LMFDB - Embedded newform 82.6.a.a.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [82,6,Mod(1,82)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(82, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("82.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 82 = 2 \cdot 41 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	82.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$13.1514732247$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.73428.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 45x - 39 $ x^3 - x^2 - 45*x - 39
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.69713$ of defining polynomial
Character	$\chi$	$=$	82.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} -8.27134 q^{3} +16.0000 q^{4} -4.76019 q^{5} -33.0854 q^{6} -76.4095 q^{7} +64.0000 q^{8} -174.585 q^{9} -19.0407 q^{10} +124.811 q^{11} -132.341 q^{12} -935.146 q^{13} -305.638 q^{14} +39.3731 q^{15} +256.000 q^{16} +1142.21 q^{17} -698.340 q^{18} -3055.02 q^{19} -76.1630 q^{20} +632.009 q^{21} +499.243 q^{22} -3108.53 q^{23} -529.366 q^{24} -3102.34 q^{25} -3740.59 q^{26} +3453.99 q^{27} -1222.55 q^{28} +2771.90 q^{29} +157.492 q^{30} +575.249 q^{31} +1024.00 q^{32} -1032.35 q^{33} +4568.85 q^{34} +363.724 q^{35} -2793.36 q^{36} -1516.57 q^{37} -12220.1 q^{38} +7734.91 q^{39} -304.652 q^{40} +1681.00 q^{41} +2528.04 q^{42} +17117.9 q^{43} +1996.97 q^{44} +831.057 q^{45} -12434.1 q^{46} +26873.2 q^{47} -2117.46 q^{48} -10968.6 q^{49} -12409.4 q^{50} -9447.64 q^{51} -14962.3 q^{52} -36955.2 q^{53} +13815.9 q^{54} -594.122 q^{55} -4890.21 q^{56} +25269.1 q^{57} +11087.6 q^{58} +7484.42 q^{59} +629.970 q^{60} +45169.2 q^{61} +2301.00 q^{62} +13339.9 q^{63} +4096.00 q^{64} +4451.47 q^{65} -4129.41 q^{66} -57071.7 q^{67} +18275.4 q^{68} +25711.7 q^{69} +1454.89 q^{70} +36439.2 q^{71} -11173.4 q^{72} +56382.1 q^{73} -6066.27 q^{74} +25660.5 q^{75} -48880.4 q^{76} -9536.72 q^{77} +30939.7 q^{78} +36053.2 q^{79} -1218.61 q^{80} +13855.0 q^{81} +6724.00 q^{82} -4558.43 q^{83} +10112.1 q^{84} -5437.15 q^{85} +68471.8 q^{86} -22927.3 q^{87} +7987.88 q^{88} -135570. q^{89} +3324.23 q^{90} +71454.1 q^{91} -49736.5 q^{92} -4758.08 q^{93} +107493. q^{94} +14542.5 q^{95} -8469.85 q^{96} -141531. q^{97} -43874.3 q^{98} -21790.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 12 q^{2} - 28 q^{3} + 48 q^{4} - 26 q^{5} - 112 q^{6} - 158 q^{7} + 192 q^{8} - 61 q^{9} - 104 q^{10} - 846 q^{11} - 448 q^{12} - 1258 q^{13} - 632 q^{14} - 682 q^{15} + 768 q^{16} - 2166 q^{17} - 244 q^{18}+ \cdots - 176672 q^{99}+O(q^{100}) $ 3 * q + 12 * q^2 - 28 * q^3 + 48 * q^4 - 26 * q^5 - 112 * q^6 - 158 * q^7 + 192 * q^8 - 61 * q^9 - 104 * q^10 - 846 * q^11 - 448 * q^12 - 1258 * q^13 - 632 * q^14 - 682 * q^15 + 768 * q^16 - 2166 * q^17 - 244 * q^18 - 932 * q^19 - 416 * q^20 - 2776 * q^21 - 3384 * q^22 - 1960 * q^23 - 1792 * q^24 - 6987 * q^25 - 5032 * q^26 - 862 * q^27 - 2528 * q^28 - 3294 * q^29 - 2728 * q^30 + 4764 * q^31 + 3072 * q^32 + 13600 * q^33 - 8664 * q^34 + 10914 * q^35 - 976 * q^36 + 970 * q^37 - 3728 * q^38 + 27232 * q^39 - 1664 * q^40 + 5043 * q^41 - 11104 * q^42 + 1520 * q^43 - 13536 * q^44 + 17990 * q^45 - 7840 * q^46 - 9544 * q^47 - 7168 * q^48 + 2571 * q^49 - 27948 * q^50 + 36952 * q^51 - 20128 * q^52 - 10178 * q^53 - 3448 * q^54 - 1906 * q^55 - 10112 * q^56 + 18016 * q^57 - 13176 * q^58 + 5756 * q^59 - 10912 * q^60 + 25514 * q^61 + 19056 * q^62 + 91820 * q^63 + 12288 * q^64 - 29616 * q^65 + 54400 * q^66 + 25342 * q^67 - 34656 * q^68 + 12468 * q^69 + 43656 * q^70 + 59272 * q^71 - 3904 * q^72 + 95626 * q^73 + 3880 * q^74 + 83752 * q^75 - 14912 * q^76 - 22528 * q^77 + 108928 * q^78 + 15448 * q^79 - 6656 * q^80 + 32231 * q^81 + 20172 * q^82 + 88536 * q^83 - 44416 * q^84 - 1948 * q^85 + 6080 * q^86 + 142596 * q^87 - 54144 * q^88 - 88446 * q^89 + 71960 * q^90 - 85068 * q^91 - 31360 * q^92 - 233788 * q^93 - 38176 * q^94 - 39470 * q^95 - 28672 * q^96 - 323962 * q^97 + 10284 * q^98 - 176672 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−8.27134	−0.530607	−0.265303	−	0.964165i	$-0.585472\pi$
$3$	−8.27134	−0.530607	−0.265303	+	0.964165i	$0.585472\pi$
$4$	16.0000	0.500000
$5$	−4.76019	−0.0851528	−0.0425764	−	0.999093i	$-0.513557\pi$
$5$	−4.76019	−0.0851528	−0.0425764	+	0.999093i	$0.513557\pi$
$6$	−33.0854	−0.375196
$7$	−76.4095	−0.589389	−0.294695	−	0.955591i	$-0.595218\pi$
$7$	−76.4095	−0.589389	−0.294695	+	0.955591i	$0.595218\pi$
$8$	64.0000	0.353553
$9$	−174.585	−0.718456
$10$	−19.0407	−0.0602121
$11$	124.811	0.311007	0.155503	−	0.987835i	$-0.450300\pi$
$11$	124.811	0.311007	0.155503	+	0.987835i	$0.450300\pi$
$12$	−132.341	−0.265303
$13$	−935.146	−1.53469	−0.767346	−	0.641233i	$-0.778423\pi$
$13$	−935.146	−1.53469	−0.767346	+	0.641233i	$0.778423\pi$
$14$	−305.638	−0.416761
$15$	39.3731	0.0451826
$16$	256.000	0.250000
$17$	1142.21	0.958572	0.479286	−	0.877659i	$-0.340896\pi$
$17$	1142.21	0.958572	0.479286	+	0.877659i	$0.340896\pi$
$18$	−698.340	−0.508025
$19$	−3055.02	−1.94147	−0.970735	−	0.240154i	$-0.922802\pi$
$19$	−3055.02	−1.94147	−0.970735	+	0.240154i	$0.922802\pi$
$20$	−76.1630	−0.0425764
$21$	632.009	0.312734
$22$	499.243	0.219915
$23$	−3108.53	−1.22528	−0.612641	−	0.790362i	$-0.709893\pi$
$23$	−3108.53	−1.22528	−0.612641	+	0.790362i	$0.709893\pi$
$24$	−529.366	−0.187598
$25$	−3102.34	−0.992749
$26$	−3740.59	−1.08519
$27$	3453.99	0.911825
$28$	−1222.55	−0.294695
$29$	2771.90	0.612044	0.306022	−	0.952024i	$-0.401002\pi$
$29$	2771.90	0.612044	0.306022	+	0.952024i	$0.401002\pi$
$30$	157.492	0.0319490
$31$	575.249	0.107511	0.0537554	−	0.998554i	$-0.482881\pi$
$31$	575.249	0.107511	0.0537554	+	0.998554i	$0.482881\pi$
$32$	1024.00	0.176777
$33$	−1032.35	−0.165022
$34$	4568.85	0.677813
$35$	363.724	0.0501882
$36$	−2793.36	−0.359228
$37$	−1516.57	−0.182120	−0.0910599	−	0.995845i	$-0.529025\pi$
$37$	−1516.57	−0.182120	−0.0910599	+	0.995845i	$0.529025\pi$
$38$	−12220.1	−1.37283
$39$	7734.91	0.814318
$40$	−304.652	−0.0301061
$41$	1681.00	0.156174
$41$	1681.00	0.156174
$42$	2528.04	0.221136
$43$	17117.9	1.41182	0.705912	−	0.708300i	$-0.250538\pi$
$43$	17117.9	1.41182	0.705912	+	0.708300i	$0.250538\pi$
$44$	1996.97	0.155503
$45$	831.057	0.0611786
$46$	−12434.1	−0.866405
$47$	26873.2	1.77449	0.887246	−	0.461296i	$-0.152615\pi$
$47$	26873.2	1.77449	0.887246	+	0.461296i	$0.152615\pi$
$48$	−2117.46	−0.132652
$49$	−10968.6	−0.652620
$50$	−12409.4	−0.701980
$51$	−9447.64	−0.508625
$52$	−14962.3	−0.767346
$53$	−36955.2	−1.80712	−0.903558	−	0.428465i	$-0.859055\pi$
$53$	−36955.2	−1.80712	−0.903558	+	0.428465i	$0.859055\pi$
$54$	13815.9	0.644757
$55$	−594.122	−0.0264831
$56$	−4890.21	−0.208381
$57$	25269.1	1.03016
$58$	11087.6	0.432780
$59$	7484.42	0.279916	0.139958	−	0.990157i	$-0.455303\pi$
$59$	7484.42	0.279916	0.139958	+	0.990157i	$0.455303\pi$
$60$	629.970	0.0225913
$61$	45169.2	1.55424	0.777120	−	0.629352i	$-0.216680\pi$
$61$	45169.2	1.55424	0.777120	+	0.629352i	$0.216680\pi$
$62$	2301.00	0.0760216
$63$	13339.9	0.423451
$64$	4096.00	0.125000
$65$	4451.47	0.130683
$66$	−4129.41	−0.116688
$67$	−57071.7	−1.55322	−0.776612	−	0.629979i	$-0.783063\pi$
$67$	−57071.7	−1.55322	−0.776612	+	0.629979i	$0.783063\pi$
$68$	18275.4	0.479286
$69$	25711.7	0.650143
$70$	1454.89	0.0354884
$71$	36439.2	0.857873	0.428936	−	0.903335i	$-0.358888\pi$
$71$	36439.2	0.857873	0.428936	+	0.903335i	$0.358888\pi$
$72$	−11173.4	−0.254013
$73$	56382.1	1.23832	0.619162	−	0.785263i	$-0.287472\pi$
$73$	56382.1	1.23832	0.619162	+	0.785263i	$0.287472\pi$
$74$	−6066.27	−0.128778
$75$	25660.5	0.526759
$76$	−48880.4	−0.970735
$77$	−9536.72	−0.183304
$78$	30939.7	0.575810
$79$	36053.2	0.649944	0.324972	−	0.945724i	$-0.394645\pi$
$79$	36053.2	0.649944	0.324972	+	0.945724i	$0.394645\pi$
$80$	−1218.61	−0.0212882
$81$	13855.0	0.234636
$82$	6724.00	0.110432
$83$	−4558.43	−0.0726306	−0.0363153	−	0.999340i	$-0.511562\pi$
$83$	−4558.43	−0.0726306	−0.0363153	+	0.999340i	$0.511562\pi$
$84$	10112.1	0.156367
$85$	−5437.15	−0.0816251
$86$	68471.8	0.998310
$87$	−22927.3	−0.324755
$88$	7987.88	0.109958
$89$	−135570.	−1.81421	−0.907105	−	0.420905i	$-0.861713\pi$
$89$	−135570.	−1.81421	−0.907105	+	0.420905i	$0.861713\pi$
$90$	3324.23	0.0432598
$91$	71454.1	0.904531
$92$	−49736.5	−0.612641
$93$	−4758.08	−0.0570459
$94$	107493.	1.25476
$95$	14542.5	0.165322
$96$	−8469.85	−0.0937989
$97$	−141531.	−1.52729	−0.763647	−	0.645634i	$-0.776593\pi$
$97$	−141531.	−1.52729	−0.763647	+	0.645634i	$0.776593\pi$
$98$	−43874.3	−0.461472
$99$	−21790.1	−0.223445
$100$	−49637.5	−0.496375
$101$	−45753.4	−0.446293	−0.223147	−	0.974785i	$-0.571633\pi$
$101$	−45753.4	−0.446293	−0.223147	+	0.974785i	$0.571633\pi$
$102$	−37790.5	−0.359652
$103$	−90208.3	−0.837825	−0.418913	−	0.908027i	$-0.637589\pi$
$103$	−90208.3	−0.837825	−0.418913	+	0.908027i	$0.637589\pi$
$104$	−59849.4	−0.542596
$105$	−3008.48	−0.0266302
$106$	−147821.	−1.27782
$107$	42059.2	0.355141	0.177571	−	0.984108i	$-0.443176\pi$
$107$	42059.2	0.355141	0.177571	+	0.984108i	$0.443176\pi$
$108$	55263.8	0.455912
$109$	−244428.	−1.97054	−0.985268	−	0.171020i	$-0.945294\pi$
$109$	−244428.	−1.97054	−0.985268	+	0.171020i	$0.945294\pi$
$110$	−2376.49	−0.0187264
$111$	12544.0	0.0966340
$112$	−19560.8	−0.147347
$113$	−68385.3	−0.503809	−0.251905	−	0.967752i	$-0.581057\pi$
$113$	−68385.3	−0.503809	−0.251905	+	0.967752i	$0.581057\pi$
$114$	101077.	0.728431
$115$	14797.2	0.104336
$116$	44350.4	0.306022
$117$	163262.	1.10261
$118$	29937.7	0.197931
$119$	−87276.0	−0.564972
$120$	2519.88	0.0159745
$121$	−145473.	−0.903275
$122$	180677.	1.09901
$123$	−13904.1	−0.0828669
$124$	9203.99	0.0537554
$125$	29643.3	0.169688
$126$	53359.8	0.299425
$127$	−186682.	−1.02705	−0.513527	−	0.858073i	$-0.671662\pi$
$127$	−186682.	−1.02705	−0.513527	+	0.858073i	$0.671662\pi$
$128$	16384.0	0.0883883
$129$	−141588.	−0.749123
$130$	17805.9	0.0924071
$131$	−2821.99	−0.0143674	−0.00718368	−	0.999974i	$-0.502287\pi$
$131$	−2821.99	−0.0143674	−0.00718368	+	0.999974i	$0.502287\pi$
$132$	−16517.6	−0.0825112
$133$	233433.	1.14428
$134$	−228287.	−1.09830
$135$	−16441.6	−0.0776444
$136$	73101.6	0.338907
$137$	55704.6	0.253565	0.126782	−	0.991931i	$-0.459535\pi$
$137$	55704.6	0.253565	0.126782	+	0.991931i	$0.459535\pi$
$138$	102847.	0.459720
$139$	417622.	1.83336	0.916678	−	0.399627i	$-0.130860\pi$
$139$	417622.	1.83336	0.916678	+	0.399627i	$0.130860\pi$
$140$	5819.58	0.0250941
$141$	−222277.	−0.941558
$142$	145757.	0.606608
$143$	−116716.	−0.477300
$144$	−44693.7	−0.179614
$145$	−13194.8	−0.0521173
$146$	225528.	0.875627
$147$	90724.9	0.346285
$148$	−24265.1	−0.0910599
$149$	159690.	0.589268	0.294634	−	0.955610i	$-0.404802\pi$
$149$	159690.	0.589268	0.294634	+	0.955610i	$0.404802\pi$
$150$	102642.	0.372475
$151$	39532.4	0.141095	0.0705474	−	0.997508i	$-0.477525\pi$
$151$	39532.4	0.141095	0.0705474	+	0.997508i	$0.477525\pi$
$152$	−195522.	−0.686413
$153$	−199413.	−0.688692
$154$	−38146.9	−0.129616
$155$	−2738.29	−0.00915484
$156$	123759.	0.407159
$157$	−154842.	−0.501347	−0.250674	−	0.968072i	$-0.580652\pi$
$157$	−154842.	−0.501347	−0.250674	+	0.968072i	$0.580652\pi$
$158$	144213.	0.459580
$159$	305669.	0.958868
$160$	−4874.43	−0.0150530
$161$	237521.	0.722168
$162$	55420.1	0.165913
$163$	−104196.	−0.307171	−0.153586	−	0.988135i	$-0.549082\pi$
$163$	−104196.	−0.307171	−0.153586	+	0.988135i	$0.549082\pi$
$164$	26896.0	0.0780869
$165$	4914.18	0.0140521
$166$	−18233.7	−0.0513576
$167$	315536.	0.875503	0.437751	−	0.899096i	$-0.355775\pi$
$167$	315536.	0.875503	0.437751	+	0.899096i	$0.355775\pi$
$168$	40448.6	0.110568
$169$	503206.	1.35528
$170$	−21748.6	−0.0577177
$171$	533361.	1.39486
$172$	273887.	0.705912
$173$	−657453.	−1.67013	−0.835064	−	0.550153i	$-0.814569\pi$
$173$	−657453.	−1.67013	−0.835064	+	0.550153i	$0.814569\pi$
$174$	−91709.3	−0.229636
$175$	237048.	0.585116
$176$	31951.5	0.0777517
$177$	−61906.1	−0.148525
$178$	−542279.	−1.28284
$179$	−545805.	−1.27322	−0.636612	−	0.771184i	$-0.719665\pi$
$179$	−545805.	−1.27322	−0.636612	+	0.771184i	$0.719665\pi$
$180$	13296.9	0.0305893
$181$	124475.	0.282413	0.141207	−	0.989980i	$-0.454902\pi$
$181$	124475.	0.282413	0.141207	+	0.989980i	$0.454902\pi$
$182$	285816.	0.639600
$183$	−373610.	−0.824690
$184$	−198946.	−0.433202
$185$	7219.14	0.0155080
$186$	−19032.3	−0.0403376
$187$	142560.	0.298123
$188$	429971.	0.887246
$189$	−263917.	−0.537420
$190$	58169.9	0.116900
$191$	460797.	0.913957	0.456979	−	0.889478i	$-0.348932\pi$
$191$	460797.	0.913957	0.456979	+	0.889478i	$0.348932\pi$
$192$	−33879.4	−0.0663258
$193$	342453.	0.661772	0.330886	−	0.943671i	$-0.392653\pi$
$193$	342453.	0.661772	0.330886	+	0.943671i	$0.392653\pi$
$194$	−566125.	−1.07996
$195$	−36819.6	−0.0693415
$196$	−175497.	−0.326310
$197$	3226.14	0.00592266	0.00296133	−	0.999996i	$-0.499057\pi$
$197$	3226.14	0.00592266	0.00296133	+	0.999996i	$0.499057\pi$
$198$	−87160.2	−0.157999
$199$	66020.1	0.118180	0.0590900	−	0.998253i	$-0.481180\pi$
$199$	66020.1	0.118180	0.0590900	+	0.998253i	$0.481180\pi$
$200$	−198550.	−0.350990
$201$	472060.	0.824151
$202$	−183014.	−0.315577
$203$	−211800.	−0.360732
$204$	−151162.	−0.254312
$205$	−8001.87	−0.0132986
$206$	−360833.	−0.592432
$207$	542703.	0.880311
$208$	−239397.	−0.383673
$209$	−381300.	−0.603810
$210$	−12033.9	−0.0188304
$211$	746065.	1.15364	0.576820	−	0.816871i	$-0.304293\pi$
$211$	746065.	1.15364	0.576820	+	0.816871i	$0.304293\pi$
$212$	−591284.	−0.903558
$213$	−301401.	−0.455193
$214$	168237.	0.251123
$215$	−81484.6	−0.120221
$216$	221055.	0.322379
$217$	−43954.5	−0.0633657
$218$	−977711.	−1.39338
$219$	−466356.	−0.657063
$220$	−9505.95	−0.0132416
$221$	−1.06814e6	−1.47111
$222$	50176.2	0.0683306
$223$	332416.	0.447630	0.223815	−	0.974632i	$-0.428149\pi$
$223$	332416.	0.447630	0.223815	+	0.974632i	$0.428149\pi$
$224$	−78243.3	−0.104190
$225$	541622.	0.713247
$226$	−273541.	−0.356247
$227$	−1.00980e6	−1.30069	−0.650343	−	0.759640i	$-0.725375\pi$
$227$	−1.00980e6	−1.30069	−0.650343	+	0.759640i	$0.725375\pi$
$228$	404306.	0.515078
$229$	32659.2	0.0411545	0.0205772	−	0.999788i	$-0.493450\pi$
$229$	32659.2	0.0411545	0.0205772	+	0.999788i	$0.493450\pi$
$230$	59188.8	0.0737768
$231$	78881.5	0.0972624
$232$	177402.	0.216390
$233$	−902680.	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$233$	−902680.	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$234$	653050.	0.779663
$235$	−127921.	−0.151103
$236$	119751.	0.139958
$237$	−298208.	−0.344865
$238$	−349104.	−0.399496
$239$	200303.	0.226826	0.113413	−	0.993548i	$-0.463822\pi$
$239$	200303.	0.226826	0.113413	+	0.993548i	$0.463822\pi$
$240$	10079.5	0.0112957
$241$	296117.	0.328414	0.164207	−	0.986426i	$-0.447494\pi$
$241$	296117.	0.328414	0.164207	+	0.986426i	$0.447494\pi$
$242$	−581893.	−0.638712
$243$	−953919.	−1.03632
$244$	722708.	0.777120
$245$	52212.5	0.0555724
$246$	−55616.5	−0.0585957
$247$	2.85689e6	2.97956
$248$	36816.0	0.0380108
$249$	37704.3	0.0385383
$250$	118573.	0.119988
$251$	215205.	0.215610	0.107805	−	0.994172i	$-0.465618\pi$
$251$	215205.	0.215610	0.107805	+	0.994172i	$0.465618\pi$
$252$	213439.	0.211725
$253$	−387978.	−0.381071
$254$	−746729.	−0.726238
$255$	44972.5	0.0433108
$256$	65536.0	0.0625000
$257$	−287466.	−0.271490	−0.135745	−	0.990744i	$-0.543343\pi$
$257$	−287466.	−0.271490	−0.135745	+	0.990744i	$0.543343\pi$
$258$	−566353.	−0.529710
$259$	115880.	0.107339
$260$	71223.5	0.0653417
$261$	−483932.	−0.439727
$262$	−11287.9	−0.0101593
$263$	−225651.	−0.201163	−0.100582	−	0.994929i	$-0.532070\pi$
$263$	−225651.	−0.201163	−0.100582	+	0.994929i	$0.532070\pi$
$264$	−66070.5	−0.0583442
$265$	175914.	0.153881
$266$	933732.	0.809129
$267$	1.12134e6	0.962632
$268$	−913148.	−0.776612
$269$	644667.	0.543193	0.271597	−	0.962411i	$-0.412448\pi$
$269$	644667.	0.543193	0.271597	+	0.962411i	$0.412448\pi$
$270$	−65766.5	−0.0549029
$271$	81510.0	0.0674199	0.0337099	−	0.999432i	$-0.489268\pi$
$271$	81510.0	0.0674199	0.0337099	+	0.999432i	$0.489268\pi$
$272$	292407.	0.239643
$273$	−591021.	−0.479950
$274$	222818.	0.179298
$275$	−387205.	−0.308752
$276$	411388.	0.325071
$277$	−748512.	−0.586138	−0.293069	−	0.956091i	$-0.594677\pi$
$277$	−748512.	−0.586138	−0.293069	+	0.956091i	$0.594677\pi$
$278$	1.67049e6	1.29638
$279$	−100430.	−0.0772418
$280$	23278.3	0.0177442
$281$	−1.80784e6	−1.36583	−0.682913	−	0.730500i	$-0.739287\pi$
$281$	−1.80784e6	−1.36583	−0.682913	+	0.730500i	$0.739287\pi$
$282$	−889109.	−0.665782
$283$	−75266.3	−0.0558643	−0.0279321	−	0.999610i	$-0.508892\pi$
$283$	−75266.3	−0.0558643	−0.0279321	+	0.999610i	$0.508892\pi$
$284$	583027.	0.428936
$285$	−120286.	−0.0877207
$286$	−466865.	−0.337502
$287$	−128444.	−0.0920472
$288$	−178775.	−0.127006
$289$	−115206.	−0.0811391
$290$	−52779.0	−0.0368525
$291$	1.17065e6	0.810393
$292$	902114.	0.619162
$293$	−100696.	−0.0685243	−0.0342621	−	0.999413i	$-0.510908\pi$
$293$	−100696.	−0.0685243	−0.0342621	+	0.999413i	$0.510908\pi$
$294$	362900.	0.244860
$295$	−35627.2	−0.0238356
$296$	−97060.3	−0.0643891
$297$	431094.	0.283584
$298$	638761.	0.416675
$299$	2.90693e6	1.88043
$300$	410568.	0.263380
$301$	−1.30797e6	−0.832114
$302$	158130.	0.0997690
$303$	378442.	0.236806
$304$	−782086.	−0.485367
$305$	−215014.	−0.132348
$306$	−797653.	−0.486979
$307$	2.30045e6	1.39305	0.696527	−	0.717531i	$-0.254728\pi$
$307$	2.30045e6	1.39305	0.696527	+	0.717531i	$0.254728\pi$
$308$	−152588.	−0.0916521
$309$	746144.	0.444556
$310$	−10953.2	−0.00647345
$311$	−2.34404e6	−1.37424	−0.687122	−	0.726542i	$-0.741126\pi$
$311$	−2.34404e6	−1.37424	−0.687122	+	0.726542i	$0.741126\pi$
$312$	495035.	0.287905
$313$	511798.	0.295283	0.147641	−	0.989041i	$-0.452832\pi$
$313$	511798.	0.295283	0.147641	+	0.989041i	$0.452832\pi$
$314$	−619367.	−0.354506
$315$	−63500.6	−0.0360580
$316$	576851.	0.324972
$317$	365311.	0.204180	0.102090	−	0.994775i	$-0.467447\pi$
$317$	365311.	0.204180	0.102090	+	0.994775i	$0.467447\pi$
$318$	1.22268e6	0.678022
$319$	345963.	0.190350
$320$	−19497.7	−0.0106441
$321$	−347886.	−0.188440
$322$	950086.	0.510650
$323$	−3.48949e6	−1.86104
$324$	221680.	0.117318
$325$	2.90114e6	1.52356
$326$	−416782.	−0.217203
$327$	2.02174e6	1.04558
$328$	107584.	0.0552158
$329$	−2.05337e6	−1.04587
$330$	19656.7	0.00993634
$331$	−3.50026e6	−1.75602	−0.878011	−	0.478640i	$-0.841130\pi$
$331$	−3.50026e6	−1.75602	−0.878011	+	0.478640i	$0.841130\pi$
$332$	−72934.9	−0.0363153
$333$	264770.	0.130845
$334$	1.26214e6	0.619074
$335$	271672.	0.132261
$336$	161794.	0.0781835
$337$	−228628.	−0.109661	−0.0548307	−	0.998496i	$-0.517462\pi$
$337$	−228628.	−0.109661	−0.0548307	+	0.998496i	$0.517462\pi$
$338$	2.01282e6	0.958327
$339$	565638.	0.267325
$340$	−86994.4	−0.0408126
$341$	71797.3	0.0334366
$342$	2.13344e6	0.986316
$343$	2.12232e6	0.974037
$344$	1.09555e6	0.499155
$345$	−122393.	−0.0553615
$346$	−2.62981e6	−1.18096
$347$	−3.56146e6	−1.58783	−0.793915	−	0.608028i	$-0.791961\pi$
$347$	−3.56146e6	−1.58783	−0.793915	+	0.608028i	$0.791961\pi$
$348$	−366837.	−0.162377
$349$	−3.23779e6	−1.42293	−0.711467	−	0.702720i	$-0.751969\pi$
$349$	−3.23779e6	−1.42293	−0.711467	+	0.702720i	$0.751969\pi$
$350$	948193.	0.413739
$351$	−3.22998e6	−1.39937
$352$	127806.	0.0549788
$353$	3.71627e6	1.58734	0.793671	−	0.608347i	$-0.208167\pi$
$353$	3.71627e6	1.58734	0.793671	+	0.608347i	$0.208167\pi$
$354$	−247625.	−0.105023
$355$	−173457.	−0.0730503
$356$	−2.16911e6	−0.907105
$357$	721889.	0.299778
$358$	−2.18322e6	−0.900306
$359$	−1.56926e6	−0.642627	−0.321314	−	0.946973i	$-0.604124\pi$
$359$	−1.56926e6	−0.642627	−0.321314	+	0.946973i	$0.604124\pi$
$360$	53187.6	0.0216299
$361$	6.85707e6	2.76930
$362$	497899.	0.199696
$363$	1.20326e6	0.479284
$364$	1.14327e6	0.452266
$365$	−268389.	−0.105447
$366$	−1.49444e6	−0.583144
$367$	2.34115e6	0.907327	0.453664	−	0.891173i	$-0.350117\pi$
$367$	2.34115e6	0.907327	0.453664	+	0.891173i	$0.350117\pi$
$368$	−795784.	−0.306320
$369$	−293477.	−0.112204
$370$	28876.6	0.0109658
$371$	2.82373e6	1.06510
$372$	−76129.3	−0.0285230
$373$	−3.55497e6	−1.32301	−0.661506	−	0.749940i	$-0.730083\pi$
$373$	−3.55497e6	−1.32301	−0.661506	+	0.749940i	$0.730083\pi$
$374$	570242.	0.210804
$375$	−245190.	−0.0900377
$376$	1.71988e6	0.627378
$377$	−2.59213e6	−0.939299
$378$	−1.05567e6	−0.380013
$379$	−117840.	−0.0421401	−0.0210700	−	0.999778i	$-0.506707\pi$
$379$	−117840.	−0.0421401	−0.0210700	+	0.999778i	$0.506707\pi$
$380$	232680.	0.0826608
$381$	1.54411e6	0.544962
$382$	1.84319e6	0.646265
$383$	−2.59595e6	−0.904274	−0.452137	−	0.891949i	$-0.649338\pi$
$383$	−2.59595e6	−0.904274	−0.452137	+	0.891949i	$0.649338\pi$
$384$	−135518.	−0.0468995
$385$	45396.6	0.0156089
$386$	1.36981e6	0.467943
$387$	−2.98853e6	−1.01433
$388$	−2.26450e6	−0.763647
$389$	−988327.	−0.331152	−0.165576	−	0.986197i	$-0.552948\pi$
$389$	−988327.	−0.331152	−0.165576	+	0.986197i	$0.552948\pi$
$390$	−147279.	−0.0490318
$391$	−3.55061e6	−1.17452
$392$	−701989.	−0.230736
$393$	23341.6	0.00762341
$394$	12904.5	0.00418796
$395$	−171620.	−0.0553445
$396$	−348641.	−0.111722
$397$	−3.82557e6	−1.21821	−0.609103	−	0.793091i	$-0.708470\pi$
$397$	−3.82557e6	−1.21821	−0.609103	+	0.793091i	$0.708470\pi$
$398$	264081.	0.0835658
$399$	−1.93080e6	−0.607164
$400$	−794199.	−0.248187
$401$	4.62686e6	1.43690	0.718449	−	0.695580i	$-0.244853\pi$
$401$	4.62686e6	1.43690	0.718449	+	0.695580i	$0.244853\pi$
$402$	1.88824e6	0.582763
$403$	−537942.	−0.164996
$404$	−732055.	−0.223147
$405$	−65952.5	−0.0199799
$406$	−847198.	−0.255076
$407$	−189284.	−0.0566405
$408$	−604649.	−0.179826
$409$	1.18106e6	0.349110	0.174555	−	0.984647i	$-0.444151\pi$
$409$	1.18106e6	0.349110	0.174555	+	0.984647i	$0.444151\pi$
$410$	−32007.5	−0.00940355
$411$	−460751.	−0.134543
$412$	−1.44333e6	−0.418913
$413$	−571881.	−0.164980
$414$	2.17081e6	0.622474
$415$	21699.0	0.00618470
$416$	−957590.	−0.271298
$417$	−3.45430e6	−0.972791
$418$	−1.52520e6	−0.426958
$419$	4.81888e6	1.34094	0.670472	−	0.741935i	$-0.266092\pi$
$419$	4.81888e6	1.34094	0.670472	+	0.741935i	$0.266092\pi$
$420$	−48135.7	−0.0133151
$421$	−4.66024e6	−1.28145	−0.640727	−	0.767769i	$-0.721367\pi$
$421$	−4.66024e6	−1.28145	−0.640727	+	0.767769i	$0.721367\pi$
$422$	2.98426e6	0.815747
$423$	−4.69165e6	−1.27490
$424$	−2.36513e6	−0.638912
$425$	−3.54353e6	−0.951622
$426$	−1.20560e6	−0.321870
$427$	−3.45136e6	−0.916053
$428$	672947.	0.177571
$429$	965400.	0.253258
$430$	−325938.	−0.0850089
$431$	2.30312e6	0.597206	0.298603	−	0.954377i	$-0.403479\pi$
$431$	2.30312e6	0.597206	0.298603	+	0.954377i	$0.403479\pi$
$432$	884221.	0.227956
$433$	4.87584e6	1.24977	0.624885	−	0.780717i	$-0.285146\pi$
$433$	4.87584e6	1.24977	0.624885	+	0.780717i	$0.285146\pi$
$434$	−175818.	−0.0448063
$435$	109138.	0.0276538
$436$	−3.91084e6	−0.985268
$437$	9.49664e6	2.37885
$438$	−1.86542e6	−0.464614
$439$	6.46368e6	1.60073	0.800365	−	0.599512i	$-0.204639\pi$
$439$	6.46368e6	1.60073	0.800365	+	0.599512i	$0.204639\pi$
$440$	−38023.8	−0.00936319
$441$	1.91495e6	0.468879
$442$	−4.27255e6	−1.04023
$443$	−6.65119e6	−1.61024	−0.805119	−	0.593114i	$-0.797898\pi$
$443$	−6.65119e6	−1.61024	−0.805119	+	0.593114i	$0.797898\pi$
$444$	200705.	0.0483170
$445$	645337.	0.154485
$446$	1.32966e6	0.316522
$447$	−1.32085e6	−0.312670
$448$	−312973.	−0.0736737
$449$	−3.46875e6	−0.812002	−0.406001	−	0.913873i	$-0.633077\pi$
$449$	−3.46875e6	−0.812002	−0.406001	+	0.913873i	$0.633077\pi$
$450$	2.16649e6	0.504342
$451$	209807.	0.0485711
$452$	−1.09416e6	−0.251905
$453$	−326986.	−0.0748658
$454$	−4.03922e6	−0.919725
$455$	−340135.	−0.0770234
$456$	1.61723e6	0.364215
$457$	1.77221e6	0.396940	0.198470	−	0.980107i	$-0.436403\pi$
$457$	1.77221e6	0.396940	0.198470	+	0.980107i	$0.436403\pi$
$458$	130637.	0.0291006
$459$	3.94519e6	0.874050
$460$	236755.	0.0521681
$461$	6.05367e6	1.32668	0.663340	−	0.748318i	$-0.269138\pi$
$461$	6.05367e6	1.32668	0.663340	+	0.748318i	$0.269138\pi$
$462$	315526.	0.0687749
$463$	1.40006e6	0.303524	0.151762	−	0.988417i	$-0.451505\pi$
$463$	1.40006e6	0.303524	0.151762	+	0.988417i	$0.451505\pi$
$464$	709607.	0.153011
$465$	22649.4	0.00485762
$466$	−3.61072e6	−0.770245
$467$	−4.16130e6	−0.882950	−0.441475	−	0.897273i	$-0.645545\pi$
$467$	−4.16130e6	−0.882950	−0.441475	+	0.897273i	$0.645545\pi$
$468$	2.61220e6	0.551305
$469$	4.36082e6	0.915454
$470$	−511685.	−0.106846
$471$	1.28075e6	0.266018
$472$	479003.	0.0989653
$473$	2.13650e6	0.439087
$474$	−1.19283e6	−0.243856
$475$	9.47772e6	1.92739
$476$	−1.39642e6	−0.282486
$477$	6.45183e6	1.29833
$478$	801212.	0.160390
$479$	4.06980e6	0.810466	0.405233	−	0.914213i	$-0.367190\pi$
$479$	4.06980e6	0.810466	0.405233	+	0.914213i	$0.367190\pi$
$480$	40318.1	0.00798724
$481$	1.41821e6	0.279498
$482$	1.18447e6	0.232224
$483$	−1.96462e6	−0.383187
$484$	−2.32757e6	−0.451637
$485$	673715.	0.130053
$486$	−3.81567e6	−0.732792
$487$	3.53330e6	0.675085	0.337542	−	0.941310i	$-0.390404\pi$
$487$	3.53330e6	0.675085	0.337542	+	0.941310i	$0.390404\pi$
$488$	2.89083e6	0.549507
$489$	861837.	0.162987
$490$	208850.	0.0392956
$491$	−3.03584e6	−0.568297	−0.284149	−	0.958780i	$-0.591711\pi$
$491$	−3.03584e6	−0.568297	−0.284149	+	0.958780i	$0.591711\pi$
$492$	−222466.	−0.0414334
$493$	3.16610e6	0.586688
$494$	1.14276e7	2.10687
$495$	103725.	0.0190270
$496$	147264.	0.0268777
$497$	−2.78430e6	−0.505621
$498$	150817.	0.0272507
$499$	−4.82818e6	−0.868025	−0.434012	−	0.900907i	$-0.642903\pi$
$499$	−4.82818e6	−0.868025	−0.434012	+	0.900907i	$0.642903\pi$
$500$	474293.	0.0848441
$501$	−2.60990e6	−0.464548
$502$	860820.	0.152459
$503$	1.00862e7	1.77750	0.888748	−	0.458395i	$-0.151576\pi$
$503$	1.00862e7	1.77750	0.888748	+	0.458395i	$0.151576\pi$
$504$	853757.	0.149712
$505$	217795.	0.0380031
$506$	−1.55191e6	−0.269458
$507$	−4.16219e6	−0.719121
$508$	−2.98692e6	−0.513527
$509$	1.03984e7	1.77898	0.889490	−	0.456954i	$-0.151060\pi$
$509$	1.03984e7	1.77898	0.889490	+	0.456954i	$0.151060\pi$
$510$	179890.	0.0306254
$511$	−4.30813e6	−0.729855
$512$	262144.	0.0441942
$513$	−1.05520e7	−1.77028
$514$	−1.14987e6	−0.191973
$515$	429408.	0.0713432
$516$	−2.26541e6	−0.374561
$517$	3.35406e6	0.551879
$518$	463521.	0.0759005
$519$	5.43802e6	0.886181
$520$	284894.	0.0462035
$521$	8.52366e6	1.37573	0.687863	−	0.725841i	$-0.258549\pi$
$521$	8.52366e6	1.37573	0.687863	+	0.725841i	$0.258549\pi$
$522$	−1.93573e6	−0.310934
$523$	−7.61908e6	−1.21800	−0.609001	−	0.793169i	$-0.708430\pi$
$523$	−7.61908e6	−1.21800	−0.609001	+	0.793169i	$0.708430\pi$
$524$	−45151.8	−0.00718368
$525$	−1.96071e6	−0.310466
$526$	−902606.	−0.142244
$527$	657058.	0.103057
$528$	−264282.	−0.0412556
$529$	3.22663e6	0.501314
$530$	703655.	0.108810
$531$	−1.30667e6	−0.201108
$532$	3.73493e6	0.572141
$533$	−1.57198e6	−0.239679
$534$	4.48537e6	0.680684
$535$	−200209.	−0.0302413
$536$	−3.65259e6	−0.549148
$537$	4.51454e6	0.675582
$538$	2.57867e6	0.384096
$539$	−1.36900e6	−0.202969
$540$	−263066.	−0.0388222
$541$	6.23795e6	0.916324	0.458162	−	0.888869i	$-0.348508\pi$
$541$	6.23795e6	0.916324	0.458162	+	0.888869i	$0.348508\pi$
$542$	326040.	0.0476730
$543$	−1.02957e6	−0.149850
$544$	1.16963e6	0.169453
$545$	1.16352e6	0.167797
$546$	−2.36408e6	−0.339376
$547$	−1.04568e7	−1.49427	−0.747135	−	0.664672i	$-0.768571\pi$
$547$	−1.04568e7	−1.49427	−0.747135	+	0.664672i	$0.768571\pi$
$548$	891273.	0.126782
$549$	−7.88587e6	−1.11665
$550$	−1.54882e6	−0.218320
$551$	−8.46822e6	−1.18826
$552$	1.64555e6	0.229860
$553$	−2.75481e6	−0.383070
$554$	−2.99405e6	−0.414462
$555$	−59712.0	−0.00822866
$556$	6.68196e6	0.916678
$557$	−358470.	−0.0489570	−0.0244785	−	0.999700i	$-0.507793\pi$
$557$	−358470.	−0.0489570	−0.0244785	+	0.999700i	$0.507793\pi$
$558$	−401720.	−0.0546182
$559$	−1.60078e7	−2.16671
$560$	93113.2	0.0125470
$561$	−1.17917e6	−0.158186
$562$	−7.23138e6	−0.965785
$563$	−1.07188e7	−1.42519	−0.712597	−	0.701573i	$-0.752481\pi$
$563$	−1.07188e7	−1.42519	−0.712597	+	0.701573i	$0.752481\pi$
$564$	−3.55643e6	−0.470779
$565$	325527.	0.0429008
$566$	−301065.	−0.0395020
$567$	−1.05866e6	−0.138292
$568$	2.33211e6	0.303304
$569$	3.31337e6	0.429032	0.214516	−	0.976721i	$-0.431183\pi$
$569$	3.31337e6	0.429032	0.214516	+	0.976721i	$0.431183\pi$
$570$	−481143.	−0.0620279
$571$	−8.57422e6	−1.10054	−0.550268	−	0.834988i	$-0.685475\pi$
$571$	−8.57422e6	−1.10054	−0.550268	+	0.834988i	$0.685475\pi$
$572$	−1.86746e6	−0.238650
$573$	−3.81141e6	−0.484952
$574$	−513778.	−0.0650872
$575$	9.64373e6	1.21640
$576$	−715100.	−0.0898071
$577$	−1.29993e7	−1.62547	−0.812737	−	0.582631i	$-0.802023\pi$
$577$	−1.29993e7	−1.62547	−0.812737	+	0.582631i	$0.802023\pi$
$578$	−460823.	−0.0573740
$579$	−2.83255e6	−0.351140
$580$	−211116.	−0.0260586
$581$	348307.	0.0428077
$582$	4.68261e6	0.573034
$583$	−4.61241e6	−0.562026
$584$	3.60846e6	0.437814
$585$	−777160.	−0.0938903
$586$	−402785.	−0.0484540
$587$	1.27659e7	1.52917	0.764584	−	0.644524i	$-0.222944\pi$
$587$	1.27659e7	1.52917	0.764584	+	0.644524i	$0.222944\pi$
$588$	1.45160e6	0.173142
$589$	−1.75740e6	−0.208729
$590$	−142509.	−0.0168543
$591$	−26684.5	−0.00314261
$592$	−388241.	−0.0455300
$593$	−1.18387e7	−1.38251	−0.691253	−	0.722613i	$-0.742941\pi$
$593$	−1.18387e7	−1.38251	−0.691253	+	0.722613i	$0.742941\pi$
$594$	1.72438e6	0.200524
$595$	415450.	0.0481090
$596$	2.55505e6	0.294634
$597$	−546075.	−0.0627071
$598$	1.16277e7	1.32966
$599$	7.06748e6	0.804818	0.402409	−	0.915460i	$-0.368173\pi$
$599$	7.06748e6	0.804818	0.402409	+	0.915460i	$0.368173\pi$
$600$	1.64227e6	0.186238
$601$	−6.07935e6	−0.686548	−0.343274	−	0.939235i	$-0.611536\pi$
$601$	−6.07935e6	−0.686548	−0.343274	+	0.939235i	$0.611536\pi$
$602$	−5.23189e6	−0.588393
$603$	9.96387e6	1.11592
$604$	632518.	0.0705474
$605$	692480.	0.0769164
$606$	1.51377e6	0.167447
$607$	1.17492e7	1.29431	0.647154	−	0.762360i	$-0.275959\pi$
$607$	1.17492e7	1.29431	0.647154	+	0.762360i	$0.275959\pi$
$608$	−3.12834e6	−0.343207
$609$	1.75187e6	0.191407
$610$	−860056.	−0.0935841
$611$	−2.51303e7	−2.72330
$612$	−3.19061e6	−0.344346
$613$	1.54849e6	0.166440	0.0832198	−	0.996531i	$-0.473480\pi$
$613$	1.54849e6	0.166440	0.0832198	+	0.996531i	$0.473480\pi$
$614$	9.20182e6	0.985038
$615$	66186.2	0.00705634
$616$	−610350.	−0.0648078
$617$	953292.	0.100812	0.0504061	−	0.998729i	$-0.483948\pi$
$617$	953292.	0.100812	0.0504061	+	0.998729i	$0.483948\pi$
$618$	2.98457e6	0.314348
$619$	−3.71124e6	−0.389307	−0.194654	−	0.980872i	$-0.562358\pi$
$619$	−3.71124e6	−0.389307	−0.194654	+	0.980872i	$0.562358\pi$
$620$	−43812.7	−0.00457742
$621$	−1.07368e7	−1.11724
$622$	−9.37616e6	−0.971738
$623$	1.03588e7	1.06928
$624$	1.98014e6	0.203579
$625$	9.55371e6	0.978300
$626$	2.04719e6	0.208796
$627$	3.15386e6	0.320386
$628$	−2.47747e6	−0.250674
$629$	−1.73224e6	−0.174575
$630$	−254003.	−0.0254969
$631$	1.37612e7	1.37589	0.687945	−	0.725763i	$-0.258513\pi$
$631$	1.37612e7	1.37589	0.687945	+	0.725763i	$0.258513\pi$
$632$	2.30740e6	0.229790
$633$	−6.17096e6	−0.612129
$634$	1.46124e6	0.144377
$635$	888642.	0.0874566
$636$	4.89071e6	0.479434
$637$	1.02572e7	1.00157
$638$	1.38385e6	0.134598
$639$	−6.36173e6	−0.616344
$640$	−77990.9	−0.00752651
$641$	−2.70568e6	−0.260094	−0.130047	−	0.991508i	$-0.541513\pi$
$641$	−2.70568e6	−0.260094	−0.130047	+	0.991508i	$0.541513\pi$
$642$	−1.39154e6	−0.133247
$643$	−8.40013e6	−0.801232	−0.400616	−	0.916246i	$-0.631204\pi$
$643$	−8.40013e6	−0.801232	−0.400616	+	0.916246i	$0.631204\pi$
$644$	3.80034e6	0.361084
$645$	673987.	0.0637899
$646$	−1.39580e7	−1.31595
$647$	−1.60069e6	−0.150331	−0.0751654	−	0.997171i	$-0.523948\pi$
$647$	−1.60069e6	−0.150331	−0.0751654	+	0.997171i	$0.523948\pi$
$648$	886722.	0.0829564
$649$	934135.	0.0870558
$650$	1.16046e7	1.07732
$651$	363563.	0.0336223
$652$	−1.66713e6	−0.153586
$653$	−1.22103e7	−1.12058	−0.560290	−	0.828297i	$-0.689310\pi$
$653$	−1.22103e7	−1.12058	−0.560290	+	0.828297i	$0.689310\pi$
$654$	8.08698e6	0.739336
$655$	13433.2	0.00122342
$656$	430336.	0.0390434
$657$	−9.84347e6	−0.889682
$658$	−8.21346e6	−0.739540
$659$	1.29323e7	1.16001	0.580004	−	0.814614i	$-0.303051\pi$
$659$	1.29323e7	1.16001	0.580004	+	0.814614i	$0.303051\pi$
$660$	78627.0	0.00702606
$661$	9.45698e6	0.841877	0.420939	−	0.907089i	$-0.361701\pi$
$661$	9.45698e6	0.841877	0.420939	+	0.907089i	$0.361701\pi$
$662$	−1.40010e7	−1.24170
$663$	8.83492e6	0.780583
$664$	−291739.	−0.0256788
$665$	−1.11118e6	−0.0974388
$666$	1.05908e6	0.0925215
$667$	−8.61654e6	−0.749926
$668$	5.04857e6	0.437751
$669$	−2.74952e6	−0.237516
$670$	1.08669e6	0.0935229
$671$	5.63760e6	0.483379
$672$	647177.	0.0552841
$673$	−3.75732e6	−0.319772	−0.159886	−	0.987135i	$-0.551113\pi$
$673$	−3.75732e6	−0.319772	−0.159886	+	0.987135i	$0.551113\pi$
$674$	−914511.	−0.0775424
$675$	−1.07154e7	−0.905213
$676$	8.05129e6	0.677640
$677$	−1.18521e7	−0.993859	−0.496930	−	0.867791i	$-0.665539\pi$
$677$	−1.18521e7	−0.993859	−0.496930	+	0.867791i	$0.665539\pi$
$678$	2.26255e6	0.189027
$679$	1.08143e7	0.900171
$680$	−347977.	−0.0288588
$681$	8.35244e6	0.690153
$682$	287189.	0.0236432
$683$	−9.53794e6	−0.782353	−0.391177	−	0.920316i	$-0.627932\pi$
$683$	−9.53794e6	−0.782353	−0.391177	+	0.920316i	$0.627932\pi$
$684$	8.53378e6	0.697431
$685$	−265164.	−0.0215918
$686$	8.48928e6	0.688748
$687$	−270136.	−0.0218368
$688$	4.38219e6	0.352956
$689$	3.45586e7	2.77337
$690$	−489570.	−0.0391465
$691$	−9.79155e6	−0.780111	−0.390055	−	0.920791i	$-0.627544\pi$
$691$	−9.79155e6	−0.780111	−0.390055	+	0.920791i	$0.627544\pi$
$692$	−1.05193e7	−0.835064
$693$	1.66497e6	0.131696
$694$	−1.42458e7	−1.12277
$695$	−1.98796e6	−0.156115
$696$	−1.46735e6	−0.114818
$697$	1.92006e6	0.149704
$698$	−1.29511e7	−1.00617
$699$	7.46638e6	0.577985
$700$	3.79277e6	0.292558
$701$	2.40504e6	0.184853	0.0924265	−	0.995720i	$-0.470538\pi$
$701$	2.40504e6	0.184853	0.0924265	+	0.995720i	$0.470538\pi$
$702$	−1.29199e7	−0.989504
$703$	4.63315e6	0.353580
$704$	511224.	0.0388759
$705$	1.05808e6	0.0801763
$706$	1.48651e7	1.12242
$707$	3.49600e6	0.263040
$708$	−990498.	−0.0742627
$709$	1.55342e7	1.16057	0.580287	−	0.814412i	$-0.302941\pi$
$709$	1.55342e7	1.16057	0.580287	+	0.814412i	$0.302941\pi$
$710$	−693829.	−0.0516543
$711$	−6.29434e6	−0.466956
$712$	−8.67646e6	−0.641420
$713$	−1.78818e6	−0.131731
$714$	2.88756e6	0.211975
$715$	555591.	0.0406434
$716$	−8.73288e6	−0.636612
$717$	−1.65677e6	−0.120355
$718$	−6.27704e6	−0.454406
$719$	1.97199e7	1.42260	0.711301	−	0.702888i	$-0.248106\pi$
$719$	1.97199e7	1.42260	0.711301	+	0.702888i	$0.248106\pi$
$720$	212751.	0.0152946
$721$	6.89277e6	0.493805
$722$	2.74283e7	1.95819
$723$	−2.44929e6	−0.174259
$724$	1.99160e6	0.141207
$725$	−8.59938e6	−0.607606
$726$	4.81304e6	0.338905
$727$	−820427.	−0.0575710	−0.0287855	−	0.999586i	$-0.509164\pi$
$727$	−820427.	−0.0575710	−0.0287855	+	0.999586i	$0.509164\pi$
$728$	4.57306e6	0.319800
$729$	4.52341e6	0.315244
$730$	−1.07356e6	−0.0745621
$731$	1.95523e7	1.35333
$732$	−5.97776e6	−0.412345
$733$	2.82300e6	0.194066	0.0970332	−	0.995281i	$-0.469065\pi$
$733$	2.82300e6	0.194066	0.0970332	+	0.995281i	$0.469065\pi$
$734$	9.36460e6	0.641577
$735$	−431867.	−0.0294871
$736$	−3.18314e6	−0.216601
$737$	−7.12316e6	−0.483063
$738$	−1.17391e6	−0.0793402
$739$	7.61633e6	0.513020	0.256510	−	0.966542i	$-0.417427\pi$
$739$	7.61633e6	0.513020	0.256510	+	0.966542i	$0.417427\pi$
$740$	115506.	0.00775401
$741$	−2.36304e7	−1.58097
$742$	1.12949e7	0.753136
$743$	−2.55640e7	−1.69886	−0.849429	−	0.527703i	$-0.823053\pi$
$743$	−2.55640e7	−1.69886	−0.849429	+	0.527703i	$0.823053\pi$
$744$	−304517.	−0.0201688
$745$	−760156.	−0.0501778
$746$	−1.42199e7	−0.935511
$747$	795833.	0.0521820
$748$	2.28097e6	0.149061
$749$	−3.21372e6	−0.209317
$750$	−980759.	−0.0636663
$751$	1.63616e7	1.05858	0.529292	−	0.848440i	$-0.322458\pi$
$751$	1.63616e7	1.05858	0.529292	+	0.848440i	$0.322458\pi$
$752$	6.87953e6	0.443623
$753$	−1.78003e6	−0.114404
$754$	−1.03685e7	−0.664185
$755$	−188182.	−0.0120146
$756$	−4.22268e6	−0.268710
$757$	1.40629e7	0.891940	0.445970	−	0.895048i	$-0.352859\pi$
$757$	1.40629e7	0.891940	0.445970	+	0.895048i	$0.352859\pi$
$758$	−471360.	−0.0297975
$759$	3.20910e6	0.202199
$760$	930719.	0.0584500
$761$	−7.75942e6	−0.485699	−0.242850	−	0.970064i	$-0.578082\pi$
$761$	−7.75942e6	−0.485699	−0.242850	+	0.970064i	$0.578082\pi$
$762$	6.17645e6	0.385347
$763$	1.86766e7	1.16141
$764$	7.37275e6	0.456979
$765$	949244.	0.0586441
$766$	−1.03838e7	−0.639418
$767$	−6.99902e6	−0.429585
$768$	−542071.	−0.0331629
$769$	−7.55369e6	−0.460620	−0.230310	−	0.973117i	$-0.573974\pi$
$769$	−7.55369e6	−0.460620	−0.230310	+	0.973117i	$0.573974\pi$
$770$	181586.	0.0110371
$771$	2.37773e6	0.144055
$772$	5.47925e6	0.330886
$773$	3.94631e6	0.237543	0.118772	−	0.992922i	$-0.462104\pi$
$773$	3.94631e6	0.237543	0.118772	+	0.992922i	$0.462104\pi$
$774$	−1.19541e7	−0.717242
$775$	−1.78462e6	−0.106731
$776$	−9.05799e6	−0.539980
$777$	−958484.	−0.0569551
$778$	−3.95331e6	−0.234159
$779$	−5.13550e6	−0.303207
$780$	−589114.	−0.0346707
$781$	4.54800e6	0.266804
$782$	−1.42024e7	−0.830512
$783$	9.57411e6	0.558077
$784$	−2.80796e6	−0.163155
$785$	737075.	0.0426911
$786$	93366.5	0.00539057
$787$	1.43242e6	0.0824393	0.0412197	−	0.999150i	$-0.486876\pi$
$787$	1.43242e6	0.0824393	0.0412197	+	0.999150i	$0.486876\pi$
$788$	51618.2	0.00296133
$789$	1.86644e6	0.106739
$790$	−686479.	−0.0391345
$791$	5.22528e6	0.296940
$792$	−1.39456e6	−0.0789997
$793$	−4.22399e7	−2.38528
$794$	−1.53023e7	−0.861401
$795$	−1.45504e6	−0.0816503
$796$	1.05632e6	0.0590900
$797$	7.99597e6	0.445887	0.222944	−	0.974831i	$-0.428433\pi$
$797$	7.99597e6	0.445887	0.222944	+	0.974831i	$0.428433\pi$
$798$	−7.72321e6	−0.429330
$799$	3.06949e7	1.70098
$800$	−3.17680e6	−0.175495
$801$	2.36684e7	1.30343
$802$	1.85075e7	1.01604
$803$	7.03709e6	0.385127
$804$	7.55296e6	0.412076
$805$	−1.13065e6	−0.0614946
$806$	−2.15177e6	−0.116670
$807$	−5.33226e6	−0.288222
$808$	−2.92822e6	−0.157788
$809$	2.49957e7	1.34275	0.671374	−	0.741119i	$-0.265705\pi$
$809$	2.49957e7	1.34275	0.671374	+	0.741119i	$0.265705\pi$
$810$	−263810.	−0.0141279
$811$	−1.69765e7	−0.906351	−0.453176	−	0.891421i	$-0.649709\pi$
$811$	−1.69765e7	−0.906351	−0.453176	+	0.891421i	$0.649709\pi$
$812$	−3.38879e6	−0.180366
$813$	−674197.	−0.0357734
$814$	−757135.	−0.0400509
$815$	495991.	0.0261565
$816$	−2.41859e6	−0.127156
$817$	−5.22957e7	−2.74101
$818$	4.72422e6	0.246858
$819$	−1.24748e7	−0.649866
$820$	−128030.	−0.00664932
$821$	2.16143e7	1.11914	0.559568	−	0.828784i	$-0.310967\pi$
$821$	2.16143e7	1.11914	0.559568	+	0.828784i	$0.310967\pi$
$822$	−1.84301e6	−0.0951365
$823$	119310.	0.00614011	0.00307005	−	0.999995i	$-0.499023\pi$
$823$	119310.	0.00614011	0.00307005	+	0.999995i	$0.499023\pi$
$824$	−5.77333e6	−0.296216
$825$	3.20271e6	0.163826
$826$	−2.28752e6	−0.116658
$827$	−3.27087e7	−1.66303	−0.831514	−	0.555504i	$-0.812526\pi$
$827$	−3.27087e7	−1.66303	−0.831514	+	0.555504i	$0.812526\pi$
$828$	8.68325e6	0.440156
$829$	1.61893e7	0.818165	0.409083	−	0.912497i	$-0.365849\pi$
$829$	1.61893e7	0.818165	0.409083	+	0.912497i	$0.365849\pi$
$830$	86795.9	0.00437325
$831$	6.19120e6	0.311009
$832$	−3.83036e6	−0.191837
$833$	−1.25285e7	−0.625584
$834$	−1.38172e7	−0.687867
$835$	−1.50201e6	−0.0745515
$836$	−6.10079e6	−0.301905
$837$	1.98690e6	0.0980310
$838$	1.92755e7	0.948191
$839$	−1.54062e7	−0.755600	−0.377800	−	0.925887i	$-0.623319\pi$
$839$	−1.54062e7	−0.755600	−0.377800	+	0.925887i	$0.623319\pi$
$840$	−192543.	−0.00941519
$841$	−1.28277e7	−0.625402
$842$	−1.86409e7	−0.906124
$843$	1.49533e7	0.724717
$844$	1.19370e7	0.576820
$845$	−2.39535e6	−0.115406
$846$	−1.87666e7	−0.901487
$847$	1.11155e7	0.532381
$848$	−9.46054e6	−0.451779
$849$	622553.	0.0296420
$850$	−1.41741e7	−0.672898
$851$	4.71430e6	0.223148
$852$	−4.82242e6	−0.227597
$853$	−2.77221e7	−1.30453	−0.652263	−	0.757993i	$-0.726180\pi$
$853$	−2.77221e7	−1.30453	−0.652263	+	0.757993i	$0.726180\pi$
$854$	−1.38054e7	−0.647747
$855$	−2.53890e6	−0.118776
$856$	2.69179e6	0.125561
$857$	−1.84328e6	−0.0857314	−0.0428657	−	0.999081i	$-0.513649\pi$
$857$	−1.84328e6	−0.0857314	−0.0428657	+	0.999081i	$0.513649\pi$
$858$	3.86160e6	0.179081
$859$	−3.78226e6	−0.174891	−0.0874456	−	0.996169i	$-0.527870\pi$
$859$	−3.78226e6	−0.174891	−0.0874456	+	0.996169i	$0.527870\pi$
$860$	−1.30375e6	−0.0601104
$861$	1.06241e6	0.0488408
$862$	9.21250e6	0.422289
$863$	−6.83390e6	−0.312350	−0.156175	−	0.987729i	$-0.549916\pi$
$863$	−6.83390e6	−0.312350	−0.156175	+	0.987729i	$0.549916\pi$
$864$	3.53688e6	0.161189
$865$	3.12960e6	0.142216
$866$	1.95034e7	0.883720
$867$	952907.	0.0430529
$868$	−703273.	−0.0316829
$869$	4.49982e6	0.202137
$870$	436553.	0.0195542
$871$	5.33704e7	2.38372
$872$	−1.56434e7	−0.696689
$873$	2.47092e7	1.09729
$874$	3.79866e7	1.68210
$875$	−2.26503e6	−0.100012
$876$	−7.46169e6	−0.328532
$877$	3.50135e7	1.53722	0.768612	−	0.639715i	$-0.220948\pi$
$877$	3.50135e7	1.53722	0.768612	+	0.639715i	$0.220948\pi$
$878$	2.58547e7	1.13189
$879$	832893.	0.0363594
$880$	−152095.	−0.00662078
$881$	−1.47812e6	−0.0641608	−0.0320804	−	0.999485i	$-0.510213\pi$
$881$	−1.47812e6	−0.0641608	−0.0320804	+	0.999485i	$0.510213\pi$
$882$	7.65980e6	0.331548
$883$	−3.43310e6	−0.148179	−0.0740893	−	0.997252i	$-0.523605\pi$
$883$	−3.43310e6	−0.148179	−0.0740893	+	0.997252i	$0.523605\pi$
$884$	−1.70902e7	−0.735557
$885$	294685.	0.0126474
$886$	−2.66047e7	−1.13861
$887$	−3.58183e7	−1.52861	−0.764304	−	0.644856i	$-0.776917\pi$
$887$	−3.58183e7	−1.52861	−0.764304	+	0.644856i	$0.776917\pi$
$888$	802819.	0.0341653
$889$	1.42643e7	0.605335
$890$	2.58135e6	0.109237
$891$	1.72926e6	0.0729735
$892$	5.31865e6	0.223815
$893$	−8.20982e7	−3.44512
$894$	−5.28341e6	−0.221091
$895$	2.59813e6	0.108419
$896$	−1.25189e6	−0.0520952
$897$	−2.40442e7	−0.997769
$898$	−1.38750e7	−0.574172
$899$	1.59453e6	0.0658013
$900$	8.66595e6	0.356623
$901$	−4.22108e7	−1.73225
$902$	839227.	0.0343450
$903$	1.08187e7	0.441525
$904$	−4.37666e6	−0.178124
$905$	−592523.	−0.0240483
$906$	−1.30794e6	−0.0529381
$907$	−2.38330e7	−0.961966	−0.480983	−	0.876730i	$-0.659720\pi$
$907$	−2.38330e7	−0.961966	−0.480983	+	0.876730i	$0.659720\pi$
$908$	−1.61569e7	−0.650343
$909$	7.98786e6	0.320642
$910$	−1.36054e6	−0.0544637
$911$	74595.9	0.00297796	0.00148898	−	0.999999i	$-0.499526\pi$
$911$	74595.9	0.00297796	0.00148898	+	0.999999i	$0.499526\pi$
$912$	6.46890e6	0.257539
$913$	−568940.	−0.0225886
$914$	7.08885e6	0.280679
$915$	1.77845e6	0.0702247
$916$	522548.	0.0205772
$917$	215627.	0.00846797
$918$	1.57808e7	0.618047
$919$	4.01030e7	1.56635	0.783174	−	0.621803i	$-0.213600\pi$
$919$	4.01030e7	1.56635	0.783174	+	0.621803i	$0.213600\pi$
$920$	947020.	0.0368884
$921$	−1.90278e7	−0.739164
$922$	2.42147e7	0.938105
$923$	−3.40760e7	−1.31657
$924$	1.26210e6	0.0486312
$925$	4.70491e6	0.180799
$926$	5.60022e6	0.214624
$927$	1.57490e7	0.601941
$928$	2.83843e6	0.108195
$929$	3.71159e7	1.41098	0.705489	−	0.708721i	$-0.250727\pi$
$929$	3.71159e7	1.41098	0.705489	+	0.708721i	$0.250727\pi$
$930$	90597.5	0.00343486
$931$	3.35093e7	1.26704
$932$	−1.44429e7	−0.544646
$933$	1.93884e7	0.729183
$934$	−1.66452e7	−0.624340
$935$	−678614.	−0.0253860
$936$	1.04488e7	0.389831
$937$	3216.86	0.000119697 0	5.98485e−5	−	1.00000i	$-0.499981\pi$
$937$	3216.86	0.000119697 0	5.98485e−5		1.00000i	$0.499981\pi$
$938$	1.74433e7	0.647324
$939$	−4.23326e6	−0.156679
$940$	−2.04674e6	−0.0755515
$941$	2.16460e7	0.796899	0.398449	−	0.917190i	$-0.369548\pi$
$941$	2.16460e7	0.796899	0.398449	+	0.917190i	$0.369548\pi$
$942$	5.12299e6	0.188103
$943$	−5.22544e6	−0.191357
$944$	1.91601e6	0.0699790
$945$	1.25630e6	0.0457628
$946$	8.54600e6	0.310481
$947$	3.47268e6	0.125831	0.0629157	−	0.998019i	$-0.479960\pi$
$947$	3.47268e6	0.125831	0.0629157	+	0.998019i	$0.479960\pi$
$948$	−4.77133e6	−0.172432
$949$	−5.27255e7	−1.90045
$950$	3.79109e7	1.36287
$951$	−3.02161e6	−0.108340
$952$	−5.58566e6	−0.199748
$953$	−3.89396e7	−1.38886	−0.694431	−	0.719559i	$-0.744344\pi$
$953$	−3.89396e7	−1.38886	−0.694431	+	0.719559i	$0.744344\pi$
$954$	2.58073e7	0.918061
$955$	−2.19348e6	−0.0778260
$956$	3.20485e6	0.113413
$957$	−2.86158e6	−0.101001
$958$	1.62792e7	0.573086
$959$	−4.25636e6	−0.149449
$960$	161272.	0.00564783
$961$	−2.82982e7	−0.988441
$962$	5.67285e6	0.197635
$963$	−7.34290e6	−0.255154
$964$	4.73788e6	0.164207
$965$	−1.63014e6	−0.0563517
$966$	−7.85848e6	−0.270954
$967$	3.11491e7	1.07122	0.535610	−	0.844465i	$-0.320082\pi$
$967$	3.11491e7	1.07122	0.535610	+	0.844465i	$0.320082\pi$
$968$	−9.31029e6	−0.319356
$969$	2.88628e7	0.987480
$970$	2.69486e6	0.0919616
$971$	8.35462e6	0.284367	0.142183	−	0.989840i	$-0.454588\pi$
$971$	8.35462e6	0.284367	0.142183	+	0.989840i	$0.454588\pi$
$972$	−1.52627e7	−0.518162
$973$	−3.19103e7	−1.08056
$974$	1.41332e7	0.477357
$975$	−2.39963e7	−0.808413
$976$	1.15633e7	0.388560
$977$	−4.38829e6	−0.147082	−0.0735409	−	0.997292i	$-0.523430\pi$
$977$	−4.38829e6	−0.147082	−0.0735409	+	0.997292i	$0.523430\pi$
$978$	3.44735e6	0.115249
$979$	−1.69205e7	−0.564232
$980$	835400.	0.0277862
$981$	4.26734e7	1.41574
$982$	−1.21434e7	−0.401847
$983$	2.17681e7	0.718516	0.359258	−	0.933238i	$-0.383030\pi$
$983$	2.17681e7	0.718516	0.359258	+	0.933238i	$0.383030\pi$
$984$	−889864.	−0.0292979
$985$	−15357.0	−0.000504331 0
$986$	1.26644e7	0.414851
$987$	1.69841e7	0.554944
$988$	4.57103e7	1.48978
$989$	−5.32117e7	−1.72988
$990$	414899.	0.0134541
$991$	3.80298e6	0.123010	0.0615050	−	0.998107i	$-0.480410\pi$
$991$	3.80298e6	0.123010	0.0615050	+	0.998107i	$0.480410\pi$
$992$	589055.	0.0190054
$993$	2.89518e7	0.931758
$994$	−1.11372e7	−0.357528
$995$	−314268.	−0.0100634
$996$	603269.	0.0192692
$997$	−1.98704e7	−0.633095	−0.316547	−	0.948577i	$-0.602524\pi$
$997$	−1.98704e7	−0.633095	−0.316547	+	0.948577i	$0.602524\pi$
$998$	−1.93127e7	−0.613786
$999$	−5.23820e6	−0.166061

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	82.6.a.a.1.2	✓	3
3.2	odd	2		738.6.a.d.1.2		3
4.3	odd	2		656.6.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
82.6.a.a.1.2	✓	3	1.1	even	1	trivial
656.6.a.a.1.2		3	4.3	odd	2
738.6.a.d.1.2		3	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:	\( N \)	\(=\)	\( 82 = 2 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	82.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -8.27134 q^{3} +16.0000 q^{4} -4.76019 q^{5} -33.0854 q^{6} -76.4095 q^{7} +64.0000 q^{8} -174.585 q^{9} -19.0407 q^{10} +124.811 q^{11} -132.341 q^{12} -935.146 q^{13} -305.638 q^{14} +39.3731 q^{15} +256.000 q^{16} +1142.21 q^{17} -698.340 q^{18} -3055.02 q^{19} -76.1630 q^{20} +632.009 q^{21} +499.243 q^{22} -3108.53 q^{23} -529.366 q^{24} -3102.34 q^{25} -3740.59 q^{26} +3453.99 q^{27} -1222.55 q^{28} +2771.90 q^{29} +157.492 q^{30} +575.249 q^{31} +1024.00 q^{32} -1032.35 q^{33} +4568.85 q^{34} +363.724 q^{35} -2793.36 q^{36} -1516.57 q^{37} -12220.1 q^{38} +7734.91 q^{39} -304.652 q^{40} +1681.00 q^{41} +2528.04 q^{42} +17117.9 q^{43} +1996.97 q^{44} +831.057 q^{45} -12434.1 q^{46} +26873.2 q^{47} -2117.46 q^{48} -10968.6 q^{49} -12409.4 q^{50} -9447.64 q^{51} -14962.3 q^{52} -36955.2 q^{53} +13815.9 q^{54} -594.122 q^{55} -4890.21 q^{56} +25269.1 q^{57} +11087.6 q^{58} +7484.42 q^{59} +629.970 q^{60} +45169.2 q^{61} +2301.00 q^{62} +13339.9 q^{63} +4096.00 q^{64} +4451.47 q^{65} -4129.41 q^{66} -57071.7 q^{67} +18275.4 q^{68} +25711.7 q^{69} +1454.89 q^{70} +36439.2 q^{71} -11173.4 q^{72} +56382.1 q^{73} -6066.27 q^{74} +25660.5 q^{75} -48880.4 q^{76} -9536.72 q^{77} +30939.7 q^{78} +36053.2 q^{79} -1218.61 q^{80} +13855.0 q^{81} +6724.00 q^{82} -4558.43 q^{83} +10112.1 q^{84} -5437.15 q^{85} +68471.8 q^{86} -22927.3 q^{87} +7987.88 q^{88} -135570. q^{89} +3324.23 q^{90} +71454.1 q^{91} -49736.5 q^{92} -4758.08 q^{93} +107493. q^{94} +14542.5 q^{95} -8469.85 q^{96} -141531. q^{97} -43874.3 q^{98} -21790.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{2} - 28 q^{3} + 48 q^{4} - 26 q^{5} - 112 q^{6} - 158 q^{7} + 192 q^{8} - 61 q^{9} - 104 q^{10} - 846 q^{11} - 448 q^{12} - 1258 q^{13} - 632 q^{14} - 682 q^{15} + 768 q^{16} - 2166 q^{17} - 244 q^{18}+ \cdots - 176672 q^{99}+O(q^{100}) \) 3 * q + 12 * q^2 - 28 * q^3 + 48 * q^4 - 26 * q^5 - 112 * q^6 - 158 * q^7 + 192 * q^8 - 61 * q^9 - 104 * q^10 - 846 * q^11 - 448 * q^12 - 1258 * q^13 - 632 * q^14 - 682 * q^15 + 768 * q^16 - 2166 * q^17 - 244 * q^18 - 932 * q^19 - 416 * q^20 - 2776 * q^21 - 3384 * q^22 - 1960 * q^23 - 1792 * q^24 - 6987 * q^25 - 5032 * q^26 - 862 * q^27 - 2528 * q^28 - 3294 * q^29 - 2728 * q^30 + 4764 * q^31 + 3072 * q^32 + 13600 * q^33 - 8664 * q^34 + 10914 * q^35 - 976 * q^36 + 970 * q^37 - 3728 * q^38 + 27232 * q^39 - 1664 * q^40 + 5043 * q^41 - 11104 * q^42 + 1520 * q^43 - 13536 * q^44 + 17990 * q^45 - 7840 * q^46 - 9544 * q^47 - 7168 * q^48 + 2571 * q^49 - 27948 * q^50 + 36952 * q^51 - 20128 * q^52 - 10178 * q^53 - 3448 * q^54 - 1906 * q^55 - 10112 * q^56 + 18016 * q^57 - 13176 * q^58 + 5756 * q^59 - 10912 * q^60 + 25514 * q^61 + 19056 * q^62 + 91820 * q^63 + 12288 * q^64 - 29616 * q^65 + 54400 * q^66 + 25342 * q^67 - 34656 * q^68 + 12468 * q^69 + 43656 * q^70 + 59272 * q^71 - 3904 * q^72 + 95626 * q^73 + 3880 * q^74 + 83752 * q^75 - 14912 * q^76 - 22528 * q^77 + 108928 * q^78 + 15448 * q^79 - 6656 * q^80 + 32231 * q^81 + 20172 * q^82 + 88536 * q^83 - 44416 * q^84 - 1948 * q^85 + 6080 * q^86 + 142596 * q^87 - 54144 * q^88 - 88446 * q^89 + 71960 * q^90 - 85068 * q^91 - 31360 * q^92 - 233788 * q^93 - 38176 * q^94 - 39470 * q^95 - 28672 * q^96 - 323962 * q^97 + 10284 * q^98 - 176672 * q^99

Introduction

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