LMFDB - Embedded newform 21.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,4,Mod(1,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("21.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	21.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:	$1.23904011012$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	21.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.27492 q^{2} +3.00000 q^{3} -2.82475 q^{4} -4.54983 q^{5} +6.82475 q^{6} +7.00000 q^{7} -24.6254 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.27492 q^{2} +3.00000 q^{3} -2.82475 q^{4} -4.54983 q^{5} +6.82475 q^{6} +7.00000 q^{7} -24.6254 q^{8} +9.00000 q^{9} -10.3505 q^{10} -40.7492 q^{11} -8.47425 q^{12} +53.2990 q^{13} +15.9244 q^{14} -13.6495 q^{15} -33.4228 q^{16} +4.54983 q^{17} +20.4743 q^{18} +122.598 q^{19} +12.8522 q^{20} +21.0000 q^{21} -92.7010 q^{22} +131.347 q^{23} -73.8762 q^{24} -104.299 q^{25} +121.251 q^{26} +27.0000 q^{27} -19.7733 q^{28} -216.598 q^{29} -31.0515 q^{30} -251.794 q^{31} +120.969 q^{32} -122.248 q^{33} +10.3505 q^{34} -31.8488 q^{35} -25.4228 q^{36} +11.8970 q^{37} +278.900 q^{38} +159.897 q^{39} +112.042 q^{40} -111.752 q^{41} +47.7733 q^{42} +369.196 q^{43} +115.106 q^{44} -40.9485 q^{45} +298.804 q^{46} -262.694 q^{47} -100.268 q^{48} +49.0000 q^{49} -237.272 q^{50} +13.6495 q^{51} -150.556 q^{52} -567.100 q^{53} +61.4228 q^{54} +185.402 q^{55} -172.378 q^{56} +367.794 q^{57} -492.743 q^{58} +839.890 q^{59} +38.5565 q^{60} -485.794 q^{61} -572.811 q^{62} +63.0000 q^{63} +542.577 q^{64} -242.502 q^{65} -278.103 q^{66} -333.691 q^{67} -12.8522 q^{68} +394.042 q^{69} -72.4535 q^{70} +590.248 q^{71} -221.629 q^{72} +490.701 q^{73} +27.0647 q^{74} -312.897 q^{75} -346.309 q^{76} -285.244 q^{77} +363.752 q^{78} +121.691 q^{79} +152.068 q^{80} +81.0000 q^{81} -254.228 q^{82} +609.608 q^{83} -59.3198 q^{84} -20.7010 q^{85} +839.890 q^{86} -649.794 q^{87} +1003.47 q^{88} +719.038 q^{89} -93.1545 q^{90} +373.093 q^{91} -371.023 q^{92} -755.382 q^{93} -597.608 q^{94} -557.801 q^{95} +362.908 q^{96} -637.877 q^{97} +111.471 q^{98} -366.743 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 6 q^{3} + 17 q^{4} + 6 q^{5} - 9 q^{6} + 14 q^{7} - 87 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 + 6 * q^3 + 17 * q^4 + 6 * q^5 - 9 * q^6 + 14 * q^7 - 87 * q^8 + 18 * q^9	\( 2 q - 3 q^{2} + 6 q^{3} + 17 q^{4} + 6 q^{5} - 9 q^{6} + 14 q^{7} - 87 q^{8} + 18 q^{9} - 66 q^{10} - 6 q^{11} + 51 q^{12} + 16 q^{13} - 21 q^{14} + 18 q^{15} + 137 q^{16} - 6 q^{17} - 27 q^{18} + 64 q^{19} + 222 q^{20} + 42 q^{21} - 276 q^{22} + 6 q^{23} - 261 q^{24} - 118 q^{25} + 318 q^{26} + 54 q^{27} + 119 q^{28} - 252 q^{29} - 198 q^{30} + 40 q^{31} - 279 q^{32} - 18 q^{33} + 66 q^{34} + 42 q^{35} + 153 q^{36} - 248 q^{37} + 588 q^{38} + 48 q^{39} - 546 q^{40} - 450 q^{41} - 63 q^{42} + 376 q^{43} + 804 q^{44} + 54 q^{45} + 960 q^{46} - 12 q^{47} + 411 q^{48} + 98 q^{49} - 165 q^{50} - 18 q^{51} - 890 q^{52} - 1104 q^{53} - 81 q^{54} + 552 q^{55} - 609 q^{56} + 192 q^{57} - 306 q^{58} + 804 q^{59} + 666 q^{60} - 428 q^{61} - 2112 q^{62} + 126 q^{63} + 1289 q^{64} - 636 q^{65} - 828 q^{66} + 148 q^{67} - 222 q^{68} + 18 q^{69} - 462 q^{70} + 954 q^{71} - 783 q^{72} + 1072 q^{73} + 1398 q^{74} - 354 q^{75} - 1508 q^{76} - 42 q^{77} + 954 q^{78} - 572 q^{79} + 1950 q^{80} + 162 q^{81} + 1530 q^{82} + 1944 q^{83} + 357 q^{84} - 132 q^{85} + 804 q^{86} - 756 q^{87} - 1164 q^{88} + 366 q^{89} - 594 q^{90} + 112 q^{91} - 2856 q^{92} + 120 q^{93} - 1920 q^{94} - 1176 q^{95} - 837 q^{96} + 808 q^{97} - 147 q^{98} - 54 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 + 6 * q^3 + 17 * q^4 + 6 * q^5 - 9 * q^6 + 14 * q^7 - 87 * q^8 + 18 * q^9 - 66 * q^10 - 6 * q^11 + 51 * q^12 + 16 * q^13 - 21 * q^14 + 18 * q^15 + 137 * q^16 - 6 * q^17 - 27 * q^18 + 64 * q^19 + 222 * q^20 + 42 * q^21 - 276 * q^22 + 6 * q^23 - 261 * q^24 - 118 * q^25 + 318 * q^26 + 54 * q^27 + 119 * q^28 - 252 * q^29 - 198 * q^30 + 40 * q^31 - 279 * q^32 - 18 * q^33 + 66 * q^34 + 42 * q^35 + 153 * q^36 - 248 * q^37 + 588 * q^38 + 48 * q^39 - 546 * q^40 - 450 * q^41 - 63 * q^42 + 376 * q^43 + 804 * q^44 + 54 * q^45 + 960 * q^46 - 12 * q^47 + 411 * q^48 + 98 * q^49 - 165 * q^50 - 18 * q^51 - 890 * q^52 - 1104 * q^53 - 81 * q^54 + 552 * q^55 - 609 * q^56 + 192 * q^57 - 306 * q^58 + 804 * q^59 + 666 * q^60 - 428 * q^61 - 2112 * q^62 + 126 * q^63 + 1289 * q^64 - 636 * q^65 - 828 * q^66 + 148 * q^67 - 222 * q^68 + 18 * q^69 - 462 * q^70 + 954 * q^71 - 783 * q^72 + 1072 * q^73 + 1398 * q^74 - 354 * q^75 - 1508 * q^76 - 42 * q^77 + 954 * q^78 - 572 * q^79 + 1950 * q^80 + 162 * q^81 + 1530 * q^82 + 1944 * q^83 + 357 * q^84 - 132 * q^85 + 804 * q^86 - 756 * q^87 - 1164 * q^88 + 366 * q^89 - 594 * q^90 + 112 * q^91 - 2856 * q^92 + 120 * q^93 - 1920 * q^94 - 1176 * q^95 - 837 * q^96 + 808 * q^97 - 147 * q^98 - 54 * 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 $

$2$	2.27492	0.804305	0.402152	−	0.915573i	$-0.368262\pi$
$2$	2.27492	0.804305	0.402152	+	0.915573i	$0.368262\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−2.82475	−0.353094
$5$	−4.54983	−0.406950	−0.203475	−	0.979080i	$-0.565223\pi$
$5$	−4.54983	−0.406950	−0.203475	+	0.979080i	$0.565223\pi$
$6$	6.82475	0.464366
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−24.6254	−1.08830
$9$	9.00000	0.333333
$10$	−10.3505	−0.327311
$11$	−40.7492	−1.11694	−0.558470	−	0.829525i	$-0.688611\pi$
$11$	−40.7492	−1.11694	−0.558470	+	0.829525i	$0.688611\pi$
$12$	−8.47425	−0.203859
$13$	53.2990	1.13711	0.568557	−	0.822644i	$-0.307502\pi$
$13$	53.2990	1.13711	0.568557	+	0.822644i	$0.307502\pi$
$14$	15.9244	0.303999
$15$	−13.6495	−0.234952
$16$	−33.4228	−0.522231
$17$	4.54983	0.0649116	0.0324558	−	0.999473i	$-0.489667\pi$
$17$	4.54983	0.0649116	0.0324558	+	0.999473i	$0.489667\pi$
$18$	20.4743	0.268102
$19$	122.598	1.48031	0.740156	−	0.672436i	$-0.234752\pi$
$19$	122.598	1.48031	0.740156	+	0.672436i	$0.234752\pi$
$20$	12.8522	0.143691
$21$	21.0000	0.218218
$22$	−92.7010	−0.898360
$23$	131.347	1.19077	0.595387	−	0.803439i	$-0.296999\pi$
$23$	131.347	1.19077	0.595387	+	0.803439i	$0.296999\pi$
$24$	−73.8762	−0.628330
$25$	−104.299	−0.834392
$26$	121.251	0.914586
$27$	27.0000	0.192450
$28$	−19.7733	−0.133457
$29$	−216.598	−1.38694	−0.693470	−	0.720486i	$-0.743919\pi$
$29$	−216.598	−1.38694	−0.693470	+	0.720486i	$0.743919\pi$
$30$	−31.0515	−0.188973
$31$	−251.794	−1.45882	−0.729412	−	0.684075i	$-0.760206\pi$
$31$	−251.794	−1.45882	−0.729412	+	0.684075i	$0.760206\pi$
$32$	120.969	0.668267
$33$	−122.248	−0.644865
$34$	10.3505	0.0522087
$35$	−31.8488	−0.153812
$36$	−25.4228	−0.117698
$37$	11.8970	0.0528610	0.0264305	−	0.999651i	$-0.491586\pi$
$37$	11.8970	0.0528610	0.0264305	+	0.999651i	$0.491586\pi$
$38$	278.900	1.19062
$39$	159.897	0.656513
$40$	112.042	0.442883
$41$	−111.752	−0.425678	−0.212839	−	0.977087i	$-0.568271\pi$
$41$	−111.752	−0.425678	−0.212839	+	0.977087i	$0.568271\pi$
$42$	47.7733	0.175514
$43$	369.196	1.30935	0.654673	−	0.755912i	$-0.272806\pi$
$43$	369.196	1.30935	0.654673	+	0.755912i	$0.272806\pi$
$44$	115.106	0.394385
$45$	−40.9485	−0.135650
$46$	298.804	0.957744
$47$	−262.694	−0.815275	−0.407637	−	0.913144i	$-0.633647\pi$
$47$	−262.694	−0.815275	−0.407637	+	0.913144i	$0.633647\pi$
$48$	−100.268	−0.301510
$49$	49.0000	0.142857
$50$	−237.272	−0.671105
$51$	13.6495	0.0374767
$52$	−150.556	−0.401508
$53$	−567.100	−1.46976	−0.734879	−	0.678199i	$-0.762761\pi$
$53$	−567.100	−1.46976	−0.734879	+	0.678199i	$0.762761\pi$
$54$	61.4228	0.154789
$55$	185.402	0.454538
$56$	−172.378	−0.411339
$57$	367.794	0.854658
$58$	−492.743	−1.11552
$59$	839.890	1.85330	0.926648	−	0.375931i	$-0.122677\pi$
$59$	839.890	1.85330	0.926648	+	0.375931i	$0.122677\pi$
$60$	38.5565	0.0829603
$61$	−485.794	−1.01966	−0.509832	−	0.860274i	$-0.670293\pi$
$61$	−485.794	−1.01966	−0.509832	+	0.860274i	$0.670293\pi$
$62$	−572.811	−1.17334
$63$	63.0000	0.125988
$64$	542.577	1.05972
$65$	−242.502	−0.462748
$66$	−278.103	−0.518668
$67$	−333.691	−0.608460	−0.304230	−	0.952599i	$-0.598399\pi$
$67$	−333.691	−0.608460	−0.304230	+	0.952599i	$0.598399\pi$
$68$	−12.8522	−0.0229199
$69$	394.042	0.687493
$70$	−72.4535	−0.123712
$71$	590.248	0.986613	0.493306	−	0.869856i	$-0.335788\pi$
$71$	590.248	0.986613	0.493306	+	0.869856i	$0.335788\pi$
$72$	−221.629	−0.362767
$73$	490.701	0.786743	0.393371	−	0.919380i	$-0.371309\pi$
$73$	490.701	0.786743	0.393371	+	0.919380i	$0.371309\pi$
$74$	27.0647	0.0425164
$75$	−312.897	−0.481736
$76$	−346.309	−0.522689
$77$	−285.244	−0.422164
$78$	363.752	0.528037
$79$	121.691	0.173308	0.0866539	−	0.996238i	$-0.472383\pi$
$79$	121.691	0.173308	0.0866539	+	0.996238i	$0.472383\pi$
$80$	152.068	0.212522
$81$	81.0000	0.111111
$82$	−254.228	−0.342375
$83$	609.608	0.806183	0.403091	−	0.915160i	$-0.367936\pi$
$83$	609.608	0.806183	0.403091	+	0.915160i	$0.367936\pi$
$84$	−59.3198	−0.0770514
$85$	−20.7010	−0.0264157
$86$	839.890	1.05311
$87$	−649.794	−0.800750
$88$	1003.47	1.21557
$89$	719.038	0.856381	0.428190	−	0.903689i	$-0.359151\pi$
$89$	719.038	0.856381	0.428190	+	0.903689i	$0.359151\pi$
$90$	−93.1545	−0.109104
$91$	373.093	0.429789
$92$	−371.023	−0.420455
$93$	−755.382	−0.842252
$94$	−597.608	−0.655729
$95$	−557.801	−0.602412
$96$	362.908	0.385824
$97$	−637.877	−0.667697	−0.333849	−	0.942627i	$-0.608347\pi$
$97$	−637.877	−0.667697	−0.333849	+	0.942627i	$0.608347\pi$
$98$	111.471	0.114901
$99$	−366.743	−0.372313
$100$	294.619	0.294619
$101$	671.148	0.661205	0.330603	−	0.943770i	$-0.392748\pi$
$101$	671.148	0.661205	0.330603	+	0.943770i	$0.392748\pi$
$102$	31.0515	0.0301427
$103$	−912.412	−0.872841	−0.436420	−	0.899743i	$-0.643754\pi$
$103$	−912.412	−0.872841	−0.436420	+	0.899743i	$0.643754\pi$
$104$	−1312.51	−1.23752
$105$	−95.5465	−0.0888037
$106$	−1290.10	−1.18213
$107$	−116.736	−0.105470	−0.0527350	−	0.998609i	$-0.516794\pi$
$107$	−116.736	−0.105470	−0.0527350	+	0.998609i	$0.516794\pi$
$108$	−76.2683	−0.0679530
$109$	837.176	0.735660	0.367830	−	0.929893i	$-0.380101\pi$
$109$	837.176	0.735660	0.367830	+	0.929893i	$0.380101\pi$
$110$	421.774	0.365587
$111$	35.6911	0.0305193
$112$	−233.959	−0.197385
$113$	−1086.58	−0.904572	−0.452286	−	0.891873i	$-0.649391\pi$
$113$	−1086.58	−0.904572	−0.452286	+	0.891873i	$0.649391\pi$
$114$	836.701	0.687406
$115$	−597.608	−0.484585
$116$	611.836	0.489720
$117$	479.691	0.379038
$118$	1910.68	1.49061
$119$	31.8488	0.0245343
$120$	336.125	0.255699
$121$	329.495	0.247554
$122$	−1105.14	−0.820121
$123$	−335.257	−0.245765
$124$	711.256	0.515102
$125$	1043.27	0.746505
$126$	143.320	0.101333
$127$	−537.113	−0.375284	−0.187642	−	0.982237i	$-0.560084\pi$
$127$	−537.113	−0.375284	−0.187642	+	0.982237i	$0.560084\pi$
$128$	266.564	0.184071
$129$	1107.59	0.755951
$130$	−551.671	−0.372190
$131$	1497.39	0.998683	0.499341	−	0.866405i	$-0.333575\pi$
$131$	1497.39	0.998683	0.499341	+	0.866405i	$0.333575\pi$
$132$	345.319	0.227698
$133$	858.186	0.559505
$134$	−759.120	−0.489388
$135$	−122.846	−0.0783175
$136$	−112.042	−0.0706433
$137$	−1380.09	−0.860650	−0.430325	−	0.902674i	$-0.641601\pi$
$137$	−1380.09	−0.860650	−0.430325	+	0.902674i	$0.641601\pi$
$138$	896.412	0.552954
$139$	−141.980	−0.0866374	−0.0433187	−	0.999061i	$-0.513793\pi$
$139$	−141.980	−0.0866374	−0.0433187	+	0.999061i	$0.513793\pi$
$140$	89.9651	0.0543103
$141$	−788.083	−0.470699
$142$	1342.76	0.793537
$143$	−2171.89	−1.27009
$144$	−300.805	−0.174077
$145$	985.485	0.564414
$146$	1116.30	0.632781
$147$	147.000	0.0824786
$148$	−33.6061	−0.0186649
$149$	−1943.87	−1.06878	−0.534390	−	0.845238i	$-0.679458\pi$
$149$	−1943.87	−1.06878	−0.534390	+	0.845238i	$0.679458\pi$
$150$	−711.815	−0.387463
$151$	−2654.76	−1.43074	−0.715370	−	0.698746i	$-0.753742\pi$
$151$	−2654.76	−1.43074	−0.715370	+	0.698746i	$0.753742\pi$
$152$	−3019.03	−1.61102
$153$	40.9485	0.0216372
$154$	−648.907	−0.339548
$155$	1145.62	0.593668
$156$	−451.669	−0.231811
$157$	1665.22	0.846489	0.423244	−	0.906016i	$-0.360891\pi$
$157$	1665.22	0.846489	0.423244	+	0.906016i	$0.360891\pi$
$158$	276.837	0.139392
$159$	−1701.30	−0.848565
$160$	−550.390	−0.271951
$161$	919.430	0.450070
$162$	184.268	0.0893672
$163$	−33.0732	−0.0158926	−0.00794629	−	0.999968i	$-0.502529\pi$
$163$	−33.0732	−0.0158926	−0.00794629	+	0.999968i	$0.502529\pi$
$164$	315.673	0.150304
$165$	556.206	0.262428
$166$	1386.81	0.648417
$167$	1654.48	0.766630	0.383315	−	0.923618i	$-0.374782\pi$
$167$	1654.48	0.766630	0.383315	+	0.923618i	$0.374782\pi$
$168$	−517.134	−0.237486
$169$	643.784	0.293029
$170$	−47.0930	−0.0212463
$171$	1103.38	0.493437
$172$	−1042.89	−0.462322
$173$	64.1909	0.0282101	0.0141050	−	0.999901i	$-0.495510\pi$
$173$	64.1909	0.0282101	0.0141050	+	0.999901i	$0.495510\pi$
$174$	−1478.23	−0.644047
$175$	−730.093	−0.315371
$176$	1361.95	0.583300
$177$	2519.67	1.07000
$178$	1635.75	0.688791
$179$	3914.68	1.63462	0.817309	−	0.576200i	$-0.195465\pi$
$179$	3914.68	1.63462	0.817309	+	0.576200i	$0.195465\pi$
$180$	115.669	0.0478971
$181$	−2058.04	−0.845156	−0.422578	−	0.906327i	$-0.638875\pi$
$181$	−2058.04	−0.845156	−0.422578	+	0.906327i	$0.638875\pi$
$182$	848.756	0.345681
$183$	−1457.38	−0.588704
$184$	−3234.48	−1.29592
$185$	−54.1295	−0.0215118
$186$	−1718.43	−0.677428
$187$	−185.402	−0.0725023
$188$	742.046	0.287869
$189$	189.000	0.0727393
$190$	−1268.95	−0.484523
$191$	428.048	0.162160	0.0810798	−	0.996708i	$-0.474163\pi$
$191$	428.048	0.162160	0.0810798	+	0.996708i	$0.474163\pi$
$192$	1627.73	0.611830
$193$	1604.93	0.598576	0.299288	−	0.954163i	$-0.403251\pi$
$193$	1604.93	0.598576	0.299288	+	0.954163i	$0.403251\pi$
$194$	−1451.12	−0.537032
$195$	−727.505	−0.267168
$196$	−138.413	−0.0504420
$197$	3738.83	1.35218	0.676092	−	0.736817i	$-0.263672\pi$
$197$	3738.83	1.35218	0.676092	+	0.736817i	$0.263672\pi$
$198$	−834.309	−0.299453
$199$	−349.030	−0.124332	−0.0621660	−	0.998066i	$-0.519801\pi$
$199$	−349.030	−0.124332	−0.0621660	+	0.998066i	$0.519801\pi$
$200$	2568.41	0.908069
$201$	−1001.07	−0.351295
$202$	1526.81	0.531810
$203$	−1516.19	−0.524214
$204$	−38.5565	−0.0132328
$205$	508.455	0.173230
$206$	−2075.66	−0.702030
$207$	1182.12	0.396924
$208$	−1781.40	−0.593836
$209$	−4995.77	−1.65342
$210$	−217.360	−0.0714252
$211$	2588.58	0.844574	0.422287	−	0.906462i	$-0.361227\pi$
$211$	2588.58	0.844574	0.422287	+	0.906462i	$0.361227\pi$
$212$	1601.92	0.518962
$213$	1770.74	0.569621
$214$	−265.565	−0.0848300
$215$	−1679.78	−0.532838
$216$	−664.886	−0.209443
$217$	−1762.56	−0.551384
$218$	1904.51	0.591695
$219$	1472.10	0.454226
$220$	−523.715	−0.160495
$221$	242.502	0.0738119
$222$	81.1942	0.0245468
$223$	−3236.21	−0.971804	−0.485902	−	0.874013i	$-0.661509\pi$
$223$	−3236.21	−0.971804	−0.485902	+	0.874013i	$0.661509\pi$
$224$	846.785	0.252581
$225$	−938.691	−0.278131
$226$	−2471.88	−0.727552
$227$	−5631.62	−1.64662	−0.823312	−	0.567589i	$-0.807876\pi$
$227$	−5631.62	−1.64662	−0.823312	+	0.567589i	$0.807876\pi$
$228$	−1038.93	−0.301775
$229$	3770.25	1.08797	0.543985	−	0.839095i	$-0.316915\pi$
$229$	3770.25	1.08797	0.543985	+	0.839095i	$0.316915\pi$
$230$	−1359.51	−0.389754
$231$	−855.733	−0.243736
$232$	5333.82	1.50941
$233$	−6560.90	−1.84472	−0.922358	−	0.386336i	$-0.873741\pi$
$233$	−6560.90	−1.84472	−0.922358	+	0.386336i	$0.873741\pi$
$234$	1091.26	0.304862
$235$	1195.22	0.331776
$236$	−2372.48	−0.654387
$237$	365.073	0.100059
$238$	72.4535	0.0197330
$239$	−771.444	−0.208789	−0.104394	−	0.994536i	$-0.533290\pi$
$239$	−771.444	−0.208789	−0.104394	+	0.994536i	$0.533290\pi$
$240$	456.204	0.122699
$241$	1252.10	0.334668	0.167334	−	0.985900i	$-0.446484\pi$
$241$	1252.10	0.334668	0.167334	+	0.985900i	$0.446484\pi$
$242$	749.574	0.199109
$243$	243.000	0.0641500
$244$	1372.25	0.360037
$245$	−222.942	−0.0581357
$246$	−762.683	−0.197670
$247$	6534.35	1.68328
$248$	6200.53	1.58764
$249$	1828.82	0.465450
$250$	2373.36	0.600418
$251$	5166.27	1.29917	0.649586	−	0.760288i	$-0.274942\pi$
$251$	5166.27	1.29917	0.649586	+	0.760288i	$0.274942\pi$
$252$	−177.959	−0.0444857
$253$	−5352.29	−1.33002
$254$	−1221.89	−0.301843
$255$	−62.1030	−0.0152511
$256$	−3734.21	−0.911672
$257$	2767.45	0.671707	0.335854	−	0.941914i	$-0.390975\pi$
$257$	2767.45	0.671707	0.335854	+	0.941914i	$0.390975\pi$
$258$	2519.67	0.608015
$259$	83.2791	0.0199796
$260$	685.007	0.163394
$261$	−1949.38	−0.462313
$262$	3406.44	0.803245
$263$	−4101.78	−0.961699	−0.480849	−	0.876803i	$-0.659672\pi$
$263$	−4101.78	−0.961699	−0.480849	+	0.876803i	$0.659672\pi$
$264$	3010.40	0.701807
$265$	2580.21	0.598117
$266$	1952.30	0.450013
$267$	2157.11	0.494432
$268$	942.594	0.214844
$269$	−6950.84	−1.57546	−0.787732	−	0.616018i	$-0.788745\pi$
$269$	−6950.84	−1.57546	−0.787732	+	0.616018i	$0.788745\pi$
$270$	−279.463	−0.0629911
$271$	−7140.29	−1.60052	−0.800262	−	0.599651i	$-0.795306\pi$
$271$	−7140.29	−1.60052	−0.800262	+	0.599651i	$0.795306\pi$
$272$	−152.068	−0.0338988
$273$	1119.28	0.248139
$274$	−3139.59	−0.692225
$275$	4250.10	0.931966
$276$	−1113.07	−0.242750
$277$	1320.51	0.286433	0.143217	−	0.989691i	$-0.454255\pi$
$277$	1320.51	0.286433	0.143217	+	0.989691i	$0.454255\pi$
$278$	−322.993	−0.0696829
$279$	−2266.15	−0.486275
$280$	784.291	0.167394
$281$	−204.309	−0.0433738	−0.0216869	−	0.999765i	$-0.506904\pi$
$281$	−204.309	−0.0433738	−0.0216869	+	0.999765i	$0.506904\pi$
$282$	−1792.82	−0.378585
$283$	975.794	0.204964	0.102482	−	0.994735i	$-0.467322\pi$
$283$	975.794	0.204964	0.102482	+	0.994735i	$0.467322\pi$
$284$	−1667.30	−0.348367
$285$	−1673.40	−0.347803
$286$	−4940.87	−1.02154
$287$	−782.267	−0.160891
$288$	1088.72	0.222756
$289$	−4892.30	−0.995786
$290$	2241.90	0.453961
$291$	−1913.63	−0.385495
$292$	−1386.11	−0.277794
$293$	607.919	0.121212	0.0606058	−	0.998162i	$-0.480697\pi$
$293$	607.919	0.121212	0.0606058	+	0.998162i	$0.480697\pi$
$294$	334.413	0.0663379
$295$	−3821.36	−0.754198
$296$	−292.969	−0.0575286
$297$	−1100.23	−0.214955
$298$	−4422.14	−0.859624
$299$	7000.67	1.35405
$300$	883.856	0.170098
$301$	2584.37	0.494886
$302$	−6039.37	−1.15075
$303$	2013.44	0.381747
$304$	−4097.56	−0.773064
$305$	2210.28	0.414952
$306$	93.1545	0.0174029
$307$	8037.08	1.49414	0.747069	−	0.664747i	$-0.231461\pi$
$307$	8037.08	1.49414	0.747069	+	0.664747i	$0.231461\pi$
$308$	805.744	0.149063
$309$	−2737.24	−0.503935
$310$	2606.19	0.477490
$311$	5311.60	0.968468	0.484234	−	0.874939i	$-0.339098\pi$
$311$	5311.60	0.968468	0.484234	+	0.874939i	$0.339098\pi$
$312$	−3937.53	−0.714483
$313$	−1531.61	−0.276587	−0.138293	−	0.990391i	$-0.544162\pi$
$313$	−1531.61	−0.276587	−0.138293	+	0.990391i	$0.544162\pi$
$314$	3788.23	0.680835
$315$	−286.640	−0.0512708
$316$	−343.747	−0.0611939
$317$	4219.19	0.747549	0.373775	−	0.927520i	$-0.378063\pi$
$317$	4219.19	0.747549	0.373775	+	0.927520i	$0.378063\pi$
$318$	−3870.31	−0.682505
$319$	8826.19	1.54913
$320$	−2468.64	−0.431253
$321$	−350.208	−0.0608931
$322$	2091.63	0.361993
$323$	557.801	0.0960893
$324$	−228.805	−0.0392327
$325$	−5559.03	−0.948799
$326$	−75.2387	−0.0127825
$327$	2511.53	0.424733
$328$	2751.95	0.463265
$329$	−1838.86	−0.308145
$330$	1265.32	0.211072
$331$	8298.19	1.37797	0.688987	−	0.724773i	$-0.258056\pi$
$331$	8298.19	1.37797	0.688987	+	0.724773i	$0.258056\pi$
$332$	−1721.99	−0.284658
$333$	107.073	0.0176203
$334$	3763.79	0.616604
$335$	1518.24	0.247613
$336$	−701.878	−0.113960
$337$	−4348.44	−0.702892	−0.351446	−	0.936208i	$-0.614310\pi$
$337$	−4348.44	−0.702892	−0.351446	+	0.936208i	$0.614310\pi$
$338$	1464.56	0.235684
$339$	−3259.73	−0.522255
$340$	58.4752	0.00932724
$341$	10260.4	1.62942
$342$	2510.10	0.396874
$343$	343.000	0.0539949
$344$	−9091.60	−1.42496
$345$	−1792.82	−0.279775
$346$	146.029	0.0226895
$347$	−8345.54	−1.29110	−0.645550	−	0.763718i	$-0.723372\pi$
$347$	−8345.54	−1.29110	−0.645550	+	0.763718i	$0.723372\pi$
$348$	1835.51	0.282740
$349$	−9982.54	−1.53110	−0.765549	−	0.643378i	$-0.777532\pi$
$349$	−9982.54	−1.53110	−0.765549	+	0.643378i	$0.777532\pi$
$350$	−1660.90	−0.253654
$351$	1439.07	0.218838
$352$	−4929.40	−0.746414
$353$	8801.59	1.32709	0.663543	−	0.748138i	$-0.269052\pi$
$353$	8801.59	1.32709	0.663543	+	0.748138i	$0.269052\pi$
$354$	5732.04	0.860606
$355$	−2685.53	−0.401502
$356$	−2031.10	−0.302383
$357$	95.5465	0.0141649
$358$	8905.56	1.31473
$359$	−524.039	−0.0770409	−0.0385205	−	0.999258i	$-0.512264\pi$
$359$	−524.039	−0.0770409	−0.0385205	+	0.999258i	$0.512264\pi$
$360$	1008.37	0.147628
$361$	8171.27	1.19132
$362$	−4681.88	−0.679763
$363$	988.485	0.142926
$364$	−1053.90	−0.151756
$365$	−2232.61	−0.320165
$366$	−3315.42	−0.473497
$367$	−6362.72	−0.904991	−0.452495	−	0.891767i	$-0.649466\pi$
$367$	−6362.72	−0.904991	−0.452495	+	0.891767i	$0.649466\pi$
$368$	−4389.99	−0.621858
$369$	−1005.77	−0.141893
$370$	−123.140	−0.0173020
$371$	−3969.70	−0.555516
$372$	2133.77	0.297394
$373$	−11265.8	−1.56387	−0.781935	−	0.623361i	$-0.785767\pi$
$373$	−11265.8	−1.56387	−0.781935	+	0.623361i	$0.785767\pi$
$374$	−421.774	−0.0583140
$375$	3129.82	0.430995
$376$	6468.96	0.887263
$377$	−11544.5	−1.57711
$378$	429.959	0.0585046
$379$	−1151.71	−0.156094	−0.0780470	−	0.996950i	$-0.524868\pi$
$379$	−1151.71	−0.156094	−0.0780470	+	0.996950i	$0.524868\pi$
$380$	1575.65	0.212708
$381$	−1611.34	−0.216670
$382$	973.774	0.130426
$383$	−151.554	−0.0202195	−0.0101097	−	0.999949i	$-0.503218\pi$
$383$	−151.554	−0.0202195	−0.0101097	+	0.999949i	$0.503218\pi$
$384$	799.692	0.106274
$385$	1297.81	0.171799
$386$	3651.08	0.481437
$387$	3322.76	0.436449
$388$	1801.84	0.235760
$389$	4794.18	0.624870	0.312435	−	0.949939i	$-0.398855\pi$
$389$	4794.18	0.624870	0.312435	+	0.949939i	$0.398855\pi$
$390$	−1655.01	−0.214884
$391$	597.608	0.0772950
$392$	−1206.65	−0.155471
$393$	4492.17	0.576590
$394$	8505.52	1.08757
$395$	−553.674	−0.0705275
$396$	1035.96	0.131462
$397$	−4623.94	−0.584556	−0.292278	−	0.956333i	$-0.594413\pi$
$397$	−4623.94	−0.584556	−0.292278	+	0.956333i	$0.594413\pi$
$398$	−794.014	−0.100001
$399$	2574.56	0.323030
$400$	3485.96	0.435745
$401$	−3610.63	−0.449642	−0.224821	−	0.974400i	$-0.572180\pi$
$401$	−3610.63	−0.449642	−0.224821	+	0.974400i	$0.572180\pi$
$402$	−2277.36	−0.282548
$403$	−13420.4	−1.65885
$404$	−1895.83	−0.233467
$405$	−368.537	−0.0452166
$406$	−3449.20	−0.421628
$407$	−484.794	−0.0590426
$408$	−336.125	−0.0407859
$409$	8959.57	1.08318	0.541592	−	0.840641i	$-0.317822\pi$
$409$	8959.57	1.08318	0.541592	+	0.840641i	$0.317822\pi$
$410$	1156.69	0.139329
$411$	−4140.27	−0.496896
$412$	2577.34	0.308195
$413$	5879.23	0.700480
$414$	2689.24	0.319248
$415$	−2773.62	−0.328076
$416$	6447.54	0.759896
$417$	−425.940	−0.0500201
$418$	−11365.0	−1.32985
$419$	−7078.28	−0.825290	−0.412645	−	0.910892i	$-0.635395\pi$
$419$	−7078.28	−0.825290	−0.412645	+	0.910892i	$0.635395\pi$
$420$	269.895	0.0313560
$421$	11551.5	1.33725	0.668626	−	0.743599i	$-0.266883\pi$
$421$	11551.5	1.33725	0.668626	+	0.743599i	$0.266883\pi$
$422$	5888.80	0.679295
$423$	−2364.25	−0.271758
$424$	13965.1	1.59954
$425$	−474.543	−0.0541617
$426$	4028.29	0.458149
$427$	−3400.56	−0.385397
$428$	329.750	0.0372408
$429$	−6515.67	−0.733286
$430$	−3821.36	−0.428564
$431$	−4064.38	−0.454232	−0.227116	−	0.973868i	$-0.572930\pi$
$431$	−4064.38	−0.454232	−0.227116	+	0.973868i	$0.572930\pi$
$432$	−902.415	−0.100503
$433$	17456.3	1.93740	0.968701	−	0.248229i	$-0.0798487\pi$
$433$	17456.3	1.93740	0.968701	+	0.248229i	$0.0798487\pi$
$434$	−4009.67	−0.443480
$435$	2956.46	0.325865
$436$	−2364.81	−0.259757
$437$	16102.9	1.76271
$438$	3348.91	0.365336
$439$	4595.39	0.499604	0.249802	−	0.968297i	$-0.419635\pi$
$439$	4595.39	0.499604	0.249802	+	0.968297i	$0.419635\pi$
$440$	−4565.60	−0.494674
$441$	441.000	0.0476190
$442$	551.671	0.0593672
$443$	−306.214	−0.0328412	−0.0164206	−	0.999865i	$-0.505227\pi$
$443$	−306.214	−0.0328412	−0.0164206	+	0.999865i	$0.505227\pi$
$444$	−100.818	−0.0107762
$445$	−3271.50	−0.348504
$446$	−7362.10	−0.781627
$447$	−5831.61	−0.617060
$448$	3798.04	0.400537
$449$	9229.22	0.970053	0.485026	−	0.874500i	$-0.338810\pi$
$449$	9229.22	0.970053	0.485026	+	0.874500i	$0.338810\pi$
$450$	−2135.44	−0.223702
$451$	4553.82	0.475457
$452$	3069.31	0.319399
$453$	−7964.29	−0.826038
$454$	−12811.5	−1.32439
$455$	−1697.51	−0.174902
$456$	−9057.08	−0.930124
$457$	−10992.2	−1.12515	−0.562577	−	0.826745i	$-0.690190\pi$
$457$	−10992.2	−1.12515	−0.562577	+	0.826745i	$0.690190\pi$
$458$	8577.01	0.875059
$459$	122.846	0.0124922
$460$	1688.09	0.171104
$461$	7387.88	0.746394	0.373197	−	0.927752i	$-0.378261\pi$
$461$	7387.88	0.746394	0.373197	+	0.927752i	$0.378261\pi$
$462$	−1946.72	−0.196038
$463$	10163.8	1.02020	0.510101	−	0.860114i	$-0.329608\pi$
$463$	10163.8	1.02020	0.510101	+	0.860114i	$0.329608\pi$
$464$	7239.30	0.724302
$465$	3436.86	0.342754
$466$	−14925.5	−1.48371
$467$	−15814.6	−1.56705	−0.783524	−	0.621362i	$-0.786580\pi$
$467$	−15814.6	−1.56705	−0.783524	+	0.621362i	$0.786580\pi$
$468$	−1355.01	−0.133836
$469$	−2335.84	−0.229976
$470$	2719.02	0.266849
$471$	4995.65	0.488720
$472$	−20682.6	−2.01694
$473$	−15044.4	−1.46246
$474$	830.511	0.0804782
$475$	−12786.9	−1.23516
$476$	−89.9651	−0.00866290
$477$	−5103.90	−0.489919
$478$	−1754.97	−0.167930
$479$	−1444.85	−0.137823	−0.0689113	−	0.997623i	$-0.521953\pi$
$479$	−1444.85	−0.137823	−0.0689113	+	0.997623i	$0.521953\pi$
$480$	−1651.17	−0.157011
$481$	634.099	0.0601090
$482$	2848.43	0.269175
$483$	2758.29	0.259848
$484$	−930.742	−0.0874100
$485$	2902.24	0.271719
$486$	552.805	0.0515962
$487$	−489.402	−0.0455378	−0.0227689	−	0.999741i	$-0.507248\pi$
$487$	−489.402	−0.0455378	−0.0227689	+	0.999741i	$0.507248\pi$
$488$	11962.9	1.10970
$489$	−99.2195	−0.00917559
$490$	−507.174	−0.0467588
$491$	−3941.30	−0.362257	−0.181129	−	0.983459i	$-0.557975\pi$
$491$	−3941.30	−0.362257	−0.181129	+	0.983459i	$0.557975\pi$
$492$	947.019	0.0867783
$493$	−985.485	−0.0900284
$494$	14865.1	1.35387
$495$	1668.62	0.151513
$496$	8415.65	0.761843
$497$	4131.73	0.372905
$498$	4160.42	0.374363
$499$	11.0894	0.000994850 0	0.000497425	−	1.00000i	$-0.499842\pi$
$499$	11.0894	0.000994850 0	0.000497425		1.00000i	$0.499842\pi$
$500$	−2946.99	−0.263586
$501$	4963.43	0.442614
$502$	11752.8	1.04493
$503$	7088.41	0.628343	0.314172	−	0.949366i	$-0.398273\pi$
$503$	7088.41	0.628343	0.314172	+	0.949366i	$0.398273\pi$
$504$	−1551.40	−0.137113
$505$	−3053.61	−0.269077
$506$	−12176.0	−1.06974
$507$	1931.35	0.169180
$508$	1517.21	0.132511
$509$	17588.4	1.53162	0.765810	−	0.643067i	$-0.222338\pi$
$509$	17588.4	1.53162	0.765810	+	0.643067i	$0.222338\pi$
$510$	−141.279	−0.0122666
$511$	3434.91	0.297361
$512$	−10627.5	−0.917333
$513$	3310.15	0.284886
$514$	6295.72	0.540257
$515$	4151.32	0.355202
$516$	−3128.66	−0.266922
$517$	10704.6	0.910613
$518$	189.453	0.0160697
$519$	192.573	0.0162871
$520$	5971.70	0.503609
$521$	−11646.6	−0.979360	−0.489680	−	0.871902i	$-0.662886\pi$
$521$	−11646.6	−0.979360	−0.489680	+	0.871902i	$0.662886\pi$
$522$	−4434.68	−0.371841
$523$	8965.82	0.749614	0.374807	−	0.927103i	$-0.377709\pi$
$523$	8965.82	0.749614	0.374807	+	0.927103i	$0.377709\pi$
$524$	−4229.75	−0.352629
$525$	−2190.28	−0.182079
$526$	−9331.22	−0.773499
$527$	−1145.62	−0.0946946
$528$	4085.85	0.336769
$529$	5085.08	0.417941
$530$	5869.76	0.481068
$531$	7559.01	0.617765
$532$	−2424.16	−0.197558
$533$	−5956.30	−0.484045
$534$	4907.26	0.397674
$535$	531.129	0.0429210
$536$	8217.28	0.662187
$537$	11744.0	0.943747
$538$	−15812.6	−1.26715
$539$	−1996.71	−0.159563
$540$	347.008	0.0276534
$541$	−195.272	−0.0155183	−0.00775914	−	0.999970i	$-0.502470\pi$
$541$	−195.272	−0.0155183	−0.00775914	+	0.999970i	$0.502470\pi$
$542$	−16243.6	−1.28731
$543$	−6174.13	−0.487951
$544$	550.390	0.0433783
$545$	−3809.01	−0.299376
$546$	2546.27	0.199579
$547$	−1399.26	−0.109375	−0.0546874	−	0.998504i	$-0.517416\pi$
$547$	−1399.26	−0.109375	−0.0546874	+	0.998504i	$0.517416\pi$
$548$	3898.41	0.303890
$549$	−4372.15	−0.339888
$550$	9668.62	0.749584
$551$	−26554.5	−2.05310
$552$	−9703.44	−0.748199
$553$	851.837	0.0655042
$554$	3004.06	0.230380
$555$	−162.388	−0.0124198
$556$	401.059	0.0305911
$557$	43.0467	0.00327459	0.00163730	−	0.999999i	$-0.499479\pi$
$557$	43.0467	0.00327459	0.00163730	+	0.999999i	$0.499479\pi$
$558$	−5155.30	−0.391113
$559$	19677.8	1.48888
$560$	1064.48	0.0803256
$561$	−556.206	−0.0418592
$562$	−464.786	−0.0348858
$563$	−19232.9	−1.43973	−0.719865	−	0.694114i	$-0.755797\pi$
$563$	−19232.9	−1.43973	−0.719865	+	0.694114i	$0.755797\pi$
$564$	2226.14	0.166201
$565$	4943.75	0.368115
$566$	2219.85	0.164854
$567$	567.000	0.0419961
$568$	−14535.1	−1.07373
$569$	5163.98	0.380466	0.190233	−	0.981739i	$-0.439076\pi$
$569$	5163.98	0.380466	0.190233	+	0.981739i	$0.439076\pi$
$570$	−3806.85	−0.279739
$571$	−10231.9	−0.749899	−0.374950	−	0.927045i	$-0.622340\pi$
$571$	−10231.9	−0.749899	−0.374950	+	0.927045i	$0.622340\pi$
$572$	6135.05	0.448460
$573$	1284.14	0.0936229
$574$	−1779.59	−0.129406
$575$	−13699.4	−0.993572
$576$	4883.20	0.353240
$577$	16563.7	1.19507	0.597537	−	0.801842i	$-0.296146\pi$
$577$	16563.7	1.19507	0.597537	+	0.801842i	$0.296146\pi$
$578$	−11129.6	−0.800916
$579$	4814.78	0.345588
$580$	−2783.75	−0.199291
$581$	4267.26	0.304708
$582$	−4353.35	−0.310055
$583$	23108.8	1.64163
$584$	−12083.7	−0.856212
$585$	−2182.51	−0.154249
$586$	1382.96	0.0974910
$587$	16020.6	1.12648	0.563239	−	0.826294i	$-0.309555\pi$
$587$	16020.6	1.12648	0.563239	+	0.826294i	$0.309555\pi$
$588$	−415.238	−0.0291227
$589$	−30869.4	−2.15951
$590$	−8693.28	−0.606605
$591$	11216.5	0.780684
$592$	−397.631	−0.0276057
$593$	−6771.14	−0.468900	−0.234450	−	0.972128i	$-0.575329\pi$
$593$	−6771.14	−0.468900	−0.234450	+	0.972128i	$0.575329\pi$
$594$	−2502.93	−0.172889
$595$	−144.907	−0.00998421
$596$	5490.95	0.377379
$597$	−1047.09	−0.0717831
$598$	15926.0	1.08906
$599$	11070.2	0.755120	0.377560	−	0.925985i	$-0.376763\pi$
$599$	11070.2	0.755120	0.377560	+	0.925985i	$0.376763\pi$
$600$	7705.22	0.524274
$601$	−24187.7	−1.64166	−0.820830	−	0.571173i	$-0.806489\pi$
$601$	−24187.7	−1.64166	−0.820830	+	0.571173i	$0.806489\pi$
$602$	5879.23	0.398039
$603$	−3003.22	−0.202820
$604$	7499.05	0.505185
$605$	−1499.15	−0.100742
$606$	4580.42	0.307041
$607$	−10074.1	−0.673631	−0.336816	−	0.941571i	$-0.609350\pi$
$607$	−10074.1	−0.673631	−0.336816	+	0.941571i	$0.609350\pi$
$608$	14830.6	0.989243
$609$	−4548.56	−0.302655
$610$	5028.21	0.333748
$611$	−14001.3	−0.927060
$612$	−115.669	−0.00763996
$613$	−11114.6	−0.732323	−0.366161	−	0.930551i	$-0.619328\pi$
$613$	−11114.6	−0.732323	−0.366161	+	0.930551i	$0.619328\pi$
$614$	18283.7	1.20174
$615$	1525.37	0.100014
$616$	7024.26	0.459441
$617$	20496.4	1.33737	0.668683	−	0.743548i	$-0.266858\pi$
$617$	20496.4	1.33737	0.668683	+	0.743548i	$0.266858\pi$
$618$	−6226.98	−0.405317
$619$	−16714.4	−1.08532	−0.542658	−	0.839954i	$-0.682582\pi$
$619$	−16714.4	−1.08532	−0.542658	+	0.839954i	$0.682582\pi$
$620$	−3236.10	−0.209621
$621$	3546.37	0.229164
$622$	12083.5	0.778943
$623$	5033.27	0.323682
$624$	−5344.20	−0.342851
$625$	8290.66	0.530602
$626$	−3484.28	−0.222460
$627$	−14987.3	−0.954602
$628$	−4703.82	−0.298890
$629$	54.1295	0.00343129
$630$	−652.081	−0.0412374
$631$	9168.53	0.578437	0.289218	−	0.957263i	$-0.406605\pi$
$631$	9168.53	0.578437	0.289218	+	0.957263i	$0.406605\pi$
$632$	−2996.69	−0.188611
$633$	7765.73	0.487615
$634$	9598.30	0.601257
$635$	2443.77	0.152722
$636$	4805.75	0.299623
$637$	2611.65	0.162445
$638$	20078.9	1.24597
$639$	5312.23	0.328871
$640$	−1212.82	−0.0749078
$641$	−4273.37	−0.263319	−0.131660	−	0.991295i	$-0.542031\pi$
$641$	−4273.37	−0.263319	−0.131660	+	0.991295i	$0.542031\pi$
$642$	−796.694	−0.0489766
$643$	2955.75	0.181281	0.0906404	−	0.995884i	$-0.471109\pi$
$643$	2955.75	0.181281	0.0906404	+	0.995884i	$0.471109\pi$
$644$	−2597.16	−0.158917
$645$	−5039.34	−0.307634
$646$	1268.95	0.0772851
$647$	22701.2	1.37941	0.689704	−	0.724091i	$-0.257741\pi$
$647$	22701.2	1.37941	0.689704	+	0.724091i	$0.257741\pi$
$648$	−1994.66	−0.120922
$649$	−34224.8	−2.07002
$650$	−12646.3	−0.763124
$651$	−5287.67	−0.318341
$652$	93.4235	0.00561158
$653$	1537.81	0.0921582	0.0460791	−	0.998938i	$-0.485327\pi$
$653$	1537.81	0.0921582	0.0460791	+	0.998938i	$0.485327\pi$
$654$	5713.52	0.341615
$655$	−6812.87	−0.406414
$656$	3735.08	0.222302
$657$	4416.31	0.262248
$658$	−4183.26	−0.247842
$659$	12338.1	0.729323	0.364661	−	0.931140i	$-0.381185\pi$
$659$	12338.1	0.729323	0.364661	+	0.931140i	$0.381185\pi$
$660$	−1571.14	−0.0926616
$661$	1845.10	0.108572	0.0542859	−	0.998525i	$-0.482712\pi$
$661$	1845.10	0.108572	0.0542859	+	0.998525i	$0.482712\pi$
$662$	18877.7	1.10831
$663$	727.505	0.0426153
$664$	−15011.8	−0.877369
$665$	−3904.60	−0.227690
$666$	243.583	0.0141721
$667$	−28449.5	−1.65153
$668$	−4673.48	−0.270692
$669$	−9708.62	−0.561072
$670$	3453.87	0.199156
$671$	19795.7	1.13890
$672$	2540.36	0.145828
$673$	23955.4	1.37208	0.686041	−	0.727563i	$-0.259347\pi$
$673$	23955.4	1.37208	0.686041	+	0.727563i	$0.259347\pi$
$674$	−9892.34	−0.565339
$675$	−2816.07	−0.160579
$676$	−1818.53	−0.103467
$677$	−3678.26	−0.208814	−0.104407	−	0.994535i	$-0.533294\pi$
$677$	−3678.26	−0.208814	−0.104407	+	0.994535i	$0.533294\pi$
$678$	−7415.63	−0.420052
$679$	−4465.14	−0.252366
$680$	509.771	0.0287482
$681$	−16894.9	−0.950679
$682$	23341.6	1.31055
$683$	4390.87	0.245991	0.122996	−	0.992407i	$-0.460750\pi$
$683$	4390.87	0.245991	0.122996	+	0.992407i	$0.460750\pi$
$684$	−3116.78	−0.174230
$685$	6279.18	0.350241
$686$	780.297	0.0434284
$687$	11310.7	0.628140
$688$	−12339.6	−0.683781
$689$	−30225.8	−1.67128
$690$	−4078.53	−0.225024
$691$	10371.7	0.570994	0.285497	−	0.958380i	$-0.407841\pi$
$691$	10371.7	0.570994	0.285497	+	0.958380i	$0.407841\pi$
$692$	−181.323	−0.00996081
$693$	−2567.20	−0.140721
$694$	−18985.4	−1.03844
$695$	645.986	0.0352570
$696$	16001.4	0.871456
$697$	−508.455	−0.0276314
$698$	−22709.4	−1.23147
$699$	−19682.7	−1.06505
$700$	2062.33	0.111355
$701$	109.675	0.00590922	0.00295461	−	0.999996i	$-0.499060\pi$
$701$	109.675	0.00590922	0.00295461	+	0.999996i	$0.499060\pi$
$702$	3273.77	0.176012
$703$	1458.55	0.0782508
$704$	−22109.6	−1.18364
$705$	3585.65	0.191551
$706$	20022.9	1.06738
$707$	4698.03	0.249912
$708$	−7117.45	−0.377811
$709$	26918.8	1.42589	0.712944	−	0.701221i	$-0.247361\pi$
$709$	26918.8	1.42589	0.712944	+	0.701221i	$0.247361\pi$
$710$	−6109.35	−0.322930
$711$	1095.22	0.0577693
$712$	−17706.6	−0.931999
$713$	−33072.4	−1.73713
$714$	217.360	0.0113929
$715$	9881.74	0.516862
$716$	−11058.0	−0.577174
$717$	−2314.33	−0.120544
$718$	−1192.14	−0.0619644
$719$	15170.8	0.786889	0.393445	−	0.919348i	$-0.371283\pi$
$719$	15170.8	0.786889	0.393445	+	0.919348i	$0.371283\pi$
$720$	1368.61	0.0708405
$721$	−6386.88	−0.329903
$722$	18589.0	0.958185
$723$	3756.31	0.193221
$724$	5813.46	0.298419
$725$	22591.0	1.15725
$726$	2248.72	0.114956
$727$	−33286.9	−1.69813	−0.849066	−	0.528288i	$-0.822834\pi$
$727$	−33286.9	−1.69813	−0.849066	+	0.528288i	$0.822834\pi$
$728$	−9187.57	−0.467739
$729$	729.000	0.0370370
$730$	−5079.00	−0.257510
$731$	1679.78	0.0849917
$732$	4116.74	0.207868
$733$	20544.0	1.03521	0.517607	−	0.855619i	$-0.326823\pi$
$733$	20544.0	1.03521	0.517607	+	0.855619i	$0.326823\pi$
$734$	−14474.7	−0.727888
$735$	−668.826	−0.0335646
$736$	15889.0	0.795755
$737$	13597.6	0.679614
$738$	−2288.05	−0.114125
$739$	34357.2	1.71022	0.855109	−	0.518449i	$-0.173490\pi$
$739$	34357.2	1.71022	0.855109	+	0.518449i	$0.173490\pi$
$740$	152.902	0.00759568
$741$	19603.1	0.971844
$742$	−9030.73	−0.446804
$743$	8166.99	0.403254	0.201627	−	0.979462i	$-0.435377\pi$
$743$	8166.99	0.403254	0.201627	+	0.979462i	$0.435377\pi$
$744$	18601.6	0.916623
$745$	8844.29	0.434939
$746$	−25628.9	−1.25783
$747$	5486.47	0.268728
$748$	523.715	0.0256001
$749$	−817.151	−0.0398639
$750$	7120.08	0.346651
$751$	17080.1	0.829909	0.414954	−	0.909842i	$-0.363798\pi$
$751$	17080.1	0.829909	0.414954	+	0.909842i	$0.363798\pi$
$752$	8779.97	0.425761
$753$	15498.8	0.750077
$754$	−26262.7	−1.26848
$755$	12078.7	0.582239
$756$	−533.878	−0.0256838
$757$	−16324.0	−0.783758	−0.391879	−	0.920017i	$-0.628175\pi$
$757$	−16324.0	−0.783758	−0.391879	+	0.920017i	$0.628175\pi$
$758$	−2620.06	−0.125547
$759$	−16056.9	−0.767888
$760$	13736.1	0.655605
$761$	−32366.2	−1.54175	−0.770875	−	0.636986i	$-0.780181\pi$
$761$	−32366.2	−1.54175	−0.770875	+	0.636986i	$0.780181\pi$
$762$	−3665.66	−0.174269
$763$	5860.23	0.278053
$764$	−1209.13	−0.0572576
$765$	−186.309	−0.00880525
$766$	−344.774	−0.0162626
$767$	44765.3	2.10741
$768$	−11202.6	−0.526354
$769$	−7948.44	−0.372728	−0.186364	−	0.982481i	$-0.559670\pi$
$769$	−7948.44	−0.372728	−0.186364	+	0.982481i	$0.559670\pi$
$770$	2952.42	0.138179
$771$	8302.35	0.387810
$772$	−4533.52	−0.211354
$773$	17819.3	0.829127	0.414564	−	0.910020i	$-0.363934\pi$
$773$	17819.3	0.829127	0.414564	+	0.910020i	$0.363934\pi$
$774$	7559.01	0.351038
$775$	26261.9	1.21723
$776$	15708.0	0.726655
$777$	249.837	0.0115352
$778$	10906.4	0.502586
$779$	−13700.6	−0.630136
$780$	2055.02	0.0943353
$781$	−24052.1	−1.10199
$782$	1359.51	0.0621687
$783$	−5848.15	−0.266917
$784$	−1637.72	−0.0746044
$785$	−7576.46	−0.344478
$786$	10219.3	0.463754
$787$	2912.38	0.131912	0.0659562	−	0.997823i	$-0.478990\pi$
$787$	2912.38	0.131912	0.0659562	+	0.997823i	$0.478990\pi$
$788$	−10561.3	−0.477448
$789$	−12305.3	−0.555237
$790$	−1259.56	−0.0567256
$791$	−7606.05	−0.341896
$792$	9031.19	0.405188
$793$	−25892.3	−1.15948
$794$	−10519.1	−0.470162
$795$	7740.63	0.345323
$796$	985.923	0.0439009
$797$	−33789.1	−1.50172	−0.750861	−	0.660460i	$-0.770361\pi$
$797$	−33789.1	−1.50172	−0.750861	+	0.660460i	$0.770361\pi$
$798$	5856.91	0.259815
$799$	−1195.22	−0.0529208
$800$	−12617.0	−0.557597
$801$	6471.34	0.285460
$802$	−8213.90	−0.361649
$803$	−19995.7	−0.878744
$804$	2827.78	0.124040
$805$	−4183.26	−0.183156
$806$	−30530.2	−1.33422
$807$	−20852.5	−0.909595
$808$	−16527.3	−0.719589
$809$	1252.13	0.0544159	0.0272079	−	0.999630i	$-0.491338\pi$
$809$	1252.13	0.0544159	0.0272079	+	0.999630i	$0.491338\pi$
$810$	−838.390	−0.0363679
$811$	−31913.1	−1.38178	−0.690889	−	0.722961i	$-0.742781\pi$
$811$	−31913.1	−1.38178	−0.690889	+	0.722961i	$0.742781\pi$
$812$	4282.85	0.185097
$813$	−21420.9	−0.924063
$814$	−1102.87	−0.0474882
$815$	150.477	0.00646748
$816$	−456.204	−0.0195715
$817$	45262.7	1.93824
$818$	20382.3	0.871210
$819$	3357.84	0.143263
$820$	−1436.26	−0.0611663
$821$	30742.4	1.30684	0.653421	−	0.756995i	$-0.273333\pi$
$821$	30742.4	1.30684	0.653421	+	0.756995i	$0.273333\pi$
$822$	−9418.77	−0.399656
$823$	−13822.6	−0.585449	−0.292724	−	0.956197i	$-0.594562\pi$
$823$	−13822.6	−0.585449	−0.292724	+	0.956197i	$0.594562\pi$
$824$	22468.5	0.949913
$825$	12750.3	0.538071
$826$	13374.8	0.563399
$827$	−42107.1	−1.77051	−0.885253	−	0.465110i	$-0.846015\pi$
$827$	−42107.1	−1.77051	−0.885253	+	0.465110i	$0.846015\pi$
$828$	−3339.21	−0.140152
$829$	−38763.8	−1.62403	−0.812015	−	0.583636i	$-0.801629\pi$
$829$	−38763.8	−1.62403	−0.812015	+	0.583636i	$0.801629\pi$
$830$	−6309.75	−0.263873
$831$	3961.54	0.165372
$832$	28918.8	1.20502
$833$	222.942	0.00927308
$834$	−968.979	−0.0402314
$835$	−7527.59	−0.311980
$836$	14111.8	0.583812
$837$	−6798.44	−0.280751
$838$	−16102.5	−0.663784
$839$	16896.3	0.695262	0.347631	−	0.937631i	$-0.386986\pi$
$839$	16896.3	0.695262	0.347631	+	0.937631i	$0.386986\pi$
$840$	2352.87	0.0966450
$841$	22525.7	0.923601
$842$	26278.6	1.07556
$843$	−612.927	−0.0250419
$844$	−7312.09	−0.298214
$845$	−2929.11	−0.119248
$846$	−5378.47	−0.218576
$847$	2306.47	0.0935668
$848$	18954.0	0.767552
$849$	2927.38	0.118336
$850$	−1079.55	−0.0435625
$851$	1562.64	0.0629455
$852$	−5001.91	−0.201130
$853$	46429.3	1.86367	0.931833	−	0.362887i	$-0.118209\pi$
$853$	46429.3	1.86367	0.931833	+	0.362887i	$0.118209\pi$
$854$	−7735.99	−0.309977
$855$	−5020.21	−0.200804
$856$	2874.67	0.114783
$857$	21206.4	0.845272	0.422636	−	0.906300i	$-0.361105\pi$
$857$	21206.4	0.845272	0.422636	+	0.906300i	$0.361105\pi$
$858$	−14822.6	−0.589785
$859$	13876.2	0.551163	0.275581	−	0.961278i	$-0.411130\pi$
$859$	13876.2	0.551163	0.275581	+	0.961278i	$0.411130\pi$
$860$	4744.96	0.188142
$861$	−2346.80	−0.0928906
$862$	−9246.12	−0.365341
$863$	14337.1	0.565515	0.282757	−	0.959191i	$-0.408751\pi$
$863$	14337.1	0.565515	0.282757	+	0.959191i	$0.408751\pi$
$864$	3266.17	0.128608
$865$	−292.058	−0.0114801
$866$	39711.6	1.55826
$867$	−14676.9	−0.574918
$868$	4978.79	0.194690
$869$	−4958.81	−0.193574
$870$	6725.69	0.262095
$871$	−17785.4	−0.691889
$872$	−20615.8	−0.800619
$873$	−5740.89	−0.222566
$874$	36632.8	1.41776
$875$	7302.91	0.282152
$876$	−4158.33	−0.160384
$877$	−24369.3	−0.938304	−0.469152	−	0.883118i	$-0.655440\pi$
$877$	−24369.3	−0.938304	−0.469152	+	0.883118i	$0.655440\pi$
$878$	10454.1	0.401834
$879$	1823.76	0.0699815
$880$	−6196.65	−0.237374
$881$	−26127.0	−0.999140	−0.499570	−	0.866273i	$-0.666509\pi$
$881$	−26127.0	−0.999140	−0.499570	+	0.866273i	$0.666509\pi$
$882$	1003.24	0.0383002
$883$	−15713.1	−0.598855	−0.299428	−	0.954119i	$-0.596796\pi$
$883$	−15713.1	−0.598855	−0.299428	+	0.954119i	$0.596796\pi$
$884$	−685.007	−0.0260625
$885$	−11464.1	−0.435436
$886$	−696.611	−0.0264143
$887$	13139.5	0.497385	0.248692	−	0.968583i	$-0.419999\pi$
$887$	13139.5	0.497385	0.248692	+	0.968583i	$0.419999\pi$
$888$	−878.907	−0.0332142
$889$	−3759.79	−0.141844
$890$	−7442.40	−0.280303
$891$	−3300.68	−0.124104
$892$	9141.48	0.343138
$893$	−32205.8	−1.20686
$894$	−13266.4	−0.496304
$895$	−17811.1	−0.665207
$896$	1865.95	0.0695725
$897$	21002.0	0.781758
$898$	20995.7	0.780218
$899$	54538.1	2.02330
$900$	2651.57	0.0982063
$901$	−2580.21	−0.0954043
$902$	10359.6	0.382412
$903$	7753.12	0.285723
$904$	26757.4	0.984446
$905$	9363.76	0.343936
$906$	−18118.1	−0.664386
$907$	−3799.71	−0.139104	−0.0695519	−	0.997578i	$-0.522157\pi$
$907$	−3799.71	−0.139104	−0.0695519	+	0.997578i	$0.522157\pi$
$908$	15907.9	0.581413
$909$	6040.33	0.220402
$910$	−3861.70	−0.140675
$911$	−51528.4	−1.87400	−0.936998	−	0.349334i	$-0.886408\pi$
$911$	−51528.4	−1.87400	−0.936998	+	0.349334i	$0.886408\pi$
$912$	−12292.7	−0.446329
$913$	−24841.0	−0.900458
$914$	−25006.4	−0.904966
$915$	6630.85	0.239573
$916$	−10650.0	−0.384156
$917$	10481.7	0.377467
$918$	279.463	0.0100476
$919$	−16984.7	−0.609657	−0.304828	−	0.952407i	$-0.598599\pi$
$919$	−16984.7	−0.609657	−0.304828	+	0.952407i	$0.598599\pi$
$920$	14716.3	0.527373
$921$	24111.2	0.862641
$922$	16806.8	0.600329
$923$	31459.6	1.12189
$924$	2417.23	0.0860618
$925$	−1240.85	−0.0441068
$926$	23121.9	0.820553
$927$	−8211.71	−0.290947
$928$	−26201.7	−0.926846
$929$	−5451.85	−0.192540	−0.0962699	−	0.995355i	$-0.530691\pi$
$929$	−5451.85	−0.192540	−0.0962699	+	0.995355i	$0.530691\pi$
$930$	7818.58	0.275679
$931$	6007.30	0.211473
$932$	18532.9	0.651358
$933$	15934.8	0.559145
$934$	−35976.8	−1.26038
$935$	843.548	0.0295048
$936$	−11812.6	−0.412507
$937$	42429.4	1.47930	0.739652	−	0.672989i	$-0.234990\pi$
$937$	42429.4	1.47930	0.739652	+	0.672989i	$0.234990\pi$
$938$	−5313.84	−0.184971
$939$	−4594.82	−0.159687
$940$	−3376.19	−0.117148
$941$	−32977.9	−1.14245	−0.571226	−	0.820793i	$-0.693532\pi$
$941$	−32977.9	−1.14245	−0.571226	+	0.820793i	$0.693532\pi$
$942$	11364.7	0.393080
$943$	−14678.4	−0.506886
$944$	−28071.5	−0.967848
$945$	−859.919	−0.0296012
$946$	−34224.8	−1.17626
$947$	−23753.4	−0.815082	−0.407541	−	0.913187i	$-0.633614\pi$
$947$	−23753.4	−0.815082	−0.407541	+	0.913187i	$0.633614\pi$
$948$	−1031.24	−0.0353303
$949$	26153.9	0.894616
$950$	−29089.0	−0.993445
$951$	12657.6	0.431598
$952$	−784.291	−0.0267006
$953$	−28074.3	−0.954267	−0.477134	−	0.878831i	$-0.658324\pi$
$953$	−28074.3	−0.954267	−0.477134	+	0.878831i	$0.658324\pi$
$954$	−11610.9	−0.394044
$955$	−1947.55	−0.0659908
$956$	2179.14	0.0737221
$957$	26478.6	0.894389
$958$	−3286.92	−0.110851
$959$	−9660.63	−0.325295
$960$	−7405.91	−0.248984
$961$	33609.2	1.12817
$962$	1442.52	0.0483460
$963$	−1050.62	−0.0351567
$964$	−3536.88	−0.118169
$965$	−7302.15	−0.243590
$966$	6274.88	0.208997
$967$	−11150.3	−0.370806	−0.185403	−	0.982663i	$-0.559359\pi$
$967$	−11150.3	−0.370806	−0.185403	+	0.982663i	$0.559359\pi$
$968$	−8113.95	−0.269414
$969$	1673.40	0.0554772
$970$	6602.35	0.218545
$971$	6059.04	0.200251	0.100126	−	0.994975i	$-0.468076\pi$
$971$	6059.04	0.200251	0.100126	+	0.994975i	$0.468076\pi$
$972$	−686.415	−0.0226510
$973$	−993.861	−0.0327459
$974$	−1113.35	−0.0366263
$975$	−16677.1	−0.547789
$976$	16236.6	0.532500
$977$	5700.49	0.186668	0.0933341	−	0.995635i	$-0.470248\pi$
$977$	5700.49	0.186668	0.0933341	+	0.995635i	$0.470248\pi$
$978$	−225.716	−0.00737997
$979$	−29300.2	−0.956526
$980$	629.755	0.0205273
$981$	7534.59	0.245220
$982$	−8966.12	−0.291365
$983$	197.480	0.00640757	0.00320378	−	0.999995i	$-0.498980\pi$
$983$	197.480	0.00640757	0.00320378	+	0.999995i	$0.498980\pi$
$984$	8255.85	0.267466
$985$	−17011.0	−0.550271
$986$	−2241.90	−0.0724103
$987$	−5516.58	−0.177908
$988$	−18457.9	−0.594357
$989$	48492.9	1.55913
$990$	3795.97	0.121862
$991$	20620.8	0.660990	0.330495	−	0.943808i	$-0.392784\pi$
$991$	20620.8	0.660990	0.330495	+	0.943808i	$0.392784\pi$
$992$	−30459.3	−0.974884
$993$	24894.6	0.795574
$994$	9399.35	0.299929
$995$	1588.03	0.0505969
$996$	−5165.97	−0.164348
$997$	19326.8	0.613928	0.306964	−	0.951721i	$-0.400687\pi$
$997$	19326.8	0.613928	0.306964	+	0.951721i	$0.400687\pi$
$998$	25.2275	0.000800162 0
$999$	321.220	0.0101731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.4.a.c.1.2	✓	2
3.2	odd	2		63.4.a.e.1.1		2
4.3	odd	2		336.4.a.m.1.1		2
5.2	odd	4		525.4.d.g.274.3		4
5.3	odd	4		525.4.d.g.274.2		4
5.4	even	2		525.4.a.n.1.1		2
7.2	even	3		147.4.e.l.67.1		4
7.3	odd	6		147.4.e.m.79.1		4
7.4	even	3		147.4.e.l.79.1		4
7.5	odd	6		147.4.e.m.67.1		4
7.6	odd	2		147.4.a.i.1.2		2
8.3	odd	2		1344.4.a.bo.1.2		2
8.5	even	2		1344.4.a.bg.1.2		2
12.11	even	2		1008.4.a.ba.1.2		2
15.14	odd	2		1575.4.a.p.1.2		2
21.2	odd	6		441.4.e.q.361.2		4
21.5	even	6		441.4.e.p.361.2		4
21.11	odd	6		441.4.e.q.226.2		4
21.17	even	6		441.4.e.p.226.2		4
21.20	even	2		441.4.a.r.1.1		2
28.27	even	2		2352.4.a.bz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.c.1.2	✓	2	1.1	even	1	trivial
63.4.a.e.1.1		2	3.2	odd	2
147.4.a.i.1.2		2	7.6	odd	2
147.4.e.l.67.1		4	7.2	even	3
147.4.e.l.79.1		4	7.4	even	3
147.4.e.m.67.1		4	7.5	odd	6
147.4.e.m.79.1		4	7.3	odd	6
336.4.a.m.1.1		2	4.3	odd	2
441.4.a.r.1.1		2	21.20	even	2
441.4.e.p.226.2		4	21.17	even	6
441.4.e.p.361.2		4	21.5	even	6
441.4.e.q.226.2		4	21.11	odd	6
441.4.e.q.361.2		4	21.2	odd	6
525.4.a.n.1.1		2	5.4	even	2
525.4.d.g.274.2		4	5.3	odd	4
525.4.d.g.274.3		4	5.2	odd	4
1008.4.a.ba.1.2		2	12.11	even	2
1344.4.a.bg.1.2		2	8.5	even	2
1344.4.a.bo.1.2		2	8.3	odd	2
1575.4.a.p.1.2		2	15.14	odd	2
2352.4.a.bz.1.2		2	28.27	even	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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)

\(f(q)\)	\(=\)	\(q+2.27492 q^{2} +3.00000 q^{3} -2.82475 q^{4} -4.54983 q^{5} +6.82475 q^{6} +7.00000 q^{7} -24.6254 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.27492 q^{2} +3.00000 q^{3} -2.82475 q^{4} -4.54983 q^{5} +6.82475 q^{6} +7.00000 q^{7} -24.6254 q^{8} +9.00000 q^{9} -10.3505 q^{10} -40.7492 q^{11} -8.47425 q^{12} +53.2990 q^{13} +15.9244 q^{14} -13.6495 q^{15} -33.4228 q^{16} +4.54983 q^{17} +20.4743 q^{18} +122.598 q^{19} +12.8522 q^{20} +21.0000 q^{21} -92.7010 q^{22} +131.347 q^{23} -73.8762 q^{24} -104.299 q^{25} +121.251 q^{26} +27.0000 q^{27} -19.7733 q^{28} -216.598 q^{29} -31.0515 q^{30} -251.794 q^{31} +120.969 q^{32} -122.248 q^{33} +10.3505 q^{34} -31.8488 q^{35} -25.4228 q^{36} +11.8970 q^{37} +278.900 q^{38} +159.897 q^{39} +112.042 q^{40} -111.752 q^{41} +47.7733 q^{42} +369.196 q^{43} +115.106 q^{44} -40.9485 q^{45} +298.804 q^{46} -262.694 q^{47} -100.268 q^{48} +49.0000 q^{49} -237.272 q^{50} +13.6495 q^{51} -150.556 q^{52} -567.100 q^{53} +61.4228 q^{54} +185.402 q^{55} -172.378 q^{56} +367.794 q^{57} -492.743 q^{58} +839.890 q^{59} +38.5565 q^{60} -485.794 q^{61} -572.811 q^{62} +63.0000 q^{63} +542.577 q^{64} -242.502 q^{65} -278.103 q^{66} -333.691 q^{67} -12.8522 q^{68} +394.042 q^{69} -72.4535 q^{70} +590.248 q^{71} -221.629 q^{72} +490.701 q^{73} +27.0647 q^{74} -312.897 q^{75} -346.309 q^{76} -285.244 q^{77} +363.752 q^{78} +121.691 q^{79} +152.068 q^{80} +81.0000 q^{81} -254.228 q^{82} +609.608 q^{83} -59.3198 q^{84} -20.7010 q^{85} +839.890 q^{86} -649.794 q^{87} +1003.47 q^{88} +719.038 q^{89} -93.1545 q^{90} +373.093 q^{91} -371.023 q^{92} -755.382 q^{93} -597.608 q^{94} -557.801 q^{95} +362.908 q^{96} -637.877 q^{97} +111.471 q^{98} -366.743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 6 q^{3} + 17 q^{4} + 6 q^{5} - 9 q^{6} + 14 q^{7} - 87 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 + 6 * q^3 + 17 * q^4 + 6 * q^5 - 9 * q^6 + 14 * q^7 - 87 * q^8 + 18 * q^9	\( 2 q - 3 q^{2} + 6 q^{3} + 17 q^{4} + 6 q^{5} - 9 q^{6} + 14 q^{7} - 87 q^{8} + 18 q^{9} - 66 q^{10} - 6 q^{11} + 51 q^{12} + 16 q^{13} - 21 q^{14} + 18 q^{15} + 137 q^{16} - 6 q^{17} - 27 q^{18} + 64 q^{19} + 222 q^{20} + 42 q^{21} - 276 q^{22} + 6 q^{23} - 261 q^{24} - 118 q^{25} + 318 q^{26} + 54 q^{27} + 119 q^{28} - 252 q^{29} - 198 q^{30} + 40 q^{31} - 279 q^{32} - 18 q^{33} + 66 q^{34} + 42 q^{35} + 153 q^{36} - 248 q^{37} + 588 q^{38} + 48 q^{39} - 546 q^{40} - 450 q^{41} - 63 q^{42} + 376 q^{43} + 804 q^{44} + 54 q^{45} + 960 q^{46} - 12 q^{47} + 411 q^{48} + 98 q^{49} - 165 q^{50} - 18 q^{51} - 890 q^{52} - 1104 q^{53} - 81 q^{54} + 552 q^{55} - 609 q^{56} + 192 q^{57} - 306 q^{58} + 804 q^{59} + 666 q^{60} - 428 q^{61} - 2112 q^{62} + 126 q^{63} + 1289 q^{64} - 636 q^{65} - 828 q^{66} + 148 q^{67} - 222 q^{68} + 18 q^{69} - 462 q^{70} + 954 q^{71} - 783 q^{72} + 1072 q^{73} + 1398 q^{74} - 354 q^{75} - 1508 q^{76} - 42 q^{77} + 954 q^{78} - 572 q^{79} + 1950 q^{80} + 162 q^{81} + 1530 q^{82} + 1944 q^{83} + 357 q^{84} - 132 q^{85} + 804 q^{86} - 756 q^{87} - 1164 q^{88} + 366 q^{89} - 594 q^{90} + 112 q^{91} - 2856 q^{92} + 120 q^{93} - 1920 q^{94} - 1176 q^{95} - 837 q^{96} + 808 q^{97} - 147 q^{98} - 54 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 + 6 * q^3 + 17 * q^4 + 6 * q^5 - 9 * q^6 + 14 * q^7 - 87 * q^8 + 18 * q^9 - 66 * q^10 - 6 * q^11 + 51 * q^12 + 16 * q^13 - 21 * q^14 + 18 * q^15 + 137 * q^16 - 6 * q^17 - 27 * q^18 + 64 * q^19 + 222 * q^20 + 42 * q^21 - 276 * q^22 + 6 * q^23 - 261 * q^24 - 118 * q^25 + 318 * q^26 + 54 * q^27 + 119 * q^28 - 252 * q^29 - 198 * q^30 + 40 * q^31 - 279 * q^32 - 18 * q^33 + 66 * q^34 + 42 * q^35 + 153 * q^36 - 248 * q^37 + 588 * q^38 + 48 * q^39 - 546 * q^40 - 450 * q^41 - 63 * q^42 + 376 * q^43 + 804 * q^44 + 54 * q^45 + 960 * q^46 - 12 * q^47 + 411 * q^48 + 98 * q^49 - 165 * q^50 - 18 * q^51 - 890 * q^52 - 1104 * q^53 - 81 * q^54 + 552 * q^55 - 609 * q^56 + 192 * q^57 - 306 * q^58 + 804 * q^59 + 666 * q^60 - 428 * q^61 - 2112 * q^62 + 126 * q^63 + 1289 * q^64 - 636 * q^65 - 828 * q^66 + 148 * q^67 - 222 * q^68 + 18 * q^69 - 462 * q^70 + 954 * q^71 - 783 * q^72 + 1072 * q^73 + 1398 * q^74 - 354 * q^75 - 1508 * q^76 - 42 * q^77 + 954 * q^78 - 572 * q^79 + 1950 * q^80 + 162 * q^81 + 1530 * q^82 + 1944 * q^83 + 357 * q^84 - 132 * q^85 + 804 * q^86 - 756 * q^87 - 1164 * q^88 + 366 * q^89 - 594 * q^90 + 112 * q^91 - 2856 * q^92 + 120 * q^93 - 1920 * q^94 - 1176 * q^95 - 837 * q^96 + 808 * q^97 - 147 * q^98 - 54 * q^99

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