LMFDB - Embedded newform 798.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [798,4,Mod(1,798)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(798, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("798.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 798 = 2 \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	798.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:	$47.0835241846$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.22397.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 37x + 28 $ x^3 - x^2 - 37*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.97577$ of defining polynomial
Character	$\chi$	$=$	798.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +7.13988 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +7.13988 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -14.2798 q^{10} -7.09142 q^{11} -12.0000 q^{12} +0.279760 q^{13} +14.0000 q^{14} -21.4196 q^{15} +16.0000 q^{16} +27.8546 q^{17} -18.0000 q^{18} +19.0000 q^{19} +28.5595 q^{20} +21.0000 q^{21} +14.1828 q^{22} +6.59019 q^{23} +24.0000 q^{24} -74.0221 q^{25} -0.559519 q^{26} -27.0000 q^{27} -28.0000 q^{28} -277.010 q^{29} +42.8393 q^{30} +207.920 q^{31} -32.0000 q^{32} +21.2743 q^{33} -55.7092 q^{34} -49.9792 q^{35} +36.0000 q^{36} -114.673 q^{37} -38.0000 q^{38} -0.839279 q^{39} -57.1190 q^{40} +326.868 q^{41} -42.0000 q^{42} -129.236 q^{43} -28.3657 q^{44} +64.2589 q^{45} -13.1804 q^{46} -296.132 q^{47} -48.0000 q^{48} +49.0000 q^{49} +148.044 q^{50} -83.5638 q^{51} +1.11904 q^{52} +148.534 q^{53} +54.0000 q^{54} -50.6319 q^{55} +56.0000 q^{56} -57.0000 q^{57} +554.020 q^{58} +455.551 q^{59} -85.6786 q^{60} +459.984 q^{61} -415.839 q^{62} -63.0000 q^{63} +64.0000 q^{64} +1.99745 q^{65} -42.5485 q^{66} +145.790 q^{67} +111.418 q^{68} -19.7706 q^{69} +99.9583 q^{70} +961.378 q^{71} -72.0000 q^{72} -889.323 q^{73} +229.346 q^{74} +222.066 q^{75} +76.0000 q^{76} +49.6399 q^{77} +1.67856 q^{78} -1254.34 q^{79} +114.238 q^{80} +81.0000 q^{81} -653.735 q^{82} -104.098 q^{83} +84.0000 q^{84} +198.879 q^{85} +258.471 q^{86} +831.030 q^{87} +56.7313 q^{88} -1301.32 q^{89} -128.518 q^{90} -1.95832 q^{91} +26.3608 q^{92} -623.759 q^{93} +592.264 q^{94} +135.658 q^{95} +96.0000 q^{96} -745.819 q^{97} -98.0000 q^{98} -63.8228 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} + 18 q^{6} - 21 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 + 18 * q^6 - 21 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} + 18 q^{6} - 21 q^{7} - 24 q^{8} + 27 q^{9} + 38 q^{11} - 36 q^{12} - 42 q^{13} + 42 q^{14} + 48 q^{16} - 30 q^{17} - 54 q^{18} + 57 q^{19} + 63 q^{21} - 76 q^{22} - 36 q^{23} + 72 q^{24} + 25 q^{25} + 84 q^{26} - 81 q^{27} - 84 q^{28} + 76 q^{29} + 166 q^{31} - 96 q^{32} - 114 q^{33} + 60 q^{34} + 108 q^{36} + 48 q^{37} - 114 q^{38} + 126 q^{39} + 102 q^{41} - 126 q^{42} - 52 q^{43} + 152 q^{44} + 72 q^{46} + 48 q^{47} - 144 q^{48} + 147 q^{49} - 50 q^{50} + 90 q^{51} - 168 q^{52} + 224 q^{53} + 162 q^{54} - 396 q^{55} + 168 q^{56} - 171 q^{57} - 152 q^{58} + 1004 q^{59} - 450 q^{61} - 332 q^{62} - 189 q^{63} + 192 q^{64} + 800 q^{65} + 228 q^{66} - 906 q^{67} - 120 q^{68} + 108 q^{69} + 946 q^{71} - 216 q^{72} - 862 q^{73} - 96 q^{74} - 75 q^{75} + 228 q^{76} - 266 q^{77} - 252 q^{78} + 148 q^{79} + 243 q^{81} - 204 q^{82} - 778 q^{83} + 252 q^{84} - 12 q^{85} + 104 q^{86} - 228 q^{87} - 304 q^{88} - 226 q^{89} + 294 q^{91} - 144 q^{92} - 498 q^{93} - 96 q^{94} + 288 q^{96} - 2200 q^{97} - 294 q^{98} + 342 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 + 18 * q^6 - 21 * q^7 - 24 * q^8 + 27 * q^9 + 38 * q^11 - 36 * q^12 - 42 * q^13 + 42 * q^14 + 48 * q^16 - 30 * q^17 - 54 * q^18 + 57 * q^19 + 63 * q^21 - 76 * q^22 - 36 * q^23 + 72 * q^24 + 25 * q^25 + 84 * q^26 - 81 * q^27 - 84 * q^28 + 76 * q^29 + 166 * q^31 - 96 * q^32 - 114 * q^33 + 60 * q^34 + 108 * q^36 + 48 * q^37 - 114 * q^38 + 126 * q^39 + 102 * q^41 - 126 * q^42 - 52 * q^43 + 152 * q^44 + 72 * q^46 + 48 * q^47 - 144 * q^48 + 147 * q^49 - 50 * q^50 + 90 * q^51 - 168 * q^52 + 224 * q^53 + 162 * q^54 - 396 * q^55 + 168 * q^56 - 171 * q^57 - 152 * q^58 + 1004 * q^59 - 450 * q^61 - 332 * q^62 - 189 * q^63 + 192 * q^64 + 800 * q^65 + 228 * q^66 - 906 * q^67 - 120 * q^68 + 108 * q^69 + 946 * q^71 - 216 * q^72 - 862 * q^73 - 96 * q^74 - 75 * q^75 + 228 * q^76 - 266 * q^77 - 252 * q^78 + 148 * q^79 + 243 * q^81 - 204 * q^82 - 778 * q^83 + 252 * q^84 - 12 * q^85 + 104 * q^86 - 228 * q^87 - 304 * q^88 - 226 * q^89 + 294 * q^91 - 144 * q^92 - 498 * q^93 - 96 * q^94 + 288 * q^96 - 2200 * q^97 - 294 * q^98 + 342 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	7.13988	0.638610	0.319305	−	0.947652i	$-0.396551\pi$
$5$	7.13988	0.638610	0.319305	+	0.947652i	$0.396551\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−14.2798	−0.451566
$11$	−7.09142	−0.194377	−0.0971883	−	0.995266i	$-0.530985\pi$
$11$	−7.09142	−0.194377	−0.0971883	+	0.995266i	$0.530985\pi$
$12$	−12.0000	−0.288675
$13$	0.279760	0.00596857	0.00298428	−	0.999996i	$-0.499050\pi$
$13$	0.279760	0.00596857	0.00298428	+	0.999996i	$0.499050\pi$
$14$	14.0000	0.267261
$15$	−21.4196	−0.368702
$16$	16.0000	0.250000
$17$	27.8546	0.397396	0.198698	−	0.980061i	$-0.436329\pi$
$17$	27.8546	0.397396	0.198698	+	0.980061i	$0.436329\pi$
$18$	−18.0000	−0.235702
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	28.5595	0.319305
$21$	21.0000	0.218218
$22$	14.1828	0.137445
$23$	6.59019	0.0597457	0.0298728	−	0.999554i	$-0.490490\pi$
$23$	6.59019	0.0597457	0.0298728	+	0.999554i	$0.490490\pi$
$24$	24.0000	0.204124
$25$	−74.0221	−0.592177
$26$	−0.559519	−0.00422041
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	−277.010	−1.77377	−0.886887	−	0.461986i	$-0.847137\pi$
$29$	−277.010	−1.77377	−0.886887	+	0.461986i	$0.847137\pi$
$30$	42.8393	0.260712
$31$	207.920	1.20463	0.602314	−	0.798259i	$-0.294245\pi$
$31$	207.920	1.20463	0.602314	+	0.798259i	$0.294245\pi$
$32$	−32.0000	−0.176777
$33$	21.2743	0.112223
$34$	−55.7092	−0.281002
$35$	−49.9792	−0.241372
$36$	36.0000	0.166667
$37$	−114.673	−0.509517	−0.254759	−	0.967005i	$-0.581996\pi$
$37$	−114.673	−0.509517	−0.254759	+	0.967005i	$0.581996\pi$
$38$	−38.0000	−0.162221
$39$	−0.839279	−0.00344595
$40$	−57.1190	−0.225783
$41$	326.868	1.24508	0.622538	−	0.782589i	$-0.286101\pi$
$41$	326.868	1.24508	0.622538	+	0.782589i	$0.286101\pi$
$42$	−42.0000	−0.154303
$43$	−129.236	−0.458331	−0.229166	−	0.973387i	$-0.573600\pi$
$43$	−129.236	−0.458331	−0.229166	+	0.973387i	$0.573600\pi$
$44$	−28.3657	−0.0971883
$45$	64.2589	0.212870
$46$	−13.1804	−0.0422466
$47$	−296.132	−0.919049	−0.459524	−	0.888165i	$-0.651980\pi$
$47$	−296.132	−0.919049	−0.459524	+	0.888165i	$0.651980\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	148.044	0.418732
$51$	−83.5638	−0.229437
$52$	1.11904	0.00298428
$53$	148.534	0.384956	0.192478	−	0.981301i	$-0.438348\pi$
$53$	148.534	0.384956	0.192478	+	0.981301i	$0.438348\pi$
$54$	54.0000	0.136083
$55$	−50.6319	−0.124131
$56$	56.0000	0.133631
$57$	−57.0000	−0.132453
$58$	554.020	1.25425
$59$	455.551	1.00521	0.502607	−	0.864515i	$-0.332374\pi$
$59$	455.551	1.00521	0.502607	+	0.864515i	$0.332374\pi$
$60$	−85.6786	−0.184351
$61$	459.984	0.965491	0.482746	−	0.875761i	$-0.339640\pi$
$61$	459.984	0.965491	0.482746	+	0.875761i	$0.339640\pi$
$62$	−415.839	−0.851801
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	1.99745	0.00381159
$66$	−42.5485	−0.0793539
$67$	145.790	0.265837	0.132918	−	0.991127i	$-0.457565\pi$
$67$	145.790	0.265837	0.132918	+	0.991127i	$0.457565\pi$
$68$	111.418	0.198698
$69$	−19.7706	−0.0344942
$70$	99.9583	0.170676
$71$	961.378	1.60697	0.803483	−	0.595328i	$-0.202978\pi$
$71$	961.378	1.60697	0.803483	+	0.595328i	$0.202978\pi$
$72$	−72.0000	−0.117851
$73$	−889.323	−1.42585	−0.712927	−	0.701238i	$-0.752631\pi$
$73$	−889.323	−1.42585	−0.712927	+	0.701238i	$0.752631\pi$
$74$	229.346	0.360283
$75$	222.066	0.341894
$76$	76.0000	0.114708
$77$	49.6399	0.0734675
$78$	1.67856	0.00243666
$79$	−1254.34	−1.78638	−0.893191	−	0.449677i	$-0.851539\pi$
$79$	−1254.34	−1.78638	−0.893191	+	0.449677i	$0.851539\pi$
$80$	114.238	0.159653
$81$	81.0000	0.111111
$82$	−653.735	−0.880402
$83$	−104.098	−0.137666	−0.0688328	−	0.997628i	$-0.521928\pi$
$83$	−104.098	−0.137666	−0.0688328	+	0.997628i	$0.521928\pi$
$84$	84.0000	0.109109
$85$	198.879	0.253781
$86$	258.471	0.324089
$87$	831.030	1.02409
$88$	56.7313	0.0687225
$89$	−1301.32	−1.54989	−0.774943	−	0.632031i	$-0.782221\pi$
$89$	−1301.32	−1.54989	−0.774943	+	0.632031i	$0.782221\pi$
$90$	−128.518	−0.150522
$91$	−1.95832	−0.00225591
$92$	26.3608	0.0298728
$93$	−623.759	−0.695492
$94$	592.264	0.649866
$95$	135.658	0.146507
$96$	96.0000	0.102062
$97$	−745.819	−0.780685	−0.390343	−	0.920670i	$-0.627643\pi$
$97$	−745.819	−0.780685	−0.390343	+	0.920670i	$0.627643\pi$
$98$	−98.0000	−0.101015
$99$	−63.8228	−0.0647922
$100$	−296.088	−0.296088
$101$	−657.808	−0.648063	−0.324031	−	0.946046i	$-0.605038\pi$
$101$	−657.808	−0.648063	−0.324031	+	0.946046i	$0.605038\pi$
$102$	167.128	0.162236
$103$	−245.398	−0.234755	−0.117378	−	0.993087i	$-0.537449\pi$
$103$	−245.398	−0.234755	−0.117378	+	0.993087i	$0.537449\pi$
$104$	−2.23808	−0.00211021
$105$	149.937	0.139356
$106$	−297.068	−0.272205
$107$	1514.94	1.36874	0.684369	−	0.729136i	$-0.260078\pi$
$107$	1514.94	1.36874	0.684369	+	0.729136i	$0.260078\pi$
$108$	−108.000	−0.0962250
$109$	−837.197	−0.735678	−0.367839	−	0.929889i	$-0.619902\pi$
$109$	−837.197	−0.735678	−0.367839	+	0.929889i	$0.619902\pi$
$110$	101.264	0.0877738
$111$	344.019	0.294170
$112$	−112.000	−0.0944911
$113$	−561.510	−0.467455	−0.233727	−	0.972302i	$-0.575092\pi$
$113$	−561.510	−0.467455	−0.233727	+	0.972302i	$0.575092\pi$
$114$	114.000	0.0936586
$115$	47.0532	0.0381542
$116$	−1108.04	−0.886887
$117$	2.51784	0.00198952
$118$	−911.102	−0.710794
$119$	−194.982	−0.150202
$120$	171.357	0.130356
$121$	−1280.71	−0.962218
$122$	−919.969	−0.682705
$123$	−980.603	−0.718845
$124$	831.679	0.602314
$125$	−1420.99	−1.01678
$126$	126.000	0.0890871
$127$	1051.15	0.734442	0.367221	−	0.930134i	$-0.380309\pi$
$127$	1051.15	0.734442	0.367221	+	0.930134i	$0.380309\pi$
$128$	−128.000	−0.0883883
$129$	387.707	0.264618
$130$	−3.99490	−0.00269520
$131$	−152.344	−0.101606	−0.0508030	−	0.998709i	$-0.516178\pi$
$131$	−152.344	−0.101606	−0.0508030	+	0.998709i	$0.516178\pi$
$132$	85.0970	0.0561117
$133$	−133.000	−0.0867110
$134$	−291.579	−0.187975
$135$	−192.777	−0.122901
$136$	−222.837	−0.140501
$137$	−2652.06	−1.65387	−0.826936	−	0.562296i	$-0.809918\pi$
$137$	−2652.06	−1.65387	−0.826936	+	0.562296i	$0.809918\pi$
$138$	39.5412	0.0243911
$139$	−1425.39	−0.869782	−0.434891	−	0.900483i	$-0.643213\pi$
$139$	−1425.39	−0.869782	−0.434891	+	0.900483i	$0.643213\pi$
$140$	−199.917	−0.120686
$141$	888.396	0.530613
$142$	−1922.76	−1.13630
$143$	−1.98389	−0.00116015
$144$	144.000	0.0833333
$145$	−1977.82	−1.13275
$146$	1778.65	1.00823
$147$	−147.000	−0.0824786
$148$	−458.692	−0.254759
$149$	245.005	0.134709	0.0673543	−	0.997729i	$-0.478544\pi$
$149$	245.005	0.134709	0.0673543	+	0.997729i	$0.478544\pi$
$150$	−444.133	−0.241755
$151$	−2684.80	−1.44692	−0.723462	−	0.690364i	$-0.757450\pi$
$151$	−2684.80	−1.44692	−0.723462	+	0.690364i	$0.757450\pi$
$152$	−152.000	−0.0811107
$153$	250.692	0.132465
$154$	−99.2799	−0.0519493
$155$	1484.52	0.769288
$156$	−3.35712	−0.00172298
$157$	−3401.75	−1.72923	−0.864615	−	0.502434i	$-0.832438\pi$
$157$	−3401.75	−1.72923	−0.864615	+	0.502434i	$0.832438\pi$
$158$	2508.68	1.26316
$159$	−445.601	−0.222255
$160$	−228.476	−0.112891
$161$	−46.1314	−0.0225817
$162$	−162.000	−0.0785674
$163$	−2342.13	−1.12546	−0.562730	−	0.826641i	$-0.690249\pi$
$163$	−2342.13	−1.12546	−0.562730	+	0.826641i	$0.690249\pi$
$164$	1307.47	0.622538
$165$	151.896	0.0716670
$166$	208.196	0.0973443
$167$	−1758.96	−0.815044	−0.407522	−	0.913195i	$-0.633607\pi$
$167$	−1758.96	−0.815044	−0.407522	+	0.913195i	$0.633607\pi$
$168$	−168.000	−0.0771517
$169$	−2196.92	−0.999964
$170$	−397.757	−0.179451
$171$	171.000	0.0764719
$172$	−516.943	−0.229166
$173$	3301.10	1.45074	0.725371	−	0.688358i	$-0.241668\pi$
$173$	3301.10	1.45074	0.725371	+	0.688358i	$0.241668\pi$
$174$	−1662.06	−0.724140
$175$	518.155	0.223822
$176$	−113.463	−0.0485942
$177$	−1366.65	−0.580361
$178$	2602.64	1.09593
$179$	912.793	0.381147	0.190574	−	0.981673i	$-0.438965\pi$
$179$	912.793	0.381147	0.190574	+	0.981673i	$0.438965\pi$
$180$	257.036	0.106435
$181$	−2502.68	−1.02775	−0.513875	−	0.857865i	$-0.671790\pi$
$181$	−2502.68	−1.02775	−0.513875	+	0.857865i	$0.671790\pi$
$182$	3.91664	0.00159517
$183$	−1379.95	−0.557427
$184$	−52.7216	−0.0211233
$185$	−818.752	−0.325383
$186$	1247.52	0.491787
$187$	−197.529	−0.0772446
$188$	−1184.53	−0.459524
$189$	189.000	0.0727393
$190$	−271.315	−0.103596
$191$	−4964.94	−1.88089	−0.940446	−	0.339944i	$-0.889592\pi$
$191$	−4964.94	−1.88089	−0.940446	+	0.339944i	$0.889592\pi$
$192$	−192.000	−0.0721688
$193$	2340.08	0.872760	0.436380	−	0.899762i	$-0.356260\pi$
$193$	2340.08	0.872760	0.436380	+	0.899762i	$0.356260\pi$
$194$	1491.64	0.552028
$195$	−5.99235	−0.00220062
$196$	196.000	0.0714286
$197$	3192.15	1.15447	0.577236	−	0.816577i	$-0.304131\pi$
$197$	3192.15	1.15447	0.577236	+	0.816577i	$0.304131\pi$
$198$	127.646	0.0458150
$199$	3178.99	1.13242	0.566212	−	0.824260i	$-0.308408\pi$
$199$	3178.99	1.13242	0.566212	+	0.824260i	$0.308408\pi$
$200$	592.177	0.209366
$201$	−437.369	−0.153481
$202$	1315.62	0.458249
$203$	1939.07	0.670424
$204$	−334.255	−0.114718
$205$	2333.80	0.795119
$206$	490.796	0.165997
$207$	59.3117	0.0199152
$208$	4.47615	0.00149214
$209$	−134.737	−0.0445931
$210$	−299.875	−0.0985397
$211$	−4295.11	−1.40136	−0.700681	−	0.713474i	$-0.747121\pi$
$211$	−4295.11	−1.40136	−0.700681	+	0.713474i	$0.747121\pi$
$212$	594.135	0.192478
$213$	−2884.13	−0.927782
$214$	−3029.88	−0.967844
$215$	−922.727	−0.292695
$216$	216.000	0.0680414
$217$	−1455.44	−0.455307
$218$	1674.39	0.520203
$219$	2667.97	0.823217
$220$	−202.528	−0.0620655
$221$	7.79260	0.00237189
$222$	−688.038	−0.208009
$223$	−2360.06	−0.708704	−0.354352	−	0.935112i	$-0.615299\pi$
$223$	−2360.06	−0.708704	−0.354352	+	0.935112i	$0.615299\pi$
$224$	224.000	0.0668153
$225$	−666.199	−0.197392
$226$	1123.02	0.330540
$227$	−2180.97	−0.637691	−0.318846	−	0.947807i	$-0.603295\pi$
$227$	−2180.97	−0.637691	−0.318846	+	0.947807i	$0.603295\pi$
$228$	−228.000	−0.0662266
$229$	−1713.65	−0.494502	−0.247251	−	0.968952i	$-0.579527\pi$
$229$	−1713.65	−0.494502	−0.247251	+	0.968952i	$0.579527\pi$
$230$	−94.1064	−0.0269791
$231$	−148.920	−0.0424165
$232$	2216.08	0.627124
$233$	−587.419	−0.165164	−0.0825818	−	0.996584i	$-0.526317\pi$
$233$	−587.419	−0.165164	−0.0825818	+	0.996584i	$0.526317\pi$
$234$	−5.03567	−0.00140680
$235$	−2114.35	−0.586914
$236$	1822.20	0.502607
$237$	3763.02	1.03137
$238$	389.965	0.106209
$239$	6817.58	1.84516	0.922579	−	0.385809i	$-0.126078\pi$
$239$	6817.58	1.84516	0.922579	+	0.385809i	$0.126078\pi$
$240$	−342.714	−0.0921755
$241$	550.676	0.147187	0.0735937	−	0.997288i	$-0.476553\pi$
$241$	550.676	0.147187	0.0735937	+	0.997288i	$0.476553\pi$
$242$	2561.42	0.680391
$243$	−243.000	−0.0641500
$244$	1839.94	0.482746
$245$	349.854	0.0912300
$246$	1961.21	0.508300
$247$	5.31543	0.00136928
$248$	−1663.36	−0.425900
$249$	312.294	0.0794813
$250$	2841.99	0.718972
$251$	5258.84	1.32245	0.661225	−	0.750188i	$-0.270037\pi$
$251$	5258.84	1.32245	0.661225	+	0.750188i	$0.270037\pi$
$252$	−252.000	−0.0629941
$253$	−46.7338	−0.0116132
$254$	−2102.29	−0.519329
$255$	−596.636	−0.146521
$256$	256.000	0.0625000
$257$	2774.20	0.673347	0.336673	−	0.941621i	$-0.390698\pi$
$257$	2774.20	0.673347	0.336673	+	0.941621i	$0.390698\pi$
$258$	−775.414	−0.187113
$259$	802.711	0.192579
$260$	7.98980	0.00190579
$261$	−2493.09	−0.591258
$262$	304.689	0.0718463
$263$	−978.024	−0.229306	−0.114653	−	0.993406i	$-0.536576\pi$
$263$	−978.024	−0.229306	−0.114653	+	0.993406i	$0.536576\pi$
$264$	−170.194	−0.0396770
$265$	1060.51	0.245837
$266$	266.000	0.0613139
$267$	3903.97	0.894827
$268$	583.159	0.132918
$269$	−3094.39	−0.701368	−0.350684	−	0.936494i	$-0.614051\pi$
$269$	−3094.39	−0.701368	−0.350684	+	0.936494i	$0.614051\pi$
$270$	385.554	0.0869038
$271$	7445.67	1.66898	0.834488	−	0.551026i	$-0.185764\pi$
$271$	7445.67	1.66898	0.834488	+	0.551026i	$0.185764\pi$
$272$	445.674	0.0993491
$273$	5.87495	0.00130245
$274$	5304.11	1.16946
$275$	524.922	0.115105
$276$	−79.0823	−0.0172471
$277$	−7497.33	−1.62625	−0.813124	−	0.582090i	$-0.802235\pi$
$277$	−7497.33	−1.62625	−0.813124	+	0.582090i	$0.802235\pi$
$278$	2850.77	0.615029
$279$	1871.28	0.401543
$280$	399.833	0.0853379
$281$	5867.27	1.24559	0.622797	−	0.782383i	$-0.285996\pi$
$281$	5867.27	1.24559	0.622797	+	0.782383i	$0.285996\pi$
$282$	−1776.79	−0.375200
$283$	−5341.28	−1.12193	−0.560965	−	0.827840i	$-0.689570\pi$
$283$	−5341.28	−1.12193	−0.560965	+	0.827840i	$0.689570\pi$
$284$	3845.51	0.803483
$285$	−406.973	−0.0845860
$286$	3.96779	0.000820350 0
$287$	−2288.07	−0.470595
$288$	−288.000	−0.0589256
$289$	−4137.12	−0.842076
$290$	3955.64	0.800976
$291$	2237.46	0.450729
$292$	−3557.29	−0.712927
$293$	−9788.62	−1.95173	−0.975866	−	0.218371i	$-0.929926\pi$
$293$	−9788.62	−1.95173	−0.975866	+	0.218371i	$0.929926\pi$
$294$	294.000	0.0583212
$295$	3252.58	0.641941
$296$	917.384	0.180142
$297$	191.468	0.0374078
$298$	−490.010	−0.0952533
$299$	1.84367	0.000356596 0
$300$	888.265	0.170947
$301$	904.649	0.173233
$302$	5369.59	1.02313
$303$	1973.42	0.374159
$304$	304.000	0.0573539
$305$	3284.23	0.616572
$306$	−501.383	−0.0936672
$307$	3603.99	0.670002	0.335001	−	0.942218i	$-0.391263\pi$
$307$	3603.99	0.670002	0.335001	+	0.942218i	$0.391263\pi$
$308$	198.560	0.0367337
$309$	736.194	0.135536
$310$	−2969.04	−0.543969
$311$	2582.45	0.470860	0.235430	−	0.971891i	$-0.424350\pi$
$311$	2582.45	0.470860	0.235430	+	0.971891i	$0.424350\pi$
$312$	6.71423	0.00121833
$313$	−8555.29	−1.54496	−0.772481	−	0.635037i	$-0.780985\pi$
$313$	−8555.29	−1.54496	−0.772481	+	0.635037i	$0.780985\pi$
$314$	6803.50	1.22275
$315$	−449.812	−0.0804573
$316$	−5017.36	−0.893191
$317$	−4302.34	−0.762282	−0.381141	−	0.924517i	$-0.624469\pi$
$317$	−4302.34	−0.762282	−0.381141	+	0.924517i	$0.624469\pi$
$318$	891.203	0.157158
$319$	1964.39	0.344780
$320$	456.952	0.0798263
$321$	−4544.82	−0.790241
$322$	92.2627	0.0159677
$323$	529.238	0.0911690
$324$	324.000	0.0555556
$325$	−20.7084	−0.00353445
$326$	4684.27	0.795820
$327$	2511.59	0.424744
$328$	−2614.94	−0.440201
$329$	2072.92	0.347368
$330$	−303.791	−0.0506762
$331$	−4287.68	−0.712001	−0.356000	−	0.934486i	$-0.615860\pi$
$331$	−4287.68	−0.712001	−0.356000	+	0.934486i	$0.615860\pi$
$332$	−416.392	−0.0688328
$333$	−1032.06	−0.169839
$334$	3517.92	0.576323
$335$	1040.92	0.169766
$336$	336.000	0.0545545
$337$	8663.67	1.40042	0.700208	−	0.713939i	$-0.253091\pi$
$337$	8663.67	1.40042	0.700208	+	0.713939i	$0.253091\pi$
$338$	4393.84	0.707082
$339$	1684.53	0.269885
$340$	795.514	0.126891
$341$	−1474.45	−0.234152
$342$	−342.000	−0.0540738
$343$	−343.000	−0.0539949
$344$	1033.89	0.162045
$345$	−141.160	−0.0220283
$346$	−6602.21	−1.02583
$347$	−7678.64	−1.18793	−0.593964	−	0.804492i	$-0.702438\pi$
$347$	−7678.64	−1.18793	−0.593964	+	0.804492i	$0.702438\pi$
$348$	3324.12	0.512045
$349$	5492.22	0.842384	0.421192	−	0.906972i	$-0.361612\pi$
$349$	5492.22	0.842384	0.421192	+	0.906972i	$0.361612\pi$
$350$	−1036.31	−0.158266
$351$	−7.55351	−0.00114865
$352$	226.925	0.0343613
$353$	−2117.72	−0.319306	−0.159653	−	0.987173i	$-0.551037\pi$
$353$	−2117.72	−0.319306	−0.159653	+	0.987173i	$0.551037\pi$
$354$	2733.31	0.410377
$355$	6864.12	1.02623
$356$	−5205.29	−0.774943
$357$	584.947	0.0867190
$358$	−1825.59	−0.269512
$359$	7627.97	1.12142	0.560709	−	0.828013i	$-0.310529\pi$
$359$	7627.97	1.12142	0.560709	+	0.828013i	$0.310529\pi$
$360$	−514.071	−0.0752609
$361$	361.000	0.0526316
$362$	5005.36	0.726729
$363$	3842.14	0.555537
$364$	−7.83327	−0.00112795
$365$	−6349.66	−0.910565
$366$	2759.91	0.394160
$367$	−2893.19	−0.411507	−0.205754	−	0.978604i	$-0.565965\pi$
$367$	−2893.19	−0.411507	−0.205754	+	0.978604i	$0.565965\pi$
$368$	105.443	0.0149364
$369$	2941.81	0.415026
$370$	1637.50	0.230080
$371$	−1039.74	−0.145500
$372$	−2495.04	−0.347746
$373$	5291.53	0.734544	0.367272	−	0.930114i	$-0.380292\pi$
$373$	5291.53	0.734544	0.367272	+	0.930114i	$0.380292\pi$
$374$	395.057	0.0546202
$375$	4262.98	0.587039
$376$	2369.06	0.324933
$377$	−77.4962	−0.0105869
$378$	−378.000	−0.0514344
$379$	−1544.28	−0.209299	−0.104649	−	0.994509i	$-0.533372\pi$
$379$	−1544.28	−0.209299	−0.104649	+	0.994509i	$0.533372\pi$
$380$	542.631	0.0732536
$381$	−3153.44	−0.424030
$382$	9929.88	1.32999
$383$	−160.209	−0.0213741	−0.0106871	−	0.999943i	$-0.503402\pi$
$383$	−160.209	−0.0213741	−0.0106871	+	0.999943i	$0.503402\pi$
$384$	384.000	0.0510310
$385$	354.423	0.0469171
$386$	−4680.16	−0.617135
$387$	−1163.12	−0.152777
$388$	−2983.28	−0.390343
$389$	2307.86	0.300805	0.150403	−	0.988625i	$-0.451943\pi$
$389$	2307.86	0.300805	0.150403	+	0.988625i	$0.451943\pi$
$390$	11.9847	0.00155607
$391$	183.567	0.0237427
$392$	−392.000	−0.0505076
$393$	457.033	0.0586623
$394$	−6384.29	−0.816335
$395$	−8955.83	−1.14080
$396$	−255.291	−0.0323961
$397$	−898.015	−0.113527	−0.0567633	−	0.998388i	$-0.518078\pi$
$397$	−898.015	−0.113527	−0.0567633	+	0.998388i	$0.518078\pi$
$398$	−6357.97	−0.800745
$399$	399.000	0.0500626
$400$	−1184.35	−0.148044
$401$	1427.76	0.177802	0.0889012	−	0.996040i	$-0.471664\pi$
$401$	1427.76	0.177802	0.0889012	+	0.996040i	$0.471664\pi$
$402$	874.738	0.108527
$403$	58.1675	0.00718990
$404$	−2631.23	−0.324031
$405$	578.330	0.0709567
$406$	−3878.14	−0.474061
$407$	813.195	0.0990382
$408$	668.511	0.0811182
$409$	8348.02	1.00925	0.504625	−	0.863339i	$-0.331631\pi$
$409$	8348.02	1.00925	0.504625	+	0.863339i	$0.331631\pi$
$410$	−4667.59	−0.562234
$411$	7956.17	0.954864
$412$	−981.592	−0.117378
$413$	−3188.86	−0.379936
$414$	−118.623	−0.0140822
$415$	−743.248	−0.0879147
$416$	−8.95231	−0.00105510
$417$	4276.16	0.502169
$418$	269.474	0.0315321
$419$	8909.98	1.03886	0.519428	−	0.854514i	$-0.326145\pi$
$419$	8909.98	1.03886	0.519428	+	0.854514i	$0.326145\pi$
$420$	599.750	0.0696781
$421$	12274.9	1.42100	0.710498	−	0.703699i	$-0.248470\pi$
$421$	12274.9	1.42100	0.710498	+	0.703699i	$0.248470\pi$
$422$	8590.22	0.990913
$423$	−2665.19	−0.306350
$424$	−1188.27	−0.136103
$425$	−2061.86	−0.235329
$426$	5768.27	0.656041
$427$	−3219.89	−0.364921
$428$	6059.77	0.684369
$429$	5.95168	0.000669813 0
$430$	1845.45	0.206967
$431$	8115.62	0.906997	0.453499	−	0.891257i	$-0.350176\pi$
$431$	8115.62	0.906997	0.453499	+	0.891257i	$0.350176\pi$
$432$	−432.000	−0.0481125
$433$	3217.73	0.357122	0.178561	−	0.983929i	$-0.442856\pi$
$433$	3217.73	0.357122	0.178561	+	0.983929i	$0.442856\pi$
$434$	2910.88	0.321950
$435$	5933.45	0.653994
$436$	−3348.79	−0.367839
$437$	125.214	0.0137066
$438$	−5335.94	−0.582103
$439$	−13875.1	−1.50848	−0.754238	−	0.656601i	$-0.771994\pi$
$439$	−13875.1	−1.50848	−0.754238	+	0.656601i	$0.771994\pi$
$440$	405.055	0.0438869
$441$	441.000	0.0476190
$442$	−15.5852	−0.00167718
$443$	7160.29	0.767936	0.383968	−	0.923346i	$-0.374557\pi$
$443$	7160.29	0.767936	0.383968	+	0.923346i	$0.374557\pi$
$444$	1376.08	0.147085
$445$	−9291.28	−0.989773
$446$	4720.11	0.501129
$447$	−735.014	−0.0777740
$448$	−448.000	−0.0472456
$449$	9692.86	1.01878	0.509392	−	0.860534i	$-0.329870\pi$
$449$	9692.86	1.01878	0.509392	+	0.860534i	$0.329870\pi$
$450$	1332.40	0.139577
$451$	−2317.96	−0.242014
$452$	−2246.04	−0.233727
$453$	8054.39	0.835382
$454$	4361.94	0.450916
$455$	−13.9822	−0.00144064
$456$	456.000	0.0468293
$457$	−7060.37	−0.722691	−0.361346	−	0.932432i	$-0.617683\pi$
$457$	−7060.37	−0.722691	−0.361346	+	0.932432i	$0.617683\pi$
$458$	3427.29	0.349665
$459$	−752.075	−0.0764789
$460$	188.213	0.0190771
$461$	3884.12	0.392411	0.196206	−	0.980563i	$-0.437138\pi$
$461$	3884.12	0.392411	0.196206	+	0.980563i	$0.437138\pi$
$462$	297.840	0.0299930
$463$	13724.9	1.37764	0.688822	−	0.724930i	$-0.258128\pi$
$463$	13724.9	1.37764	0.688822	+	0.724930i	$0.258128\pi$
$464$	−4432.16	−0.443444
$465$	−4453.56	−0.444149
$466$	1174.84	0.116788
$467$	−10644.9	−1.05479	−0.527394	−	0.849621i	$-0.676831\pi$
$467$	−10644.9	−1.05479	−0.527394	+	0.849621i	$0.676831\pi$
$468$	10.0713	0.000994761 0
$469$	−1020.53	−0.100477
$470$	4228.69	0.415011
$471$	10205.2	0.998372
$472$	−3644.41	−0.355397
$473$	916.464	0.0890889
$474$	−7526.03	−0.729287
$475$	−1406.42	−0.135855
$476$	−779.929	−0.0751008
$477$	1336.80	0.128319
$478$	−13635.2	−1.30472
$479$	−12293.0	−1.17261	−0.586307	−	0.810089i	$-0.699419\pi$
$479$	−12293.0	−1.17261	−0.586307	+	0.810089i	$0.699419\pi$
$480$	685.428	0.0651779
$481$	−32.0809	−0.00304109
$482$	−1101.35	−0.104077
$483$	138.394	0.0130376
$484$	−5122.85	−0.481109
$485$	−5325.06	−0.498553
$486$	486.000	0.0453609
$487$	12826.8	1.19350	0.596752	−	0.802426i	$-0.296458\pi$
$487$	12826.8	1.19350	0.596752	+	0.802426i	$0.296458\pi$
$488$	−3679.87	−0.341353
$489$	7026.40	0.649785
$490$	−699.708	−0.0645094
$491$	16093.9	1.47924	0.739620	−	0.673025i	$-0.235005\pi$
$491$	16093.9	1.47924	0.739620	+	0.673025i	$0.235005\pi$
$492$	−3922.41	−0.359423
$493$	−7716.01	−0.704891
$494$	−10.6309	−0.000968229 0
$495$	−455.687	−0.0413770
$496$	3326.72	0.301157
$497$	−6729.65	−0.607376
$498$	−624.589	−0.0562018
$499$	−10467.6	−0.939069	−0.469535	−	0.882914i	$-0.655578\pi$
$499$	−10467.6	−0.939069	−0.469535	+	0.882914i	$0.655578\pi$
$500$	−5683.98	−0.508390
$501$	5276.88	0.470566
$502$	−10517.7	−0.935114
$503$	−11361.1	−1.00709	−0.503545	−	0.863969i	$-0.667971\pi$
$503$	−11361.1	−1.00709	−0.503545	+	0.863969i	$0.667971\pi$
$504$	504.000	0.0445435
$505$	−4696.67	−0.413859
$506$	93.4676	0.00821175
$507$	6590.77	0.577330
$508$	4204.59	0.367221
$509$	18021.8	1.56936	0.784678	−	0.619904i	$-0.212828\pi$
$509$	18021.8	1.56936	0.784678	+	0.619904i	$0.212828\pi$
$510$	1193.27	0.103606
$511$	6225.26	0.538922
$512$	−512.000	−0.0441942
$513$	−513.000	−0.0441511
$514$	−5548.41	−0.476128
$515$	−1752.11	−0.149917
$516$	1550.83	0.132309
$517$	2100.00	0.178642
$518$	−1605.42	−0.136174
$519$	−9903.31	−0.837586
$520$	−15.9796	−0.00134760
$521$	−4151.33	−0.349085	−0.174542	−	0.984650i	$-0.555845\pi$
$521$	−4151.33	−0.349085	−0.174542	+	0.984650i	$0.555845\pi$
$522$	4986.18	0.418083
$523$	119.349	0.00997851	0.00498926	−	0.999988i	$-0.498412\pi$
$523$	119.349	0.00997851	0.00498926	+	0.999988i	$0.498412\pi$
$524$	−609.378	−0.0508030
$525$	−1554.46	−0.129224
$526$	1956.05	0.162144
$527$	5791.52	0.478715
$528$	340.388	0.0280559
$529$	−12123.6	−0.996430
$530$	−2121.03	−0.173833
$531$	4099.96	0.335072
$532$	−532.000	−0.0433555
$533$	91.4444	0.00743132
$534$	−7807.93	−0.632738
$535$	10816.5	0.874090
$536$	−1166.32	−0.0939874
$537$	−2738.38	−0.220056
$538$	6188.77	0.495942
$539$	−347.480	−0.0277681
$540$	−771.107	−0.0614503
$541$	6924.87	0.550320	0.275160	−	0.961398i	$-0.411269\pi$
$541$	6924.87	0.550320	0.275160	+	0.961398i	$0.411269\pi$
$542$	−14891.3	−1.18014
$543$	7508.04	0.593372
$544$	−891.348	−0.0702504
$545$	−5977.48	−0.469812
$546$	−11.7499	−0.000920970 0
$547$	3624.87	0.283342	0.141671	−	0.989914i	$-0.454752\pi$
$547$	3624.87	0.283342	0.141671	+	0.989914i	$0.454752\pi$
$548$	−10608.2	−0.826936
$549$	4139.86	0.321830
$550$	−1049.84	−0.0813918
$551$	−5263.19	−0.406932
$552$	158.165	0.0121955
$553$	8780.37	0.675189
$554$	14994.7	1.14993
$555$	2456.26	0.187860
$556$	−5701.55	−0.434891
$557$	−16286.1	−1.23890	−0.619448	−	0.785038i	$-0.712643\pi$
$557$	−16286.1	−1.23890	−0.619448	+	0.785038i	$0.712643\pi$
$558$	−3742.55	−0.283934
$559$	−36.1549	−0.00273558
$560$	−799.667	−0.0603430
$561$	592.586	0.0445972
$562$	−11734.5	−0.880768
$563$	−16084.3	−1.20403	−0.602017	−	0.798484i	$-0.705636\pi$
$563$	−16084.3	−1.20403	−0.602017	+	0.798484i	$0.705636\pi$
$564$	3553.58	0.265307
$565$	−4009.11	−0.298521
$566$	10682.6	0.793324
$567$	−567.000	−0.0419961
$568$	−7691.03	−0.568148
$569$	3896.25	0.287064	0.143532	−	0.989646i	$-0.454154\pi$
$569$	3896.25	0.287064	0.143532	+	0.989646i	$0.454154\pi$
$570$	813.946	0.0598113
$571$	3105.72	0.227618	0.113809	−	0.993503i	$-0.463695\pi$
$571$	3105.72	0.227618	0.113809	+	0.993503i	$0.463695\pi$
$572$	−7.93557	−0.000580075 0
$573$	14894.8	1.08593
$574$	4576.15	0.332761
$575$	−487.820	−0.0353800
$576$	576.000	0.0416667
$577$	−27370.0	−1.97475	−0.987374	−	0.158407i	$-0.949364\pi$
$577$	−27370.0	−1.97475	−0.987374	+	0.158407i	$0.949364\pi$
$578$	8274.24	0.595438
$579$	−7020.24	−0.503888
$580$	−7911.27	−0.566375
$581$	728.687	0.0520327
$582$	−4474.91	−0.318713
$583$	−1053.32	−0.0748265
$584$	7114.58	0.504116
$585$	17.9771	0.00127053
$586$	19577.2	1.38008
$587$	8734.93	0.614189	0.307095	−	0.951679i	$-0.400643\pi$
$587$	8734.93	0.614189	0.307095	+	0.951679i	$0.400643\pi$
$588$	−588.000	−0.0412393
$589$	3950.47	0.276361
$590$	−6505.16	−0.453921
$591$	−9576.44	−0.666535
$592$	−1834.77	−0.127379
$593$	−15233.9	−1.05495	−0.527473	−	0.849572i	$-0.676860\pi$
$593$	−15233.9	−1.05495	−0.527473	+	0.849572i	$0.676860\pi$
$594$	−382.937	−0.0264513
$595$	−1392.15	−0.0959203
$596$	980.019	0.0673543
$597$	−9536.96	−0.653805
$598$	−3.68734	−0.000252151 0
$599$	−3859.56	−0.263268	−0.131634	−	0.991298i	$-0.542022\pi$
$599$	−3859.56	−0.263268	−0.131634	+	0.991298i	$0.542022\pi$
$600$	−1776.53	−0.120878
$601$	8576.92	0.582129	0.291065	−	0.956703i	$-0.405991\pi$
$601$	8576.92	0.582129	0.291065	+	0.956703i	$0.405991\pi$
$602$	−1809.30	−0.122494
$603$	1312.11	0.0886122
$604$	−10739.2	−0.723462
$605$	−9144.13	−0.614482
$606$	−3946.85	−0.264570
$607$	6535.58	0.437019	0.218510	−	0.975835i	$-0.429880\pi$
$607$	6535.58	0.437019	0.218510	+	0.975835i	$0.429880\pi$
$608$	−608.000	−0.0405554
$609$	−5817.21	−0.387069
$610$	−6568.47	−0.435983
$611$	−82.8458	−0.00548540
$612$	1002.77	0.0662327
$613$	6624.07	0.436450	0.218225	−	0.975898i	$-0.429973\pi$
$613$	6624.07	0.436450	0.218225	+	0.975898i	$0.429973\pi$
$614$	−7207.98	−0.473763
$615$	−7001.39	−0.459062
$616$	−397.119	−0.0259747
$617$	6462.51	0.421671	0.210835	−	0.977522i	$-0.432382\pi$
$617$	6462.51	0.421671	0.210835	+	0.977522i	$0.432382\pi$
$618$	−1472.39	−0.0958384
$619$	−1946.64	−0.126400	−0.0632002	−	0.998001i	$-0.520131\pi$
$619$	−1946.64	−0.126400	−0.0632002	+	0.998001i	$0.520131\pi$
$620$	5938.09	0.384644
$621$	−177.935	−0.0114981
$622$	−5164.90	−0.332948
$623$	9109.25	0.585802
$624$	−13.4285	−0.000861488 0
$625$	−892.962	−0.0571496
$626$	17110.6	1.09245
$627$	404.211	0.0257458
$628$	−13607.0	−0.864615
$629$	−3194.17	−0.202480
$630$	899.625	0.0568919
$631$	−15677.8	−0.989103	−0.494552	−	0.869148i	$-0.664668\pi$
$631$	−15677.8	−0.989103	−0.494552	+	0.869148i	$0.664668\pi$
$632$	10034.7	0.631581
$633$	12885.3	0.809077
$634$	8604.67	0.539015
$635$	7505.06	0.469022
$636$	−1782.41	−0.111127
$637$	13.7082	0.000852652 0
$638$	−3928.79	−0.243796
$639$	8652.40	0.535655
$640$	−913.905	−0.0564457
$641$	−22837.0	−1.40719	−0.703595	−	0.710601i	$-0.748423\pi$
$641$	−22837.0	−1.40719	−0.703595	+	0.710601i	$0.748423\pi$
$642$	9089.65	0.558785
$643$	20126.3	1.23438	0.617188	−	0.786816i	$-0.288272\pi$
$643$	20126.3	1.23438	0.617188	+	0.786816i	$0.288272\pi$
$644$	−184.525	−0.0112909
$645$	2768.18	0.168988
$646$	−1058.48	−0.0644662
$647$	−10752.5	−0.653363	−0.326682	−	0.945134i	$-0.605930\pi$
$647$	−10752.5	−0.653363	−0.326682	+	0.945134i	$0.605930\pi$
$648$	−648.000	−0.0392837
$649$	−3230.50	−0.195390
$650$	41.4168	0.00249923
$651$	4366.31	0.262871
$652$	−9368.53	−0.562730
$653$	−16425.8	−0.984364	−0.492182	−	0.870492i	$-0.663801\pi$
$653$	−16425.8	−0.984364	−0.492182	+	0.870492i	$0.663801\pi$
$654$	−5023.18	−0.300339
$655$	−1087.72	−0.0648867
$656$	5229.88	0.311269
$657$	−8003.91	−0.475285
$658$	−4145.85	−0.245626
$659$	2803.51	0.165720	0.0828598	−	0.996561i	$-0.473595\pi$
$659$	2803.51	0.165720	0.0828598	+	0.996561i	$0.473595\pi$
$660$	607.583	0.0358335
$661$	−22916.9	−1.34851	−0.674254	−	0.738500i	$-0.735534\pi$
$661$	−22916.9	−1.34851	−0.674254	+	0.738500i	$0.735534\pi$
$662$	8575.36	0.503461
$663$	−23.3778	−0.00136941
$664$	832.785	0.0486722
$665$	−949.604	−0.0553745
$666$	2064.11	0.120094
$667$	−1825.55	−0.105975
$668$	−7035.84	−0.407522
$669$	7080.17	0.409170
$670$	−2081.84	−0.120043
$671$	−3261.94	−0.187669
$672$	−672.000	−0.0385758
$673$	−5130.17	−0.293839	−0.146919	−	0.989148i	$-0.546936\pi$
$673$	−5130.17	−0.293839	−0.146919	+	0.989148i	$0.546936\pi$
$674$	−17327.3	−0.990243
$675$	1998.60	0.113965
$676$	−8787.69	−0.499982
$677$	−3817.87	−0.216740	−0.108370	−	0.994111i	$-0.534563\pi$
$677$	−3817.87	−0.216740	−0.108370	+	0.994111i	$0.534563\pi$
$678$	−3369.06	−0.190838
$679$	5220.73	0.295071
$680$	−1591.03	−0.0897253
$681$	6542.90	0.368171
$682$	2948.89	0.165570
$683$	11076.5	0.620545	0.310273	−	0.950648i	$-0.399580\pi$
$683$	11076.5	0.620545	0.310273	+	0.950648i	$0.399580\pi$
$684$	684.000	0.0382360
$685$	−18935.4	−1.05618
$686$	686.000	0.0381802
$687$	5140.94	0.285501
$688$	−2067.77	−0.114583
$689$	41.5538	0.00229764
$690$	282.319	0.0155764
$691$	24486.4	1.34806	0.674028	−	0.738706i	$-0.264563\pi$
$691$	24486.4	1.34806	0.674028	+	0.738706i	$0.264563\pi$
$692$	13204.4	0.725371
$693$	446.759	0.0244892
$694$	15357.3	0.839992
$695$	−10177.1	−0.555452
$696$	−6648.24	−0.362070
$697$	9104.77	0.494789
$698$	−10984.4	−0.595655
$699$	1762.26	0.0953572
$700$	2072.62	0.111911
$701$	15545.5	0.837584	0.418792	−	0.908082i	$-0.362454\pi$
$701$	15545.5	0.837584	0.418792	+	0.908082i	$0.362454\pi$
$702$	15.1070	0.000812219 0
$703$	−2178.79	−0.116891
$704$	−453.851	−0.0242971
$705$	6343.04	0.338855
$706$	4235.44	0.225783
$707$	4604.65	0.244945
$708$	−5466.61	−0.290181
$709$	−118.318	−0.00626733	−0.00313367	−	0.999995i	$-0.500997\pi$
$709$	−118.318	−0.00626733	−0.00313367	+	0.999995i	$0.500997\pi$
$710$	−13728.2	−0.725651
$711$	−11289.1	−0.595461
$712$	10410.6	0.547967
$713$	1370.23	0.0719713
$714$	−1169.89	−0.0613196
$715$	−14.1648	−0.000740884 0
$716$	3651.17	0.190574
$717$	−20452.7	−1.06530
$718$	−15255.9	−0.792962
$719$	628.050	0.0325762	0.0162881	−	0.999867i	$-0.494815\pi$
$719$	628.050	0.0325762	0.0162881	+	0.999867i	$0.494815\pi$
$720$	1028.14	0.0532175
$721$	1717.79	0.0887291
$722$	−722.000	−0.0372161
$723$	−1652.03	−0.0849786
$724$	−10010.7	−0.513875
$725$	20504.9	1.05039
$726$	−7684.27	−0.392824
$727$	3248.73	0.165734	0.0828672	−	0.996561i	$-0.473592\pi$
$727$	3248.73	0.165734	0.0828672	+	0.996561i	$0.473592\pi$
$728$	15.6665	0.000797583 0
$729$	729.000	0.0370370
$730$	12699.3	0.643867
$731$	−3599.81	−0.182139
$732$	−5519.81	−0.278713
$733$	−10878.4	−0.548162	−0.274081	−	0.961707i	$-0.588374\pi$
$733$	−10878.4	−0.548162	−0.274081	+	0.961707i	$0.588374\pi$
$734$	5786.37	0.290979
$735$	−1049.56	−0.0526717
$736$	−210.886	−0.0105616
$737$	−1033.86	−0.0516724
$738$	−5883.62	−0.293467
$739$	−4355.44	−0.216803	−0.108402	−	0.994107i	$-0.534573\pi$
$739$	−4355.44	−0.216803	−0.108402	+	0.994107i	$0.534573\pi$
$740$	−3275.01	−0.162691
$741$	−15.9463	−0.000790556 0
$742$	2079.47	0.102884
$743$	−18664.5	−0.921579	−0.460789	−	0.887510i	$-0.652434\pi$
$743$	−18664.5	−0.921579	−0.460789	+	0.887510i	$0.652434\pi$
$744$	4990.07	0.245894
$745$	1749.30	0.0860263
$746$	−10583.1	−0.519401
$747$	−936.883	−0.0458886
$748$	−790.115	−0.0386223
$749$	−10604.6	−0.517334
$750$	−8525.96	−0.415099
$751$	−26317.9	−1.27877	−0.639383	−	0.768889i	$-0.720810\pi$
$751$	−26317.9	−1.27877	−0.639383	+	0.768889i	$0.720810\pi$
$752$	−4738.11	−0.229762
$753$	−15776.5	−0.763517
$754$	154.992	0.00748606
$755$	−19169.1	−0.924021
$756$	756.000	0.0363696
$757$	−34363.5	−1.64988	−0.824942	−	0.565217i	$-0.808792\pi$
$757$	−34363.5	−1.64988	−0.824942	+	0.565217i	$0.808792\pi$
$758$	3088.55	0.147996
$759$	140.201	0.00670486
$760$	−1085.26	−0.0517981
$761$	−18506.3	−0.881543	−0.440771	−	0.897619i	$-0.645295\pi$
$761$	−18506.3	−0.881543	−0.440771	+	0.897619i	$0.645295\pi$
$762$	6306.88	0.299835
$763$	5860.38	0.278060
$764$	−19859.8	−0.940446
$765$	1789.91	0.0845938
$766$	320.418	0.0151138
$767$	127.445	0.00599969
$768$	−768.000	−0.0360844
$769$	17916.3	0.840152	0.420076	−	0.907489i	$-0.362003\pi$
$769$	17916.3	0.840152	0.420076	+	0.907489i	$0.362003\pi$
$770$	−708.846	−0.0331754
$771$	−8322.61	−0.388757
$772$	9360.32	0.436380
$773$	30924.7	1.43892	0.719460	−	0.694534i	$-0.244389\pi$
$773$	30924.7	1.43892	0.719460	+	0.694534i	$0.244389\pi$
$774$	2326.24	0.108030
$775$	−15390.7	−0.713353
$776$	5966.55	0.276014
$777$	−2408.13	−0.111186
$778$	−4615.73	−0.212702
$779$	6210.48	0.285640
$780$	−23.9694	−0.00110031
$781$	−6817.53	−0.312357
$782$	−367.135	−0.0167886
$783$	7479.27	0.341363
$784$	784.000	0.0357143
$785$	−24288.1	−1.10430
$786$	−914.067	−0.0414805
$787$	3172.74	0.143705	0.0718526	−	0.997415i	$-0.477109\pi$
$787$	3172.74	0.143705	0.0718526	+	0.997415i	$0.477109\pi$
$788$	12768.6	0.577236
$789$	2934.07	0.132390
$790$	17911.7	0.806669
$791$	3930.57	0.176681
$792$	510.582	0.0229075
$793$	128.685	0.00576260
$794$	1796.03	0.0802755
$795$	−3181.54	−0.141934
$796$	12715.9	0.566212
$797$	2998.57	0.133268	0.0666341	−	0.997777i	$-0.478774\pi$
$797$	2998.57	0.133268	0.0666341	+	0.997777i	$0.478774\pi$
$798$	−798.000	−0.0353996
$799$	−8248.64	−0.365227
$800$	2368.71	0.104683
$801$	−11711.9	−0.516629
$802$	−2855.51	−0.125725
$803$	6306.56	0.277153
$804$	−1749.48	−0.0767404
$805$	−329.372	−0.0144209
$806$	−116.335	−0.00508403
$807$	9283.16	0.404935
$808$	5262.46	0.229125
$809$	−487.256	−0.0211755	−0.0105878	−	0.999944i	$-0.503370\pi$
$809$	−487.256	−0.0211755	−0.0105878	+	0.999944i	$0.503370\pi$
$810$	−1156.66	−0.0501740
$811$	18226.4	0.789169	0.394585	−	0.918860i	$-0.370889\pi$
$811$	18226.4	0.789169	0.394585	+	0.918860i	$0.370889\pi$
$812$	7756.28	0.335212
$813$	−22337.0	−0.963584
$814$	−1626.39	−0.0700306
$815$	−16722.5	−0.718730
$816$	−1337.02	−0.0573592
$817$	−2455.48	−0.105148
$818$	−16696.0	−0.713647
$819$	−17.6249	−0.000751969 0
$820$	9335.18	0.397559
$821$	9781.11	0.415789	0.207895	−	0.978151i	$-0.433339\pi$
$821$	9781.11	0.415789	0.207895	+	0.978151i	$0.433339\pi$
$822$	−15912.3	−0.675191
$823$	7626.18	0.323003	0.161502	−	0.986872i	$-0.448366\pi$
$823$	7626.18	0.323003	0.161502	+	0.986872i	$0.448366\pi$
$824$	1963.18	0.0829985
$825$	−1574.77	−0.0664561
$826$	6377.71	0.268655
$827$	9432.19	0.396602	0.198301	−	0.980141i	$-0.436458\pi$
$827$	9432.19	0.396602	0.198301	+	0.980141i	$0.436458\pi$
$828$	237.247	0.00995761
$829$	24612.9	1.03117	0.515585	−	0.856839i	$-0.327575\pi$
$829$	24612.9	1.03117	0.515585	+	0.856839i	$0.327575\pi$
$830$	1486.50	0.0621651
$831$	22492.0	0.938915
$832$	17.9046	0.000746071 0
$833$	1364.88	0.0567709
$834$	−8552.32	−0.355087
$835$	−12558.8	−0.520496
$836$	−538.948	−0.0222965
$837$	−5613.83	−0.231831
$838$	−17820.0	−0.734582
$839$	−2812.58	−0.115734	−0.0578671	−	0.998324i	$-0.518430\pi$
$839$	−2812.58	−0.115734	−0.0578671	+	0.998324i	$0.518430\pi$
$840$	−1199.50	−0.0492699
$841$	52345.5	2.14628
$842$	−24549.7	−1.00480
$843$	−17601.8	−0.719144
$844$	−17180.4	−0.700681
$845$	−15685.8	−0.638588
$846$	5330.38	0.216622
$847$	8964.98	0.363684
$848$	2376.54	0.0962391
$849$	16023.8	0.647746
$850$	4123.72	0.166403
$851$	−755.718	−0.0304414
$852$	−11536.5	−0.463891
$853$	10475.0	0.420465	0.210233	−	0.977651i	$-0.432578\pi$
$853$	10475.0	0.420465	0.210233	+	0.977651i	$0.432578\pi$
$854$	6439.78	0.258038
$855$	1220.92	0.0488357
$856$	−12119.5	−0.483922
$857$	2002.45	0.0798160	0.0399080	−	0.999203i	$-0.487294\pi$
$857$	2002.45	0.0798160	0.0399080	+	0.999203i	$0.487294\pi$
$858$	−11.9034	−0.000473629 0
$859$	23396.4	0.929306	0.464653	−	0.885493i	$-0.346179\pi$
$859$	23396.4	0.929306	0.464653	+	0.885493i	$0.346179\pi$
$860$	−3690.91	−0.146348
$861$	6864.22	0.271698
$862$	−16231.2	−0.641344
$863$	−44422.8	−1.75222	−0.876112	−	0.482108i	$-0.839871\pi$
$863$	−44422.8	−1.75222	−0.876112	+	0.482108i	$0.839871\pi$
$864$	864.000	0.0340207
$865$	23569.5	0.926459
$866$	−6435.45	−0.252524
$867$	12411.4	0.486173
$868$	−5821.75	−0.227653
$869$	8895.04	0.347231
$870$	−11866.9	−0.462443
$871$	40.7861	0.00158666
$872$	6697.57	0.260101
$873$	−6712.37	−0.260228
$874$	−250.427	−0.00969203
$875$	9946.96	0.384307
$876$	10671.9	0.411609
$877$	24254.0	0.933863	0.466932	−	0.884293i	$-0.345359\pi$
$877$	24254.0	0.933863	0.466932	+	0.884293i	$0.345359\pi$
$878$	27750.2	1.06665
$879$	29365.9	1.12683
$880$	−810.110	−0.0310327
$881$	−18563.8	−0.709910	−0.354955	−	0.934883i	$-0.615504\pi$
$881$	−18563.8	−0.709910	−0.354955	+	0.934883i	$0.615504\pi$
$882$	−882.000	−0.0336718
$883$	35254.3	1.34360	0.671802	−	0.740731i	$-0.265521\pi$
$883$	35254.3	1.34360	0.671802	+	0.740731i	$0.265521\pi$
$884$	31.1704	0.00118594
$885$	−9757.74	−0.370625
$886$	−14320.6	−0.543012
$887$	27633.1	1.04603	0.523016	−	0.852323i	$-0.324807\pi$
$887$	27633.1	1.04603	0.523016	+	0.852323i	$0.324807\pi$
$888$	−2752.15	−0.104005
$889$	−7358.03	−0.277593
$890$	18582.6	0.699875
$891$	−574.405	−0.0215974
$892$	−9440.22	−0.354352
$893$	−5626.51	−0.210844
$894$	1470.03	0.0549945
$895$	6517.23	0.243405
$896$	896.000	0.0334077
$897$	−5.53101	−0.000205881 0
$898$	−19385.7	−0.720389
$899$	−57595.8	−2.13674
$900$	−2664.80	−0.0986962
$901$	4137.35	0.152980
$902$	4635.91	0.171130
$903$	−2713.95	−0.100016
$904$	4492.08	0.165270
$905$	−17868.8	−0.656331
$906$	−16108.8	−0.590704
$907$	46796.3	1.71317	0.856585	−	0.516007i	$-0.172582\pi$
$907$	46796.3	1.71317	0.856585	+	0.516007i	$0.172582\pi$
$908$	−8723.87	−0.318846
$909$	−5920.27	−0.216021
$910$	27.9643	0.00101869
$911$	49812.0	1.81157	0.905787	−	0.423733i	$-0.139281\pi$
$911$	49812.0	1.81157	0.905787	+	0.423733i	$0.139281\pi$
$912$	−912.000	−0.0331133
$913$	738.203	0.0267590
$914$	14120.7	0.511020
$915$	−9852.70	−0.355978
$916$	−6854.58	−0.247251
$917$	1066.41	0.0384035
$918$	1504.15	0.0540788
$919$	19157.6	0.687651	0.343825	−	0.939034i	$-0.388277\pi$
$919$	19157.6	0.687651	0.343825	+	0.939034i	$0.388277\pi$
$920$	−376.426	−0.0134895
$921$	−10812.0	−0.386826
$922$	−7768.24	−0.277477
$923$	268.955	0.00959128
$924$	−595.679	−0.0212082
$925$	8488.34	0.301724
$926$	−27449.8	−0.974142
$927$	−2208.58	−0.0782517
$928$	8864.32	0.313562
$929$	283.154	0.0100000	0.00500000	−	0.999987i	$-0.498408\pi$
$929$	283.154	0.0100000	0.00500000	+	0.999987i	$0.498408\pi$
$930$	8907.13	0.314061
$931$	931.000	0.0327737
$932$	−2349.68	−0.0825818
$933$	−7747.35	−0.271851
$934$	21289.7	0.745847
$935$	−1410.33	−0.0493292
$936$	−20.1427	−0.000703402 0
$937$	−17269.0	−0.602086	−0.301043	−	0.953611i	$-0.597335\pi$
$937$	−17269.0	−0.602086	−0.301043	+	0.953611i	$0.597335\pi$
$938$	2041.06	0.0710478
$939$	25665.9	0.891985
$940$	−8457.39	−0.293457
$941$	−48487.3	−1.67975	−0.839874	−	0.542782i	$-0.817371\pi$
$941$	−48487.3	−1.67975	−0.839874	+	0.542782i	$0.817371\pi$
$942$	−20410.5	−0.705955
$943$	2154.12	0.0743879
$944$	7288.82	0.251304
$945$	1349.44	0.0464521
$946$	−1832.93	−0.0629954
$947$	11141.2	0.382302	0.191151	−	0.981561i	$-0.438778\pi$
$947$	11141.2	0.382302	0.191151	+	0.981561i	$0.438778\pi$
$948$	15052.1	0.515684
$949$	−248.797	−0.00851031
$950$	2812.84	0.0960638
$951$	12907.0	0.440104
$952$	1559.86	0.0531043
$953$	27131.8	0.922229	0.461114	−	0.887341i	$-0.347450\pi$
$953$	27131.8	0.922229	0.461114	+	0.887341i	$0.347450\pi$
$954$	−2673.61	−0.0907351
$955$	−35449.1	−1.20116
$956$	27270.3	0.922579
$957$	−5893.18	−0.199059
$958$	24586.0	0.829164
$959$	18564.4	0.625105
$960$	−1370.86	−0.0460877
$961$	13439.6	0.451129
$962$	64.1618	0.00215037
$963$	13634.5	0.456246
$964$	2202.70	0.0735937
$965$	16707.9	0.557354
$966$	−276.788	−0.00921896
$967$	−16864.3	−0.560825	−0.280413	−	0.959880i	$-0.590471\pi$
$967$	−16864.3	−0.560825	−0.280413	+	0.959880i	$0.590471\pi$
$968$	10245.7	0.340195
$969$	−1587.71	−0.0526364
$970$	10650.1	0.352531
$971$	15338.3	0.506931	0.253465	−	0.967344i	$-0.418430\pi$
$971$	15338.3	0.506931	0.253465	+	0.967344i	$0.418430\pi$
$972$	−972.000	−0.0320750
$973$	9977.71	0.328747
$974$	−25653.5	−0.843935
$975$	62.1252	0.00204061
$976$	7359.75	0.241373
$977$	2090.32	0.0684495	0.0342248	−	0.999414i	$-0.489104\pi$
$977$	2090.32	0.0684495	0.0342248	+	0.999414i	$0.489104\pi$
$978$	−14052.8	−0.459467
$979$	9228.22	0.301262
$980$	1399.42	0.0456150
$981$	−7534.77	−0.245226
$982$	−32187.8	−1.04598
$983$	22789.3	0.739436	0.369718	−	0.929144i	$-0.379454\pi$
$983$	22789.3	0.739436	0.369718	+	0.929144i	$0.379454\pi$
$984$	7844.82	0.254150
$985$	22791.5	0.737258
$986$	15432.0	0.498433
$987$	−6218.77	−0.200553
$988$	21.2617	0.000684642 0
$989$	−851.688	−0.0273833
$990$	911.374	0.0292579
$991$	14643.9	0.469402	0.234701	−	0.972068i	$-0.424589\pi$
$991$	14643.9	0.469402	0.234701	+	0.972068i	$0.424589\pi$
$992$	−6653.43	−0.212950
$993$	12863.0	0.411074
$994$	13459.3	0.429480
$995$	22697.6	0.723178
$996$	1249.18	0.0397407
$997$	−42873.9	−1.36192	−0.680958	−	0.732322i	$-0.738436\pi$
$997$	−42873.9	−1.36192	−0.680958	+	0.732322i	$0.738436\pi$
$998$	20935.3	0.664022
$999$	3096.17	0.0980566

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	798.4.a.e.1.2	✓	3
3.2	odd	2		2394.4.a.o.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.4.a.e.1.2	✓	3	1.1	even	1	trivial
2394.4.a.o.1.2		3	3.2	odd	2

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:	\( N \)	\(=\)	\( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	798.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +7.13988 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +7.13988 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -14.2798 q^{10} -7.09142 q^{11} -12.0000 q^{12} +0.279760 q^{13} +14.0000 q^{14} -21.4196 q^{15} +16.0000 q^{16} +27.8546 q^{17} -18.0000 q^{18} +19.0000 q^{19} +28.5595 q^{20} +21.0000 q^{21} +14.1828 q^{22} +6.59019 q^{23} +24.0000 q^{24} -74.0221 q^{25} -0.559519 q^{26} -27.0000 q^{27} -28.0000 q^{28} -277.010 q^{29} +42.8393 q^{30} +207.920 q^{31} -32.0000 q^{32} +21.2743 q^{33} -55.7092 q^{34} -49.9792 q^{35} +36.0000 q^{36} -114.673 q^{37} -38.0000 q^{38} -0.839279 q^{39} -57.1190 q^{40} +326.868 q^{41} -42.0000 q^{42} -129.236 q^{43} -28.3657 q^{44} +64.2589 q^{45} -13.1804 q^{46} -296.132 q^{47} -48.0000 q^{48} +49.0000 q^{49} +148.044 q^{50} -83.5638 q^{51} +1.11904 q^{52} +148.534 q^{53} +54.0000 q^{54} -50.6319 q^{55} +56.0000 q^{56} -57.0000 q^{57} +554.020 q^{58} +455.551 q^{59} -85.6786 q^{60} +459.984 q^{61} -415.839 q^{62} -63.0000 q^{63} +64.0000 q^{64} +1.99745 q^{65} -42.5485 q^{66} +145.790 q^{67} +111.418 q^{68} -19.7706 q^{69} +99.9583 q^{70} +961.378 q^{71} -72.0000 q^{72} -889.323 q^{73} +229.346 q^{74} +222.066 q^{75} +76.0000 q^{76} +49.6399 q^{77} +1.67856 q^{78} -1254.34 q^{79} +114.238 q^{80} +81.0000 q^{81} -653.735 q^{82} -104.098 q^{83} +84.0000 q^{84} +198.879 q^{85} +258.471 q^{86} +831.030 q^{87} +56.7313 q^{88} -1301.32 q^{89} -128.518 q^{90} -1.95832 q^{91} +26.3608 q^{92} -623.759 q^{93} +592.264 q^{94} +135.658 q^{95} +96.0000 q^{96} -745.819 q^{97} -98.0000 q^{98} -63.8228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} + 18 q^{6} - 21 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 + 18 * q^6 - 21 * q^7 - 24 * q^8 + 27 * q^9	\( 3 q - 6 q^{2} - 9 q^{3} + 12 q^{4} + 18 q^{6} - 21 q^{7} - 24 q^{8} + 27 q^{9} + 38 q^{11} - 36 q^{12} - 42 q^{13} + 42 q^{14} + 48 q^{16} - 30 q^{17} - 54 q^{18} + 57 q^{19} + 63 q^{21} - 76 q^{22} - 36 q^{23} + 72 q^{24} + 25 q^{25} + 84 q^{26} - 81 q^{27} - 84 q^{28} + 76 q^{29} + 166 q^{31} - 96 q^{32} - 114 q^{33} + 60 q^{34} + 108 q^{36} + 48 q^{37} - 114 q^{38} + 126 q^{39} + 102 q^{41} - 126 q^{42} - 52 q^{43} + 152 q^{44} + 72 q^{46} + 48 q^{47} - 144 q^{48} + 147 q^{49} - 50 q^{50} + 90 q^{51} - 168 q^{52} + 224 q^{53} + 162 q^{54} - 396 q^{55} + 168 q^{56} - 171 q^{57} - 152 q^{58} + 1004 q^{59} - 450 q^{61} - 332 q^{62} - 189 q^{63} + 192 q^{64} + 800 q^{65} + 228 q^{66} - 906 q^{67} - 120 q^{68} + 108 q^{69} + 946 q^{71} - 216 q^{72} - 862 q^{73} - 96 q^{74} - 75 q^{75} + 228 q^{76} - 266 q^{77} - 252 q^{78} + 148 q^{79} + 243 q^{81} - 204 q^{82} - 778 q^{83} + 252 q^{84} - 12 q^{85} + 104 q^{86} - 228 q^{87} - 304 q^{88} - 226 q^{89} + 294 q^{91} - 144 q^{92} - 498 q^{93} - 96 q^{94} + 288 q^{96} - 2200 q^{97} - 294 q^{98} + 342 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 - 9 * q^3 + 12 * q^4 + 18 * q^6 - 21 * q^7 - 24 * q^8 + 27 * q^9 + 38 * q^11 - 36 * q^12 - 42 * q^13 + 42 * q^14 + 48 * q^16 - 30 * q^17 - 54 * q^18 + 57 * q^19 + 63 * q^21 - 76 * q^22 - 36 * q^23 + 72 * q^24 + 25 * q^25 + 84 * q^26 - 81 * q^27 - 84 * q^28 + 76 * q^29 + 166 * q^31 - 96 * q^32 - 114 * q^33 + 60 * q^34 + 108 * q^36 + 48 * q^37 - 114 * q^38 + 126 * q^39 + 102 * q^41 - 126 * q^42 - 52 * q^43 + 152 * q^44 + 72 * q^46 + 48 * q^47 - 144 * q^48 + 147 * q^49 - 50 * q^50 + 90 * q^51 - 168 * q^52 + 224 * q^53 + 162 * q^54 - 396 * q^55 + 168 * q^56 - 171 * q^57 - 152 * q^58 + 1004 * q^59 - 450 * q^61 - 332 * q^62 - 189 * q^63 + 192 * q^64 + 800 * q^65 + 228 * q^66 - 906 * q^67 - 120 * q^68 + 108 * q^69 + 946 * q^71 - 216 * q^72 - 862 * q^73 - 96 * q^74 - 75 * q^75 + 228 * q^76 - 266 * q^77 - 252 * q^78 + 148 * q^79 + 243 * q^81 - 204 * q^82 - 778 * q^83 + 252 * q^84 - 12 * q^85 + 104 * q^86 - 228 * q^87 - 304 * q^88 - 226 * q^89 + 294 * q^91 - 144 * q^92 - 498 * q^93 - 96 * q^94 + 288 * q^96 - 2200 * q^97 - 294 * q^98 + 342 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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