LMFDB - Embedded newform 867.6.a.t.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,6,Mod(1,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("867.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	867.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:	$139.052771778$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	867.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.85553 q^{2} -9.00000 q^{3} -8.42383 q^{4} -82.3928 q^{5} +43.6998 q^{6} -35.3683 q^{7} +196.279 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-4.85553 q^{2} -9.00000 q^{3} -8.42383 q^{4} -82.3928 q^{5} +43.6998 q^{6} -35.3683 q^{7} +196.279 q^{8} +81.0000 q^{9} +400.061 q^{10} +88.9161 q^{11} +75.8144 q^{12} -607.679 q^{13} +171.732 q^{14} +741.535 q^{15} -683.477 q^{16} -393.298 q^{18} +728.381 q^{19} +694.062 q^{20} +318.315 q^{21} -431.735 q^{22} -842.921 q^{23} -1766.51 q^{24} +3663.57 q^{25} +2950.60 q^{26} -729.000 q^{27} +297.937 q^{28} +521.365 q^{29} -3600.55 q^{30} +1459.76 q^{31} -2962.29 q^{32} -800.245 q^{33} +2914.10 q^{35} -682.330 q^{36} +11600.8 q^{37} -3536.68 q^{38} +5469.11 q^{39} -16172.0 q^{40} +14400.6 q^{41} -1545.59 q^{42} +2446.99 q^{43} -749.014 q^{44} -6673.81 q^{45} +4092.83 q^{46} -25631.6 q^{47} +6151.29 q^{48} -15556.1 q^{49} -17788.6 q^{50} +5118.98 q^{52} -2320.07 q^{53} +3539.68 q^{54} -7326.05 q^{55} -6942.07 q^{56} -6555.43 q^{57} -2531.50 q^{58} -37043.5 q^{59} -6246.56 q^{60} -11489.3 q^{61} -7087.93 q^{62} -2864.84 q^{63} +36254.7 q^{64} +50068.3 q^{65} +3885.62 q^{66} -57126.2 q^{67} +7586.29 q^{69} -14149.5 q^{70} -62663.3 q^{71} +15898.6 q^{72} -31928.3 q^{73} -56328.0 q^{74} -32972.1 q^{75} -6135.76 q^{76} -3144.82 q^{77} -26555.4 q^{78} -51843.7 q^{79} +56313.5 q^{80} +6561.00 q^{81} -69922.4 q^{82} -68331.8 q^{83} -2681.43 q^{84} -11881.4 q^{86} -4692.29 q^{87} +17452.4 q^{88} -130403. q^{89} +32404.9 q^{90} +21492.6 q^{91} +7100.62 q^{92} -13137.9 q^{93} +124455. q^{94} -60013.4 q^{95} +26660.6 q^{96} -53915.3 q^{97} +75533.0 q^{98} +7202.21 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 252 q^{3} + 384 q^{4} + 244 q^{5} + 392 q^{7} + 2268 q^{9}+O(q^{10}) $ 28 * q - 252 * q^3 + 384 * q^4 + 244 * q^5 + 392 * q^7 + 2268 * q^9	\( 28 q - 252 q^{3} + 384 q^{4} + 244 q^{5} + 392 q^{7} + 2268 q^{9} + 800 q^{10} + 1132 q^{11} - 3456 q^{12} - 828 q^{13} + 1568 q^{14} - 2196 q^{15} + 4096 q^{16} - 7532 q^{19} + 9216 q^{20} - 3528 q^{21} + 664 q^{22} + 4548 q^{23} + 9904 q^{25} - 20412 q^{27} + 36936 q^{28} + 36680 q^{29} - 7200 q^{30} + 7688 q^{31} - 10188 q^{33} + 13136 q^{35} + 31104 q^{36} + 42360 q^{37} - 1984 q^{38} + 7452 q^{39} + 81000 q^{40} + 6060 q^{41} - 14112 q^{42} - 50044 q^{43} + 56960 q^{44} + 19764 q^{45} + 51488 q^{46} + 16552 q^{47} - 36864 q^{48} + 38084 q^{49} - 48064 q^{50} - 31360 q^{52} - 4328 q^{53} - 71740 q^{55} - 9248 q^{56} + 67788 q^{57} + 95256 q^{58} - 89832 q^{59} - 82944 q^{60} + 154104 q^{61} + 106624 q^{62} + 31752 q^{63} + 93720 q^{64} + 13260 q^{65} - 5976 q^{66} - 182136 q^{67} - 40932 q^{69} - 145464 q^{70} - 24504 q^{71} + 58184 q^{73} + 308992 q^{74} - 89136 q^{75} - 602544 q^{76} - 240576 q^{77} + 136080 q^{79} + 317440 q^{80} + 183708 q^{81} + 46648 q^{82} - 203576 q^{83} - 332424 q^{84} - 421280 q^{86} - 330120 q^{87} + 145152 q^{88} - 120880 q^{89} + 64800 q^{90} + 290152 q^{91} + 491616 q^{92} - 69192 q^{93} - 799168 q^{94} + 543084 q^{95} + 409328 q^{97} + 193072 q^{98} + 91692 q^{99}+O(q^{100}) \) 28 * q - 252 * q^3 + 384 * q^4 + 244 * q^5 + 392 * q^7 + 2268 * q^9 + 800 * q^10 + 1132 * q^11 - 3456 * q^12 - 828 * q^13 + 1568 * q^14 - 2196 * q^15 + 4096 * q^16 - 7532 * q^19 + 9216 * q^20 - 3528 * q^21 + 664 * q^22 + 4548 * q^23 + 9904 * q^25 - 20412 * q^27 + 36936 * q^28 + 36680 * q^29 - 7200 * q^30 + 7688 * q^31 - 10188 * q^33 + 13136 * q^35 + 31104 * q^36 + 42360 * q^37 - 1984 * q^38 + 7452 * q^39 + 81000 * q^40 + 6060 * q^41 - 14112 * q^42 - 50044 * q^43 + 56960 * q^44 + 19764 * q^45 + 51488 * q^46 + 16552 * q^47 - 36864 * q^48 + 38084 * q^49 - 48064 * q^50 - 31360 * q^52 - 4328 * q^53 - 71740 * q^55 - 9248 * q^56 + 67788 * q^57 + 95256 * q^58 - 89832 * q^59 - 82944 * q^60 + 154104 * q^61 + 106624 * q^62 + 31752 * q^63 + 93720 * q^64 + 13260 * q^65 - 5976 * q^66 - 182136 * q^67 - 40932 * q^69 - 145464 * q^70 - 24504 * q^71 + 58184 * q^73 + 308992 * q^74 - 89136 * q^75 - 602544 * q^76 - 240576 * q^77 + 136080 * q^79 + 317440 * q^80 + 183708 * q^81 + 46648 * q^82 - 203576 * q^83 - 332424 * q^84 - 421280 * q^86 - 330120 * q^87 + 145152 * q^88 - 120880 * q^89 + 64800 * q^90 + 290152 * q^91 + 491616 * q^92 - 69192 * q^93 - 799168 * q^94 + 543084 * q^95 + 409328 * q^97 + 193072 * q^98 + 91692 * 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 $

$2$	−4.85553	−0.858345	−0.429172	−	0.903223i	$-0.641195\pi$
$2$	−4.85553	−0.858345	−0.429172	+	0.903223i	$0.641195\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	−8.42383	−0.263245
$5$	−82.3928	−1.47389	−0.736943	−	0.675955i	$-0.763732\pi$
$5$	−82.3928	−1.47389	−0.736943	+	0.675955i	$0.763732\pi$
$6$	43.6998	0.495565
$7$	−35.3683	−0.272816	−0.136408	−	0.990653i	$-0.543556\pi$
$7$	−35.3683	−0.272816	−0.136408	+	0.990653i	$0.543556\pi$
$8$	196.279	1.08430
$9$	81.0000	0.333333
$10$	400.061	1.26510
$11$	88.9161	0.221564	0.110782	−	0.993845i	$-0.464664\pi$
$11$	88.9161	0.221564	0.110782	+	0.993845i	$0.464664\pi$
$12$	75.8144	0.151984
$13$	−607.679	−0.997276	−0.498638	−	0.866810i	$-0.666166\pi$
$13$	−607.679	−0.997276	−0.498638	+	0.866810i	$0.666166\pi$
$14$	171.732	0.234170
$15$	741.535	0.850949
$16$	−683.477	−0.667458
$17$	0	0
$17$	0	0
$18$	−393.298	−0.286115
$19$	728.381	0.462887	0.231443	−	0.972848i	$-0.425655\pi$
$19$	728.381	0.462887	0.231443	+	0.972848i	$0.425655\pi$
$20$	694.062	0.387993
$21$	318.315	0.157510
$22$	−431.735	−0.190178
$23$	−842.921	−0.332252	−0.166126	−	0.986105i	$-0.553126\pi$
$23$	−842.921	−0.332252	−0.166126	+	0.986105i	$0.553126\pi$
$24$	−1766.51	−0.626020
$25$	3663.57	1.17234
$26$	2950.60	0.856007
$27$	−729.000	−0.192450
$28$	297.937	0.0718173
$29$	521.365	0.115119	0.0575595	−	0.998342i	$-0.481668\pi$
$29$	521.365	0.115119	0.0575595	+	0.998342i	$0.481668\pi$
$30$	−3600.55	−0.730407
$31$	1459.76	0.272822	0.136411	−	0.990652i	$-0.456443\pi$
$31$	1459.76	0.272822	0.136411	+	0.990652i	$0.456443\pi$
$32$	−2962.29	−0.511390
$33$	−800.245	−0.127920
$34$	0	0
$35$	2914.10	0.402100
$36$	−682.330	−0.0877482
$37$	11600.8	1.39310	0.696552	−	0.717506i	$-0.254717\pi$
$37$	11600.8	1.39310	0.696552	+	0.717506i	$0.254717\pi$
$38$	−3536.68	−0.397316
$39$	5469.11	0.575778
$40$	−16172.0	−1.59813
$41$	14400.6	1.33789	0.668944	−	0.743313i	$-0.266747\pi$
$41$	14400.6	1.33789	0.668944	+	0.743313i	$0.266747\pi$
$42$	−1545.59	−0.135198
$43$	2446.99	0.201818	0.100909	−	0.994896i	$-0.467825\pi$
$43$	2446.99	0.201818	0.100909	+	0.994896i	$0.467825\pi$
$44$	−749.014	−0.0583255
$45$	−6673.81	−0.491296
$46$	4092.83	0.285186
$47$	−25631.6	−1.69251	−0.846255	−	0.532778i	$-0.821148\pi$
$47$	−25631.6	−1.69251	−0.846255	+	0.532778i	$0.821148\pi$
$48$	6151.29	0.385357
$49$	−15556.1	−0.925572
$50$	−17788.6	−1.00627
$51$	0	0
$52$	5118.98	0.262528
$53$	−2320.07	−0.113452	−0.0567260	−	0.998390i	$-0.518066\pi$
$53$	−2320.07	−0.113452	−0.0567260	+	0.998390i	$0.518066\pi$
$54$	3539.68	0.165188
$55$	−7326.05	−0.326560
$56$	−6942.07	−0.295814
$57$	−6555.43	−0.267248
$58$	−2531.50	−0.0988117
$59$	−37043.5	−1.38542	−0.692711	−	0.721215i	$-0.743584\pi$
$59$	−37043.5	−1.38542	−0.692711	+	0.721215i	$0.743584\pi$
$60$	−6246.56	−0.224008
$61$	−11489.3	−0.395338	−0.197669	−	0.980269i	$-0.563337\pi$
$61$	−11489.3	−0.395338	−0.197669	+	0.980269i	$0.563337\pi$
$62$	−7087.93	−0.234175
$63$	−2864.84	−0.0909386
$64$	36254.7	1.10641
$65$	50068.3	1.46987
$66$	3885.62	0.109799
$67$	−57126.2	−1.55471	−0.777353	−	0.629065i	$-0.783438\pi$
$67$	−57126.2	−1.55471	−0.777353	+	0.629065i	$0.783438\pi$
$68$	0	0
$69$	7586.29	0.191826
$70$	−14149.5	−0.345140
$71$	−62663.3	−1.47526	−0.737628	−	0.675208i	$-0.764054\pi$
$71$	−62663.3	−1.47526	−0.737628	+	0.675208i	$0.764054\pi$
$72$	15898.6	0.361433
$73$	−31928.3	−0.701243	−0.350621	−	0.936517i	$-0.614030\pi$
$73$	−31928.3	−0.701243	−0.350621	+	0.936517i	$0.614030\pi$
$74$	−56328.0	−1.19576
$75$	−32972.1	−0.676852
$76$	−6135.76	−0.121852
$77$	−3144.82	−0.0604461
$78$	−26555.4	−0.494216
$79$	−51843.7	−0.934606	−0.467303	−	0.884097i	$-0.654774\pi$
$79$	−51843.7	−0.934606	−0.467303	+	0.884097i	$0.654774\pi$
$80$	56313.5	0.983757
$81$	6561.00	0.111111
$82$	−69922.4	−1.14837
$83$	−68331.8	−1.08875	−0.544374	−	0.838843i	$-0.683233\pi$
$83$	−68331.8	−1.08875	−0.544374	+	0.838843i	$0.683233\pi$
$84$	−2681.43	−0.0414637
$85$	0	0
$86$	−11881.4	−0.173230
$87$	−4692.29	−0.0664640
$88$	17452.4	0.240241
$89$	−130403.	−1.74506	−0.872531	−	0.488558i	$-0.837523\pi$
$89$	−130403.	−1.74506	−0.872531	+	0.488558i	$0.837523\pi$
$90$	32404.9	0.421701
$91$	21492.6	0.272073
$92$	7100.62	0.0874635
$93$	−13137.9	−0.157514
$94$	124455.	1.45276
$95$	−60013.4	−0.682243
$96$	26660.6	0.295251
$97$	−53915.3	−0.581812	−0.290906	−	0.956752i	$-0.593957\pi$
$97$	−53915.3	−0.581812	−0.290906	+	0.956752i	$0.593957\pi$
$98$	75533.0	0.794459
$99$	7202.21	0.0738546
$100$	−30861.3	−0.308613
$101$	178516.	1.74130	0.870651	−	0.491901i	$-0.163698\pi$
$101$	178516.	1.74130	0.870651	+	0.491901i	$0.163698\pi$
$102$	0	0
$103$	25684.7	0.238551	0.119275	−	0.992861i	$-0.461943\pi$
$103$	25684.7	0.238551	0.119275	+	0.992861i	$0.461943\pi$
$104$	−119275.	−1.08135
$105$	−26226.9	−0.232152
$106$	11265.2	0.0973808
$107$	146355.	1.23580	0.617902	−	0.786255i	$-0.287983\pi$
$107$	146355.	1.23580	0.617902	+	0.786255i	$0.287983\pi$
$108$	6140.97	0.0506614
$109$	−135166.	−1.08968	−0.544842	−	0.838539i	$-0.683410\pi$
$109$	−135166.	−1.08968	−0.544842	+	0.838539i	$0.683410\pi$
$110$	35571.8	0.280301
$111$	−104407.	−0.804309
$112$	24173.4	0.182093
$113$	116529.	0.858495	0.429247	−	0.903187i	$-0.358779\pi$
$113$	116529.	0.858495	0.429247	+	0.903187i	$0.358779\pi$
$114$	31830.1	0.229391
$115$	69450.6	0.489701
$116$	−4391.89	−0.0303044
$117$	−49222.0	−0.332425
$118$	179866.	1.18917
$119$	0	0
$120$	145548.	0.922683
$121$	−153145.	−0.950909
$122$	55786.6	0.339336
$123$	−129605.	−0.772430
$124$	−12296.8	−0.0718188
$125$	−44374.2	−0.254013
$126$	13910.3	0.0780567
$127$	234366.	1.28939	0.644695	−	0.764440i	$-0.276984\pi$
$127$	234366.	1.28939	0.644695	+	0.764440i	$0.276984\pi$
$128$	−81242.7	−0.438288
$129$	−22022.9	−0.116520
$130$	−243108.	−1.26166
$131$	35395.7	0.180207	0.0901037	−	0.995932i	$-0.471280\pi$
$131$	35395.7	0.180207	0.0901037	+	0.995932i	$0.471280\pi$
$132$	6741.13	0.0336742
$133$	−25761.6	−0.126283
$134$	277378.	1.33447
$135$	60064.3	0.283650
$136$	0	0
$137$	292411.	1.33104	0.665522	−	0.746378i	$-0.268209\pi$
$137$	292411.	1.33104	0.665522	+	0.746378i	$0.268209\pi$
$138$	−36835.5	−0.164652
$139$	−281862.	−1.23737	−0.618685	−	0.785639i	$-0.712334\pi$
$139$	−281862.	−1.23737	−0.618685	+	0.785639i	$0.712334\pi$
$140$	−24547.8	−0.105851
$141$	230685.	0.977171
$142$	304263.	1.26628
$143$	−54032.4	−0.220960
$144$	−55361.6	−0.222486
$145$	−42956.7	−0.169672
$146$	155029.	0.601908
$147$	140005.	0.534379
$148$	−97723.1	−0.366727
$149$	124400.	0.459046	0.229523	−	0.973303i	$-0.426283\pi$
$149$	124400.	0.459046	0.229523	+	0.973303i	$0.426283\pi$
$150$	160097.	0.580972
$151$	−497288.	−1.77487	−0.887433	−	0.460936i	$-0.847514\pi$
$151$	−497288.	−1.77487	−0.887433	+	0.460936i	$0.847514\pi$
$152$	142966.	0.501908
$153$	0	0
$154$	15269.7	0.0518836
$155$	−120274.	−0.402108
$156$	−46070.8	−0.151570
$157$	−471249.	−1.52581	−0.762906	−	0.646510i	$-0.776228\pi$
$157$	−471249.	−1.52581	−0.762906	+	0.646510i	$0.776228\pi$
$158$	251729.	0.802214
$159$	20880.7	0.0655015
$160$	244071.	0.753732
$161$	29812.7	0.0906435
$162$	−31857.1	−0.0953716
$163$	429530.	1.26626	0.633132	−	0.774044i	$-0.281769\pi$
$163$	429530.	1.26626	0.633132	+	0.774044i	$0.281769\pi$
$164$	−121308.	−0.352192
$165$	65934.4	0.188540
$166$	331787.	0.934521
$167$	251418.	0.697599	0.348799	−	0.937197i	$-0.386589\pi$
$167$	251418.	0.697599	0.348799	+	0.937197i	$0.386589\pi$
$168$	62478.6	0.170788
$169$	−2019.82	−0.00543997
$170$	0	0
$171$	58998.9	0.154296
$172$	−20613.0	−0.0531276
$173$	−364784.	−0.926659	−0.463330	−	0.886186i	$-0.653345\pi$
$173$	−364784.	−0.926659	−0.463330	+	0.886186i	$0.653345\pi$
$174$	22783.5	0.0570490
$175$	−129574.	−0.319833
$176$	−60772.1	−0.147884
$177$	333392.	0.799874
$178$	633174.	1.49787
$179$	−512444.	−1.19540	−0.597701	−	0.801719i	$-0.703919\pi$
$179$	−512444.	−1.19540	−0.597701	+	0.801719i	$0.703919\pi$
$180$	56219.1	0.129331
$181$	−593596.	−1.34677	−0.673387	−	0.739290i	$-0.735161\pi$
$181$	−593596.	−1.34677	−0.673387	+	0.739290i	$0.735161\pi$
$182$	−104358.	−0.233532
$183$	103404.	0.228248
$184$	−165448.	−0.360260
$185$	−955822.	−2.05328
$186$	63791.4	0.135201
$187$	0	0
$188$	215916.	0.445544
$189$	25783.5	0.0525034
$190$	291397.	0.585599
$191$	853915.	1.69368	0.846840	−	0.531848i	$-0.178502\pi$
$191$	853915.	1.69368	0.846840	+	0.531848i	$0.178502\pi$
$192$	−326293.	−0.638784
$193$	−619608.	−1.19736	−0.598679	−	0.800989i	$-0.704308\pi$
$193$	−619608.	−1.19736	−0.598679	+	0.800989i	$0.704308\pi$
$194$	261787.	0.499395
$195$	−450615.	−0.848631
$196$	131042.	0.243652
$197$	−483204.	−0.887085	−0.443542	−	0.896253i	$-0.646278\pi$
$197$	−483204.	−0.887085	−0.443542	+	0.896253i	$0.646278\pi$
$198$	−34970.5	−0.0633927
$199$	987693.	1.76803	0.884015	−	0.467459i	$-0.154831\pi$
$199$	987693.	1.76803	0.884015	+	0.467459i	$0.154831\pi$
$200$	719082.	1.27117
$201$	514136.	0.897610
$202$	−866791.	−1.49464
$203$	−18439.8	−0.0314063
$204$	0	0
$205$	−1.18650e6	−1.97190
$206$	−124713.	−0.204759
$207$	−68276.6	−0.110751
$208$	415334.	0.665640
$209$	64764.9	0.102559
$210$	127345.	0.199267
$211$	−350297.	−0.541665	−0.270833	−	0.962626i	$-0.587299\pi$
$211$	−350297.	−0.541665	−0.270833	+	0.962626i	$0.587299\pi$
$212$	19543.9	0.0298656
$213$	563969.	0.851739
$214$	−710633.	−1.06074
$215$	−201614.	−0.297458
$216$	−143087.	−0.208673
$217$	−51629.5	−0.0744300
$218$	656302.	0.935324
$219$	287355.	0.404863
$220$	61713.4	0.0859652
$221$	0	0
$222$	506952.	0.690374
$223$	−460117.	−0.619592	−0.309796	−	0.950803i	$-0.600261\pi$
$223$	−460117.	−0.619592	−0.309796	+	0.950803i	$0.600261\pi$
$224$	104771.	0.139515
$225$	296749.	0.390781
$226$	−565810.	−0.736884
$227$	−180122.	−0.232008	−0.116004	−	0.993249i	$-0.537009\pi$
$227$	−180122.	−0.232008	−0.116004	+	0.993249i	$0.537009\pi$
$228$	55221.8	0.0703516
$229$	854469.	1.07673	0.538366	−	0.842711i	$-0.319042\pi$
$229$	854469.	1.07673	0.538366	+	0.842711i	$0.319042\pi$
$230$	−337219.	−0.420333
$231$	28303.3	0.0348986
$232$	102333.	0.124823
$233$	851701.	1.02777	0.513887	−	0.857858i	$-0.328205\pi$
$233$	851701.	1.02777	0.513887	+	0.857858i	$0.328205\pi$
$234$	238999.	0.285336
$235$	2.11186e6	2.49457
$236$	312048.	0.364705
$237$	466593.	0.539595
$238$	0	0
$239$	53163.9	0.0602036	0.0301018	−	0.999547i	$-0.490417\pi$
$239$	53163.9	0.0602036	0.0301018	+	0.999547i	$0.490417\pi$
$240$	−506822.	−0.567972
$241$	−1839.35	−0.00203996	−0.00101998	−	0.999999i	$-0.500325\pi$
$241$	−1839.35	−0.00203996	−0.00101998	+	0.999999i	$0.500325\pi$
$242$	743600.	0.816208
$243$	−59049.0	−0.0641500
$244$	96783.8	0.104071
$245$	1.28171e6	1.36419
$246$	629301.	0.663011
$247$	−442622.	−0.461626
$248$	286521.	0.295820
$249$	614986.	0.628589
$250$	215460.	0.218030
$251$	−510288.	−0.511247	−0.255624	−	0.966776i	$-0.582281\pi$
$251$	−510288.	−0.511247	−0.255624	+	0.966776i	$0.582281\pi$
$252$	24132.9	0.0239391
$253$	−74949.3	−0.0736150
$254$	−1.13797e6	−1.10674
$255$	0	0
$256$	−765675.	−0.730205
$257$	−635023.	−0.599731	−0.299866	−	0.953981i	$-0.596942\pi$
$257$	−635023.	−0.599731	−0.299866	+	0.953981i	$0.596942\pi$
$258$	106933.	0.100014
$259$	−410301.	−0.380061
$260$	−421767.	−0.386936
$261$	42230.6	0.0383730
$262$	−171865.	−0.154680
$263$	658997.	0.587481	0.293741	−	0.955885i	$-0.405100\pi$
$263$	658997.	0.587481	0.293741	+	0.955885i	$0.405100\pi$
$264$	−157071.	−0.138703
$265$	191157.	0.167215
$266$	125086.	0.108394
$267$	1.17362e6	1.00751
$268$	481221.	0.409268
$269$	755566.	0.636637	0.318318	−	0.947984i	$-0.396882\pi$
$269$	755566.	0.636637	0.318318	+	0.947984i	$0.396882\pi$
$270$	−291644.	−0.243469
$271$	−171275.	−0.141668	−0.0708340	−	0.997488i	$-0.522566\pi$
$271$	−171275.	−0.141668	−0.0708340	+	0.997488i	$0.522566\pi$
$272$	0	0
$273$	−193433.	−0.157081
$274$	−1.41981e6	−1.14249
$275$	325750.	0.259749
$276$	−63905.6	−0.0504971
$277$	1.49471e6	1.17046	0.585230	−	0.810867i	$-0.301004\pi$
$277$	1.49471e6	1.17046	0.585230	+	0.810867i	$0.301004\pi$
$278$	1.36859e6	1.06209
$279$	118241.	0.0909405
$280$	571976.	0.435996
$281$	690053.	0.521335	0.260667	−	0.965429i	$-0.416057\pi$
$281$	690053.	0.521335	0.260667	+	0.965429i	$0.416057\pi$
$282$	−1.12010e6	−0.838750
$283$	−361859.	−0.268580	−0.134290	−	0.990942i	$-0.542875\pi$
$283$	−361859.	−0.268580	−0.134290	+	0.990942i	$0.542875\pi$
$284$	527864.	0.388353
$285$	540120.	0.393893
$286$	262356.	0.189660
$287$	−509324.	−0.364997
$288$	−239945.	−0.170463
$289$	0	0
$290$	208578.	0.145637
$291$	485238.	0.335909
$292$	268958.	0.184598
$293$	−817433.	−0.556267	−0.278133	−	0.960542i	$-0.589716\pi$
$293$	−817433.	−0.556267	−0.278133	+	0.960542i	$0.589716\pi$
$294$	−679797.	−0.458681
$295$	3.05212e6	2.04196
$296$	2.27699e6	1.51054
$297$	−64819.9	−0.0426400
$298$	−604030.	−0.394020
$299$	512225.	0.331347
$300$	277751.	0.178178
$301$	−86545.9	−0.0550593
$302$	2.41460e6	1.52345
$303$	−1.60665e6	−1.00534
$304$	−497832.	−0.308957
$305$	946634.	0.582683
$306$	0	0
$307$	−2.22091e6	−1.34488	−0.672442	−	0.740149i	$-0.734755\pi$
$307$	−2.22091e6	−1.34488	−0.672442	+	0.740149i	$0.734755\pi$
$308$	26491.4	0.0159121
$309$	−231162.	−0.137727
$310$	583994.	0.345147
$311$	−22862.3	−0.0134035	−0.00670177	−	0.999978i	$-0.502133\pi$
$311$	−22862.3	−0.0134035	−0.00670177	+	0.999978i	$0.502133\pi$
$312$	1.07347e6	0.624315
$313$	1.65461e6	0.954632	0.477316	−	0.878732i	$-0.341610\pi$
$313$	1.65461e6	0.954632	0.477316	+	0.878732i	$0.341610\pi$
$314$	2.28816e6	1.30967
$315$	236042.	0.134033
$316$	436722.	0.246030
$317$	−1.06021e6	−0.592574	−0.296287	−	0.955099i	$-0.595748\pi$
$317$	−1.06021e6	−0.592574	−0.296287	+	0.955099i	$0.595748\pi$
$318$	−101387.	−0.0562229
$319$	46357.8	0.0255062
$320$	−2.98713e6	−1.63072
$321$	−1.31720e6	−0.713491
$322$	−144757.	−0.0778034
$323$	0	0
$324$	−55268.7	−0.0292494
$325$	−2.22627e6	−1.16915
$326$	−2.08559e6	−1.08689
$327$	1.21649e6	0.629129
$328$	2.82653e6	1.45067
$329$	906548.	0.461744
$330$	−320147.	−0.161832
$331$	532449.	0.267121	0.133561	−	0.991041i	$-0.457359\pi$
$331$	532449.	0.267121	0.133561	+	0.991041i	$0.457359\pi$
$332$	575615.	0.286607
$333$	939665.	0.464368
$334$	−1.22077e6	−0.598780
$335$	4.70678e6	2.29146
$336$	−217561.	−0.105131
$337$	1.17270e6	0.562489	0.281244	−	0.959636i	$-0.409253\pi$
$337$	1.17270e6	0.562489	0.281244	+	0.959636i	$0.409253\pi$
$338$	9807.31	0.00466937
$339$	−1.04876e6	−0.495652
$340$	0	0
$341$	129797.	0.0604474
$342$	−286471.	−0.132439
$343$	1.14463e6	0.525326
$344$	480293.	0.218832
$345$	−625055.	−0.282729
$346$	1.77122e6	0.795393
$347$	3.27941e6	1.46208	0.731042	−	0.682332i	$-0.239034\pi$
$347$	3.27941e6	1.46208	0.731042	+	0.682332i	$0.239034\pi$
$348$	39527.0	0.0174963
$349$	−3.56553e6	−1.56697	−0.783485	−	0.621411i	$-0.786560\pi$
$349$	−3.56553e6	−1.56697	−0.783485	+	0.621411i	$0.786560\pi$
$350$	629152.	0.274527
$351$	442998.	0.191926
$352$	−263395.	−0.113306
$353$	−2.40109e6	−1.02559	−0.512793	−	0.858512i	$-0.671389\pi$
$353$	−2.40109e6	−1.02559	−0.512793	+	0.858512i	$0.671389\pi$
$354$	−1.61879e6	−0.686568
$355$	5.16300e6	2.17436
$356$	1.09849e6	0.459378
$357$	0	0
$358$	2.48819e6	1.02607
$359$	18532.7	0.00758930	0.00379465	−	0.999993i	$-0.498792\pi$
$359$	18532.7	0.00758930	0.00379465	+	0.999993i	$0.498792\pi$
$360$	−1.30993e6	−0.532711
$361$	−1.94556e6	−0.785736
$362$	2.88223e6	1.15600
$363$	1.37830e6	0.549008
$364$	−181050.	−0.0716217
$365$	2.63066e6	1.03355
$366$	−502079.	−0.195916
$367$	−3.01002e6	−1.16655	−0.583275	−	0.812275i	$-0.698229\pi$
$367$	−3.01002e6	−1.16655	−0.583275	+	0.812275i	$0.698229\pi$
$368$	576117.	0.221764
$369$	1.16645e6	0.445963
$370$	4.64102e6	1.76242
$371$	82057.1	0.0309515
$372$	110671.	0.0414646
$373$	−1.79099e6	−0.666534	−0.333267	−	0.942833i	$-0.608151\pi$
$373$	−1.79099e6	−0.666534	−0.333267	+	0.942833i	$0.608151\pi$
$374$	0	0
$375$	399367.	0.146654
$376$	−5.03095e6	−1.83519
$377$	−316822.	−0.114805
$378$	−125193.	−0.0450660
$379$	−1.04095e6	−0.372248	−0.186124	−	0.982526i	$-0.559593\pi$
$379$	−1.04095e6	−0.372248	−0.186124	+	0.982526i	$0.559593\pi$
$380$	505542.	0.179597
$381$	−2.10929e6	−0.744430
$382$	−4.14621e6	−1.45376
$383$	315801.	0.110006	0.0550030	−	0.998486i	$-0.482483\pi$
$383$	315801.	0.110006	0.0550030	+	0.998486i	$0.482483\pi$
$384$	731184.	0.253046
$385$	259110.	0.0890907
$386$	3.00853e6	1.02775
$387$	198206.	0.0672728
$388$	454173.	0.153159
$389$	2.24538e6	0.752343	0.376172	−	0.926550i	$-0.377240\pi$
$389$	2.24538e6	0.752343	0.376172	+	0.926550i	$0.377240\pi$
$390$	2.18797e6	0.728418
$391$	0	0
$392$	−3.05333e6	−1.00360
$393$	−318562.	−0.104043
$394$	2.34621e6	0.761425
$395$	4.27155e6	1.37750
$396$	−60670.1	−0.0194418
$397$	−1.40448e6	−0.447239	−0.223619	−	0.974677i	$-0.571787\pi$
$397$	−1.40448e6	−0.447239	−0.223619	+	0.974677i	$0.571787\pi$
$398$	−4.79577e6	−1.51758
$399$	231855.	0.0729094
$400$	−2.50396e6	−0.782489
$401$	−6.09175e6	−1.89182	−0.945912	−	0.324423i	$-0.894830\pi$
$401$	−6.09175e6	−1.89182	−0.945912	+	0.324423i	$0.894830\pi$
$402$	−2.49640e6	−0.770459
$403$	−887068.	−0.272078
$404$	−1.50379e6	−0.458388
$405$	−540579.	−0.163765
$406$	89535.1	0.0269574
$407$	1.03150e6	0.308661
$408$	0	0
$409$	−2.15158e6	−0.635989	−0.317995	−	0.948093i	$-0.603009\pi$
$409$	−2.15158e6	−0.635989	−0.317995	+	0.948093i	$0.603009\pi$
$410$	5.76110e6	1.69257
$411$	−2.63170e6	−0.768479
$412$	−216363.	−0.0627973
$413$	1.31017e6	0.377965
$414$	331519.	0.0950622
$415$	5.63004e6	1.60469
$416$	1.80012e6	0.509998
$417$	2.53676e6	0.714396
$418$	−314468.	−0.0880310
$419$	3.76613e6	1.04800	0.523999	−	0.851719i	$-0.324439\pi$
$419$	3.76613e6	1.04800	0.523999	+	0.851719i	$0.324439\pi$
$420$	220931.	0.0611128
$421$	5.15676e6	1.41798	0.708992	−	0.705217i	$-0.249150\pi$
$421$	5.15676e6	1.41798	0.708992	+	0.705217i	$0.249150\pi$
$422$	1.70088e6	0.464935
$423$	−2.07616e6	−0.564170
$424$	−455382.	−0.123016
$425$	0	0
$426$	−2.73837e6	−0.731086
$427$	406357.	0.107854
$428$	−1.23287e6	−0.325318
$429$	486292.	0.127572
$430$	978944.	0.255321
$431$	−4.63311e6	−1.20138	−0.600689	−	0.799483i	$-0.705107\pi$
$431$	−4.63311e6	−1.20138	−0.600689	+	0.799483i	$0.705107\pi$
$432$	498255.	0.128452
$433$	1485.21	0.000380686 0	0.000190343	−	1.00000i	$-0.499939\pi$
$433$	1485.21	0.000380686 0	0.000190343		1.00000i	$0.499939\pi$
$434$	250688.	0.0638866
$435$	386610.	0.0979604
$436$	1.13861e6	0.286853
$437$	−613968.	−0.153795
$438$	−1.39526e6	−0.347512
$439$	7.02203e6	1.73901	0.869504	−	0.493926i	$-0.164439\pi$
$439$	7.02203e6	1.73901	0.869504	+	0.493926i	$0.164439\pi$
$440$	−1.43795e6	−0.354089
$441$	−1.26004e6	−0.308524
$442$	0	0
$443$	5.06390e6	1.22596	0.612979	−	0.790099i	$-0.289971\pi$
$443$	5.06390e6	1.22596	0.612979	+	0.790099i	$0.289971\pi$
$444$	879508.	0.211730
$445$	1.07442e7	2.57203
$446$	2.23411e6	0.531824
$447$	−1.11960e6	−0.265030
$448$	−1.28227e6	−0.301845
$449$	1.36157e6	0.318731	0.159366	−	0.987220i	$-0.449055\pi$
$449$	1.36157e6	0.318731	0.159366	+	0.987220i	$0.449055\pi$
$450$	−1.44087e6	−0.335424
$451$	1.28044e6	0.296428
$452$	−981620.	−0.225994
$453$	4.47559e6	1.02472
$454$	874589.	0.199143
$455$	−1.77083e6	−0.401004
$456$	−1.28669e6	−0.289777
$457$	−41544.6	−0.00930515	−0.00465258	−	0.999989i	$-0.501481\pi$
$457$	−41544.6	−0.00930515	−0.00465258	+	0.999989i	$0.501481\pi$
$458$	−4.14890e6	−0.924207
$459$	0	0
$460$	−585040.	−0.128911
$461$	349314.	0.0765534	0.0382767	−	0.999267i	$-0.487813\pi$
$461$	349314.	0.0765534	0.0382767	+	0.999267i	$0.487813\pi$
$462$	−137428.	−0.0299550
$463$	−7.95491e6	−1.72458	−0.862289	−	0.506417i	$-0.830970\pi$
$463$	−7.95491e6	−1.72458	−0.862289	+	0.506417i	$0.830970\pi$
$464$	−356341.	−0.0768370
$465$	1.08247e6	0.232157
$466$	−4.13546e6	−0.882184
$467$	−3.16371e6	−0.671281	−0.335641	−	0.941990i	$-0.608953\pi$
$467$	−3.16371e6	−0.671281	−0.335641	+	0.941990i	$0.608953\pi$
$468$	414637.	0.0875092
$469$	2.02046e6	0.424148
$470$	−1.02542e7	−2.14120
$471$	4.24124e6	0.880928
$472$	−7.27087e6	−1.50221
$473$	217577.	0.0447157
$474$	−2.26556e6	−0.463158
$475$	2.66848e6	0.542662
$476$	0	0
$477$	−187926.	−0.0378173
$478$	−258139.	−0.0516754
$479$	−1.20473e6	−0.239912	−0.119956	−	0.992779i	$-0.538275\pi$
$479$	−1.20473e6	−0.239912	−0.119956	+	0.992779i	$0.538275\pi$
$480$	−2.19664e6	−0.435167
$481$	−7.04955e6	−1.38931
$482$	8931.03	0.00175099
$483$	−268314.	−0.0523331
$484$	1.29007e6	0.250322
$485$	4.44223e6	0.857525
$486$	286714.	0.0550628
$487$	855818.	0.163516	0.0817578	−	0.996652i	$-0.473947\pi$
$487$	855818.	0.163516	0.0817578	+	0.996652i	$0.473947\pi$
$488$	−2.25511e6	−0.428665
$489$	−3.86577e6	−0.731078
$490$	−6.22337e6	−1.17094
$491$	−7.91330e6	−1.48134	−0.740668	−	0.671871i	$-0.765491\pi$
$491$	−7.91330e6	−1.48134	−0.740668	+	0.671871i	$0.765491\pi$
$492$	1.09177e6	0.203338
$493$	0	0
$494$	2.14916e6	0.396234
$495$	−593410.	−0.108853
$496$	−997715.	−0.182097
$497$	2.21630e6	0.402473
$498$	−2.98608e6	−0.539546
$499$	−2.98471e6	−0.536600	−0.268300	−	0.963335i	$-0.586462\pi$
$499$	−2.98471e6	−0.536600	−0.268300	+	0.963335i	$0.586462\pi$
$500$	373800.	0.0668674
$501$	−2.26276e6	−0.402759
$502$	2.47772e6	0.438826
$503$	4.87638e6	0.859366	0.429683	−	0.902980i	$-0.358625\pi$
$503$	4.87638e6	0.859366	0.429683	+	0.902980i	$0.358625\pi$
$504$	−562307.	−0.0986047
$505$	−1.47084e7	−2.56648
$506$	363918.	0.0631870
$507$	18178.4	0.00314077
$508$	−1.97426e6	−0.339425
$509$	−6.74014e6	−1.15312	−0.576560	−	0.817055i	$-0.695605\pi$
$509$	−6.74014e6	−1.15312	−0.576560	+	0.817055i	$0.695605\pi$
$510$	0	0
$511$	1.12925e6	0.191310
$512$	6.31753e6	1.06506
$513$	−530990.	−0.0890826
$514$	3.08337e6	0.514776
$515$	−2.11623e6	−0.351597
$516$	185517.	0.0306732
$517$	−2.27906e6	−0.374999
$518$	1.99223e6	0.326223
$519$	3.28305e6	0.535007
$520$	9.82736e6	1.59378
$521$	−2.60824e6	−0.420973	−0.210486	−	0.977597i	$-0.567505\pi$
$521$	−2.60824e6	−0.420973	−0.210486	+	0.977597i	$0.567505\pi$
$522$	−205052.	−0.0329372
$523$	−5.70267e6	−0.911641	−0.455821	−	0.890072i	$-0.650654\pi$
$523$	−5.70267e6	−0.911641	−0.455821	+	0.890072i	$0.650654\pi$
$524$	−298167.	−0.0474386
$525$	1.16617e6	0.184656
$526$	−3.19978e6	−0.504261
$527$	0	0
$528$	546949.	0.0853812
$529$	−5.72583e6	−0.889609
$530$	−928170.	−0.143528
$531$	−3.00053e6	−0.461808
$532$	217012.	0.0332433
$533$	−8.75091e6	−1.33424
$534$	−5.69856e6	−0.864793
$535$	−1.20586e7	−1.82143
$536$	−1.12127e7	−1.68577
$537$	4.61200e6	0.690166
$538$	−3.66867e6	−0.546454
$539$	−1.38319e6	−0.205073
$540$	−505972.	−0.0746692
$541$	1.21066e7	1.77840	0.889202	−	0.457514i	$-0.151260\pi$
$541$	1.21066e7	1.77840	0.889202	+	0.457514i	$0.151260\pi$
$542$	831633.	0.121600
$543$	5.34237e6	0.777560
$544$	0	0
$545$	1.11367e7	1.60607
$546$	939221.	0.134830
$547$	−6.64964e6	−0.950232	−0.475116	−	0.879923i	$-0.657594\pi$
$547$	−6.64964e6	−0.950232	−0.475116	+	0.879923i	$0.657594\pi$
$548$	−2.46322e6	−0.350390
$549$	−930632.	−0.131779
$550$	−1.58169e6	−0.222954
$551$	379753.	0.0532871
$552$	1.48903e6	0.207996
$553$	1.83363e6	0.254975
$554$	−7.25759e6	−1.00466
$555$	8.60240e6	1.18546
$556$	2.37436e6	0.325731
$557$	3.95921e6	0.540718	0.270359	−	0.962760i	$-0.412858\pi$
$557$	3.95921e6	0.540718	0.270359	+	0.962760i	$0.412858\pi$
$558$	−574123.	−0.0780583
$559$	−1.48698e6	−0.201269
$560$	−1.99172e6	−0.268384
$561$	0	0
$562$	−3.35057e6	−0.447485
$563$	1.31190e7	1.74433	0.872164	−	0.489213i	$-0.162716\pi$
$563$	1.31190e7	1.74433	0.872164	+	0.489213i	$0.162716\pi$
$564$	−1.94325e6	−0.257235
$565$	−9.60114e6	−1.26532
$566$	1.75702e6	0.230534
$567$	−232052.	−0.0303129
$568$	−1.22995e7	−1.59962
$569$	−3.91300e6	−0.506674	−0.253337	−	0.967378i	$-0.581528\pi$
$569$	−3.91300e6	−0.506674	−0.253337	+	0.967378i	$0.581528\pi$
$570$	−2.62257e6	−0.338096
$571$	−2.09705e6	−0.269165	−0.134583	−	0.990902i	$-0.542969\pi$
$571$	−2.09705e6	−0.269165	−0.134583	+	0.990902i	$0.542969\pi$
$572$	455160.	0.0581666
$573$	−7.68523e6	−0.977846
$574$	2.47304e6	0.313293
$575$	−3.08810e6	−0.389513
$576$	2.93663e6	0.368802
$577$	8.68859e6	1.08645	0.543225	−	0.839587i	$-0.317203\pi$
$577$	8.68859e6	1.08645	0.543225	+	0.839587i	$0.317203\pi$
$578$	0	0
$579$	5.57648e6	0.691295
$580$	361860.	0.0446653
$581$	2.41678e6	0.297028
$582$	−2.35609e6	−0.288326
$583$	−206292.	−0.0251368
$584$	−6.26685e6	−0.760357
$585$	4.05553e6	0.489957
$586$	3.96907e6	0.477469
$587$	6.01409e6	0.720402	0.360201	−	0.932875i	$-0.382708\pi$
$587$	6.01409e6	0.720402	0.360201	+	0.932875i	$0.382708\pi$
$588$	−1.17938e6	−0.140672
$589$	1.06327e6	0.126286
$590$	−1.48197e7	−1.75270
$591$	4.34884e6	0.512159
$592$	−7.92887e6	−0.929838
$593$	1.55176e7	1.81212	0.906062	−	0.423144i	$-0.139074\pi$
$593$	1.55176e7	1.81212	0.906062	+	0.423144i	$0.139074\pi$
$594$	314735.	0.0365998
$595$	0	0
$596$	−1.04793e6	−0.120841
$597$	−8.88924e6	−1.02077
$598$	−2.48712e6	−0.284410
$599$	5.72200e6	0.651600	0.325800	−	0.945439i	$-0.394366\pi$
$599$	5.72200e6	0.651600	0.325800	+	0.945439i	$0.394366\pi$
$600$	−6.47174e6	−0.733910
$601$	4.06450e6	0.459009	0.229505	−	0.973308i	$-0.426289\pi$
$601$	4.06450e6	0.459009	0.229505	+	0.973308i	$0.426289\pi$
$602$	420226.	0.0472598
$603$	−4.62722e6	−0.518235
$604$	4.18907e6	0.467224
$605$	1.26180e7	1.40153
$606$	7.80111e6	0.862929
$607$	−1.44495e7	−1.59177	−0.795887	−	0.605445i	$-0.792995\pi$
$607$	−1.44495e7	−1.59177	−0.795887	+	0.605445i	$0.792995\pi$
$608$	−2.15768e6	−0.236716
$609$	165958.	0.0181324
$610$	−4.59641e6	−0.500143
$611$	1.55758e7	1.68790
$612$	0	0
$613$	−8.87340e6	−0.953759	−0.476880	−	0.878969i	$-0.658232\pi$
$613$	−8.87340e6	−0.953759	−0.476880	+	0.878969i	$0.658232\pi$
$614$	1.07837e7	1.15437
$615$	1.06785e7	1.13847
$616$	−617262.	−0.0655417
$617$	1.67988e7	1.77650	0.888252	−	0.459357i	$-0.151920\pi$
$617$	1.67988e7	1.77650	0.888252	+	0.459357i	$0.151920\pi$
$618$	1.12241e6	0.118218
$619$	−8.69167e6	−0.911751	−0.455876	−	0.890043i	$-0.650674\pi$
$619$	−8.69167e6	−0.911751	−0.455876	+	0.890043i	$0.650674\pi$
$620$	1.01317e6	0.105853
$621$	614489.	0.0639419
$622$	111009.	0.0115049
$623$	4.61212e6	0.476081
$624$	−3.73801e6	−0.384307
$625$	−7.79254e6	−0.797956
$626$	−8.03403e6	−0.819403
$627$	−582884.	−0.0592125
$628$	3.96972e6	0.401662
$629$	0	0
$630$	−1.14611e6	−0.115047
$631$	5.83156e6	0.583058	0.291529	−	0.956562i	$-0.405836\pi$
$631$	5.83156e6	0.583058	0.291529	+	0.956562i	$0.405836\pi$
$632$	−1.01758e7	−1.01339
$633$	3.15268e6	0.312731
$634$	5.14786e6	0.508632
$635$	−1.93100e7	−1.90042
$636$	−175895.	−0.0172429
$637$	9.45310e6	0.923051
$638$	−225092.	−0.0218931
$639$	−5.07572e6	−0.491752
$640$	6.69381e6	0.645987
$641$	−3.83410e6	−0.368569	−0.184284	−	0.982873i	$-0.558997\pi$
$641$	−3.83410e6	−0.368569	−0.184284	+	0.982873i	$0.558997\pi$
$642$	6.39570e6	0.612421
$643$	−1.83408e7	−1.74941	−0.874703	−	0.484660i	$-0.838943\pi$
$643$	−1.83408e7	−1.74941	−0.874703	+	0.484660i	$0.838943\pi$
$644$	−251137.	−0.0238614
$645$	1.81453e6	0.171737
$646$	0	0
$647$	1.29974e7	1.22066	0.610330	−	0.792148i	$-0.291037\pi$
$647$	1.29974e7	1.22066	0.610330	+	0.792148i	$0.291037\pi$
$648$	1.28779e6	0.120478
$649$	−3.29377e6	−0.306960
$650$	1.08097e7	1.00353
$651$	464665.	0.0429722
$652$	−3.61828e6	−0.333337
$653$	2.70030e6	0.247816	0.123908	−	0.992294i	$-0.460457\pi$
$653$	2.70030e6	0.247816	0.123908	+	0.992294i	$0.460457\pi$
$654$	−5.90672e6	−0.540010
$655$	−2.91635e6	−0.265605
$656$	−9.84245e6	−0.892984
$657$	−2.58619e6	−0.233748
$658$	−4.40177e6	−0.396335
$659$	1.32077e6	0.118472	0.0592358	−	0.998244i	$-0.481134\pi$
$659$	1.32077e6	0.118472	0.0592358	+	0.998244i	$0.481134\pi$
$660$	−555420.	−0.0496320
$661$	1.21919e7	1.08535	0.542673	−	0.839944i	$-0.317412\pi$
$661$	1.21919e7	1.08535	0.542673	+	0.839944i	$0.317412\pi$
$662$	−2.58532e6	−0.229282
$663$	0	0
$664$	−1.34121e7	−1.18053
$665$	2.12257e6	0.186127
$666$	−4.56257e6	−0.398588
$667$	−439470.	−0.0382485
$668$	−2.11790e6	−0.183639
$669$	4.14105e6	0.357722
$670$	−2.28539e7	−1.96686
$671$	−1.02158e6	−0.0875926
$672$	−942941.	−0.0805493
$673$	3.40439e6	0.289735	0.144868	−	0.989451i	$-0.453724\pi$
$673$	3.40439e6	0.289735	0.144868	+	0.989451i	$0.453724\pi$
$674$	−5.69410e6	−0.482809
$675$	−2.67074e6	−0.225617
$676$	17014.6	0.00143204
$677$	1.16605e7	0.977791	0.488895	−	0.872342i	$-0.337400\pi$
$677$	1.16605e7	0.977791	0.488895	+	0.872342i	$0.337400\pi$
$678$	5.09229e6	0.425440
$679$	1.90689e6	0.158728
$680$	0	0
$681$	1.62110e6	0.133950
$682$	−630232.	−0.0518847
$683$	−5.56138e6	−0.456174	−0.228087	−	0.973641i	$-0.573247\pi$
$683$	−5.56138e6	−0.456174	−0.228087	+	0.973641i	$0.573247\pi$
$684$	−496996.	−0.0406175
$685$	−2.40926e7	−1.96181
$686$	−5.55778e6	−0.450911
$687$	−7.69022e6	−0.621652
$688$	−1.67246e6	−0.134705
$689$	1.40986e6	0.113143
$690$	3.03497e6	0.242679
$691$	−2.67455e6	−0.213086	−0.106543	−	0.994308i	$-0.533978\pi$
$691$	−2.67455e6	−0.213086	−0.106543	+	0.994308i	$0.533978\pi$
$692$	3.07287e6	0.243938
$693$	−254730.	−0.0201487
$694$	−1.59233e7	−1.25497
$695$	2.32234e7	1.82374
$696$	−920998.	−0.0720668
$697$	0	0
$698$	1.73125e7	1.34500
$699$	−7.66531e6	−0.593385
$700$	1.09151e6	0.0841944
$701$	−1.56726e7	−1.20460	−0.602302	−	0.798268i	$-0.705750\pi$
$701$	−1.56726e7	−1.20460	−0.602302	+	0.798268i	$0.705750\pi$
$702$	−2.15099e6	−0.164739
$703$	8.44980e6	0.644849
$704$	3.22363e6	0.245140
$705$	−1.90067e7	−1.44024
$706$	1.16586e7	0.880306
$707$	−6.31382e6	−0.475055
$708$	−2.80843e6	−0.210563
$709$	1.49150e7	1.11431	0.557156	−	0.830408i	$-0.311892\pi$
$709$	1.49150e7	1.11431	0.557156	+	0.830408i	$0.311892\pi$
$710$	−2.50691e7	−1.86635
$711$	−4.19934e6	−0.311535
$712$	−2.55953e7	−1.89217
$713$	−1.23047e6	−0.0906454
$714$	0	0
$715$	4.45188e6	0.325671
$716$	4.31674e6	0.314683
$717$	−478475.	−0.0347586
$718$	−89985.9	−0.00651423
$719$	1.26695e7	0.913978	0.456989	−	0.889472i	$-0.348928\pi$
$719$	1.26695e7	0.913978	0.456989	+	0.889472i	$0.348928\pi$
$720$	4.56140e6	0.327919
$721$	−908425.	−0.0650805
$722$	9.44672e6	0.674432
$723$	16554.2	0.00117777
$724$	5.00035e6	0.354531
$725$	1.91006e6	0.134959
$726$	−6.69240e6	−0.471238
$727$	1.07708e6	0.0755810	0.0377905	−	0.999286i	$-0.487968\pi$
$727$	1.07708e6	0.0755810	0.0377905	+	0.999286i	$0.487968\pi$
$728$	4.21854e6	0.295008
$729$	531441.	0.0370370
$730$	−1.27732e7	−0.887144
$731$	0	0
$732$	−871054.	−0.0600852
$733$	2.29858e7	1.58015	0.790077	−	0.613008i	$-0.210041\pi$
$733$	2.29858e7	1.58015	0.790077	+	0.613008i	$0.210041\pi$
$734$	1.46152e7	1.00130
$735$	−1.15354e7	−0.787614
$736$	2.49698e6	0.169910
$737$	−5.07944e6	−0.344467
$738$	−5.66371e6	−0.382790
$739$	−7.07195e6	−0.476352	−0.238176	−	0.971222i	$-0.576550\pi$
$739$	−7.07195e6	−0.476352	−0.238176	+	0.971222i	$0.576550\pi$
$740$	8.05168e6	0.540514
$741$	3.98360e6	0.266520
$742$	−398431.	−0.0265670
$743$	−5.88199e6	−0.390888	−0.195444	−	0.980715i	$-0.562615\pi$
$743$	−5.88199e6	−0.390888	−0.195444	+	0.980715i	$0.562615\pi$
$744$	−2.57869e6	−0.170792
$745$	−1.02497e7	−0.676582
$746$	8.69622e6	0.572115
$747$	−5.53487e6	−0.362916
$748$	0	0
$749$	−5.17635e6	−0.337147
$750$	−1.93914e6	−0.125880
$751$	−2.60946e7	−1.68831	−0.844153	−	0.536102i	$-0.819896\pi$
$751$	−2.60946e7	−1.68831	−0.844153	+	0.536102i	$0.819896\pi$
$752$	1.75186e7	1.12968
$753$	4.59259e6	0.295169
$754$	1.53834e6	0.0985426
$755$	4.09729e7	2.61595
$756$	−217196.	−0.0138212
$757$	−2.16589e7	−1.37372	−0.686859	−	0.726791i	$-0.741011\pi$
$757$	−2.16589e7	−1.37372	−0.686859	+	0.726791i	$0.741011\pi$
$758$	5.05438e6	0.319517
$759$	674544.	0.0425016
$760$	−1.17794e7	−0.739755
$761$	−1.16283e7	−0.727869	−0.363934	−	0.931425i	$-0.618567\pi$
$761$	−1.16283e7	−0.727869	−0.363934	+	0.931425i	$0.618567\pi$
$762$	1.02417e7	0.638978
$763$	4.78059e6	0.297283
$764$	−7.19323e6	−0.445852
$765$	0	0
$766$	−1.53338e6	−0.0944231
$767$	2.25106e7	1.38165
$768$	6.89108e6	0.421584
$769$	−1.99396e7	−1.21591	−0.607954	−	0.793972i	$-0.708009\pi$
$769$	−1.99396e7	−1.21591	−0.607954	+	0.793972i	$0.708009\pi$
$770$	−1.25812e6	−0.0764705
$771$	5.71521e6	0.346255
$772$	5.21947e6	0.315198
$773$	−2.52638e6	−0.152072	−0.0760360	−	0.997105i	$-0.524226\pi$
$773$	−2.52638e6	−0.152072	−0.0760360	+	0.997105i	$0.524226\pi$
$774$	−962396.	−0.0577433
$775$	5.34795e6	0.319840
$776$	−1.05824e7	−0.630858
$777$	3.69271e6	0.219428
$778$	−1.09025e7	−0.645770
$779$	1.04891e7	0.619291
$780$	3.79590e6	0.223398
$781$	−5.57178e6	−0.326863
$782$	0	0
$783$	−380075.	−0.0221547
$784$	1.06322e7	0.617780
$785$	3.88275e7	2.24887
$786$	1.54679e6	0.0893046
$787$	−1.49359e7	−0.859597	−0.429799	−	0.902925i	$-0.641415\pi$
$787$	−1.49359e7	−0.859597	−0.429799	+	0.902925i	$0.641415\pi$
$788$	4.07043e6	0.233520
$789$	−5.93097e6	−0.339182
$790$	−2.07406e7	−1.18237
$791$	−4.12144e6	−0.234211
$792$	1.41364e6	0.0800805
$793$	6.98179e6	0.394261
$794$	6.81950e6	0.383885
$795$	−1.72041e6	−0.0965418
$796$	−8.32016e6	−0.465424
$797$	2.93554e6	0.163697	0.0818487	−	0.996645i	$-0.473918\pi$
$797$	2.93554e6	0.163697	0.0818487	+	0.996645i	$0.473918\pi$
$798$	−1.12578e6	−0.0625814
$799$	0	0
$800$	−1.08526e7	−0.599524
$801$	−1.05626e7	−0.581688
$802$	2.95787e7	1.62384
$803$	−2.83894e6	−0.155370
$804$	−4.33099e6	−0.236291
$805$	−2.45635e6	−0.133598
$806$	4.30718e6	0.233537
$807$	−6.80009e6	−0.367562
$808$	3.50390e7	1.88809
$809$	−9.29691e6	−0.499422	−0.249711	−	0.968320i	$-0.580336\pi$
$809$	−9.29691e6	−0.499422	−0.249711	+	0.968320i	$0.580336\pi$
$810$	2.62480e6	0.140567
$811$	1.86033e7	0.993203	0.496601	−	0.867979i	$-0.334581\pi$
$811$	1.86033e7	0.993203	0.496601	+	0.867979i	$0.334581\pi$
$812$	155334.	0.00826753
$813$	1.54148e6	0.0817921
$814$	−5.00847e6	−0.264938
$815$	−3.53901e7	−1.86633
$816$	0	0
$817$	1.78234e6	0.0934191
$818$	1.04471e7	0.545898
$819$	1.74090e6	0.0906909
$820$	9.99489e6	0.519091
$821$	3.38436e7	1.75234	0.876170	−	0.482002i	$-0.160090\pi$
$821$	3.38436e7	1.75234	0.876170	+	0.482002i	$0.160090\pi$
$822$	1.27783e7	0.659620
$823$	−1.55672e7	−0.801146	−0.400573	−	0.916265i	$-0.631189\pi$
$823$	−1.55672e7	−0.801146	−0.400573	+	0.916265i	$0.631189\pi$
$824$	5.04137e6	0.258661
$825$	−2.93175e6	−0.149966
$826$	−6.36156e6	−0.324424
$827$	1.89667e7	0.964337	0.482169	−	0.876078i	$-0.339849\pi$
$827$	1.89667e7	0.964337	0.482169	+	0.876078i	$0.339849\pi$
$828$	575150.	0.0291545
$829$	−6.48044e6	−0.327505	−0.163752	−	0.986501i	$-0.552360\pi$
$829$	−6.48044e6	−0.327505	−0.163752	+	0.986501i	$0.552360\pi$
$830$	−2.73368e7	−1.37738
$831$	−1.34524e7	−0.675765
$832$	−2.20312e7	−1.10339
$833$	0	0
$834$	−1.23173e7	−0.613198
$835$	−2.07150e7	−1.02818
$836$	−545568.	−0.0269981
$837$	−1.06417e6	−0.0525045
$838$	−1.82866e7	−0.899544
$839$	−1.33849e7	−0.656461	−0.328230	−	0.944598i	$-0.606452\pi$
$839$	−1.33849e7	−0.656461	−0.328230	+	0.944598i	$0.606452\pi$
$840$	−5.14778e6	−0.251723
$841$	−2.02393e7	−0.986748
$842$	−2.50388e7	−1.21712
$843$	−6.21048e6	−0.300993
$844$	2.95085e6	0.142590
$845$	166419.	0.00801790
$846$	1.00809e7	0.484252
$847$	5.41648e6	0.259423
$848$	1.58572e6	0.0757244
$849$	3.25673e6	0.155065
$850$	0	0
$851$	−9.77855e6	−0.462861
$852$	−4.75078e6	−0.224216
$853$	2.08372e7	0.980542	0.490271	−	0.871570i	$-0.336898\pi$
$853$	2.08372e7	0.980542	0.490271	+	0.871570i	$0.336898\pi$
$854$	−1.97308e6	−0.0925763
$855$	−4.86108e6	−0.227414
$856$	2.87265e7	1.33998
$857$	−1.49036e7	−0.693168	−0.346584	−	0.938019i	$-0.612658\pi$
$857$	−1.49036e7	−0.693168	−0.346584	+	0.938019i	$0.612658\pi$
$858$	−2.36120e6	−0.109500
$859$	2.27420e7	1.05159	0.525795	−	0.850611i	$-0.323768\pi$
$859$	2.27420e7	1.05159	0.525795	+	0.850611i	$0.323768\pi$
$860$	1.69836e6	0.0783041
$861$	4.58392e6	0.210731
$862$	2.24962e7	1.03120
$863$	−2.75601e7	−1.25966	−0.629830	−	0.776733i	$-0.716875\pi$
$863$	−2.75601e7	−1.25966	−0.629830	+	0.776733i	$0.716875\pi$
$864$	2.15951e6	0.0984171
$865$	3.00555e7	1.36579
$866$	−7211.46	−0.000326760 0
$867$	0	0
$868$	434918.	0.0195933
$869$	−4.60974e6	−0.207075
$870$	−1.87720e6	−0.0840837
$871$	3.47144e7	1.55047
$872$	−2.65302e7	−1.18154
$873$	−4.36714e6	−0.193937
$874$	2.98114e6	0.132009
$875$	1.56944e6	0.0692986
$876$	−2.42062e6	−0.106578
$877$	−1.37826e7	−0.605105	−0.302553	−	0.953133i	$-0.597839\pi$
$877$	−1.37826e7	−0.605105	−0.302553	+	0.953133i	$0.597839\pi$
$878$	−3.40957e7	−1.49267
$879$	7.35690e6	0.321161
$880$	5.00718e6	0.217965
$881$	3.19822e7	1.38825	0.694127	−	0.719853i	$-0.255791\pi$
$881$	3.19822e7	1.38825	0.694127	+	0.719853i	$0.255791\pi$
$882$	6.11817e6	0.264820
$883$	1.86859e7	0.806512	0.403256	−	0.915087i	$-0.367878\pi$
$883$	1.86859e7	0.806512	0.403256	+	0.915087i	$0.367878\pi$
$884$	0	0
$885$	−2.74691e7	−1.17892
$886$	−2.45879e7	−1.05229
$887$	−2.25039e7	−0.960394	−0.480197	−	0.877161i	$-0.659435\pi$
$887$	−2.25039e7	−0.960394	−0.480197	+	0.877161i	$0.659435\pi$
$888$	−2.04929e7	−0.872111
$889$	−8.28912e6	−0.351766
$890$	−5.21689e7	−2.20768
$891$	583379.	0.0246182
$892$	3.87594e6	0.163104
$893$	−1.86696e7	−0.783441
$894$	5.43627e6	0.227487
$895$	4.22217e7	1.76189
$896$	2.87342e6	0.119572
$897$	−4.61002e6	−0.191303
$898$	−6.61115e6	−0.273581
$899$	761070.	0.0314069
$900$	−2.49976e6	−0.102871
$901$	0	0
$902$	−6.21723e6	−0.254437
$903$	778913.	0.0317885
$904$	2.28722e7	0.930865
$905$	4.89080e7	1.98499
$906$	−2.17314e7	−0.879562
$907$	2.78151e7	1.12270	0.561348	−	0.827580i	$-0.310283\pi$
$907$	2.78151e7	1.12270	0.561348	+	0.827580i	$0.310283\pi$
$908$	1.51732e6	0.0610748
$909$	1.44598e7	0.580434
$910$	8.59833e6	0.344200
$911$	−2.46855e7	−0.985476	−0.492738	−	0.870178i	$-0.664004\pi$
$911$	−2.46855e7	−0.985476	−0.492738	+	0.870178i	$0.664004\pi$
$912$	4.48049e6	0.178377
$913$	−6.07580e6	−0.241227
$914$	201721.	0.00798703
$915$	−8.51971e6	−0.336412
$916$	−7.19790e6	−0.283444
$917$	−1.25189e6	−0.0491634
$918$	0	0
$919$	3.12446e7	1.22036	0.610178	−	0.792264i	$-0.291098\pi$
$919$	3.12446e7	1.22036	0.610178	+	0.792264i	$0.291098\pi$
$920$	1.36317e7	0.530983
$921$	1.99882e7	0.776470
$922$	−1.69611e6	−0.0657092
$923$	3.80791e7	1.47124
$924$	−238422.	−0.00918686
$925$	4.25003e7	1.63319
$926$	3.86253e7	1.48028
$927$	2.08046e6	0.0795170
$928$	−1.54443e6	−0.0588707
$929$	−4.42427e7	−1.68191	−0.840955	−	0.541106i	$-0.818006\pi$
$929$	−4.42427e7	−1.68191	−0.840955	+	0.541106i	$0.818006\pi$
$930$	−5.25595e6	−0.199271
$931$	−1.13308e7	−0.428435
$932$	−7.17458e6	−0.270556
$933$	205761.	0.00773854
$934$	1.53615e7	0.576191
$935$	0	0
$936$	−9.66124e6	−0.360449
$937$	−2.59918e7	−0.967137	−0.483568	−	0.875307i	$-0.660660\pi$
$937$	−2.59918e7	−0.967137	−0.483568	+	0.875307i	$0.660660\pi$
$938$	−9.81040e6	−0.364065
$939$	−1.48915e7	−0.551157
$940$	−1.77899e7	−0.656682
$941$	3.49032e7	1.28496	0.642482	−	0.766301i	$-0.277905\pi$
$941$	3.49032e7	1.28496	0.642482	+	0.766301i	$0.277905\pi$
$942$	−2.05935e7	−0.756140
$943$	−1.21385e7	−0.444516
$944$	2.53184e7	0.924711
$945$	−2.12438e6	−0.0773841
$946$	−1.05645e6	−0.0383815
$947$	4.52987e7	1.64139	0.820693	−	0.571370i	$-0.193588\pi$
$947$	4.52987e7	1.64139	0.820693	+	0.571370i	$0.193588\pi$
$948$	−3.93050e6	−0.142045
$949$	1.94021e7	0.699333
$950$	−1.29569e7	−0.465791
$951$	9.54186e6	0.342123
$952$	0	0
$953$	−2.68954e7	−0.959281	−0.479641	−	0.877465i	$-0.659233\pi$
$953$	−2.68954e7	−0.959281	−0.479641	+	0.877465i	$0.659233\pi$
$954$	912480.	0.0324603
$955$	−7.03564e7	−2.49629
$956$	−447844.	−0.0158483
$957$	−417220.	−0.0147260
$958$	5.84962e6	0.205927
$959$	−1.03421e7	−0.363130
$960$	2.68842e7	0.941496
$961$	−2.64982e7	−0.925568
$962$	3.42293e7	1.19251
$963$	1.18548e7	0.411934
$964$	15494.4	0.000537009 0
$965$	5.10513e7	1.76477
$966$	1.30281e6	0.0449198
$967$	1.29142e7	0.444120	0.222060	−	0.975033i	$-0.428722\pi$
$967$	1.29142e7	0.444120	0.222060	+	0.975033i	$0.428722\pi$
$968$	−3.00591e7	−1.03107
$969$	0	0
$970$	−2.15694e7	−0.736052
$971$	1.33715e7	0.455127	0.227564	−	0.973763i	$-0.426924\pi$
$971$	1.33715e7	0.455127	0.227564	+	0.973763i	$0.426924\pi$
$972$	497419.	0.0168871
$973$	9.96900e6	0.337574
$974$	−4.15545e6	−0.140353
$975$	2.00364e7	0.675008
$976$	7.85266e6	0.263871
$977$	1.85797e7	0.622733	0.311367	−	0.950290i	$-0.399213\pi$
$977$	1.85797e7	0.622733	0.311367	+	0.950290i	$0.399213\pi$
$978$	1.87703e7	0.627516
$979$	−1.15949e7	−0.386643
$980$	−1.07969e7	−0.359115
$981$	−1.09484e7	−0.363228
$982$	3.84232e7	1.27150
$983$	−3.17048e7	−1.04650	−0.523252	−	0.852178i	$-0.675281\pi$
$983$	−3.17048e7	−1.04650	−0.523252	+	0.852178i	$0.675281\pi$
$984$	−2.54388e7	−0.837545
$985$	3.98125e7	1.30746
$986$	0	0
$987$	−8.15893e6	−0.266588
$988$	3.72857e6	0.121521
$989$	−2.06262e6	−0.0670545
$990$	2.88132e6	0.0934337
$991$	2.75365e7	0.890687	0.445344	−	0.895360i	$-0.353082\pi$
$991$	2.75365e7	0.890687	0.445344	+	0.895360i	$0.353082\pi$
$992$	−4.32425e6	−0.139518
$993$	−4.79204e6	−0.154222
$994$	−1.07613e7	−0.345460
$995$	−8.13788e7	−2.60587
$996$	−5.18053e6	−0.165473
$997$	4.65005e7	1.48156	0.740781	−	0.671747i	$-0.234456\pi$
$997$	4.65005e7	1.48156	0.740781	+	0.671747i	$0.234456\pi$
$998$	1.44924e7	0.460588
$999$	−8.45698e6	−0.268103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.6.a.t.1.10		28
17.5	odd	16		51.6.h.a.25.10	✓	56
17.7	odd	16		51.6.h.a.49.10	yes	56
17.16	even	2		867.6.a.u.1.10		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.6.h.a.25.10	✓	56	17.5	odd	16
51.6.h.a.49.10	yes	56	17.7	odd	16
867.6.a.t.1.10		28	1.1	even	1	trivial
867.6.a.u.1.10		28	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

\(f(q)\)	\(=\)	\(q-4.85553 q^{2} -9.00000 q^{3} -8.42383 q^{4} -82.3928 q^{5} +43.6998 q^{6} -35.3683 q^{7} +196.279 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-4.85553 q^{2} -9.00000 q^{3} -8.42383 q^{4} -82.3928 q^{5} +43.6998 q^{6} -35.3683 q^{7} +196.279 q^{8} +81.0000 q^{9} +400.061 q^{10} +88.9161 q^{11} +75.8144 q^{12} -607.679 q^{13} +171.732 q^{14} +741.535 q^{15} -683.477 q^{16} -393.298 q^{18} +728.381 q^{19} +694.062 q^{20} +318.315 q^{21} -431.735 q^{22} -842.921 q^{23} -1766.51 q^{24} +3663.57 q^{25} +2950.60 q^{26} -729.000 q^{27} +297.937 q^{28} +521.365 q^{29} -3600.55 q^{30} +1459.76 q^{31} -2962.29 q^{32} -800.245 q^{33} +2914.10 q^{35} -682.330 q^{36} +11600.8 q^{37} -3536.68 q^{38} +5469.11 q^{39} -16172.0 q^{40} +14400.6 q^{41} -1545.59 q^{42} +2446.99 q^{43} -749.014 q^{44} -6673.81 q^{45} +4092.83 q^{46} -25631.6 q^{47} +6151.29 q^{48} -15556.1 q^{49} -17788.6 q^{50} +5118.98 q^{52} -2320.07 q^{53} +3539.68 q^{54} -7326.05 q^{55} -6942.07 q^{56} -6555.43 q^{57} -2531.50 q^{58} -37043.5 q^{59} -6246.56 q^{60} -11489.3 q^{61} -7087.93 q^{62} -2864.84 q^{63} +36254.7 q^{64} +50068.3 q^{65} +3885.62 q^{66} -57126.2 q^{67} +7586.29 q^{69} -14149.5 q^{70} -62663.3 q^{71} +15898.6 q^{72} -31928.3 q^{73} -56328.0 q^{74} -32972.1 q^{75} -6135.76 q^{76} -3144.82 q^{77} -26555.4 q^{78} -51843.7 q^{79} +56313.5 q^{80} +6561.00 q^{81} -69922.4 q^{82} -68331.8 q^{83} -2681.43 q^{84} -11881.4 q^{86} -4692.29 q^{87} +17452.4 q^{88} -130403. q^{89} +32404.9 q^{90} +21492.6 q^{91} +7100.62 q^{92} -13137.9 q^{93} +124455. q^{94} -60013.4 q^{95} +26660.6 q^{96} -53915.3 q^{97} +75533.0 q^{98} +7202.21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 252 q^{3} + 384 q^{4} + 244 q^{5} + 392 q^{7} + 2268 q^{9}+O(q^{10}) \) 28 * q - 252 * q^3 + 384 * q^4 + 244 * q^5 + 392 * q^7 + 2268 * q^9	\( 28 q - 252 q^{3} + 384 q^{4} + 244 q^{5} + 392 q^{7} + 2268 q^{9} + 800 q^{10} + 1132 q^{11} - 3456 q^{12} - 828 q^{13} + 1568 q^{14} - 2196 q^{15} + 4096 q^{16} - 7532 q^{19} + 9216 q^{20} - 3528 q^{21} + 664 q^{22} + 4548 q^{23} + 9904 q^{25} - 20412 q^{27} + 36936 q^{28} + 36680 q^{29} - 7200 q^{30} + 7688 q^{31} - 10188 q^{33} + 13136 q^{35} + 31104 q^{36} + 42360 q^{37} - 1984 q^{38} + 7452 q^{39} + 81000 q^{40} + 6060 q^{41} - 14112 q^{42} - 50044 q^{43} + 56960 q^{44} + 19764 q^{45} + 51488 q^{46} + 16552 q^{47} - 36864 q^{48} + 38084 q^{49} - 48064 q^{50} - 31360 q^{52} - 4328 q^{53} - 71740 q^{55} - 9248 q^{56} + 67788 q^{57} + 95256 q^{58} - 89832 q^{59} - 82944 q^{60} + 154104 q^{61} + 106624 q^{62} + 31752 q^{63} + 93720 q^{64} + 13260 q^{65} - 5976 q^{66} - 182136 q^{67} - 40932 q^{69} - 145464 q^{70} - 24504 q^{71} + 58184 q^{73} + 308992 q^{74} - 89136 q^{75} - 602544 q^{76} - 240576 q^{77} + 136080 q^{79} + 317440 q^{80} + 183708 q^{81} + 46648 q^{82} - 203576 q^{83} - 332424 q^{84} - 421280 q^{86} - 330120 q^{87} + 145152 q^{88} - 120880 q^{89} + 64800 q^{90} + 290152 q^{91} + 491616 q^{92} - 69192 q^{93} - 799168 q^{94} + 543084 q^{95} + 409328 q^{97} + 193072 q^{98} + 91692 q^{99}+O(q^{100}) \) 28 * q - 252 * q^3 + 384 * q^4 + 244 * q^5 + 392 * q^7 + 2268 * q^9 + 800 * q^10 + 1132 * q^11 - 3456 * q^12 - 828 * q^13 + 1568 * q^14 - 2196 * q^15 + 4096 * q^16 - 7532 * q^19 + 9216 * q^20 - 3528 * q^21 + 664 * q^22 + 4548 * q^23 + 9904 * q^25 - 20412 * q^27 + 36936 * q^28 + 36680 * q^29 - 7200 * q^30 + 7688 * q^31 - 10188 * q^33 + 13136 * q^35 + 31104 * q^36 + 42360 * q^37 - 1984 * q^38 + 7452 * q^39 + 81000 * q^40 + 6060 * q^41 - 14112 * q^42 - 50044 * q^43 + 56960 * q^44 + 19764 * q^45 + 51488 * q^46 + 16552 * q^47 - 36864 * q^48 + 38084 * q^49 - 48064 * q^50 - 31360 * q^52 - 4328 * q^53 - 71740 * q^55 - 9248 * q^56 + 67788 * q^57 + 95256 * q^58 - 89832 * q^59 - 82944 * q^60 + 154104 * q^61 + 106624 * q^62 + 31752 * q^63 + 93720 * q^64 + 13260 * q^65 - 5976 * q^66 - 182136 * q^67 - 40932 * q^69 - 145464 * q^70 - 24504 * q^71 + 58184 * q^73 + 308992 * q^74 - 89136 * q^75 - 602544 * q^76 - 240576 * q^77 + 136080 * q^79 + 317440 * q^80 + 183708 * q^81 + 46648 * q^82 - 203576 * q^83 - 332424 * q^84 - 421280 * q^86 - 330120 * q^87 + 145152 * q^88 - 120880 * q^89 + 64800 * q^90 + 290152 * q^91 + 491616 * q^92 - 69192 * q^93 - 799168 * q^94 + 543084 * q^95 + 409328 * q^97 + 193072 * q^98 + 91692 * q^99

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