LMFDB - Embedded newform 35.6.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [35,6,Mod(1,35)] 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("35.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 35 = 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	35.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2] 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:	$5.61343369345$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.53113$ of defining polynomial
Character	$\chi$	$=$	35.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.53113 q^{2} -10.5934 q^{3} -11.4689 q^{4} -25.0000 q^{5} -48.0000 q^{6} -49.0000 q^{7} -196.963 q^{8} -130.780 q^{9} -113.278 q^{10} +90.5195 q^{11} +121.494 q^{12} -74.8502 q^{13} -222.025 q^{14} +264.835 q^{15} -525.461 q^{16} +1032.31 q^{17} -592.582 q^{18} -31.6771 q^{19} +286.722 q^{20} +519.076 q^{21} +410.156 q^{22} -3857.08 q^{23} +2086.51 q^{24} +625.000 q^{25} -339.156 q^{26} +3959.60 q^{27} +561.975 q^{28} -866.917 q^{29} +1200.00 q^{30} +3526.76 q^{31} +3921.89 q^{32} -958.908 q^{33} +4677.54 q^{34} +1225.00 q^{35} +1499.90 q^{36} -9531.22 q^{37} -143.533 q^{38} +792.917 q^{39} +4924.08 q^{40} -14503.9 q^{41} +2352.00 q^{42} -9844.30 q^{43} -1038.16 q^{44} +3269.50 q^{45} -17476.9 q^{46} -16993.5 q^{47} +5566.41 q^{48} +2401.00 q^{49} +2831.96 q^{50} -10935.7 q^{51} +858.447 q^{52} -29621.1 q^{53} +17941.4 q^{54} -2262.99 q^{55} +9651.19 q^{56} +335.568 q^{57} -3928.11 q^{58} +50697.4 q^{59} -3037.35 q^{60} -2921.38 q^{61} +15980.2 q^{62} +6408.23 q^{63} +34585.3 q^{64} +1871.25 q^{65} -4344.94 q^{66} -41086.2 q^{67} -11839.5 q^{68} +40859.5 q^{69} +5550.63 q^{70} +61753.1 q^{71} +25758.9 q^{72} -23664.5 q^{73} -43187.2 q^{74} -6620.87 q^{75} +363.300 q^{76} -4435.46 q^{77} +3592.81 q^{78} +45191.5 q^{79} +13136.5 q^{80} -10166.0 q^{81} -65718.8 q^{82} +39095.9 q^{83} -5953.22 q^{84} -25807.8 q^{85} -44605.8 q^{86} +9183.58 q^{87} -17829.0 q^{88} -41891.1 q^{89} +14814.5 q^{90} +3667.66 q^{91} +44236.3 q^{92} -37360.4 q^{93} -76999.9 q^{94} +791.927 q^{95} -41546.1 q^{96} +8036.53 q^{97} +10879.2 q^{98} -11838.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 3 q^{3} - 31 q^{4} - 50 q^{5} - 96 q^{6} - 98 q^{7} - 15 q^{8} - 189 q^{9} - 25 q^{10} - 601 q^{11} - 144 q^{12} - 577 q^{13} - 49 q^{14} - 75 q^{15} - 543 q^{16} + 41 q^{17} - 387 q^{18}+ \cdots + 28422 q^{99}+O(q^{100}) $ 2 * q + q^2 + 3 * q^3 - 31 * q^4 - 50 * q^5 - 96 * q^6 - 98 * q^7 - 15 * q^8 - 189 * q^9 - 25 * q^10 - 601 * q^11 - 144 * q^12 - 577 * q^13 - 49 * q^14 - 75 * q^15 - 543 * q^16 + 41 * q^17 - 387 * q^18 + 630 * q^19 + 775 * q^20 - 147 * q^21 + 2852 * q^22 - 442 * q^23 + 4560 * q^24 + 1250 * q^25 + 1434 * q^26 - 135 * q^27 + 1519 * q^28 + 5885 * q^29 + 2400 * q^30 - 396 * q^31 - 1839 * q^32 - 10359 * q^33 + 8178 * q^34 + 2450 * q^35 + 2637 * q^36 - 8904 * q^37 - 2480 * q^38 - 6033 * q^39 + 375 * q^40 + 1774 * q^41 + 4704 * q^42 - 27122 * q^43 + 12468 * q^44 + 4725 * q^45 - 29536 * q^46 - 21289 * q^47 + 5328 * q^48 + 4802 * q^49 + 625 * q^50 - 24411 * q^51 + 10666 * q^52 - 55582 * q^53 + 32400 * q^54 + 15025 * q^55 + 735 * q^56 + 9330 * q^57 - 27770 * q^58 + 59600 * q^59 + 3600 * q^60 - 51846 * q^61 + 29832 * q^62 + 9261 * q^63 + 55489 * q^64 + 14425 * q^65 + 28848 * q^66 - 45344 * q^67 + 7522 * q^68 + 87282 * q^69 + 1225 * q^70 + 80744 * q^71 + 15165 * q^72 - 13532 * q^73 - 45402 * q^74 + 1875 * q^75 - 12560 * q^76 + 29449 * q^77 + 27696 * q^78 - 51795 * q^79 + 13575 * q^80 - 51678 * q^81 - 123198 * q^82 + 109828 * q^83 + 7056 * q^84 - 1025 * q^85 + 16404 * q^86 + 100965 * q^87 - 143660 * q^88 - 37650 * q^89 + 9675 * q^90 + 28273 * q^91 - 22464 * q^92 - 90684 * q^93 - 61832 * q^94 - 15750 * q^95 - 119856 * q^96 - 96339 * q^97 + 2401 * q^98 + 28422 * 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 $

$2$	4.53113	0.800998	0.400499	−	0.916297i	$-0.368837\pi$
$2$	4.53113	0.800998	0.400499	+	0.916297i	$0.368837\pi$
$3$	−10.5934	−0.679566	−0.339783	−	0.940504i	$-0.610354\pi$
$3$	−10.5934	−0.679566	−0.339783	+	0.940504i	$0.610354\pi$
$4$	−11.4689	−0.358402
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	−48.0000	−0.544331
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	−196.963	−1.08808
$9$	−130.780	−0.538190
$10$	−113.278	−0.358217
$11$	90.5195	0.225559	0.112780	−	0.993620i	$-0.464025\pi$
$11$	90.5195	0.225559	0.112780	+	0.993620i	$0.464025\pi$
$12$	121.494	0.243558
$13$	−74.8502	−0.122838	−0.0614192	−	0.998112i	$-0.519563\pi$
$13$	−74.8502	−0.122838	−0.0614192	+	0.998112i	$0.519563\pi$
$14$	−222.025	−0.302749
$15$	264.835	0.303911
$16$	−525.461	−0.513146
$17$	1032.31	0.866342	0.433171	−	0.901312i	$-0.357395\pi$
$17$	1032.31	0.866342	0.433171	+	0.901312i	$0.357395\pi$
$18$	−592.582	−0.431089
$19$	−31.6771	−0.0201308	−0.0100654	−	0.999949i	$-0.503204\pi$
$19$	−31.6771	−0.0201308	−0.0100654	+	0.999949i	$0.503204\pi$
$20$	286.722	0.160282
$21$	519.076	0.256852
$22$	410.156	0.180672
$23$	−3857.08	−1.52033	−0.760167	−	0.649728i	$-0.774883\pi$
$23$	−3857.08	−1.52033	−0.760167	+	0.649728i	$0.774883\pi$
$24$	2086.51	0.739421
$25$	625.000	0.200000
$26$	−339.156	−0.0983934
$27$	3959.60	1.04530
$28$	561.975	0.135463
$29$	−866.917	−0.191418	−0.0957089	−	0.995409i	$-0.530512\pi$
$29$	−866.917	−0.191418	−0.0957089	+	0.995409i	$0.530512\pi$
$30$	1200.00	0.243432
$31$	3526.76	0.659131	0.329566	−	0.944133i	$-0.393098\pi$
$31$	3526.76	0.659131	0.329566	+	0.944133i	$0.393098\pi$
$32$	3921.89	0.677049
$33$	−958.908	−0.153282
$34$	4677.54	0.693938
$35$	1225.00	0.169031
$36$	1499.90	0.192888
$37$	−9531.22	−1.14458	−0.572288	−	0.820053i	$-0.693944\pi$
$37$	−9531.22	−1.14458	−0.572288	+	0.820053i	$0.693944\pi$
$38$	−143.533	−0.0161247
$39$	792.917	0.0834769
$40$	4924.08	0.486603
$41$	−14503.9	−1.34748	−0.673742	−	0.738967i	$-0.735314\pi$
$41$	−14503.9	−1.34748	−0.673742	+	0.738967i	$0.735314\pi$
$42$	2352.00	0.205738
$43$	−9844.30	−0.811921	−0.405960	−	0.913891i	$-0.633063\pi$
$43$	−9844.30	−0.811921	−0.405960	+	0.913891i	$0.633063\pi$
$44$	−1038.16	−0.0808409
$45$	3269.50	0.240686
$46$	−17476.9	−1.21778
$47$	−16993.5	−1.12212	−0.561059	−	0.827775i	$-0.689606\pi$
$47$	−16993.5	−1.12212	−0.561059	+	0.827775i	$0.689606\pi$
$48$	5566.41	0.348716
$49$	2401.00	0.142857
$50$	2831.96	0.160200
$51$	−10935.7	−0.588736
$52$	858.447	0.0440256
$53$	−29621.1	−1.44848	−0.724239	−	0.689549i	$-0.757809\pi$
$53$	−29621.1	−1.44848	−0.724239	+	0.689549i	$0.757809\pi$
$54$	17941.4	0.837285
$55$	−2262.99	−0.100873
$56$	9651.19	0.411255
$57$	335.568	0.0136802
$58$	−3928.11	−0.153325
$59$	50697.4	1.89607	0.948037	−	0.318159i	$-0.103065\pi$
$59$	50697.4	1.89607	0.948037	+	0.318159i	$0.103065\pi$
$60$	−3037.35	−0.108922
$61$	−2921.38	−0.100522	−0.0502612	−	0.998736i	$-0.516005\pi$
$61$	−2921.38	−0.100522	−0.0502612	+	0.998736i	$0.516005\pi$
$62$	15980.2	0.527963
$63$	6408.23	0.203417
$64$	34585.3	1.05546
$65$	1871.25	0.0549350
$66$	−4344.94	−0.122779
$67$	−41086.2	−1.11817	−0.559086	−	0.829109i	$-0.688848\pi$
$67$	−41086.2	−1.11817	−0.559086	+	0.829109i	$0.688848\pi$
$68$	−11839.5	−0.310499
$69$	40859.5	1.03317
$70$	5550.63	0.135393
$71$	61753.1	1.45383	0.726914	−	0.686729i	$-0.240954\pi$
$71$	61753.1	1.45383	0.726914	+	0.686729i	$0.240954\pi$
$72$	25758.9	0.585592
$73$	−23664.5	−0.519745	−0.259872	−	0.965643i	$-0.583680\pi$
$73$	−23664.5	−0.519745	−0.259872	+	0.965643i	$0.583680\pi$
$74$	−43187.2	−0.916802
$75$	−6620.87	−0.135913
$76$	363.300	0.00721493
$77$	−4435.46	−0.0852533
$78$	3592.81	0.0668648
$79$	45191.5	0.814683	0.407341	−	0.913276i	$-0.366456\pi$
$79$	45191.5	0.814683	0.407341	+	0.913276i	$0.366456\pi$
$80$	13136.5	0.229486
$81$	−10166.0	−0.172162
$82$	−65718.8	−1.07933
$83$	39095.9	0.622925	0.311462	−	0.950259i	$-0.399181\pi$
$83$	39095.9	0.622925	0.311462	+	0.950259i	$0.399181\pi$
$84$	−5953.22	−0.0920563
$85$	−25807.8	−0.387440
$86$	−44605.8	−0.650347
$87$	9183.58	0.130081
$88$	−17829.0	−0.245426
$89$	−41891.1	−0.560592	−0.280296	−	0.959914i	$-0.590433\pi$
$89$	−41891.1	−0.560592	−0.280296	+	0.959914i	$0.590433\pi$
$90$	14814.5	0.192789
$91$	3667.66	0.0464286
$92$	44236.3	0.544891
$93$	−37360.4	−0.447923
$94$	−76999.9	−0.898815
$95$	791.927	0.00900277
$96$	−41546.1	−0.460099
$97$	8036.53	0.0867240	0.0433620	−	0.999059i	$-0.486193\pi$
$97$	8036.53	0.0867240	0.0433620	+	0.999059i	$0.486193\pi$
$98$	10879.2	0.114428
$99$	−11838.2	−0.121394
$100$	−7168.04	−0.0716804
$101$	124979.	1.21908	0.609542	−	0.792754i	$-0.291353\pi$
$101$	124979.	1.21908	0.609542	+	0.792754i	$0.291353\pi$
$102$	−49551.0	−0.471577
$103$	−76635.1	−0.711762	−0.355881	−	0.934531i	$-0.615819\pi$
$103$	−76635.1	−0.711762	−0.355881	+	0.934531i	$0.615819\pi$
$104$	14742.7	0.133658
$105$	−12976.9	−0.114868
$106$	−134217.	−1.16023
$107$	135296.	1.14242	0.571209	−	0.820804i	$-0.306474\pi$
$107$	135296.	1.14242	0.571209	+	0.820804i	$0.306474\pi$
$108$	−45412.1	−0.374638
$109$	−199508.	−1.60840	−0.804202	−	0.594356i	$-0.797407\pi$
$109$	−199508.	−1.60840	−0.804202	+	0.594356i	$0.797407\pi$
$110$	−10253.9	−0.0807992
$111$	100968.	0.777814
$112$	25747.6	0.193951
$113$	122569.	0.902991	0.451496	−	0.892273i	$-0.350891\pi$
$113$	122569.	0.902991	0.451496	+	0.892273i	$0.350891\pi$
$114$	1520.50	0.0109578
$115$	96427.0	0.679914
$116$	9942.56	0.0686046
$117$	9788.92	0.0661104
$118$	229716.	1.51875
$119$	−50583.4	−0.327446
$120$	−52162.6	−0.330679
$121$	−152857.	−0.949123
$122$	−13237.1	−0.0805183
$123$	153645.	0.915704
$124$	−40448.0	−0.236234
$125$	−15625.0	−0.0894427
$126$	29036.5	0.162936
$127$	40046.2	0.220319	0.110160	−	0.993914i	$-0.464864\pi$
$127$	40046.2	0.220319	0.110160	+	0.993914i	$0.464864\pi$
$128$	31210.2	0.168373
$129$	104284.	0.551754
$130$	8478.89	0.0440029
$131$	211963.	1.07915	0.539576	−	0.841937i	$-0.318585\pi$
$131$	211963.	1.07915	0.539576	+	0.841937i	$0.318585\pi$
$132$	10997.6	0.0549367
$133$	1552.18	0.00760873
$134$	−186167.	−0.895654
$135$	−98989.9	−0.467473
$136$	−203328.	−0.942647
$137$	40980.5	0.186542	0.0932709	−	0.995641i	$-0.470268\pi$
$137$	40980.5	0.186542	0.0932709	+	0.995641i	$0.470268\pi$
$138$	185140.	0.827565
$139$	−418948.	−1.83917	−0.919587	−	0.392886i	$-0.871477\pi$
$139$	−418948.	−1.83917	−0.919587	+	0.392886i	$0.871477\pi$
$140$	−14049.4	−0.0605810
$141$	180019.	0.762554
$142$	279811.	1.16451
$143$	−6775.40	−0.0277073
$144$	68719.9	0.276170
$145$	21672.9	0.0856047
$146$	−107227.	−0.416314
$147$	−25434.7	−0.0970809
$148$	109312.	0.410218
$149$	−64962.7	−0.239717	−0.119858	−	0.992791i	$-0.538244\pi$
$149$	−64962.7	−0.239717	−0.119858	+	0.992791i	$0.538244\pi$
$150$	−30000.0	−0.108866
$151$	379801.	1.35554	0.677772	−	0.735272i	$-0.262946\pi$
$151$	379801.	1.35554	0.677772	+	0.735272i	$0.262946\pi$
$152$	6239.22	0.0219039
$153$	−135006.	−0.466256
$154$	−20097.6	−0.0682878
$155$	−88169.1	−0.294773
$156$	−9093.86	−0.0299183
$157$	−281546.	−0.911590	−0.455795	−	0.890085i	$-0.650645\pi$
$157$	−281546.	−0.911590	−0.455795	+	0.890085i	$0.650645\pi$
$158$	204768.	0.652559
$159$	313788.	0.984337
$160$	−98047.1	−0.302785
$161$	188997.	0.574632
$162$	−46063.3	−0.137901
$163$	−382587.	−1.12788	−0.563938	−	0.825817i	$-0.690714\pi$
$163$	−382587.	−1.12788	−0.563938	+	0.825817i	$0.690714\pi$
$164$	166343.	0.482941
$165$	23972.7	0.0685499
$166$	177148.	0.498961
$167$	−388635.	−1.07833	−0.539164	−	0.842201i	$-0.681260\pi$
$167$	−388635.	−1.07833	−0.539164	+	0.842201i	$0.681260\pi$
$168$	−102239.	−0.279475
$169$	−365690.	−0.984911
$170$	−116939.	−0.310339
$171$	4142.73	0.0108342
$172$	112903.	0.290994
$173$	−389693.	−0.989936	−0.494968	−	0.868911i	$-0.664820\pi$
$173$	−389693.	−0.989936	−0.494968	+	0.868911i	$0.664820\pi$
$174$	41612.0	0.104195
$175$	−30625.0	−0.0755929
$176$	−47564.5	−0.115745
$177$	−537057.	−1.28851
$178$	−189814.	−0.449033
$179$	−149812.	−0.349473	−0.174737	−	0.984615i	$-0.555907\pi$
$179$	−149812.	−0.349473	−0.174737	+	0.984615i	$0.555907\pi$
$180$	−37497.5	−0.0862623
$181$	−821019.	−1.86276	−0.931380	−	0.364049i	$-0.881394\pi$
$181$	−821019.	−1.86276	−0.931380	+	0.364049i	$0.881394\pi$
$182$	16618.6	0.0371892
$183$	30947.3	0.0683117
$184$	759702.	1.65424
$185$	238281.	0.511870
$186$	−169285.	−0.358786
$187$	93444.5	0.195411
$188$	194897.	0.402170
$189$	−194020.	−0.395087
$190$	3588.32	0.00721120
$191$	286619.	0.568487	0.284244	−	0.958752i	$-0.408258\pi$
$191$	286619.	0.568487	0.284244	+	0.958752i	$0.408258\pi$
$192$	−366376.	−0.717255
$193$	993573.	1.92002	0.960012	−	0.279959i	$-0.0903209\pi$
$193$	993573.	1.92002	0.960012	+	0.279959i	$0.0903209\pi$
$194$	36414.6	0.0694657
$195$	−19822.9	−0.0373320
$196$	−27536.8	−0.0512003
$197$	−38209.0	−0.0701456	−0.0350728	−	0.999385i	$-0.511166\pi$
$197$	−38209.0	−0.0701456	−0.0350728	+	0.999385i	$0.511166\pi$
$198$	−53640.2	−0.0972361
$199$	730487.	1.30761	0.653807	−	0.756661i	$-0.273171\pi$
$199$	730487.	1.30761	0.653807	+	0.756661i	$0.273171\pi$
$200$	−123102.	−0.217615
$201$	435242.	0.759872
$202$	566296.	0.976484
$203$	42478.9	0.0723491
$204$	125420.	0.211004
$205$	362596.	0.602613
$206$	−347244.	−0.570120
$207$	504429.	0.818228
$208$	39330.9	0.0630340
$209$	−2867.39	−0.00454069
$210$	−58800.0	−0.0920087
$211$	194286.	0.300424	0.150212	−	0.988654i	$-0.452004\pi$
$211$	194286.	0.300424	0.150212	+	0.988654i	$0.452004\pi$
$212$	339721.	0.519138
$213$	−654175.	−0.987972
$214$	613043.	0.915075
$215$	246107.	0.363102
$216$	−779894.	−1.13737
$217$	−172811.	−0.249128
$218$	−903999.	−1.28833
$219$	250687.	0.353201
$220$	25953.9	0.0361531
$221$	−77268.8	−0.106420
$222$	457499.	0.623028
$223$	1.13569e6	1.52931	0.764656	−	0.644438i	$-0.222909\pi$
$223$	1.13569e6	1.52931	0.764656	+	0.644438i	$0.222909\pi$
$224$	−192172.	−0.255900
$225$	−81737.6	−0.107638
$226$	555375.	0.723294
$227$	−143806.	−0.185231	−0.0926155	−	0.995702i	$-0.529523\pi$
$227$	−143806.	−0.185231	−0.0926155	+	0.995702i	$0.529523\pi$
$228$	−3848.58	−0.00490302
$229$	−3832.50	−0.00482940	−0.00241470	−	0.999997i	$-0.500769\pi$
$229$	−3832.50	−0.00482940	−0.00241470	+	0.999997i	$0.500769\pi$
$230$	436923.	0.544610
$231$	46986.5	0.0579353
$232$	170751.	0.208277
$233$	−1.35599e6	−1.63631	−0.818157	−	0.574995i	$-0.805004\pi$
$233$	−1.35599e6	−1.63631	−0.818157	+	0.574995i	$0.805004\pi$
$234$	44354.8	0.0529543
$235$	424838.	0.501827
$236$	−581442.	−0.679557
$237$	−478731.	−0.553631
$238$	−229200.	−0.262284
$239$	−478372.	−0.541716	−0.270858	−	0.962619i	$-0.587307\pi$
$239$	−478372.	−0.541716	−0.270858	+	0.962619i	$0.587307\pi$
$240$	−139160.	−0.155951
$241$	−1.31082e6	−1.45379	−0.726894	−	0.686750i	$-0.759037\pi$
$241$	−1.31082e6	−1.45379	−0.726894	+	0.686750i	$0.759037\pi$
$242$	−692616.	−0.760246
$243$	−854490.	−0.928307
$244$	33504.9	0.0360275
$245$	−60025.0	−0.0638877
$246$	696185.	0.733477
$247$	2371.04	0.00247284
$248$	−694642.	−0.717186
$249$	−414157.	−0.423318
$250$	−70798.9	−0.0716434
$251$	340113.	0.340753	0.170376	−	0.985379i	$-0.445502\pi$
$251$	340113.	0.340753	0.170376	+	0.985379i	$0.445502\pi$
$252$	−73495.1	−0.0729050
$253$	−349141.	−0.342925
$254$	181454.	0.176475
$255$	273392.	0.263291
$256$	−965313.	−0.920594
$257$	−58839.6	−0.0555696	−0.0277848	−	0.999614i	$-0.508845\pi$
$257$	−58839.6	−0.0555696	−0.0277848	+	0.999614i	$0.508845\pi$
$258$	472526.	0.441954
$259$	467030.	0.432609
$260$	−21461.2	−0.0196888
$261$	113376.	0.103019
$262$	960433.	0.864398
$263$	−1380.79	−0.00123095	−0.000615473	−	1.00000i	$-0.500196\pi$
$263$	−1380.79	−0.00123095	−0.000615473		1.00000i	$0.500196\pi$
$264$	188869.	0.166783
$265$	740528.	0.647779
$266$	7033.11	0.00609458
$267$	443769.	0.380959
$268$	471212.	0.400756
$269$	886839.	0.747247	0.373624	−	0.927580i	$-0.378115\pi$
$269$	886839.	0.747247	0.373624	+	0.927580i	$0.378115\pi$
$270$	−448536.	−0.374445
$271$	−376955.	−0.311793	−0.155897	−	0.987773i	$-0.549827\pi$
$271$	−376955.	−0.311793	−0.155897	+	0.987773i	$0.549827\pi$
$272$	−542441.	−0.444559
$273$	−38852.9	−0.0315513
$274$	185688.	0.149420
$275$	56574.7	0.0451118
$276$	−468613.	−0.370289
$277$	−880363.	−0.689386	−0.344693	−	0.938716i	$-0.612017\pi$
$277$	−880363.	−0.689386	−0.344693	+	0.938716i	$0.612017\pi$
$278$	−1.89831e6	−1.47317
$279$	−461231.	−0.354738
$280$	−241280.	−0.183919
$281$	2.08492e6	1.57515	0.787577	−	0.616217i	$-0.211335\pi$
$281$	2.08492e6	1.57515	0.787577	+	0.616217i	$0.211335\pi$
$282$	815689.	0.610804
$283$	692401.	0.513915	0.256958	−	0.966423i	$-0.417280\pi$
$283$	692401.	0.513915	0.256958	+	0.966423i	$0.417280\pi$
$284$	−708238.	−0.521055
$285$	−8389.19	−0.00611798
$286$	−30700.2	−0.0221935
$287$	710689.	0.509301
$288$	−512905.	−0.364381
$289$	−354186.	−0.249452
$290$	98202.8	0.0685692
$291$	−85134.1	−0.0589347
$292$	271405.	0.186278
$293$	2.27624e6	1.54899	0.774495	−	0.632580i	$-0.218004\pi$
$293$	2.27624e6	1.54899	0.774495	+	0.632580i	$0.218004\pi$
$294$	−115248.	−0.0777616
$295$	−1.26743e6	−0.847950
$296$	1.87730e6	1.24539
$297$	358421.	0.235777
$298$	−294354.	−0.192013
$299$	288703.	0.186755
$300$	75933.9	0.0487116
$301$	482371.	0.306877
$302$	1.72093e6	1.08579
$303$	−1.32395e6	−0.828448
$304$	16645.1	0.0103300
$305$	73034.5	0.0449550
$306$	−611730.	−0.373470
$307$	−2.65187e6	−1.60586	−0.802928	−	0.596076i	$-0.796726\pi$
$307$	−2.65187e6	−1.60586	−0.802928	+	0.596076i	$0.796726\pi$
$308$	50869.7	0.0305550
$309$	811825.	0.483689
$310$	−399505.	−0.236112
$311$	2.57626e6	1.51039	0.755195	−	0.655501i	$-0.227542\pi$
$311$	2.57626e6	1.51039	0.755195	+	0.655501i	$0.227542\pi$
$312$	−156175.	−0.0908293
$313$	2.51549e6	1.45131	0.725657	−	0.688057i	$-0.241536\pi$
$313$	2.51549e6	1.45131	0.725657	+	0.688057i	$0.241536\pi$
$314$	−1.27572e6	−0.730182
$315$	−160206.	−0.0909707
$316$	−518295.	−0.291984
$317$	−2.30397e6	−1.28774	−0.643870	−	0.765135i	$-0.722672\pi$
$317$	−2.30397e6	−1.28774	−0.643870	+	0.765135i	$0.722672\pi$
$318$	1.42181e6	0.788452
$319$	−78472.9	−0.0431760
$320$	−864633.	−0.472016
$321$	−1.43324e6	−0.776349
$322$	856369.	0.460279
$323$	−32700.7	−0.0174402
$324$	116592.	0.0617031
$325$	−46781.4	−0.0245677
$326$	−1.73355e6	−0.903427
$327$	2.11347e6	1.09302
$328$	2.85672e6	1.46617
$329$	832683.	0.424121
$330$	108623.	0.0549084
$331$	697305.	0.349827	0.174913	−	0.984584i	$-0.444035\pi$
$331$	697305.	0.349827	0.174913	+	0.984584i	$0.444035\pi$
$332$	−448385.	−0.223258
$333$	1.24649e6	0.615999
$334$	−1.76096e6	−0.863739
$335$	1.02715e6	0.500062
$336$	−272754.	−0.131802
$337$	−1.14848e6	−0.550868	−0.275434	−	0.961320i	$-0.588822\pi$
$337$	−1.14848e6	−0.550868	−0.275434	+	0.961320i	$0.588822\pi$
$338$	−1.65699e6	−0.788911
$339$	−1.29842e6	−0.613642
$340$	295987.	0.138859
$341$	319241.	0.148673
$342$	18771.3	0.00867817
$343$	−117649.	−0.0539949
$344$	1.93896e6	0.883433
$345$	−1.02149e6	−0.462046
$346$	−1.76575e6	−0.792937
$347$	−2.43716e6	−1.08658	−0.543289	−	0.839546i	$-0.682821\pi$
$347$	−2.43716e6	−1.08658	−0.543289	+	0.839546i	$0.682821\pi$
$348$	−105325.	−0.0466213
$349$	896801.	0.394124	0.197062	−	0.980391i	$-0.436860\pi$
$349$	896801.	0.394124	0.197062	+	0.980391i	$0.436860\pi$
$350$	−138766.	−0.0605498
$351$	−296377.	−0.128403
$352$	355007.	0.152715
$353$	4.33422e6	1.85129	0.925644	−	0.378396i	$-0.123524\pi$
$353$	4.33422e6	1.85129	0.925644	+	0.378396i	$0.123524\pi$
$354$	−2.43347e6	−1.03209
$355$	−1.54383e6	−0.650172
$356$	480444.	0.200917
$357$	535849.	0.222521
$358$	−678817.	−0.279927
$359$	−3.59167e6	−1.47082	−0.735411	−	0.677621i	$-0.763011\pi$
$359$	−3.59167e6	−1.47082	−0.735411	+	0.677621i	$0.763011\pi$
$360$	−643972.	−0.261885
$361$	−2.47510e6	−0.999595
$362$	−3.72014e6	−1.49207
$363$	1.61928e6	0.644992
$364$	−42063.9	−0.0166401
$365$	591612.	0.232437
$366$	140226.	0.0547175
$367$	361791.	0.140215	0.0701073	−	0.997539i	$-0.477666\pi$
$367$	361791.	0.140215	0.0701073	+	0.997539i	$0.477666\pi$
$368$	2.02674e6	0.780152
$369$	1.89682e6	0.725202
$370$	1.07968e6	0.410006
$371$	1.45144e6	0.547473
$372$	428481.	0.160537
$373$	−3.08325e6	−1.14746	−0.573729	−	0.819045i	$-0.694504\pi$
$373$	−3.08325e6	−1.14746	−0.573729	+	0.819045i	$0.694504\pi$
$374$	423409.	0.156524
$375$	165522.	0.0607822
$376$	3.34710e6	1.22095
$377$	64888.9	0.0235135
$378$	−879131.	−0.316464
$379$	−5.04227e6	−1.80313	−0.901567	−	0.432639i	$-0.857582\pi$
$379$	−5.04227e6	−1.80313	−0.901567	+	0.432639i	$0.857582\pi$
$380$	−9082.51	−0.00322661
$381$	−424225.	−0.149721
$382$	1.29871e6	0.455357
$383$	−3.11928e6	−1.08657	−0.543285	−	0.839548i	$-0.682820\pi$
$383$	−3.11928e6	−1.08657	−0.543285	+	0.839548i	$0.682820\pi$
$384$	−330622.	−0.114420
$385$	110886.	0.0381265
$386$	4.50201e6	1.53794
$387$	1.28744e6	0.436968
$388$	−92169.9	−0.0310821
$389$	−1.64560e6	−0.551378	−0.275689	−	0.961247i	$-0.588906\pi$
$389$	−1.64560e6	−0.551378	−0.275689	+	0.961247i	$0.588906\pi$
$390$	−89820.2	−0.0299028
$391$	−3.98171e6	−1.31713
$392$	−472908.	−0.155440
$393$	−2.24541e6	−0.733355
$394$	−173130.	−0.0561865
$395$	−1.12979e6	−0.364337
$396$	135770.	0.0435078
$397$	3.72727e6	1.18690	0.593451	−	0.804870i	$-0.297765\pi$
$397$	3.72727e6	1.18690	0.593451	+	0.804870i	$0.297765\pi$
$398$	3.30993e6	1.04740
$399$	−16442.8	−0.00517063
$400$	−328413.	−0.102629
$401$	4.84820e6	1.50564	0.752818	−	0.658229i	$-0.228694\pi$
$401$	4.84820e6	1.50564	0.752818	+	0.658229i	$0.228694\pi$
$402$	1.97214e6	0.608656
$403$	−263979.	−0.0809667
$404$	−1.43337e6	−0.436923
$405$	254149.	0.0769930
$406$	192477.	0.0579515
$407$	−862761.	−0.258169
$408$	2.15393e6	0.640591
$409$	−3.03126e6	−0.896014	−0.448007	−	0.894030i	$-0.647866\pi$
$409$	−3.03126e6	−0.896014	−0.448007	+	0.894030i	$0.647866\pi$
$410$	1.64297e6	0.482692
$411$	−434123.	−0.126768
$412$	878918.	0.255097
$413$	−2.48417e6	−0.716649
$414$	2.28563e6	0.655399
$415$	−977396.	−0.278580
$416$	−293554.	−0.0831676
$417$	4.43808e6	1.24984
$418$	−12992.5	−0.00363708
$419$	1.66905e6	0.464444	0.232222	−	0.972663i	$-0.425400\pi$
$419$	1.66905e6	0.464444	0.232222	+	0.972663i	$0.425400\pi$
$420$	148830.	0.0411688
$421$	−1.76031e6	−0.484043	−0.242022	−	0.970271i	$-0.577810\pi$
$421$	−1.76031e6	−0.484043	−0.242022	+	0.970271i	$0.577810\pi$
$422$	880335.	0.240639
$423$	2.22242e6	0.603913
$424$	5.83427e6	1.57606
$425$	645196.	0.173268
$426$	−2.96415e6	−0.791364
$427$	143148.	0.0379939
$428$	−1.55169e6	−0.409445
$429$	71774.4	0.0188290
$430$	1.11514e6	0.290844
$431$	−648832.	−0.168244	−0.0841219	−	0.996455i	$-0.526809\pi$
$431$	−648832.	−0.168244	−0.0841219	+	0.996455i	$0.526809\pi$
$432$	−2.08061e6	−0.536392
$433$	−360993.	−0.0925293	−0.0462646	−	0.998929i	$-0.514732\pi$
$433$	−360993.	−0.0925293	−0.0462646	+	0.998929i	$0.514732\pi$
$434$	−783031.	−0.199551
$435$	−229590.	−0.0581740
$436$	2.28814e6	0.576455
$437$	122181.	0.0306055
$438$	1.13590e6	0.282913
$439$	1.07021e6	0.265038	0.132519	−	0.991180i	$-0.457693\pi$
$439$	1.07021e6	0.265038	0.132519	+	0.991180i	$0.457693\pi$
$440$	445725.	0.109758
$441$	−314003.	−0.0768843
$442$	−350115.	−0.0852423
$443$	−797145.	−0.192987	−0.0964935	−	0.995334i	$-0.530763\pi$
$443$	−797145.	−0.192987	−0.0964935	+	0.995334i	$0.530763\pi$
$444$	−1.15799e6	−0.278770
$445$	1.04728e6	0.250704
$446$	5.14594e6	1.22498
$447$	688175.	0.162903
$448$	−1.69468e6	−0.398927
$449$	6.40103e6	1.49842	0.749211	−	0.662331i	$-0.230433\pi$
$449$	6.40103e6	1.49842	0.749211	+	0.662331i	$0.230433\pi$
$450$	−370364.	−0.0862178
$451$	−1.31288e6	−0.303937
$452$	−1.40572e6	−0.323634
$453$	−4.02338e6	−0.921181
$454$	−651605.	−0.148370
$455$	−91691.5	−0.0207635
$456$	−66094.4	−0.0148851
$457$	5.91843e6	1.32561	0.662806	−	0.748791i	$-0.269366\pi$
$457$	5.91843e6	1.32561	0.662806	+	0.748791i	$0.269366\pi$
$458$	−17365.5	−0.00386834
$459$	4.08755e6	0.905588
$460$	−1.10591e6	−0.243683
$461$	−8.84337e6	−1.93805	−0.969026	−	0.246960i	$-0.920569\pi$
$461$	−8.84337e6	−1.93805	−0.969026	+	0.246960i	$0.920569\pi$
$462$	212902.	0.0464060
$463$	−6.76858e6	−1.46739	−0.733695	−	0.679479i	$-0.762206\pi$
$463$	−6.76858e6	−1.46739	−0.733695	+	0.679479i	$0.762206\pi$
$464$	455531.	0.0982252
$465$	934009.	0.200317
$466$	−6.14416e6	−1.31068
$467$	−1.04740e6	−0.222239	−0.111120	−	0.993807i	$-0.535444\pi$
$467$	−1.04740e6	−0.222239	−0.111120	+	0.993807i	$0.535444\pi$
$468$	−112268.	−0.0236941
$469$	2.01322e6	0.422630
$470$	1.92500e6	0.401962
$471$	2.98252e6	0.619486
$472$	−9.98551e6	−2.06308
$473$	−891101.	−0.183136
$474$	−2.16919e6	−0.443457
$475$	−19798.2	−0.00402616
$476$	580134.	0.117358
$477$	3.87386e6	0.779556
$478$	−2.16757e6	−0.433913
$479$	−9.37725e6	−1.86740	−0.933699	−	0.358060i	$-0.883438\pi$
$479$	−9.37725e6	−1.86740	−0.933699	+	0.358060i	$0.883438\pi$
$480$	1.03865e6	0.205763
$481$	713414.	0.140598
$482$	−5.93950e6	−1.16448
$483$	−2.00212e6	−0.390500
$484$	1.75310e6	0.340168
$485$	−200913.	−0.0387841
$486$	−3.87181e6	−0.743572
$487$	−2.88960e6	−0.552098	−0.276049	−	0.961144i	$-0.589025\pi$
$487$	−2.88960e6	−0.552098	−0.276049	+	0.961144i	$0.589025\pi$
$488$	575404.	0.109376
$489$	4.05290e6	0.766467
$490$	−271981.	−0.0511739
$491$	−1.04845e6	−0.196265	−0.0981324	−	0.995173i	$-0.531287\pi$
$491$	−1.04845e6	−0.196265	−0.0981324	+	0.995173i	$0.531287\pi$
$492$	−1.76213e6	−0.328191
$493$	−894930.	−0.165833
$494$	10743.5	0.00198074
$495$	295954.	0.0542889
$496$	−1.85318e6	−0.338230
$497$	−3.02590e6	−0.549495
$498$	−1.87660e6	−0.339077
$499$	6.38524e6	1.14796	0.573979	−	0.818870i	$-0.305399\pi$
$499$	6.38524e6	1.14796	0.573979	+	0.818870i	$0.305399\pi$
$500$	179201.	0.0320565
$501$	4.11696e6	0.732795
$502$	1.54110e6	0.272942
$503$	1.35497e6	0.238787	0.119394	−	0.992847i	$-0.461905\pi$
$503$	1.35497e6	0.238787	0.119394	+	0.992847i	$0.461905\pi$
$504$	−1.26218e6	−0.221333
$505$	−3.12448e6	−0.545191
$506$	−1.58200e6	−0.274682
$507$	3.87390e6	0.669312
$508$	−459285.	−0.0789628
$509$	6.43410e6	1.10076	0.550381	−	0.834914i	$-0.314482\pi$
$509$	6.43410e6	1.10076	0.550381	+	0.834914i	$0.314482\pi$
$510$	1.23878e6	0.210896
$511$	1.15956e6	0.196445
$512$	−5.37268e6	−0.905767
$513$	−125429.	−0.0210428
$514$	−266610.	−0.0445111
$515$	1.91588e6	0.318310
$516$	−1.19603e6	−0.197750
$517$	−1.53825e6	−0.253104
$518$	2.11617e6	0.346519
$519$	4.12817e6	0.672727
$520$	−368568.	−0.0597736
$521$	6.16552e6	0.995120	0.497560	−	0.867430i	$-0.334229\pi$
$521$	6.16552e6	0.995120	0.497560	+	0.867430i	$0.334229\pi$
$522$	513719.	0.0825181
$523$	7.12400e6	1.13886	0.569429	−	0.822040i	$-0.307164\pi$
$523$	7.12400e6	1.13886	0.569429	+	0.822040i	$0.307164\pi$
$524$	−2.43098e6	−0.386770
$525$	324422.	0.0513704
$526$	−6256.55	−0.000985985 0
$527$	3.64072e6	0.571033
$528$	503869.	0.0786562
$529$	8.44071e6	1.31141
$530$	3.35543e6	0.518870
$531$	−6.63021e6	−1.02045
$532$	−17801.7	−0.00272699
$533$	1.08562e6	0.165523
$534$	2.01077e6	0.305148
$535$	−3.38240e6	−0.510905
$536$	8.09246e6	1.21666
$537$	1.58702e6	0.237490
$538$	4.01838e6	0.598544
$539$	217337.	0.0322227
$540$	1.13530e6	0.167543
$541$	−9.51129e6	−1.39716	−0.698580	−	0.715532i	$-0.746184\pi$
$541$	−9.51129e6	−1.39716	−0.698580	+	0.715532i	$0.746184\pi$
$542$	−1.70803e6	−0.249746
$543$	8.69737e6	1.26587
$544$	4.04862e6	0.586556
$545$	4.98771e6	0.719300
$546$	−176048.	−0.0252725
$547$	1.48406e6	0.212071	0.106036	−	0.994362i	$-0.466184\pi$
$547$	1.48406e6	0.212071	0.106036	+	0.994362i	$0.466184\pi$
$548$	−470001.	−0.0668570
$549$	382058.	0.0541002
$550$	256347.	0.0361345
$551$	27461.4	0.00385340
$552$	−8.04782e6	−1.12417
$553$	−2.21438e6	−0.307921
$554$	−3.98904e6	−0.552197
$555$	−2.52420e6	−0.347849
$556$	4.80486e6	0.659164
$557$	−7.72192e6	−1.05460	−0.527299	−	0.849680i	$-0.676795\pi$
$557$	−7.72192e6	−1.05460	−0.527299	+	0.849680i	$0.676795\pi$
$558$	−2.08990e6	−0.284144
$559$	736847.	0.0997351
$560$	−643690.	−0.0867374
$561$	−989894.	−0.132795
$562$	9.44703e6	1.26169
$563$	3.41747e6	0.454395	0.227198	−	0.973849i	$-0.427044\pi$
$563$	3.41747e6	0.454395	0.227198	+	0.973849i	$0.427044\pi$
$564$	−2.06461e6	−0.273301
$565$	−3.06422e6	−0.403830
$566$	3.13736e6	0.411645
$567$	498133.	0.0650710
$568$	−1.21631e7	−1.58188
$569$	−2.42117e6	−0.313505	−0.156752	−	0.987638i	$-0.550102\pi$
$569$	−2.42117e6	−0.313505	−0.156752	+	0.987638i	$0.550102\pi$
$570$	−38012.5	−0.00490049
$571$	2.31941e6	0.297705	0.148853	−	0.988859i	$-0.452442\pi$
$571$	2.31941e6	0.297705	0.148853	+	0.988859i	$0.452442\pi$
$572$	77706.2	0.00993037
$573$	−3.03626e6	−0.386325
$574$	3.22022e6	0.407949
$575$	−2.41067e6	−0.304067
$576$	−4.52307e6	−0.568038
$577$	2.56645e6	0.320917	0.160459	−	0.987043i	$-0.448703\pi$
$577$	2.56645e6	0.320917	0.160459	+	0.987043i	$0.448703\pi$
$578$	−1.60486e6	−0.199811
$579$	−1.05253e7	−1.30478
$580$	−248564.	−0.0306809
$581$	−1.91570e6	−0.235443
$582$	−385753.	−0.0472066
$583$	−2.68129e6	−0.326718
$584$	4.66103e6	0.565522
$585$	−244723.	−0.0295655
$586$	1.03139e7	1.24074
$587$	2.09851e6	0.251371	0.125685	−	0.992070i	$-0.459887\pi$
$587$	2.09851e6	0.251371	0.125685	+	0.992070i	$0.459887\pi$
$588$	291708.	0.0347940
$589$	−111718.	−0.0132688
$590$	−5.74291e6	−0.679207
$591$	404763.	0.0476686
$592$	5.00829e6	0.587334
$593$	−5.11831e6	−0.597709	−0.298854	−	0.954299i	$-0.596604\pi$
$593$	−5.11831e6	−0.597709	−0.298854	+	0.954299i	$0.596604\pi$
$594$	1.62405e6	0.188857
$595$	1.26458e6	0.146438
$596$	745048.	0.0859150
$597$	−7.73833e6	−0.888610
$598$	1.30815e6	0.149591
$599$	−2.61775e6	−0.298099	−0.149049	−	0.988830i	$-0.547621\pi$
$599$	−2.61775e6	−0.298099	−0.149049	+	0.988830i	$0.547621\pi$
$600$	1.30407e6	0.147884
$601$	9.49925e6	1.07276	0.536381	−	0.843976i	$-0.319791\pi$
$601$	9.49925e6	1.07276	0.536381	+	0.843976i	$0.319791\pi$
$602$	2.18568e6	0.245808
$603$	5.37326e6	0.601789
$604$	−4.35589e6	−0.485830
$605$	3.82143e6	0.424461
$606$	−5.99899e6	−0.663586
$607$	6.07366e6	0.669081	0.334541	−	0.942381i	$-0.391419\pi$
$607$	6.07366e6	0.669081	0.334541	+	0.942381i	$0.391419\pi$
$608$	−124234.	−0.0136295
$609$	−449996.	−0.0491660
$610$	330929.	0.0360089
$611$	1.27197e6	0.137839
$612$	1.54837e6	0.167107
$613$	−9.20907e6	−0.989839	−0.494920	−	0.868939i	$-0.664802\pi$
$613$	−9.20907e6	−0.989839	−0.494920	+	0.868939i	$0.664802\pi$
$614$	−1.20160e7	−1.28629
$615$	−3.84112e6	−0.409515
$616$	873621.	0.0927622
$617$	1.37224e7	1.45117	0.725585	−	0.688133i	$-0.241569\pi$
$617$	1.37224e7	1.45117	0.725585	+	0.688133i	$0.241569\pi$
$618$	3.67848e6	0.387434
$619$	−6.80356e6	−0.713690	−0.356845	−	0.934164i	$-0.616148\pi$
$619$	−6.80356e6	−0.713690	−0.356845	+	0.934164i	$0.616148\pi$
$620$	1.01120e6	0.105647
$621$	−1.52725e7	−1.58921
$622$	1.16734e7	1.20982
$623$	2.05266e6	0.211884
$624$	−416647.	−0.0428358
$625$	390625.	0.0400000
$626$	1.13980e7	1.16250
$627$	30375.4	0.00308570
$628$	3.22901e6	0.326716
$629$	−9.83921e6	−0.991593
$630$	−725913.	−0.0728674
$631$	−2.80897e6	−0.280850	−0.140425	−	0.990091i	$-0.544847\pi$
$631$	−2.80897e6	−0.280850	−0.140425	+	0.990091i	$0.544847\pi$
$632$	−8.90105e6	−0.886438
$633$	−2.05815e6	−0.204158
$634$	−1.04396e7	−1.03148
$635$	−1.00115e6	−0.0985297
$636$	−3.59880e6	−0.352788
$637$	−179715.	−0.0175484
$638$	−355571.	−0.0345839
$639$	−8.07608e6	−0.782435
$640$	−780256.	−0.0752986
$641$	−3.83494e6	−0.368649	−0.184325	−	0.982865i	$-0.559010\pi$
$641$	−3.83494e6	−0.368649	−0.184325	+	0.982865i	$0.559010\pi$
$642$	−6.49420e6	−0.621854
$643$	1.59562e7	1.52195	0.760977	−	0.648779i	$-0.224720\pi$
$643$	1.59562e7	1.52195	0.760977	+	0.648779i	$0.224720\pi$
$644$	−2.16758e6	−0.205949
$645$	−2.60711e6	−0.246752
$646$	−148171.	−0.0139695
$647$	−7.48025e6	−0.702514	−0.351257	−	0.936279i	$-0.614246\pi$
$647$	−7.48025e6	−0.702514	−0.351257	+	0.936279i	$0.614246\pi$
$648$	2.00232e6	0.187325
$649$	4.58910e6	0.427677
$650$	−211972.	−0.0196787
$651$	1.83066e6	0.169299
$652$	4.38785e6	0.404234
$653$	2.62102e6	0.240540	0.120270	−	0.992741i	$-0.461624\pi$
$653$	2.62102e6	0.240540	0.120270	+	0.992741i	$0.461624\pi$
$654$	9.57641e6	0.875504
$655$	−5.29908e6	−0.482611
$656$	7.62121e6	0.691456
$657$	3.09485e6	0.279721
$658$	3.77299e6	0.339720
$659$	−8.01276e6	−0.718734	−0.359367	−	0.933196i	$-0.617007\pi$
$659$	−8.01276e6	−0.718734	−0.359367	+	0.933196i	$0.617007\pi$
$660$	−274940.	−0.0245685
$661$	−7.21439e6	−0.642238	−0.321119	−	0.947039i	$-0.604059\pi$
$661$	−7.21439e6	−0.642238	−0.321119	+	0.947039i	$0.604059\pi$
$662$	3.15958e6	0.280210
$663$	818539.	0.0723195
$664$	−7.70044e6	−0.677790
$665$	−38804.4	−0.00340273
$666$	5.64803e6	0.493414
$667$	3.34377e6	0.291019
$668$	4.45721e6	0.386475
$669$	−1.20308e7	−1.03927
$670$	4.65417e6	0.400549
$671$	−264442.	−0.0226738
$672$	2.03576e6	0.173901
$673$	1.43323e7	1.21977	0.609887	−	0.792488i	$-0.291215\pi$
$673$	1.43323e7	1.21977	0.609887	+	0.792488i	$0.291215\pi$
$674$	−5.20390e6	−0.441244
$675$	2.47475e6	0.209060
$676$	4.19406e6	0.352994
$677$	−4.94781e6	−0.414898	−0.207449	−	0.978246i	$-0.566516\pi$
$677$	−4.94781e6	−0.414898	−0.207449	+	0.978246i	$0.566516\pi$
$678$	−5.88330e6	−0.491526
$679$	−393790.	−0.0327786
$680$	5.08319e6	0.421565
$681$	1.52340e6	0.125877
$682$	1.44652e6	0.119087
$683$	7.77331e6	0.637608	0.318804	−	0.947821i	$-0.396719\pi$
$683$	7.77331e6	0.637608	0.318804	+	0.947821i	$0.396719\pi$
$684$	−47512.5	−0.00388300
$685$	−1.02451e6	−0.0834241
$686$	−533083.	−0.0432498
$687$	40599.1	0.00328190
$688$	5.17280e6	0.416634
$689$	2.21715e6	0.177929
$690$	−4.62849e6	−0.370098
$691$	−7.69151e6	−0.612797	−0.306398	−	0.951903i	$-0.599124\pi$
$691$	−7.69151e6	−0.612797	−0.306398	+	0.951903i	$0.599124\pi$
$692$	4.46934e6	0.354795
$693$	580070.	0.0458825
$694$	−1.10431e7	−0.870347
$695$	1.04737e7	0.822504
$696$	−1.80883e6	−0.141538
$697$	−1.49725e7	−1.16738
$698$	4.06352e6	0.315692
$699$	1.43645e7	1.11198
$700$	351234.	0.0270927
$701$	1.53139e7	1.17704	0.588521	−	0.808482i	$-0.299711\pi$
$701$	1.53139e7	1.17704	0.588521	+	0.808482i	$0.299711\pi$
$702$	−1.34292e6	−0.102851
$703$	301921.	0.0230412
$704$	3.13065e6	0.238069
$705$	−4.50048e6	−0.341025
$706$	1.96389e7	1.48288
$707$	−6.12397e6	−0.460771
$708$	6.15944e6	0.461804
$709$	1.84381e7	1.37753	0.688764	−	0.724986i	$-0.258154\pi$
$709$	1.84381e7	1.37753	0.688764	+	0.724986i	$0.258154\pi$
$710$	−6.99528e6	−0.520786
$711$	−5.91015e6	−0.438454
$712$	8.25100e6	0.609968
$713$	−1.36030e7	−1.00210
$714$	2.42800e6	0.178239
$715$	169385.	0.0123911
$716$	1.71817e6	0.125252
$717$	5.06758e6	0.368132
$718$	−1.62743e7	−1.17813
$719$	−99951.2	−0.00721051	−0.00360525	−	0.999994i	$-0.501148\pi$
$719$	−99951.2	−0.00721051	−0.00360525	+	0.999994i	$0.501148\pi$
$720$	−1.71800e6	−0.123507
$721$	3.75512e6	0.269021
$722$	−1.12150e7	−0.800673
$723$	1.38860e7	0.987945
$724$	9.41616e6	0.667617
$725$	−541823.	−0.0382836
$726$	7.33715e6	0.516637
$727$	1.25198e7	0.878538	0.439269	−	0.898356i	$-0.355238\pi$
$727$	1.25198e7	0.878538	0.439269	+	0.898356i	$0.355238\pi$
$728$	−722393.	−0.0505179
$729$	1.15223e7	0.803007
$730$	2.68067e6	0.186181
$731$	−1.01624e7	−0.703401
$732$	−354931.	−0.0244831
$733$	−1.72341e7	−1.18476	−0.592378	−	0.805660i	$-0.701811\pi$
$733$	−1.72341e7	−1.18476	−0.592378	+	0.805660i	$0.701811\pi$
$734$	1.63932e6	0.112312
$735$	635868.	0.0434159
$736$	−1.51270e7	−1.02934
$737$	−3.71910e6	−0.252214
$738$	8.59472e6	0.580886
$739$	−1.83842e7	−1.23832	−0.619160	−	0.785265i	$-0.712527\pi$
$739$	−1.83842e7	−1.23832	−0.619160	+	0.785265i	$0.712527\pi$
$740$	−2.73281e6	−0.183455
$741$	−25117.3	−0.00168046
$742$	6.57664e6	0.438525
$743$	2.79507e7	1.85746	0.928732	−	0.370752i	$-0.120900\pi$
$743$	2.79507e7	1.85746	0.928732	+	0.370752i	$0.120900\pi$
$744$	7.35861e6	0.487375
$745$	1.62407e6	0.107205
$746$	−1.39706e7	−0.919111
$747$	−5.11296e6	−0.335252
$748$	−1.07170e6	−0.0700358
$749$	−6.62950e6	−0.431794
$750$	750000.	0.0486864
$751$	1.70136e7	1.10077	0.550383	−	0.834912i	$-0.314482\pi$
$751$	1.70136e7	1.10077	0.550383	+	0.834912i	$0.314482\pi$
$752$	8.92944e6	0.575810
$753$	−3.60295e6	−0.231564
$754$	294020.	0.0188342
$755$	−9.49502e6	−0.606218
$756$	2.22519e6	0.141600
$757$	−1.91600e7	−1.21522	−0.607611	−	0.794234i	$-0.707872\pi$
$757$	−1.91600e7	−1.21522	−0.607611	+	0.794234i	$0.707872\pi$
$758$	−2.28472e7	−1.44431
$759$	3.69858e6	0.233040
$760$	−155980.	−0.00979571
$761$	−2.06404e7	−1.29198	−0.645992	−	0.763344i	$-0.723556\pi$
$761$	−2.06404e7	−1.29198	−0.645992	+	0.763344i	$0.723556\pi$
$762$	−1.92222e6	−0.119926
$763$	9.77592e6	0.607919
$764$	−3.28719e6	−0.203747
$765$	3.37515e6	0.208516
$766$	−1.41339e7	−0.870341
$767$	−3.79471e6	−0.232911
$768$	1.02259e7	0.625605
$769$	1.11748e7	0.681432	0.340716	−	0.940166i	$-0.389331\pi$
$769$	1.11748e7	0.681432	0.340716	+	0.940166i	$0.389331\pi$
$770$	502441.	0.0305392
$771$	623311.	0.0377632
$772$	−1.13952e7	−0.688141
$773$	−3.07555e7	−1.85129	−0.925645	−	0.378394i	$-0.876477\pi$
$773$	−3.07555e7	−1.85129	−0.925645	+	0.378394i	$0.876477\pi$
$774$	5.83355e6	0.350010
$775$	2.20423e6	0.131826
$776$	−1.58290e6	−0.0943624
$777$	−4.94743e6	−0.293986
$778$	−7.45641e6	−0.441653
$779$	459440.	0.0271259
$780$	227347.	0.0133799
$781$	5.58986e6	0.327924
$782$	−1.80417e7	−1.05502
$783$	−3.43264e6	−0.200089
$784$	−1.26163e6	−0.0733065
$785$	7.03864e6	0.407675
$786$	−1.01742e7	−0.587416
$787$	3.52357e6	0.202790	0.101395	−	0.994846i	$-0.467669\pi$
$787$	3.52357e6	0.202790	0.101395	+	0.994846i	$0.467669\pi$
$788$	438214.	0.0251403
$789$	14627.3	0.000836509 0
$790$	−5.11921e6	−0.291833
$791$	−6.00587e6	−0.341299
$792$	2.33168e6	0.132086
$793$	218666.	0.0123480
$794$	1.68887e7	0.950706
$795$	−7.84470e6	−0.440209
$796$	−8.37786e6	−0.468652
$797$	−3.57469e6	−0.199339	−0.0996695	−	0.995021i	$-0.531779\pi$
$797$	−3.57469e6	−0.199339	−0.0996695	+	0.995021i	$0.531779\pi$
$798$	−74504.5	−0.00414167
$799$	−1.75426e7	−0.972139
$800$	2.45118e6	0.135410
$801$	5.47853e6	0.301705
$802$	2.19678e7	1.20601
$803$	−2.14210e6	−0.117233
$804$	−4.99173e6	−0.272340
$805$	−4.72492e6	−0.256983
$806$	−1.19612e6	−0.0648542
$807$	−9.39463e6	−0.507804
$808$	−2.46163e7	−1.32646
$809$	3.38466e7	1.81821	0.909104	−	0.416569i	$-0.136767\pi$
$809$	3.38466e7	1.81821	0.909104	+	0.416569i	$0.136767\pi$
$810$	1.15158e6	0.0616712
$811$	3.21058e7	1.71408	0.857042	−	0.515247i	$-0.172300\pi$
$811$	3.21058e7	1.71408	0.857042	+	0.515247i	$0.172300\pi$
$812$	−487185.	−0.0259301
$813$	3.99323e6	0.211884
$814$	−3.90928e6	−0.206793
$815$	9.56468e6	0.504402
$816$	5.74628e6	0.302108
$817$	311839.	0.0163446
$818$	−1.37350e7	−0.717706
$819$	−479657.	−0.0249874
$820$	−4.15857e6	−0.215978
$821$	3.04076e6	0.157443	0.0787216	−	0.996897i	$-0.474916\pi$
$821$	3.04076e6	0.157443	0.0787216	+	0.996897i	$0.474916\pi$
$822$	−1.96707e6	−0.101541
$823$	3.72147e6	0.191520	0.0957601	−	0.995404i	$-0.469472\pi$
$823$	3.72147e6	0.191520	0.0957601	+	0.995404i	$0.469472\pi$
$824$	1.50943e7	0.774452
$825$	−599318.	−0.0306565
$826$	−1.12561e7	−0.574034
$827$	−8.77895e6	−0.446354	−0.223177	−	0.974778i	$-0.571643\pi$
$827$	−8.77895e6	−0.446354	−0.223177	+	0.974778i	$0.571643\pi$
$828$	−5.78523e6	−0.293255
$829$	6.61553e6	0.334332	0.167166	−	0.985929i	$-0.446538\pi$
$829$	6.61553e6	0.334332	0.167166	+	0.985929i	$0.446538\pi$
$830$	−4.42871e6	−0.223142
$831$	9.32602e6	0.468483
$832$	−2.58872e6	−0.129651
$833$	2.47858e6	0.123763
$834$	2.01095e7	1.00112
$835$	9.71588e6	0.482243
$836$	32885.8	0.00162739
$837$	1.39646e7	0.688991
$838$	7.56267e6	0.372019
$839$	−4.67110e6	−0.229094	−0.114547	−	0.993418i	$-0.536542\pi$
$839$	−4.67110e6	−0.229094	−0.114547	+	0.993418i	$0.536542\pi$
$840$	2.55597e6	0.124985
$841$	−1.97596e7	−0.963359
$842$	−7.97619e6	−0.387718
$843$	−2.20863e7	−1.07042
$844$	−2.22824e6	−0.107673
$845$	9.14226e6	0.440465
$846$	1.00701e7	0.483733
$847$	7.49000e6	0.358735
$848$	1.55648e7	0.743280
$849$	−7.33487e6	−0.349239
$850$	2.92347e6	0.138788
$851$	3.67627e7	1.74014
$852$	7.50264e6	0.354091
$853$	−3.66146e7	−1.72299	−0.861494	−	0.507768i	$-0.830471\pi$
$853$	−3.66146e7	−1.72299	−0.861494	+	0.507768i	$0.830471\pi$
$854$	648620.	0.0304331
$855$	−103568.	−0.00484520
$856$	−2.66483e7	−1.24304
$857$	1.15520e7	0.537288	0.268644	−	0.963240i	$-0.413425\pi$
$857$	1.15520e7	0.537288	0.268644	+	0.963240i	$0.413425\pi$
$858$	325219.	0.0150820
$859$	3.53878e7	1.63633	0.818165	−	0.574983i	$-0.194991\pi$
$859$	3.53878e7	1.63633	0.818165	+	0.574983i	$0.194991\pi$
$860$	−2.82257e6	−0.130137
$861$	−7.52860e6	−0.346104
$862$	−2.93994e6	−0.134763
$863$	1.07433e7	0.491034	0.245517	−	0.969392i	$-0.421042\pi$
$863$	1.07433e7	0.491034	0.245517	+	0.969392i	$0.421042\pi$
$864$	1.55291e7	0.707720
$865$	9.74232e6	0.442713
$866$	−1.63571e6	−0.0741158
$867$	3.75203e6	0.169519
$868$	1.98195e6	0.0892881
$869$	4.09071e6	0.183759
$870$	−1.04030e6	−0.0465973
$871$	3.07531e6	0.137355
$872$	3.92958e7	1.75007
$873$	−1.05102e6	−0.0466740
$874$	553618.	0.0245150
$875$	765625.	0.0338062
$876$	−2.87510e6	−0.126588
$877$	4.58293e6	0.201208	0.100604	−	0.994927i	$-0.467923\pi$
$877$	4.58293e6	0.201208	0.100604	+	0.994927i	$0.467923\pi$
$878$	4.84927e6	0.212295
$879$	−2.41131e7	−1.05264
$880$	1.18911e6	0.0517626
$881$	1.68070e7	0.729543	0.364772	−	0.931097i	$-0.381147\pi$
$881$	1.68070e7	0.729543	0.364772	+	0.931097i	$0.381147\pi$
$882$	−1.42279e6	−0.0615842
$883$	−5.28260e6	−0.228006	−0.114003	−	0.993480i	$-0.536367\pi$
$883$	−5.28260e6	−0.228006	−0.114003	+	0.993480i	$0.536367\pi$
$884$	886186.	0.0381412
$885$	1.34264e7	0.576238
$886$	−3.61197e6	−0.154582
$887$	4.43107e7	1.89104	0.945518	−	0.325569i	$-0.105556\pi$
$887$	4.43107e7	1.89104	0.945518	+	0.325569i	$0.105556\pi$
$888$	−1.98870e7	−0.846322
$889$	−1.96226e6	−0.0832728
$890$	4.74535e6	0.200814
$891$	−920219.	−0.0388326
$892$	−1.30250e7	−0.548109
$893$	538305.	0.0225892
$894$	3.11821e6	0.130485
$895$	3.74530e6	0.156289
$896$	−1.52930e6	−0.0636389
$897$	−3.05834e6	−0.126913
$898$	2.90039e7	1.20023
$899$	−3.05741e6	−0.126170
$900$	937438.	0.0385777
$901$	−3.05783e7	−1.25488
$902$	−5.94883e6	−0.243453
$903$	−5.10994e6	−0.208543
$904$	−2.41415e7	−0.982524
$905$	2.05255e7	0.833051
$906$	−1.82304e7	−0.737865
$907$	−3.95608e7	−1.59679	−0.798394	−	0.602135i	$-0.794317\pi$
$907$	−3.95608e7	−1.59679	−0.798394	+	0.602135i	$0.794317\pi$
$908$	1.64930e6	0.0663872
$909$	−1.63448e7	−0.656099
$910$	−415466.	−0.0166315
$911$	−2.40982e7	−0.962030	−0.481015	−	0.876712i	$-0.659732\pi$
$911$	−2.40982e7	−0.962030	−0.481015	+	0.876712i	$0.659732\pi$
$912$	−176328.	−0.00701994
$913$	3.53894e6	0.140506
$914$	2.68172e7	1.06181
$915$	−773682.	−0.0305499
$916$	43954.4	0.00173087
$917$	−1.03862e7	−0.407881
$918$	1.85212e7	0.725375
$919$	−2.24660e7	−0.877480	−0.438740	−	0.898614i	$-0.644575\pi$
$919$	−2.24660e7	−0.877480	−0.438740	+	0.898614i	$0.644575\pi$
$920$	−1.89925e7	−0.739799
$921$	2.80923e7	1.09129
$922$	−4.00704e7	−1.55238
$923$	−4.62223e6	−0.178586
$924$	−538882.	−0.0207641
$925$	−5.95701e6	−0.228915
$926$	−3.06693e7	−1.17538
$927$	1.00224e7	0.383063
$928$	−3.39995e6	−0.129599
$929$	−1.16280e7	−0.442043	−0.221021	−	0.975269i	$-0.570939\pi$
$929$	−1.16280e7	−0.442043	−0.221021	+	0.975269i	$0.570939\pi$
$930$	4.23212e6	0.160454
$931$	−76056.7	−0.00287583
$932$	1.55517e7	0.586458
$933$	−2.72913e7	−1.02641
$934$	−4.74591e6	−0.178013
$935$	−2.33611e6	−0.0873906
$936$	−1.92806e6	−0.0719333
$937$	−2.40956e7	−0.896580	−0.448290	−	0.893888i	$-0.647967\pi$
$937$	−2.40956e7	−0.896580	−0.448290	+	0.893888i	$0.647967\pi$
$938$	9.12218e6	0.338525
$939$	−2.66475e7	−0.986264
$940$	−4.87241e6	−0.179856
$941$	−567590.	−0.0208959	−0.0104479	−	0.999945i	$-0.503326\pi$
$941$	−567590.	−0.0208959	−0.0104479	+	0.999945i	$0.503326\pi$
$942$	1.35142e7	0.496207
$943$	5.59425e7	2.04863
$944$	−2.66395e7	−0.972962
$945$	4.85051e6	0.176688
$946$	−4.03769e6	−0.146692
$947$	9.04501e6	0.327743	0.163872	−	0.986482i	$-0.447602\pi$
$947$	9.04501e6	0.327743	0.163872	+	0.986482i	$0.447602\pi$
$948$	5.49050e6	0.198423
$949$	1.77129e6	0.0638446
$950$	−89708.1	−0.00322495
$951$	2.44068e7	0.875105
$952$	9.96305e6	0.356287
$953$	−1.48053e7	−0.528063	−0.264032	−	0.964514i	$-0.585052\pi$
$953$	−1.48053e7	−0.528063	−0.264032	+	0.964514i	$0.585052\pi$
$954$	1.75529e7	0.624423
$955$	−7.16546e6	−0.254235
$956$	5.48639e6	0.194152
$957$	831294.	0.0293410
$958$	−4.24895e7	−1.49578
$959$	−2.00805e6	−0.0705062
$960$	9.15939e6	0.320766
$961$	−1.61911e7	−0.565546
$962$	3.23257e6	0.112619
$963$	−1.76940e7	−0.614838
$964$	1.50336e7	0.521041
$965$	−2.48393e7	−0.858661
$966$	−9.07185e6	−0.312790
$967$	−2.02304e7	−0.695726	−0.347863	−	0.937545i	$-0.613093\pi$
$967$	−2.02304e7	−0.695726	−0.347863	+	0.937545i	$0.613093\pi$
$968$	3.01072e7	1.03272
$969$	346411.	0.0118517
$970$	−910364.	−0.0310660
$971$	−255216.	−0.00868679	−0.00434340	−	0.999991i	$-0.501383\pi$
$971$	−255216.	−0.00868679	−0.00434340	+	0.999991i	$0.501383\pi$
$972$	9.80004e6	0.332707
$973$	2.05284e7	0.695142
$974$	−1.30932e7	−0.442229
$975$	495573.	0.0166954
$976$	1.53507e6	0.0515827
$977$	3.15610e7	1.05783	0.528914	−	0.848675i	$-0.322599\pi$
$977$	3.15610e7	1.05783	0.528914	+	0.848675i	$0.322599\pi$
$978$	1.83642e7	0.613938
$979$	−3.79196e6	−0.126447
$980$	688419.	0.0228975
$981$	2.60917e7	0.865627
$982$	−4.75064e6	−0.157208
$983$	−5.24054e7	−1.72978	−0.864892	−	0.501958i	$-0.832613\pi$
$983$	−5.24054e7	−1.72978	−0.864892	+	0.501958i	$0.832613\pi$
$984$	−3.02624e7	−0.996357
$985$	955225.	0.0313701
$986$	−4.05504e6	−0.132832
$987$	−8.82093e6	−0.288218
$988$	−27193.1	−0.000886270 0
$989$	3.79702e7	1.23439
$990$	1.34101e6	0.0434853
$991$	3.73103e7	1.20683	0.603413	−	0.797429i	$-0.293807\pi$
$991$	3.73103e7	1.20683	0.603413	+	0.797429i	$0.293807\pi$
$992$	1.38316e7	0.446264
$993$	−7.38682e6	−0.237730
$994$	−1.37108e7	−0.440145
$995$	−1.82622e7	−0.584783
$996$	4.74992e6	0.151718
$997$	−2.27918e7	−0.726174	−0.363087	−	0.931755i	$-0.618277\pi$
$997$	−2.27918e7	−0.726174	−0.363087	+	0.931755i	$0.618277\pi$
$998$	2.89323e7	0.919512
$999$	−3.77398e7	−1.19643

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	35.6.a.b.1.2	✓	2
3.2	odd	2		315.6.a.c.1.1		2
4.3	odd	2		560.6.a.l.1.2		2
5.2	odd	4		175.6.b.d.99.4		4
5.3	odd	4		175.6.b.d.99.1		4
5.4	even	2		175.6.a.d.1.1		2
7.6	odd	2		245.6.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.a.b.1.2	✓	2	1.1	even	1	trivial
175.6.a.d.1.1		2	5.4	even	2
175.6.b.d.99.1		4	5.3	odd	4
175.6.b.d.99.4		4	5.2	odd	4
245.6.a.c.1.2		2	7.6	odd	2
315.6.a.c.1.1		2	3.2	odd	2
560.6.a.l.1.2		2	4.3	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:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	35.a (trivial)

\(f(q)\)	\(=\)	\(q+4.53113 q^{2} -10.5934 q^{3} -11.4689 q^{4} -25.0000 q^{5} -48.0000 q^{6} -49.0000 q^{7} -196.963 q^{8} -130.780 q^{9} -113.278 q^{10} +90.5195 q^{11} +121.494 q^{12} -74.8502 q^{13} -222.025 q^{14} +264.835 q^{15} -525.461 q^{16} +1032.31 q^{17} -592.582 q^{18} -31.6771 q^{19} +286.722 q^{20} +519.076 q^{21} +410.156 q^{22} -3857.08 q^{23} +2086.51 q^{24} +625.000 q^{25} -339.156 q^{26} +3959.60 q^{27} +561.975 q^{28} -866.917 q^{29} +1200.00 q^{30} +3526.76 q^{31} +3921.89 q^{32} -958.908 q^{33} +4677.54 q^{34} +1225.00 q^{35} +1499.90 q^{36} -9531.22 q^{37} -143.533 q^{38} +792.917 q^{39} +4924.08 q^{40} -14503.9 q^{41} +2352.00 q^{42} -9844.30 q^{43} -1038.16 q^{44} +3269.50 q^{45} -17476.9 q^{46} -16993.5 q^{47} +5566.41 q^{48} +2401.00 q^{49} +2831.96 q^{50} -10935.7 q^{51} +858.447 q^{52} -29621.1 q^{53} +17941.4 q^{54} -2262.99 q^{55} +9651.19 q^{56} +335.568 q^{57} -3928.11 q^{58} +50697.4 q^{59} -3037.35 q^{60} -2921.38 q^{61} +15980.2 q^{62} +6408.23 q^{63} +34585.3 q^{64} +1871.25 q^{65} -4344.94 q^{66} -41086.2 q^{67} -11839.5 q^{68} +40859.5 q^{69} +5550.63 q^{70} +61753.1 q^{71} +25758.9 q^{72} -23664.5 q^{73} -43187.2 q^{74} -6620.87 q^{75} +363.300 q^{76} -4435.46 q^{77} +3592.81 q^{78} +45191.5 q^{79} +13136.5 q^{80} -10166.0 q^{81} -65718.8 q^{82} +39095.9 q^{83} -5953.22 q^{84} -25807.8 q^{85} -44605.8 q^{86} +9183.58 q^{87} -17829.0 q^{88} -41891.1 q^{89} +14814.5 q^{90} +3667.66 q^{91} +44236.3 q^{92} -37360.4 q^{93} -76999.9 q^{94} +791.927 q^{95} -41546.1 q^{96} +8036.53 q^{97} +10879.2 q^{98} -11838.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{3} - 31 q^{4} - 50 q^{5} - 96 q^{6} - 98 q^{7} - 15 q^{8} - 189 q^{9} - 25 q^{10} - 601 q^{11} - 144 q^{12} - 577 q^{13} - 49 q^{14} - 75 q^{15} - 543 q^{16} + 41 q^{17} - 387 q^{18}+ \cdots + 28422 q^{99}+O(q^{100}) \) 2 * q + q^2 + 3 * q^3 - 31 * q^4 - 50 * q^5 - 96 * q^6 - 98 * q^7 - 15 * q^8 - 189 * q^9 - 25 * q^10 - 601 * q^11 - 144 * q^12 - 577 * q^13 - 49 * q^14 - 75 * q^15 - 543 * q^16 + 41 * q^17 - 387 * q^18 + 630 * q^19 + 775 * q^20 - 147 * q^21 + 2852 * q^22 - 442 * q^23 + 4560 * q^24 + 1250 * q^25 + 1434 * q^26 - 135 * q^27 + 1519 * q^28 + 5885 * q^29 + 2400 * q^30 - 396 * q^31 - 1839 * q^32 - 10359 * q^33 + 8178 * q^34 + 2450 * q^35 + 2637 * q^36 - 8904 * q^37 - 2480 * q^38 - 6033 * q^39 + 375 * q^40 + 1774 * q^41 + 4704 * q^42 - 27122 * q^43 + 12468 * q^44 + 4725 * q^45 - 29536 * q^46 - 21289 * q^47 + 5328 * q^48 + 4802 * q^49 + 625 * q^50 - 24411 * q^51 + 10666 * q^52 - 55582 * q^53 + 32400 * q^54 + 15025 * q^55 + 735 * q^56 + 9330 * q^57 - 27770 * q^58 + 59600 * q^59 + 3600 * q^60 - 51846 * q^61 + 29832 * q^62 + 9261 * q^63 + 55489 * q^64 + 14425 * q^65 + 28848 * q^66 - 45344 * q^67 + 7522 * q^68 + 87282 * q^69 + 1225 * q^70 + 80744 * q^71 + 15165 * q^72 - 13532 * q^73 - 45402 * q^74 + 1875 * q^75 - 12560 * q^76 + 29449 * q^77 + 27696 * q^78 - 51795 * q^79 + 13575 * q^80 - 51678 * q^81 - 123198 * q^82 + 109828 * q^83 + 7056 * q^84 - 1025 * q^85 + 16404 * q^86 + 100965 * q^87 - 143660 * q^88 - 37650 * q^89 + 9675 * q^90 + 28273 * q^91 - 22464 * q^92 - 90684 * q^93 - 61832 * q^94 - 15750 * q^95 - 119856 * q^96 - 96339 * q^97 + 2401 * q^98 + 28422 * q^99

Introduction

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