LMFDB - Embedded newform 722.4.a.p.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [722,4,Mod(1,722)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(722, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("722.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 722 = 2 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	722.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:	$42.5993790241$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.6719782761.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 75x^{4} + 135x^{3} + 1857x^{2} - 1425x - 14797 $ x^6 - 3x^5 - 75x^4 + 135x^3 + 1857x^2 - 1425*x - 14797
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 19 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-4.51546$ of defining polynomial
Character	$\chi$	$=$	722.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +6.57964 q^{3} +4.00000 q^{4} -11.3499 q^{5} +13.1593 q^{6} -8.71985 q^{7} +8.00000 q^{8} +16.2917 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +6.57964 q^{3} +4.00000 q^{4} -11.3499 q^{5} +13.1593 q^{6} -8.71985 q^{7} +8.00000 q^{8} +16.2917 q^{9} -22.6998 q^{10} -51.7082 q^{11} +26.3186 q^{12} -27.7227 q^{13} -17.4397 q^{14} -74.6782 q^{15} +16.0000 q^{16} +117.423 q^{17} +32.5833 q^{18} -45.3995 q^{20} -57.3735 q^{21} -103.416 q^{22} -99.5795 q^{23} +52.6371 q^{24} +3.81990 q^{25} -55.4454 q^{26} -70.4570 q^{27} -34.8794 q^{28} -139.561 q^{29} -149.356 q^{30} -3.10303 q^{31} +32.0000 q^{32} -340.221 q^{33} +234.845 q^{34} +98.9694 q^{35} +65.1666 q^{36} -91.5180 q^{37} -182.405 q^{39} -90.7991 q^{40} -250.053 q^{41} -114.747 q^{42} -413.919 q^{43} -206.833 q^{44} -184.908 q^{45} -199.159 q^{46} +611.805 q^{47} +105.274 q^{48} -266.964 q^{49} +7.63981 q^{50} +772.598 q^{51} -110.891 q^{52} -608.545 q^{53} -140.914 q^{54} +586.882 q^{55} -69.7588 q^{56} -279.121 q^{58} -572.277 q^{59} -298.713 q^{60} +572.684 q^{61} -6.20606 q^{62} -142.061 q^{63} +64.0000 q^{64} +314.649 q^{65} -680.442 q^{66} +298.501 q^{67} +469.690 q^{68} -655.197 q^{69} +197.939 q^{70} +253.057 q^{71} +130.333 q^{72} -520.754 q^{73} -183.036 q^{74} +25.1336 q^{75} +450.888 q^{77} -364.810 q^{78} +226.426 q^{79} -181.598 q^{80} -903.457 q^{81} -500.107 q^{82} -380.939 q^{83} -229.494 q^{84} -1332.73 q^{85} -827.839 q^{86} -918.259 q^{87} -413.665 q^{88} +1010.11 q^{89} -369.817 q^{90} +241.738 q^{91} -398.318 q^{92} -20.4168 q^{93} +1223.61 q^{94} +210.548 q^{96} -235.144 q^{97} -533.928 q^{98} -842.412 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{2} - 9 q^{3} + 24 q^{4} - 27 q^{5} - 18 q^{6} - 21 q^{7} + 48 q^{8} + 33 q^{9}+O(q^{10}) $ 6 * q + 12 * q^2 - 9 * q^3 + 24 * q^4 - 27 * q^5 - 18 * q^6 - 21 * q^7 + 48 * q^8 + 33 * q^9	\( 6 q + 12 q^{2} - 9 q^{3} + 24 q^{4} - 27 q^{5} - 18 q^{6} - 21 q^{7} + 48 q^{8} + 33 q^{9} - 54 q^{10} + 9 q^{11} - 36 q^{12} - 24 q^{13} - 42 q^{14} + 96 q^{16} - 102 q^{17} + 66 q^{18} - 108 q^{20} - 51 q^{21} + 18 q^{22} - 264 q^{23} - 72 q^{24} + 177 q^{25} - 48 q^{26} - 189 q^{27} - 84 q^{28} - 483 q^{29} - 72 q^{31} + 192 q^{32} - 387 q^{33} - 204 q^{34} + 135 q^{35} + 132 q^{36} + 558 q^{37} - 624 q^{39} - 216 q^{40} - 396 q^{41} - 102 q^{42} - 2064 q^{43} + 36 q^{44} - 1296 q^{45} - 528 q^{46} + 858 q^{47} - 144 q^{48} - 1413 q^{49} + 354 q^{50} + 1272 q^{51} - 96 q^{52} - 762 q^{53} - 378 q^{54} - 1107 q^{55} - 168 q^{56} - 966 q^{58} - 393 q^{59} - 627 q^{61} - 144 q^{62} - 84 q^{63} + 384 q^{64} - 495 q^{65} - 774 q^{66} - 2028 q^{67} - 408 q^{68} + 237 q^{69} + 270 q^{70} - 1284 q^{71} + 264 q^{72} - 2688 q^{73} + 1116 q^{74} - 927 q^{75} - 708 q^{77} - 1248 q^{78} - 969 q^{79} - 432 q^{80} - 1398 q^{81} - 792 q^{82} - 927 q^{83} - 204 q^{84} + 396 q^{85} - 4128 q^{86} - 2892 q^{87} + 72 q^{88} + 1257 q^{89} - 2592 q^{90} + 1323 q^{91} - 1056 q^{92} - 1368 q^{93} + 1716 q^{94} - 288 q^{96} + 2403 q^{97} - 2826 q^{98} + 567 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 - 9 * q^3 + 24 * q^4 - 27 * q^5 - 18 * q^6 - 21 * q^7 + 48 * q^8 + 33 * q^9 - 54 * q^10 + 9 * q^11 - 36 * q^12 - 24 * q^13 - 42 * q^14 + 96 * q^16 - 102 * q^17 + 66 * q^18 - 108 * q^20 - 51 * q^21 + 18 * q^22 - 264 * q^23 - 72 * q^24 + 177 * q^25 - 48 * q^26 - 189 * q^27 - 84 * q^28 - 483 * q^29 - 72 * q^31 + 192 * q^32 - 387 * q^33 - 204 * q^34 + 135 * q^35 + 132 * q^36 + 558 * q^37 - 624 * q^39 - 216 * q^40 - 396 * q^41 - 102 * q^42 - 2064 * q^43 + 36 * q^44 - 1296 * q^45 - 528 * q^46 + 858 * q^47 - 144 * q^48 - 1413 * q^49 + 354 * q^50 + 1272 * q^51 - 96 * q^52 - 762 * q^53 - 378 * q^54 - 1107 * q^55 - 168 * q^56 - 966 * q^58 - 393 * q^59 - 627 * q^61 - 144 * q^62 - 84 * q^63 + 384 * q^64 - 495 * q^65 - 774 * q^66 - 2028 * q^67 - 408 * q^68 + 237 * q^69 + 270 * q^70 - 1284 * q^71 + 264 * q^72 - 2688 * q^73 + 1116 * q^74 - 927 * q^75 - 708 * q^77 - 1248 * q^78 - 969 * q^79 - 432 * q^80 - 1398 * q^81 - 792 * q^82 - 927 * q^83 - 204 * q^84 + 396 * q^85 - 4128 * q^86 - 2892 * q^87 + 72 * q^88 + 1257 * q^89 - 2592 * q^90 + 1323 * q^91 - 1056 * q^92 - 1368 * q^93 + 1716 * q^94 - 288 * q^96 + 2403 * q^97 - 2826 * q^98 + 567 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	6.57964	1.26625	0.633126	−	0.774049i	$-0.281771\pi$
$3$	6.57964	1.26625	0.633126	+	0.774049i	$0.281771\pi$
$4$	4.00000	0.500000
$5$	−11.3499	−1.01516	−0.507582	−	0.861603i	$-0.669461\pi$
$5$	−11.3499	−1.01516	−0.507582	+	0.861603i	$0.669461\pi$
$6$	13.1593	0.895376
$7$	−8.71985	−0.470828	−0.235414	−	0.971895i	$-0.575645\pi$
$7$	−8.71985	−0.470828	−0.235414	+	0.971895i	$0.575645\pi$
$8$	8.00000	0.353553
$9$	16.2917	0.603395
$10$	−22.6998	−0.717830
$11$	−51.7082	−1.41733	−0.708663	−	0.705547i	$-0.750702\pi$
$11$	−51.7082	−1.41733	−0.708663	+	0.705547i	$0.750702\pi$
$12$	26.3186	0.633126
$13$	−27.7227	−0.591453	−0.295726	−	0.955273i	$-0.595562\pi$
$13$	−27.7227	−0.591453	−0.295726	+	0.955273i	$0.595562\pi$
$14$	−17.4397	−0.332926
$15$	−74.6782	−1.28545
$16$	16.0000	0.250000
$17$	117.423	1.67524	0.837622	−	0.546250i	$-0.183945\pi$
$17$	117.423	1.67524	0.837622	+	0.546250i	$0.183945\pi$
$18$	32.5833	0.426665
$19$	0	0
$19$	0	0
$20$	−45.3995	−0.507582
$21$	−57.3735	−0.596187
$22$	−103.416	−1.00220
$23$	−99.5795	−0.902772	−0.451386	−	0.892329i	$-0.649070\pi$
$23$	−99.5795	−0.902772	−0.451386	+	0.892329i	$0.649070\pi$
$24$	52.6371	0.447688
$25$	3.81990	0.0305592
$26$	−55.4454	−0.418220
$27$	−70.4570	−0.502202
$28$	−34.8794	−0.235414
$29$	−139.561	−0.893647	−0.446823	−	0.894622i	$-0.647445\pi$
$29$	−139.561	−0.893647	−0.446823	+	0.894622i	$0.647445\pi$
$30$	−149.356	−0.908954
$31$	−3.10303	−0.0179781	−0.00898905	−	0.999960i	$-0.502861\pi$
$31$	−3.10303	−0.0179781	−0.00898905	+	0.999960i	$0.502861\pi$
$32$	32.0000	0.176777
$33$	−340.221	−1.79469
$34$	234.845	1.18458
$35$	98.9694	0.477968
$36$	65.1666	0.301697
$37$	−91.5180	−0.406634	−0.203317	−	0.979113i	$-0.565172\pi$
$37$	−91.5180	−0.406634	−0.203317	+	0.979113i	$0.565172\pi$
$38$	0	0
$39$	−182.405	−0.748929
$40$	−90.7991	−0.358915
$41$	−250.053	−0.952482	−0.476241	−	0.879315i	$-0.658001\pi$
$41$	−250.053	−0.952482	−0.476241	+	0.879315i	$0.658001\pi$
$42$	−114.747	−0.421568
$43$	−413.919	−1.46796	−0.733978	−	0.679173i	$-0.762339\pi$
$43$	−413.919	−1.46796	−0.733978	+	0.679173i	$0.762339\pi$
$44$	−206.833	−0.708663
$45$	−184.908	−0.612545
$46$	−199.159	−0.638356
$47$	611.805	1.89874	0.949371	−	0.314157i	$-0.101722\pi$
$47$	611.805	1.89874	0.949371	+	0.314157i	$0.101722\pi$
$48$	105.274	0.316563
$49$	−266.964	−0.778321
$50$	7.63981	0.0216086
$51$	772.598	2.12128
$52$	−110.891	−0.295726
$53$	−608.545	−1.57717	−0.788586	−	0.614924i	$-0.789187\pi$
$53$	−608.545	−1.57717	−0.788586	+	0.614924i	$0.789187\pi$
$54$	−140.914	−0.355111
$55$	586.882	1.43882
$56$	−69.7588	−0.166463
$57$	0	0
$58$	−279.121	−0.631904
$59$	−572.277	−1.26278	−0.631390	−	0.775465i	$-0.717515\pi$
$59$	−572.277	−1.26278	−0.631390	+	0.775465i	$0.717515\pi$
$60$	−298.713	−0.642727
$61$	572.684	1.20204	0.601022	−	0.799232i	$-0.294760\pi$
$61$	572.684	1.20204	0.601022	+	0.799232i	$0.294760\pi$
$62$	−6.20606	−0.0127124
$63$	−142.061	−0.284095
$64$	64.0000	0.125000
$65$	314.649	0.600422
$66$	−680.442	−1.26904
$67$	298.501	0.544295	0.272147	−	0.962256i	$-0.412266\pi$
$67$	298.501	0.544295	0.272147	+	0.962256i	$0.412266\pi$
$68$	469.690	0.837622
$69$	−655.197	−1.14314
$70$	197.939	0.337974
$71$	253.057	0.422991	0.211495	−	0.977379i	$-0.432167\pi$
$71$	253.057	0.422991	0.211495	+	0.977379i	$0.432167\pi$
$72$	130.333	0.213332
$73$	−520.754	−0.834927	−0.417463	−	0.908694i	$-0.637081\pi$
$73$	−520.754	−0.834927	−0.417463	+	0.908694i	$0.637081\pi$
$74$	−183.036	−0.287534
$75$	25.1336	0.0386957
$76$	0	0
$77$	450.888	0.667317
$78$	−364.810	−0.529572
$79$	226.426	0.322468	0.161234	−	0.986916i	$-0.448453\pi$
$79$	226.426	0.322468	0.161234	+	0.986916i	$0.448453\pi$
$80$	−181.598	−0.253791
$81$	−903.457	−1.23931
$82$	−500.107	−0.673507
$83$	−380.939	−0.503776	−0.251888	−	0.967756i	$-0.581052\pi$
$83$	−380.939	−0.503776	−0.251888	+	0.967756i	$0.581052\pi$
$84$	−229.494	−0.298093
$85$	−1332.73	−1.70065
$86$	−827.839	−1.03800
$87$	−918.259	−1.13158
$88$	−413.665	−0.501101
$89$	1010.11	1.20305	0.601523	−	0.798856i	$-0.294561\pi$
$89$	1010.11	1.20305	0.601523	+	0.798856i	$0.294561\pi$
$90$	−369.817	−0.433135
$91$	241.738	0.278473
$92$	−398.318	−0.451386
$93$	−20.4168	−0.0227648
$94$	1223.61	1.34261
$95$	0	0
$96$	210.548	0.223844
$97$	−235.144	−0.246137	−0.123068	−	0.992398i	$-0.539273\pi$
$97$	−235.144	−0.246137	−0.123068	+	0.992398i	$0.539273\pi$
$98$	−533.928	−0.550356
$99$	−842.412	−0.855208
$100$	15.2796	0.0152796
$101$	940.599	0.926664	0.463332	−	0.886185i	$-0.346654\pi$
$101$	940.599	0.926664	0.463332	+	0.886185i	$0.346654\pi$
$102$	1545.20	1.49997
$103$	1772.39	1.69552	0.847762	−	0.530378i	$-0.177950\pi$
$103$	1772.39	1.69552	0.847762	+	0.530378i	$0.177950\pi$
$104$	−221.781	−0.209110
$105$	651.183	0.605228
$106$	−1217.09	−1.11523
$107$	−525.174	−0.474491	−0.237245	−	0.971450i	$-0.576245\pi$
$107$	−525.174	−0.474491	−0.237245	+	0.971450i	$0.576245\pi$
$108$	−281.828	−0.251101
$109$	819.142	0.719813	0.359906	−	0.932988i	$-0.382809\pi$
$109$	819.142	0.719813	0.359906	+	0.932988i	$0.382809\pi$
$110$	1173.76	1.01740
$111$	−602.156	−0.514902
$112$	−139.518	−0.117707
$113$	1939.84	1.61491	0.807457	−	0.589927i	$-0.200843\pi$
$113$	1939.84	1.61491	0.807457	+	0.589927i	$0.200843\pi$
$114$	0	0
$115$	1130.22	0.916463
$116$	−558.243	−0.446823
$117$	−451.648	−0.356880
$118$	−1144.55	−0.892921
$119$	−1023.91	−0.788752
$120$	−597.425	−0.454477
$121$	1342.73	1.00882
$122$	1145.37	0.849974
$123$	−1645.26	−1.20608
$124$	−12.4121	−0.00898905
$125$	1375.38	0.984142
$126$	−284.122	−0.200886
$127$	672.004	0.469533	0.234766	−	0.972052i	$-0.424567\pi$
$127$	672.004	0.469533	0.234766	+	0.972052i	$0.424567\pi$
$128$	128.000	0.0883883
$129$	−2723.44	−1.85880
$130$	629.298	0.424562
$131$	1840.91	1.22779	0.613897	−	0.789386i	$-0.289601\pi$
$131$	1840.91	1.22779	0.613897	+	0.789386i	$0.289601\pi$
$132$	−1360.88	−0.897347
$133$	0	0
$134$	597.003	0.384875
$135$	799.679	0.509818
$136$	939.380	0.592288
$137$	−942.201	−0.587574	−0.293787	−	0.955871i	$-0.594916\pi$
$137$	−942.201	−0.587574	−0.293787	+	0.955871i	$0.594916\pi$
$138$	−1310.39	−0.808320
$139$	257.490	0.157122	0.0785612	−	0.996909i	$-0.474967\pi$
$139$	257.490	0.157122	0.0785612	+	0.996909i	$0.474967\pi$
$140$	395.877	0.238984
$141$	4025.45	2.40429
$142$	506.114	0.299100
$143$	1433.49	0.838282
$144$	260.667	0.150849
$145$	1584.00	0.907199
$146$	−1041.51	−0.590382
$147$	−1756.53	−0.985551
$148$	−366.072	−0.203317
$149$	−1880.16	−1.03375	−0.516875	−	0.856061i	$-0.672905\pi$
$149$	−1880.16	−1.03375	−0.516875	+	0.856061i	$0.672905\pi$
$150$	50.2672	0.0273620
$151$	−1342.24	−0.723375	−0.361688	−	0.932299i	$-0.617799\pi$
$151$	−1342.24	−0.723375	−0.361688	+	0.932299i	$0.617799\pi$
$152$	0	0
$153$	1913.01	1.01083
$154$	901.775	0.471864
$155$	35.2191	0.0182507
$156$	−729.621	−0.374464
$157$	1339.58	0.680958	0.340479	−	0.940252i	$-0.389411\pi$
$157$	1339.58	0.680958	0.340479	+	0.940252i	$0.389411\pi$
$158$	452.852	0.228019
$159$	−4004.01	−1.99710
$160$	−363.196	−0.179457
$161$	868.319	0.425050
$162$	−1806.91	−0.876324
$163$	−2328.59	−1.11895	−0.559476	−	0.828846i	$-0.688998\pi$
$163$	−2328.59	−1.11895	−0.559476	+	0.828846i	$0.688998\pi$
$164$	−1000.21	−0.476241
$165$	3861.47	1.82191
$166$	−761.877	−0.356224
$167$	2761.83	1.27974	0.639870	−	0.768483i	$-0.278988\pi$
$167$	2761.83	1.27974	0.639870	+	0.768483i	$0.278988\pi$
$168$	−458.988	−0.210784
$169$	−1428.45	−0.650184
$170$	−2665.46	−1.20254
$171$	0	0
$172$	−1655.68	−0.733978
$173$	−643.127	−0.282636	−0.141318	−	0.989964i	$-0.545134\pi$
$173$	−643.127	−0.282636	−0.141318	+	0.989964i	$0.545134\pi$
$174$	−1836.52	−0.800150
$175$	−33.3090	−0.0143881
$176$	−827.331	−0.354332
$177$	−3765.37	−1.59900
$178$	2020.21	0.850682
$179$	−2129.36	−0.889137	−0.444569	−	0.895745i	$-0.646643\pi$
$179$	−2129.36	−0.889137	−0.444569	+	0.895745i	$0.646643\pi$
$180$	−739.634	−0.306273
$181$	−1047.99	−0.430368	−0.215184	−	0.976573i	$-0.569035\pi$
$181$	−1047.99	−0.430368	−0.215184	+	0.976573i	$0.569035\pi$
$182$	483.475	0.196910
$183$	3768.06	1.52209
$184$	−796.636	−0.319178
$185$	1038.72	0.412801
$186$	−40.8337	−0.0160971
$187$	−6071.70	−2.37437
$188$	2447.22	0.949371
$189$	614.375	0.236451
$190$	0	0
$191$	−2438.71	−0.923869	−0.461934	−	0.886914i	$-0.652844\pi$
$191$	−2438.71	−0.923869	−0.461934	+	0.886914i	$0.652844\pi$
$192$	421.097	0.158282
$193$	−840.778	−0.313578	−0.156789	−	0.987632i	$-0.550114\pi$
$193$	−840.778	−0.313578	−0.156789	+	0.987632i	$0.550114\pi$
$194$	−470.288	−0.174045
$195$	2070.28	0.760286
$196$	−1067.86	−0.389161
$197$	1426.01	0.515732	0.257866	−	0.966181i	$-0.416981\pi$
$197$	1426.01	0.515732	0.257866	+	0.966181i	$0.416981\pi$
$198$	−1684.82	−0.604723
$199$	2466.17	0.878502	0.439251	−	0.898364i	$-0.355244\pi$
$199$	2466.17	0.878502	0.439251	+	0.898364i	$0.355244\pi$
$200$	30.5592	0.0108043
$201$	1964.03	0.689215
$202$	1881.20	0.655251
$203$	1216.95	0.420754
$204$	3090.39	1.06064
$205$	2838.08	0.966926
$206$	3544.78	1.19892
$207$	−1622.32	−0.544728
$208$	−443.563	−0.147863
$209$	0	0
$210$	1302.37	0.427961
$211$	−2146.65	−0.700386	−0.350193	−	0.936677i	$-0.613884\pi$
$211$	−2146.65	−0.700386	−0.350193	+	0.936677i	$0.613884\pi$
$212$	−2434.18	−0.788586
$213$	1665.02	0.535613
$214$	−1050.35	−0.335516
$215$	4697.94	1.49022
$216$	−563.656	−0.177555
$217$	27.0580	0.00846459
$218$	1638.28	0.508985
$219$	−3426.37	−1.05723
$220$	2347.53	0.719410
$221$	−3255.27	−0.990828
$222$	−1204.31	−0.364091
$223$	−3116.98	−0.936002	−0.468001	−	0.883728i	$-0.655026\pi$
$223$	−3116.98	−0.936002	−0.468001	+	0.883728i	$0.655026\pi$
$224$	−279.035	−0.0832314
$225$	62.2326	0.0184393
$226$	3879.69	1.14192
$227$	−2700.49	−0.789595	−0.394798	−	0.918768i	$-0.629185\pi$
$227$	−2700.49	−0.789595	−0.394798	+	0.918768i	$0.629185\pi$
$228$	0	0
$229$	−6126.53	−1.76791	−0.883957	−	0.467568i	$-0.845130\pi$
$229$	−6126.53	−1.76791	−0.883957	+	0.467568i	$0.845130\pi$
$230$	2260.43	0.648037
$231$	2966.68	0.844992
$232$	−1116.49	−0.315952
$233$	−265.975	−0.0747837	−0.0373918	−	0.999301i	$-0.511905\pi$
$233$	−265.975	−0.0747837	−0.0373918	+	0.999301i	$0.511905\pi$
$234$	−903.297	−0.252352
$235$	−6943.91	−1.92754
$236$	−2289.11	−0.631390
$237$	1489.80	0.408325
$238$	−2047.81	−0.557732
$239$	1767.95	0.478491	0.239245	−	0.970959i	$-0.423100\pi$
$239$	1767.95	0.478491	0.239245	+	0.970959i	$0.423100\pi$
$240$	−1194.85	−0.321364
$241$	−4987.74	−1.33315	−0.666573	−	0.745439i	$-0.732240\pi$
$241$	−4987.74	−1.33315	−0.666573	+	0.745439i	$0.732240\pi$
$242$	2685.47	0.713340
$243$	−4042.08	−1.06708
$244$	2290.74	0.601022
$245$	3030.01	0.790124
$246$	−3290.52	−0.852829
$247$	0	0
$248$	−24.8243	−0.00635622
$249$	−2506.44	−0.637908
$250$	2750.76	0.695893
$251$	−1194.93	−0.300492	−0.150246	−	0.988649i	$-0.548007\pi$
$251$	−1194.93	−0.300492	−0.150246	+	0.988649i	$0.548007\pi$
$252$	−568.244	−0.142048
$253$	5149.07	1.27952
$254$	1344.01	0.332010
$255$	−8768.90	−2.15345
$256$	256.000	0.0625000
$257$	7131.58	1.73096	0.865479	−	0.500946i	$-0.167014\pi$
$257$	7131.58	1.73096	0.865479	+	0.500946i	$0.167014\pi$
$258$	−5446.88	−1.31437
$259$	798.024	0.191455
$260$	1258.60	0.300211
$261$	−2273.67	−0.539222
$262$	3681.82	0.868181
$263$	1436.59	0.336820	0.168410	−	0.985717i	$-0.446137\pi$
$263$	1436.59	0.336820	0.168410	+	0.985717i	$0.446137\pi$
$264$	−2721.77	−0.634520
$265$	6906.92	1.60109
$266$	0	0
$267$	6646.14	1.52336
$268$	1194.01	0.272147
$269$	1201.62	0.272358	0.136179	−	0.990684i	$-0.456518\pi$
$269$	1201.62	0.272358	0.136179	+	0.990684i	$0.456518\pi$
$270$	1599.36	0.360496
$271$	835.458	0.187271	0.0936355	−	0.995607i	$-0.470151\pi$
$271$	835.458	0.187271	0.0936355	+	0.995607i	$0.470151\pi$
$272$	1878.76	0.418811
$273$	1590.55	0.352616
$274$	−1884.40	−0.415478
$275$	−197.520	−0.0433124
$276$	−2620.79	−0.571569
$277$	−571.329	−0.123927	−0.0619636	−	0.998078i	$-0.519736\pi$
$277$	−571.329	−0.123927	−0.0619636	+	0.998078i	$0.519736\pi$
$278$	514.980	0.111102
$279$	−50.5535	−0.0108479
$280$	791.755	0.168987
$281$	−6617.20	−1.40480	−0.702401	−	0.711782i	$-0.747889\pi$
$281$	−6617.20	−1.40480	−0.702401	+	0.711782i	$0.747889\pi$
$282$	8050.91	1.70009
$283$	−7080.88	−1.48733	−0.743665	−	0.668552i	$-0.766914\pi$
$283$	−7080.88	−1.48733	−0.743665	+	0.668552i	$0.766914\pi$
$284$	1012.23	0.211495
$285$	0	0
$286$	2866.98	0.592755
$287$	2180.43	0.448455
$288$	521.333	0.106666
$289$	8875.05	1.80644
$290$	3167.99	0.641486
$291$	−1547.16	−0.311671
$292$	−2083.02	−0.417463
$293$	6634.31	1.32280	0.661400	−	0.750033i	$-0.269963\pi$
$293$	6634.31	1.32280	0.661400	+	0.750033i	$0.269963\pi$
$294$	−3513.06	−0.696890
$295$	6495.27	1.28193
$296$	−732.144	−0.143767
$297$	3643.20	0.711785
$298$	−3760.32	−0.730972
$299$	2760.61	0.533947
$300$	100.534	0.0193479
$301$	3609.32	0.691155
$302$	−2684.47	−0.511504
$303$	6188.80	1.17339
$304$	0	0
$305$	−6499.90	−1.22027
$306$	3826.02	0.714767
$307$	3907.31	0.726391	0.363195	−	0.931713i	$-0.381686\pi$
$307$	3907.31	0.726391	0.363195	+	0.931713i	$0.381686\pi$
$308$	1803.55	0.333659
$309$	11661.7	2.14696
$310$	70.4381	0.0129052
$311$	2733.72	0.498440	0.249220	−	0.968447i	$-0.419826\pi$
$311$	2733.72	0.498440	0.249220	+	0.968447i	$0.419826\pi$
$312$	−1459.24	−0.264786
$313$	1480.79	0.267410	0.133705	−	0.991021i	$-0.457312\pi$
$313$	1480.79	0.267410	0.133705	+	0.991021i	$0.457312\pi$
$314$	2679.17	0.481510
$315$	1612.38	0.288403
$316$	905.705	0.161234
$317$	−10537.7	−1.86705	−0.933527	−	0.358508i	$-0.883286\pi$
$317$	−10537.7	−1.86705	−0.933527	+	0.358508i	$0.883286\pi$
$318$	−8008.02	−1.41216
$319$	7216.42	1.26659
$320$	−726.393	−0.126896
$321$	−3455.46	−0.600825
$322$	1736.64	0.300556
$323$	0	0
$324$	−3613.83	−0.619655
$325$	−105.898	−0.0180743
$326$	−4657.18	−0.791219
$327$	5389.66	0.911465
$328$	−2000.43	−0.336753
$329$	−5334.85	−0.893981
$330$	7722.94	1.28828
$331$	−1833.79	−0.304515	−0.152257	−	0.988341i	$-0.548654\pi$
$331$	−1833.79	−0.304515	−0.152257	+	0.988341i	$0.548654\pi$
$332$	−1523.75	−0.251888
$333$	−1490.98	−0.245361
$334$	5523.66	0.904914
$335$	−3387.96	−0.552549
$336$	−917.976	−0.149047
$337$	−1449.70	−0.234332	−0.117166	−	0.993112i	$-0.537381\pi$
$337$	−1449.70	−0.234332	−0.117166	+	0.993112i	$0.537381\pi$
$338$	−2856.91	−0.459749
$339$	12763.5	2.04489
$340$	−5330.93	−0.850324
$341$	160.452	0.0254808
$342$	0	0
$343$	5318.80	0.837283
$344$	−3311.35	−0.519001
$345$	7436.41	1.16047
$346$	−1286.25	−0.199854
$347$	−3123.67	−0.483249	−0.241624	−	0.970370i	$-0.577680\pi$
$347$	−3123.67	−0.483249	−0.241624	+	0.970370i	$0.577680\pi$
$348$	−3673.04	−0.565791
$349$	533.538	0.0818327	0.0409164	−	0.999163i	$-0.486972\pi$
$349$	533.538	0.0818327	0.0409164	+	0.999163i	$0.486972\pi$
$350$	−66.6180	−0.0101740
$351$	1953.26	0.297029
$352$	−1654.66	−0.250550
$353$	7494.42	1.12999	0.564997	−	0.825093i	$-0.308877\pi$
$353$	7494.42	1.12999	0.564997	+	0.825093i	$0.308877\pi$
$354$	−7530.75	−1.13066
$355$	−2872.17	−0.429405
$356$	4040.43	0.601523
$357$	−6736.94	−0.998759
$358$	−4258.71	−0.628715
$359$	−8677.70	−1.27574	−0.637871	−	0.770143i	$-0.720185\pi$
$359$	−8677.70	−1.27574	−0.637871	+	0.770143i	$0.720185\pi$
$360$	−1479.27	−0.216567
$361$	0	0
$362$	−2095.98	−0.304316
$363$	8834.70	1.27742
$364$	966.951	0.139236
$365$	5910.50	0.847588
$366$	7536.11	1.07628
$367$	2967.60	0.422091	0.211046	−	0.977476i	$-0.432313\pi$
$367$	2967.60	0.422091	0.211046	+	0.977476i	$0.432313\pi$
$368$	−1593.27	−0.225693
$369$	−4073.78	−0.574723
$370$	2077.44	0.291894
$371$	5306.43	0.742577
$372$	−81.6673	−0.0113824
$373$	−5481.32	−0.760890	−0.380445	−	0.924803i	$-0.624229\pi$
$373$	−5481.32	−0.760890	−0.380445	+	0.924803i	$0.624229\pi$
$374$	−12143.4	−1.67893
$375$	9049.51	1.24617
$376$	4894.44	0.671307
$377$	3868.99	0.528550
$378$	1228.75	0.167196
$379$	−9315.11	−1.26249	−0.631246	−	0.775582i	$-0.717456\pi$
$379$	−9315.11	−1.26249	−0.631246	+	0.775582i	$0.717456\pi$
$380$	0	0
$381$	4421.54	0.594547
$382$	−4877.42	−0.653274
$383$	3276.63	0.437149	0.218575	−	0.975820i	$-0.429859\pi$
$383$	3276.63	0.437149	0.218575	+	0.975820i	$0.429859\pi$
$384$	842.194	0.111922
$385$	−5117.52	−0.677437
$386$	−1681.56	−0.221733
$387$	−6743.43	−0.885757
$388$	−940.576	−0.123068
$389$	8311.58	1.08333	0.541663	−	0.840596i	$-0.317795\pi$
$389$	8311.58	1.08333	0.541663	+	0.840596i	$0.317795\pi$
$390$	4140.56	0.537603
$391$	−11692.9	−1.51236
$392$	−2135.71	−0.275178
$393$	12112.5	1.55470
$394$	2852.02	0.364677
$395$	−2569.91	−0.327358
$396$	−3369.65	−0.427604
$397$	−6835.99	−0.864203	−0.432101	−	0.901825i	$-0.642228\pi$
$397$	−6835.99	−0.864203	−0.432101	+	0.901825i	$0.642228\pi$
$398$	4932.33	0.621195
$399$	0	0
$400$	61.1185	0.00763981
$401$	7323.83	0.912056	0.456028	−	0.889965i	$-0.349272\pi$
$401$	7323.83	0.912056	0.456028	+	0.889965i	$0.349272\pi$
$402$	3928.06	0.487348
$403$	86.0243	0.0106332
$404$	3762.40	0.463332
$405$	10254.1	1.25810
$406$	2433.90	0.297518
$407$	4732.23	0.576334
$408$	6180.78	0.749986
$409$	−5514.51	−0.666687	−0.333344	−	0.942805i	$-0.608177\pi$
$409$	−5514.51	−0.666687	−0.333344	+	0.942805i	$0.608177\pi$
$410$	5676.15	0.683720
$411$	−6199.34	−0.744017
$412$	7089.56	0.847762
$413$	4990.17	0.594552
$414$	−3244.63	−0.385181
$415$	4323.61	0.511416
$416$	−887.126	−0.104555
$417$	1694.19	0.198957
$418$	0	0
$419$	−1125.88	−0.131272	−0.0656359	−	0.997844i	$-0.520908\pi$
$419$	−1125.88	−0.131272	−0.0656359	+	0.997844i	$0.520908\pi$
$420$	2604.73	0.302614
$421$	−6256.55	−0.724288	−0.362144	−	0.932122i	$-0.617955\pi$
$421$	−6256.55	−0.724288	−0.362144	+	0.932122i	$0.617955\pi$
$422$	−4293.30	−0.495248
$423$	9967.31	1.14569
$424$	−4868.36	−0.557615
$425$	448.543	0.0511942
$426$	3330.05	0.378736
$427$	−4993.72	−0.565956
$428$	−2100.70	−0.237245
$429$	9431.84	1.06148
$430$	9395.88	1.05374
$431$	−686.819	−0.0767585	−0.0383792	−	0.999263i	$-0.512219\pi$
$431$	−686.819	−0.0767585	−0.0383792	+	0.999263i	$0.512219\pi$
$432$	−1127.31	−0.125551
$433$	−4094.21	−0.454400	−0.227200	−	0.973848i	$-0.572957\pi$
$433$	−4094.21	−0.454400	−0.227200	+	0.973848i	$0.572957\pi$
$434$	54.1160	0.00598537
$435$	10422.1	1.14874
$436$	3276.57	0.359906
$437$	0	0
$438$	−6852.75	−0.747573
$439$	12331.9	1.34071	0.670355	−	0.742041i	$-0.266142\pi$
$439$	12331.9	1.34071	0.670355	+	0.742041i	$0.266142\pi$
$440$	4695.05	0.508700
$441$	−4349.29	−0.469635
$442$	−6510.53	−0.700621
$443$	−5690.36	−0.610287	−0.305144	−	0.952306i	$-0.598704\pi$
$443$	−5690.36	−0.610287	−0.305144	+	0.952306i	$0.598704\pi$
$444$	−2408.62	−0.257451
$445$	−11464.6	−1.22129
$446$	−6233.96	−0.661853
$447$	−12370.8	−1.30899
$448$	−558.071	−0.0588535
$449$	14973.3	1.57380	0.786898	−	0.617083i	$-0.211686\pi$
$449$	14973.3	1.57380	0.786898	+	0.617083i	$0.211686\pi$
$450$	124.465	0.0130385
$451$	12929.8	1.34998
$452$	7759.38	0.807457
$453$	−8831.43	−0.915976
$454$	−5400.99	−0.558328
$455$	−2743.70	−0.282695
$456$	0	0
$457$	−13039.1	−1.33466	−0.667332	−	0.744760i	$-0.732564\pi$
$457$	−13039.1	−1.33466	−0.667332	+	0.744760i	$0.732564\pi$
$458$	−12253.1	−1.25010
$459$	−8273.24	−0.841311
$460$	4520.86	0.458231
$461$	−5776.83	−0.583631	−0.291816	−	0.956475i	$-0.594259\pi$
$461$	−5776.83	−0.583631	−0.291816	+	0.956475i	$0.594259\pi$
$462$	5933.36	0.597499
$463$	−16838.6	−1.69019	−0.845094	−	0.534617i	$-0.820456\pi$
$463$	−16838.6	−1.69019	−0.845094	+	0.534617i	$0.820456\pi$
$464$	−2232.97	−0.223412
$465$	231.729	0.0231100
$466$	−531.950	−0.0528801
$467$	4389.55	0.434956	0.217478	−	0.976065i	$-0.430217\pi$
$467$	4389.55	0.434956	0.217478	+	0.976065i	$0.430217\pi$
$468$	−1806.59	−0.178440
$469$	−2602.89	−0.256269
$470$	−13887.8	−1.36297
$471$	8813.98	0.862265
$472$	−4578.21	−0.446460
$473$	21403.0	2.08057
$474$	2979.61	0.288730
$475$	0	0
$476$	−4095.63	−0.394376
$477$	−9914.21	−0.951658
$478$	3535.90	0.338344
$479$	−2993.90	−0.285584	−0.142792	−	0.989753i	$-0.545608\pi$
$479$	−2993.90	−0.285584	−0.142792	+	0.989753i	$0.545608\pi$
$480$	−2389.70	−0.227238
$481$	2537.13	0.240505
$482$	−9975.47	−0.942677
$483$	5713.23	0.538221
$484$	5370.93	0.504408
$485$	2668.86	0.249869
$486$	−8084.16	−0.754537
$487$	5532.20	0.514760	0.257380	−	0.966310i	$-0.417141\pi$
$487$	5532.20	0.514760	0.257380	+	0.966310i	$0.417141\pi$
$488$	4581.47	0.424987
$489$	−15321.3	−1.41688
$490$	6060.02	0.558702
$491$	−4521.54	−0.415589	−0.207794	−	0.978173i	$-0.566628\pi$
$491$	−4521.54	−0.415589	−0.207794	+	0.978173i	$0.566628\pi$
$492$	−6581.04	−0.603042
$493$	−16387.6	−1.49708
$494$	0	0
$495$	9561.28	0.868177
$496$	−49.6485	−0.00449452
$497$	−2206.62	−0.199156
$498$	−5012.88	−0.451069
$499$	−1983.44	−0.177938	−0.0889691	−	0.996034i	$-0.528357\pi$
$499$	−1983.44	−0.177938	−0.0889691	+	0.996034i	$0.528357\pi$
$500$	5501.52	0.492071
$501$	18171.8	1.62047
$502$	−2389.86	−0.212480
$503$	−18851.4	−1.67106	−0.835530	−	0.549445i	$-0.814839\pi$
$503$	−18851.4	−1.67106	−0.835530	+	0.549445i	$0.814839\pi$
$504$	−1136.49	−0.100443
$505$	−10675.7	−0.940717
$506$	10298.1	0.904760
$507$	−9398.71	−0.823296
$508$	2688.01	0.234766
$509$	−4854.17	−0.422706	−0.211353	−	0.977410i	$-0.567787\pi$
$509$	−4854.17	−0.422706	−0.211353	+	0.977410i	$0.567787\pi$
$510$	−17537.8	−1.52272
$511$	4540.90	0.393107
$512$	512.000	0.0441942
$513$	0	0
$514$	14263.2	1.22397
$515$	−20116.4	−1.72124
$516$	−10893.8	−0.929401
$517$	−31635.3	−2.69114
$518$	1596.05	0.135379
$519$	−4231.55	−0.357889
$520$	2517.19	0.212281
$521$	−7893.73	−0.663782	−0.331891	−	0.943318i	$-0.607687\pi$
$521$	−7893.73	−0.663782	−0.331891	+	0.943318i	$0.607687\pi$
$522$	−4547.35	−0.381287
$523$	17108.2	1.43038	0.715189	−	0.698931i	$-0.246341\pi$
$523$	17108.2	1.43038	0.715189	+	0.698931i	$0.246341\pi$
$524$	7363.64	0.613897
$525$	−219.161	−0.0182190
$526$	2873.17	0.238168
$527$	−364.366	−0.0301177
$528$	−5443.54	−0.448673
$529$	−2250.92	−0.185002
$530$	13813.8	1.13214
$531$	−9323.33	−0.761955
$532$	0	0
$533$	6932.15	0.563348
$534$	13292.3	1.07718
$535$	5960.67	0.481686
$536$	2388.01	0.192437
$537$	−14010.4	−1.12587
$538$	2403.25	0.192586
$539$	13804.2	1.10314
$540$	3198.72	0.254909
$541$	23093.5	1.83525	0.917623	−	0.397451i	$-0.130105\pi$
$541$	23093.5	1.83525	0.917623	+	0.397451i	$0.130105\pi$
$542$	1670.92	0.132421
$543$	−6895.41	−0.544955
$544$	3757.52	0.296144
$545$	−9297.17	−0.730729
$546$	3181.09	0.249337
$547$	20004.9	1.56371	0.781854	−	0.623461i	$-0.214274\pi$
$547$	20004.9	1.56371	0.781854	+	0.623461i	$0.214274\pi$
$548$	−3768.80	−0.293787
$549$	9329.98	0.725307
$550$	−395.040	−0.0306265
$551$	0	0
$552$	−5241.58	−0.404160
$553$	−1974.40	−0.151827
$554$	−1142.66	−0.0876298
$555$	6834.40	0.522710
$556$	1029.96	0.0785612
$557$	−7313.79	−0.556365	−0.278183	−	0.960528i	$-0.589732\pi$
$557$	−7313.79	−0.556365	−0.278183	+	0.960528i	$0.589732\pi$
$558$	−101.107	−0.00767062
$559$	11475.0	0.868227
$560$	1583.51	0.119492
$561$	−39949.6	−3.00655
$562$	−13234.4	−0.993345
$563$	14546.0	1.08888	0.544442	−	0.838799i	$-0.316742\pi$
$563$	14546.0	1.08888	0.544442	+	0.838799i	$0.316742\pi$
$564$	16101.8	1.20214
$565$	−22017.0	−1.63940
$566$	−14161.8	−1.05170
$567$	7878.01	0.583502
$568$	2024.46	0.149550
$569$	−9607.67	−0.707864	−0.353932	−	0.935271i	$-0.615156\pi$
$569$	−9607.67	−0.707864	−0.353932	+	0.935271i	$0.615156\pi$
$570$	0	0
$571$	4066.67	0.298047	0.149024	−	0.988834i	$-0.452387\pi$
$571$	4066.67	0.298047	0.149024	+	0.988834i	$0.452387\pi$
$572$	5733.95	0.419141
$573$	−16045.8	−1.16985
$574$	4360.86	0.317106
$575$	−380.384	−0.0275880
$576$	1042.67	0.0754243
$577$	8766.39	0.632495	0.316247	−	0.948677i	$-0.397577\pi$
$577$	8766.39	0.632495	0.316247	+	0.948677i	$0.397577\pi$
$578$	17750.1	1.27735
$579$	−5532.02	−0.397069
$580$	6335.99	0.453599
$581$	3321.73	0.237192
$582$	−3094.33	−0.220385
$583$	31466.8	2.23537
$584$	−4166.03	−0.295191
$585$	5126.16	0.362292
$586$	13268.6	0.935361
$587$	16508.1	1.16075	0.580376	−	0.814348i	$-0.302905\pi$
$587$	16508.1	1.16075	0.580376	+	0.814348i	$0.302905\pi$
$588$	−7026.11	−0.492775
$589$	0	0
$590$	12990.5	0.906462
$591$	9382.65	0.653047
$592$	−1464.29	−0.101659
$593$	5650.33	0.391283	0.195642	−	0.980675i	$-0.437321\pi$
$593$	5650.33	0.391283	0.195642	+	0.980675i	$0.437321\pi$
$594$	7286.41	0.503308
$595$	11621.2	0.800713
$596$	−7520.64	−0.516875
$597$	16226.5	1.11241
$598$	5521.22	0.377558
$599$	−1625.81	−0.110899	−0.0554496	−	0.998461i	$-0.517659\pi$
$599$	−1625.81	−0.110899	−0.0554496	+	0.998461i	$0.517659\pi$
$600$	201.069	0.0136810
$601$	−24547.1	−1.66605	−0.833024	−	0.553237i	$-0.813393\pi$
$601$	−24547.1	−1.66605	−0.833024	+	0.553237i	$0.813393\pi$
$602$	7218.63	0.488720
$603$	4863.08	0.328425
$604$	−5368.95	−0.361688
$605$	−15239.9	−1.02411
$606$	12377.6	0.829713
$607$	6347.64	0.424452	0.212226	−	0.977221i	$-0.431929\pi$
$607$	6347.64	0.424452	0.212226	+	0.977221i	$0.431929\pi$
$608$	0	0
$609$	8007.08	0.532781
$610$	−12999.8	−0.862863
$611$	−16960.9	−1.12302
$612$	7652.03	0.505417
$613$	−24336.3	−1.60348	−0.801742	−	0.597671i	$-0.796093\pi$
$613$	−24336.3	−1.60348	−0.801742	+	0.597671i	$0.796093\pi$
$614$	7814.62	0.513636
$615$	18673.5	1.22437
$616$	3607.10	0.235932
$617$	−6074.70	−0.396367	−0.198183	−	0.980165i	$-0.563504\pi$
$617$	−6074.70	−0.396367	−0.198183	+	0.980165i	$0.563504\pi$
$618$	23323.4	1.51813
$619$	116.247	0.00754826	0.00377413	−	0.999993i	$-0.498799\pi$
$619$	116.247	0.00754826	0.00377413	+	0.999993i	$0.498799\pi$
$620$	140.876	0.00912536
$621$	7016.08	0.453374
$622$	5467.43	0.352450
$623$	−8807.98	−0.566427
$624$	−2918.48	−0.187232
$625$	−16087.9	−1.02963
$626$	2961.59	0.189088
$627$	0	0
$628$	5358.33	0.340479
$629$	−10746.3	−0.681212
$630$	3224.75	0.203932
$631$	−16646.4	−1.05021	−0.525105	−	0.851038i	$-0.675974\pi$
$631$	−16646.4	−1.05021	−0.525105	+	0.851038i	$0.675974\pi$
$632$	1811.41	0.114010
$633$	−14124.2	−0.886866
$634$	−21075.4	−1.32021
$635$	−7627.16	−0.476653
$636$	−16016.0	−0.998549
$637$	7400.96	0.460340
$638$	14432.8	0.895614
$639$	4122.72	0.255230
$640$	−1452.79	−0.0897287
$641$	−5283.00	−0.325532	−0.162766	−	0.986665i	$-0.552042\pi$
$641$	−5283.00	−0.325532	−0.162766	+	0.986665i	$0.552042\pi$
$642$	−6910.91	−0.424847
$643$	−24191.9	−1.48373	−0.741863	−	0.670552i	$-0.766057\pi$
$643$	−24191.9	−1.48373	−0.741863	+	0.670552i	$0.766057\pi$
$644$	3473.28	0.212525
$645$	30910.7	1.88699
$646$	0	0
$647$	−22736.9	−1.38158	−0.690789	−	0.723056i	$-0.742737\pi$
$647$	−22736.9	−1.38158	−0.690789	+	0.723056i	$0.742737\pi$
$648$	−7227.65	−0.438162
$649$	29591.4	1.78977
$650$	−211.796	−0.0127805
$651$	178.032	0.0107183
$652$	−9314.36	−0.559476
$653$	1973.42	0.118263	0.0591317	−	0.998250i	$-0.481167\pi$
$653$	1973.42	0.118263	0.0591317	+	0.998250i	$0.481167\pi$
$654$	10779.3	0.644503
$655$	−20894.1	−1.24641
$656$	−4000.85	−0.238121
$657$	−8483.95	−0.503791
$658$	−10669.7	−0.632140
$659$	−1187.82	−0.0702140	−0.0351070	−	0.999384i	$-0.511177\pi$
$659$	−1187.82	−0.0702140	−0.0351070	+	0.999384i	$0.511177\pi$
$660$	15445.9	0.910955
$661$	28902.3	1.70071	0.850356	−	0.526208i	$-0.176387\pi$
$661$	28902.3	1.70071	0.850356	+	0.526208i	$0.176387\pi$
$662$	−3667.58	−0.215324
$663$	−21418.5	−1.25464
$664$	−3047.51	−0.178112
$665$	0	0
$666$	−2981.96	−0.173496
$667$	13897.4	0.806760
$668$	11047.3	0.639870
$669$	−20508.6	−1.18521
$670$	−6775.91	−0.390711
$671$	−29612.5	−1.70369
$672$	−1835.95	−0.105392
$673$	10684.5	0.611974	0.305987	−	0.952036i	$-0.401014\pi$
$673$	10684.5	0.611974	0.305987	+	0.952036i	$0.401014\pi$
$674$	−2899.39	−0.165698
$675$	−269.139	−0.0153469
$676$	−5713.81	−0.325092
$677$	−24902.1	−1.41369	−0.706844	−	0.707369i	$-0.749882\pi$
$677$	−24902.1	−1.41369	−0.706844	+	0.707369i	$0.749882\pi$
$678$	25527.0	1.44595
$679$	2050.42	0.115888
$680$	−10661.9	−0.601270
$681$	−17768.3	−0.999827
$682$	320.904	0.0180177
$683$	24754.8	1.38685	0.693423	−	0.720531i	$-0.256102\pi$
$683$	24754.8	1.38685	0.693423	+	0.720531i	$0.256102\pi$
$684$	0	0
$685$	10693.9	0.596484
$686$	10637.6	0.592049
$687$	−40310.3	−2.23863
$688$	−6622.71	−0.366989
$689$	16870.5	0.932823
$690$	14872.8	0.820578
$691$	8123.33	0.447215	0.223608	−	0.974679i	$-0.428217\pi$
$691$	8123.33	0.447215	0.223608	+	0.974679i	$0.428217\pi$
$692$	−2572.51	−0.141318
$693$	7345.71	0.402656
$694$	−6247.34	−0.341709
$695$	−2922.48	−0.159505
$696$	−7346.07	−0.400075
$697$	−29361.9	−1.59564
$698$	1067.08	0.0578645
$699$	−1750.02	−0.0946950
$700$	−133.236	−0.00719407
$701$	−15182.5	−0.818027	−0.409014	−	0.912528i	$-0.634127\pi$
$701$	−15182.5	−0.818027	−0.409014	+	0.912528i	$0.634127\pi$
$702$	3906.51	0.210031
$703$	0	0
$704$	−3309.32	−0.177166
$705$	−45688.4	−2.44075
$706$	14988.8	0.799026
$707$	−8201.89	−0.436299
$708$	−15061.5	−0.799499
$709$	−16735.5	−0.886482	−0.443241	−	0.896403i	$-0.646171\pi$
$709$	−16735.5	−0.886482	−0.443241	+	0.896403i	$0.646171\pi$
$710$	−5744.34	−0.303635
$711$	3688.86	0.194575
$712$	8080.85	0.425341
$713$	308.998	0.0162301
$714$	−13473.9	−0.706229
$715$	−16269.9	−0.850994
$716$	−8517.42	−0.444569
$717$	11632.5	0.605890
$718$	−17355.4	−0.902086
$719$	26590.6	1.37922	0.689612	−	0.724179i	$-0.257781\pi$
$719$	26590.6	1.37922	0.689612	+	0.724179i	$0.257781\pi$
$720$	−2958.54	−0.153136
$721$	−15455.0	−0.798300
$722$	0	0
$723$	−32817.5	−1.68810
$724$	−4191.97	−0.215184
$725$	−533.108	−0.0273092
$726$	17669.4	0.903269
$727$	−14273.5	−0.728163	−0.364082	−	0.931367i	$-0.618617\pi$
$727$	−14273.5	−0.728163	−0.364082	+	0.931367i	$0.618617\pi$
$728$	1933.90	0.0984549
$729$	−2202.10	−0.111878
$730$	11821.0	0.599335
$731$	−48603.5	−2.45918
$732$	15072.2	0.761046
$733$	−26295.0	−1.32501	−0.662503	−	0.749060i	$-0.730506\pi$
$733$	−26295.0	−1.32501	−0.662503	+	0.749060i	$0.730506\pi$
$734$	5935.20	0.298464
$735$	19936.4	1.00050
$736$	−3186.54	−0.159589
$737$	−15435.0	−0.771444
$738$	−8147.57	−0.406390
$739$	4066.72	0.202431	0.101216	−	0.994865i	$-0.467727\pi$
$739$	4066.72	0.202431	0.101216	+	0.994865i	$0.467727\pi$
$740$	4154.88	0.206400
$741$	0	0
$742$	10612.9	0.525081
$743$	15714.7	0.775929	0.387964	−	0.921674i	$-0.373178\pi$
$743$	15714.7	0.775929	0.387964	+	0.921674i	$0.373178\pi$
$744$	−163.335	−0.00804857
$745$	21339.6	1.04943
$746$	−10962.6	−0.538031
$747$	−6206.12	−0.303976
$748$	−24286.8	−1.18718
$749$	4579.44	0.223403
$750$	18099.0	0.881177
$751$	−32914.0	−1.59927	−0.799633	−	0.600489i	$-0.794973\pi$
$751$	−32914.0	−1.59927	−0.799633	+	0.600489i	$0.794973\pi$
$752$	9788.87	0.474685
$753$	−7862.22	−0.380498
$754$	7737.99	0.373741
$755$	15234.2	0.734345
$756$	2457.50	0.118225
$757$	23538.5	1.13015	0.565074	−	0.825040i	$-0.308848\pi$
$757$	23538.5	1.13015	0.565074	+	0.825040i	$0.308848\pi$
$758$	−18630.2	−0.892717
$759$	33879.0	1.62020
$760$	0	0
$761$	−32174.2	−1.53260	−0.766302	−	0.642480i	$-0.777905\pi$
$761$	−32174.2	−1.53260	−0.766302	+	0.642480i	$0.777905\pi$
$762$	8843.08	0.420408
$763$	−7142.80	−0.338908
$764$	−9754.84	−0.461934
$765$	−21712.4	−1.02616
$766$	6553.27	0.309111
$767$	15865.0	0.746875
$768$	1684.39	0.0791408
$769$	−10351.7	−0.485426	−0.242713	−	0.970098i	$-0.578037\pi$
$769$	−10351.7	−0.485426	−0.242713	+	0.970098i	$0.578037\pi$
$770$	−10235.0	−0.479020
$771$	46923.3	2.19183
$772$	−3363.11	−0.156789
$773$	18509.2	0.861231	0.430615	−	0.902536i	$-0.358297\pi$
$773$	18509.2	0.861231	0.430615	+	0.902536i	$0.358297\pi$
$774$	−13486.9	−0.626325
$775$	−11.8533	−0.000549397 0
$776$	−1881.15	−0.0870224
$777$	5250.71	0.242430
$778$	16623.2	0.766027
$779$	0	0
$780$	8281.11	0.380143
$781$	−13085.1	−0.599516
$782$	−23385.8	−1.06940
$783$	9833.03	0.448791
$784$	−4271.43	−0.194580
$785$	−15204.1	−0.691284
$786$	24225.0	1.09934
$787$	−28804.0	−1.30464	−0.652320	−	0.757944i	$-0.726204\pi$
$787$	−28804.0	−1.30464	−0.652320	+	0.757944i	$0.726204\pi$
$788$	5704.05	0.257866
$789$	9452.22	0.426499
$790$	−5139.82	−0.231477
$791$	−16915.2	−0.760346
$792$	−6739.29	−0.302362
$793$	−15876.3	−0.710953
$794$	−13672.0	−0.611084
$795$	45445.1	2.02738
$796$	9864.67	0.439251
$797$	−27046.4	−1.20205	−0.601025	−	0.799230i	$-0.705241\pi$
$797$	−27046.4	−1.20205	−0.601025	+	0.799230i	$0.705241\pi$
$798$	0	0
$799$	71839.6	3.18086
$800$	122.237	0.00540216
$801$	16456.3	0.725911
$802$	14647.7	0.644921
$803$	26927.2	1.18336
$804$	7856.13	0.344607
$805$	−9855.32	−0.431496
$806$	172.049	0.00751880
$807$	7906.25	0.344874
$808$	7524.79	0.327625
$809$	−37290.5	−1.62060	−0.810300	−	0.586015i	$-0.800696\pi$
$809$	−37290.5	−1.62060	−0.810300	+	0.586015i	$0.800696\pi$
$810$	20508.3	0.889613
$811$	−4392.89	−0.190204	−0.0951018	−	0.995468i	$-0.530318\pi$
$811$	−4392.89	−0.190204	−0.0951018	+	0.995468i	$0.530318\pi$
$812$	4867.79	0.210377
$813$	5497.01	0.237132
$814$	9464.46	0.407530
$815$	26429.2	1.13592
$816$	12361.6	0.530320
$817$	0	0
$818$	−11029.0	−0.471419
$819$	3938.31	0.168029
$820$	11352.3	0.483463
$821$	−18524.5	−0.787467	−0.393733	−	0.919225i	$-0.628817\pi$
$821$	−18524.5	−0.787467	−0.393733	+	0.919225i	$0.628817\pi$
$822$	−12398.7	−0.526099
$823$	37288.1	1.57932	0.789661	−	0.613543i	$-0.210256\pi$
$823$	37288.1	1.57932	0.789661	+	0.613543i	$0.210256\pi$
$824$	14179.1	0.599458
$825$	−1299.61	−0.0548445
$826$	9980.34	0.420412
$827$	18175.7	0.764245	0.382122	−	0.924112i	$-0.375193\pi$
$827$	18175.7	0.764245	0.382122	+	0.924112i	$0.375193\pi$
$828$	−6489.26	−0.272364
$829$	−4062.26	−0.170191	−0.0850953	−	0.996373i	$-0.527119\pi$
$829$	−4062.26	−0.170191	−0.0850953	+	0.996373i	$0.527119\pi$
$830$	8647.22	0.361626
$831$	−3759.14	−0.156923
$832$	−1774.25	−0.0739316
$833$	−31347.6	−1.30388
$834$	3388.38	0.140684
$835$	−31346.4	−1.29915
$836$	0	0
$837$	218.630	0.00902864
$838$	−2251.76	−0.0928232
$839$	−35263.6	−1.45106	−0.725528	−	0.688193i	$-0.758404\pi$
$839$	−35263.6	−1.45106	−0.725528	+	0.688193i	$0.758404\pi$
$840$	5209.46	0.213980
$841$	−4911.83	−0.201395
$842$	−12513.1	−0.512149
$843$	−43538.8	−1.77883
$844$	−8586.60	−0.350193
$845$	16212.8	0.660043
$846$	19934.6	0.810126
$847$	−11708.4	−0.474979
$848$	−9736.73	−0.394293
$849$	−46589.6	−1.88334
$850$	897.086	0.0361997
$851$	9113.32	0.367098
$852$	6660.10	0.267807
$853$	−27894.0	−1.11966	−0.559831	−	0.828607i	$-0.689134\pi$
$853$	−27894.0	−1.11966	−0.559831	+	0.828607i	$0.689134\pi$
$854$	−9987.45	−0.400191
$855$	0	0
$856$	−4201.39	−0.167758
$857$	−1383.45	−0.0551432	−0.0275716	−	0.999620i	$-0.508777\pi$
$857$	−1383.45	−0.0551432	−0.0275716	+	0.999620i	$0.508777\pi$
$858$	18863.7	0.750577
$859$	−14283.6	−0.567348	−0.283674	−	0.958921i	$-0.591553\pi$
$859$	−14283.6	−0.567348	−0.283674	+	0.958921i	$0.591553\pi$
$860$	18791.8	0.745109
$861$	14346.4	0.567858
$862$	−1373.64	−0.0542764
$863$	1103.14	0.0435126	0.0217563	−	0.999763i	$-0.493074\pi$
$863$	1103.14	0.0435126	0.0217563	+	0.999763i	$0.493074\pi$
$864$	−2254.62	−0.0887777
$865$	7299.42	0.286922
$866$	−8188.43	−0.321310
$867$	58394.6	2.28741
$868$	108.232	0.00423229
$869$	−11708.1	−0.457042
$870$	20844.3	0.812284
$871$	−8275.26	−0.321925
$872$	6553.14	0.254492
$873$	−3830.89	−0.148518
$874$	0	0
$875$	−11993.1	−0.463362
$876$	−13705.5	−0.528614
$877$	33220.4	1.27910	0.639551	−	0.768749i	$-0.279120\pi$
$877$	33220.4	1.27910	0.639551	+	0.768749i	$0.279120\pi$
$878$	24663.9	0.948025
$879$	43651.4	1.67500
$880$	9390.11	0.359705
$881$	28675.9	1.09661	0.548306	−	0.836278i	$-0.315273\pi$
$881$	28675.9	1.09661	0.548306	+	0.836278i	$0.315273\pi$
$882$	−8698.58	−0.332082
$883$	15476.5	0.589836	0.294918	−	0.955523i	$-0.404708\pi$
$883$	15476.5	0.589836	0.294918	+	0.955523i	$0.404708\pi$
$884$	−13021.1	−0.495414
$885$	42736.6	1.62325
$886$	−11380.7	−0.431538
$887$	13372.3	0.506197	0.253099	−	0.967440i	$-0.418550\pi$
$887$	13372.3	0.506197	0.253099	+	0.967440i	$0.418550\pi$
$888$	−4817.25	−0.182045
$889$	−5859.77	−0.221069
$890$	−22929.2	−0.863582
$891$	46716.1	1.75651
$892$	−12467.9	−0.468001
$893$	0	0
$894$	−24741.6	−0.925595
$895$	24167.9	0.902621
$896$	−1116.14	−0.0416157
$897$	18163.8	0.676112
$898$	29946.6	1.11284
$899$	433.061	0.0160661
$900$	248.930	0.00921964
$901$	−71456.9	−2.64215
$902$	25859.6	0.954579
$903$	23748.0	0.875176
$904$	15518.8	0.570958
$905$	11894.6	0.436895
$906$	−17662.9	−0.647693
$907$	−32210.6	−1.17920	−0.589600	−	0.807695i	$-0.700715\pi$
$907$	−32210.6	−1.17920	−0.589600	+	0.807695i	$0.700715\pi$
$908$	−10802.0	−0.394798
$909$	15323.9	0.559144
$910$	−5487.39	−0.199896
$911$	−9415.47	−0.342424	−0.171212	−	0.985234i	$-0.554768\pi$
$911$	−9415.47	−0.342424	−0.171212	+	0.985234i	$0.554768\pi$
$912$	0	0
$913$	19697.6	0.714016
$914$	−26078.1	−0.943750
$915$	−42767.0	−1.54517
$916$	−24506.1	−0.883957
$917$	−16052.5	−0.578080
$918$	−16546.5	−0.594897
$919$	−15367.4	−0.551604	−0.275802	−	0.961215i	$-0.588943\pi$
$919$	−15367.4	−0.551604	−0.275802	+	0.961215i	$0.588943\pi$
$920$	9041.73	0.324018
$921$	25708.7	0.919794
$922$	−11553.7	−0.412690
$923$	−7015.42	−0.250179
$924$	11866.7	0.422496
$925$	−349.590	−0.0124264
$926$	−33677.3	−1.19514
$927$	28875.2	1.02307
$928$	−4465.94	−0.157976
$929$	3284.09	0.115982	0.0579911	−	0.998317i	$-0.481531\pi$
$929$	3284.09	0.115982	0.0579911	+	0.998317i	$0.481531\pi$
$930$	463.457	0.0163413
$931$	0	0
$932$	−1063.90	−0.0373918
$933$	17986.9	0.631151
$934$	8779.11	0.307560
$935$	68913.1	2.41037
$936$	−3613.19	−0.126176
$937$	15378.8	0.536182	0.268091	−	0.963394i	$-0.413607\pi$
$937$	15378.8	0.536182	0.268091	+	0.963394i	$0.413607\pi$
$938$	−5205.78	−0.181210
$939$	9743.09	0.338609
$940$	−27775.6	−0.963768
$941$	−31637.1	−1.09600	−0.548002	−	0.836477i	$-0.684611\pi$
$941$	−31637.1	−1.09600	−0.548002	+	0.836477i	$0.684611\pi$
$942$	17628.0	0.609713
$943$	24900.2	0.859875
$944$	−9156.42	−0.315695
$945$	−6973.09	−0.240037
$946$	42806.0	1.47119
$947$	33955.0	1.16514	0.582571	−	0.812780i	$-0.302047\pi$
$947$	33955.0	1.16514	0.582571	+	0.812780i	$0.302047\pi$
$948$	5959.21	0.204163
$949$	14436.7	0.493820
$950$	0	0
$951$	−69334.2	−2.36416
$952$	−8191.26	−0.278866
$953$	50434.5	1.71431	0.857153	−	0.515061i	$-0.172231\pi$
$953$	50434.5	1.71431	0.857153	+	0.515061i	$0.172231\pi$
$954$	−19828.4	−0.672924
$955$	27679.1	0.937879
$956$	7071.81	0.239245
$957$	47481.5	1.60382
$958$	−5987.80	−0.201939
$959$	8215.85	0.276646
$960$	−4779.40	−0.160682
$961$	−29781.4	−0.999677
$962$	5074.25	0.170063
$963$	−8555.96	−0.286305
$964$	−19950.9	−0.666573
$965$	9542.74	0.318333
$966$	11426.5	0.380580
$967$	−1231.13	−0.0409414	−0.0204707	−	0.999790i	$-0.506516\pi$
$967$	−1231.13	−0.0409414	−0.0204707	+	0.999790i	$0.506516\pi$
$968$	10741.9	0.356670
$969$	0	0
$970$	5337.71	0.176684
$971$	36327.4	1.20062	0.600310	−	0.799767i	$-0.295044\pi$
$971$	36327.4	1.20062	0.600310	+	0.799767i	$0.295044\pi$
$972$	−16168.3	−0.533538
$973$	−2245.28	−0.0739776
$974$	11064.4	0.363990
$975$	−696.771	−0.0228867
$976$	9162.95	0.300511
$977$	10566.3	0.346004	0.173002	−	0.984922i	$-0.444653\pi$
$977$	10566.3	0.346004	0.173002	+	0.984922i	$0.444653\pi$
$978$	−30642.6	−1.00188
$979$	−52230.7	−1.70511
$980$	12120.0	0.395062
$981$	13345.2	0.434331
$982$	−9043.07	−0.293866
$983$	46648.8	1.51360	0.756798	−	0.653649i	$-0.226763\pi$
$983$	46648.8	1.51360	0.756798	+	0.653649i	$0.226763\pi$
$984$	−13162.1	−0.426415
$985$	−16185.1	−0.523553
$986$	−32775.1	−1.05859
$987$	−35101.4	−1.13201
$988$	0	0
$989$	41217.9	1.32523
$990$	19122.6	0.613894
$991$	12812.9	0.410711	0.205356	−	0.978687i	$-0.434165\pi$
$991$	12812.9	0.410711	0.205356	+	0.978687i	$0.434165\pi$
$992$	−99.2970	−0.00317811
$993$	−12065.7	−0.385592
$994$	−4413.24	−0.140824
$995$	−27990.7	−0.891824
$996$	−10025.8	−0.318954
$997$	50083.8	1.59094	0.795471	−	0.605992i	$-0.207224\pi$
$997$	50083.8	1.59094	0.795471	+	0.605992i	$0.207224\pi$
$998$	−3966.89	−0.125821
$999$	6448.09	0.204213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	722.4.a.p.1.6		6
19.4	even	9		38.4.e.a.35.1	yes	12
19.5	even	9		38.4.e.a.25.1	✓	12
19.18	odd	2		722.4.a.o.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.e.a.25.1	✓	12	19.5	even	9
38.4.e.a.35.1	yes	12	19.4	even	9
722.4.a.o.1.1		6	19.18	odd	2
722.4.a.p.1.6		6	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:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	722.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +6.57964 q^{3} +4.00000 q^{4} -11.3499 q^{5} +13.1593 q^{6} -8.71985 q^{7} +8.00000 q^{8} +16.2917 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +6.57964 q^{3} +4.00000 q^{4} -11.3499 q^{5} +13.1593 q^{6} -8.71985 q^{7} +8.00000 q^{8} +16.2917 q^{9} -22.6998 q^{10} -51.7082 q^{11} +26.3186 q^{12} -27.7227 q^{13} -17.4397 q^{14} -74.6782 q^{15} +16.0000 q^{16} +117.423 q^{17} +32.5833 q^{18} -45.3995 q^{20} -57.3735 q^{21} -103.416 q^{22} -99.5795 q^{23} +52.6371 q^{24} +3.81990 q^{25} -55.4454 q^{26} -70.4570 q^{27} -34.8794 q^{28} -139.561 q^{29} -149.356 q^{30} -3.10303 q^{31} +32.0000 q^{32} -340.221 q^{33} +234.845 q^{34} +98.9694 q^{35} +65.1666 q^{36} -91.5180 q^{37} -182.405 q^{39} -90.7991 q^{40} -250.053 q^{41} -114.747 q^{42} -413.919 q^{43} -206.833 q^{44} -184.908 q^{45} -199.159 q^{46} +611.805 q^{47} +105.274 q^{48} -266.964 q^{49} +7.63981 q^{50} +772.598 q^{51} -110.891 q^{52} -608.545 q^{53} -140.914 q^{54} +586.882 q^{55} -69.7588 q^{56} -279.121 q^{58} -572.277 q^{59} -298.713 q^{60} +572.684 q^{61} -6.20606 q^{62} -142.061 q^{63} +64.0000 q^{64} +314.649 q^{65} -680.442 q^{66} +298.501 q^{67} +469.690 q^{68} -655.197 q^{69} +197.939 q^{70} +253.057 q^{71} +130.333 q^{72} -520.754 q^{73} -183.036 q^{74} +25.1336 q^{75} +450.888 q^{77} -364.810 q^{78} +226.426 q^{79} -181.598 q^{80} -903.457 q^{81} -500.107 q^{82} -380.939 q^{83} -229.494 q^{84} -1332.73 q^{85} -827.839 q^{86} -918.259 q^{87} -413.665 q^{88} +1010.11 q^{89} -369.817 q^{90} +241.738 q^{91} -398.318 q^{92} -20.4168 q^{93} +1223.61 q^{94} +210.548 q^{96} -235.144 q^{97} -533.928 q^{98} -842.412 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{2} - 9 q^{3} + 24 q^{4} - 27 q^{5} - 18 q^{6} - 21 q^{7} + 48 q^{8} + 33 q^{9}+O(q^{10}) \) 6 * q + 12 * q^2 - 9 * q^3 + 24 * q^4 - 27 * q^5 - 18 * q^6 - 21 * q^7 + 48 * q^8 + 33 * q^9	\( 6 q + 12 q^{2} - 9 q^{3} + 24 q^{4} - 27 q^{5} - 18 q^{6} - 21 q^{7} + 48 q^{8} + 33 q^{9} - 54 q^{10} + 9 q^{11} - 36 q^{12} - 24 q^{13} - 42 q^{14} + 96 q^{16} - 102 q^{17} + 66 q^{18} - 108 q^{20} - 51 q^{21} + 18 q^{22} - 264 q^{23} - 72 q^{24} + 177 q^{25} - 48 q^{26} - 189 q^{27} - 84 q^{28} - 483 q^{29} - 72 q^{31} + 192 q^{32} - 387 q^{33} - 204 q^{34} + 135 q^{35} + 132 q^{36} + 558 q^{37} - 624 q^{39} - 216 q^{40} - 396 q^{41} - 102 q^{42} - 2064 q^{43} + 36 q^{44} - 1296 q^{45} - 528 q^{46} + 858 q^{47} - 144 q^{48} - 1413 q^{49} + 354 q^{50} + 1272 q^{51} - 96 q^{52} - 762 q^{53} - 378 q^{54} - 1107 q^{55} - 168 q^{56} - 966 q^{58} - 393 q^{59} - 627 q^{61} - 144 q^{62} - 84 q^{63} + 384 q^{64} - 495 q^{65} - 774 q^{66} - 2028 q^{67} - 408 q^{68} + 237 q^{69} + 270 q^{70} - 1284 q^{71} + 264 q^{72} - 2688 q^{73} + 1116 q^{74} - 927 q^{75} - 708 q^{77} - 1248 q^{78} - 969 q^{79} - 432 q^{80} - 1398 q^{81} - 792 q^{82} - 927 q^{83} - 204 q^{84} + 396 q^{85} - 4128 q^{86} - 2892 q^{87} + 72 q^{88} + 1257 q^{89} - 2592 q^{90} + 1323 q^{91} - 1056 q^{92} - 1368 q^{93} + 1716 q^{94} - 288 q^{96} + 2403 q^{97} - 2826 q^{98} + 567 q^{99}+O(q^{100}) \) 6 * q + 12 * q^2 - 9 * q^3 + 24 * q^4 - 27 * q^5 - 18 * q^6 - 21 * q^7 + 48 * q^8 + 33 * q^9 - 54 * q^10 + 9 * q^11 - 36 * q^12 - 24 * q^13 - 42 * q^14 + 96 * q^16 - 102 * q^17 + 66 * q^18 - 108 * q^20 - 51 * q^21 + 18 * q^22 - 264 * q^23 - 72 * q^24 + 177 * q^25 - 48 * q^26 - 189 * q^27 - 84 * q^28 - 483 * q^29 - 72 * q^31 + 192 * q^32 - 387 * q^33 - 204 * q^34 + 135 * q^35 + 132 * q^36 + 558 * q^37 - 624 * q^39 - 216 * q^40 - 396 * q^41 - 102 * q^42 - 2064 * q^43 + 36 * q^44 - 1296 * q^45 - 528 * q^46 + 858 * q^47 - 144 * q^48 - 1413 * q^49 + 354 * q^50 + 1272 * q^51 - 96 * q^52 - 762 * q^53 - 378 * q^54 - 1107 * q^55 - 168 * q^56 - 966 * q^58 - 393 * q^59 - 627 * q^61 - 144 * q^62 - 84 * q^63 + 384 * q^64 - 495 * q^65 - 774 * q^66 - 2028 * q^67 - 408 * q^68 + 237 * q^69 + 270 * q^70 - 1284 * q^71 + 264 * q^72 - 2688 * q^73 + 1116 * q^74 - 927 * q^75 - 708 * q^77 - 1248 * q^78 - 969 * q^79 - 432 * q^80 - 1398 * q^81 - 792 * q^82 - 927 * q^83 - 204 * q^84 + 396 * q^85 - 4128 * q^86 - 2892 * q^87 + 72 * q^88 + 1257 * q^89 - 2592 * q^90 + 1323 * q^91 - 1056 * q^92 - 1368 * q^93 + 1716 * q^94 - 288 * q^96 + 2403 * q^97 - 2826 * q^98 + 567 * q^99

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