LMFDB - Embedded newform 798.4.a.k.1.3

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:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 310x^{2} - 1444x - 1712 $ x^4 - 310x^2 - 1444x - 1712
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			1.3
Root			$-2.35201$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.35201 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.35201 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +4.70401 q^{10} -65.1379 q^{11} -12.0000 q^{12} -31.4764 q^{13} +14.0000 q^{14} +7.05602 q^{15} +16.0000 q^{16} -43.0831 q^{17} -18.0000 q^{18} -19.0000 q^{19} -9.40803 q^{20} +21.0000 q^{21} +130.276 q^{22} +92.8218 q^{23} +24.0000 q^{24} -119.468 q^{25} +62.9528 q^{26} -27.0000 q^{27} -28.0000 q^{28} -95.7140 q^{29} -14.1120 q^{30} +161.087 q^{31} -32.0000 q^{32} +195.414 q^{33} +86.1662 q^{34} +16.4641 q^{35} +36.0000 q^{36} +5.04480 q^{37} +38.0000 q^{38} +94.4291 q^{39} +18.8161 q^{40} -434.908 q^{41} -42.0000 q^{42} +54.7476 q^{43} -260.552 q^{44} -21.1681 q^{45} -185.644 q^{46} -220.621 q^{47} -48.0000 q^{48} +49.0000 q^{49} +238.936 q^{50} +129.249 q^{51} -125.906 q^{52} +175.142 q^{53} +54.0000 q^{54} +153.205 q^{55} +56.0000 q^{56} +57.0000 q^{57} +191.428 q^{58} -383.026 q^{59} +28.2241 q^{60} +431.625 q^{61} -322.174 q^{62} -63.0000 q^{63} +64.0000 q^{64} +74.0327 q^{65} -390.828 q^{66} -989.711 q^{67} -172.332 q^{68} -278.465 q^{69} -32.9281 q^{70} -456.366 q^{71} -72.0000 q^{72} +92.3693 q^{73} -10.0896 q^{74} +358.404 q^{75} -76.0000 q^{76} +455.965 q^{77} -188.858 q^{78} +620.873 q^{79} -37.6321 q^{80} +81.0000 q^{81} +869.816 q^{82} +519.728 q^{83} +84.0000 q^{84} +101.332 q^{85} -109.495 q^{86} +287.142 q^{87} +521.103 q^{88} +1049.89 q^{89} +42.3361 q^{90} +220.335 q^{91} +371.287 q^{92} -483.261 q^{93} +441.241 q^{94} +44.6881 q^{95} +96.0000 q^{96} -166.305 q^{97} -98.0000 q^{98} -586.241 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} - 66 q^{11} - 48 q^{12} + 10 q^{13} + 56 q^{14} + 64 q^{16} - 8 q^{17} - 72 q^{18} - 76 q^{19} + 84 q^{21} + 132 q^{22} - 250 q^{23} + 96 q^{24} + 120 q^{25} - 20 q^{26} - 108 q^{27} - 112 q^{28} - 66 q^{29} - 52 q^{31} - 128 q^{32} + 198 q^{33} + 16 q^{34} + 144 q^{36} + 288 q^{37} + 152 q^{38} - 30 q^{39} - 196 q^{41} - 168 q^{42} - 280 q^{43} - 264 q^{44} + 500 q^{46} + 498 q^{47} - 192 q^{48} + 196 q^{49} - 240 q^{50} + 24 q^{51} + 40 q^{52} - 250 q^{53} + 216 q^{54} - 1432 q^{55} + 224 q^{56} + 228 q^{57} + 132 q^{58} - 872 q^{59} + 600 q^{61} + 104 q^{62} - 252 q^{63} + 256 q^{64} - 908 q^{65} - 396 q^{66} + 1174 q^{67} - 32 q^{68} + 750 q^{69} - 1766 q^{71} - 288 q^{72} + 556 q^{73} - 576 q^{74} - 360 q^{75} - 304 q^{76} + 462 q^{77} + 60 q^{78} + 26 q^{79} + 324 q^{81} + 392 q^{82} + 674 q^{83} + 336 q^{84} + 2052 q^{85} + 560 q^{86} + 198 q^{87} + 528 q^{88} - 324 q^{89} - 70 q^{91} - 1000 q^{92} + 156 q^{93} - 996 q^{94} + 384 q^{96} + 2902 q^{97} - 392 q^{98} - 594 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9 - 66 * q^11 - 48 * q^12 + 10 * q^13 + 56 * q^14 + 64 * q^16 - 8 * q^17 - 72 * q^18 - 76 * q^19 + 84 * q^21 + 132 * q^22 - 250 * q^23 + 96 * q^24 + 120 * q^25 - 20 * q^26 - 108 * q^27 - 112 * q^28 - 66 * q^29 - 52 * q^31 - 128 * q^32 + 198 * q^33 + 16 * q^34 + 144 * q^36 + 288 * q^37 + 152 * q^38 - 30 * q^39 - 196 * q^41 - 168 * q^42 - 280 * q^43 - 264 * q^44 + 500 * q^46 + 498 * q^47 - 192 * q^48 + 196 * q^49 - 240 * q^50 + 24 * q^51 + 40 * q^52 - 250 * q^53 + 216 * q^54 - 1432 * q^55 + 224 * q^56 + 228 * q^57 + 132 * q^58 - 872 * q^59 + 600 * q^61 + 104 * q^62 - 252 * q^63 + 256 * q^64 - 908 * q^65 - 396 * q^66 + 1174 * q^67 - 32 * q^68 + 750 * q^69 - 1766 * q^71 - 288 * q^72 + 556 * q^73 - 576 * q^74 - 360 * q^75 - 304 * q^76 + 462 * q^77 + 60 * q^78 + 26 * q^79 + 324 * q^81 + 392 * q^82 + 674 * q^83 + 336 * q^84 + 2052 * q^85 + 560 * q^86 + 198 * q^87 + 528 * q^88 - 324 * q^89 - 70 * q^91 - 1000 * q^92 + 156 * q^93 - 996 * q^94 + 384 * q^96 + 2902 * q^97 - 392 * q^98 - 594 * 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
$5$	−2.35201	−0.210370	−0.105185	−	0.994453i	$-0.533543\pi$
$5$	−2.35201	−0.210370	−0.105185	+	0.994453i	$0.533543\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	4.70401	0.148754
$11$	−65.1379	−1.78544	−0.892719	−	0.450613i	$-0.851205\pi$
$11$	−65.1379	−1.78544	−0.892719	+	0.450613i	$0.851205\pi$
$12$	−12.0000	−0.288675
$13$	−31.4764	−0.671537	−0.335768	−	0.941945i	$-0.608996\pi$
$13$	−31.4764	−0.671537	−0.335768	+	0.941945i	$0.608996\pi$
$14$	14.0000	0.267261
$15$	7.05602	0.121457
$16$	16.0000	0.250000
$17$	−43.0831	−0.614658	−0.307329	−	0.951603i	$-0.599435\pi$
$17$	−43.0831	−0.614658	−0.307329	+	0.951603i	$0.599435\pi$
$18$	−18.0000	−0.235702
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−9.40803	−0.105185
$21$	21.0000	0.218218
$22$	130.276	1.26250
$23$	92.8218	0.841508	0.420754	−	0.907175i	$-0.361766\pi$
$23$	92.8218	0.841508	0.420754	+	0.907175i	$0.361766\pi$
$24$	24.0000	0.204124
$25$	−119.468	−0.955744
$26$	62.9528	0.474848
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	−95.7140	−0.612884	−0.306442	−	0.951889i	$-0.599139\pi$
$29$	−95.7140	−0.612884	−0.306442	+	0.951889i	$0.599139\pi$
$30$	−14.1120	−0.0858832
$31$	161.087	0.933294	0.466647	−	0.884444i	$-0.345462\pi$
$31$	161.087	0.933294	0.466647	+	0.884444i	$0.345462\pi$
$32$	−32.0000	−0.176777
$33$	195.414	1.03082
$34$	86.1662	0.434629
$35$	16.4641	0.0795124
$36$	36.0000	0.166667
$37$	5.04480	0.0224151	0.0112076	−	0.999937i	$-0.496432\pi$
$37$	5.04480	0.0224151	0.0112076	+	0.999937i	$0.496432\pi$
$38$	38.0000	0.162221
$39$	94.4291	0.387712
$40$	18.8161	0.0743770
$41$	−434.908	−1.65661	−0.828307	−	0.560274i	$-0.810696\pi$
$41$	−434.908	−1.65661	−0.828307	+	0.560274i	$0.810696\pi$
$42$	−42.0000	−0.154303
$43$	54.7476	0.194161	0.0970805	−	0.995277i	$-0.469050\pi$
$43$	54.7476	0.194161	0.0970805	+	0.995277i	$0.469050\pi$
$44$	−260.552	−0.892719
$45$	−21.1681	−0.0701233
$46$	−185.644	−0.595036
$47$	−220.621	−0.684698	−0.342349	−	0.939573i	$-0.611223\pi$
$47$	−220.621	−0.684698	−0.342349	+	0.939573i	$0.611223\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	238.936	0.675813
$51$	129.249	0.354873
$52$	−125.906	−0.335768
$53$	175.142	0.453917	0.226959	−	0.973904i	$-0.427122\pi$
$53$	175.142	0.453917	0.226959	+	0.973904i	$0.427122\pi$
$54$	54.0000	0.136083
$55$	153.205	0.375603
$56$	56.0000	0.133631
$57$	57.0000	0.132453
$58$	191.428	0.433375
$59$	−383.026	−0.845181	−0.422591	−	0.906321i	$-0.638879\pi$
$59$	−383.026	−0.845181	−0.422591	+	0.906321i	$0.638879\pi$
$60$	28.2241	0.0607286
$61$	431.625	0.905965	0.452983	−	0.891519i	$-0.350360\pi$
$61$	431.625	0.905965	0.452983	+	0.891519i	$0.350360\pi$
$62$	−322.174	−0.659938
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	74.0327	0.141271
$66$	−390.828	−0.728902
$67$	−989.711	−1.80466	−0.902332	−	0.431042i	$-0.858146\pi$
$67$	−989.711	−1.80466	−0.902332	+	0.431042i	$0.858146\pi$
$68$	−172.332	−0.307329
$69$	−278.465	−0.485845
$70$	−32.9281	−0.0562237
$71$	−456.366	−0.762827	−0.381413	−	0.924405i	$-0.624563\pi$
$71$	−456.366	−0.762827	−0.381413	+	0.924405i	$0.624563\pi$
$72$	−72.0000	−0.117851
$73$	92.3693	0.148096	0.0740480	−	0.997255i	$-0.476408\pi$
$73$	92.3693	0.148096	0.0740480	+	0.997255i	$0.476408\pi$
$74$	−10.0896	−0.0158499
$75$	358.404	0.551799
$76$	−76.0000	−0.114708
$77$	455.965	0.674832
$78$	−188.858	−0.274154
$79$	620.873	0.884224	0.442112	−	0.896960i	$-0.354229\pi$
$79$	620.873	0.884224	0.442112	+	0.896960i	$0.354229\pi$
$80$	−37.6321	−0.0525925
$81$	81.0000	0.111111
$82$	869.816	1.17140
$83$	519.728	0.687320	0.343660	−	0.939094i	$-0.388333\pi$
$83$	519.728	0.687320	0.343660	+	0.939094i	$0.388333\pi$
$84$	84.0000	0.109109
$85$	101.332	0.129306
$86$	−109.495	−0.137293
$87$	287.142	0.353849
$88$	521.103	0.631248
$89$	1049.89	1.25042	0.625212	−	0.780455i	$-0.285013\pi$
$89$	1049.89	1.25042	0.625212	+	0.780455i	$0.285013\pi$
$90$	42.3361	0.0495847
$91$	220.335	0.253817
$92$	371.287	0.420754
$93$	−483.261	−0.538837
$94$	441.241	0.484155
$95$	44.6881	0.0482622
$96$	96.0000	0.102062
$97$	−166.305	−0.174079	−0.0870396	−	0.996205i	$-0.527741\pi$
$97$	−166.305	−0.174079	−0.0870396	+	0.996205i	$0.527741\pi$
$98$	−98.0000	−0.101015
$99$	−586.241	−0.595146
$100$	−477.872	−0.477872
$101$	1504.58	1.48229	0.741147	−	0.671343i	$-0.234282\pi$
$101$	1504.58	1.48229	0.741147	+	0.671343i	$0.234282\pi$
$102$	−258.499	−0.250933
$103$	−1145.68	−1.09599	−0.547994	−	0.836482i	$-0.684608\pi$
$103$	−1145.68	−1.09599	−0.547994	+	0.836482i	$0.684608\pi$
$104$	251.811	0.237424
$105$	−49.3922	−0.0459065
$106$	−350.284	−0.320968
$107$	−804.550	−0.726905	−0.363452	−	0.931613i	$-0.618402\pi$
$107$	−804.550	−0.726905	−0.363452	+	0.931613i	$0.618402\pi$
$108$	−108.000	−0.0962250
$109$	2146.46	1.88618	0.943089	−	0.332541i	$-0.107906\pi$
$109$	2146.46	1.88618	0.943089	+	0.332541i	$0.107906\pi$
$110$	−306.410	−0.265591
$111$	−15.1344	−0.0129414
$112$	−112.000	−0.0944911
$113$	−456.239	−0.379818	−0.189909	−	0.981802i	$-0.560819\pi$
$113$	−456.239	−0.379818	−0.189909	+	0.981802i	$0.560819\pi$
$114$	−114.000	−0.0936586
$115$	−218.318	−0.177028
$116$	−382.856	−0.306442
$117$	−283.287	−0.223846
$118$	766.051	0.597633
$119$	301.582	0.232319
$120$	−56.4482	−0.0429416
$121$	2911.95	2.18779
$122$	−863.250	−0.640614
$123$	1304.72	0.956447
$124$	644.349	0.466647
$125$	574.991	0.411430
$126$	126.000	0.0890871
$127$	1257.44	0.878582	0.439291	−	0.898345i	$-0.355230\pi$
$127$	1257.44	0.878582	0.439291	+	0.898345i	$0.355230\pi$
$128$	−128.000	−0.0883883
$129$	−164.243	−0.112099
$130$	−148.065	−0.0998938
$131$	−2329.46	−1.55363	−0.776817	−	0.629726i	$-0.783167\pi$
$131$	−2329.46	−1.55363	−0.776817	+	0.629726i	$0.783167\pi$
$132$	781.655	0.515412
$133$	133.000	0.0867110
$134$	1979.42	1.27609
$135$	63.5042	0.0404857
$136$	344.665	0.217315
$137$	1728.59	1.07798	0.538990	−	0.842312i	$-0.318806\pi$
$137$	1728.59	1.07798	0.538990	+	0.842312i	$0.318806\pi$
$138$	556.931	0.343544
$139$	−691.205	−0.421779	−0.210889	−	0.977510i	$-0.567636\pi$
$139$	−691.205	−0.421779	−0.210889	+	0.977510i	$0.567636\pi$
$140$	65.8562	0.0397562
$141$	661.862	0.395311
$142$	912.733	0.539400
$143$	2050.31	1.19899
$144$	144.000	0.0833333
$145$	225.120	0.128932
$146$	−184.739	−0.104720
$147$	−147.000	−0.0824786
$148$	20.1792	0.0112076
$149$	−2337.76	−1.28535	−0.642674	−	0.766140i	$-0.722175\pi$
$149$	−2337.76	−1.28535	−0.642674	+	0.766140i	$0.722175\pi$
$150$	−716.808	−0.390181
$151$	2681.77	1.44529	0.722646	−	0.691218i	$-0.242926\pi$
$151$	2681.77	1.44529	0.722646	+	0.691218i	$0.242926\pi$
$152$	152.000	0.0811107
$153$	−387.748	−0.204886
$154$	−911.931	−0.477178
$155$	−378.878	−0.196337
$156$	377.717	0.193856
$157$	762.000	0.387352	0.193676	−	0.981066i	$-0.437959\pi$
$157$	762.000	0.387352	0.193676	+	0.981066i	$0.437959\pi$
$158$	−1241.75	−0.625241
$159$	−525.426	−0.262069
$160$	75.2642	0.0371885
$161$	−649.752	−0.318060
$162$	−162.000	−0.0785674
$163$	3452.06	1.65881	0.829406	−	0.558647i	$-0.188679\pi$
$163$	3452.06	1.65881	0.829406	+	0.558647i	$0.188679\pi$
$164$	−1739.63	−0.828307
$165$	−459.615	−0.216854
$166$	−1039.46	−0.486009
$167$	3589.50	1.66326	0.831628	−	0.555333i	$-0.187409\pi$
$167$	3589.50	1.66326	0.831628	+	0.555333i	$0.187409\pi$
$168$	−168.000	−0.0771517
$169$	−1206.24	−0.549039
$170$	−202.664	−0.0914329
$171$	−171.000	−0.0764719
$172$	218.990	0.0970805
$173$	1795.76	0.789187	0.394593	−	0.918856i	$-0.370885\pi$
$173$	1795.76	0.789187	0.394593	+	0.918856i	$0.370885\pi$
$174$	−574.284	−0.250209
$175$	836.276	0.361237
$176$	−1042.21	−0.446360
$177$	1149.08	0.487966
$178$	−2099.77	−0.884183
$179$	−2476.29	−1.03400	−0.517002	−	0.855984i	$-0.672952\pi$
$179$	−2476.29	−1.03400	−0.517002	+	0.855984i	$0.672952\pi$
$180$	−84.6723	−0.0350617
$181$	2402.09	0.986441	0.493220	−	0.869904i	$-0.335820\pi$
$181$	2402.09	0.986441	0.493220	+	0.869904i	$0.335820\pi$
$182$	−440.669	−0.179476
$183$	−1294.87	−0.523059
$184$	−742.574	−0.297518
$185$	−11.8654	−0.00471547
$186$	966.523	0.381016
$187$	2806.34	1.09743
$188$	−882.482	−0.342349
$189$	189.000	0.0727393
$190$	−89.3763	−0.0341265
$191$	−1635.57	−0.619610	−0.309805	−	0.950800i	$-0.600264\pi$
$191$	−1635.57	−0.619610	−0.309805	+	0.950800i	$0.600264\pi$
$192$	−192.000	−0.0721688
$193$	4432.79	1.65326	0.826630	−	0.562746i	$-0.190255\pi$
$193$	4432.79	1.65326	0.826630	+	0.562746i	$0.190255\pi$
$194$	332.609	0.123093
$195$	−222.098	−0.0815629
$196$	196.000	0.0714286
$197$	2601.65	0.940914	0.470457	−	0.882423i	$-0.344089\pi$
$197$	2601.65	0.940914	0.470457	+	0.882423i	$0.344089\pi$
$198$	1172.48	0.420832
$199$	1975.71	0.703791	0.351895	−	0.936039i	$-0.385537\pi$
$199$	1975.71	0.703791	0.351895	+	0.936039i	$0.385537\pi$
$200$	955.744	0.337907
$201$	2969.13	1.04192
$202$	−3009.17	−1.04814
$203$	669.998	0.231649
$204$	516.997	0.177437
$205$	1022.91	0.348502
$206$	2291.35	0.774980
$207$	835.396	0.280503
$208$	−503.622	−0.167884
$209$	1237.62	0.409608
$210$	98.7843	0.0324608
$211$	−3735.55	−1.21880	−0.609398	−	0.792864i	$-0.708589\pi$
$211$	−3735.55	−1.21880	−0.609398	+	0.792864i	$0.708589\pi$
$212$	700.568	0.226959
$213$	1369.10	0.440418
$214$	1609.10	0.513999
$215$	−128.767	−0.0408456
$216$	216.000	0.0680414
$217$	−1127.61	−0.352752
$218$	−4292.91	−1.33373
$219$	−277.108	−0.0855033
$220$	612.819	0.187801
$221$	1356.10	0.412766
$222$	30.2688	0.00915094
$223$	5621.09	1.68796	0.843982	−	0.536372i	$-0.180205\pi$
$223$	5621.09	1.68796	0.843982	+	0.536372i	$0.180205\pi$
$224$	224.000	0.0668153
$225$	−1075.21	−0.318581
$226$	912.479	0.268572
$227$	−88.9122	−0.0259970	−0.0129985	−	0.999916i	$-0.504138\pi$
$227$	−88.9122	−0.0259970	−0.0129985	+	0.999916i	$0.504138\pi$
$228$	228.000	0.0662266
$229$	−3591.62	−1.03642	−0.518211	−	0.855253i	$-0.673402\pi$
$229$	−3591.62	−1.03642	−0.518211	+	0.855253i	$0.673402\pi$
$230$	436.635	0.125178
$231$	−1367.90	−0.389615
$232$	765.712	0.216687
$233$	−1783.34	−0.501417	−0.250709	−	0.968063i	$-0.580664\pi$
$233$	−1783.34	−0.501417	−0.250709	+	0.968063i	$0.580664\pi$
$234$	566.575	0.158283
$235$	518.901	0.144040
$236$	−1532.10	−0.422591
$237$	−1862.62	−0.510507
$238$	−603.164	−0.164274
$239$	−5280.63	−1.42919	−0.714594	−	0.699540i	$-0.753388\pi$
$239$	−5280.63	−1.42919	−0.714594	+	0.699540i	$0.753388\pi$
$240$	112.896	0.0303643
$241$	7111.18	1.90071	0.950355	−	0.311167i	$-0.100720\pi$
$241$	7111.18	1.90071	0.950355	+	0.311167i	$0.100720\pi$
$242$	−5823.90	−1.54700
$243$	−243.000	−0.0641500
$244$	1726.50	0.452983
$245$	−115.248	−0.0300528
$246$	−2609.45	−0.676310
$247$	598.051	0.154061
$248$	−1288.70	−0.329969
$249$	−1559.18	−0.396824
$250$	−1149.98	−0.290925
$251$	−2522.95	−0.634450	−0.317225	−	0.948350i	$-0.602751\pi$
$251$	−2522.95	−0.634450	−0.317225	+	0.948350i	$0.602751\pi$
$252$	−252.000	−0.0629941
$253$	−6046.22	−1.50246
$254$	−2514.88	−0.621251
$255$	−303.995	−0.0746546
$256$	256.000	0.0625000
$257$	−2126.61	−0.516165	−0.258082	−	0.966123i	$-0.583091\pi$
$257$	−2126.61	−0.516165	−0.258082	+	0.966123i	$0.583091\pi$
$258$	328.485	0.0792659
$259$	−35.3136	−0.00847213
$260$	296.131	0.0706356
$261$	−861.426	−0.204295
$262$	4658.92	1.09859
$263$	−2709.59	−0.635287	−0.317644	−	0.948210i	$-0.602892\pi$
$263$	−2709.59	−0.635287	−0.317644	+	0.948210i	$0.602892\pi$
$264$	−1563.31	−0.364451
$265$	−411.935	−0.0954905
$266$	−266.000	−0.0613139
$267$	−3149.66	−0.721932
$268$	−3958.84	−0.902332
$269$	−1063.61	−0.241076	−0.120538	−	0.992709i	$-0.538462\pi$
$269$	−1063.61	−0.241076	−0.120538	+	0.992709i	$0.538462\pi$
$270$	−127.008	−0.0286277
$271$	2415.84	0.541519	0.270759	−	0.962647i	$-0.412725\pi$
$271$	2415.84	0.541519	0.270759	+	0.962647i	$0.412725\pi$
$272$	−689.330	−0.153665
$273$	−661.004	−0.146541
$274$	−3457.18	−0.762247
$275$	7781.90	1.70642
$276$	−1113.86	−0.242922
$277$	−2923.91	−0.634226	−0.317113	−	0.948388i	$-0.602713\pi$
$277$	−2923.91	−0.634226	−0.317113	+	0.948388i	$0.602713\pi$
$278$	1382.41	0.298243
$279$	1449.78	0.311098
$280$	−131.712	−0.0281119
$281$	−7064.52	−1.49976	−0.749882	−	0.661571i	$-0.769890\pi$
$281$	−7064.52	−1.49976	−0.749882	+	0.661571i	$0.769890\pi$
$282$	−1323.72	−0.279527
$283$	−7199.32	−1.51221	−0.756104	−	0.654451i	$-0.772900\pi$
$283$	−7199.32	−1.51221	−0.756104	+	0.654451i	$0.772900\pi$
$284$	−1825.47	−0.381413
$285$	−134.064	−0.0278642
$286$	−4100.61	−0.847812
$287$	3044.35	0.626141
$288$	−288.000	−0.0589256
$289$	−3056.85	−0.622195
$290$	−450.240	−0.0911690
$291$	498.914	0.100505
$292$	369.477	0.0740480
$293$	3001.16	0.598395	0.299198	−	0.954191i	$-0.403281\pi$
$293$	3001.16	0.598395	0.299198	+	0.954191i	$0.403281\pi$
$294$	294.000	0.0583212
$295$	900.879	0.177801
$296$	−40.3584	−0.00792495
$297$	1758.72	0.343608
$298$	4675.52	0.908878
$299$	−2921.69	−0.565103
$300$	1433.62	0.275900
$301$	−383.233	−0.0733860
$302$	−5363.54	−1.02198
$303$	−4513.75	−0.855803
$304$	−304.000	−0.0573539
$305$	−1015.18	−0.190588
$306$	775.496	0.144876
$307$	1737.38	0.322988	0.161494	−	0.986874i	$-0.448369\pi$
$307$	1737.38	0.322988	0.161494	+	0.986874i	$0.448369\pi$
$308$	1823.86	0.337416
$309$	3437.03	0.632769
$310$	757.756	0.138831
$311$	3951.77	0.720529	0.360264	−	0.932850i	$-0.382686\pi$
$311$	3951.77	0.720529	0.360264	+	0.932850i	$0.382686\pi$
$312$	−755.433	−0.137077
$313$	2496.86	0.450898	0.225449	−	0.974255i	$-0.427615\pi$
$313$	2496.86	0.450898	0.225449	+	0.974255i	$0.427615\pi$
$314$	−1524.00	−0.273899
$315$	148.176	0.0265041
$316$	2483.49	0.442112
$317$	2515.69	0.445725	0.222863	−	0.974850i	$-0.428460\pi$
$317$	2515.69	0.445725	0.222863	+	0.974850i	$0.428460\pi$
$318$	1050.85	0.185311
$319$	6234.61	1.09427
$320$	−150.528	−0.0262962
$321$	2413.65	0.419679
$322$	1299.50	0.224902
$323$	818.579	0.141012
$324$	324.000	0.0555556
$325$	3760.42	0.641817
$326$	−6904.12	−1.17296
$327$	−6439.37	−1.08898
$328$	3479.26	0.585702
$329$	1544.34	0.258792
$330$	919.229	0.153339
$331$	980.008	0.162738	0.0813688	−	0.996684i	$-0.474071\pi$
$331$	980.008	0.162738	0.0813688	+	0.996684i	$0.474071\pi$
$332$	2078.91	0.343660
$333$	45.4032	0.00747171
$334$	−7179.00	−1.17610
$335$	2327.81	0.379647
$336$	336.000	0.0545545
$337$	−8806.36	−1.42348	−0.711740	−	0.702443i	$-0.752093\pi$
$337$	−8806.36	−1.42348	−0.711740	+	0.702443i	$0.752093\pi$
$338$	2412.48	0.388229
$339$	1368.72	0.219288
$340$	405.327	0.0646528
$341$	−10492.9	−1.66634
$342$	342.000	0.0540738
$343$	−343.000	−0.0539949
$344$	−437.980	−0.0686463
$345$	654.953	0.102207
$346$	−3591.53	−0.558039
$347$	−4506.05	−0.697110	−0.348555	−	0.937288i	$-0.613328\pi$
$347$	−4506.05	−0.697110	−0.348555	+	0.937288i	$0.613328\pi$
$348$	1148.57	0.176924
$349$	−11683.9	−1.79204	−0.896021	−	0.444011i	$-0.853555\pi$
$349$	−11683.9	−1.79204	−0.896021	+	0.444011i	$0.853555\pi$
$350$	−1672.55	−0.255433
$351$	849.862	0.129237
$352$	2084.41	0.315624
$353$	10542.7	1.58960	0.794802	−	0.606868i	$-0.207574\pi$
$353$	10542.7	1.58960	0.794802	+	0.606868i	$0.207574\pi$
$354$	−2298.15	−0.345044
$355$	1073.38	0.160476
$356$	4199.54	0.625212
$357$	−904.745	−0.134129
$358$	4952.58	0.731151
$359$	−365.565	−0.0537432	−0.0268716	−	0.999639i	$-0.508555\pi$
$359$	−365.565	−0.0537432	−0.0268716	+	0.999639i	$0.508555\pi$
$360$	169.345	0.0247923
$361$	361.000	0.0526316
$362$	−4804.18	−0.697519
$363$	−8735.85	−1.26312
$364$	881.339	0.126909
$365$	−217.253	−0.0311550
$366$	2589.75	0.369859
$367$	−3287.08	−0.467532	−0.233766	−	0.972293i	$-0.575105\pi$
$367$	−3287.08	−0.467532	−0.233766	+	0.972293i	$0.575105\pi$
$368$	1485.15	0.210377
$369$	−3914.17	−0.552205
$370$	23.7308	0.00333434
$371$	−1225.99	−0.171565
$372$	−1933.05	−0.269419
$373$	8899.17	1.23534	0.617669	−	0.786438i	$-0.288077\pi$
$373$	8899.17	1.23534	0.617669	+	0.786438i	$0.288077\pi$
$374$	−5612.69	−0.776003
$375$	−1724.97	−0.237539
$376$	1764.96	0.242077
$377$	3012.73	0.411574
$378$	−378.000	−0.0514344
$379$	−11650.9	−1.57907	−0.789536	−	0.613705i	$-0.789679\pi$
$379$	−11650.9	−1.57907	−0.789536	+	0.613705i	$0.789679\pi$
$380$	178.753	0.0241311
$381$	−3772.32	−0.507249
$382$	3271.14	0.438131
$383$	3378.50	0.450739	0.225370	−	0.974273i	$-0.427641\pi$
$383$	3378.50	0.450739	0.225370	+	0.974273i	$0.427641\pi$
$384$	384.000	0.0510310
$385$	−1072.43	−0.141964
$386$	−8865.58	−1.16903
$387$	492.728	0.0647203
$388$	−665.219	−0.0870396
$389$	−4069.59	−0.530428	−0.265214	−	0.964190i	$-0.585443\pi$
$389$	−4069.59	−0.530428	−0.265214	+	0.964190i	$0.585443\pi$
$390$	444.196	0.0576737
$391$	−3999.05	−0.517240
$392$	−392.000	−0.0505076
$393$	6988.39	0.896991
$394$	−5203.30	−0.665327
$395$	−1460.30	−0.186014
$396$	−2344.97	−0.297573
$397$	9273.75	1.17238	0.586191	−	0.810173i	$-0.300627\pi$
$397$	9273.75	1.17238	0.586191	+	0.810173i	$0.300627\pi$
$398$	−3951.42	−0.497655
$399$	−399.000	−0.0500626
$400$	−1911.49	−0.238936
$401$	5544.49	0.690471	0.345235	−	0.938516i	$-0.387799\pi$
$401$	5544.49	0.690471	0.345235	+	0.938516i	$0.387799\pi$
$402$	−5938.27	−0.736751
$403$	−5070.44	−0.626741
$404$	6018.34	0.741147
$405$	−190.513	−0.0233744
$406$	−1340.00	−0.163800
$407$	−328.608	−0.0400209
$408$	−1033.99	−0.125467
$409$	−4863.21	−0.587947	−0.293973	−	0.955814i	$-0.594978\pi$
$409$	−4863.21	−0.587947	−0.293973	+	0.955814i	$0.594978\pi$
$410$	−2045.81	−0.246428
$411$	−5185.77	−0.622372
$412$	−4582.70	−0.547994
$413$	2681.18	0.319449
$414$	−1670.79	−0.198345
$415$	−1222.40	−0.144591
$416$	1007.24	0.118712
$417$	2073.61	0.243514
$418$	−2475.24	−0.289636
$419$	−2633.77	−0.307084	−0.153542	−	0.988142i	$-0.549068\pi$
$419$	−2633.77	−0.307084	−0.153542	+	0.988142i	$0.549068\pi$
$420$	−197.569	−0.0229532
$421$	12763.2	1.47753	0.738767	−	0.673961i	$-0.235409\pi$
$421$	12763.2	1.47753	0.738767	+	0.673961i	$0.235409\pi$
$422$	7471.10	0.861819
$423$	−1985.59	−0.228233
$424$	−1401.14	−0.160484
$425$	5147.06	0.587456
$426$	−2738.20	−0.311423
$427$	−3021.37	−0.342423
$428$	−3218.20	−0.363452
$429$	−6150.92	−0.692236
$430$	257.533	0.0288822
$431$	−3351.91	−0.374607	−0.187304	−	0.982302i	$-0.559975\pi$
$431$	−3351.91	−0.374607	−0.187304	+	0.982302i	$0.559975\pi$
$432$	−432.000	−0.0481125
$433$	−3364.43	−0.373405	−0.186702	−	0.982417i	$-0.559780\pi$
$433$	−3364.43	−0.373405	−0.186702	+	0.982417i	$0.559780\pi$
$434$	2255.22	0.249433
$435$	−675.360	−0.0744392
$436$	8585.83	0.943089
$437$	−1763.61	−0.193055
$438$	554.216	0.0604600
$439$	−4991.62	−0.542682	−0.271341	−	0.962483i	$-0.587467\pi$
$439$	−4991.62	−0.542682	−0.271341	+	0.962483i	$0.587467\pi$
$440$	−1225.64	−0.132796
$441$	441.000	0.0476190
$442$	−2712.20	−0.291869
$443$	−13437.4	−1.44115	−0.720574	−	0.693378i	$-0.756122\pi$
$443$	−13437.4	−1.44115	−0.720574	+	0.693378i	$0.756122\pi$
$444$	−60.5376	−0.00647069
$445$	−2469.34	−0.263051
$446$	−11242.2	−1.19357
$447$	7013.28	0.742096
$448$	−448.000	−0.0472456
$449$	3090.05	0.324785	0.162392	−	0.986726i	$-0.448079\pi$
$449$	3090.05	0.324785	0.162392	+	0.986726i	$0.448079\pi$
$450$	2150.43	0.225271
$451$	28329.0	2.95778
$452$	−1824.96	−0.189909
$453$	−8045.30	−0.834440
$454$	177.824	0.0183826
$455$	−518.229	−0.0533955
$456$	−456.000	−0.0468293
$457$	9282.31	0.950127	0.475063	−	0.879952i	$-0.342425\pi$
$457$	9282.31	0.950127	0.475063	+	0.879952i	$0.342425\pi$
$458$	7183.23	0.732861
$459$	1163.24	0.118291
$460$	−873.270	−0.0885140
$461$	−14936.7	−1.50904	−0.754522	−	0.656275i	$-0.772131\pi$
$461$	−14936.7	−1.50904	−0.754522	+	0.656275i	$0.772131\pi$
$462$	2735.79	0.275499
$463$	7024.39	0.705078	0.352539	−	0.935797i	$-0.385318\pi$
$463$	7024.39	0.705078	0.352539	+	0.935797i	$0.385318\pi$
$464$	−1531.42	−0.153221
$465$	1136.63	0.113355
$466$	3566.67	0.354556
$467$	800.205	0.0792914	0.0396457	−	0.999214i	$-0.487377\pi$
$467$	800.205	0.0792914	0.0396457	+	0.999214i	$0.487377\pi$
$468$	−1133.15	−0.111923
$469$	6927.98	0.682099
$470$	−1037.80	−0.101852
$471$	−2286.00	−0.223638
$472$	3064.21	0.298817
$473$	−3566.14	−0.346663
$474$	3725.24	0.360983
$475$	2269.89	0.219263
$476$	1206.33	0.116159
$477$	1576.28	0.151306
$478$	10561.3	1.01059
$479$	3573.45	0.340867	0.170433	−	0.985369i	$-0.445483\pi$
$479$	3573.45	0.340867	0.170433	+	0.985369i	$0.445483\pi$
$480$	−225.793	−0.0214708
$481$	−158.792	−0.0150526
$482$	−14222.4	−1.34401
$483$	1949.26	0.183632
$484$	11647.8	1.09389
$485$	391.150	0.0366210
$486$	486.000	0.0453609
$487$	−13085.1	−1.21755	−0.608773	−	0.793345i	$-0.708338\pi$
$487$	−13085.1	−1.21755	−0.608773	+	0.793345i	$0.708338\pi$
$488$	−3453.00	−0.320307
$489$	−10356.2	−0.957715
$490$	230.497	0.0212506
$491$	−9072.15	−0.833850	−0.416925	−	0.908941i	$-0.636892\pi$
$491$	−9072.15	−0.833850	−0.416925	+	0.908941i	$0.636892\pi$
$492$	5218.89	0.478223
$493$	4123.66	0.376714
$494$	−1196.10	−0.108938
$495$	1378.84	0.125201
$496$	2577.39	0.233323
$497$	3194.56	0.288321
$498$	3118.37	0.280597
$499$	−1920.38	−0.172281	−0.0861403	−	0.996283i	$-0.527453\pi$
$499$	−1920.38	−0.172281	−0.0861403	+	0.996283i	$0.527453\pi$
$500$	2299.96	0.205715
$501$	−10768.5	−0.960281
$502$	5045.89	0.448624
$503$	−14852.4	−1.31657	−0.658285	−	0.752769i	$-0.728718\pi$
$503$	−14852.4	−1.31657	−0.658285	+	0.752769i	$0.728718\pi$
$504$	504.000	0.0445435
$505$	−3538.79	−0.311830
$506$	12092.4	1.06240
$507$	3618.71	0.316988
$508$	5029.76	0.439291
$509$	14779.7	1.28703	0.643514	−	0.765434i	$-0.277476\pi$
$509$	14779.7	1.28703	0.643514	+	0.765434i	$0.277476\pi$
$510$	607.991	0.0527888
$511$	−646.585	−0.0559751
$512$	−512.000	−0.0441942
$513$	513.000	0.0441511
$514$	4253.22	0.364983
$515$	2694.64	0.230563
$516$	−656.971	−0.0560495
$517$	14370.8	1.22249
$518$	70.6272	0.00599070
$519$	−5387.29	−0.455637
$520$	−592.261	−0.0499469
$521$	6548.34	0.550648	0.275324	−	0.961351i	$-0.411215\pi$
$521$	6548.34	0.550648	0.275324	+	0.961351i	$0.411215\pi$
$522$	1722.85	0.144458
$523$	783.385	0.0654972	0.0327486	−	0.999464i	$-0.489574\pi$
$523$	783.385	0.0654972	0.0327486	+	0.999464i	$0.489574\pi$
$524$	−9317.85	−0.776817
$525$	−2508.83	−0.208561
$526$	5419.18	0.449216
$527$	−6940.14	−0.573657
$528$	3126.62	0.257706
$529$	−3551.12	−0.291865
$530$	823.871	0.0675220
$531$	−3447.23	−0.281727
$532$	532.000	0.0433555
$533$	13689.3	1.11248
$534$	6299.32	0.510483
$535$	1892.31	0.152919
$536$	7917.69	0.638045
$537$	7428.88	0.596983
$538$	2127.22	0.170466
$539$	−3191.76	−0.255063
$540$	254.017	0.0202429
$541$	11120.2	0.883720	0.441860	−	0.897084i	$-0.354319\pi$
$541$	11120.2	0.883720	0.441860	+	0.897084i	$0.354319\pi$
$542$	−4831.67	−0.382912
$543$	−7206.26	−0.569522
$544$	1378.66	0.108657
$545$	−5048.48	−0.396795
$546$	1322.01	0.103620
$547$	−16006.1	−1.25114	−0.625569	−	0.780169i	$-0.715133\pi$
$547$	−16006.1	−1.25114	−0.625569	+	0.780169i	$0.715133\pi$
$548$	6914.36	0.538990
$549$	3884.62	0.301988
$550$	−15563.8	−1.20662
$551$	1818.57	0.140605
$552$	2227.72	0.171772
$553$	−4346.11	−0.334205
$554$	5847.82	0.448466
$555$	35.5962	0.00272248
$556$	−2764.82	−0.210889
$557$	−13422.4	−1.02105	−0.510524	−	0.859863i	$-0.670549\pi$
$557$	−13422.4	−1.02105	−0.510524	+	0.859863i	$0.670549\pi$
$558$	−2899.57	−0.219979
$559$	−1723.25	−0.130386
$560$	263.425	0.0198781
$561$	−8419.03	−0.633604
$562$	14129.0	1.06049
$563$	18317.0	1.37117	0.685586	−	0.727991i	$-0.259546\pi$
$563$	18317.0	1.37117	0.685586	+	0.727991i	$0.259546\pi$
$564$	2647.45	0.197655
$565$	1073.08	0.0799022
$566$	14398.6	1.06929
$567$	−567.000	−0.0419961
$568$	3650.93	0.269700
$569$	10513.3	0.774589	0.387295	−	0.921956i	$-0.373410\pi$
$569$	10513.3	0.774589	0.387295	+	0.921956i	$0.373410\pi$
$570$	268.129	0.0197029
$571$	−12420.2	−0.910277	−0.455138	−	0.890421i	$-0.650410\pi$
$571$	−12420.2	−0.910277	−0.455138	+	0.890421i	$0.650410\pi$
$572$	8201.22	0.599494
$573$	4906.70	0.357732
$574$	−6088.71	−0.442749
$575$	−11089.2	−0.804266
$576$	576.000	0.0416667
$577$	7149.46	0.515833	0.257917	−	0.966167i	$-0.416964\pi$
$577$	7149.46	0.515833	0.257917	+	0.966167i	$0.416964\pi$
$578$	6113.69	0.439958
$579$	−13298.4	−0.954510
$580$	900.480	0.0644662
$581$	−3638.10	−0.259782
$582$	−997.828	−0.0710675
$583$	−11408.4	−0.810441
$584$	−738.955	−0.0523599
$585$	666.294	0.0470904
$586$	−6002.32	−0.423129
$587$	25413.4	1.78692	0.893461	−	0.449140i	$-0.148270\pi$
$587$	25413.4	1.78692	0.893461	+	0.449140i	$0.148270\pi$
$588$	−588.000	−0.0412393
$589$	−3060.66	−0.214112
$590$	−1801.76	−0.125724
$591$	−7804.96	−0.543237
$592$	80.7168	0.00560379
$593$	11905.5	0.824452	0.412226	−	0.911082i	$-0.364751\pi$
$593$	11905.5	0.824452	0.412226	+	0.911082i	$0.364751\pi$
$594$	−3517.45	−0.242967
$595$	−709.323	−0.0488729
$596$	−9351.04	−0.642674
$597$	−5927.13	−0.406334
$598$	5843.39	0.399588
$599$	22430.9	1.53005	0.765027	−	0.643998i	$-0.222725\pi$
$599$	22430.9	1.53005	0.765027	+	0.643998i	$0.222725\pi$
$600$	−2867.23	−0.195091
$601$	−9645.59	−0.654662	−0.327331	−	0.944910i	$-0.606149\pi$
$601$	−9645.59	−0.654662	−0.327331	+	0.944910i	$0.606149\pi$
$602$	766.466	0.0518917
$603$	−8907.40	−0.601555
$604$	10727.1	0.722646
$605$	−6848.92	−0.460245
$606$	9027.51	0.605144
$607$	−2535.39	−0.169536	−0.0847679	−	0.996401i	$-0.527015\pi$
$607$	−2535.39	−0.169536	−0.0847679	+	0.996401i	$0.527015\pi$
$608$	608.000	0.0405554
$609$	−2009.99	−0.133742
$610$	2030.37	0.134766
$611$	6944.34	0.459800
$612$	−1550.99	−0.102443
$613$	17163.0	1.13084	0.565421	−	0.824802i	$-0.308714\pi$
$613$	17163.0	1.13084	0.565421	+	0.824802i	$0.308714\pi$
$614$	−3474.75	−0.228387
$615$	−3068.72	−0.201208
$616$	−3647.72	−0.238589
$617$	2852.64	0.186131	0.0930656	−	0.995660i	$-0.470333\pi$
$617$	2852.64	0.186131	0.0930656	+	0.995660i	$0.470333\pi$
$618$	−6874.05	−0.447435
$619$	−15912.1	−1.03322	−0.516608	−	0.856222i	$-0.672805\pi$
$619$	−15912.1	−1.03322	−0.516608	+	0.856222i	$0.672805\pi$
$620$	−1515.51	−0.0981685
$621$	−2506.19	−0.161948
$622$	−7903.54	−0.509491
$623$	−7349.20	−0.472616
$624$	1510.87	0.0969280
$625$	13581.1	0.869192
$626$	−4993.73	−0.318833
$627$	−3712.86	−0.236487
$628$	3048.00	0.193676
$629$	−217.346	−0.0137777
$630$	−296.353	−0.0187412
$631$	−6717.70	−0.423815	−0.211908	−	0.977290i	$-0.567968\pi$
$631$	−6717.70	−0.423815	−0.211908	+	0.977290i	$0.567968\pi$
$632$	−4966.98	−0.312620
$633$	11206.7	0.703672
$634$	−5031.37	−0.315175
$635$	−2957.51	−0.184827
$636$	−2101.70	−0.131035
$637$	−1542.34	−0.0959338
$638$	−12469.2	−0.773764
$639$	−4107.30	−0.254276
$640$	301.057	0.0185943
$641$	23563.0	1.45192	0.725961	−	0.687736i	$-0.241395\pi$
$641$	23563.0	1.45192	0.725961	+	0.687736i	$0.241395\pi$
$642$	−4827.30	−0.296758
$643$	12722.7	0.780303	0.390152	−	0.920751i	$-0.372423\pi$
$643$	12722.7	0.780303	0.390152	+	0.920751i	$0.372423\pi$
$644$	−2599.01	−0.159030
$645$	386.300	0.0235822
$646$	−1637.16	−0.0997107
$647$	20322.2	1.23485	0.617424	−	0.786631i	$-0.288176\pi$
$647$	20322.2	1.23485	0.617424	+	0.786631i	$0.288176\pi$
$648$	−648.000	−0.0392837
$649$	24949.5	1.50902
$650$	−7520.84	−0.453833
$651$	3382.83	0.203661
$652$	13808.2	0.829406
$653$	−19287.3	−1.15585	−0.577924	−	0.816091i	$-0.696137\pi$
$653$	−19287.3	−1.15585	−0.577924	+	0.816091i	$0.696137\pi$
$654$	12878.7	0.770029
$655$	5478.91	0.326838
$656$	−6958.52	−0.414154
$657$	831.324	0.0493654
$658$	−3088.69	−0.182993
$659$	−7186.76	−0.424820	−0.212410	−	0.977181i	$-0.568131\pi$
$659$	−7186.76	−0.424820	−0.212410	+	0.977181i	$0.568131\pi$
$660$	−1838.46	−0.108427
$661$	−23799.5	−1.40044	−0.700220	−	0.713927i	$-0.746915\pi$
$661$	−23799.5	−1.40044	−0.700220	+	0.713927i	$0.746915\pi$
$662$	−1960.02	−0.115073
$663$	−4068.30	−0.238310
$664$	−4157.82	−0.243004
$665$	−312.817	−0.0182414
$666$	−90.8064	−0.00528330
$667$	−8884.35	−0.515747
$668$	14358.0	0.831628
$669$	−16863.3	−0.974546
$670$	−4655.62	−0.268451
$671$	−28115.1	−1.61755
$672$	−672.000	−0.0385758
$673$	−6426.09	−0.368065	−0.184032	−	0.982920i	$-0.558915\pi$
$673$	−6426.09	−0.368065	−0.184032	+	0.982920i	$0.558915\pi$
$674$	17612.7	1.00655
$675$	3225.64	0.183933
$676$	−4824.95	−0.274519
$677$	5295.09	0.300601	0.150301	−	0.988640i	$-0.451976\pi$
$677$	5295.09	0.300601	0.150301	+	0.988640i	$0.451976\pi$
$678$	−2737.44	−0.155060
$679$	1164.13	0.0657958
$680$	−810.654	−0.0457164
$681$	266.737	0.0150094
$682$	20985.8	1.17828
$683$	27838.6	1.55961	0.779807	−	0.626020i	$-0.215317\pi$
$683$	27838.6	1.55961	0.779807	+	0.626020i	$0.215317\pi$
$684$	−684.000	−0.0382360
$685$	−4065.65	−0.226775
$686$	686.000	0.0381802
$687$	10774.9	0.598379
$688$	875.961	0.0485403
$689$	−5512.84	−0.304822
$690$	−1309.91	−0.0722714
$691$	−19503.3	−1.07372	−0.536859	−	0.843672i	$-0.680389\pi$
$691$	−19503.3	−1.07372	−0.536859	+	0.843672i	$0.680389\pi$
$692$	7183.05	0.394593
$693$	4103.69	0.224944
$694$	9012.10	0.492932
$695$	1625.72	0.0887296
$696$	−2297.14	−0.125104
$697$	18737.2	1.01825
$698$	23367.7	1.26717
$699$	5350.01	0.289494
$700$	3345.11	0.180619
$701$	6387.15	0.344136	0.172068	−	0.985085i	$-0.444955\pi$
$701$	6387.15	0.344136	0.172068	+	0.985085i	$0.444955\pi$
$702$	−1699.72	−0.0913846
$703$	−95.8512	−0.00514239
$704$	−4168.83	−0.223180
$705$	−1556.70	−0.0831615
$706$	−21085.4	−1.12402
$707$	−10532.1	−0.560255
$708$	4596.31	0.243983
$709$	24946.9	1.32144	0.660720	−	0.750632i	$-0.270251\pi$
$709$	24946.9	1.32144	0.660720	+	0.750632i	$0.270251\pi$
$710$	−2146.75	−0.113474
$711$	5587.86	0.294741
$712$	−8399.09	−0.442091
$713$	14952.4	0.785374
$714$	1809.49	0.0948438
$715$	−4822.33	−0.252231
$716$	−9905.17	−0.517002
$717$	15841.9	0.825142
$718$	731.130	0.0380022
$719$	32315.2	1.67615	0.838077	−	0.545552i	$-0.183680\pi$
$719$	32315.2	1.67615	0.838077	+	0.545552i	$0.183680\pi$
$720$	−338.689	−0.0175308
$721$	8019.73	0.414244
$722$	−722.000	−0.0372161
$723$	−21333.5	−1.09738
$724$	9608.35	0.493220
$725$	11434.8	0.585761
$726$	17471.7	0.893162
$727$	21195.9	1.08131	0.540656	−	0.841244i	$-0.318176\pi$
$727$	21195.9	1.08131	0.540656	+	0.841244i	$0.318176\pi$
$728$	−1762.68	−0.0897379
$729$	729.000	0.0370370
$730$	434.507	0.0220299
$731$	−2358.70	−0.119343
$732$	−5179.50	−0.261530
$733$	−11952.5	−0.602287	−0.301144	−	0.953579i	$-0.597368\pi$
$733$	−11952.5	−0.602287	−0.301144	+	0.953579i	$0.597368\pi$
$734$	6574.16	0.330595
$735$	345.745	0.0173510
$736$	−2970.30	−0.148759
$737$	64467.7	3.22212
$738$	7828.34	0.390468
$739$	−12355.2	−0.615010	−0.307505	−	0.951546i	$-0.599494\pi$
$739$	−12355.2	−0.615010	−0.307505	+	0.951546i	$0.599494\pi$
$740$	−47.4616	−0.00235774
$741$	−1794.15	−0.0889472
$742$	2451.99	0.121314
$743$	545.630	0.0269411	0.0134705	−	0.999909i	$-0.495712\pi$
$743$	545.630	0.0269411	0.0134705	+	0.999909i	$0.495712\pi$
$744$	3866.09	0.190508
$745$	5498.43	0.270399
$746$	−17798.3	−0.873516
$747$	4677.55	0.229107
$748$	11225.4	0.548717
$749$	5631.85	0.274744
$750$	3449.94	0.167966
$751$	−24657.5	−1.19809	−0.599044	−	0.800716i	$-0.704453\pi$
$751$	−24657.5	−1.19809	−0.599044	+	0.800716i	$0.704453\pi$
$752$	−3529.93	−0.171175
$753$	7568.84	0.366300
$754$	−6025.46	−0.291027
$755$	−6307.54	−0.304046
$756$	756.000	0.0363696
$757$	−9802.82	−0.470660	−0.235330	−	0.971916i	$-0.575617\pi$
$757$	−9802.82	−0.470660	−0.235330	+	0.971916i	$0.575617\pi$
$758$	23301.9	1.11657
$759$	18138.7	0.867446
$760$	−357.505	−0.0170633
$761$	−38995.6	−1.85754	−0.928770	−	0.370656i	$-0.879133\pi$
$761$	−38995.6	−1.85754	−0.928770	+	0.370656i	$0.879133\pi$
$762$	7544.65	0.358679
$763$	−15025.2	−0.712908
$764$	−6542.27	−0.309805
$765$	911.986	0.0431019
$766$	−6756.99	−0.318721
$767$	12056.3	0.567570
$768$	−768.000	−0.0360844
$769$	11656.9	0.546631	0.273315	−	0.961924i	$-0.411880\pi$
$769$	11656.9	0.546631	0.273315	+	0.961924i	$0.411880\pi$
$770$	2144.87	0.100384
$771$	6379.83	0.298008
$772$	17731.2	0.826630
$773$	18534.7	0.862414	0.431207	−	0.902253i	$-0.358088\pi$
$773$	18534.7	0.862414	0.431207	+	0.902253i	$0.358088\pi$
$774$	−985.456	−0.0457642
$775$	−19244.8	−0.891990
$776$	1330.44	0.0615463
$777$	105.941	0.00489139
$778$	8139.19	0.375069
$779$	8263.25	0.380053
$780$	−888.392	−0.0407815
$781$	29726.7	1.36198
$782$	7998.10	0.365744
$783$	2584.28	0.117950
$784$	784.000	0.0357143
$785$	−1792.23	−0.0814872
$786$	−13976.8	−0.634268
$787$	21029.9	0.952521	0.476261	−	0.879304i	$-0.341992\pi$
$787$	21029.9	0.952521	0.476261	+	0.879304i	$0.341992\pi$
$788$	10406.6	0.470457
$789$	8128.77	0.366783
$790$	2920.60	0.131532
$791$	3193.68	0.143558
$792$	4689.93	0.210416
$793$	−13586.0	−0.608389
$794$	−18547.5	−0.829000
$795$	1235.81	0.0551315
$796$	7902.84	0.351895
$797$	−16890.1	−0.750661	−0.375330	−	0.926891i	$-0.622471\pi$
$797$	−16890.1	−0.750661	−0.375330	+	0.926891i	$0.622471\pi$
$798$	798.000	0.0353996
$799$	9505.02	0.420856
$800$	3822.98	0.168953
$801$	9448.97	0.416808
$802$	−11089.0	−0.488237
$803$	−6016.75	−0.264416
$804$	11876.5	0.520962
$805$	1528.22	0.0669103
$806$	10140.9	0.443173
$807$	3190.83	0.139185
$808$	−12036.7	−0.524070
$809$	−26270.6	−1.14169	−0.570843	−	0.821059i	$-0.693384\pi$
$809$	−26270.6	−1.14169	−0.570843	+	0.821059i	$0.693384\pi$
$810$	381.025	0.0165282
$811$	−886.827	−0.0383979	−0.0191990	−	0.999816i	$-0.506112\pi$
$811$	−886.827	−0.0383979	−0.0191990	+	0.999816i	$0.506112\pi$
$812$	2679.99	0.115824
$813$	−7247.51	−0.312646
$814$	657.216	0.0282990
$815$	−8119.27	−0.348964
$816$	2067.99	0.0887183
$817$	−1040.20	−0.0445436
$818$	9726.42	0.415741
$819$	1983.01	0.0846057
$820$	4091.63	0.174251
$821$	−16810.5	−0.714605	−0.357303	−	0.933989i	$-0.616304\pi$
$821$	−16810.5	−0.714605	−0.357303	+	0.933989i	$0.616304\pi$
$822$	10371.5	0.440084
$823$	−40374.7	−1.71005	−0.855027	−	0.518583i	$-0.826460\pi$
$823$	−40374.7	−1.71005	−0.855027	+	0.518583i	$0.826460\pi$
$824$	9165.40	0.387490
$825$	−23345.7	−0.985204
$826$	−5362.36	−0.225884
$827$	−21653.2	−0.910468	−0.455234	−	0.890372i	$-0.650444\pi$
$827$	−21653.2	−0.910468	−0.455234	+	0.890372i	$0.650444\pi$
$828$	3341.58	0.140251
$829$	9284.49	0.388979	0.194490	−	0.980905i	$-0.437695\pi$
$829$	9284.49	0.388979	0.194490	+	0.980905i	$0.437695\pi$
$830$	2444.81	0.102242
$831$	8771.73	0.366171
$832$	−2014.49	−0.0839421
$833$	−2111.07	−0.0878083
$834$	−4147.23	−0.172190
$835$	−8442.53	−0.349899
$836$	4950.48	0.204804
$837$	−4349.35	−0.179612
$838$	5267.54	0.217141
$839$	30110.7	1.23902	0.619509	−	0.784989i	$-0.287332\pi$
$839$	30110.7	1.23902	0.619509	+	0.784989i	$0.287332\pi$
$840$	395.137	0.0162304
$841$	−15227.8	−0.624373
$842$	−25526.4	−1.04477
$843$	21193.6	0.865890
$844$	−14942.2	−0.609398
$845$	2837.08	0.115501
$846$	3971.17	0.161385
$847$	−20383.6	−0.826907
$848$	2802.27	0.113479
$849$	21598.0	0.873074
$850$	−10294.1	−0.415394
$851$	468.268	0.0188625
$852$	5476.40	0.220209
$853$	−28220.7	−1.13278	−0.566389	−	0.824138i	$-0.691660\pi$
$853$	−28220.7	−1.13278	−0.566389	+	0.824138i	$0.691660\pi$
$854$	6042.75	0.242129
$855$	402.193	0.0160874
$856$	6436.40	0.257000
$857$	37317.9	1.48746	0.743731	−	0.668479i	$-0.233054\pi$
$857$	37317.9	1.48746	0.743731	+	0.668479i	$0.233054\pi$
$858$	12301.8	0.489485
$859$	−15643.5	−0.621363	−0.310681	−	0.950514i	$-0.600557\pi$
$859$	−15643.5	−0.621363	−0.310681	+	0.950514i	$0.600557\pi$
$860$	−515.067	−0.0204228
$861$	−9133.06	−0.361503
$862$	6703.82	0.264887
$863$	−17616.3	−0.694860	−0.347430	−	0.937706i	$-0.612946\pi$
$863$	−17616.3	−0.694860	−0.347430	+	0.937706i	$0.612946\pi$
$864$	864.000	0.0340207
$865$	−4223.65	−0.166021
$866$	6728.87	0.264037
$867$	9170.54	0.359225
$868$	−4510.44	−0.176376
$869$	−40442.4	−1.57873
$870$	1350.72	0.0526364
$871$	31152.5	1.21190
$872$	−17171.7	−0.666864
$873$	−1496.74	−0.0580264
$874$	3527.23	0.136511
$875$	−4024.93	−0.155506
$876$	−1108.43	−0.0427517
$877$	12573.4	0.484121	0.242060	−	0.970261i	$-0.422177\pi$
$877$	12573.4	0.484121	0.242060	+	0.970261i	$0.422177\pi$
$878$	9983.25	0.383734
$879$	−9003.49	−0.345484
$880$	2451.28	0.0939006
$881$	−9079.75	−0.347224	−0.173612	−	0.984814i	$-0.555544\pi$
$881$	−9079.75	−0.347224	−0.173612	+	0.984814i	$0.555544\pi$
$882$	−882.000	−0.0336718
$883$	30666.4	1.16875	0.584376	−	0.811483i	$-0.301340\pi$
$883$	30666.4	1.16875	0.584376	+	0.811483i	$0.301340\pi$
$884$	5424.40	0.206383
$885$	−2702.64	−0.102653
$886$	26874.7	1.01905
$887$	24420.8	0.924431	0.462216	−	0.886768i	$-0.347055\pi$
$887$	24420.8	0.924431	0.462216	+	0.886768i	$0.347055\pi$
$888$	121.075	0.00457547
$889$	−8802.09	−0.332073
$890$	4938.68	0.186005
$891$	−5276.17	−0.198382
$892$	22484.4	0.843982
$893$	4191.79	0.157081
$894$	−14026.6	−0.524741
$895$	5824.26	0.217523
$896$	896.000	0.0334077
$897$	8765.08	0.326263
$898$	−6180.09	−0.229657
$899$	−15418.3	−0.572001
$900$	−4300.85	−0.159291
$901$	−7545.67	−0.279004
$902$	−56658.0	−2.09147
$903$	1149.70	0.0423694
$904$	3649.92	0.134286
$905$	−5649.73	−0.207517
$906$	16090.6	0.590038
$907$	−23548.1	−0.862075	−0.431037	−	0.902334i	$-0.641852\pi$
$907$	−23548.1	−0.862075	−0.431037	+	0.902334i	$0.641852\pi$
$908$	−355.649	−0.0129985
$909$	13541.3	0.494098
$910$	1036.46	0.0377563
$911$	−26956.7	−0.980368	−0.490184	−	0.871619i	$-0.663070\pi$
$911$	−26956.7	−0.980368	−0.490184	+	0.871619i	$0.663070\pi$
$912$	912.000	0.0331133
$913$	−33854.0	−1.22717
$914$	−18564.6	−0.671841
$915$	3045.55	0.110036
$916$	−14366.5	−0.518211
$917$	16306.2	0.587218
$918$	−2326.49	−0.0836444
$919$	−8314.91	−0.298459	−0.149229	−	0.988803i	$-0.547679\pi$
$919$	−8314.91	−0.298459	−0.149229	+	0.988803i	$0.547679\pi$
$920$	1746.54	0.0625888
$921$	−5212.13	−0.186477
$922$	29873.3	1.06706
$923$	14364.8	0.512266
$924$	−5471.59	−0.194807
$925$	−602.693	−0.0214231
$926$	−14048.8	−0.498566
$927$	−10311.1	−0.365329
$928$	3062.85	0.108344
$929$	−716.122	−0.0252908	−0.0126454	−	0.999920i	$-0.504025\pi$
$929$	−716.122	−0.0252908	−0.0126454	+	0.999920i	$0.504025\pi$
$930$	−2273.27	−0.0801542
$931$	−931.000	−0.0327737
$932$	−7133.35	−0.250709
$933$	−11855.3	−0.415997
$934$	−1600.41	−0.0560675
$935$	−6600.54	−0.230867
$936$	2266.30	0.0791414
$937$	−29545.2	−1.03010	−0.515048	−	0.857161i	$-0.672226\pi$
$937$	−29545.2	−1.03010	−0.515048	+	0.857161i	$0.672226\pi$
$938$	−13856.0	−0.482317
$939$	−7490.59	−0.260326
$940$	2075.61	0.0720200
$941$	−12021.2	−0.416452	−0.208226	−	0.978081i	$-0.566769\pi$
$941$	−12021.2	−0.416452	−0.208226	+	0.978081i	$0.566769\pi$
$942$	4572.00	0.158136
$943$	−40368.9	−1.39405
$944$	−6128.41	−0.211295
$945$	−444.529	−0.0153022
$946$	7132.28	0.245127
$947$	32892.4	1.12868	0.564340	−	0.825542i	$-0.309131\pi$
$947$	32892.4	1.12868	0.564340	+	0.825542i	$0.309131\pi$
$948$	−7450.48	−0.255253
$949$	−2907.45	−0.0994519
$950$	−4539.79	−0.155042
$951$	−7547.06	−0.257340
$952$	−2412.65	−0.0821372
$953$	−25757.2	−0.875505	−0.437752	−	0.899096i	$-0.644225\pi$
$953$	−25757.2	−0.875505	−0.437752	+	0.899096i	$0.644225\pi$
$954$	−3152.56	−0.106989
$955$	3846.87	0.130347
$956$	−21122.5	−0.714594
$957$	−18703.8	−0.631775
$958$	−7146.91	−0.241029
$959$	−12100.1	−0.407438
$960$	451.585	0.0151821
$961$	−3841.94	−0.128963
$962$	317.584	0.0106438
$963$	−7240.95	−0.242302
$964$	28444.7	0.950355
$965$	−10426.0	−0.347796
$966$	−3898.51	−0.129847
$967$	−18681.0	−0.621243	−0.310621	−	0.950534i	$-0.600537\pi$
$967$	−18681.0	−0.621243	−0.310621	+	0.950534i	$0.600537\pi$
$968$	−23295.6	−0.773501
$969$	−2455.74	−0.0814135
$970$	−782.300	−0.0258950
$971$	20470.4	0.676546	0.338273	−	0.941048i	$-0.390157\pi$
$971$	20470.4	0.676546	0.338273	+	0.941048i	$0.390157\pi$
$972$	−972.000	−0.0320750
$973$	4838.43	0.159417
$974$	26170.3	0.860934
$975$	−11281.3	−0.370553
$976$	6906.00	0.226491
$977$	−57784.2	−1.89220	−0.946100	−	0.323874i	$-0.895015\pi$
$977$	−57784.2	−1.89220	−0.946100	+	0.323874i	$0.895015\pi$
$978$	20712.4	0.677207
$979$	−68387.4	−2.23255
$980$	−460.993	−0.0150264
$981$	19318.1	0.628726
$982$	18144.3	0.589621
$983$	37476.2	1.21598	0.607989	−	0.793945i	$-0.291976\pi$
$983$	37476.2	1.21598	0.607989	+	0.793945i	$0.291976\pi$
$984$	−10437.8	−0.338155
$985$	−6119.10	−0.197940
$986$	−8247.32	−0.266377
$987$	−4633.03	−0.149413
$988$	2392.20	0.0770305
$989$	5081.77	0.163388
$990$	−2757.69	−0.0885304
$991$	−20663.9	−0.662373	−0.331186	−	0.943565i	$-0.607449\pi$
$991$	−20663.9	−0.662373	−0.331186	+	0.943565i	$0.607449\pi$
$992$	−5154.79	−0.164985
$993$	−2940.02	−0.0939566
$994$	−6389.13	−0.203874
$995$	−4646.88	−0.148056
$996$	−6236.73	−0.198412
$997$	41487.0	1.31786	0.658930	−	0.752205i	$-0.271009\pi$
$997$	41487.0	1.31786	0.658930	+	0.752205i	$0.271009\pi$
$998$	3840.76	0.121821
$999$	−136.210	−0.00431380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	798.4.a.k.1.3	✓	4
3.2	odd	2		2394.4.a.x.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.4.a.k.1.3	✓	4	1.1	even	1	trivial
2394.4.a.x.1.2		4	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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.35201 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -2.35201 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +4.70401 q^{10} -65.1379 q^{11} -12.0000 q^{12} -31.4764 q^{13} +14.0000 q^{14} +7.05602 q^{15} +16.0000 q^{16} -43.0831 q^{17} -18.0000 q^{18} -19.0000 q^{19} -9.40803 q^{20} +21.0000 q^{21} +130.276 q^{22} +92.8218 q^{23} +24.0000 q^{24} -119.468 q^{25} +62.9528 q^{26} -27.0000 q^{27} -28.0000 q^{28} -95.7140 q^{29} -14.1120 q^{30} +161.087 q^{31} -32.0000 q^{32} +195.414 q^{33} +86.1662 q^{34} +16.4641 q^{35} +36.0000 q^{36} +5.04480 q^{37} +38.0000 q^{38} +94.4291 q^{39} +18.8161 q^{40} -434.908 q^{41} -42.0000 q^{42} +54.7476 q^{43} -260.552 q^{44} -21.1681 q^{45} -185.644 q^{46} -220.621 q^{47} -48.0000 q^{48} +49.0000 q^{49} +238.936 q^{50} +129.249 q^{51} -125.906 q^{52} +175.142 q^{53} +54.0000 q^{54} +153.205 q^{55} +56.0000 q^{56} +57.0000 q^{57} +191.428 q^{58} -383.026 q^{59} +28.2241 q^{60} +431.625 q^{61} -322.174 q^{62} -63.0000 q^{63} +64.0000 q^{64} +74.0327 q^{65} -390.828 q^{66} -989.711 q^{67} -172.332 q^{68} -278.465 q^{69} -32.9281 q^{70} -456.366 q^{71} -72.0000 q^{72} +92.3693 q^{73} -10.0896 q^{74} +358.404 q^{75} -76.0000 q^{76} +455.965 q^{77} -188.858 q^{78} +620.873 q^{79} -37.6321 q^{80} +81.0000 q^{81} +869.816 q^{82} +519.728 q^{83} +84.0000 q^{84} +101.332 q^{85} -109.495 q^{86} +287.142 q^{87} +521.103 q^{88} +1049.89 q^{89} +42.3361 q^{90} +220.335 q^{91} +371.287 q^{92} -483.261 q^{93} +441.241 q^{94} +44.6881 q^{95} +96.0000 q^{96} -166.305 q^{97} -98.0000 q^{98} -586.241 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} - 66 q^{11} - 48 q^{12} + 10 q^{13} + 56 q^{14} + 64 q^{16} - 8 q^{17} - 72 q^{18} - 76 q^{19} + 84 q^{21} + 132 q^{22} - 250 q^{23} + 96 q^{24} + 120 q^{25} - 20 q^{26} - 108 q^{27} - 112 q^{28} - 66 q^{29} - 52 q^{31} - 128 q^{32} + 198 q^{33} + 16 q^{34} + 144 q^{36} + 288 q^{37} + 152 q^{38} - 30 q^{39} - 196 q^{41} - 168 q^{42} - 280 q^{43} - 264 q^{44} + 500 q^{46} + 498 q^{47} - 192 q^{48} + 196 q^{49} - 240 q^{50} + 24 q^{51} + 40 q^{52} - 250 q^{53} + 216 q^{54} - 1432 q^{55} + 224 q^{56} + 228 q^{57} + 132 q^{58} - 872 q^{59} + 600 q^{61} + 104 q^{62} - 252 q^{63} + 256 q^{64} - 908 q^{65} - 396 q^{66} + 1174 q^{67} - 32 q^{68} + 750 q^{69} - 1766 q^{71} - 288 q^{72} + 556 q^{73} - 576 q^{74} - 360 q^{75} - 304 q^{76} + 462 q^{77} + 60 q^{78} + 26 q^{79} + 324 q^{81} + 392 q^{82} + 674 q^{83} + 336 q^{84} + 2052 q^{85} + 560 q^{86} + 198 q^{87} + 528 q^{88} - 324 q^{89} - 70 q^{91} - 1000 q^{92} + 156 q^{93} - 996 q^{94} + 384 q^{96} + 2902 q^{97} - 392 q^{98} - 594 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9 - 66 * q^11 - 48 * q^12 + 10 * q^13 + 56 * q^14 + 64 * q^16 - 8 * q^17 - 72 * q^18 - 76 * q^19 + 84 * q^21 + 132 * q^22 - 250 * q^23 + 96 * q^24 + 120 * q^25 - 20 * q^26 - 108 * q^27 - 112 * q^28 - 66 * q^29 - 52 * q^31 - 128 * q^32 + 198 * q^33 + 16 * q^34 + 144 * q^36 + 288 * q^37 + 152 * q^38 - 30 * q^39 - 196 * q^41 - 168 * q^42 - 280 * q^43 - 264 * q^44 + 500 * q^46 + 498 * q^47 - 192 * q^48 + 196 * q^49 - 240 * q^50 + 24 * q^51 + 40 * q^52 - 250 * q^53 + 216 * q^54 - 1432 * q^55 + 224 * q^56 + 228 * q^57 + 132 * q^58 - 872 * q^59 + 600 * q^61 + 104 * q^62 - 252 * q^63 + 256 * q^64 - 908 * q^65 - 396 * q^66 + 1174 * q^67 - 32 * q^68 + 750 * q^69 - 1766 * q^71 - 288 * q^72 + 556 * q^73 - 576 * q^74 - 360 * q^75 - 304 * q^76 + 462 * q^77 + 60 * q^78 + 26 * q^79 + 324 * q^81 + 392 * q^82 + 674 * q^83 + 336 * q^84 + 2052 * q^85 + 560 * q^86 + 198 * q^87 + 528 * q^88 - 324 * q^89 - 70 * q^91 - 1000 * q^92 + 156 * q^93 - 996 * q^94 + 384 * q^96 + 2902 * q^97 - 392 * q^98 - 594 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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