LMFDB - Embedded newform 546.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,4,Mod(1,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, 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("546.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	546.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:	$32.2150428631$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 32 $ x^2 - x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.17891$ of defining polynomial
Character	$\chi$	$=$	546.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.17891 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.17891 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.3578 q^{10} +14.7156 q^{11} +12.0000 q^{12} -13.0000 q^{13} +14.0000 q^{14} -15.5367 q^{15} +16.0000 q^{16} -70.5047 q^{17} -18.0000 q^{18} +128.041 q^{19} -20.7156 q^{20} -21.0000 q^{21} -29.4313 q^{22} +83.1149 q^{23} -24.0000 q^{24} -98.1789 q^{25} +26.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} -169.748 q^{29} +31.0735 q^{30} -30.8211 q^{31} -32.0000 q^{32} +44.1469 q^{33} +141.009 q^{34} +36.2524 q^{35} +36.0000 q^{36} -216.230 q^{37} -256.083 q^{38} -39.0000 q^{39} +41.4313 q^{40} -236.377 q^{41} +42.0000 q^{42} +119.831 q^{43} +58.8625 q^{44} -46.6102 q^{45} -166.230 q^{46} -204.821 q^{47} +48.0000 q^{48} +49.0000 q^{49} +196.358 q^{50} -211.514 q^{51} -52.0000 q^{52} -142.591 q^{53} -54.0000 q^{54} -76.2109 q^{55} +56.0000 q^{56} +384.124 q^{57} +339.495 q^{58} -314.927 q^{59} -62.1469 q^{60} -511.175 q^{61} +61.6422 q^{62} -63.0000 q^{63} +64.0000 q^{64} +67.3258 q^{65} -88.2938 q^{66} +375.789 q^{67} -282.019 q^{68} +249.345 q^{69} -72.5047 q^{70} -416.652 q^{71} -72.0000 q^{72} +1082.58 q^{73} +432.460 q^{74} -294.537 q^{75} +512.166 q^{76} -103.009 q^{77} +78.0000 q^{78} -140.591 q^{79} -82.8625 q^{80} +81.0000 q^{81} +472.753 q^{82} +60.3164 q^{83} -84.0000 q^{84} +365.137 q^{85} -239.661 q^{86} -509.243 q^{87} -117.725 q^{88} -1602.60 q^{89} +93.2204 q^{90} +91.0000 q^{91} +332.460 q^{92} -92.4633 q^{93} +409.642 q^{94} -663.115 q^{95} -96.0000 q^{96} -448.776 q^{97} -98.0000 q^{98} +132.441 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + q^{5} - 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 + q^5 - 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + q^{5} - 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9} - 2 q^{10} - 16 q^{11} + 24 q^{12} - 26 q^{13} + 28 q^{14} + 3 q^{15} + 32 q^{16} + 18 q^{17} - 36 q^{18} + 63 q^{19} + 4 q^{20} - 42 q^{21} + 32 q^{22} - 95 q^{23} - 48 q^{24} - 185 q^{25} + 52 q^{26} + 54 q^{27} - 56 q^{28} - 419 q^{29} - 6 q^{30} - 73 q^{31} - 64 q^{32} - 48 q^{33} - 36 q^{34} - 7 q^{35} + 72 q^{36} + 90 q^{37} - 126 q^{38} - 78 q^{39} - 8 q^{40} + 186 q^{41} + 84 q^{42} - 67 q^{43} - 64 q^{44} + 9 q^{45} + 190 q^{46} - 421 q^{47} + 96 q^{48} + 98 q^{49} + 370 q^{50} + 54 q^{51} - 104 q^{52} - 819 q^{53} - 108 q^{54} - 266 q^{55} + 112 q^{56} + 189 q^{57} + 838 q^{58} - 698 q^{59} + 12 q^{60} + 68 q^{61} + 146 q^{62} - 126 q^{63} + 128 q^{64} - 13 q^{65} + 96 q^{66} + 638 q^{67} + 72 q^{68} - 285 q^{69} + 14 q^{70} - 538 q^{71} - 144 q^{72} + 541 q^{73} - 180 q^{74} - 555 q^{75} + 252 q^{76} + 112 q^{77} + 156 q^{78} - 815 q^{79} + 16 q^{80} + 162 q^{81} - 372 q^{82} + 291 q^{83} - 168 q^{84} + 912 q^{85} + 134 q^{86} - 1257 q^{87} + 128 q^{88} - 945 q^{89} - 18 q^{90} + 182 q^{91} - 380 q^{92} - 219 q^{93} + 842 q^{94} - 1065 q^{95} - 192 q^{96} - 23 q^{97} - 196 q^{98} - 144 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 + q^5 - 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9 - 2 * q^10 - 16 * q^11 + 24 * q^12 - 26 * q^13 + 28 * q^14 + 3 * q^15 + 32 * q^16 + 18 * q^17 - 36 * q^18 + 63 * q^19 + 4 * q^20 - 42 * q^21 + 32 * q^22 - 95 * q^23 - 48 * q^24 - 185 * q^25 + 52 * q^26 + 54 * q^27 - 56 * q^28 - 419 * q^29 - 6 * q^30 - 73 * q^31 - 64 * q^32 - 48 * q^33 - 36 * q^34 - 7 * q^35 + 72 * q^36 + 90 * q^37 - 126 * q^38 - 78 * q^39 - 8 * q^40 + 186 * q^41 + 84 * q^42 - 67 * q^43 - 64 * q^44 + 9 * q^45 + 190 * q^46 - 421 * q^47 + 96 * q^48 + 98 * q^49 + 370 * q^50 + 54 * q^51 - 104 * q^52 - 819 * q^53 - 108 * q^54 - 266 * q^55 + 112 * q^56 + 189 * q^57 + 838 * q^58 - 698 * q^59 + 12 * q^60 + 68 * q^61 + 146 * q^62 - 126 * q^63 + 128 * q^64 - 13 * q^65 + 96 * q^66 + 638 * q^67 + 72 * q^68 - 285 * q^69 + 14 * q^70 - 538 * q^71 - 144 * q^72 + 541 * q^73 - 180 * q^74 - 555 * q^75 + 252 * q^76 + 112 * q^77 + 156 * q^78 - 815 * q^79 + 16 * q^80 + 162 * q^81 - 372 * q^82 + 291 * q^83 - 168 * q^84 + 912 * q^85 + 134 * q^86 - 1257 * q^87 + 128 * q^88 - 945 * q^89 - 18 * q^90 + 182 * q^91 - 380 * q^92 - 219 * q^93 + 842 * q^94 - 1065 * q^95 - 192 * q^96 - 23 * q^97 - 196 * q^98 - 144 * 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
$5$	−5.17891	−0.463216	−0.231608	−	0.972809i	$-0.574399\pi$
$5$	−5.17891	−0.463216	−0.231608	+	0.972809i	$0.574399\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	10.3578	0.327543
$11$	14.7156	0.403357	0.201679	−	0.979452i	$-0.435360\pi$
$11$	14.7156	0.403357	0.201679	+	0.979452i	$0.435360\pi$
$12$	12.0000	0.288675
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	14.0000	0.267261
$15$	−15.5367	−0.267438
$16$	16.0000	0.250000
$17$	−70.5047	−1.00588	−0.502938	−	0.864322i	$-0.667748\pi$
$17$	−70.5047	−1.00588	−0.502938	+	0.864322i	$0.667748\pi$
$18$	−18.0000	−0.235702
$19$	128.041	1.54604	0.773019	−	0.634383i	$-0.218746\pi$
$19$	128.041	1.54604	0.773019	+	0.634383i	$0.218746\pi$
$20$	−20.7156	−0.231608
$21$	−21.0000	−0.218218
$22$	−29.4313	−0.285217
$23$	83.1149	0.753507	0.376753	−	0.926314i	$-0.377040\pi$
$23$	83.1149	0.753507	0.376753	+	0.926314i	$0.377040\pi$
$24$	−24.0000	−0.204124
$25$	−98.1789	−0.785431
$26$	26.0000	0.196116
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	−169.748	−1.08694	−0.543471	−	0.839428i	$-0.682890\pi$
$29$	−169.748	−1.08694	−0.543471	+	0.839428i	$0.682890\pi$
$30$	31.0735	0.189107
$31$	−30.8211	−0.178569	−0.0892844	−	0.996006i	$-0.528458\pi$
$31$	−30.8211	−0.178569	−0.0892844	+	0.996006i	$0.528458\pi$
$32$	−32.0000	−0.176777
$33$	44.1469	0.232878
$34$	141.009	0.711262
$35$	36.2524	0.175079
$36$	36.0000	0.166667
$37$	−216.230	−0.960756	−0.480378	−	0.877062i	$-0.659500\pi$
$37$	−216.230	−0.960756	−0.480378	+	0.877062i	$0.659500\pi$
$38$	−256.083	−1.09321
$39$	−39.0000	−0.160128
$40$	41.4313	0.163771
$41$	−236.377	−0.900386	−0.450193	−	0.892931i	$-0.648645\pi$
$41$	−236.377	−0.900386	−0.450193	+	0.892931i	$0.648645\pi$
$42$	42.0000	0.154303
$43$	119.831	0.424976	0.212488	−	0.977164i	$-0.431843\pi$
$43$	119.831	0.424976	0.212488	+	0.977164i	$0.431843\pi$
$44$	58.8625	0.201679
$45$	−46.6102	−0.154405
$46$	−166.230	−0.532810
$47$	−204.821	−0.635664	−0.317832	−	0.948147i	$-0.602955\pi$
$47$	−204.821	−0.635664	−0.317832	+	0.948147i	$0.602955\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	196.358	0.555384
$51$	−211.514	−0.580743
$52$	−52.0000	−0.138675
$53$	−142.591	−0.369555	−0.184778	−	0.982780i	$-0.559156\pi$
$53$	−142.591	−0.369555	−0.184778	+	0.982780i	$0.559156\pi$
$54$	−54.0000	−0.136083
$55$	−76.2109	−0.186841
$56$	56.0000	0.133631
$57$	384.124	0.892605
$58$	339.495	0.768585
$59$	−314.927	−0.694914	−0.347457	−	0.937696i	$-0.612955\pi$
$59$	−314.927	−0.694914	−0.347457	+	0.937696i	$0.612955\pi$
$60$	−62.1469	−0.133719
$61$	−511.175	−1.07294	−0.536469	−	0.843920i	$-0.680242\pi$
$61$	−511.175	−1.07294	−0.536469	+	0.843920i	$0.680242\pi$
$62$	61.6422	0.126267
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	67.3258	0.128473
$66$	−88.2938	−0.164670
$67$	375.789	0.685223	0.342612	−	0.939477i	$-0.388689\pi$
$67$	375.789	0.685223	0.342612	+	0.939477i	$0.388689\pi$
$68$	−282.019	−0.502938
$69$	249.345	0.435037
$70$	−72.5047	−0.123800
$71$	−416.652	−0.696443	−0.348221	−	0.937412i	$-0.613214\pi$
$71$	−416.652	−0.696443	−0.348221	+	0.937412i	$0.613214\pi$
$72$	−72.0000	−0.117851
$73$	1082.58	1.73571	0.867855	−	0.496817i	$-0.165498\pi$
$73$	1082.58	1.73571	0.867855	+	0.496817i	$0.165498\pi$
$74$	432.460	0.679357
$75$	−294.537	−0.453469
$76$	512.166	0.773019
$77$	−103.009	−0.152455
$78$	78.0000	0.113228
$79$	−140.591	−0.200225	−0.100112	−	0.994976i	$-0.531920\pi$
$79$	−140.591	−0.200225	−0.100112	+	0.994976i	$0.531920\pi$
$80$	−82.8625	−0.115804
$81$	81.0000	0.111111
$82$	472.753	0.636669
$83$	60.3164	0.0797661	0.0398830	−	0.999204i	$-0.487301\pi$
$83$	60.3164	0.0797661	0.0398830	+	0.999204i	$0.487301\pi$
$84$	−84.0000	−0.109109
$85$	365.137	0.465938
$86$	−239.661	−0.300504
$87$	−509.243	−0.627547
$88$	−117.725	−0.142608
$89$	−1602.60	−1.90871	−0.954357	−	0.298668i	$-0.903458\pi$
$89$	−1602.60	−1.90871	−0.954357	+	0.298668i	$0.903458\pi$
$90$	93.2204	0.109181
$91$	91.0000	0.104828
$92$	332.460	0.376753
$93$	−92.4633	−0.103097
$94$	409.642	0.449483
$95$	−663.115	−0.716149
$96$	−96.0000	−0.102062
$97$	−448.776	−0.469756	−0.234878	−	0.972025i	$-0.575469\pi$
$97$	−448.776	−0.469756	−0.234878	+	0.972025i	$0.575469\pi$
$98$	−98.0000	−0.101015
$99$	132.441	0.134452
$100$	−392.716	−0.392716
$101$	−1276.15	−1.25724	−0.628621	−	0.777712i	$-0.716380\pi$
$101$	−1276.15	−1.25724	−0.628621	+	0.777712i	$0.716380\pi$
$102$	423.028	0.410647
$103$	−277.137	−0.265118	−0.132559	−	0.991175i	$-0.542319\pi$
$103$	−277.137	−0.265118	−0.132559	+	0.991175i	$0.542319\pi$
$104$	104.000	0.0980581
$105$	108.757	0.101082
$106$	285.183	0.261315
$107$	−1675.03	−1.51337	−0.756687	−	0.653777i	$-0.773183\pi$
$107$	−1675.03	−1.51337	−0.756687	+	0.653777i	$0.773183\pi$
$108$	108.000	0.0962250
$109$	−1283.05	−1.12746	−0.563732	−	0.825958i	$-0.690635\pi$
$109$	−1283.05	−1.12746	−0.563732	+	0.825958i	$0.690635\pi$
$110$	152.422	0.132117
$111$	−648.689	−0.554693
$112$	−112.000	−0.0944911
$113$	−95.7739	−0.0797314	−0.0398657	−	0.999205i	$-0.512693\pi$
$113$	−95.7739	−0.0797314	−0.0398657	+	0.999205i	$0.512693\pi$
$114$	−768.249	−0.631167
$115$	−430.444	−0.349036
$116$	−678.991	−0.543471
$117$	−117.000	−0.0924500
$118$	629.853	0.491379
$119$	493.533	0.380186
$120$	124.294	0.0945535
$121$	−1114.45	−0.837303
$122$	1022.35	0.758682
$123$	−709.130	−0.519838
$124$	−123.284	−0.0892844
$125$	1155.82	0.827040
$126$	126.000	0.0890871
$127$	380.791	0.266061	0.133030	−	0.991112i	$-0.457529\pi$
$127$	380.791	0.266061	0.133030	+	0.991112i	$0.457529\pi$
$128$	−128.000	−0.0883883
$129$	359.492	0.245360
$130$	−134.652	−0.0908441
$131$	−1715.71	−1.14429	−0.572145	−	0.820153i	$-0.693888\pi$
$131$	−1715.71	−1.14429	−0.572145	+	0.820153i	$0.693888\pi$
$132$	176.588	0.116439
$133$	−896.290	−0.584347
$134$	−751.578	−0.484526
$135$	−139.831	−0.0891459
$136$	564.038	0.355631
$137$	542.644	0.338403	0.169202	−	0.985581i	$-0.445881\pi$
$137$	542.644	0.338403	0.169202	+	0.985581i	$0.445881\pi$
$138$	−498.689	−0.307618
$139$	−1618.05	−0.987346	−0.493673	−	0.869648i	$-0.664346\pi$
$139$	−1618.05	−0.987346	−0.493673	+	0.869648i	$0.664346\pi$
$140$	145.009	0.0875395
$141$	−614.463	−0.367001
$142$	833.303	0.492460
$143$	−191.303	−0.111871
$144$	144.000	0.0833333
$145$	879.107	0.503489
$146$	−2165.17	−1.22733
$147$	147.000	0.0824786
$148$	−864.919	−0.480378
$149$	1191.77	0.655261	0.327631	−	0.944806i	$-0.393750\pi$
$149$	1191.77	0.655261	0.327631	+	0.944806i	$0.393750\pi$
$150$	589.073	0.320651
$151$	733.265	0.395181	0.197590	−	0.980285i	$-0.436688\pi$
$151$	733.265	0.395181	0.197590	+	0.980285i	$0.436688\pi$
$152$	−1024.33	−0.546607
$153$	−634.542	−0.335292
$154$	206.019	0.107802
$155$	159.620	0.0827158
$156$	−156.000	−0.0800641
$157$	−2636.25	−1.34010	−0.670051	−	0.742315i	$-0.733728\pi$
$157$	−2636.25	−1.34010	−0.670051	+	0.742315i	$0.733728\pi$
$158$	281.183	0.141580
$159$	−427.774	−0.213363
$160$	165.725	0.0818857
$161$	−581.804	−0.284799
$162$	−162.000	−0.0785674
$163$	3809.08	1.83037	0.915186	−	0.403032i	$-0.132044\pi$
$163$	3809.08	1.83037	0.915186	+	0.403032i	$0.132044\pi$
$164$	−945.507	−0.450193
$165$	−228.633	−0.107873
$166$	−120.633	−0.0564031
$167$	3628.09	1.68114	0.840569	−	0.541704i	$-0.182221\pi$
$167$	3628.09	1.68114	0.840569	+	0.541704i	$0.182221\pi$
$168$	168.000	0.0771517
$169$	169.000	0.0769231
$170$	−730.275	−0.329468
$171$	1152.37	0.515346
$172$	479.322	0.212488
$173$	−2300.49	−1.01100	−0.505499	−	0.862827i	$-0.668692\pi$
$173$	−2300.49	−1.01100	−0.505499	+	0.862827i	$0.668692\pi$
$174$	1018.49	0.443743
$175$	687.252	0.296865
$176$	235.450	0.100839
$177$	−944.780	−0.401209
$178$	3205.21	1.34966
$179$	−961.330	−0.401414	−0.200707	−	0.979651i	$-0.564324\pi$
$179$	−961.330	−0.401414	−0.200707	+	0.979651i	$0.564324\pi$
$180$	−186.441	−0.0772026
$181$	1865.96	0.766276	0.383138	−	0.923691i	$-0.374843\pi$
$181$	1865.96	0.766276	0.383138	+	0.923691i	$0.374843\pi$
$182$	−182.000	−0.0741249
$183$	−1533.53	−0.619462
$184$	−664.919	−0.266405
$185$	1119.83	0.445037
$186$	184.927	0.0729004
$187$	−1037.52	−0.405728
$188$	−819.284	−0.317832
$189$	−189.000	−0.0727393
$190$	1326.23	0.506394
$191$	−1454.82	−0.551138	−0.275569	−	0.961281i	$-0.588866\pi$
$191$	−1454.82	−0.551138	−0.275569	+	0.961281i	$0.588866\pi$
$192$	192.000	0.0721688
$193$	3561.82	1.32842	0.664211	−	0.747545i	$-0.268768\pi$
$193$	3561.82	1.32842	0.664211	+	0.747545i	$0.268768\pi$
$194$	897.552	0.332167
$195$	201.977	0.0741739
$196$	196.000	0.0714286
$197$	−4103.21	−1.48397	−0.741984	−	0.670418i	$-0.766115\pi$
$197$	−4103.21	−1.48397	−0.741984	+	0.670418i	$0.766115\pi$
$198$	−264.881	−0.0950722
$199$	−346.094	−0.123286	−0.0616432	−	0.998098i	$-0.519634\pi$
$199$	−346.094	−0.123286	−0.0616432	+	0.998098i	$0.519634\pi$
$200$	785.431	0.277692
$201$	1127.37	0.395614
$202$	2552.29	0.889004
$203$	1188.23	0.410826
$204$	−846.057	−0.290372
$205$	1224.17	0.417073
$206$	554.275	0.187467
$207$	748.034	0.251169
$208$	−208.000	−0.0693375
$209$	1884.21	0.623606
$210$	−217.514	−0.0714757
$211$	412.366	0.134542	0.0672711	−	0.997735i	$-0.478571\pi$
$211$	412.366	0.134542	0.0672711	+	0.997735i	$0.478571\pi$
$212$	−570.365	−0.184778
$213$	−1249.95	−0.402092
$214$	3350.06	1.07012
$215$	−620.591	−0.196856
$216$	−216.000	−0.0680414
$217$	215.748	0.0674926
$218$	2566.09	0.797238
$219$	3247.75	1.00211
$220$	−304.844	−0.0934207
$221$	916.561	0.278980
$222$	1297.38	0.392227
$223$	4868.79	1.46205	0.731027	−	0.682348i	$-0.239041\pi$
$223$	4868.79	1.46205	0.731027	+	0.682348i	$0.239041\pi$
$224$	224.000	0.0668153
$225$	−883.610	−0.261810
$226$	191.548	0.0563786
$227$	590.263	0.172587	0.0862933	−	0.996270i	$-0.472498\pi$
$227$	590.263	0.172587	0.0862933	+	0.996270i	$0.472498\pi$
$228$	1536.50	0.446303
$229$	−55.2655	−0.0159478	−0.00797390	−	0.999968i	$-0.502538\pi$
$229$	−55.2655	−0.0159478	−0.00797390	+	0.999968i	$0.502538\pi$
$230$	860.889	0.246806
$231$	−309.028	−0.0880198
$232$	1357.98	0.384292
$233$	994.855	0.279721	0.139861	−	0.990171i	$-0.455335\pi$
$233$	994.855	0.279721	0.139861	+	0.990171i	$0.455335\pi$
$234$	234.000	0.0653720
$235$	1060.75	0.294450
$236$	−1259.71	−0.347457
$237$	−421.774	−0.115600
$238$	−987.066	−0.268832
$239$	−93.9697	−0.0254326	−0.0127163	−	0.999919i	$-0.504048\pi$
$239$	−93.9697	−0.0254326	−0.0127163	+	0.999919i	$0.504048\pi$
$240$	−248.588	−0.0668594
$241$	341.356	0.0912393	0.0456197	−	0.998959i	$-0.485474\pi$
$241$	341.356	0.0912393	0.0456197	+	0.998959i	$0.485474\pi$
$242$	2228.90	0.592063
$243$	243.000	0.0641500
$244$	−2044.70	−0.536469
$245$	−253.767	−0.0661737
$246$	1418.26	0.367581
$247$	−1664.54	−0.428794
$248$	246.569	0.0631336
$249$	180.949	0.0460530
$250$	−2311.65	−0.584805
$251$	3797.96	0.955080	0.477540	−	0.878610i	$-0.341529\pi$
$251$	3797.96	0.955080	0.477540	+	0.878610i	$0.341529\pi$
$252$	−252.000	−0.0629941
$253$	1223.09	0.303932
$254$	−761.582	−0.188134
$255$	1095.41	0.269009
$256$	256.000	0.0625000
$257$	−1498.48	−0.363707	−0.181854	−	0.983326i	$-0.558210\pi$
$257$	−1498.48	−0.363707	−0.181854	+	0.983326i	$0.558210\pi$
$258$	−718.983	−0.173496
$259$	1513.61	0.363131
$260$	269.303	0.0642365
$261$	−1527.73	−0.362314
$262$	3431.41	0.809135
$263$	2295.33	0.538159	0.269080	−	0.963118i	$-0.413281\pi$
$263$	2295.33	0.538159	0.269080	+	0.963118i	$0.413281\pi$
$264$	−353.175	−0.0823350
$265$	738.467	0.171184
$266$	1792.58	0.413196
$267$	−4807.81	−1.10200
$268$	1503.16	0.342612
$269$	7689.96	1.74299	0.871497	−	0.490401i	$-0.163150\pi$
$269$	7689.96	1.74299	0.871497	+	0.490401i	$0.163150\pi$
$270$	279.661	0.0630357
$271$	−3045.88	−0.682745	−0.341372	−	0.939928i	$-0.610892\pi$
$271$	−3045.88	−0.682745	−0.341372	+	0.939928i	$0.610892\pi$
$272$	−1128.08	−0.251469
$273$	273.000	0.0605228
$274$	−1085.29	−0.239287
$275$	−1444.76	−0.316809
$276$	997.379	0.217519
$277$	2812.69	0.610101	0.305050	−	0.952336i	$-0.401327\pi$
$277$	2812.69	0.610101	0.305050	+	0.952336i	$0.401327\pi$
$278$	3236.10	0.698159
$279$	−277.390	−0.0595229
$280$	−290.019	−0.0618998
$281$	4064.38	0.862850	0.431425	−	0.902149i	$-0.358011\pi$
$281$	4064.38	0.862850	0.431425	+	0.902149i	$0.358011\pi$
$282$	1228.93	0.259509
$283$	5361.14	1.12610	0.563050	−	0.826423i	$-0.309628\pi$
$283$	5361.14	1.12610	0.563050	+	0.826423i	$0.309628\pi$
$284$	−1666.61	−0.348221
$285$	−1989.34	−0.413469
$286$	382.606	0.0791049
$287$	1654.64	0.340314
$288$	−288.000	−0.0589256
$289$	57.9151	0.0117881
$290$	−1758.21	−0.356020
$291$	−1346.33	−0.271214
$292$	4330.34	0.867855
$293$	1951.35	0.389075	0.194537	−	0.980895i	$-0.437679\pi$
$293$	1951.35	0.389075	0.194537	+	0.980895i	$0.437679\pi$
$294$	−294.000	−0.0583212
$295$	1630.98	0.321895
$296$	1729.84	0.339678
$297$	397.322	0.0776262
$298$	−2383.55	−0.463340
$299$	−1080.49	−0.208985
$300$	−1178.15	−0.226734
$301$	−838.814	−0.160626
$302$	−1466.53	−0.279435
$303$	−3828.44	−0.725869
$304$	2048.66	0.386509
$305$	2647.33	0.497002
$306$	1269.08	0.237087
$307$	9761.10	1.81464	0.907321	−	0.420438i	$-0.138124\pi$
$307$	9761.10	1.81464	0.907321	+	0.420438i	$0.138124\pi$
$308$	−412.038	−0.0762274
$309$	−831.412	−0.153066
$310$	−319.239	−0.0584889
$311$	−7585.16	−1.38301	−0.691503	−	0.722373i	$-0.743051\pi$
$311$	−7585.16	−1.38301	−0.691503	+	0.722373i	$0.743051\pi$
$312$	312.000	0.0566139
$313$	−6867.60	−1.24019	−0.620095	−	0.784527i	$-0.712906\pi$
$313$	−6867.60	−1.24019	−0.620095	+	0.784527i	$0.712906\pi$
$314$	5272.51	0.947595
$315$	326.271	0.0583597
$316$	−562.365	−0.100112
$317$	459.947	0.0814928	0.0407464	−	0.999170i	$-0.487026\pi$
$317$	459.947	0.0814928	0.0407464	+	0.999170i	$0.487026\pi$
$318$	855.548	0.150870
$319$	−2497.94	−0.438426
$320$	−331.450	−0.0579020
$321$	−5025.08	−0.873747
$322$	1163.61	0.201383
$323$	−9027.53	−1.55512
$324$	324.000	0.0555556
$325$	1276.33	0.217839
$326$	−7618.17	−1.29427
$327$	−3849.14	−0.650942
$328$	1891.01	0.318335
$329$	1433.75	0.240259
$330$	457.265	0.0762777
$331$	1958.49	0.325222	0.162611	−	0.986690i	$-0.448008\pi$
$331$	1958.49	0.325222	0.162611	+	0.986690i	$0.448008\pi$
$332$	241.265	0.0398830
$333$	−1946.07	−0.320252
$334$	−7256.18	−1.18874
$335$	−1946.18	−0.317406
$336$	−336.000	−0.0545545
$337$	7832.20	1.26601	0.633007	−	0.774146i	$-0.281820\pi$
$337$	7832.20	1.26601	0.633007	+	0.774146i	$0.281820\pi$
$338$	−338.000	−0.0543928
$339$	−287.322	−0.0460330
$340$	1460.55	0.232969
$341$	−453.552	−0.0720270
$342$	−2304.75	−0.364405
$343$	−343.000	−0.0539949
$344$	−958.644	−0.150252
$345$	−1291.33	−0.201516
$346$	4600.97	0.714884
$347$	−6323.04	−0.978209	−0.489104	−	0.872225i	$-0.662676\pi$
$347$	−6323.04	−0.978209	−0.489104	+	0.872225i	$0.662676\pi$
$348$	−2036.97	−0.313773
$349$	1363.52	0.209133	0.104567	−	0.994518i	$-0.466654\pi$
$349$	1363.52	0.209133	0.104567	+	0.994518i	$0.466654\pi$
$350$	−1374.50	−0.209915
$351$	−351.000	−0.0533761
$352$	−470.900	−0.0713042
$353$	3571.43	0.538493	0.269246	−	0.963071i	$-0.413225\pi$
$353$	3571.43	0.538493	0.269246	+	0.963071i	$0.413225\pi$
$354$	1889.56	0.283698
$355$	2157.80	0.322603
$356$	−6410.41	−0.954357
$357$	1480.60	0.219500
$358$	1922.66	0.283843
$359$	−2143.78	−0.315165	−0.157583	−	0.987506i	$-0.550370\pi$
$359$	−2143.78	−0.315165	−0.157583	+	0.987506i	$0.550370\pi$
$360$	372.881	0.0545905
$361$	9535.61	1.39023
$362$	−3731.92	−0.541839
$363$	−3343.35	−0.483417
$364$	364.000	0.0524142
$365$	−5606.60	−0.804008
$366$	3067.05	0.438026
$367$	−5966.64	−0.848654	−0.424327	−	0.905509i	$-0.639489\pi$
$367$	−5966.64	−0.848654	−0.424327	+	0.905509i	$0.639489\pi$
$368$	1329.84	0.188377
$369$	−2127.39	−0.300129
$370$	−2239.67	−0.314689
$371$	998.139	0.139679
$372$	−369.853	−0.0515484
$373$	3371.93	0.468075	0.234037	−	0.972228i	$-0.424806\pi$
$373$	3371.93	0.468075	0.234037	+	0.972228i	$0.424806\pi$
$374$	2075.04	0.286893
$375$	3467.47	0.477492
$376$	1638.57	0.224741
$377$	2206.72	0.301464
$378$	378.000	0.0514344
$379$	−9927.10	−1.34544	−0.672719	−	0.739898i	$-0.734874\pi$
$379$	−9927.10	−1.34544	−0.672719	+	0.739898i	$0.734874\pi$
$380$	−2652.46	−0.358074
$381$	1142.37	0.153610
$382$	2909.65	0.389714
$383$	4677.16	0.623999	0.311999	−	0.950082i	$-0.399001\pi$
$383$	4677.16	0.623999	0.311999	+	0.950082i	$0.399001\pi$
$384$	−384.000	−0.0510310
$385$	533.476	0.0706194
$386$	−7123.64	−0.939336
$387$	1078.47	0.141659
$388$	−1795.10	−0.234878
$389$	3817.86	0.497617	0.248809	−	0.968553i	$-0.419961\pi$
$389$	3817.86	0.497617	0.248809	+	0.968553i	$0.419961\pi$
$390$	−403.955	−0.0524488
$391$	−5859.99	−0.757935
$392$	−392.000	−0.0505076
$393$	−5147.12	−0.660656
$394$	8206.42	1.04932
$395$	728.109	0.0927473
$396$	529.763	0.0672262
$397$	−11927.6	−1.50788	−0.753941	−	0.656942i	$-0.771850\pi$
$397$	−11927.6	−1.50788	−0.753941	+	0.656942i	$0.771850\pi$
$398$	692.189	0.0871766
$399$	−2688.87	−0.337373
$400$	−1570.86	−0.196358
$401$	8210.49	1.02247	0.511237	−	0.859439i	$-0.329187\pi$
$401$	8210.49	1.02247	0.511237	+	0.859439i	$0.329187\pi$
$402$	−2254.73	−0.279741
$403$	400.674	0.0495261
$404$	−5104.59	−0.628621
$405$	−419.492	−0.0514684
$406$	−2376.47	−0.290498
$407$	−3181.96	−0.387528
$408$	1692.11	0.205324
$409$	−5803.16	−0.701584	−0.350792	−	0.936453i	$-0.614088\pi$
$409$	−5803.16	−0.701584	−0.350792	+	0.936453i	$0.614088\pi$
$410$	−2448.35	−0.294915
$411$	1627.93	0.195377
$412$	−1108.55	−0.132559
$413$	2204.49	0.262653
$414$	−1496.07	−0.177603
$415$	−312.373	−0.0369489
$416$	416.000	0.0490290
$417$	−4854.15	−0.570045
$418$	−3768.42	−0.440956
$419$	13826.8	1.61214	0.806069	−	0.591822i	$-0.201591\pi$
$419$	13826.8	1.61214	0.806069	+	0.591822i	$0.201591\pi$
$420$	435.028	0.0505410
$421$	−12991.0	−1.50391	−0.751953	−	0.659217i	$-0.770888\pi$
$421$	−12991.0	−1.50391	−0.751953	+	0.659217i	$0.770888\pi$
$422$	−824.731	−0.0951357
$423$	−1843.39	−0.211888
$424$	1140.73	0.130657
$425$	6922.08	0.790047
$426$	2499.91	0.284322
$427$	3578.23	0.405533
$428$	−6700.11	−0.756687
$429$	−573.910	−0.0645889
$430$	1241.18	0.139198
$431$	10419.2	1.16444	0.582221	−	0.813030i	$-0.302184\pi$
$431$	10419.2	1.16444	0.582221	+	0.813030i	$0.302184\pi$
$432$	432.000	0.0481125
$433$	−7523.56	−0.835010	−0.417505	−	0.908675i	$-0.637095\pi$
$433$	−7523.56	−0.835010	−0.417505	+	0.908675i	$0.637095\pi$
$434$	−431.495	−0.0477245
$435$	2637.32	0.290689
$436$	−5132.19	−0.563732
$437$	10642.2	1.16495
$438$	−6495.50	−0.708601
$439$	4710.29	0.512095	0.256048	−	0.966664i	$-0.417580\pi$
$439$	4710.29	0.512095	0.256048	+	0.966664i	$0.417580\pi$
$440$	609.687	0.0660584
$441$	441.000	0.0476190
$442$	−1833.12	−0.197269
$443$	−11561.7	−1.23998	−0.619992	−	0.784608i	$-0.712864\pi$
$443$	−11561.7	−1.23998	−0.619992	+	0.784608i	$0.712864\pi$
$444$	−2594.76	−0.277346
$445$	8299.73	0.884146
$446$	−9737.57	−1.03383
$447$	3575.32	0.378315
$448$	−448.000	−0.0472456
$449$	4298.31	0.451782	0.225891	−	0.974153i	$-0.427471\pi$
$449$	4298.31	0.451782	0.225891	+	0.974153i	$0.427471\pi$
$450$	1767.22	0.185128
$451$	−3478.43	−0.363177
$452$	−383.096	−0.0398657
$453$	2199.80	0.228158
$454$	−1180.53	−0.122037
$455$	−471.281	−0.0485582
$456$	−3072.99	−0.315584
$457$	4444.08	0.454891	0.227446	−	0.973791i	$-0.426963\pi$
$457$	4444.08	0.454891	0.227446	+	0.973791i	$0.426963\pi$
$458$	110.531	0.0112768
$459$	−1903.63	−0.193581
$460$	−1721.78	−0.174518
$461$	−14349.1	−1.44968	−0.724840	−	0.688918i	$-0.758086\pi$
$461$	−14349.1	−1.44968	−0.724840	+	0.688918i	$0.758086\pi$
$462$	618.057	0.0622394
$463$	−16269.5	−1.63307	−0.816533	−	0.577299i	$-0.804107\pi$
$463$	−16269.5	−1.63307	−0.816533	+	0.577299i	$0.804107\pi$
$464$	−2715.96	−0.271736
$465$	478.859	0.0477560
$466$	−1989.71	−0.197793
$467$	−11214.0	−1.11118	−0.555590	−	0.831457i	$-0.687507\pi$
$467$	−11214.0	−1.11118	−0.555590	+	0.831457i	$0.687507\pi$
$468$	−468.000	−0.0462250
$469$	−2630.52	−0.258990
$470$	−2121.50	−0.208207
$471$	−7908.76	−0.773708
$472$	2519.41	0.245689
$473$	1763.38	0.171417
$474$	843.548	0.0817414
$475$	−12571.0	−1.21431
$476$	1974.13	0.190093
$477$	−1283.32	−0.123185
$478$	187.939	0.0179836
$479$	−9944.09	−0.948554	−0.474277	−	0.880376i	$-0.657290\pi$
$479$	−9944.09	−0.948554	−0.474277	+	0.880376i	$0.657290\pi$
$480$	497.175	0.0472767
$481$	2810.99	0.266466
$482$	−682.712	−0.0645160
$483$	−1745.41	−0.164429
$484$	−4457.80	−0.418651
$485$	2324.17	0.217598
$486$	−486.000	−0.0453609
$487$	−9340.97	−0.869158	−0.434579	−	0.900634i	$-0.643103\pi$
$487$	−9340.97	−0.869158	−0.434579	+	0.900634i	$0.643103\pi$
$488$	4089.40	0.379341
$489$	11427.3	1.05677
$490$	507.533	0.0467918
$491$	20488.6	1.88318	0.941588	−	0.336766i	$-0.109333\pi$
$491$	20488.6	1.88318	0.941588	+	0.336766i	$0.109333\pi$
$492$	−2836.52	−0.259919
$493$	11968.0	1.09333
$494$	3329.08	0.303203
$495$	−685.898	−0.0622805
$496$	−493.137	−0.0446422
$497$	2916.56	0.263231
$498$	−361.898	−0.0325644
$499$	15203.5	1.36393	0.681964	−	0.731386i	$-0.261126\pi$
$499$	15203.5	1.36393	0.681964	+	0.731386i	$0.261126\pi$
$500$	4623.29	0.413520
$501$	10884.3	0.970606
$502$	−7595.92	−0.675343
$503$	−744.618	−0.0660057	−0.0330029	−	0.999455i	$-0.510507\pi$
$503$	−744.618	−0.0660057	−0.0330029	+	0.999455i	$0.510507\pi$
$504$	504.000	0.0445435
$505$	6609.05	0.582374
$506$	−2446.18	−0.214913
$507$	507.000	0.0444116
$508$	1523.16	0.133030
$509$	7216.23	0.628396	0.314198	−	0.949357i	$-0.398264\pi$
$509$	7216.23	0.628396	0.314198	+	0.949357i	$0.398264\pi$
$510$	−2190.82	−0.190218
$511$	−7578.09	−0.656037
$512$	−512.000	−0.0441942
$513$	3457.12	0.297535
$514$	2996.96	0.257180
$515$	1435.27	0.122807
$516$	1437.97	0.122680
$517$	−3014.07	−0.256400
$518$	−3027.22	−0.256773
$519$	−6901.46	−0.583700
$520$	−538.606	−0.0454220
$521$	−11085.2	−0.932149	−0.466074	−	0.884746i	$-0.654332\pi$
$521$	−11085.2	−0.932149	−0.466074	+	0.884746i	$0.654332\pi$
$522$	3055.46	0.256195
$523$	−14284.4	−1.19429	−0.597144	−	0.802134i	$-0.703698\pi$
$523$	−14284.4	−1.19429	−0.597144	+	0.802134i	$0.703698\pi$
$524$	−6862.82	−0.572145
$525$	2061.76	0.171395
$526$	−4590.65	−0.380536
$527$	2173.03	0.179618
$528$	706.350	0.0582196
$529$	−5258.91	−0.432228
$530$	−1476.93	−0.121045
$531$	−2834.34	−0.231638
$532$	−3585.16	−0.292174
$533$	3072.90	0.249722
$534$	9615.62	0.779229
$535$	8674.82	0.701019
$536$	−3006.31	−0.242263
$537$	−2883.99	−0.231757
$538$	−15379.9	−1.23248
$539$	721.066	0.0576225
$540$	−559.322	−0.0445729
$541$	10860.3	0.863069	0.431534	−	0.902097i	$-0.357972\pi$
$541$	10860.3	0.863069	0.431534	+	0.902097i	$0.357972\pi$
$542$	6091.75	0.482773
$543$	5597.89	0.442409
$544$	2256.15	0.177816
$545$	6644.78	0.522259
$546$	−546.000	−0.0427960
$547$	2695.09	0.210665	0.105332	−	0.994437i	$-0.466409\pi$
$547$	2695.09	0.210665	0.105332	+	0.994437i	$0.466409\pi$
$548$	2170.58	0.169202
$549$	−4600.58	−0.357646
$550$	2889.53	0.224018
$551$	−21734.7	−1.68046
$552$	−1994.76	−0.153809
$553$	984.139	0.0756779
$554$	−5625.37	−0.431406
$555$	3359.50	0.256942
$556$	−6472.20	−0.493673
$557$	21261.0	1.61734	0.808670	−	0.588263i	$-0.200188\pi$
$557$	21261.0	1.61734	0.808670	+	0.588263i	$0.200188\pi$
$558$	554.780	0.0420891
$559$	−1557.80	−0.117867
$560$	580.038	0.0437698
$561$	−3112.56	−0.234247
$562$	−8128.77	−0.610127
$563$	1747.50	0.130814	0.0654072	−	0.997859i	$-0.479165\pi$
$563$	1747.50	0.130814	0.0654072	+	0.997859i	$0.479165\pi$
$564$	−2457.85	−0.183501
$565$	496.004	0.0369329
$566$	−10722.3	−0.796274
$567$	−567.000	−0.0419961
$568$	3333.21	0.246230
$569$	24818.5	1.82855	0.914275	−	0.405093i	$-0.132761\pi$
$569$	24818.5	1.82855	0.914275	+	0.405093i	$0.132761\pi$
$570$	3978.69	0.292367
$571$	4251.54	0.311596	0.155798	−	0.987789i	$-0.450205\pi$
$571$	4251.54	0.311596	0.155798	+	0.987789i	$0.450205\pi$
$572$	−765.213	−0.0559356
$573$	−4364.47	−0.318200
$574$	−3309.27	−0.240638
$575$	−8160.13	−0.591828
$576$	576.000	0.0416667
$577$	−1286.89	−0.0928491	−0.0464245	−	0.998922i	$-0.514783\pi$
$577$	−1286.89	−0.0928491	−0.0464245	+	0.998922i	$0.514783\pi$
$578$	−115.830	−0.00833547
$579$	10685.5	0.766965
$580$	3516.43	0.251744
$581$	−422.215	−0.0301487
$582$	2692.66	0.191777
$583$	−2098.32	−0.149063
$584$	−8660.67	−0.613666
$585$	605.932	0.0428243
$586$	−3902.70	−0.275118
$587$	−1211.11	−0.0851583	−0.0425792	−	0.999093i	$-0.513557\pi$
$587$	−1211.11	−0.0851583	−0.0425792	+	0.999093i	$0.513557\pi$
$588$	588.000	0.0412393
$589$	−3946.38	−0.276074
$590$	−3261.95	−0.227614
$591$	−12309.6	−0.856769
$592$	−3459.68	−0.240189
$593$	−6325.41	−0.438033	−0.219017	−	0.975721i	$-0.570285\pi$
$593$	−6325.41	−0.438033	−0.219017	+	0.975721i	$0.570285\pi$
$594$	−794.644	−0.0548900
$595$	−2555.96	−0.176108
$596$	4767.10	0.327631
$597$	−1038.28	−0.0711794
$598$	2160.99	0.147775
$599$	6755.90	0.460832	0.230416	−	0.973092i	$-0.425991\pi$
$599$	6755.90	0.460832	0.230416	+	0.973092i	$0.425991\pi$
$600$	2356.29	0.160325
$601$	−15007.6	−1.01859	−0.509294	−	0.860593i	$-0.670093\pi$
$601$	−15007.6	−1.01859	−0.509294	+	0.860593i	$0.670093\pi$
$602$	1677.63	0.113580
$603$	3382.10	0.228408
$604$	2933.06	0.197590
$605$	5771.64	0.387852
$606$	7656.88	0.513267
$607$	−13185.9	−0.881710	−0.440855	−	0.897578i	$-0.645325\pi$
$607$	−13185.9	−0.881710	−0.440855	+	0.897578i	$0.645325\pi$
$608$	−4097.33	−0.273303
$609$	3564.70	0.237190
$610$	−5294.66	−0.351434
$611$	2662.67	0.176302
$612$	−2538.17	−0.167646
$613$	13168.6	0.867661	0.433830	−	0.900995i	$-0.357162\pi$
$613$	13168.6	0.867661	0.433830	+	0.900995i	$0.357162\pi$
$614$	−19522.2	−1.28315
$615$	3672.52	0.240797
$616$	824.075	0.0539009
$617$	13436.6	0.876720	0.438360	−	0.898800i	$-0.355560\pi$
$617$	13436.6	0.876720	0.438360	+	0.898800i	$0.355560\pi$
$618$	1662.82	0.108234
$619$	17464.3	1.13400	0.567001	−	0.823717i	$-0.308103\pi$
$619$	17464.3	1.13400	0.567001	+	0.823717i	$0.308103\pi$
$620$	638.478	0.0413579
$621$	2244.10	0.145012
$622$	15170.3	0.977933
$623$	11218.2	0.721426
$624$	−624.000	−0.0400320
$625$	6286.46	0.402334
$626$	13735.2	0.876947
$627$	5652.63	0.360039
$628$	−10545.0	−0.670051
$629$	15245.2	0.966402
$630$	−652.542	−0.0412665
$631$	−11483.6	−0.724490	−0.362245	−	0.932083i	$-0.617990\pi$
$631$	−11483.6	−0.724490	−0.362245	+	0.932083i	$0.617990\pi$
$632$	1124.73	0.0707902
$633$	1237.10	0.0776780
$634$	−919.895	−0.0576241
$635$	−1972.08	−0.123244
$636$	−1711.10	−0.106681
$637$	−637.000	−0.0396214
$638$	4995.89	0.310014
$639$	−3749.86	−0.232148
$640$	662.900	0.0409429
$641$	−6774.76	−0.417452	−0.208726	−	0.977974i	$-0.566932\pi$
$641$	−6774.76	−0.417452	−0.208726	+	0.977974i	$0.566932\pi$
$642$	10050.2	0.617833
$643$	3329.27	0.204189	0.102095	−	0.994775i	$-0.467446\pi$
$643$	3329.27	0.204189	0.102095	+	0.994775i	$0.467446\pi$
$644$	−2327.22	−0.142399
$645$	−1861.77	−0.113655
$646$	18055.1	1.09964
$647$	18857.6	1.14586	0.572929	−	0.819605i	$-0.305807\pi$
$647$	18857.6	1.14586	0.572929	+	0.819605i	$0.305807\pi$
$648$	−648.000	−0.0392837
$649$	−4634.34	−0.280299
$650$	−2552.65	−0.154036
$651$	647.243	0.0389669
$652$	15236.3	0.915186
$653$	17441.6	1.04524	0.522621	−	0.852565i	$-0.324954\pi$
$653$	17441.6	1.04524	0.522621	+	0.852565i	$0.324954\pi$
$654$	7698.28	0.460285
$655$	8885.49	0.530053
$656$	−3782.03	−0.225097
$657$	9743.26	0.578570
$658$	−2867.50	−0.169888
$659$	−7278.41	−0.430238	−0.215119	−	0.976588i	$-0.569014\pi$
$659$	−7278.41	−0.430238	−0.215119	+	0.976588i	$0.569014\pi$
$660$	−914.531	−0.0539365
$661$	−24157.2	−1.42149	−0.710746	−	0.703449i	$-0.751642\pi$
$661$	−24157.2	−1.42149	−0.710746	+	0.703449i	$0.751642\pi$
$662$	−3916.99	−0.229967
$663$	2749.68	0.161069
$664$	−482.531	−0.0282016
$665$	4641.80	0.270679
$666$	3892.14	0.226452
$667$	−14108.6	−0.819019
$668$	14512.4	0.840569
$669$	14606.4	0.844117
$670$	3892.35	0.224440
$671$	−7522.27	−0.432778
$672$	672.000	0.0385758
$673$	18280.1	1.04702	0.523510	−	0.852020i	$-0.324622\pi$
$673$	18280.1	1.04702	0.523510	+	0.852020i	$0.324622\pi$
$674$	−15664.4	−0.895208
$675$	−2650.83	−0.151156
$676$	676.000	0.0384615
$677$	−9725.74	−0.552128	−0.276064	−	0.961139i	$-0.589030\pi$
$677$	−9725.74	−0.552128	−0.276064	+	0.961139i	$0.589030\pi$
$678$	574.644	0.0325502
$679$	3141.43	0.177551
$680$	−2921.10	−0.164734
$681$	1770.79	0.0996429
$682$	907.104	0.0509308
$683$	−12973.5	−0.726818	−0.363409	−	0.931630i	$-0.618387\pi$
$683$	−12973.5	−0.726818	−0.363409	+	0.931630i	$0.618387\pi$
$684$	4609.49	0.257673
$685$	−2810.30	−0.156754
$686$	686.000	0.0381802
$687$	−165.796	−0.00920747
$688$	1917.29	0.106244
$689$	1853.69	0.102496
$690$	2582.67	0.142493
$691$	−30769.7	−1.69397	−0.846986	−	0.531616i	$-0.821585\pi$
$691$	−30769.7	−1.69397	−0.846986	+	0.531616i	$0.821585\pi$
$692$	−9201.94	−0.505499
$693$	−927.085	−0.0508182
$694$	12646.1	0.691698
$695$	8379.73	0.457354
$696$	4073.94	0.221871
$697$	16665.7	0.905678
$698$	−2727.04	−0.147879
$699$	2984.56	0.161497
$700$	2749.01	0.148433
$701$	−21289.0	−1.14704	−0.573519	−	0.819192i	$-0.694422\pi$
$701$	−21289.0	−1.14704	−0.573519	+	0.819192i	$0.694422\pi$
$702$	702.000	0.0377426
$703$	−27686.4	−1.48536
$704$	941.801	0.0504197
$705$	3182.25	0.170001
$706$	−7142.86	−0.380772
$707$	8933.03	0.475193
$708$	−3779.12	−0.200604
$709$	−11968.8	−0.633990	−0.316995	−	0.948427i	$-0.602674\pi$
$709$	−11968.8	−0.633990	−0.316995	+	0.948427i	$0.602674\pi$
$710$	−4315.60	−0.228115
$711$	−1265.32	−0.0667416
$712$	12820.8	0.674832
$713$	−2561.69	−0.134553
$714$	−2961.20	−0.155210
$715$	990.742	0.0518205
$716$	−3845.32	−0.200707
$717$	−281.909	−0.0146835
$718$	4287.56	0.222855
$719$	−9750.75	−0.505760	−0.252880	−	0.967498i	$-0.581378\pi$
$719$	−9750.75	−0.505760	−0.252880	+	0.967498i	$0.581378\pi$
$720$	−745.763	−0.0386013
$721$	1939.96	0.100205
$722$	−19071.2	−0.983043
$723$	1024.07	0.0526771
$724$	7463.85	0.383138
$725$	16665.6	0.853719
$726$	6686.70	0.341827
$727$	−25300.1	−1.29068	−0.645342	−	0.763894i	$-0.723285\pi$
$727$	−25300.1	−1.29068	−0.645342	+	0.763894i	$0.723285\pi$
$728$	−728.000	−0.0370625
$729$	729.000	0.0370370
$730$	11213.2	0.568520
$731$	−8448.62	−0.427474
$732$	−6134.10	−0.309731
$733$	−9573.00	−0.482383	−0.241192	−	0.970478i	$-0.577538\pi$
$733$	−9573.00	−0.482383	−0.241192	+	0.970478i	$0.577538\pi$
$734$	11933.3	0.600089
$735$	−761.300	−0.0382054
$736$	−2659.68	−0.133202
$737$	5529.97	0.276390
$738$	4254.78	0.212223
$739$	1975.43	0.0983319	0.0491659	−	0.998791i	$-0.484344\pi$
$739$	1975.43	0.0983319	0.0491659	+	0.998791i	$0.484344\pi$
$740$	4479.34	0.222519
$741$	−4993.62	−0.247564
$742$	−1996.28	−0.0987678
$743$	14933.1	0.737339	0.368670	−	0.929561i	$-0.379813\pi$
$743$	14933.1	0.737339	0.368670	+	0.929561i	$0.379813\pi$
$744$	739.706	0.0364502
$745$	−6172.09	−0.303527
$746$	−6743.86	−0.330979
$747$	542.847	0.0265887
$748$	−4150.09	−0.202864
$749$	11725.2	0.572002
$750$	−6934.94	−0.337638
$751$	6451.12	0.313455	0.156728	−	0.987642i	$-0.449906\pi$
$751$	6451.12	0.313455	0.156728	+	0.987642i	$0.449906\pi$
$752$	−3277.14	−0.158916
$753$	11393.9	0.551416
$754$	−4413.44	−0.213167
$755$	−3797.51	−0.183054
$756$	−756.000	−0.0363696
$757$	20335.8	0.976377	0.488189	−	0.872738i	$-0.337658\pi$
$757$	20335.8	0.976377	0.488189	+	0.872738i	$0.337658\pi$
$758$	19854.2	0.951369
$759$	3669.26	0.175475
$760$	5304.92	0.253197
$761$	−21653.3	−1.03145	−0.515725	−	0.856754i	$-0.672477\pi$
$761$	−21653.3	−1.03145	−0.515725	+	0.856754i	$0.672477\pi$
$762$	−2284.75	−0.108619
$763$	8981.33	0.426142
$764$	−5819.30	−0.275569
$765$	3286.24	0.155313
$766$	−9354.31	−0.441234
$767$	4094.05	0.192735
$768$	768.000	0.0360844
$769$	3490.62	0.163686	0.0818432	−	0.996645i	$-0.473919\pi$
$769$	3490.62	0.163686	0.0818432	+	0.996645i	$0.473919\pi$
$770$	−1066.95	−0.0499355
$771$	−4495.45	−0.209986
$772$	14247.3	0.664211
$773$	−4490.15	−0.208926	−0.104463	−	0.994529i	$-0.533312\pi$
$773$	−4490.15	−0.208926	−0.104463	+	0.994529i	$0.533312\pi$
$774$	−2156.95	−0.100168
$775$	3025.98	0.140253
$776$	3590.21	0.166084
$777$	4540.83	0.209654
$778$	−7635.71	−0.351868
$779$	−30266.0	−1.39203
$780$	807.910	0.0370869
$781$	−6131.29	−0.280915
$782$	11720.0	0.535941
$783$	−4583.19	−0.209182
$784$	784.000	0.0357143
$785$	13652.9	0.620756
$786$	10294.2	0.467154
$787$	−5121.64	−0.231978	−0.115989	−	0.993250i	$-0.537004\pi$
$787$	−5121.64	−0.231978	−0.115989	+	0.993250i	$0.537004\pi$
$788$	−16412.8	−0.741984
$789$	6885.98	0.310706
$790$	−1456.22	−0.0655822
$791$	670.417	0.0301357
$792$	−1059.53	−0.0475361
$793$	6645.28	0.297580
$794$	23855.2	1.06623
$795$	2215.40	0.0988330
$796$	−1384.38	−0.0616432
$797$	29255.5	1.30023	0.650114	−	0.759837i	$-0.274721\pi$
$797$	29255.5	1.30023	0.650114	+	0.759837i	$0.274721\pi$
$798$	5377.74	0.238559
$799$	14440.9	0.639400
$800$	3141.73	0.138846
$801$	−14423.4	−0.636238
$802$	−16421.0	−0.722999
$803$	15930.9	0.700111
$804$	4509.47	0.197807
$805$	3013.11	0.131923
$806$	−801.348	−0.0350202
$807$	23069.9	1.00632
$808$	10209.2	0.444502
$809$	−24292.8	−1.05574	−0.527868	−	0.849327i	$-0.677008\pi$
$809$	−24292.8	−1.05574	−0.527868	+	0.849327i	$0.677008\pi$
$810$	838.983	0.0363937
$811$	−20068.2	−0.868913	−0.434456	−	0.900693i	$-0.643060\pi$
$811$	−20068.2	−0.868913	−0.434456	+	0.900693i	$0.643060\pi$
$812$	4752.93	0.205413
$813$	−9137.63	−0.394183
$814$	6363.92	0.274024
$815$	−19726.9	−0.847857
$816$	−3384.23	−0.145186
$817$	15343.3	0.657030
$818$	11606.3	0.496094
$819$	819.000	0.0349428
$820$	4896.69	0.208536
$821$	−19555.9	−0.831308	−0.415654	−	0.909523i	$-0.636447\pi$
$821$	−19555.9	−0.831308	−0.415654	+	0.909523i	$0.636447\pi$
$822$	−3255.87	−0.138152
$823$	45137.8	1.91179	0.955895	−	0.293709i	$-0.0948896\pi$
$823$	45137.8	1.91179	0.955895	+	0.293709i	$0.0948896\pi$
$824$	2217.10	0.0937334
$825$	−4334.29	−0.182910
$826$	−4408.97	−0.185724
$827$	4734.97	0.199094	0.0995471	−	0.995033i	$-0.468261\pi$
$827$	4734.97	0.199094	0.0995471	+	0.995033i	$0.468261\pi$
$828$	2992.14	0.125584
$829$	−9647.64	−0.404194	−0.202097	−	0.979366i	$-0.564776\pi$
$829$	−9647.64	−0.404194	−0.202097	+	0.979366i	$0.564776\pi$
$830$	624.746	0.0261268
$831$	8438.06	0.352242
$832$	−832.000	−0.0346688
$833$	−3454.73	−0.143697
$834$	9708.30	0.403082
$835$	−18789.5	−0.778730
$836$	7536.84	0.311803
$837$	−832.169	−0.0343656
$838$	−27653.7	−1.13995
$839$	−25674.6	−1.05648	−0.528238	−	0.849096i	$-0.677147\pi$
$839$	−25674.6	−1.05648	−0.528238	+	0.849096i	$0.677147\pi$
$840$	−870.057	−0.0357379
$841$	4425.26	0.181445
$842$	25982.1	1.06342
$843$	12193.2	0.498167
$844$	1649.46	0.0672711
$845$	−875.236	−0.0356320
$846$	3686.78	0.149828
$847$	7801.15	0.316471
$848$	−2281.46	−0.0923888
$849$	16083.4	0.650155
$850$	−13844.2	−0.558648
$851$	−17971.9	−0.723936
$852$	−4999.82	−0.201046
$853$	−28092.0	−1.12761	−0.563805	−	0.825908i	$-0.690663\pi$
$853$	−28092.0	−1.12761	−0.563805	+	0.825908i	$0.690663\pi$
$854$	−7156.45	−0.286755
$855$	−5968.03	−0.238716
$856$	13400.2	0.535059
$857$	5192.46	0.206967	0.103484	−	0.994631i	$-0.467001\pi$
$857$	5192.46	0.206967	0.103484	+	0.994631i	$0.467001\pi$
$858$	1147.82	0.0456712
$859$	−32655.5	−1.29708	−0.648540	−	0.761180i	$-0.724620\pi$
$859$	−32655.5	−1.29708	−0.648540	+	0.761180i	$0.724620\pi$
$860$	−2482.37	−0.0984279
$861$	4963.91	0.196480
$862$	−20838.4	−0.823385
$863$	−5454.55	−0.215151	−0.107575	−	0.994197i	$-0.534309\pi$
$863$	−5454.55	−0.215151	−0.107575	+	0.994197i	$0.534309\pi$
$864$	−864.000	−0.0340207
$865$	11914.0	0.468310
$866$	15047.1	0.590441
$867$	173.745	0.00680588
$868$	862.991	0.0337463
$869$	−2068.89	−0.0807621
$870$	−5274.64	−0.205549
$871$	−4885.26	−0.190047
$872$	10264.4	0.398619
$873$	−4038.98	−0.156585
$874$	−21284.3	−0.823744
$875$	−8090.76	−0.312592
$876$	12991.0	0.501056
$877$	−40218.9	−1.54857	−0.774284	−	0.632838i	$-0.781890\pi$
$877$	−40218.9	−1.54857	−0.774284	+	0.632838i	$0.781890\pi$
$878$	−9420.58	−0.362106
$879$	5854.04	0.224633
$880$	−1219.37	−0.0467104
$881$	39235.0	1.50041	0.750204	−	0.661206i	$-0.229955\pi$
$881$	39235.0	1.50041	0.750204	+	0.661206i	$0.229955\pi$
$882$	−882.000	−0.0336718
$883$	7267.89	0.276992	0.138496	−	0.990363i	$-0.455773\pi$
$883$	7267.89	0.276992	0.138496	+	0.990363i	$0.455773\pi$
$884$	3666.25	0.139490
$885$	4892.93	0.185846
$886$	23123.4	0.876801
$887$	931.739	0.0352703	0.0176351	−	0.999844i	$-0.494386\pi$
$887$	931.739	0.0352703	0.0176351	+	0.999844i	$0.494386\pi$
$888$	5189.51	0.196113
$889$	−2665.54	−0.100562
$890$	−16599.5	−0.625186
$891$	1191.97	0.0448175
$892$	19475.1	0.731027
$893$	−26225.6	−0.982761
$894$	−7150.65	−0.267509
$895$	4978.64	0.185941
$896$	896.000	0.0334077
$897$	−3241.48	−0.120658
$898$	−8596.63	−0.319458
$899$	5231.81	0.194094
$900$	−3534.44	−0.130905
$901$	10053.4	0.371727
$902$	6956.87	0.256805
$903$	−2516.44	−0.0927374
$904$	766.191	0.0281893
$905$	−9663.65	−0.354951
$906$	−4399.59	−0.161332
$907$	30944.4	1.13285	0.566424	−	0.824114i	$-0.308326\pi$
$907$	30944.4	1.13285	0.566424	+	0.824114i	$0.308326\pi$
$908$	2361.05	0.0862933
$909$	−11485.3	−0.419080
$910$	942.561	0.0343358
$911$	17873.1	0.650015	0.325008	−	0.945711i	$-0.394633\pi$
$911$	17873.1	0.650015	0.325008	+	0.945711i	$0.394633\pi$
$912$	6145.99	0.223151
$913$	887.594	0.0321742
$914$	−8888.16	−0.321657
$915$	7941.99	0.286944
$916$	−221.062	−0.00797390
$917$	12009.9	0.432501
$918$	3807.25	0.136882
$919$	−12524.9	−0.449574	−0.224787	−	0.974408i	$-0.572169\pi$
$919$	−12524.9	−0.449574	−0.224787	+	0.974408i	$0.572169\pi$
$920$	3443.56	0.123403
$921$	29283.3	1.04768
$922$	28698.1	1.02508
$923$	5416.47	0.193159
$924$	−1236.11	−0.0440099
$925$	21229.2	0.754607
$926$	32539.1	1.15475
$927$	−2494.24	−0.0883727
$928$	5431.92	0.192146
$929$	24457.2	0.863740	0.431870	−	0.901936i	$-0.357854\pi$
$929$	24457.2	0.863740	0.431870	+	0.901936i	$0.357854\pi$
$930$	−957.718	−0.0337686
$931$	6274.03	0.220863
$932$	3979.42	0.139861
$933$	−22755.5	−0.798479
$934$	22427.9	0.785723
$935$	5373.23	0.187939
$936$	936.000	0.0326860
$937$	−38856.4	−1.35473	−0.677365	−	0.735647i	$-0.736878\pi$
$937$	−38856.4	−1.35473	−0.677365	+	0.735647i	$0.736878\pi$
$938$	5261.05	0.183134
$939$	−20602.8	−0.716024
$940$	4243.00	0.147225
$941$	51724.5	1.79189	0.895946	−	0.444163i	$-0.146499\pi$
$941$	51724.5	1.79189	0.895946	+	0.444163i	$0.146499\pi$
$942$	15817.5	0.547094
$943$	−19646.4	−0.678447
$944$	−5038.82	−0.173729
$945$	978.814	0.0336940
$946$	−3526.76	−0.121210
$947$	50763.1	1.74190	0.870950	−	0.491372i	$-0.163504\pi$
$947$	50763.1	1.74190	0.870950	+	0.491372i	$0.163504\pi$
$948$	−1687.10	−0.0577999
$949$	−14073.6	−0.481399
$950$	25141.9	0.858644
$951$	1379.84	0.0470499
$952$	−3948.26	−0.134416
$953$	−17560.5	−0.596895	−0.298447	−	0.954426i	$-0.596469\pi$
$953$	−17560.5	−0.596895	−0.298447	+	0.954426i	$0.596469\pi$
$954$	2566.64	0.0871050
$955$	7534.40	0.255296
$956$	−375.879	−0.0127163
$957$	−7493.83	−0.253126
$958$	19888.2	0.670729
$959$	−3798.51	−0.127904
$960$	−994.350	−0.0334297
$961$	−28841.1	−0.968113
$962$	−5621.97	−0.188420
$963$	−15075.3	−0.504458
$964$	1365.42	0.0456197
$965$	−18446.3	−0.615346
$966$	3490.83	0.116269
$967$	41742.4	1.38815	0.694077	−	0.719901i	$-0.255813\pi$
$967$	41742.4	1.38815	0.694077	+	0.719901i	$0.255813\pi$
$968$	8915.60	0.296031
$969$	−27082.6	−0.897851
$970$	−4648.34	−0.153865
$971$	6923.84	0.228833	0.114416	−	0.993433i	$-0.463500\pi$
$971$	6923.84	0.228833	0.114416	+	0.993433i	$0.463500\pi$
$972$	972.000	0.0320750
$973$	11326.3	0.373182
$974$	18681.9	0.614587
$975$	3828.98	0.125770
$976$	−8178.80	−0.268235
$977$	20282.0	0.664155	0.332078	−	0.943252i	$-0.392250\pi$
$977$	20282.0	0.664155	0.332078	+	0.943252i	$0.392250\pi$
$978$	−22854.5	−0.747246
$979$	−23583.3	−0.769894
$980$	−1015.07	−0.0330868
$981$	−11547.4	−0.375821
$982$	−40977.3	−1.33161
$983$	−48097.7	−1.56061	−0.780305	−	0.625400i	$-0.784936\pi$
$983$	−48097.7	−1.56061	−0.780305	+	0.625400i	$0.784936\pi$
$984$	5673.04	0.183791
$985$	21250.1	0.687397
$986$	−23936.0	−0.773102
$987$	4301.24	0.138713
$988$	−6658.15	−0.214397
$989$	9959.70	0.320223
$990$	1371.80	0.0440389
$991$	−22925.2	−0.734858	−0.367429	−	0.930052i	$-0.619762\pi$
$991$	−22925.2	−0.734858	−0.367429	+	0.930052i	$0.619762\pi$
$992$	986.275	0.0315668
$993$	5875.48	0.187767
$994$	−5833.12	−0.186132
$995$	1792.39	0.0571082
$996$	723.796	0.0230265
$997$	46115.6	1.46489	0.732445	−	0.680826i	$-0.238379\pi$
$997$	46115.6	1.46489	0.732445	+	0.680826i	$0.238379\pi$
$998$	−30406.9	−0.964443
$999$	−5838.20	−0.184898

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.4.a.h.1.1	✓	2
3.2	odd	2		1638.4.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.h.1.1	✓	2	1.1	even	1	trivial
1638.4.a.r.1.2		2	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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.17891 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -5.17891 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +10.3578 q^{10} +14.7156 q^{11} +12.0000 q^{12} -13.0000 q^{13} +14.0000 q^{14} -15.5367 q^{15} +16.0000 q^{16} -70.5047 q^{17} -18.0000 q^{18} +128.041 q^{19} -20.7156 q^{20} -21.0000 q^{21} -29.4313 q^{22} +83.1149 q^{23} -24.0000 q^{24} -98.1789 q^{25} +26.0000 q^{26} +27.0000 q^{27} -28.0000 q^{28} -169.748 q^{29} +31.0735 q^{30} -30.8211 q^{31} -32.0000 q^{32} +44.1469 q^{33} +141.009 q^{34} +36.2524 q^{35} +36.0000 q^{36} -216.230 q^{37} -256.083 q^{38} -39.0000 q^{39} +41.4313 q^{40} -236.377 q^{41} +42.0000 q^{42} +119.831 q^{43} +58.8625 q^{44} -46.6102 q^{45} -166.230 q^{46} -204.821 q^{47} +48.0000 q^{48} +49.0000 q^{49} +196.358 q^{50} -211.514 q^{51} -52.0000 q^{52} -142.591 q^{53} -54.0000 q^{54} -76.2109 q^{55} +56.0000 q^{56} +384.124 q^{57} +339.495 q^{58} -314.927 q^{59} -62.1469 q^{60} -511.175 q^{61} +61.6422 q^{62} -63.0000 q^{63} +64.0000 q^{64} +67.3258 q^{65} -88.2938 q^{66} +375.789 q^{67} -282.019 q^{68} +249.345 q^{69} -72.5047 q^{70} -416.652 q^{71} -72.0000 q^{72} +1082.58 q^{73} +432.460 q^{74} -294.537 q^{75} +512.166 q^{76} -103.009 q^{77} +78.0000 q^{78} -140.591 q^{79} -82.8625 q^{80} +81.0000 q^{81} +472.753 q^{82} +60.3164 q^{83} -84.0000 q^{84} +365.137 q^{85} -239.661 q^{86} -509.243 q^{87} -117.725 q^{88} -1602.60 q^{89} +93.2204 q^{90} +91.0000 q^{91} +332.460 q^{92} -92.4633 q^{93} +409.642 q^{94} -663.115 q^{95} -96.0000 q^{96} -448.776 q^{97} -98.0000 q^{98} +132.441 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + q^{5} - 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 + q^5 - 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9	\( 2 q - 4 q^{2} + 6 q^{3} + 8 q^{4} + q^{5} - 12 q^{6} - 14 q^{7} - 16 q^{8} + 18 q^{9} - 2 q^{10} - 16 q^{11} + 24 q^{12} - 26 q^{13} + 28 q^{14} + 3 q^{15} + 32 q^{16} + 18 q^{17} - 36 q^{18} + 63 q^{19} + 4 q^{20} - 42 q^{21} + 32 q^{22} - 95 q^{23} - 48 q^{24} - 185 q^{25} + 52 q^{26} + 54 q^{27} - 56 q^{28} - 419 q^{29} - 6 q^{30} - 73 q^{31} - 64 q^{32} - 48 q^{33} - 36 q^{34} - 7 q^{35} + 72 q^{36} + 90 q^{37} - 126 q^{38} - 78 q^{39} - 8 q^{40} + 186 q^{41} + 84 q^{42} - 67 q^{43} - 64 q^{44} + 9 q^{45} + 190 q^{46} - 421 q^{47} + 96 q^{48} + 98 q^{49} + 370 q^{50} + 54 q^{51} - 104 q^{52} - 819 q^{53} - 108 q^{54} - 266 q^{55} + 112 q^{56} + 189 q^{57} + 838 q^{58} - 698 q^{59} + 12 q^{60} + 68 q^{61} + 146 q^{62} - 126 q^{63} + 128 q^{64} - 13 q^{65} + 96 q^{66} + 638 q^{67} + 72 q^{68} - 285 q^{69} + 14 q^{70} - 538 q^{71} - 144 q^{72} + 541 q^{73} - 180 q^{74} - 555 q^{75} + 252 q^{76} + 112 q^{77} + 156 q^{78} - 815 q^{79} + 16 q^{80} + 162 q^{81} - 372 q^{82} + 291 q^{83} - 168 q^{84} + 912 q^{85} + 134 q^{86} - 1257 q^{87} + 128 q^{88} - 945 q^{89} - 18 q^{90} + 182 q^{91} - 380 q^{92} - 219 q^{93} + 842 q^{94} - 1065 q^{95} - 192 q^{96} - 23 q^{97} - 196 q^{98} - 144 q^{99}+O(q^{100}) \) 2 * q - 4 * q^2 + 6 * q^3 + 8 * q^4 + q^5 - 12 * q^6 - 14 * q^7 - 16 * q^8 + 18 * q^9 - 2 * q^10 - 16 * q^11 + 24 * q^12 - 26 * q^13 + 28 * q^14 + 3 * q^15 + 32 * q^16 + 18 * q^17 - 36 * q^18 + 63 * q^19 + 4 * q^20 - 42 * q^21 + 32 * q^22 - 95 * q^23 - 48 * q^24 - 185 * q^25 + 52 * q^26 + 54 * q^27 - 56 * q^28 - 419 * q^29 - 6 * q^30 - 73 * q^31 - 64 * q^32 - 48 * q^33 - 36 * q^34 - 7 * q^35 + 72 * q^36 + 90 * q^37 - 126 * q^38 - 78 * q^39 - 8 * q^40 + 186 * q^41 + 84 * q^42 - 67 * q^43 - 64 * q^44 + 9 * q^45 + 190 * q^46 - 421 * q^47 + 96 * q^48 + 98 * q^49 + 370 * q^50 + 54 * q^51 - 104 * q^52 - 819 * q^53 - 108 * q^54 - 266 * q^55 + 112 * q^56 + 189 * q^57 + 838 * q^58 - 698 * q^59 + 12 * q^60 + 68 * q^61 + 146 * q^62 - 126 * q^63 + 128 * q^64 - 13 * q^65 + 96 * q^66 + 638 * q^67 + 72 * q^68 - 285 * q^69 + 14 * q^70 - 538 * q^71 - 144 * q^72 + 541 * q^73 - 180 * q^74 - 555 * q^75 + 252 * q^76 + 112 * q^77 + 156 * q^78 - 815 * q^79 + 16 * q^80 + 162 * q^81 - 372 * q^82 + 291 * q^83 - 168 * q^84 + 912 * q^85 + 134 * q^86 - 1257 * q^87 + 128 * q^88 - 945 * q^89 - 18 * q^90 + 182 * q^91 - 380 * q^92 - 219 * q^93 + 842 * q^94 - 1065 * q^95 - 192 * q^96 - 23 * q^97 - 196 * q^98 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more