LMFDB - Embedded newform 1045.4.a.c.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, 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("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + 1317797 x^{12} - 1160501 x^{11} - 9914845 x^{10} + 7570653 x^{9} + \cdots + 150528 $ x^20 - x^19 - 105x^18 + 103x^17 + 4500x^16 - 4345x^15 - 101844x^14 + 95592x^13 + 1317797x^12 - 1160501x^11 - 9914845x^10 + 7570653x^9 + 42786958x^8 - 23777633x^7 - 102801526x^6 + 28436356x^5 + 122325928x^4 + 411232x^3 - 47350496x^2 - 4782848x + 150528
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.783853$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.783853 q^{2} +0.0603595 q^{3} -7.38557 q^{4} -5.00000 q^{5} -0.0473130 q^{6} -1.09978 q^{7} +12.0600 q^{8} -26.9964 q^{9} +O(q^{10})$	\(q-0.783853 q^{2} +0.0603595 q^{3} -7.38557 q^{4} -5.00000 q^{5} -0.0473130 q^{6} -1.09978 q^{7} +12.0600 q^{8} -26.9964 q^{9} +3.91926 q^{10} -11.0000 q^{11} -0.445790 q^{12} -22.1285 q^{13} +0.862068 q^{14} -0.301798 q^{15} +49.6313 q^{16} +88.1083 q^{17} +21.1612 q^{18} +19.0000 q^{19} +36.9279 q^{20} -0.0663823 q^{21} +8.62238 q^{22} +168.327 q^{23} +0.727938 q^{24} +25.0000 q^{25} +17.3455 q^{26} -3.25919 q^{27} +8.12252 q^{28} +214.076 q^{29} +0.236565 q^{30} -36.7173 q^{31} -135.384 q^{32} -0.663955 q^{33} -69.0640 q^{34} +5.49891 q^{35} +199.384 q^{36} -218.260 q^{37} -14.8932 q^{38} -1.33567 q^{39} -60.3001 q^{40} +28.0068 q^{41} +0.0520340 q^{42} -376.154 q^{43} +81.2413 q^{44} +134.982 q^{45} -131.944 q^{46} -249.449 q^{47} +2.99572 q^{48} -341.790 q^{49} -19.5963 q^{50} +5.31818 q^{51} +163.432 q^{52} +446.371 q^{53} +2.55473 q^{54} +55.0000 q^{55} -13.2634 q^{56} +1.14683 q^{57} -167.804 q^{58} +826.710 q^{59} +2.22895 q^{60} +349.824 q^{61} +28.7809 q^{62} +29.6901 q^{63} -290.929 q^{64} +110.643 q^{65} +0.520443 q^{66} -340.516 q^{67} -650.731 q^{68} +10.1602 q^{69} -4.31034 q^{70} -1068.42 q^{71} -325.577 q^{72} -99.5645 q^{73} +171.084 q^{74} +1.50899 q^{75} -140.326 q^{76} +12.0976 q^{77} +1.04697 q^{78} +1027.99 q^{79} -248.157 q^{80} +728.705 q^{81} -21.9532 q^{82} +609.027 q^{83} +0.490272 q^{84} -440.542 q^{85} +294.849 q^{86} +12.9215 q^{87} -132.660 q^{88} +175.527 q^{89} -105.806 q^{90} +24.3366 q^{91} -1243.19 q^{92} -2.21624 q^{93} +195.531 q^{94} -95.0000 q^{95} -8.17171 q^{96} +38.6274 q^{97} +267.913 q^{98} +296.960 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) $ 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * 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 $

$2$	−0.783853	−0.277134	−0.138567	−	0.990353i	$-0.544250\pi$
$2$	−0.783853	−0.277134	−0.138567	+	0.990353i	$0.544250\pi$
$3$	0.0603595	0.0116162	0.00580810	−	0.999983i	$-0.498151\pi$
$3$	0.0603595	0.0116162	0.00580810	+	0.999983i	$0.498151\pi$
$4$	−7.38557	−0.923197
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−0.0473130	−0.00321924
$7$	−1.09978	−0.0593827	−0.0296913	−	0.999559i	$-0.509452\pi$
$7$	−1.09978	−0.0593827	−0.0296913	+	0.999559i	$0.509452\pi$
$8$	12.0600	0.532983
$9$	−26.9964	−0.999865
$10$	3.91926	0.123938
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−0.445790	−0.0107240
$13$	−22.1285	−0.472104	−0.236052	−	0.971740i	$-0.575854\pi$
$13$	−22.1285	−0.472104	−0.236052	+	0.971740i	$0.575854\pi$
$14$	0.862068	0.0164569
$15$	−0.301798	−0.00519492
$16$	49.6313	0.775489
$17$	88.1083	1.25702	0.628512	−	0.777800i	$-0.283664\pi$
$17$	88.1083	1.25702	0.628512	+	0.777800i	$0.283664\pi$
$18$	21.1612	0.277096
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	36.9279	0.412866
$21$	−0.0663823	−0.000689801 0
$22$	8.62238	0.0835590
$23$	168.327	1.52603	0.763015	−	0.646381i	$-0.223718\pi$
$23$	168.327	1.52603	0.763015	+	0.646381i	$0.223718\pi$
$24$	0.727938	0.00619123
$25$	25.0000	0.200000
$26$	17.3455	0.130836
$27$	−3.25919	−0.0232308
$28$	8.12252	0.0548219
$29$	214.076	1.37079	0.685395	−	0.728171i	$-0.259629\pi$
$29$	214.076	1.37079	0.685395	+	0.728171i	$0.259629\pi$
$30$	0.236565	0.00143969
$31$	−36.7173	−0.212730	−0.106365	−	0.994327i	$-0.533921\pi$
$31$	−36.7173	−0.212730	−0.106365	+	0.994327i	$0.533921\pi$
$32$	−135.384	−0.747897
$33$	−0.663955	−0.00350241
$34$	−69.0640	−0.348364
$35$	5.49891	0.0265567
$36$	199.384	0.923072
$37$	−218.260	−0.969778	−0.484889	−	0.874576i	$-0.661140\pi$
$37$	−218.260	−0.969778	−0.484889	+	0.874576i	$0.661140\pi$
$38$	−14.8932	−0.0635789
$39$	−1.33567	−0.00548405
$40$	−60.3001	−0.238357
$41$	28.0068	0.106681	0.0533406	−	0.998576i	$-0.483013\pi$
$41$	28.0068	0.106681	0.0533406	+	0.998576i	$0.483013\pi$
$42$	0.0520340	0.000191167 0
$43$	−376.154	−1.33402	−0.667011	−	0.745048i	$-0.732427\pi$
$43$	−376.154	−1.33402	−0.667011	+	0.745048i	$0.732427\pi$
$44$	81.2413	0.278354
$45$	134.982	0.447153
$46$	−131.944	−0.422915
$47$	−249.449	−0.774168	−0.387084	−	0.922044i	$-0.626518\pi$
$47$	−249.449	−0.774168	−0.387084	+	0.922044i	$0.626518\pi$
$48$	2.99572	0.00900823
$49$	−341.790	−0.996474
$50$	−19.5963	−0.0554268
$51$	5.31818	0.0146018
$52$	163.432	0.435845
$53$	446.371	1.15686	0.578432	−	0.815731i	$-0.303665\pi$
$53$	446.371	1.15686	0.578432	+	0.815731i	$0.303665\pi$
$54$	2.55473	0.00643805
$55$	55.0000	0.134840
$56$	−13.2634	−0.0316499
$57$	1.14683	0.00266494
$58$	−167.804	−0.379893
$59$	826.710	1.82421	0.912106	−	0.409955i	$-0.134456\pi$
$59$	826.710	1.82421	0.912106	+	0.409955i	$0.134456\pi$
$60$	2.22895	0.00479593
$61$	349.824	0.734269	0.367134	−	0.930168i	$-0.380339\pi$
$61$	349.824	0.734269	0.367134	+	0.930168i	$0.380339\pi$
$62$	28.7809	0.0589546
$63$	29.6901	0.0593746
$64$	−290.929	−0.568222
$65$	110.643	0.211131
$66$	0.520443	0.000970638 0
$67$	−340.516	−0.620906	−0.310453	−	0.950589i	$-0.600481\pi$
$67$	−340.516	−0.620906	−0.310453	+	0.950589i	$0.600481\pi$
$68$	−650.731	−1.16048
$69$	10.1602	0.0177267
$70$	−4.31034	−0.00735977
$71$	−1068.42	−1.78588	−0.892942	−	0.450172i	$-0.851363\pi$
$71$	−1068.42	−1.78588	−0.892942	+	0.450172i	$0.851363\pi$
$72$	−325.577	−0.532911
$73$	−99.5645	−0.159632	−0.0798161	−	0.996810i	$-0.525433\pi$
$73$	−99.5645	−0.159632	−0.0798161	+	0.996810i	$0.525433\pi$
$74$	171.084	0.268758
$75$	1.50899	0.00232324
$76$	−140.326	−0.211796
$77$	12.0976	0.0179045
$78$	1.04697	0.00151982
$79$	1027.99	1.46402	0.732012	−	0.681292i	$-0.238582\pi$
$79$	1027.99	1.46402	0.732012	+	0.681292i	$0.238582\pi$
$80$	−248.157	−0.346809
$81$	728.705	0.999595
$82$	−21.9532	−0.0295650
$83$	609.027	0.805414	0.402707	−	0.915329i	$-0.368069\pi$
$83$	609.027	0.805414	0.402707	+	0.915329i	$0.368069\pi$
$84$	0.490272	0.000636822 0
$85$	−440.542	−0.562158
$86$	294.849	0.369703
$87$	12.9215	0.0159234
$88$	−132.660	−0.160700
$89$	175.527	0.209055	0.104527	−	0.994522i	$-0.466667\pi$
$89$	175.527	0.209055	0.104527	+	0.994522i	$0.466667\pi$
$90$	−105.806	−0.123921
$91$	24.3366	0.0280348
$92$	−1243.19	−1.40883
$93$	−2.21624	−0.00247111
$94$	195.531	0.214548
$95$	−95.0000	−0.102598
$96$	−8.17171	−0.00868772
$97$	38.6274	0.0404332	0.0202166	−	0.999796i	$-0.493564\pi$
$97$	38.6274	0.0404332	0.0202166	+	0.999796i	$0.493564\pi$
$98$	267.913	0.276157
$99$	296.960	0.301471
$100$	−184.639	−0.184639
$101$	−592.060	−0.583289	−0.291644	−	0.956527i	$-0.594202\pi$
$101$	−592.060	−0.583289	−0.291644	+	0.956527i	$0.594202\pi$
$102$	−4.16867	−0.00404666
$103$	886.258	0.847821	0.423910	−	0.905704i	$-0.360657\pi$
$103$	886.258	0.847821	0.423910	+	0.905704i	$0.360657\pi$
$104$	−266.871	−0.251623
$105$	0.331912	0.000308488 0
$106$	−349.889	−0.320606
$107$	−1745.46	−1.57701	−0.788506	−	0.615027i	$-0.789145\pi$
$107$	−1745.46	−1.57701	−0.788506	+	0.615027i	$0.789145\pi$
$108$	24.0710	0.0214466
$109$	−1923.93	−1.69063	−0.845316	−	0.534266i	$-0.820588\pi$
$109$	−1923.93	−1.69063	−0.845316	+	0.534266i	$0.820588\pi$
$110$	−43.1119	−0.0373687
$111$	−13.1741	−0.0112651
$112$	−54.5836	−0.0460506
$113$	−720.292	−0.599641	−0.299820	−	0.953996i	$-0.596927\pi$
$113$	−720.292	−0.599641	−0.299820	+	0.953996i	$0.596927\pi$
$114$	−0.898947	−0.000738545 0
$115$	−841.637	−0.682461
$116$	−1581.08	−1.26551
$117$	597.390	0.472040
$118$	−648.019	−0.505551
$119$	−96.9000	−0.0746454
$120$	−3.63969	−0.00276880
$121$	121.000	0.0909091
$122$	−274.211	−0.203491
$123$	1.69048	0.00123923
$124$	271.178	0.196391
$125$	−125.000	−0.0894427
$126$	−23.2727	−0.0164547
$127$	−1879.77	−1.31341	−0.656704	−	0.754149i	$-0.728050\pi$
$127$	−1879.77	−1.31341	−0.656704	+	0.754149i	$0.728050\pi$
$128$	1311.12	0.905371
$129$	−22.7045	−0.0154963
$130$	−86.7276	−0.0585117
$131$	−77.2212	−0.0515027	−0.0257513	−	0.999668i	$-0.508198\pi$
$131$	−77.2212	−0.0515027	−0.0257513	+	0.999668i	$0.508198\pi$
$132$	4.90369	0.00323342
$133$	−20.8959	−0.0136233
$134$	266.915	0.172074
$135$	16.2960	0.0103891
$136$	1062.59	0.669972
$137$	−244.748	−0.152629	−0.0763147	−	0.997084i	$-0.524315\pi$
$137$	−244.748	−0.152629	−0.0763147	+	0.997084i	$0.524315\pi$
$138$	−7.96407	−0.00491266
$139$	−1554.03	−0.948279	−0.474139	−	0.880450i	$-0.657241\pi$
$139$	−1554.03	−0.948279	−0.474139	+	0.880450i	$0.657241\pi$
$140$	−40.6126	−0.0245171
$141$	−15.0566	−0.00899289
$142$	837.482	0.494929
$143$	243.414	0.142345
$144$	−1339.86	−0.775385
$145$	−1070.38	−0.613036
$146$	78.0440	0.0442395
$147$	−20.6303	−0.0115752
$148$	1611.98	0.895296
$149$	−1846.74	−1.01538	−0.507688	−	0.861541i	$-0.669500\pi$
$149$	−1846.74	−1.01538	−0.507688	+	0.861541i	$0.669500\pi$
$150$	−1.18282	−0.000643848 0
$151$	−1013.96	−0.546458	−0.273229	−	0.961949i	$-0.588092\pi$
$151$	−1013.96	−0.546458	−0.273229	+	0.961949i	$0.588092\pi$
$152$	229.141	0.122275
$153$	−2378.60	−1.25685
$154$	−9.48274	−0.00496196
$155$	183.586	0.0951355
$156$	9.86468	0.00506286
$157$	−3340.56	−1.69812	−0.849062	−	0.528293i	$-0.822832\pi$
$157$	−3340.56	−1.69812	−0.849062	+	0.528293i	$0.822832\pi$
$158$	−805.793	−0.405731
$159$	26.9428	0.0134384
$160$	676.919	0.334470
$161$	−185.123	−0.0906197
$162$	−571.197	−0.277022
$163$	720.474	0.346208	0.173104	−	0.984904i	$-0.444620\pi$
$163$	720.474	0.346208	0.173104	+	0.984904i	$0.444620\pi$
$164$	−206.846	−0.0984878
$165$	3.31977	0.00156633
$166$	−477.388	−0.223208
$167$	−278.976	−0.129268	−0.0646342	−	0.997909i	$-0.520588\pi$
$167$	−278.976	−0.129268	−0.0646342	+	0.997909i	$0.520588\pi$
$168$	−0.800573	−0.000367652 0
$169$	−1707.33	−0.777118
$170$	345.320	0.155793
$171$	−512.931	−0.229385
$172$	2778.11	1.23157
$173$	−228.531	−0.100433	−0.0502165	−	0.998738i	$-0.515991\pi$
$173$	−228.531	−0.100433	−0.0502165	+	0.998738i	$0.515991\pi$
$174$	−10.1286	−0.00441291
$175$	−27.4946	−0.0118765
$176$	−545.944	−0.233819
$177$	49.8998	0.0211904
$178$	−137.588	−0.0579361
$179$	3443.47	1.43786	0.718930	−	0.695083i	$-0.244632\pi$
$179$	3443.47	1.43786	0.718930	+	0.695083i	$0.244632\pi$
$180$	−996.918	−0.412810
$181$	3327.36	1.36641	0.683207	−	0.730225i	$-0.260585\pi$
$181$	3327.36	1.36641	0.683207	+	0.730225i	$0.260585\pi$
$182$	−19.0763	−0.00776939
$183$	21.1152	0.00852941
$184$	2030.03	0.813348
$185$	1091.30	0.433698
$186$	1.73720	0.000684828 0
$187$	−969.192	−0.379007
$188$	1842.32	0.714709
$189$	3.58440	0.00137951
$190$	74.4660	0.0284333
$191$	1998.23	0.757000	0.378500	−	0.925601i	$-0.376440\pi$
$191$	1998.23	0.757000	0.378500	+	0.925601i	$0.376440\pi$
$192$	−17.5604	−0.00660057
$193$	−1524.15	−0.568448	−0.284224	−	0.958758i	$-0.591736\pi$
$193$	−1524.15	−0.568448	−0.284224	+	0.958758i	$0.591736\pi$
$194$	−30.2782	−0.0112054
$195$	6.67834	0.00245254
$196$	2524.32	0.919941
$197$	−281.245	−0.101715	−0.0508575	−	0.998706i	$-0.516195\pi$
$197$	−281.245	−0.101715	−0.0508575	+	0.998706i	$0.516195\pi$
$198$	−232.773	−0.0835477
$199$	−3911.67	−1.39342	−0.696711	−	0.717352i	$-0.745354\pi$
$199$	−3911.67	−1.39342	−0.696711	+	0.717352i	$0.745354\pi$
$200$	301.501	0.106597
$201$	−20.5534	−0.00721256
$202$	464.088	0.161649
$203$	−235.437	−0.0814012
$204$	−39.2778	−0.0134804
$205$	−140.034	−0.0477093
$206$	−694.696	−0.234960
$207$	−4544.23	−1.52582
$208$	−1098.27	−0.366112
$209$	−209.000	−0.0691714
$210$	−0.260170	−8.54925e−5 0
$211$	297.386	0.0970279	0.0485139	−	0.998823i	$-0.484551\pi$
$211$	297.386	0.0970279	0.0485139	+	0.998823i	$0.484551\pi$
$212$	−3296.71	−1.06801
$213$	−64.4891	−0.0207452
$214$	1368.19	0.437043
$215$	1880.77	0.596593
$216$	−39.3060	−0.0123816
$217$	40.3810	0.0126324
$218$	1508.08	0.468532
$219$	−6.00967	−0.00185432
$220$	−406.207	−0.124484
$221$	−1949.71	−0.593446
$222$	10.3266	0.00312195
$223$	2419.03	0.726413	0.363206	−	0.931709i	$-0.381682\pi$
$223$	2419.03	0.726413	0.363206	+	0.931709i	$0.381682\pi$
$224$	148.893	0.0444121
$225$	−674.909	−0.199973
$226$	564.603	0.166181
$227$	−3083.26	−0.901512	−0.450756	−	0.892647i	$-0.648846\pi$
$227$	−3083.26	−0.901512	−0.450756	+	0.892647i	$0.648846\pi$
$228$	−8.47001	−0.00246026
$229$	1552.11	0.447887	0.223944	−	0.974602i	$-0.428107\pi$
$229$	1552.11	0.447887	0.223944	+	0.974602i	$0.428107\pi$
$230$	659.720	0.189133
$231$	0.730206	0.000207983 0
$232$	2581.76	0.730608
$233$	1396.79	0.392733	0.196366	−	0.980531i	$-0.437086\pi$
$233$	1396.79	0.392733	0.196366	+	0.980531i	$0.437086\pi$
$234$	−468.266	−0.130818
$235$	1247.25	0.346218
$236$	−6105.73	−1.68411
$237$	62.0490	0.0170064
$238$	75.9553	0.0206868
$239$	2918.42	0.789861	0.394930	−	0.918711i	$-0.370769\pi$
$239$	2918.42	0.789861	0.394930	+	0.918711i	$0.370769\pi$
$240$	−14.9786	−0.00402861
$241$	−2402.93	−0.642267	−0.321133	−	0.947034i	$-0.604064\pi$
$241$	−2402.93	−0.642267	−0.321133	+	0.947034i	$0.604064\pi$
$242$	−94.8462	−0.0251940
$243$	131.983	0.0348423
$244$	−2583.65	−0.677875
$245$	1708.95	0.445637
$246$	−1.32509	−0.000343433 0
$247$	−420.442	−0.108308
$248$	−442.811	−0.113381
$249$	36.7606	0.00935585
$250$	97.9816	0.0247876
$251$	4142.81	1.04180	0.520900	−	0.853617i	$-0.325596\pi$
$251$	4142.81	1.04180	0.520900	+	0.853617i	$0.325596\pi$
$252$	−219.279	−0.0548145
$253$	−1851.60	−0.460115
$254$	1473.46	0.363990
$255$	−26.5909	−0.00653014
$256$	1299.71	0.317313
$257$	−2926.38	−0.710283	−0.355142	−	0.934813i	$-0.615567\pi$
$257$	−2926.38	−0.710283	−0.355142	+	0.934813i	$0.615567\pi$
$258$	17.7970	0.00429454
$259$	240.039	0.0575880
$260$	−817.160	−0.194916
$261$	−5779.27	−1.37061
$262$	60.5301	0.0142731
$263$	−2387.23	−0.559707	−0.279853	−	0.960043i	$-0.590286\pi$
$263$	−2387.23	−0.559707	−0.279853	+	0.960043i	$0.590286\pi$
$264$	−8.00731	−0.00186673
$265$	−2231.86	−0.517365
$266$	16.3793	0.00377548
$267$	10.5947	0.00242842
$268$	2514.91	0.573218
$269$	−5830.08	−1.32144	−0.660718	−	0.750634i	$-0.729748\pi$
$269$	−5830.08	−1.32144	−0.660718	+	0.750634i	$0.729748\pi$
$270$	−12.7736	−0.00287918
$271$	−6575.21	−1.47386	−0.736930	−	0.675970i	$-0.763725\pi$
$271$	−6575.21	−1.47386	−0.736930	+	0.675970i	$0.763725\pi$
$272$	4372.93	0.974809
$273$	1.46894	0.000325658 0
$274$	191.846	0.0422988
$275$	−275.000	−0.0603023
$276$	−75.0386	−0.0163652
$277$	5770.55	1.25169	0.625846	−	0.779947i	$-0.284754\pi$
$277$	5770.55	1.25169	0.625846	+	0.779947i	$0.284754\pi$
$278$	1218.13	0.262800
$279$	991.232	0.212701
$280$	66.3170	0.0141543
$281$	372.824	0.0791488	0.0395744	−	0.999217i	$-0.487400\pi$
$281$	372.824	0.0791488	0.0395744	+	0.999217i	$0.487400\pi$
$282$	11.8022	0.00249223
$283$	1853.03	0.389226	0.194613	−	0.980880i	$-0.437655\pi$
$283$	1853.03	0.389226	0.194613	+	0.980880i	$0.437655\pi$
$284$	7890.87	1.64872
$285$	−5.73416	−0.00119180
$286$	−190.801	−0.0394486
$287$	−30.8014	−0.00633501
$288$	3654.87	0.747796
$289$	2850.08	0.580109
$290$	839.021	0.169893
$291$	2.33153	0.000469680 0
$292$	735.341	0.147372
$293$	−3682.91	−0.734328	−0.367164	−	0.930156i	$-0.619671\pi$
$293$	−3682.91	−0.734328	−0.367164	+	0.930156i	$0.619671\pi$
$294$	16.1711	0.00320789
$295$	−4133.55	−0.815812
$296$	−2632.23	−0.516875
$297$	35.8511	0.00700436
$298$	1447.57	0.281395
$299$	−3724.84	−0.720445
$300$	−11.1447	−0.00214481
$301$	413.688	0.0792178
$302$	794.798	0.151442
$303$	−35.7364	−0.00677559
$304$	942.995	0.177909
$305$	−1749.12	−0.328375
$306$	1864.48	0.348317
$307$	−4358.02	−0.810180	−0.405090	−	0.914277i	$-0.632760\pi$
$307$	−4358.02	−0.810180	−0.405090	+	0.914277i	$0.632760\pi$
$308$	−89.3478	−0.0165294
$309$	53.4941	0.00984845
$310$	−143.905	−0.0263653
$311$	−467.150	−0.0851756	−0.0425878	−	0.999093i	$-0.513560\pi$
$311$	−467.150	−0.0851756	−0.0425878	+	0.999093i	$0.513560\pi$
$312$	−16.1082	−0.00292291
$313$	8790.72	1.58748	0.793740	−	0.608258i	$-0.208131\pi$
$313$	8790.72	1.58748	0.793740	+	0.608258i	$0.208131\pi$
$314$	2618.51	0.470608
$315$	−148.451	−0.0265531
$316$	−7592.29	−1.35158
$317$	636.066	0.112697	0.0563486	−	0.998411i	$-0.482054\pi$
$317$	636.066	0.112697	0.0563486	+	0.998411i	$0.482054\pi$
$318$	−21.1192	−0.00372422
$319$	−2354.84	−0.413309
$320$	1454.65	0.254116
$321$	−105.355	−0.0183189
$322$	145.110	0.0251138
$323$	1674.06	0.288381
$324$	−5381.90	−0.922823
$325$	−553.214	−0.0944208
$326$	−564.746	−0.0959459
$327$	−116.127	−0.0196387
$328$	337.763	0.0568593
$329$	274.340	0.0459721
$330$	−2.60221	−0.000434082 0
$331$	−8584.94	−1.42559	−0.712796	−	0.701371i	$-0.752572\pi$
$331$	−8584.94	−1.42559	−0.712796	+	0.701371i	$0.752572\pi$
$332$	−4498.01	−0.743556
$333$	5892.24	0.969647
$334$	218.676	0.0358246
$335$	1702.58	0.277677
$336$	−3.29464	−0.000534933 0
$337$	−1382.13	−0.223411	−0.111706	−	0.993741i	$-0.535631\pi$
$337$	−1382.13	−0.223411	−0.111706	+	0.993741i	$0.535631\pi$
$338$	1338.29	0.215366
$339$	−43.4765	−0.00696554
$340$	3253.65	0.518983
$341$	403.890	0.0641404
$342$	402.062	0.0635703
$343$	753.120	0.118556
$344$	−4536.43	−0.711011
$345$	−50.8008	−0.00792760
$346$	179.135	0.0278334
$347$	11869.6	1.83629	0.918143	−	0.396249i	$-0.129688\pi$
$347$	11869.6	1.83629	0.918143	+	0.396249i	$0.129688\pi$
$348$	−95.4329	−0.0147004
$349$	−413.020	−0.0633481	−0.0316740	−	0.999498i	$-0.510084\pi$
$349$	−413.020	−0.0633481	−0.0316740	+	0.999498i	$0.510084\pi$
$350$	21.5517	0.00329139
$351$	72.1212	0.0109674
$352$	1489.22	0.225500
$353$	−3256.96	−0.491078	−0.245539	−	0.969387i	$-0.578965\pi$
$353$	−3256.96	−0.491078	−0.245539	+	0.969387i	$0.578965\pi$
$354$	−39.1141	−0.00587258
$355$	5342.08	0.798672
$356$	−1296.37	−0.192999
$357$	−5.84884	−0.000867096 0
$358$	−2699.17	−0.398479
$359$	−5256.97	−0.772848	−0.386424	−	0.922321i	$-0.626290\pi$
$359$	−5256.97	−0.772848	−0.386424	+	0.922321i	$0.626290\pi$
$360$	1627.88	0.238325
$361$	361.000	0.0526316
$362$	−2608.16	−0.378680
$363$	7.30350	0.00105602
$364$	−179.740	−0.0258816
$365$	497.823	0.0713897
$366$	−16.5512	−0.00236379
$367$	1958.55	0.278571	0.139286	−	0.990252i	$-0.455519\pi$
$367$	1958.55	0.278571	0.139286	+	0.990252i	$0.455519\pi$
$368$	8354.31	1.18342
$369$	−756.082	−0.106667
$370$	−855.420	−0.120192
$371$	−490.911	−0.0686977
$372$	16.3682	0.00228132
$373$	−5965.31	−0.828076	−0.414038	−	0.910260i	$-0.635882\pi$
$373$	−5965.31	−0.828076	−0.414038	+	0.910260i	$0.635882\pi$
$374$	759.704	0.105036
$375$	−7.54494	−0.00103898
$376$	−3008.36	−0.412618
$377$	−4737.19	−0.647156
$378$	−2.80965	−0.000382308 0
$379$	375.676	0.0509161	0.0254580	−	0.999676i	$-0.491896\pi$
$379$	375.676	0.0509161	0.0254580	+	0.999676i	$0.491896\pi$
$380$	701.630	0.0947180
$381$	−113.462	−0.0152568
$382$	−1566.32	−0.209790
$383$	292.289	0.0389954	0.0194977	−	0.999810i	$-0.493793\pi$
$383$	292.289	0.0389954	0.0194977	+	0.999810i	$0.493793\pi$
$384$	79.1384	0.0105170
$385$	−60.4880	−0.00800716
$386$	1194.71	0.157536
$387$	10154.8	1.33384
$388$	−285.286	−0.0373278
$389$	−6104.86	−0.795704	−0.397852	−	0.917450i	$-0.630244\pi$
$389$	−6104.86	−0.795704	−0.397852	+	0.917450i	$0.630244\pi$
$390$	−5.23484	−0.000679683 0
$391$	14831.0	1.91826
$392$	−4122.00	−0.531104
$393$	−4.66104	−0.000598265 0
$394$	220.455	0.0281887
$395$	−5139.95	−0.654731
$396$	−2193.22	−0.278317
$397$	2537.19	0.320751	0.160375	−	0.987056i	$-0.448730\pi$
$397$	2537.19	0.320751	0.160375	+	0.987056i	$0.448730\pi$
$398$	3066.17	0.386164
$399$	−1.26126	−0.000158251 0
$400$	1240.78	0.155098
$401$	−2019.82	−0.251534	−0.125767	−	0.992060i	$-0.540139\pi$
$401$	−2019.82	−0.251534	−0.125767	+	0.992060i	$0.540139\pi$
$402$	16.1108	0.00199884
$403$	812.500	0.100430
$404$	4372.70	0.538490
$405$	−3643.52	−0.447033
$406$	184.548	0.0225590
$407$	2400.86	0.292399
$408$	64.1374	0.00778253
$409$	−5930.91	−0.717029	−0.358514	−	0.933524i	$-0.616717\pi$
$409$	−5930.91	−0.717029	−0.358514	+	0.933524i	$0.616717\pi$
$410$	109.766	0.0132219
$411$	−14.7729	−0.00177297
$412$	−6545.52	−0.782706
$413$	−909.201	−0.108327
$414$	3562.00	0.422857
$415$	−3045.14	−0.360192
$416$	2995.85	0.353085
$417$	−93.8003	−0.0110154
$418$	163.825	0.0191697
$419$	13939.9	1.62532	0.812658	−	0.582741i	$-0.198020\pi$
$419$	13939.9	1.62532	0.812658	+	0.582741i	$0.198020\pi$
$420$	−2.45136	−0.000284795 0
$421$	−11207.9	−1.29748	−0.648738	−	0.761012i	$-0.724703\pi$
$421$	−11207.9	−1.29748	−0.648738	+	0.761012i	$0.724703\pi$
$422$	−233.107	−0.0268897
$423$	6734.22	0.774063
$424$	5383.25	0.616589
$425$	2202.71	0.251405
$426$	50.5500	0.00574919
$427$	−384.730	−0.0436028
$428$	12891.2	1.45589
$429$	14.6924	0.00165350
$430$	−1474.25	−0.165336
$431$	2686.39	0.300230	0.150115	−	0.988669i	$-0.452036\pi$
$431$	2686.39	0.300230	0.150115	+	0.988669i	$0.452036\pi$
$432$	−161.758	−0.0180153
$433$	5824.77	0.646468	0.323234	−	0.946319i	$-0.395230\pi$
$433$	5824.77	0.646468	0.323234	+	0.946319i	$0.395230\pi$
$434$	−31.6528	−0.00350088
$435$	−64.6077	−0.00712115
$436$	14209.3	1.56079
$437$	3198.22	0.350095
$438$	4.71070	0.000513894 0
$439$	−16128.4	−1.75346	−0.876728	−	0.480986i	$-0.840279\pi$
$439$	−16128.4	−1.75346	−0.876728	+	0.480986i	$0.840279\pi$
$440$	663.302	0.0718674
$441$	9227.10	0.996339
$442$	1528.29	0.164464
$443$	−16147.4	−1.73179	−0.865896	−	0.500224i	$-0.833251\pi$
$443$	−16147.4	−1.73179	−0.865896	+	0.500224i	$0.833251\pi$
$444$	97.2983	0.0103999
$445$	−877.637	−0.0934921
$446$	−1896.16	−0.201314
$447$	−111.468	−0.0117948
$448$	319.959	0.0337425
$449$	10233.4	1.07560	0.537801	−	0.843072i	$-0.319255\pi$
$449$	10233.4	1.07560	0.537801	+	0.843072i	$0.319255\pi$
$450$	529.029	0.0554193
$451$	−308.075	−0.0321656
$452$	5319.77	0.553586
$453$	−61.2023	−0.00634776
$454$	2416.82	0.249840
$455$	−121.683	−0.0125375
$456$	13.8308	0.00142037
$457$	6256.80	0.640439	0.320220	−	0.947343i	$-0.396243\pi$
$457$	6256.80	0.640439	0.320220	+	0.947343i	$0.396243\pi$
$458$	−1216.62	−0.124125
$459$	−287.162	−0.0292017
$460$	6215.97	0.630046
$461$	11010.4	1.11238	0.556189	−	0.831056i	$-0.312263\pi$
$461$	11010.4	1.11238	0.556189	+	0.831056i	$0.312263\pi$
$462$	−0.572374	−5.76391e−5 0
$463$	16642.0	1.67046	0.835228	−	0.549904i	$-0.185336\pi$
$463$	16642.0	1.67046	0.835228	+	0.549904i	$0.185336\pi$
$464$	10624.9	1.06303
$465$	11.0812	0.00110511
$466$	−1094.88	−0.108840
$467$	−6140.06	−0.608411	−0.304206	−	0.952606i	$-0.598391\pi$
$467$	−6140.06	−0.608411	−0.304206	+	0.952606i	$0.598391\pi$
$468$	−4412.07	−0.435786
$469$	374.494	0.0368710
$470$	−977.657	−0.0959488
$471$	−201.635	−0.0197258
$472$	9970.15	0.972274
$473$	4137.70	0.402223
$474$	−48.6373	−0.00471305
$475$	475.000	0.0458831
$476$	715.662	0.0689124
$477$	−12050.4	−1.15671
$478$	−2287.61	−0.218897
$479$	−2999.01	−0.286072	−0.143036	−	0.989718i	$-0.545686\pi$
$479$	−2999.01	−0.286072	−0.143036	+	0.989718i	$0.545686\pi$
$480$	40.8585	0.00388527
$481$	4829.79	0.457836
$482$	1883.54	0.177994
$483$	−11.1740	−0.00105266
$484$	−893.655	−0.0839270
$485$	−193.137	−0.0180823
$486$	−103.455	−0.00965599
$487$	13804.2	1.28445	0.642224	−	0.766517i	$-0.278012\pi$
$487$	13804.2	1.28445	0.642224	+	0.766517i	$0.278012\pi$
$488$	4218.89	0.391353
$489$	43.4875	0.00402162
$490$	−1339.57	−0.123501
$491$	1097.44	0.100869	0.0504346	−	0.998727i	$-0.483939\pi$
$491$	1097.44	0.100869	0.0504346	+	0.998727i	$0.483939\pi$
$492$	−12.4852	−0.00114405
$493$	18861.9	1.72312
$494$	329.565	0.0300158
$495$	−1484.80	−0.134822
$496$	−1822.33	−0.164969
$497$	1175.03	0.106051
$498$	−28.8149	−0.00259282
$499$	−11098.7	−0.995679	−0.497840	−	0.867269i	$-0.665873\pi$
$499$	−11098.7	−0.995679	−0.497840	+	0.867269i	$0.665873\pi$
$500$	923.197	0.0825732
$501$	−16.8389	−0.00150161
$502$	−3247.36	−0.288718
$503$	6753.57	0.598662	0.299331	−	0.954149i	$-0.403237\pi$
$503$	6753.57	0.598662	0.299331	+	0.954149i	$0.403237\pi$
$504$	358.064	0.0316457
$505$	2960.30	0.260855
$506$	1451.38	0.127514
$507$	−103.053	−0.00902715
$508$	13883.2	1.21253
$509$	−10002.2	−0.870997	−0.435499	−	0.900189i	$-0.643428\pi$
$509$	−10002.2	−0.870997	−0.435499	+	0.900189i	$0.643428\pi$
$510$	20.8433	0.00180972
$511$	109.499	0.00947938
$512$	−11507.7	−0.993309
$513$	−61.9247	−0.00532952
$514$	2293.85	0.196843
$515$	−4431.29	−0.379157
$516$	167.686	0.0143061
$517$	2743.94	0.233420
$518$	−188.155	−0.0159596
$519$	−13.7940	−0.00116665
$520$	1334.35	0.112529
$521$	−107.338	−0.00902602	−0.00451301	−	0.999990i	$-0.501437\pi$
$521$	−107.338	−0.00902602	−0.00451301	+	0.999990i	$0.501437\pi$
$522$	4530.10	0.379841
$523$	9371.01	0.783491	0.391745	−	0.920074i	$-0.371871\pi$
$523$	9371.01	0.783491	0.391745	+	0.920074i	$0.371871\pi$
$524$	570.323	0.0475471
$525$	−1.65956	−0.000137960 0
$526$	1871.24	0.155114
$527$	−3235.10	−0.267406
$528$	−32.9529	−0.00271608
$529$	16167.1	1.32877
$530$	1749.45	0.143379
$531$	−22318.2	−1.82397
$532$	154.328	0.0125770
$533$	−619.750	−0.0503646
$534$	−8.30472	−0.000672997 0
$535$	8727.31	0.705261
$536$	−4106.63	−0.330932
$537$	207.846	0.0167025
$538$	4569.92	0.366215
$539$	3759.70	0.300448
$540$	−120.355	−0.00959122
$541$	−3851.36	−0.306068	−0.153034	−	0.988221i	$-0.548904\pi$
$541$	−3851.36	−0.306068	−0.153034	+	0.988221i	$0.548904\pi$
$542$	5154.00	0.408456
$543$	200.838	0.0158725
$544$	−11928.4	−0.940125
$545$	9619.64	0.756074
$546$	−1.15144	−9.02508e−5 0
$547$	572.812	0.0447746	0.0223873	−	0.999749i	$-0.492873\pi$
$547$	572.812	0.0447746	0.0223873	+	0.999749i	$0.492873\pi$
$548$	1807.60	0.140907
$549$	−9443.98	−0.734170
$550$	215.560	0.0167118
$551$	4067.45	0.314481
$552$	122.532	0.00944801
$553$	−1130.56	−0.0869376
$554$	−4523.26	−0.346886
$555$	65.8705	0.00503792
$556$	11477.4	0.875448
$557$	−21782.7	−1.65703	−0.828513	−	0.559971i	$-0.810812\pi$
$557$	−21782.7	−1.65703	−0.828513	+	0.559971i	$0.810812\pi$
$558$	−776.980	−0.0589466
$559$	8323.74	0.629798
$560$	272.918	0.0205945
$561$	−58.4999	−0.00440262
$562$	−292.239	−0.0219348
$563$	7408.62	0.554593	0.277297	−	0.960784i	$-0.410562\pi$
$563$	7408.62	0.554593	0.277297	+	0.960784i	$0.410562\pi$
$564$	111.202	0.00830220
$565$	3601.46	0.268168
$566$	−1452.50	−0.107868
$567$	−801.417	−0.0593586
$568$	−12885.1	−0.951846
$569$	2367.61	0.174438	0.0872190	−	0.996189i	$-0.472202\pi$
$569$	2367.61	0.174438	0.0872190	+	0.996189i	$0.472202\pi$
$570$	4.49473	0.000330287 0
$571$	−22777.5	−1.66937	−0.834683	−	0.550731i	$-0.814349\pi$
$571$	−22777.5	−1.66937	−0.834683	+	0.550731i	$0.814349\pi$
$572$	−1797.75	−0.131412
$573$	120.612	0.00879346
$574$	24.1438	0.00175565
$575$	4208.18	0.305206
$576$	7854.03	0.568145
$577$	−16073.0	−1.15967	−0.579833	−	0.814735i	$-0.696882\pi$
$577$	−16073.0	−1.15967	−0.579833	+	0.814735i	$0.696882\pi$
$578$	−2234.04	−0.160768
$579$	−91.9967	−0.00660320
$580$	7905.38	0.565953
$581$	−669.797	−0.0478276
$582$	−1.82758	−0.000130164 0
$583$	−4910.08	−0.348808
$584$	−1200.75	−0.0850812
$585$	−2986.95	−0.211103
$586$	2886.86	0.203507
$587$	−14991.8	−1.05413	−0.527067	−	0.849823i	$-0.676708\pi$
$587$	−14991.8	−1.05413	−0.527067	+	0.849823i	$0.676708\pi$
$588$	152.367	0.0106862
$589$	−697.628	−0.0488035
$590$	3240.10	0.226089
$591$	−16.9758	−0.00118154
$592$	−10832.6	−0.752053
$593$	8315.72	0.575861	0.287930	−	0.957651i	$-0.407033\pi$
$593$	8315.72	0.575861	0.287930	+	0.957651i	$0.407033\pi$
$594$	−28.1020	−0.00194114
$595$	484.500	0.0333824
$596$	13639.2	0.937391
$597$	−236.106	−0.0161863
$598$	2919.73	0.199660
$599$	−18674.7	−1.27383	−0.636916	−	0.770933i	$-0.719790\pi$
$599$	−18674.7	−1.27383	−0.636916	+	0.770933i	$0.719790\pi$
$600$	18.1984	0.00123825
$601$	−807.103	−0.0547794	−0.0273897	−	0.999625i	$-0.508719\pi$
$601$	−807.103	−0.0547794	−0.0273897	+	0.999625i	$0.508719\pi$
$602$	−324.270	−0.0219539
$603$	9192.70	0.620822
$604$	7488.70	0.504488
$605$	−605.000	−0.0406558
$606$	28.0121	0.00187775
$607$	16324.2	1.09156	0.545782	−	0.837927i	$-0.316233\pi$
$607$	16324.2	1.09156	0.545782	+	0.837927i	$0.316233\pi$
$608$	−2572.29	−0.171579
$609$	−14.2109	−0.000945572 0
$610$	1371.05	0.0910038
$611$	5519.95	0.365488
$612$	17567.4	1.16032
$613$	−824.080	−0.0542973	−0.0271487	−	0.999631i	$-0.508643\pi$
$613$	−824.080	−0.0542973	−0.0271487	+	0.999631i	$0.508643\pi$
$614$	3416.04	0.224528
$615$	−8.45239	−0.000554200 0
$616$	145.897	0.00954282
$617$	−15112.5	−0.986070	−0.493035	−	0.870009i	$-0.664113\pi$
$617$	−15112.5	−0.986070	−0.493035	+	0.870009i	$0.664113\pi$
$618$	−41.9315	−0.00272934
$619$	3574.05	0.232073	0.116036	−	0.993245i	$-0.462981\pi$
$619$	3574.05	0.232073	0.116036	+	0.993245i	$0.462981\pi$
$620$	−1355.89	−0.0878288
$621$	−548.612	−0.0354509
$622$	366.177	0.0236051
$623$	−193.042	−0.0124142
$624$	−66.2910	−0.00425283
$625$	625.000	0.0400000
$626$	−6890.63	−0.439944
$627$	−12.6151	−0.000803509 0
$628$	24671.9	1.56770
$629$	−19230.6	−1.21903
$630$	116.363	0.00735878
$631$	1678.56	0.105899	0.0529495	−	0.998597i	$-0.483138\pi$
$631$	1678.56	0.105899	0.0529495	+	0.998597i	$0.483138\pi$
$632$	12397.6	0.780300
$633$	17.9501	0.00112709
$634$	−498.582	−0.0312322
$635$	9398.86	0.587374
$636$	−198.988	−0.0124063
$637$	7563.33	0.470439
$638$	1845.85	0.114542
$639$	28843.4	1.78564
$640$	−6555.58	−0.404894
$641$	2186.08	0.134704	0.0673518	−	0.997729i	$-0.478545\pi$
$641$	2186.08	0.134704	0.0673518	+	0.997729i	$0.478545\pi$
$642$	82.5831	0.00507678
$643$	−22353.0	−1.37094	−0.685472	−	0.728099i	$-0.740404\pi$
$643$	−22353.0	−1.37094	−0.685472	+	0.728099i	$0.740404\pi$
$644$	1367.24	0.0836598
$645$	113.522	0.00693014
$646$	−1312.22	−0.0799202
$647$	848.196	0.0515394	0.0257697	−	0.999668i	$-0.491796\pi$
$647$	848.196	0.0515394	0.0257697	+	0.999668i	$0.491796\pi$
$648$	8788.20	0.532767
$649$	−9093.81	−0.550021
$650$	433.638	0.0261672
$651$	2.43738	0.000146741 0
$652$	−5321.12	−0.319618
$653$	20748.5	1.24341	0.621707	−	0.783250i	$-0.286439\pi$
$653$	20748.5	1.24341	0.621707	+	0.783250i	$0.286439\pi$
$654$	91.0268	0.00544255
$655$	386.106	0.0230327
$656$	1390.01	0.0827301
$657$	2687.88	0.159611
$658$	−215.042	−0.0127404
$659$	−5565.49	−0.328984	−0.164492	−	0.986378i	$-0.552599\pi$
$659$	−5565.49	−0.328984	−0.164492	+	0.986378i	$0.552599\pi$
$660$	−24.5184	−0.00144603
$661$	−6981.39	−0.410809	−0.205404	−	0.978677i	$-0.565851\pi$
$661$	−6981.39	−0.410809	−0.205404	+	0.978677i	$0.565851\pi$
$662$	6729.33	0.395080
$663$	−117.684	−0.00689359
$664$	7344.88	0.429272
$665$	104.479	0.00609253
$666$	−4618.65	−0.268722
$667$	36034.9	2.09187
$668$	2060.40	0.119340
$669$	146.011	0.00843816
$670$	−1334.57	−0.0769538
$671$	−3848.07	−0.221390
$672$	8.98710	0.000515900 0
$673$	15367.4	0.880192	0.440096	−	0.897951i	$-0.354944\pi$
$673$	15367.4	0.880192	0.440096	+	0.897951i	$0.354944\pi$
$674$	1083.39	0.0619148
$675$	−81.4799	−0.00464617
$676$	12609.6	0.717433
$677$	−20345.3	−1.15500	−0.577500	−	0.816391i	$-0.695972\pi$
$677$	−20345.3	−1.15500	−0.577500	+	0.816391i	$0.695972\pi$
$678$	34.0792	0.00193039
$679$	−42.4818	−0.00240103
$680$	−5312.94	−0.299621
$681$	−186.104	−0.0104721
$682$	−316.590	−0.0177755
$683$	−17653.0	−0.988982	−0.494491	−	0.869183i	$-0.664645\pi$
$683$	−17653.0	−0.988982	−0.494491	+	0.869183i	$0.664645\pi$
$684$	3788.29	0.211767
$685$	1223.74	0.0682579
$686$	−590.336	−0.0328559
$687$	93.6845	0.00520275
$688$	−18669.0	−1.03452
$689$	−9877.54	−0.546160
$690$	39.8204	0.00219701
$691$	−27619.5	−1.52054	−0.760272	−	0.649605i	$-0.774934\pi$
$691$	−27619.5	−1.52054	−0.760272	+	0.649605i	$0.774934\pi$
$692$	1687.83	0.0927194
$693$	−326.591	−0.0179021
$694$	−9303.99	−0.508897
$695$	7770.13	0.424083
$696$	155.834	0.00848689
$697$	2467.63	0.134101
$698$	323.747	0.0175559
$699$	84.3095	0.00456206
$700$	203.063	0.0109644
$701$	−10809.2	−0.582393	−0.291196	−	0.956663i	$-0.594053\pi$
$701$	−10809.2	−0.582393	−0.291196	+	0.956663i	$0.594053\pi$
$702$	−56.5324	−0.00303943
$703$	−4146.95	−0.222482
$704$	3200.22	0.171325
$705$	75.2831	0.00402174
$706$	2552.98	0.136094
$707$	651.137	0.0346372
$708$	−368.539	−0.0195629
$709$	−14360.2	−0.760663	−0.380331	−	0.924850i	$-0.624190\pi$
$709$	−14360.2	−0.760663	−0.380331	+	0.924850i	$0.624190\pi$
$710$	−4187.41	−0.221339
$711$	−27752.0	−1.46383
$712$	2116.86	0.111423
$713$	−6180.52	−0.324632
$714$	4.58463	0.000240302 0
$715$	−1217.07	−0.0636585
$716$	−25432.0	−1.32743
$717$	176.154	0.00917518
$718$	4120.69	0.214182
$719$	19614.5	1.01738	0.508690	−	0.860950i	$-0.330130\pi$
$719$	19614.5	1.01738	0.508690	+	0.860950i	$0.330130\pi$
$720$	6699.32	0.346763
$721$	−974.690	−0.0503459
$722$	−282.971	−0.0145860
$723$	−145.040	−0.00746070
$724$	−24574.5	−1.26147
$725$	5351.90	0.274158
$726$	−5.72487	−0.000292658 0
$727$	7497.02	0.382461	0.191231	−	0.981545i	$-0.438752\pi$
$727$	7497.02	0.382461	0.191231	+	0.981545i	$0.438752\pi$
$728$	293.500	0.0149421
$729$	−19667.1	−0.999190
$730$	−390.220	−0.0197845
$731$	−33142.3	−1.67690
$732$	−155.948	−0.00787433
$733$	21328.5	1.07474	0.537371	−	0.843346i	$-0.319418\pi$
$733$	21328.5	1.07474	0.537371	+	0.843346i	$0.319418\pi$
$734$	−1535.22	−0.0772016
$735$	103.152	0.00517660
$736$	−22788.8	−1.14131
$737$	3745.68	0.187210
$738$	592.657	0.0295610
$739$	13437.4	0.668879	0.334440	−	0.942417i	$-0.391453\pi$
$739$	13437.4	0.668879	0.334440	+	0.942417i	$0.391453\pi$
$740$	−8059.89	−0.400389
$741$	−25.3777	−0.00125813
$742$	384.802	0.0190384
$743$	−3760.57	−0.185682	−0.0928411	−	0.995681i	$-0.529595\pi$
$743$	−3760.57	−0.185682	−0.0928411	+	0.995681i	$0.529595\pi$
$744$	−26.7279	−0.00131706
$745$	9233.70	0.454090
$746$	4675.93	0.229488
$747$	−16441.5	−0.805306
$748$	7158.04	0.349898
$749$	1919.63	0.0936471
$750$	5.91412	0.000287938 0
$751$	−27004.2	−1.31211	−0.656056	−	0.754712i	$-0.727777\pi$
$751$	−27004.2	−1.31211	−0.656056	+	0.754712i	$0.727777\pi$
$752$	−12380.5	−0.600359
$753$	250.058	0.0121018
$754$	3713.26	0.179349
$755$	5069.82	0.244383
$756$	−26.4729	−0.00127356
$757$	4049.74	0.194439	0.0972195	−	0.995263i	$-0.469005\pi$
$757$	4049.74	0.194439	0.0972195	+	0.995263i	$0.469005\pi$
$758$	−294.475	−0.0141106
$759$	−111.762	−0.00534479
$760$	−1145.70	−0.0546829
$761$	−25094.8	−1.19538	−0.597692	−	0.801726i	$-0.703915\pi$
$761$	−25094.8	−1.19538	−0.597692	+	0.801726i	$0.703915\pi$
$762$	88.9376	0.00422818
$763$	2115.90	0.100394
$764$	−14758.1	−0.698860
$765$	11893.0	0.562082
$766$	−229.111	−0.0108070
$767$	−18293.9	−0.861218
$768$	78.4500	0.00368597
$769$	12245.3	0.574224	0.287112	−	0.957897i	$-0.407305\pi$
$769$	12245.3	0.574224	0.287112	+	0.957897i	$0.407305\pi$
$770$	47.4137	0.00221905
$771$	−176.635	−0.00825079
$772$	11256.7	0.524789
$773$	4554.29	0.211910	0.105955	−	0.994371i	$-0.466210\pi$
$773$	4554.29	0.211910	0.105955	+	0.994371i	$0.466210\pi$
$774$	−7959.86	−0.369653
$775$	−917.932	−0.0425459
$776$	465.848	0.0215502
$777$	14.4886	0.000668954 0
$778$	4785.32	0.220517
$779$	532.129	0.0244743
$780$	−49.3234	−0.00226418
$781$	11752.6	0.538464
$782$	−11625.4	−0.531614
$783$	−697.716	−0.0318446
$784$	−16963.5	−0.772755
$785$	16702.8	0.759424
$786$	3.65357	0.000165800 0
$787$	4110.23	0.186167	0.0930837	−	0.995658i	$-0.470328\pi$
$787$	4110.23	0.186167	0.0930837	+	0.995658i	$0.470328\pi$
$788$	2077.15	0.0939030
$789$	−144.092	−0.00650167
$790$	4028.96	0.181448
$791$	792.165	0.0356083
$792$	3581.34	0.160679
$793$	−7741.10	−0.346651
$794$	−1988.79	−0.0888909
$795$	−134.714	−0.00600982
$796$	28889.9	1.28640
$797$	28046.0	1.24647	0.623237	−	0.782033i	$-0.285817\pi$
$797$	28046.0	1.24647	0.623237	+	0.782033i	$0.285817\pi$
$798$	0.988646	4.38567e−5 0
$799$	−21978.5	−0.973148
$800$	−3384.60	−0.149579
$801$	−4738.60	−0.209026
$802$	1583.25	0.0697087
$803$	1095.21	0.0481309
$804$	151.799	0.00665861
$805$	925.617	0.0405264
$806$	−636.880	−0.0278327
$807$	−351.901	−0.0153501
$808$	−7140.26	−0.310883
$809$	−12876.7	−0.559607	−0.279803	−	0.960057i	$-0.590269\pi$
$809$	−12876.7	−0.559607	−0.279803	+	0.960057i	$0.590269\pi$
$810$	2855.99	0.123888
$811$	2515.04	0.108896	0.0544482	−	0.998517i	$-0.482660\pi$
$811$	2515.04	0.108896	0.0544482	+	0.998517i	$0.482660\pi$
$812$	1738.84	0.0751493
$813$	−396.877	−0.0171206
$814$	−1881.92	−0.0810337
$815$	−3602.37	−0.154829
$816$	263.948	0.0113236
$817$	−7146.93	−0.306046
$818$	4648.96	0.198713
$819$	−656.999	−0.0280310
$820$	1034.23	0.0440451
$821$	−39018.8	−1.65867	−0.829333	−	0.558754i	$-0.811280\pi$
$821$	−39018.8	−1.65867	−0.829333	+	0.558754i	$0.811280\pi$
$822$	11.5798	0.000491351 0
$823$	−3391.84	−0.143660	−0.0718299	−	0.997417i	$-0.522884\pi$
$823$	−3391.84	−0.143660	−0.0718299	+	0.997417i	$0.522884\pi$
$824$	10688.3	0.451874
$825$	−16.5989	−0.000700483 0
$826$	712.680	0.0300210
$827$	−36397.7	−1.53044	−0.765220	−	0.643769i	$-0.777370\pi$
$827$	−36397.7	−1.53044	−0.765220	+	0.643769i	$0.777370\pi$
$828$	33561.7	1.40864
$829$	−25754.4	−1.07899	−0.539497	−	0.841987i	$-0.681386\pi$
$829$	−25754.4	−1.07899	−0.539497	+	0.841987i	$0.681386\pi$
$830$	2386.94	0.0998215
$831$	348.308	0.0145399
$832$	6437.85	0.268260
$833$	−30114.6	−1.25259
$834$	73.5256	0.00305274
$835$	1394.88	0.0578105
$836$	1543.59	0.0638589
$837$	119.669	0.00494188
$838$	−10926.8	−0.450430
$839$	−26723.0	−1.09962	−0.549809	−	0.835290i	$-0.685300\pi$
$839$	−26723.0	−1.09962	−0.549809	+	0.835290i	$0.685300\pi$
$840$	4.00286	0.000164419 0
$841$	21439.6	0.879067
$842$	8785.31	0.359575
$843$	22.5035	0.000919408 0
$844$	−2196.36	−0.0895758
$845$	8536.64	0.347538
$846$	−5278.64	−0.214519
$847$	−133.074	−0.00539842
$848$	22154.0	0.897136
$849$	111.848	0.00452133
$850$	−1726.60	−0.0696728
$851$	−36739.2	−1.47991
$852$	476.289	0.0191519
$853$	31404.1	1.26056	0.630279	−	0.776368i	$-0.282940\pi$
$853$	31404.1	1.26056	0.630279	+	0.776368i	$0.282940\pi$
$854$	301.572	0.0120838
$855$	2564.65	0.102584
$856$	−21050.3	−0.840520
$857$	−19490.7	−0.776884	−0.388442	−	0.921473i	$-0.626987\pi$
$857$	−19490.7	−0.776884	−0.388442	+	0.921473i	$0.626987\pi$
$858$	−11.5166	−0.000458242 0
$859$	−2031.11	−0.0806758	−0.0403379	−	0.999186i	$-0.512843\pi$
$859$	−2031.11	−0.0806758	−0.0403379	+	0.999186i	$0.512843\pi$
$860$	−13890.6	−0.550773
$861$	−1.85916	−7.35888e−5 0
$862$	−2105.74	−0.0832038
$863$	−31486.2	−1.24195	−0.620975	−	0.783830i	$-0.713263\pi$
$863$	−31486.2	−1.24195	−0.620975	+	0.783830i	$0.713263\pi$
$864$	441.242	0.0173743
$865$	1142.66	0.0449150
$866$	−4565.77	−0.179158
$867$	172.029	0.00673866
$868$	−298.237	−0.0116622
$869$	−11307.9	−0.441420
$870$	50.6429	0.00197351
$871$	7535.13	0.293132
$872$	−23202.6	−0.901078
$873$	−1042.80	−0.0404278
$874$	−2506.93	−0.0970232
$875$	137.473	0.00531135
$876$	44.3849	0.00171190
$877$	−16898.1	−0.650638	−0.325319	−	0.945604i	$-0.605472\pi$
$877$	−16898.1	−0.650638	−0.325319	+	0.945604i	$0.605472\pi$
$878$	12642.3	0.485942
$879$	−222.299	−0.00853009
$880$	2729.72	0.104567
$881$	25394.8	0.971138	0.485569	−	0.874198i	$-0.338612\pi$
$881$	25394.8	0.971138	0.485569	+	0.874198i	$0.338612\pi$
$882$	−7232.69	−0.276119
$883$	−19200.4	−0.731762	−0.365881	−	0.930662i	$-0.619232\pi$
$883$	−19200.4	−0.731762	−0.365881	+	0.930662i	$0.619232\pi$
$884$	14399.7	0.547868
$885$	−249.499	−0.00947664
$886$	12657.2	0.479938
$887$	41952.3	1.58807	0.794036	−	0.607871i	$-0.207976\pi$
$887$	41952.3	1.58807	0.794036	+	0.607871i	$0.207976\pi$
$888$	−158.880	−0.00600412
$889$	2067.34	0.0779936
$890$	687.938	0.0259098
$891$	−8015.75	−0.301389
$892$	−17865.9	−0.670622
$893$	−4739.53	−0.177606
$894$	87.3748	0.00326874
$895$	−17217.3	−0.643030
$896$	−1441.94	−0.0537633
$897$	−224.830	−0.00836883
$898$	−8021.50	−0.298086
$899$	−7860.29	−0.291608
$900$	4984.59	0.184614
$901$	39329.0	1.45421
$902$	241.485	0.00891418
$903$	24.9700	0.000920209 0
$904$	−8686.75	−0.319598
$905$	−16636.8	−0.611079
$906$	47.9736	0.00175918
$907$	7143.32	0.261511	0.130755	−	0.991415i	$-0.458260\pi$
$907$	7143.32	0.261511	0.130755	+	0.991415i	$0.458260\pi$
$908$	22771.7	0.832273
$909$	15983.5	0.583210
$910$	95.3815	0.00347458
$911$	−19674.4	−0.715522	−0.357761	−	0.933813i	$-0.616460\pi$
$911$	−19674.4	−0.715522	−0.357761	+	0.933813i	$0.616460\pi$
$912$	56.9187	0.00206663
$913$	−6699.30	−0.242842
$914$	−4904.41	−0.177487
$915$	−105.576	−0.00381447
$916$	−11463.2	−0.413488
$917$	84.9265	0.00305837
$918$	225.093	0.00809278
$919$	30418.3	1.09185	0.545924	−	0.837835i	$-0.316179\pi$
$919$	30418.3	1.09185	0.545924	+	0.837835i	$0.316179\pi$
$920$	−10150.2	−0.363740
$921$	−263.048	−0.00941121
$922$	−8630.54	−0.308277
$923$	23642.5	0.843123
$924$	−5.39299	−0.000192009 0
$925$	−5456.51	−0.193956
$926$	−13044.9	−0.462940
$927$	−23925.7	−0.847707
$928$	−28982.5	−1.02521
$929$	20882.8	0.737504	0.368752	−	0.929528i	$-0.379785\pi$
$929$	20882.8	0.737504	0.368752	+	0.929528i	$0.379785\pi$
$930$	−8.68602	−0.000306264 0
$931$	−6494.02	−0.228607
$932$	−10316.1	−0.362570
$933$	−28.1969	−0.000989417 0
$934$	4812.90	0.168611
$935$	4845.96	0.169497
$936$	7204.54	0.251590
$937$	−49312.2	−1.71927	−0.859636	−	0.510907i	$-0.829310\pi$
$937$	−49312.2	−1.71927	−0.859636	+	0.510907i	$0.829310\pi$
$938$	−293.548	−0.0102182
$939$	530.604	0.0184405
$940$	−9211.62	−0.319628
$941$	20987.9	0.727083	0.363542	−	0.931578i	$-0.381567\pi$
$941$	20987.9	0.727083	0.363542	+	0.931578i	$0.381567\pi$
$942$	158.052	0.00546667
$943$	4714.31	0.162799
$944$	41030.7	1.41466
$945$	−17.9220	−0.000616935 0
$946$	−3243.34	−0.111470
$947$	−50485.8	−1.73238	−0.866192	−	0.499711i	$-0.833440\pi$
$947$	−50485.8	−1.73238	−0.866192	+	0.499711i	$0.833440\pi$
$948$	−458.267	−0.0157002
$949$	2203.22	0.0753630
$950$	−372.330	−0.0127158
$951$	38.3926	0.00130911
$952$	−1168.62	−0.0397847
$953$	35925.7	1.22114	0.610570	−	0.791962i	$-0.290940\pi$
$953$	35925.7	1.22114	0.610570	+	0.791962i	$0.290940\pi$
$954$	9445.74	0.320563
$955$	−9991.16	−0.338541
$956$	−21554.2	−0.729197
$957$	−142.137	−0.00480108
$958$	2350.78	0.0792802
$959$	269.169	0.00906354
$960$	87.8018	0.00295187
$961$	−28442.8	−0.954746
$962$	−3785.84	−0.126882
$963$	47121.1	1.57680
$964$	17747.0	0.592939
$965$	7620.73	0.254218
$966$	8.75874	0.000291727 0
$967$	−13309.2	−0.442600	−0.221300	−	0.975206i	$-0.571030\pi$
$967$	−13309.2	−0.442600	−0.221300	+	0.975206i	$0.571030\pi$
$968$	1459.26	0.0484530
$969$	101.045	0.00334989
$970$	151.391	0.00501121
$971$	−15424.5	−0.509781	−0.254890	−	0.966970i	$-0.582039\pi$
$971$	−15424.5	−0.509781	−0.254890	+	0.966970i	$0.582039\pi$
$972$	−974.767	−0.0321663
$973$	1709.09	0.0563113
$974$	−10820.4	−0.355964
$975$	−33.3917	−0.00109681
$976$	17362.2	0.569418
$977$	31101.5	1.01845	0.509224	−	0.860634i	$-0.329932\pi$
$977$	31101.5	1.01845	0.509224	+	0.860634i	$0.329932\pi$
$978$	−34.0878	−0.00111453
$979$	−1930.80	−0.0630323
$980$	−12621.6	−0.411410
$981$	51939.1	1.69040
$982$	−860.231	−0.0279543
$983$	6010.71	0.195027	0.0975136	−	0.995234i	$-0.468911\pi$
$983$	6010.71	0.195027	0.0975136	+	0.995234i	$0.468911\pi$
$984$	20.3872	0.000660488 0
$985$	1406.22	0.0454884
$986$	−14784.9	−0.477534
$987$	16.5590	0.000534021 0
$988$	3105.21	0.0999897
$989$	−63317.0	−2.03576
$990$	1163.86	0.0373637
$991$	−19192.5	−0.615208	−0.307604	−	0.951515i	$-0.599527\pi$
$991$	−19192.5	−0.615208	−0.307604	+	0.951515i	$0.599527\pi$
$992$	4970.93	0.159100
$993$	−518.183	−0.0165600
$994$	−921.047	−0.0293902
$995$	19558.3	0.623157
$996$	−271.498	−0.00863729
$997$	53233.1	1.69098	0.845490	−	0.533990i	$-0.179308\pi$
$997$	53233.1	1.69098	0.845490	+	0.533990i	$0.179308\pi$
$998$	8699.71	0.275936
$999$	711.353	0.0225287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.c.1.9	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.c.1.9	✓	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-0.783853 q^{2} +0.0603595 q^{3} -7.38557 q^{4} -5.00000 q^{5} -0.0473130 q^{6} -1.09978 q^{7} +12.0600 q^{8} -26.9964 q^{9} +O(q^{10})\)	\(q-0.783853 q^{2} +0.0603595 q^{3} -7.38557 q^{4} -5.00000 q^{5} -0.0473130 q^{6} -1.09978 q^{7} +12.0600 q^{8} -26.9964 q^{9} +3.91926 q^{10} -11.0000 q^{11} -0.445790 q^{12} -22.1285 q^{13} +0.862068 q^{14} -0.301798 q^{15} +49.6313 q^{16} +88.1083 q^{17} +21.1612 q^{18} +19.0000 q^{19} +36.9279 q^{20} -0.0663823 q^{21} +8.62238 q^{22} +168.327 q^{23} +0.727938 q^{24} +25.0000 q^{25} +17.3455 q^{26} -3.25919 q^{27} +8.12252 q^{28} +214.076 q^{29} +0.236565 q^{30} -36.7173 q^{31} -135.384 q^{32} -0.663955 q^{33} -69.0640 q^{34} +5.49891 q^{35} +199.384 q^{36} -218.260 q^{37} -14.8932 q^{38} -1.33567 q^{39} -60.3001 q^{40} +28.0068 q^{41} +0.0520340 q^{42} -376.154 q^{43} +81.2413 q^{44} +134.982 q^{45} -131.944 q^{46} -249.449 q^{47} +2.99572 q^{48} -341.790 q^{49} -19.5963 q^{50} +5.31818 q^{51} +163.432 q^{52} +446.371 q^{53} +2.55473 q^{54} +55.0000 q^{55} -13.2634 q^{56} +1.14683 q^{57} -167.804 q^{58} +826.710 q^{59} +2.22895 q^{60} +349.824 q^{61} +28.7809 q^{62} +29.6901 q^{63} -290.929 q^{64} +110.643 q^{65} +0.520443 q^{66} -340.516 q^{67} -650.731 q^{68} +10.1602 q^{69} -4.31034 q^{70} -1068.42 q^{71} -325.577 q^{72} -99.5645 q^{73} +171.084 q^{74} +1.50899 q^{75} -140.326 q^{76} +12.0976 q^{77} +1.04697 q^{78} +1027.99 q^{79} -248.157 q^{80} +728.705 q^{81} -21.9532 q^{82} +609.027 q^{83} +0.490272 q^{84} -440.542 q^{85} +294.849 q^{86} +12.9215 q^{87} -132.660 q^{88} +175.527 q^{89} -105.806 q^{90} +24.3366 q^{91} -1243.19 q^{92} -2.21624 q^{93} +195.531 q^{94} -95.0000 q^{95} -8.17171 q^{96} +38.6274 q^{97} +267.913 q^{98} +296.960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

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