LMFDB - Embedded newform 1045.4.a.c.1.4

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} + \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			1.4
Root			$3.91893$ 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)

$f(q)$	$=$	$q-3.91893 q^{2} -3.32066 q^{3} +7.35803 q^{4} -5.00000 q^{5} +13.0134 q^{6} -28.1182 q^{7} +2.51584 q^{8} -15.9732 q^{9} +O(q^{10})$	\(q-3.91893 q^{2} -3.32066 q^{3} +7.35803 q^{4} -5.00000 q^{5} +13.0134 q^{6} -28.1182 q^{7} +2.51584 q^{8} -15.9732 q^{9} +19.5947 q^{10} -11.0000 q^{11} -24.4335 q^{12} -57.3328 q^{13} +110.193 q^{14} +16.6033 q^{15} -68.7236 q^{16} +77.5195 q^{17} +62.5980 q^{18} +19.0000 q^{19} -36.7901 q^{20} +93.3709 q^{21} +43.1083 q^{22} +25.3511 q^{23} -8.35424 q^{24} +25.0000 q^{25} +224.683 q^{26} +142.699 q^{27} -206.895 q^{28} +66.9421 q^{29} -65.0671 q^{30} -288.879 q^{31} +249.197 q^{32} +36.5272 q^{33} -303.794 q^{34} +140.591 q^{35} -117.532 q^{36} +287.719 q^{37} -74.4597 q^{38} +190.383 q^{39} -12.5792 q^{40} -140.535 q^{41} -365.914 q^{42} -203.180 q^{43} -80.9383 q^{44} +79.8662 q^{45} -99.3493 q^{46} +190.852 q^{47} +228.208 q^{48} +447.633 q^{49} -97.9733 q^{50} -257.416 q^{51} -421.857 q^{52} +585.208 q^{53} -559.229 q^{54} +55.0000 q^{55} -70.7409 q^{56} -63.0925 q^{57} -262.342 q^{58} -369.162 q^{59} +122.167 q^{60} +11.4860 q^{61} +1132.10 q^{62} +449.139 q^{63} -426.795 q^{64} +286.664 q^{65} -143.148 q^{66} -291.136 q^{67} +570.391 q^{68} -84.1823 q^{69} -550.967 q^{70} +921.785 q^{71} -40.1861 q^{72} +1188.93 q^{73} -1127.55 q^{74} -83.0164 q^{75} +139.803 q^{76} +309.300 q^{77} -746.096 q^{78} -179.464 q^{79} +343.618 q^{80} -42.5781 q^{81} +550.746 q^{82} -184.309 q^{83} +687.026 q^{84} -387.597 q^{85} +796.248 q^{86} -222.292 q^{87} -27.6742 q^{88} -335.864 q^{89} -312.990 q^{90} +1612.10 q^{91} +186.534 q^{92} +959.266 q^{93} -747.938 q^{94} -95.0000 q^{95} -827.496 q^{96} -1155.10 q^{97} -1754.24 q^{98} +175.706 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.91893	−1.38555	−0.692776	−	0.721153i	$-0.743612\pi$
$2$	−3.91893	−1.38555	−0.692776	+	0.721153i	$0.743612\pi$
$3$	−3.32066	−0.639061	−0.319530	−	0.947576i	$-0.603525\pi$
$3$	−3.32066	−0.639061	−0.319530	+	0.947576i	$0.603525\pi$
$4$	7.35803	0.919754
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	13.0134	0.885452
$7$	−28.1182	−1.51824	−0.759120	−	0.650951i	$-0.774370\pi$
$7$	−28.1182	−1.51824	−0.759120	+	0.650951i	$0.774370\pi$
$8$	2.51584	0.111186
$9$	−15.9732	−0.591601
$10$	19.5947	0.619638
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−24.4335	−0.587778
$13$	−57.3328	−1.22317	−0.611587	−	0.791177i	$-0.709469\pi$
$13$	−57.3328	−1.22317	−0.611587	+	0.791177i	$0.709469\pi$
$14$	110.193	2.10360
$15$	16.6033	0.285797
$16$	−68.7236	−1.07381
$17$	77.5195	1.10596	0.552978	−	0.833196i	$-0.313492\pi$
$17$	77.5195	1.10596	0.552978	+	0.833196i	$0.313492\pi$
$18$	62.5980	0.819694
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−36.7901	−0.411326
$21$	93.3709	0.970247
$22$	43.1083	0.417760
$23$	25.3511	0.229829	0.114915	−	0.993375i	$-0.463341\pi$
$23$	25.3511	0.229829	0.114915	+	0.993375i	$0.463341\pi$
$24$	−8.35424	−0.0710543
$25$	25.0000	0.200000
$26$	224.683	1.69477
$27$	142.699	1.01713
$28$	−206.895	−1.39641
$29$	66.9421	0.428650	0.214325	−	0.976762i	$-0.431245\pi$
$29$	66.9421	0.428650	0.214325	+	0.976762i	$0.431245\pi$
$30$	−65.0671	−0.395986
$31$	−288.879	−1.67368	−0.836841	−	0.547447i	$-0.815600\pi$
$31$	−288.879	−1.67368	−0.836841	+	0.547447i	$0.815600\pi$
$32$	249.197	1.37663
$33$	36.5272	0.192684
$34$	−303.794	−1.53236
$35$	140.591	0.678978
$36$	−117.532	−0.544128
$37$	287.719	1.27840	0.639199	−	0.769042i	$-0.279266\pi$
$37$	287.719	1.27840	0.639199	+	0.769042i	$0.279266\pi$
$38$	−74.4597	−0.317867
$39$	190.383	0.781682
$40$	−12.5792	−0.0497237
$41$	−140.535	−0.535313	−0.267656	−	0.963514i	$-0.586249\pi$
$41$	−140.535	−0.535313	−0.267656	+	0.963514i	$0.586249\pi$
$42$	−365.914	−1.34433
$43$	−203.180	−0.720573	−0.360286	−	0.932842i	$-0.617321\pi$
$43$	−203.180	−0.720573	−0.360286	+	0.932842i	$0.617321\pi$
$44$	−80.9383	−0.277316
$45$	79.8662	0.264572
$46$	−99.3493	−0.318440
$47$	190.852	0.592313	0.296156	−	0.955139i	$-0.404295\pi$
$47$	190.852	0.592313	0.296156	+	0.955139i	$0.404295\pi$
$48$	228.208	0.686228
$49$	447.633	1.30505
$50$	−97.9733	−0.277110
$51$	−257.416	−0.706772
$52$	−421.857	−1.12502
$53$	585.208	1.51669	0.758345	−	0.651854i	$-0.226008\pi$
$53$	585.208	1.51669	0.758345	+	0.651854i	$0.226008\pi$
$54$	−559.229	−1.40929
$55$	55.0000	0.134840
$56$	−70.7409	−0.168806
$57$	−63.0925	−0.146611
$58$	−262.342	−0.593916
$59$	−369.162	−0.814590	−0.407295	−	0.913297i	$-0.633528\pi$
$59$	−369.162	−0.814590	−0.407295	+	0.913297i	$0.633528\pi$
$60$	122.167	0.262862
$61$	11.4860	0.0241088	0.0120544	−	0.999927i	$-0.496163\pi$
$61$	11.4860	0.0241088	0.0120544	+	0.999927i	$0.496163\pi$
$62$	1132.10	2.31897
$63$	449.139	0.898193
$64$	−426.795	−0.833585
$65$	286.664	0.547020
$66$	−143.148	−0.266974
$67$	−291.136	−0.530864	−0.265432	−	0.964130i	$-0.585515\pi$
$67$	−291.136	−0.530864	−0.265432	+	0.964130i	$0.585515\pi$
$68$	570.391	1.01721
$69$	−84.1823	−0.146875
$70$	−550.967	−0.940759
$71$	921.785	1.54078	0.770392	−	0.637570i	$-0.220060\pi$
$71$	921.785	1.54078	0.770392	+	0.637570i	$0.220060\pi$
$72$	−40.1861	−0.0657775
$73$	1188.93	1.90622	0.953111	−	0.302622i	$-0.0978621\pi$
$73$	1188.93	1.90622	0.953111	+	0.302622i	$0.0978621\pi$
$74$	−1127.55	−1.77129
$75$	−83.0164	−0.127812
$76$	139.803	0.211006
$77$	309.300	0.457767
$78$	−746.096	−1.08306
$79$	−179.464	−0.255586	−0.127793	−	0.991801i	$-0.540789\pi$
$79$	−179.464	−0.255586	−0.127793	+	0.991801i	$0.540789\pi$
$80$	343.618	0.480221
$81$	−42.5781	−0.0584062
$82$	550.746	0.741704
$83$	−184.309	−0.243741	−0.121871	−	0.992546i	$-0.538889\pi$
$83$	−184.309	−0.243741	−0.121871	+	0.992546i	$0.538889\pi$
$84$	687.026	0.892389
$85$	−387.597	−0.494598
$86$	796.248	0.998391
$87$	−222.292	−0.273933
$88$	−27.6742	−0.0335237
$89$	−335.864	−0.400017	−0.200008	−	0.979794i	$-0.564097\pi$
$89$	−335.864	−0.400017	−0.200008	+	0.979794i	$0.564097\pi$
$90$	−312.990	−0.366578
$91$	1612.10	1.85707
$92$	186.534	0.211386
$93$	959.266	1.06958
$94$	−747.938	−0.820680
$95$	−95.0000	−0.102598
$96$	−827.496	−0.879750
$97$	−1155.10	−1.20910	−0.604551	−	0.796566i	$-0.706648\pi$
$97$	−1155.10	−1.20910	−0.604551	+	0.796566i	$0.706648\pi$
$98$	−1754.24	−1.80822
$99$	175.706	0.178375
$100$	183.951	0.183951
$101$	1827.61	1.80053	0.900265	−	0.435342i	$-0.143372\pi$
$101$	1827.61	1.80053	0.900265	+	0.435342i	$0.143372\pi$
$102$	1008.79	0.979270
$103$	902.201	0.863073	0.431537	−	0.902095i	$-0.357972\pi$
$103$	902.201	0.863073	0.431537	+	0.902095i	$0.357972\pi$
$104$	−144.240	−0.135999
$105$	−466.854	−0.433908
$106$	−2293.39	−2.10145
$107$	253.661	0.229181	0.114590	−	0.993413i	$-0.463444\pi$
$107$	253.661	0.229181	0.114590	+	0.993413i	$0.463444\pi$
$108$	1049.99	0.935509
$109$	−1165.42	−1.02410	−0.512050	−	0.858956i	$-0.671114\pi$
$109$	−1165.42	−1.02410	−0.512050	+	0.858956i	$0.671114\pi$
$110$	−215.541	−0.186828
$111$	−955.416	−0.816974
$112$	1932.38	1.63030
$113$	1107.31	0.921829	0.460914	−	0.887445i	$-0.347522\pi$
$113$	1107.31	0.921829	0.460914	+	0.887445i	$0.347522\pi$
$114$	247.255	0.203137
$115$	−126.756	−0.102783
$116$	492.562	0.394252
$117$	915.791	0.723632
$118$	1446.72	1.12866
$119$	−2179.71	−1.67911
$120$	41.7712	0.0317764
$121$	121.000	0.0909091
$122$	−45.0129	−0.0334039
$123$	466.667	0.342097
$124$	−2125.58	−1.53937
$125$	−125.000	−0.0894427
$126$	−1760.14	−1.24449
$127$	1360.23	0.950401	0.475200	−	0.879878i	$-0.342376\pi$
$127$	1360.23	0.950401	0.475200	+	0.879878i	$0.342376\pi$
$128$	−320.991	−0.221655
$129$	674.691	0.460490
$130$	−1123.42	−0.757925
$131$	1370.14	0.913815	0.456908	−	0.889514i	$-0.348957\pi$
$131$	1370.14	0.913815	0.456908	+	0.889514i	$0.348957\pi$
$132$	268.768	0.177222
$133$	−534.246	−0.348308
$134$	1140.94	0.735540
$135$	−713.497	−0.454874
$136$	195.027	0.122966
$137$	−1593.95	−0.994014	−0.497007	−	0.867747i	$-0.665568\pi$
$137$	−1593.95	−0.994014	−0.497007	+	0.867747i	$0.665568\pi$
$138$	329.905	0.203503
$139$	2001.81	1.22152	0.610760	−	0.791816i	$-0.290864\pi$
$139$	2001.81	1.22152	0.610760	+	0.791816i	$0.290864\pi$
$140$	1034.47	0.624492
$141$	−633.756	−0.378524
$142$	−3612.41	−2.13484
$143$	630.661	0.368801
$144$	1097.74	0.635266
$145$	−334.711	−0.191698
$146$	−4659.35	−2.64117
$147$	−1486.44	−0.834008
$148$	2117.04	1.17581
$149$	−1191.38	−0.655047	−0.327523	−	0.944843i	$-0.606214\pi$
$149$	−1191.38	−0.655047	−0.327523	+	0.944843i	$0.606214\pi$
$150$	325.336	0.177090
$151$	−1632.57	−0.879845	−0.439923	−	0.898036i	$-0.644994\pi$
$151$	−1632.57	−0.879845	−0.439923	+	0.898036i	$0.644994\pi$
$152$	47.8010	0.0255077
$153$	−1238.24	−0.654285
$154$	−1212.13	−0.634259
$155$	1444.39	0.748493
$156$	1400.84	0.718955
$157$	1812.99	0.921606	0.460803	−	0.887503i	$-0.347562\pi$
$157$	1812.99	0.921606	0.460803	+	0.887503i	$0.347562\pi$
$158$	703.306	0.354127
$159$	−1943.28	−0.969256
$160$	−1245.98	−0.615647
$161$	−712.828	−0.348936
$162$	166.861	0.0809249
$163$	785.746	0.377573	0.188787	−	0.982018i	$-0.439545\pi$
$163$	785.746	0.377573	0.188787	+	0.982018i	$0.439545\pi$
$164$	−1034.06	−0.492356
$165$	−182.636	−0.0861709
$166$	722.294	0.337716
$167$	−2673.35	−1.23874	−0.619372	−	0.785098i	$-0.712613\pi$
$167$	−2673.35	−1.23874	−0.619372	+	0.785098i	$0.712613\pi$
$168$	234.906	0.107877
$169$	1090.05	0.496155
$170$	1518.97	0.685291
$171$	−303.492	−0.135723
$172$	−1495.00	−0.662750
$173$	−2380.16	−1.04601	−0.523007	−	0.852328i	$-0.675190\pi$
$173$	−2380.16	−1.04601	−0.523007	+	0.852328i	$0.675190\pi$
$174$	871.146	0.379548
$175$	−702.955	−0.303648
$176$	755.960	0.323765
$177$	1225.86	0.520572
$178$	1316.23	0.554244
$179$	−1090.90	−0.455520	−0.227760	−	0.973717i	$-0.573140\pi$
$179$	−1090.90	−0.455520	−0.227760	+	0.973717i	$0.573140\pi$
$180$	587.658	0.243341
$181$	−3398.71	−1.39572	−0.697858	−	0.716237i	$-0.745863\pi$
$181$	−3398.71	−1.39572	−0.697858	+	0.716237i	$0.745863\pi$
$182$	−6317.69	−2.57307
$183$	−38.1411	−0.0154070
$184$	63.7794	0.0255537
$185$	−1438.59	−0.571717
$186$	−3759.30	−1.48196
$187$	−852.714	−0.333458
$188$	1404.30	0.544782
$189$	−4012.45	−1.54425
$190$	372.299	0.142155
$191$	−1008.83	−0.382181	−0.191091	−	0.981572i	$-0.561202\pi$
$191$	−1008.83	−0.382181	−0.191091	+	0.981572i	$0.561202\pi$
$192$	1417.24	0.532711
$193$	809.763	0.302011	0.151005	−	0.988533i	$-0.451749\pi$
$193$	809.763	0.302011	0.151005	+	0.988533i	$0.451749\pi$
$194$	4526.77	1.67527
$195$	−951.913	−0.349579
$196$	3293.70	1.20033
$197$	−2965.40	−1.07247	−0.536233	−	0.844070i	$-0.680153\pi$
$197$	−2965.40	−1.07247	−0.536233	+	0.844070i	$0.680153\pi$
$198$	−688.578	−0.247147
$199$	−1714.62	−0.610783	−0.305392	−	0.952227i	$-0.598787\pi$
$199$	−1714.62	−0.610783	−0.305392	+	0.952227i	$0.598787\pi$
$200$	62.8960	0.0222371
$201$	966.762	0.339254
$202$	−7162.26	−2.49473
$203$	−1882.29	−0.650793
$204$	−1894.07	−0.650057
$205$	702.673	0.239399
$206$	−3535.67	−1.19583
$207$	−404.939	−0.135967
$208$	3940.12	1.31345
$209$	−209.000	−0.0691714
$210$	1829.57	0.601202
$211$	3132.39	1.02200	0.511002	−	0.859580i	$-0.329275\pi$
$211$	3132.39	1.02200	0.511002	+	0.859580i	$0.329275\pi$
$212$	4305.98	1.39498
$213$	−3060.93	−0.984655
$214$	−994.080	−0.317542
$215$	1015.90	0.322250
$216$	359.009	0.113090
$217$	8122.74	2.54105
$218$	4567.19	1.41894
$219$	−3948.04	−1.21819
$220$	404.692	0.124020
$221$	−4444.41	−1.35278
$222$	3744.21	1.13196
$223$	−908.542	−0.272827	−0.136414	−	0.990652i	$-0.543558\pi$
$223$	−908.542	−0.272827	−0.136414	+	0.990652i	$0.543558\pi$
$224$	−7006.96	−2.09005
$225$	−399.331	−0.118320
$226$	−4339.46	−1.27724
$227$	774.695	0.226513	0.113256	−	0.993566i	$-0.463872\pi$
$227$	774.695	0.226513	0.113256	+	0.993566i	$0.463872\pi$
$228$	−464.236	−0.134846
$229$	−5095.67	−1.47044	−0.735221	−	0.677827i	$-0.762922\pi$
$229$	−5095.67	−1.47044	−0.735221	+	0.677827i	$0.762922\pi$
$230$	496.747	0.142411
$231$	−1027.08	−0.292541
$232$	168.416	0.0476596
$233$	−1911.29	−0.537395	−0.268697	−	0.963225i	$-0.586593\pi$
$233$	−1911.29	−0.537395	−0.268697	+	0.963225i	$0.586593\pi$
$234$	−3588.92	−1.00263
$235$	−954.262	−0.264890
$236$	−2716.31	−0.749222
$237$	595.938	0.163335
$238$	8542.13	2.32649
$239$	−4410.63	−1.19372	−0.596861	−	0.802344i	$-0.703586\pi$
$239$	−4410.63	−1.19372	−0.596861	+	0.802344i	$0.703586\pi$
$240$	−1141.04	−0.306890
$241$	250.078	0.0668421	0.0334211	−	0.999441i	$-0.489360\pi$
$241$	250.078	0.0668421	0.0334211	+	0.999441i	$0.489360\pi$
$242$	−474.191	−0.125959
$243$	−3711.50	−0.979805
$244$	84.5144	0.0221741
$245$	−2238.17	−0.583637
$246$	−1828.84	−0.473994
$247$	−1089.32	−0.280615
$248$	−726.772	−0.186089
$249$	612.026	0.155765
$250$	489.867	0.123928
$251$	−3445.73	−0.866503	−0.433252	−	0.901273i	$-0.642634\pi$
$251$	−3445.73	−0.866503	−0.433252	+	0.901273i	$0.642634\pi$
$252$	3304.78	0.826116
$253$	−278.862	−0.0692961
$254$	−5330.65	−1.31683
$255$	1287.08	0.316078
$256$	4672.30	1.14070
$257$	−3970.68	−0.963751	−0.481876	−	0.876240i	$-0.660044\pi$
$257$	−3970.68	−0.963751	−0.481876	+	0.876240i	$0.660044\pi$
$258$	−2644.07	−0.638033
$259$	−8090.14	−1.94091
$260$	2109.28	0.503124
$261$	−1069.28	−0.253590
$262$	−5369.49	−1.26614
$263$	4618.72	1.08290	0.541450	−	0.840733i	$-0.317876\pi$
$263$	4618.72	1.08290	0.541450	+	0.840733i	$0.317876\pi$
$264$	91.8967	0.0214237
$265$	−2926.04	−0.678284
$266$	2093.67	0.482599
$267$	1115.29	0.255635
$268$	−2142.18	−0.488264
$269$	4579.37	1.03795	0.518976	−	0.854789i	$-0.326313\pi$
$269$	4579.37	1.03795	0.518976	+	0.854789i	$0.326313\pi$
$270$	2796.15	0.630252
$271$	196.774	0.0441077	0.0220539	−	0.999757i	$-0.492979\pi$
$271$	196.774	0.0441077	0.0220539	+	0.999757i	$0.492979\pi$
$272$	−5327.42	−1.18758
$273$	−5353.22	−1.18678
$274$	6246.56	1.37726
$275$	−275.000	−0.0603023
$276$	−619.416	−0.135089
$277$	1689.16	0.366396	0.183198	−	0.983076i	$-0.441355\pi$
$277$	1689.16	0.366396	0.183198	+	0.983076i	$0.441355\pi$
$278$	−7844.95	−1.69248
$279$	4614.33	0.990152
$280$	353.705	0.0754925
$281$	1164.31	0.247177	0.123588	−	0.992334i	$-0.460560\pi$
$281$	1164.31	0.247177	0.123588	+	0.992334i	$0.460560\pi$
$282$	2483.65	0.524464
$283$	−6879.58	−1.44505	−0.722523	−	0.691346i	$-0.757018\pi$
$283$	−6879.58	−1.44505	−0.722523	+	0.691346i	$0.757018\pi$
$284$	6782.52	1.41714
$285$	315.462	0.0655662
$286$	−2471.52	−0.510993
$287$	3951.58	0.812733
$288$	−3980.48	−0.814416
$289$	1096.27	0.223137
$290$	1311.71	0.265607
$291$	3835.70	0.772690
$292$	8748.21	1.75325
$293$	7796.11	1.55445	0.777225	−	0.629223i	$-0.216627\pi$
$293$	7796.11	1.55445	0.777225	+	0.629223i	$0.216627\pi$
$294$	5825.24	1.15556
$295$	1845.81	0.364296
$296$	723.855	0.142139
$297$	−1569.69	−0.306676
$298$	4668.95	0.907601
$299$	−1453.45	−0.281121
$300$	−610.837	−0.117556
$301$	5713.05	1.09400
$302$	6397.93	1.21907
$303$	−6068.85	−1.15065
$304$	−1305.75	−0.246348
$305$	−57.4301	−0.0107818
$306$	4852.57	0.906545
$307$	1410.97	0.262307	0.131153	−	0.991362i	$-0.458132\pi$
$307$	1410.97	0.262307	0.131153	+	0.991362i	$0.458132\pi$
$308$	2275.84	0.421032
$309$	−2995.90	−0.551556
$310$	−5660.48	−1.03708
$311$	−2507.83	−0.457254	−0.228627	−	0.973514i	$-0.573424\pi$
$311$	−2507.83	−0.457254	−0.228627	+	0.973514i	$0.573424\pi$
$312$	478.972	0.0869118
$313$	3904.38	0.705075	0.352537	−	0.935798i	$-0.385319\pi$
$313$	3904.38	0.705075	0.352537	+	0.935798i	$0.385319\pi$
$314$	−7104.97	−1.27693
$315$	−2245.69	−0.401684
$316$	−1320.50	−0.235076
$317$	8171.74	1.44786	0.723928	−	0.689875i	$-0.242335\pi$
$317$	8171.74	1.44786	0.723928	+	0.689875i	$0.242335\pi$
$318$	7615.56	1.34295
$319$	−736.363	−0.129243
$320$	2133.98	0.372790
$321$	−842.321	−0.146460
$322$	2793.52	0.483469
$323$	1472.87	0.253724
$324$	−313.291	−0.0537193
$325$	−1433.32	−0.244635
$326$	−3079.29	−0.523147
$327$	3869.95	0.654461
$328$	−353.563	−0.0595190
$329$	−5366.43	−0.899273
$330$	715.739	0.119394
$331$	7359.36	1.22208	0.611038	−	0.791602i	$-0.290752\pi$
$331$	7359.36	1.22208	0.611038	+	0.791602i	$0.290752\pi$
$332$	−1356.15	−0.224182
$333$	−4595.80	−0.756302
$334$	10476.7	1.71634
$335$	1455.68	0.237410
$336$	−6416.79	−1.04186
$337$	−1322.59	−0.213787	−0.106894	−	0.994270i	$-0.534090\pi$
$337$	−1322.59	−0.213787	−0.106894	+	0.994270i	$0.534090\pi$
$338$	−4271.84	−0.687448
$339$	−3676.98	−0.589104
$340$	−2851.95	−0.454908
$341$	3177.66	0.504634
$342$	1189.36	0.188051
$343$	−2942.09	−0.463143
$344$	−511.168	−0.0801173
$345$	420.912	0.0656844
$346$	9327.70	1.44931
$347$	1027.87	0.159017	0.0795083	−	0.996834i	$-0.474665\pi$
$347$	1027.87	0.159017	0.0795083	+	0.996834i	$0.474665\pi$
$348$	−1635.63	−0.251951
$349$	−511.102	−0.0783917	−0.0391958	−	0.999232i	$-0.512480\pi$
$349$	−511.102	−0.0783917	−0.0391958	+	0.999232i	$0.512480\pi$
$350$	2754.83	0.420720
$351$	−8181.36	−1.24413
$352$	−2741.16	−0.415069
$353$	−5115.08	−0.771242	−0.385621	−	0.922657i	$-0.626013\pi$
$353$	−5115.08	−0.771242	−0.385621	+	0.922657i	$0.626013\pi$
$354$	−4804.06	−0.721280
$355$	−4608.92	−0.689060
$356$	−2471.30	−0.367917
$357$	7238.06	1.07305
$358$	4275.18	0.631146
$359$	6241.86	0.917640	0.458820	−	0.888529i	$-0.348272\pi$
$359$	6241.86	0.917640	0.458820	+	0.888529i	$0.348272\pi$
$360$	200.931	0.0294166
$361$	361.000	0.0526316
$362$	13319.3	1.93384
$363$	−401.799	−0.0580964
$364$	11861.8	1.70805
$365$	−5944.67	−0.852488
$366$	149.472	0.0213471
$367$	−2128.98	−0.302811	−0.151406	−	0.988472i	$-0.548380\pi$
$367$	−2128.98	−0.302811	−0.151406	+	0.988472i	$0.548380\pi$
$368$	−1742.22	−0.246792
$369$	2244.79	0.316692
$370$	5637.76	0.792143
$371$	−16455.0	−2.30270
$372$	7058.31	0.983754
$373$	11718.8	1.62674	0.813370	−	0.581746i	$-0.197630\pi$
$373$	11718.8	1.62674	0.813370	+	0.581746i	$0.197630\pi$
$374$	3341.73	0.462023
$375$	415.082	0.0571593
$376$	480.154	0.0658566
$377$	−3837.98	−0.524313
$378$	15724.5	2.13963
$379$	551.770	0.0747824	0.0373912	−	0.999301i	$-0.488095\pi$
$379$	551.770	0.0747824	0.0373912	+	0.999301i	$0.488095\pi$
$380$	−699.013	−0.0943647
$381$	−4516.86	−0.607364
$382$	3953.55	0.529532
$383$	2797.83	0.373270	0.186635	−	0.982429i	$-0.440242\pi$
$383$	2797.83	0.373270	0.186635	+	0.982429i	$0.440242\pi$
$384$	1065.90	0.141651
$385$	−1546.50	−0.204719
$386$	−3173.41	−0.418451
$387$	3245.44	0.426292
$388$	−8499.28	−1.11208
$389$	−14952.0	−1.94883	−0.974415	−	0.224755i	$-0.927842\pi$
$389$	−14952.0	−1.94883	−0.974415	+	0.224755i	$0.927842\pi$
$390$	3730.48	0.484360
$391$	1965.21	0.254181
$392$	1126.17	0.145103
$393$	−4549.77	−0.583983
$394$	11621.2	1.48596
$395$	897.319	0.114301
$396$	1292.85	0.164061
$397$	1991.30	0.251740	0.125870	−	0.992047i	$-0.459828\pi$
$397$	1991.30	0.251740	0.125870	+	0.992047i	$0.459828\pi$
$398$	6719.46	0.846272
$399$	1774.05	0.222590
$400$	−1718.09	−0.214761
$401$	14488.0	1.80422	0.902112	−	0.431502i	$-0.142016\pi$
$401$	14488.0	1.80422	0.902112	+	0.431502i	$0.142016\pi$
$402$	−3788.67	−0.470054
$403$	16562.2	2.04720
$404$	13447.6	1.65604
$405$	212.891	0.0261201
$406$	7376.57	0.901707
$407$	−3164.91	−0.385451
$408$	−647.617	−0.0785829
$409$	−8769.62	−1.06022	−0.530110	−	0.847929i	$-0.677849\pi$
$409$	−8769.62	−1.06022	−0.530110	+	0.847929i	$0.677849\pi$
$410$	−2753.73	−0.331700
$411$	5292.94	0.635235
$412$	6638.42	0.793815
$413$	10380.2	1.23674
$414$	1586.93	0.188390
$415$	921.544	0.109004
$416$	−14287.1	−1.68386
$417$	−6647.32	−0.780625
$418$	819.057	0.0958406
$419$	−9576.93	−1.11662	−0.558310	−	0.829633i	$-0.688550\pi$
$419$	−9576.93	−1.11662	−0.558310	+	0.829633i	$0.688550\pi$
$420$	−3435.13	−0.399088
$421$	15818.9	1.83127	0.915637	−	0.402005i	$-0.131687\pi$
$421$	15818.9	1.83127	0.915637	+	0.402005i	$0.131687\pi$
$422$	−12275.6	−1.41604
$423$	−3048.53	−0.350413
$424$	1472.29	0.168634
$425$	1937.99	0.221191
$426$	11995.6	1.36429
$427$	−322.966	−0.0366029
$428$	1866.44	0.210790
$429$	−2094.21	−0.235686
$430$	−3981.24	−0.446494
$431$	−12782.7	−1.42859	−0.714293	−	0.699846i	$-0.753252\pi$
$431$	−12782.7	−1.42859	−0.714293	+	0.699846i	$0.753252\pi$
$432$	−9806.82	−1.09220
$433$	−14205.7	−1.57663	−0.788315	−	0.615272i	$-0.789046\pi$
$433$	−14205.7	−1.57663	−0.788315	+	0.615272i	$0.789046\pi$
$434$	−31832.5	−3.52076
$435$	1111.46	0.122507
$436$	−8575.18	−0.941919
$437$	481.671	0.0527264
$438$	15472.1	1.68787
$439$	1601.33	0.174094	0.0870471	−	0.996204i	$-0.472257\pi$
$439$	1601.33	0.174094	0.0870471	+	0.996204i	$0.472257\pi$
$440$	138.371	0.0149923
$441$	−7150.15	−0.772071
$442$	17417.3	1.87434
$443$	−8342.54	−0.894732	−0.447366	−	0.894351i	$-0.647638\pi$
$443$	−8342.54	−0.894732	−0.447366	+	0.894351i	$0.647638\pi$
$444$	−7029.98	−0.751414
$445$	1679.32	0.178893
$446$	3560.52	0.378016
$447$	3956.18	0.418615
$448$	12000.7	1.26558
$449$	−752.975	−0.0791428	−0.0395714	−	0.999217i	$-0.512599\pi$
$449$	−752.975	−0.0791428	−0.0395714	+	0.999217i	$0.512599\pi$
$450$	1564.95	0.163939
$451$	1545.88	0.161403
$452$	8147.59	0.847855
$453$	5421.20	0.562274
$454$	−3035.98	−0.313845
$455$	−8060.48	−0.830508
$456$	−158.731	−0.0163010
$457$	9358.93	0.957970	0.478985	−	0.877823i	$-0.341005\pi$
$457$	9358.93	0.957970	0.478985	+	0.877823i	$0.341005\pi$
$458$	19969.6	2.03737
$459$	11062.0	1.12490
$460$	−932.671	−0.0945348
$461$	4841.74	0.489159	0.244579	−	0.969629i	$-0.421350\pi$
$461$	4841.74	0.489159	0.244579	+	0.969629i	$0.421350\pi$
$462$	4025.06	0.405330
$463$	17472.4	1.75381	0.876903	−	0.480668i	$-0.159606\pi$
$463$	17472.4	1.75381	0.876903	+	0.480668i	$0.159606\pi$
$464$	−4600.50	−0.460287
$465$	−4796.33	−0.478332
$466$	7490.23	0.744588
$467$	−5971.48	−0.591707	−0.295854	−	0.955233i	$-0.595604\pi$
$467$	−5971.48	−0.591707	−0.295854	+	0.955233i	$0.595604\pi$
$468$	6738.42	0.665563
$469$	8186.21	0.805979
$470$	3739.69	0.367019
$471$	−6020.31	−0.588962
$472$	−928.753	−0.0905706
$473$	2234.98	0.217261
$474$	−2335.44	−0.226309
$475$	475.000	0.0458831
$476$	−16038.4	−1.54436
$477$	−9347.67	−0.897275
$478$	17284.9	1.65396
$479$	−5237.71	−0.499618	−0.249809	−	0.968295i	$-0.580368\pi$
$479$	−5237.71	−0.499618	−0.249809	+	0.968295i	$0.580368\pi$
$480$	4137.48	0.393436
$481$	−16495.7	−1.56370
$482$	−980.040	−0.0926133
$483$	2367.06	0.222991
$484$	890.322	0.0836140
$485$	5775.52	0.540727
$486$	14545.1	1.35757
$487$	−14711.5	−1.36888	−0.684438	−	0.729071i	$-0.739952\pi$
$487$	−14711.5	−1.36888	−0.684438	+	0.729071i	$0.739952\pi$
$488$	28.8970	0.00268054
$489$	−2609.19	−0.241292
$490$	8771.22	0.808660
$491$	−6658.64	−0.612016	−0.306008	−	0.952029i	$-0.598994\pi$
$491$	−6658.64	−0.612016	−0.306008	+	0.952029i	$0.598994\pi$
$492$	3433.75	0.314645
$493$	5189.32	0.474067
$494$	4268.98	0.388807
$495$	−878.528	−0.0797715
$496$	19852.8	1.79721
$497$	−25918.9	−2.33928
$498$	−2398.49	−0.215821
$499$	10280.8	0.922306	0.461153	−	0.887321i	$-0.347436\pi$
$499$	10280.8	0.922306	0.461153	+	0.887321i	$0.347436\pi$
$500$	−919.754	−0.0822653
$501$	8877.28	0.791632
$502$	13503.6	1.20058
$503$	5869.41	0.520286	0.260143	−	0.965570i	$-0.416230\pi$
$503$	5869.41	0.520286	0.260143	+	0.965570i	$0.416230\pi$
$504$	1129.96	0.0998660
$505$	−9138.03	−0.805222
$506$	1092.84	0.0960134
$507$	−3619.69	−0.317073
$508$	10008.6	0.874134
$509$	4264.71	0.371375	0.185688	−	0.982609i	$-0.440549\pi$
$509$	4264.71	0.371375	0.185688	+	0.982609i	$0.440549\pi$
$510$	−5043.97	−0.437943
$511$	−33430.7	−2.89410
$512$	−15742.5	−1.35884
$513$	2711.29	0.233346
$514$	15560.8	1.33533
$515$	−4511.01	−0.385978
$516$	4964.39	0.423537
$517$	−2099.38	−0.178589
$518$	31704.7	2.68924
$519$	7903.70	0.668467
$520$	721.201	0.0608207
$521$	21955.6	1.84624	0.923122	−	0.384508i	$-0.125629\pi$
$521$	21955.6	1.84624	0.923122	+	0.384508i	$0.125629\pi$
$522$	4190.44	0.351362
$523$	22503.6	1.88148	0.940742	−	0.339124i	$-0.110131\pi$
$523$	22503.6	1.88148	0.940742	+	0.339124i	$0.110131\pi$
$524$	10081.5	0.840485
$525$	2334.27	0.194049
$526$	−18100.5	−1.50041
$527$	−22393.7	−1.85102
$528$	−2510.28	−0.206905
$529$	−11524.3	−0.947179
$530$	11467.0	0.939798
$531$	5896.71	0.481913
$532$	−3931.00	−0.320358
$533$	8057.25	0.654781
$534$	−4370.74	−0.354196
$535$	−1268.30	−0.102493
$536$	−732.451	−0.0590244
$537$	3622.52	0.291105
$538$	−17946.2	−1.43814
$539$	−4923.96	−0.393488
$540$	−5249.93	−0.418372
$541$	1278.17	0.101577	0.0507883	−	0.998709i	$-0.483827\pi$
$541$	1278.17	0.101577	0.0507883	+	0.998709i	$0.483827\pi$
$542$	−771.145	−0.0611135
$543$	11286.0	0.891947
$544$	19317.6	1.52249
$545$	5827.09	0.457991
$546$	20978.9	1.64435
$547$	−20428.7	−1.59684	−0.798418	−	0.602103i	$-0.794330\pi$
$547$	−20428.7	−1.59684	−0.798418	+	0.602103i	$0.794330\pi$
$548$	−11728.3	−0.914248
$549$	−183.469	−0.0142628
$550$	1077.71	0.0835519
$551$	1271.90	0.0983389
$552$	−211.789	−0.0163304
$553$	5046.20	0.388040
$554$	−6619.69	−0.507660
$555$	4777.08	0.365362
$556$	14729.4	1.12350
$557$	11799.0	0.897561	0.448781	−	0.893642i	$-0.351858\pi$
$557$	11799.0	0.897561	0.448781	+	0.893642i	$0.351858\pi$
$558$	−18083.2	−1.37191
$559$	11648.9	0.881386
$560$	−9661.92	−0.729091
$561$	2831.57	0.213100
$562$	−4562.84	−0.342476
$563$	402.180	0.0301064	0.0150532	−	0.999887i	$-0.495208\pi$
$563$	402.180	0.0301064	0.0150532	+	0.999887i	$0.495208\pi$
$564$	−4663.19	−0.348149
$565$	−5536.53	−0.412254
$566$	26960.6	2.00219
$567$	1197.22	0.0886747
$568$	2319.06	0.171313
$569$	25457.3	1.87561	0.937807	−	0.347157i	$-0.112853\pi$
$569$	25457.3	1.87561	0.937807	+	0.347157i	$0.112853\pi$
$570$	−1236.28	−0.0908454
$571$	9738.70	0.713751	0.356876	−	0.934152i	$-0.383842\pi$
$571$	9738.70	0.713751	0.356876	+	0.934152i	$0.383842\pi$
$572$	4640.42	0.339206
$573$	3349.99	0.244237
$574$	−15486.0	−1.12608
$575$	633.778	0.0459659
$576$	6817.30	0.493150
$577$	2334.56	0.168438	0.0842192	−	0.996447i	$-0.473160\pi$
$577$	2334.56	0.168438	0.0842192	+	0.996447i	$0.473160\pi$
$578$	−4296.21	−0.309168
$579$	−2688.95	−0.193003
$580$	−2462.81	−0.176315
$581$	5182.43	0.370058
$582$	−15031.9	−1.07060
$583$	−6437.29	−0.457299
$584$	2991.17	0.211944
$585$	−4578.95	−0.323618
$586$	−30552.4	−2.15377
$587$	−2351.89	−0.165371	−0.0826856	−	0.996576i	$-0.526350\pi$
$587$	−2351.89	−0.165371	−0.0826856	+	0.996576i	$0.526350\pi$
$588$	−10937.2	−0.767082
$589$	−5488.69	−0.383969
$590$	−7233.61	−0.504751
$591$	9847.07	0.685371
$592$	−19773.1	−1.37275
$593$	2161.67	0.149695	0.0748476	−	0.997195i	$-0.476153\pi$
$593$	2161.67	0.149695	0.0748476	+	0.997195i	$0.476153\pi$
$594$	6151.52	0.424916
$595$	10898.5	0.750919
$596$	−8766.24	−0.602482
$597$	5693.65	0.390328
$598$	5695.98	0.389508
$599$	596.820	0.0407102	0.0203551	−	0.999793i	$-0.493520\pi$
$599$	596.820	0.0407102	0.0203551	+	0.999793i	$0.493520\pi$
$600$	−208.856	−0.0142109
$601$	22746.4	1.54384	0.771918	−	0.635722i	$-0.219298\pi$
$601$	22746.4	1.54384	0.771918	+	0.635722i	$0.219298\pi$
$602$	−22389.1	−1.51580
$603$	4650.38	0.314060
$604$	−12012.5	−0.809241
$605$	−605.000	−0.0406558
$606$	23783.4	1.59428
$607$	−669.883	−0.0447936	−0.0223968	−	0.999749i	$-0.507130\pi$
$607$	−669.883	−0.0447936	−0.0223968	+	0.999749i	$0.507130\pi$
$608$	4734.73	0.315820
$609$	6250.44	0.415896
$610$	225.065	0.0149387
$611$	−10942.1	−0.724502
$612$	−9110.99	−0.601781
$613$	11588.7	0.763562	0.381781	−	0.924253i	$-0.375311\pi$
$613$	11588.7	0.763562	0.381781	+	0.924253i	$0.375311\pi$
$614$	−5529.49	−0.363440
$615$	−2333.34	−0.152991
$616$	778.150	0.0508970
$617$	−3803.57	−0.248178	−0.124089	−	0.992271i	$-0.539601\pi$
$617$	−3803.57	−0.248178	−0.124089	+	0.992271i	$0.539601\pi$
$618$	11740.7	0.764210
$619$	−16334.9	−1.06067	−0.530337	−	0.847787i	$-0.677934\pi$
$619$	−16334.9	−1.06067	−0.530337	+	0.847787i	$0.677934\pi$
$620$	10627.9	0.688429
$621$	3617.59	0.233766
$622$	9828.02	0.633550
$623$	9443.89	0.607321
$624$	−13083.8	−0.839376
$625$	625.000	0.0400000
$626$	−15301.0	−0.976917
$627$	694.017	0.0442048
$628$	13340.0	0.847650
$629$	22303.8	1.41385
$630$	8800.72	0.556554
$631$	−10215.1	−0.644466	−0.322233	−	0.946660i	$-0.604433\pi$
$631$	−10215.1	−0.644466	−0.322233	+	0.946660i	$0.604433\pi$
$632$	−451.502	−0.0284174
$633$	−10401.6	−0.653122
$634$	−32024.5	−2.00608
$635$	−6801.15	−0.425032
$636$	−14298.7	−0.891477
$637$	−25664.1	−1.59631
$638$	2885.76	0.179072
$639$	−14723.9	−0.911531
$640$	1604.95	0.0991271
$641$	5783.20	0.356353	0.178177	−	0.983999i	$-0.442980\pi$
$641$	5783.20	0.356353	0.178177	+	0.983999i	$0.442980\pi$
$642$	3301.00	0.202928
$643$	−22633.8	−1.38816	−0.694081	−	0.719897i	$-0.744189\pi$
$643$	−22633.8	−1.38816	−0.694081	+	0.719897i	$0.744189\pi$
$644$	−5245.01	−0.320935
$645$	−3373.45	−0.205937
$646$	−5772.08	−0.351547
$647$	−17532.2	−1.06532	−0.532660	−	0.846330i	$-0.678807\pi$
$647$	−17532.2	−1.06532	−0.532660	+	0.846330i	$0.678807\pi$
$648$	−107.120	−0.00649393
$649$	4060.78	0.245608
$650$	5617.09	0.338954
$651$	−26972.8	−1.62388
$652$	5781.55	0.347274
$653$	−13881.6	−0.831898	−0.415949	−	0.909388i	$-0.636551\pi$
$653$	−13881.6	−0.831898	−0.415949	+	0.909388i	$0.636551\pi$
$654$	−15166.1	−0.906790
$655$	−6850.70	−0.408670
$656$	9658.05	0.574823
$657$	−18991.1	−1.12772
$658$	21030.7	1.24599
$659$	−10563.2	−0.624407	−0.312203	−	0.950015i	$-0.601067\pi$
$659$	−10563.2	−0.624407	−0.312203	+	0.950015i	$0.601067\pi$
$660$	−1343.84	−0.0792560
$661$	−28208.8	−1.65990	−0.829952	−	0.557835i	$-0.811632\pi$
$661$	−28208.8	−1.65990	−0.829952	+	0.557835i	$0.811632\pi$
$662$	−28840.8	−1.69325
$663$	14758.4	0.864506
$664$	−463.692	−0.0271005
$665$	2671.23	0.155768
$666$	18010.6	1.04790
$667$	1697.06	0.0985162
$668$	−19670.6	−1.13934
$669$	3016.96	0.174353
$670$	−5704.71	−0.328943
$671$	−126.346	−0.00726906
$672$	23267.7	1.33567
$673$	−869.218	−0.0497859	−0.0248929	−	0.999690i	$-0.507924\pi$
$673$	−869.218	−0.0497859	−0.0248929	+	0.999690i	$0.507924\pi$
$674$	5183.15	0.296213
$675$	3567.48	0.203426
$676$	8020.63	0.456340
$677$	2529.80	0.143616	0.0718082	−	0.997418i	$-0.477123\pi$
$677$	2529.80	0.143616	0.0718082	+	0.997418i	$0.477123\pi$
$678$	14409.9	0.816235
$679$	32479.4	1.83571
$680$	−975.133	−0.0549921
$681$	−2572.50	−0.144755
$682$	−12453.0	−0.699196
$683$	−26204.9	−1.46808	−0.734042	−	0.679104i	$-0.762368\pi$
$683$	−26204.9	−1.46808	−0.734042	+	0.679104i	$0.762368\pi$
$684$	−2233.10	−0.124831
$685$	7969.73	0.444537
$686$	11529.9	0.641708
$687$	16921.0	0.939702
$688$	13963.3	0.773756
$689$	−33551.6	−1.85517
$690$	−1649.52	−0.0910092
$691$	−24451.0	−1.34610	−0.673052	−	0.739595i	$-0.735017\pi$
$691$	−24451.0	−1.34610	−0.673052	+	0.739595i	$0.735017\pi$
$692$	−17513.3	−0.962075
$693$	−4940.53	−0.270815
$694$	−4028.14	−0.220326
$695$	−10009.0	−0.546280
$696$	−559.251	−0.0304574
$697$	−10894.2	−0.592032
$698$	2002.98	0.108616
$699$	6346.75	0.343428
$700$	−5172.36	−0.279281
$701$	27478.4	1.48052	0.740260	−	0.672320i	$-0.234702\pi$
$701$	27478.4	1.48052	0.740260	+	0.672320i	$0.234702\pi$
$702$	32062.2	1.72380
$703$	5466.66	0.293285
$704$	4694.75	0.251335
$705$	3168.78	0.169281
$706$	20045.7	1.06860
$707$	−51389.0	−2.73364
$708$	9019.92	0.478798
$709$	−20299.2	−1.07525	−0.537625	−	0.843184i	$-0.680678\pi$
$709$	−20299.2	−1.07525	−0.537625	+	0.843184i	$0.680678\pi$
$710$	18062.1	0.954728
$711$	2866.62	0.151205
$712$	−844.980	−0.0444761
$713$	−7323.39	−0.384661
$714$	−28365.5	−1.48677
$715$	−3153.30	−0.164933
$716$	−8026.90	−0.418966
$717$	14646.2	0.762861
$718$	−24461.4	−1.27144
$719$	36000.9	1.86732	0.933662	−	0.358155i	$-0.116594\pi$
$719$	36000.9	1.86732	0.933662	+	0.358155i	$0.116594\pi$
$720$	−5488.70	−0.284099
$721$	−25368.3	−1.31035
$722$	−1414.73	−0.0729238
$723$	−830.424	−0.0427162
$724$	−25007.8	−1.28371
$725$	1673.55	0.0857299
$726$	1574.62	0.0804956
$727$	14499.3	0.739683	0.369842	−	0.929095i	$-0.379412\pi$
$727$	14499.3	0.739683	0.369842	+	0.929095i	$0.379412\pi$
$728$	4055.78	0.206479
$729$	13474.2	0.684561
$730$	23296.7	1.18117
$731$	−15750.4	−0.796921
$732$	−280.643	−0.0141706
$733$	12240.5	0.616801	0.308400	−	0.951257i	$-0.400206\pi$
$733$	12240.5	0.616801	0.308400	+	0.951257i	$0.400206\pi$
$734$	8343.32	0.419561
$735$	7432.18	0.372980
$736$	6317.41	0.316390
$737$	3202.49	0.160062
$738$	−8797.19	−0.438793
$739$	−27133.2	−1.35062	−0.675311	−	0.737533i	$-0.735991\pi$
$739$	−27133.2	−1.35062	−0.675311	+	0.737533i	$0.735991\pi$
$740$	−10585.2	−0.525839
$741$	3617.27	0.179330
$742$	64486.0	3.19051
$743$	34391.1	1.69810	0.849049	−	0.528314i	$-0.177176\pi$
$743$	34391.1	1.69810	0.849049	+	0.528314i	$0.177176\pi$
$744$	2413.36	0.118922
$745$	5956.92	0.292946
$746$	−45925.0	−2.25393
$747$	2944.01	0.144198
$748$	−6274.30	−0.306699
$749$	−7132.49	−0.347951
$750$	−1626.68	−0.0791972
$751$	−14379.3	−0.698680	−0.349340	−	0.936996i	$-0.613594\pi$
$751$	−14379.3	−0.698680	−0.349340	+	0.936996i	$0.613594\pi$
$752$	−13116.1	−0.636029
$753$	11442.1	0.553748
$754$	15040.8	0.726463
$755$	8162.85	0.393479
$756$	−29523.7	−1.42033
$757$	30444.4	1.46172	0.730859	−	0.682528i	$-0.239120\pi$
$757$	30444.4	1.46172	0.730859	+	0.682528i	$0.239120\pi$
$758$	−2162.35	−0.103615
$759$	926.006	0.0442844
$760$	−239.005	−0.0114074
$761$	−5965.31	−0.284156	−0.142078	−	0.989855i	$-0.545378\pi$
$761$	−5965.31	−0.284156	−0.142078	+	0.989855i	$0.545378\pi$
$762$	17701.3	0.841534
$763$	32769.5	1.55483
$764$	−7423.03	−0.351513
$765$	6191.19	0.292605
$766$	−10964.5	−0.517185
$767$	21165.1	0.996385
$768$	−15515.1	−0.728976
$769$	14686.6	0.688701	0.344351	−	0.938841i	$-0.388099\pi$
$769$	14686.6	0.688701	0.344351	+	0.938841i	$0.388099\pi$
$770$	6060.63	0.283649
$771$	13185.3	0.615895
$772$	5958.26	0.277775
$773$	−15133.4	−0.704155	−0.352077	−	0.935971i	$-0.614525\pi$
$773$	−15133.4	−0.704155	−0.352077	+	0.935971i	$0.614525\pi$
$774$	−12718.7	−0.590650
$775$	−7221.96	−0.334736
$776$	−2906.06	−0.134435
$777$	26864.6	1.24036
$778$	58595.8	2.70021
$779$	−2670.16	−0.122809
$780$	−7004.20	−0.321527
$781$	−10139.6	−0.464564
$782$	−7701.51	−0.352181
$783$	9552.60	0.435992
$784$	−30763.0	−1.40137
$785$	−9064.93	−0.412155
$786$	17830.2	0.809139
$787$	32972.8	1.49346	0.746730	−	0.665127i	$-0.231623\pi$
$787$	32972.8	1.49346	0.746730	+	0.665127i	$0.231623\pi$
$788$	−21819.5	−0.986404
$789$	−15337.2	−0.692039
$790$	−3516.53	−0.158370
$791$	−31135.5	−1.39956
$792$	442.047	0.0198327
$793$	−658.526	−0.0294892
$794$	−7803.79	−0.348798
$795$	9716.38	0.433465
$796$	−12616.2	−0.561770
$797$	18614.0	0.827279	0.413640	−	0.910441i	$-0.364257\pi$
$797$	18614.0	0.827279	0.413640	+	0.910441i	$0.364257\pi$
$798$	−6952.37	−0.308410
$799$	14794.8	0.655071
$800$	6229.91	0.275326
$801$	5364.83	0.236651
$802$	−56777.3	−2.49985
$803$	−13078.3	−0.574747
$804$	7113.46	0.312030
$805$	3564.14	0.156049
$806$	−64906.2	−2.83651
$807$	−15206.5	−0.663314
$808$	4597.97	0.200193
$809$	−14371.3	−0.624560	−0.312280	−	0.949990i	$-0.601093\pi$
$809$	−14371.3	−0.624560	−0.312280	+	0.949990i	$0.601093\pi$
$810$	−834.304	−0.0361907
$811$	−17584.7	−0.761385	−0.380693	−	0.924702i	$-0.624314\pi$
$811$	−17584.7	−0.761385	−0.380693	+	0.924702i	$0.624314\pi$
$812$	−13850.0	−0.598569
$813$	−653.420	−0.0281875
$814$	12403.1	0.534063
$815$	−3928.73	−0.168856
$816$	17690.5	0.758937
$817$	−3860.42	−0.165311
$818$	34367.6	1.46899
$819$	−25750.4	−1.09865
$820$	5170.29	0.220188
$821$	7026.99	0.298713	0.149357	−	0.988783i	$-0.452280\pi$
$821$	7026.99	0.298713	0.149357	+	0.988783i	$0.452280\pi$
$822$	−20742.7	−0.880151
$823$	−37636.6	−1.59408	−0.797040	−	0.603926i	$-0.793602\pi$
$823$	−37636.6	−1.59408	−0.797040	+	0.603926i	$0.793602\pi$
$824$	2269.79	0.0959612
$825$	913.181	0.0385368
$826$	−40679.2	−1.71357
$827$	−4436.77	−0.186556	−0.0932778	−	0.995640i	$-0.529734\pi$
$827$	−4436.77	−0.186556	−0.0932778	+	0.995640i	$0.529734\pi$
$828$	−2979.56	−0.125056
$829$	−8504.57	−0.356304	−0.178152	−	0.984003i	$-0.557012\pi$
$829$	−8504.57	−0.356304	−0.178152	+	0.984003i	$0.557012\pi$
$830$	−3611.47	−0.151031
$831$	−5609.11	−0.234149
$832$	24469.4	1.01962
$833$	34700.3	1.44333
$834$	26050.4	1.08160
$835$	13366.8	0.553983
$836$	−1537.83	−0.0636207
$837$	−41222.8	−1.70235
$838$	37531.3	1.54713
$839$	−18663.1	−0.767965	−0.383983	−	0.923340i	$-0.625448\pi$
$839$	−18663.1	−0.767965	−0.383983	+	0.923340i	$0.625448\pi$
$840$	−1174.53	−0.0482443
$841$	−19907.8	−0.816260
$842$	−61993.3	−2.53733
$843$	−3866.26	−0.157961
$844$	23048.2	0.939991
$845$	−5450.26	−0.221887
$846$	11947.0	0.485515
$847$	−3402.30	−0.138022
$848$	−40217.6	−1.62863
$849$	22844.7	0.923473
$850$	−7594.84	−0.306472
$851$	7294.00	0.293813
$852$	−22522.4	−0.905640
$853$	24903.2	0.999611	0.499806	−	0.866138i	$-0.333405\pi$
$853$	24903.2	0.999611	0.499806	+	0.866138i	$0.333405\pi$
$854$	1265.68	0.0507152
$855$	1517.46	0.0606970
$856$	638.170	0.0254816
$857$	31032.0	1.23691	0.618456	−	0.785820i	$-0.287759\pi$
$857$	31032.0	1.23691	0.618456	+	0.785820i	$0.287759\pi$
$858$	8207.06	0.326555
$859$	35115.5	1.39479	0.697396	−	0.716686i	$-0.254342\pi$
$859$	35115.5	1.39479	0.697396	+	0.716686i	$0.254342\pi$
$860$	7475.02	0.296391
$861$	−13121.8	−0.519386
$862$	50094.5	1.97938
$863$	25464.6	1.00443	0.502217	−	0.864742i	$-0.332518\pi$
$863$	25464.6	1.00443	0.502217	+	0.864742i	$0.332518\pi$
$864$	35560.2	1.40021
$865$	11900.8	0.467792
$866$	55671.1	2.18450
$867$	−3640.34	−0.142598
$868$	59767.4	2.33714
$869$	1974.10	0.0770619
$870$	−4355.73	−0.169739
$871$	16691.6	0.649339
$872$	−2932.01	−0.113865
$873$	18450.7	0.715307
$874$	−1887.64	−0.0730552
$875$	3514.77	0.135796
$876$	−29049.8	−1.12044
$877$	−22982.2	−0.884894	−0.442447	−	0.896795i	$-0.645890\pi$
$877$	−22982.2	−0.884894	−0.442447	+	0.896795i	$0.645890\pi$
$878$	−6275.50	−0.241216
$879$	−25888.2	−0.993387
$880$	−3779.80	−0.144792
$881$	−9389.36	−0.359064	−0.179532	−	0.983752i	$-0.557458\pi$
$881$	−9389.36	−0.359064	−0.179532	+	0.983752i	$0.557458\pi$
$882$	28020.9	1.06974
$883$	−27191.0	−1.03630	−0.518148	−	0.855291i	$-0.673378\pi$
$883$	−27191.0	−1.03630	−0.518148	+	0.855291i	$0.673378\pi$
$884$	−32702.1	−1.24422
$885$	−6129.30	−0.232807
$886$	32693.9	1.23970
$887$	−36800.3	−1.39305	−0.696523	−	0.717534i	$-0.745271\pi$
$887$	−36800.3	−1.39305	−0.696523	+	0.717534i	$0.745271\pi$
$888$	−2403.67	−0.0908356
$889$	−38247.2	−1.44294
$890$	−6581.14	−0.247865
$891$	468.360	0.0176101
$892$	−6685.08	−0.250934
$893$	3626.20	0.135886
$894$	−15504.0	−0.580012
$895$	5454.52	0.203715
$896$	9025.68	0.336526
$897$	4826.41	0.179653
$898$	2950.86	0.109656
$899$	−19338.1	−0.717423
$900$	−2938.29	−0.108826
$901$	45365.0	1.67739
$902$	−6058.20	−0.223632
$903$	−18971.1	−0.699134
$904$	2785.81	0.102494
$905$	16993.6	0.624183
$906$	−21245.3	−0.779060
$907$	−2340.27	−0.0856753	−0.0428376	−	0.999082i	$-0.513640\pi$
$907$	−2340.27	−0.0856753	−0.0428376	+	0.999082i	$0.513640\pi$
$908$	5700.23	0.208336
$909$	−29192.8	−1.06520
$910$	31588.5	1.15071
$911$	14580.2	0.530256	0.265128	−	0.964213i	$-0.414586\pi$
$911$	14580.2	0.530256	0.265128	+	0.964213i	$0.414586\pi$
$912$	4335.94	0.157431
$913$	2027.40	0.0734908
$914$	−36677.0	−1.32732
$915$	190.706	0.00689020
$916$	−37494.1	−1.35244
$917$	−38525.9	−1.38739
$918$	−43351.2	−1.55861
$919$	−13094.4	−0.470016	−0.235008	−	0.971993i	$-0.575512\pi$
$919$	−13094.4	−0.470016	−0.235008	+	0.971993i	$0.575512\pi$
$920$	−318.897	−0.0114280
$921$	−4685.34	−0.167630
$922$	−18974.4	−0.677755
$923$	−52848.5	−1.88465
$924$	−7557.28	−0.269065
$925$	7192.97	0.255680
$926$	−68473.2	−2.42999
$927$	−14411.1	−0.510595
$928$	16681.7	0.590092
$929$	−38888.5	−1.37340	−0.686701	−	0.726940i	$-0.740942\pi$
$929$	−38888.5	−1.37340	−0.686701	+	0.726940i	$0.740942\pi$
$930$	18796.5	0.662754
$931$	8505.03	0.299400
$932$	−14063.4	−0.494271
$933$	8327.65	0.292213
$934$	23401.8	0.819841
$935$	4263.57	0.149127
$936$	2303.98	0.0804573
$937$	10205.2	0.355806	0.177903	−	0.984048i	$-0.443069\pi$
$937$	10205.2	0.355806	0.177903	+	0.984048i	$0.443069\pi$
$938$	−32081.2	−1.11673
$939$	−12965.1	−0.450585
$940$	−7021.49	−0.243634
$941$	−11322.2	−0.392234	−0.196117	−	0.980580i	$-0.562833\pi$
$941$	−11322.2	−0.392234	−0.196117	+	0.980580i	$0.562833\pi$
$942$	23593.2	0.816037
$943$	−3562.71	−0.123031
$944$	25370.2	0.874712
$945$	20062.2	0.690608
$946$	−8758.73	−0.301026
$947$	−52171.9	−1.79024	−0.895122	−	0.445822i	$-0.852911\pi$
$947$	−52171.9	−1.79024	−0.895122	+	0.445822i	$0.852911\pi$
$948$	4384.93	0.150228
$949$	−68164.9	−2.33164
$950$	−1861.49	−0.0635735
$951$	−27135.5	−0.925268
$952$	−5483.80	−0.186692
$953$	47269.8	1.60673	0.803367	−	0.595484i	$-0.203040\pi$
$953$	47269.8	1.60673	0.803367	+	0.595484i	$0.203040\pi$
$954$	36632.9	1.24322
$955$	5044.17	0.170917
$956$	−32453.5	−1.09793
$957$	2445.21	0.0825939
$958$	20526.2	0.692247
$959$	44818.9	1.50915
$960$	−7086.20	−0.238236
$961$	53659.8	1.80121
$962$	64645.7	2.16659
$963$	−4051.79	−0.135584
$964$	1840.08	0.0614783
$965$	−4048.82	−0.135063
$966$	−9276.33	−0.308966
$967$	−9102.45	−0.302704	−0.151352	−	0.988480i	$-0.548363\pi$
$967$	−9102.45	−0.302704	−0.151352	+	0.988480i	$0.548363\pi$
$968$	304.417	0.0101078
$969$	−4890.90	−0.162145
$970$	−22633.9	−0.749205
$971$	−35620.1	−1.17724	−0.588622	−	0.808409i	$-0.700329\pi$
$971$	−35620.1	−1.17724	−0.588622	+	0.808409i	$0.700329\pi$
$972$	−27309.3	−0.901179
$973$	−56287.2	−1.85456
$974$	57653.5	1.89665
$975$	4759.56	0.156336
$976$	−789.361	−0.0258881
$977$	−49668.6	−1.62645	−0.813224	−	0.581950i	$-0.802290\pi$
$977$	−49668.6	−1.62645	−0.813224	+	0.581950i	$0.802290\pi$
$978$	10225.3	0.334323
$979$	3694.50	0.120610
$980$	−16468.5	−0.536802
$981$	18615.5	0.605859
$982$	26094.7	0.847980
$983$	−17171.1	−0.557145	−0.278573	−	0.960415i	$-0.589861\pi$
$983$	−17171.1	−0.557145	−0.278573	+	0.960415i	$0.589861\pi$
$984$	1174.06	0.0380363
$985$	14827.0	0.479621
$986$	−20336.6	−0.656845
$987$	17820.1	0.574690
$988$	−8015.27	−0.258097
$989$	−5150.84	−0.165609
$990$	3442.89	0.110528
$991$	43853.7	1.40571	0.702855	−	0.711333i	$-0.251908\pi$
$991$	43853.7	1.40571	0.702855	+	0.711333i	$0.251908\pi$
$992$	−71987.5	−2.30404
$993$	−24437.9	−0.780980
$994$	101575.	3.24120
$995$	8573.08	0.273151
$996$	4503.31	0.143266
$997$	37818.2	1.20132	0.600659	−	0.799505i	$-0.294905\pi$
$997$	37818.2	1.20132	0.600659	+	0.799505i	$0.294905\pi$
$998$	−40289.6	−1.27790
$999$	41057.3	1.30030

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.c.1.4	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.c.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-3.91893 q^{2} -3.32066 q^{3} +7.35803 q^{4} -5.00000 q^{5} +13.0134 q^{6} -28.1182 q^{7} +2.51584 q^{8} -15.9732 q^{9} +O(q^{10})\)	\(q-3.91893 q^{2} -3.32066 q^{3} +7.35803 q^{4} -5.00000 q^{5} +13.0134 q^{6} -28.1182 q^{7} +2.51584 q^{8} -15.9732 q^{9} +19.5947 q^{10} -11.0000 q^{11} -24.4335 q^{12} -57.3328 q^{13} +110.193 q^{14} +16.6033 q^{15} -68.7236 q^{16} +77.5195 q^{17} +62.5980 q^{18} +19.0000 q^{19} -36.7901 q^{20} +93.3709 q^{21} +43.1083 q^{22} +25.3511 q^{23} -8.35424 q^{24} +25.0000 q^{25} +224.683 q^{26} +142.699 q^{27} -206.895 q^{28} +66.9421 q^{29} -65.0671 q^{30} -288.879 q^{31} +249.197 q^{32} +36.5272 q^{33} -303.794 q^{34} +140.591 q^{35} -117.532 q^{36} +287.719 q^{37} -74.4597 q^{38} +190.383 q^{39} -12.5792 q^{40} -140.535 q^{41} -365.914 q^{42} -203.180 q^{43} -80.9383 q^{44} +79.8662 q^{45} -99.3493 q^{46} +190.852 q^{47} +228.208 q^{48} +447.633 q^{49} -97.9733 q^{50} -257.416 q^{51} -421.857 q^{52} +585.208 q^{53} -559.229 q^{54} +55.0000 q^{55} -70.7409 q^{56} -63.0925 q^{57} -262.342 q^{58} -369.162 q^{59} +122.167 q^{60} +11.4860 q^{61} +1132.10 q^{62} +449.139 q^{63} -426.795 q^{64} +286.664 q^{65} -143.148 q^{66} -291.136 q^{67} +570.391 q^{68} -84.1823 q^{69} -550.967 q^{70} +921.785 q^{71} -40.1861 q^{72} +1188.93 q^{73} -1127.55 q^{74} -83.0164 q^{75} +139.803 q^{76} +309.300 q^{77} -746.096 q^{78} -179.464 q^{79} +343.618 q^{80} -42.5781 q^{81} +550.746 q^{82} -184.309 q^{83} +687.026 q^{84} -387.597 q^{85} +796.248 q^{86} -222.292 q^{87} -27.6742 q^{88} -335.864 q^{89} -312.990 q^{90} +1612.10 q^{91} +186.534 q^{92} +959.266 q^{93} -747.938 q^{94} -95.0000 q^{95} -827.496 q^{96} -1155.10 q^{97} -1754.24 q^{98} +175.706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) 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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