LMFDB - Embedded newform 1339.4.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,4,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1339.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1339.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:	$79.0035574977$
Analytic rank:	$1$
Dimension:	$72$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.11041 q^{2} +0.0856874 q^{3} +18.1162 q^{4} +6.41522 q^{5} -0.437897 q^{6} +31.2200 q^{7} -51.6981 q^{8} -26.9927 q^{9} +O(q^{10})$	\(q-5.11041 q^{2} +0.0856874 q^{3} +18.1162 q^{4} +6.41522 q^{5} -0.437897 q^{6} +31.2200 q^{7} -51.6981 q^{8} -26.9927 q^{9} -32.7844 q^{10} -69.2864 q^{11} +1.55233 q^{12} -13.0000 q^{13} -159.547 q^{14} +0.549703 q^{15} +119.268 q^{16} +79.0099 q^{17} +137.943 q^{18} +8.87840 q^{19} +116.220 q^{20} +2.67516 q^{21} +354.082 q^{22} -23.3977 q^{23} -4.42988 q^{24} -83.8450 q^{25} +66.4353 q^{26} -4.62649 q^{27} +565.590 q^{28} +233.073 q^{29} -2.80921 q^{30} +210.738 q^{31} -195.925 q^{32} -5.93697 q^{33} -403.773 q^{34} +200.283 q^{35} -489.006 q^{36} -50.3550 q^{37} -45.3722 q^{38} -1.11394 q^{39} -331.655 q^{40} -257.806 q^{41} -13.6712 q^{42} -356.848 q^{43} -1255.21 q^{44} -173.164 q^{45} +119.572 q^{46} -167.426 q^{47} +10.2198 q^{48} +631.690 q^{49} +428.482 q^{50} +6.77015 q^{51} -235.511 q^{52} -512.786 q^{53} +23.6432 q^{54} -444.487 q^{55} -1614.02 q^{56} +0.760767 q^{57} -1191.10 q^{58} +435.840 q^{59} +9.95856 q^{60} +472.814 q^{61} -1076.95 q^{62} -842.712 q^{63} +47.1080 q^{64} -83.3978 q^{65} +30.3403 q^{66} +841.081 q^{67} +1431.36 q^{68} -2.00488 q^{69} -1023.53 q^{70} -667.610 q^{71} +1395.47 q^{72} +73.0807 q^{73} +257.335 q^{74} -7.18446 q^{75} +160.843 q^{76} -2163.12 q^{77} +5.69267 q^{78} +231.340 q^{79} +765.132 q^{80} +728.405 q^{81} +1317.49 q^{82} -130.029 q^{83} +48.4639 q^{84} +506.866 q^{85} +1823.64 q^{86} +19.9714 q^{87} +3581.98 q^{88} +82.3784 q^{89} +884.937 q^{90} -405.860 q^{91} -423.878 q^{92} +18.0576 q^{93} +855.614 q^{94} +56.9569 q^{95} -16.7883 q^{96} -1524.28 q^{97} -3228.19 q^{98} +1870.22 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q - q^{2} - 19 q^{3} + 261 q^{4} - 73 q^{5} - 52 q^{6} + 3 q^{7} - 18 q^{8} + 517 q^{9}+O(q^{10}) $ 72 * q - q^2 - 19 * q^3 + 261 * q^4 - 73 * q^5 - 52 * q^6 + 3 * q^7 - 18 * q^8 + 517 * q^9	\( 72 q - q^{2} - 19 q^{3} + 261 q^{4} - 73 q^{5} - 52 q^{6} + 3 q^{7} - 18 q^{8} + 517 q^{9} - 100 q^{10} - 104 q^{11} - 228 q^{12} - 936 q^{13} - 337 q^{14} + 63 q^{15} + 833 q^{16} - 307 q^{17} - 110 q^{18} - 182 q^{19} - 428 q^{20} - 177 q^{21} - 74 q^{22} - 516 q^{23} - 901 q^{24} + 1285 q^{25} + 13 q^{26} - 817 q^{27} + 187 q^{28} - 1458 q^{29} - 178 q^{30} - 620 q^{31} - 35 q^{32} - 196 q^{33} - 1619 q^{34} - 1155 q^{35} - 615 q^{36} - 1031 q^{37} - 1614 q^{38} + 247 q^{39} - 3300 q^{40} - 1124 q^{41} - 2190 q^{42} - 383 q^{43} - 1902 q^{44} - 1828 q^{45} - 1109 q^{46} - 1531 q^{47} - 4464 q^{48} + 1191 q^{49} - 352 q^{50} - 1267 q^{51} - 3393 q^{52} - 3092 q^{53} - 1581 q^{54} - 2264 q^{55} - 1470 q^{56} - 502 q^{57} - 2003 q^{58} - 2452 q^{59} - 1909 q^{60} - 1764 q^{61} - 2232 q^{62} - 1086 q^{63} + 2272 q^{64} + 949 q^{65} - 2539 q^{66} + 832 q^{67} - 1796 q^{68} - 4738 q^{69} - 3251 q^{70} - 2995 q^{71} - 2534 q^{72} + 16 q^{73} - 3408 q^{74} - 3060 q^{75} - 3729 q^{76} - 5454 q^{77} + 676 q^{78} - 3594 q^{79} - 1631 q^{80} + 16 q^{81} - 1293 q^{82} - 2832 q^{83} - 4565 q^{84} - 887 q^{85} - 3982 q^{86} - 1530 q^{87} - 1757 q^{88} - 8104 q^{89} - 5932 q^{90} - 39 q^{91} - 5371 q^{92} - 3042 q^{93} - 3464 q^{94} - 4582 q^{95} - 3991 q^{96} - 2514 q^{97} - 2886 q^{98} - 2752 q^{99}+O(q^{100}) \) 72 * q - q^2 - 19 * q^3 + 261 * q^4 - 73 * q^5 - 52 * q^6 + 3 * q^7 - 18 * q^8 + 517 * q^9 - 100 * q^10 - 104 * q^11 - 228 * q^12 - 936 * q^13 - 337 * q^14 + 63 * q^15 + 833 * q^16 - 307 * q^17 - 110 * q^18 - 182 * q^19 - 428 * q^20 - 177 * q^21 - 74 * q^22 - 516 * q^23 - 901 * q^24 + 1285 * q^25 + 13 * q^26 - 817 * q^27 + 187 * q^28 - 1458 * q^29 - 178 * q^30 - 620 * q^31 - 35 * q^32 - 196 * q^33 - 1619 * q^34 - 1155 * q^35 - 615 * q^36 - 1031 * q^37 - 1614 * q^38 + 247 * q^39 - 3300 * q^40 - 1124 * q^41 - 2190 * q^42 - 383 * q^43 - 1902 * q^44 - 1828 * q^45 - 1109 * q^46 - 1531 * q^47 - 4464 * q^48 + 1191 * q^49 - 352 * q^50 - 1267 * q^51 - 3393 * q^52 - 3092 * q^53 - 1581 * q^54 - 2264 * q^55 - 1470 * q^56 - 502 * q^57 - 2003 * q^58 - 2452 * q^59 - 1909 * q^60 - 1764 * q^61 - 2232 * q^62 - 1086 * q^63 + 2272 * q^64 + 949 * q^65 - 2539 * q^66 + 832 * q^67 - 1796 * q^68 - 4738 * q^69 - 3251 * q^70 - 2995 * q^71 - 2534 * q^72 + 16 * q^73 - 3408 * q^74 - 3060 * q^75 - 3729 * q^76 - 5454 * q^77 + 676 * q^78 - 3594 * q^79 - 1631 * q^80 + 16 * q^81 - 1293 * q^82 - 2832 * q^83 - 4565 * q^84 - 887 * q^85 - 3982 * q^86 - 1530 * q^87 - 1757 * q^88 - 8104 * q^89 - 5932 * q^90 - 39 * q^91 - 5371 * q^92 - 3042 * q^93 - 3464 * q^94 - 4582 * q^95 - 3991 * q^96 - 2514 * q^97 - 2886 * q^98 - 2752 * 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 $

$2$	−5.11041	−1.80680	−0.903401	−	0.428798i	$-0.858937\pi$
$2$	−5.11041	−1.80680	−0.903401	+	0.428798i	$0.858937\pi$
$3$	0.0856874	0.0164905	0.00824527	−	0.999966i	$-0.497375\pi$
$3$	0.0856874	0.0164905	0.00824527	+	0.999966i	$0.497375\pi$
$4$	18.1162	2.26453
$5$	6.41522	0.573794	0.286897	−	0.957961i	$-0.407376\pi$
$5$	6.41522	0.573794	0.286897	+	0.957961i	$0.407376\pi$
$6$	−0.437897	−0.0297951
$7$	31.2200	1.68572	0.842862	−	0.538130i	$-0.180869\pi$
$7$	31.2200	1.68572	0.842862	+	0.538130i	$0.180869\pi$
$8$	−51.6981	−2.28476
$9$	−26.9927	−0.999728
$10$	−32.7844	−1.03673
$11$	−69.2864	−1.89915	−0.949574	−	0.313542i	$-0.898484\pi$
$11$	−69.2864	−1.89915	−0.949574	+	0.313542i	$0.898484\pi$
$12$	1.55233	0.0373434
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−159.547	−3.04577
$15$	0.549703	0.00946219
$16$	119.268	1.86357
$17$	79.0099	1.12722	0.563609	−	0.826042i	$-0.309412\pi$
$17$	79.0099	1.12722	0.563609	+	0.826042i	$0.309412\pi$
$18$	137.943	1.80631
$19$	8.87840	0.107202	0.0536012	−	0.998562i	$-0.482930\pi$
$19$	8.87840	0.107202	0.0536012	+	0.998562i	$0.482930\pi$
$20$	116.220	1.29938
$21$	2.67516	0.0277985
$22$	354.082	3.43138
$23$	−23.3977	−0.212120	−0.106060	−	0.994360i	$-0.533823\pi$
$23$	−23.3977	−0.212120	−0.106060	+	0.994360i	$0.533823\pi$
$24$	−4.42988	−0.0376769
$25$	−83.8450	−0.670760
$26$	66.4353	0.501116
$27$	−4.62649	−0.0329766
$28$	565.590	3.81737
$29$	233.073	1.49243	0.746216	−	0.665703i	$-0.231868\pi$
$29$	233.073	1.49243	0.746216	+	0.665703i	$0.231868\pi$
$30$	−2.80921	−0.0170963
$31$	210.738	1.22095	0.610477	−	0.792034i	$-0.290978\pi$
$31$	210.738	1.22095	0.610477	+	0.792034i	$0.290978\pi$
$32$	−195.925	−1.08234
$33$	−5.93697	−0.0313180
$34$	−403.773	−2.03666
$35$	200.283	0.967259
$36$	−489.006	−2.26391
$37$	−50.3550	−0.223738	−0.111869	−	0.993723i	$-0.535684\pi$
$37$	−50.3550	−0.223738	−0.111869	+	0.993723i	$0.535684\pi$
$38$	−45.3722	−0.193693
$39$	−1.11394	−0.00457366
$40$	−331.655	−1.31098
$41$	−257.806	−0.982013	−0.491006	−	0.871156i	$-0.663371\pi$
$41$	−257.806	−0.982013	−0.491006	+	0.871156i	$0.663371\pi$
$42$	−13.6712	−0.0502264
$43$	−356.848	−1.26555	−0.632776	−	0.774335i	$-0.718085\pi$
$43$	−356.848	−1.26555	−0.632776	+	0.774335i	$0.718085\pi$
$44$	−1255.21	−4.30068
$45$	−173.164	−0.573638
$46$	119.572	0.383258
$47$	−167.426	−0.519608	−0.259804	−	0.965661i	$-0.583658\pi$
$47$	−167.426	−0.519608	−0.259804	+	0.965661i	$0.583658\pi$
$48$	10.2198	0.0307313
$49$	631.690	1.84166
$50$	428.482	1.21193
$51$	6.77015	0.0185885
$52$	−235.511	−0.628068
$53$	−512.786	−1.32899	−0.664496	−	0.747292i	$-0.731354\pi$
$53$	−512.786	−1.32899	−0.664496	+	0.747292i	$0.731354\pi$
$54$	23.6432	0.0595822
$55$	−444.487	−1.08972
$56$	−1614.02	−3.85146
$57$	0.760767	0.00176783
$58$	−1191.10	−2.69653
$59$	435.840	0.961722	0.480861	−	0.876797i	$-0.340324\pi$
$59$	435.840	0.961722	0.480861	+	0.876797i	$0.340324\pi$
$60$	9.95856	0.0214274
$61$	472.814	0.992421	0.496210	−	0.868202i	$-0.334725\pi$
$61$	472.814	0.992421	0.496210	+	0.868202i	$0.334725\pi$
$62$	−1076.95	−2.20602
$63$	−842.712	−1.68526
$64$	47.1080	0.0920077
$65$	−83.3978	−0.159142
$66$	30.3403	0.0565854
$67$	841.081	1.53365	0.766824	−	0.641858i	$-0.221836\pi$
$67$	841.081	1.53365	0.766824	+	0.641858i	$0.221836\pi$
$68$	1431.36	2.55262
$69$	−2.00488	−0.00349797
$70$	−1023.53	−1.74764
$71$	−667.610	−1.11593	−0.557963	−	0.829866i	$-0.688417\pi$
$71$	−667.610	−1.11593	−0.557963	+	0.829866i	$0.688417\pi$
$72$	1395.47	2.28413
$73$	73.0807	0.117171	0.0585853	−	0.998282i	$-0.481341\pi$
$73$	73.0807	0.117171	0.0585853	+	0.998282i	$0.481341\pi$
$74$	257.335	0.404251
$75$	−7.18446	−0.0110612
$76$	160.843	0.242763
$77$	−2163.12	−3.20144
$78$	5.69267	0.00826369
$79$	231.340	0.329466	0.164733	−	0.986338i	$-0.447324\pi$
$79$	231.340	0.329466	0.164733	+	0.986338i	$0.447324\pi$
$80$	765.132	1.06930
$81$	728.405	0.999184
$82$	1317.49	1.77430
$83$	−130.029	−0.171958	−0.0859791	−	0.996297i	$-0.527402\pi$
$83$	−130.029	−0.171958	−0.0859791	+	0.996297i	$0.527402\pi$
$84$	48.4639	0.0629506
$85$	506.866	0.646792
$86$	1823.64	2.28660
$87$	19.9714	0.0246110
$88$	3581.98	4.33909
$89$	82.3784	0.0981134	0.0490567	−	0.998796i	$-0.484379\pi$
$89$	82.3784	0.0981134	0.0490567	+	0.998796i	$0.484379\pi$
$90$	884.937	1.03645
$91$	−405.860	−0.467535
$92$	−423.878	−0.480351
$93$	18.0576	0.0201342
$94$	855.614	0.938828
$95$	56.9569	0.0615121
$96$	−16.7883	−0.0178484
$97$	−1524.28	−1.59554	−0.797768	−	0.602965i	$-0.793986\pi$
$97$	−1524.28	−1.59554	−0.797768	+	0.602965i	$0.793986\pi$
$98$	−3228.19	−3.32752
$99$	1870.22	1.89863
$100$	−1518.96	−1.51896
$101$	−1560.26	−1.53714	−0.768571	−	0.639765i	$-0.779032\pi$
$101$	−1560.26	−1.53714	−0.768571	+	0.639765i	$0.779032\pi$
$102$	−34.5982	−0.0335856
$103$	103.000	0.0985329
$103$	103.000	0.0985329
$104$	672.075	0.633677
$105$	17.1618	0.0159506
$106$	2620.55	2.40122
$107$	−1978.97	−1.78798	−0.893992	−	0.448084i	$-0.852107\pi$
$107$	−1978.97	−1.78798	−0.893992	+	0.448084i	$0.852107\pi$
$108$	−83.8146	−0.0746765
$109$	−903.042	−0.793539	−0.396769	−	0.917918i	$-0.629869\pi$
$109$	−903.042	−0.793539	−0.396769	+	0.917918i	$0.629869\pi$
$110$	2271.51	1.96891
$111$	−4.31479	−0.00368957
$112$	3723.56	3.14146
$113$	1247.75	1.03875	0.519373	−	0.854548i	$-0.326165\pi$
$113$	1247.75	1.03875	0.519373	+	0.854548i	$0.326165\pi$
$114$	−3.88783	−0.00319411
$115$	−150.101	−0.121713
$116$	4222.41	3.37966
$117$	350.905	0.277275
$118$	−2227.32	−1.73764
$119$	2466.69	1.90018
$120$	−28.4186	−0.0216188
$121$	3469.60	2.60676
$122$	−2416.27	−1.79311
$123$	−22.0907	−0.0161939
$124$	3817.77	2.76489
$125$	−1339.79	−0.958673
$126$	4306.60	3.04494
$127$	1070.32	0.747837	0.373919	−	0.927462i	$-0.378014\pi$
$127$	1070.32	0.747837	0.373919	+	0.927462i	$0.378014\pi$
$128$	1326.66	0.916102
$129$	−30.5774	−0.0208697
$130$	426.197	0.287538
$131$	−1649.39	−1.10006	−0.550029	−	0.835146i	$-0.685383\pi$
$131$	−1649.39	−1.10006	−0.550029	+	0.835146i	$0.685383\pi$
$132$	−107.556	−0.0709206
$133$	277.184	0.180714
$134$	−4298.26	−2.77100
$135$	−29.6799	−0.0189218
$136$	−4084.66	−2.57542
$137$	−2266.07	−1.41316	−0.706581	−	0.707632i	$-0.749764\pi$
$137$	−2266.07	−1.41316	−0.706581	+	0.707632i	$0.749764\pi$
$138$	10.2458	0.00632013
$139$	−1528.04	−0.932422	−0.466211	−	0.884674i	$-0.654381\pi$
$139$	−1528.04	−0.932422	−0.466211	+	0.884674i	$0.654381\pi$
$140$	3628.38	2.19039
$141$	−14.3463	−0.00856862
$142$	3411.76	2.01625
$143$	900.723	0.526729
$144$	−3219.37	−1.86306
$145$	1495.21	0.856350
$146$	−373.472	−0.211704
$147$	54.1279	0.0303700
$148$	−912.244	−0.506662
$149$	155.763	0.0856414	0.0428207	−	0.999083i	$-0.486366\pi$
$149$	155.763	0.0856414	0.0428207	+	0.999083i	$0.486366\pi$
$150$	36.7155	0.0199854
$151$	−627.940	−0.338418	−0.169209	−	0.985580i	$-0.554121\pi$
$151$	−627.940	−0.338418	−0.169209	+	0.985580i	$0.554121\pi$
$152$	−458.997	−0.244931
$153$	−2132.69	−1.12691
$154$	11054.4	5.78436
$155$	1351.93	0.700577
$156$	−20.1803	−0.0103572
$157$	2762.01	1.40403	0.702014	−	0.712163i	$-0.252284\pi$
$157$	2762.01	1.40403	0.702014	+	0.712163i	$0.252284\pi$
$158$	−1182.24	−0.595279
$159$	−43.9393	−0.0219158
$160$	−1256.90	−0.621041
$161$	−730.476	−0.357575
$162$	−3722.45	−1.80533
$163$	−342.411	−0.164538	−0.0822690	−	0.996610i	$-0.526217\pi$
$163$	−342.411	−0.164538	−0.0822690	+	0.996610i	$0.526217\pi$
$164$	−4670.48	−2.22380
$165$	−38.0870	−0.0179701
$166$	664.501	0.310694
$167$	−1203.39	−0.557610	−0.278805	−	0.960348i	$-0.589938\pi$
$167$	−1203.39	−0.557610	−0.278805	+	0.960348i	$0.589938\pi$
$168$	−138.301	−0.0635128
$169$	169.000	0.0769231
$170$	−2590.29	−1.16862
$171$	−239.652	−0.107173
$172$	−6464.74	−2.86588
$173$	−2356.06	−1.03542	−0.517712	−	0.855555i	$-0.673216\pi$
$173$	−2356.06	−1.03542	−0.517712	+	0.855555i	$0.673216\pi$
$174$	−102.062	−0.0444673
$175$	−2617.64	−1.13072
$176$	−8263.67	−3.53919
$177$	37.3460	0.0158593
$178$	−420.987	−0.177271
$179$	2863.69	1.19577	0.597884	−	0.801583i	$-0.296008\pi$
$179$	2863.69	1.19577	0.597884	+	0.801583i	$0.296008\pi$
$180$	−3137.08	−1.29902
$181$	−510.132	−0.209491	−0.104745	−	0.994499i	$-0.533403\pi$
$181$	−510.132	−0.209491	−0.104745	+	0.994499i	$0.533403\pi$
$182$	2074.11	0.844744
$183$	40.5142	0.0163656
$184$	1209.61	0.484641
$185$	−323.039	−0.128380
$186$	−92.2815	−0.0363785
$187$	−5474.31	−2.14076
$188$	−3033.13	−1.17667
$189$	−144.439	−0.0555894
$190$	−291.073	−0.111140
$191$	−3620.75	−1.37167	−0.685833	−	0.727759i	$-0.740562\pi$
$191$	−3620.75	−1.37167	−0.685833	+	0.727759i	$0.740562\pi$
$192$	4.03656	0.00151726
$193$	2173.54	0.810648	0.405324	−	0.914173i	$-0.367159\pi$
$193$	2173.54	0.810648	0.405324	+	0.914173i	$0.367159\pi$
$194$	7789.68	2.88282
$195$	−7.14614	−0.00262434
$196$	11443.9	4.17050
$197$	−18.4727	−0.00668084	−0.00334042	−	0.999994i	$-0.501063\pi$
$197$	−18.4727	−0.00668084	−0.00334042	+	0.999994i	$0.501063\pi$
$198$	−9557.60	−3.43045
$199$	−748.460	−0.266618	−0.133309	−	0.991075i	$-0.542560\pi$
$199$	−748.460	−0.266618	−0.133309	+	0.991075i	$0.542560\pi$
$200$	4334.63	1.53252
$201$	72.0700	0.0252907
$202$	7973.54	2.77731
$203$	7276.54	2.51583
$204$	122.650	0.0420941
$205$	−1653.88	−0.563474
$206$	−526.372	−0.178029
$207$	631.565	0.212062
$208$	−1550.49	−0.516861
$209$	−615.153	−0.203593
$210$	−87.7035	−0.0288196
$211$	410.215	0.133841	0.0669203	−	0.997758i	$-0.478683\pi$
$211$	410.215	0.133841	0.0669203	+	0.997758i	$0.478683\pi$
$212$	−9289.76	−3.00954
$213$	−57.2057	−0.0184022
$214$	10113.3	3.23053
$215$	−2289.26	−0.726167
$216$	239.181	0.0753435
$217$	6579.24	2.05819
$218$	4614.91	1.43377
$219$	6.26210	0.00193221
$220$	−8052.44	−2.46771
$221$	−1027.13	−0.312634
$222$	22.0503	0.00666632
$223$	2882.67	0.865640	0.432820	−	0.901480i	$-0.357519\pi$
$223$	2882.67	0.865640	0.432820	+	0.901480i	$0.357519\pi$
$224$	−6116.78	−1.82453
$225$	2263.20	0.670577
$226$	−6376.50	−1.87681
$227$	−946.471	−0.276738	−0.138369	−	0.990381i	$-0.544186\pi$
$227$	−946.471	−0.276738	−0.138369	+	0.990381i	$0.544186\pi$
$228$	13.7822	0.00400330
$229$	1902.35	0.548955	0.274477	−	0.961594i	$-0.411495\pi$
$229$	1902.35	0.548955	0.274477	+	0.961594i	$0.411495\pi$
$230$	767.077	0.219911
$231$	−185.352	−0.0527935
$232$	−12049.4	−3.40984
$233$	−1857.96	−0.522399	−0.261199	−	0.965285i	$-0.584118\pi$
$233$	−1857.96	−0.522399	−0.261199	+	0.965285i	$0.584118\pi$
$234$	−1793.26	−0.500980
$235$	−1074.07	−0.298148
$236$	7895.79	2.17785
$237$	19.8229	0.00543307
$238$	−12605.8	−3.43324
$239$	−514.749	−0.139315	−0.0696577	−	0.997571i	$-0.522191\pi$
$239$	−514.749	−0.139315	−0.0696577	+	0.997571i	$0.522191\pi$
$240$	65.5622	0.0176334
$241$	454.418	0.121459	0.0607296	−	0.998154i	$-0.480657\pi$
$241$	454.418	0.121459	0.0607296	+	0.998154i	$0.480657\pi$
$242$	−17731.1	−4.70991
$243$	187.330	0.0494537
$244$	8565.62	2.24737
$245$	4052.43	1.05674
$246$	112.893	0.0292592
$247$	−115.419	−0.0297326
$248$	−10894.7	−2.78958
$249$	−11.1418	−0.00283569
$250$	6846.85	1.73213
$251$	4459.69	1.12149	0.560743	−	0.827990i	$-0.310515\pi$
$251$	4459.69	1.12149	0.560743	+	0.827990i	$0.310515\pi$
$252$	−15266.8	−3.81633
$253$	1621.14	0.402846
$254$	−5469.76	−1.35119
$255$	43.4320	0.0106660
$256$	−7156.62	−1.74722
$257$	−6353.43	−1.54208	−0.771042	−	0.636784i	$-0.780264\pi$
$257$	−6353.43	−1.54208	−0.771042	+	0.636784i	$0.780264\pi$
$258$	156.263	0.0377073
$259$	−1572.09	−0.377161
$260$	−1510.86	−0.360382
$261$	−6291.26	−1.49203
$262$	8429.03	1.98758
$263$	3426.16	0.803293	0.401646	−	0.915795i	$-0.368438\pi$
$263$	3426.16	0.803293	0.401646	+	0.915795i	$0.368438\pi$
$264$	306.930	0.0715540
$265$	−3289.63	−0.762568
$266$	−1416.52	−0.326513
$267$	7.05879	0.00161794
$268$	15237.2	3.47299
$269$	−3708.85	−0.840642	−0.420321	−	0.907375i	$-0.638083\pi$
$269$	−3708.85	−0.840642	−0.420321	+	0.907375i	$0.638083\pi$
$270$	151.677	0.0341879
$271$	−4745.64	−1.06375	−0.531876	−	0.846822i	$-0.678513\pi$
$271$	−4745.64	−1.06375	−0.531876	+	0.846822i	$0.678513\pi$
$272$	9423.38	2.10065
$273$	−34.7771	−0.00770992
$274$	11580.5	2.55330
$275$	5809.32	1.27387
$276$	−36.3210	−0.00792125
$277$	−8476.36	−1.83861	−0.919305	−	0.393545i	$-0.871249\pi$
$277$	−8476.36	−1.83861	−0.919305	+	0.393545i	$0.871249\pi$
$278$	7808.90	1.68470
$279$	−5688.37	−1.22062
$280$	−10354.3	−2.20995
$281$	−7835.32	−1.66340	−0.831701	−	0.555223i	$-0.812633\pi$
$281$	−7835.32	−1.66340	−0.831701	+	0.555223i	$0.812633\pi$
$282$	73.3154	0.0154818
$283$	2581.77	0.542299	0.271149	−	0.962537i	$-0.412596\pi$
$283$	2581.77	0.542299	0.271149	+	0.962537i	$0.412596\pi$
$284$	−12094.6	−2.52705
$285$	4.88049	0.00101437
$286$	−4603.06	−0.951695
$287$	−8048.71	−1.65540
$288$	5288.53	1.08205
$289$	1329.56	0.270621
$290$	−7641.15	−1.54725
$291$	−130.611	−0.0263113
$292$	1323.95	0.265336
$293$	9352.94	1.86486	0.932431	−	0.361348i	$-0.117683\pi$
$293$	9352.94	1.86486	0.932431	+	0.361348i	$0.117683\pi$
$294$	−276.616	−0.0548726
$295$	2796.01	0.551831
$296$	2603.26	0.511187
$297$	320.553	0.0626275
$298$	−796.010	−0.154737
$299$	304.170	0.0588314
$300$	−130.155	−0.0250484
$301$	−11140.8	−2.13337
$302$	3209.03	0.611453
$303$	−133.694	−0.0253483
$304$	1058.91	0.199779
$305$	3033.21	0.569445
$306$	10898.9	2.03611
$307$	−8854.48	−1.64610	−0.823049	−	0.567970i	$-0.807729\pi$
$307$	−8854.48	−1.64610	−0.823049	+	0.567970i	$0.807729\pi$
$308$	−39187.7	−7.24976
$309$	8.82580	0.00162486
$310$	−6908.90	−1.26580
$311$	−4668.51	−0.851212	−0.425606	−	0.904909i	$-0.639939\pi$
$311$	−4668.51	−0.851212	−0.425606	+	0.904909i	$0.639939\pi$
$312$	57.5884	0.0104497
$313$	−3700.68	−0.668290	−0.334145	−	0.942522i	$-0.608448\pi$
$313$	−3700.68	−0.668290	−0.334145	+	0.942522i	$0.608448\pi$
$314$	−14115.0	−2.53680
$315$	−5406.18	−0.966996
$316$	4191.02	0.746086
$317$	3654.37	0.647477	0.323738	−	0.946147i	$-0.395060\pi$
$317$	3654.37	0.647477	0.323738	+	0.946147i	$0.395060\pi$
$318$	224.548	0.0395975
$319$	−16148.8	−2.83435
$320$	302.208	0.0527935
$321$	−169.573	−0.0294848
$322$	3733.03	0.646067
$323$	701.482	0.120841
$324$	13196.0	2.26268
$325$	1089.98	0.186035
$326$	1749.86	0.297288
$327$	−77.3793	−0.0130859
$328$	13328.1	2.24366
$329$	−5227.04	−0.875915
$330$	194.640	0.0324684
$331$	−1662.10	−0.276004	−0.138002	−	0.990432i	$-0.544068\pi$
$331$	−1662.10	−0.276004	−0.138002	+	0.990432i	$0.544068\pi$
$332$	−2355.64	−0.389405
$333$	1359.22	0.223677
$334$	6149.79	1.00749
$335$	5395.72	0.879999
$336$	319.062	0.0518044
$337$	−10314.2	−1.66721	−0.833604	−	0.552362i	$-0.813727\pi$
$337$	−10314.2	−1.66721	−0.833604	+	0.552362i	$0.813727\pi$
$338$	−863.659	−0.138985
$339$	106.916	0.0171295
$340$	9182.50	1.46468
$341$	−14601.3	−2.31877
$342$	1224.72	0.193641
$343$	9012.92	1.41881
$344$	18448.4	2.89148
$345$	−12.8618	−0.00200711
$346$	12040.4	1.87080
$347$	1247.58	0.193007	0.0965035	−	0.995333i	$-0.469234\pi$
$347$	1247.58	0.193007	0.0965035	+	0.995333i	$0.469234\pi$
$348$	361.807	0.0557324
$349$	8207.83	1.25890	0.629448	−	0.777042i	$-0.283281\pi$
$349$	8207.83	1.25890	0.629448	+	0.777042i	$0.283281\pi$
$350$	13377.2	2.04298
$351$	60.1444	0.00914607
$352$	13574.9	2.05553
$353$	1307.40	0.197126	0.0985632	−	0.995131i	$-0.468575\pi$
$353$	1307.40	0.197126	0.0985632	+	0.995131i	$0.468575\pi$
$354$	−190.853	−0.0286546
$355$	−4282.86	−0.640312
$356$	1492.39	0.222181
$357$	211.364	0.0313350
$358$	−14634.6	−2.16051
$359$	3478.65	0.511410	0.255705	−	0.966755i	$-0.417692\pi$
$359$	3478.65	0.511410	0.255705	+	0.966755i	$0.417692\pi$
$360$	8952.24	1.31062
$361$	−6780.17	−0.988508
$362$	2606.98	0.378508
$363$	297.301	0.0429870
$364$	−7352.67	−1.05875
$365$	468.829	0.0672318
$366$	−207.044	−0.0295693
$367$	−7216.14	−1.02637	−0.513187	−	0.858277i	$-0.671535\pi$
$367$	−7216.14	−1.02637	−0.513187	+	0.858277i	$0.671535\pi$
$368$	−2790.60	−0.395299
$369$	6958.87	0.981746
$370$	1650.86	0.231957
$371$	−16009.2	−2.24031
$372$	327.135	0.0455945
$373$	−10910.0	−1.51447	−0.757234	−	0.653144i	$-0.773450\pi$
$373$	−10910.0	−1.51447	−0.757234	+	0.653144i	$0.773450\pi$
$374$	27975.9	3.86792
$375$	−114.803	−0.0158090
$376$	8655.60	1.18718
$377$	−3029.95	−0.413926
$378$	738.143	0.100439
$379$	5663.24	0.767549	0.383775	−	0.923427i	$-0.374624\pi$
$379$	5663.24	0.767549	0.383775	+	0.923427i	$0.374624\pi$
$380$	1031.84	0.139296
$381$	91.7127	0.0123322
$382$	18503.5	2.47833
$383$	4290.38	0.572398	0.286199	−	0.958170i	$-0.407608\pi$
$383$	4290.38	0.572398	0.286199	+	0.958170i	$0.407608\pi$
$384$	113.678	0.0151070
$385$	−13876.9	−1.83697
$386$	−11107.7	−1.46468
$387$	9632.27	1.26521
$388$	−27614.2	−3.61314
$389$	−6922.94	−0.902332	−0.451166	−	0.892440i	$-0.648992\pi$
$389$	−6922.94	−0.902332	−0.451166	+	0.892440i	$0.648992\pi$
$390$	36.5197	0.00474166
$391$	−1848.65	−0.239105
$392$	−32657.2	−4.20775
$393$	−141.332	−0.0181406
$394$	94.4030	0.0120709
$395$	1484.10	0.189046
$396$	33881.4	4.29951
$397$	14013.7	1.77160	0.885801	−	0.464064i	$-0.153609\pi$
$397$	14013.7	1.77160	0.885801	+	0.464064i	$0.153609\pi$
$398$	3824.93	0.481725
$399$	23.7512	0.00298007
$400$	−10000.1	−1.25001
$401$	15125.1	1.88356	0.941782	−	0.336224i	$-0.109150\pi$
$401$	15125.1	1.88356	0.941782	+	0.336224i	$0.109150\pi$
$402$	−368.307	−0.0456953
$403$	−2739.59	−0.338632
$404$	−28266.0	−3.48090
$405$	4672.88	0.573326
$406$	−37186.1	−4.54560
$407$	3488.92	0.424912
$408$	−350.004	−0.0424701
$409$	−6834.62	−0.826285	−0.413142	−	0.910666i	$-0.635569\pi$
$409$	−6834.62	−0.826285	−0.413142	+	0.910666i	$0.635569\pi$
$410$	8452.00	1.01808
$411$	−194.174	−0.0233038
$412$	1865.97	0.223131
$413$	13607.0	1.62120
$414$	−3227.55	−0.383154
$415$	−834.164	−0.0986687
$416$	2547.02	0.300187
$417$	−130.934	−0.0153761
$418$	3143.68	0.367853
$419$	−4102.35	−0.478312	−0.239156	−	0.970981i	$-0.576871\pi$
$419$	−4102.35	−0.478312	−0.239156	+	0.970981i	$0.576871\pi$
$420$	310.907	0.0361207
$421$	−11610.7	−1.34411	−0.672057	−	0.740499i	$-0.734589\pi$
$421$	−11610.7	−1.34411	−0.672057	+	0.740499i	$0.734589\pi$
$422$	−2096.37	−0.241823
$423$	4519.27	0.519467
$424$	26510.1	3.03642
$425$	−6624.58	−0.756093
$426$	292.345	0.0332492
$427$	14761.3	1.67295
$428$	−35851.5	−4.04894
$429$	77.1806	0.00868605
$430$	11699.0	1.31204
$431$	3929.38	0.439145	0.219573	−	0.975596i	$-0.429534\pi$
$431$	3929.38	0.439145	0.219573	+	0.975596i	$0.429534\pi$
$432$	−551.794	−0.0614542
$433$	11073.4	1.22899	0.614495	−	0.788921i	$-0.289360\pi$
$433$	11073.4	1.22899	0.614495	+	0.788921i	$0.289360\pi$
$434$	−33622.6	−3.71874
$435$	128.121	0.0141217
$436$	−16359.7	−1.79699
$437$	−207.734	−0.0227397
$438$	−32.0019	−0.00349111
$439$	−2269.41	−0.246727	−0.123363	−	0.992362i	$-0.539368\pi$
$439$	−2269.41	−0.246727	−0.123363	+	0.992362i	$0.539368\pi$
$440$	22979.2	2.48975
$441$	−17051.0	−1.84116
$442$	5249.04	0.564868
$443$	−13413.2	−1.43856	−0.719278	−	0.694722i	$-0.755527\pi$
$443$	−13413.2	−1.43856	−0.719278	+	0.694722i	$0.755527\pi$
$444$	−78.1678	−0.00835514
$445$	528.475	0.0562969
$446$	−14731.6	−1.56404
$447$	13.3469	0.00141227
$448$	1470.71	0.155100
$449$	7556.94	0.794285	0.397143	−	0.917757i	$-0.370002\pi$
$449$	7556.94	0.794285	0.397143	+	0.917757i	$0.370002\pi$
$450$	−11565.9	−1.21160
$451$	17862.4	1.86499
$452$	22604.5	2.35227
$453$	−53.8066	−0.00558069
$454$	4836.85	0.500010
$455$	−2603.68	−0.268269
$456$	−39.3302	−0.00403905
$457$	6863.59	0.702550	0.351275	−	0.936272i	$-0.385748\pi$
$457$	6863.59	0.702550	0.351275	+	0.936272i	$0.385748\pi$
$458$	−9721.77	−0.991852
$459$	−365.539	−0.0371718
$460$	−2719.27	−0.275623
$461$	6656.47	0.672500	0.336250	−	0.941773i	$-0.390841\pi$
$461$	6656.47	0.672500	0.336250	+	0.941773i	$0.390841\pi$
$462$	947.226	0.0953873
$463$	−12584.6	−1.26319	−0.631594	−	0.775299i	$-0.717599\pi$
$463$	−12584.6	−1.26319	−0.631594	+	0.775299i	$0.717599\pi$
$464$	27798.2	2.78125
$465$	115.843	0.0115529
$466$	9494.92	0.943871
$467$	−19839.0	−1.96582	−0.982911	−	0.184082i	$-0.941069\pi$
$467$	−19839.0	−1.96582	−0.982911	+	0.184082i	$0.941069\pi$
$468$	6357.07	0.627897
$469$	26258.6	2.58531
$470$	5488.95	0.538694
$471$	236.670	0.0231532
$472$	−22532.1	−2.19730
$473$	24724.7	2.40347
$474$	−101.303	−0.00981648
$475$	−744.410	−0.0719071
$476$	44687.2	4.30301
$477$	13841.5	1.32863
$478$	2630.58	0.251715
$479$	945.042	0.0901463	0.0450732	−	0.998984i	$-0.485648\pi$
$479$	945.042	0.0901463	0.0450732	+	0.998984i	$0.485648\pi$
$480$	−107.700	−0.0102413
$481$	654.616	0.0620539
$482$	−2322.26	−0.219453
$483$	−62.5926	−0.00589660
$484$	62856.2	5.90310
$485$	−9778.57	−0.915510
$486$	−957.334	−0.0893530
$487$	11812.5	1.09913	0.549564	−	0.835452i	$-0.314794\pi$
$487$	11812.5	1.09913	0.549564	+	0.835452i	$0.314794\pi$
$488$	−24443.6	−2.26744
$489$	−29.3403	−0.00271332
$490$	−20709.6	−1.90931
$491$	10090.5	0.927446	0.463723	−	0.885980i	$-0.346513\pi$
$491$	10090.5	0.927446	0.463723	+	0.885980i	$0.346513\pi$
$492$	−400.201	−0.0366717
$493$	18415.1	1.68230
$494$	589.839	0.0537209
$495$	11997.9	1.08942
$496$	25134.3	2.27533
$497$	−20842.8	−1.88114
$498$	56.9394	0.00512352
$499$	−15067.1	−1.35169	−0.675846	−	0.737043i	$-0.736221\pi$
$499$	−15067.1	−1.35169	−0.675846	+	0.737043i	$0.736221\pi$
$500$	−24271.9	−2.17094
$501$	−103.115	−0.00919529
$502$	−22790.8	−2.02630
$503$	−8194.49	−0.726390	−0.363195	−	0.931713i	$-0.618314\pi$
$503$	−8194.49	−0.726390	−0.363195	+	0.931713i	$0.618314\pi$
$504$	43566.6	3.85042
$505$	−10009.4	−0.882004
$506$	−8284.68	−0.727863
$507$	14.4812	0.00126850
$508$	19390.1	1.69350
$509$	−3586.94	−0.312354	−0.156177	−	0.987729i	$-0.549917\pi$
$509$	−3586.94	−0.312354	−0.156177	+	0.987729i	$0.549917\pi$
$510$	−221.955	−0.0192713
$511$	2281.58	0.197517
$512$	25960.0	2.24078
$513$	−41.0759	−0.00353517
$514$	32468.6	2.78624
$515$	660.767	0.0565377
$516$	−553.947	−0.0472600
$517$	11600.3	0.986813
$518$	8034.00	0.681455
$519$	−201.885	−0.0170747
$520$	4311.51	0.363600
$521$	11216.7	0.943206	0.471603	−	0.881811i	$-0.343675\pi$
$521$	11216.7	0.943206	0.471603	+	0.881811i	$0.343675\pi$
$522$	32150.9	2.69580
$523$	−14293.1	−1.19502	−0.597510	−	0.801862i	$-0.703843\pi$
$523$	−14293.1	−1.19502	−0.597510	+	0.801862i	$0.703843\pi$
$524$	−29880.7	−2.49111
$525$	−224.299	−0.0186461
$526$	−17509.1	−1.45139
$527$	16650.4	1.37628
$528$	−708.093	−0.0583632
$529$	−11619.5	−0.955005
$530$	16811.4	1.37781
$531$	−11764.5	−0.961460
$532$	5021.53	0.409231
$533$	3351.48	0.272361
$534$	−36.0733	−0.00292330
$535$	−12695.5	−1.02593
$536$	−43482.3	−3.50401
$537$	245.382	0.0197189
$538$	18953.7	1.51887
$539$	−43767.5	−3.49759
$540$	−537.689	−0.0428490
$541$	4629.02	0.367869	0.183934	−	0.982939i	$-0.441117\pi$
$541$	4629.02	0.367869	0.183934	+	0.982939i	$0.441117\pi$
$542$	24252.1	1.92199
$543$	−43.7119	−0.00345461
$544$	−15480.0	−1.22004
$545$	−5793.21	−0.455328
$546$	177.725	0.0139303
$547$	−10829.1	−0.846471	−0.423236	−	0.906020i	$-0.639106\pi$
$547$	−10829.1	−0.846471	−0.423236	+	0.906020i	$0.639106\pi$
$548$	−41052.6	−3.20015
$549$	−12762.5	−0.992151
$550$	−29688.0	−2.30163
$551$	2069.32	0.159992
$552$	103.649	0.00799200
$553$	7222.45	0.555388
$554$	43317.6	3.32200
$555$	−27.6803	−0.00211705
$556$	−27682.3	−2.11150
$557$	6358.02	0.483659	0.241829	−	0.970319i	$-0.422253\pi$
$557$	6358.02	0.483659	0.241829	+	0.970319i	$0.422253\pi$
$558$	29069.9	2.20542
$559$	4639.02	0.351001
$560$	23887.5	1.80255
$561$	−469.079	−0.0353022
$562$	40041.7	3.00544
$563$	−4862.00	−0.363959	−0.181979	−	0.983302i	$-0.558250\pi$
$563$	−4862.00	−0.363959	−0.181979	+	0.983302i	$0.558250\pi$
$564$	−259.901	−0.0194039
$565$	8004.58	0.596027
$566$	−13193.9	−0.979826
$567$	22740.8	1.68435
$568$	34514.2	2.54962
$569$	−10141.4	−0.747187	−0.373594	−	0.927593i	$-0.621874\pi$
$569$	−10141.4	−0.747187	−0.373594	+	0.927593i	$0.621874\pi$
$570$	−24.9413	−0.00183276
$571$	−14001.7	−1.02619	−0.513094	−	0.858333i	$-0.671501\pi$
$571$	−14001.7	−1.02619	−0.513094	+	0.858333i	$0.671501\pi$
$572$	16317.7	1.19279
$573$	−310.253	−0.0226195
$574$	41132.2	2.99098
$575$	1961.78	0.142281
$576$	−1271.57	−0.0919827
$577$	25969.7	1.87371	0.936857	−	0.349713i	$-0.113721\pi$
$577$	25969.7	1.87371	0.936857	+	0.349713i	$0.113721\pi$
$578$	−6794.61	−0.488959
$579$	186.245	0.0133680
$580$	27087.7	1.93923
$581$	−4059.51	−0.289874
$582$	667.477	0.0475392
$583$	35529.1	2.52395
$584$	−3778.13	−0.267706
$585$	2251.13	0.159099
$586$	−47797.3	−3.36943
$587$	−23924.7	−1.68224	−0.841122	−	0.540845i	$-0.818105\pi$
$587$	−23924.7	−1.68224	−0.841122	+	0.540845i	$0.818105\pi$
$588$	980.594	0.0687739
$589$	1871.01	0.130889
$590$	−14288.8	−0.997048
$591$	−1.58288	−0.000110171 0
$592$	−6005.76	−0.416952
$593$	13872.8	0.960690	0.480345	−	0.877080i	$-0.340511\pi$
$593$	13872.8	0.960690	0.480345	+	0.877080i	$0.340511\pi$
$594$	−1638.16	−0.113155
$595$	15824.4	1.09031
$596$	2821.83	0.193937
$597$	−64.1336	−0.00439667
$598$	−1554.43	−0.106297
$599$	17167.2	1.17101	0.585503	−	0.810670i	$-0.300897\pi$
$599$	17167.2	1.17101	0.585503	+	0.810670i	$0.300897\pi$
$600$	371.423	0.0252721
$601$	−14990.6	−1.01744	−0.508718	−	0.860933i	$-0.669880\pi$
$601$	−14990.6	−1.01744	−0.508718	+	0.860933i	$0.669880\pi$
$602$	56934.0	3.85458
$603$	−22703.0	−1.53323
$604$	−11375.9	−0.766357
$605$	22258.3	1.49575
$606$	683.232	0.0457994
$607$	12790.7	0.855289	0.427644	−	0.903947i	$-0.359344\pi$
$607$	12790.7	0.855289	0.427644	+	0.903947i	$0.359344\pi$
$608$	−1739.50	−0.116030
$609$	623.508	0.0414874
$610$	−15500.9	−1.02887
$611$	2176.54	0.144113
$612$	−38636.3	−2.55193
$613$	−24656.7	−1.62459	−0.812296	−	0.583246i	$-0.801782\pi$
$613$	−24656.7	−1.62459	−0.812296	+	0.583246i	$0.801782\pi$
$614$	45250.0	2.97417
$615$	−141.717	−0.00929199
$616$	111829.	7.31450
$617$	8700.06	0.567668	0.283834	−	0.958873i	$-0.408394\pi$
$617$	8700.06	0.567668	0.283834	+	0.958873i	$0.408394\pi$
$618$	−45.1034	−0.00293580
$619$	4242.69	0.275490	0.137745	−	0.990468i	$-0.456015\pi$
$619$	4242.69	0.275490	0.137745	+	0.990468i	$0.456015\pi$
$620$	24491.9	1.58648
$621$	108.249	0.00699498
$622$	23858.0	1.53797
$623$	2571.86	0.165392
$624$	−132.857	−0.00852332
$625$	1885.61	0.120679
$626$	18912.0	1.20747
$627$	−52.7108	−0.00335736
$628$	50037.3	3.17946
$629$	−3978.55	−0.252202
$630$	27627.8	1.74717
$631$	−18858.7	−1.18979	−0.594893	−	0.803805i	$-0.702805\pi$
$631$	−18858.7	−1.18979	−0.594893	+	0.803805i	$0.702805\pi$
$632$	−11959.9	−0.752749
$633$	35.1503	0.00220711
$634$	−18675.3	−1.16986
$635$	6866.32	0.429105
$636$	−796.015	−0.0496290
$637$	−8211.97	−0.510785
$638$	82526.8	5.12111
$639$	18020.6	1.11562
$640$	8510.79	0.525654
$641$	−25281.1	−1.55779	−0.778894	−	0.627155i	$-0.784219\pi$
$641$	−25281.1	−1.55779	−0.778894	+	0.627155i	$0.784219\pi$
$642$	866.586	0.0532732
$643$	−24132.1	−1.48006	−0.740030	−	0.672574i	$-0.765189\pi$
$643$	−24132.1	−1.48006	−0.740030	+	0.672574i	$0.765189\pi$
$644$	−13233.5	−0.809739
$645$	−196.160	−0.0119749
$646$	−3584.86	−0.218335
$647$	27068.9	1.64480	0.822401	−	0.568908i	$-0.192634\pi$
$647$	27068.9	1.64480	0.822401	+	0.568908i	$0.192634\pi$
$648$	−37657.2	−2.28289
$649$	−30197.8	−1.82645
$650$	−5570.26	−0.336129
$651$	563.758	0.0339407
$652$	−6203.20	−0.372601
$653$	1315.09	0.0788110	0.0394055	−	0.999223i	$-0.487454\pi$
$653$	1315.09	0.0788110	0.0394055	+	0.999223i	$0.487454\pi$
$654$	395.440	0.0236436
$655$	−10581.2	−0.631207
$656$	−30748.1	−1.83005
$657$	−1972.64	−0.117139
$658$	26712.3	1.58260
$659$	−27994.3	−1.65478	−0.827392	−	0.561626i	$-0.810176\pi$
$659$	−27994.3	−1.65478	−0.827392	+	0.561626i	$0.810176\pi$
$660$	−689.993	−0.0406938
$661$	−32294.7	−1.90033	−0.950166	−	0.311745i	$-0.899087\pi$
$661$	−32294.7	−1.90033	−0.950166	+	0.311745i	$0.899087\pi$
$662$	8493.99	0.498684
$663$	−88.0120	−0.00515551
$664$	6722.25	0.392882
$665$	1778.20	0.103692
$666$	−6946.15	−0.404141
$667$	−5453.36	−0.316574
$668$	−21800.8	−1.26272
$669$	247.008	0.0142749
$670$	−27574.3	−1.58998
$671$	−32759.6	−1.88475
$672$	−524.131	−0.0300875
$673$	15024.1	0.860532	0.430266	−	0.902702i	$-0.358420\pi$
$673$	15024.1	0.860532	0.430266	+	0.902702i	$0.358420\pi$
$674$	52709.7	3.01231
$675$	387.908	0.0221194
$676$	3061.65	0.174195
$677$	2280.70	0.129475	0.0647374	−	0.997902i	$-0.479379\pi$
$677$	2280.70	0.129475	0.0647374	+	0.997902i	$0.479379\pi$
$678$	−546.386	−0.0309496
$679$	−47588.0	−2.68963
$680$	−26204.0	−1.47776
$681$	−81.1006	−0.00456356
$682$	74618.3	4.18956
$683$	−22228.9	−1.24534	−0.622668	−	0.782486i	$-0.713951\pi$
$683$	−22228.9	−1.24534	−0.622668	+	0.782486i	$0.713951\pi$
$684$	−4341.59	−0.242697
$685$	−14537.3	−0.810865
$686$	−46059.7	−2.56351
$687$	163.007	0.00905257
$688$	−42560.6	−2.35844
$689$	6666.22	0.368596
$690$	65.7289	0.00362646
$691$	14908.0	0.820732	0.410366	−	0.911921i	$-0.365401\pi$
$691$	14908.0	0.820732	0.410366	+	0.911921i	$0.365401\pi$
$692$	−42683.0	−2.34475
$693$	58388.4	3.20057
$694$	−6375.62	−0.348725
$695$	−9802.70	−0.535018
$696$	−1032.48	−0.0562302
$697$	−20369.2	−1.10694
$698$	−41945.3	−2.27458
$699$	−159.204	−0.00861464
$700$	−47421.9	−2.56054
$701$	−1134.57	−0.0611299	−0.0305649	−	0.999533i	$-0.509731\pi$
$701$	−1134.57	−0.0611299	−0.0305649	+	0.999533i	$0.509731\pi$
$702$	−307.362	−0.0165251
$703$	−447.072	−0.0239853
$704$	−3263.94	−0.174736
$705$	−92.0346	−0.00491663
$706$	−6681.32	−0.356168
$707$	−48711.3	−2.59120
$708$	676.570	0.0359139
$709$	−246.159	−0.0130390	−0.00651952	−	0.999979i	$-0.502075\pi$
$709$	−246.159	−0.0130390	−0.00651952	+	0.999979i	$0.502075\pi$
$710$	21887.2	1.15692
$711$	−6244.49	−0.329376
$712$	−4258.81	−0.224165
$713$	−4930.77	−0.258988
$714$	−1080.16	−0.0566161
$715$	5778.33	0.302234
$716$	51879.3	2.70785
$717$	−44.1075	−0.00229739
$718$	−17777.3	−0.924015
$719$	28297.1	1.46774	0.733869	−	0.679291i	$-0.237712\pi$
$719$	28297.1	1.46774	0.733869	+	0.679291i	$0.237712\pi$
$720$	−20653.0	−1.06901
$721$	3215.66	0.166099
$722$	34649.4	1.78604
$723$	38.9379	0.00200293
$724$	−9241.67	−0.474398
$725$	−19542.0	−1.00106
$726$	−1519.33	−0.0776689
$727$	14940.3	0.762178	0.381089	−	0.924538i	$-0.375549\pi$
$727$	14940.3	0.762178	0.381089	+	0.924538i	$0.375549\pi$
$728$	20982.2	1.06820
$729$	−19650.9	−0.998369
$730$	−2395.90	−0.121475
$731$	−28194.5	−1.42655
$732$	733.966	0.0370603
$733$	22293.5	1.12337	0.561684	−	0.827352i	$-0.310154\pi$
$733$	22293.5	1.12337	0.561684	+	0.827352i	$0.310154\pi$
$734$	36877.4	1.85445
$735$	347.242	0.0174262
$736$	4584.18	0.229586
$737$	−58275.5	−2.91262
$738$	−35562.6	−1.77382
$739$	18752.2	0.933437	0.466718	−	0.884406i	$-0.345436\pi$
$739$	18752.2	0.933437	0.466718	+	0.884406i	$0.345436\pi$
$740$	−5852.25	−0.290720
$741$	−9.88998	−0.000490307 0
$742$	81813.5	4.04780
$743$	−1774.70	−0.0876278	−0.0438139	−	0.999040i	$-0.513951\pi$
$743$	−1774.70	−0.0876278	−0.0438139	+	0.999040i	$0.513951\pi$
$744$	−933.542	−0.0460018
$745$	999.251	0.0491405
$746$	55754.4	2.73634
$747$	3509.83	0.171911
$748$	−99173.9	−4.84781
$749$	−61783.5	−3.01404
$750$	586.689	0.0285638
$751$	−21559.5	−1.04756	−0.523779	−	0.851854i	$-0.675478\pi$
$751$	−21559.5	−1.04756	−0.523779	+	0.851854i	$0.675478\pi$
$752$	−19968.6	−0.968325
$753$	382.139	0.0184939
$754$	15484.3	0.747883
$755$	−4028.37	−0.194182
$756$	−2616.70	−0.125884
$757$	−6313.99	−0.303152	−0.151576	−	0.988446i	$-0.548435\pi$
$757$	−6313.99	−0.303152	−0.151576	+	0.988446i	$0.548435\pi$
$758$	−28941.5	−1.38681
$759$	138.911	0.00664316
$760$	−2944.56	−0.140540
$761$	−10324.5	−0.491803	−0.245901	−	0.969295i	$-0.579084\pi$
$761$	−10324.5	−0.491803	−0.245901	+	0.969295i	$0.579084\pi$
$762$	−468.689	−0.0222819
$763$	−28193.0	−1.33769
$764$	−65594.4	−3.10618
$765$	−13681.7	−0.646616
$766$	−21925.6	−1.03421
$767$	−5665.93	−0.266734
$768$	−613.232	−0.0288126
$769$	17425.7	0.817146	0.408573	−	0.912726i	$-0.366027\pi$
$769$	17425.7	0.817146	0.408573	+	0.912726i	$0.366027\pi$
$770$	70916.6	3.31904
$771$	−544.409	−0.0254298
$772$	39376.4	1.83574
$773$	−10078.6	−0.468955	−0.234477	−	0.972122i	$-0.575338\pi$
$773$	−10078.6	−0.468955	−0.234477	+	0.972122i	$0.575338\pi$
$774$	−49224.8	−2.28598
$775$	−17669.3	−0.818968
$776$	78802.3	3.64541
$777$	−134.708	−0.00621959
$778$	35379.0	1.63033
$779$	−2288.91	−0.105274
$780$	−129.461	−0.00594289
$781$	46256.3	2.11931
$782$	9447.33	0.432015
$783$	−1078.31	−0.0492154
$784$	75340.7	3.43206
$785$	17718.9	0.805624
$786$	722.262	0.0327764
$787$	25803.8	1.16875	0.584374	−	0.811484i	$-0.301340\pi$
$787$	25803.8	1.16875	0.584374	+	0.811484i	$0.301340\pi$
$788$	−334.656	−0.0151290
$789$	293.579	0.0132467
$790$	−7584.34	−0.341568
$791$	38954.8	1.75104
$792$	−96687.0	−4.33791
$793$	−6146.58	−0.275248
$794$	−71615.6	−3.20093
$795$	−281.880	−0.0125752
$796$	−13559.3	−0.603764
$797$	−31104.7	−1.38242	−0.691209	−	0.722655i	$-0.742921\pi$
$797$	−31104.7	−1.38242	−0.691209	+	0.722655i	$0.742921\pi$
$798$	−121.378	−0.00538439
$799$	−13228.3	−0.585712
$800$	16427.3	0.725991
$801$	−2223.61	−0.0980867
$802$	−77295.2	−3.40323
$803$	−5063.50	−0.222524
$804$	1305.64	0.0572715
$805$	−4686.16	−0.205174
$806$	14000.4	0.611841
$807$	−317.802	−0.0138626
$808$	80662.3	3.51199
$809$	−15841.6	−0.688457	−0.344229	−	0.938886i	$-0.611859\pi$
$809$	−15841.6	−0.688457	−0.344229	+	0.938886i	$0.611859\pi$
$810$	−23880.3	−1.03589
$811$	15500.1	0.671126	0.335563	−	0.942018i	$-0.391073\pi$
$811$	15500.1	0.671126	0.335563	+	0.942018i	$0.391073\pi$
$812$	131824.	5.69717
$813$	−406.641	−0.0175419
$814$	−17829.8	−0.767732
$815$	−2196.64	−0.0944110
$816$	807.465	0.0346408
$817$	−3168.24	−0.135670
$818$	34927.7	1.49293
$819$	10955.3	0.467408
$820$	−29962.1	−1.27600
$821$	8828.23	0.375283	0.187641	−	0.982238i	$-0.439916\pi$
$821$	8828.23	0.375283	0.187641	+	0.982238i	$0.439916\pi$
$822$	992.305	0.0421054
$823$	−4772.49	−0.202137	−0.101068	−	0.994879i	$-0.532226\pi$
$823$	−4772.49	−0.202137	−0.101068	+	0.994879i	$0.532226\pi$
$824$	−5324.91	−0.225124
$825$	497.785	0.0210069
$826$	−69537.0	−2.92918
$827$	152.788	0.00642439	0.00321219	−	0.999995i	$-0.498978\pi$
$827$	152.788	0.00642439	0.00321219	+	0.999995i	$0.498978\pi$
$828$	11441.6	0.480220
$829$	−30151.8	−1.26323	−0.631614	−	0.775283i	$-0.717607\pi$
$829$	−30151.8	−1.26323	−0.631614	+	0.775283i	$0.717607\pi$
$830$	4262.92	0.178275
$831$	−726.317	−0.0303197
$832$	−612.404	−0.0255184
$833$	49909.8	2.07596
$834$	669.124	0.0277816
$835$	−7719.99	−0.319953
$836$	−11144.3	−0.461043
$837$	−974.976	−0.0402630
$838$	20964.7	0.864215
$839$	−18110.5	−0.745224	−0.372612	−	0.927987i	$-0.621538\pi$
$839$	−18110.5	−0.745224	−0.372612	+	0.927987i	$0.621538\pi$
$840$	−887.230	−0.0364433
$841$	29934.0	1.22736
$842$	59335.5	2.42855
$843$	−671.389	−0.0274304
$844$	7431.56	0.303086
$845$	1084.17	0.0441380
$846$	−23095.3	−0.938573
$847$	108321.	4.39428
$848$	−61159.2	−2.47667
$849$	221.226	0.00894281
$850$	33854.3	1.36611
$851$	1178.19	0.0474593
$852$	−1036.35	−0.0416724
$853$	−32201.5	−1.29257	−0.646283	−	0.763098i	$-0.723678\pi$
$853$	−32201.5	−1.29257	−0.646283	+	0.763098i	$0.723678\pi$
$854$	−75436.1	−3.02268
$855$	−1537.42	−0.0614954
$856$	102309.	4.08510
$857$	−30668.8	−1.22243	−0.611217	−	0.791463i	$-0.709320\pi$
$857$	−30668.8	−1.22243	−0.611217	+	0.791463i	$0.709320\pi$
$858$	−394.424	−0.0156940
$859$	−10371.4	−0.411954	−0.205977	−	0.978557i	$-0.566037\pi$
$859$	−10371.4	−0.411954	−0.205977	+	0.978557i	$0.566037\pi$
$860$	−41472.7	−1.64443
$861$	−689.673	−0.0272985
$862$	−20080.7	−0.793448
$863$	−12465.8	−0.491702	−0.245851	−	0.969308i	$-0.579067\pi$
$863$	−12465.8	−0.491702	−0.245851	+	0.969308i	$0.579067\pi$
$864$	906.444	0.0356920
$865$	−15114.7	−0.594120
$866$	−56589.5	−2.22054
$867$	113.927	0.00446270
$868$	119191.	4.66084
$869$	−16028.7	−0.625705
$870$	−654.750	−0.0255151
$871$	−10934.1	−0.425357
$872$	46685.5	1.81304
$873$	41144.3	1.59510
$874$	1061.60	0.0410862
$875$	−41828.2	−1.61606
$876$	113.446	0.00437554
$877$	25900.0	0.997242	0.498621	−	0.866820i	$-0.333840\pi$
$877$	25900.0	0.997242	0.498621	+	0.866820i	$0.333840\pi$
$878$	11597.6	0.445787
$879$	801.429	0.0307526
$880$	−53013.3	−2.03077
$881$	−13907.9	−0.531860	−0.265930	−	0.963992i	$-0.585679\pi$
$881$	−13907.9	−0.531860	−0.265930	+	0.963992i	$0.585679\pi$
$882$	87137.5	3.32661
$883$	34648.2	1.32050	0.660252	−	0.751044i	$-0.270450\pi$
$883$	34648.2	1.32050	0.660252	+	0.751044i	$0.270450\pi$
$884$	−18607.7	−0.707970
$885$	239.583	0.00909999
$886$	68546.9	2.59919
$887$	35821.3	1.35599	0.677994	−	0.735067i	$-0.262849\pi$
$887$	35821.3	1.35599	0.677994	+	0.735067i	$0.262849\pi$
$888$	223.067	0.00842976
$889$	33415.3	1.26065
$890$	−2700.72	−0.101717
$891$	−50468.6	−1.89760
$892$	52223.1	1.96027
$893$	−1486.47	−0.0557032
$894$	−68.2080	−0.00255170
$895$	18371.2	0.686125
$896$	41418.3	1.54429
$897$	26.0635	0.000970162 0
$898$	−38619.0	−1.43512
$899$	49117.2	1.82219
$900$	41000.7	1.51854
$901$	−40515.2	−1.49806
$902$	−91284.3	−3.36966
$903$	−954.626	−0.0351805
$904$	−64506.3	−2.37328
$905$	−3272.61	−0.120205
$906$	274.973	0.0100832
$907$	−35413.7	−1.29646	−0.648231	−	0.761443i	$-0.724491\pi$
$907$	−35413.7	−1.29646	−0.648231	+	0.761443i	$0.724491\pi$
$908$	−17146.5	−0.626681
$909$	42115.5	1.53672
$910$	13305.9	0.484709
$911$	11670.6	0.424440	0.212220	−	0.977222i	$-0.431931\pi$
$911$	11670.6	0.424440	0.212220	+	0.977222i	$0.431931\pi$
$912$	90.7355	0.00329446
$913$	9009.24	0.326574
$914$	−35075.7	−1.26937
$915$	259.908	0.00939047
$916$	34463.4	1.24313
$917$	−51493.9	−1.85439
$918$	1868.05	0.0671621
$919$	27431.4	0.984635	0.492317	−	0.870416i	$-0.336150\pi$
$919$	27431.4	0.984635	0.492317	+	0.870416i	$0.336150\pi$
$920$	7759.94	0.278084
$921$	−758.718	−0.0271451
$922$	−34017.2	−1.21507
$923$	8678.93	0.309502
$924$	−3357.89	−0.119552
$925$	4222.02	0.150075
$926$	64312.4	2.28233
$927$	−2780.24	−0.0985061
$928$	−45664.7	−1.61532
$929$	−7201.80	−0.254342	−0.127171	−	0.991881i	$-0.540590\pi$
$929$	−7201.80	−0.254342	−0.127171	+	0.991881i	$0.540590\pi$
$930$	−592.006	−0.0208738
$931$	5608.40	0.197431
$932$	−33659.2	−1.18299
$933$	−400.032	−0.0140370
$934$	101385.	3.55185
$935$	−35118.9	−1.22835
$936$	−18141.1	−0.633505
$937$	−43461.3	−1.51528	−0.757641	−	0.652671i	$-0.773648\pi$
$937$	−43461.3	−1.51528	−0.757641	+	0.652671i	$0.773648\pi$
$938$	−134192.	−4.67113
$939$	−317.102	−0.0110205
$940$	−19458.2	−0.675166
$941$	12377.4	0.428792	0.214396	−	0.976747i	$-0.431222\pi$
$941$	12377.4	0.428792	0.214396	+	0.976747i	$0.431222\pi$
$942$	−1209.48	−0.0418332
$943$	6032.06	0.208304
$944$	51982.0	1.79223
$945$	−926.609	−0.0318969
$946$	−126353.	−4.34260
$947$	5424.04	0.186122	0.0930610	−	0.995660i	$-0.470335\pi$
$947$	5424.04	0.186122	0.0930610	+	0.995660i	$0.470335\pi$
$948$	359.117	0.0123034
$949$	−950.049	−0.0324973
$950$	3804.23	0.129922
$951$	313.134	0.0106772
$952$	−127523.	−4.34144
$953$	−9225.79	−0.313591	−0.156796	−	0.987631i	$-0.550116\pi$
$953$	−9225.79	−0.313591	−0.156796	+	0.987631i	$0.550116\pi$
$954$	−70735.5	−2.40057
$955$	−23227.9	−0.787055
$956$	−9325.32	−0.315484
$957$	−1383.75	−0.0467400
$958$	−4829.55	−0.162876
$959$	−70746.7	−2.38220
$960$	25.8954	0.000870594 0
$961$	14619.4	0.490731
$962$	−3345.35	−0.112119
$963$	53417.6	1.78750
$964$	8232.36	0.275048
$965$	13943.8	0.465145
$966$	319.873	0.0106540
$967$	41590.3	1.38310	0.691548	−	0.722330i	$-0.256929\pi$
$967$	41590.3	1.38310	0.691548	+	0.722330i	$0.256929\pi$
$968$	−179372.	−5.95582
$969$	60.1081	0.00199273
$970$	49972.5	1.65414
$971$	27225.3	0.899795	0.449897	−	0.893080i	$-0.351461\pi$
$971$	27225.3	0.899795	0.449897	+	0.893080i	$0.351461\pi$
$972$	3393.72	0.111989
$973$	−47705.4	−1.57180
$974$	−60366.7	−1.98591
$975$	93.3980	0.00306782
$976$	56391.8	1.84944
$977$	−11808.5	−0.386681	−0.193341	−	0.981132i	$-0.561932\pi$
$977$	−11808.5	−0.386681	−0.193341	+	0.981132i	$0.561932\pi$
$978$	149.941	0.00490244
$979$	−5707.70	−0.186332
$980$	73414.8	2.39301
$981$	24375.5	0.793323
$982$	−51566.3	−1.67571
$983$	32286.9	1.04760	0.523801	−	0.851841i	$-0.324513\pi$
$983$	32286.9	1.04760	0.523801	+	0.851841i	$0.324513\pi$
$984$	1142.05	0.0369992
$985$	−118.506	−0.00383343
$986$	−94108.5	−3.03958
$987$	−447.892	−0.0144443
$988$	−2090.96	−0.0673304
$989$	8349.40	0.268448
$990$	−61314.1	−1.96837
$991$	−38020.9	−1.21874	−0.609372	−	0.792885i	$-0.708578\pi$
$991$	−38020.9	−1.21874	−0.609372	+	0.792885i	$0.708578\pi$
$992$	−41288.7	−1.32149
$993$	−142.421	−0.00455145
$994$	106515.	3.39885
$995$	−4801.53	−0.152984
$996$	−201.848	−0.00642150
$997$	55997.0	1.77878	0.889390	−	0.457149i	$-0.151130\pi$
$997$	55997.0	1.77878	0.889390	+	0.457149i	$0.151130\pi$
$998$	76998.8	2.44224
$999$	232.967	0.00737813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.4.a.a.1.5	✓	72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.4.a.a.1.5	✓	72	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-5.11041 q^{2} +0.0856874 q^{3} +18.1162 q^{4} +6.41522 q^{5} -0.437897 q^{6} +31.2200 q^{7} -51.6981 q^{8} -26.9927 q^{9} +O(q^{10})\)	\(q-5.11041 q^{2} +0.0856874 q^{3} +18.1162 q^{4} +6.41522 q^{5} -0.437897 q^{6} +31.2200 q^{7} -51.6981 q^{8} -26.9927 q^{9} -32.7844 q^{10} -69.2864 q^{11} +1.55233 q^{12} -13.0000 q^{13} -159.547 q^{14} +0.549703 q^{15} +119.268 q^{16} +79.0099 q^{17} +137.943 q^{18} +8.87840 q^{19} +116.220 q^{20} +2.67516 q^{21} +354.082 q^{22} -23.3977 q^{23} -4.42988 q^{24} -83.8450 q^{25} +66.4353 q^{26} -4.62649 q^{27} +565.590 q^{28} +233.073 q^{29} -2.80921 q^{30} +210.738 q^{31} -195.925 q^{32} -5.93697 q^{33} -403.773 q^{34} +200.283 q^{35} -489.006 q^{36} -50.3550 q^{37} -45.3722 q^{38} -1.11394 q^{39} -331.655 q^{40} -257.806 q^{41} -13.6712 q^{42} -356.848 q^{43} -1255.21 q^{44} -173.164 q^{45} +119.572 q^{46} -167.426 q^{47} +10.2198 q^{48} +631.690 q^{49} +428.482 q^{50} +6.77015 q^{51} -235.511 q^{52} -512.786 q^{53} +23.6432 q^{54} -444.487 q^{55} -1614.02 q^{56} +0.760767 q^{57} -1191.10 q^{58} +435.840 q^{59} +9.95856 q^{60} +472.814 q^{61} -1076.95 q^{62} -842.712 q^{63} +47.1080 q^{64} -83.3978 q^{65} +30.3403 q^{66} +841.081 q^{67} +1431.36 q^{68} -2.00488 q^{69} -1023.53 q^{70} -667.610 q^{71} +1395.47 q^{72} +73.0807 q^{73} +257.335 q^{74} -7.18446 q^{75} +160.843 q^{76} -2163.12 q^{77} +5.69267 q^{78} +231.340 q^{79} +765.132 q^{80} +728.405 q^{81} +1317.49 q^{82} -130.029 q^{83} +48.4639 q^{84} +506.866 q^{85} +1823.64 q^{86} +19.9714 q^{87} +3581.98 q^{88} +82.3784 q^{89} +884.937 q^{90} -405.860 q^{91} -423.878 q^{92} +18.0576 q^{93} +855.614 q^{94} +56.9569 q^{95} -16.7883 q^{96} -1524.28 q^{97} -3228.19 q^{98} +1870.22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - q^{2} - 19 q^{3} + 261 q^{4} - 73 q^{5} - 52 q^{6} + 3 q^{7} - 18 q^{8} + 517 q^{9}+O(q^{10}) \) 72 * q - q^2 - 19 * q^3 + 261 * q^4 - 73 * q^5 - 52 * q^6 + 3 * q^7 - 18 * q^8 + 517 * q^9	\( 72 q - q^{2} - 19 q^{3} + 261 q^{4} - 73 q^{5} - 52 q^{6} + 3 q^{7} - 18 q^{8} + 517 q^{9} - 100 q^{10} - 104 q^{11} - 228 q^{12} - 936 q^{13} - 337 q^{14} + 63 q^{15} + 833 q^{16} - 307 q^{17} - 110 q^{18} - 182 q^{19} - 428 q^{20} - 177 q^{21} - 74 q^{22} - 516 q^{23} - 901 q^{24} + 1285 q^{25} + 13 q^{26} - 817 q^{27} + 187 q^{28} - 1458 q^{29} - 178 q^{30} - 620 q^{31} - 35 q^{32} - 196 q^{33} - 1619 q^{34} - 1155 q^{35} - 615 q^{36} - 1031 q^{37} - 1614 q^{38} + 247 q^{39} - 3300 q^{40} - 1124 q^{41} - 2190 q^{42} - 383 q^{43} - 1902 q^{44} - 1828 q^{45} - 1109 q^{46} - 1531 q^{47} - 4464 q^{48} + 1191 q^{49} - 352 q^{50} - 1267 q^{51} - 3393 q^{52} - 3092 q^{53} - 1581 q^{54} - 2264 q^{55} - 1470 q^{56} - 502 q^{57} - 2003 q^{58} - 2452 q^{59} - 1909 q^{60} - 1764 q^{61} - 2232 q^{62} - 1086 q^{63} + 2272 q^{64} + 949 q^{65} - 2539 q^{66} + 832 q^{67} - 1796 q^{68} - 4738 q^{69} - 3251 q^{70} - 2995 q^{71} - 2534 q^{72} + 16 q^{73} - 3408 q^{74} - 3060 q^{75} - 3729 q^{76} - 5454 q^{77} + 676 q^{78} - 3594 q^{79} - 1631 q^{80} + 16 q^{81} - 1293 q^{82} - 2832 q^{83} - 4565 q^{84} - 887 q^{85} - 3982 q^{86} - 1530 q^{87} - 1757 q^{88} - 8104 q^{89} - 5932 q^{90} - 39 q^{91} - 5371 q^{92} - 3042 q^{93} - 3464 q^{94} - 4582 q^{95} - 3991 q^{96} - 2514 q^{97} - 2886 q^{98} - 2752 q^{99}+O(q^{100}) \) 72 * q - q^2 - 19 * q^3 + 261 * q^4 - 73 * q^5 - 52 * q^6 + 3 * q^7 - 18 * q^8 + 517 * q^9 - 100 * q^10 - 104 * q^11 - 228 * q^12 - 936 * q^13 - 337 * q^14 + 63 * q^15 + 833 * q^16 - 307 * q^17 - 110 * q^18 - 182 * q^19 - 428 * q^20 - 177 * q^21 - 74 * q^22 - 516 * q^23 - 901 * q^24 + 1285 * q^25 + 13 * q^26 - 817 * q^27 + 187 * q^28 - 1458 * q^29 - 178 * q^30 - 620 * q^31 - 35 * q^32 - 196 * q^33 - 1619 * q^34 - 1155 * q^35 - 615 * q^36 - 1031 * q^37 - 1614 * q^38 + 247 * q^39 - 3300 * q^40 - 1124 * q^41 - 2190 * q^42 - 383 * q^43 - 1902 * q^44 - 1828 * q^45 - 1109 * q^46 - 1531 * q^47 - 4464 * q^48 + 1191 * q^49 - 352 * q^50 - 1267 * q^51 - 3393 * q^52 - 3092 * q^53 - 1581 * q^54 - 2264 * q^55 - 1470 * q^56 - 502 * q^57 - 2003 * q^58 - 2452 * q^59 - 1909 * q^60 - 1764 * q^61 - 2232 * q^62 - 1086 * q^63 + 2272 * q^64 + 949 * q^65 - 2539 * q^66 + 832 * q^67 - 1796 * q^68 - 4738 * q^69 - 3251 * q^70 - 2995 * q^71 - 2534 * q^72 + 16 * q^73 - 3408 * q^74 - 3060 * q^75 - 3729 * q^76 - 5454 * q^77 + 676 * q^78 - 3594 * q^79 - 1631 * q^80 + 16 * q^81 - 1293 * q^82 - 2832 * q^83 - 4565 * q^84 - 887 * q^85 - 3982 * q^86 - 1530 * q^87 - 1757 * q^88 - 8104 * q^89 - 5932 * q^90 - 39 * q^91 - 5371 * q^92 - 3042 * q^93 - 3464 * q^94 - 4582 * q^95 - 3991 * q^96 - 2514 * q^97 - 2886 * q^98 - 2752 * q^99

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