LMFDB - Embedded newform 8042.2.a.a.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -2.51956 q^{3} +1.00000 q^{4} -0.909270 q^{5} -2.51956 q^{6} +0.00482385 q^{7} +1.00000 q^{8} +3.34819 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.51956 q^{3} +1.00000 q^{4} -0.909270 q^{5} -2.51956 q^{6} +0.00482385 q^{7} +1.00000 q^{8} +3.34819 q^{9} -0.909270 q^{10} +3.05966 q^{11} -2.51956 q^{12} +1.45203 q^{13} +0.00482385 q^{14} +2.29096 q^{15} +1.00000 q^{16} +2.08149 q^{17} +3.34819 q^{18} -5.78760 q^{19} -0.909270 q^{20} -0.0121540 q^{21} +3.05966 q^{22} +2.17008 q^{23} -2.51956 q^{24} -4.17323 q^{25} +1.45203 q^{26} -0.877281 q^{27} +0.00482385 q^{28} -3.09812 q^{29} +2.29096 q^{30} +4.31573 q^{31} +1.00000 q^{32} -7.70900 q^{33} +2.08149 q^{34} -0.00438618 q^{35} +3.34819 q^{36} -3.76423 q^{37} -5.78760 q^{38} -3.65848 q^{39} -0.909270 q^{40} -12.6995 q^{41} -0.0121540 q^{42} -11.8078 q^{43} +3.05966 q^{44} -3.04441 q^{45} +2.17008 q^{46} -7.36640 q^{47} -2.51956 q^{48} -6.99998 q^{49} -4.17323 q^{50} -5.24444 q^{51} +1.45203 q^{52} +10.1099 q^{53} -0.877281 q^{54} -2.78206 q^{55} +0.00482385 q^{56} +14.5822 q^{57} -3.09812 q^{58} +8.55161 q^{59} +2.29096 q^{60} +13.6179 q^{61} +4.31573 q^{62} +0.0161512 q^{63} +1.00000 q^{64} -1.32029 q^{65} -7.70900 q^{66} +11.6157 q^{67} +2.08149 q^{68} -5.46765 q^{69} -0.00438618 q^{70} +4.96073 q^{71} +3.34819 q^{72} +15.4576 q^{73} -3.76423 q^{74} +10.5147 q^{75} -5.78760 q^{76} +0.0147593 q^{77} -3.65848 q^{78} -6.72488 q^{79} -0.909270 q^{80} -7.83420 q^{81} -12.6995 q^{82} -1.12077 q^{83} -0.0121540 q^{84} -1.89264 q^{85} -11.8078 q^{86} +7.80591 q^{87} +3.05966 q^{88} +2.64423 q^{89} -3.04441 q^{90} +0.00700439 q^{91} +2.17008 q^{92} -10.8737 q^{93} -7.36640 q^{94} +5.26249 q^{95} -2.51956 q^{96} +12.7829 q^{97} -6.99998 q^{98} +10.2443 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	−2.51956	−1.45467	−0.727335	−	0.686283i	$-0.759241\pi$
$3$	−2.51956	−1.45467	−0.727335	+	0.686283i	$0.759241\pi$
$4$	1.00000	0.500000
$5$	−0.909270	−0.406638	−0.203319	−	0.979113i	$-0.565173\pi$
$5$	−0.909270	−0.406638	−0.203319	+	0.979113i	$0.565173\pi$
$6$	−2.51956	−1.02861
$7$	0.00482385	0.00182324	0.000911622	−	1.00000i	$-0.499710\pi$
$7$	0.00482385	0.00182324	0.000911622		1.00000i	$0.499710\pi$
$8$	1.00000	0.353553
$9$	3.34819	1.11606
$10$	−0.909270	−0.287536
$11$	3.05966	0.922522	0.461261	−	0.887265i	$-0.347397\pi$
$11$	3.05966	0.922522	0.461261	+	0.887265i	$0.347397\pi$
$12$	−2.51956	−0.727335
$13$	1.45203	0.402721	0.201361	−	0.979517i	$-0.435464\pi$
$13$	1.45203	0.402721	0.201361	+	0.979517i	$0.435464\pi$
$14$	0.00482385	0.00128923
$15$	2.29096	0.591524
$16$	1.00000	0.250000
$17$	2.08149	0.504836	0.252418	−	0.967618i	$-0.418774\pi$
$17$	2.08149	0.504836	0.252418	+	0.967618i	$0.418774\pi$
$18$	3.34819	0.789175
$19$	−5.78760	−1.32777	−0.663883	−	0.747836i	$-0.731093\pi$
$19$	−5.78760	−1.32777	−0.663883	+	0.747836i	$0.731093\pi$
$20$	−0.909270	−0.203319
$21$	−0.0121540	−0.00265222
$22$	3.05966	0.652322
$23$	2.17008	0.452493	0.226246	−	0.974070i	$-0.427355\pi$
$23$	2.17008	0.452493	0.226246	+	0.974070i	$0.427355\pi$
$24$	−2.51956	−0.514303
$25$	−4.17323	−0.834646
$26$	1.45203	0.284767
$27$	−0.877281	−0.168833
$28$	0.00482385	0.000911622 0
$29$	−3.09812	−0.575307	−0.287654	−	0.957735i	$-0.592875\pi$
$29$	−3.09812	−0.575307	−0.287654	+	0.957735i	$0.592875\pi$
$30$	2.29096	0.418270
$31$	4.31573	0.775128	0.387564	−	0.921843i	$-0.373317\pi$
$31$	4.31573	0.775128	0.387564	+	0.921843i	$0.373317\pi$
$32$	1.00000	0.176777
$33$	−7.70900	−1.34196
$34$	2.08149	0.356973
$35$	−0.00438618	−0.000741400 0
$36$	3.34819	0.558031
$37$	−3.76423	−0.618835	−0.309418	−	0.950926i	$-0.600134\pi$
$37$	−3.76423	−0.618835	−0.309418	+	0.950926i	$0.600134\pi$
$38$	−5.78760	−0.938873
$39$	−3.65848	−0.585826
$40$	−0.909270	−0.143768
$41$	−12.6995	−1.98333	−0.991663	−	0.128855i	$-0.958870\pi$
$41$	−12.6995	−1.98333	−0.991663	+	0.128855i	$0.958870\pi$
$42$	−0.0121540	−0.00187540
$43$	−11.8078	−1.80067	−0.900335	−	0.435198i	$-0.856678\pi$
$43$	−11.8078	−1.80067	−0.900335	+	0.435198i	$0.856678\pi$
$44$	3.05966	0.461261
$45$	−3.04441	−0.453833
$46$	2.17008	0.319961
$47$	−7.36640	−1.07450	−0.537250	−	0.843423i	$-0.680537\pi$
$47$	−7.36640	−1.07450	−0.537250	+	0.843423i	$0.680537\pi$
$48$	−2.51956	−0.363667
$49$	−6.99998	−0.999997
$50$	−4.17323	−0.590184
$51$	−5.24444	−0.734369
$52$	1.45203	0.201361
$53$	10.1099	1.38870	0.694352	−	0.719636i	$-0.255691\pi$
$53$	10.1099	1.38870	0.694352	+	0.719636i	$0.255691\pi$
$54$	−0.877281	−0.119383
$55$	−2.78206	−0.375132
$56$	0.00482385	0.000644614 0
$57$	14.5822	1.93146
$58$	−3.09812	−0.406804
$59$	8.55161	1.11332	0.556662	−	0.830739i	$-0.312082\pi$
$59$	8.55161	1.11332	0.556662	+	0.830739i	$0.312082\pi$
$60$	2.29096	0.295762
$61$	13.6179	1.74360	0.871798	−	0.489866i	$-0.162954\pi$
$61$	13.6179	1.74360	0.871798	+	0.489866i	$0.162954\pi$
$62$	4.31573	0.548098
$63$	0.0161512	0.00203485
$64$	1.00000	0.125000
$65$	−1.32029	−0.163762
$66$	−7.70900	−0.948912
$67$	11.6157	1.41908	0.709540	−	0.704665i	$-0.248903\pi$
$67$	11.6157	1.41908	0.709540	+	0.704665i	$0.248903\pi$
$68$	2.08149	0.252418
$69$	−5.46765	−0.658227
$70$	−0.00438618	−0.000524249 0
$71$	4.96073	0.588730	0.294365	−	0.955693i	$-0.404892\pi$
$71$	4.96073	0.588730	0.294365	+	0.955693i	$0.404892\pi$
$72$	3.34819	0.394588
$73$	15.4576	1.80917	0.904585	−	0.426294i	$-0.140181\pi$
$73$	15.4576	1.80917	0.904585	+	0.426294i	$0.140181\pi$
$74$	−3.76423	−0.437582
$75$	10.5147	1.21413
$76$	−5.78760	−0.663883
$77$	0.0147593	0.00168198
$78$	−3.65848	−0.414242
$79$	−6.72488	−0.756608	−0.378304	−	0.925681i	$-0.623493\pi$
$79$	−6.72488	−0.756608	−0.378304	+	0.925681i	$0.623493\pi$
$80$	−0.909270	−0.101659
$81$	−7.83420	−0.870467
$82$	−12.6995	−1.40242
$83$	−1.12077	−0.123020	−0.0615102	−	0.998106i	$-0.519592\pi$
$83$	−1.12077	−0.123020	−0.0615102	+	0.998106i	$0.519592\pi$
$84$	−0.0121540	−0.00132611
$85$	−1.89264	−0.205285
$86$	−11.8078	−1.27327
$87$	7.80591	0.836882
$88$	3.05966	0.326161
$89$	2.64423	0.280288	0.140144	−	0.990131i	$-0.455243\pi$
$89$	2.64423	0.280288	0.140144	+	0.990131i	$0.455243\pi$
$90$	−3.04441	−0.320909
$91$	0.00700439	0.000734259 0
$92$	2.17008	0.226246
$93$	−10.8737	−1.12756
$94$	−7.36640	−0.759786
$95$	5.26249	0.539920
$96$	−2.51956	−0.257152
$97$	12.7829	1.29790	0.648952	−	0.760829i	$-0.275208\pi$
$97$	12.7829	1.29790	0.648952	+	0.760829i	$0.275208\pi$
$98$	−6.99998	−0.707104
$99$	10.2443	1.02959
$100$	−4.17323	−0.417323
$101$	4.31481	0.429340	0.214670	−	0.976687i	$-0.431132\pi$
$101$	4.31481	0.429340	0.214670	+	0.976687i	$0.431132\pi$
$102$	−5.24444	−0.519277
$103$	3.31105	0.326248	0.163124	−	0.986606i	$-0.447843\pi$
$103$	3.31105	0.326248	0.163124	+	0.986606i	$0.447843\pi$
$104$	1.45203	0.142383
$105$	0.0110513	0.00107849
$106$	10.1099	0.981962
$107$	−15.5758	−1.50577	−0.752884	−	0.658153i	$-0.771338\pi$
$107$	−15.5758	−1.50577	−0.752884	+	0.658153i	$0.771338\pi$
$108$	−0.877281	−0.0844164
$109$	−9.38773	−0.899182	−0.449591	−	0.893235i	$-0.648430\pi$
$109$	−9.38773	−0.899182	−0.449591	+	0.893235i	$0.648430\pi$
$110$	−2.78206	−0.265259
$111$	9.48420	0.900200
$112$	0.00482385	0.000455811 0
$113$	−4.89596	−0.460573	−0.230287	−	0.973123i	$-0.573966\pi$
$113$	−4.89596	−0.460573	−0.230287	+	0.973123i	$0.573966\pi$
$114$	14.5822	1.36575
$115$	−1.97319	−0.184001
$116$	−3.09812	−0.287654
$117$	4.86168	0.449462
$118$	8.55161	0.787240
$119$	0.0100408	0.000920438 0
$120$	2.29096	0.209135
$121$	−1.63849	−0.148953
$122$	13.6179	1.23291
$123$	31.9971	2.88508
$124$	4.31573	0.387564
$125$	8.34094	0.746036
$126$	0.0161512	0.00143886
$127$	12.3403	1.09502	0.547512	−	0.836798i	$-0.315575\pi$
$127$	12.3403	1.09502	0.547512	+	0.836798i	$0.315575\pi$
$128$	1.00000	0.0883883
$129$	29.7504	2.61938
$130$	−1.32029	−0.115797
$131$	−12.4298	−1.08600	−0.542998	−	0.839734i	$-0.682711\pi$
$131$	−12.4298	−1.08600	−0.542998	+	0.839734i	$0.682711\pi$
$132$	−7.70900	−0.670982
$133$	−0.0279185	−0.00242084
$134$	11.6157	1.00344
$135$	0.797685	0.0686538
$136$	2.08149	0.178486
$137$	0.530161	0.0452947	0.0226474	−	0.999744i	$-0.492791\pi$
$137$	0.530161	0.0452947	0.0226474	+	0.999744i	$0.492791\pi$
$138$	−5.46765	−0.465437
$139$	2.48719	0.210960	0.105480	−	0.994421i	$-0.466362\pi$
$139$	2.48719	0.210960	0.105480	+	0.994421i	$0.466362\pi$
$140$	−0.00438618	−0.000370700 0
$141$	18.5601	1.56304
$142$	4.96073	0.416295
$143$	4.44272	0.371519
$144$	3.34819	0.279016
$145$	2.81703	0.233942
$146$	15.4576	1.27928
$147$	17.6369	1.45466
$148$	−3.76423	−0.309418
$149$	−20.6331	−1.69033	−0.845164	−	0.534508i	$-0.820497\pi$
$149$	−20.6331	−1.69033	−0.845164	+	0.534508i	$0.820497\pi$
$150$	10.5147	0.858522
$151$	−0.941015	−0.0765787	−0.0382893	−	0.999267i	$-0.512191\pi$
$151$	−0.941015	−0.0765787	−0.0382893	+	0.999267i	$0.512191\pi$
$152$	−5.78760	−0.469436
$153$	6.96922	0.563428
$154$	0.0147593	0.00118934
$155$	−3.92416	−0.315196
$156$	−3.65848	−0.292913
$157$	−3.93743	−0.314241	−0.157121	−	0.987579i	$-0.550221\pi$
$157$	−3.93743	−0.314241	−0.157121	+	0.987579i	$0.550221\pi$
$158$	−6.72488	−0.535003
$159$	−25.4725	−2.02010
$160$	−0.909270	−0.0718841
$161$	0.0104681	0.000825005 0
$162$	−7.83420	−0.615513
$163$	−21.0093	−1.64557	−0.822786	−	0.568351i	$-0.807582\pi$
$163$	−21.0093	−1.64557	−0.822786	+	0.568351i	$0.807582\pi$
$164$	−12.6995	−0.991663
$165$	7.00956	0.545693
$166$	−1.12077	−0.0869885
$167$	−24.2258	−1.87465	−0.937325	−	0.348456i	$-0.886706\pi$
$167$	−24.2258	−1.87465	−0.937325	+	0.348456i	$0.886706\pi$
$168$	−0.0121540	−0.000937700 0
$169$	−10.8916	−0.837816
$170$	−1.89264	−0.145159
$171$	−19.3780	−1.48187
$172$	−11.8078	−0.900335
$173$	−12.0624	−0.917090	−0.458545	−	0.888671i	$-0.651629\pi$
$173$	−12.0624	−0.917090	−0.458545	+	0.888671i	$0.651629\pi$
$174$	7.80591	0.591765
$175$	−0.0201310	−0.00152176
$176$	3.05966	0.230630
$177$	−21.5463	−1.61952
$178$	2.64423	0.198194
$179$	0.744495	0.0556462	0.0278231	−	0.999613i	$-0.491142\pi$
$179$	0.744495	0.0556462	0.0278231	+	0.999613i	$0.491142\pi$
$180$	−3.04441	−0.226917
$181$	−14.5856	−1.08414	−0.542070	−	0.840333i	$-0.682359\pi$
$181$	−14.5856	−1.08414	−0.542070	+	0.840333i	$0.682359\pi$
$182$	0.00700439	0.000519200 0
$183$	−34.3112	−2.53635
$184$	2.17008	0.159980
$185$	3.42270	0.251642
$186$	−10.8737	−0.797302
$187$	6.36865	0.465722
$188$	−7.36640	−0.537250
$189$	−0.00423187	−0.000307823 0
$190$	5.26249	0.381781
$191$	9.11568	0.659588	0.329794	−	0.944053i	$-0.393021\pi$
$191$	9.11568	0.659588	0.329794	+	0.944053i	$0.393021\pi$
$192$	−2.51956	−0.181834
$193$	−13.1995	−0.950121	−0.475061	−	0.879953i	$-0.657574\pi$
$193$	−13.1995	−0.950121	−0.475061	+	0.879953i	$0.657574\pi$
$194$	12.7829	0.917757
$195$	3.32655	0.238219
$196$	−6.99998	−0.499998
$197$	−9.82199	−0.699788	−0.349894	−	0.936789i	$-0.613782\pi$
$197$	−9.82199	−0.699788	−0.349894	+	0.936789i	$0.613782\pi$
$198$	10.2443	0.728032
$199$	−3.36120	−0.238269	−0.119135	−	0.992878i	$-0.538012\pi$
$199$	−3.36120	−0.238269	−0.119135	+	0.992878i	$0.538012\pi$
$200$	−4.17323	−0.295092
$201$	−29.2664	−2.06429
$202$	4.31481	0.303589
$203$	−0.0149449	−0.00104893
$204$	−5.24444	−0.367184
$205$	11.5473	0.806496
$206$	3.31105	0.230692
$207$	7.26583	0.505010
$208$	1.45203	0.100680
$209$	−17.7081	−1.22489
$210$	0.0110513	0.000762609 0
$211$	16.7356	1.15213	0.576063	−	0.817405i	$-0.304588\pi$
$211$	16.7356	1.15213	0.576063	+	0.817405i	$0.304588\pi$
$212$	10.1099	0.694352
$213$	−12.4989	−0.856408
$214$	−15.5758	−1.06474
$215$	10.7365	0.732220
$216$	−0.877281	−0.0596914
$217$	0.0208184	0.00141325
$218$	−9.38773	−0.635818
$219$	−38.9462	−2.63174
$220$	−2.78206	−0.187566
$221$	3.02239	0.203308
$222$	9.48420	0.636538
$223$	9.21293	0.616943	0.308472	−	0.951234i	$-0.400183\pi$
$223$	9.21293	0.616943	0.308472	+	0.951234i	$0.400183\pi$
$224$	0.00482385	0.000322307 0
$225$	−13.9728	−0.931517
$226$	−4.89596	−0.325674
$227$	−6.50751	−0.431919	−0.215959	−	0.976402i	$-0.569288\pi$
$227$	−6.50751	−0.431919	−0.215959	+	0.976402i	$0.569288\pi$
$228$	14.5822	0.965731
$229$	5.19735	0.343451	0.171725	−	0.985145i	$-0.445066\pi$
$229$	5.19735	0.343451	0.171725	+	0.985145i	$0.445066\pi$
$230$	−1.97319	−0.130108
$231$	−0.0371871	−0.00244673
$232$	−3.09812	−0.203402
$233$	−16.1368	−1.05716	−0.528579	−	0.848884i	$-0.677275\pi$
$233$	−16.1368	−1.05716	−0.528579	+	0.848884i	$0.677275\pi$
$234$	4.86168	0.317818
$235$	6.69805	0.436932
$236$	8.55161	0.556662
$237$	16.9437	1.10061
$238$	0.0100408	0.000650848 0
$239$	17.1520	1.10947	0.554735	−	0.832027i	$-0.312820\pi$
$239$	17.1520	1.10947	0.554735	+	0.832027i	$0.312820\pi$
$240$	2.29096	0.147881
$241$	−15.1272	−0.974428	−0.487214	−	0.873283i	$-0.661987\pi$
$241$	−15.1272	−0.974428	−0.487214	+	0.873283i	$0.661987\pi$
$242$	−1.63849	−0.105326
$243$	22.3706	1.43507
$244$	13.6179	0.871798
$245$	6.36487	0.406636
$246$	31.9971	2.04006
$247$	−8.40378	−0.534720
$248$	4.31573	0.274049
$249$	2.82385	0.178954
$250$	8.34094	0.527527
$251$	6.19085	0.390763	0.195382	−	0.980727i	$-0.437405\pi$
$251$	6.19085	0.390763	0.195382	+	0.980727i	$0.437405\pi$
$252$	0.0161512	0.00101743
$253$	6.63970	0.417434
$254$	12.3403	0.774299
$255$	4.76861	0.298622
$256$	1.00000	0.0625000
$257$	4.43236	0.276483	0.138242	−	0.990399i	$-0.455855\pi$
$257$	4.43236	0.276483	0.138242	+	0.990399i	$0.455855\pi$
$258$	29.7504	1.85218
$259$	−0.0181581	−0.00112829
$260$	−1.32029	−0.0818808
$261$	−10.3731	−0.642079
$262$	−12.4298	−0.767915
$263$	−30.6784	−1.89171	−0.945855	−	0.324588i	$-0.894774\pi$
$263$	−30.6784	−1.89171	−0.945855	+	0.324588i	$0.894774\pi$
$264$	−7.70900	−0.474456
$265$	−9.19264	−0.564699
$266$	−0.0279185	−0.00171179
$267$	−6.66231	−0.407727
$268$	11.6157	0.709540
$269$	−17.5567	−1.07045	−0.535227	−	0.844709i	$-0.679774\pi$
$269$	−17.5567	−1.07045	−0.535227	+	0.844709i	$0.679774\pi$
$270$	0.797685	0.0485456
$271$	−13.8240	−0.839747	−0.419873	−	0.907583i	$-0.637926\pi$
$271$	−13.8240	−0.839747	−0.419873	+	0.907583i	$0.637926\pi$
$272$	2.08149	0.126209
$273$	−0.0176480	−0.00106810
$274$	0.530161	0.0320282
$275$	−12.7687	−0.769979
$276$	−5.46765	−0.329114
$277$	18.9453	1.13831	0.569156	−	0.822229i	$-0.307270\pi$
$277$	18.9453	1.13831	0.569156	+	0.822229i	$0.307270\pi$
$278$	2.48719	0.149171
$279$	14.4499	0.865092
$280$	−0.00438618	−0.000262124 0
$281$	3.09691	0.184746	0.0923730	−	0.995724i	$-0.470555\pi$
$281$	3.09691	0.184746	0.0923730	+	0.995724i	$0.470555\pi$
$282$	18.5601	1.10524
$283$	13.7990	0.820265	0.410133	−	0.912026i	$-0.365482\pi$
$283$	13.7990	0.820265	0.410133	+	0.912026i	$0.365482\pi$
$284$	4.96073	0.294365
$285$	−13.2592	−0.785405
$286$	4.44272	0.262704
$287$	−0.0612604	−0.00361609
$288$	3.34819	0.197294
$289$	−12.6674	−0.745141
$290$	2.81703	0.165422
$291$	−32.2072	−1.88802
$292$	15.4576	0.904585
$293$	−29.6938	−1.73473	−0.867364	−	0.497674i	$-0.834188\pi$
$293$	−29.6938	−1.73473	−0.867364	+	0.497674i	$0.834188\pi$
$294$	17.6369	1.02860
$295$	−7.77572	−0.452720
$296$	−3.76423	−0.218791
$297$	−2.68418	−0.155752
$298$	−20.6331	−1.19524
$299$	3.15102	0.182228
$300$	10.5147	0.607067
$301$	−0.0569590	−0.00328306
$302$	−0.941015	−0.0541493
$303$	−10.8714	−0.624548
$304$	−5.78760	−0.331942
$305$	−12.3824	−0.709012
$306$	6.96922	0.398404
$307$	10.1992	0.582097	0.291048	−	0.956708i	$-0.405996\pi$
$307$	10.1992	0.582097	0.291048	+	0.956708i	$0.405996\pi$
$308$	0.0147593	0.000840991 0
$309$	−8.34240	−0.474583
$310$	−3.92416	−0.222878
$311$	27.9822	1.58673	0.793364	−	0.608748i	$-0.208328\pi$
$311$	27.9822	1.58673	0.793364	+	0.608748i	$0.208328\pi$
$312$	−3.65848	−0.207121
$313$	3.27472	0.185098	0.0925492	−	0.995708i	$-0.470498\pi$
$313$	3.27472	0.185098	0.0925492	+	0.995708i	$0.470498\pi$
$314$	−3.93743	−0.222202
$315$	−0.0146858	−0.000827449 0
$316$	−6.72488	−0.378304
$317$	5.36778	0.301484	0.150742	−	0.988573i	$-0.451834\pi$
$317$	5.36778	0.301484	0.150742	+	0.988573i	$0.451834\pi$
$318$	−25.4725	−1.42843
$319$	−9.47920	−0.530733
$320$	−0.909270	−0.0508297
$321$	39.2441	2.19039
$322$	0.0104681	0.000583366 0
$323$	−12.0468	−0.670304
$324$	−7.83420	−0.435233
$325$	−6.05966	−0.336130
$326$	−21.0093	−1.16360
$327$	23.6530	1.30801
$328$	−12.6995	−0.701212
$329$	−0.0355344	−0.00195908
$330$	7.00956	0.385864
$331$	−22.9150	−1.25952	−0.629761	−	0.776789i	$-0.716847\pi$
$331$	−22.9150	−1.25952	−0.629761	+	0.776789i	$0.716847\pi$
$332$	−1.12077	−0.0615102
$333$	−12.6033	−0.690659
$334$	−24.2258	−1.32558
$335$	−10.5618	−0.577051
$336$	−0.0121540	−0.000663054 0
$337$	−30.9426	−1.68555	−0.842776	−	0.538265i	$-0.819080\pi$
$337$	−30.9426	−1.68555	−0.842776	+	0.538265i	$0.819080\pi$
$338$	−10.8916	−0.592425
$339$	12.3357	0.669981
$340$	−1.89264	−0.102643
$341$	13.2047	0.715073
$342$	−19.3780	−1.04784
$343$	−0.0675338	−0.00364648
$344$	−11.8078	−0.636633
$345$	4.97157	0.267660
$346$	−12.0624	−0.648481
$347$	7.59180	0.407549	0.203775	−	0.979018i	$-0.434679\pi$
$347$	7.59180	0.407549	0.203775	+	0.979018i	$0.434679\pi$
$348$	7.80591	0.418441
$349$	24.4006	1.30613	0.653066	−	0.757301i	$-0.273482\pi$
$349$	24.4006	1.30613	0.653066	+	0.757301i	$0.273482\pi$
$350$	−0.0201310	−0.00107605
$351$	−1.27384	−0.0679925
$352$	3.05966	0.163080
$353$	17.3119	0.921417	0.460709	−	0.887551i	$-0.347595\pi$
$353$	17.3119	0.921417	0.460709	+	0.887551i	$0.347595\pi$
$354$	−21.5463	−1.14517
$355$	−4.51064	−0.239400
$356$	2.64423	0.140144
$357$	−0.0252984	−0.00133893
$358$	0.744495	0.0393478
$359$	24.9744	1.31810	0.659050	−	0.752099i	$-0.270959\pi$
$359$	24.9744	1.31810	0.659050	+	0.752099i	$0.270959\pi$
$360$	−3.04441	−0.160454
$361$	14.4963	0.762964
$362$	−14.5856	−0.766602
$363$	4.12826	0.216678
$364$	0.00700439	0.000367130 0
$365$	−14.0551	−0.735677
$366$	−34.3112	−1.79347
$367$	−4.24439	−0.221555	−0.110778	−	0.993845i	$-0.535334\pi$
$367$	−4.24439	−0.221555	−0.110778	+	0.993845i	$0.535334\pi$
$368$	2.17008	0.113123
$369$	−42.5203	−2.21352
$370$	3.42270	0.177938
$371$	0.0487687	0.00253195
$372$	−10.8737	−0.563778
$373$	−5.26135	−0.272423	−0.136211	−	0.990680i	$-0.543493\pi$
$373$	−5.26135	−0.272423	−0.136211	+	0.990680i	$0.543493\pi$
$374$	6.36865	0.329315
$375$	−21.0155	−1.08524
$376$	−7.36640	−0.379893
$377$	−4.49857	−0.231688
$378$	−0.00423187	−0.000217664 0
$379$	6.69263	0.343777	0.171889	−	0.985116i	$-0.445013\pi$
$379$	6.69263	0.343777	0.171889	+	0.985116i	$0.445013\pi$
$380$	5.26249	0.269960
$381$	−31.0921	−1.59290
$382$	9.11568	0.466399
$383$	4.64020	0.237103	0.118552	−	0.992948i	$-0.462175\pi$
$383$	4.64020	0.237103	0.118552	+	0.992948i	$0.462175\pi$
$384$	−2.51956	−0.128576
$385$	−0.0134202	−0.000683958 0
$386$	−13.1995	−0.671837
$387$	−39.5347	−2.00966
$388$	12.7829	0.648952
$389$	12.0354	0.610217	0.305108	−	0.952318i	$-0.401307\pi$
$389$	12.0354	0.610217	0.305108	+	0.952318i	$0.401307\pi$
$390$	3.32655	0.168446
$391$	4.51700	0.228434
$392$	−6.99998	−0.353552
$393$	31.3176	1.57977
$394$	−9.82199	−0.494825
$395$	6.11473	0.307665
$396$	10.2443	0.514796
$397$	−1.63041	−0.0818277	−0.0409138	−	0.999163i	$-0.513027\pi$
$397$	−1.63041	−0.0818277	−0.0409138	+	0.999163i	$0.513027\pi$
$398$	−3.36120	−0.168482
$399$	0.0703424	0.00352153
$400$	−4.17323	−0.208661
$401$	−33.0363	−1.64975	−0.824876	−	0.565313i	$-0.808755\pi$
$401$	−33.0363	−1.64975	−0.824876	+	0.565313i	$0.808755\pi$
$402$	−29.2664	−1.45967
$403$	6.26658	0.312161
$404$	4.31481	0.214670
$405$	7.12340	0.353965
$406$	−0.0149449	−0.000741702 0
$407$	−11.5173	−0.570889
$408$	−5.24444	−0.259639
$409$	−14.6760	−0.725680	−0.362840	−	0.931851i	$-0.618193\pi$
$409$	−14.6760	−0.725680	−0.362840	+	0.931851i	$0.618193\pi$
$410$	11.5473	0.570279
$411$	−1.33577	−0.0658889
$412$	3.31105	0.163124
$413$	0.0412517	0.00202986
$414$	7.26583	0.357096
$415$	1.01908	0.0500247
$416$	1.45203	0.0711917
$417$	−6.26662	−0.306877
$418$	−17.7081	−0.866131
$419$	−3.01023	−0.147059	−0.0735297	−	0.997293i	$-0.523426\pi$
$419$	−3.01023	−0.147059	−0.0735297	+	0.997293i	$0.523426\pi$
$420$	0.0110513	0.000539246 0
$421$	30.4712	1.48508	0.742539	−	0.669803i	$-0.233622\pi$
$421$	30.4712	1.48508	0.742539	+	0.669803i	$0.233622\pi$
$422$	16.7356	0.814676
$423$	−24.6641	−1.19921
$424$	10.1099	0.490981
$425$	−8.68654	−0.421359
$426$	−12.4989	−0.605572
$427$	0.0656908	0.00317900
$428$	−15.5758	−0.752884
$429$	−11.1937	−0.540438
$430$	10.7365	0.517758
$431$	−28.1569	−1.35627	−0.678135	−	0.734938i	$-0.737211\pi$
$431$	−28.1569	−1.35627	−0.678135	+	0.734938i	$0.737211\pi$
$432$	−0.877281	−0.0422082
$433$	−24.9466	−1.19886	−0.599429	−	0.800428i	$-0.704606\pi$
$433$	−24.9466	−1.19886	−0.599429	+	0.800428i	$0.704606\pi$
$434$	0.0208184	0.000999317 0
$435$	−7.09768	−0.340308
$436$	−9.38773	−0.449591
$437$	−12.5595	−0.600805
$438$	−38.9462	−1.86092
$439$	2.28144	0.108887	0.0544437	−	0.998517i	$-0.482661\pi$
$439$	2.28144	0.108887	0.0544437	+	0.998517i	$0.482661\pi$
$440$	−2.78206	−0.132629
$441$	−23.4372	−1.11606
$442$	3.02239	0.143760
$443$	10.8977	0.517765	0.258882	−	0.965909i	$-0.416646\pi$
$443$	10.8977	0.517765	0.258882	+	0.965909i	$0.416646\pi$
$444$	9.48420	0.450100
$445$	−2.40432	−0.113976
$446$	9.21293	0.436245
$447$	51.9863	2.45887
$448$	0.00482385	0.000227906 0
$449$	−25.2566	−1.19193	−0.595966	−	0.803010i	$-0.703231\pi$
$449$	−25.2566	−1.19193	−0.595966	+	0.803010i	$0.703231\pi$
$450$	−13.9728	−0.658682
$451$	−38.8561	−1.82966
$452$	−4.89596	−0.230287
$453$	2.37094	0.111397
$454$	−6.50751	−0.305413
$455$	−0.00636888	−0.000298578 0
$456$	14.5822	0.682875
$457$	−28.0112	−1.31031	−0.655155	−	0.755494i	$-0.727397\pi$
$457$	−28.0112	−1.31031	−0.655155	+	0.755494i	$0.727397\pi$
$458$	5.19735	0.242856
$459$	−1.82605	−0.0852328
$460$	−1.97319	−0.0920003
$461$	−42.3588	−1.97285	−0.986424	−	0.164220i	$-0.947489\pi$
$461$	−42.3588	−1.97285	−0.986424	+	0.164220i	$0.947489\pi$
$462$	−0.0371871	−0.00173010
$463$	−9.55174	−0.443907	−0.221953	−	0.975057i	$-0.571243\pi$
$463$	−9.55174	−0.443907	−0.221953	+	0.975057i	$0.571243\pi$
$464$	−3.09812	−0.143827
$465$	9.88717	0.458507
$466$	−16.1368	−0.747523
$467$	22.0439	1.02007	0.510034	−	0.860154i	$-0.329633\pi$
$467$	22.0439	1.02007	0.510034	+	0.860154i	$0.329633\pi$
$468$	4.86168	0.224731
$469$	0.0560322	0.00258733
$470$	6.69805	0.308958
$471$	9.92061	0.457117
$472$	8.55161	0.393620
$473$	−36.1278	−1.66116
$474$	16.9437	0.778252
$475$	24.1530	1.10821
$476$	0.0100408	0.000460219 0
$477$	33.8499	1.54988
$478$	17.1520	0.784514
$479$	11.1450	0.509227	0.254614	−	0.967043i	$-0.418052\pi$
$479$	11.1450	0.509227	0.254614	+	0.967043i	$0.418052\pi$
$480$	2.29096	0.104568
$481$	−5.46578	−0.249218
$482$	−15.1272	−0.689025
$483$	−0.0263751	−0.00120011
$484$	−1.63849	−0.0744766
$485$	−11.6231	−0.527777
$486$	22.3706	1.01475
$487$	−21.3065	−0.965488	−0.482744	−	0.875762i	$-0.660360\pi$
$487$	−21.3065	−0.965488	−0.482744	+	0.875762i	$0.660360\pi$
$488$	13.6179	0.616454
$489$	52.9341	2.39376
$490$	6.36487	0.287535
$491$	−16.8422	−0.760080	−0.380040	−	0.924970i	$-0.624090\pi$
$491$	−16.8422	−0.760080	−0.380040	+	0.924970i	$0.624090\pi$
$492$	31.9971	1.44254
$493$	−6.44871	−0.290435
$494$	−8.40378	−0.378104
$495$	−9.31485	−0.418671
$496$	4.31573	0.193782
$497$	0.0239298	0.00107340
$498$	2.82385	0.126540
$499$	−3.01941	−0.135167	−0.0675837	−	0.997714i	$-0.521529\pi$
$499$	−3.01941	−0.135167	−0.0675837	+	0.997714i	$0.521529\pi$
$500$	8.34094	0.373018
$501$	61.0384	2.72700
$502$	6.19085	0.276311
$503$	2.51457	0.112119	0.0560595	−	0.998427i	$-0.482146\pi$
$503$	2.51457	0.112119	0.0560595	+	0.998427i	$0.482146\pi$
$504$	0.0161512	0.000719430 0
$505$	−3.92333	−0.174586
$506$	6.63970	0.295171
$507$	27.4421	1.21874
$508$	12.3403	0.547512
$509$	33.0177	1.46349	0.731743	−	0.681581i	$-0.238707\pi$
$509$	33.0177	1.46349	0.731743	+	0.681581i	$0.238707\pi$
$510$	4.76861	0.211158
$511$	0.0745649	0.00329856
$512$	1.00000	0.0441942
$513$	5.07735	0.224171
$514$	4.43236	0.195503
$515$	−3.01064	−0.132665
$516$	29.7504	1.30969
$517$	−22.5387	−0.991250
$518$	−0.0181581	−0.000797820 0
$519$	30.3921	1.33406
$520$	−1.32029	−0.0578985
$521$	19.7960	0.867280	0.433640	−	0.901086i	$-0.357229\pi$
$521$	19.7960	0.867280	0.433640	+	0.901086i	$0.357229\pi$
$522$	−10.3731	−0.454018
$523$	−34.9545	−1.52846	−0.764228	−	0.644946i	$-0.776880\pi$
$523$	−34.9545	−1.52846	−0.764228	+	0.644946i	$0.776880\pi$
$524$	−12.4298	−0.542998
$525$	0.0507214	0.00221366
$526$	−30.6784	−1.33764
$527$	8.98315	0.391312
$528$	−7.70900	−0.335491
$529$	−18.2908	−0.795250
$530$	−9.19264	−0.399303
$531$	28.6324	1.24254
$532$	−0.0279185	−0.00121042
$533$	−18.4401	−0.798728
$534$	−6.66231	−0.288306
$535$	14.1626	0.612302
$536$	11.6157	0.501720
$537$	−1.87580	−0.0809467
$538$	−17.5567	−0.756925
$539$	−21.4175	−0.922519
$540$	0.797685	0.0343269
$541$	−18.2568	−0.784921	−0.392460	−	0.919769i	$-0.628376\pi$
$541$	−18.2568	−0.784921	−0.392460	+	0.919769i	$0.628376\pi$
$542$	−13.8240	−0.593791
$543$	36.7493	1.57706
$544$	2.08149	0.0892432
$545$	8.53598	0.365641
$546$	−0.0176480	−0.000755264 0
$547$	−0.439760	−0.0188028	−0.00940140	−	0.999956i	$-0.502993\pi$
$547$	−0.439760	−0.0188028	−0.00940140	+	0.999956i	$0.502993\pi$
$548$	0.530161	0.0226474
$549$	45.5953	1.94596
$550$	−12.7687	−0.544457
$551$	17.9307	0.763874
$552$	−5.46765	−0.232718
$553$	−0.0324398	−0.00137948
$554$	18.9453	0.804909
$555$	−8.62370	−0.366055
$556$	2.48719	0.105480
$557$	−10.4775	−0.443946	−0.221973	−	0.975053i	$-0.571250\pi$
$557$	−10.4775	−0.443946	−0.221973	+	0.975053i	$0.571250\pi$
$558$	14.4499	0.611712
$559$	−17.1453	−0.725168
$560$	−0.00438618	−0.000185350 0
$561$	−16.0462	−0.677471
$562$	3.09691	0.130635
$563$	6.47141	0.272737	0.136369	−	0.990658i	$-0.456457\pi$
$563$	6.47141	0.272737	0.136369	+	0.990658i	$0.456457\pi$
$564$	18.5601	0.781521
$565$	4.45175	0.187286
$566$	13.7990	0.580015
$567$	−0.0377910	−0.00158707
$568$	4.96073	0.208148
$569$	−14.6877	−0.615739	−0.307870	−	0.951429i	$-0.599616\pi$
$569$	−14.6877	−0.615739	−0.307870	+	0.951429i	$0.599616\pi$
$570$	−13.2592	−0.555365
$571$	45.6164	1.90898	0.954492	−	0.298235i	$-0.0963979\pi$
$571$	45.6164	1.90898	0.954492	+	0.298235i	$0.0963979\pi$
$572$	4.44272	0.185760
$573$	−22.9675	−0.959482
$574$	−0.0612604	−0.00255696
$575$	−9.05623	−0.377671
$576$	3.34819	0.139508
$577$	−39.7662	−1.65549	−0.827744	−	0.561107i	$-0.810376\pi$
$577$	−39.7662	−1.65549	−0.827744	+	0.561107i	$0.810376\pi$
$578$	−12.6674	−0.526894
$579$	33.2570	1.38211
$580$	2.81703	0.116971
$581$	−0.00540642	−0.000224296 0
$582$	−32.2072	−1.33503
$583$	30.9329	1.28111
$584$	15.4576	0.639638
$585$	−4.42058	−0.182768
$586$	−29.6938	−1.22664
$587$	−31.7503	−1.31048	−0.655238	−	0.755423i	$-0.727432\pi$
$587$	−31.7503	−1.31048	−0.655238	+	0.755423i	$0.727432\pi$
$588$	17.6369	0.727332
$589$	−24.9777	−1.02919
$590$	−7.77572	−0.320121
$591$	24.7471	1.01796
$592$	−3.76423	−0.154709
$593$	−36.0972	−1.48233	−0.741166	−	0.671322i	$-0.765727\pi$
$593$	−36.0972	−1.48233	−0.741166	+	0.671322i	$0.765727\pi$
$594$	−2.68418	−0.110133
$595$	−0.00912980	−0.000374285 0
$596$	−20.6331	−0.845164
$597$	8.46876	0.346603
$598$	3.15102	0.128855
$599$	−14.9244	−0.609794	−0.304897	−	0.952385i	$-0.598622\pi$
$599$	−14.9244	−0.609794	−0.304897	+	0.952385i	$0.598622\pi$
$600$	10.5147	0.429261
$601$	7.89652	0.322106	0.161053	−	0.986946i	$-0.448511\pi$
$601$	7.89652	0.322106	0.161053	+	0.986946i	$0.448511\pi$
$602$	−0.0569590	−0.00232147
$603$	38.8914	1.58378
$604$	−0.941015	−0.0382893
$605$	1.48983	0.0605700
$606$	−10.8714	−0.441622
$607$	−4.08812	−0.165932	−0.0829658	−	0.996552i	$-0.526439\pi$
$607$	−4.08812	−0.165932	−0.0829658	+	0.996552i	$0.526439\pi$
$608$	−5.78760	−0.234718
$609$	0.0376545	0.00152584
$610$	−12.3824	−0.501347
$611$	−10.6962	−0.432724
$612$	6.96922	0.281714
$613$	−27.2340	−1.09997	−0.549986	−	0.835174i	$-0.685367\pi$
$613$	−27.2340	−1.09997	−0.549986	+	0.835174i	$0.685367\pi$
$614$	10.1992	0.411605
$615$	−29.0940	−1.17318
$616$	0.0147593	0.000594671 0
$617$	−39.7334	−1.59961	−0.799803	−	0.600262i	$-0.795063\pi$
$617$	−39.7334	−1.59961	−0.799803	+	0.600262i	$0.795063\pi$
$618$	−8.34240	−0.335581
$619$	−12.8849	−0.517887	−0.258943	−	0.965892i	$-0.583374\pi$
$619$	−12.8849	−0.517887	−0.258943	+	0.965892i	$0.583374\pi$
$620$	−3.92416	−0.157598
$621$	−1.90377	−0.0763956
$622$	27.9822	1.12199
$623$	0.0127554	0.000511034 0
$624$	−3.65848	−0.146457
$625$	13.2820	0.531279
$626$	3.27472	0.130884
$627$	44.6166	1.78182
$628$	−3.93743	−0.157121
$629$	−7.83520	−0.312410
$630$	−0.0146858	−0.000585095 0
$631$	43.1708	1.71860	0.859301	−	0.511470i	$-0.170899\pi$
$631$	43.1708	1.71860	0.859301	+	0.511470i	$0.170899\pi$
$632$	−6.72488	−0.267501
$633$	−42.1664	−1.67596
$634$	5.36778	0.213182
$635$	−11.2207	−0.445278
$636$	−25.4725	−1.01005
$637$	−10.1642	−0.402720
$638$	−9.47920	−0.375285
$639$	16.6095	0.657060
$640$	−0.909270	−0.0359420
$641$	−0.858958	−0.0339268	−0.0169634	−	0.999856i	$-0.505400\pi$
$641$	−0.858958	−0.0339268	−0.0169634	+	0.999856i	$0.505400\pi$
$642$	39.2441	1.54884
$643$	−18.9536	−0.747455	−0.373728	−	0.927539i	$-0.621920\pi$
$643$	−18.9536	−0.747455	−0.373728	+	0.927539i	$0.621920\pi$
$644$	0.0104681	0.000412502 0
$645$	−27.0512	−1.06514
$646$	−12.0468	−0.473976
$647$	43.9905	1.72944	0.864722	−	0.502251i	$-0.167495\pi$
$647$	43.9905	1.72944	0.864722	+	0.502251i	$0.167495\pi$
$648$	−7.83420	−0.307756
$649$	26.1650	1.02707
$650$	−6.05966	−0.237679
$651$	−0.0524533	−0.00205581
$652$	−21.0093	−0.822786
$653$	13.5271	0.529356	0.264678	−	0.964337i	$-0.414734\pi$
$653$	13.5271	0.529356	0.264678	+	0.964337i	$0.414734\pi$
$654$	23.6530	0.924904
$655$	11.3020	0.441607
$656$	−12.6995	−0.495832
$657$	51.7548	2.01915
$658$	−0.0355344	−0.00138528
$659$	15.5526	0.605844	0.302922	−	0.953015i	$-0.402038\pi$
$659$	15.5526	0.605844	0.302922	+	0.953015i	$0.402038\pi$
$660$	7.00956	0.272847
$661$	5.45183	0.212051	0.106026	−	0.994363i	$-0.466187\pi$
$661$	5.45183	0.212051	0.106026	+	0.994363i	$0.466187\pi$
$662$	−22.9150	−0.890616
$663$	−7.61510	−0.295746
$664$	−1.12077	−0.0434943
$665$	0.0253855	0.000984406 0
$666$	−12.6033	−0.488369
$667$	−6.72317	−0.260322
$668$	−24.2258	−0.937325
$669$	−23.2125	−0.897448
$670$	−10.5618	−0.408037
$671$	41.6662	1.60850
$672$	−0.0121540	−0.000468850 0
$673$	3.47368	0.133901	0.0669504	−	0.997756i	$-0.478673\pi$
$673$	3.47368	0.133901	0.0669504	+	0.997756i	$0.478673\pi$
$674$	−30.9426	−1.19186
$675$	3.66109	0.140916
$676$	−10.8916	−0.418908
$677$	−37.2915	−1.43323	−0.716614	−	0.697470i	$-0.754309\pi$
$677$	−37.2915	−1.43323	−0.716614	+	0.697470i	$0.754309\pi$
$678$	12.3357	0.473748
$679$	0.0616627	0.00236640
$680$	−1.89264	−0.0725793
$681$	16.3961	0.628299
$682$	13.2047	0.505633
$683$	33.9930	1.30071	0.650353	−	0.759632i	$-0.274621\pi$
$683$	33.9930	1.30071	0.650353	+	0.759632i	$0.274621\pi$
$684$	−19.3780	−0.740935
$685$	−0.482060	−0.0184186
$686$	−0.0675338	−0.00257845
$687$	−13.0950	−0.499607
$688$	−11.8078	−0.450167
$689$	14.6799	0.559260
$690$	4.97157	0.189264
$691$	−1.37786	−0.0524164	−0.0262082	−	0.999657i	$-0.508343\pi$
$691$	−1.37786	−0.0524164	−0.0262082	+	0.999657i	$0.508343\pi$
$692$	−12.0624	−0.458545
$693$	0.0494170	0.00187720
$694$	7.59180	0.288181
$695$	−2.26152	−0.0857844
$696$	7.80591	0.295882
$697$	−26.4339	−1.00125
$698$	24.4006	0.923575
$699$	40.6577	1.53781
$700$	−0.0201310	−0.000760881 0
$701$	17.4152	0.657762	0.328881	−	0.944371i	$-0.393329\pi$
$701$	17.4152	0.657762	0.328881	+	0.944371i	$0.393329\pi$
$702$	−1.27384	−0.0480780
$703$	21.7858	0.821668
$704$	3.05966	0.115315
$705$	−16.8761	−0.635592
$706$	17.3119	0.651540
$707$	0.0208140	0.000782792 0
$708$	−21.5463	−0.809760
$709$	7.29196	0.273855	0.136928	−	0.990581i	$-0.456277\pi$
$709$	7.29196	0.273855	0.136928	+	0.990581i	$0.456277\pi$
$710$	−4.51064	−0.169281
$711$	−22.5162	−0.844422
$712$	2.64423	0.0990969
$713$	9.36548	0.350740
$714$	−0.0252984	−0.000946769 0
$715$	−4.03963	−0.151074
$716$	0.744495	0.0278231
$717$	−43.2155	−1.61391
$718$	24.9744	0.932038
$719$	2.69508	0.100509	0.0502547	−	0.998736i	$-0.483997\pi$
$719$	2.69508	0.100509	0.0502547	+	0.998736i	$0.483997\pi$
$720$	−3.04441	−0.113458
$721$	0.0159720	0.000594830 0
$722$	14.4963	0.539497
$723$	38.1139	1.41747
$724$	−14.5856	−0.542070
$725$	12.9292	0.480178
$726$	4.12826	0.153214
$727$	−2.85675	−0.105951	−0.0529754	−	0.998596i	$-0.516871\pi$
$727$	−2.85675	−0.105951	−0.0529754	+	0.998596i	$0.516871\pi$
$728$	0.00700439	0.000259600 0
$729$	−32.8615	−1.21709
$730$	−14.0551	−0.520202
$731$	−24.5778	−0.909042
$732$	−34.3112	−1.26818
$733$	50.9565	1.88212	0.941060	−	0.338240i	$-0.109832\pi$
$733$	50.9565	1.88212	0.941060	+	0.338240i	$0.109832\pi$
$734$	−4.24439	−0.156663
$735$	−16.0367	−0.591522
$736$	2.17008	0.0799902
$737$	35.5400	1.30913
$738$	−42.5203	−1.56519
$739$	−49.0010	−1.80253	−0.901265	−	0.433268i	$-0.857360\pi$
$739$	−49.0010	−1.80253	−0.901265	+	0.433268i	$0.857360\pi$
$740$	3.42270	0.125821
$741$	21.1738	0.777840
$742$	0.0487687	0.00179036
$743$	−21.4421	−0.786634	−0.393317	−	0.919403i	$-0.628672\pi$
$743$	−21.4421	−0.786634	−0.393317	+	0.919403i	$0.628672\pi$
$744$	−10.8737	−0.398651
$745$	18.7610	0.687351
$746$	−5.26135	−0.192632
$747$	−3.75254	−0.137298
$748$	6.36865	0.232861
$749$	−0.0751352	−0.00274538
$750$	−21.0155	−0.767378
$751$	13.3972	0.488872	0.244436	−	0.969665i	$-0.421397\pi$
$751$	13.3972	0.488872	0.244436	+	0.969665i	$0.421397\pi$
$752$	−7.36640	−0.268625
$753$	−15.5982	−0.568431
$754$	−4.49857	−0.163828
$755$	0.855636	0.0311398
$756$	−0.00423187	−0.000153912 0
$757$	−53.9187	−1.95971	−0.979854	−	0.199714i	$-0.935999\pi$
$757$	−53.9187	−1.95971	−0.979854	+	0.199714i	$0.935999\pi$
$758$	6.69263	0.243087
$759$	−16.7291	−0.607229
$760$	5.26249	0.190891
$761$	−39.4432	−1.42981	−0.714907	−	0.699220i	$-0.753531\pi$
$761$	−39.4432	−1.42981	−0.714907	+	0.699220i	$0.753531\pi$
$762$	−31.0921	−1.12635
$763$	−0.0452850	−0.00163943
$764$	9.11568	0.329794
$765$	−6.33690	−0.229111
$766$	4.64020	0.167657
$767$	12.4172	0.448360
$768$	−2.51956	−0.0909168
$769$	−3.68605	−0.132922	−0.0664611	−	0.997789i	$-0.521171\pi$
$769$	−3.68605	−0.132922	−0.0664611	+	0.997789i	$0.521171\pi$
$770$	−0.0134202	−0.000483631 0
$771$	−11.1676	−0.402192
$772$	−13.1995	−0.475061
$773$	23.9425	0.861153	0.430577	−	0.902554i	$-0.358310\pi$
$773$	23.9425	0.861153	0.430577	+	0.902554i	$0.358310\pi$
$774$	−39.5347	−1.42104
$775$	−18.0105	−0.646957
$776$	12.7829	0.458878
$777$	0.0457504	0.00164128
$778$	12.0354	0.431489
$779$	73.4996	2.63340
$780$	3.32655	0.119110
$781$	15.1781	0.543117
$782$	4.51700	0.161528
$783$	2.71792	0.0971307
$784$	−6.99998	−0.249999
$785$	3.58019	0.127782
$786$	31.3176	1.11706
$787$	−6.64066	−0.236714	−0.118357	−	0.992971i	$-0.537763\pi$
$787$	−6.64066	−0.236714	−0.118357	+	0.992971i	$0.537763\pi$
$788$	−9.82199	−0.349894
$789$	77.2961	2.75181
$790$	6.11473	0.217552
$791$	−0.0236174	−0.000839737 0
$792$	10.2443	0.364016
$793$	19.7736	0.702183
$794$	−1.63041	−0.0578609
$795$	23.1614	0.821451
$796$	−3.36120	−0.119135
$797$	17.0863	0.605229	0.302615	−	0.953113i	$-0.402140\pi$
$797$	17.0863	0.605229	0.302615	+	0.953113i	$0.402140\pi$
$798$	0.0703424	0.00249009
$799$	−15.3331	−0.542446
$800$	−4.17323	−0.147546
$801$	8.85339	0.312819
$802$	−33.0363	−1.16655
$803$	47.2948	1.66900
$804$	−29.2664	−1.03215
$805$	−0.00951836	−0.000335478 0
$806$	6.26658	0.220731
$807$	44.2353	1.55716
$808$	4.31481	0.151795
$809$	1.20514	0.0423706	0.0211853	−	0.999776i	$-0.493256\pi$
$809$	1.20514	0.0423706	0.0211853	+	0.999776i	$0.493256\pi$
$810$	7.12340	0.250291
$811$	25.0280	0.878851	0.439425	−	0.898279i	$-0.355182\pi$
$811$	25.0280	0.878851	0.439425	+	0.898279i	$0.355182\pi$
$812$	−0.0149449	−0.000524463 0
$813$	34.8304	1.22155
$814$	−11.5173	−0.403679
$815$	19.1031	0.669152
$816$	−5.24444	−0.183592
$817$	68.3387	2.39087
$818$	−14.6760	−0.513133
$819$	0.0234520	0.000819479 0
$820$	11.5473	0.403248
$821$	49.9093	1.74185	0.870923	−	0.491420i	$-0.163522\pi$
$821$	49.9093	1.74185	0.870923	+	0.491420i	$0.163522\pi$
$822$	−1.33577	−0.0465905
$823$	−16.2421	−0.566165	−0.283082	−	0.959096i	$-0.591357\pi$
$823$	−16.2421	−0.566165	−0.283082	+	0.959096i	$0.591357\pi$
$824$	3.31105	0.115346
$825$	32.1714	1.12006
$826$	0.0412517	0.00143533
$827$	−27.6952	−0.963055	−0.481528	−	0.876431i	$-0.659918\pi$
$827$	−27.6952	−0.963055	−0.481528	+	0.876431i	$0.659918\pi$
$828$	7.26583	0.252505
$829$	22.8577	0.793882	0.396941	−	0.917844i	$-0.370072\pi$
$829$	22.8577	0.793882	0.396941	+	0.917844i	$0.370072\pi$
$830$	1.01908	0.0353728
$831$	−47.7338	−1.65587
$832$	1.45203	0.0503402
$833$	−14.5704	−0.504834
$834$	−6.26662	−0.216995
$835$	22.0278	0.762304
$836$	−17.7081	−0.612447
$837$	−3.78611	−0.130867
$838$	−3.01023	−0.103987
$839$	21.9530	0.757902	0.378951	−	0.925417i	$-0.376285\pi$
$839$	21.9530	0.757902	0.378951	+	0.925417i	$0.376285\pi$
$840$	0.0110513	0.000381304 0
$841$	−19.4016	−0.669022
$842$	30.4712	1.05011
$843$	−7.80285	−0.268744
$844$	16.7356	0.576063
$845$	9.90341	0.340688
$846$	−24.6641	−0.847969
$847$	−0.00790381	−0.000271578 0
$848$	10.1099	0.347176
$849$	−34.7674	−1.19321
$850$	−8.68654	−0.297946
$851$	−8.16867	−0.280018
$852$	−12.4989	−0.428204
$853$	7.52913	0.257792	0.128896	−	0.991658i	$-0.458857\pi$
$853$	7.52913	0.257792	0.128896	+	0.991658i	$0.458857\pi$
$854$	0.0656908	0.00224789
$855$	17.6198	0.602585
$856$	−15.5758	−0.532369
$857$	−30.7690	−1.05105	−0.525524	−	0.850779i	$-0.676131\pi$
$857$	−30.7690	−1.05105	−0.525524	+	0.850779i	$0.676131\pi$
$858$	−11.1937	−0.382147
$859$	−49.0163	−1.67242	−0.836208	−	0.548413i	$-0.815232\pi$
$859$	−49.0163	−1.67242	−0.836208	+	0.548413i	$0.815232\pi$
$860$	10.7365	0.366110
$861$	0.154349	0.00526021
$862$	−28.1569	−0.959027
$863$	19.5655	0.666019	0.333009	−	0.942924i	$-0.391936\pi$
$863$	19.5655	0.666019	0.333009	+	0.942924i	$0.391936\pi$
$864$	−0.877281	−0.0298457
$865$	10.9680	0.372924
$866$	−24.9466	−0.847720
$867$	31.9163	1.08393
$868$	0.0208184	0.000706624 0
$869$	−20.5758	−0.697987
$870$	−7.09768	−0.240634
$871$	16.8663	0.571493
$872$	−9.38773	−0.317909
$873$	42.7995	1.44854
$874$	−12.5595	−0.424833
$875$	0.0402354	0.00136021
$876$	−38.9462	−1.31587
$877$	−15.8734	−0.536006	−0.268003	−	0.963418i	$-0.586364\pi$
$877$	−15.8734	−0.536006	−0.268003	+	0.963418i	$0.586364\pi$
$878$	2.28144	0.0769949
$879$	74.8153	2.52346
$880$	−2.78206	−0.0937831
$881$	−4.84391	−0.163195	−0.0815977	−	0.996665i	$-0.526002\pi$
$881$	−4.84391	−0.163195	−0.0815977	+	0.996665i	$0.526002\pi$
$882$	−23.4372	−0.789173
$883$	−30.0985	−1.01290	−0.506448	−	0.862270i	$-0.669042\pi$
$883$	−30.0985	−1.01290	−0.506448	+	0.862270i	$0.669042\pi$
$884$	3.02239	0.101654
$885$	19.5914	0.658558
$886$	10.8977	0.366115
$887$	49.4499	1.66037	0.830183	−	0.557490i	$-0.188236\pi$
$887$	49.4499	1.66037	0.830183	+	0.557490i	$0.188236\pi$
$888$	9.48420	0.318269
$889$	0.0595278	0.00199650
$890$	−2.40432	−0.0805931
$891$	−23.9700	−0.803025
$892$	9.21293	0.308472
$893$	42.6338	1.42669
$894$	51.9863	1.73868
$895$	−0.676947	−0.0226278
$896$	0.00482385	0.000161154 0
$897$	−7.93920	−0.265082
$898$	−25.2566	−0.842823
$899$	−13.3707	−0.445937
$900$	−13.9728	−0.465758
$901$	21.0437	0.701067
$902$	−38.8561	−1.29377
$903$	0.143512	0.00477577
$904$	−4.89596	−0.162837
$905$	13.2623	0.440852
$906$	2.37094	0.0787693
$907$	−19.0460	−0.632411	−0.316205	−	0.948691i	$-0.602409\pi$
$907$	−19.0460	−0.632411	−0.316205	+	0.948691i	$0.602409\pi$
$908$	−6.50751	−0.215959
$909$	14.4468	0.479170
$910$	−0.00636888	−0.000211126 0
$911$	38.7959	1.28537	0.642683	−	0.766132i	$-0.277821\pi$
$911$	38.7959	1.28537	0.642683	+	0.766132i	$0.277821\pi$
$912$	14.5822	0.482865
$913$	−3.42917	−0.113489
$914$	−28.0112	−0.926529
$915$	31.1981	1.03138
$916$	5.19735	0.171725
$917$	−0.0599595	−0.00198004
$918$	−1.82605	−0.0602687
$919$	−25.4502	−0.839523	−0.419762	−	0.907634i	$-0.637886\pi$
$919$	−25.4502	−0.839523	−0.419762	+	0.907634i	$0.637886\pi$
$920$	−1.97319	−0.0650541
$921$	−25.6974	−0.846759
$922$	−42.3588	−1.39501
$923$	7.20314	0.237094
$924$	−0.0371871	−0.00122336
$925$	15.7090	0.516508
$926$	−9.55174	−0.313890
$927$	11.0860	0.364113
$928$	−3.09812	−0.101701
$929$	9.23192	0.302890	0.151445	−	0.988466i	$-0.451607\pi$
$929$	9.23192	0.302890	0.151445	+	0.988466i	$0.451607\pi$
$930$	9.88717	0.324213
$931$	40.5131	1.32776
$932$	−16.1368	−0.528579
$933$	−70.5030	−2.30816
$934$	22.0439	0.721297
$935$	−5.79082	−0.189380
$936$	4.86168	0.158909
$937$	2.92926	0.0956947	0.0478473	−	0.998855i	$-0.484764\pi$
$937$	2.92926	0.0956947	0.0478473	+	0.998855i	$0.484764\pi$
$938$	0.0560322	0.00182952
$939$	−8.25087	−0.269257
$940$	6.69805	0.218466
$941$	39.0516	1.27305	0.636523	−	0.771258i	$-0.280372\pi$
$941$	39.0516	1.27305	0.636523	+	0.771258i	$0.280372\pi$
$942$	9.92061	0.323231
$943$	−27.5589	−0.897441
$944$	8.55161	0.278331
$945$	0.00384791	0.000125173 0
$946$	−36.1278	−1.17462
$947$	50.0727	1.62714	0.813572	−	0.581464i	$-0.197520\pi$
$947$	50.0727	1.62714	0.813572	+	0.581464i	$0.197520\pi$
$948$	16.9437	0.550307
$949$	22.4449	0.728591
$950$	24.1530	0.783626
$951$	−13.5244	−0.438560
$952$	0.0100408	0.000325424 0
$953$	−29.3196	−0.949754	−0.474877	−	0.880052i	$-0.657507\pi$
$953$	−29.3196	−0.949754	−0.474877	+	0.880052i	$0.657507\pi$
$954$	33.8499	1.09593
$955$	−8.28861	−0.268213
$956$	17.1520	0.554735
$957$	23.8834	0.772042
$958$	11.1450	0.360078
$959$	0.00255742	8.25834e−5 0
$960$	2.29096	0.0739404
$961$	−12.3745	−0.399176
$962$	−5.46578	−0.176224
$963$	−52.1506	−1.68053
$964$	−15.1272	−0.487214
$965$	12.0019	0.386355
$966$	−0.0263751	−0.000848605 0
$967$	−31.9281	−1.02674	−0.513370	−	0.858168i	$-0.671603\pi$
$967$	−31.9281	−1.02674	−0.513370	+	0.858168i	$0.671603\pi$
$968$	−1.63849	−0.0526629
$969$	30.3527	0.975070
$970$	−11.6231	−0.373195
$971$	47.1387	1.51275	0.756376	−	0.654137i	$-0.226968\pi$
$971$	47.1387	1.51275	0.756376	+	0.654137i	$0.226968\pi$
$972$	22.3706	0.717537
$973$	0.0119978	0.000384632 0
$974$	−21.3065	−0.682703
$975$	15.2677	0.488957
$976$	13.6179	0.435899
$977$	8.47638	0.271183	0.135592	−	0.990765i	$-0.456706\pi$
$977$	8.47638	0.271183	0.135592	+	0.990765i	$0.456706\pi$
$978$	52.9341	1.69265
$979$	8.09046	0.258572
$980$	6.36487	0.203318
$981$	−31.4319	−1.00354
$982$	−16.8422	−0.537457
$983$	−45.1205	−1.43912	−0.719560	−	0.694430i	$-0.755657\pi$
$983$	−45.1205	−1.43912	−0.719560	+	0.694430i	$0.755657\pi$
$984$	31.9971	1.02003
$985$	8.93084	0.284560
$986$	−6.44871	−0.205369
$987$	0.0895311	0.00284981
$988$	−8.40378	−0.267360
$989$	−25.6238	−0.814790
$990$	−9.31485	−0.296045
$991$	−23.0820	−0.733223	−0.366611	−	0.930374i	$-0.619482\pi$
$991$	−23.0820	−0.733223	−0.366611	+	0.930374i	$0.619482\pi$
$992$	4.31573	0.137025
$993$	57.7357	1.83219
$994$	0.0239298	0.000759008 0
$995$	3.05624	0.0968894
$996$	2.82385	0.0894770
$997$	−12.8296	−0.406317	−0.203159	−	0.979146i	$-0.565121\pi$
$997$	−12.8296	−0.406317	−0.203159	+	0.979146i	$0.565121\pi$
$998$	−3.01941	−0.0955778
$999$	3.30228	0.104480

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.8	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.8	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.51956 q^{3} +1.00000 q^{4} -0.909270 q^{5} -2.51956 q^{6} +0.00482385 q^{7} +1.00000 q^{8} +3.34819 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.51956 q^{3} +1.00000 q^{4} -0.909270 q^{5} -2.51956 q^{6} +0.00482385 q^{7} +1.00000 q^{8} +3.34819 q^{9} -0.909270 q^{10} +3.05966 q^{11} -2.51956 q^{12} +1.45203 q^{13} +0.00482385 q^{14} +2.29096 q^{15} +1.00000 q^{16} +2.08149 q^{17} +3.34819 q^{18} -5.78760 q^{19} -0.909270 q^{20} -0.0121540 q^{21} +3.05966 q^{22} +2.17008 q^{23} -2.51956 q^{24} -4.17323 q^{25} +1.45203 q^{26} -0.877281 q^{27} +0.00482385 q^{28} -3.09812 q^{29} +2.29096 q^{30} +4.31573 q^{31} +1.00000 q^{32} -7.70900 q^{33} +2.08149 q^{34} -0.00438618 q^{35} +3.34819 q^{36} -3.76423 q^{37} -5.78760 q^{38} -3.65848 q^{39} -0.909270 q^{40} -12.6995 q^{41} -0.0121540 q^{42} -11.8078 q^{43} +3.05966 q^{44} -3.04441 q^{45} +2.17008 q^{46} -7.36640 q^{47} -2.51956 q^{48} -6.99998 q^{49} -4.17323 q^{50} -5.24444 q^{51} +1.45203 q^{52} +10.1099 q^{53} -0.877281 q^{54} -2.78206 q^{55} +0.00482385 q^{56} +14.5822 q^{57} -3.09812 q^{58} +8.55161 q^{59} +2.29096 q^{60} +13.6179 q^{61} +4.31573 q^{62} +0.0161512 q^{63} +1.00000 q^{64} -1.32029 q^{65} -7.70900 q^{66} +11.6157 q^{67} +2.08149 q^{68} -5.46765 q^{69} -0.00438618 q^{70} +4.96073 q^{71} +3.34819 q^{72} +15.4576 q^{73} -3.76423 q^{74} +10.5147 q^{75} -5.78760 q^{76} +0.0147593 q^{77} -3.65848 q^{78} -6.72488 q^{79} -0.909270 q^{80} -7.83420 q^{81} -12.6995 q^{82} -1.12077 q^{83} -0.0121540 q^{84} -1.89264 q^{85} -11.8078 q^{86} +7.80591 q^{87} +3.05966 q^{88} +2.64423 q^{89} -3.04441 q^{90} +0.00700439 q^{91} +2.17008 q^{92} -10.8737 q^{93} -7.36640 q^{94} +5.26249 q^{95} -2.51956 q^{96} +12.7829 q^{97} -6.99998 q^{98} +10.2443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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