566	−0.454783	+	0.890602i	$0.650283\pi$
$402$	−0.491069	−0.0244923
$403$	2.81519	0.140234
$404$	1.43584	0.0714358
$405$	−31.3744	−1.55900
$406$	−4.53757	−0.225196
$407$	4.63037	0.229519
$408$	0.285145	0.0141168
$409$	24.2767	1.20041	0.600203	−	0.799848i	$-0.295086\pi$
$409$	24.2767	1.20041	0.600203	+	0.799848i	$0.295086\pi$
$410$	28.7164	1.41820
$411$	1.11124	0.0548133
$412$	−2.93501	−0.144597
$413$	51.6618	2.54211
$414$	−3.83356	−0.188409
$415$	−57.1018	−2.80302
$416$	−0.685830	−0.0336256
$417$	−0.659224	−0.0322823
$418$	3.60107	0.176134
$419$	−3.50317	−0.171141	−0.0855706	−	0.996332i	$-0.527271\pi$
$419$	−3.50317	−0.171141	−0.0855706	+	0.996332i	$0.527271\pi$
$420$	−1.08801	−0.0530894
$421$	14.3544	0.699589	0.349794	−	0.936827i	$-0.386251\pi$
$421$	14.3544	0.699589	0.349794	+	0.936827i	$0.386251\pi$
$422$	26.0947	1.27027
$423$	−12.3018	−0.598135
$424$	−9.92221	−0.481865
$425$	33.7549	1.63735
$426$	0.0933150	0.00452113
$427$	59.6598	2.88714
$428$	−5.29527	−0.255956
$429$	0.0182876	0.000882935 0
$430$	−10.6683	−0.514472
$431$	25.1471	1.21129	0.605646	−	0.795734i	$-0.292915\pi$
$431$	25.1471	1.21129	0.605646	+	0.795734i	$0.292915\pi$
$432$	0.366932	0.0176540
$433$	−17.7019	−0.850697	−0.425348	−	0.905030i	$-0.639848\pi$
$433$	−17.7019	−0.850697	−0.425348	+	0.905030i	$0.639848\pi$
$434$	20.8572	1.00118
$435$	−0.191216	−0.00916808
$436$	−9.98409	−0.478151
$437$	−10.5735	−0.505800
$438$	0.238503	0.0113961
$439$	−23.5557	−1.12425	−0.562127	−	0.827051i	$-0.690017\pi$
$439$	−23.5557	−1.12425	−0.562127	+	0.827051i	$0.690017\pi$
$440$	−1.52474	−0.0726893
$441$	−56.3853	−2.68501
$442$	−3.19578	−0.152008
$443$	9.43668	0.448350	0.224175	−	0.974549i	$-0.428031\pi$
$443$	9.43668	0.448350	0.224175	+	0.974549i	$0.428031\pi$
$444$	0.650256	0.0308598
$445$	7.40959	0.351248
$446$	5.41806	0.256552
$447$	0.617231	0.0291940
$448$	−5.08120	−0.240064
$449$	0.654226	0.0308748	0.0154374	−	0.999881i	$-0.495086\pi$
$449$	0.654226	0.0308748	0.0154374	+	0.999881i	$0.495086\pi$
$450$	21.7048	1.02317
$451$	−3.57606	−0.168390
$452$	2.51508	0.118299
$453$	−0.687216	−0.0322882
$454$	4.42350	0.207605
$455$	12.1939	0.571660
$456$	0.505709	0.0236820
$457$	−37.5655	−1.75724	−0.878619	−	0.477523i	$-0.841535\pi$
$457$	−37.5655	−1.75724	−0.878619	+	0.477523i	$0.841535\pi$
$458$	0.821632	0.0383924
$459$	1.70980	0.0798067
$460$	4.47697	0.208740
$461$	−14.4018	−0.670758	−0.335379	−	0.942083i	$-0.608864\pi$
$461$	−14.4018	−0.670758	−0.335379	+	0.942083i	$0.608864\pi$
$462$	0.135490	0.00630356
$463$	−13.4904	−0.626951	−0.313475	−	0.949596i	$-0.601493\pi$
$463$	−13.4904	−0.626951	−0.313475	+	0.949596i	$0.601493\pi$
$464$	−0.893012	−0.0414570
$465$	0.878935	0.0407596
$466$	−15.1979	−0.704027
$467$	28.0903	1.29986	0.649932	−	0.759993i	$-0.274797\pi$
$467$	28.0903	1.29986	0.649932	+	0.759993i	$0.274797\pi$
$468$	−2.05492	−0.0949888
$469$	40.7759	1.88286
$470$	14.3665	0.662678
$471$	−1.12564	−0.0518667
$472$	10.1672	0.467985
$473$	1.32853	0.0610858
$474$	0.0538491	0.00247337
$475$	59.8648	2.74679
$476$	−23.6770	−1.08523
$477$	−29.7295	−1.36122
$478$	−4.92727	−0.225368
$479$	31.3288	1.43145	0.715724	−	0.698383i	$-0.246097\pi$
$479$	31.3288	1.43145	0.715724	+	0.698383i	$0.246097\pi$
$480$	−0.214124	−0.00977339
$481$	−7.28780	−0.332295
$482$	−18.4369	−0.839776
$483$	−0.397828	−0.0181018
$484$	−10.8101	−0.491369
$485$	−6.22104	−0.282483
$486$	1.64947	0.0748217
$487$	−24.3563	−1.10369	−0.551844	−	0.833947i	$-0.686076\pi$
$487$	−24.3563	−1.10369	−0.551844	+	0.833947i	$0.686076\pi$
$488$	11.7413	0.531503
$489$	1.15704	0.0523231
$490$	65.8488	2.97475
$491$	−6.03591	−0.272397	−0.136198	−	0.990682i	$-0.543488\pi$
$491$	−6.03591	−0.272397	−0.136198	+	0.990682i	$0.543488\pi$
$492$	−0.502197	−0.0226408
$493$	−4.16119	−0.187411
$494$	−5.66777	−0.255005
$495$	−4.56852	−0.205340
$496$	4.10479	0.184310
$497$	−7.74842	−0.347564
$498$	0.998606	0.0447486
$499$	−28.7559	−1.28729	−0.643647	−	0.765323i	$-0.722579\pi$
$499$	−28.7559	−1.28729	−0.643647	+	0.765323i	$0.722579\pi$
$500$	−7.85192	−0.351149
$501$	0.507296	0.0226643
$502$	−22.6972	−1.01303
$503$	−21.9100	−0.976919	−0.488460	−	0.872586i	$-0.662441\pi$
$503$	−21.9100	−0.976919	−0.488460	+	0.872586i	$0.662441\pi$
$504$	−15.2246	−0.678156
$505$	−5.02421	−0.223574
$506$	−0.557519	−0.0247847
$507$	0.766732	0.0340518
$508$	−18.2688	−0.810548
$509$	−8.26373	−0.366283	−0.183142	−	0.983087i	$-0.558627\pi$
$509$	−8.26373	−0.366283	−0.183142	+	0.983087i	$0.558627\pi$
$510$	−0.997762	−0.0441816
$511$	−19.8041	−0.876082
$512$	−1.00000	−0.0441942
$513$	3.03236	0.133882
$514$	12.2930	0.542221
$515$	10.2700	0.452550
$516$	0.186569	0.00821325
$517$	−1.78907	−0.0786831
$518$	−53.9940	−2.37236
$519$	−0.202433	−0.00888581
$520$	2.39981	0.105239
$521$	35.1737	1.54099	0.770495	−	0.637446i	$-0.220009\pi$
$521$	35.1737	1.54099	0.770495	+	0.637446i	$0.220009\pi$
$522$	−2.67569	−0.117112
$523$	3.65078	0.159638	0.0798188	−	0.996809i	$-0.474566\pi$
$523$	3.65078	0.159638	0.0798188	+	0.996809i	$0.474566\pi$
$524$	6.84595	0.299067
$525$	2.25241	0.0983032
$526$	28.1035	1.22537
$527$	19.1272	0.833194
$528$	0.0266650	0.00116044
$529$	−21.3630	−0.928826
$530$	34.7192	1.50810
$531$	30.4636	1.32201
$532$	−41.9916	−1.82056
$533$	5.62841	0.243794
$534$	−0.129580	−0.00560748
$535$	18.5289	0.801072
$536$	8.02486	0.346621
$537$	−1.06167	−0.0458144
$538$	14.3762	0.619803
$539$	−8.20017	−0.353206
$540$	−1.28394	−0.0552521
$541$	4.10374	0.176433	0.0882167	−	0.996101i	$-0.471883\pi$
$541$	4.10374	0.176433	0.0882167	+	0.996101i	$0.471883\pi$
$542$	10.6452	0.457251
$543$	−1.56036	−0.0669616
$544$	−4.65973	−0.199784
$545$	34.9357	1.49648
$546$	−0.213249	−0.00912623
$547$	−32.7998	−1.40242	−0.701208	−	0.712957i	$-0.747356\pi$
$547$	−32.7998	−1.40242	−0.701208	+	0.712957i	$0.747356\pi$
$548$	−18.1594	−0.775732
$549$	35.1799	1.50144
$550$	3.15654	0.134595
$551$	−7.37994	−0.314396
$552$	−0.0782940	−0.00333241
$553$	−4.47136	−0.190142
$554$	23.5976	1.00257
$555$	−2.27534	−0.0965826
$556$	10.7728	0.456868
$557$	−23.4568	−0.993897	−0.496949	−	0.867780i	$-0.665546\pi$
$557$	−23.4568	−0.993897	−0.496949	+	0.867780i	$0.665546\pi$
$558$	12.2990	0.520658
$559$	−2.09099	−0.0884393
$560$	17.7798	0.751334
$561$	0.124251	0.00524590
$562$	4.46587	0.188381
$563$	11.8134	0.497876	0.248938	−	0.968519i	$-0.419918\pi$
$563$	11.8134	0.497876	0.248938	+	0.968519i	$0.419918\pi$
$564$	−0.251244	−0.0105793
$565$	−8.80061	−0.370244
$566$	−30.4143	−1.27841
$567$	−45.5596	−1.91332
$568$	−1.52492	−0.0639842
$569$	2.62999	0.110255	0.0551274	−	0.998479i	$-0.482444\pi$
$569$	2.62999	0.110255	0.0551274	+	0.998479i	$0.482444\pi$
$570$	−1.76954	−0.0741181
$571$	−32.6436	−1.36609	−0.683046	−	0.730375i	$-0.739345\pi$
$571$	−32.6436	−1.36609	−0.683046	+	0.730375i	$0.739345\pi$
$572$	−0.298849	−0.0124955
$573$	−1.39994	−0.0584833
$574$	41.7000	1.74052
$575$	−9.26829	−0.386514
$576$	−2.99626	−0.124844
$577$	−11.0412	−0.459650	−0.229825	−	0.973232i	$-0.573815\pi$
$577$	−11.0412	−0.459650	−0.229825	+	0.973232i	$0.573815\pi$
$578$	−4.71309	−0.196039
$579$	−0.427756	−0.0177769
$580$	3.12477	0.129749
$581$	−82.9192	−3.44007
$582$	0.108795	0.00450968
$583$	−4.32359	−0.179065
$584$	−3.89753	−0.161281
$585$	7.19045	0.297289
$586$	29.8683	1.23385
$587$	−37.3438	−1.54134	−0.770671	−	0.637233i	$-0.780079\pi$
$587$	−37.3438	−1.54134	−0.770671	+	0.637233i	$0.780079\pi$
$588$	−1.15157	−0.0474901
$589$	33.9224	1.39775
$590$	−35.5766	−1.46466
$591$	0.153054	0.00629580
$592$	−10.6262	−0.436736
$593$	13.2150	0.542675	0.271337	−	0.962484i	$-0.412534\pi$
$593$	13.2150	0.542675	0.271337	+	0.962484i	$0.412534\pi$
$594$	0.159890	0.00656036
$595$	82.8491	3.39648
$596$	−10.0866	−0.413161
$597$	−0.409752	−0.0167700
$598$	0.877486	0.0358831
$599$	−25.9499	−1.06028	−0.530142	−	0.847909i	$-0.677861\pi$
$599$	−25.9499	−1.06028	−0.530142	+	0.847909i	$0.677861\pi$
$600$	0.443283	0.0180969
$601$	−34.2177	−1.39577	−0.697885	−	0.716210i	$-0.745875\pi$
$601$	−34.2177	−1.39577	−0.697885	+	0.716210i	$0.745875\pi$
$602$	−15.4918	−0.631397
$603$	24.0445	0.979169
$604$	11.2302	0.456952
$605$	37.8261	1.53785
$606$	0.0878641	0.00356924
$607$	−22.1999	−0.901066	−0.450533	−	0.892760i	$-0.648766\pi$
$607$	−22.1999	−0.901066	−0.450533	+	0.892760i	$0.648766\pi$
$608$	−8.26410	−0.335154
$609$	−0.277670	−0.0112517
$610$	−41.0844	−1.66346
$611$	2.81584	0.113917
$612$	−13.9617	−0.564370
$613$	25.2622	1.02033	0.510166	−	0.860076i	$-0.329584\pi$
$613$	25.2622	1.02033	0.510166	+	0.860076i	$0.329584\pi$
$614$	−21.2830	−0.858911
$615$	1.75726	0.0708594
$616$	−2.21412	−0.0892096
$617$	6.32614	0.254681	0.127340	−	0.991859i	$-0.459356\pi$
$617$	6.32614	0.254681	0.127340	+	0.991859i	$0.459356\pi$
$618$	−0.179603	−0.00722470
$619$	−12.0759	−0.485370	−0.242685	−	0.970105i	$-0.578028\pi$
$619$	−12.0759	−0.485370	−0.242685	+	0.970105i	$0.578028\pi$
$620$	−14.3632	−0.576841
$621$	−0.469471	−0.0188392
$622$	−22.1697	−0.888923
$623$	10.7597	0.431078
$624$	−0.0419683	−0.00168008
$625$	−8.74485	−0.349794
$626$	25.9576	1.03747
$627$	0.220362	0.00880041
$628$	18.3948	0.734031
$629$	−49.5154	−1.97431
$630$	53.2729	2.12244
$631$	−25.0928	−0.998929	−0.499464	−	0.866334i	$-0.666470\pi$
$631$	−25.0928	−0.998929	−0.499464	+	0.866334i	$0.666470\pi$
$632$	−0.879981	−0.0350038
$633$	1.59682	0.0634681
$634$	−26.5281	−1.05357
$635$	63.9251	2.53679
$636$	−0.607174	−0.0240760
$637$	12.9064	0.511369
$638$	−0.389128	−0.0154057
$639$	−4.56905	−0.180749
$640$	3.49914	0.138316
$641$	9.64622	0.381003	0.190501	−	0.981687i	$-0.438989\pi$
$641$	9.64622	0.381003	0.190501	+	0.981687i	$0.438989\pi$
$642$	−0.324036	−0.0127887
$643$	1.36328	0.0537624	0.0268812	−	0.999639i	$-0.491442\pi$
$643$	1.36328	0.0537624	0.0268812	+	0.999639i	$0.491442\pi$
$644$	6.50115	0.256181
$645$	−0.652831	−0.0257052
$646$	−38.5085	−1.51510
$647$	31.6755	1.24529	0.622646	−	0.782504i	$-0.286058\pi$
$647$	31.6755	1.24529	0.622646	+	0.782504i	$0.286058\pi$
$648$	−8.96631	−0.352230
$649$	4.43036	0.173907
$650$	−4.96813	−0.194866
$651$	1.27633	0.0500232
$652$	−18.9079	−0.740490
$653$	31.5563	1.23489	0.617447	−	0.786612i	$-0.288167\pi$
$653$	31.5563	1.23489	0.617447	+	0.786612i	$0.288167\pi$
$654$	−0.610961	−0.0238905
$655$	−23.9549	−0.935996
$656$	8.20672	0.320418
$657$	−11.6780	−0.455602
$658$	20.8621	0.813287
$659$	7.24458	0.282209	0.141104	−	0.989995i	$-0.454935\pi$
$659$	7.24458	0.282209	0.141104	+	0.989995i	$0.454935\pi$
$660$	−0.0933043	−0.00363187
$661$	−5.77347	−0.224562	−0.112281	−	0.993677i	$-0.535816\pi$
$661$	−5.77347	−0.224562	−0.112281	+	0.993677i	$0.535816\pi$
$662$	−7.32483	−0.284687
$663$	−0.195561	−0.00759496
$664$	−16.3188	−0.633293
$665$	146.934	5.69786
$666$	−31.8389	−1.23373
$667$	1.14256	0.0442403
$668$	−8.29004	−0.320751
$669$	0.331550	0.0128184
$670$	−28.0801	−1.08483
$671$	5.11625	0.197511
$672$	−0.310936	−0.0119946
$673$	−16.6691	−0.642547	−0.321273	−	0.946986i	$-0.604111\pi$
$673$	−16.6691	−0.642547	−0.321273	+	0.946986i	$0.604111\pi$
$674$	−28.2581	−1.08846
$675$	2.65804	0.102308
$676$	−12.5296	−0.481909
$677$	−43.1623	−1.65886	−0.829430	−	0.558610i	$-0.811335\pi$
$677$	−43.1623	−1.65886	−0.829430	+	0.558610i	$0.811335\pi$
$678$	0.153906	0.00591074
$679$	−9.03376	−0.346684
$680$	16.3050	0.625270
$681$	0.270689	0.0103728
$682$	1.78865	0.0684911
$683$	40.4684	1.54848	0.774239	−	0.632893i	$-0.218133\pi$
$683$	40.4684	1.54848	0.774239	+	0.632893i	$0.218133\pi$
$684$	−24.7614	−0.946774
$685$	63.5423	2.42783
$686$	60.0526	2.29282
$687$	0.0502785	0.00191825
$688$	−3.04884	−0.116236
$689$	6.80495	0.259248
$690$	0.273961	0.0104295
$691$	−49.2510	−1.87359	−0.936797	−	0.349873i	$-0.886225\pi$
$691$	−49.2510	−1.87359	−0.936797	+	0.349873i	$0.886225\pi$
$692$	3.30808	0.125754
$693$	−6.63408	−0.252008
$694$	9.50897	0.360955
$695$	−37.6954	−1.42987
$696$	−0.0546465	−0.00207137
$697$	38.2411	1.44848
$698$	17.2692	0.653647
$699$	−0.930009	−0.0351762
$700$	−36.8080	−1.39121
$701$	−23.0826	−0.871818	−0.435909	−	0.899991i	$-0.643573\pi$
$701$	−23.0826	−0.871818	−0.435909	+	0.899991i	$0.643573\pi$
$702$	−0.251653	−0.00949802
$703$	−87.8163	−3.31206
$704$	−0.435748	−0.0164229
$705$	0.879137	0.0331102
$706$	6.93260	0.260912
$707$	−7.29580	−0.274387
$708$	0.622168	0.0233825
$709$	−29.4652	−1.10659	−0.553294	−	0.832986i	$-0.686629\pi$
$709$	−29.4652	−1.10659	−0.553294	+	0.832986i	$0.686629\pi$
$710$	5.33590	0.200253
$711$	−2.63665	−0.0988821
$712$	2.11755	0.0793585
$713$	−5.25187	−0.196684
$714$	−1.44888	−0.0542229
$715$	1.04571	0.0391075
$716$	17.3494	0.648378
$717$	−0.301517	−0.0112604
$718$	−1.92339	−0.0717802
$719$	−16.3129	−0.608370	−0.304185	−	0.952613i	$-0.598384\pi$
$719$	−16.3129	−0.608370	−0.304185	+	0.952613i	$0.598384\pi$
$720$	10.4843	0.390727
$721$	14.9134	0.555402
$722$	−49.2954	−1.83458
$723$	−1.12822	−0.0419588
$724$	25.4989	0.947658
$725$	−6.46894	−0.240250
$726$	−0.661509	−0.0245509
$727$	18.1167	0.671910	0.335955	−	0.941878i	$-0.390941\pi$
$727$	18.1167	0.671910	0.335955	+	0.941878i	$0.390941\pi$
$728$	3.48484	0.129157
$729$	−26.7980	−0.992519
$730$	13.6380	0.504764
$731$	−14.2068	−0.525457
$732$	0.718490	0.0265562
$733$	18.9057	0.698298	0.349149	−	0.937067i	$-0.386471\pi$
$733$	18.9057	0.698298	0.349149	+	0.937067i	$0.386471\pi$
$734$	−9.66206	−0.356633
$735$	4.02952	0.148631
$736$	1.27945	0.0471612
$737$	3.49682	0.128807
$738$	24.5894	0.905149
$739$	53.3588	1.96283	0.981417	−	0.191887i	$-0.0614607\pi$
$739$	53.3588	1.96283	0.981417	+	0.191887i	$0.0614607\pi$
$740$	37.1827	1.36686
$741$	−0.346830	−0.0127411
$742$	50.4167	1.85086
$743$	26.4758	0.971304	0.485652	−	0.874152i	$-0.338582\pi$
$743$	26.4758	0.971304	0.485652	+	0.874152i	$0.338582\pi$
$744$	0.251186	0.00920893
$745$	35.2942	1.29308
$746$	−1.27250	−0.0465896
$747$	−48.8954	−1.78899
$748$	−2.03047	−0.0742413
$749$	26.9063	0.983135
$750$	−0.480486	−0.0175449
$751$	−7.00119	−0.255477	−0.127738	−	0.991808i	$-0.540772\pi$
$751$	−7.00119	−0.255477	−0.127738	+	0.991808i	$0.540772\pi$
$752$	4.10573	0.149721
$753$	−1.38892	−0.0506151
$754$	0.612454	0.0223043
$755$	−39.2961	−1.43013
$756$	−1.86445	−0.0678095
$757$	−17.7201	−0.644047	−0.322024	−	0.946732i	$-0.604363\pi$
$757$	−17.7201	−0.644047	−0.322024	+	0.946732i	$0.604363\pi$
$758$	25.8845	0.940167
$759$	−0.0341165	−0.00123835
$760$	28.9172	1.04894
$761$	−41.4532	−1.50268	−0.751339	−	0.659916i	$-0.770592\pi$
$761$	−41.4532	−1.50268	−0.751339	+	0.659916i	$0.770592\pi$
$762$	−1.11793	−0.0404984
$763$	50.7311	1.83659
$764$	22.8773	0.827671
$765$	48.8541	1.76632
$766$	7.70473	0.278383
$767$	−6.97300	−0.251780
$768$	−0.0611935	−0.00220813
$769$	32.3244	1.16565	0.582824	−	0.812599i	$-0.301948\pi$
$769$	32.3244	1.16565	0.582824	+	0.812599i	$0.301948\pi$
$770$	7.74753	0.279201
$771$	0.752251	0.0270917
$772$	6.99022	0.251584
$773$	7.51575	0.270323	0.135161	−	0.990824i	$-0.456845\pi$
$773$	7.51575	0.270323	0.135161	+	0.990824i	$0.456845\pi$
$774$	−9.13511	−0.328355
$775$	29.7349	1.06811
$776$	−1.77788	−0.0638221
$777$	−3.30408	−0.118533
$778$	7.85027	0.281446
$779$	67.8211	2.42994
$780$	0.146853	0.00525818
$781$	−0.664481	−0.0237770
$782$	5.96190	0.213197
$783$	−0.327674	−0.0117101
$784$	18.8186	0.672093
$785$	−64.3658	−2.29731
$786$	0.418927	0.0149426
$787$	−24.0165	−0.856095	−0.428048	−	0.903756i	$-0.640798\pi$
$787$	−24.0165	−0.856095	−0.428048	+	0.903756i	$0.640798\pi$
$788$	−2.50115	−0.0890998
$789$	1.71975	0.0612247
$790$	3.07918	0.109552
$791$	−12.7796	−0.454391
$792$	−1.30561	−0.0463930
$793$	−8.05253	−0.285954
$794$	−6.42948	−0.228174
$795$	2.12459	0.0753513
$796$	6.69601	0.237334
$797$	1.82374	0.0646002	0.0323001	−	0.999478i	$-0.489717\pi$
$797$	1.82374	0.0646002	0.0323001	+	0.999478i	$0.489717\pi$
$798$	−2.56961	−0.0909632
$799$	19.1316	0.676828
$800$	−7.24396	−0.256113
$801$	6.34472	0.224180
$802$	18.2141	0.643161
$803$	−1.69834	−0.0599332
$804$	0.491069	0.0173187
$805$	−22.7484	−0.801776
$806$	−2.81519	−0.0991607
$807$	0.879731	0.0309680
$808$	−1.43584	−0.0505127
$809$	−37.5568	−1.32043	−0.660213	−	0.751079i	$-0.729534\pi$
$809$	−37.5568	−1.32043	−0.660213	+	0.751079i	$0.729534\pi$
$810$	31.3744	1.10238
$811$	−40.5144	−1.42265	−0.711326	−	0.702862i	$-0.751905\pi$
$811$	−40.5144	−1.42265	−0.711326	+	0.702862i	$0.751905\pi$
$812$	4.53757	0.159238
$813$	0.651418	0.0228462
$814$	−4.63037	−0.162294
$815$	66.1613	2.31753
$816$	−0.285145	−0.00998207
$817$	−25.1959	−0.881494
$818$	−24.2767	−0.848815
$819$	10.4415	0.364855
$820$	−28.7164	−1.00282
$821$	−40.1395	−1.40088	−0.700439	−	0.713712i	$-0.747013\pi$
$821$	−40.1395	−1.40088	−0.700439	+	0.713712i	$0.747013\pi$
$822$	−1.11124	−0.0387589
$823$	0.209763	0.00731188	0.00365594	−	0.999993i	$-0.498836\pi$
$823$	0.209763	0.00731188	0.00365594	+	0.999993i	$0.498836\pi$
$824$	2.93501	0.102246
$825$	0.193160	0.00672496
$826$	−51.6618	−1.79754
$827$	30.6362	1.06532	0.532662	−	0.846328i	$-0.321192\pi$
$827$	30.6362	1.06532	0.532662	+	0.846328i	$0.321192\pi$
$828$	3.83356	0.133225
$829$	0.925440	0.0321419	0.0160709	−	0.999871i	$-0.494884\pi$
$829$	0.925440	0.0321419	0.0160709	+	0.999871i	$0.494884\pi$
$830$	57.1018	1.98203
$831$	1.44402	0.0500925
$832$	0.685830	0.0237769
$833$	87.6896	3.03826
$834$	0.659224	0.0228270
$835$	29.0080	1.00386
$836$	−3.60107	−0.124546
$837$	1.50618	0.0520611
$838$	3.50317	0.121015
$839$	30.0803	1.03849	0.519244	−	0.854626i	$-0.326213\pi$
$839$	30.0803	1.03849	0.519244	+	0.854626i	$0.326213\pi$
$840$	1.08801	0.0375399
$841$	−28.2025	−0.972501
$842$	−14.3544	−0.494684
$843$	0.273282	0.00941233
$844$	−26.0947	−0.898217
$845$	43.8429	1.50824
$846$	12.3018	0.422945
$847$	54.9284	1.88736
$848$	9.92221	0.340730
$849$	−1.86116	−0.0638747
$850$	−33.7549	−1.15778
$851$	13.5958	0.466056
$852$	−0.0933150	−0.00319692
$853$	−37.1863	−1.27323	−0.636617	−	0.771180i	$-0.719667\pi$
$853$	−37.1863	−1.27323	−0.636617	+	0.771180i	$0.719667\pi$
$854$	−59.6598	−2.04152
$855$	86.6434	2.96314
$856$	5.29527	0.180988
$857$	−55.4575	−1.89439	−0.947197	−	0.320653i	$-0.896098\pi$
$857$	−55.4575	−1.89439	−0.947197	+	0.320653i	$0.896098\pi$
$858$	−0.0182876	−0.000624329 0
$859$	−2.98923	−0.101991	−0.0509957	−	0.998699i	$-0.516239\pi$
$859$	−2.98923	−0.101991	−0.0509957	+	0.998699i	$0.516239\pi$
$860$	10.6683	0.363786
$861$	2.55176	0.0869639
$862$	−25.1471	−0.856513
$863$	4.99496	0.170030	0.0850152	−	0.996380i	$-0.472906\pi$
$863$	4.99496	0.170030	0.0850152	+	0.996380i	$0.472906\pi$
$864$	−0.366932	−0.0124833
$865$	−11.5754	−0.393576
$866$	17.7019	0.601533
$867$	−0.288410	−0.00979493
$868$	−20.8572	−0.707941
$869$	−0.383451	−0.0130077
$870$	0.191216	0.00648281
$871$	−5.50369	−0.186485
$872$	9.98409	0.338104
$873$	−5.32698	−0.180291
$874$	10.5735	0.357654
$875$	39.8972	1.34877
$876$	−0.238503	−0.00805827
$877$	−14.5310	−0.490678	−0.245339	−	0.969437i	$-0.578899\pi$
$877$	−14.5310	−0.490678	−0.245339	+	0.969437i	$0.578899\pi$
$878$	23.5557	0.794967
$879$	1.82775	0.0616483
$880$	1.52474	0.0513991
$881$	−5.88818	−0.198378	−0.0991890	−	0.995069i	$-0.531625\pi$
$881$	−5.88818	−0.198378	−0.0991890	+	0.995069i	$0.531625\pi$
$882$	56.3853	1.89859
$883$	39.3783	1.32519	0.662593	−	0.748980i	$-0.269456\pi$
$883$	39.3783	1.32519	0.662593	+	0.748980i	$0.269456\pi$
$884$	3.19578	0.107486
$885$	−2.17705	−0.0731808
$886$	−9.43668	−0.317031
$887$	16.5659	0.556228	0.278114	−	0.960548i	$-0.410291\pi$
$887$	16.5659	0.556228	0.278114	+	0.960548i	$0.410291\pi$
$888$	−0.650256	−0.0218212
$889$	92.8276	3.11334
$890$	−7.40959	−0.248370
$891$	−3.90706	−0.130891
$892$	−5.41806	−0.181410
$893$	33.9302	1.13543
$894$	−0.617231	−0.0206433
$895$	−60.7079	−2.02924
$896$	5.08120	0.169751
$897$	0.0536964	0.00179287
$898$	−0.654226	−0.0218318
$899$	−3.66562	−0.122255
$900$	−21.7048	−0.723492
$901$	46.2348	1.54030
$902$	3.57606	0.119070
$903$	−0.947995	−0.0315473
$904$	−2.51508	−0.0836503
$905$	−89.2240	−2.96591
$906$	0.687216	0.0228312
$907$	−55.0964	−1.82945	−0.914723	−	0.404082i	$-0.867591\pi$
$907$	−55.0964	−1.82945	−0.914723	+	0.404082i	$0.867591\pi$
$908$	−4.42350	−0.146799
$909$	−4.30215	−0.142693
$910$	−12.1939	−0.404225
$911$	50.1208	1.66058	0.830288	−	0.557335i	$-0.188176\pi$
$911$	50.1208	1.66058	0.830288	+	0.557335i	$0.188176\pi$
$912$	−0.505709	−0.0167457
$913$	−7.11090	−0.235337
$914$	37.5655	1.24255
$915$	−2.51409	−0.0831134
$916$	−0.821632	−0.0271475
$917$	−34.7856	−1.14872
$918$	−1.70980	−0.0564319
$919$	34.7091	1.14495	0.572473	−	0.819923i	$-0.305984\pi$
$919$	34.7091	1.14495	0.572473	+	0.819923i	$0.305984\pi$
$920$	−4.47697	−0.147601
$921$	−1.30238	−0.0429148
$922$	14.4018	0.474298
$923$	1.04584	0.0344241
$924$	−0.135490	−0.00445729
$925$	−76.9760	−2.53096
$926$	13.4904	0.443321
$927$	8.79403	0.288834
$928$	0.893012	0.0293145
$929$	50.9562	1.67182	0.835910	−	0.548866i	$-0.184940\pi$
$929$	50.9562	1.67182	0.835910	+	0.548866i	$0.184940\pi$
$930$	−0.878935	−0.0288214
$931$	155.519	5.09692
$932$	15.1979	0.497822
$933$	−1.35664	−0.0444144
$934$	−28.0903	−0.919142
$935$	7.10489	0.232355
$936$	2.05492	0.0671672
$937$	−33.7795	−1.10353	−0.551765	−	0.834000i	$-0.686045\pi$
$937$	−33.7795	−1.10353	−0.551765	+	0.834000i	$0.686045\pi$
$938$	−40.7759	−1.33138
$939$	1.58843	0.0518365
$940$	−14.3665	−0.468584
$941$	−8.77729	−0.286131	−0.143066	−	0.989713i	$-0.545696\pi$
$941$	−8.77729	−0.286131	−0.143066	+	0.989713i	$0.545696\pi$
$942$	1.12564	0.0366753
$943$	−10.5001	−0.341930
$944$	−10.1672	−0.330915
$945$	6.52398	0.212225
$946$	−1.32853	−0.0431942
$947$	−6.23151	−0.202497	−0.101248	−	0.994861i	$-0.532284\pi$
$947$	−6.23151	−0.202497	−0.101248	+	0.994861i	$0.532284\pi$
$948$	−0.0538491	−0.00174894
$949$	2.67304	0.0867706
$950$	−59.8648	−1.94227
$951$	−1.62335	−0.0526407
$952$	23.6770	0.767377
$953$	43.4502	1.40749	0.703745	−	0.710452i	$-0.251510\pi$
$953$	43.4502	1.40749	0.703745	+	0.710452i	$0.251510\pi$
$954$	29.7295	0.962527
$955$	−80.0507	−2.59038
$956$	4.92727	0.159359
$957$	−0.0238121	−0.000769736 0
$958$	−31.3288	−1.01219
$959$	92.2716	2.97961
$960$	0.214124	0.00691083
$961$	−14.1507	−0.456475
$962$	7.28780	0.234968
$963$	15.8660	0.511274
$964$	18.4369	0.593811
$965$	−24.4597	−0.787387
$966$	0.397828	0.0127999
$967$	55.5146	1.78523	0.892614	−	0.450821i	$-0.148869\pi$
$967$	55.5146	1.78523	0.892614	+	0.450821i	$0.148869\pi$
$968$	10.8101	0.347451
$969$	−2.35647	−0.0757007
$970$	6.22104	0.199746
$971$	11.4729	0.368184	0.184092	−	0.982909i	$-0.441066\pi$
$971$	11.4729	0.368184	0.184092	+	0.982909i	$0.441066\pi$
$972$	−1.64947	−0.0529069
$973$	−54.7387	−1.75484
$974$	24.3563	0.780425
$975$	−0.304017	−0.00973633
$976$	−11.7413	−0.375829
$977$	28.5973	0.914910	0.457455	−	0.889233i	$-0.348761\pi$
$977$	28.5973	0.914910	0.457455	+	0.889233i	$0.348761\pi$
$978$	−1.15704	−0.0369980
$979$	0.922719	0.0294902
$980$	−65.8488	−2.10346
$981$	29.9149	0.955109
$982$	6.03591	0.192614
$983$	−2.36378	−0.0753927	−0.0376964	−	0.999289i	$-0.512002\pi$
$983$	−2.36378	−0.0753927	−0.0376964	+	0.999289i	$0.512002\pi$
$984$	0.502197	0.0160095
$985$	8.75187	0.278858
$986$	4.16119	0.132519
$987$	1.27662	0.0406353
$988$	5.66777	0.180316
$989$	3.90084	0.124040
$990$	4.56852	0.145197
$991$	18.1790	0.577474	0.288737	−	0.957408i	$-0.406765\pi$
$991$	18.1790	0.577474	0.288737	+	0.957408i	$0.406765\pi$
$992$	−4.10479	−0.130327
$993$	−0.448231	−0.0142242
$994$	7.74842	0.245765
$995$	−23.4302	−0.742789
$996$	−0.998606	−0.0316420
$997$	11.4428	0.362396	0.181198	−	0.983447i	$-0.442002\pi$
$997$	11.4428	0.362396	0.181198	+	0.983447i	$0.442002\pi$
$998$	28.7559	0.910254
$999$	−3.89910	−0.123362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.d.1.18	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.d.1.18	✓	37	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:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.0611935 q^{3} +1.00000 q^{4} -3.49914 q^{5} +0.0611935 q^{6} -5.08120 q^{7} -1.00000 q^{8} -2.99626 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.0611935 q^{3} +1.00000 q^{4} -3.49914 q^{5} +0.0611935 q^{6} -5.08120 q^{7} -1.00000 q^{8} -2.99626 q^{9} +3.49914 q^{10} -0.435748 q^{11} -0.0611935 q^{12} +0.685830 q^{13} +5.08120 q^{14} +0.214124 q^{15} +1.00000 q^{16} +4.65973 q^{17} +2.99626 q^{18} +8.26410 q^{19} -3.49914 q^{20} +0.310936 q^{21} +0.435748 q^{22} -1.27945 q^{23} +0.0611935 q^{24} +7.24396 q^{25} -0.685830 q^{26} +0.366932 q^{27} -5.08120 q^{28} -0.893012 q^{29} -0.214124 q^{30} +4.10479 q^{31} -1.00000 q^{32} +0.0266650 q^{33} -4.65973 q^{34} +17.7798 q^{35} -2.99626 q^{36} -10.6262 q^{37} -8.26410 q^{38} -0.0419683 q^{39} +3.49914 q^{40} +8.20672 q^{41} -0.310936 q^{42} -3.04884 q^{43} -0.435748 q^{44} +10.4843 q^{45} +1.27945 q^{46} +4.10573 q^{47} -0.0611935 q^{48} +18.8186 q^{49} -7.24396 q^{50} -0.285145 q^{51} +0.685830 q^{52} +9.92221 q^{53} -0.366932 q^{54} +1.52474 q^{55} +5.08120 q^{56} -0.505709 q^{57} +0.893012 q^{58} -10.1672 q^{59} +0.214124 q^{60} -11.7413 q^{61} -4.10479 q^{62} +15.2246 q^{63} +1.00000 q^{64} -2.39981 q^{65} -0.0266650 q^{66} -8.02486 q^{67} +4.65973 q^{68} +0.0782940 q^{69} -17.7798 q^{70} +1.52492 q^{71} +2.99626 q^{72} +3.89753 q^{73} +10.6262 q^{74} -0.443283 q^{75} +8.26410 q^{76} +2.21412 q^{77} +0.0419683 q^{78} +0.879981 q^{79} -3.49914 q^{80} +8.96631 q^{81} -8.20672 q^{82} +16.3188 q^{83} +0.310936 q^{84} -16.3050 q^{85} +3.04884 q^{86} +0.0546465 q^{87} +0.435748 q^{88} -2.11755 q^{89} -10.4843 q^{90} -3.48484 q^{91} -1.27945 q^{92} -0.251186 q^{93} -4.10573 q^{94} -28.9172 q^{95} +0.0611935 q^{96} +1.77788 q^{97} -18.8186 q^{98} +1.30561 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9}+O(q^{10}) \) 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9	\( 37 q - 37 q^{2} - 5 q^{3} + 37 q^{4} - 13 q^{5} + 5 q^{6} - 22 q^{7} - 37 q^{8} + 32 q^{9} + 13 q^{10} + 12 q^{11} - 5 q^{12} - 36 q^{13} + 22 q^{14} - 7 q^{15} + 37 q^{16} + 4 q^{17} - 32 q^{18} - 10 q^{19} - 13 q^{20} - 11 q^{21} - 12 q^{22} - q^{23} + 5 q^{24} + 8 q^{25} + 36 q^{26} - 20 q^{27} - 22 q^{28} - 6 q^{29} + 7 q^{30} - 23 q^{31} - 37 q^{32} - 27 q^{33} - 4 q^{34} + 24 q^{35} + 32 q^{36} - 46 q^{37} + 10 q^{38} + 5 q^{39} + 13 q^{40} + 11 q^{41} + 11 q^{42} - 22 q^{43} + 12 q^{44} - 57 q^{45} + q^{46} - 18 q^{47} - 5 q^{48} - q^{49} - 8 q^{50} + 18 q^{51} - 36 q^{52} - 25 q^{53} + 20 q^{54} - 25 q^{55} + 22 q^{56} - 25 q^{57} + 6 q^{58} + 24 q^{59} - 7 q^{60} - 40 q^{61} + 23 q^{62} - 38 q^{63} + 37 q^{64} + 14 q^{65} + 27 q^{66} - 49 q^{67} + 4 q^{68} - 19 q^{69} - 24 q^{70} + 27 q^{71} - 32 q^{72} - 87 q^{73} + 46 q^{74} - 5 q^{75} - 10 q^{76} - 50 q^{77} - 5 q^{78} + 11 q^{79} - 13 q^{80} + 5 q^{81} - 11 q^{82} - 9 q^{83} - 11 q^{84} - 68 q^{85} + 22 q^{86} - 12 q^{87} - 12 q^{88} - 3 q^{89} + 57 q^{90} - 19 q^{91} - q^{92} - 69 q^{93} + 18 q^{94} + 27 q^{95} + 5 q^{96} - 98 q^{97} + q^{98} + 33 q^{99}+O(q^{100}) \) 37 * q - 37 * q^2 - 5 * q^3 + 37 * q^4 - 13 * q^5 + 5 * q^6 - 22 * q^7 - 37 * q^8 + 32 * q^9 + 13 * q^10 + 12 * q^11 - 5 * q^12 - 36 * q^13 + 22 * q^14 - 7 * q^15 + 37 * q^16 + 4 * q^17 - 32 * q^18 - 10 * q^19 - 13 * q^20 - 11 * q^21 - 12 * q^22 - q^23 + 5 * q^24 + 8 * q^25 + 36 * q^26 - 20 * q^27 - 22 * q^28 - 6 * q^29 + 7 * q^30 - 23 * q^31 - 37 * q^32 - 27 * q^33 - 4 * q^34 + 24 * q^35 + 32 * q^36 - 46 * q^37 + 10 * q^38 + 5 * q^39 + 13 * q^40 + 11 * q^41 + 11 * q^42 - 22 * q^43 + 12 * q^44 - 57 * q^45 + q^46 - 18 * q^47 - 5 * q^48 - q^49 - 8 * q^50 + 18 * q^51 - 36 * q^52 - 25 * q^53 + 20 * q^54 - 25 * q^55 + 22 * q^56 - 25 * q^57 + 6 * q^58 + 24 * q^59 - 7 * q^60 - 40 * q^61 + 23 * q^62 - 38 * q^63 + 37 * q^64 + 14 * q^65 + 27 * q^66 - 49 * q^67 + 4 * q^68 - 19 * q^69 - 24 * q^70 + 27 * q^71 - 32 * q^72 - 87 * q^73 + 46 * q^74 - 5 * q^75 - 10 * q^76 - 50 * q^77 - 5 * q^78 + 11 * q^79 - 13 * q^80 + 5 * q^81 - 11 * q^82 - 9 * q^83 - 11 * q^84 - 68 * q^85 + 22 * q^86 - 12 * q^87 - 12 * q^88 - 3 * q^89 + 57 * q^90 - 19 * q^91 - q^92 - 69 * q^93 + 18 * q^94 + 27 * q^95 + 5 * q^96 - 98 * q^97 + q^98 + 33 * q^99

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