LMFDB - Embedded newform 85.6.a.a.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [85,6,Mod(1,85)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("85.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(85, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 85 = 5 \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	85.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.6326246841$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 $ x^5 - 2x^4 - 95x^3 + 220x^2 + 1668x - 4640
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.29890$ of defining polynomial
Character	$\chi$	$=$	85.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.29890 q^{2} -29.9706 q^{3} -13.5195 q^{4} +25.0000 q^{5} +128.840 q^{6} +45.8012 q^{7} +195.684 q^{8} +655.235 q^{9} -107.472 q^{10} -504.686 q^{11} +405.186 q^{12} +513.142 q^{13} -196.895 q^{14} -749.264 q^{15} -408.601 q^{16} +289.000 q^{17} -2816.79 q^{18} -2084.02 q^{19} -337.987 q^{20} -1372.69 q^{21} +2169.60 q^{22} +4718.18 q^{23} -5864.75 q^{24} +625.000 q^{25} -2205.94 q^{26} -12354.9 q^{27} -619.207 q^{28} -3092.07 q^{29} +3221.01 q^{30} +4157.17 q^{31} -4505.34 q^{32} +15125.7 q^{33} -1242.38 q^{34} +1145.03 q^{35} -8858.42 q^{36} +4514.40 q^{37} +8958.99 q^{38} -15379.1 q^{39} +4892.09 q^{40} +7375.94 q^{41} +5901.04 q^{42} -17592.2 q^{43} +6823.09 q^{44} +16380.9 q^{45} -20283.0 q^{46} -14737.7 q^{47} +12246.0 q^{48} -14709.3 q^{49} -2686.81 q^{50} -8661.49 q^{51} -6937.40 q^{52} -1008.07 q^{53} +53112.5 q^{54} -12617.2 q^{55} +8962.53 q^{56} +62459.2 q^{57} +13292.5 q^{58} +496.132 q^{59} +10129.6 q^{60} -13046.2 q^{61} -17871.2 q^{62} +30010.5 q^{63} +32443.2 q^{64} +12828.5 q^{65} -65024.0 q^{66} +21128.0 q^{67} -3907.12 q^{68} -141406. q^{69} -4922.36 q^{70} -41206.2 q^{71} +128219. q^{72} +53173.8 q^{73} -19406.9 q^{74} -18731.6 q^{75} +28174.8 q^{76} -23115.2 q^{77} +66113.4 q^{78} +22083.6 q^{79} -10215.0 q^{80} +211061. q^{81} -31708.4 q^{82} -63967.7 q^{83} +18558.0 q^{84} +7225.00 q^{85} +75627.1 q^{86} +92671.1 q^{87} -98758.8 q^{88} -145034. q^{89} -70419.7 q^{90} +23502.5 q^{91} -63787.2 q^{92} -124593. q^{93} +63356.1 q^{94} -52100.5 q^{95} +135028. q^{96} -131574. q^{97} +63233.6 q^{98} -330688. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 7 q^{2} - 36 q^{3} + 43 q^{4} + 125 q^{5} - 105 q^{6} - 204 q^{7} - 63 q^{8} + 531 q^{9} - 175 q^{10} - 792 q^{11} + 785 q^{12} + 88 q^{13} + 860 q^{14} - 900 q^{15} - 2365 q^{16} + 1445 q^{17} - 2052 q^{18}+ \cdots - 535112 q^{99}+O(q^{100}) $ 5 * q - 7 * q^2 - 36 * q^3 + 43 * q^4 + 125 * q^5 - 105 * q^6 - 204 * q^7 - 63 * q^8 + 531 * q^9 - 175 * q^10 - 792 * q^11 + 785 * q^12 + 88 * q^13 + 860 * q^14 - 900 * q^15 - 2365 * q^16 + 1445 * q^17 - 2052 * q^18 - 5160 * q^19 + 1075 * q^20 - 6428 * q^21 - 3058 * q^22 - 6140 * q^23 - 11751 * q^24 + 3125 * q^25 - 9643 * q^26 - 13380 * q^27 - 4568 * q^28 - 7328 * q^29 - 2625 * q^30 - 8968 * q^31 - 4703 * q^32 + 13072 * q^33 - 2023 * q^34 - 5100 * q^35 + 2048 * q^36 + 9834 * q^37 + 10625 * q^38 + 10108 * q^39 - 1575 * q^40 - 5302 * q^41 + 16222 * q^42 - 4916 * q^43 - 31514 * q^44 + 13275 * q^45 - 16768 * q^46 - 20364 * q^47 + 45557 * q^48 - 26483 * q^49 - 4375 * q^50 - 10404 * q^51 - 11525 * q^52 + 56492 * q^53 + 64635 * q^54 - 19800 * q^55 + 73228 * q^56 + 67554 * q^57 + 54213 * q^58 - 96200 * q^59 + 19625 * q^60 - 7284 * q^61 + 122335 * q^62 + 64108 * q^63 + 28483 * q^64 + 2200 * q^65 + 115232 * q^66 - 60308 * q^67 + 12427 * q^68 - 89076 * q^69 + 21500 * q^70 - 25304 * q^71 + 118722 * q^72 + 163328 * q^73 - 82768 * q^74 - 22500 * q^75 + 92027 * q^76 + 19912 * q^77 + 134895 * q^78 - 50940 * q^79 - 59125 * q^80 + 92445 * q^81 + 191390 * q^82 - 179060 * q^83 - 128054 * q^84 + 36125 * q^85 + 87086 * q^86 + 55764 * q^87 + 167634 * q^88 - 387356 * q^89 - 51300 * q^90 - 115200 * q^91 - 293228 * q^92 - 202558 * q^93 + 100515 * q^94 - 129000 * q^95 + 428113 * q^96 - 106076 * q^97 + 49329 * q^98 - 535112 * q^99

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 $

$2$	−4.29890	−0.759945	−0.379973	−	0.924998i	$-0.624067\pi$
$2$	−4.29890	−0.759945	−0.379973	+	0.924998i	$0.624067\pi$
$3$	−29.9706	−1.92261	−0.961306	−	0.275482i	$-0.911163\pi$
$3$	−29.9706	−1.92261	−0.961306	+	0.275482i	$0.911163\pi$
$4$	−13.5195	−0.422483
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	128.840	1.46108
$7$	45.8012	0.353290	0.176645	−	0.984275i	$-0.443476\pi$
$7$	45.8012	0.353290	0.176645	+	0.984275i	$0.443476\pi$
$8$	195.684	1.08101
$9$	655.235	2.69644
$10$	−107.472	−0.339858
$11$	−504.686	−1.25759	−0.628796	−	0.777570i	$-0.716452\pi$
$11$	−504.686	−1.25759	−0.628796	+	0.777570i	$0.716452\pi$
$12$	405.186	0.812271
$13$	513.142	0.842130	0.421065	−	0.907031i	$-0.361656\pi$
$13$	513.142	0.842130	0.421065	+	0.907031i	$0.361656\pi$
$14$	−196.895	−0.268481
$15$	−749.264	−0.859818
$16$	−408.601	−0.399025
$17$	289.000	0.242536
$17$	289.000	0.242536
$18$	−2816.79	−2.04915
$19$	−2084.02	−1.32440	−0.662198	−	0.749329i	$-0.730376\pi$
$19$	−2084.02	−1.32440	−0.662198	+	0.749329i	$0.730376\pi$
$20$	−337.987	−0.188940
$21$	−1372.69	−0.679240
$22$	2169.60	0.955701
$23$	4718.18	1.85975	0.929875	−	0.367876i	$-0.119915\pi$
$23$	4718.18	1.85975	0.929875	+	0.367876i	$0.119915\pi$
$24$	−5864.75	−2.07836
$25$	625.000	0.200000
$26$	−2205.94	−0.639972
$27$	−12354.9	−3.26159
$28$	−619.207	−0.149259
$29$	−3092.07	−0.682739	−0.341369	−	0.939929i	$-0.610891\pi$
$29$	−3092.07	−0.682739	−0.341369	+	0.939929i	$0.610891\pi$
$30$	3221.01	0.653415
$31$	4157.17	0.776950	0.388475	−	0.921459i	$-0.373002\pi$
$31$	4157.17	0.776950	0.388475	+	0.921459i	$0.373002\pi$
$32$	−4505.34	−0.777772
$33$	15125.7	2.41786
$34$	−1242.38	−0.184314
$35$	1145.03	0.157996
$36$	−8858.42	−1.13920
$37$	4514.40	0.542120	0.271060	−	0.962562i	$-0.412626\pi$
$37$	4514.40	0.542120	0.271060	+	0.962562i	$0.412626\pi$
$38$	8958.99	1.00647
$39$	−15379.1	−1.61909
$40$	4892.09	0.483442
$41$	7375.94	0.685264	0.342632	−	0.939470i	$-0.388682\pi$
$41$	7375.94	0.685264	0.342632	+	0.939470i	$0.388682\pi$
$42$	5901.04	0.516185
$43$	−17592.2	−1.45094	−0.725469	−	0.688255i	$-0.758377\pi$
$43$	−17592.2	−1.45094	−0.725469	+	0.688255i	$0.758377\pi$
$44$	6823.09	0.531311
$45$	16380.9	1.20588
$46$	−20283.0	−1.41331
$47$	−14737.7	−0.973164	−0.486582	−	0.873635i	$-0.661757\pi$
$47$	−14737.7	−0.973164	−0.486582	+	0.873635i	$0.661757\pi$
$48$	12246.0	0.767170
$49$	−14709.3	−0.875186
$50$	−2686.81	−0.151989
$51$	−8661.49	−0.466302
$52$	−6937.40	−0.355786
$53$	−1008.07	−0.0492946	−0.0246473	−	0.999696i	$-0.507846\pi$
$53$	−1008.07	−0.0492946	−0.0246473	+	0.999696i	$0.507846\pi$
$54$	53112.5	2.47863
$55$	−12617.2	−0.562412
$56$	8962.53	0.381910
$57$	62459.2	2.54630
$58$	13292.5	0.518844
$59$	496.132	0.0185553	0.00927764	−	0.999957i	$-0.497047\pi$
$59$	496.132	0.0185553	0.00927764	+	0.999957i	$0.497047\pi$
$60$	10129.6	0.363259
$61$	−13046.2	−0.448910	−0.224455	−	0.974484i	$-0.572060\pi$
$61$	−13046.2	−0.448910	−0.224455	+	0.974484i	$0.572060\pi$
$62$	−17871.2	−0.590439
$63$	30010.5	0.952625
$64$	32443.2	0.990089
$65$	12828.5	0.376612
$66$	−65024.0	−1.83744
$67$	21128.0	0.575004	0.287502	−	0.957780i	$-0.407175\pi$
$67$	21128.0	0.575004	0.287502	+	0.957780i	$0.407175\pi$
$68$	−3907.12	−0.102467
$69$	−141406.	−3.57558
$70$	−4922.36	−0.120068
$71$	−41206.2	−0.970100	−0.485050	−	0.874487i	$-0.661199\pi$
$71$	−41206.2	−0.970100	−0.485050	+	0.874487i	$0.661199\pi$
$72$	128219.	2.91488
$73$	53173.8	1.16786	0.583929	−	0.811805i	$-0.301515\pi$
$73$	53173.8	1.16786	0.583929	+	0.811805i	$0.301515\pi$
$74$	−19406.9	−0.411981
$75$	−18731.6	−0.384523
$76$	28174.8	0.559535
$77$	−23115.2	−0.444295
$78$	66113.4	1.23042
$79$	22083.6	0.398110	0.199055	−	0.979988i	$-0.436213\pi$
$79$	22083.6	0.398110	0.199055	+	0.979988i	$0.436213\pi$
$80$	−10215.0	−0.178449
$81$	211061.	3.57434
$82$	−31708.4	−0.520763
$83$	−63967.7	−1.01922	−0.509608	−	0.860407i	$-0.670209\pi$
$83$	−63967.7	−1.01922	−0.509608	+	0.860407i	$0.670209\pi$
$84$	18558.0	0.286967
$85$	7225.00	0.108465
$86$	75627.1	1.10263
$87$	92671.1	1.31264
$88$	−98758.8	−1.35947
$89$	−145034.	−1.94087	−0.970433	−	0.241369i	$-0.922404\pi$
$89$	−145034.	−1.94087	−0.970433	+	0.241369i	$0.922404\pi$
$90$	−70419.7	−0.916406
$91$	23502.5	0.297516
$92$	−63787.2	−0.785713
$93$	−124593.	−1.49377
$94$	63356.1	0.739552
$95$	−52100.5	−0.592288
$96$	135028.	1.49535
$97$	−131574.	−1.41984	−0.709921	−	0.704282i	$-0.751269\pi$
$97$	−131574.	−1.41984	−0.709921	+	0.704282i	$0.751269\pi$
$98$	63233.6	0.665094
$99$	−330688.	−3.39102
$100$	−8449.66	−0.0844966
$101$	−101611.	−0.991149	−0.495575	−	0.868565i	$-0.665043\pi$
$101$	−101611.	−0.991149	−0.495575	+	0.868565i	$0.665043\pi$
$102$	37234.9	0.354364
$103$	72080.6	0.669461	0.334731	−	0.942314i	$-0.391355\pi$
$103$	72080.6	0.669461	0.334731	+	0.942314i	$0.391355\pi$
$104$	100413.	0.910350
$105$	−34317.2	−0.303765
$106$	4333.57	0.0374612
$107$	−175657.	−1.48322	−0.741610	−	0.670831i	$-0.765938\pi$
$107$	−175657.	−1.48322	−0.741610	+	0.670831i	$0.765938\pi$
$108$	167032.	1.37797
$109$	−16726.2	−0.134844	−0.0674220	−	0.997725i	$-0.521477\pi$
$109$	−16726.2	−0.134844	−0.0674220	+	0.997725i	$0.521477\pi$
$110$	54239.9	0.427402
$111$	−135299.	−1.04229
$112$	−18714.4	−0.140971
$113$	−106978.	−0.788134	−0.394067	−	0.919082i	$-0.628932\pi$
$113$	−106978.	−0.788134	−0.394067	+	0.919082i	$0.628932\pi$
$114$	−268506.	−1.93505
$115$	117954.	0.831706
$116$	41803.2	0.288446
$117$	336228.	2.27075
$118$	−2132.82	−0.0141010
$119$	13236.5	0.0856854
$120$	−146619.	−0.929472
$121$	93657.2	0.581537
$122$	56084.3	0.341147
$123$	−221061.	−1.31750
$124$	−56202.6	−0.328248
$125$	15625.0	0.0894427
$126$	−129012.	−0.723943
$127$	156823.	0.862781	0.431390	−	0.902165i	$-0.358023\pi$
$127$	156823.	0.862781	0.431390	+	0.902165i	$0.358023\pi$
$128$	4700.61	0.0253588
$129$	527248.	2.78959
$130$	−55148.6	−0.286204
$131$	124664.	0.634690	0.317345	−	0.948310i	$-0.397209\pi$
$131$	124664.	0.634690	0.317345	+	0.948310i	$0.397209\pi$
$132$	−204492.	−1.02151
$133$	−95450.5	−0.467896
$134$	−90827.1	−0.436972
$135$	−308873.	−1.45863
$136$	56552.6	0.262183
$137$	−49378.2	−0.224767	−0.112384	−	0.993665i	$-0.535849\pi$
$137$	−49378.2	−0.224767	−0.112384	+	0.993665i	$0.535849\pi$
$138$	607892.	2.71724
$139$	41751.8	0.183290	0.0916449	−	0.995792i	$-0.470788\pi$
$139$	41751.8	0.183290	0.0916449	+	0.995792i	$0.470788\pi$
$140$	−15480.2	−0.0667507
$141$	441698.	1.87102
$142$	177141.	0.737223
$143$	−258976.	−1.05906
$144$	−267730.	−1.07595
$145$	−77301.8	−0.305330
$146$	−228589.	−0.887509
$147$	440845.	1.68264
$148$	−61032.2	−0.229037
$149$	−46782.8	−0.172632	−0.0863159	−	0.996268i	$-0.527509\pi$
$149$	−46782.8	−0.172632	−0.0863159	+	0.996268i	$0.527509\pi$
$150$	80525.3	0.292216
$151$	−55545.1	−0.198246	−0.0991228	−	0.995075i	$-0.531604\pi$
$151$	−55545.1	−0.198246	−0.0991228	+	0.995075i	$0.531604\pi$
$152$	−407808.	−1.43168
$153$	189363.	0.653983
$154$	99370.0	0.337640
$155$	103929.	0.347463
$156$	207918.	0.684038
$157$	64212.8	0.207909	0.103954	−	0.994582i	$-0.466850\pi$
$157$	64212.8	0.207909	0.103954	+	0.994582i	$0.466850\pi$
$158$	−94935.4	−0.302542
$159$	30212.3	0.0947744
$160$	−112633.	−0.347830
$161$	216098.	0.657031
$162$	−907332.	−2.71631
$163$	595853.	1.75659	0.878294	−	0.478121i	$-0.158682\pi$
$163$	595853.	1.75659	0.878294	+	0.478121i	$0.158682\pi$
$164$	−99718.8	−0.289513
$165$	378143.	1.08130
$166$	274991.	0.774548
$167$	−533223.	−1.47951	−0.739755	−	0.672876i	$-0.765059\pi$
$167$	−533223.	−1.47951	−0.739755	+	0.672876i	$0.765059\pi$
$168$	−268612.	−0.734265
$169$	−107979.	−0.290818
$170$	−31059.5	−0.0824276
$171$	−1.36552e6	−3.57115
$172$	237837.	0.612997
$173$	743259.	1.88810	0.944050	−	0.329804i	$-0.106983\pi$
$173$	743259.	1.88810	0.944050	+	0.329804i	$0.106983\pi$
$174$	−398384.	−0.997536
$175$	28625.7	0.0706580
$176$	206215.	0.501810
$177$	−14869.4	−0.0356746
$178$	623488.	1.47495
$179$	−369567.	−0.862105	−0.431053	−	0.902327i	$-0.641858\pi$
$179$	−369567.	−0.862105	−0.431053	+	0.902327i	$0.641858\pi$
$180$	−221461.	−0.509466
$181$	490167.	1.11211	0.556055	−	0.831145i	$-0.312314\pi$
$181$	490167.	1.11211	0.556055	+	0.831145i	$0.312314\pi$
$182$	−101035.	−0.226096
$183$	391002.	0.863080
$184$	923270.	2.01041
$185$	112860.	0.242443
$186$	535611.	1.13519
$187$	−145854.	−0.305011
$188$	199246.	0.411146
$189$	−565869.	−1.15229
$190$	223975.	0.450106
$191$	−7647.56	−0.0151684	−0.00758420	−	0.999971i	$-0.502414\pi$
$191$	−7647.56	−0.0151684	−0.00758420	+	0.999971i	$0.502414\pi$
$192$	−972342.	−1.90356
$193$	−195606.	−0.377998	−0.188999	−	0.981977i	$-0.560524\pi$
$193$	−195606.	−0.377998	−0.188999	+	0.981977i	$0.560524\pi$
$194$	565622.	1.07900
$195$	−384479.	−0.724079
$196$	198861.	0.369751
$197$	−885254.	−1.62518	−0.812591	−	0.582834i	$-0.801944\pi$
$197$	−885254.	−1.62518	−0.812591	+	0.582834i	$0.801944\pi$
$198$	1.42159e6	2.57699
$199$	−877413.	−1.57062	−0.785310	−	0.619102i	$-0.787497\pi$
$199$	−877413.	−1.57062	−0.785310	+	0.619102i	$0.787497\pi$
$200$	122302.	0.216202
$201$	−633218.	−1.10551
$202$	436817.	0.753219
$203$	−141620.	−0.241205
$204$	117099.	0.197005
$205$	184399.	0.306459
$206$	−309867.	−0.508754
$207$	3.09151e6	5.01470
$208$	−209670.	−0.336031
$209$	1.05178e6	1.66555
$210$	147526.	0.230845
$211$	−325844.	−0.503852	−0.251926	−	0.967746i	$-0.581064\pi$
$211$	−325844.	−0.503852	−0.251926	+	0.967746i	$0.581064\pi$
$212$	13628.5	0.0208261
$213$	1.23497e6	1.86513
$214$	755131.	1.12717
$215$	−439805.	−0.648879
$216$	−2.41765e6	−3.52581
$217$	190403.	0.274489
$218$	71904.4	0.102474
$219$	−1.59365e6	−2.24534
$220$	170577.	0.237610
$221$	148298.	0.204246
$222$	581637.	0.792081
$223$	−758258.	−1.02107	−0.510534	−	0.859858i	$-0.670552\pi$
$223$	−758258.	−1.02107	−0.510534	+	0.859858i	$0.670552\pi$
$224$	−206350.	−0.274779
$225$	409522.	0.539288
$226$	459890.	0.598939
$227$	487718.	0.628209	0.314105	−	0.949388i	$-0.398296\pi$
$227$	487718.	0.628209	0.314105	+	0.949388i	$0.398296\pi$
$228$	−844415.	−1.07577
$229$	−413467.	−0.521017	−0.260509	−	0.965472i	$-0.583890\pi$
$229$	−413467.	−0.521017	−0.260509	+	0.965472i	$0.583890\pi$
$230$	−507074.	−0.632051
$231$	692776.	0.854206
$232$	−605068.	−0.738047
$233$	750445.	0.905584	0.452792	−	0.891616i	$-0.350428\pi$
$233$	750445.	0.905584	0.452792	+	0.891616i	$0.350428\pi$
$234$	−1.44541e6	−1.72565
$235$	−368443.	−0.435212
$236$	−6707.44	−0.00783929
$237$	−661859.	−0.765411
$238$	−56902.5	−0.0651162
$239$	−421206.	−0.476980	−0.238490	−	0.971145i	$-0.576652\pi$
$239$	−421206.	−0.476980	−0.238490	+	0.971145i	$0.576652\pi$
$240$	306150.	0.343089
$241$	212219.	0.235365	0.117682	−	0.993051i	$-0.462454\pi$
$241$	212219.	0.235365	0.117682	+	0.993051i	$0.462454\pi$
$242$	−402623.	−0.441937
$243$	−3.32339e6	−3.61048
$244$	176377.	0.189657
$245$	−367731.	−0.391395
$246$	950320.	1.00123
$247$	−1.06940e6	−1.11531
$248$	813489.	0.839890
$249$	1.91715e6	1.95956
$250$	−67170.3	−0.0679716
$251$	−937463.	−0.939226	−0.469613	−	0.882872i	$-0.655606\pi$
$251$	−937463.	−0.939226	−0.469613	+	0.882872i	$0.655606\pi$
$252$	−405726.	−0.402468
$253$	−2.38120e6	−2.33881
$254$	−674166.	−0.655666
$255$	−216537.	−0.208537
$256$	−1.05839e6	−1.00936
$257$	−1.48510e6	−1.40256	−0.701281	−	0.712885i	$-0.747388\pi$
$257$	−1.48510e6	−1.40256	−0.701281	+	0.712885i	$0.747388\pi$
$258$	−2.26659e6	−2.11994
$259$	206765.	0.191526
$260$	−173435.	−0.159112
$261$	−2.02603e6	−1.84096
$262$	−535917.	−0.482330
$263$	−420914.	−0.375235	−0.187618	−	0.982242i	$-0.560077\pi$
$263$	−420914.	−0.375235	−0.187618	+	0.982242i	$0.560077\pi$
$264$	2.95986e6	2.61373
$265$	−25201.6	−0.0220452
$266$	410332.	0.355575
$267$	4.34676e6	3.73153
$268$	−285639.	−0.242930
$269$	−1.19320e6	−1.00538	−0.502691	−	0.864466i	$-0.667657\pi$
$269$	−1.19320e6	−1.00538	−0.502691	+	0.864466i	$0.667657\pi$
$270$	1.32781e6	1.10848
$271$	2.07801e6	1.71880	0.859398	−	0.511308i	$-0.170839\pi$
$271$	2.07801e6	1.71880	0.859398	+	0.511308i	$0.170839\pi$
$272$	−118086.	−0.0967777
$273$	−704383.	−0.572008
$274$	212272.	0.170811
$275$	−315429.	−0.251518
$276$	1.91174e6	1.51062
$277$	883691.	0.691992	0.345996	−	0.938236i	$-0.387541\pi$
$277$	883691.	0.691992	0.345996	+	0.938236i	$0.387541\pi$
$278$	−179487.	−0.139290
$279$	2.72392e6	2.09500
$280$	224063.	0.170795
$281$	353265.	0.266892	0.133446	−	0.991056i	$-0.457396\pi$
$281$	353265.	0.266892	0.133446	+	0.991056i	$0.457396\pi$
$282$	−1.89882e6	−1.42187
$283$	−78060.8	−0.0579384	−0.0289692	−	0.999580i	$-0.509222\pi$
$283$	−78060.8	−0.0579384	−0.0289692	+	0.999580i	$0.509222\pi$
$284$	557085.	0.409851
$285$	1.56148e6	1.13874
$286$	1.11331e6	0.804824
$287$	337827.	0.242097
$288$	−2.95205e6	−2.09722
$289$	83521.0	0.0588235
$290$	332313.	0.232034
$291$	3.94334e6	2.72981
$292$	−718881.	−0.493401
$293$	−1.79083e6	−1.21867	−0.609335	−	0.792913i	$-0.708564\pi$
$293$	−1.79083e6	−1.21867	−0.609335	+	0.792913i	$0.708564\pi$
$294$	−1.89515e6	−1.27872
$295$	12403.3	0.00829817
$296$	883393.	0.586037
$297$	6.23535e6	4.10176
$298$	201115.	0.131191
$299$	2.42109e6	1.56615
$300$	253241.	0.162454
$301$	−805743.	−0.512602
$302$	238783.	0.150656
$303$	3.04535e6	1.90560
$304$	851533.	0.528466
$305$	−326155.	−0.200759
$306$	−814052.	−0.496991
$307$	−1.87896e6	−1.13781	−0.568907	−	0.822402i	$-0.692634\pi$
$307$	−1.87896e6	−1.13781	−0.568907	+	0.822402i	$0.692634\pi$
$308$	312505.	0.187707
$309$	−2.16030e6	−1.28711
$310$	−446781.	−0.264053
$311$	−2.65990e6	−1.55943	−0.779714	−	0.626136i	$-0.784635\pi$
$311$	−2.65990e6	−1.55943	−0.779714	+	0.626136i	$0.784635\pi$
$312$	−3.00945e6	−1.75025
$313$	576430.	0.332572	0.166286	−	0.986078i	$-0.446822\pi$
$313$	576430.	0.332572	0.166286	+	0.986078i	$0.446822\pi$
$314$	−276044.	−0.157999
$315$	750263.	0.426027
$316$	−298559.	−0.168195
$317$	−760504.	−0.425063	−0.212532	−	0.977154i	$-0.568171\pi$
$317$	−760504.	−0.425063	−0.212532	+	0.977154i	$0.568171\pi$
$318$	−129880.	−0.0720233
$319$	1.56053e6	0.858607
$320$	811081.	0.442781
$321$	5.26454e6	2.85166
$322$	−928983.	−0.499308
$323$	−602281.	−0.321213
$324$	−2.85344e6	−1.51010
$325$	320714.	0.168426
$326$	−2.56151e6	−1.33491
$327$	501295.	0.259253
$328$	1.44335e6	0.740777
$329$	−675005.	−0.343809
$330$	−1.62560e6	−0.821729
$331$	−915317.	−0.459200	−0.229600	−	0.973285i	$-0.573742\pi$
$331$	−915317.	−0.459200	−0.229600	+	0.973285i	$0.573742\pi$
$332$	864810.	0.430601
$333$	2.95799e6	1.46179
$334$	2.29227e6	1.12435
$335$	528200.	0.257150
$336$	560881.	0.271033
$337$	−1.58665e6	−0.761040	−0.380520	−	0.924773i	$-0.624255\pi$
$337$	−1.58665e6	−0.761040	−0.380520	+	0.924773i	$0.624255\pi$
$338$	464189.	0.221006
$339$	3.20620e6	1.51528
$340$	−97678.1	−0.0458247
$341$	−2.09806e6	−0.977086
$342$	5.87024e6	2.71388
$343$	−1.44348e6	−0.662484
$344$	−3.44250e6	−1.56848
$345$	−3.53516e6	−1.59905
$346$	−3.19519e6	−1.43485
$347$	3.71862e6	1.65790	0.828950	−	0.559322i	$-0.188939\pi$
$347$	3.71862e6	1.65790	0.828950	+	0.559322i	$0.188939\pi$
$348$	−1.25286e6	−0.554569
$349$	2.20157e6	0.967540	0.483770	−	0.875195i	$-0.339267\pi$
$349$	2.20157e6	0.967540	0.483770	+	0.875195i	$0.339267\pi$
$350$	−123059.	−0.0536962
$351$	−6.33982e6	−2.74669
$352$	2.27378e6	0.978120
$353$	−2.24412e6	−0.958539	−0.479269	−	0.877668i	$-0.659098\pi$
$353$	−2.24412e6	−0.958539	−0.479269	+	0.877668i	$0.659098\pi$
$354$	63921.9	0.0271107
$355$	−1.03015e6	−0.433842
$356$	1.96079e6	0.819984
$357$	−396706.	−0.164740
$358$	1.58873e6	0.655153
$359$	−2.48856e6	−1.01909	−0.509544	−	0.860444i	$-0.670186\pi$
$359$	−2.48856e6	−1.01909	−0.509544	+	0.860444i	$0.670186\pi$
$360$	3.20547e6	1.30357
$361$	1.86703e6	0.754023
$362$	−2.10718e6	−0.845143
$363$	−2.80696e6	−1.11807
$364$	−317741.	−0.125695
$365$	1.32934e6	0.522282
$366$	−1.68088e6	−0.655893
$367$	−4.14031e6	−1.60460	−0.802302	−	0.596919i	$-0.796391\pi$
$367$	−4.14031e6	−1.60460	−0.802302	+	0.596919i	$0.796391\pi$
$368$	−1.92785e6	−0.742086
$369$	4.83297e6	1.84777
$370$	−485173.	−0.184244
$371$	−46170.6	−0.0174153
$372$	1.68442e6	0.631094
$373$	1.79738e6	0.668910	0.334455	−	0.942412i	$-0.391448\pi$
$373$	1.79738e6	0.668910	0.334455	+	0.942412i	$0.391448\pi$
$374$	627013.	0.231792
$375$	−468290.	−0.171964
$376$	−2.88393e6	−1.05200
$377$	−1.58667e6	−0.574955
$378$	2.43261e6	0.875676
$379$	90472.6	0.0323533	0.0161767	−	0.999869i	$-0.494851\pi$
$379$	90472.6	0.0323533	0.0161767	+	0.999869i	$0.494851\pi$
$380$	704370.	0.250232
$381$	−4.70007e6	−1.65879
$382$	32876.1	0.0115272
$383$	2.92745e6	1.01975	0.509873	−	0.860250i	$-0.329692\pi$
$383$	2.92745e6	1.01975	0.509873	+	0.860250i	$0.329692\pi$
$384$	−140880.	−0.0487552
$385$	−577880.	−0.198695
$386$	840892.	0.287258
$387$	−1.15270e7	−3.91237
$388$	1.77881e6	0.599859
$389$	1.35743e6	0.454823	0.227411	−	0.973799i	$-0.426974\pi$
$389$	1.35743e6	0.454823	0.227411	+	0.973799i	$0.426974\pi$
$390$	1.65284e6	0.550260
$391$	1.36355e6	0.451056
$392$	−2.87836e6	−0.946084
$393$	−3.73624e6	−1.22026
$394$	3.80562e6	1.23505
$395$	552091.	0.178040
$396$	4.47072e6	1.43265
$397$	−223344.	−0.0711210	−0.0355605	−	0.999368i	$-0.511322\pi$
$397$	−223344.	−0.0711210	−0.0355605	+	0.999368i	$0.511322\pi$
$398$	3.77191e6	1.19359
$399$	2.86070e6	0.899582
$400$	−255376.	−0.0798049
$401$	3.43768e6	1.06759	0.533796	−	0.845613i	$-0.320765\pi$
$401$	3.43768e6	1.06759	0.533796	+	0.845613i	$0.320765\pi$
$402$	2.72214e6	0.840127
$403$	2.13322e6	0.654293
$404$	1.37373e6	0.418744
$405$	5.27654e6	1.59850
$406$	608812.	0.183302
$407$	−2.27835e6	−0.681766
$408$	−1.69491e6	−0.504077
$409$	−2.01964e6	−0.596987	−0.298494	−	0.954412i	$-0.596484\pi$
$409$	−2.01964e6	−0.596987	−0.298494	+	0.954412i	$0.596484\pi$
$410$	−792711.	−0.232892
$411$	1.47989e6	0.432141
$412$	−974491.	−0.282836
$413$	22723.4	0.00655539
$414$	−1.32901e7	−3.81090
$415$	−1.59919e6	−0.455807
$416$	−2.31188e6	−0.654985
$417$	−1.25133e6	−0.352395
$418$	−4.52148e6	−1.26573
$419$	2.77100e6	0.771085	0.385542	−	0.922690i	$-0.374014\pi$
$419$	2.77100e6	0.771085	0.385542	+	0.922690i	$0.374014\pi$
$420$	463950.	0.128336
$421$	−1.08269e6	−0.297714	−0.148857	−	0.988859i	$-0.547559\pi$
$421$	−1.08269e6	−0.297714	−0.148857	+	0.988859i	$0.547559\pi$
$422$	1.40077e6	0.382900
$423$	−9.65668e6	−2.62408
$424$	−197262.	−0.0532879
$425$	180625.	0.0485071
$426$	−5.30902e6	−1.41739
$427$	−597531.	−0.158595
$428$	2.37479e6	0.626636
$429$	7.76164e6	2.03615
$430$	1.89068e6	0.493113
$431$	−3.08443e6	−0.799800	−0.399900	−	0.916559i	$-0.630955\pi$
$431$	−3.08443e6	−0.799800	−0.399900	+	0.916559i	$0.630955\pi$
$432$	5.04823e6	1.30146
$433$	3.83729e6	0.983569	0.491784	−	0.870717i	$-0.336345\pi$
$433$	3.83729e6	0.983569	0.491784	+	0.870717i	$0.336345\pi$
$434$	−818523.	−0.208596
$435$	2.31678e6	0.587031
$436$	226130.	0.0569694
$437$	−9.83277e6	−2.46304
$438$	6.85093e6	1.70634
$439$	−1.77903e6	−0.440577	−0.220289	−	0.975435i	$-0.570700\pi$
$439$	−1.77903e6	−0.440577	−0.220289	+	0.975435i	$0.570700\pi$
$440$	−2.46897e6	−0.607973
$441$	−9.63801e6	−2.35989
$442$	−637518.	−0.155216
$443$	2.29093e6	0.554629	0.277315	−	0.960779i	$-0.410556\pi$
$443$	2.29093e6	0.554629	0.277315	+	0.960779i	$0.410556\pi$
$444$	1.82917e6	0.440349
$445$	−3.62586e6	−0.867982
$446$	3.25967e6	0.775956
$447$	1.40211e6	0.331904
$448$	1.48594e6	0.349789
$449$	6.34594e6	1.48552	0.742762	−	0.669555i	$-0.233515\pi$
$449$	6.34594e6	1.48552	0.742762	+	0.669555i	$0.233515\pi$
$450$	−1.76049e6	−0.409829
$451$	−3.72254e6	−0.861782
$452$	1.44629e6	0.332973
$453$	1.66472e6	0.381150
$454$	−2.09665e6	−0.477405
$455$	587562.	0.133053
$456$	1.22222e7	2.75257
$457$	1.39090e6	0.311535	0.155767	−	0.987794i	$-0.450215\pi$
$457$	1.39090e6	0.311535	0.155767	+	0.987794i	$0.450215\pi$
$458$	1.77745e6	0.395944
$459$	−3.57057e6	−0.791053
$460$	−1.59468e6	−0.351382
$461$	−4.36882e6	−0.957441	−0.478721	−	0.877967i	$-0.658899\pi$
$461$	−4.36882e6	−0.957441	−0.478721	+	0.877967i	$0.658899\pi$
$462$	−2.97817e6	−0.649150
$463$	737398.	0.159864	0.0799318	−	0.996800i	$-0.474530\pi$
$463$	737398.	0.159864	0.0799318	+	0.996800i	$0.474530\pi$
$464$	1.26342e6	0.272430
$465$	−3.11481e6	−0.668036
$466$	−3.22609e6	−0.688194
$467$	1.95716e6	0.415273	0.207636	−	0.978206i	$-0.433423\pi$
$467$	1.95716e6	0.415273	0.207636	+	0.978206i	$0.433423\pi$
$468$	−4.54563e6	−0.959354
$469$	967686.	0.203143
$470$	1.58390e6	0.330738
$471$	−1.92449e6	−0.399728
$472$	97084.9	0.0200584
$473$	8.87854e6	1.82469
$474$	2.84527e6	0.581670
$475$	−1.30251e6	−0.264879
$476$	−178951.	−0.0362006
$477$	−660520.	−0.132920
$478$	1.81072e6	0.362478
$479$	−3.22574e6	−0.642377	−0.321189	−	0.947015i	$-0.604082\pi$
$479$	−3.22574e6	−0.642377	−0.321189	+	0.947015i	$0.604082\pi$
$480$	3.37569e6	0.668743
$481$	2.31652e6	0.456535
$482$	−912308.	−0.178864
$483$	−6.47658e6	−1.26322
$484$	−1.26619e6	−0.245690
$485$	−3.28934e6	−0.634972
$486$	1.42869e7	2.74377
$487$	5.66306e6	1.08200	0.541001	−	0.841022i	$-0.318045\pi$
$487$	5.66306e6	1.08200	0.541001	+	0.841022i	$0.318045\pi$
$488$	−2.55293e6	−0.485276
$489$	−1.78580e7	−3.37724
$490$	1.58084e6	0.297439
$491$	290042.	0.0542947	0.0271474	−	0.999631i	$-0.491358\pi$
$491$	290042.	0.0542947	0.0271474	+	0.999631i	$0.491358\pi$
$492$	2.98863e6	0.556620
$493$	−893609.	−0.165588
$494$	4.59723e6	0.847576
$495$	−8.26720e6	−1.51651
$496$	−1.69862e6	−0.310022
$497$	−1.88729e6	−0.342726
$498$	−8.24163e6	−1.48916
$499$	7.87433e6	1.41567	0.707835	−	0.706377i	$-0.249672\pi$
$499$	7.87433e6	1.41567	0.707835	+	0.706377i	$0.249672\pi$
$500$	−211242.	−0.0377880
$501$	1.59810e7	2.84452
$502$	4.03006e6	0.713760
$503$	7.91006e6	1.39399	0.696995	−	0.717076i	$-0.254520\pi$
$503$	7.91006e6	1.39399	0.696995	+	0.717076i	$0.254520\pi$
$504$	5.87256e6	1.02980
$505$	−2.54029e6	−0.443255
$506$	1.02365e7	1.77736
$507$	3.23618e6	0.559130
$508$	−2.12016e6	−0.364510
$509$	−4.88210e6	−0.835241	−0.417621	−	0.908621i	$-0.637136\pi$
$509$	−4.88210e6	−0.835241	−0.417621	+	0.908621i	$0.637136\pi$
$510$	930872.	0.158476
$511$	2.43542e6	0.412593
$512$	4.39950e6	0.741700
$513$	2.57479e7	4.31964
$514$	6.38428e6	1.06587
$515$	1.80201e6	0.299392
$516$	−7.12811e6	−1.17856
$517$	7.43793e6	1.22384
$518$	−888860.	−0.145549
$519$	−2.22759e7	−3.63008
$520$	2.51034e6	0.407121
$521$	−4.67038e6	−0.753803	−0.376901	−	0.926253i	$-0.623010\pi$
$521$	−4.67038e6	−0.753803	−0.376901	+	0.926253i	$0.623010\pi$
$522$	8.70971e6	1.39903
$523$	−8.87695e6	−1.41909	−0.709544	−	0.704661i	$-0.751099\pi$
$523$	−8.87695e6	−1.41909	−0.709544	+	0.704661i	$0.751099\pi$
$524$	−1.68539e6	−0.268146
$525$	−857929.	−0.135848
$526$	1.80947e6	0.285158
$527$	1.20142e6	0.188438
$528$	−6.18039e6	−0.964787
$529$	1.58248e7	2.45867
$530$	108339.	0.0167531
$531$	325083.	0.0500332
$532$	1.29044e6	0.197678
$533$	3.78490e6	0.577081
$534$	−1.86863e7	−2.83576
$535$	−4.39142e6	−0.663316
$536$	4.13440e6	0.621585
$537$	1.10761e7	1.65749
$538$	5.12943e6	0.764035
$539$	7.42356e6	1.10063
$540$	4.17579e6	0.616246
$541$	−9.68582e6	−1.42280	−0.711399	−	0.702789i	$-0.751938\pi$
$541$	−9.68582e6	−1.42280	−0.711399	+	0.702789i	$0.751938\pi$
$542$	−8.93315e6	−1.30619
$543$	−1.46906e7	−2.13816
$544$	−1.30204e6	−0.188638
$545$	−418156.	−0.0603041
$546$	3.02807e6	0.434695
$547$	−1.98238e6	−0.283282	−0.141641	−	0.989918i	$-0.545238\pi$
$547$	−1.98238e6	−0.283282	−0.141641	+	0.989918i	$0.545238\pi$
$548$	667566.	0.0949605
$549$	−8.54832e6	−1.21046
$550$	1.35600e6	0.191140
$551$	6.44394e6	0.904216
$552$	−2.76709e7	−3.86523
$553$	1.01146e6	0.140648
$554$	−3.79890e6	−0.525876
$555$	−3.38247e6	−0.466125
$556$	−564462.	−0.0774369
$557$	−7.73137e6	−1.05589	−0.527945	−	0.849279i	$-0.677037\pi$
$557$	−7.73137e6	−1.05589	−0.527945	+	0.849279i	$0.677037\pi$
$558$	−1.17099e7	−1.59208
$559$	−9.02729e6	−1.22188
$560$	−467860.	−0.0630443
$561$	4.37134e6	0.586418
$562$	−1.51865e6	−0.202823
$563$	−6.73841e6	−0.895955	−0.447978	−	0.894045i	$-0.647856\pi$
$563$	−6.73841e6	−0.895955	−0.447978	+	0.894045i	$0.647856\pi$
$564$	−5.97152e6	−0.790474
$565$	−2.67446e6	−0.352464
$566$	335575.	0.0440300
$567$	9.66686e6	1.26278
$568$	−8.06337e6	−1.04869
$569$	1.38027e6	0.178724	0.0893622	−	0.995999i	$-0.471517\pi$
$569$	1.38027e6	0.178724	0.0893622	+	0.995999i	$0.471517\pi$
$570$	−6.71265e6	−0.865380
$571$	7.93779e6	1.01885	0.509424	−	0.860516i	$-0.329859\pi$
$571$	7.93779e6	1.01885	0.509424	+	0.860516i	$0.329859\pi$
$572$	3.50121e6	0.447433
$573$	229202.	0.0291629
$574$	−1.45228e6	−0.183980
$575$	2.94886e6	0.371950
$576$	2.12579e7	2.66972
$577$	8.26007e6	1.03287	0.516433	−	0.856327i	$-0.327259\pi$
$577$	8.26007e6	1.03287	0.516433	+	0.856327i	$0.327259\pi$
$578$	−359048.	−0.0447027
$579$	5.86243e6	0.726744
$580$	1.04508e6	0.128997
$581$	−2.92980e6	−0.360078
$582$	−1.69520e7	−2.07450
$583$	508757.	0.0619925
$584$	1.04052e7	1.26247
$585$	8.40571e6	1.01551
$586$	7.69862e6	0.926123
$587$	1.42684e7	1.70915	0.854573	−	0.519331i	$-0.173819\pi$
$587$	1.42684e7	1.70915	0.854573	+	0.519331i	$0.173819\pi$
$588$	−5.95998e6	−0.710889
$589$	−8.66361e6	−1.02899
$590$	−53320.6	−0.00630616
$591$	2.65316e7	3.12460
$592$	−1.84459e6	−0.216319
$593$	1.26496e7	1.47720	0.738601	−	0.674143i	$-0.235487\pi$
$593$	1.26496e7	1.47720	0.738601	+	0.674143i	$0.235487\pi$
$594$	−2.68051e7	−3.11711
$595$	330913.	0.0383197
$596$	632478.	0.0729340
$597$	2.62966e7	3.01970
$598$	−1.04080e7	−1.19019
$599$	−4.60166e6	−0.524019	−0.262010	−	0.965065i	$-0.584385\pi$
$599$	−4.60166e6	−0.524019	−0.262010	+	0.965065i	$0.584385\pi$
$600$	−3.66547e6	−0.415672
$601$	−8.69808e6	−0.982284	−0.491142	−	0.871080i	$-0.663420\pi$
$601$	−8.69808e6	−0.982284	−0.491142	+	0.871080i	$0.663420\pi$
$602$	3.46381e6	0.389549
$603$	1.38438e7	1.55046
$604$	750940.	0.0837555
$605$	2.34143e6	0.260071
$606$	−1.30917e7	−1.44815
$607$	8.58642e6	0.945889	0.472945	−	0.881092i	$-0.343191\pi$
$607$	8.58642e6	0.945889	0.472945	+	0.881092i	$0.343191\pi$
$608$	9.38921e6	1.03008
$609$	4.24445e6	0.463743
$610$	1.40211e6	0.152566
$611$	−7.56255e6	−0.819531
$612$	−2.56008e6	−0.276297
$613$	1.80976e6	0.194522	0.0972611	−	0.995259i	$-0.468992\pi$
$613$	1.80976e6	0.194522	0.0972611	+	0.995259i	$0.468992\pi$
$614$	8.07745e6	0.864676
$615$	−5.52653e6	−0.589203
$616$	−4.52327e6	−0.480287
$617$	428692.	0.0453348	0.0226674	−	0.999743i	$-0.492784\pi$
$617$	428692.	0.0453348	0.0226674	+	0.999743i	$0.492784\pi$
$618$	9.28690e6	0.978136
$619$	1.35257e7	1.41883	0.709417	−	0.704789i	$-0.248958\pi$
$619$	1.35257e7	1.41883	0.709417	+	0.704789i	$0.248958\pi$
$620$	−1.40507e6	−0.146797
$621$	−5.82926e7	−6.06575
$622$	1.14347e7	1.18508
$623$	−6.64274e6	−0.685689
$624$	6.28394e6	0.646057
$625$	390625.	0.0400000
$626$	−2.47802e6	−0.252737
$627$	−3.15223e7	−3.20220
$628$	−868123.	−0.0878379
$629$	1.30466e6	0.131483
$630$	−3.22530e6	−0.323757
$631$	943645.	0.0943486	0.0471743	−	0.998887i	$-0.484978\pi$
$631$	943645.	0.0943486	0.0471743	+	0.998887i	$0.484978\pi$
$632$	4.32141e6	0.430360
$633$	9.76572e6	0.968712
$634$	3.26933e6	0.323025
$635$	3.92057e6	0.385847
$636$	−408454.	−0.0400406
$637$	−7.54793e6	−0.737020
$638$	−6.70855e6	−0.652494
$639$	−2.69997e7	−2.61581
$640$	117515.	0.0113408
$641$	−1.91645e7	−1.84227	−0.921134	−	0.389247i	$-0.872735\pi$
$641$	−1.91645e7	−1.84227	−0.921134	+	0.389247i	$0.872735\pi$
$642$	−2.26317e7	−2.16710
$643$	−5.39663e6	−0.514748	−0.257374	−	0.966312i	$-0.582857\pi$
$643$	−5.39663e6	−0.514748	−0.257374	+	0.966312i	$0.582857\pi$
$644$	−2.92153e6	−0.277585
$645$	1.31812e7	1.24754
$646$	2.58915e6	0.244104
$647$	−7.22529e6	−0.678570	−0.339285	−	0.940684i	$-0.610185\pi$
$647$	−7.22529e6	−0.678570	−0.339285	+	0.940684i	$0.610185\pi$
$648$	4.13013e7	3.86390
$649$	−250391.	−0.0233350
$650$	−1.37872e6	−0.127994
$651$	−5.70648e6	−0.527735
$652$	−8.05561e6	−0.742129
$653$	1.67534e7	1.53752	0.768758	−	0.639540i	$-0.220875\pi$
$653$	1.67534e7	1.53752	0.768758	+	0.639540i	$0.220875\pi$
$654$	−2.15502e6	−0.197018
$655$	3.11659e6	0.283842
$656$	−3.01382e6	−0.273437
$657$	3.48413e7	3.14906
$658$	2.90178e6	0.261276
$659$	1.70752e7	1.53162	0.765810	−	0.643067i	$-0.222338\pi$
$659$	1.70752e7	1.53162	0.765810	+	0.643067i	$0.222338\pi$
$660$	−5.11229e6	−0.456831
$661$	−1.45834e7	−1.29824	−0.649122	−	0.760685i	$-0.724863\pi$
$661$	−1.45834e7	−1.29824	−0.649122	+	0.760685i	$0.724863\pi$
$662$	3.93486e6	0.348967
$663$	−4.44457e6	−0.392687
$664$	−1.25174e7	−1.10178
$665$	−2.38626e6	−0.209249
$666$	−1.27161e7	−1.11088
$667$	−1.45889e7	−1.26972
$668$	7.20889e6	0.625068
$669$	2.27254e7	1.96312
$670$	−2.27068e6	−0.195420
$671$	6.58423e6	0.564545
$672$	6.18442e6	0.528294
$673$	9.19877e6	0.782874	0.391437	−	0.920205i	$-0.371978\pi$
$673$	9.19877e6	0.782874	0.391437	+	0.920205i	$0.371978\pi$
$674$	6.82086e6	0.578348
$675$	−7.72182e6	−0.652319
$676$	1.45981e6	0.122866
$677$	−1.16519e7	−0.977068	−0.488534	−	0.872545i	$-0.662468\pi$
$677$	−1.16519e7	−0.977068	−0.488534	+	0.872545i	$0.662468\pi$
$678$	−1.37831e7	−1.15153
$679$	−6.02623e6	−0.501616
$680$	1.41381e6	0.117252
$681$	−1.46172e7	−1.20780
$682$	9.01937e6	0.742532
$683$	−1.79480e7	−1.47219	−0.736094	−	0.676879i	$-0.763332\pi$
$683$	−1.79480e7	−1.47219	−0.736094	+	0.676879i	$0.763332\pi$
$684$	1.84611e7	1.50875
$685$	−1.23445e6	−0.100519
$686$	6.20538e6	0.503452
$687$	1.23918e7	1.00171
$688$	7.18820e6	0.578960
$689$	−517281.	−0.0415124
$690$	1.51973e7	1.21519
$691$	−1.29794e7	−1.03410	−0.517048	−	0.855956i	$-0.672969\pi$
$691$	−1.29794e7	−1.03410	−0.517048	+	0.855956i	$0.672969\pi$
$692$	−1.00485e7	−0.797690
$693$	−1.51459e7	−1.19801
$694$	−1.59860e7	−1.25991
$695$	1.04380e6	0.0819697
$696$	1.81342e7	1.41898
$697$	2.13165e6	0.166201
$698$	−9.46433e6	−0.735278
$699$	−2.24912e7	−1.74109
$700$	−387004.	−0.0298518
$701$	1.99415e7	1.53272	0.766358	−	0.642413i	$-0.222067\pi$
$701$	1.99415e7	1.53272	0.766358	+	0.642413i	$0.222067\pi$
$702$	2.72542e7	2.08733
$703$	−9.40808e6	−0.717981
$704$	−1.63737e7	−1.24513
$705$	1.10425e7	0.836745
$706$	9.64726e6	0.728437
$707$	−4.65392e6	−0.350163
$708$	201026.	0.0150719
$709$	−1.53176e7	−1.14439	−0.572196	−	0.820117i	$-0.693908\pi$
$709$	−1.53176e7	−1.14439	−0.572196	+	0.820117i	$0.693908\pi$
$710$	4.42853e6	0.329696
$711$	1.44700e7	1.07348
$712$	−2.83808e7	−2.09810
$713$	1.96142e7	1.44493
$714$	1.70540e6	0.125193
$715$	−6.47439e6	−0.473624
$716$	4.99634e6	0.364225
$717$	1.26238e7	0.917047
$718$	1.06981e7	0.774452
$719$	−1.24260e7	−0.896413	−0.448207	−	0.893930i	$-0.647937\pi$
$719$	−1.24260e7	−0.896413	−0.448207	+	0.893930i	$0.647937\pi$
$720$	−6.69324e6	−0.481178
$721$	3.30137e6	0.236514
$722$	−8.02619e6	−0.573016
$723$	−6.36032e6	−0.452515
$724$	−6.62680e6	−0.469848
$725$	−1.93255e6	−0.136548
$726$	1.20668e7	0.849673
$727$	1.09417e7	0.767803	0.383902	−	0.923374i	$-0.374580\pi$
$727$	1.09417e7	0.767803	0.383902	+	0.923374i	$0.374580\pi$
$728$	4.59905e6	0.321618
$729$	4.83159e7	3.36722
$730$	−5.71472e6	−0.396906
$731$	−5.08415e6	−0.351904
$732$	−5.28613e6	−0.364637
$733$	−8.63822e6	−0.593833	−0.296916	−	0.954903i	$-0.595958\pi$
$733$	−8.63822e6	−0.593833	−0.296916	+	0.954903i	$0.595958\pi$
$734$	1.77988e7	1.21941
$735$	1.10211e7	0.752501
$736$	−2.12570e7	−1.44646
$737$	−1.06630e7	−0.723121
$738$	−2.07765e7	−1.40421
$739$	−1.28244e7	−0.863825	−0.431913	−	0.901915i	$-0.642161\pi$
$739$	−1.28244e7	−0.863825	−0.431913	+	0.901915i	$0.642161\pi$
$740$	−1.52581e6	−0.102428
$741$	3.20504e7	2.14431
$742$	198483.	0.0132347
$743$	1.16623e7	0.775020	0.387510	−	0.921865i	$-0.373335\pi$
$743$	1.16623e7	0.775020	0.387510	+	0.921865i	$0.373335\pi$
$744$	−2.43807e7	−1.61478
$745$	−1.16957e6	−0.0772032
$746$	−7.72675e6	−0.508335
$747$	−4.19139e7	−2.74825
$748$	1.97187e6	0.128862
$749$	−8.04529e6	−0.524007
$750$	2.01313e6	0.130683
$751$	2.02785e7	1.31200	0.656002	−	0.754759i	$-0.272246\pi$
$751$	2.02785e7	1.31200	0.656002	+	0.754759i	$0.272246\pi$
$752$	6.02186e6	0.388317
$753$	2.80963e7	1.80577
$754$	6.82094e6	0.436934
$755$	−1.38863e6	−0.0886582
$756$	7.65024e6	0.486823
$757$	4.98405e6	0.316114	0.158057	−	0.987430i	$-0.449477\pi$
$757$	4.98405e6	0.316114	0.158057	+	0.987430i	$0.449477\pi$
$758$	−388933.	−0.0245868
$759$	7.13659e7	4.49662
$760$	−1.01952e7	−0.640268
$761$	1.63795e7	1.02527	0.512635	−	0.858607i	$-0.328669\pi$
$761$	1.63795e7	1.02527	0.512635	+	0.858607i	$0.328669\pi$
$762$	2.02051e7	1.26059
$763$	−766081.	−0.0476391
$764$	103391.	0.00640839
$765$	4.73407e6	0.292470
$766$	−1.25848e7	−0.774951
$767$	254586.	0.0156259
$768$	3.17206e7	1.94061
$769$	−2.09430e7	−1.27709	−0.638547	−	0.769582i	$-0.720464\pi$
$769$	−2.09430e7	−1.27709	−0.638547	+	0.769582i	$0.720464\pi$
$770$	2.48425e6	0.150997
$771$	4.45092e7	2.69658
$772$	2.64449e6	0.159698
$773$	−1.56215e7	−0.940318	−0.470159	−	0.882582i	$-0.655803\pi$
$773$	−1.56215e7	−0.940318	−0.470159	+	0.882582i	$0.655803\pi$
$774$	4.95535e7	2.97318
$775$	2.59823e6	0.155390
$776$	−2.57468e7	−1.53486
$777$	−6.19685e6	−0.368229
$778$	−5.83544e6	−0.345640
$779$	−1.53716e7	−0.907560
$780$	5.19794e6	0.305911
$781$	2.07962e7	1.21999
$782$	−5.86178e6	−0.342778
$783$	3.82023e7	2.22682
$784$	6.01022e6	0.349221
$785$	1.60532e6	0.0929796
$786$	1.60617e7	0.927333
$787$	2.34939e7	1.35213	0.676065	−	0.736842i	$-0.263684\pi$
$787$	2.34939e7	1.35213	0.676065	+	0.736842i	$0.263684\pi$
$788$	1.19682e7	0.686613
$789$	1.26150e7	0.721432
$790$	−2.37338e6	−0.135301
$791$	−4.89974e6	−0.278440
$792$	−6.47102e7	−3.66572
$793$	−6.69454e6	−0.378040
$794$	960133.	0.0540480
$795$	755307.	0.0423844
$796$	1.18622e7	0.663561
$797$	−2.03038e6	−0.113222	−0.0566111	−	0.998396i	$-0.518030\pi$
$797$	−2.03038e6	−0.113222	−0.0566111	+	0.998396i	$0.518030\pi$
$798$	−1.22979e7	−0.683633
$799$	−4.25921e6	−0.236027
$800$	−2.81584e6	−0.155554
$801$	−9.50315e7	−5.23343
$802$	−1.47783e7	−0.811311
$803$	−2.68361e7	−1.46869
$804$	8.56076e6	0.467060
$805$	5.40245e6	0.293833
$806$	−9.17048e6	−0.497227
$807$	3.57607e7	1.93296
$808$	−1.98837e7	−1.07144
$809$	−1.53040e7	−0.822120	−0.411060	−	0.911608i	$-0.634841\pi$
$809$	−1.53040e7	−0.822120	−0.411060	+	0.911608i	$0.634841\pi$
$810$	−2.26833e7	−1.21477
$811$	−1.82768e6	−0.0975769	−0.0487884	−	0.998809i	$-0.515536\pi$
$811$	−1.82768e6	−0.0975769	−0.0487884	+	0.998809i	$0.515536\pi$
$812$	1.91463e6	0.101905
$813$	−6.22791e7	−3.30458
$814$	9.79441e6	0.518104
$815$	1.48963e7	0.785570
$816$	3.53910e6	0.186066
$817$	3.66625e7	1.92162
$818$	8.68222e6	0.453678
$819$	1.53996e7	0.802234
$820$	−2.49297e6	−0.129474
$821$	3.84838e6	0.199260	0.0996301	−	0.995025i	$-0.468234\pi$
$821$	3.84838e6	0.199260	0.0996301	+	0.995025i	$0.468234\pi$
$822$	−6.36190e6	−0.328403
$823$	8.89074e6	0.457550	0.228775	−	0.973479i	$-0.426528\pi$
$823$	8.89074e6	0.457550	0.228775	+	0.973479i	$0.426528\pi$
$824$	1.41050e7	0.723694
$825$	9.45358e6	0.483572
$826$	−97685.7	−0.00498174
$827$	2.28605e7	1.16231	0.581155	−	0.813793i	$-0.302601\pi$
$827$	2.28605e7	1.16231	0.581155	+	0.813793i	$0.302601\pi$
$828$	−4.17956e7	−2.11863
$829$	1.27813e7	0.645934	0.322967	−	0.946410i	$-0.395320\pi$
$829$	1.27813e7	0.645934	0.322967	+	0.946410i	$0.395320\pi$
$830$	6.87477e6	0.346388
$831$	−2.64847e7	−1.33043
$832$	1.66480e7	0.833783
$833$	−4.25097e6	−0.212264
$834$	5.37932e6	0.267801
$835$	−1.33306e7	−0.661657
$836$	−1.42194e7	−0.703666
$837$	−5.13614e7	−2.53410
$838$	−1.19123e7	−0.585982
$839$	−2.44397e7	−1.19865	−0.599323	−	0.800508i	$-0.704563\pi$
$839$	−2.44397e7	−1.19865	−0.599323	+	0.800508i	$0.704563\pi$
$840$	−6.71531e6	−0.328373
$841$	−1.09502e7	−0.533868
$842$	4.65438e6	0.226247
$843$	−1.05876e7	−0.513130
$844$	4.40523e6	0.212869
$845$	−2.69946e6	−0.130058
$846$	4.15131e7	1.99416
$847$	4.28961e6	0.205451
$848$	411897.	0.0196698
$849$	2.33953e6	0.111393
$850$	−776489.	−0.0368628
$851$	2.12997e7	1.00821
$852$	−1.66962e7	−0.787984
$853$	−3.88293e7	−1.82720	−0.913601	−	0.406612i	$-0.866710\pi$
$853$	−3.88293e7	−1.82720	−0.913601	+	0.406612i	$0.866710\pi$
$854$	2.56872e6	0.120524
$855$	−3.41380e7	−1.59707
$856$	−3.43732e7	−1.60338
$857$	−2.51074e7	−1.16775	−0.583875	−	0.811843i	$-0.698464\pi$
$857$	−2.51074e7	−1.16775	−0.583875	+	0.811843i	$0.698464\pi$
$858$	−3.33665e7	−1.54736
$859$	−1.46587e6	−0.0677817	−0.0338909	−	0.999426i	$-0.510790\pi$
$859$	−1.46587e6	−0.0677817	−0.0338909	+	0.999426i	$0.510790\pi$
$860$	5.94593e6	0.274141
$861$	−1.01249e7	−0.465458
$862$	1.32596e7	0.607804
$863$	−1.85564e7	−0.848138	−0.424069	−	0.905630i	$-0.639399\pi$
$863$	−1.85564e7	−0.848138	−0.424069	+	0.905630i	$0.639399\pi$
$864$	5.56630e7	2.53678
$865$	1.85815e7	0.844384
$866$	−1.64961e7	−0.747459
$867$	−2.50317e6	−0.113095
$868$	−2.57415e6	−0.115967
$869$	−1.11453e7	−0.500660
$870$	−9.95960e6	−0.446112
$871$	1.08417e7	0.484228
$872$	−3.27305e6	−0.145768
$873$	−8.62117e7	−3.82852
$874$	4.22701e7	1.87178
$875$	715643.	0.0315992
$876$	2.15453e7	0.948618
$877$	−4.32523e7	−1.89894	−0.949468	−	0.313864i	$-0.898377\pi$
$877$	−4.32523e7	−1.89894	−0.949468	+	0.313864i	$0.898377\pi$
$878$	7.64787e6	0.334814
$879$	5.36723e7	2.34303
$880$	5.15539e6	0.224416
$881$	2.75182e7	1.19448	0.597241	−	0.802062i	$-0.296264\pi$
$881$	2.75182e7	1.19448	0.597241	+	0.802062i	$0.296264\pi$
$882$	4.14329e7	1.79338
$883$	3.80483e7	1.64223	0.821115	−	0.570763i	$-0.193352\pi$
$883$	3.80483e7	1.64223	0.821115	+	0.570763i	$0.193352\pi$
$884$	−2.00491e6	−0.0862907
$885$	−371734.	−0.0159542
$886$	−9.84849e6	−0.421488
$887$	1.82142e7	0.777322	0.388661	−	0.921381i	$-0.372938\pi$
$887$	1.82142e7	0.777322	0.388661	+	0.921381i	$0.372938\pi$
$888$	−2.64758e7	−1.12672
$889$	7.18267e6	0.304812
$890$	1.55872e7	0.659619
$891$	−1.06520e8	−4.49507
$892$	1.02512e7	0.431384
$893$	3.07137e7	1.28885
$894$	−6.02752e6	−0.252229
$895$	−9.23917e6	−0.385545
$896$	215293.	0.00895902
$897$	−7.25615e7	−3.01110
$898$	−2.72805e7	−1.12892
$899$	−1.28543e7	−0.530454
$900$	−5.53651e6	−0.227840
$901$	−291331.	−0.0119557
$902$	1.60028e7	0.654907
$903$	2.41486e7	0.985535
$904$	−2.09339e7	−0.851980
$905$	1.22542e7	0.497351
$906$	−7.15646e6	−0.289653
$907$	3.24189e7	1.30852	0.654259	−	0.756271i	$-0.272981\pi$
$907$	3.24189e7	1.30852	0.654259	+	0.756271i	$0.272981\pi$
$908$	−6.59369e6	−0.265408
$909$	−6.65793e7	−2.67257
$910$	−2.52587e6	−0.101113
$911$	4.17461e7	1.66655	0.833277	−	0.552855i	$-0.186462\pi$
$911$	4.17461e7	1.66655	0.833277	+	0.552855i	$0.186462\pi$
$912$	−2.55209e7	−1.01604
$913$	3.22836e7	1.28176
$914$	−5.97935e6	−0.236749
$915$	9.77504e6	0.385981
$916$	5.58985e6	0.220121
$917$	5.70974e6	0.224230
$918$	1.53495e7	0.601157
$919$	−5.75989e6	−0.224970	−0.112485	−	0.993653i	$-0.535881\pi$
$919$	−5.75989e6	−0.224970	−0.112485	+	0.993653i	$0.535881\pi$
$920$	2.30817e7	0.899081
$921$	5.63134e7	2.18757
$922$	1.87811e7	0.727603
$923$	−2.11446e7	−0.816950
$924$	−9.36596e6	−0.360888
$925$	2.82150e6	0.108424
$926$	−3.17000e6	−0.121488
$927$	4.72297e7	1.80516
$928$	1.39308e7	0.531015
$929$	−3.14157e6	−0.119428	−0.0597142	−	0.998216i	$-0.519019\pi$
$929$	−3.14157e6	−0.119428	−0.0597142	+	0.998216i	$0.519019\pi$
$930$	1.33903e7	0.507671
$931$	3.06544e7	1.15909
$932$	−1.01456e7	−0.382594
$933$	7.97188e7	2.99817
$934$	−8.41362e6	−0.315585
$935$	−3.64636e6	−0.136405
$936$	6.57944e7	2.45470
$937$	2.19517e7	0.816808	0.408404	−	0.912801i	$-0.366085\pi$
$937$	2.19517e7	0.816808	0.408404	+	0.912801i	$0.366085\pi$
$938$	−4.15999e6	−0.154378
$939$	−1.72759e7	−0.639408
$940$	4.98116e6	0.183870
$941$	580958.	0.0213880	0.0106940	−	0.999943i	$-0.496596\pi$
$941$	580958.	0.0213880	0.0106940	+	0.999943i	$0.496596\pi$
$942$	8.27321e6	0.303771
$943$	3.48010e7	1.27442
$944$	−202720.	−0.00740401
$945$	−1.41467e7	−0.515319
$946$	−3.81680e7	−1.38666
$947$	1.94273e7	0.703945	0.351972	−	0.936010i	$-0.385511\pi$
$947$	1.94273e7	0.703945	0.351972	+	0.936010i	$0.385511\pi$
$948$	8.94798e6	0.323373
$949$	2.72857e7	0.983488
$950$	5.59937e6	0.201294
$951$	2.27927e7	0.817232
$952$	2.59017e6	0.0926267
$953$	−5.06290e7	−1.80579	−0.902895	−	0.429860i	$-0.858563\pi$
$953$	−5.06290e7	−1.80579	−0.902895	+	0.429860i	$0.858563\pi$
$954$	2.83951e6	0.101012
$955$	−191189.	−0.00678351
$956$	5.69448e6	0.201516
$957$	−4.67698e7	−1.65077
$958$	1.38671e7	0.488172
$959$	−2.26158e6	−0.0794081
$960$	−2.43086e7	−0.851297
$961$	−1.13471e7	−0.396349
$962$	−9.95851e6	−0.346942
$963$	−1.15096e8	−3.99941
$964$	−2.86909e6	−0.0994377
$965$	−4.89016e6	−0.169046
$966$	2.78422e7	0.959975
$967$	2.24825e7	0.773175	0.386587	−	0.922253i	$-0.373654\pi$
$967$	2.24825e7	0.773175	0.386587	+	0.922253i	$0.373654\pi$
$968$	1.83272e7	0.628647
$969$	1.80507e7	0.617568
$970$	1.41406e7	0.482544
$971$	−2.03554e7	−0.692836	−0.346418	−	0.938080i	$-0.612602\pi$
$971$	−2.03554e7	−0.692836	−0.346418	+	0.938080i	$0.612602\pi$
$972$	4.49304e7	1.52537
$973$	1.91228e6	0.0647545
$974$	−2.43449e7	−0.822263
$975$	−9.61197e6	−0.323818
$976$	5.33069e6	0.179126
$977$	−1.70028e7	−0.569880	−0.284940	−	0.958545i	$-0.591974\pi$
$977$	−1.70028e7	−0.569880	−0.284940	+	0.958545i	$0.591974\pi$
$978$	7.67699e7	2.56652
$979$	7.31968e7	2.44082
$980$	4.97153e6	0.165358
$981$	−1.09596e7	−0.363599
$982$	−1.24686e6	−0.0412610
$983$	4.26423e7	1.40753	0.703763	−	0.710435i	$-0.251502\pi$
$983$	4.26423e7	1.40753	0.703763	+	0.710435i	$0.251502\pi$
$984$	−4.32580e7	−1.42423
$985$	−2.21313e7	−0.726804
$986$	3.84153e6	0.125838
$987$	2.02303e7	0.661012
$988$	1.44577e7	0.471201
$989$	−8.30031e7	−2.69838
$990$	3.55399e7	1.15246
$991$	−2.07849e7	−0.672302	−0.336151	−	0.941808i	$-0.609125\pi$
$991$	−2.07849e7	−0.672302	−0.336151	+	0.941808i	$0.609125\pi$
$992$	−1.87294e7	−0.604290
$993$	2.74326e7	0.882863
$994$	8.11327e6	0.260453
$995$	−2.19353e7	−0.702403
$996$	−2.59188e7	−0.827879
$997$	−2.54002e6	−0.0809282	−0.0404641	−	0.999181i	$-0.512884\pi$
$997$	−2.54002e6	−0.0809282	−0.0404641	+	0.999181i	$0.512884\pi$
$998$	−3.38510e7	−1.07583
$999$	−5.57749e7	−1.76818

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.6.a.a.1.3	✓	5
3.2	odd	2		765.6.a.g.1.3		5
5.4	even	2		425.6.a.d.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.6.a.a.1.3	✓	5	1.1	even	1	trivial
425.6.a.d.1.3		5	5.4	even	2
765.6.a.g.1.3		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	85.a (trivial)

\(f(q)\)	\(=\)	\(q-4.29890 q^{2} -29.9706 q^{3} -13.5195 q^{4} +25.0000 q^{5} +128.840 q^{6} +45.8012 q^{7} +195.684 q^{8} +655.235 q^{9} -107.472 q^{10} -504.686 q^{11} +405.186 q^{12} +513.142 q^{13} -196.895 q^{14} -749.264 q^{15} -408.601 q^{16} +289.000 q^{17} -2816.79 q^{18} -2084.02 q^{19} -337.987 q^{20} -1372.69 q^{21} +2169.60 q^{22} +4718.18 q^{23} -5864.75 q^{24} +625.000 q^{25} -2205.94 q^{26} -12354.9 q^{27} -619.207 q^{28} -3092.07 q^{29} +3221.01 q^{30} +4157.17 q^{31} -4505.34 q^{32} +15125.7 q^{33} -1242.38 q^{34} +1145.03 q^{35} -8858.42 q^{36} +4514.40 q^{37} +8958.99 q^{38} -15379.1 q^{39} +4892.09 q^{40} +7375.94 q^{41} +5901.04 q^{42} -17592.2 q^{43} +6823.09 q^{44} +16380.9 q^{45} -20283.0 q^{46} -14737.7 q^{47} +12246.0 q^{48} -14709.3 q^{49} -2686.81 q^{50} -8661.49 q^{51} -6937.40 q^{52} -1008.07 q^{53} +53112.5 q^{54} -12617.2 q^{55} +8962.53 q^{56} +62459.2 q^{57} +13292.5 q^{58} +496.132 q^{59} +10129.6 q^{60} -13046.2 q^{61} -17871.2 q^{62} +30010.5 q^{63} +32443.2 q^{64} +12828.5 q^{65} -65024.0 q^{66} +21128.0 q^{67} -3907.12 q^{68} -141406. q^{69} -4922.36 q^{70} -41206.2 q^{71} +128219. q^{72} +53173.8 q^{73} -19406.9 q^{74} -18731.6 q^{75} +28174.8 q^{76} -23115.2 q^{77} +66113.4 q^{78} +22083.6 q^{79} -10215.0 q^{80} +211061. q^{81} -31708.4 q^{82} -63967.7 q^{83} +18558.0 q^{84} +7225.00 q^{85} +75627.1 q^{86} +92671.1 q^{87} -98758.8 q^{88} -145034. q^{89} -70419.7 q^{90} +23502.5 q^{91} -63787.2 q^{92} -124593. q^{93} +63356.1 q^{94} -52100.5 q^{95} +135028. q^{96} -131574. q^{97} +63233.6 q^{98} -330688. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 7 q^{2} - 36 q^{3} + 43 q^{4} + 125 q^{5} - 105 q^{6} - 204 q^{7} - 63 q^{8} + 531 q^{9} - 175 q^{10} - 792 q^{11} + 785 q^{12} + 88 q^{13} + 860 q^{14} - 900 q^{15} - 2365 q^{16} + 1445 q^{17} - 2052 q^{18}+ \cdots - 535112 q^{99}+O(q^{100}) \) 5 * q - 7 * q^2 - 36 * q^3 + 43 * q^4 + 125 * q^5 - 105 * q^6 - 204 * q^7 - 63 * q^8 + 531 * q^9 - 175 * q^10 - 792 * q^11 + 785 * q^12 + 88 * q^13 + 860 * q^14 - 900 * q^15 - 2365 * q^16 + 1445 * q^17 - 2052 * q^18 - 5160 * q^19 + 1075 * q^20 - 6428 * q^21 - 3058 * q^22 - 6140 * q^23 - 11751 * q^24 + 3125 * q^25 - 9643 * q^26 - 13380 * q^27 - 4568 * q^28 - 7328 * q^29 - 2625 * q^30 - 8968 * q^31 - 4703 * q^32 + 13072 * q^33 - 2023 * q^34 - 5100 * q^35 + 2048 * q^36 + 9834 * q^37 + 10625 * q^38 + 10108 * q^39 - 1575 * q^40 - 5302 * q^41 + 16222 * q^42 - 4916 * q^43 - 31514 * q^44 + 13275 * q^45 - 16768 * q^46 - 20364 * q^47 + 45557 * q^48 - 26483 * q^49 - 4375 * q^50 - 10404 * q^51 - 11525 * q^52 + 56492 * q^53 + 64635 * q^54 - 19800 * q^55 + 73228 * q^56 + 67554 * q^57 + 54213 * q^58 - 96200 * q^59 + 19625 * q^60 - 7284 * q^61 + 122335 * q^62 + 64108 * q^63 + 28483 * q^64 + 2200 * q^65 + 115232 * q^66 - 60308 * q^67 + 12427 * q^68 - 89076 * q^69 + 21500 * q^70 - 25304 * q^71 + 118722 * q^72 + 163328 * q^73 - 82768 * q^74 - 22500 * q^75 + 92027 * q^76 + 19912 * q^77 + 134895 * q^78 - 50940 * q^79 - 59125 * q^80 + 92445 * q^81 + 191390 * q^82 - 179060 * q^83 - 128054 * q^84 + 36125 * q^85 + 87086 * q^86 + 55764 * q^87 + 167634 * q^88 - 387356 * q^89 - 51300 * q^90 - 115200 * q^91 - 293228 * q^92 - 202558 * q^93 + 100515 * q^94 - 129000 * q^95 + 428113 * q^96 - 106076 * q^97 + 49329 * q^98 - 535112 * q^99

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