LMFDB - Embedded newform 147.6.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, 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("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 147 = 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	147.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:	$23.5764215125$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 97x^{2} + 7x + 294 $ x^4 - x^3 - 97x^2 + 7x + 294
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.74818$ of defining polynomial
Character	$\chi$	$=$	147.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.74818 q^{2} +9.00000 q^{3} -24.4475 q^{4} +58.3673 q^{5} +24.7336 q^{6} -155.128 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+2.74818 q^{2} +9.00000 q^{3} -24.4475 q^{4} +58.3673 q^{5} +24.7336 q^{6} -155.128 q^{8} +81.0000 q^{9} +160.404 q^{10} +17.4241 q^{11} -220.027 q^{12} +889.933 q^{13} +525.305 q^{15} +356.000 q^{16} +1026.64 q^{17} +222.603 q^{18} -1739.40 q^{19} -1426.93 q^{20} +47.8846 q^{22} +3936.22 q^{23} -1396.15 q^{24} +281.739 q^{25} +2445.70 q^{26} +729.000 q^{27} +5633.53 q^{29} +1443.63 q^{30} -3096.53 q^{31} +5942.45 q^{32} +156.817 q^{33} +2821.38 q^{34} -1980.25 q^{36} +5026.86 q^{37} -4780.20 q^{38} +8009.40 q^{39} -9054.40 q^{40} +18367.0 q^{41} -1630.91 q^{43} -425.976 q^{44} +4727.75 q^{45} +10817.4 q^{46} -9605.23 q^{47} +3204.00 q^{48} +774.269 q^{50} +9239.72 q^{51} -21756.6 q^{52} -23256.5 q^{53} +2003.42 q^{54} +1017.00 q^{55} -15654.6 q^{57} +15482.0 q^{58} -3603.24 q^{59} -12842.4 q^{60} +22876.6 q^{61} -8509.83 q^{62} +4938.92 q^{64} +51943.0 q^{65} +430.961 q^{66} +47012.8 q^{67} -25098.7 q^{68} +35426.0 q^{69} -1599.63 q^{71} -12565.4 q^{72} +5931.35 q^{73} +13814.7 q^{74} +2535.65 q^{75} +42524.1 q^{76} +22011.3 q^{78} -88468.9 q^{79} +20778.8 q^{80} +6561.00 q^{81} +50475.8 q^{82} -95823.9 q^{83} +59921.9 q^{85} -4482.04 q^{86} +50701.7 q^{87} -2702.96 q^{88} -46507.9 q^{89} +12992.7 q^{90} -96230.8 q^{92} -27868.8 q^{93} -26396.9 q^{94} -101524. q^{95} +53482.0 q^{96} -75981.8 q^{97} +1411.35 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9}+O(q^{10}) $ 4 * q + 3 * q^2 + 36 * q^3 + 69 * q^4 + 27 * q^6 + 123 * q^8 + 324 * q^9	\( 4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9} + 283 q^{10} + 402 q^{11} + 621 q^{12} + 462 q^{13} + 3273 q^{16} + 276 q^{17} + 243 q^{18} + 510 q^{19} + 4719 q^{20} + 1375 q^{22} + 6900 q^{23} + 1107 q^{24} + 2814 q^{25} - 15138 q^{26} + 2916 q^{27} + 540 q^{29} + 2547 q^{30} - 6410 q^{31} + 15519 q^{32} + 3618 q^{33} + 21144 q^{34} + 5589 q^{36} + 15250 q^{37} - 41250 q^{38} + 4158 q^{39} - 8547 q^{40} + 4308 q^{41} + 29198 q^{43} + 70743 q^{44} + 61800 q^{46} - 15060 q^{47} + 29457 q^{48} - 7302 q^{50} + 2484 q^{51} - 47476 q^{52} + 13692 q^{53} + 2187 q^{54} + 73124 q^{55} + 4590 q^{57} + 52309 q^{58} + 34830 q^{59} + 42471 q^{60} - 5364 q^{61} + 16029 q^{62} - 73487 q^{64} + 66864 q^{65} + 12375 q^{66} - 5994 q^{67} - 58272 q^{68} + 62100 q^{69} + 89268 q^{71} + 9963 q^{72} + 59638 q^{73} - 185442 q^{74} + 25326 q^{75} + 21308 q^{76} - 136242 q^{78} - 44062 q^{79} - 33381 q^{80} + 26244 q^{81} + 57596 q^{82} - 208446 q^{83} + 36324 q^{85} - 136968 q^{86} + 4860 q^{87} + 87597 q^{88} - 77520 q^{89} + 22923 q^{90} + 158256 q^{92} - 57690 q^{93} - 73722 q^{94} - 221376 q^{95} + 139671 q^{96} - 188630 q^{97} + 32562 q^{99}+O(q^{100}) \) 4 * q + 3 * q^2 + 36 * q^3 + 69 * q^4 + 27 * q^6 + 123 * q^8 + 324 * q^9 + 283 * q^10 + 402 * q^11 + 621 * q^12 + 462 * q^13 + 3273 * q^16 + 276 * q^17 + 243 * q^18 + 510 * q^19 + 4719 * q^20 + 1375 * q^22 + 6900 * q^23 + 1107 * q^24 + 2814 * q^25 - 15138 * q^26 + 2916 * q^27 + 540 * q^29 + 2547 * q^30 - 6410 * q^31 + 15519 * q^32 + 3618 * q^33 + 21144 * q^34 + 5589 * q^36 + 15250 * q^37 - 41250 * q^38 + 4158 * q^39 - 8547 * q^40 + 4308 * q^41 + 29198 * q^43 + 70743 * q^44 + 61800 * q^46 - 15060 * q^47 + 29457 * q^48 - 7302 * q^50 + 2484 * q^51 - 47476 * q^52 + 13692 * q^53 + 2187 * q^54 + 73124 * q^55 + 4590 * q^57 + 52309 * q^58 + 34830 * q^59 + 42471 * q^60 - 5364 * q^61 + 16029 * q^62 - 73487 * q^64 + 66864 * q^65 + 12375 * q^66 - 5994 * q^67 - 58272 * q^68 + 62100 * q^69 + 89268 * q^71 + 9963 * q^72 + 59638 * q^73 - 185442 * q^74 + 25326 * q^75 + 21308 * q^76 - 136242 * q^78 - 44062 * q^79 - 33381 * q^80 + 26244 * q^81 + 57596 * q^82 - 208446 * q^83 + 36324 * q^85 - 136968 * q^86 + 4860 * q^87 + 87597 * q^88 - 77520 * q^89 + 22923 * q^90 + 158256 * q^92 - 57690 * q^93 - 73722 * q^94 - 221376 * q^95 + 139671 * q^96 - 188630 * q^97 + 32562 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.74818	0.485814	0.242907	−	0.970050i	$-0.421899\pi$
$2$	2.74818	0.485814	0.242907	+	0.970050i	$0.421899\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−24.4475	−0.763984
$5$	58.3673	1.04411	0.522053	−	0.852913i	$-0.325166\pi$
$5$	58.3673	1.04411	0.522053	+	0.852913i	$0.325166\pi$
$6$	24.7336	0.280485
$7$	0	0
$7$	0	0
$8$	−155.128	−0.856969
$9$	81.0000	0.333333
$10$	160.404	0.507242
$11$	17.4241	0.0434179	0.0217089	−	0.999764i	$-0.493089\pi$
$11$	17.4241	0.0434179	0.0217089	+	0.999764i	$0.493089\pi$
$12$	−220.027	−0.441087
$13$	889.933	1.46049	0.730246	−	0.683185i	$-0.239406\pi$
$13$	889.933	1.46049	0.730246	+	0.683185i	$0.239406\pi$
$14$	0	0
$15$	525.305	0.602815
$16$	356.000	0.347656
$17$	1026.64	0.861577	0.430788	−	0.902453i	$-0.358236\pi$
$17$	1026.64	0.861577	0.430788	+	0.902453i	$0.358236\pi$
$18$	222.603	0.161938
$19$	−1739.40	−1.10539	−0.552696	−	0.833383i	$-0.686401\pi$
$19$	−1739.40	−1.10539	−0.552696	+	0.833383i	$0.686401\pi$
$20$	−1426.93	−0.797680
$21$	0	0
$22$	47.8846	0.0210930
$23$	3936.22	1.55153	0.775764	−	0.631023i	$-0.217364\pi$
$23$	3936.22	1.55153	0.775764	+	0.631023i	$0.217364\pi$
$24$	−1396.15	−0.494771
$25$	281.739	0.0901563
$26$	2445.70	0.709528
$27$	729.000	0.192450
$28$	0	0
$29$	5633.53	1.24390	0.621950	−	0.783057i	$-0.286341\pi$
$29$	5633.53	1.24390	0.621950	+	0.783057i	$0.286341\pi$
$30$	1443.63	0.292856
$31$	−3096.53	−0.578724	−0.289362	−	0.957220i	$-0.593443\pi$
$31$	−3096.53	−0.578724	−0.289362	+	0.957220i	$0.593443\pi$
$32$	5942.45	1.02587
$33$	156.817	0.0250673
$34$	2821.38	0.418566
$35$	0	0
$36$	−1980.25	−0.254661
$37$	5026.86	0.603660	0.301830	−	0.953362i	$-0.402402\pi$
$37$	5026.86	0.603660	0.301830	+	0.953362i	$0.402402\pi$
$38$	−4780.20	−0.537015
$39$	8009.40	0.843215
$40$	−9054.40	−0.894766
$41$	18367.0	1.70639	0.853195	−	0.521592i	$-0.174662\pi$
$41$	18367.0	1.70639	0.853195	+	0.521592i	$0.174662\pi$
$42$	0	0
$43$	−1630.91	−0.134511	−0.0672557	−	0.997736i	$-0.521424\pi$
$43$	−1630.91	−0.134511	−0.0672557	+	0.997736i	$0.521424\pi$
$44$	−425.976	−0.0331706
$45$	4727.75	0.348035
$46$	10817.4	0.753755
$47$	−9605.23	−0.634254	−0.317127	−	0.948383i	$-0.602718\pi$
$47$	−9605.23	−0.634254	−0.317127	+	0.948383i	$0.602718\pi$
$48$	3204.00	0.200720
$49$	0	0
$50$	774.269	0.0437992
$51$	9239.72	0.497432
$52$	−21756.6	−1.11579
$53$	−23256.5	−1.13725	−0.568624	−	0.822598i	$-0.692524\pi$
$53$	−23256.5	−1.13725	−0.568624	+	0.822598i	$0.692524\pi$
$54$	2003.42	0.0934950
$55$	1017.00	0.0453328
$56$	0	0
$57$	−15654.6	−0.638198
$58$	15482.0	0.604304
$59$	−3603.24	−0.134761	−0.0673803	−	0.997727i	$-0.521464\pi$
$59$	−3603.24	−0.134761	−0.0673803	+	0.997727i	$0.521464\pi$
$60$	−12842.4	−0.460541
$61$	22876.6	0.787167	0.393584	−	0.919289i	$-0.371235\pi$
$61$	22876.6	0.787167	0.393584	+	0.919289i	$0.371235\pi$
$62$	−8509.83	−0.281152
$63$	0	0
$64$	4938.92	0.150724
$65$	51943.0	1.52491
$66$	430.961	0.0121781
$67$	47012.8	1.27947	0.639733	−	0.768597i	$-0.279045\pi$
$67$	47012.8	1.27947	0.639733	+	0.768597i	$0.279045\pi$
$68$	−25098.7	−0.658231
$69$	35426.0	0.895776
$70$	0	0
$71$	−1599.63	−0.0376595	−0.0188298	−	0.999823i	$-0.505994\pi$
$71$	−1599.63	−0.0376595	−0.0188298	+	0.999823i	$0.505994\pi$
$72$	−12565.4	−0.285656
$73$	5931.35	0.130271	0.0651353	−	0.997876i	$-0.479252\pi$
$73$	5931.35	0.130271	0.0651353	+	0.997876i	$0.479252\pi$
$74$	13814.7	0.293267
$75$	2535.65	0.0520518
$76$	42524.1	0.844502
$77$	0	0
$78$	22011.3	0.409646
$79$	−88468.9	−1.59486	−0.797431	−	0.603411i	$-0.793808\pi$
$79$	−88468.9	−1.59486	−0.797431	+	0.603411i	$0.793808\pi$
$80$	20778.8	0.362990
$81$	6561.00	0.111111
$82$	50475.8	0.828989
$83$	−95823.9	−1.52679	−0.763394	−	0.645933i	$-0.776469\pi$
$83$	−95823.9	−1.52679	−0.763394	+	0.645933i	$0.776469\pi$
$84$	0	0
$85$	59921.9	0.899577
$86$	−4482.04	−0.0653476
$87$	50701.7	0.718166
$88$	−2702.96	−0.0372078
$89$	−46507.9	−0.622374	−0.311187	−	0.950349i	$-0.600727\pi$
$89$	−46507.9	−0.622374	−0.311187	+	0.950349i	$0.600727\pi$
$90$	12992.7	0.169081
$91$	0	0
$92$	−96230.8	−1.18534
$93$	−27868.8	−0.334126
$94$	−26396.9	−0.308130
$95$	−101524.	−1.15415
$96$	53482.0	0.592284
$97$	−75981.8	−0.819937	−0.409968	−	0.912100i	$-0.634460\pi$
$97$	−75981.8	−0.819937	−0.409968	+	0.912100i	$0.634460\pi$
$98$	0	0
$99$	1411.35	0.0144726
$100$	−6887.80	−0.0688780
$101$	−46157.4	−0.450234	−0.225117	−	0.974332i	$-0.572276\pi$
$101$	−46157.4	−0.450234	−0.225117	+	0.974332i	$0.572276\pi$
$102$	25392.4	0.241659
$103$	−81973.6	−0.761344	−0.380672	−	0.924710i	$-0.624307\pi$
$103$	−81973.6	−0.761344	−0.380672	+	0.924710i	$0.624307\pi$
$104$	−138054.	−1.25160
$105$	0	0
$106$	−63913.2	−0.552491
$107$	−2853.67	−0.0240960	−0.0120480	−	0.999927i	$-0.503835\pi$
$107$	−2853.67	−0.0240960	−0.0120480	+	0.999927i	$0.503835\pi$
$108$	−17822.2	−0.147029
$109$	−166278.	−1.34051	−0.670254	−	0.742132i	$-0.733815\pi$
$109$	−166278.	−1.34051	−0.670254	+	0.742132i	$0.733815\pi$
$110$	2794.89	0.0220234
$111$	45241.7	0.348523
$112$	0	0
$113$	260304.	1.91772	0.958858	−	0.283886i	$-0.0916237\pi$
$113$	260304.	1.91772	0.958858	+	0.283886i	$0.0916237\pi$
$114$	−43021.8	−0.310046
$115$	229746.	1.61996
$116$	−137726.	−0.950320
$117$	72084.6	0.486831
$118$	−9902.35	−0.0654687
$119$	0	0
$120$	−81489.6	−0.516593
$121$	−160747.	−0.998115
$122$	62869.1	0.382417
$123$	165303.	0.985185
$124$	75702.5	0.442136
$125$	−165953.	−0.949973
$126$	0	0
$127$	−233743.	−1.28596	−0.642982	−	0.765882i	$-0.722303\pi$
$127$	−233743.	−1.28596	−0.642982	+	0.765882i	$0.722303\pi$
$128$	−176585.	−0.952642
$129$	−14678.2	−0.0776602
$130$	142749.	0.740822
$131$	−157290.	−0.800797	−0.400398	−	0.916341i	$-0.631128\pi$
$131$	−157290.	−0.800797	−0.400398	+	0.916341i	$0.631128\pi$
$132$	−3833.78	−0.0191510
$133$	0	0
$134$	129200.	0.621583
$135$	42549.7	0.200938
$136$	−159260.	−0.738345
$137$	−171548.	−0.780878	−0.390439	−	0.920629i	$-0.627677\pi$
$137$	−171548.	−0.780878	−0.390439	+	0.920629i	$0.627677\pi$
$138$	97357.0	0.435181
$139$	210625.	0.924642	0.462321	−	0.886713i	$-0.347017\pi$
$139$	210625.	0.924642	0.462321	+	0.886713i	$0.347017\pi$
$140$	0	0
$141$	−86447.1	−0.366187
$142$	−4396.08	−0.0182955
$143$	15506.3	0.0634114
$144$	28836.0	0.115885
$145$	328814.	1.29876
$146$	16300.4	0.0632873
$147$	0	0
$148$	−122894.	−0.461187
$149$	239413.	0.883450	0.441725	−	0.897151i	$-0.354367\pi$
$149$	239413.	0.883450	0.441725	+	0.897151i	$0.354367\pi$
$150$	6968.42	0.0252875
$151$	217039.	0.774634	0.387317	−	0.921947i	$-0.373402\pi$
$151$	217039.	0.774634	0.387317	+	0.921947i	$0.373402\pi$
$152$	269830.	0.947287
$153$	83157.5	0.287192
$154$	0	0
$155$	−180736.	−0.604249
$156$	−195810.	−0.644203
$157$	166904.	0.540404	0.270202	−	0.962804i	$-0.412909\pi$
$157$	166904.	0.540404	0.270202	+	0.962804i	$0.412909\pi$
$158$	−243129.	−0.774807
$159$	−209309.	−0.656590
$160$	346844.	1.07111
$161$	0	0
$162$	18030.8	0.0539794
$163$	506172.	1.49221	0.746104	−	0.665830i	$-0.231922\pi$
$163$	506172.	1.49221	0.746104	+	0.665830i	$0.231922\pi$
$164$	−449027.	−1.30366
$165$	9152.97	0.0261729
$166$	−263342.	−0.741736
$167$	−565560.	−1.56923	−0.784616	−	0.619982i	$-0.787140\pi$
$167$	−565560.	−1.56923	−0.784616	+	0.619982i	$0.787140\pi$
$168$	0	0
$169$	420688.	1.13304
$170$	164676.	0.437027
$171$	−140892.	−0.368464
$172$	39871.7	0.102765
$173$	−659435.	−1.67516	−0.837581	−	0.546313i	$-0.816031\pi$
$173$	−659435.	−1.67516	−0.837581	+	0.546313i	$0.816031\pi$
$174$	139338.	0.348895
$175$	0	0
$176$	6202.98	0.0150945
$177$	−32429.1	−0.0778041
$178$	−127812.	−0.302358
$179$	165290.	0.385580	0.192790	−	0.981240i	$-0.438246\pi$
$179$	165290.	0.385580	0.192790	+	0.981240i	$0.438246\pi$
$180$	−115582.	−0.265893
$181$	148492.	0.336904	0.168452	−	0.985710i	$-0.446123\pi$
$181$	148492.	0.336904	0.168452	+	0.985710i	$0.446123\pi$
$182$	0	0
$183$	205889.	0.454471
$184$	−610618.	−1.32961
$185$	293404.	0.630285
$186$	−76588.5	−0.162323
$187$	17888.2	0.0374078
$188$	234824.	0.484560
$189$	0	0
$190$	−279007.	−0.560701
$191$	385911.	0.765426	0.382713	−	0.923867i	$-0.374990\pi$
$191$	385911.	0.765426	0.382713	+	0.923867i	$0.374990\pi$
$192$	44450.3	0.0870204
$193$	496296.	0.959063	0.479531	−	0.877525i	$-0.340807\pi$
$193$	496296.	0.959063	0.479531	+	0.877525i	$0.340807\pi$
$194$	−208812.	−0.398337
$195$	467487.	0.880406
$196$	0	0
$197$	441439.	0.810411	0.405206	−	0.914226i	$-0.367200\pi$
$197$	441439.	0.810411	0.405206	+	0.914226i	$0.367200\pi$
$198$	3878.65	0.00703101
$199$	−75838.8	−0.135756	−0.0678779	−	0.997694i	$-0.521623\pi$
$199$	−75838.8	−0.135756	−0.0678779	+	0.997694i	$0.521623\pi$
$200$	−43705.5	−0.0772612
$201$	423115.	0.738700
$202$	−126849.	−0.218730
$203$	0	0
$204$	−225888.	−0.380030
$205$	1.07203e6	1.78165
$206$	−225278.	−0.369872
$207$	318834.	0.517176
$208$	316816.	0.507749
$209$	−30307.5	−0.0479938
$210$	0	0
$211$	778704.	1.20411	0.602055	−	0.798454i	$-0.294349\pi$
$211$	778704.	1.20411	0.602055	+	0.798454i	$0.294349\pi$
$212$	568564.	0.868840
$213$	−14396.7	−0.0217427
$214$	−7842.41	−0.0117062
$215$	−95191.9	−0.140444
$216$	−113088.	−0.164924
$217$	0	0
$218$	−456963.	−0.651238
$219$	53382.1	0.0752117
$220$	−24863.0	−0.0346336
$221$	913637.	1.25833
$222$	124333.	0.169318
$223$	738085.	0.993903	0.496951	−	0.867778i	$-0.334453\pi$
$223$	738085.	0.993903	0.496951	+	0.867778i	$0.334453\pi$
$224$	0	0
$225$	22820.8	0.0300521
$226$	715362.	0.931654
$227$	−544115.	−0.700852	−0.350426	−	0.936590i	$-0.613963\pi$
$227$	−544115.	−0.700852	−0.350426	+	0.936590i	$0.613963\pi$
$228$	382716.	0.487573
$229$	−233562.	−0.294316	−0.147158	−	0.989113i	$-0.547013\pi$
$229$	−233562.	−0.294316	−0.147158	+	0.989113i	$0.547013\pi$
$230$	631385.	0.787000
$231$	0	0
$232$	−873917.	−1.06598
$233$	−618100.	−0.745880	−0.372940	−	0.927856i	$-0.621650\pi$
$233$	−618100.	−0.745880	−0.372940	+	0.927856i	$0.621650\pi$
$234$	198102.	0.236509
$235$	−560631.	−0.662228
$236$	88090.2	0.102955
$237$	−796220.	−0.920794
$238$	0	0
$239$	937500.	1.06164	0.530819	−	0.847485i	$-0.321884\pi$
$239$	937500.	1.06164	0.530819	+	0.847485i	$0.321884\pi$
$240$	187009.	0.209572
$241$	932036.	1.03369	0.516845	−	0.856079i	$-0.327106\pi$
$241$	932036.	1.03369	0.516845	+	0.856079i	$0.327106\pi$
$242$	−441763.	−0.484899
$243$	59049.0	0.0641500
$244$	−559276.	−0.601383
$245$	0	0
$246$	454282.	0.478617
$247$	−1.54795e6	−1.61442
$248$	480359.	0.495948
$249$	−862416.	−0.881492
$250$	−456070.	−0.461510
$251$	214975.	0.215379	0.107690	−	0.994185i	$-0.465655\pi$
$251$	214975.	0.215379	0.107690	+	0.994185i	$0.465655\pi$
$252$	0	0
$253$	68585.1	0.0673641
$254$	−642367.	−0.624740
$255$	539297.	0.519371
$256$	−643334.	−0.613531
$257$	79187.0	0.0747862	0.0373931	−	0.999301i	$-0.488095\pi$
$257$	79187.0	0.0747862	0.0373931	+	0.999301i	$0.488095\pi$
$258$	−40338.4	−0.0377285
$259$	0	0
$260$	−1.26988e6	−1.16501
$261$	456316.	0.414633
$262$	−432261.	−0.389039
$263$	−432828.	−0.385856	−0.192928	−	0.981213i	$-0.561798\pi$
$263$	−432828.	−0.385856	−0.192928	+	0.981213i	$0.561798\pi$
$264$	−24326.7	−0.0214819
$265$	−1.35742e6	−1.18741
$266$	0	0
$267$	−418571.	−0.359328
$268$	−1.14934e6	−0.977492
$269$	−4691.86	−0.00395334	−0.00197667	−	0.999998i	$-0.500629\pi$
$269$	−4691.86	−0.00395334	−0.00197667	+	0.999998i	$0.500629\pi$
$270$	116934.	0.0976187
$271$	105264.	0.0870678	0.0435339	−	0.999052i	$-0.486138\pi$
$271$	105264.	0.0870678	0.0435339	+	0.999052i	$0.486138\pi$
$272$	365482.	0.299533
$273$	0	0
$274$	−471444.	−0.379362
$275$	4909.04	0.00391440
$276$	−866077.	−0.684359
$277$	−763117.	−0.597574	−0.298787	−	0.954320i	$-0.596582\pi$
$277$	−763117.	−0.597574	−0.298787	+	0.954320i	$0.596582\pi$
$278$	578837.	0.449205
$279$	−250819.	−0.192908
$280$	0	0
$281$	−729540.	−0.551167	−0.275584	−	0.961277i	$-0.588871\pi$
$281$	−729540.	−0.551167	−0.275584	+	0.961277i	$0.588871\pi$
$282$	−237572.	−0.177899
$283$	−1.19043e6	−0.883563	−0.441781	−	0.897123i	$-0.645653\pi$
$283$	−1.19043e6	−0.883563	−0.441781	+	0.897123i	$0.645653\pi$
$284$	39107.0	0.0287713
$285$	−913718.	−0.666346
$286$	42614.1	0.0308062
$287$	0	0
$288$	481338.	0.341955
$289$	−365877.	−0.257686
$290$	903639.	0.630958
$291$	−683836.	−0.473391
$292$	−145007.	−0.0995246
$293$	−1.02503e6	−0.697537	−0.348769	−	0.937209i	$-0.613400\pi$
$293$	−1.02503e6	−0.697537	−0.348769	+	0.937209i	$0.613400\pi$
$294$	0	0
$295$	−210311.	−0.140704
$296$	−779807.	−0.517318
$297$	12702.2	0.00835577
$298$	657950.	0.429193
$299$	3.50297e6	2.26599
$300$	−61990.2	−0.0397667
$301$	0	0
$302$	596464.	0.376328
$303$	−415417.	−0.259943
$304$	−619228.	−0.384297
$305$	1.33525e6	0.821886
$306$	228532.	0.139522
$307$	−709845.	−0.429850	−0.214925	−	0.976631i	$-0.568951\pi$
$307$	−709845.	−0.429850	−0.214925	+	0.976631i	$0.568951\pi$
$308$	0	0
$309$	−737762.	−0.439562
$310$	−496696.	−0.293553
$311$	1.56763e6	0.919058	0.459529	−	0.888163i	$-0.348018\pi$
$311$	1.56763e6	0.919058	0.459529	+	0.888163i	$0.348018\pi$
$312$	−1.24248e6	−0.722609
$313$	372952.	0.215175	0.107588	−	0.994196i	$-0.465687\pi$
$313$	372952.	0.215175	0.107588	+	0.994196i	$0.465687\pi$
$314$	458684.	0.262536
$315$	0	0
$316$	2.16284e6	1.21845
$317$	−3.02816e6	−1.69251	−0.846253	−	0.532781i	$-0.821147\pi$
$317$	−3.02816e6	−1.69251	−0.846253	+	0.532781i	$0.821147\pi$
$318$	−575218.	−0.318981
$319$	98159.1	0.0540075
$320$	288271.	0.157372
$321$	−25683.1	−0.0139118
$322$	0	0
$323$	−1.78573e6	−0.952380
$324$	−160400.	−0.0848871
$325$	250728.	0.131673
$326$	1.39105e6	0.724936
$327$	−1.49651e6	−0.773943
$328$	−2.84923e6	−1.46232
$329$	0	0
$330$	25154.0	0.0127152
$331$	1.06690e6	0.535244	0.267622	−	0.963524i	$-0.413762\pi$
$331$	1.06690e6	0.535244	0.267622	+	0.963524i	$0.413762\pi$
$332$	2.34266e6	1.16644
$333$	407176.	0.201220
$334$	−1.55426e6	−0.762356
$335$	2.74401e6	1.33590
$336$	0	0
$337$	1.55734e6	0.746981	0.373490	−	0.927634i	$-0.378161\pi$
$337$	1.55734e6	0.746981	0.373490	+	0.927634i	$0.378161\pi$
$338$	1.15613e6	0.550445
$339$	2.34273e6	1.10719
$340$	−1.46494e6	−0.687263
$341$	−53954.3	−0.0251270
$342$	−387196.	−0.179005
$343$	0	0
$344$	253000.	0.115272
$345$	2.06772e6	0.935284
$346$	−1.81225e6	−0.813818
$347$	−2.23496e6	−0.996428	−0.498214	−	0.867054i	$-0.666011\pi$
$347$	−2.23496e6	−0.996428	−0.498214	+	0.867054i	$0.666011\pi$
$348$	−1.23953e6	−0.548667
$349$	−1.72982e6	−0.760218	−0.380109	−	0.924942i	$-0.624114\pi$
$349$	−1.72982e6	−0.760218	−0.380109	+	0.924942i	$0.624114\pi$
$350$	0	0
$351$	648761.	0.281072
$352$	103542.	0.0445409
$353$	−2.36574e6	−1.01048	−0.505242	−	0.862978i	$-0.668597\pi$
$353$	−2.36574e6	−1.01048	−0.505242	+	0.862978i	$0.668597\pi$
$354$	−89121.2	−0.0377983
$355$	−93366.3	−0.0393205
$356$	1.13700e6	0.475484
$357$	0	0
$358$	454247.	0.187320
$359$	51028.7	0.0208967	0.0104484	−	0.999945i	$-0.496674\pi$
$359$	51028.7	0.0208967	0.0104484	+	0.999945i	$0.496674\pi$
$360$	−733406.	−0.298255
$361$	549424.	0.221891
$362$	408083.	0.163673
$363$	−1.44673e6	−0.576262
$364$	0	0
$365$	346197.	0.136016
$366$	565822.	0.220789
$367$	3.76823e6	1.46040	0.730200	−	0.683233i	$-0.239427\pi$
$367$	3.76823e6	1.46040	0.730200	+	0.683233i	$0.239427\pi$
$368$	1.40130e6	0.539399
$369$	1.48773e6	0.568797
$370$	806328.	0.306201
$371$	0	0
$372$	681322.	0.255267
$373$	4.63440e6	1.72473	0.862366	−	0.506285i	$-0.168982\pi$
$373$	4.63440e6	1.72473	0.862366	+	0.506285i	$0.168982\pi$
$374$	49160.0	0.0181733
$375$	−1.49358e6	−0.548467
$376$	1.49004e6	0.543536
$377$	5.01346e6	1.81670
$378$	0	0
$379$	−4.17169e6	−1.49181	−0.745905	−	0.666052i	$-0.767983\pi$
$379$	−4.17169e6	−1.49181	−0.745905	+	0.666052i	$0.767983\pi$
$380$	2.48201e6	0.881749
$381$	−2.10368e6	−0.742451
$382$	1.06055e6	0.371855
$383$	−4.17172e6	−1.45318	−0.726588	−	0.687074i	$-0.758895\pi$
$383$	−4.17172e6	−1.45318	−0.726588	+	0.687074i	$0.758895\pi$
$384$	−1.58927e6	−0.550008
$385$	0	0
$386$	1.36391e6	0.465927
$387$	−132104.	−0.0448372
$388$	1.85757e6	0.626419
$389$	2.75394e6	0.922743	0.461371	−	0.887207i	$-0.347358\pi$
$389$	2.75394e6	0.922743	0.461371	+	0.887207i	$0.347358\pi$
$390$	1.28474e6	0.427714
$391$	4.04106e6	1.33676
$392$	0	0
$393$	−1.41561e6	−0.462340
$394$	1.21316e6	0.393709
$395$	−5.16369e6	−1.66520
$396$	−34504.0	−0.0110569
$397$	−2.53204e6	−0.806297	−0.403148	−	0.915135i	$-0.632084\pi$
$397$	−2.53204e6	−0.806297	−0.403148	+	0.915135i	$0.632084\pi$
$398$	−208419.	−0.0659522
$399$	0	0
$400$	100299.	0.0313434
$401$	−2.15747e6	−0.670014	−0.335007	−	0.942216i	$-0.608739\pi$
$401$	−2.15747e6	−0.670014	−0.335007	+	0.942216i	$0.608739\pi$
$402$	1.16280e6	0.358871
$403$	−2.75571e6	−0.845221
$404$	1.12843e6	0.343972
$405$	382948.	0.116012
$406$	0	0
$407$	87588.5	0.0262096
$408$	−1.43334e6	−0.426283
$409$	−4.32804e6	−1.27933	−0.639665	−	0.768653i	$-0.720927\pi$
$409$	−4.32804e6	−1.27933	−0.639665	+	0.768653i	$0.720927\pi$
$410$	2.94614e6	0.865552
$411$	−1.54393e6	−0.450840
$412$	2.00405e6	0.581655
$413$	0	0
$414$	876213.	0.251252
$415$	−5.59298e6	−1.59413
$416$	5.28838e6	1.49827
$417$	1.89563e6	0.533843
$418$	−83290.6	−0.0233161
$419$	−1.51129e6	−0.420544	−0.210272	−	0.977643i	$-0.567435\pi$
$419$	−1.51129e6	−0.420544	−0.210272	+	0.977643i	$0.567435\pi$
$420$	0	0
$421$	1.11586e6	0.306835	0.153418	−	0.988161i	$-0.450972\pi$
$421$	1.11586e6	0.306835	0.153418	+	0.988161i	$0.450972\pi$
$422$	2.14002e6	0.584974
$423$	−778024.	−0.211418
$424$	3.60774e6	0.974586
$425$	289243.	0.0776766
$426$	−39564.8	−0.0105629
$427$	0	0
$428$	69765.2	0.0184090
$429$	139557.	0.0366106
$430$	−261605.	−0.0682298
$431$	6.21345e6	1.61116	0.805581	−	0.592485i	$-0.201853\pi$
$431$	6.21345e6	1.61116	0.805581	+	0.592485i	$0.201853\pi$
$432$	259524.	0.0669065
$433$	3.24118e6	0.830775	0.415388	−	0.909644i	$-0.363646\pi$
$433$	3.24118e6	0.830775	0.415388	+	0.909644i	$0.363646\pi$
$434$	0	0
$435$	2.95932e6	0.749841
$436$	4.06509e6	1.02413
$437$	−6.84667e6	−1.71505
$438$	146704.	0.0365389
$439$	2.30497e6	0.570825	0.285413	−	0.958405i	$-0.407869\pi$
$439$	2.30497e6	0.570825	0.285413	+	0.958405i	$0.407869\pi$
$440$	−157765.	−0.0388488
$441$	0	0
$442$	2.51084e6	0.611313
$443$	1.53266e6	0.371052	0.185526	−	0.982639i	$-0.440601\pi$
$443$	1.53266e6	0.371052	0.185526	+	0.982639i	$0.440601\pi$
$444$	−1.10605e6	−0.266266
$445$	−2.71454e6	−0.649824
$446$	2.02839e6	0.482852
$447$	2.15472e6	0.510060
$448$	0	0
$449$	−3.55718e6	−0.832702	−0.416351	−	0.909204i	$-0.636691\pi$
$449$	−3.55718e6	−0.832702	−0.416351	+	0.909204i	$0.636691\pi$
$450$	62715.8	0.0145997
$451$	320028.	0.0740878
$452$	−6.36378e6	−1.46511
$453$	1.95336e6	0.447235
$454$	−1.49533e6	−0.340484
$455$	0	0
$456$	2.42847e6	0.546916
$457$	2.51883e6	0.564167	0.282083	−	0.959390i	$-0.408974\pi$
$457$	2.51883e6	0.564167	0.282083	+	0.959390i	$0.408974\pi$
$458$	−641872.	−0.142983
$459$	748417.	0.165811
$460$	−5.61673e6	−1.23762
$461$	−6.63271e6	−1.45358	−0.726789	−	0.686861i	$-0.758988\pi$
$461$	−6.63271e6	−1.45358	−0.726789	+	0.686861i	$0.758988\pi$
$462$	0	0
$463$	−4.40432e6	−0.954830	−0.477415	−	0.878678i	$-0.658426\pi$
$463$	−4.40432e6	−0.954830	−0.477415	+	0.878678i	$0.658426\pi$
$464$	2.00554e6	0.432450
$465$	−1.62663e6	−0.348863
$466$	−1.69865e6	−0.362359
$467$	245845.	0.0521637	0.0260819	−	0.999660i	$-0.491697\pi$
$467$	245845.	0.0521637	0.0260819	+	0.999660i	$0.491697\pi$
$468$	−1.76229e6	−0.371931
$469$	0	0
$470$	−1.54072e6	−0.321720
$471$	1.50214e6	0.312002
$472$	558963.	0.115486
$473$	−28417.2	−0.00584020
$474$	−2.18816e6	−0.447335
$475$	−490057.	−0.0996581
$476$	0	0
$477$	−1.88378e6	−0.379083
$478$	2.57642e6	0.515759
$479$	−40351.1	−0.00803557	−0.00401779	−	0.999992i	$-0.501279\pi$
$479$	−40351.1	−0.00803557	−0.00401779	+	0.999992i	$0.501279\pi$
$480$	3.12160e6	0.618407
$481$	4.47357e6	0.881640
$482$	2.56140e6	0.502181
$483$	0	0
$484$	3.92987e6	0.762544
$485$	−4.43485e6	−0.856101
$486$	162277.	0.0311650
$487$	−3.30968e6	−0.632359	−0.316180	−	0.948699i	$-0.602400\pi$
$487$	−3.30968e6	−0.632359	−0.316180	+	0.948699i	$0.602400\pi$
$488$	−3.54880e6	−0.674578
$489$	4.55555e6	0.861526
$490$	0	0
$491$	−1.97959e6	−0.370570	−0.185285	−	0.982685i	$-0.559321\pi$
$491$	−1.97959e6	−0.370570	−0.185285	+	0.982685i	$0.559321\pi$
$492$	−4.04124e6	−0.752666
$493$	5.78358e6	1.07171
$494$	−4.25405e6	−0.784306
$495$	82376.8	0.0151109
$496$	−1.10237e6	−0.201197
$497$	0	0
$498$	−2.37007e6	−0.428241
$499$	−3.16995e6	−0.569904	−0.284952	−	0.958542i	$-0.591978\pi$
$499$	−3.16995e6	−0.569904	−0.284952	+	0.958542i	$0.591978\pi$
$500$	4.05715e6	0.725764
$501$	−5.09004e6	−0.905997
$502$	590791.	0.104634
$503$	−4.01273e6	−0.707164	−0.353582	−	0.935403i	$-0.615037\pi$
$503$	−4.01273e6	−0.707164	−0.353582	+	0.935403i	$0.615037\pi$
$504$	0	0
$505$	−2.69408e6	−0.470092
$506$	188484.	0.0327264
$507$	3.78619e6	0.654158
$508$	5.71442e6	0.982456
$509$	4.11257e6	0.703590	0.351795	−	0.936077i	$-0.385571\pi$
$509$	4.11257e6	0.703590	0.351795	+	0.936077i	$0.385571\pi$
$510$	1.48209e6	0.252318
$511$	0	0
$512$	3.88273e6	0.654580
$513$	−1.26802e6	−0.212733
$514$	217620.	0.0363322
$515$	−4.78457e6	−0.794923
$516$	358845.	0.0593312
$517$	−167363.	−0.0275380
$518$	0	0
$519$	−5.93492e6	−0.967155
$520$	−8.05781e6	−1.30680
$521$	8.55515e6	1.38081	0.690404	−	0.723424i	$-0.257433\pi$
$521$	8.55515e6	1.38081	0.690404	+	0.723424i	$0.257433\pi$
$522$	1.25404e6	0.201435
$523$	−1.79364e6	−0.286735	−0.143368	−	0.989670i	$-0.545793\pi$
$523$	−1.79364e6	−0.286735	−0.143368	+	0.989670i	$0.545793\pi$
$524$	3.84534e6	0.611796
$525$	0	0
$526$	−1.18949e6	−0.187455
$527$	−3.17901e6	−0.498615
$528$	55826.8	0.00871482
$529$	9.05749e6	1.40724
$530$	−3.73044e6	−0.576859
$531$	−291862.	−0.0449202
$532$	0	0
$533$	1.63454e7	2.49217
$534$	−1.15031e6	−0.174567
$535$	−166561.	−0.0251588
$536$	−7.29299e6	−1.09646
$537$	1.48761e6	0.222614
$538$	−12894.1	−0.00192059
$539$	0	0
$540$	−1.04023e6	−0.153514
$541$	−357561.	−0.0525239	−0.0262619	−	0.999655i	$-0.508360\pi$
$541$	−357561.	−0.0525239	−0.0262619	+	0.999655i	$0.508360\pi$
$542$	289285.	0.0422988
$543$	1.33643e6	0.194512
$544$	6.10073e6	0.883862
$545$	−9.70522e6	−1.39963
$546$	0	0
$547$	−3.79404e6	−0.542167	−0.271084	−	0.962556i	$-0.587382\pi$
$547$	−3.79404e6	−0.542167	−0.271084	+	0.962556i	$0.587382\pi$
$548$	4.19391e6	0.596578
$549$	1.85300e6	0.262389
$550$	13490.9	0.00190167
$551$	−9.79897e6	−1.37500
$552$	−5.49556e6	−0.767652
$553$	0	0
$554$	−2.09718e6	−0.290310
$555$	2.64064e6	0.363895
$556$	−5.14926e6	−0.706412
$557$	−4.89599e6	−0.668655	−0.334328	−	0.942457i	$-0.608509\pi$
$557$	−4.89599e6	−0.668655	−0.334328	+	0.942457i	$0.608509\pi$
$558$	−689296.	−0.0937175
$559$	−1.45140e6	−0.196453
$560$	0	0
$561$	160994.	0.0215974
$562$	−2.00491e6	−0.267765
$563$	−4.33167e6	−0.575949	−0.287974	−	0.957638i	$-0.592982\pi$
$563$	−4.33167e6	−0.575949	−0.287974	+	0.957638i	$0.592982\pi$
$564$	2.11342e6	0.279761
$565$	1.51932e7	2.00230
$566$	−3.27151e6	−0.429248
$567$	0	0
$568$	248148.	0.0322730
$569$	−2.19696e6	−0.284473	−0.142236	−	0.989833i	$-0.545429\pi$
$569$	−2.19696e6	−0.284473	−0.142236	+	0.989833i	$0.545429\pi$
$570$	−2.51106e6	−0.323721
$571$	−7.47692e6	−0.959693	−0.479846	−	0.877353i	$-0.659308\pi$
$571$	−7.47692e6	−0.959693	−0.479846	+	0.877353i	$0.659308\pi$
$572$	−379090.	−0.0484454
$573$	3.47320e6	0.441919
$574$	0	0
$575$	1.10899e6	0.139880
$576$	400052.	0.0502413
$577$	−1.36622e6	−0.170837	−0.0854183	−	0.996345i	$-0.527223\pi$
$577$	−1.36622e6	−0.170837	−0.0854183	+	0.996345i	$0.527223\pi$
$578$	−1.00550e6	−0.125187
$579$	4.46666e6	0.553715
$580$	−8.03867e6	−0.992234
$581$	0	0
$582$	−1.87931e6	−0.229980
$583$	−405224.	−0.0493769
$584$	−920118.	−0.111638
$585$	4.20738e6	0.508302
$586$	−2.81697e6	−0.338874
$587$	−9.27217e6	−1.11067	−0.555336	−	0.831626i	$-0.687410\pi$
$587$	−9.27217e6	−1.11067	−0.555336	+	0.831626i	$0.687410\pi$
$588$	0	0
$589$	5.38612e6	0.639717
$590$	−577973.	−0.0683562
$591$	3.97295e6	0.467891
$592$	1.78956e6	0.209866
$593$	9.21045e6	1.07558	0.537792	−	0.843078i	$-0.319259\pi$
$593$	9.21045e6	1.07558	0.537792	+	0.843078i	$0.319259\pi$
$594$	34907.9	0.00405936
$595$	0	0
$596$	−5.85305e6	−0.674942
$597$	−682549.	−0.0783787
$598$	9.62681e6	1.10085
$599$	−1.36916e7	−1.55915	−0.779575	−	0.626309i	$-0.784565\pi$
$599$	−1.36916e7	−1.55915	−0.779575	+	0.626309i	$0.784565\pi$
$600$	−393350.	−0.0446068
$601$	1.61113e6	0.181946	0.0909732	−	0.995853i	$-0.471002\pi$
$601$	1.61113e6	0.181946	0.0909732	+	0.995853i	$0.471002\pi$
$602$	0	0
$603$	3.80803e6	0.426489
$604$	−5.30607e6	−0.591808
$605$	−9.38239e6	−1.04214
$606$	−1.14164e6	−0.126284
$607$	−1.40656e7	−1.54948	−0.774742	−	0.632278i	$-0.782120\pi$
$607$	−1.40656e7	−1.54948	−0.774742	+	0.632278i	$0.782120\pi$
$608$	−1.03363e7	−1.13398
$609$	0	0
$610$	3.66950e6	0.399284
$611$	−8.54802e6	−0.926323
$612$	−2.03299e6	−0.219410
$613$	1.35968e7	1.46146	0.730729	−	0.682667i	$-0.239180\pi$
$613$	1.35968e7	1.46146	0.730729	+	0.682667i	$0.239180\pi$
$614$	−1.95078e6	−0.208828
$615$	9.64828e6	1.02864
$616$	0	0
$617$	−5.74287e6	−0.607318	−0.303659	−	0.952781i	$-0.598208\pi$
$617$	−5.74287e6	−0.607318	−0.303659	+	0.952781i	$0.598208\pi$
$618$	−2.02750e6	−0.213546
$619$	−6.02598e6	−0.632122	−0.316061	−	0.948739i	$-0.602360\pi$
$619$	−6.02598e6	−0.632122	−0.316061	+	0.948739i	$0.602360\pi$
$620$	4.41855e6	0.461637
$621$	2.86950e6	0.298592
$622$	4.30814e6	0.446492
$623$	0	0
$624$	2.85135e6	0.293149
$625$	−1.05667e7	−1.08203
$626$	1.02494e6	0.104535
$627$	−272768.	−0.0277092
$628$	−4.08039e6	−0.412860
$629$	5.16075e6	0.520099
$630$	0	0
$631$	−6.90670e6	−0.690554	−0.345277	−	0.938501i	$-0.612215\pi$
$631$	−6.90670e6	−0.690554	−0.345277	+	0.938501i	$0.612215\pi$
$632$	1.37240e7	1.36675
$633$	7.00834e6	0.695194
$634$	−8.32193e6	−0.822244
$635$	−1.36429e7	−1.34268
$636$	5.11708e6	0.501625
$637$	0	0
$638$	269759.	0.0262376
$639$	−129570.	−0.0125532
$640$	−1.03068e7	−0.994658
$641$	1.64966e7	1.58580	0.792900	−	0.609352i	$-0.208570\pi$
$641$	1.64966e7	1.58580	0.792900	+	0.609352i	$0.208570\pi$
$642$	−70581.7	−0.00675857
$643$	1.70171e7	1.62315	0.811576	−	0.584247i	$-0.198610\pi$
$643$	1.70171e7	1.62315	0.811576	+	0.584247i	$0.198610\pi$
$644$	0	0
$645$	−856727.	−0.0810855
$646$	−4.90752e6	−0.462680
$647$	3.48573e6	0.327366	0.163683	−	0.986513i	$-0.447663\pi$
$647$	3.48573e6	0.327366	0.163683	+	0.986513i	$0.447663\pi$
$648$	−1.01779e6	−0.0952188
$649$	−62783.2	−0.00585102
$650$	689047.	0.0639684
$651$	0	0
$652$	−1.23746e7	−1.14002
$653$	1.54449e7	1.41743	0.708716	−	0.705494i	$-0.249275\pi$
$653$	1.54449e7	1.41743	0.708716	+	0.705494i	$0.249275\pi$
$654$	−4.11267e6	−0.375993
$655$	−9.18058e6	−0.836116
$656$	6.53865e6	0.593238
$657$	480439.	0.0434235
$658$	0	0
$659$	−3.11193e6	−0.279136	−0.139568	−	0.990212i	$-0.544571\pi$
$659$	−3.11193e6	−0.279136	−0.139568	+	0.990212i	$0.544571\pi$
$660$	−223767.	−0.0199957
$661$	8.17220e6	0.727504	0.363752	−	0.931496i	$-0.381496\pi$
$661$	8.17220e6	0.727504	0.363752	+	0.931496i	$0.381496\pi$
$662$	2.93202e6	0.260029
$663$	8.22273e6	0.726495
$664$	1.48650e7	1.30841
$665$	0	0
$666$	1.11899e6	0.0977556
$667$	2.21748e7	1.92995
$668$	1.38265e7	1.19887
$669$	6.64276e6	0.573830
$670$	7.54103e6	0.648998
$671$	398604.	0.0341771
$672$	0	0
$673$	1.60182e7	1.36325	0.681627	−	0.731700i	$-0.261273\pi$
$673$	1.60182e7	1.36325	0.681627	+	0.731700i	$0.261273\pi$
$674$	4.27986e6	0.362894
$675$	205387.	0.0173506
$676$	−1.02848e7	−0.865621
$677$	1.02437e6	0.0858984	0.0429492	−	0.999077i	$-0.486325\pi$
$677$	1.02437e6	0.0858984	0.0429492	+	0.999077i	$0.486325\pi$
$678$	6.43826e6	0.537891
$679$	0	0
$680$	−9.29556e6	−0.770910
$681$	−4.89704e6	−0.404637
$682$	−148276.	−0.0122070
$683$	−5.39377e6	−0.442426	−0.221213	−	0.975226i	$-0.571002\pi$
$683$	−5.39377e6	−0.442426	−0.221213	+	0.975226i	$0.571002\pi$
$684$	3.44445e6	0.281501
$685$	−1.00128e7	−0.815319
$686$	0	0
$687$	−2.10206e6	−0.169924
$688$	−580605.	−0.0467638
$689$	−2.06968e7	−1.66094
$690$	5.68246e6	0.454375
$691$	−905653.	−0.0721550	−0.0360775	−	0.999349i	$-0.511486\pi$
$691$	−905653.	−0.0721550	−0.0360775	+	0.999349i	$0.511486\pi$
$692$	1.61215e7	1.27980
$693$	0	0
$694$	−6.14207e6	−0.484079
$695$	1.22936e7	0.965424
$696$	−7.86526e6	−0.615446
$697$	1.88562e7	1.47019
$698$	−4.75387e6	−0.369325
$699$	−5.56290e6	−0.430634
$700$	0	0
$701$	1.12573e7	0.865246	0.432623	−	0.901575i	$-0.357588\pi$
$701$	1.12573e7	0.865246	0.432623	+	0.901575i	$0.357588\pi$
$702$	1.78291e6	0.136549
$703$	−8.74374e6	−0.667281
$704$	86056.2	0.00654411
$705$	−5.04568e6	−0.382338
$706$	−6.50147e6	−0.490908
$707$	0	0
$708$	792812.	0.0594411
$709$	4.35509e6	0.325373	0.162687	−	0.986678i	$-0.447984\pi$
$709$	4.35509e6	0.325373	0.162687	+	0.986678i	$0.447984\pi$
$710$	−256587.	−0.0191025
$711$	−7.16598e6	−0.531620
$712$	7.21467e6	0.533355
$713$	−1.21886e7	−0.897907
$714$	0	0
$715$	905060.	0.0662082
$716$	−4.04093e6	−0.294577
$717$	8.43750e6	0.612937
$718$	140236.	0.0101519
$719$	−1.41444e7	−1.02038	−0.510191	−	0.860061i	$-0.670425\pi$
$719$	−1.41444e7	−1.02038	−0.510191	+	0.860061i	$0.670425\pi$
$720$	1.68308e6	0.120997
$721$	0	0
$722$	1.50992e6	0.107798
$723$	8.38833e6	0.596801
$724$	−3.63025e6	−0.257389
$725$	1.58718e6	0.112145
$726$	−3.97587e6	−0.279956
$727$	−6.26406e6	−0.439561	−0.219781	−	0.975549i	$-0.570534\pi$
$727$	−6.26406e6	−0.439561	−0.219781	+	0.975549i	$0.570534\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	951411.	0.0660786
$731$	−1.67435e6	−0.115892
$732$	−5.03348e6	−0.347209
$733$	−2.04177e7	−1.40361	−0.701806	−	0.712368i	$-0.747623\pi$
$733$	−2.04177e7	−1.40361	−0.701806	+	0.712368i	$0.747623\pi$
$734$	1.03558e7	0.709484
$735$	0	0
$736$	2.33908e7	1.59166
$737$	819155.	0.0555517
$738$	4.08854e6	0.276330
$739$	1.48451e7	0.999937	0.499969	−	0.866043i	$-0.333345\pi$
$739$	1.48451e7	0.999937	0.499969	+	0.866043i	$0.333345\pi$
$740$	−7.17300e6	−0.481528
$741$	−1.39316e7	−0.932083
$742$	0	0
$743$	2.36601e7	1.57234	0.786168	−	0.618013i	$-0.212062\pi$
$743$	2.36601e7	1.57234	0.786168	+	0.618013i	$0.212062\pi$
$744$	4.32323e6	0.286336
$745$	1.39739e7	0.922415
$746$	1.27362e7	0.837900
$747$	−7.76174e6	−0.508929
$748$	−437322.	−0.0285790
$749$	0	0
$750$	−4.10463e6	−0.266453
$751$	2.06573e7	1.33652	0.668258	−	0.743929i	$-0.267040\pi$
$751$	2.06573e7	1.33652	0.668258	+	0.743929i	$0.267040\pi$
$752$	−3.41946e6	−0.220503
$753$	1.93478e6	0.124349
$754$	1.37779e7	0.882581
$755$	1.26680e7	0.808799
$756$	0	0
$757$	−1.26697e7	−0.803573	−0.401787	−	0.915733i	$-0.631611\pi$
$757$	−1.26697e7	−0.803573	−0.401787	+	0.915733i	$0.631611\pi$
$758$	−1.14646e7	−0.724743
$759$	617266.	0.0388927
$760$	1.57492e7	0.989067
$761$	1.62752e7	1.01874	0.509372	−	0.860546i	$-0.329878\pi$
$761$	1.62752e7	1.01874	0.509372	+	0.860546i	$0.329878\pi$
$762$	−5.78130e6	−0.360694
$763$	0	0
$764$	−9.43455e6	−0.584774
$765$	4.85367e6	0.299859
$766$	−1.14646e7	−0.705974
$767$	−3.20664e6	−0.196817
$768$	−5.79000e6	−0.354222
$769$	1.60471e7	0.978547	0.489273	−	0.872130i	$-0.337262\pi$
$769$	1.60471e7	0.978547	0.489273	+	0.872130i	$0.337262\pi$
$770$	0	0
$771$	712683.	0.0431778
$772$	−1.21332e7	−0.732709
$773$	2.15677e6	0.129824	0.0649121	−	0.997891i	$-0.479323\pi$
$773$	2.15677e6	0.129824	0.0649121	+	0.997891i	$0.479323\pi$
$774$	−363045.	−0.0217825
$775$	−872413.	−0.0521756
$776$	1.17869e7	0.702661
$777$	0	0
$778$	7.56833e6	0.448282
$779$	−3.19476e7	−1.88623
$780$	−1.14289e7	−0.672616
$781$	−27872.2	−0.00163510
$782$	1.11056e7	0.649418
$783$	4.10684e6	0.239389
$784$	0	0
$785$	9.74175e6	0.564239
$786$	−3.89035e6	−0.224612
$787$	1.40101e7	0.806312	0.403156	−	0.915131i	$-0.367913\pi$
$787$	1.40101e7	0.806312	0.403156	+	0.915131i	$0.367913\pi$
$788$	−1.07921e7	−0.619142
$789$	−3.89545e6	−0.222774
$790$	−1.41908e7	−0.808980
$791$	0	0
$792$	−218940.	−0.0124026
$793$	2.03587e7	1.14965
$794$	−6.95852e6	−0.391711
$795$	−1.22168e7	−0.685550
$796$	1.85407e6	0.103715
$797$	4.73289e6	0.263925	0.131963	−	0.991255i	$-0.457872\pi$
$797$	4.73289e6	0.263925	0.131963	+	0.991255i	$0.457872\pi$
$798$	0	0
$799$	−9.86107e6	−0.546459
$800$	1.67422e6	0.0924883
$801$	−3.76714e6	−0.207458
$802$	−5.92912e6	−0.325502
$803$	103348.	0.00565607
$804$	−1.03441e7	−0.564355
$805$	0	0
$806$	−7.57318e6	−0.410621
$807$	−42226.7	−0.00228246
$808$	7.16031e6	0.385836
$809$	8.10892e6	0.435604	0.217802	−	0.975993i	$-0.430111\pi$
$809$	8.10892e6	0.435604	0.217802	+	0.975993i	$0.430111\pi$
$810$	1.05241e6	0.0563602
$811$	−9.13444e6	−0.487674	−0.243837	−	0.969816i	$-0.578406\pi$
$811$	−9.13444e6	−0.487674	−0.243837	+	0.969816i	$0.578406\pi$
$812$	0	0
$813$	947378.	0.0502686
$814$	240709.	0.0127330
$815$	2.95439e7	1.55802
$816$	3.28934e6	0.172935
$817$	2.83681e6	0.148688
$818$	−1.18942e7	−0.621517
$819$	0	0
$820$	−2.62085e7	−1.36115
$821$	1.70155e6	0.0881023	0.0440511	−	0.999029i	$-0.485974\pi$
$821$	1.70155e6	0.0881023	0.0440511	+	0.999029i	$0.485974\pi$
$822$	−4.24299e6	−0.219025
$823$	−2.49207e7	−1.28251	−0.641256	−	0.767327i	$-0.721586\pi$
$823$	−2.49207e7	−1.28251	−0.641256	+	0.767327i	$0.721586\pi$
$824$	1.27164e7	0.652448
$825$	44181.4	0.00225998
$826$	0	0
$827$	−1.91232e7	−0.972292	−0.486146	−	0.873878i	$-0.661598\pi$
$827$	−1.91232e7	−0.972292	−0.486146	+	0.873878i	$0.661598\pi$
$828$	−7.79469e6	−0.395115
$829$	−2.20267e7	−1.11318	−0.556588	−	0.830789i	$-0.687890\pi$
$829$	−2.20267e7	−1.11318	−0.556588	+	0.830789i	$0.687890\pi$
$830$	−1.53705e7	−0.774450
$831$	−6.86805e6	−0.345009
$832$	4.39531e6	0.220131
$833$	0	0
$834$	5.20953e6	0.259348
$835$	−3.30102e7	−1.63844
$836$	740943.	0.0366665
$837$	−2.25737e6	−0.111375
$838$	−4.15329e6	−0.204306
$839$	4.28039e6	0.209932	0.104966	−	0.994476i	$-0.466527\pi$
$839$	4.28039e6	0.209932	0.104966	+	0.994476i	$0.466527\pi$
$840$	0	0
$841$	1.12255e7	0.547286
$842$	3.06659e6	0.149065
$843$	−6.56586e6	−0.318216
$844$	−1.90374e7	−0.919922
$845$	2.45544e7	1.18301
$846$	−2.13815e6	−0.102710
$847$	0	0
$848$	−8.27933e6	−0.395372
$849$	−1.07139e7	−0.510125
$850$	794892.	0.0377364
$851$	1.97868e7	0.936596
$852$	351963.	0.0166111
$853$	−1.72415e7	−0.811341	−0.405670	−	0.914019i	$-0.632962\pi$
$853$	−1.72415e7	−0.811341	−0.405670	+	0.914019i	$0.632962\pi$
$854$	0	0
$855$	−8.22346e6	−0.384715
$856$	442685.	0.0206495
$857$	2.08340e7	0.968991	0.484495	−	0.874794i	$-0.339003\pi$
$857$	2.08340e7	0.968991	0.484495	+	0.874794i	$0.339003\pi$
$858$	383527.	0.0177860
$859$	−1.58356e7	−0.732238	−0.366119	−	0.930568i	$-0.619314\pi$
$859$	−1.58356e7	−0.732238	−0.366119	+	0.930568i	$0.619314\pi$
$860$	2.32720e6	0.107297
$861$	0	0
$862$	1.70757e7	0.782726
$863$	−1.59282e7	−0.728012	−0.364006	−	0.931397i	$-0.618591\pi$
$863$	−1.59282e7	−0.728012	−0.364006	+	0.931397i	$0.618591\pi$
$864$	4.33204e6	0.197428
$865$	−3.84894e7	−1.74905
$866$	8.90735e6	0.403603
$867$	−3.29289e6	−0.148775
$868$	0	0
$869$	−1.54149e6	−0.0692455
$870$	8.13275e6	0.364283
$871$	4.18382e7	1.86865
$872$	2.57944e7	1.14877
$873$	−6.15453e6	−0.273312
$874$	−1.88159e7	−0.833195
$875$	0	0
$876$	−1.30506e6	−0.0574606
$877$	5.20734e6	0.228621	0.114311	−	0.993445i	$-0.463534\pi$
$877$	5.20734e6	0.228621	0.114311	+	0.993445i	$0.463534\pi$
$878$	6.33446e6	0.277315
$879$	−9.22527e6	−0.402723
$880$	362051.	0.0157603
$881$	−932829.	−0.0404913	−0.0202457	−	0.999795i	$-0.506445\pi$
$881$	−932829.	−0.0404913	−0.0202457	+	0.999795i	$0.506445\pi$
$882$	0	0
$883$	1.33789e7	0.577457	0.288728	−	0.957411i	$-0.406768\pi$
$883$	1.33789e7	0.577457	0.288728	+	0.957411i	$0.406768\pi$
$884$	−2.23361e7	−0.961341
$885$	−1.89280e6	−0.0812357
$886$	4.21202e6	0.180263
$887$	−6.96900e6	−0.297414	−0.148707	−	0.988881i	$-0.547511\pi$
$887$	−6.96900e6	−0.297414	−0.148707	+	0.988881i	$0.547511\pi$
$888$	−7.01826e6	−0.298674
$889$	0	0
$890$	−7.46004e6	−0.315694
$891$	114320.	0.00482421
$892$	−1.80443e7	−0.759326
$893$	1.67074e7	0.701099
$894$	5.92155e6	0.247794
$895$	9.64753e6	0.402586
$896$	0	0
$897$	3.15268e7	1.30827
$898$	−9.77577e6	−0.404539
$899$	−1.74444e7	−0.719874
$900$	−557912.	−0.0229593
$901$	−2.38760e7	−0.979826
$902$	879496.	0.0359929
$903$	0	0
$904$	−4.03804e7	−1.64342
$905$	8.66706e6	0.351763
$906$	5.36817e6	0.217273
$907$	−3.76563e7	−1.51992	−0.759959	−	0.649972i	$-0.774781\pi$
$907$	−3.76563e7	−1.51992	−0.759959	+	0.649972i	$0.774781\pi$
$908$	1.33023e7	0.535440
$909$	−3.73875e6	−0.150078
$910$	0	0
$911$	−3.09942e7	−1.23733	−0.618663	−	0.785657i	$-0.712325\pi$
$911$	−3.09942e7	−1.23733	−0.618663	+	0.785657i	$0.712325\pi$
$912$	−5.57305e6	−0.221874
$913$	−1.66965e6	−0.0662899
$914$	6.92219e6	0.274080
$915$	1.20172e7	0.474516
$916$	5.71002e6	0.224853
$917$	0	0
$918$	2.05679e6	0.0805531
$919$	716014.	0.0279662	0.0139831	−	0.999902i	$-0.495549\pi$
$919$	716014.	0.0279662	0.0139831	+	0.999902i	$0.495549\pi$
$920$	−3.56401e7	−1.38826
$921$	−6.38860e6	−0.248174
$922$	−1.82279e7	−0.706169
$923$	−1.42357e6	−0.0550014
$924$	0	0
$925$	1.41626e6	0.0544238
$926$	−1.21039e7	−0.463870
$927$	−6.63986e6	−0.253781
$928$	3.34769e7	1.27607
$929$	4.17630e7	1.58764	0.793820	−	0.608153i	$-0.208089\pi$
$929$	4.17630e7	1.58764	0.793820	+	0.608153i	$0.208089\pi$
$930$	−4.47026e6	−0.169483
$931$	0	0
$932$	1.51110e7	0.569840
$933$	1.41087e7	0.530618
$934$	675626.	0.0253419
$935$	1.04409e6	0.0390577
$936$	−1.11823e7	−0.417199
$937$	−2.17917e7	−0.810854	−0.405427	−	0.914127i	$-0.632877\pi$
$937$	−2.17917e7	−0.810854	−0.405427	+	0.914127i	$0.632877\pi$
$938$	0	0
$939$	3.35657e6	0.124232
$940$	1.37060e7	0.505932
$941$	−2.15218e6	−0.0792329	−0.0396164	−	0.999215i	$-0.512614\pi$
$941$	−2.15218e6	−0.0792329	−0.0396164	+	0.999215i	$0.512614\pi$
$942$	4.12815e6	0.151575
$943$	7.22965e7	2.64751
$944$	−1.28275e6	−0.0468504
$945$	0	0
$946$	−78095.5	−0.00283725
$947$	−2.64249e7	−0.957501	−0.478750	−	0.877951i	$-0.658910\pi$
$947$	−2.64249e7	−0.957501	−0.478750	+	0.877951i	$0.658910\pi$
$948$	1.94656e7	0.703472
$949$	5.27850e6	0.190259
$950$	−1.34677e6	−0.0484153
$951$	−2.72534e7	−0.977169
$952$	0	0
$953$	9.07051e6	0.323519	0.161759	−	0.986830i	$-0.448283\pi$
$953$	9.07051e6	0.323519	0.161759	+	0.986830i	$0.448283\pi$
$954$	−5.17697e6	−0.184164
$955$	2.25246e7	0.799186
$956$	−2.29195e7	−0.811075
$957$	883432.	0.0311812
$958$	−110892.	−0.00390380
$959$	0	0
$960$	2.59444e6	0.0908585
$961$	−1.90406e7	−0.665079
$962$	1.22942e7	0.428314
$963$	−231148.	−0.00803200
$964$	−2.27860e7	−0.789723
$965$	2.89674e7	1.00136
$966$	0	0
$967$	1.80108e7	0.619395	0.309698	−	0.950835i	$-0.399772\pi$
$967$	1.80108e7	0.619395	0.309698	+	0.950835i	$0.399772\pi$
$968$	2.49364e7	0.855354
$969$	−1.60716e7	−0.549857
$970$	−1.21878e7	−0.415906
$971$	−4.67631e7	−1.59168	−0.795838	−	0.605509i	$-0.792970\pi$
$971$	−4.67631e7	−1.59168	−0.795838	+	0.605509i	$0.792970\pi$
$972$	−1.44360e6	−0.0490096
$973$	0	0
$974$	−9.09561e6	−0.307209
$975$	2.25656e6	0.0760212
$976$	8.14408e6	0.273664
$977$	4.32433e6	0.144938	0.0724690	−	0.997371i	$-0.476912\pi$
$977$	4.32433e6	0.144938	0.0724690	+	0.997371i	$0.476912\pi$
$978$	1.25195e7	0.418542
$979$	−810357.	−0.0270221
$980$	0	0
$981$	−1.34685e7	−0.446836
$982$	−5.44026e6	−0.180028
$983$	2.08733e7	0.688982	0.344491	−	0.938790i	$-0.388051\pi$
$983$	2.08733e7	0.688982	0.344491	+	0.938790i	$0.388051\pi$
$984$	−2.56431e7	−0.844273
$985$	2.57656e7	0.846155
$986$	1.58943e7	0.520655
$987$	0	0
$988$	3.78436e7	1.23339
$989$	−6.41963e6	−0.208698
$990$	226386.	0.00734112
$991$	4.12209e7	1.33332	0.666658	−	0.745364i	$-0.267724\pi$
$991$	4.12209e7	1.33332	0.666658	+	0.745364i	$0.267724\pi$
$992$	−1.84010e7	−0.593693
$993$	9.60206e6	0.309023
$994$	0	0
$995$	−4.42650e6	−0.141743
$996$	2.10839e7	0.673446
$997$	−5.34615e7	−1.70335	−0.851674	−	0.524073i	$-0.824412\pi$
$997$	−5.34615e7	−1.70335	−0.851674	+	0.524073i	$0.824412\pi$
$998$	−8.71160e6	−0.276867
$999$	3.66458e6	0.116174

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	147.6.a.m.1.3		4
3.2	odd	2		441.6.a.w.1.2		4
7.2	even	3		21.6.e.c.4.2	✓	8
7.3	odd	6		147.6.e.o.79.2		8
7.4	even	3		21.6.e.c.16.2	yes	8
7.5	odd	6		147.6.e.o.67.2		8
7.6	odd	2		147.6.a.l.1.3		4
21.2	odd	6		63.6.e.e.46.3		8
21.11	odd	6		63.6.e.e.37.3		8
21.20	even	2		441.6.a.v.1.2		4
28.11	odd	6		336.6.q.j.289.1		8
28.23	odd	6		336.6.q.j.193.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.e.c.4.2	✓	8	7.2	even	3
21.6.e.c.16.2	yes	8	7.4	even	3
63.6.e.e.37.3		8	21.11	odd	6
63.6.e.e.46.3		8	21.2	odd	6
147.6.a.l.1.3		4	7.6	odd	2
147.6.a.m.1.3		4	1.1	even	1	trivial
147.6.e.o.67.2		8	7.5	odd	6
147.6.e.o.79.2		8	7.3	odd	6
336.6.q.j.193.1		8	28.23	odd	6
336.6.q.j.289.1		8	28.11	odd	6
441.6.a.v.1.2		4	21.20	even	2
441.6.a.w.1.2		4	3.2	odd	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:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)

\(f(q)\)	\(=\)	\(q+2.74818 q^{2} +9.00000 q^{3} -24.4475 q^{4} +58.3673 q^{5} +24.7336 q^{6} -155.128 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+2.74818 q^{2} +9.00000 q^{3} -24.4475 q^{4} +58.3673 q^{5} +24.7336 q^{6} -155.128 q^{8} +81.0000 q^{9} +160.404 q^{10} +17.4241 q^{11} -220.027 q^{12} +889.933 q^{13} +525.305 q^{15} +356.000 q^{16} +1026.64 q^{17} +222.603 q^{18} -1739.40 q^{19} -1426.93 q^{20} +47.8846 q^{22} +3936.22 q^{23} -1396.15 q^{24} +281.739 q^{25} +2445.70 q^{26} +729.000 q^{27} +5633.53 q^{29} +1443.63 q^{30} -3096.53 q^{31} +5942.45 q^{32} +156.817 q^{33} +2821.38 q^{34} -1980.25 q^{36} +5026.86 q^{37} -4780.20 q^{38} +8009.40 q^{39} -9054.40 q^{40} +18367.0 q^{41} -1630.91 q^{43} -425.976 q^{44} +4727.75 q^{45} +10817.4 q^{46} -9605.23 q^{47} +3204.00 q^{48} +774.269 q^{50} +9239.72 q^{51} -21756.6 q^{52} -23256.5 q^{53} +2003.42 q^{54} +1017.00 q^{55} -15654.6 q^{57} +15482.0 q^{58} -3603.24 q^{59} -12842.4 q^{60} +22876.6 q^{61} -8509.83 q^{62} +4938.92 q^{64} +51943.0 q^{65} +430.961 q^{66} +47012.8 q^{67} -25098.7 q^{68} +35426.0 q^{69} -1599.63 q^{71} -12565.4 q^{72} +5931.35 q^{73} +13814.7 q^{74} +2535.65 q^{75} +42524.1 q^{76} +22011.3 q^{78} -88468.9 q^{79} +20778.8 q^{80} +6561.00 q^{81} +50475.8 q^{82} -95823.9 q^{83} +59921.9 q^{85} -4482.04 q^{86} +50701.7 q^{87} -2702.96 q^{88} -46507.9 q^{89} +12992.7 q^{90} -96230.8 q^{92} -27868.8 q^{93} -26396.9 q^{94} -101524. q^{95} +53482.0 q^{96} -75981.8 q^{97} +1411.35 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9}+O(q^{10}) \) 4 * q + 3 * q^2 + 36 * q^3 + 69 * q^4 + 27 * q^6 + 123 * q^8 + 324 * q^9	\( 4 q + 3 q^{2} + 36 q^{3} + 69 q^{4} + 27 q^{6} + 123 q^{8} + 324 q^{9} + 283 q^{10} + 402 q^{11} + 621 q^{12} + 462 q^{13} + 3273 q^{16} + 276 q^{17} + 243 q^{18} + 510 q^{19} + 4719 q^{20} + 1375 q^{22} + 6900 q^{23} + 1107 q^{24} + 2814 q^{25} - 15138 q^{26} + 2916 q^{27} + 540 q^{29} + 2547 q^{30} - 6410 q^{31} + 15519 q^{32} + 3618 q^{33} + 21144 q^{34} + 5589 q^{36} + 15250 q^{37} - 41250 q^{38} + 4158 q^{39} - 8547 q^{40} + 4308 q^{41} + 29198 q^{43} + 70743 q^{44} + 61800 q^{46} - 15060 q^{47} + 29457 q^{48} - 7302 q^{50} + 2484 q^{51} - 47476 q^{52} + 13692 q^{53} + 2187 q^{54} + 73124 q^{55} + 4590 q^{57} + 52309 q^{58} + 34830 q^{59} + 42471 q^{60} - 5364 q^{61} + 16029 q^{62} - 73487 q^{64} + 66864 q^{65} + 12375 q^{66} - 5994 q^{67} - 58272 q^{68} + 62100 q^{69} + 89268 q^{71} + 9963 q^{72} + 59638 q^{73} - 185442 q^{74} + 25326 q^{75} + 21308 q^{76} - 136242 q^{78} - 44062 q^{79} - 33381 q^{80} + 26244 q^{81} + 57596 q^{82} - 208446 q^{83} + 36324 q^{85} - 136968 q^{86} + 4860 q^{87} + 87597 q^{88} - 77520 q^{89} + 22923 q^{90} + 158256 q^{92} - 57690 q^{93} - 73722 q^{94} - 221376 q^{95} + 139671 q^{96} - 188630 q^{97} + 32562 q^{99}+O(q^{100}) \) 4 * q + 3 * q^2 + 36 * q^3 + 69 * q^4 + 27 * q^6 + 123 * q^8 + 324 * q^9 + 283 * q^10 + 402 * q^11 + 621 * q^12 + 462 * q^13 + 3273 * q^16 + 276 * q^17 + 243 * q^18 + 510 * q^19 + 4719 * q^20 + 1375 * q^22 + 6900 * q^23 + 1107 * q^24 + 2814 * q^25 - 15138 * q^26 + 2916 * q^27 + 540 * q^29 + 2547 * q^30 - 6410 * q^31 + 15519 * q^32 + 3618 * q^33 + 21144 * q^34 + 5589 * q^36 + 15250 * q^37 - 41250 * q^38 + 4158 * q^39 - 8547 * q^40 + 4308 * q^41 + 29198 * q^43 + 70743 * q^44 + 61800 * q^46 - 15060 * q^47 + 29457 * q^48 - 7302 * q^50 + 2484 * q^51 - 47476 * q^52 + 13692 * q^53 + 2187 * q^54 + 73124 * q^55 + 4590 * q^57 + 52309 * q^58 + 34830 * q^59 + 42471 * q^60 - 5364 * q^61 + 16029 * q^62 - 73487 * q^64 + 66864 * q^65 + 12375 * q^66 - 5994 * q^67 - 58272 * q^68 + 62100 * q^69 + 89268 * q^71 + 9963 * q^72 + 59638 * q^73 - 185442 * q^74 + 25326 * q^75 + 21308 * q^76 - 136242 * q^78 - 44062 * q^79 - 33381 * q^80 + 26244 * q^81 + 57596 * q^82 - 208446 * q^83 + 36324 * q^85 - 136968 * q^86 + 4860 * q^87 + 87597 * q^88 - 77520 * q^89 + 22923 * q^90 + 158256 * q^92 - 57690 * q^93 - 73722 * q^94 - 221376 * q^95 + 139671 * q^96 - 188630 * q^97 + 32562 * q^99

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