LMFDB - Embedded newform 1045.4.a.b.1.2

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} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + \cdots + 17756160 $ x^20 - 8x^19 - 82x^18 + 700x^17 + 2826x^16 - 25467x^15 - 53768x^14 + 498499x^13 + 623613x^12 - 5673747x^11 - 4539454x^10 + 37893109x^9 + 19879768x^8 - 143049638x^7 - 45064360x^6 + 280461480x^5 + 43097920x^4 - 240447168x^3 - 36068096x^2 + 74703872*x + 17756160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.19424$ 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)

$f(q)$	$=$	$q-5.19424 q^{2} +8.38824 q^{3} +18.9801 q^{4} +5.00000 q^{5} -43.5706 q^{6} -4.76095 q^{7} -57.0335 q^{8} +43.3626 q^{9} +O(q^{10})$	\(q-5.19424 q^{2} +8.38824 q^{3} +18.9801 q^{4} +5.00000 q^{5} -43.5706 q^{6} -4.76095 q^{7} -57.0335 q^{8} +43.3626 q^{9} -25.9712 q^{10} +11.0000 q^{11} +159.210 q^{12} -70.7652 q^{13} +24.7295 q^{14} +41.9412 q^{15} +144.405 q^{16} -99.7845 q^{17} -225.236 q^{18} +19.0000 q^{19} +94.9007 q^{20} -39.9360 q^{21} -57.1367 q^{22} -0.0718409 q^{23} -478.411 q^{24} +25.0000 q^{25} +367.572 q^{26} +137.254 q^{27} -90.3635 q^{28} -102.894 q^{29} -217.853 q^{30} +210.813 q^{31} -293.805 q^{32} +92.2707 q^{33} +518.305 q^{34} -23.8047 q^{35} +823.029 q^{36} +10.5127 q^{37} -98.6906 q^{38} -593.596 q^{39} -285.168 q^{40} +249.884 q^{41} +207.437 q^{42} -105.462 q^{43} +208.782 q^{44} +216.813 q^{45} +0.373159 q^{46} -167.931 q^{47} +1211.30 q^{48} -320.333 q^{49} -129.856 q^{50} -837.017 q^{51} -1343.13 q^{52} -442.160 q^{53} -712.928 q^{54} +55.0000 q^{55} +271.534 q^{56} +159.377 q^{57} +534.456 q^{58} -339.288 q^{59} +796.050 q^{60} +116.908 q^{61} -1095.02 q^{62} -206.447 q^{63} +370.855 q^{64} -353.826 q^{65} -479.276 q^{66} -999.959 q^{67} -1893.92 q^{68} -0.602619 q^{69} +123.648 q^{70} -465.452 q^{71} -2473.12 q^{72} -207.074 q^{73} -54.6054 q^{74} +209.706 q^{75} +360.623 q^{76} -52.3704 q^{77} +3083.28 q^{78} +597.983 q^{79} +722.024 q^{80} -19.4747 q^{81} -1297.96 q^{82} +614.320 q^{83} -757.991 q^{84} -498.923 q^{85} +547.794 q^{86} -863.099 q^{87} -627.369 q^{88} -405.032 q^{89} -1126.18 q^{90} +336.909 q^{91} -1.36355 q^{92} +1768.35 q^{93} +872.276 q^{94} +95.0000 q^{95} -2464.51 q^{96} -890.374 q^{97} +1663.89 q^{98} +476.989 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) $ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * 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 $

$2$	−5.19424	−1.83644	−0.918221	−	0.396069i	$-0.870374\pi$
$2$	−5.19424	−1.83644	−0.918221	+	0.396069i	$0.870374\pi$
$3$	8.38824	1.61432	0.807159	−	0.590334i	$-0.201004\pi$
$3$	8.38824	1.61432	0.807159	+	0.590334i	$0.201004\pi$
$4$	18.9801	2.37252
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−43.5706	−2.96460
$7$	−4.76095	−0.257067	−0.128534	−	0.991705i	$-0.541027\pi$
$7$	−4.76095	−0.257067	−0.128534	+	0.991705i	$0.541027\pi$
$8$	−57.0335	−2.52055
$9$	43.3626	1.60602
$10$	−25.9712	−0.821282
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	159.210	3.83000
$13$	−70.7652	−1.50975	−0.754874	−	0.655869i	$-0.772302\pi$
$13$	−70.7652	−1.50975	−0.754874	+	0.655869i	$0.772302\pi$
$14$	24.7295	0.472089
$15$	41.9412	0.721945
$16$	144.405	2.25632
$17$	−99.7845	−1.42361	−0.711803	−	0.702379i	$-0.752121\pi$
$17$	−99.7845	−1.42361	−0.711803	+	0.702379i	$0.752121\pi$
$18$	−225.236	−2.94937
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	94.9007	1.06102
$21$	−39.9360	−0.414988
$22$	−57.1367	−0.553708
$23$	−0.0718409	−0.000651299 0	−0.000325649	−	1.00000i	$-0.500104\pi$
$23$	−0.0718409	−0.000651299 0	−0.000325649		1.00000i	$0.500104\pi$
$24$	−478.411	−4.06897
$25$	25.0000	0.200000
$26$	367.572	2.77257
$27$	137.254	0.978313
$28$	−90.3635	−0.609896
$29$	−102.894	−0.658859	−0.329430	−	0.944180i	$-0.606856\pi$
$29$	−102.894	−0.658859	−0.329430	+	0.944180i	$0.606856\pi$
$30$	−217.853	−1.32581
$31$	210.813	1.22139	0.610697	−	0.791864i	$-0.290889\pi$
$31$	210.813	1.22139	0.610697	+	0.791864i	$0.290889\pi$
$32$	−293.805	−1.62306
$33$	92.2707	0.486735
$34$	518.305	2.61437
$35$	−23.8047	−0.114964
$36$	823.029	3.81032
$37$	10.5127	0.0467101	0.0233551	−	0.999727i	$-0.492565\pi$
$37$	10.5127	0.0467101	0.0233551	+	0.999727i	$0.492565\pi$
$38$	−98.6906	−0.421309
$39$	−593.596	−2.43721
$40$	−285.168	−1.12722
$41$	249.884	0.951836	0.475918	−	0.879490i	$-0.342116\pi$
$41$	249.884	0.951836	0.475918	+	0.879490i	$0.342116\pi$
$42$	207.437	0.762101
$43$	−105.462	−0.374018	−0.187009	−	0.982358i	$-0.559879\pi$
$43$	−105.462	−0.374018	−0.187009	+	0.982358i	$0.559879\pi$
$44$	208.782	0.715341
$45$	216.813	0.718235
$46$	0.373159	0.00119607
$47$	−167.931	−0.521177	−0.260588	−	0.965450i	$-0.583917\pi$
$47$	−167.931	−0.521177	−0.260588	+	0.965450i	$0.583917\pi$
$48$	1211.30	3.64242
$49$	−320.333	−0.933917
$50$	−129.856	−0.367288
$51$	−837.017	−2.29815
$52$	−1343.13	−3.58191
$53$	−442.160	−1.14595	−0.572975	−	0.819573i	$-0.694211\pi$
$53$	−442.160	−1.14595	−0.572975	+	0.819573i	$0.694211\pi$
$54$	−712.928	−1.79661
$55$	55.0000	0.134840
$56$	271.534	0.647950
$57$	159.377	0.370350
$58$	534.456	1.20996
$59$	−339.288	−0.748669	−0.374335	−	0.927294i	$-0.622129\pi$
$59$	−339.288	−0.748669	−0.374335	+	0.927294i	$0.622129\pi$
$60$	796.050	1.71283
$61$	116.908	0.245385	0.122692	−	0.992445i	$-0.460847\pi$
$61$	116.908	0.245385	0.122692	+	0.992445i	$0.460847\pi$
$62$	−1095.02	−2.24302
$63$	−206.447	−0.412855
$64$	370.855	0.724327
$65$	−353.826	−0.675180
$66$	−479.276	−0.893861
$67$	−999.959	−1.82335	−0.911675	−	0.410912i	$-0.865210\pi$
$67$	−999.959	−1.82335	−0.911675	+	0.410912i	$0.865210\pi$
$68$	−1893.92	−3.37753
$69$	−0.602619	−0.00105140
$70$	123.648	0.211124
$71$	−465.452	−0.778014	−0.389007	−	0.921235i	$-0.627182\pi$
$71$	−465.452	−0.778014	−0.389007	+	0.921235i	$0.627182\pi$
$72$	−2473.12	−4.04806
$73$	−207.074	−0.332002	−0.166001	−	0.986126i	$-0.553085\pi$
$73$	−207.074	−0.332002	−0.166001	+	0.986126i	$0.553085\pi$
$74$	−54.6054	−0.0857804
$75$	209.706	0.322864
$76$	360.623	0.544293
$77$	−52.3704	−0.0775086
$78$	3083.28	4.47580
$79$	597.983	0.851625	0.425812	−	0.904812i	$-0.359988\pi$
$79$	597.983	0.851625	0.425812	+	0.904812i	$0.359988\pi$
$80$	722.024	1.00906
$81$	−19.4747	−0.0267143
$82$	−1297.96	−1.74799
$83$	614.320	0.812415	0.406207	−	0.913781i	$-0.366851\pi$
$83$	614.320	0.812415	0.406207	+	0.913781i	$0.366851\pi$
$84$	−757.991	−0.984566
$85$	−498.923	−0.636656
$86$	547.794	0.686863
$87$	−863.099	−1.06361
$88$	−627.369	−0.759974
$89$	−405.032	−0.482397	−0.241199	−	0.970476i	$-0.577541\pi$
$89$	−405.032	−0.482397	−0.241199	+	0.970476i	$0.577541\pi$
$90$	−1126.18	−1.31900
$91$	336.909	0.388107
$92$	−1.36355	−0.00154522
$93$	1768.35	1.97172
$94$	872.276	0.957111
$95$	95.0000	0.102598
$96$	−2464.51	−2.62013
$97$	−890.374	−0.931998	−0.465999	−	0.884785i	$-0.654305\pi$
$97$	−890.374	−0.931998	−0.465999	+	0.884785i	$0.654305\pi$
$98$	1663.89	1.71508
$99$	476.989	0.484234
$100$	474.504	0.474504
$101$	−1807.36	−1.78059	−0.890293	−	0.455389i	$-0.849500\pi$
$101$	−1807.36	−1.78059	−0.890293	+	0.455389i	$0.849500\pi$
$102$	4347.67	4.22042
$103$	753.261	0.720593	0.360296	−	0.932838i	$-0.382676\pi$
$103$	753.261	0.720593	0.360296	+	0.932838i	$0.382676\pi$
$104$	4035.99	3.80540
$105$	−199.680	−0.185588
$106$	2296.68	2.10447
$107$	526.375	0.475575	0.237788	−	0.971317i	$-0.423578\pi$
$107$	526.375	0.475575	0.237788	+	0.971317i	$0.423578\pi$
$108$	2605.09	2.32107
$109$	−1498.60	−1.31688	−0.658441	−	0.752633i	$-0.728784\pi$
$109$	−1498.60	−1.31688	−0.658441	+	0.752633i	$0.728784\pi$
$110$	−285.683	−0.247626
$111$	88.1829	0.0754050
$112$	−687.503	−0.580026
$113$	1582.48	1.31741	0.658706	−	0.752400i	$-0.271104\pi$
$113$	1582.48	1.31741	0.658706	+	0.752400i	$0.271104\pi$
$114$	−827.841	−0.680126
$115$	−0.359205	−0.000291270 0
$116$	−1952.94	−1.56316
$117$	−3068.56	−2.42469
$118$	1762.34	1.37489
$119$	475.069	0.365962
$120$	−2392.05	−1.81970
$121$	121.000	0.0909091
$122$	−607.246	−0.450635
$123$	2096.08	1.53657
$124$	4001.27	2.89778
$125$	125.000	0.0894427
$126$	1072.34	0.758185
$127$	1801.37	1.25863	0.629315	−	0.777150i	$-0.283335\pi$
$127$	1801.37	1.25863	0.629315	+	0.777150i	$0.283335\pi$
$128$	424.126	0.292874
$129$	−884.640	−0.603784
$130$	1837.86	1.23993
$131$	−390.005	−0.260114	−0.130057	−	0.991507i	$-0.541516\pi$
$131$	−390.005	−0.260114	−0.130057	+	0.991507i	$0.541516\pi$
$132$	1751.31	1.15479
$133$	−90.4580	−0.0589752
$134$	5194.03	3.34848
$135$	686.268	0.437515
$136$	5691.06	3.58827
$137$	−2688.68	−1.67671	−0.838356	−	0.545123i	$-0.816483\pi$
$137$	−2688.68	−1.67671	−0.838356	+	0.545123i	$0.816483\pi$
$138$	3.13015	0.00193084
$139$	935.699	0.570971	0.285486	−	0.958383i	$-0.407845\pi$
$139$	935.699	0.570971	0.285486	+	0.958383i	$0.407845\pi$
$140$	−451.817	−0.272754
$141$	−1408.65	−0.841345
$142$	2417.67	1.42878
$143$	−778.417	−0.455206
$144$	6261.76	3.62371
$145$	−514.470	−0.294651
$146$	1075.59	0.609702
$147$	−2687.03	−1.50764
$148$	199.532	0.110821
$149$	−3207.39	−1.76349	−0.881743	−	0.471730i	$-0.843630\pi$
$149$	−3207.39	−1.76349	−0.881743	+	0.471730i	$0.843630\pi$
$150$	−1089.26	−0.592920
$151$	3006.91	1.62052	0.810262	−	0.586068i	$-0.199325\pi$
$151$	3006.91	1.62052	0.810262	+	0.586068i	$0.199325\pi$
$152$	−1083.64	−0.578254
$153$	−4326.92	−2.28634
$154$	272.025	0.142340
$155$	1054.07	0.546224
$156$	−11266.5	−5.78234
$157$	1875.45	0.953357	0.476679	−	0.879078i	$-0.341841\pi$
$157$	1875.45	0.953357	0.476679	+	0.879078i	$0.341841\pi$
$158$	−3106.07	−1.56396
$159$	−3708.94	−1.84993
$160$	−1469.02	−0.725853
$161$	0.342031	0.000167427 0
$162$	101.156	0.0490592
$163$	−2754.52	−1.32362	−0.661812	−	0.749670i	$-0.730212\pi$
$163$	−2754.52	−1.32362	−0.661812	+	0.749670i	$0.730212\pi$
$164$	4742.83	2.25825
$165$	461.353	0.217675
$166$	−3190.93	−1.49195
$167$	1388.61	0.643437	0.321718	−	0.946835i	$-0.395740\pi$
$167$	1388.61	0.643437	0.321718	+	0.946835i	$0.395740\pi$
$168$	2277.69	1.04600
$169$	2810.71	1.27934
$170$	2591.52	1.16918
$171$	823.890	0.368447
$172$	−2001.68	−0.887365
$173$	1511.52	0.664271	0.332135	−	0.943232i	$-0.392231\pi$
$173$	1511.52	0.664271	0.332135	+	0.943232i	$0.392231\pi$
$174$	4483.14	1.95325
$175$	−119.024	−0.0514134
$176$	1588.45	0.680307
$177$	−2846.03	−1.20859
$178$	2103.84	0.885894
$179$	−1593.38	−0.665334	−0.332667	−	0.943044i	$-0.607948\pi$
$179$	−1593.38	−0.665334	−0.332667	+	0.943044i	$0.607948\pi$
$180$	4115.14	1.70403
$181$	−485.376	−0.199324	−0.0996622	−	0.995021i	$-0.531776\pi$
$181$	−485.376	−0.199324	−0.0996622	+	0.995021i	$0.531776\pi$
$182$	−1749.99	−0.712735
$183$	980.649	0.396129
$184$	4.09734	0.00164163
$185$	52.5634	0.0208894
$186$	−9185.26	−3.62095
$187$	−1097.63	−0.429233
$188$	−3187.36	−1.23650
$189$	−653.457	−0.251492
$190$	−493.453	−0.188415
$191$	−1912.48	−0.724512	−0.362256	−	0.932079i	$-0.617993\pi$
$191$	−1912.48	−0.724512	−0.362256	+	0.932079i	$0.617993\pi$
$192$	3110.82	1.16929
$193$	409.308	0.152656	0.0763281	−	0.997083i	$-0.475680\pi$
$193$	409.308	0.152656	0.0763281	+	0.997083i	$0.475680\pi$
$194$	4624.82	1.71156
$195$	−2967.98	−1.08996
$196$	−6079.97	−2.21573
$197$	−5137.03	−1.85786	−0.928929	−	0.370258i	$-0.879269\pi$
$197$	−5137.03	−1.85786	−0.928929	+	0.370258i	$0.879269\pi$
$198$	−2477.59	−0.889267
$199$	1679.84	0.598394	0.299197	−	0.954191i	$-0.403281\pi$
$199$	1679.84	0.598394	0.299197	+	0.954191i	$0.403281\pi$
$200$	−1425.84	−0.504110
$201$	−8387.90	−2.94347
$202$	9387.87	3.26994
$203$	489.873	0.169371
$204$	−15886.7	−5.45241
$205$	1249.42	0.425674
$206$	−3912.62	−1.32333
$207$	−3.11521	−0.00104600
$208$	−10218.8	−3.40648
$209$	209.000	0.0691714
$210$	1037.19	0.340822
$211$	2124.20	0.693061	0.346531	−	0.938039i	$-0.387360\pi$
$211$	2124.20	0.693061	0.346531	+	0.938039i	$0.387360\pi$
$212$	−8392.26	−2.71879
$213$	−3904.33	−1.25596
$214$	−2734.12	−0.873366
$215$	−527.309	−0.167266
$216$	−7828.05	−2.46589
$217$	−1003.67	−0.313980
$218$	7784.10	2.41838
$219$	−1736.98	−0.535956
$220$	1043.91	0.319910
$221$	7061.27	2.14929
$222$	−458.043	−0.138477
$223$	−2467.70	−0.741030	−0.370515	−	0.928827i	$-0.620819\pi$
$223$	−2467.70	−0.741030	−0.370515	+	0.928827i	$0.620819\pi$
$224$	1398.79	0.417235
$225$	1084.07	0.321204
$226$	−8219.81	−2.41935
$227$	−1488.91	−0.435341	−0.217670	−	0.976022i	$-0.569846\pi$
$227$	−1488.91	−0.435341	−0.217670	+	0.976022i	$0.569846\pi$
$228$	3024.99	0.878662
$229$	2519.66	0.727092	0.363546	−	0.931576i	$-0.381566\pi$
$229$	2519.66	0.727092	0.363546	+	0.931576i	$0.381566\pi$
$230$	1.86580	0.000534900 0
$231$	−439.296	−0.125124
$232$	5868.40	1.66069
$233$	−4504.21	−1.26644	−0.633220	−	0.773972i	$-0.718267\pi$
$233$	−4504.21	−1.26644	−0.633220	+	0.773972i	$0.718267\pi$
$234$	15938.9	4.45280
$235$	−839.657	−0.233077
$236$	−6439.73	−1.77623
$237$	5016.03	1.37479
$238$	−2467.62	−0.672068
$239$	2576.32	0.697273	0.348637	−	0.937258i	$-0.386645\pi$
$239$	2576.32	0.697273	0.348637	+	0.937258i	$0.386645\pi$
$240$	6056.51	1.62894
$241$	3648.47	0.975181	0.487590	−	0.873073i	$-0.337876\pi$
$241$	3648.47	0.975181	0.487590	+	0.873073i	$0.337876\pi$
$242$	−628.503	−0.166949
$243$	−3869.20	−1.02144
$244$	2218.92	0.582180
$245$	−1601.67	−0.417660
$246$	−10887.6	−2.82181
$247$	−1344.54	−0.346360
$248$	−12023.4	−3.07858
$249$	5153.07	1.31150
$250$	−649.280	−0.164256
$251$	−552.533	−0.138946	−0.0694732	−	0.997584i	$-0.522132\pi$
$251$	−552.533	−0.138946	−0.0694732	+	0.997584i	$0.522132\pi$
$252$	−3918.40	−0.979507
$253$	−0.790250	−0.000196374 0
$254$	−9356.77	−2.31140
$255$	−4185.08	−1.02776
$256$	−5169.86	−1.26217
$257$	4438.68	1.07734	0.538672	−	0.842516i	$-0.318926\pi$
$257$	4438.68	1.07734	0.538672	+	0.842516i	$0.318926\pi$
$258$	4595.03	1.10881
$259$	−50.0503	−0.0120076
$260$	−6715.67	−1.60188
$261$	−4461.75	−1.05814
$262$	2025.78	0.477684
$263$	−6077.98	−1.42503	−0.712517	−	0.701655i	$-0.752445\pi$
$263$	−6077.98	−1.42503	−0.712517	+	0.701655i	$0.752445\pi$
$264$	−5262.52	−1.22684
$265$	−2210.80	−0.512484
$266$	469.861	0.108305
$267$	−3397.51	−0.778742
$268$	−18979.4	−4.32593
$269$	−2921.61	−0.662207	−0.331104	−	0.943594i	$-0.607421\pi$
$269$	−2921.61	−0.662207	−0.331104	+	0.943594i	$0.607421\pi$
$270$	−3564.64	−0.803471
$271$	7317.96	1.64035	0.820174	−	0.572114i	$-0.193876\pi$
$271$	7317.96	1.64035	0.820174	+	0.572114i	$0.193876\pi$
$272$	−14409.4	−3.21212
$273$	2826.08	0.626528
$274$	13965.7	3.07919
$275$	275.000	0.0603023
$276$	−11.4378	−0.00249447
$277$	−4768.50	−1.03434	−0.517169	−	0.855884i	$-0.673014\pi$
$277$	−4768.50	−1.03434	−0.517169	+	0.855884i	$0.673014\pi$
$278$	−4860.25	−1.04856
$279$	9141.42	1.96159
$280$	1357.67	0.289772
$281$	−5961.67	−1.26563	−0.632817	−	0.774301i	$-0.718102\pi$
$281$	−5961.67	−1.26563	−0.632817	+	0.774301i	$0.718102\pi$
$282$	7316.86	1.54508
$283$	8854.46	1.85987	0.929934	−	0.367726i	$-0.119863\pi$
$283$	8854.46	1.85987	0.929934	+	0.367726i	$0.119863\pi$
$284$	−8834.35	−1.84585
$285$	796.883	0.165626
$286$	4043.29	0.835960
$287$	−1189.68	−0.244686
$288$	−12740.1	−2.60667
$289$	5043.95	1.02665
$290$	2672.28	0.541109
$291$	−7468.67	−1.50454
$292$	−3930.29	−0.787680
$293$	−5271.63	−1.05110	−0.525549	−	0.850763i	$-0.676140\pi$
$293$	−5271.63	−1.05110	−0.525549	+	0.850763i	$0.676140\pi$
$294$	13957.1	2.76869
$295$	−1696.44	−0.334815
$296$	−599.575	−0.117735
$297$	1509.79	0.294972
$298$	16659.9	3.23854
$299$	5.08384	0.000983297 0
$300$	3980.25	0.766000
$301$	502.099	0.0961478
$302$	−15618.6	−2.97600
$303$	−15160.6	−2.87443
$304$	2743.69	0.517636
$305$	584.538	0.109739
$306$	22475.0	4.19873
$307$	−2453.15	−0.456054	−0.228027	−	0.973655i	$-0.573228\pi$
$307$	−2453.15	−0.456054	−0.228027	+	0.973655i	$0.573228\pi$
$308$	−993.998	−0.183891
$309$	6318.54	1.16327
$310$	−5475.08	−1.00311
$311$	7206.59	1.31398	0.656991	−	0.753898i	$-0.271829\pi$
$311$	7206.59	1.31398	0.656991	+	0.753898i	$0.271829\pi$
$312$	33854.8	6.14312
$313$	7493.66	1.35325	0.676624	−	0.736329i	$-0.263442\pi$
$313$	7493.66	1.35325	0.676624	+	0.736329i	$0.263442\pi$
$314$	−9741.53	−1.75079
$315$	−1032.24	−0.184635
$316$	11349.8	2.02049
$317$	−1027.21	−0.181999	−0.0909996	−	0.995851i	$-0.529006\pi$
$317$	−1027.21	−0.181999	−0.0909996	+	0.995851i	$0.529006\pi$
$318$	19265.1	3.39728
$319$	−1131.83	−0.198654
$320$	1854.28	0.323929
$321$	4415.36	0.767730
$322$	−1.77659	−0.000307471 0
$323$	−1895.91	−0.326598
$324$	−369.633	−0.0633801
$325$	−1769.13	−0.301950
$326$	14307.6	2.43076
$327$	−12570.6	−2.12586
$328$	−14251.7	−2.39915
$329$	799.513	0.133977
$330$	−2396.38	−0.399747
$331$	6598.19	1.09568	0.547839	−	0.836584i	$-0.315451\pi$
$331$	6598.19	1.09568	0.547839	+	0.836584i	$0.315451\pi$
$332$	11659.9	1.92747
$333$	455.857	0.0750175
$334$	−7212.78	−1.18163
$335$	−4999.79	−0.815427
$336$	−5766.94	−0.936347
$337$	−7232.91	−1.16915	−0.584573	−	0.811341i	$-0.698738\pi$
$337$	−7232.91	−1.16915	−0.584573	+	0.811341i	$0.698738\pi$
$338$	−14599.5	−2.34944
$339$	13274.3	2.12672
$340$	−9469.62	−1.51048
$341$	2318.95	0.368264
$342$	−4279.48	−0.676631
$343$	3158.10	0.497146
$344$	6014.86	0.942732
$345$	−3.01310	−0.000470202 0
$346$	−7851.21	−1.21989
$347$	−4548.55	−0.703686	−0.351843	−	0.936059i	$-0.614445\pi$
$347$	−4548.55	−0.703686	−0.351843	+	0.936059i	$0.614445\pi$
$348$	−16381.7	−2.52343
$349$	−1886.05	−0.289277	−0.144639	−	0.989485i	$-0.546202\pi$
$349$	−1886.05	−0.289277	−0.144639	+	0.989485i	$0.546202\pi$
$350$	618.238	0.0944177
$351$	−9712.77	−1.47701
$352$	−3231.85	−0.489370
$353$	−603.058	−0.0909279	−0.0454639	−	0.998966i	$-0.514477\pi$
$353$	−603.058	−0.0909279	−0.0454639	+	0.998966i	$0.514477\pi$
$354$	14782.9	2.21951
$355$	−2327.26	−0.347938
$356$	−7687.57	−1.14450
$357$	3984.99	0.590779
$358$	8276.40	1.22185
$359$	719.637	0.105797	0.0528983	−	0.998600i	$-0.483154\pi$
$359$	719.637	0.105797	0.0528983	+	0.998600i	$0.483154\pi$
$360$	−12365.6	−1.81035
$361$	361.000	0.0526316
$362$	2521.16	0.366048
$363$	1014.98	0.146756
$364$	6394.59	0.920790
$365$	−1035.37	−0.148476
$366$	−5093.73	−0.727468
$367$	1477.38	0.210133	0.105066	−	0.994465i	$-0.466494\pi$
$367$	1477.38	0.210133	0.105066	+	0.994465i	$0.466494\pi$
$368$	−10.3742	−0.00146954
$369$	10835.6	1.52867
$370$	−273.027	−0.0383622
$371$	2105.10	0.294586
$372$	33563.6	4.67794
$373$	−5584.27	−0.775181	−0.387591	−	0.921832i	$-0.626693\pi$
$373$	−5584.27	−0.775181	−0.387591	+	0.921832i	$0.626693\pi$
$374$	5701.35	0.788262
$375$	1048.53	0.144389
$376$	9577.72	1.31365
$377$	7281.31	0.994712
$378$	3394.21	0.461850
$379$	−9600.22	−1.30114	−0.650568	−	0.759448i	$-0.725469\pi$
$379$	−9600.22	−1.30114	−0.650568	+	0.759448i	$0.725469\pi$
$380$	1803.11	0.243415
$381$	15110.4	2.03183
$382$	9933.86	1.33052
$383$	11237.3	1.49921	0.749605	−	0.661886i	$-0.230244\pi$
$383$	11237.3	1.49921	0.749605	+	0.661886i	$0.230244\pi$
$384$	3557.67	0.472791
$385$	−261.852	−0.0346629
$386$	−2126.05	−0.280344
$387$	−4573.10	−0.600682
$388$	−16899.4	−2.21118
$389$	−5666.74	−0.738599	−0.369299	−	0.929310i	$-0.620402\pi$
$389$	−5666.74	−0.738599	−0.369299	+	0.929310i	$0.620402\pi$
$390$	15416.4	2.00164
$391$	7.16861	0.000927192 0
$392$	18269.7	2.35398
$393$	−3271.46	−0.419906
$394$	26683.0	3.41185
$395$	2989.92	0.380858
$396$	9053.31	1.14885
$397$	11390.4	1.43997	0.719983	−	0.693992i	$-0.244150\pi$
$397$	11390.4	1.43997	0.719983	+	0.693992i	$0.244150\pi$
$398$	−8725.48	−1.09892
$399$	−758.784	−0.0952048
$400$	3610.12	0.451265
$401$	−8220.82	−1.02376	−0.511880	−	0.859057i	$-0.671051\pi$
$401$	−8220.82	−1.02376	−0.511880	+	0.859057i	$0.671051\pi$
$402$	43568.8	5.40550
$403$	−14918.3	−1.84400
$404$	−34304.0	−4.22447
$405$	−97.3736	−0.0119470
$406$	−2544.52	−0.311040
$407$	115.639	0.0140836
$408$	47738.0	5.79261
$409$	−9406.32	−1.13720	−0.568598	−	0.822616i	$-0.692514\pi$
$409$	−9406.32	−1.13720	−0.568598	+	0.822616i	$0.692514\pi$
$410$	−6489.78	−0.781725
$411$	−22553.3	−2.70675
$412$	14297.0	1.70962
$413$	1615.33	0.192458
$414$	16.1812	0.00192092
$415$	3071.60	0.363323
$416$	20791.2	2.45041
$417$	7848.87	0.921729
$418$	−1085.60	−0.127029
$419$	−3383.53	−0.394502	−0.197251	−	0.980353i	$-0.563201\pi$
$419$	−3383.53	−0.394502	−0.197251	+	0.980353i	$0.563201\pi$
$420$	−3789.95	−0.440311
$421$	9123.91	1.05623	0.528114	−	0.849173i	$-0.322899\pi$
$421$	9123.91	1.05623	0.528114	+	0.849173i	$0.322899\pi$
$422$	−11033.6	−1.27277
$423$	−7281.94	−0.837022
$424$	25217.9	2.88842
$425$	−2494.61	−0.284721
$426$	20280.0	2.30650
$427$	−556.591	−0.0630804
$428$	9990.67	1.12831
$429$	−6529.55	−0.734848
$430$	2738.97	0.307174
$431$	8480.86	0.947816	0.473908	−	0.880574i	$-0.342843\pi$
$431$	8480.86	0.947816	0.473908	+	0.880574i	$0.342843\pi$
$432$	19820.1	2.20739
$433$	−5952.61	−0.660656	−0.330328	−	0.943866i	$-0.607159\pi$
$433$	−5952.61	−0.660656	−0.330328	+	0.943866i	$0.607159\pi$
$434$	5213.31	0.576606
$435$	−4315.50	−0.475660
$436$	−28443.7	−3.12432
$437$	−1.36498	−0.000149418 0
$438$	9022.31	0.984253
$439$	6043.05	0.656991	0.328496	−	0.944505i	$-0.393458\pi$
$439$	6043.05	0.656991	0.328496	+	0.944505i	$0.393458\pi$
$440$	−3136.84	−0.339871
$441$	−13890.5	−1.49989
$442$	−36677.9	−3.94704
$443$	3666.05	0.393182	0.196591	−	0.980486i	$-0.437013\pi$
$443$	3666.05	0.393182	0.196591	+	0.980486i	$0.437013\pi$
$444$	1673.72	0.178900
$445$	−2025.16	−0.215735
$446$	12817.8	1.36086
$447$	−26904.3	−2.84683
$448$	−1765.62	−0.186201
$449$	−3753.64	−0.394533	−0.197267	−	0.980350i	$-0.563206\pi$
$449$	−3753.64	−0.394533	−0.197267	+	0.980350i	$0.563206\pi$
$450$	−5630.90	−0.589873
$451$	2748.72	0.286989
$452$	30035.8	3.12559
$453$	25222.7	2.61604
$454$	7733.75	0.799478
$455$	1684.55	0.173567
$456$	−9089.81	−0.933485
$457$	5398.34	0.552568	0.276284	−	0.961076i	$-0.410897\pi$
$457$	5398.34	0.552568	0.276284	+	0.961076i	$0.410897\pi$
$458$	−13087.7	−1.33526
$459$	−13695.8	−1.39273
$460$	−6.81775	−0.000691042 0
$461$	−16362.0	−1.65305	−0.826523	−	0.562903i	$-0.809684\pi$
$461$	−16362.0	−1.65305	−0.826523	+	0.562903i	$0.809684\pi$
$462$	2281.81	0.229782
$463$	6453.99	0.647824	0.323912	−	0.946087i	$-0.395002\pi$
$463$	6453.99	0.647824	0.323912	+	0.946087i	$0.395002\pi$
$464$	−14858.4	−1.48660
$465$	8841.77	0.881779
$466$	23395.9	2.32574
$467$	−19704.1	−1.95246	−0.976228	−	0.216745i	$-0.930456\pi$
$467$	−19704.1	−1.95246	−0.976228	+	0.216745i	$0.930456\pi$
$468$	−58241.8	−5.75262
$469$	4760.75	0.468723
$470$	4361.38	0.428033
$471$	15731.7	1.53902
$472$	19350.8	1.88706
$473$	−1160.08	−0.112771
$474$	−26054.5	−2.52473
$475$	475.000	0.0458831
$476$	9016.88	0.868252
$477$	−19173.2	−1.84042
$478$	−13382.0	−1.28050
$479$	16548.8	1.57857	0.789284	−	0.614028i	$-0.210452\pi$
$479$	16548.8	1.57857	0.789284	+	0.614028i	$0.210452\pi$
$480$	−12322.5	−1.17176
$481$	−743.932	−0.0705205
$482$	−18951.0	−1.79086
$483$	2.86904	0.000270281 0
$484$	2296.60	0.215683
$485$	−4451.87	−0.416802
$486$	20097.6	1.87581
$487$	−1035.81	−0.0963803	−0.0481901	−	0.998838i	$-0.515345\pi$
$487$	−1035.81	−0.0963803	−0.0481901	+	0.998838i	$0.515345\pi$
$488$	−6667.65	−0.618505
$489$	−23105.6	−2.13675
$490$	8319.44	0.767009
$491$	−12613.6	−1.15935	−0.579677	−	0.814847i	$-0.696821\pi$
$491$	−12613.6	−1.15935	−0.579677	+	0.814847i	$0.696821\pi$
$492$	39784.0	3.64553
$493$	10267.2	0.937956
$494$	6983.86	0.636070
$495$	2384.94	0.216556
$496$	30442.5	2.75586
$497$	2215.99	0.200002
$498$	−26766.3	−2.40848
$499$	4289.82	0.384847	0.192424	−	0.981312i	$-0.438365\pi$
$499$	4289.82	0.384847	0.192424	+	0.981312i	$0.438365\pi$
$500$	2372.52	0.212204
$501$	11648.0	1.03871
$502$	2869.99	0.255167
$503$	17148.9	1.52015	0.760073	−	0.649838i	$-0.225163\pi$
$503$	17148.9	1.52015	0.760073	+	0.649838i	$0.225163\pi$
$504$	11774.4	1.04062
$505$	−9036.80	−0.796302
$506$	4.10475	0.000360629 0
$507$	23576.9	2.06526
$508$	34190.3	2.98612
$509$	4423.56	0.385208	0.192604	−	0.981277i	$-0.438307\pi$
$509$	4423.56	0.385208	0.192604	+	0.981277i	$0.438307\pi$
$510$	21738.3	1.88743
$511$	985.867	0.0853467
$512$	23460.5	2.02503
$513$	2607.82	0.224440
$514$	−23055.6	−1.97848
$515$	3766.31	0.322259
$516$	−16790.6	−1.43249
$517$	−1847.25	−0.157141
$518$	259.973	0.0220513
$519$	12679.0	1.07234
$520$	20179.9	1.70182
$521$	6856.17	0.576534	0.288267	−	0.957550i	$-0.406921\pi$
$521$	6856.17	0.576534	0.288267	+	0.957550i	$0.406921\pi$
$522$	23175.4	1.94322
$523$	−6230.19	−0.520893	−0.260447	−	0.965488i	$-0.583870\pi$
$523$	−6230.19	−0.520893	−0.260447	+	0.965488i	$0.583870\pi$
$524$	−7402.35	−0.617125
$525$	−998.400	−0.0829976
$526$	31570.5	2.61699
$527$	−21035.9	−1.73878
$528$	13324.3	1.09823
$529$	−12167.0	−1.00000
$530$	11483.4	0.941147
$531$	−14712.4	−1.20238
$532$	−1716.91	−0.139920
$533$	−17683.1	−1.43703
$534$	17647.5	1.43011
$535$	2631.87	0.212684
$536$	57031.2	4.59584
$537$	−13365.7	−1.07406
$538$	15175.5	1.21610
$539$	−3523.67	−0.281586
$540$	13025.5	1.03801
$541$	−18628.0	−1.48037	−0.740186	−	0.672402i	$-0.765263\pi$
$541$	−18628.0	−1.48037	−0.740186	+	0.672402i	$0.765263\pi$
$542$	−38011.3	−3.01240
$543$	−4071.45	−0.321773
$544$	29317.2	2.31059
$545$	−7493.01	−0.588927
$546$	−14679.3	−1.15058
$547$	−4779.31	−0.373581	−0.186790	−	0.982400i	$-0.559808\pi$
$547$	−4779.31	−0.373581	−0.186790	+	0.982400i	$0.559808\pi$
$548$	−51031.6	−3.97803
$549$	5069.42	0.394094
$550$	−1428.42	−0.110742
$551$	−1954.98	−0.151153
$552$	34.3695	0.00265011
$553$	−2846.97	−0.218925
$554$	24768.7	1.89950
$555$	440.915	0.0337221
$556$	17759.7	1.35464
$557$	−6027.55	−0.458520	−0.229260	−	0.973365i	$-0.573631\pi$
$557$	−6027.55	−0.458520	−0.229260	+	0.973365i	$0.573631\pi$
$558$	−47482.7	−3.60234
$559$	7463.03	0.564674
$560$	−3437.52	−0.259396
$561$	−9207.18	−0.692919
$562$	30966.3	2.32426
$563$	−6509.38	−0.487278	−0.243639	−	0.969866i	$-0.578341\pi$
$563$	−6509.38	−0.487278	−0.243639	+	0.969866i	$0.578341\pi$
$564$	−26736.4	−1.99611
$565$	7912.42	0.589165
$566$	−45992.2	−3.41554
$567$	92.7181	0.00686736
$568$	26546.4	1.96102
$569$	3001.34	0.221130	0.110565	−	0.993869i	$-0.464734\pi$
$569$	3001.34	0.221130	0.110565	+	0.993869i	$0.464734\pi$
$570$	−4139.20	−0.304162
$571$	1462.38	0.107178	0.0535889	−	0.998563i	$-0.482934\pi$
$571$	1462.38	0.107178	0.0535889	+	0.998563i	$0.482934\pi$
$572$	−14774.5	−1.07999
$573$	−16042.3	−1.16959
$574$	6179.50	0.449351
$575$	−1.79602	−0.000130260 0
$576$	16081.3	1.16329
$577$	−20222.6	−1.45906	−0.729531	−	0.683948i	$-0.760261\pi$
$577$	−20222.6	−1.45906	−0.729531	+	0.683948i	$0.760261\pi$
$578$	−26199.5	−1.88539
$579$	3433.38	0.246436
$580$	−9764.71	−0.699064
$581$	−2924.75	−0.208845
$582$	38794.1	2.76300
$583$	−4863.76	−0.345517
$584$	11810.1	0.836827
$585$	−15342.8	−1.08435
$586$	27382.1	1.93028
$587$	−7285.14	−0.512249	−0.256124	−	0.966644i	$-0.582446\pi$
$587$	−7285.14	−0.512249	−0.256124	+	0.966644i	$0.582446\pi$
$588$	−51000.3	−3.57690
$589$	4005.46	0.280207
$590$	8811.71	0.614868
$591$	−43090.6	−2.99917
$592$	1518.08	0.105393
$593$	20830.9	1.44254	0.721268	−	0.692657i	$-0.243560\pi$
$593$	20830.9	1.44254	0.721268	+	0.692657i	$0.243560\pi$
$594$	−7842.21	−0.541700
$595$	2375.34	0.163663
$596$	−60876.7	−4.18390
$597$	14090.9	0.965999
$598$	−26.4067	−0.00180577
$599$	26655.5	1.81822	0.909111	−	0.416554i	$-0.136762\pi$
$599$	26655.5	1.81822	0.909111	+	0.416554i	$0.136762\pi$
$600$	−11960.3	−0.813794
$601$	26158.8	1.77544	0.887722	−	0.460380i	$-0.152287\pi$
$601$	26158.8	1.77544	0.887722	+	0.460380i	$0.152287\pi$
$602$	−2608.02	−0.176570
$603$	−43360.8	−2.92834
$604$	57071.6	3.84472
$605$	605.000	0.0406558
$606$	78747.7	5.27872
$607$	11393.3	0.761847	0.380924	−	0.924607i	$-0.375606\pi$
$607$	11393.3	0.761847	0.380924	+	0.924607i	$0.375606\pi$
$608$	−5582.29	−0.372355
$609$	4109.17	0.273419
$610$	−3036.23	−0.201530
$611$	11883.7	0.786846
$612$	−82125.5	−5.42439
$613$	−9187.63	−0.605359	−0.302679	−	0.953092i	$-0.597881\pi$
$613$	−9187.63	−0.605359	−0.302679	+	0.953092i	$0.597881\pi$
$614$	12742.3	0.837517
$615$	10480.4	0.687173
$616$	2986.87	0.195364
$617$	2388.36	0.155838	0.0779188	−	0.996960i	$-0.475173\pi$
$617$	2388.36	0.155838	0.0779188	+	0.996960i	$0.475173\pi$
$618$	−32820.0	−2.13627
$619$	17087.0	1.10951	0.554754	−	0.832015i	$-0.312812\pi$
$619$	17087.0	1.10951	0.554754	+	0.832015i	$0.312812\pi$
$620$	20006.3	1.29593
$621$	−9.86042	−0.000637174 0
$622$	−37432.8	−2.41305
$623$	1928.34	0.124008
$624$	−85718.0	−5.49914
$625$	625.000	0.0400000
$626$	−38923.9	−2.48516
$627$	1753.14	0.111665
$628$	35596.3	2.26186
$629$	−1049.00	−0.0664968
$630$	5361.68	0.339071
$631$	−22299.6	−1.40686	−0.703432	−	0.710762i	$-0.748350\pi$
$631$	−22299.6	−1.40686	−0.703432	+	0.710762i	$0.748350\pi$
$632$	−34105.1	−2.14656
$633$	17818.3	1.11882
$634$	5335.57	0.334231
$635$	9006.87	0.562877
$636$	−70396.3	−4.38898
$637$	22668.5	1.40998
$638$	5879.01	0.364816
$639$	−20183.2	−1.24951
$640$	2120.63	0.130977
$641$	16268.0	1.00242	0.501208	−	0.865327i	$-0.332889\pi$
$641$	16268.0	1.00242	0.501208	+	0.865327i	$0.332889\pi$
$642$	−22934.4	−1.40989
$643$	17487.5	1.07253	0.536267	−	0.844048i	$-0.319834\pi$
$643$	17487.5	1.07253	0.536267	+	0.844048i	$0.319834\pi$
$644$	6.49180	0.000397225 0
$645$	−4423.20	−0.270021
$646$	9847.79	0.599777
$647$	−6482.11	−0.393876	−0.196938	−	0.980416i	$-0.563100\pi$
$647$	−6482.11	−0.393876	−0.196938	+	0.980416i	$0.563100\pi$
$648$	1110.71	0.0673347
$649$	−3732.16	−0.225732
$650$	9189.29	0.554513
$651$	−8419.04	−0.506864
$652$	−52281.2	−3.14032
$653$	104.942	0.00628899	0.00314449	−	0.999995i	$-0.498999\pi$
$653$	104.942	0.00628899	0.00314449	+	0.999995i	$0.498999\pi$
$654$	65294.9	3.90403
$655$	−1950.03	−0.116326
$656$	36084.4	2.14765
$657$	−8979.25	−0.533202
$658$	−4152.86	−0.246042
$659$	29257.3	1.72944	0.864722	−	0.502251i	$-0.167495\pi$
$659$	29257.3	1.72944	0.864722	+	0.502251i	$0.167495\pi$
$660$	8756.55	0.516437
$661$	19123.1	1.12527	0.562634	−	0.826706i	$-0.309788\pi$
$661$	19123.1	1.12527	0.562634	+	0.826706i	$0.309788\pi$
$662$	−34272.6	−2.01215
$663$	59231.6	3.46963
$664$	−35036.8	−2.04773
$665$	−452.290	−0.0263745
$666$	−2367.83	−0.137765
$667$	7.39199	0.000429114 0
$668$	26356.0	1.52657
$669$	−20699.7	−1.19626
$670$	25970.1	1.49748
$671$	1285.98	0.0739863
$672$	11733.4	0.673549
$673$	189.617	0.0108606	0.00543030	−	0.999985i	$-0.498271\pi$
$673$	189.617	0.0108606	0.00543030	+	0.999985i	$0.498271\pi$
$674$	37569.5	2.14707
$675$	3431.34	0.195663
$676$	53347.7	3.03526
$677$	23223.9	1.31841	0.659207	−	0.751962i	$-0.270892\pi$
$677$	23223.9	1.31841	0.659207	+	0.751962i	$0.270892\pi$
$678$	−68949.7	−3.90560
$679$	4239.02	0.239586
$680$	28455.3	1.60472
$681$	−12489.3	−0.702778
$682$	−12045.2	−0.676296
$683$	22431.0	1.25666	0.628329	−	0.777948i	$-0.283739\pi$
$683$	22431.0	1.25666	0.628329	+	0.777948i	$0.283739\pi$
$684$	15637.5	0.874147
$685$	−13443.4	−0.749849
$686$	−16403.9	−0.912980
$687$	21135.5	1.17376
$688$	−15229.2	−0.843906
$689$	31289.5	1.73010
$690$	15.6507	0.000863498 0
$691$	4628.20	0.254797	0.127399	−	0.991852i	$-0.459337\pi$
$691$	4628.20	0.254797	0.127399	+	0.991852i	$0.459337\pi$
$692$	28688.9	1.57599
$693$	−2270.92	−0.124481
$694$	23626.3	1.29228
$695$	4678.50	0.255346
$696$	49225.6	2.68088
$697$	−24934.5	−1.35504
$698$	9796.59	0.531241
$699$	−37782.4	−2.04444
$700$	−2259.09	−0.121979
$701$	8112.08	0.437074	0.218537	−	0.975829i	$-0.429872\pi$
$701$	8112.08	0.437074	0.218537	+	0.975829i	$0.429872\pi$
$702$	50450.5	2.71244
$703$	199.741	0.0107160
$704$	4079.41	0.218393
$705$	−7043.25	−0.376261
$706$	3132.43	0.166984
$707$	8604.75	0.457730
$708$	−54018.0	−2.86740
$709$	27173.6	1.43939	0.719695	−	0.694291i	$-0.244282\pi$
$709$	27173.6	1.43939	0.719695	+	0.694291i	$0.244282\pi$
$710$	12088.4	0.638969
$711$	25930.1	1.36773
$712$	23100.4	1.21591
$713$	−15.1450	−0.000795492 0
$714$	−20699.0	−1.08493
$715$	−3892.09	−0.203574
$716$	−30242.6	−1.57852
$717$	21610.8	1.12562
$718$	−3737.97	−0.194289
$719$	14605.2	0.757557	0.378779	−	0.925487i	$-0.376344\pi$
$719$	14605.2	0.757557	0.378779	+	0.925487i	$0.376344\pi$
$720$	31308.8	1.62057
$721$	−3586.24	−0.185241
$722$	−1875.12	−0.0966548
$723$	30604.2	1.57425
$724$	−9212.51	−0.472901
$725$	−2572.35	−0.131772
$726$	−5272.04	−0.269509
$727$	8049.40	0.410640	0.205320	−	0.978695i	$-0.434176\pi$
$727$	8049.40	0.410640	0.205320	+	0.978695i	$0.434176\pi$
$728$	−19215.1	−0.978242
$729$	−31930.0	−1.62221
$730$	5377.95	0.272667
$731$	10523.5	0.532455
$732$	18612.9	0.939824
$733$	−12872.8	−0.648658	−0.324329	−	0.945944i	$-0.605138\pi$
$733$	−12872.8	−0.648658	−0.324329	+	0.945944i	$0.605138\pi$
$734$	−7673.88	−0.385897
$735$	−13435.2	−0.674236
$736$	21.1072	0.00105710
$737$	−10999.5	−0.549761
$738$	−56282.8	−2.80731
$739$	−13766.3	−0.685251	−0.342626	−	0.939472i	$-0.611316\pi$
$739$	−13766.3	−0.685251	−0.342626	+	0.939472i	$0.611316\pi$
$740$	997.661	0.0495605
$741$	−11278.3	−0.559135
$742$	−10934.4	−0.540990
$743$	−26474.4	−1.30720	−0.653602	−	0.756838i	$-0.726743\pi$
$743$	−26474.4	−1.30720	−0.653602	+	0.756838i	$0.726743\pi$
$744$	−100855.	−4.96981
$745$	−16036.9	−0.788655
$746$	29006.1	1.42358
$747$	26638.5	1.30476
$748$	−20833.2	−1.01836
$749$	−2506.04	−0.122255
$750$	−5446.32	−0.265162
$751$	−13161.6	−0.639510	−0.319755	−	0.947500i	$-0.603601\pi$
$751$	−13161.6	−0.639510	−0.319755	+	0.947500i	$0.603601\pi$
$752$	−24250.1	−1.17594
$753$	−4634.78	−0.224304
$754$	−37820.9	−1.82673
$755$	15034.6	0.724720
$756$	−12402.7	−0.596669
$757$	7269.55	0.349031	0.174515	−	0.984654i	$-0.444164\pi$
$757$	7269.55	0.349031	0.174515	+	0.984654i	$0.444164\pi$
$758$	49865.9	2.38946
$759$	−6.62881	−0.000317010 0
$760$	−5418.18	−0.258603
$761$	−7716.42	−0.367569	−0.183784	−	0.982967i	$-0.558835\pi$
$761$	−7716.42	−0.367569	−0.183784	+	0.982967i	$0.558835\pi$
$762$	−78486.8	−3.73134
$763$	7134.77	0.338527
$764$	−36299.1	−1.71892
$765$	−21634.6	−1.02248
$766$	−58369.0	−2.75321
$767$	24009.8	1.13030
$768$	−43366.0	−2.03755
$769$	−24974.0	−1.17111	−0.585555	−	0.810632i	$-0.699124\pi$
$769$	−24974.0	−1.17111	−0.585555	+	0.810632i	$0.699124\pi$
$770$	1360.12	0.0636564
$771$	37232.7	1.73918
$772$	7768.73	0.362180
$773$	−10451.7	−0.486316	−0.243158	−	0.969987i	$-0.578183\pi$
$773$	−10451.7	−0.486316	−0.243158	+	0.969987i	$0.578183\pi$
$774$	23753.8	1.10312
$775$	5270.34	0.244279
$776$	50781.2	2.34915
$777$	−419.834	−0.0193841
$778$	29434.4	1.35639
$779$	4747.79	0.218366
$780$	−56332.6	−2.58594
$781$	−5119.97	−0.234580
$782$	−37.2355	−0.00170273
$783$	−14122.5	−0.644571
$784$	−46257.6	−2.10722
$785$	9377.25	0.426354
$786$	16992.7	0.771133
$787$	−21836.5	−0.989054	−0.494527	−	0.869162i	$-0.664659\pi$
$787$	−21836.5	−0.989054	−0.494527	+	0.869162i	$0.664659\pi$
$788$	−97501.5	−4.40780
$789$	−50983.5	−2.30046
$790$	−15530.3	−0.699424
$791$	−7534.13	−0.338663
$792$	−27204.3	−1.22054
$793$	−8272.99	−0.370470
$794$	−59164.3	−2.64441
$795$	−18544.7	−0.827312
$796$	31883.5	1.41970
$797$	40879.4	1.81684	0.908422	−	0.418055i	$-0.137288\pi$
$797$	40879.4	1.81684	0.908422	+	0.418055i	$0.137288\pi$
$798$	3941.31	0.174838
$799$	16756.9	0.741950
$800$	−7345.12	−0.324612
$801$	−17563.3	−0.774741
$802$	42700.9	1.88008
$803$	−2277.81	−0.100102
$804$	−159204.	−6.98343
$805$	1.71015	7.48758e−5 0
$806$	77489.0	3.38639
$807$	−24507.2	−1.06901
$808$	103080.	4.48805
$809$	9624.88	0.418285	0.209143	−	0.977885i	$-0.432933\pi$
$809$	9624.88	0.418285	0.209143	+	0.977885i	$0.432933\pi$
$810$	505.782	0.0219400
$811$	−31612.2	−1.36875	−0.684375	−	0.729130i	$-0.739925\pi$
$811$	−31612.2	−1.36875	−0.684375	+	0.729130i	$0.739925\pi$
$812$	9297.85	0.401836
$813$	61384.8	2.64804
$814$	−600.659	−0.0258638
$815$	−13772.6	−0.591943
$816$	−120869.	−5.18538
$817$	−2003.78	−0.0858057
$818$	48858.7	2.08839
$819$	14609.3	0.623308
$820$	23714.1	1.00992
$821$	−29247.9	−1.24331	−0.621656	−	0.783290i	$-0.713540\pi$
$821$	−29247.9	−1.24331	−0.621656	+	0.783290i	$0.713540\pi$
$822$	117147.	4.97078
$823$	5629.01	0.238414	0.119207	−	0.992869i	$-0.461965\pi$
$823$	5629.01	0.238414	0.119207	+	0.992869i	$0.461965\pi$
$824$	−42961.1	−1.81629
$825$	2306.77	0.0973470
$826$	−8390.42	−0.353438
$827$	4550.38	0.191333	0.0956664	−	0.995413i	$-0.469502\pi$
$827$	4550.38	0.191333	0.0956664	+	0.995413i	$0.469502\pi$
$828$	−59.1271	−0.00248165
$829$	−30438.3	−1.27523	−0.637615	−	0.770356i	$-0.720079\pi$
$829$	−30438.3	−1.27523	−0.637615	+	0.770356i	$0.720079\pi$
$830$	−15954.6	−0.667221
$831$	−39999.3	−1.66975
$832$	−26243.7	−1.09355
$833$	31964.3	1.32953
$834$	−40768.9	−1.69270
$835$	6943.06	0.287754
$836$	3966.85	0.164111
$837$	28934.9	1.19491
$838$	17574.9	0.724480
$839$	−21788.2	−0.896558	−0.448279	−	0.893894i	$-0.647963\pi$
$839$	−21788.2	−0.896558	−0.448279	+	0.893894i	$0.647963\pi$
$840$	11388.4	0.467784
$841$	−13801.8	−0.565904
$842$	−47391.8	−1.93970
$843$	−50007.9	−2.04314
$844$	40317.6	1.64430
$845$	14053.6	0.572139
$846$	37824.2	1.53714
$847$	−576.075	−0.0233697
$848$	−63850.0	−2.58563
$849$	74273.3	3.00242
$850$	12957.6	0.522874
$851$	−0.755241	−3.04222e−5 0
$852$	−74104.7	−2.97979
$853$	−15663.9	−0.628746	−0.314373	−	0.949300i	$-0.601794\pi$
$853$	−15663.9	−0.628746	−0.314373	+	0.949300i	$0.601794\pi$
$854$	2891.07	0.115843
$855$	4119.45	0.164774
$856$	−30021.0	−1.19871
$857$	48697.5	1.94105	0.970523	−	0.241008i	$-0.0774781\pi$
$857$	48697.5	1.94105	0.970523	+	0.241008i	$0.0774781\pi$
$858$	33916.1	1.34951
$859$	−40311.5	−1.60118	−0.800588	−	0.599216i	$-0.795479\pi$
$859$	−40311.5	−1.60118	−0.800588	+	0.599216i	$0.795479\pi$
$860$	−10008.4	−0.396842
$861$	−9979.35	−0.395000
$862$	−44051.6	−1.74061
$863$	45800.3	1.80656	0.903280	−	0.429052i	$-0.141152\pi$
$863$	45800.3	1.80656	0.903280	+	0.429052i	$0.141152\pi$
$864$	−40325.7	−1.58786
$865$	7557.61	0.297071
$866$	30919.3	1.21326
$867$	42309.8	1.65734
$868$	−19049.8	−0.744923
$869$	6577.81	0.256774
$870$	22415.7	0.873522
$871$	70762.3	2.75280
$872$	85470.5	3.31926
$873$	−38608.9	−1.49681
$874$	7.09002	0.000274398 0
$875$	−595.119	−0.0229928
$876$	−32968.2	−1.27157
$877$	−42566.1	−1.63894	−0.819472	−	0.573119i	$-0.805733\pi$
$877$	−42566.1	−1.63894	−0.819472	+	0.573119i	$0.805733\pi$
$878$	−31389.1	−1.20653
$879$	−44219.7	−1.69681
$880$	7942.26	0.304243
$881$	6635.41	0.253749	0.126874	−	0.991919i	$-0.459505\pi$
$881$	6635.41	0.253749	0.126874	+	0.991919i	$0.459505\pi$
$882$	72150.6	2.75446
$883$	15511.1	0.591154	0.295577	−	0.955319i	$-0.404488\pi$
$883$	15511.1	0.591154	0.295577	+	0.955319i	$0.404488\pi$
$884$	134024.	5.09922
$885$	−14230.1	−0.540498
$886$	−19042.4	−0.722055
$887$	−27257.7	−1.03182	−0.515910	−	0.856643i	$-0.672546\pi$
$887$	−27257.7	−1.03182	−0.515910	+	0.856643i	$0.672546\pi$
$888$	−5029.38	−0.190062
$889$	−8576.25	−0.323552
$890$	10519.2	0.396184
$891$	−214.222	−0.00805466
$892$	−46837.4	−1.75811
$893$	−3190.70	−0.119566
$894$	139748.	5.22803
$895$	−7966.90	−0.297546
$896$	−2019.24	−0.0752881
$897$	42.6445	0.00158735
$898$	19497.3	0.724537
$899$	−21691.4	−0.804727
$900$	20575.7	0.762063
$901$	44120.7	1.63138
$902$	−14277.5	−0.527039
$903$	4211.72	0.155213
$904$	−90254.7	−3.32060
$905$	−2426.88	−0.0891406
$906$	−131013.	−4.80421
$907$	187.398	0.00686046	0.00343023	−	0.999994i	$-0.498908\pi$
$907$	187.398	0.00686046	0.00343023	+	0.999994i	$0.498908\pi$
$908$	−28259.7	−1.03285
$909$	−78371.9	−2.85966
$910$	−8749.94	−0.318745
$911$	−1814.93	−0.0660057	−0.0330029	−	0.999455i	$-0.510507\pi$
$911$	−1814.93	−0.0660057	−0.0330029	+	0.999455i	$0.510507\pi$
$912$	23014.7	0.835629
$913$	6757.52	0.244952
$914$	−28040.3	−1.01476
$915$	4903.24	0.177154
$916$	47823.6	1.72504
$917$	1856.79	0.0668667
$918$	71139.2	2.55767
$919$	−8089.20	−0.290357	−0.145179	−	0.989405i	$-0.546376\pi$
$919$	−8089.20	−0.290357	−0.145179	+	0.989405i	$0.546376\pi$
$920$	20.4867	0.000734159 0
$921$	−20577.6	−0.736217
$922$	84988.1	3.03572
$923$	32937.8	1.17461
$924$	−8337.90	−0.296858
$925$	262.817	0.00934202
$926$	−33523.6	−1.18969
$927$	32663.4	1.15729
$928$	30230.7	1.06937
$929$	−38914.3	−1.37431	−0.687157	−	0.726509i	$-0.741142\pi$
$929$	−38914.3	−1.37431	−0.687157	+	0.726509i	$0.741142\pi$
$930$	−45926.3	−1.61934
$931$	−6086.33	−0.214255
$932$	−85490.5	−3.00465
$933$	60450.7	2.12118
$934$	102348.	3.58557
$935$	−5488.15	−0.191959
$936$	175011.	6.11155
$937$	20677.1	0.720907	0.360454	−	0.932777i	$-0.382622\pi$
$937$	20677.1	0.720907	0.360454	+	0.932777i	$0.382622\pi$
$938$	−24728.5	−0.860783
$939$	62858.6	2.18457
$940$	−15936.8	−0.552980
$941$	−41251.7	−1.42908	−0.714541	−	0.699593i	$-0.753365\pi$
$941$	−41251.7	−1.42908	−0.714541	+	0.699593i	$0.753365\pi$
$942$	−81714.4	−2.82632
$943$	−17.9519	−0.000619929 0
$944$	−48994.7	−1.68924
$945$	−3267.28	−0.112471
$946$	6025.74	0.207097
$947$	−44750.6	−1.53559	−0.767793	−	0.640698i	$-0.778645\pi$
$947$	−44750.6	−1.53559	−0.767793	+	0.640698i	$0.778645\pi$
$948$	95204.9	3.26172
$949$	14653.6	0.501239
$950$	−2467.26	−0.0842617
$951$	−8616.47	−0.293805
$952$	−27094.8	−0.922426
$953$	−10007.5	−0.340162	−0.170081	−	0.985430i	$-0.554403\pi$
$953$	−10007.5	−0.340162	−0.170081	+	0.985430i	$0.554403\pi$
$954$	99590.2	3.37982
$955$	−9562.38	−0.324012
$956$	48898.9	1.65429
$957$	−9494.09	−0.320690
$958$	−85958.5	−2.89895
$959$	12800.7	0.431028
$960$	15554.1	0.522924
$961$	14651.3	0.491803
$962$	3864.16	0.129507
$963$	22825.0	0.763785
$964$	69248.5	2.31363
$965$	2046.54	0.0682699
$966$	−14.9025	−0.000496355 0
$967$	30830.5	1.02528	0.512638	−	0.858605i	$-0.328668\pi$
$967$	30830.5	1.02528	0.512638	+	0.858605i	$0.328668\pi$
$968$	−6901.06	−0.229141
$969$	−15903.3	−0.527232
$970$	23124.1	0.765433
$971$	−3791.31	−0.125303	−0.0626513	−	0.998035i	$-0.519956\pi$
$971$	−3791.31	−0.125303	−0.0626513	+	0.998035i	$0.519956\pi$
$972$	−73438.0	−2.42338
$973$	−4454.82	−0.146778
$974$	5380.27	0.176997
$975$	−14839.9	−0.487443
$976$	16882.0	0.553668
$977$	−45588.5	−1.49284	−0.746420	−	0.665476i	$-0.768229\pi$
$977$	−45588.5	−1.49284	−0.746420	+	0.665476i	$0.768229\pi$
$978$	120016.	3.92402
$979$	−4455.36	−0.145448
$980$	−30399.9	−0.990906
$981$	−64983.3	−2.11494
$982$	65517.9	2.12908
$983$	57214.4	1.85641	0.928207	−	0.372065i	$-0.121350\pi$
$983$	57214.4	1.85641	0.928207	+	0.372065i	$0.121350\pi$
$984$	−119547.	−3.87299
$985$	−25685.1	−0.830859
$986$	−53330.4	−1.72250
$987$	6706.51	0.216282
$988$	−25519.5	−0.821746
$989$	7.57648	0.000243598 0
$990$	−12388.0	−0.397693
$991$	40216.8	1.28913	0.644565	−	0.764550i	$-0.277039\pi$
$991$	40216.8	1.28913	0.644565	+	0.764550i	$0.277039\pi$
$992$	−61938.0	−1.98239
$993$	55347.2	1.76877
$994$	−11510.4	−0.367292
$995$	8399.19	0.267610
$996$	97806.0	3.11155
$997$	17006.8	0.540232	0.270116	−	0.962828i	$-0.412938\pi$
$997$	17006.8	0.540232	0.270116	+	0.962828i	$0.412938\pi$
$998$	−22282.4	−0.706749
$999$	1442.90	0.0456971

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.2	✓	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)

\(f(q)\)	\(=\)	\(q-5.19424 q^{2} +8.38824 q^{3} +18.9801 q^{4} +5.00000 q^{5} -43.5706 q^{6} -4.76095 q^{7} -57.0335 q^{8} +43.3626 q^{9} +O(q^{10})\)	\(q-5.19424 q^{2} +8.38824 q^{3} +18.9801 q^{4} +5.00000 q^{5} -43.5706 q^{6} -4.76095 q^{7} -57.0335 q^{8} +43.3626 q^{9} -25.9712 q^{10} +11.0000 q^{11} +159.210 q^{12} -70.7652 q^{13} +24.7295 q^{14} +41.9412 q^{15} +144.405 q^{16} -99.7845 q^{17} -225.236 q^{18} +19.0000 q^{19} +94.9007 q^{20} -39.9360 q^{21} -57.1367 q^{22} -0.0718409 q^{23} -478.411 q^{24} +25.0000 q^{25} +367.572 q^{26} +137.254 q^{27} -90.3635 q^{28} -102.894 q^{29} -217.853 q^{30} +210.813 q^{31} -293.805 q^{32} +92.2707 q^{33} +518.305 q^{34} -23.8047 q^{35} +823.029 q^{36} +10.5127 q^{37} -98.6906 q^{38} -593.596 q^{39} -285.168 q^{40} +249.884 q^{41} +207.437 q^{42} -105.462 q^{43} +208.782 q^{44} +216.813 q^{45} +0.373159 q^{46} -167.931 q^{47} +1211.30 q^{48} -320.333 q^{49} -129.856 q^{50} -837.017 q^{51} -1343.13 q^{52} -442.160 q^{53} -712.928 q^{54} +55.0000 q^{55} +271.534 q^{56} +159.377 q^{57} +534.456 q^{58} -339.288 q^{59} +796.050 q^{60} +116.908 q^{61} -1095.02 q^{62} -206.447 q^{63} +370.855 q^{64} -353.826 q^{65} -479.276 q^{66} -999.959 q^{67} -1893.92 q^{68} -0.602619 q^{69} +123.648 q^{70} -465.452 q^{71} -2473.12 q^{72} -207.074 q^{73} -54.6054 q^{74} +209.706 q^{75} +360.623 q^{76} -52.3704 q^{77} +3083.28 q^{78} +597.983 q^{79} +722.024 q^{80} -19.4747 q^{81} -1297.96 q^{82} +614.320 q^{83} -757.991 q^{84} -498.923 q^{85} +547.794 q^{86} -863.099 q^{87} -627.369 q^{88} -405.032 q^{89} -1126.18 q^{90} +336.909 q^{91} -1.36355 q^{92} +1768.35 q^{93} +872.276 q^{94} +95.0000 q^{95} -2464.51 q^{96} -890.374 q^{97} +1663.89 q^{98} +476.989 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

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