LMFDB - Embedded newform 105.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [105,4,Mod(1,105)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(105, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("105.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 105 = 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	105.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$6.19520055060$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	105.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.23607 q^{2} +3.00000 q^{3} +9.94427 q^{4} -5.00000 q^{5} -12.7082 q^{6} -7.00000 q^{7} -8.23607 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.23607 q^{2} +3.00000 q^{3} +9.94427 q^{4} -5.00000 q^{5} -12.7082 q^{6} -7.00000 q^{7} -8.23607 q^{8} +9.00000 q^{9} +21.1803 q^{10} -41.5279 q^{11} +29.8328 q^{12} +88.9706 q^{13} +29.6525 q^{14} -15.0000 q^{15} -44.6656 q^{16} -120.387 q^{17} -38.1246 q^{18} -112.138 q^{19} -49.7214 q^{20} -21.0000 q^{21} +175.915 q^{22} -115.279 q^{23} -24.7082 q^{24} +25.0000 q^{25} -376.885 q^{26} +27.0000 q^{27} -69.6099 q^{28} -144.833 q^{29} +63.5410 q^{30} -258.079 q^{31} +255.095 q^{32} -124.584 q^{33} +509.967 q^{34} +35.0000 q^{35} +89.4984 q^{36} +48.3344 q^{37} +475.023 q^{38} +266.912 q^{39} +41.1803 q^{40} +200.885 q^{41} +88.9574 q^{42} -218.217 q^{43} -412.964 q^{44} -45.0000 q^{45} +488.328 q^{46} +575.659 q^{47} -133.997 q^{48} +49.0000 q^{49} -105.902 q^{50} -361.161 q^{51} +884.748 q^{52} -184.302 q^{53} -114.374 q^{54} +207.639 q^{55} +57.6525 q^{56} -336.413 q^{57} +613.522 q^{58} -151.502 q^{59} -149.164 q^{60} -529.830 q^{61} +1093.24 q^{62} -63.0000 q^{63} -723.276 q^{64} -444.853 q^{65} +527.745 q^{66} +1.28485 q^{67} -1197.16 q^{68} -345.836 q^{69} -148.262 q^{70} -61.4226 q^{71} -74.1246 q^{72} +484.800 q^{73} -204.748 q^{74} +75.0000 q^{75} -1115.13 q^{76} +290.695 q^{77} -1130.66 q^{78} +878.257 q^{79} +223.328 q^{80} +81.0000 q^{81} -850.964 q^{82} +491.830 q^{83} -208.830 q^{84} +601.935 q^{85} +924.381 q^{86} -434.498 q^{87} +342.026 q^{88} -415.560 q^{89} +190.623 q^{90} -622.794 q^{91} -1146.36 q^{92} -774.237 q^{93} -2438.53 q^{94} +560.689 q^{95} +765.286 q^{96} -1031.70 q^{97} -207.567 q^{98} -373.751 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 12 q^{6} - 14 q^{7} - 12 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 + 6 * q^3 + 2 * q^4 - 10 * q^5 - 12 * q^6 - 14 * q^7 - 12 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 12 q^{6} - 14 q^{7} - 12 q^{8} + 18 q^{9} + 20 q^{10} - 92 q^{11} + 6 q^{12} + 8 q^{13} + 28 q^{14} - 30 q^{15} + 18 q^{16} - 44 q^{17} - 36 q^{18} - 108 q^{19} - 10 q^{20} - 42 q^{21} + 164 q^{22} - 320 q^{23} - 36 q^{24} + 50 q^{25} - 396 q^{26} + 54 q^{27} - 14 q^{28} - 236 q^{29} + 60 q^{30} - 60 q^{31} + 300 q^{32} - 276 q^{33} + 528 q^{34} + 70 q^{35} + 18 q^{36} + 204 q^{37} + 476 q^{38} + 24 q^{39} + 60 q^{40} + 44 q^{41} + 84 q^{42} + 136 q^{43} - 12 q^{44} - 90 q^{45} + 440 q^{46} + 400 q^{47} + 54 q^{48} + 98 q^{49} - 100 q^{50} - 132 q^{51} + 1528 q^{52} + 16 q^{53} - 108 q^{54} + 460 q^{55} + 84 q^{56} - 324 q^{57} + 592 q^{58} - 464 q^{59} - 30 q^{60} - 684 q^{61} + 1140 q^{62} - 126 q^{63} - 1214 q^{64} - 40 q^{65} + 492 q^{66} + 736 q^{67} - 1804 q^{68} - 960 q^{69} - 140 q^{70} - 740 q^{71} - 108 q^{72} + 424 q^{73} - 168 q^{74} + 150 q^{75} - 1148 q^{76} + 644 q^{77} - 1188 q^{78} - 408 q^{79} - 90 q^{80} + 162 q^{81} - 888 q^{82} + 608 q^{83} - 42 q^{84} + 220 q^{85} + 1008 q^{86} - 708 q^{87} + 532 q^{88} - 1332 q^{89} + 180 q^{90} - 56 q^{91} + 480 q^{92} - 180 q^{93} - 2480 q^{94} + 540 q^{95} + 900 q^{96} - 2448 q^{97} - 196 q^{98} - 828 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 6 * q^3 + 2 * q^4 - 10 * q^5 - 12 * q^6 - 14 * q^7 - 12 * q^8 + 18 * q^9 + 20 * q^10 - 92 * q^11 + 6 * q^12 + 8 * q^13 + 28 * q^14 - 30 * q^15 + 18 * q^16 - 44 * q^17 - 36 * q^18 - 108 * q^19 - 10 * q^20 - 42 * q^21 + 164 * q^22 - 320 * q^23 - 36 * q^24 + 50 * q^25 - 396 * q^26 + 54 * q^27 - 14 * q^28 - 236 * q^29 + 60 * q^30 - 60 * q^31 + 300 * q^32 - 276 * q^33 + 528 * q^34 + 70 * q^35 + 18 * q^36 + 204 * q^37 + 476 * q^38 + 24 * q^39 + 60 * q^40 + 44 * q^41 + 84 * q^42 + 136 * q^43 - 12 * q^44 - 90 * q^45 + 440 * q^46 + 400 * q^47 + 54 * q^48 + 98 * q^49 - 100 * q^50 - 132 * q^51 + 1528 * q^52 + 16 * q^53 - 108 * q^54 + 460 * q^55 + 84 * q^56 - 324 * q^57 + 592 * q^58 - 464 * q^59 - 30 * q^60 - 684 * q^61 + 1140 * q^62 - 126 * q^63 - 1214 * q^64 - 40 * q^65 + 492 * q^66 + 736 * q^67 - 1804 * q^68 - 960 * q^69 - 140 * q^70 - 740 * q^71 - 108 * q^72 + 424 * q^73 - 168 * q^74 + 150 * q^75 - 1148 * q^76 + 644 * q^77 - 1188 * q^78 - 408 * q^79 - 90 * q^80 + 162 * q^81 - 888 * q^82 + 608 * q^83 - 42 * q^84 + 220 * q^85 + 1008 * q^86 - 708 * q^87 + 532 * q^88 - 1332 * q^89 + 180 * q^90 - 56 * q^91 + 480 * q^92 - 180 * q^93 - 2480 * q^94 + 540 * q^95 + 900 * q^96 - 2448 * q^97 - 196 * q^98 - 828 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.23607	−1.49768	−0.748838	−	0.662753i	$-0.769388\pi$
$2$	−4.23607	−1.49768	−0.748838	+	0.662753i	$0.769388\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	9.94427	1.24303
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−12.7082	−0.864684
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.23607	−0.363986
$9$	9.00000	0.333333
$10$	21.1803	0.669781
$11$	−41.5279	−1.13828	−0.569142	−	0.822239i	$-0.692725\pi$
$11$	−41.5279	−1.13828	−0.569142	+	0.822239i	$0.692725\pi$
$12$	29.8328	0.717666
$13$	88.9706	1.89815	0.949077	−	0.315044i	$-0.102019\pi$
$13$	88.9706	1.89815	0.949077	+	0.315044i	$0.102019\pi$
$14$	29.6525	0.566068
$15$	−15.0000	−0.258199
$16$	−44.6656	−0.697900
$17$	−120.387	−1.71754	−0.858769	−	0.512364i	$-0.828770\pi$
$17$	−120.387	−1.71754	−0.858769	+	0.512364i	$0.828770\pi$
$18$	−38.1246	−0.499225
$19$	−112.138	−1.35401	−0.677004	−	0.735979i	$-0.736722\pi$
$19$	−112.138	−1.35401	−0.677004	+	0.735979i	$0.736722\pi$
$20$	−49.7214	−0.555902
$21$	−21.0000	−0.218218
$22$	175.915	1.70478
$23$	−115.279	−1.04510	−0.522549	−	0.852609i	$-0.675019\pi$
$23$	−115.279	−1.04510	−0.522549	+	0.852609i	$0.675019\pi$
$24$	−24.7082	−0.210148
$25$	25.0000	0.200000
$26$	−376.885	−2.84282
$27$	27.0000	0.192450
$28$	−69.6099	−0.469823
$29$	−144.833	−0.927406	−0.463703	−	0.885991i	$-0.653480\pi$
$29$	−144.833	−0.927406	−0.463703	+	0.885991i	$0.653480\pi$
$30$	63.5410	0.386698
$31$	−258.079	−1.49524	−0.747618	−	0.664128i	$-0.768803\pi$
$31$	−258.079	−1.49524	−0.747618	+	0.664128i	$0.768803\pi$
$32$	255.095	1.40922
$33$	−124.584	−0.657188
$34$	509.967	2.57231
$35$	35.0000	0.169031
$36$	89.4984	0.414345
$37$	48.3344	0.214760	0.107380	−	0.994218i	$-0.465754\pi$
$37$	48.3344	0.214760	0.107380	+	0.994218i	$0.465754\pi$
$38$	475.023	2.02787
$39$	266.912	1.09590
$40$	41.1803	0.162780
$41$	200.885	0.765196	0.382598	−	0.923915i	$-0.375029\pi$
$41$	200.885	0.765196	0.382598	+	0.923915i	$0.375029\pi$
$42$	88.9574	0.326820
$43$	−218.217	−0.773901	−0.386950	−	0.922101i	$-0.626472\pi$
$43$	−218.217	−0.773901	−0.386950	+	0.922101i	$0.626472\pi$
$44$	−412.964	−1.41493
$45$	−45.0000	−0.149071
$46$	488.328	1.56522
$47$	575.659	1.78657	0.893283	−	0.449496i	$-0.148396\pi$
$47$	575.659	1.78657	0.893283	+	0.449496i	$0.148396\pi$
$48$	−133.997	−0.402933
$49$	49.0000	0.142857
$50$	−105.902	−0.299535
$51$	−361.161	−0.991621
$52$	884.748	2.35947
$53$	−184.302	−0.477657	−0.238828	−	0.971062i	$-0.576763\pi$
$53$	−184.302	−0.477657	−0.238828	+	0.971062i	$0.576763\pi$
$54$	−114.374	−0.288228
$55$	207.639	0.509056
$56$	57.6525	0.137574
$57$	−336.413	−0.781737
$58$	613.522	1.38895
$59$	−151.502	−0.334302	−0.167151	−	0.985931i	$-0.553457\pi$
$59$	−151.502	−0.334302	−0.167151	+	0.985931i	$0.553457\pi$
$60$	−149.164	−0.320950
$61$	−529.830	−1.11209	−0.556047	−	0.831151i	$-0.687683\pi$
$61$	−529.830	−1.11209	−0.556047	+	0.831151i	$0.687683\pi$
$62$	1093.24	2.23938
$63$	−63.0000	−0.125988
$64$	−723.276	−1.41265
$65$	−444.853	−0.848880
$66$	527.745	0.984256
$67$	1.28485	0.00234283	0.00117142	−	0.999999i	$-0.499627\pi$
$67$	1.28485	0.00234283	0.00117142	+	0.999999i	$0.499627\pi$
$68$	−1197.16	−2.13496
$69$	−345.836	−0.603388
$70$	−148.262	−0.253153
$71$	−61.4226	−0.102669	−0.0513347	−	0.998682i	$-0.516348\pi$
$71$	−61.4226	−0.102669	−0.0513347	+	0.998682i	$0.516348\pi$
$72$	−74.1246	−0.121329
$73$	484.800	0.777282	0.388641	−	0.921389i	$-0.372945\pi$
$73$	484.800	0.777282	0.388641	+	0.921389i	$0.372945\pi$
$74$	−204.748	−0.321641
$75$	75.0000	0.115470
$76$	−1115.13	−1.68308
$77$	290.695	0.430231
$78$	−1130.66	−1.64130
$79$	878.257	1.25078	0.625390	−	0.780312i	$-0.284940\pi$
$79$	878.257	1.25078	0.625390	+	0.780312i	$0.284940\pi$
$80$	223.328	0.312111
$81$	81.0000	0.111111
$82$	−850.964	−1.14602
$83$	491.830	0.650426	0.325213	−	0.945641i	$-0.394564\pi$
$83$	491.830	0.650426	0.325213	+	0.945641i	$0.394564\pi$
$84$	−208.830	−0.271252
$85$	601.935	0.768106
$86$	924.381	1.15905
$87$	−434.498	−0.535438
$88$	342.026	0.414320
$89$	−415.560	−0.494936	−0.247468	−	0.968896i	$-0.579599\pi$
$89$	−415.560	−0.494936	−0.247468	+	0.968896i	$0.579599\pi$
$90$	190.623	0.223260
$91$	−622.794	−0.717435
$92$	−1146.36	−1.29909
$93$	−774.237	−0.863275
$94$	−2438.53	−2.67570
$95$	560.689	0.605531
$96$	765.286	0.813611
$97$	−1031.70	−1.07993	−0.539964	−	0.841688i	$-0.681562\pi$
$97$	−1031.70	−1.07993	−0.539964	+	0.841688i	$0.681562\pi$
$98$	−207.567	−0.213954
$99$	−373.751	−0.379428
$100$	248.607	0.248607
$101$	1447.19	1.42576	0.712878	−	0.701288i	$-0.247391\pi$
$101$	1447.19	1.42576	0.712878	+	0.701288i	$0.247391\pi$
$102$	1529.90	1.48513
$103$	−163.567	−0.156473	−0.0782364	−	0.996935i	$-0.524929\pi$
$103$	−163.567	−0.156473	−0.0782364	+	0.996935i	$0.524929\pi$
$104$	−732.768	−0.690902
$105$	105.000	0.0975900
$106$	780.715	0.715375
$107$	−129.653	−0.117141	−0.0585703	−	0.998283i	$-0.518654\pi$
$107$	−129.653	−0.117141	−0.0585703	+	0.998283i	$0.518654\pi$
$108$	268.495	0.239222
$109$	566.681	0.497965	0.248983	−	0.968508i	$-0.419904\pi$
$109$	566.681	0.497965	0.248983	+	0.968508i	$0.419904\pi$
$110$	−879.574	−0.762401
$111$	145.003	0.123992
$112$	312.659	0.263782
$113$	809.890	0.674230	0.337115	−	0.941463i	$-0.390549\pi$
$113$	809.890	0.674230	0.337115	+	0.941463i	$0.390549\pi$
$114$	1425.07	1.17079
$115$	576.393	0.467382
$116$	−1440.26	−1.15280
$117$	800.735	0.632718
$118$	641.771	0.500676
$119$	842.709	0.649168
$120$	123.541	0.0939808
$121$	393.563	0.295690
$122$	2244.39	1.66556
$123$	602.656	0.441786
$124$	−2566.41	−1.85863
$125$	−125.000	−0.0894427
$126$	266.872	0.188689
$127$	−2584.25	−1.80563	−0.902816	−	0.430028i	$-0.858504\pi$
$127$	−2584.25	−1.80563	−0.902816	+	0.430028i	$0.858504\pi$
$128$	1023.08	0.706473
$129$	−654.650	−0.446812
$130$	1884.43	1.27135
$131$	−1421.10	−0.947804	−0.473902	−	0.880578i	$-0.657155\pi$
$131$	−1421.10	−0.947804	−0.473902	+	0.880578i	$0.657155\pi$
$132$	−1238.89	−0.816908
$133$	784.964	0.511767
$134$	−5.44272	−0.00350880
$135$	−135.000	−0.0860663
$136$	991.515	0.625160
$137$	104.878	0.0654037	0.0327019	−	0.999465i	$-0.489589\pi$
$137$	104.878	0.0654037	0.0327019	+	0.999465i	$0.489589\pi$
$138$	1464.98	0.903679
$139$	−913.160	−0.557217	−0.278609	−	0.960405i	$-0.589873\pi$
$139$	−913.160	−0.557217	−0.278609	+	0.960405i	$0.589873\pi$
$140$	348.050	0.210111
$141$	1726.98	1.03147
$142$	260.190	0.153765
$143$	−3694.76	−2.16064
$144$	−401.991	−0.232633
$145$	724.164	0.414749
$146$	−2053.65	−1.16412
$147$	147.000	0.0824786
$148$	480.650	0.266954
$149$	1781.45	0.979476	0.489738	−	0.871870i	$-0.337092\pi$
$149$	1781.45	0.979476	0.489738	+	0.871870i	$0.337092\pi$
$150$	−317.705	−0.172937
$151$	1407.53	0.758564	0.379282	−	0.925281i	$-0.376171\pi$
$151$	1407.53	0.758564	0.379282	+	0.925281i	$0.376171\pi$
$152$	923.574	0.492841
$153$	−1083.48	−0.572512
$154$	−1231.40	−0.644346
$155$	1290.39	0.668690
$156$	2654.24	1.36224
$157$	−1598.94	−0.812798	−0.406399	−	0.913696i	$-0.633216\pi$
$157$	−1598.94	−0.812798	−0.406399	+	0.913696i	$0.633216\pi$
$158$	−3720.36	−1.87326
$159$	−552.906	−0.275775
$160$	−1275.48	−0.630220
$161$	806.950	0.395010
$162$	−343.122	−0.166408
$163$	−204.892	−0.0984562	−0.0492281	−	0.998788i	$-0.515676\pi$
$163$	−204.892	−0.0984562	−0.0492281	+	0.998788i	$0.515676\pi$
$164$	1997.66	0.951165
$165$	622.918	0.293904
$166$	−2083.42	−0.974127
$167$	−1165.94	−0.540259	−0.270129	−	0.962824i	$-0.587067\pi$
$167$	−1165.94	−0.540259	−0.270129	+	0.962824i	$0.587067\pi$
$168$	172.957	0.0794283
$169$	5718.76	2.60299
$170$	−2549.84	−1.15037
$171$	−1009.24	−0.451336
$172$	−2170.01	−0.961985
$173$	−2538.00	−1.11538	−0.557690	−	0.830049i	$-0.688312\pi$
$173$	−2538.00	−1.11538	−0.557690	+	0.830049i	$0.688312\pi$
$174$	1840.56	0.801913
$175$	−175.000	−0.0755929
$176$	1854.87	0.794409
$177$	−454.505	−0.193009
$178$	1760.34	0.741254
$179$	392.255	0.163791	0.0818954	−	0.996641i	$-0.473903\pi$
$179$	392.255	0.163791	0.0818954	+	0.996641i	$0.473903\pi$
$180$	−447.492	−0.185301
$181$	−2978.08	−1.22298	−0.611489	−	0.791253i	$-0.709429\pi$
$181$	−2978.08	−1.22298	−0.611489	+	0.791253i	$0.709429\pi$
$182$	2638.20	1.07448
$183$	−1589.49	−0.642068
$184$	949.443	0.380401
$185$	−241.672	−0.0960436
$186$	3279.72	1.29291
$187$	4999.41	1.95504
$188$	5724.51	2.22076
$189$	−189.000	−0.0727393
$190$	−2375.12	−0.906890
$191$	−1097.37	−0.415722	−0.207861	−	0.978158i	$-0.566650\pi$
$191$	−1097.37	−0.415722	−0.207861	+	0.978158i	$0.566650\pi$
$192$	−2169.83	−0.815592
$193$	3500.31	1.30548	0.652740	−	0.757582i	$-0.273619\pi$
$193$	3500.31	1.30548	0.652740	+	0.757582i	$0.273619\pi$
$194$	4370.34	1.61738
$195$	−1334.56	−0.490101
$196$	487.269	0.177576
$197$	1573.96	0.569237	0.284618	−	0.958641i	$-0.408133\pi$
$197$	1573.96	0.569237	0.284618	+	0.958641i	$0.408133\pi$
$198$	1583.23	0.568260
$199$	−3396.62	−1.20995	−0.604976	−	0.796244i	$-0.706817\pi$
$199$	−3396.62	−1.20995	−0.604976	+	0.796244i	$0.706817\pi$
$200$	−205.902	−0.0727972
$201$	3.85456	0.00135263
$202$	−6130.42	−2.13532
$203$	1013.83	0.350527
$204$	−3591.48	−1.23262
$205$	−1004.43	−0.342206
$206$	692.879	0.234346
$207$	−1037.51	−0.348366
$208$	−3973.93	−1.32472
$209$	4656.84	1.54125
$210$	−444.787	−0.146158
$211$	3337.81	1.08903	0.544513	−	0.838753i	$-0.316715\pi$
$211$	3337.81	1.08903	0.544513	+	0.838753i	$0.316715\pi$
$212$	−1832.75	−0.593744
$213$	−184.268	−0.0592762
$214$	549.220	0.175439
$215$	1091.08	0.346099
$216$	−222.374	−0.0700492
$217$	1806.55	0.565146
$218$	−2400.50	−0.745791
$219$	1454.40	0.448764
$220$	2064.82	0.632774
$221$	−10710.9	−3.26015
$222$	−614.243	−0.185700
$223$	127.328	0.0382356	0.0191178	−	0.999817i	$-0.493914\pi$
$223$	127.328	0.0382356	0.0191178	+	0.999817i	$0.493914\pi$
$224$	−1785.67	−0.532633
$225$	225.000	0.0666667
$226$	−3430.75	−1.00978
$227$	3844.12	1.12398	0.561990	−	0.827144i	$-0.310036\pi$
$227$	3844.12	1.12398	0.561990	+	0.827144i	$0.310036\pi$
$228$	−3345.39	−0.971726
$229$	2536.95	0.732080	0.366040	−	0.930599i	$-0.380713\pi$
$229$	2536.95	0.732080	0.366040	+	0.930599i	$0.380713\pi$
$230$	−2441.64	−0.699987
$231$	872.085	0.248394
$232$	1192.85	0.337563
$233$	3987.44	1.12114	0.560570	−	0.828107i	$-0.310582\pi$
$233$	3987.44	1.12114	0.560570	+	0.828107i	$0.310582\pi$
$234$	−3391.97	−0.947607
$235$	−2878.30	−0.798976
$236$	−1506.57	−0.415549
$237$	2634.77	0.722138
$238$	−3569.77	−0.972244
$239$	−3367.18	−0.911317	−0.455659	−	0.890155i	$-0.650596\pi$
$239$	−3367.18	−0.911317	−0.455659	+	0.890155i	$0.650596\pi$
$240$	669.984	0.180197
$241$	−939.551	−0.251128	−0.125564	−	0.992086i	$-0.540074\pi$
$241$	−939.551	−0.251128	−0.125564	+	0.992086i	$0.540074\pi$
$242$	−1667.16	−0.442848
$243$	243.000	0.0641500
$244$	−5268.77	−1.38237
$245$	−245.000	−0.0638877
$246$	−2552.89	−0.661653
$247$	−9976.96	−2.57012
$248$	2125.56	0.544246
$249$	1475.49	0.375523
$250$	529.508	0.133956
$251$	−1403.96	−0.353056	−0.176528	−	0.984296i	$-0.556487\pi$
$251$	−1403.96	−0.353056	−0.176528	+	0.984296i	$0.556487\pi$
$252$	−626.489	−0.156608
$253$	4787.28	1.18962
$254$	10947.1	2.70425
$255$	1805.80	0.443466
$256$	1452.36	0.354579
$257$	−1964.86	−0.476905	−0.238453	−	0.971154i	$-0.576640\pi$
$257$	−1964.86	−0.476905	−0.238453	+	0.971154i	$0.576640\pi$
$258$	2773.14	0.669179
$259$	−338.341	−0.0811717
$260$	−4423.74	−1.05519
$261$	−1303.50	−0.309135
$262$	6019.89	1.41950
$263$	−393.821	−0.0923347	−0.0461673	−	0.998934i	$-0.514701\pi$
$263$	−393.821	−0.0923347	−0.0461673	+	0.998934i	$0.514701\pi$
$264$	1026.08	0.239208
$265$	921.509	0.213615
$266$	−3325.16	−0.766462
$267$	−1246.68	−0.285751
$268$	12.7769	0.00291222
$269$	−1877.03	−0.425444	−0.212722	−	0.977113i	$-0.568233\pi$
$269$	−1877.03	−0.425444	−0.212722	+	0.977113i	$0.568233\pi$
$270$	571.869	0.128899
$271$	−689.909	−0.154646	−0.0773228	−	0.997006i	$-0.524637\pi$
$271$	−689.909	−0.154646	−0.0773228	+	0.997006i	$0.524637\pi$
$272$	5377.16	1.19867
$273$	−1868.38	−0.414211
$274$	−444.269	−0.0979536
$275$	−1038.20	−0.227657
$276$	−3439.09	−0.750031
$277$	6289.13	1.36418	0.682088	−	0.731270i	$-0.261072\pi$
$277$	6289.13	1.36418	0.682088	+	0.731270i	$0.261072\pi$
$278$	3868.21	0.834531
$279$	−2322.71	−0.498412
$280$	−288.262	−0.0615249
$281$	−1954.87	−0.415010	−0.207505	−	0.978234i	$-0.566534\pi$
$281$	−1954.87	−0.415010	−0.207505	+	0.978234i	$0.566534\pi$
$282$	−7315.60	−1.54481
$283$	5033.96	1.05738	0.528688	−	0.848816i	$-0.322684\pi$
$283$	5033.96	1.05738	0.528688	+	0.848816i	$0.322684\pi$
$284$	−610.803	−0.127621
$285$	1682.07	0.349604
$286$	15651.2	3.23594
$287$	−1406.20	−0.289217
$288$	2295.86	0.469738
$289$	9580.03	1.94993
$290$	−3067.61	−0.621159
$291$	−3095.09	−0.623497
$292$	4820.99	0.966188
$293$	6369.12	1.26993	0.634963	−	0.772543i	$-0.281015\pi$
$293$	6369.12	1.26993	0.634963	+	0.772543i	$0.281015\pi$
$294$	−622.702	−0.123526
$295$	757.508	0.149504
$296$	−398.085	−0.0781697
$297$	−1121.25	−0.219063
$298$	−7546.34	−1.46694
$299$	−10256.4	−1.98376
$300$	745.820	0.143533
$301$	1527.52	0.292507
$302$	−5962.39	−1.13608
$303$	4341.58	0.823160
$304$	5008.70	0.944963
$305$	2649.15	0.497344
$306$	4589.71	0.857438
$307$	−6619.83	−1.23066	−0.615332	−	0.788268i	$-0.710978\pi$
$307$	−6619.83	−1.23066	−0.615332	+	0.788268i	$0.710978\pi$
$308$	2890.75	0.534792
$309$	−490.700	−0.0903396
$310$	−5466.20	−1.00148
$311$	−9909.22	−1.80675	−0.903377	−	0.428848i	$-0.858920\pi$
$311$	−9909.22	−1.80675	−0.903377	+	0.428848i	$0.858920\pi$
$312$	−2198.30	−0.398892
$313$	−422.336	−0.0762678	−0.0381339	−	0.999273i	$-0.512141\pi$
$313$	−422.336	−0.0762678	−0.0381339	+	0.999273i	$0.512141\pi$
$314$	6773.22	1.21731
$315$	315.000	0.0563436
$316$	8733.63	1.55476
$317$	−4902.78	−0.868668	−0.434334	−	0.900752i	$-0.643016\pi$
$317$	−4902.78	−0.868668	−0.434334	+	0.900752i	$0.643016\pi$
$318$	2342.15	0.413022
$319$	6014.60	1.05565
$320$	3616.38	0.631755
$321$	−388.960	−0.0676312
$322$	−3418.30	−0.591597
$323$	13499.9	2.32556
$324$	805.486	0.138115
$325$	2224.26	0.379631
$326$	867.935	0.147455
$327$	1700.04	0.287500
$328$	−1654.51	−0.278521
$329$	−4029.62	−0.675258
$330$	−2638.72	−0.440172
$331$	−5281.74	−0.877071	−0.438535	−	0.898714i	$-0.644503\pi$
$331$	−5281.74	−0.877071	−0.438535	+	0.898714i	$0.644503\pi$
$332$	4890.89	0.808501
$333$	435.009	0.0715867
$334$	4939.01	0.809133
$335$	−6.42426	−0.00104775
$336$	937.978	0.152294
$337$	4459.60	0.720860	0.360430	−	0.932786i	$-0.382630\pi$
$337$	4459.60	0.720860	0.360430	+	0.932786i	$0.382630\pi$
$338$	−24225.1	−3.89843
$339$	2429.67	0.389267
$340$	5985.80	0.954782
$341$	10717.5	1.70200
$342$	4275.21	0.675956
$343$	−343.000	−0.0539949
$344$	1797.25	0.281689
$345$	1729.18	0.269843
$346$	10751.2	1.67048
$347$	5261.97	0.814056	0.407028	−	0.913416i	$-0.366565\pi$
$347$	5261.97	0.814056	0.407028	+	0.913416i	$0.366565\pi$
$348$	−4320.77	−0.665568
$349$	960.325	0.147292	0.0736461	−	0.997284i	$-0.476536\pi$
$349$	960.325	0.147292	0.0736461	+	0.997284i	$0.476536\pi$
$350$	741.312	0.113214
$351$	2402.21	0.365300
$352$	−10593.6	−1.60409
$353$	−8925.80	−1.34581	−0.672907	−	0.739727i	$-0.734955\pi$
$353$	−8925.80	−1.34581	−0.672907	+	0.739727i	$0.734955\pi$
$354$	1925.31	0.289066
$355$	307.113	0.0459151
$356$	−4132.45	−0.615222
$357$	2528.13	0.374797
$358$	−1661.62	−0.245306
$359$	−3056.27	−0.449314	−0.224657	−	0.974438i	$-0.572126\pi$
$359$	−3056.27	−0.449314	−0.224657	+	0.974438i	$0.572126\pi$
$360$	370.623	0.0542599
$361$	5715.88	0.833340
$362$	12615.4	1.83162
$363$	1180.69	0.170717
$364$	−6193.23	−0.891796
$365$	−2424.00	−0.347611
$366$	6733.18	0.961610
$367$	−1813.52	−0.257943	−0.128971	−	0.991648i	$-0.541168\pi$
$367$	−1813.52	−0.257943	−0.128971	+	0.991648i	$0.541168\pi$
$368$	5148.99	0.729375
$369$	1807.97	0.255065
$370$	1023.74	0.143842
$371$	1290.11	0.180537
$372$	−7699.22	−1.07308
$373$	−4517.48	−0.627094	−0.313547	−	0.949573i	$-0.601517\pi$
$373$	−4517.48	−0.627094	−0.313547	+	0.949573i	$0.601517\pi$
$374$	−21177.9	−2.92802
$375$	−375.000	−0.0516398
$376$	−4741.17	−0.650285
$377$	−12885.9	−1.76036
$378$	800.617	0.108940
$379$	−4931.24	−0.668340	−0.334170	−	0.942513i	$-0.608456\pi$
$379$	−4931.24	−0.668340	−0.334170	+	0.942513i	$0.608456\pi$
$380$	5575.64	0.752696
$381$	−7752.75	−1.04248
$382$	4648.53	0.622617
$383$	−1482.37	−0.197770	−0.0988849	−	0.995099i	$-0.531528\pi$
$383$	−1482.37	−0.197770	−0.0988849	+	0.995099i	$0.531528\pi$
$384$	3069.25	0.407883
$385$	−1453.48	−0.192405
$386$	−14827.5	−1.95519
$387$	−1963.95	−0.257967
$388$	−10259.5	−1.34239
$389$	−5448.98	−0.710217	−0.355109	−	0.934825i	$-0.615556\pi$
$389$	−5448.98	−0.710217	−0.355109	+	0.934825i	$0.615556\pi$
$390$	5653.28	0.734013
$391$	13878.0	1.79500
$392$	−403.567	−0.0519980
$393$	−4263.31	−0.547215
$394$	−6667.38	−0.852532
$395$	−4391.28	−0.559366
$396$	−3716.68	−0.471642
$397$	−13675.9	−1.72891	−0.864453	−	0.502713i	$-0.832335\pi$
$397$	−13675.9	−1.72891	−0.864453	+	0.502713i	$0.832335\pi$
$398$	14388.3	1.81212
$399$	2354.89	0.295469
$400$	−1116.64	−0.139580
$401$	14109.9	1.75714	0.878570	−	0.477613i	$-0.158498\pi$
$401$	14109.9	1.75714	0.878570	+	0.477613i	$0.158498\pi$
$402$	−16.3282	−0.00202581
$403$	−22961.4	−2.83819
$404$	14391.3	1.77226
$405$	−405.000	−0.0496904
$406$	−4294.65	−0.524975
$407$	−2007.22	−0.244458
$408$	2974.55	0.360936
$409$	−13995.6	−1.69203	−0.846015	−	0.533159i	$-0.821005\pi$
$409$	−13995.6	−1.69203	−0.846015	+	0.533159i	$0.821005\pi$
$410$	4254.82	0.512514
$411$	314.633	0.0377609
$412$	−1626.55	−0.194501
$413$	1060.51	0.126354
$414$	4394.95	0.521740
$415$	−2459.15	−0.290879
$416$	22696.0	2.67491
$417$	−2739.48	−0.321709
$418$	−19726.7	−2.30829
$419$	−9840.61	−1.14736	−0.573682	−	0.819078i	$-0.694485\pi$
$419$	−9840.61	−1.14736	−0.573682	+	0.819078i	$0.694485\pi$
$420$	1044.15	0.121308
$421$	−12660.5	−1.46564	−0.732822	−	0.680420i	$-0.761797\pi$
$421$	−12660.5	−1.46564	−0.732822	+	0.680420i	$0.761797\pi$
$422$	−14139.2	−1.63101
$423$	5180.93	0.595522
$424$	1517.92	0.173860
$425$	−3009.67	−0.343507
$426$	780.571	0.0887765
$427$	3708.81	0.420332
$428$	−1289.31	−0.145610
$429$	−11084.3	−1.24744
$430$	−4621.90	−0.518344
$431$	−4578.91	−0.511736	−0.255868	−	0.966712i	$-0.582361\pi$
$431$	−4578.91	−0.511736	−0.255868	+	0.966712i	$0.582361\pi$
$432$	−1205.97	−0.134311
$433$	−3279.88	−0.364020	−0.182010	−	0.983297i	$-0.558260\pi$
$433$	−3279.88	−0.364020	−0.182010	+	0.983297i	$0.558260\pi$
$434$	−7652.68	−0.846406
$435$	2172.49	0.239455
$436$	5635.23	0.618988
$437$	12927.1	1.41507
$438$	−6160.94	−0.672103
$439$	−427.807	−0.0465105	−0.0232552	−	0.999730i	$-0.507403\pi$
$439$	−427.807	−0.0465105	−0.0232552	+	0.999730i	$0.507403\pi$
$440$	−1710.13	−0.185289
$441$	441.000	0.0476190
$442$	45372.1	4.88265
$443$	15441.2	1.65605	0.828027	−	0.560688i	$-0.189463\pi$
$443$	15441.2	1.65605	0.828027	+	0.560688i	$0.189463\pi$
$444$	1441.95	0.154126
$445$	2077.80	0.221342
$446$	−539.371	−0.0572645
$447$	5344.35	0.565501
$448$	5062.93	0.533931
$449$	9382.02	0.986113	0.493057	−	0.869997i	$-0.335880\pi$
$449$	9382.02	0.986113	0.493057	+	0.869997i	$0.335880\pi$
$450$	−953.115	−0.0998451
$451$	−8342.34	−0.871010
$452$	8053.77	0.838091
$453$	4222.59	0.437957
$454$	−16284.0	−1.68336
$455$	3113.97	0.320847
$456$	2770.72	0.284542
$457$	13570.4	1.38905	0.694524	−	0.719469i	$-0.255615\pi$
$457$	13570.4	1.38905	0.694524	+	0.719469i	$0.255615\pi$
$458$	−10746.7	−1.09642
$459$	−3250.45	−0.330540
$460$	5731.81	0.580972
$461$	1251.88	0.126477	0.0632386	−	0.997998i	$-0.479857\pi$
$461$	1251.88	0.126477	0.0632386	+	0.997998i	$0.479857\pi$
$462$	−3694.21	−0.372014
$463$	7934.36	0.796417	0.398209	−	0.917295i	$-0.369632\pi$
$463$	7934.36	0.796417	0.398209	+	0.917295i	$0.369632\pi$
$464$	6469.05	0.647237
$465$	3871.18	0.386069
$466$	−16891.1	−1.67911
$467$	7583.76	0.751466	0.375733	−	0.926728i	$-0.377391\pi$
$467$	7583.76	0.751466	0.375733	+	0.926728i	$0.377391\pi$
$468$	7962.73	0.786490
$469$	−8.99396	−0.000885507 0
$470$	12192.7	1.19661
$471$	−4796.82	−0.469269
$472$	1247.78	0.121681
$473$	9062.07	0.880919
$474$	−11161.1	−1.08153
$475$	−2803.44	−0.270802
$476$	8380.13	0.806938
$477$	−1658.72	−0.159219
$478$	14263.6	1.36486
$479$	−5829.34	−0.556053	−0.278027	−	0.960573i	$-0.589680\pi$
$479$	−5829.34	−0.556053	−0.278027	+	0.960573i	$0.589680\pi$
$480$	−3826.43	−0.363858
$481$	4300.34	0.407648
$482$	3980.00	0.376108
$483$	2420.85	0.228059
$484$	3913.70	0.367553
$485$	5158.49	0.482959
$486$	−1029.36	−0.0960760
$487$	−19902.1	−1.85185	−0.925925	−	0.377708i	$-0.876712\pi$
$487$	−19902.1	−1.85185	−0.925925	+	0.377708i	$0.876712\pi$
$488$	4363.71	0.404787
$489$	−614.675	−0.0568437
$490$	1037.84	0.0956830
$491$	−16821.6	−1.54613	−0.773065	−	0.634327i	$-0.781277\pi$
$491$	−16821.6	−1.54613	−0.773065	+	0.634327i	$0.781277\pi$
$492$	5992.98	0.549155
$493$	17436.0	1.59285
$494$	42263.1	3.84920
$495$	1868.75	0.169685
$496$	11527.3	1.04353
$497$	429.958	0.0388054
$498$	−6250.27	−0.562412
$499$	6031.83	0.541126	0.270563	−	0.962702i	$-0.412790\pi$
$499$	6031.83	0.541126	0.270563	+	0.962702i	$0.412790\pi$
$500$	−1243.03	−0.111180
$501$	−3497.82	−0.311919
$502$	5947.27	0.528764
$503$	−17176.4	−1.52258	−0.761290	−	0.648412i	$-0.775434\pi$
$503$	−17176.4	−1.52258	−0.761290	+	0.648412i	$0.775434\pi$
$504$	518.872	0.0458580
$505$	−7235.97	−0.637617
$506$	−20279.2	−1.78166
$507$	17156.3	1.50284
$508$	−25698.5	−2.24446
$509$	4706.59	0.409854	0.204927	−	0.978777i	$-0.434304\pi$
$509$	4706.59	0.409854	0.204927	+	0.978777i	$0.434304\pi$
$510$	−7649.51	−0.664169
$511$	−3393.60	−0.293785
$512$	−14336.9	−1.23752
$513$	−3027.72	−0.260579
$514$	8323.28	0.714250
$515$	817.833	0.0699768
$516$	−6510.02	−0.555402
$517$	−23905.9	−2.03362
$518$	1433.23	0.121569
$519$	−7614.01	−0.643965
$520$	3663.84	0.308981
$521$	−8557.18	−0.719572	−0.359786	−	0.933035i	$-0.617150\pi$
$521$	−8557.18	−0.719572	−0.359786	+	0.933035i	$0.617150\pi$
$522$	5521.69	0.462985
$523$	18248.5	1.52572	0.762858	−	0.646566i	$-0.223796\pi$
$523$	18248.5	1.52572	0.762858	+	0.646566i	$0.223796\pi$
$524$	−14131.8	−1.17815
$525$	−525.000	−0.0436436
$526$	1668.25	0.138287
$527$	31069.3	2.56813
$528$	5564.60	0.458652
$529$	1122.16	0.0922302
$530$	−3903.58	−0.319925
$531$	−1363.51	−0.111434
$532$	7805.90	0.636144
$533$	17872.9	1.45246
$534$	5281.03	0.427963
$535$	648.266	0.0523869
$536$	−10.5821	−0.000852758 0
$537$	1176.77	0.0945646
$538$	7951.22	0.637177
$539$	−2034.87	−0.162612
$540$	−1342.48	−0.106983
$541$	−5734.17	−0.455696	−0.227848	−	0.973697i	$-0.573169\pi$
$541$	−5734.17	−0.455696	−0.227848	+	0.973697i	$0.573169\pi$
$542$	2922.50	0.231609
$543$	−8934.24	−0.706087
$544$	−30710.1	−2.42038
$545$	−2833.41	−0.222697
$546$	7914.59	0.620354
$547$	−8002.52	−0.625527	−0.312763	−	0.949831i	$-0.601255\pi$
$547$	−8002.52	−0.625527	−0.312763	+	0.949831i	$0.601255\pi$
$548$	1042.93	0.0812991
$549$	−4768.47	−0.370698
$550$	4397.87	0.340956
$551$	16241.2	1.25572
$552$	2848.33	0.219625
$553$	−6147.80	−0.472750
$554$	−26641.2	−2.04310
$555$	−725.016	−0.0554508
$556$	−9080.71	−0.692640
$557$	1276.82	0.0971289	0.0485644	−	0.998820i	$-0.484535\pi$
$557$	1276.82	0.0971289	0.0485644	+	0.998820i	$0.484535\pi$
$558$	9839.16	0.746460
$559$	−19414.9	−1.46898
$560$	−1563.30	−0.117967
$561$	14998.2	1.12875
$562$	8280.96	0.621550
$563$	11027.7	0.825507	0.412753	−	0.910843i	$-0.364567\pi$
$563$	11027.7	0.825507	0.412753	+	0.910843i	$0.364567\pi$
$564$	17173.5	1.28216
$565$	−4049.45	−0.301525
$566$	−21324.2	−1.58361
$567$	−567.000	−0.0419961
$568$	505.881	0.0373702
$569$	−4519.03	−0.332948	−0.166474	−	0.986046i	$-0.553238\pi$
$569$	−4519.03	−0.332948	−0.166474	+	0.986046i	$0.553238\pi$
$570$	−7125.35	−0.523593
$571$	3598.81	0.263758	0.131879	−	0.991266i	$-0.457899\pi$
$571$	3598.81	0.263758	0.131879	+	0.991266i	$0.457899\pi$
$572$	−36741.7	−2.68575
$573$	−3292.11	−0.240017
$574$	5956.75	0.433153
$575$	−2881.97	−0.209020
$576$	−6509.48	−0.470883
$577$	−3439.23	−0.248140	−0.124070	−	0.992273i	$-0.539595\pi$
$577$	−3439.23	−0.248140	−0.124070	+	0.992273i	$0.539595\pi$
$578$	−40581.6	−2.92037
$579$	10500.9	0.753720
$580$	7201.28	0.515547
$581$	−3442.81	−0.245838
$582$	13111.0	0.933797
$583$	7653.66	0.543709
$584$	−3992.85	−0.282920
$585$	−4003.68	−0.282960
$586$	−26980.0	−1.90194
$587$	21285.2	1.49665	0.748327	−	0.663330i	$-0.230857\pi$
$587$	21285.2	1.49665	0.748327	+	0.663330i	$0.230857\pi$
$588$	1461.81	0.102524
$589$	28940.4	2.02456
$590$	−3208.85	−0.223909
$591$	4721.87	0.328649
$592$	−2158.89	−0.149881
$593$	−14200.8	−0.983404	−0.491702	−	0.870764i	$-0.663625\pi$
$593$	−14200.8	−0.983404	−0.491702	+	0.870764i	$0.663625\pi$
$594$	4749.70	0.328085
$595$	−4213.54	−0.290317
$596$	17715.2	1.21752
$597$	−10189.9	−0.698566
$598$	43446.8	2.97103
$599$	−8885.05	−0.606065	−0.303033	−	0.952980i	$-0.597999\pi$
$599$	−8885.05	−0.606065	−0.303033	+	0.952980i	$0.597999\pi$
$600$	−617.705	−0.0420295
$601$	−2052.89	−0.139333	−0.0696664	−	0.997570i	$-0.522193\pi$
$601$	−2052.89	−0.139333	−0.0696664	+	0.997570i	$0.522193\pi$
$602$	−6470.67	−0.438081
$603$	11.5637	0.000780943 0
$604$	13996.9	0.942920
$605$	−1967.82	−0.132237
$606$	−18391.2	−1.23283
$607$	10280.0	0.687404	0.343702	−	0.939079i	$-0.388319\pi$
$607$	10280.0	0.687404	0.343702	+	0.939079i	$0.388319\pi$
$608$	−28605.8	−1.90809
$609$	3041.49	0.202377
$610$	−11222.0	−0.744860
$611$	51216.8	3.39118
$612$	−10774.4	−0.711652
$613$	23409.5	1.54242	0.771208	−	0.636584i	$-0.219653\pi$
$613$	23409.5	1.54242	0.771208	+	0.636584i	$0.219653\pi$
$614$	28042.0	1.84314
$615$	−3013.28	−0.197573
$616$	−2394.18	−0.156598
$617$	−6632.75	−0.432779	−0.216389	−	0.976307i	$-0.569428\pi$
$617$	−6632.75	−0.432779	−0.216389	+	0.976307i	$0.569428\pi$
$618$	2078.64	0.135299
$619$	10734.0	0.696990	0.348495	−	0.937311i	$-0.386693\pi$
$619$	10734.0	0.696990	0.348495	+	0.937311i	$0.386693\pi$
$620$	12832.0	0.831205
$621$	−3112.52	−0.201129
$622$	41976.1	2.70593
$623$	2908.92	0.187068
$624$	−11921.8	−0.764829
$625$	625.000	0.0400000
$626$	1789.04	0.114225
$627$	13970.5	0.889839
$628$	−15900.3	−1.01034
$629$	−5818.83	−0.368858
$630$	−1334.36	−0.0843845
$631$	−17071.0	−1.07700	−0.538499	−	0.842626i	$-0.681008\pi$
$631$	−17071.0	−1.07700	−0.538499	+	0.842626i	$0.681008\pi$
$632$	−7233.38	−0.455267
$633$	10013.4	0.628749
$634$	20768.5	1.30098
$635$	12921.3	0.807503
$636$	−5498.24	−0.342798
$637$	4359.56	0.271165
$638$	−25478.2	−1.58102
$639$	−552.804	−0.0342231
$640$	−5115.41	−0.315945
$641$	−19389.7	−1.19477	−0.597386	−	0.801954i	$-0.703794\pi$
$641$	−19389.7	−1.19477	−0.597386	+	0.801954i	$0.703794\pi$
$642$	1647.66	0.101290
$643$	−25409.3	−1.55839	−0.779196	−	0.626780i	$-0.784372\pi$
$643$	−25409.3	−1.55839	−0.779196	+	0.626780i	$0.784372\pi$
$644$	8024.54	0.491011
$645$	3273.25	0.199820
$646$	−57186.6	−3.48294
$647$	−6039.08	−0.366956	−0.183478	−	0.983024i	$-0.558736\pi$
$647$	−6039.08	−0.366956	−0.183478	+	0.983024i	$0.558736\pi$
$648$	−667.122	−0.0404429
$649$	6291.54	0.380531
$650$	−9422.14	−0.568564
$651$	5419.66	0.326287
$652$	−2037.50	−0.122384
$653$	−30666.2	−1.83776	−0.918882	−	0.394532i	$-0.870907\pi$
$653$	−30666.2	−1.83776	−0.918882	+	0.394532i	$0.870907\pi$
$654$	−7201.50	−0.430582
$655$	7105.51	0.423871
$656$	−8972.67	−0.534031
$657$	4363.20	0.259094
$658$	17069.7	1.01132
$659$	−2765.96	−0.163500	−0.0817500	−	0.996653i	$-0.526051\pi$
$659$	−2765.96	−0.163500	−0.0817500	+	0.996653i	$0.526051\pi$
$660$	6194.47	0.365332
$661$	27261.8	1.60418	0.802089	−	0.597204i	$-0.203722\pi$
$661$	27261.8	1.60418	0.802089	+	0.597204i	$0.203722\pi$
$662$	22373.8	1.31357
$663$	−32132.7	−1.88225
$664$	−4050.74	−0.236746
$665$	−3924.82	−0.228869
$666$	−1842.73	−0.107214
$667$	16696.1	0.969230
$668$	−11594.4	−0.671560
$669$	381.985	0.0220753
$670$	27.2136	0.00156918
$671$	22002.7	1.26588
$672$	−5357.00	−0.307516
$673$	−1048.17	−0.0600356	−0.0300178	−	0.999549i	$-0.509556\pi$
$673$	−1048.17	−0.0600356	−0.0300178	+	0.999549i	$0.509556\pi$
$674$	−18891.2	−1.07961
$675$	675.000	0.0384900
$676$	56869.0	3.23560
$677$	−34554.7	−1.96166	−0.980831	−	0.194860i	$-0.937575\pi$
$677$	−34554.7	−1.96166	−0.980831	+	0.194860i	$0.937575\pi$
$678$	−10292.2	−0.582996
$679$	7221.89	0.408175
$680$	−4957.58	−0.279580
$681$	11532.4	0.648930
$682$	−45399.9	−2.54905
$683$	14711.6	0.824192	0.412096	−	0.911140i	$-0.364797\pi$
$683$	14711.6	0.824192	0.412096	+	0.911140i	$0.364797\pi$
$684$	−10036.2	−0.561026
$685$	−524.389	−0.0292494
$686$	1452.97	0.0808669
$687$	7610.85	0.422667
$688$	9746.79	0.540106
$689$	−16397.4	−0.906666
$690$	−7324.92	−0.404138
$691$	−24522.6	−1.35005	−0.675024	−	0.737796i	$-0.735867\pi$
$691$	−24522.6	−1.35005	−0.675024	+	0.737796i	$0.735867\pi$
$692$	−25238.6	−1.38646
$693$	2616.26	0.143410
$694$	−22290.1	−1.21919
$695$	4565.80	0.249195
$696$	3578.56	0.194892
$697$	−24184.0	−1.31425
$698$	−4068.00	−0.220596
$699$	11962.3	0.647291
$700$	−1740.25	−0.0939645
$701$	19912.2	1.07286	0.536429	−	0.843946i	$-0.319773\pi$
$701$	19912.2	1.07286	0.536429	+	0.843946i	$0.319773\pi$
$702$	−10175.9	−0.547101
$703$	−5420.11	−0.290787
$704$	30036.1	1.60799
$705$	−8634.89	−0.461289
$706$	37810.3	2.01559
$707$	−10130.4	−0.538885
$708$	−4519.72	−0.239917
$709$	6208.79	0.328880	0.164440	−	0.986387i	$-0.447418\pi$
$709$	6208.79	0.328880	0.164440	+	0.986387i	$0.447418\pi$
$710$	−1300.95	−0.0687660
$711$	7904.31	0.416927
$712$	3422.58	0.180150
$713$	29751.0	1.56267
$714$	−10709.3	−0.561325
$715$	18473.8	0.966267
$716$	3900.69	0.203597
$717$	−10101.5	−0.526149
$718$	12946.6	0.672927
$719$	13063.6	0.677593	0.338797	−	0.940860i	$-0.389980\pi$
$719$	13063.6	0.677593	0.338797	+	0.940860i	$0.389980\pi$
$720$	2009.95	0.104037
$721$	1144.97	0.0591411
$722$	−24212.9	−1.24807
$723$	−2818.65	−0.144989
$724$	−29614.8	−1.52020
$725$	−3620.82	−0.185481
$726$	−5001.49	−0.255678
$727$	−12897.0	−0.657940	−0.328970	−	0.944340i	$-0.606702\pi$
$727$	−12897.0	−0.657940	−0.328970	+	0.944340i	$0.606702\pi$
$728$	5129.37	0.261136
$729$	729.000	0.0370370
$730$	10268.2	0.520609
$731$	26270.5	1.32920
$732$	−15806.3	−0.798112
$733$	11699.6	0.589540	0.294770	−	0.955568i	$-0.404757\pi$
$733$	11699.6	0.589540	0.294770	+	0.955568i	$0.404757\pi$
$734$	7682.19	0.386315
$735$	−735.000	−0.0368856
$736$	−29407.0	−1.47277
$737$	−53.3571	−0.00266681
$738$	−7658.68	−0.382005
$739$	14974.0	0.745368	0.372684	−	0.927958i	$-0.378438\pi$
$739$	14974.0	0.745368	0.372684	+	0.927958i	$0.378438\pi$
$740$	−2403.25	−0.119385
$741$	−29930.9	−1.48386
$742$	−5465.01	−0.270386
$743$	18500.7	0.913492	0.456746	−	0.889597i	$-0.349015\pi$
$743$	18500.7	0.913492	0.456746	+	0.889597i	$0.349015\pi$
$744$	6376.67	0.314220
$745$	−8907.24	−0.438035
$746$	19136.3	0.939184
$747$	4426.47	0.216809
$748$	49715.5	2.43019
$749$	907.572	0.0442750
$750$	1588.53	0.0773397
$751$	−26348.4	−1.28025	−0.640125	−	0.768271i	$-0.721117\pi$
$751$	−26348.4	−1.28025	−0.640125	+	0.768271i	$0.721117\pi$
$752$	−25712.2	−1.24684
$753$	−4211.88	−0.203837
$754$	54585.4	2.63645
$755$	−7037.65	−0.339240
$756$	−1879.47	−0.0904174
$757$	−28061.7	−1.34732	−0.673659	−	0.739042i	$-0.735278\pi$
$757$	−28061.7	−1.34732	−0.673659	+	0.739042i	$0.735278\pi$
$758$	20889.1	1.00096
$759$	14361.8	0.686826
$760$	−4617.87	−0.220405
$761$	−3579.22	−0.170495	−0.0852476	−	0.996360i	$-0.527168\pi$
$761$	−3579.22	−0.170495	−0.0852476	+	0.996360i	$0.527168\pi$
$762$	32841.2	1.56130
$763$	−3966.77	−0.188213
$764$	−10912.5	−0.516757
$765$	5417.41	0.256035
$766$	6279.44	0.296195
$767$	−13479.2	−0.634557
$768$	4357.07	0.204716
$769$	4339.61	0.203499	0.101749	−	0.994810i	$-0.467556\pi$
$769$	4339.61	0.203499	0.101749	+	0.994810i	$0.467556\pi$
$770$	6157.02	0.288161
$771$	−5894.58	−0.275341
$772$	34808.0	1.62276
$773$	10005.1	0.465537	0.232769	−	0.972532i	$-0.425222\pi$
$773$	10005.1	0.465537	0.232769	+	0.972532i	$0.425222\pi$
$774$	8319.43	0.386351
$775$	−6451.97	−0.299047
$776$	8497.14	0.393079
$777$	−1015.02	−0.0468645
$778$	23082.3	1.06368
$779$	−22526.8	−1.03608
$780$	−13271.2	−0.609212
$781$	2550.75	0.116867
$782$	−58788.4	−2.68832
$783$	−3910.49	−0.178479
$784$	−2188.62	−0.0997001
$785$	7994.70	0.363494
$786$	18059.7	0.819550
$787$	17826.8	0.807443	0.403721	−	0.914882i	$-0.367716\pi$
$787$	17826.8	0.807443	0.403721	+	0.914882i	$0.367716\pi$
$788$	15651.8	0.707581
$789$	−1181.46	−0.0533094
$790$	18601.8	0.837749
$791$	−5669.23	−0.254835
$792$	3078.24	0.138107
$793$	−47139.3	−2.11093
$794$	57932.2	2.58934
$795$	2764.53	0.123330
$796$	−33777.0	−1.50401
$797$	36723.0	1.63211	0.816057	−	0.577971i	$-0.196155\pi$
$797$	36723.0	1.63211	0.816057	+	0.577971i	$0.196155\pi$
$798$	−9975.49	−0.442517
$799$	−69301.9	−3.06849
$800$	6377.38	0.281843
$801$	−3740.04	−0.164979
$802$	−59770.4	−2.63163
$803$	−20132.7	−0.884767
$804$	38.3307	0.00168137
$805$	−4034.75	−0.176654
$806$	97266.2	4.25069
$807$	−5631.08	−0.245630
$808$	−11919.2	−0.518955
$809$	5657.55	0.245870	0.122935	−	0.992415i	$-0.460769\pi$
$809$	5657.55	0.245870	0.122935	+	0.992415i	$0.460769\pi$
$810$	1715.61	0.0744201
$811$	7532.41	0.326139	0.163070	−	0.986615i	$-0.447861\pi$
$811$	7532.41	0.326139	0.163070	+	0.986615i	$0.447861\pi$
$812$	10081.8	0.435716
$813$	−2069.73	−0.0892847
$814$	8502.73	0.366119
$815$	1024.46	0.0440309
$816$	16131.5	0.692053
$817$	24470.3	1.04787
$818$	59286.5	2.53411
$819$	−5605.15	−0.239145
$820$	−9988.30	−0.425374
$821$	−6489.25	−0.275854	−0.137927	−	0.990442i	$-0.544044\pi$
$821$	−6489.25	−0.275854	−0.137927	+	0.990442i	$0.544044\pi$
$822$	−1332.81	−0.0565535
$823$	7901.57	0.334668	0.167334	−	0.985900i	$-0.446484\pi$
$823$	7901.57	0.334668	0.167334	+	0.985900i	$0.446484\pi$
$824$	1347.15	0.0569539
$825$	−3114.59	−0.131438
$826$	−4492.40	−0.189238
$827$	−37815.8	−1.59007	−0.795033	−	0.606566i	$-0.792547\pi$
$827$	−37815.8	−1.59007	−0.795033	+	0.606566i	$0.792547\pi$
$828$	−10317.3	−0.433031
$829$	26073.5	1.09236	0.546182	−	0.837667i	$-0.316081\pi$
$829$	26073.5	1.09236	0.546182	+	0.837667i	$0.316081\pi$
$830$	10417.1	0.435643
$831$	18867.4	0.787608
$832$	−64350.2	−2.68142
$833$	−5898.96	−0.245362
$834$	11604.6	0.481817
$835$	5829.71	0.241611
$836$	46308.9	1.91582
$837$	−6968.13	−0.287758
$838$	41685.5	1.71838
$839$	−15590.3	−0.641523	−0.320762	−	0.947160i	$-0.603939\pi$
$839$	−15590.3	−0.641523	−0.320762	+	0.947160i	$0.603939\pi$
$840$	−864.787	−0.0355214
$841$	−3412.46	−0.139918
$842$	53630.8	2.19506
$843$	−5864.61	−0.239606
$844$	33192.1	1.35370
$845$	−28593.8	−1.16409
$846$	−21946.8	−0.891899
$847$	−2754.94	−0.111760
$848$	8231.96	0.333357
$849$	15101.9	0.610477
$850$	12749.2	0.514463
$851$	−5571.92	−0.224445
$852$	−1832.41	−0.0736823
$853$	−17476.1	−0.701488	−0.350744	−	0.936471i	$-0.614071\pi$
$853$	−17476.1	−0.701488	−0.350744	+	0.936471i	$0.614071\pi$
$854$	−15710.8	−0.629521
$855$	5046.20	0.201844
$856$	1067.83	0.0426376
$857$	−5694.54	−0.226980	−0.113490	−	0.993539i	$-0.536203\pi$
$857$	−5694.54	−0.226980	−0.113490	+	0.993539i	$0.536203\pi$
$858$	46953.7	1.86827
$859$	27313.6	1.08490	0.542448	−	0.840089i	$-0.317497\pi$
$859$	27313.6	1.08490	0.542448	+	0.840089i	$0.317497\pi$
$860$	10850.0	0.430213
$861$	−4218.59	−0.166979
$862$	19396.6	0.766415
$863$	−9046.07	−0.356815	−0.178408	−	0.983957i	$-0.557095\pi$
$863$	−9046.07	−0.356815	−0.178408	+	0.983957i	$0.557095\pi$
$864$	6887.57	0.271204
$865$	12690.0	0.498813
$866$	13893.8	0.545185
$867$	28740.1	1.12580
$868$	17964.8	0.702496
$869$	−36472.1	−1.42374
$870$	−9202.82	−0.358626
$871$	114.314	0.00444705
$872$	−4667.22	−0.181252
$873$	−9285.28	−0.359976
$874$	−54760.0	−2.11932
$875$	875.000	0.0338062
$876$	14463.0	0.557829
$877$	−2104.29	−0.0810224	−0.0405112	−	0.999179i	$-0.512899\pi$
$877$	−2104.29	−0.0810224	−0.0405112	+	0.999179i	$0.512899\pi$
$878$	1812.22	0.0696576
$879$	19107.4	0.733192
$880$	−9274.34	−0.355270
$881$	−22589.6	−0.863861	−0.431931	−	0.901907i	$-0.642167\pi$
$881$	−22589.6	−0.863861	−0.431931	+	0.901907i	$0.642167\pi$
$882$	−1868.11	−0.0713179
$883$	−2419.71	−0.0922193	−0.0461096	−	0.998936i	$-0.514682\pi$
$883$	−2419.71	−0.0922193	−0.0461096	+	0.998936i	$0.514682\pi$
$884$	−106512.	−4.05248
$885$	2272.52	0.0863164
$886$	−65409.8	−2.48023
$887$	13177.0	0.498806	0.249403	−	0.968400i	$-0.419766\pi$
$887$	13177.0	0.498806	0.249403	+	0.968400i	$0.419766\pi$
$888$	−1194.26	−0.0451313
$889$	18089.8	0.682464
$890$	−8801.71	−0.331499
$891$	−3363.76	−0.126476
$892$	1266.19	0.0475281
$893$	−64553.2	−2.41902
$894$	−22639.0	−0.846937
$895$	−1961.28	−0.0732495
$896$	−7161.58	−0.267022
$897$	−30769.2	−1.14532
$898$	−39742.9	−1.47688
$899$	37378.3	1.38669
$900$	2237.46	0.0828689
$901$	22187.5	0.820393
$902$	35338.7	1.30449
$903$	4582.55	0.168879
$904$	−6670.31	−0.245411
$905$	14890.4	0.546932
$906$	−17887.2	−0.655918
$907$	9189.14	0.336406	0.168203	−	0.985752i	$-0.446204\pi$
$907$	9189.14	0.336406	0.168203	+	0.985752i	$0.446204\pi$
$908$	38227.0	1.39714
$909$	13024.8	0.475252
$910$	−13191.0	−0.480524
$911$	−17045.8	−0.619928	−0.309964	−	0.950748i	$-0.600317\pi$
$911$	−17045.8	−0.619928	−0.309964	+	0.950748i	$0.600317\pi$
$912$	15026.1	0.545575
$913$	−20424.6	−0.740369
$914$	−57485.0	−2.08034
$915$	7947.45	0.287141
$916$	25228.1	0.910001
$917$	9947.71	0.358236
$918$	13769.1	0.495042
$919$	−30825.0	−1.10645	−0.553223	−	0.833033i	$-0.686602\pi$
$919$	−30825.0	−1.10645	−0.553223	+	0.833033i	$0.686602\pi$
$920$	−4747.21	−0.170121
$921$	−19859.5	−0.710524
$922$	−5303.06	−0.189422
$923$	−5464.81	−0.194882
$924$	8672.25	0.308762
$925$	1208.36	0.0429520
$926$	−33610.5	−1.19277
$927$	−1472.10	−0.0521576
$928$	−36946.2	−1.30691
$929$	−5785.88	−0.204336	−0.102168	−	0.994767i	$-0.532578\pi$
$929$	−5785.88	−0.204336	−0.102168	+	0.994767i	$0.532578\pi$
$930$	−16398.6	−0.578206
$931$	−5494.75	−0.193430
$932$	39652.2	1.39362
$933$	−29727.7	−1.04313
$934$	−32125.3	−1.12545
$935$	−24997.1	−0.874323
$936$	−6594.91	−0.230301
$937$	−13680.9	−0.476986	−0.238493	−	0.971144i	$-0.576653\pi$
$937$	−13680.9	−0.476986	−0.238493	+	0.971144i	$0.576653\pi$
$938$	38.0990	0.00132620
$939$	−1267.01	−0.0440333
$940$	−28622.6	−0.993155
$941$	−45448.8	−1.57448	−0.787242	−	0.616644i	$-0.788492\pi$
$941$	−45448.8	−1.57448	−0.787242	+	0.616644i	$0.788492\pi$
$942$	20319.6	0.702813
$943$	−23157.8	−0.799705
$944$	6766.91	0.233310
$945$	945.000	0.0325300
$946$	−38387.6	−1.31933
$947$	−7788.45	−0.267255	−0.133628	−	0.991032i	$-0.542663\pi$
$947$	−7788.45	−0.267255	−0.133628	+	0.991032i	$0.542663\pi$
$948$	26200.9	0.897642
$949$	43133.0	1.47540
$950$	11875.6	0.405573
$951$	−14708.4	−0.501526
$952$	−6940.61	−0.236288
$953$	6149.43	0.209024	0.104512	−	0.994524i	$-0.466672\pi$
$953$	6149.43	0.209024	0.104512	+	0.994524i	$0.466672\pi$
$954$	7026.44	0.238458
$955$	5486.85	0.185917
$956$	−33484.2	−1.13280
$957$	18043.8	0.609481
$958$	24693.5	0.832788
$959$	−734.144	−0.0247203
$960$	10849.1	0.364744
$961$	36813.7	1.23573
$962$	−18216.5	−0.610524
$963$	−1166.88	−0.0390469
$964$	−9343.15	−0.312160
$965$	−17501.5	−0.583829
$966$	−10254.9	−0.341559
$967$	23902.9	0.794896	0.397448	−	0.917625i	$-0.369896\pi$
$967$	23902.9	0.794896	0.397448	+	0.917625i	$0.369896\pi$
$968$	−3241.42	−0.107627
$969$	40499.8	1.34266
$970$	−21851.7	−0.723316
$971$	−8015.06	−0.264898	−0.132449	−	0.991190i	$-0.542284\pi$
$971$	−8015.06	−0.264898	−0.132449	+	0.991190i	$0.542284\pi$
$972$	2416.46	0.0797407
$973$	6392.12	0.210608
$974$	84306.7	2.77347
$975$	6672.79	0.219180
$976$	23665.2	0.776131
$977$	−34861.1	−1.14156	−0.570780	−	0.821103i	$-0.693359\pi$
$977$	−34861.1	−1.14156	−0.570780	+	0.821103i	$0.693359\pi$
$978$	2603.80	0.0851334
$979$	17257.3	0.563378
$980$	−2436.35	−0.0794145
$981$	5100.13	0.165988
$982$	71257.6	2.31560
$983$	6620.83	0.214824	0.107412	−	0.994215i	$-0.465744\pi$
$983$	6620.83	0.214824	0.107412	+	0.994215i	$0.465744\pi$
$984$	−4963.52	−0.160804
$985$	−7869.78	−0.254570
$986$	−73860.0	−2.38558
$987$	−12088.8	−0.389860
$988$	−99213.6	−3.19474
$989$	25155.7	0.808802
$990$	−7916.17	−0.254134
$991$	10360.1	0.332089	0.166045	−	0.986118i	$-0.446900\pi$
$991$	10360.1	0.332089	0.166045	+	0.986118i	$0.446900\pi$
$992$	−65834.7	−2.10711
$993$	−15845.2	−0.506377
$994$	−1821.33	−0.0581179
$995$	16983.1	0.541107
$996$	14672.7	0.466788
$997$	−40309.3	−1.28045	−0.640225	−	0.768188i	$-0.721159\pi$
$997$	−40309.3	−1.28045	−0.640225	+	0.768188i	$0.721159\pi$
$998$	−25551.3	−0.810432
$999$	1305.03	0.0413306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	105.4.a.d.1.1	✓	2
3.2	odd	2		315.4.a.l.1.2		2
4.3	odd	2		1680.4.a.bd.1.1		2
5.2	odd	4		525.4.d.k.274.1		4
5.3	odd	4		525.4.d.k.274.4		4
5.4	even	2		525.4.a.o.1.2		2
7.6	odd	2		735.4.a.m.1.1		2
15.14	odd	2		1575.4.a.n.1.1		2
21.20	even	2		2205.4.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.d.1.1	✓	2	1.1	even	1	trivial
315.4.a.l.1.2		2	3.2	odd	2
525.4.a.o.1.2		2	5.4	even	2
525.4.d.k.274.1		4	5.2	odd	4
525.4.d.k.274.4		4	5.3	odd	4
735.4.a.m.1.1		2	7.6	odd	2
1575.4.a.n.1.1		2	15.14	odd	2
1680.4.a.bd.1.1		2	4.3	odd	2
2205.4.a.be.1.2		2	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 105 = 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	105.a (trivial)

\(f(q)\)	\(=\)	\(q-4.23607 q^{2} +3.00000 q^{3} +9.94427 q^{4} -5.00000 q^{5} -12.7082 q^{6} -7.00000 q^{7} -8.23607 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.23607 q^{2} +3.00000 q^{3} +9.94427 q^{4} -5.00000 q^{5} -12.7082 q^{6} -7.00000 q^{7} -8.23607 q^{8} +9.00000 q^{9} +21.1803 q^{10} -41.5279 q^{11} +29.8328 q^{12} +88.9706 q^{13} +29.6525 q^{14} -15.0000 q^{15} -44.6656 q^{16} -120.387 q^{17} -38.1246 q^{18} -112.138 q^{19} -49.7214 q^{20} -21.0000 q^{21} +175.915 q^{22} -115.279 q^{23} -24.7082 q^{24} +25.0000 q^{25} -376.885 q^{26} +27.0000 q^{27} -69.6099 q^{28} -144.833 q^{29} +63.5410 q^{30} -258.079 q^{31} +255.095 q^{32} -124.584 q^{33} +509.967 q^{34} +35.0000 q^{35} +89.4984 q^{36} +48.3344 q^{37} +475.023 q^{38} +266.912 q^{39} +41.1803 q^{40} +200.885 q^{41} +88.9574 q^{42} -218.217 q^{43} -412.964 q^{44} -45.0000 q^{45} +488.328 q^{46} +575.659 q^{47} -133.997 q^{48} +49.0000 q^{49} -105.902 q^{50} -361.161 q^{51} +884.748 q^{52} -184.302 q^{53} -114.374 q^{54} +207.639 q^{55} +57.6525 q^{56} -336.413 q^{57} +613.522 q^{58} -151.502 q^{59} -149.164 q^{60} -529.830 q^{61} +1093.24 q^{62} -63.0000 q^{63} -723.276 q^{64} -444.853 q^{65} +527.745 q^{66} +1.28485 q^{67} -1197.16 q^{68} -345.836 q^{69} -148.262 q^{70} -61.4226 q^{71} -74.1246 q^{72} +484.800 q^{73} -204.748 q^{74} +75.0000 q^{75} -1115.13 q^{76} +290.695 q^{77} -1130.66 q^{78} +878.257 q^{79} +223.328 q^{80} +81.0000 q^{81} -850.964 q^{82} +491.830 q^{83} -208.830 q^{84} +601.935 q^{85} +924.381 q^{86} -434.498 q^{87} +342.026 q^{88} -415.560 q^{89} +190.623 q^{90} -622.794 q^{91} -1146.36 q^{92} -774.237 q^{93} -2438.53 q^{94} +560.689 q^{95} +765.286 q^{96} -1031.70 q^{97} -207.567 q^{98} -373.751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 12 q^{6} - 14 q^{7} - 12 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 + 6 * q^3 + 2 * q^4 - 10 * q^5 - 12 * q^6 - 14 * q^7 - 12 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 12 q^{6} - 14 q^{7} - 12 q^{8} + 18 q^{9} + 20 q^{10} - 92 q^{11} + 6 q^{12} + 8 q^{13} + 28 q^{14} - 30 q^{15} + 18 q^{16} - 44 q^{17} - 36 q^{18} - 108 q^{19} - 10 q^{20} - 42 q^{21} + 164 q^{22} - 320 q^{23} - 36 q^{24} + 50 q^{25} - 396 q^{26} + 54 q^{27} - 14 q^{28} - 236 q^{29} + 60 q^{30} - 60 q^{31} + 300 q^{32} - 276 q^{33} + 528 q^{34} + 70 q^{35} + 18 q^{36} + 204 q^{37} + 476 q^{38} + 24 q^{39} + 60 q^{40} + 44 q^{41} + 84 q^{42} + 136 q^{43} - 12 q^{44} - 90 q^{45} + 440 q^{46} + 400 q^{47} + 54 q^{48} + 98 q^{49} - 100 q^{50} - 132 q^{51} + 1528 q^{52} + 16 q^{53} - 108 q^{54} + 460 q^{55} + 84 q^{56} - 324 q^{57} + 592 q^{58} - 464 q^{59} - 30 q^{60} - 684 q^{61} + 1140 q^{62} - 126 q^{63} - 1214 q^{64} - 40 q^{65} + 492 q^{66} + 736 q^{67} - 1804 q^{68} - 960 q^{69} - 140 q^{70} - 740 q^{71} - 108 q^{72} + 424 q^{73} - 168 q^{74} + 150 q^{75} - 1148 q^{76} + 644 q^{77} - 1188 q^{78} - 408 q^{79} - 90 q^{80} + 162 q^{81} - 888 q^{82} + 608 q^{83} - 42 q^{84} + 220 q^{85} + 1008 q^{86} - 708 q^{87} + 532 q^{88} - 1332 q^{89} + 180 q^{90} - 56 q^{91} + 480 q^{92} - 180 q^{93} - 2480 q^{94} + 540 q^{95} + 900 q^{96} - 2448 q^{97} - 196 q^{98} - 828 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 6 * q^3 + 2 * q^4 - 10 * q^5 - 12 * q^6 - 14 * q^7 - 12 * q^8 + 18 * q^9 + 20 * q^10 - 92 * q^11 + 6 * q^12 + 8 * q^13 + 28 * q^14 - 30 * q^15 + 18 * q^16 - 44 * q^17 - 36 * q^18 - 108 * q^19 - 10 * q^20 - 42 * q^21 + 164 * q^22 - 320 * q^23 - 36 * q^24 + 50 * q^25 - 396 * q^26 + 54 * q^27 - 14 * q^28 - 236 * q^29 + 60 * q^30 - 60 * q^31 + 300 * q^32 - 276 * q^33 + 528 * q^34 + 70 * q^35 + 18 * q^36 + 204 * q^37 + 476 * q^38 + 24 * q^39 + 60 * q^40 + 44 * q^41 + 84 * q^42 + 136 * q^43 - 12 * q^44 - 90 * q^45 + 440 * q^46 + 400 * q^47 + 54 * q^48 + 98 * q^49 - 100 * q^50 - 132 * q^51 + 1528 * q^52 + 16 * q^53 - 108 * q^54 + 460 * q^55 + 84 * q^56 - 324 * q^57 + 592 * q^58 - 464 * q^59 - 30 * q^60 - 684 * q^61 + 1140 * q^62 - 126 * q^63 - 1214 * q^64 - 40 * q^65 + 492 * q^66 + 736 * q^67 - 1804 * q^68 - 960 * q^69 - 140 * q^70 - 740 * q^71 - 108 * q^72 + 424 * q^73 - 168 * q^74 + 150 * q^75 - 1148 * q^76 + 644 * q^77 - 1188 * q^78 - 408 * q^79 - 90 * q^80 + 162 * q^81 - 888 * q^82 + 608 * q^83 - 42 * q^84 + 220 * q^85 + 1008 * q^86 - 708 * q^87 + 532 * q^88 - 1332 * q^89 + 180 * q^90 - 56 * q^91 + 480 * q^92 - 180 * q^93 - 2480 * q^94 + 540 * q^95 + 900 * q^96 - 2448 * q^97 - 196 * q^98 - 828 * q^99

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