LMFDB - Embedded newform 8045.2.a.b.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$1$
Dimension:	$126$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44573 q^{2} +0.659690 q^{3} +3.98158 q^{4} -1.00000 q^{5} -1.61342 q^{6} -0.125305 q^{7} -4.84641 q^{8} -2.56481 q^{9} +O(q^{10})$	\(q-2.44573 q^{2} +0.659690 q^{3} +3.98158 q^{4} -1.00000 q^{5} -1.61342 q^{6} -0.125305 q^{7} -4.84641 q^{8} -2.56481 q^{9} +2.44573 q^{10} -2.92758 q^{11} +2.62661 q^{12} -3.36219 q^{13} +0.306462 q^{14} -0.659690 q^{15} +3.88984 q^{16} +5.31063 q^{17} +6.27283 q^{18} +6.64611 q^{19} -3.98158 q^{20} -0.0826623 q^{21} +7.16007 q^{22} +2.69231 q^{23} -3.19713 q^{24} +1.00000 q^{25} +8.22301 q^{26} -3.67105 q^{27} -0.498912 q^{28} +9.50936 q^{29} +1.61342 q^{30} -2.34411 q^{31} +0.179335 q^{32} -1.93130 q^{33} -12.9883 q^{34} +0.125305 q^{35} -10.2120 q^{36} -3.49279 q^{37} -16.2546 q^{38} -2.21800 q^{39} +4.84641 q^{40} -12.5528 q^{41} +0.202170 q^{42} -4.12734 q^{43} -11.6564 q^{44} +2.56481 q^{45} -6.58466 q^{46} -11.1642 q^{47} +2.56609 q^{48} -6.98430 q^{49} -2.44573 q^{50} +3.50336 q^{51} -13.3869 q^{52} +9.64515 q^{53} +8.97838 q^{54} +2.92758 q^{55} +0.607279 q^{56} +4.38437 q^{57} -23.2573 q^{58} +11.3945 q^{59} -2.62661 q^{60} -0.106124 q^{61} +5.73305 q^{62} +0.321383 q^{63} -8.21829 q^{64} +3.36219 q^{65} +4.72343 q^{66} -7.38211 q^{67} +21.1447 q^{68} +1.77609 q^{69} -0.306462 q^{70} -9.06796 q^{71} +12.4301 q^{72} +16.6356 q^{73} +8.54240 q^{74} +0.659690 q^{75} +26.4620 q^{76} +0.366841 q^{77} +5.42464 q^{78} -4.24904 q^{79} -3.88984 q^{80} +5.27268 q^{81} +30.7008 q^{82} +0.247639 q^{83} -0.329127 q^{84} -5.31063 q^{85} +10.0943 q^{86} +6.27323 q^{87} +14.1883 q^{88} -4.52621 q^{89} -6.27283 q^{90} +0.421300 q^{91} +10.7197 q^{92} -1.54638 q^{93} +27.3047 q^{94} -6.64611 q^{95} +0.118305 q^{96} +17.6304 q^{97} +17.0817 q^{98} +7.50870 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) $ 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.44573	−1.72939	−0.864695	−	0.502297i	$-0.832489\pi$
$2$	−2.44573	−1.72939	−0.864695	+	0.502297i	$0.832489\pi$
$3$	0.659690	0.380872	0.190436	−	0.981700i	$-0.439010\pi$
$3$	0.659690	0.380872	0.190436	+	0.981700i	$0.439010\pi$
$4$	3.98158	1.99079
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.61342	−0.658676
$7$	−0.125305	−0.0473608	−0.0236804	−	0.999720i	$-0.507538\pi$
$7$	−0.125305	−0.0473608	−0.0236804	+	0.999720i	$0.507538\pi$
$8$	−4.84641	−1.71347
$9$	−2.56481	−0.854937
$10$	2.44573	0.773407
$11$	−2.92758	−0.882700	−0.441350	−	0.897335i	$-0.645500\pi$
$11$	−2.92758	−0.882700	−0.441350	+	0.897335i	$0.645500\pi$
$12$	2.62661	0.758237
$13$	−3.36219	−0.932505	−0.466253	−	0.884652i	$-0.654396\pi$
$13$	−3.36219	−0.932505	−0.466253	+	0.884652i	$0.654396\pi$
$14$	0.306462	0.0819053
$15$	−0.659690	−0.170331
$16$	3.88984	0.972460
$17$	5.31063	1.28802	0.644008	−	0.765019i	$-0.277270\pi$
$17$	5.31063	1.28802	0.644008	+	0.765019i	$0.277270\pi$
$18$	6.27283	1.47852
$19$	6.64611	1.52472	0.762361	−	0.647152i	$-0.224040\pi$
$19$	6.64611	1.52472	0.762361	+	0.647152i	$0.224040\pi$
$20$	−3.98158	−0.890309
$21$	−0.0826623	−0.0180384
$22$	7.16007	1.52653
$23$	2.69231	0.561386	0.280693	−	0.959798i	$-0.409436\pi$
$23$	2.69231	0.561386	0.280693	+	0.959798i	$0.409436\pi$
$24$	−3.19713	−0.652611
$25$	1.00000	0.200000
$26$	8.22301	1.61267
$27$	−3.67105	−0.706493
$28$	−0.498912	−0.0942855
$29$	9.50936	1.76584	0.882922	−	0.469520i	$-0.155573\pi$
$29$	9.50936	1.76584	0.882922	+	0.469520i	$0.155573\pi$
$30$	1.61342	0.294569
$31$	−2.34411	−0.421014	−0.210507	−	0.977592i	$-0.567511\pi$
$31$	−2.34411	−0.421014	−0.210507	+	0.977592i	$0.567511\pi$
$32$	0.179335	0.0317022
$33$	−1.93130	−0.336196
$34$	−12.9883	−2.22748
$35$	0.125305	0.0211804
$36$	−10.2120	−1.70200
$37$	−3.49279	−0.574211	−0.287105	−	0.957899i	$-0.592693\pi$
$37$	−3.49279	−0.574211	−0.287105	+	0.957899i	$0.592693\pi$
$38$	−16.2546	−2.63684
$39$	−2.21800	−0.355165
$40$	4.84641	0.766285
$41$	−12.5528	−1.96042	−0.980212	−	0.197949i	$-0.936572\pi$
$41$	−12.5528	−1.96042	−0.980212	+	0.197949i	$0.936572\pi$
$42$	0.202170	0.0311954
$43$	−4.12734	−0.629413	−0.314706	−	0.949189i	$-0.601906\pi$
$43$	−4.12734	−0.629413	−0.314706	+	0.949189i	$0.601906\pi$
$44$	−11.6564	−1.75727
$45$	2.56481	0.382339
$46$	−6.58466	−0.970856
$47$	−11.1642	−1.62847	−0.814236	−	0.580534i	$-0.802844\pi$
$47$	−11.1642	−1.62847	−0.814236	+	0.580534i	$0.802844\pi$
$48$	2.56609	0.370383
$49$	−6.98430	−0.997757
$50$	−2.44573	−0.345878
$51$	3.50336	0.490569
$52$	−13.3869	−1.85642
$53$	9.64515	1.32486	0.662432	−	0.749122i	$-0.269524\pi$
$53$	9.64515	1.32486	0.662432	+	0.749122i	$0.269524\pi$
$54$	8.97838	1.22180
$55$	2.92758	0.394755
$56$	0.607279	0.0811511
$57$	4.38437	0.580724
$58$	−23.2573	−3.05383
$59$	11.3945	1.48344	0.741719	−	0.670711i	$-0.234011\pi$
$59$	11.3945	1.48344	0.741719	+	0.670711i	$0.234011\pi$
$60$	−2.62661	−0.339094
$61$	−0.106124	−0.0135877	−0.00679387	−	0.999977i	$-0.502163\pi$
$61$	−0.106124	−0.0135877	−0.00679387	+	0.999977i	$0.502163\pi$
$62$	5.73305	0.728098
$63$	0.321383	0.0404905
$64$	−8.21829	−1.02729
$65$	3.36219	0.417029
$66$	4.72343	0.581414
$67$	−7.38211	−0.901869	−0.450934	−	0.892557i	$-0.648909\pi$
$67$	−7.38211	−0.901869	−0.450934	+	0.892557i	$0.648909\pi$
$68$	21.1447	2.56417
$69$	1.77609	0.213816
$70$	−0.306462	−0.0366292
$71$	−9.06796	−1.07617	−0.538084	−	0.842891i	$-0.680852\pi$
$71$	−9.06796	−1.07617	−0.538084	+	0.842891i	$0.680852\pi$
$72$	12.4301	1.46490
$73$	16.6356	1.94705	0.973524	−	0.228586i	$-0.0734103\pi$
$73$	16.6356	1.94705	0.973524	+	0.228586i	$0.0734103\pi$
$74$	8.54240	0.993034
$75$	0.659690	0.0761744
$76$	26.4620	3.03540
$77$	0.366841	0.0418054
$78$	5.42464	0.614219
$79$	−4.24904	−0.478054	−0.239027	−	0.971013i	$-0.576828\pi$
$79$	−4.24904	−0.478054	−0.239027	+	0.971013i	$0.576828\pi$
$80$	−3.88984	−0.434898
$81$	5.27268	0.585853
$82$	30.7008	3.39034
$83$	0.247639	0.0271819	0.0135910	−	0.999908i	$-0.495674\pi$
$83$	0.247639	0.0271819	0.0135910	+	0.999908i	$0.495674\pi$
$84$	−0.329127	−0.0359107
$85$	−5.31063	−0.576018
$86$	10.0943	1.08850
$87$	6.27323	0.672561
$88$	14.1883	1.51248
$89$	−4.52621	−0.479778	−0.239889	−	0.970800i	$-0.577111\pi$
$89$	−4.52621	−0.479778	−0.239889	+	0.970800i	$0.577111\pi$
$90$	−6.27283	−0.661214
$91$	0.421300	0.0441642
$92$	10.7197	1.11760
$93$	−1.54638	−0.160352
$94$	27.3047	2.81626
$95$	−6.64611	−0.681876
$96$	0.118305	0.0120745
$97$	17.6304	1.79009	0.895047	−	0.445972i	$-0.147142\pi$
$97$	17.6304	1.79009	0.895047	+	0.445972i	$0.147142\pi$
$98$	17.0817	1.72551
$99$	7.50870	0.754652
$100$	3.98158	0.398158
$101$	6.61669	0.658386	0.329193	−	0.944263i	$-0.393223\pi$
$101$	6.61669	0.658386	0.329193	+	0.944263i	$0.393223\pi$
$102$	−8.56827	−0.848386
$103$	−7.65712	−0.754479	−0.377239	−	0.926116i	$-0.623127\pi$
$103$	−7.65712	−0.754479	−0.377239	+	0.926116i	$0.623127\pi$
$104$	16.2946	1.59782
$105$	0.0826623	0.00806702
$106$	−23.5894	−2.29121
$107$	3.93654	0.380559	0.190280	−	0.981730i	$-0.439060\pi$
$107$	3.93654	0.380559	0.190280	+	0.981730i	$0.439060\pi$
$108$	−14.6166	−1.40648
$109$	7.37753	0.706639	0.353320	−	0.935503i	$-0.385053\pi$
$109$	7.37753	0.706639	0.353320	+	0.935503i	$0.385053\pi$
$110$	−7.16007	−0.682686
$111$	−2.30415	−0.218701
$112$	−0.487416	−0.0460565
$113$	5.06398	0.476379	0.238190	−	0.971219i	$-0.423446\pi$
$113$	5.06398	0.476379	0.238190	+	0.971219i	$0.423446\pi$
$114$	−10.7230	−1.00430
$115$	−2.69231	−0.251059
$116$	37.8623	3.51543
$117$	8.62339	0.797233
$118$	−27.8678	−2.56544
$119$	−0.665447	−0.0610015
$120$	3.19713	0.291857
$121$	−2.42925	−0.220841
$122$	0.259550	0.0234985
$123$	−8.28098	−0.746671
$124$	−9.33326	−0.838151
$125$	−1.00000	−0.0894427
$126$	−0.786016	−0.0700239
$127$	1.13098	0.100359	0.0501793	−	0.998740i	$-0.484021\pi$
$127$	1.13098	0.100359	0.0501793	+	0.998740i	$0.484021\pi$
$128$	19.7410	1.74488
$129$	−2.72276	−0.239726
$130$	−8.22301	−0.721206
$131$	4.91727	0.429624	0.214812	−	0.976655i	$-0.431086\pi$
$131$	4.91727	0.429624	0.214812	+	0.976655i	$0.431086\pi$
$132$	−7.68962	−0.669296
$133$	−0.832790	−0.0722121
$134$	18.0546	1.55968
$135$	3.67105	0.315953
$136$	−25.7375	−2.20697
$137$	−1.86266	−0.159138	−0.0795690	−	0.996829i	$-0.525354\pi$
$137$	−1.86266	−0.159138	−0.0795690	+	0.996829i	$0.525354\pi$
$138$	−4.34383	−0.369772
$139$	15.1122	1.28180	0.640899	−	0.767626i	$-0.278562\pi$
$139$	15.1122	1.28180	0.640899	+	0.767626i	$0.278562\pi$
$140$	0.498912	0.0421658
$141$	−7.36494	−0.620239
$142$	22.1778	1.86112
$143$	9.84311	0.823122
$144$	−9.97670	−0.831392
$145$	−9.50936	−0.789710
$146$	−40.6861	−3.36721
$147$	−4.60747	−0.380018
$148$	−13.9068	−1.14313
$149$	−18.6852	−1.53075	−0.765376	−	0.643583i	$-0.777447\pi$
$149$	−18.6852	−1.53075	−0.765376	+	0.643583i	$0.777447\pi$
$150$	−1.61342	−0.131735
$151$	10.4262	0.848469	0.424234	−	0.905552i	$-0.360543\pi$
$151$	10.4262	0.848469	0.424234	+	0.905552i	$0.360543\pi$
$152$	−32.2098	−2.61256
$153$	−13.6207	−1.10117
$154$	−0.897192	−0.0722978
$155$	2.34411	0.188283
$156$	−8.83117	−0.707060
$157$	4.85653	0.387593	0.193797	−	0.981042i	$-0.437920\pi$
$157$	4.85653	0.387593	0.193797	+	0.981042i	$0.437920\pi$
$158$	10.3920	0.826743
$159$	6.36281	0.504603
$160$	−0.179335	−0.0141777
$161$	−0.337360	−0.0265877
$162$	−12.8955	−1.01317
$163$	−22.8883	−1.79275	−0.896377	−	0.443293i	$-0.853810\pi$
$163$	−22.8883	−1.79275	−0.896377	+	0.443293i	$0.853810\pi$
$164$	−49.9802	−3.90280
$165$	1.93130	0.150351
$166$	−0.605658	−0.0470081
$167$	6.83359	0.528799	0.264400	−	0.964413i	$-0.414826\pi$
$167$	6.83359	0.528799	0.264400	+	0.964413i	$0.414826\pi$
$168$	0.400616	0.0309082
$169$	−1.69564	−0.130434
$170$	12.9883	0.996160
$171$	−17.0460	−1.30354
$172$	−16.4333	−1.25303
$173$	21.8315	1.65982	0.829908	−	0.557901i	$-0.188393\pi$
$173$	21.8315	1.65982	0.829908	+	0.557901i	$0.188393\pi$
$174$	−15.3426	−1.16312
$175$	−0.125305	−0.00947216
$176$	−11.3878	−0.858391
$177$	7.51683	0.565000
$178$	11.0699	0.829723
$179$	−2.60976	−0.195063	−0.0975314	−	0.995232i	$-0.531095\pi$
$179$	−2.60976	−0.195063	−0.0975314	+	0.995232i	$0.531095\pi$
$180$	10.2120	0.761158
$181$	4.57052	0.339724	0.169862	−	0.985468i	$-0.445668\pi$
$181$	4.57052	0.339724	0.169862	+	0.985468i	$0.445668\pi$
$182$	−1.03038	−0.0763771
$183$	−0.0700087	−0.00517519
$184$	−13.0481	−0.961916
$185$	3.49279	0.256795
$186$	3.78203	0.277312
$187$	−15.5473	−1.13693
$188$	−44.4514	−3.24195
$189$	0.460000	0.0334601
$190$	16.2546	1.17923
$191$	−14.1040	−1.02053	−0.510265	−	0.860017i	$-0.670453\pi$
$191$	−14.1040	−1.02053	−0.510265	+	0.860017i	$0.670453\pi$
$192$	−5.42152	−0.391264
$193$	−17.7206	−1.27556	−0.637778	−	0.770220i	$-0.720146\pi$
$193$	−17.7206	−1.27556	−0.637778	+	0.770220i	$0.720146\pi$
$194$	−43.1191	−3.09577
$195$	2.21800	0.158835
$196$	−27.8086	−1.98633
$197$	20.9495	1.49259	0.746295	−	0.665615i	$-0.231831\pi$
$197$	20.9495	1.49259	0.746295	+	0.665615i	$0.231831\pi$
$198$	−18.3642	−1.30509
$199$	17.7032	1.25495	0.627474	−	0.778638i	$-0.284089\pi$
$199$	17.7032	1.25495	0.627474	+	0.778638i	$0.284089\pi$
$200$	−4.84641	−0.342693
$201$	−4.86990	−0.343496
$202$	−16.1826	−1.13861
$203$	−1.19157	−0.0836318
$204$	13.9489	0.976621
$205$	12.5528	0.876729
$206$	18.7272	1.30479
$207$	−6.90527	−0.479949
$208$	−13.0784	−0.906824
$209$	−19.4570	−1.34587
$210$	−0.202170	−0.0139510
$211$	−16.2916	−1.12156	−0.560779	−	0.827966i	$-0.689498\pi$
$211$	−16.2916	−1.12156	−0.560779	+	0.827966i	$0.689498\pi$
$212$	38.4030	2.63753
$213$	−5.98204	−0.409882
$214$	−9.62770	−0.658136
$215$	4.12734	0.281482
$216$	17.7914	1.21055
$217$	0.293728	0.0199396
$218$	−18.0434	−1.22206
$219$	10.9743	0.741576
$220$	11.6564	0.785876
$221$	−17.8554	−1.20108
$222$	5.63533	0.378219
$223$	−15.7066	−1.05179	−0.525897	−	0.850548i	$-0.676270\pi$
$223$	−15.7066	−1.05179	−0.525897	+	0.850548i	$0.676270\pi$
$224$	−0.0224715	−0.00150144
$225$	−2.56481	−0.170987
$226$	−12.3851	−0.823846
$227$	24.4568	1.62325	0.811627	−	0.584175i	$-0.198582\pi$
$227$	24.4568	1.62325	0.811627	+	0.584175i	$0.198582\pi$
$228$	17.4567	1.15610
$229$	−1.49990	−0.0991165	−0.0495582	−	0.998771i	$-0.515781\pi$
$229$	−1.49990	−0.0991165	−0.0495582	+	0.998771i	$0.515781\pi$
$230$	6.58466	0.434180
$231$	0.242001	0.0159225
$232$	−46.0863	−3.02571
$233$	21.0056	1.37612	0.688061	−	0.725652i	$-0.258462\pi$
$233$	21.0056	1.37612	0.688061	+	0.725652i	$0.258462\pi$
$234$	−21.0905	−1.37873
$235$	11.1642	0.728275
$236$	45.3682	2.95322
$237$	−2.80305	−0.182077
$238$	1.62750	0.105495
$239$	23.9580	1.54971	0.774857	−	0.632136i	$-0.217822\pi$
$239$	23.9580	1.54971	0.774857	+	0.632136i	$0.217822\pi$
$240$	−2.56609	−0.165640
$241$	−17.1788	−1.10658	−0.553292	−	0.832987i	$-0.686629\pi$
$241$	−17.1788	−1.10658	−0.553292	+	0.832987i	$0.686629\pi$
$242$	5.94128	0.381920
$243$	14.4915	0.929628
$244$	−0.422540	−0.0270504
$245$	6.98430	0.446210
$246$	20.2530	1.29129
$247$	−22.3455	−1.42181
$248$	11.3605	0.721393
$249$	0.163365	0.0103528
$250$	2.44573	0.154681
$251$	−5.75755	−0.363413	−0.181707	−	0.983353i	$-0.558162\pi$
$251$	−5.75755	−0.363413	−0.181707	+	0.983353i	$0.558162\pi$
$252$	1.27961	0.0806081
$253$	−7.88197	−0.495535
$254$	−2.76608	−0.173559
$255$	−3.50336	−0.219389
$256$	−31.8446	−1.99029
$257$	11.2102	0.699273	0.349636	−	0.936885i	$-0.386305\pi$
$257$	11.2102	0.699273	0.349636	+	0.936885i	$0.386305\pi$
$258$	6.65913	0.414579
$259$	0.437663	0.0271951
$260$	13.3869	0.830218
$261$	−24.3897	−1.50968
$262$	−12.0263	−0.742988
$263$	20.4237	1.25938	0.629690	−	0.776847i	$-0.283182\pi$
$263$	20.4237	1.25938	0.629690	+	0.776847i	$0.283182\pi$
$264$	9.35987	0.576060
$265$	−9.64515	−0.592497
$266$	2.03678	0.124883
$267$	−2.98590	−0.182734
$268$	−29.3925	−1.79543
$269$	−20.1015	−1.22561	−0.612804	−	0.790235i	$-0.709958\pi$
$269$	−20.1015	−1.22561	−0.612804	+	0.790235i	$0.709958\pi$
$270$	−8.97838	−0.546407
$271$	−24.3671	−1.48020	−0.740099	−	0.672498i	$-0.765221\pi$
$271$	−24.3671	−1.48020	−0.740099	+	0.672498i	$0.765221\pi$
$272$	20.6575	1.25254
$273$	0.277927	0.0168209
$274$	4.55556	0.275212
$275$	−2.92758	−0.176540
$276$	7.07165	0.425664
$277$	−15.1522	−0.910405	−0.455202	−	0.890388i	$-0.650433\pi$
$277$	−15.1522	−0.910405	−0.455202	+	0.890388i	$0.650433\pi$
$278$	−36.9602	−2.21673
$279$	6.01219	0.359940
$280$	−0.607279	−0.0362919
$281$	10.1483	0.605396	0.302698	−	0.953087i	$-0.402113\pi$
$281$	10.1483	0.605396	0.302698	+	0.953087i	$0.402113\pi$
$282$	18.0126	1.07264
$283$	9.65261	0.573788	0.286894	−	0.957962i	$-0.407377\pi$
$283$	9.65261	0.573788	0.286894	+	0.957962i	$0.407377\pi$
$284$	−36.1048	−2.14243
$285$	−4.38437	−0.259708
$286$	−24.0736	−1.42350
$287$	1.57293	0.0928473
$288$	−0.459960	−0.0271034
$289$	11.2027	0.658985
$290$	23.2573	1.36572
$291$	11.6306	0.681796
$292$	66.2360	3.87617
$293$	19.5472	1.14196	0.570980	−	0.820964i	$-0.306563\pi$
$293$	19.5472	1.14196	0.570980	+	0.820964i	$0.306563\pi$
$294$	11.2686	0.657199
$295$	−11.3945	−0.663414
$296$	16.9275	0.983890
$297$	10.7473	0.623622
$298$	45.6989	2.64727
$299$	−9.05208	−0.523495
$300$	2.62661	0.151647
$301$	0.517175	0.0298095
$302$	−25.4995	−1.46733
$303$	4.36496	0.250761
$304$	25.8523	1.48273
$305$	0.106124	0.00607662
$306$	33.3126	1.90436
$307$	−32.9125	−1.87841	−0.939207	−	0.343351i	$-0.888438\pi$
$307$	−32.9125	−1.87841	−0.939207	+	0.343351i	$0.888438\pi$
$308$	1.46061	0.0832258
$309$	−5.05132	−0.287360
$310$	−5.73305	−0.325615
$311$	8.28114	0.469580	0.234790	−	0.972046i	$-0.424560\pi$
$311$	8.28114	0.469580	0.234790	+	0.972046i	$0.424560\pi$
$312$	10.7494	0.608563
$313$	−29.7612	−1.68221	−0.841103	−	0.540876i	$-0.818093\pi$
$313$	−29.7612	−1.68221	−0.841103	+	0.540876i	$0.818093\pi$
$314$	−11.8777	−0.670300
$315$	−0.321383	−0.0181079
$316$	−16.9179	−0.951707
$317$	−1.86033	−0.104486	−0.0522431	−	0.998634i	$-0.516637\pi$
$317$	−1.86033	−0.104486	−0.0522431	+	0.998634i	$0.516637\pi$
$318$	−15.5617	−0.872656
$319$	−27.8395	−1.55871
$320$	8.21829	0.459416
$321$	2.59689	0.144944
$322$	0.825091	0.0459805
$323$	35.2950	1.96387
$324$	20.9936	1.16631
$325$	−3.36219	−0.186501
$326$	55.9787	3.10037
$327$	4.86688	0.269139
$328$	60.8363	3.35912
$329$	1.39893	0.0771258
$330$	−4.72343	−0.260016
$331$	2.13980	0.117614	0.0588072	−	0.998269i	$-0.481270\pi$
$331$	2.13980	0.117614	0.0588072	+	0.998269i	$0.481270\pi$
$332$	0.985995	0.0541135
$333$	8.95833	0.490914
$334$	−16.7131	−0.914500
$335$	7.38211	0.403328
$336$	−0.321543	−0.0175416
$337$	−25.9216	−1.41204	−0.706020	−	0.708192i	$-0.749511\pi$
$337$	−25.9216	−1.41204	−0.706020	+	0.708192i	$0.749511\pi$
$338$	4.14709	0.225572
$339$	3.34066	0.181440
$340$	−21.1447	−1.14673
$341$	6.86257	0.371629
$342$	41.6899	2.25433
$343$	1.75230	0.0946154
$344$	20.0028	1.07848
$345$	−1.77609	−0.0956215
$346$	−53.3938	−2.87047
$347$	−5.94469	−0.319127	−0.159564	−	0.987188i	$-0.551009\pi$
$347$	−5.94469	−0.319127	−0.159564	+	0.987188i	$0.551009\pi$
$348$	24.9774	1.33893
$349$	−28.9902	−1.55181	−0.775904	−	0.630850i	$-0.782706\pi$
$349$	−28.9902	−1.55181	−0.775904	+	0.630850i	$0.782706\pi$
$350$	0.306462	0.0163811
$351$	12.3428	0.658809
$352$	−0.525018	−0.0279836
$353$	−24.3375	−1.29536	−0.647678	−	0.761914i	$-0.724260\pi$
$353$	−24.3375	−1.29536	−0.647678	+	0.761914i	$0.724260\pi$
$354$	−18.3841	−0.977105
$355$	9.06796	0.481277
$356$	−18.0215	−0.955138
$357$	−0.438989	−0.0232337
$358$	6.38277	0.337340
$359$	−13.9940	−0.738574	−0.369287	−	0.929315i	$-0.620398\pi$
$359$	−13.9940	−0.738574	−0.369287	+	0.929315i	$0.620398\pi$
$360$	−12.4301	−0.655125
$361$	25.1708	1.32478
$362$	−11.1782	−0.587516
$363$	−1.60255	−0.0841121
$364$	1.67744	0.0879217
$365$	−16.6356	−0.870746
$366$	0.171222	0.00894993
$367$	−4.12449	−0.215297	−0.107648	−	0.994189i	$-0.534332\pi$
$367$	−4.12449	−0.215297	−0.107648	+	0.994189i	$0.534332\pi$
$368$	10.4727	0.545926
$369$	32.1957	1.67604
$370$	−8.54240	−0.444098
$371$	−1.20858	−0.0627466
$372$	−6.15705	−0.319228
$373$	−10.2556	−0.531015	−0.265508	−	0.964109i	$-0.585540\pi$
$373$	−10.2556	−0.531015	−0.265508	+	0.964109i	$0.585540\pi$
$374$	38.0245	1.96620
$375$	−0.659690	−0.0340662
$376$	54.1065	2.79033
$377$	−31.9723	−1.64666
$378$	−1.12504	−0.0578656
$379$	−13.0087	−0.668211	−0.334106	−	0.942536i	$-0.608434\pi$
$379$	−13.0087	−0.668211	−0.334106	+	0.942536i	$0.608434\pi$
$380$	−26.4620	−1.35747
$381$	0.746098	0.0382238
$382$	34.4946	1.76490
$383$	−21.9232	−1.12022	−0.560111	−	0.828418i	$-0.689241\pi$
$383$	−21.9232	−1.12022	−0.560111	+	0.828418i	$0.689241\pi$
$384$	13.0229	0.664575
$385$	−0.366841	−0.0186959
$386$	43.3397	2.20593
$387$	10.5858	0.538108
$388$	70.1968	3.56370
$389$	5.37653	0.272601	0.136301	−	0.990668i	$-0.456479\pi$
$389$	5.37653	0.272601	0.136301	+	0.990668i	$0.456479\pi$
$390$	−5.42464	−0.274687
$391$	14.2979	0.723074
$392$	33.8488	1.70962
$393$	3.24387	0.163632
$394$	−51.2368	−2.58127
$395$	4.24904	0.213792
$396$	29.8965	1.50236
$397$	18.2597	0.916428	0.458214	−	0.888842i	$-0.348489\pi$
$397$	18.2597	0.916428	0.458214	+	0.888842i	$0.348489\pi$
$398$	−43.2972	−2.17029
$399$	−0.549383	−0.0275035
$400$	3.88984	0.194492
$401$	−29.0141	−1.44890	−0.724448	−	0.689329i	$-0.757905\pi$
$401$	−29.0141	−1.44890	−0.724448	+	0.689329i	$0.757905\pi$
$402$	11.9105	0.594040
$403$	7.88134	0.392598
$404$	26.3449	1.31071
$405$	−5.27268	−0.262001
$406$	2.91425	0.144632
$407$	10.2254	0.506856
$408$	−16.9788	−0.840574
$409$	−28.3276	−1.40071	−0.700355	−	0.713795i	$-0.746975\pi$
$409$	−28.3276	−1.40071	−0.700355	+	0.713795i	$0.746975\pi$
$410$	−30.7008	−1.51621
$411$	−1.22878	−0.0606112
$412$	−30.4875	−1.50201
$413$	−1.42779	−0.0702568
$414$	16.8884	0.830020
$415$	−0.247639	−0.0121561
$416$	−0.602959	−0.0295625
$417$	9.96934	0.488201
$418$	47.5866	2.32754
$419$	11.3549	0.554721	0.277360	−	0.960766i	$-0.410540\pi$
$419$	11.3549	0.554721	0.277360	+	0.960766i	$0.410540\pi$
$420$	0.329127	0.0160598
$421$	−28.5021	−1.38911	−0.694554	−	0.719441i	$-0.744398\pi$
$421$	−28.5021	−1.38911	−0.694554	+	0.719441i	$0.744398\pi$
$422$	39.8447	1.93961
$423$	28.6342	1.39224
$424$	−46.7444	−2.27011
$425$	5.31063	0.257603
$426$	14.6304	0.708847
$427$	0.0132978	0.000643526 0
$428$	15.6737	0.757615
$429$	6.49340	0.313504
$430$	−10.0943	−0.486792
$431$	−26.8023	−1.29102	−0.645512	−	0.763750i	$-0.723356\pi$
$431$	−26.8023	−1.29102	−0.645512	+	0.763750i	$0.723356\pi$
$432$	−14.2798	−0.687037
$433$	11.9501	0.574287	0.287144	−	0.957888i	$-0.407294\pi$
$433$	11.9501	0.574287	0.287144	+	0.957888i	$0.407294\pi$
$434$	−0.718379	−0.0344833
$435$	−6.27323	−0.300778
$436$	29.3743	1.40677
$437$	17.8934	0.855958
$438$	−26.8402	−1.28247
$439$	−25.6894	−1.22609	−0.613045	−	0.790048i	$-0.710056\pi$
$439$	−25.6894	−1.22609	−0.613045	+	0.790048i	$0.710056\pi$
$440$	−14.1883	−0.676400
$441$	17.9134	0.853019
$442$	43.6693	2.07714
$443$	−39.2473	−1.86470	−0.932348	−	0.361561i	$-0.882244\pi$
$443$	−39.2473	−1.86470	−0.932348	+	0.361561i	$0.882244\pi$
$444$	−9.17418	−0.435388
$445$	4.52621	0.214563
$446$	38.4142	1.81896
$447$	−12.3264	−0.583021
$448$	1.02979	0.0486531
$449$	−30.7377	−1.45060	−0.725300	−	0.688432i	$-0.758299\pi$
$449$	−30.7377	−1.45060	−0.725300	+	0.688432i	$0.758299\pi$
$450$	6.27283	0.295704
$451$	36.7495	1.73047
$452$	20.1627	0.948372
$453$	6.87803	0.323158
$454$	−59.8147	−2.80724
$455$	−0.421300	−0.0197508
$456$	−21.2485	−0.995051
$457$	−5.15170	−0.240986	−0.120493	−	0.992714i	$-0.538448\pi$
$457$	−5.15170	−0.240986	−0.120493	+	0.992714i	$0.538448\pi$
$458$	3.66836	0.171411
$459$	−19.4956	−0.909975
$460$	−10.7197	−0.499807
$461$	15.0241	0.699744	0.349872	−	0.936797i	$-0.386225\pi$
$461$	15.0241	0.699744	0.349872	+	0.936797i	$0.386225\pi$
$462$	−0.591868	−0.0275362
$463$	13.4829	0.626602	0.313301	−	0.949654i	$-0.398565\pi$
$463$	13.4829	0.626602	0.313301	+	0.949654i	$0.398565\pi$
$464$	36.9899	1.71721
$465$	1.54638	0.0717118
$466$	−51.3740	−2.37985
$467$	−3.91927	−0.181362	−0.0906812	−	0.995880i	$-0.528904\pi$
$467$	−3.91927	−0.181362	−0.0906812	+	0.995880i	$0.528904\pi$
$468$	34.3348	1.58712
$469$	0.925015	0.0427132
$470$	−27.3047	−1.25947
$471$	3.20380	0.147623
$472$	−55.2225	−2.54182
$473$	12.0831	0.555583
$474$	6.85549	0.314883
$475$	6.64611	0.304944
$476$	−2.64953	−0.121441
$477$	−24.7380	−1.13267
$478$	−58.5948	−2.68006
$479$	−30.5912	−1.39775	−0.698874	−	0.715245i	$-0.746315\pi$
$479$	−30.5912	−1.39775	−0.698874	+	0.715245i	$0.746315\pi$
$480$	−0.118305	−0.00539988
$481$	11.7434	0.535454
$482$	42.0147	1.91372
$483$	−0.222553	−0.0101265
$484$	−9.67226	−0.439648
$485$	−17.6304	−0.800554
$486$	−35.4422	−1.60769
$487$	12.9145	0.585211	0.292606	−	0.956233i	$-0.405478\pi$
$487$	12.9145	0.585211	0.292606	+	0.956233i	$0.405478\pi$
$488$	0.514319	0.0232821
$489$	−15.0992	−0.682809
$490$	−17.0817	−0.771672
$491$	−9.60430	−0.433436	−0.216718	−	0.976234i	$-0.569535\pi$
$491$	−9.60430	−0.433436	−0.216718	+	0.976234i	$0.569535\pi$
$492$	−32.9714	−1.48647
$493$	50.5007	2.27444
$494$	54.6510	2.45887
$495$	−7.50870	−0.337491
$496$	−9.11820	−0.409419
$497$	1.13626	0.0509682
$498$	−0.399546	−0.0179041
$499$	−4.06768	−0.182094	−0.0910472	−	0.995847i	$-0.529021\pi$
$499$	−4.06768	−0.182094	−0.0910472	+	0.995847i	$0.529021\pi$
$500$	−3.98158	−0.178062
$501$	4.50805	0.201405
$502$	14.0814	0.628484
$503$	−24.3104	−1.08395	−0.541974	−	0.840395i	$-0.682323\pi$
$503$	−24.3104	−1.08395	−0.541974	+	0.840395i	$0.682323\pi$
$504$	−1.55756	−0.0693791
$505$	−6.61669	−0.294439
$506$	19.2772	0.856974
$507$	−1.11860	−0.0496787
$508$	4.50311	0.199793
$509$	−19.4229	−0.860906	−0.430453	−	0.902613i	$-0.641646\pi$
$509$	−19.4229	−0.860906	−0.430453	+	0.902613i	$0.641646\pi$
$510$	8.56827	0.379410
$511$	−2.08452	−0.0922137
$512$	38.4012	1.69711
$513$	−24.3982	−1.07721
$514$	−27.4171	−1.20932
$515$	7.65712	0.337413
$516$	−10.8409	−0.477244
$517$	32.6843	1.43745
$518$	−1.07041	−0.0470309
$519$	14.4020	0.632177
$520$	−16.2946	−0.714565
$521$	−19.4609	−0.852598	−0.426299	−	0.904582i	$-0.640183\pi$
$521$	−19.4609	−0.852598	−0.426299	+	0.904582i	$0.640183\pi$
$522$	59.6506	2.61083
$523$	17.8225	0.779324	0.389662	−	0.920958i	$-0.372592\pi$
$523$	17.8225	0.779324	0.389662	+	0.920958i	$0.372592\pi$
$524$	19.5785	0.855292
$525$	−0.0826623	−0.00360768
$526$	−49.9508	−2.17796
$527$	−12.4487	−0.542273
$528$	−7.51244	−0.326937
$529$	−15.7515	−0.684846
$530$	23.5894	1.02466
$531$	−29.2247	−1.26825
$532$	−3.31582	−0.143759
$533$	42.2051	1.82811
$534$	7.30269	0.316018
$535$	−3.93654	−0.170191
$536$	35.7768	1.54532
$537$	−1.72163	−0.0742940
$538$	49.1627	2.11955
$539$	20.4471	0.880720
$540$	14.6166	0.628997
$541$	−10.7870	−0.463770	−0.231885	−	0.972743i	$-0.574489\pi$
$541$	−10.7870	−0.463770	−0.231885	+	0.972743i	$0.574489\pi$
$542$	59.5954	2.55984
$543$	3.01512	0.129391
$544$	0.952380	0.0408330
$545$	−7.37753	−0.316019
$546$	−0.679734	−0.0290899
$547$	−0.291458	−0.0124619	−0.00623093	−	0.999981i	$-0.501983\pi$
$547$	−0.291458	−0.0124619	−0.00623093	+	0.999981i	$0.501983\pi$
$548$	−7.41634	−0.316811
$549$	0.272187	0.0116167
$550$	7.16007	0.305307
$551$	63.2003	2.69242
$552$	−8.60767	−0.366367
$553$	0.532425	0.0226410
$554$	37.0580	1.57445
$555$	2.30415	0.0978059
$556$	60.1704	2.55179
$557$	19.4104	0.822445	0.411222	−	0.911535i	$-0.365102\pi$
$557$	19.4104	0.822445	0.411222	+	0.911535i	$0.365102\pi$
$558$	−14.7042	−0.622477
$559$	13.8769	0.586931
$560$	0.487416	0.0205971
$561$	−10.2564	−0.433025
$562$	−24.8199	−1.04697
$563$	5.74656	0.242189	0.121094	−	0.992641i	$-0.461360\pi$
$563$	5.74656	0.242189	0.121094	+	0.992641i	$0.461360\pi$
$564$	−29.3241	−1.23477
$565$	−5.06398	−0.213043
$566$	−23.6077	−0.992304
$567$	−0.660692	−0.0277465
$568$	43.9471	1.84398
$569$	−41.8547	−1.75464	−0.877321	−	0.479905i	$-0.840671\pi$
$569$	−41.8547	−1.75464	−0.877321	+	0.479905i	$0.840671\pi$
$570$	10.7230	0.449136
$571$	−0.307659	−0.0128751	−0.00643757	−	0.999979i	$-0.502049\pi$
$571$	−0.307659	−0.0128751	−0.00643757	+	0.999979i	$0.502049\pi$
$572$	39.1912	1.63867
$573$	−9.30427	−0.388692
$574$	−3.84697	−0.160569
$575$	2.69231	0.112277
$576$	21.0783	0.878264
$577$	40.8907	1.70230	0.851152	−	0.524919i	$-0.175904\pi$
$577$	40.8907	1.70230	0.851152	+	0.524919i	$0.175904\pi$
$578$	−27.3988	−1.13964
$579$	−11.6901	−0.485823
$580$	−37.8623	−1.57215
$581$	−0.0310304	−0.00128736
$582$	−28.4452	−1.17909
$583$	−28.2370	−1.16946
$584$	−80.6229	−3.33620
$585$	−8.62339	−0.356533
$586$	−47.8072	−1.97490
$587$	−2.59899	−0.107272	−0.0536359	−	0.998561i	$-0.517081\pi$
$587$	−2.59899	−0.107272	−0.0536359	+	0.998561i	$0.517081\pi$
$588$	−18.3450	−0.756536
$589$	−15.5792	−0.641929
$590$	27.8678	1.14730
$591$	13.8202	0.568486
$592$	−13.5864	−0.558397
$593$	27.8862	1.14515	0.572575	−	0.819853i	$-0.305945\pi$
$593$	27.8862	1.14515	0.572575	+	0.819853i	$0.305945\pi$
$594$	−26.2850	−1.07849
$595$	0.665447	0.0272807
$596$	−74.3967	−3.04741
$597$	11.6786	0.477974
$598$	22.1389	0.905328
$599$	−27.1681	−1.11006	−0.555029	−	0.831831i	$-0.687293\pi$
$599$	−27.1681	−1.11006	−0.555029	+	0.831831i	$0.687293\pi$
$600$	−3.19713	−0.130522
$601$	−26.6476	−1.08698	−0.543489	−	0.839417i	$-0.682897\pi$
$601$	−26.6476	−1.08698	−0.543489	+	0.839417i	$0.682897\pi$
$602$	−1.26487	−0.0515523
$603$	18.9337	0.771041
$604$	41.5126	1.68912
$605$	2.42925	0.0987630
$606$	−10.6755	−0.433663
$607$	27.1179	1.10068	0.550340	−	0.834941i	$-0.314498\pi$
$607$	27.1179	1.10068	0.550340	+	0.834941i	$0.314498\pi$
$608$	1.19188	0.0483371
$609$	−0.786066	−0.0318530
$610$	−0.259550	−0.0105089
$611$	37.5364	1.51856
$612$	−54.2321	−2.19220
$613$	−13.1161	−0.529754	−0.264877	−	0.964282i	$-0.585331\pi$
$613$	−13.1161	−0.529754	−0.264877	+	0.964282i	$0.585331\pi$
$614$	80.4950	3.24851
$615$	8.28098	0.333921
$616$	−1.77786	−0.0716321
$617$	16.0400	0.645747	0.322873	−	0.946442i	$-0.395351\pi$
$617$	16.0400	0.645747	0.322873	+	0.946442i	$0.395351\pi$
$618$	12.3542	0.496957
$619$	2.52034	0.101301	0.0506505	−	0.998716i	$-0.483871\pi$
$619$	2.52034	0.101301	0.0506505	+	0.998716i	$0.483871\pi$
$620$	9.33326	0.374833
$621$	−9.88361	−0.396615
$622$	−20.2534	−0.812088
$623$	0.567157	0.0227227
$624$	−8.62769	−0.345384
$625$	1.00000	0.0400000
$626$	72.7879	2.90919
$627$	−12.8356	−0.512605
$628$	19.3367	0.771617
$629$	−18.5489	−0.739592
$630$	0.786016	0.0313156
$631$	28.7324	1.14382	0.571910	−	0.820317i	$-0.306203\pi$
$631$	28.7324	1.14382	0.571910	+	0.820317i	$0.306203\pi$
$632$	20.5926	0.819130
$633$	−10.7474	−0.427170
$634$	4.54985	0.180698
$635$	−1.13098	−0.0448817
$636$	25.3340	1.00456
$637$	23.4826	0.930413
$638$	68.0877	2.69562
$639$	23.2576	0.920056
$640$	−19.7410	−0.780332
$641$	34.7415	1.37221	0.686103	−	0.727504i	$-0.259320\pi$
$641$	34.7415	1.37221	0.686103	+	0.727504i	$0.259320\pi$
$642$	−6.35129	−0.250666
$643$	−42.5991	−1.67995	−0.839973	−	0.542629i	$-0.817429\pi$
$643$	−42.5991	−1.67995	−0.839973	+	0.542629i	$0.817429\pi$
$644$	−1.34323	−0.0529306
$645$	2.72276	0.107209
$646$	−86.3219	−3.39629
$647$	−14.5510	−0.572059	−0.286030	−	0.958221i	$-0.592336\pi$
$647$	−14.5510	−0.572059	−0.286030	+	0.958221i	$0.592336\pi$
$648$	−25.5536	−1.00384
$649$	−33.3584	−1.30943
$650$	8.22301	0.322533
$651$	0.193769	0.00759442
$652$	−91.1319	−3.56900
$653$	−24.2386	−0.948531	−0.474266	−	0.880382i	$-0.657286\pi$
$653$	−24.2386	−0.948531	−0.474266	+	0.880382i	$0.657286\pi$
$654$	−11.9031	−0.465447
$655$	−4.91727	−0.192134
$656$	−48.8286	−1.90644
$657$	−42.6671	−1.66460
$658$	−3.42141	−0.133381
$659$	13.7990	0.537534	0.268767	−	0.963205i	$-0.413384\pi$
$659$	13.7990	0.537534	0.268767	+	0.963205i	$0.413384\pi$
$660$	7.68962	0.299318
$661$	20.7094	0.805502	0.402751	−	0.915309i	$-0.368054\pi$
$661$	20.7094	0.805502	0.402751	+	0.915309i	$0.368054\pi$
$662$	−5.23338	−0.203401
$663$	−11.7790	−0.457458
$664$	−1.20016	−0.0465753
$665$	0.832790	0.0322942
$666$	−21.9096	−0.848981
$667$	25.6022	0.991320
$668$	27.2085	1.05273
$669$	−10.3615	−0.400599
$670$	−18.0546	−0.697512
$671$	0.310686	0.0119939
$672$	−0.0148242	−0.000571858 0
$673$	−4.82115	−0.185842	−0.0929209	−	0.995673i	$-0.529620\pi$
$673$	−4.82115	−0.185842	−0.0929209	+	0.995673i	$0.529620\pi$
$674$	63.3972	2.44197
$675$	−3.67105	−0.141299
$676$	−6.75135	−0.259667
$677$	−0.589505	−0.0226565	−0.0113283	−	0.999936i	$-0.503606\pi$
$677$	−0.589505	−0.0226565	−0.0113283	+	0.999936i	$0.503606\pi$
$678$	−8.17034	−0.313780
$679$	−2.20917	−0.0847803
$680$	25.7375	0.986988
$681$	16.1339	0.618252
$682$	−16.7840	−0.642692
$683$	23.6805	0.906108	0.453054	−	0.891483i	$-0.350335\pi$
$683$	23.6805	0.906108	0.453054	+	0.891483i	$0.350335\pi$
$684$	−67.8701	−2.59508
$685$	1.86266	0.0711687
$686$	−4.28565	−0.163627
$687$	−0.989471	−0.0377507
$688$	−16.0547	−0.612079
$689$	−32.4289	−1.23544
$690$	4.34383	0.165367
$691$	3.34038	0.127074	0.0635371	−	0.997979i	$-0.479762\pi$
$691$	3.34038	0.127074	0.0635371	+	0.997979i	$0.479762\pi$
$692$	86.9238	3.30435
$693$	−0.940877	−0.0357409
$694$	14.5391	0.551896
$695$	−15.1122	−0.573237
$696$	−30.4027	−1.15241
$697$	−66.6634	−2.52506
$698$	70.9021	2.68368
$699$	13.8572	0.524127
$700$	−0.498912	−0.0188571
$701$	3.93419	0.148592	0.0742962	−	0.997236i	$-0.476329\pi$
$701$	3.93419	0.148592	0.0742962	+	0.997236i	$0.476329\pi$
$702$	−30.1871	−1.13934
$703$	−23.2134	−0.875511
$704$	24.0597	0.906785
$705$	7.36494	0.277380
$706$	59.5230	2.24018
$707$	−0.829104	−0.0311817
$708$	29.9289	1.12480
$709$	−44.1376	−1.65762	−0.828811	−	0.559529i	$-0.810982\pi$
$709$	−44.1376	−1.65762	−0.828811	+	0.559529i	$0.810982\pi$
$710$	−22.1778	−0.832316
$711$	10.8980	0.408706
$712$	21.9359	0.822083
$713$	−6.31107	−0.236351
$714$	1.07365	0.0401802
$715$	−9.84311	−0.368111
$716$	−10.3910	−0.388330
$717$	15.8048	0.590243
$718$	34.2255	1.27728
$719$	−26.2999	−0.980819	−0.490410	−	0.871492i	$-0.663153\pi$
$719$	−26.2999	−0.980819	−0.490410	+	0.871492i	$0.663153\pi$
$720$	9.97670	0.371810
$721$	0.959475	0.0357327
$722$	−61.5608	−2.29106
$723$	−11.3327	−0.421467
$724$	18.1979	0.676320
$725$	9.50936	0.353169
$726$	3.91940	0.145463
$727$	16.6824	0.618717	0.309359	−	0.950945i	$-0.399886\pi$
$727$	16.6824	0.618717	0.309359	+	0.950945i	$0.399886\pi$
$728$	−2.04179	−0.0756738
$729$	−6.25816	−0.231784
$730$	40.6861	1.50586
$731$	−21.9187	−0.810694
$732$	−0.278745	−0.0103027
$733$	28.0993	1.03787	0.518936	−	0.854813i	$-0.326328\pi$
$733$	28.0993	1.03787	0.518936	+	0.854813i	$0.326328\pi$
$734$	10.0874	0.372332
$735$	4.60747	0.169949
$736$	0.482826	0.0177972
$737$	21.6118	0.796079
$738$	−78.7418	−2.89853
$739$	−3.15222	−0.115956	−0.0579781	−	0.998318i	$-0.518465\pi$
$739$	−3.15222	−0.115956	−0.0579781	+	0.998318i	$0.518465\pi$
$740$	13.9068	0.511225
$741$	−14.7411	−0.541528
$742$	2.95587	0.108513
$743$	−9.95390	−0.365173	−0.182587	−	0.983190i	$-0.558447\pi$
$743$	−9.95390	−0.365173	−0.182587	+	0.983190i	$0.558447\pi$
$744$	7.49441	0.274758
$745$	18.6852	0.684573
$746$	25.0824	0.918333
$747$	−0.635147	−0.0232388
$748$	−61.9029	−2.26339
$749$	−0.493268	−0.0180236
$750$	1.61342	0.0589138
$751$	−10.7612	−0.392681	−0.196341	−	0.980536i	$-0.562906\pi$
$751$	−10.7612	−0.392681	−0.196341	+	0.980536i	$0.562906\pi$
$752$	−43.4271	−1.58362
$753$	−3.79820	−0.138414
$754$	78.1956	2.84772
$755$	−10.4262	−0.379447
$756$	1.83153	0.0666121
$757$	−23.3432	−0.848422	−0.424211	−	0.905563i	$-0.639448\pi$
$757$	−23.3432	−0.848422	−0.424211	+	0.905563i	$0.639448\pi$
$758$	31.8157	1.15560
$759$	−5.19966	−0.188736
$760$	32.2098	1.16837
$761$	7.57851	0.274721	0.137360	−	0.990521i	$-0.456138\pi$
$761$	7.57851	0.274721	0.137360	+	0.990521i	$0.456138\pi$
$762$	−1.82475	−0.0661038
$763$	−0.924441	−0.0334670
$764$	−56.1563	−2.03166
$765$	13.6207	0.492459
$766$	53.6181	1.93730
$767$	−38.3105	−1.38331
$768$	−21.0075	−0.758045
$769$	−12.9084	−0.465490	−0.232745	−	0.972538i	$-0.574771\pi$
$769$	−12.9084	−0.465490	−0.232745	+	0.972538i	$0.574771\pi$
$770$	0.897192	0.0323326
$771$	7.39525	0.266333
$772$	−70.5560	−2.53937
$773$	24.6703	0.887331	0.443665	−	0.896193i	$-0.353678\pi$
$773$	24.6703	0.887331	0.443665	+	0.896193i	$0.353678\pi$
$774$	−25.8901	−0.930599
$775$	−2.34411	−0.0842028
$776$	−85.4441	−3.06726
$777$	0.288722	0.0103578
$778$	−13.1495	−0.471434
$779$	−83.4276	−2.98910
$780$	8.83117	0.316207
$781$	26.5472	0.949934
$782$	−34.9687	−1.25048
$783$	−34.9093	−1.24756
$784$	−27.1678	−0.970279
$785$	−4.85653	−0.173337
$786$	−7.93363	−0.282983
$787$	23.0714	0.822407	0.411203	−	0.911544i	$-0.365109\pi$
$787$	23.0714	0.822407	0.411203	+	0.911544i	$0.365109\pi$
$788$	83.4122	2.97144
$789$	13.4733	0.479662
$790$	−10.3920	−0.369731
$791$	−0.634542	−0.0225617
$792$	−36.3903	−1.29307
$793$	0.356808	0.0126706
$794$	−44.6583	−1.58486
$795$	−6.36281	−0.225665
$796$	70.4868	2.49834
$797$	−55.0099	−1.94855	−0.974275	−	0.225361i	$-0.927644\pi$
$797$	−55.0099	−1.94855	−0.974275	+	0.225361i	$0.927644\pi$
$798$	1.34364	0.0475644
$799$	−59.2891	−2.09750
$800$	0.179335	0.00634045
$801$	11.6089	0.410180
$802$	70.9607	2.50571
$803$	−48.7021	−1.71866
$804$	−19.3899	−0.683830
$805$	0.337360	0.0118904
$806$	−19.2756	−0.678955
$807$	−13.2607	−0.466800
$808$	−32.0672	−1.12812
$809$	36.4628	1.28196	0.640982	−	0.767556i	$-0.278528\pi$
$809$	36.4628	1.28196	0.640982	+	0.767556i	$0.278528\pi$
$810$	12.8955	0.453103
$811$	−28.3647	−0.996018	−0.498009	−	0.867172i	$-0.665935\pi$
$811$	−28.3647	−0.996018	−0.498009	+	0.867172i	$0.665935\pi$
$812$	−4.74433	−0.166493
$813$	−16.0747	−0.563766
$814$	−25.0086	−0.876551
$815$	22.8883	0.801744
$816$	13.6275	0.477059
$817$	−27.4307	−0.959680
$818$	69.2816	2.42237
$819$	−1.08055	−0.0377576
$820$	49.9802	1.74538
$821$	−1.83309	−0.0639755	−0.0319877	−	0.999488i	$-0.510184\pi$
$821$	−1.83309	−0.0639755	−0.0319877	+	0.999488i	$0.510184\pi$
$822$	3.00526	0.104820
$823$	7.81607	0.272451	0.136226	−	0.990678i	$-0.456503\pi$
$823$	7.81607	0.272451	0.136226	+	0.990678i	$0.456503\pi$
$824$	37.1096	1.29277
$825$	−1.93130	−0.0672391
$826$	3.49198	0.121501
$827$	−48.7317	−1.69457	−0.847284	−	0.531139i	$-0.821764\pi$
$827$	−48.7317	−1.69457	−0.847284	+	0.531139i	$0.821764\pi$
$828$	−27.4939	−0.955479
$829$	−26.1318	−0.907595	−0.453797	−	0.891105i	$-0.649931\pi$
$829$	−26.1318	−0.907595	−0.453797	+	0.891105i	$0.649931\pi$
$830$	0.605658	0.0210227
$831$	−9.99572	−0.346748
$832$	27.6315	0.957949
$833$	−37.0910	−1.28513
$834$	−24.3823	−0.844290
$835$	−6.83359	−0.236486
$836$	−77.4699	−2.67935
$837$	8.60532	0.297444
$838$	−27.7709	−0.959329
$839$	−57.1880	−1.97435	−0.987175	−	0.159643i	$-0.948966\pi$
$839$	−57.1880	−1.97435	−0.987175	+	0.159643i	$0.948966\pi$
$840$	−0.400616	−0.0138226
$841$	61.4280	2.11821
$842$	69.7084	2.40231
$843$	6.69472	0.230578
$844$	−64.8662	−2.23279
$845$	1.69564	0.0583320
$846$	−70.0314	−2.40773
$847$	0.304397	0.0104592
$848$	37.5181	1.28838
$849$	6.36773	0.218540
$850$	−12.9883	−0.445496
$851$	−9.40367	−0.322354
$852$	−23.8180	−0.815991
$853$	0.209163	0.00716162	0.00358081	−	0.999994i	$-0.498860\pi$
$853$	0.209163	0.00716162	0.00358081	+	0.999994i	$0.498860\pi$
$854$	−0.0325228	−0.00111291
$855$	17.0460	0.582961
$856$	−19.0781	−0.652076
$857$	10.4095	0.355582	0.177791	−	0.984068i	$-0.443105\pi$
$857$	10.4095	0.355582	0.177791	+	0.984068i	$0.443105\pi$
$858$	−15.8811	−0.542171
$859$	−51.1041	−1.74365	−0.871825	−	0.489817i	$-0.837064\pi$
$859$	−51.1041	−1.74365	−0.871825	+	0.489817i	$0.837064\pi$
$860$	16.4333	0.560372
$861$	1.03765	0.0353629
$862$	65.5512	2.23268
$863$	12.3445	0.420213	0.210107	−	0.977678i	$-0.432619\pi$
$863$	12.3445	0.420213	0.210107	+	0.977678i	$0.432619\pi$
$864$	−0.658347	−0.0223974
$865$	−21.8315	−0.742292
$866$	−29.2268	−0.993167
$867$	7.39033	0.250989
$868$	1.16950	0.0396955
$869$	12.4394	0.421978
$870$	15.3426	0.520163
$871$	24.8201	0.840997
$872$	−35.7546	−1.21080
$873$	−45.2186	−1.53042
$874$	−43.7624	−1.48028
$875$	0.125305	0.00423608
$876$	43.6952	1.47632
$877$	−14.9672	−0.505408	−0.252704	−	0.967544i	$-0.581320\pi$
$877$	−14.9672	−0.505408	−0.252704	+	0.967544i	$0.581320\pi$
$878$	62.8294	2.12039
$879$	12.8951	0.434941
$880$	11.3878	0.383884
$881$	−36.2838	−1.22243	−0.611217	−	0.791463i	$-0.709320\pi$
$881$	−36.2838	−1.22243	−0.611217	+	0.791463i	$0.709320\pi$
$882$	−43.8113	−1.47520
$883$	−41.1519	−1.38487	−0.692435	−	0.721480i	$-0.743462\pi$
$883$	−41.1519	−1.38487	−0.692435	+	0.721480i	$0.743462\pi$
$884$	−71.0926	−2.39110
$885$	−7.51683	−0.252676
$886$	95.9883	3.22479
$887$	−27.3666	−0.918882	−0.459441	−	0.888208i	$-0.651950\pi$
$887$	−27.3666	−0.918882	−0.459441	+	0.888208i	$0.651950\pi$
$888$	11.1669	0.374736
$889$	−0.141718	−0.00475306
$890$	−11.0699	−0.371063
$891$	−15.4362	−0.517132
$892$	−62.5373	−2.09390
$893$	−74.1988	−2.48297
$894$	30.1471	1.00827
$895$	2.60976	0.0872348
$896$	−2.47365	−0.0826388
$897$	−5.97156	−0.199385
$898$	75.1760	2.50866
$899$	−22.2910	−0.743445
$900$	−10.2120	−0.340400
$901$	51.2218	1.70644
$902$	−89.8793	−2.99265
$903$	0.341175	0.0113536
$904$	−24.5422	−0.816260
$905$	−4.57052	−0.151929
$906$	−16.8218	−0.558866
$907$	2.58923	0.0859739	0.0429870	−	0.999076i	$-0.486313\pi$
$907$	2.58923	0.0859739	0.0429870	+	0.999076i	$0.486313\pi$
$908$	97.3768	3.23156
$909$	−16.9706	−0.562878
$910$	1.03038	0.0341569
$911$	15.2467	0.505147	0.252574	−	0.967578i	$-0.418723\pi$
$911$	15.2467	0.505147	0.252574	+	0.967578i	$0.418723\pi$
$912$	17.0545	0.564731
$913$	−0.724984	−0.0239935
$914$	12.5996	0.416759
$915$	0.0700087	0.00231442
$916$	−5.97199	−0.197320
$917$	−0.616158	−0.0203473
$918$	47.6808	1.57370
$919$	1.31837	0.0434889	0.0217445	−	0.999764i	$-0.493078\pi$
$919$	1.31837	0.0434889	0.0217445	+	0.999764i	$0.493078\pi$
$920$	13.0481	0.430182
$921$	−21.7120	−0.715435
$922$	−36.7450	−1.21013
$923$	30.4882	1.00353
$924$	0.963547	0.0316984
$925$	−3.49279	−0.114842
$926$	−32.9754	−1.08364
$927$	19.6391	0.645031
$928$	1.70536	0.0559812
$929$	5.23390	0.171719	0.0858594	−	0.996307i	$-0.472636\pi$
$929$	5.23390	0.171719	0.0858594	+	0.996307i	$0.472636\pi$
$930$	−3.78203	−0.124018
$931$	−46.4184	−1.52130
$932$	83.6356	2.73957
$933$	5.46298	0.178850
$934$	9.58547	0.313646
$935$	15.5473	0.508451
$936$	−41.7925	−1.36603
$937$	12.5878	0.411227	0.205613	−	0.978633i	$-0.434081\pi$
$937$	12.5878	0.411227	0.205613	+	0.978633i	$0.434081\pi$
$938$	−2.26233	−0.0738679
$939$	−19.6332	−0.640705
$940$	44.4514	1.44984
$941$	19.5781	0.638228	0.319114	−	0.947716i	$-0.396615\pi$
$941$	19.5781	0.638228	0.319114	+	0.947716i	$0.396615\pi$
$942$	−7.83563	−0.255298
$943$	−33.7962	−1.10055
$944$	44.3228	1.44258
$945$	−0.460000	−0.0149638
$946$	−29.5520	−0.960819
$947$	24.9741	0.811548	0.405774	−	0.913973i	$-0.367002\pi$
$947$	24.9741	0.811548	0.405774	+	0.913973i	$0.367002\pi$
$948$	−11.1606	−0.362478
$949$	−55.9321	−1.81563
$950$	−16.2546	−0.527368
$951$	−1.22724	−0.0397959
$952$	3.22503	0.104524
$953$	17.5749	0.569306	0.284653	−	0.958631i	$-0.408122\pi$
$953$	17.5749	0.569306	0.284653	+	0.958631i	$0.408122\pi$
$954$	60.5024	1.95884
$955$	14.1040	0.456395
$956$	95.3908	3.08516
$957$	−18.3654	−0.593669
$958$	74.8177	2.41725
$959$	0.233401	0.00753690
$960$	5.42152	0.174979
$961$	−25.5052	−0.822747
$962$	−28.7212	−0.926010
$963$	−10.0965	−0.325354
$964$	−68.3989	−2.20298
$965$	17.7206	0.570446
$966$	0.544304	0.0175127
$967$	−22.7756	−0.732413	−0.366206	−	0.930534i	$-0.619344\pi$
$967$	−22.7756	−0.732413	−0.366206	+	0.930534i	$0.619344\pi$
$968$	11.7731	0.378403
$969$	23.2837	0.747981
$970$	43.1191	1.38447
$971$	−20.8904	−0.670403	−0.335202	−	0.942146i	$-0.608804\pi$
$971$	−20.8904	−0.670403	−0.335202	+	0.942146i	$0.608804\pi$
$972$	57.6990	1.85070
$973$	−1.89363	−0.0607069
$974$	−31.5853	−1.01206
$975$	−2.21800	−0.0710330
$976$	−0.412804	−0.0132135
$977$	−51.7695	−1.65625	−0.828126	−	0.560542i	$-0.810593\pi$
$977$	−51.7695	−1.65625	−0.828126	+	0.560542i	$0.810593\pi$
$978$	36.9285	1.18084
$979$	13.2509	0.423500
$980$	27.8086	0.888312
$981$	−18.9220	−0.604132
$982$	23.4895	0.749580
$983$	−20.9950	−0.669636	−0.334818	−	0.942283i	$-0.608675\pi$
$983$	−20.9950	−0.669636	−0.334818	+	0.942283i	$0.608675\pi$
$984$	40.1331	1.27940
$985$	−20.9495	−0.667507
$986$	−123.511	−3.93339
$987$	0.922862	0.0293750
$988$	−88.9705	−2.83053
$989$	−11.1121	−0.353344
$990$	18.3642	0.583653
$991$	22.9866	0.730193	0.365097	−	0.930970i	$-0.381036\pi$
$991$	22.9866	0.730193	0.365097	+	0.930970i	$0.381036\pi$
$992$	−0.420380	−0.0133471
$993$	1.41161	0.0447960
$994$	−2.77898	−0.0881439
$995$	−17.7032	−0.561229
$996$	0.650451	0.0206103
$997$	−55.7374	−1.76522	−0.882611	−	0.470104i	$-0.844216\pi$
$997$	−55.7374	−1.76522	−0.882611	+	0.470104i	$0.844216\pi$
$998$	9.94844	0.314912
$999$	12.8222	0.405676

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.b.1.9	✓	126

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.b.1.9	✓	126	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44573 q^{2} +0.659690 q^{3} +3.98158 q^{4} -1.00000 q^{5} -1.61342 q^{6} -0.125305 q^{7} -4.84641 q^{8} -2.56481 q^{9} +O(q^{10})\)	\(q-2.44573 q^{2} +0.659690 q^{3} +3.98158 q^{4} -1.00000 q^{5} -1.61342 q^{6} -0.125305 q^{7} -4.84641 q^{8} -2.56481 q^{9} +2.44573 q^{10} -2.92758 q^{11} +2.62661 q^{12} -3.36219 q^{13} +0.306462 q^{14} -0.659690 q^{15} +3.88984 q^{16} +5.31063 q^{17} +6.27283 q^{18} +6.64611 q^{19} -3.98158 q^{20} -0.0826623 q^{21} +7.16007 q^{22} +2.69231 q^{23} -3.19713 q^{24} +1.00000 q^{25} +8.22301 q^{26} -3.67105 q^{27} -0.498912 q^{28} +9.50936 q^{29} +1.61342 q^{30} -2.34411 q^{31} +0.179335 q^{32} -1.93130 q^{33} -12.9883 q^{34} +0.125305 q^{35} -10.2120 q^{36} -3.49279 q^{37} -16.2546 q^{38} -2.21800 q^{39} +4.84641 q^{40} -12.5528 q^{41} +0.202170 q^{42} -4.12734 q^{43} -11.6564 q^{44} +2.56481 q^{45} -6.58466 q^{46} -11.1642 q^{47} +2.56609 q^{48} -6.98430 q^{49} -2.44573 q^{50} +3.50336 q^{51} -13.3869 q^{52} +9.64515 q^{53} +8.97838 q^{54} +2.92758 q^{55} +0.607279 q^{56} +4.38437 q^{57} -23.2573 q^{58} +11.3945 q^{59} -2.62661 q^{60} -0.106124 q^{61} +5.73305 q^{62} +0.321383 q^{63} -8.21829 q^{64} +3.36219 q^{65} +4.72343 q^{66} -7.38211 q^{67} +21.1447 q^{68} +1.77609 q^{69} -0.306462 q^{70} -9.06796 q^{71} +12.4301 q^{72} +16.6356 q^{73} +8.54240 q^{74} +0.659690 q^{75} +26.4620 q^{76} +0.366841 q^{77} +5.42464 q^{78} -4.24904 q^{79} -3.88984 q^{80} +5.27268 q^{81} +30.7008 q^{82} +0.247639 q^{83} -0.329127 q^{84} -5.31063 q^{85} +10.0943 q^{86} +6.27323 q^{87} +14.1883 q^{88} -4.52621 q^{89} -6.27283 q^{90} +0.421300 q^{91} +10.7197 q^{92} -1.54638 q^{93} +27.3047 q^{94} -6.64611 q^{95} +0.118305 q^{96} +17.6304 q^{97} +17.0817 q^{98} +7.50870 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * q^99

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