LMFDB - Embedded newform 867.6.a.u.1.9

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:	$1$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-4.87620 q^{2} +9.00000 q^{3} -8.22272 q^{4} -88.7638 q^{5} -43.8858 q^{6} +118.568 q^{7} +196.134 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-4.87620 q^{2} +9.00000 q^{3} -8.22272 q^{4} -88.7638 q^{5} -43.8858 q^{6} +118.568 q^{7} +196.134 q^{8} +81.0000 q^{9} +432.830 q^{10} -98.7365 q^{11} -74.0045 q^{12} -783.191 q^{13} -578.162 q^{14} -798.875 q^{15} -693.260 q^{16} -394.972 q^{18} -659.862 q^{19} +729.880 q^{20} +1067.11 q^{21} +481.459 q^{22} +4297.87 q^{23} +1765.20 q^{24} +4754.02 q^{25} +3818.99 q^{26} +729.000 q^{27} -974.953 q^{28} -7838.54 q^{29} +3895.47 q^{30} +5176.75 q^{31} -2895.81 q^{32} -888.629 q^{33} -10524.6 q^{35} -666.040 q^{36} -7735.88 q^{37} +3217.62 q^{38} -7048.72 q^{39} -17409.6 q^{40} +9335.56 q^{41} -5203.46 q^{42} +14679.4 q^{43} +811.883 q^{44} -7189.87 q^{45} -20957.2 q^{46} -4979.13 q^{47} -6239.34 q^{48} -2748.58 q^{49} -23181.5 q^{50} +6439.96 q^{52} +15767.6 q^{53} -3554.75 q^{54} +8764.23 q^{55} +23255.2 q^{56} -5938.76 q^{57} +38222.3 q^{58} -12686.0 q^{59} +6568.92 q^{60} -28695.1 q^{61} -25242.9 q^{62} +9604.03 q^{63} +36304.9 q^{64} +69519.0 q^{65} +4333.13 q^{66} -266.677 q^{67} +38680.8 q^{69} +51319.9 q^{70} +45320.4 q^{71} +15886.8 q^{72} +73064.5 q^{73} +37721.7 q^{74} +42786.2 q^{75} +5425.86 q^{76} -11707.0 q^{77} +34370.9 q^{78} +6893.08 q^{79} +61536.4 q^{80} +6561.00 q^{81} -45522.0 q^{82} +60939.8 q^{83} -8774.58 q^{84} -71579.6 q^{86} -70546.9 q^{87} -19365.6 q^{88} +82549.9 q^{89} +35059.2 q^{90} -92861.5 q^{91} -35340.1 q^{92} +46590.8 q^{93} +24279.2 q^{94} +58571.9 q^{95} -26062.3 q^{96} -45571.7 q^{97} +13402.6 q^{98} -7997.66 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.87620	−0.861998	−0.430999	−	0.902352i	$-0.641839\pi$
$2$	−4.87620	−0.861998	−0.430999	+	0.902352i	$0.641839\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−8.22272	−0.256960
$5$	−88.7638	−1.58786	−0.793928	−	0.608012i	$-0.791967\pi$
$5$	−88.7638	−1.58786	−0.793928	+	0.608012i	$0.791967\pi$
$6$	−43.8858	−0.497675
$7$	118.568	0.914583	0.457292	−	0.889317i	$-0.348820\pi$
$7$	118.568	0.914583	0.457292	+	0.889317i	$0.348820\pi$
$8$	196.134	1.08350
$9$	81.0000	0.333333
$10$	432.830	1.36873
$11$	−98.7365	−0.246035	−0.123017	−	0.992405i	$-0.539257\pi$
$11$	−98.7365	−0.246035	−0.123017	+	0.992405i	$0.539257\pi$
$12$	−74.0045	−0.148356
$13$	−783.191	−1.28531	−0.642657	−	0.766154i	$-0.722168\pi$
$13$	−783.191	−1.28531	−0.642657	+	0.766154i	$0.722168\pi$
$14$	−578.162	−0.788369
$15$	−798.875	−0.916749
$16$	−693.260	−0.677012
$17$	0	0
$17$	0	0
$18$	−394.972	−0.287333
$19$	−659.862	−0.419343	−0.209672	−	0.977772i	$-0.567239\pi$
$19$	−659.862	−0.419343	−0.209672	+	0.977772i	$0.567239\pi$
$20$	729.880	0.408015
$21$	1067.11	0.528035
$22$	481.459	0.212081
$23$	4297.87	1.69408	0.847039	−	0.531531i	$-0.178383\pi$
$23$	4297.87	1.69408	0.847039	+	0.531531i	$0.178383\pi$
$24$	1765.20	0.625557
$25$	4754.02	1.52129
$26$	3818.99	1.10794
$27$	729.000	0.192450
$28$	−974.953	−0.235011
$29$	−7838.54	−1.73077	−0.865387	−	0.501105i	$-0.832927\pi$
$29$	−7838.54	−1.73077	−0.865387	+	0.501105i	$0.832927\pi$
$30$	3895.47	0.790235
$31$	5176.75	0.967505	0.483752	−	0.875205i	$-0.339274\pi$
$31$	5176.75	0.967505	0.483752	+	0.875205i	$0.339274\pi$
$32$	−2895.81	−0.499914
$33$	−888.629	−0.142048
$34$	0	0
$35$	−10524.6	−1.45223
$36$	−666.040	−0.0856533
$37$	−7735.88	−0.928978	−0.464489	−	0.885579i	$-0.653762\pi$
$37$	−7735.88	−0.928978	−0.464489	+	0.885579i	$0.653762\pi$
$38$	3217.62	0.361473
$39$	−7048.72	−0.742076
$40$	−17409.6	−1.72044
$41$	9335.56	0.867323	0.433661	−	0.901076i	$-0.357221\pi$
$41$	9335.56	0.867323	0.433661	+	0.901076i	$0.357221\pi$
$42$	−5203.46	−0.455165
$43$	14679.4	1.21070	0.605351	−	0.795959i	$-0.293033\pi$
$43$	14679.4	1.21070	0.605351	+	0.795959i	$0.293033\pi$
$44$	811.883	0.0632210
$45$	−7189.87	−0.529285
$46$	−20957.2	−1.46029
$47$	−4979.13	−0.328783	−0.164391	−	0.986395i	$-0.552566\pi$
$47$	−4979.13	−0.328783	−0.164391	+	0.986395i	$0.552566\pi$
$48$	−6239.34	−0.390873
$49$	−2748.58	−0.163538
$50$	−23181.5	−1.31134
$51$	0	0
$52$	6439.96	0.330274
$53$	15767.6	0.771038	0.385519	−	0.922700i	$-0.374022\pi$
$53$	15767.6	0.771038	0.385519	+	0.922700i	$0.374022\pi$
$54$	−3554.75	−0.165892
$55$	8764.23	0.390667
$56$	23255.2	0.990948
$57$	−5938.76	−0.242108
$58$	38222.3	1.49192
$59$	−12686.0	−0.474456	−0.237228	−	0.971454i	$-0.576239\pi$
$59$	−12686.0	−0.474456	−0.237228	+	0.971454i	$0.576239\pi$
$60$	6568.92	0.235568
$61$	−28695.1	−0.987378	−0.493689	−	0.869638i	$-0.664352\pi$
$61$	−28695.1	−0.987378	−0.493689	+	0.869638i	$0.664352\pi$
$62$	−25242.9	−0.833987
$63$	9604.03	0.304861
$64$	36304.9	1.10794
$65$	69519.0	2.04089
$66$	4333.13	0.122445
$67$	−266.677	−0.00725769	−0.00362885	−	0.999993i	$-0.501155\pi$
$67$	−266.677	−0.00725769	−0.00362885	+	0.999993i	$0.501155\pi$
$68$	0	0
$69$	38680.8	0.978076
$70$	51319.9	1.25182
$71$	45320.4	1.06696	0.533480	−	0.845813i	$-0.320884\pi$
$71$	45320.4	1.06696	0.533480	+	0.845813i	$0.320884\pi$
$72$	15886.8	0.361166
$73$	73064.5	1.60472	0.802360	−	0.596840i	$-0.203577\pi$
$73$	73064.5	1.60472	0.802360	+	0.596840i	$0.203577\pi$
$74$	37721.7	0.800777
$75$	42786.2	0.878315
$76$	5425.86	0.107754
$77$	−11707.0	−0.225019
$78$	34370.9	0.639668
$79$	6893.08	0.124264	0.0621320	−	0.998068i	$-0.480210\pi$
$79$	6893.08	0.124264	0.0621320	+	0.998068i	$0.480210\pi$
$80$	61536.4	1.07500
$81$	6561.00	0.111111
$82$	−45522.0	−0.747630
$83$	60939.8	0.970970	0.485485	−	0.874245i	$-0.338643\pi$
$83$	60939.8	0.970970	0.485485	+	0.874245i	$0.338643\pi$
$84$	−8774.58	−0.135684
$85$	0	0
$86$	−71579.6	−1.04362
$87$	−70546.9	−0.999262
$88$	−19365.6	−0.266578
$89$	82549.9	1.10469	0.552346	−	0.833615i	$-0.313733\pi$
$89$	82549.9	1.10469	0.552346	+	0.833615i	$0.313733\pi$
$90$	35059.2	0.456243
$91$	−92861.5	−1.17553
$92$	−35340.1	−0.435310
$93$	46590.8	0.558589
$94$	24279.2	0.283410
$95$	58571.9	0.665856
$96$	−26062.3	−0.288626
$97$	−45571.7	−0.491775	−0.245887	−	0.969298i	$-0.579079\pi$
$97$	−45571.7	−0.491775	−0.245887	+	0.969298i	$0.579079\pi$
$98$	13402.6	0.140969
$99$	−7997.66	−0.0820115
$100$	−39091.0	−0.390910
$101$	−138878.	−1.35465	−0.677327	−	0.735682i	$-0.736862\pi$
$101$	−138878.	−1.35465	−0.677327	+	0.735682i	$0.736862\pi$
$102$	0	0
$103$	94745.1	0.879961	0.439981	−	0.898007i	$-0.354985\pi$
$103$	94745.1	0.879961	0.439981	+	0.898007i	$0.354985\pi$
$104$	−153610.	−1.39263
$105$	−94721.1	−0.838443
$106$	−76885.9	−0.664633
$107$	168463.	1.42247	0.711237	−	0.702952i	$-0.248135\pi$
$107$	168463.	1.42247	0.711237	+	0.702952i	$0.248135\pi$
$108$	−5994.36	−0.0494520
$109$	−157053.	−1.26614	−0.633069	−	0.774095i	$-0.718205\pi$
$109$	−157053.	−1.26614	−0.633069	+	0.774095i	$0.718205\pi$
$110$	−42736.1	−0.336754
$111$	−69622.9	−0.536346
$112$	−82198.6	−0.619183
$113$	119202.	0.878191	0.439095	−	0.898440i	$-0.355299\pi$
$113$	119202.	0.878191	0.439095	+	0.898440i	$0.355299\pi$
$114$	28958.6	0.208696
$115$	−381495.	−2.68995
$116$	64454.1	0.444739
$117$	−63438.4	−0.428438
$118$	61859.6	0.408980
$119$	0	0
$120$	−156686.	−0.993294
$121$	−151302.	−0.939467
$122$	139923.	0.851118
$123$	84020.1	0.500749
$124$	−42567.0	−0.248610
$125$	−144598.	−0.827727
$126$	−46831.1	−0.262790
$127$	−160052.	−0.880544	−0.440272	−	0.897865i	$-0.645118\pi$
$127$	−160052.	−0.880544	−0.440272	+	0.897865i	$0.645118\pi$
$128$	−84363.6	−0.455124
$129$	132115.	0.698999
$130$	−338988.	−1.75924
$131$	207669.	1.05729	0.528644	−	0.848843i	$-0.322701\pi$
$131$	207669.	1.05729	0.528644	+	0.848843i	$0.322701\pi$
$132$	7306.94	0.0365007
$133$	−78238.7	−0.383524
$134$	1300.37	0.00625611
$135$	−64708.8	−0.305583
$136$	0	0
$137$	225046.	1.02440	0.512200	−	0.858866i	$-0.328831\pi$
$137$	225046.	1.02440	0.512200	+	0.858866i	$0.328831\pi$
$138$	−188615.	−0.843099
$139$	242730.	1.06558	0.532791	−	0.846247i	$-0.321143\pi$
$139$	242730.	1.06558	0.532791	+	0.846247i	$0.321143\pi$
$140$	86540.6	0.373164
$141$	−44812.2	−0.189823
$142$	−220991.	−0.919717
$143$	77329.5	0.316232
$144$	−56154.0	−0.225671
$145$	695779.	2.74822
$146$	−356277.	−1.38327
$147$	−24737.2	−0.0944185
$148$	63610.0	0.238710
$149$	−61901.5	−0.228421	−0.114210	−	0.993457i	$-0.536434\pi$
$149$	−61901.5	−0.228421	−0.114210	+	0.993457i	$0.536434\pi$
$150$	−208634.	−0.757105
$151$	−174025.	−0.621112	−0.310556	−	0.950555i	$-0.600515\pi$
$151$	−174025.	−0.621112	−0.310556	+	0.950555i	$0.600515\pi$
$152$	−129421.	−0.454357
$153$	0	0
$154$	57085.7	0.193966
$155$	−459508.	−1.53626
$156$	57959.6	0.190684
$157$	−205459.	−0.665235	−0.332618	−	0.943062i	$-0.607932\pi$
$157$	−205459.	−0.665235	−0.332618	+	0.943062i	$0.607932\pi$
$158$	−33612.0	−0.107115
$159$	141908.	0.445159
$160$	257043.	0.793792
$161$	509590.	1.54937
$162$	−31992.7	−0.0957775
$163$	−446008.	−1.31484	−0.657421	−	0.753523i	$-0.728353\pi$
$163$	−446008.	−1.31484	−0.657421	+	0.753523i	$0.728353\pi$
$164$	−76763.7	−0.222867
$165$	78878.1	0.225552
$166$	−297154.	−0.836974
$167$	−631529.	−1.75227	−0.876137	−	0.482063i	$-0.839888\pi$
$167$	−631529.	−1.75227	−0.876137	+	0.482063i	$0.839888\pi$
$168$	209297.	0.572124
$169$	242095.	0.652031
$170$	0	0
$171$	−53448.9	−0.139781
$172$	−120705.	−0.311102
$173$	−364605.	−0.926205	−0.463102	−	0.886305i	$-0.653264\pi$
$173$	−364605.	−0.926205	−0.463102	+	0.886305i	$0.653264\pi$
$174$	344000.	0.861362
$175$	563675.	1.39134
$176$	68450.1	0.166568
$177$	−114174.	−0.273927
$178$	−402529.	−0.952243
$179$	287201.	0.669967	0.334984	−	0.942224i	$-0.391269\pi$
$179$	287201.	0.669967	0.334984	+	0.942224i	$0.391269\pi$
$180$	59120.3	0.136005
$181$	292756.	0.664216	0.332108	−	0.943241i	$-0.392240\pi$
$181$	292756.	0.664216	0.332108	+	0.943241i	$0.392240\pi$
$182$	452811.	1.01330
$183$	−258256.	−0.570063
$184$	842957.	1.83553
$185$	686666.	1.47508
$186$	−227186.	−0.481503
$187$	0	0
$188$	40942.0	0.0844840
$189$	86436.2	0.176012
$190$	−285608.	−0.573967
$191$	22140.3	0.0439136	0.0219568	−	0.999759i	$-0.493010\pi$
$191$	22140.3	0.0439136	0.0219568	+	0.999759i	$0.493010\pi$
$192$	326744.	0.639667
$193$	−633448.	−1.22410	−0.612052	−	0.790818i	$-0.709656\pi$
$193$	−633448.	−1.22410	−0.612052	+	0.790818i	$0.709656\pi$
$194$	222217.	0.423909
$195$	625671.	1.17831
$196$	22600.8	0.0420226
$197$	−418890.	−0.769014	−0.384507	−	0.923122i	$-0.625629\pi$
$197$	−418890.	−0.769014	−0.384507	+	0.923122i	$0.625629\pi$
$198$	38998.1	0.0706937
$199$	12547.2	0.0224601	0.0112301	−	0.999937i	$-0.496425\pi$
$199$	12547.2	0.0224601	0.0112301	+	0.999937i	$0.496425\pi$
$200$	932424.	1.64831
$201$	−2400.09	−0.00419023
$202$	677194.	1.16771
$203$	−929402.	−1.58294
$204$	0	0
$205$	−828660.	−1.37718
$206$	−461995.	−0.758525
$207$	348127.	0.564692
$208$	542955.	0.870172
$209$	65152.5	0.103173
$210$	461879.	0.722736
$211$	−991842.	−1.53369	−0.766843	−	0.641835i	$-0.778173\pi$
$211$	−991842.	−1.53369	−0.766843	+	0.641835i	$0.778173\pi$
$212$	−129652.	−0.198126
$213$	407884.	0.616010
$214$	−821457.	−1.22617
$215$	−1.30300e6	−1.92242
$216$	142982.	0.208519
$217$	613798.	0.884864
$218$	765823.	1.09141
$219$	657581.	0.926486
$220$	−72065.8	−0.100386
$221$	0	0
$222$	339495.	0.462329
$223$	693678.	0.934105	0.467053	−	0.884230i	$-0.345316\pi$
$223$	693678.	0.934105	0.467053	+	0.884230i	$0.345316\pi$
$224$	−343351.	−0.457213
$225$	385075.	0.507095
$226$	−581254.	−0.756998
$227$	982273.	1.26522	0.632612	−	0.774468i	$-0.281983\pi$
$227$	982273.	1.26522	0.632612	+	0.774468i	$0.281983\pi$
$228$	48832.8	0.0622120
$229$	−477926.	−0.602244	−0.301122	−	0.953586i	$-0.597361\pi$
$229$	−477926.	−0.602244	−0.301122	+	0.953586i	$0.597361\pi$
$230$	1.86024e6	2.31873
$231$	−105363.	−0.129915
$232$	−1.53740e6	−1.87529
$233$	−937221.	−1.13097	−0.565486	−	0.824758i	$-0.691311\pi$
$233$	−937221.	−1.13097	−0.565486	+	0.824758i	$0.691311\pi$
$234$	309338.	0.369312
$235$	441967.	0.522059
$236$	104314.	0.121916
$237$	62037.7	0.0717439
$238$	0	0
$239$	−707183.	−0.800824	−0.400412	−	0.916335i	$-0.631133\pi$
$239$	−707183.	−0.800824	−0.400412	+	0.916335i	$0.631133\pi$
$240$	553828.	0.620650
$241$	−1.74744e6	−1.93802	−0.969011	−	0.247018i	$-0.920549\pi$
$241$	−1.74744e6	−1.93802	−0.969011	+	0.247018i	$0.920549\pi$
$242$	737779.	0.809818
$243$	59049.0	0.0641500
$244$	235952.	0.253717
$245$	243974.	0.259674
$246$	−409698.	−0.431645
$247$	516798.	0.538987
$248$	1.01534e6	1.04829
$249$	548458.	0.560590
$250$	705088.	0.713499
$251$	−1.43455e6	−1.43725	−0.718626	−	0.695397i	$-0.755229\pi$
$251$	−1.43455e6	−1.43725	−0.718626	+	0.695397i	$0.755229\pi$
$252$	−78971.2	−0.0783371
$253$	−424356.	−0.416802
$254$	780443.	0.759027
$255$	0	0
$256$	−750382.	−0.715620
$257$	−18459.5	−0.0174336	−0.00871682	−	0.999962i	$-0.502775\pi$
$257$	−18459.5	−0.0174336	−0.00871682	+	0.999962i	$0.502775\pi$
$258$	−644217.	−0.602536
$259$	−917230.	−0.849628
$260$	−571635.	−0.524428
$261$	−634922.	−0.576924
$262$	−1.01263e6	−0.911380
$263$	920085.	0.820235	0.410118	−	0.912033i	$-0.365488\pi$
$263$	920085.	0.820235	0.410118	+	0.912033i	$0.365488\pi$
$264$	−174290.	−0.153909
$265$	−1.39959e6	−1.22430
$266$	381507.	0.330597
$267$	742949.	0.637795
$268$	2192.81	0.00186494
$269$	−2.02094e6	−1.70283	−0.851417	−	0.524489i	$-0.824256\pi$
$269$	−2.02094e6	−1.70283	−0.851417	+	0.524489i	$0.824256\pi$
$270$	315533.	0.263412
$271$	38431.2	0.0317878	0.0158939	−	0.999874i	$-0.494941\pi$
$271$	38431.2	0.0317878	0.0158939	+	0.999874i	$0.494941\pi$
$272$	0	0
$273$	−835754.	−0.678690
$274$	−1.09737e6	−0.883030
$275$	−469395.	−0.374289
$276$	−318061.	−0.251326
$277$	405402.	0.317458	0.158729	−	0.987322i	$-0.449260\pi$
$277$	405402.	0.317458	0.158729	+	0.987322i	$0.449260\pi$
$278$	−1.18360e6	−0.918529
$279$	419317.	0.322502
$280$	−2.06422e6	−1.57348
$281$	158557.	0.119790	0.0598949	−	0.998205i	$-0.480923\pi$
$281$	158557.	0.119790	0.0598949	+	0.998205i	$0.480923\pi$
$282$	218513.	0.163627
$283$	−946197.	−0.702288	−0.351144	−	0.936321i	$-0.614207\pi$
$283$	−946197.	−0.702288	−0.351144	+	0.936321i	$0.614207\pi$
$284$	−372657.	−0.274166
$285$	527147.	0.384432
$286$	−377074.	−0.272591
$287$	1.10690e6	0.793239
$288$	−234561.	−0.166638
$289$	0	0
$290$	−3.39275e6	−2.36896
$291$	−410145.	−0.283926
$292$	−600789.	−0.412349
$293$	721184.	0.490769	0.245385	−	0.969426i	$-0.421086\pi$
$293$	721184.	0.490769	0.245385	+	0.969426i	$0.421086\pi$
$294$	120623.	0.0813886
$295$	1.12606e6	0.753367
$296$	−1.51727e6	−1.00654
$297$	−71978.9	−0.0473494
$298$	301844.	0.196898
$299$	−3.36605e6	−2.17742
$300$	−351819.	−0.225692
$301$	1.74051e6	1.10729
$302$	848582.	0.535398
$303$	−1.24990e6	−0.782110
$304$	457456.	0.283900
$305$	2.54709e6	1.56781
$306$	0	0
$307$	1.60726e6	0.973285	0.486643	−	0.873601i	$-0.338221\pi$
$307$	1.60726e6	0.973285	0.486643	+	0.873601i	$0.338221\pi$
$308$	96263.5	0.0578209
$309$	852706.	0.508046
$310$	2.24065e6	1.32425
$311$	−613205.	−0.359505	−0.179753	−	0.983712i	$-0.557530\pi$
$311$	−613205.	−0.359505	−0.179753	+	0.983712i	$0.557530\pi$
$312$	−1.38249e6	−0.804037
$313$	560660.	0.323473	0.161737	−	0.986834i	$-0.448290\pi$
$313$	560660.	0.323473	0.161737	+	0.986834i	$0.448290\pi$
$314$	1.00186e6	0.573431
$315$	−852490.	−0.484075
$316$	−56679.8	−0.0319309
$317$	1.06297e6	0.594116	0.297058	−	0.954859i	$-0.403995\pi$
$317$	1.06297e6	0.594116	0.297058	+	0.954859i	$0.403995\pi$
$318$	−691973.	−0.383726
$319$	773950.	0.425830
$320$	−3.22256e6	−1.75924
$321$	1.51616e6	0.821266
$322$	−2.48486e6	−1.33556
$323$	0	0
$324$	−53949.3	−0.0285511
$325$	−3.72330e6	−1.95533
$326$	2.17482e6	1.13339
$327$	−1.41348e6	−0.731005
$328$	1.83102e6	0.939741
$329$	−590367.	−0.300699
$330$	−384625.	−0.194425
$331$	56561.1	0.0283758	0.0141879	−	0.999899i	$-0.495484\pi$
$331$	56561.1	0.0283758	0.0141879	+	0.999899i	$0.495484\pi$
$332$	−501091.	−0.249501
$333$	−626606.	−0.309659
$334$	3.07946e6	1.51046
$335$	23671.3	0.0115242
$336$	−739787.	−0.357486
$337$	−3.87001e6	−1.85626	−0.928128	−	0.372262i	$-0.878582\pi$
$337$	−3.87001e6	−1.85626	−0.928128	+	0.372262i	$0.878582\pi$
$338$	−1.18050e6	−0.562050
$339$	1.07282e6	0.507024
$340$	0	0
$341$	−511135.	−0.238040
$342$	260627.	0.120491
$343$	−2.31867e6	−1.06415
$344$	2.87913e6	1.31179
$345$	−3.43346e6	−1.55304
$346$	1.77788e6	0.798386
$347$	−1.27525e6	−0.568555	−0.284277	−	0.958742i	$-0.591754\pi$
$347$	−1.27525e6	−0.568555	−0.284277	+	0.958742i	$0.591754\pi$
$348$	580087.	0.256770
$349$	−3.51542e6	−1.54495	−0.772473	−	0.635048i	$-0.780980\pi$
$349$	−3.51542e6	−1.54495	−0.772473	+	0.635048i	$0.780980\pi$
$350$	−2.74859e6	−1.19933
$351$	−570946.	−0.247359
$352$	285922.	0.122996
$353$	2.31834e6	0.990240	0.495120	−	0.868825i	$-0.335124\pi$
$353$	2.31834e6	0.990240	0.495120	+	0.868825i	$0.335124\pi$
$354$	556736.	0.236125
$355$	−4.02282e6	−1.69418
$356$	−678785.	−0.283862
$357$	0	0
$358$	−1.40045e6	−0.577510
$359$	−2.68615e6	−1.10000	−0.550001	−	0.835164i	$-0.685373\pi$
$359$	−2.68615e6	−1.10000	−0.550001	+	0.835164i	$0.685373\pi$
$360$	−1.41018e6	−0.573479
$361$	−2.04068e6	−0.824151
$362$	−1.42754e6	−0.572553
$363$	−1.36172e6	−0.542402
$364$	763574.	0.302063
$365$	−6.48549e6	−2.54807
$366$	1.25931e6	0.491393
$367$	−2.38232e6	−0.923284	−0.461642	−	0.887066i	$-0.652740\pi$
$367$	−2.38232e6	−0.923284	−0.461642	+	0.887066i	$0.652740\pi$
$368$	−2.97954e6	−1.14691
$369$	756181.	0.289108
$370$	−3.34832e6	−1.27152
$371$	1.86954e6	0.705178
$372$	−383103.	−0.143535
$373$	619300.	0.230478	0.115239	−	0.993338i	$-0.463237\pi$
$373$	619300.	0.230478	0.115239	+	0.993338i	$0.463237\pi$
$374$	0	0
$375$	−1.30138e6	−0.477888
$376$	−976576.	−0.356235
$377$	6.13907e6	2.22459
$378$	−421480.	−0.151722
$379$	−1.32437e6	−0.473600	−0.236800	−	0.971558i	$-0.576099\pi$
$379$	−1.32437e6	−0.473600	−0.236800	+	0.971558i	$0.576099\pi$
$380$	−481620.	−0.171098
$381$	−1.44046e6	−0.508382
$382$	−107960.	−0.0378534
$383$	2.16302e6	0.753466	0.376733	−	0.926322i	$-0.377047\pi$
$383$	2.16302e6	0.753466	0.376733	+	0.926322i	$0.377047\pi$
$384$	−759272.	−0.262766
$385$	1.03916e6	0.357298
$386$	3.08882e6	1.05517
$387$	1.18903e6	0.403567
$388$	374723.	0.126366
$389$	2.10452e6	0.705145	0.352572	−	0.935785i	$-0.385307\pi$
$389$	2.10452e6	0.705145	0.352572	+	0.935785i	$0.385307\pi$
$390$	−3.05089e6	−1.01570
$391$	0	0
$392$	−539089.	−0.177193
$393$	1.86902e6	0.610426
$394$	2.04259e6	0.662888
$395$	−611856.	−0.197313
$396$	65762.5	0.0210737
$397$	3.07058e6	0.977785	0.488893	−	0.872344i	$-0.337401\pi$
$397$	3.07058e6	0.977785	0.488893	+	0.872344i	$0.337401\pi$
$398$	−61182.4	−0.0193606
$399$	−704148.	−0.221428
$400$	−3.29577e6	−1.02993
$401$	−721462.	−0.224054	−0.112027	−	0.993705i	$-0.535734\pi$
$401$	−721462.	−0.224054	−0.112027	+	0.993705i	$0.535734\pi$
$402$	11703.3	0.00361197
$403$	−4.05438e6	−1.24355
$404$	1.14195e6	0.348092
$405$	−582380.	−0.176428
$406$	4.53194e6	1.36449
$407$	763814.	0.228561
$408$	0	0
$409$	−3.62826e6	−1.07248	−0.536241	−	0.844065i	$-0.680156\pi$
$409$	−3.62826e6	−1.07248	−0.536241	+	0.844065i	$0.680156\pi$
$410$	4.04071e6	1.18713
$411$	2.02541e6	0.591437
$412$	−779062.	−0.226115
$413$	−1.50416e6	−0.433929
$414$	−1.69754e6	−0.486764
$415$	−5.40925e6	−1.54176
$416$	2.26797e6	0.642547
$417$	2.18457e6	0.615214
$418$	−317696.	−0.0889348
$419$	−4.44859e6	−1.23791	−0.618953	−	0.785428i	$-0.712443\pi$
$419$	−4.44859e6	−1.23791	−0.618953	+	0.785428i	$0.712443\pi$
$420$	778865.	0.215446
$421$	2.44854e6	0.673291	0.336645	−	0.941632i	$-0.390708\pi$
$421$	2.44854e6	0.673291	0.336645	+	0.941632i	$0.390708\pi$
$422$	4.83641e6	1.32203
$423$	−403310.	−0.109594
$424$	3.09256e6	0.835417
$425$	0	0
$426$	−1.98892e6	−0.530999
$427$	−3.40233e6	−0.903040
$428$	−1.38522e6	−0.365519
$429$	695966.	0.182576
$430$	6.35368e6	1.65712
$431$	−2.03173e6	−0.526832	−0.263416	−	0.964682i	$-0.584849\pi$
$431$	−2.03173e6	−0.526832	−0.263416	+	0.964682i	$0.584849\pi$
$432$	−505386.	−0.130291
$433$	7.21200e6	1.84857	0.924285	−	0.381702i	$-0.124662\pi$
$433$	7.21200e6	1.84857	0.924285	+	0.381702i	$0.124662\pi$
$434$	−2.99300e6	−0.762750
$435$	6.26201e6	1.58668
$436$	1.29141e6	0.325347
$437$	−2.83600e6	−0.710399
$438$	−3.20649e6	−0.798629
$439$	3.65684e6	0.905618	0.452809	−	0.891608i	$-0.350422\pi$
$439$	3.65684e6	0.905618	0.452809	+	0.891608i	$0.350422\pi$
$440$	1.71896e6	0.423287
$441$	−222635.	−0.0545126
$442$	0	0
$443$	4.65092e6	1.12598	0.562989	−	0.826465i	$-0.309651\pi$
$443$	4.65092e6	1.12598	0.562989	+	0.826465i	$0.309651\pi$
$444$	572490.	0.137819
$445$	−7.32744e6	−1.75409
$446$	−3.38251e6	−0.805197
$447$	−557113.	−0.131879
$448$	4.30460e6	1.01330
$449$	4.82682e6	1.12991	0.564957	−	0.825121i	$-0.308893\pi$
$449$	4.82682e6	1.12991	0.564957	+	0.825121i	$0.308893\pi$
$450$	−1.87770e6	−0.437115
$451$	−921761.	−0.213391
$452$	−980168.	−0.225660
$453$	−1.56623e6	−0.358599
$454$	−4.78976e6	−1.09062
$455$	8.24275e6	1.86657
$456$	−1.16479e6	−0.262323
$457$	−5.39718e6	−1.20886	−0.604431	−	0.796658i	$-0.706599\pi$
$457$	−5.39718e6	−1.20886	−0.604431	+	0.796658i	$0.706599\pi$
$458$	2.33046e6	0.519133
$459$	0	0
$460$	3.13693e6	0.691210
$461$	−560624.	−0.122863	−0.0614313	−	0.998111i	$-0.519567\pi$
$461$	−560624.	−0.122863	−0.0614313	+	0.998111i	$0.519567\pi$
$462$	513771.	0.111986
$463$	4.64423e6	1.00684	0.503421	−	0.864041i	$-0.332074\pi$
$463$	4.64423e6	1.00684	0.503421	+	0.864041i	$0.332074\pi$
$464$	5.43415e6	1.17175
$465$	−4.13558e6	−0.886959
$466$	4.57007e6	0.974896
$467$	3.35911e6	0.712742	0.356371	−	0.934345i	$-0.384014\pi$
$467$	3.35911e6	0.712742	0.356371	+	0.934345i	$0.384014\pi$
$468$	521637.	0.110091
$469$	−31619.4	−0.00663776
$470$	−2.15512e6	−0.450014
$471$	−1.84913e6	−0.384074
$472$	−2.48816e6	−0.514071
$473$	−1.44939e6	−0.297875
$474$	−302508.	−0.0618430
$475$	−3.13700e6	−0.637941
$476$	0	0
$477$	1.27717e6	0.257013
$478$	3.44836e6	0.690308
$479$	−4.10363e6	−0.817202	−0.408601	−	0.912713i	$-0.633983\pi$
$479$	−4.10363e6	−0.817202	−0.408601	+	0.912713i	$0.633983\pi$
$480$	2.31339e6	0.458296
$481$	6.05867e6	1.19403
$482$	8.52084e6	1.67057
$483$	4.58631e6	0.894532
$484$	1.24411e6	0.241405
$485$	4.04512e6	0.780867
$486$	−287934.	−0.0552972
$487$	−2.88568e6	−0.551347	−0.275674	−	0.961251i	$-0.588901\pi$
$487$	−2.88568e6	−0.551347	−0.275674	+	0.961251i	$0.588901\pi$
$488$	−5.62809e6	−1.06982
$489$	−4.01407e6	−0.759124
$490$	−1.18967e6	−0.223839
$491$	1.43373e6	0.268387	0.134194	−	0.990955i	$-0.457156\pi$
$491$	1.43373e6	0.268387	0.134194	+	0.990955i	$0.457156\pi$
$492$	−690873.	−0.128672
$493$	0	0
$494$	−2.52001e6	−0.464606
$495$	709903.	0.130222
$496$	−3.58883e6	−0.655012
$497$	5.37356e6	0.975824
$498$	−2.67439e6	−0.483227
$499$	−7.24412e6	−1.30237	−0.651185	−	0.758919i	$-0.725728\pi$
$499$	−7.24412e6	−1.30237	−0.651185	+	0.758919i	$0.725728\pi$
$500$	1.18899e6	0.212693
$501$	−5.68376e6	−1.01168
$502$	6.99517e6	1.23891
$503$	5.94693e6	1.04803	0.524014	−	0.851710i	$-0.324434\pi$
$503$	5.94693e6	1.04803	0.524014	+	0.851710i	$0.324434\pi$
$504$	1.88367e6	0.330316
$505$	1.23273e7	2.15100
$506$	2.06924e6	0.359282
$507$	2.17885e6	0.376450
$508$	1.31606e6	0.226264
$509$	−5.10561e6	−0.873481	−0.436740	−	0.899588i	$-0.643867\pi$
$509$	−5.10561e6	−0.873481	−0.436740	+	0.899588i	$0.643867\pi$
$510$	0	0
$511$	8.66313e6	1.46765
$512$	6.35865e6	1.07199
$513$	−481040.	−0.0807026
$514$	90012.3	0.0150278
$515$	−8.40994e6	−1.39725
$516$	−1.08634e6	−0.179615
$517$	491622.	0.0808919
$518$	4.47259e6	0.732377
$519$	−3.28144e6	−0.534745
$520$	1.36350e7	2.21130
$521$	−2.69951e6	−0.435702	−0.217851	−	0.975982i	$-0.569905\pi$
$521$	−2.69951e6	−0.435702	−0.217851	+	0.975982i	$0.569905\pi$
$522$	3.09600e6	0.497308
$523$	−1.21306e7	−1.93922	−0.969609	−	0.244661i	$-0.921323\pi$
$523$	−1.21306e7	−1.93922	−0.969609	+	0.244661i	$0.921323\pi$
$524$	−1.70760e6	−0.271681
$525$	5.07308e6	0.803292
$526$	−4.48651e6	−0.707041
$527$	0	0
$528$	616051.	0.0961682
$529$	1.20353e7	1.86990
$530$	6.82468e6	1.05534
$531$	−1.02757e6	−0.158152
$532$	643335.	0.0985503
$533$	−7.31153e6	−1.11478
$534$	−3.62276e6	−0.549777
$535$	−1.49534e7	−2.25868
$536$	−52304.4	−0.00786369
$537$	2.58481e6	0.386806
$538$	9.85449e6	1.46784
$539$	271385.	0.0402359
$540$	532083.	0.0785226
$541$	2.79928e6	0.411200	0.205600	−	0.978636i	$-0.434085\pi$
$541$	2.79928e6	0.411200	0.205600	+	0.978636i	$0.434085\pi$
$542$	−187398.	−0.0274010
$543$	2.63480e6	0.383485
$544$	0	0
$545$	1.39407e7	2.01044
$546$	4.07530e6	0.585030
$547$	−4.88545e6	−0.698130	−0.349065	−	0.937099i	$-0.613501\pi$
$547$	−4.88545e6	−0.698130	−0.349065	+	0.937099i	$0.613501\pi$
$548$	−1.85049e6	−0.263230
$549$	−2.32431e6	−0.329126
$550$	2.28886e6	0.322636
$551$	5.17236e6	0.725788
$552$	7.58661e6	1.05974
$553$	817300.	0.113650
$554$	−1.97682e6	−0.273648
$555$	6.18000e6	0.851640
$556$	−1.99590e6	−0.273812
$557$	2.67518e6	0.365355	0.182678	−	0.983173i	$-0.441524\pi$
$557$	2.67518e6	0.365355	0.182678	+	0.983173i	$0.441524\pi$
$558$	−2.04467e6	−0.277996
$559$	−1.14968e7	−1.55613
$560$	7.29626e6	0.983174
$561$	0	0
$562$	−773155.	−0.103258
$563$	−1.35933e7	−1.80740	−0.903699	−	0.428169i	$-0.859159\pi$
$563$	−1.35933e7	−1.80740	−0.903699	+	0.428169i	$0.859159\pi$
$564$	368478.	0.0487769
$565$	−1.05809e7	−1.39444
$566$	4.61384e6	0.605371
$567$	777926.	0.101620
$568$	8.88887e6	1.15605
$569$	1.15910e7	1.50086	0.750429	−	0.660951i	$-0.229847\pi$
$569$	1.15910e7	1.50086	0.750429	+	0.660951i	$0.229847\pi$
$570$	−2.57047e6	−0.331380
$571$	−7.93750e6	−1.01881	−0.509405	−	0.860527i	$-0.670135\pi$
$571$	−7.93750e6	−1.01881	−0.509405	+	0.860527i	$0.670135\pi$
$572$	−635859.	−0.0812589
$573$	199262.	0.0253535
$574$	−5.39746e6	−0.683770
$575$	2.04321e7	2.57718
$576$	2.94069e6	0.369312
$577$	−776797.	−0.0971333	−0.0485666	−	0.998820i	$-0.515465\pi$
$577$	−776797.	−0.0971333	−0.0485666	+	0.998820i	$0.515465\pi$
$578$	0	0
$579$	−5.70104e6	−0.706736
$580$	−5.72119e6	−0.706182
$581$	7.22553e6	0.888033
$582$	1.99995e6	0.244744
$583$	−1.55684e6	−0.189702
$584$	1.43304e7	1.73871
$585$	5.63104e6	0.680298
$586$	−3.51664e6	−0.423042
$587$	7.20601e6	0.863176	0.431588	−	0.902071i	$-0.357953\pi$
$587$	7.20601e6	0.863176	0.431588	+	0.902071i	$0.357953\pi$
$588$	203407.	0.0242618
$589$	−3.41594e6	−0.405716
$590$	−5.49089e6	−0.649401
$591$	−3.77001e6	−0.443991
$592$	5.36298e6	0.628929
$593$	−6.54724e6	−0.764577	−0.382289	−	0.924043i	$-0.624864\pi$
$593$	−6.54724e6	−0.764577	−0.382289	+	0.924043i	$0.624864\pi$
$594$	350983.	0.0408151
$595$	0	0
$596$	508999.	0.0586950
$597$	112924.	0.0129674
$598$	1.64135e7	1.87693
$599$	−1.44144e7	−1.64145	−0.820726	−	0.571322i	$-0.806431\pi$
$599$	−1.44144e7	−1.64145	−0.820726	+	0.571322i	$0.806431\pi$
$600$	8.39181e6	0.951651
$601$	−1.63097e7	−1.84188	−0.920939	−	0.389708i	$-0.872576\pi$
$601$	−1.63097e7	−1.84188	−0.920939	+	0.389708i	$0.872576\pi$
$602$	−8.48707e6	−0.954479
$603$	−21600.8	−0.00241923
$604$	1.43096e6	0.159601
$605$	1.34302e7	1.49174
$606$	6.09475e6	0.674177
$607$	703387.	0.0774859	0.0387430	−	0.999249i	$-0.487665\pi$
$607$	703387.	0.0774859	0.0387430	+	0.999249i	$0.487665\pi$
$608$	1.91084e6	0.209636
$609$	−8.36462e6	−0.913909
$610$	−1.24201e7	−1.35145
$611$	3.89961e6	0.422589
$612$	0	0
$613$	5.70854e6	0.613583	0.306792	−	0.951777i	$-0.400745\pi$
$613$	5.70854e6	0.613583	0.306792	+	0.951777i	$0.400745\pi$
$614$	−7.83731e6	−0.838970
$615$	−7.45794e6	−0.795117
$616$	−2.29614e6	−0.243807
$617$	−4.69756e6	−0.496774	−0.248387	−	0.968661i	$-0.579901\pi$
$617$	−4.69756e6	−0.496774	−0.248387	+	0.968661i	$0.579901\pi$
$618$	−4.15796e6	−0.437934
$619$	−1.67075e7	−1.75261	−0.876306	−	0.481755i	$-0.839999\pi$
$619$	−1.67075e7	−1.75261	−0.876306	+	0.481755i	$0.839999\pi$
$620$	3.77841e6	0.394757
$621$	3.13314e6	0.326025
$622$	2.99011e6	0.309892
$623$	9.78779e6	1.01033
$624$	4.88659e6	0.502394
$625$	−2.02124e6	−0.206975
$626$	−2.73389e6	−0.278833
$627$	586373.	0.0595669
$628$	1.68943e6	0.170939
$629$	0	0
$630$	4.15691e6	0.417272
$631$	5.89792e6	0.589692	0.294846	−	0.955545i	$-0.404732\pi$
$631$	5.89792e6	0.589692	0.294846	+	0.955545i	$0.404732\pi$
$632$	1.35197e6	0.134640
$633$	−8.92657e6	−0.885473
$634$	−5.18323e6	−0.512127
$635$	1.42068e7	1.39818
$636$	−1.16687e6	−0.114388
$637$	2.15266e6	0.210197
$638$	−3.77393e6	−0.367065
$639$	3.67095e6	0.355653
$640$	7.48844e6	0.722672
$641$	2.87699e6	0.276562	0.138281	−	0.990393i	$-0.455842\pi$
$641$	2.87699e6	0.276562	0.138281	+	0.990393i	$0.455842\pi$
$642$	−7.39311e6	−0.707929
$643$	7.73999e6	0.738266	0.369133	−	0.929377i	$-0.379655\pi$
$643$	7.73999e6	0.738266	0.369133	+	0.929377i	$0.379655\pi$
$644$	−4.19022e6	−0.398127
$645$	−1.17270e7	−1.10991
$646$	0	0
$647$	−3.92608e6	−0.368722	−0.184361	−	0.982859i	$-0.559021\pi$
$647$	−3.92608e6	−0.368722	−0.184361	+	0.982859i	$0.559021\pi$
$648$	1.28683e6	0.120389
$649$	1.25257e6	0.116733
$650$	1.81556e7	1.68549
$651$	5.52418e6	0.510876
$652$	3.66740e6	0.337862
$653$	9.91758e6	0.910170	0.455085	−	0.890448i	$-0.349609\pi$
$653$	9.91758e6	0.910170	0.455085	+	0.890448i	$0.349609\pi$
$654$	6.89241e6	0.630125
$655$	−1.84335e7	−1.67882
$656$	−6.47197e6	−0.587188
$657$	5.91823e6	0.534907
$658$	2.87874e6	0.259202
$659$	−625321.	−0.0560905	−0.0280453	−	0.999607i	$-0.508928\pi$
$659$	−625321.	−0.0560905	−0.0280453	+	0.999607i	$0.508928\pi$
$660$	−648592.	−0.0579578
$661$	1.15812e7	1.03098	0.515490	−	0.856896i	$-0.327610\pi$
$661$	1.15812e7	1.03098	0.515490	+	0.856896i	$0.327610\pi$
$662$	−275803.	−0.0244599
$663$	0	0
$664$	1.19524e7	1.05204
$665$	6.94477e6	0.608981
$666$	3.05545e6	0.266926
$667$	−3.36890e7	−2.93206
$668$	5.19288e6	0.450264
$669$	6.24310e6	0.539306
$670$	−115426.	−0.00993381
$671$	2.83326e6	0.242929
$672$	−3.09016e6	−0.263972
$673$	1.79839e7	1.53054	0.765272	−	0.643708i	$-0.222605\pi$
$673$	1.79839e7	1.53054	0.765272	+	0.643708i	$0.222605\pi$
$674$	1.88709e7	1.60009
$675$	3.46568e6	0.292772
$676$	−1.99068e6	−0.167546
$677$	−4.04700e6	−0.339361	−0.169680	−	0.985499i	$-0.554274\pi$
$677$	−4.04700e6	−0.339361	−0.169680	+	0.985499i	$0.554274\pi$
$678$	−5.23129e6	−0.437053
$679$	−5.40336e6	−0.449769
$680$	0	0
$681$	8.84046e6	0.730478
$682$	2.49239e6	0.205190
$683$	1.83686e7	1.50669	0.753345	−	0.657626i	$-0.228439\pi$
$683$	1.83686e7	1.50669	0.753345	+	0.657626i	$0.228439\pi$
$684$	439495.	0.0359181
$685$	−1.99759e7	−1.62660
$686$	1.13063e7	0.917297
$687$	−4.30134e6	−0.347706
$688$	−1.01766e7	−0.819659
$689$	−1.23490e7	−0.991026
$690$	1.67422e7	1.33872
$691$	1.97816e6	0.157604	0.0788019	−	0.996890i	$-0.474891\pi$
$691$	1.97816e6	0.157604	0.0788019	+	0.996890i	$0.474891\pi$
$692$	2.99804e6	0.237998
$693$	−948268.	−0.0750064
$694$	6.21838e6	0.490093
$695$	−2.15457e7	−1.69199
$696$	−1.38366e7	−1.08270
$697$	0	0
$698$	1.71419e7	1.33174
$699$	−8.43499e6	−0.652967
$700$	−4.63495e6	−0.357519
$701$	1.50720e7	1.15845	0.579224	−	0.815169i	$-0.303356\pi$
$701$	1.50720e7	1.15845	0.579224	+	0.815169i	$0.303356\pi$
$702$	2.78404e6	0.213223
$703$	5.10462e6	0.389560
$704$	−3.58462e6	−0.272591
$705$	3.97770e6	0.301411
$706$	−1.13047e7	−0.853585
$707$	−1.64665e7	−1.23894
$708$	938823.	0.0703883
$709$	−1.01812e7	−0.760645	−0.380323	−	0.924854i	$-0.624187\pi$
$709$	−1.01812e7	−0.760645	−0.380323	+	0.924854i	$0.624187\pi$
$710$	1.96160e7	1.46038
$711$	558339.	0.0414213
$712$	1.61908e7	1.19693
$713$	2.22490e7	1.63903
$714$	0	0
$715$	−6.86407e6	−0.502130
$716$	−2.36157e6	−0.172155
$717$	−6.36465e6	−0.462356
$718$	1.30982e7	0.948200
$719$	2.15368e7	1.55367	0.776834	−	0.629705i	$-0.216824\pi$
$719$	2.15368e7	1.55367	0.776834	+	0.629705i	$0.216824\pi$
$720$	4.98445e6	0.358332
$721$	1.12338e7	0.804798
$722$	9.95076e6	0.710417
$723$	−1.57269e7	−1.11892
$724$	−2.40725e6	−0.170677
$725$	−3.72646e7	−2.63300
$726$	6.64001e6	0.467549
$727$	6.05970e6	0.425221	0.212611	−	0.977137i	$-0.431803\pi$
$727$	6.05970e6	0.425221	0.212611	+	0.977137i	$0.431803\pi$
$728$	−1.82133e7	−1.27368
$729$	531441.	0.0370370
$730$	3.16245e7	2.19643
$731$	0	0
$732$	2.12357e6	0.146483
$733$	1.17620e7	0.808573	0.404287	−	0.914632i	$-0.367520\pi$
$733$	1.17620e7	0.808573	0.404287	+	0.914632i	$0.367520\pi$
$734$	1.16167e7	0.795869
$735$	2.19577e6	0.149923
$736$	−1.24458e7	−0.846893
$737$	26330.8	0.00178564
$738$	−3.68728e6	−0.249210
$739$	−1.92644e7	−1.29761	−0.648806	−	0.760954i	$-0.724731\pi$
$739$	−1.92644e7	−1.29761	−0.648806	+	0.760954i	$0.724731\pi$
$740$	−5.64627e6	−0.379037
$741$	4.65118e6	0.311184
$742$	−9.11622e6	−0.607862
$743$	−2.73199e7	−1.81554	−0.907771	−	0.419466i	$-0.862217\pi$
$743$	−2.73199e7	−1.81554	−0.907771	+	0.419466i	$0.862217\pi$
$744$	9.13803e6	0.605229
$745$	5.49461e6	0.362699
$746$	−3.01983e6	−0.198671
$747$	4.93613e6	0.323657
$748$	0	0
$749$	1.99743e7	1.30097
$750$	6.34579e6	0.411939
$751$	−2.30140e7	−1.48899	−0.744497	−	0.667626i	$-0.767311\pi$
$751$	−2.30140e7	−1.48899	−0.744497	+	0.667626i	$0.767311\pi$
$752$	3.45183e6	0.222590
$753$	−1.29110e7	−0.829798
$754$	−2.99353e7	−1.91759
$755$	1.54472e7	0.986237
$756$	−710741.	−0.0452279
$757$	2.39874e7	1.52140	0.760699	−	0.649105i	$-0.224856\pi$
$757$	2.39874e7	1.52140	0.760699	+	0.649105i	$0.224856\pi$
$758$	6.45790e6	0.408243
$759$	−3.81921e6	−0.240641
$760$	1.14879e7	0.721453
$761$	7.31885e6	0.458122	0.229061	−	0.973412i	$-0.426434\pi$
$761$	7.31885e6	0.458122	0.229061	+	0.973412i	$0.426434\pi$
$762$	7.02399e6	0.438224
$763$	−1.86215e7	−1.15799
$764$	−182053.	−0.0112840
$765$	0	0
$766$	−1.05473e7	−0.649486
$767$	9.93558e6	0.609825
$768$	−6.75344e6	−0.413164
$769$	−6.38666e6	−0.389456	−0.194728	−	0.980857i	$-0.562382\pi$
$769$	−6.38666e6	−0.389456	−0.194728	+	0.980857i	$0.562382\pi$
$770$	−5.06714e6	−0.307990
$771$	−166136.	−0.0100653
$772$	5.20867e6	0.314546
$773$	−1.13879e7	−0.685478	−0.342739	−	0.939431i	$-0.611355\pi$
$773$	−1.13879e7	−0.685478	−0.342739	+	0.939431i	$0.611355\pi$
$774$	−5.79795e6	−0.347874
$775$	2.46104e7	1.47185
$776$	−8.93816e6	−0.532836
$777$	−8.25507e6	−0.490533
$778$	−1.02620e7	−0.607833
$779$	−6.16019e6	−0.363706
$780$	−5.14472e6	−0.302779
$781$	−4.47478e6	−0.262509
$782$	0	0
$783$	−5.71430e6	−0.333087
$784$	1.90548e6	0.110717
$785$	1.82373e7	1.05630
$786$	−9.11371e6	−0.526186
$787$	5.48803e6	0.315849	0.157925	−	0.987451i	$-0.449520\pi$
$787$	5.48803e6	0.315849	0.157925	+	0.987451i	$0.449520\pi$
$788$	3.44441e6	0.197606
$789$	8.28076e6	0.473563
$790$	2.98353e6	0.170084
$791$	1.41336e7	0.803178
$792$	−1.56861e6	−0.0888592
$793$	2.24738e7	1.26909
$794$	−1.49727e7	−0.842849
$795$	−1.25963e7	−0.706848
$796$	−103172.	−0.00577136
$797$	1.38122e7	0.770226	0.385113	−	0.922870i	$-0.374163\pi$
$797$	1.38122e7	0.770226	0.385113	+	0.922870i	$0.374163\pi$
$798$	3.43357e6	0.190870
$799$	0	0
$800$	−1.37667e7	−0.760512
$801$	6.68654e6	0.368231
$802$	3.51799e6	0.193134
$803$	−7.21414e6	−0.394817
$804$	19735.3	0.00107672
$805$	−4.52332e7	−2.46018
$806$	1.97700e7	1.07193
$807$	−1.81885e7	−0.983132
$808$	−2.72386e7	−1.46776
$809$	−1.58448e7	−0.851166	−0.425583	−	0.904919i	$-0.639931\pi$
$809$	−1.58448e7	−0.851166	−0.425583	+	0.904919i	$0.639931\pi$
$810$	2.83980e6	0.152081
$811$	−2.65872e7	−1.41945	−0.709725	−	0.704479i	$-0.751181\pi$
$811$	−2.65872e7	−1.41945	−0.709725	+	0.704479i	$0.751181\pi$
$812$	7.64221e6	0.406751
$813$	345881.	0.0183527
$814$	−3.72451e6	−0.197019
$815$	3.95894e7	2.08778
$816$	0	0
$817$	−9.68639e6	−0.507699
$818$	1.76921e7	0.924477
$819$	−7.52178e6	−0.391842
$820$	6.81384e6	0.353881
$821$	−7.35242e6	−0.380691	−0.190345	−	0.981717i	$-0.560961\pi$
$821$	−7.35242e6	−0.380691	−0.190345	+	0.981717i	$0.560961\pi$
$822$	−9.87630e6	−0.509818
$823$	−1.00732e7	−0.518405	−0.259202	−	0.965823i	$-0.583460\pi$
$823$	−1.00732e7	−0.518405	−0.259202	+	0.965823i	$0.583460\pi$
$824$	1.85827e7	0.953435
$825$	−4.22456e6	−0.216096
$826$	7.33458e6	0.374046
$827$	1.28162e7	0.651620	0.325810	−	0.945435i	$-0.394363\pi$
$827$	1.28162e7	0.651620	0.325810	+	0.945435i	$0.394363\pi$
$828$	−2.86255e6	−0.145103
$829$	5.12320e6	0.258914	0.129457	−	0.991585i	$-0.458677\pi$
$829$	5.12320e6	0.258914	0.129457	+	0.991585i	$0.458677\pi$
$830$	2.63766e7	1.32899
$831$	3.64862e6	0.183285
$832$	−2.84336e7	−1.42405
$833$	0	0
$834$	−1.06524e7	−0.530313
$835$	5.60569e7	2.78236
$836$	−535731.	−0.0265113
$837$	3.77385e6	0.186196
$838$	2.16922e7	1.06707
$839$	1.33833e7	0.656383	0.328191	−	0.944611i	$-0.393561\pi$
$839$	1.33833e7	0.656383	0.328191	+	0.944611i	$0.393561\pi$
$840$	−1.85780e7	−0.908450
$841$	4.09316e7	1.99558
$842$	−1.19396e7	−0.580375
$843$	1.42701e6	0.0691606
$844$	8.15564e6	0.394096
$845$	−2.14893e7	−1.03533
$846$	1.96662e6	0.0944700
$847$	−1.79396e7	−0.859221
$848$	−1.09310e7	−0.522002
$849$	−8.51577e6	−0.405466
$850$	0	0
$851$	−3.32478e7	−1.57376
$852$	−3.35391e6	−0.158290
$853$	1.92346e7	0.905127	0.452564	−	0.891732i	$-0.350510\pi$
$853$	1.92346e7	0.905127	0.452564	+	0.891732i	$0.350510\pi$
$854$	1.65904e7	0.778418
$855$	4.74433e6	0.221952
$856$	3.30412e7	1.54125
$857$	−1.13293e7	−0.526929	−0.263464	−	0.964669i	$-0.584865\pi$
$857$	−1.13293e7	−0.526929	−0.263464	+	0.964669i	$0.584865\pi$
$858$	−3.39366e6	−0.157380
$859$	2.45042e7	1.13307	0.566537	−	0.824036i	$-0.308283\pi$
$859$	2.45042e7	1.13307	0.566537	+	0.824036i	$0.308283\pi$
$860$	1.07142e7	0.493985
$861$	9.96211e6	0.457977
$862$	9.90710e6	0.454128
$863$	1.49919e7	0.685219	0.342610	−	0.939478i	$-0.388689\pi$
$863$	1.49919e7	0.685219	0.342610	+	0.939478i	$0.388689\pi$
$864$	−2.11105e6	−0.0962085
$865$	3.23637e7	1.47068
$866$	−3.51671e7	−1.59346
$867$	0	0
$868$	−5.04709e6	−0.227375
$869$	−680598.	−0.0305732
$870$	−3.05348e7	−1.36772
$871$	208859.	0.00932841
$872$	−3.08035e7	−1.37186
$873$	−3.69131e6	−0.163925
$874$	1.38289e7	0.612363
$875$	−1.71447e7	−0.757025
$876$	−5.40710e6	−0.238070
$877$	−1.74825e7	−0.767547	−0.383773	−	0.923427i	$-0.625376\pi$
$877$	−1.74825e7	−0.767547	−0.383773	+	0.923427i	$0.625376\pi$
$878$	−1.78315e7	−0.780641
$879$	6.49066e6	0.283346
$880$	−6.07589e6	−0.264486
$881$	1.38605e7	0.601644	0.300822	−	0.953680i	$-0.402739\pi$
$881$	1.38605e7	0.601644	0.300822	+	0.953680i	$0.402739\pi$
$882$	1.08561e6	0.0469897
$883$	−2.80065e7	−1.20881	−0.604404	−	0.796678i	$-0.706589\pi$
$883$	−2.80065e7	−1.20881	−0.604404	+	0.796678i	$0.706589\pi$
$884$	0	0
$885$	1.01345e7	0.434957
$886$	−2.26788e7	−0.970590
$887$	−4.02108e7	−1.71607	−0.858033	−	0.513594i	$-0.828314\pi$
$887$	−4.02108e7	−1.71607	−0.858033	+	0.513594i	$0.828314\pi$
$888$	−1.36554e7	−0.581129
$889$	−1.89770e7	−0.805330
$890$	3.57301e7	1.51202
$891$	−647810.	−0.0273372
$892$	−5.70392e6	−0.240028
$893$	3.28554e6	0.137873
$894$	2.71659e6	0.113679
$895$	−2.54931e7	−1.06381
$896$	−1.00028e7	−0.416249
$897$	−3.02944e7	−1.25713
$898$	−2.35365e7	−0.973983
$899$	−4.05782e7	−1.67453
$900$	−3.16637e6	−0.130303
$901$	0	0
$902$	4.49469e6	0.183943
$903$	1.56646e7	0.639293
$904$	2.33796e7	0.951517
$905$	−2.59861e7	−1.05468
$906$	7.63724e6	0.309112
$907$	1.97977e7	0.799093	0.399546	−	0.916713i	$-0.369168\pi$
$907$	1.97977e7	0.799093	0.399546	+	0.916713i	$0.369168\pi$
$908$	−8.07696e6	−0.325112
$909$	−1.12491e7	−0.451552
$910$	−4.01932e7	−1.60898
$911$	−3.82583e7	−1.52732	−0.763660	−	0.645619i	$-0.776599\pi$
$911$	−3.82583e7	−1.52732	−0.763660	+	0.645619i	$0.776599\pi$
$912$	4.11711e6	0.163910
$913$	−6.01699e6	−0.238892
$914$	2.63177e7	1.04204
$915$	2.29238e7	0.905178
$916$	3.92985e6	0.154753
$917$	2.46229e7	0.966978
$918$	0	0
$919$	6.82747e6	0.266668	0.133334	−	0.991071i	$-0.457432\pi$
$919$	6.82747e6	0.266668	0.133334	+	0.991071i	$0.457432\pi$
$920$	−7.48241e7	−2.91455
$921$	1.44653e7	0.561927
$922$	2.73371e6	0.105907
$923$	−3.54945e7	−1.37138
$924$	866371.	0.0333829
$925$	−3.67765e7	−1.41324
$926$	−2.26462e7	−0.867896
$927$	7.67435e6	0.293320
$928$	2.26989e7	0.865238
$929$	−7.56872e6	−0.287729	−0.143864	−	0.989597i	$-0.545953\pi$
$929$	−7.56872e6	−0.287729	−0.143864	+	0.989597i	$0.545953\pi$
$930$	2.01659e7	0.764557
$931$	1.81368e6	0.0685784
$932$	7.70650e6	0.290615
$933$	−5.51885e6	−0.207560
$934$	−1.63797e7	−0.614382
$935$	0	0
$936$	−1.24424e7	−0.464211
$937$	3.09224e7	1.15060	0.575300	−	0.817943i	$-0.304885\pi$
$937$	3.09224e7	1.15060	0.575300	+	0.817943i	$0.304885\pi$
$938$	154182.	0.00572174
$939$	5.04594e6	0.186757
$940$	−3.63417e6	−0.134148
$941$	−972677.	−0.0358092	−0.0179046	−	0.999840i	$-0.505700\pi$
$941$	−972677.	−0.0358092	−0.0179046	+	0.999840i	$0.505700\pi$
$942$	9.01670e6	0.331071
$943$	4.01230e7	1.46931
$944$	8.79472e6	0.321212
$945$	−7.67241e6	−0.279481
$946$	7.06752e6	0.256767
$947$	−1.01868e7	−0.369116	−0.184558	−	0.982822i	$-0.559085\pi$
$947$	−1.01868e7	−0.369116	−0.184558	+	0.982822i	$0.559085\pi$
$948$	−510118.	−0.0184353
$949$	−5.72235e7	−2.06257
$950$	1.52966e7	0.549903
$951$	9.56669e6	0.343013
$952$	0	0
$953$	−1.60340e7	−0.571887	−0.285943	−	0.958247i	$-0.592307\pi$
$953$	−1.60340e7	−0.571887	−0.285943	+	0.958247i	$0.592307\pi$
$954$	−6.22775e6	−0.221544
$955$	−1.96525e6	−0.0697285
$956$	5.81497e6	0.205780
$957$	6.96555e6	0.245853
$958$	2.00101e7	0.704426
$959$	2.66833e7	0.936899
$960$	−2.90030e7	−1.01570
$961$	−1.83039e6	−0.0639344
$962$	−2.95433e7	−1.02925
$963$	1.36455e7	0.474158
$964$	1.43687e7	0.497994
$965$	5.62273e7	1.94370
$966$	−2.23638e7	−0.771084
$967$	3.80310e7	1.30789	0.653945	−	0.756542i	$-0.273113\pi$
$967$	3.80310e7	1.30789	0.653945	+	0.756542i	$0.273113\pi$
$968$	−2.96755e7	−1.01791
$969$	0	0
$970$	−1.97248e7	−0.673106
$971$	−1.96312e7	−0.668187	−0.334094	−	0.942540i	$-0.608430\pi$
$971$	−1.96312e7	−0.668187	−0.334094	+	0.942540i	$0.608430\pi$
$972$	−485543.	−0.0164840
$973$	2.87801e7	0.974563
$974$	1.40711e7	0.475260
$975$	−3.35097e7	−1.12891
$976$	1.98932e7	0.668467
$977$	3.91788e7	1.31315	0.656576	−	0.754260i	$-0.272004\pi$
$977$	3.91788e7	1.31315	0.656576	+	0.754260i	$0.272004\pi$
$978$	1.95734e7	0.654363
$979$	−8.15069e6	−0.271793
$980$	−2.00613e6	−0.0667259
$981$	−1.27213e7	−0.422046
$982$	−6.99112e6	−0.231349
$983$	4.73277e7	1.56218	0.781090	−	0.624418i	$-0.214664\pi$
$983$	4.73277e7	1.56218	0.781090	+	0.624418i	$0.214664\pi$
$984$	1.64792e7	0.542560
$985$	3.71823e7	1.22108
$986$	0	0
$987$	−5.31330e6	−0.173609
$988$	−4.24949e6	−0.138498
$989$	6.30901e7	2.05102
$990$	−3.46162e6	−0.112251
$991$	−2.01479e7	−0.651698	−0.325849	−	0.945422i	$-0.605650\pi$
$991$	−2.01479e7	−0.651698	−0.325849	+	0.945422i	$0.605650\pi$
$992$	−1.49909e7	−0.483669
$993$	509050.	0.0163828
$994$	−2.62025e7	−0.841158
$995$	−1.11373e6	−0.0356635
$996$	−4.50982e6	−0.144049
$997$	5.16814e7	1.64663	0.823315	−	0.567584i	$-0.192122\pi$
$997$	5.16814e7	1.64663	0.823315	+	0.567584i	$0.192122\pi$
$998$	3.53237e7	1.12264
$999$	−5.63946e6	−0.178782

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.6.a.u.1.9		28
17.3	odd	16		51.6.h.a.43.5	yes	56
17.6	odd	16		51.6.h.a.19.5	✓	56
17.16	even	2		867.6.a.t.1.9		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.6.h.a.19.5	✓	56	17.6	odd	16
51.6.h.a.43.5	yes	56	17.3	odd	16
867.6.a.t.1.9		28	17.16	even	2
867.6.a.u.1.9		28	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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	867.a (trivial)

\(f(q)\)	\(=\)	\(q-4.87620 q^{2} +9.00000 q^{3} -8.22272 q^{4} -88.7638 q^{5} -43.8858 q^{6} +118.568 q^{7} +196.134 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-4.87620 q^{2} +9.00000 q^{3} -8.22272 q^{4} -88.7638 q^{5} -43.8858 q^{6} +118.568 q^{7} +196.134 q^{8} +81.0000 q^{9} +432.830 q^{10} -98.7365 q^{11} -74.0045 q^{12} -783.191 q^{13} -578.162 q^{14} -798.875 q^{15} -693.260 q^{16} -394.972 q^{18} -659.862 q^{19} +729.880 q^{20} +1067.11 q^{21} +481.459 q^{22} +4297.87 q^{23} +1765.20 q^{24} +4754.02 q^{25} +3818.99 q^{26} +729.000 q^{27} -974.953 q^{28} -7838.54 q^{29} +3895.47 q^{30} +5176.75 q^{31} -2895.81 q^{32} -888.629 q^{33} -10524.6 q^{35} -666.040 q^{36} -7735.88 q^{37} +3217.62 q^{38} -7048.72 q^{39} -17409.6 q^{40} +9335.56 q^{41} -5203.46 q^{42} +14679.4 q^{43} +811.883 q^{44} -7189.87 q^{45} -20957.2 q^{46} -4979.13 q^{47} -6239.34 q^{48} -2748.58 q^{49} -23181.5 q^{50} +6439.96 q^{52} +15767.6 q^{53} -3554.75 q^{54} +8764.23 q^{55} +23255.2 q^{56} -5938.76 q^{57} +38222.3 q^{58} -12686.0 q^{59} +6568.92 q^{60} -28695.1 q^{61} -25242.9 q^{62} +9604.03 q^{63} +36304.9 q^{64} +69519.0 q^{65} +4333.13 q^{66} -266.677 q^{67} +38680.8 q^{69} +51319.9 q^{70} +45320.4 q^{71} +15886.8 q^{72} +73064.5 q^{73} +37721.7 q^{74} +42786.2 q^{75} +5425.86 q^{76} -11707.0 q^{77} +34370.9 q^{78} +6893.08 q^{79} +61536.4 q^{80} +6561.00 q^{81} -45522.0 q^{82} +60939.8 q^{83} -8774.58 q^{84} -71579.6 q^{86} -70546.9 q^{87} -19365.6 q^{88} +82549.9 q^{89} +35059.2 q^{90} -92861.5 q^{91} -35340.1 q^{92} +46590.8 q^{93} +24279.2 q^{94} +58571.9 q^{95} -26062.3 q^{96} -45571.7 q^{97} +13402.6 q^{98} -7997.66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 252 q^{3} + 384 q^{4} - 244 q^{5} - 392 q^{7} + 2268 q^{9}+O(q^{10}) \) 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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