LMFDB - Embedded newform 462.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [462,4,Mod(1,462)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(462, 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("462.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 462 = 2 \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	462.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:	$27.2588824227$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	462.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +11.0000 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +11.0000 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -22.0000 q^{10} +11.0000 q^{11} -12.0000 q^{12} -37.0000 q^{13} +14.0000 q^{14} -33.0000 q^{15} +16.0000 q^{16} -46.0000 q^{17} -18.0000 q^{18} +15.0000 q^{19} +44.0000 q^{20} +21.0000 q^{21} -22.0000 q^{22} -92.0000 q^{23} +24.0000 q^{24} -4.00000 q^{25} +74.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +205.000 q^{29} +66.0000 q^{30} +142.000 q^{31} -32.0000 q^{32} -33.0000 q^{33} +92.0000 q^{34} -77.0000 q^{35} +36.0000 q^{36} -431.000 q^{37} -30.0000 q^{38} +111.000 q^{39} -88.0000 q^{40} -8.00000 q^{41} -42.0000 q^{42} +448.000 q^{43} +44.0000 q^{44} +99.0000 q^{45} +184.000 q^{46} +149.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} +8.00000 q^{50} +138.000 q^{51} -148.000 q^{52} -672.000 q^{53} +54.0000 q^{54} +121.000 q^{55} +56.0000 q^{56} -45.0000 q^{57} -410.000 q^{58} -615.000 q^{59} -132.000 q^{60} +322.000 q^{61} -284.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} -407.000 q^{65} +66.0000 q^{66} -411.000 q^{67} -184.000 q^{68} +276.000 q^{69} +154.000 q^{70} -968.000 q^{71} -72.0000 q^{72} -227.000 q^{73} +862.000 q^{74} +12.0000 q^{75} +60.0000 q^{76} -77.0000 q^{77} -222.000 q^{78} +176.000 q^{80} +81.0000 q^{81} +16.0000 q^{82} -1302.00 q^{83} +84.0000 q^{84} -506.000 q^{85} -896.000 q^{86} -615.000 q^{87} -88.0000 q^{88} -870.000 q^{89} -198.000 q^{90} +259.000 q^{91} -368.000 q^{92} -426.000 q^{93} -298.000 q^{94} +165.000 q^{95} +96.0000 q^{96} -1736.00 q^{97} -98.0000 q^{98} +99.0000 q^{99} +O(q^{100})\)

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
$5$	11.0000	0.983870	0.491935	−	0.870632i	$-0.336290\pi$
$5$	11.0000	0.983870	0.491935	+	0.870632i	$0.336290\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−22.0000	−0.695701
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−12.0000	−0.288675
$13$	−37.0000	−0.789381	−0.394691	−	0.918814i	$-0.629148\pi$
$13$	−37.0000	−0.789381	−0.394691	+	0.918814i	$0.629148\pi$
$14$	14.0000	0.267261
$15$	−33.0000	−0.568038
$16$	16.0000	0.250000
$17$	−46.0000	−0.656273	−0.328136	−	0.944630i	$-0.606421\pi$
$17$	−46.0000	−0.656273	−0.328136	+	0.944630i	$0.606421\pi$
$18$	−18.0000	−0.235702
$19$	15.0000	0.181118	0.0905588	−	0.995891i	$-0.471135\pi$
$19$	15.0000	0.181118	0.0905588	+	0.995891i	$0.471135\pi$
$20$	44.0000	0.491935
$21$	21.0000	0.218218
$22$	−22.0000	−0.213201
$23$	−92.0000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−92.0000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	24.0000	0.204124
$25$	−4.00000	−0.0320000
$26$	74.0000	0.558177
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	205.000	1.31267	0.656337	−	0.754468i	$-0.272105\pi$
$29$	205.000	1.31267	0.656337	+	0.754468i	$0.272105\pi$
$30$	66.0000	0.401663
$31$	142.000	0.822708	0.411354	−	0.911476i	$-0.365056\pi$
$31$	142.000	0.822708	0.411354	+	0.911476i	$0.365056\pi$
$32$	−32.0000	−0.176777
$33$	−33.0000	−0.174078
$34$	92.0000	0.464055
$35$	−77.0000	−0.371868
$36$	36.0000	0.166667
$37$	−431.000	−1.91503	−0.957513	−	0.288390i	$-0.906880\pi$
$37$	−431.000	−1.91503	−0.957513	+	0.288390i	$0.906880\pi$
$38$	−30.0000	−0.128070
$39$	111.000	0.455749
$40$	−88.0000	−0.347851
$41$	−8.00000	−0.0304729	−0.0152365	−	0.999884i	$-0.504850\pi$
$41$	−8.00000	−0.0304729	−0.0152365	+	0.999884i	$0.504850\pi$
$42$	−42.0000	−0.154303
$43$	448.000	1.58882	0.794411	−	0.607380i	$-0.207780\pi$
$43$	448.000	1.58882	0.794411	+	0.607380i	$0.207780\pi$
$44$	44.0000	0.150756
$45$	99.0000	0.327957
$46$	184.000	0.589768
$47$	149.000	0.462423	0.231212	−	0.972904i	$-0.425731\pi$
$47$	149.000	0.462423	0.231212	+	0.972904i	$0.425731\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	8.00000	0.0226274
$51$	138.000	0.378899
$52$	−148.000	−0.394691
$53$	−672.000	−1.74163	−0.870814	−	0.491612i	$-0.836408\pi$
$53$	−672.000	−1.74163	−0.870814	+	0.491612i	$0.836408\pi$
$54$	54.0000	0.136083
$55$	121.000	0.296648
$56$	56.0000	0.133631
$57$	−45.0000	−0.104568
$58$	−410.000	−0.928201
$59$	−615.000	−1.35705	−0.678527	−	0.734576i	$-0.737381\pi$
$59$	−615.000	−1.35705	−0.678527	+	0.734576i	$0.737381\pi$
$60$	−132.000	−0.284019
$61$	322.000	0.675867	0.337933	−	0.941170i	$-0.390272\pi$
$61$	322.000	0.675867	0.337933	+	0.941170i	$0.390272\pi$
$62$	−284.000	−0.581743
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−407.000	−0.776648
$66$	66.0000	0.123091
$67$	−411.000	−0.749427	−0.374714	−	0.927141i	$-0.622259\pi$
$67$	−411.000	−0.749427	−0.374714	+	0.927141i	$0.622259\pi$
$68$	−184.000	−0.328136
$69$	276.000	0.481543
$70$	154.000	0.262950
$71$	−968.000	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$71$	−968.000	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$72$	−72.0000	−0.117851
$73$	−227.000	−0.363950	−0.181975	−	0.983303i	$-0.558249\pi$
$73$	−227.000	−0.363950	−0.181975	+	0.983303i	$0.558249\pi$
$74$	862.000	1.35413
$75$	12.0000	0.0184752
$76$	60.0000	0.0905588
$77$	−77.0000	−0.113961
$78$	−222.000	−0.322263
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	176.000	0.245967
$81$	81.0000	0.111111
$82$	16.0000	0.0215476
$83$	−1302.00	−1.72184	−0.860922	−	0.508737i	$-0.830113\pi$
$83$	−1302.00	−1.72184	−0.860922	+	0.508737i	$0.830113\pi$
$84$	84.0000	0.109109
$85$	−506.000	−0.645687
$86$	−896.000	−1.12347
$87$	−615.000	−0.757873
$88$	−88.0000	−0.106600
$89$	−870.000	−1.03618	−0.518089	−	0.855327i	$-0.673356\pi$
$89$	−870.000	−1.03618	−0.518089	+	0.855327i	$0.673356\pi$
$90$	−198.000	−0.231900
$91$	259.000	0.298358
$92$	−368.000	−0.417029
$93$	−426.000	−0.474991
$94$	−298.000	−0.326982
$95$	165.000	0.178196
$96$	96.0000	0.102062
$97$	−1736.00	−1.81716	−0.908578	−	0.417716i	$-0.862831\pi$
$97$	−1736.00	−1.81716	−0.908578	+	0.417716i	$0.862831\pi$
$98$	−98.0000	−0.101015
$99$	99.0000	0.100504
$100$	−16.0000	−0.0160000
$101$	−1398.00	−1.37729	−0.688645	−	0.725099i	$-0.741794\pi$
$101$	−1398.00	−1.37729	−0.688645	+	0.725099i	$0.741794\pi$
$102$	−276.000	−0.267922
$103$	1338.00	1.27997	0.639986	−	0.768387i	$-0.278940\pi$
$103$	1338.00	1.27997	0.639986	+	0.768387i	$0.278940\pi$
$104$	296.000	0.279088
$105$	231.000	0.214698
$106$	1344.00	1.23152
$107$	1109.00	1.00197	0.500986	−	0.865455i	$-0.332971\pi$
$107$	1109.00	1.00197	0.500986	+	0.865455i	$0.332971\pi$
$108$	−108.000	−0.0962250
$109$	−590.000	−0.518456	−0.259228	−	0.965816i	$-0.583468\pi$
$109$	−590.000	−0.518456	−0.259228	+	0.965816i	$0.583468\pi$
$110$	−242.000	−0.209762
$111$	1293.00	1.10564
$112$	−112.000	−0.0944911
$113$	−622.000	−0.517813	−0.258906	−	0.965902i	$-0.583362\pi$
$113$	−622.000	−0.517813	−0.258906	+	0.965902i	$0.583362\pi$
$114$	90.0000	0.0739410
$115$	−1012.00	−0.820604
$116$	820.000	0.656337
$117$	−333.000	−0.263127
$118$	1230.00	0.959582
$119$	322.000	0.248048
$120$	264.000	0.200832
$121$	121.000	0.0909091
$122$	−644.000	−0.477910
$123$	24.0000	0.0175936
$124$	568.000	0.411354
$125$	−1419.00	−1.01535
$126$	126.000	0.0890871
$127$	674.000	0.470928	0.235464	−	0.971883i	$-0.424339\pi$
$127$	674.000	0.470928	0.235464	+	0.971883i	$0.424339\pi$
$128$	−128.000	−0.0883883
$129$	−1344.00	−0.917307
$130$	814.000	0.549173
$131$	672.000	0.448190	0.224095	−	0.974567i	$-0.428057\pi$
$131$	672.000	0.448190	0.224095	+	0.974567i	$0.428057\pi$
$132$	−132.000	−0.0870388
$133$	−105.000	−0.0684561
$134$	822.000	0.529925
$135$	−297.000	−0.189346
$136$	368.000	0.232027
$137$	654.000	0.407847	0.203923	−	0.978987i	$-0.434631\pi$
$137$	654.000	0.407847	0.203923	+	0.978987i	$0.434631\pi$
$138$	−552.000	−0.340503
$139$	1640.00	1.00074	0.500370	−	0.865811i	$-0.333197\pi$
$139$	1640.00	1.00074	0.500370	+	0.865811i	$0.333197\pi$
$140$	−308.000	−0.185934
$141$	−447.000	−0.266980
$142$	1936.00	1.14412
$143$	−407.000	−0.238007
$144$	144.000	0.0833333
$145$	2255.00	1.29150
$146$	454.000	0.257351
$147$	−147.000	−0.0824786
$148$	−1724.00	−0.957513
$149$	−1975.00	−1.08589	−0.542947	−	0.839767i	$-0.682692\pi$
$149$	−1975.00	−1.08589	−0.542947	+	0.839767i	$0.682692\pi$
$150$	−24.0000	−0.0130639
$151$	1302.00	0.701690	0.350845	−	0.936433i	$-0.385894\pi$
$151$	1302.00	0.701690	0.350845	+	0.936433i	$0.385894\pi$
$152$	−120.000	−0.0640348
$153$	−414.000	−0.218758
$154$	154.000	0.0805823
$155$	1562.00	0.809438
$156$	444.000	0.227875
$157$	2764.00	1.40504	0.702520	−	0.711664i	$-0.252058\pi$
$157$	2764.00	1.40504	0.702520	+	0.711664i	$0.252058\pi$
$158$	0	0
$159$	2016.00	1.00553
$160$	−352.000	−0.173925
$161$	644.000	0.315244
$162$	−162.000	−0.0785674
$163$	−2147.00	−1.03169	−0.515847	−	0.856681i	$-0.672523\pi$
$163$	−2147.00	−1.03169	−0.515847	+	0.856681i	$0.672523\pi$
$164$	−32.0000	−0.0152365
$165$	−363.000	−0.171270
$166$	2604.00	1.21753
$167$	−1736.00	−0.804405	−0.402203	−	0.915551i	$-0.631755\pi$
$167$	−1736.00	−0.804405	−0.402203	+	0.915551i	$0.631755\pi$
$168$	−168.000	−0.0771517
$169$	−828.000	−0.376878
$170$	1012.00	0.456570
$171$	135.000	0.0603726
$172$	1792.00	0.794411
$173$	−1112.00	−0.488692	−0.244346	−	0.969688i	$-0.578573\pi$
$173$	−1112.00	−0.488692	−0.244346	+	0.969688i	$0.578573\pi$
$174$	1230.00	0.535897
$175$	28.0000	0.0120949
$176$	176.000	0.0753778
$177$	1845.00	0.783495
$178$	1740.00	0.732688
$179$	−380.000	−0.158673	−0.0793367	−	0.996848i	$-0.525280\pi$
$179$	−380.000	−0.158673	−0.0793367	+	0.996848i	$0.525280\pi$
$180$	396.000	0.163978
$181$	−2548.00	−1.04636	−0.523181	−	0.852222i	$-0.675255\pi$
$181$	−2548.00	−1.04636	−0.523181	+	0.852222i	$0.675255\pi$
$182$	−518.000	−0.210971
$183$	−966.000	−0.390212
$184$	736.000	0.294884
$185$	−4741.00	−1.88414
$186$	852.000	0.335869
$187$	−506.000	−0.197874
$188$	596.000	0.231212
$189$	189.000	0.0727393
$190$	−330.000	−0.126004
$191$	722.000	0.273519	0.136759	−	0.990604i	$-0.456331\pi$
$191$	722.000	0.273519	0.136759	+	0.990604i	$0.456331\pi$
$192$	−192.000	−0.0721688
$193$	1628.00	0.607181	0.303591	−	0.952803i	$-0.401814\pi$
$193$	1628.00	0.607181	0.303591	+	0.952803i	$0.401814\pi$
$194$	3472.00	1.28492
$195$	1221.00	0.448398
$196$	196.000	0.0714286
$197$	−6.00000	−0.00216996	−0.00108498	−	0.999999i	$-0.500345\pi$
$197$	−6.00000	−0.00216996	−0.00108498	+	0.999999i	$0.500345\pi$
$198$	−198.000	−0.0710669
$199$	1000.00	0.356222	0.178111	−	0.984010i	$-0.443001\pi$
$199$	1000.00	0.356222	0.178111	+	0.984010i	$0.443001\pi$
$200$	32.0000	0.0113137
$201$	1233.00	0.432682
$202$	2796.00	0.973890
$203$	−1435.00	−0.496144
$204$	552.000	0.189450
$205$	−88.0000	−0.0299814
$206$	−2676.00	−0.905076
$207$	−828.000	−0.278019
$208$	−592.000	−0.197345
$209$	165.000	0.0546090
$210$	−462.000	−0.151814
$211$	−5978.00	−1.95044	−0.975219	−	0.221241i	$-0.928989\pi$
$211$	−5978.00	−1.95044	−0.975219	+	0.221241i	$0.928989\pi$
$212$	−2688.00	−0.870814
$213$	2904.00	0.934173
$214$	−2218.00	−0.708502
$215$	4928.00	1.56319
$216$	216.000	0.0680414
$217$	−994.000	−0.310954
$218$	1180.00	0.366604
$219$	681.000	0.210127
$220$	484.000	0.148324
$221$	1702.00	0.518049
$222$	−2586.00	−0.781806
$223$	3778.00	1.13450	0.567250	−	0.823546i	$-0.308007\pi$
$223$	3778.00	1.13450	0.567250	+	0.823546i	$0.308007\pi$
$224$	224.000	0.0668153
$225$	−36.0000	−0.0106667
$226$	1244.00	0.366149
$227$	2674.00	0.781849	0.390924	−	0.920423i	$-0.372155\pi$
$227$	2674.00	0.781849	0.390924	+	0.920423i	$0.372155\pi$
$228$	−180.000	−0.0522842
$229$	2240.00	0.646390	0.323195	−	0.946332i	$-0.395243\pi$
$229$	2240.00	0.646390	0.323195	+	0.946332i	$0.395243\pi$
$230$	2024.00	0.580255
$231$	231.000	0.0657952
$232$	−1640.00	−0.464100
$233$	5778.00	1.62459	0.812295	−	0.583247i	$-0.198218\pi$
$233$	5778.00	1.62459	0.812295	+	0.583247i	$0.198218\pi$
$234$	666.000	0.186059
$235$	1639.00	0.454964
$236$	−2460.00	−0.678527
$237$	0	0
$238$	−644.000	−0.175396
$239$	−2825.00	−0.764578	−0.382289	−	0.924043i	$-0.624864\pi$
$239$	−2825.00	−0.764578	−0.382289	+	0.924043i	$0.624864\pi$
$240$	−528.000	−0.142009
$241$	−2073.00	−0.554082	−0.277041	−	0.960858i	$-0.589354\pi$
$241$	−2073.00	−0.554082	−0.277041	+	0.960858i	$0.589354\pi$
$242$	−242.000	−0.0642824
$243$	−243.000	−0.0641500
$244$	1288.00	0.337933
$245$	539.000	0.140553
$246$	−48.0000	−0.0124405
$247$	−555.000	−0.142971
$248$	−1136.00	−0.290871
$249$	3906.00	0.994107
$250$	2838.00	0.717964
$251$	3207.00	0.806470	0.403235	−	0.915096i	$-0.367886\pi$
$251$	3207.00	0.806470	0.403235	+	0.915096i	$0.367886\pi$
$252$	−252.000	−0.0629941
$253$	−1012.00	−0.251478
$254$	−1348.00	−0.332996
$255$	1518.00	0.372788
$256$	256.000	0.0625000
$257$	−981.000	−0.238105	−0.119053	−	0.992888i	$-0.537986\pi$
$257$	−981.000	−0.238105	−0.119053	+	0.992888i	$0.537986\pi$
$258$	2688.00	0.648634
$259$	3017.00	0.723812
$260$	−1628.00	−0.388324
$261$	1845.00	0.437558
$262$	−1344.00	−0.316918
$263$	−7287.00	−1.70850	−0.854250	−	0.519862i	$-0.825983\pi$
$263$	−7287.00	−1.70850	−0.854250	+	0.519862i	$0.825983\pi$
$264$	264.000	0.0615457
$265$	−7392.00	−1.71354
$266$	210.000	0.0484057
$267$	2610.00	0.598237
$268$	−1644.00	−0.374714
$269$	6390.00	1.44835	0.724173	−	0.689618i	$-0.242222\pi$
$269$	6390.00	1.44835	0.724173	+	0.689618i	$0.242222\pi$
$270$	594.000	0.133888
$271$	8827.00	1.97861	0.989303	−	0.145877i	$-0.0466005\pi$
$271$	8827.00	1.97861	0.989303	+	0.145877i	$0.0466005\pi$
$272$	−736.000	−0.164068
$273$	−777.000	−0.172257
$274$	−1308.00	−0.288391
$275$	−44.0000	−0.00964836
$276$	1104.00	0.240772
$277$	−1716.00	−0.372218	−0.186109	−	0.982529i	$-0.559588\pi$
$277$	−1716.00	−0.372218	−0.186109	+	0.982529i	$0.559588\pi$
$278$	−3280.00	−0.707631
$279$	1278.00	0.274236
$280$	616.000	0.131475
$281$	−5023.00	−1.06636	−0.533180	−	0.846002i	$-0.679003\pi$
$281$	−5023.00	−1.06636	−0.533180	+	0.846002i	$0.679003\pi$
$282$	894.000	0.188783
$283$	−2217.00	−0.465678	−0.232839	−	0.972515i	$-0.574802\pi$
$283$	−2217.00	−0.465678	−0.232839	+	0.972515i	$0.574802\pi$
$284$	−3872.00	−0.809017
$285$	−495.000	−0.102882
$286$	814.000	0.168297
$287$	56.0000	0.0115177
$288$	−288.000	−0.0589256
$289$	−2797.00	−0.569306
$290$	−4510.00	−0.913229
$291$	5208.00	1.04914
$292$	−908.000	−0.181975
$293$	6048.00	1.20590	0.602949	−	0.797780i	$-0.293992\pi$
$293$	6048.00	1.20590	0.602949	+	0.797780i	$0.293992\pi$
$294$	294.000	0.0583212
$295$	−6765.00	−1.33516
$296$	3448.00	0.677064
$297$	−297.000	−0.0580259
$298$	3950.00	0.767843
$299$	3404.00	0.658389
$300$	48.0000	0.00923760
$301$	−3136.00	−0.600518
$302$	−2604.00	−0.496170
$303$	4194.00	0.795178
$304$	240.000	0.0452794
$305$	3542.00	0.664965
$306$	828.000	0.154685
$307$	−2956.00	−0.549537	−0.274768	−	0.961510i	$-0.588601\pi$
$307$	−2956.00	−0.549537	−0.274768	+	0.961510i	$0.588601\pi$
$308$	−308.000	−0.0569803
$309$	−4014.00	−0.738992
$310$	−3124.00	−0.572359
$311$	−1768.00	−0.322360	−0.161180	−	0.986925i	$-0.551530\pi$
$311$	−1768.00	−0.322360	−0.161180	+	0.986925i	$0.551530\pi$
$312$	−888.000	−0.161132
$313$	7378.00	1.33236	0.666181	−	0.745790i	$-0.267928\pi$
$313$	7378.00	1.33236	0.666181	+	0.745790i	$0.267928\pi$
$314$	−5528.00	−0.993513
$315$	−693.000	−0.123956
$316$	0	0
$317$	1224.00	0.216867	0.108433	−	0.994104i	$-0.465417\pi$
$317$	1224.00	0.216867	0.108433	+	0.994104i	$0.465417\pi$
$318$	−4032.00	−0.711017
$319$	2255.00	0.395786
$320$	704.000	0.122984
$321$	−3327.00	−0.578489
$322$	−1288.00	−0.222911
$323$	−690.000	−0.118863
$324$	324.000	0.0555556
$325$	148.000	0.0252602
$326$	4294.00	0.729517
$327$	1770.00	0.299331
$328$	64.0000	0.0107738
$329$	−1043.00	−0.174779
$330$	726.000	0.121106
$331$	5392.00	0.895381	0.447691	−	0.894189i	$-0.352247\pi$
$331$	5392.00	0.895381	0.447691	+	0.894189i	$0.352247\pi$
$332$	−5208.00	−0.860922
$333$	−3879.00	−0.638342
$334$	3472.00	0.568801
$335$	−4521.00	−0.737339
$336$	336.000	0.0545545
$337$	924.000	0.149358	0.0746788	−	0.997208i	$-0.476207\pi$
$337$	924.000	0.149358	0.0746788	+	0.997208i	$0.476207\pi$
$338$	1656.00	0.266493
$339$	1866.00	0.298959
$340$	−2024.00	−0.322844
$341$	1562.00	0.248056
$342$	−270.000	−0.0426898
$343$	−343.000	−0.0539949
$344$	−3584.00	−0.561734
$345$	3036.00	0.473776
$346$	2224.00	0.345558
$347$	−8176.00	−1.26487	−0.632436	−	0.774613i	$-0.717945\pi$
$347$	−8176.00	−1.26487	−0.632436	+	0.774613i	$0.717945\pi$
$348$	−2460.00	−0.378936
$349$	−3625.00	−0.555994	−0.277997	−	0.960582i	$-0.589670\pi$
$349$	−3625.00	−0.555994	−0.277997	+	0.960582i	$0.589670\pi$
$350$	−56.0000	−0.00855236
$351$	999.000	0.151916
$352$	−352.000	−0.0533002
$353$	−6447.00	−0.972066	−0.486033	−	0.873941i	$-0.661556\pi$
$353$	−6447.00	−0.972066	−0.486033	+	0.873941i	$0.661556\pi$
$354$	−3690.00	−0.554015
$355$	−10648.0	−1.59194
$356$	−3480.00	−0.518089
$357$	−966.000	−0.143210
$358$	760.000	0.112199
$359$	3620.00	0.532190	0.266095	−	0.963947i	$-0.414266\pi$
$359$	3620.00	0.532190	0.266095	+	0.963947i	$0.414266\pi$
$360$	−792.000	−0.115950
$361$	−6634.00	−0.967196
$362$	5096.00	0.739889
$363$	−363.000	−0.0524864
$364$	1036.00	0.149179
$365$	−2497.00	−0.358079
$366$	1932.00	0.275921
$367$	−3626.00	−0.515737	−0.257869	−	0.966180i	$-0.583020\pi$
$367$	−3626.00	−0.515737	−0.257869	+	0.966180i	$0.583020\pi$
$368$	−1472.00	−0.208514
$369$	−72.0000	−0.0101576
$370$	9482.00	1.33229
$371$	4704.00	0.658274
$372$	−1704.00	−0.237495
$373$	−3322.00	−0.461144	−0.230572	−	0.973055i	$-0.574060\pi$
$373$	−3322.00	−0.461144	−0.230572	+	0.973055i	$0.574060\pi$
$374$	1012.00	0.139918
$375$	4257.00	0.586215
$376$	−1192.00	−0.163491
$377$	−7585.00	−1.03620
$378$	−378.000	−0.0514344
$379$	5755.00	0.779985	0.389993	−	0.920818i	$-0.372478\pi$
$379$	5755.00	0.779985	0.389993	+	0.920818i	$0.372478\pi$
$380$	660.000	0.0890981
$381$	−2022.00	−0.271890
$382$	−1444.00	−0.193407
$383$	−10672.0	−1.42380	−0.711898	−	0.702283i	$-0.752164\pi$
$383$	−10672.0	−1.42380	−0.711898	+	0.702283i	$0.752164\pi$
$384$	384.000	0.0510310
$385$	−847.000	−0.112122
$386$	−3256.00	−0.429342
$387$	4032.00	0.529607
$388$	−6944.00	−0.908578
$389$	−6900.00	−0.899342	−0.449671	−	0.893194i	$-0.648459\pi$
$389$	−6900.00	−0.899342	−0.449671	+	0.893194i	$0.648459\pi$
$390$	−2442.00	−0.317065
$391$	4232.00	0.547369
$392$	−392.000	−0.0505076
$393$	−2016.00	−0.258763
$394$	12.0000	0.00153439
$395$	0	0
$396$	396.000	0.0502519
$397$	8994.00	1.13702	0.568509	−	0.822677i	$-0.307521\pi$
$397$	8994.00	1.13702	0.568509	+	0.822677i	$0.307521\pi$
$398$	−2000.00	−0.251887
$399$	315.000	0.0395231
$400$	−64.0000	−0.00800000
$401$	8652.00	1.07746	0.538729	−	0.842479i	$-0.318905\pi$
$401$	8652.00	1.07746	0.538729	+	0.842479i	$0.318905\pi$
$402$	−2466.00	−0.305952
$403$	−5254.00	−0.649430
$404$	−5592.00	−0.688645
$405$	891.000	0.109319
$406$	2870.00	0.350827
$407$	−4741.00	−0.577402
$408$	−1104.00	−0.133961
$409$	9970.00	1.20534	0.602671	−	0.797990i	$-0.294103\pi$
$409$	9970.00	1.20534	0.602671	+	0.797990i	$0.294103\pi$
$410$	176.000	0.0212000
$411$	−1962.00	−0.235470
$412$	5352.00	0.639986
$413$	4305.00	0.512918
$414$	1656.00	0.196589
$415$	−14322.0	−1.69407
$416$	1184.00	0.139544
$417$	−4920.00	−0.577778
$418$	−330.000	−0.0386144
$419$	12535.0	1.46152	0.730758	−	0.682637i	$-0.239167\pi$
$419$	12535.0	1.46152	0.730758	+	0.682637i	$0.239167\pi$
$420$	924.000	0.107349
$421$	−13193.0	−1.52729	−0.763643	−	0.645639i	$-0.776591\pi$
$421$	−13193.0	−1.52729	−0.763643	+	0.645639i	$0.776591\pi$
$422$	11956.0	1.37917
$423$	1341.00	0.154141
$424$	5376.00	0.615759
$425$	184.000	0.0210007
$426$	−5808.00	−0.660560
$427$	−2254.00	−0.255454
$428$	4436.00	0.500986
$429$	1221.00	0.137414
$430$	−9856.00	−1.10535
$431$	9817.00	1.09714	0.548571	−	0.836104i	$-0.315172\pi$
$431$	9817.00	1.09714	0.548571	+	0.836104i	$0.315172\pi$
$432$	−432.000	−0.0481125
$433$	2948.00	0.327187	0.163593	−	0.986528i	$-0.447692\pi$
$433$	2948.00	0.327187	0.163593	+	0.986528i	$0.447692\pi$
$434$	1988.00	0.219878
$435$	−6765.00	−0.745648
$436$	−2360.00	−0.259228
$437$	−1380.00	−0.151063
$438$	−1362.00	−0.148582
$439$	115.000	0.0125026	0.00625131	−	0.999980i	$-0.498010\pi$
$439$	115.000	0.0125026	0.00625131	+	0.999980i	$0.498010\pi$
$440$	−968.000	−0.104881
$441$	441.000	0.0476190
$442$	−3404.00	−0.366316
$443$	8088.00	0.867432	0.433716	−	0.901050i	$-0.357202\pi$
$443$	8088.00	0.867432	0.433716	+	0.901050i	$0.357202\pi$
$444$	5172.00	0.552820
$445$	−9570.00	−1.01946
$446$	−7556.00	−0.802213
$447$	5925.00	0.626942
$448$	−448.000	−0.0472456
$449$	−1800.00	−0.189192	−0.0945960	−	0.995516i	$-0.530156\pi$
$449$	−1800.00	−0.189192	−0.0945960	+	0.995516i	$0.530156\pi$
$450$	72.0000	0.00754247
$451$	−88.0000	−0.00918793
$452$	−2488.00	−0.258906
$453$	−3906.00	−0.405121
$454$	−5348.00	−0.552850
$455$	2849.00	0.293545
$456$	360.000	0.0369705
$457$	−256.000	−0.0262039	−0.0131019	−	0.999914i	$-0.504171\pi$
$457$	−256.000	−0.0262039	−0.0131019	+	0.999914i	$0.504171\pi$
$458$	−4480.00	−0.457067
$459$	1242.00	0.126300
$460$	−4048.00	−0.410302
$461$	72.0000	0.00727413	0.00363707	−	0.999993i	$-0.498842\pi$
$461$	72.0000	0.00727413	0.00363707	+	0.999993i	$0.498842\pi$
$462$	−462.000	−0.0465242
$463$	−13547.0	−1.35979	−0.679895	−	0.733310i	$-0.737975\pi$
$463$	−13547.0	−1.35979	−0.679895	+	0.733310i	$0.737975\pi$
$464$	3280.00	0.328168
$465$	−4686.00	−0.467329
$466$	−11556.0	−1.14876
$467$	4629.00	0.458682	0.229341	−	0.973346i	$-0.426343\pi$
$467$	4629.00	0.458682	0.229341	+	0.973346i	$0.426343\pi$
$468$	−1332.00	−0.131564
$469$	2877.00	0.283257
$470$	−3278.00	−0.321708
$471$	−8292.00	−0.811200
$472$	4920.00	0.479791
$473$	4928.00	0.479048
$474$	0	0
$475$	−60.0000	−0.00579577
$476$	1288.00	0.124024
$477$	−6048.00	−0.580543
$478$	5650.00	0.540638
$479$	−19210.0	−1.83242	−0.916208	−	0.400703i	$-0.868766\pi$
$479$	−19210.0	−1.83242	−0.916208	+	0.400703i	$0.868766\pi$
$480$	1056.00	0.100416
$481$	15947.0	1.51169
$482$	4146.00	0.391795
$483$	−1932.00	−0.182006
$484$	484.000	0.0454545
$485$	−19096.0	−1.78784
$486$	486.000	0.0453609
$487$	18704.0	1.74037	0.870184	−	0.492727i	$-0.164000\pi$
$487$	18704.0	1.74037	0.870184	+	0.492727i	$0.164000\pi$
$488$	−2576.00	−0.238955
$489$	6441.00	0.595648
$490$	−1078.00	−0.0993859
$491$	5397.00	0.496055	0.248028	−	0.968753i	$-0.420218\pi$
$491$	5397.00	0.496055	0.248028	+	0.968753i	$0.420218\pi$
$492$	96.0000	0.00879678
$493$	−9430.00	−0.861472
$494$	1110.00	0.101096
$495$	1089.00	0.0988826
$496$	2272.00	0.205677
$497$	6776.00	0.611560
$498$	−7812.00	−0.702940
$499$	15515.0	1.39188	0.695939	−	0.718101i	$-0.254989\pi$
$499$	15515.0	1.39188	0.695939	+	0.718101i	$0.254989\pi$
$500$	−5676.00	−0.507677
$501$	5208.00	0.464424
$502$	−6414.00	−0.570261
$503$	18888.0	1.67430	0.837151	−	0.546971i	$-0.184219\pi$
$503$	18888.0	1.67430	0.837151	+	0.546971i	$0.184219\pi$
$504$	504.000	0.0445435
$505$	−15378.0	−1.35507
$506$	2024.00	0.177822
$507$	2484.00	0.217590
$508$	2696.00	0.235464
$509$	−2890.00	−0.251664	−0.125832	−	0.992052i	$-0.540160\pi$
$509$	−2890.00	−0.251664	−0.125832	+	0.992052i	$0.540160\pi$
$510$	−3036.00	−0.263601
$511$	1589.00	0.137560
$512$	−512.000	−0.0441942
$513$	−405.000	−0.0348561
$514$	1962.00	0.168366
$515$	14718.0	1.25933
$516$	−5376.00	−0.458653
$517$	1639.00	0.139426
$518$	−6034.00	−0.511812
$519$	3336.00	0.282147
$520$	3256.00	0.274587
$521$	6667.00	0.560627	0.280313	−	0.959909i	$-0.409562\pi$
$521$	6667.00	0.560627	0.280313	+	0.959909i	$0.409562\pi$
$522$	−3690.00	−0.309400
$523$	4433.00	0.370634	0.185317	−	0.982679i	$-0.440669\pi$
$523$	4433.00	0.370634	0.185317	+	0.982679i	$0.440669\pi$
$524$	2688.00	0.224095
$525$	−84.0000	−0.00698297
$526$	14574.0	1.20809
$527$	−6532.00	−0.539921
$528$	−528.000	−0.0435194
$529$	−3703.00	−0.304348
$530$	14784.0	1.21165
$531$	−5535.00	−0.452351
$532$	−420.000	−0.0342280
$533$	296.000	0.0240548
$534$	−5220.00	−0.423018
$535$	12199.0	0.985811
$536$	3288.00	0.264963
$537$	1140.00	0.0916101
$538$	−12780.0	−1.02414
$539$	539.000	0.0430730
$540$	−1188.00	−0.0946729
$541$	−1828.00	−0.145271	−0.0726357	−	0.997359i	$-0.523141\pi$
$541$	−1828.00	−0.145271	−0.0726357	+	0.997359i	$0.523141\pi$
$542$	−17654.0	−1.39909
$543$	7644.00	0.604117
$544$	1472.00	0.116014
$545$	−6490.00	−0.510094
$546$	1554.00	0.121804
$547$	3084.00	0.241065	0.120532	−	0.992709i	$-0.461540\pi$
$547$	3084.00	0.241065	0.120532	+	0.992709i	$0.461540\pi$
$548$	2616.00	0.203923
$549$	2898.00	0.225289
$550$	88.0000	0.00682242
$551$	3075.00	0.237748
$552$	−2208.00	−0.170251
$553$	0	0
$554$	3432.00	0.263198
$555$	14223.0	1.08781
$556$	6560.00	0.500370
$557$	9329.00	0.709663	0.354832	−	0.934930i	$-0.384538\pi$
$557$	9329.00	0.709663	0.354832	+	0.934930i	$0.384538\pi$
$558$	−2556.00	−0.193914
$559$	−16576.0	−1.25419
$560$	−1232.00	−0.0929670
$561$	1518.00	0.114242
$562$	10046.0	0.754030
$563$	16928.0	1.26719	0.633597	−	0.773663i	$-0.281578\pi$
$563$	16928.0	1.26719	0.633597	+	0.773663i	$0.281578\pi$
$564$	−1788.00	−0.133490
$565$	−6842.00	−0.509460
$566$	4434.00	0.329284
$567$	−567.000	−0.0419961
$568$	7744.00	0.572062
$569$	10950.0	0.806763	0.403381	−	0.915032i	$-0.367835\pi$
$569$	10950.0	0.806763	0.403381	+	0.915032i	$0.367835\pi$
$570$	990.000	0.0727483
$571$	−13508.0	−0.990004	−0.495002	−	0.868892i	$-0.664833\pi$
$571$	−13508.0	−0.990004	−0.495002	+	0.868892i	$0.664833\pi$
$572$	−1628.00	−0.119004
$573$	−2166.00	−0.157916
$574$	−112.000	−0.00814423
$575$	368.000	0.0266898
$576$	576.000	0.0416667
$577$	−8026.00	−0.579076	−0.289538	−	0.957167i	$-0.593502\pi$
$577$	−8026.00	−0.579076	−0.289538	+	0.957167i	$0.593502\pi$
$578$	5594.00	0.402560
$579$	−4884.00	−0.350556
$580$	9020.00	0.645750
$581$	9114.00	0.650796
$582$	−10416.0	−0.741851
$583$	−7392.00	−0.525121
$584$	1816.00	0.128676
$585$	−3663.00	−0.258883
$586$	−12096.0	−0.852698
$587$	10179.0	0.715728	0.357864	−	0.933774i	$-0.383505\pi$
$587$	10179.0	0.715728	0.357864	+	0.933774i	$0.383505\pi$
$588$	−588.000	−0.0412393
$589$	2130.00	0.149007
$590$	13530.0	0.944104
$591$	18.0000	0.00125283
$592$	−6896.00	−0.478757
$593$	23148.0	1.60299	0.801496	−	0.598000i	$-0.204038\pi$
$593$	23148.0	1.60299	0.801496	+	0.598000i	$0.204038\pi$
$594$	594.000	0.0410305
$595$	3542.00	0.244047
$596$	−7900.00	−0.542947
$597$	−3000.00	−0.205665
$598$	−6808.00	−0.465552
$599$	17520.0	1.19507	0.597536	−	0.801842i	$-0.296147\pi$
$599$	17520.0	1.19507	0.597536	+	0.801842i	$0.296147\pi$
$600$	−96.0000	−0.00653197
$601$	24617.0	1.67080	0.835398	−	0.549646i	$-0.185237\pi$
$601$	24617.0	1.67080	0.835398	+	0.549646i	$0.185237\pi$
$602$	6272.00	0.424631
$603$	−3699.00	−0.249809
$604$	5208.00	0.350845
$605$	1331.00	0.0894427
$606$	−8388.00	−0.562276
$607$	−3901.00	−0.260851	−0.130426	−	0.991458i	$-0.541634\pi$
$607$	−3901.00	−0.260851	−0.130426	+	0.991458i	$0.541634\pi$
$608$	−480.000	−0.0320174
$609$	4305.00	0.286449
$610$	−7084.00	−0.470201
$611$	−5513.00	−0.365028
$612$	−1656.00	−0.109379
$613$	418.000	0.0275414	0.0137707	−	0.999905i	$-0.495617\pi$
$613$	418.000	0.0275414	0.0137707	+	0.999905i	$0.495617\pi$
$614$	5912.00	0.388581
$615$	264.000	0.0173098
$616$	616.000	0.0402911
$617$	−15486.0	−1.01044	−0.505221	−	0.862990i	$-0.668589\pi$
$617$	−15486.0	−1.01044	−0.505221	+	0.862990i	$0.668589\pi$
$618$	8028.00	0.522546
$619$	−6190.00	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−6190.00	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	6248.00	0.404719
$621$	2484.00	0.160514
$622$	3536.00	0.227943
$623$	6090.00	0.391638
$624$	1776.00	0.113937
$625$	−15109.0	−0.966976
$626$	−14756.0	−0.942122
$627$	−495.000	−0.0315285
$628$	11056.0	0.702520
$629$	19826.0	1.25678
$630$	1386.00	0.0876501
$631$	8052.00	0.507995	0.253998	−	0.967205i	$-0.418254\pi$
$631$	8052.00	0.507995	0.253998	+	0.967205i	$0.418254\pi$
$632$	0	0
$633$	17934.0	1.12609
$634$	−2448.00	−0.153348
$635$	7414.00	0.463332
$636$	8064.00	0.502765
$637$	−1813.00	−0.112769
$638$	−4510.00	−0.279863
$639$	−8712.00	−0.539345
$640$	−1408.00	−0.0869626
$641$	−24138.0	−1.48735	−0.743677	−	0.668539i	$-0.766920\pi$
$641$	−24138.0	−1.48735	−0.743677	+	0.668539i	$0.766920\pi$
$642$	6654.00	0.409054
$643$	16288.0	0.998967	0.499484	−	0.866323i	$-0.333523\pi$
$643$	16288.0	0.998967	0.499484	+	0.866323i	$0.333523\pi$
$644$	2576.00	0.157622
$645$	−14784.0	−0.902511
$646$	1380.00	0.0840486
$647$	−321.000	−0.0195051	−0.00975256	−	0.999952i	$-0.503104\pi$
$647$	−321.000	−0.0195051	−0.00975256	+	0.999952i	$0.503104\pi$
$648$	−648.000	−0.0392837
$649$	−6765.00	−0.409167
$650$	−296.000	−0.0178617
$651$	2982.00	0.179530
$652$	−8588.00	−0.515847
$653$	−19472.0	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$653$	−19472.0	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$654$	−3540.00	−0.211659
$655$	7392.00	0.440961
$656$	−128.000	−0.00761823
$657$	−2043.00	−0.121317
$658$	2086.00	0.123588
$659$	−2855.00	−0.168763	−0.0843816	−	0.996434i	$-0.526891\pi$
$659$	−2855.00	−0.168763	−0.0843816	+	0.996434i	$0.526891\pi$
$660$	−1452.00	−0.0856349
$661$	−6148.00	−0.361769	−0.180885	−	0.983504i	$-0.557896\pi$
$661$	−6148.00	−0.361769	−0.180885	+	0.983504i	$0.557896\pi$
$662$	−10784.0	−0.633130
$663$	−5106.00	−0.299096
$664$	10416.0	0.608764
$665$	−1155.00	−0.0673518
$666$	7758.00	0.451376
$667$	−18860.0	−1.09485
$668$	−6944.00	−0.402203
$669$	−11334.0	−0.655004
$670$	9042.00	0.521378
$671$	3542.00	0.203782
$672$	−672.000	−0.0385758
$673$	−5432.00	−0.311127	−0.155563	−	0.987826i	$-0.549719\pi$
$673$	−5432.00	−0.311127	−0.155563	+	0.987826i	$0.549719\pi$
$674$	−1848.00	−0.105612
$675$	108.000	0.00615840
$676$	−3312.00	−0.188439
$677$	−11866.0	−0.673630	−0.336815	−	0.941571i	$-0.609350\pi$
$677$	−11866.0	−0.673630	−0.336815	+	0.941571i	$0.609350\pi$
$678$	−3732.00	−0.211396
$679$	12152.0	0.686820
$680$	4048.00	0.228285
$681$	−8022.00	−0.451400
$682$	−3124.00	−0.175402
$683$	−26082.0	−1.46120	−0.730600	−	0.682805i	$-0.760760\pi$
$683$	−26082.0	−1.46120	−0.730600	+	0.682805i	$0.760760\pi$
$684$	540.000	0.0301863
$685$	7194.00	0.401268
$686$	686.000	0.0381802
$687$	−6720.00	−0.373194
$688$	7168.00	0.397206
$689$	24864.0	1.37481
$690$	−6072.00	−0.335010
$691$	32442.0	1.78604	0.893019	−	0.450020i	$-0.148583\pi$
$691$	32442.0	1.78604	0.893019	+	0.450020i	$0.148583\pi$
$692$	−4448.00	−0.244346
$693$	−693.000	−0.0379869
$694$	16352.0	0.894400
$695$	18040.0	0.984599
$696$	4920.00	0.267948
$697$	368.000	0.0199986
$698$	7250.00	0.393147
$699$	−17334.0	−0.937957
$700$	112.000	0.00604743
$701$	11202.0	0.603557	0.301779	−	0.953378i	$-0.402420\pi$
$701$	11202.0	0.603557	0.301779	+	0.953378i	$0.402420\pi$
$702$	−1998.00	−0.107421
$703$	−6465.00	−0.346845
$704$	704.000	0.0376889
$705$	−4917.00	−0.262674
$706$	12894.0	0.687354
$707$	9786.00	0.520566
$708$	7380.00	0.391748
$709$	11355.0	0.601475	0.300738	−	0.953707i	$-0.402767\pi$
$709$	11355.0	0.601475	0.300738	+	0.953707i	$0.402767\pi$
$710$	21296.0	1.12567
$711$	0	0
$712$	6960.00	0.366344
$713$	−13064.0	−0.686186
$714$	1932.00	0.101265
$715$	−4477.00	−0.234168
$716$	−1520.00	−0.0793367
$717$	8475.00	0.441429
$718$	−7240.00	−0.376315
$719$	23395.0	1.21347	0.606736	−	0.794903i	$-0.292479\pi$
$719$	23395.0	1.21347	0.606736	+	0.794903i	$0.292479\pi$
$720$	1584.00	0.0819892
$721$	−9366.00	−0.483784
$722$	13268.0	0.683911
$723$	6219.00	0.319899
$724$	−10192.0	−0.523181
$725$	−820.000	−0.0420056
$726$	726.000	0.0371135
$727$	6104.00	0.311396	0.155698	−	0.987805i	$-0.450237\pi$
$727$	6104.00	0.311396	0.155698	+	0.987805i	$0.450237\pi$
$728$	−2072.00	−0.105485
$729$	729.000	0.0370370
$730$	4994.00	0.253200
$731$	−20608.0	−1.04270
$732$	−3864.00	−0.195106
$733$	14618.0	0.736600	0.368300	−	0.929707i	$-0.379940\pi$
$733$	14618.0	0.736600	0.368300	+	0.929707i	$0.379940\pi$
$734$	7252.00	0.364681
$735$	−1617.00	−0.0811482
$736$	2944.00	0.147442
$737$	−4521.00	−0.225961
$738$	144.000	0.00718254
$739$	−14640.0	−0.728743	−0.364372	−	0.931254i	$-0.618716\pi$
$739$	−14640.0	−0.728743	−0.364372	+	0.931254i	$0.618716\pi$
$740$	−18964.0	−0.942068
$741$	1665.00	0.0825443
$742$	−9408.00	−0.465470
$743$	−7887.00	−0.389429	−0.194715	−	0.980860i	$-0.562378\pi$
$743$	−7887.00	−0.389429	−0.194715	+	0.980860i	$0.562378\pi$
$744$	3408.00	0.167935
$745$	−21725.0	−1.06838
$746$	6644.00	0.326078
$747$	−11718.0	−0.573948
$748$	−2024.00	−0.0989369
$749$	−7763.00	−0.378710
$750$	−8514.00	−0.414516
$751$	−14003.0	−0.680395	−0.340198	−	0.940354i	$-0.610494\pi$
$751$	−14003.0	−0.680395	−0.340198	+	0.940354i	$0.610494\pi$
$752$	2384.00	0.115606
$753$	−9621.00	−0.465616
$754$	15170.0	0.732704
$755$	14322.0	0.690372
$756$	756.000	0.0363696
$757$	−14191.0	−0.681348	−0.340674	−	0.940181i	$-0.610655\pi$
$757$	−14191.0	−0.681348	−0.340674	+	0.940181i	$0.610655\pi$
$758$	−11510.0	−0.551533
$759$	3036.00	0.145191
$760$	−1320.00	−0.0630019
$761$	−39168.0	−1.86575	−0.932877	−	0.360195i	$-0.882710\pi$
$761$	−39168.0	−1.86575	−0.932877	+	0.360195i	$0.882710\pi$
$762$	4044.00	0.192255
$763$	4130.00	0.195958
$764$	2888.00	0.136759
$765$	−4554.00	−0.215229
$766$	21344.0	1.00678
$767$	22755.0	1.07123
$768$	−768.000	−0.0360844
$769$	3175.00	0.148886	0.0744431	−	0.997225i	$-0.476282\pi$
$769$	3175.00	0.148886	0.0744431	+	0.997225i	$0.476282\pi$
$770$	1694.00	0.0792825
$771$	2943.00	0.137470
$772$	6512.00	0.303591
$773$	8463.00	0.393781	0.196891	−	0.980425i	$-0.436916\pi$
$773$	8463.00	0.393781	0.196891	+	0.980425i	$0.436916\pi$
$774$	−8064.00	−0.374489
$775$	−568.000	−0.0263267
$776$	13888.0	0.642462
$777$	−9051.00	−0.417893
$778$	13800.0	0.635931
$779$	−120.000	−0.00551919
$780$	4884.00	0.224199
$781$	−10648.0	−0.487856
$782$	−8464.00	−0.387049
$783$	−5535.00	−0.252624
$784$	784.000	0.0357143
$785$	30404.0	1.38238
$786$	4032.00	0.182973
$787$	−3751.00	−0.169897	−0.0849484	−	0.996385i	$-0.527073\pi$
$787$	−3751.00	−0.169897	−0.0849484	+	0.996385i	$0.527073\pi$
$788$	−24.0000	−0.00108498
$789$	21861.0	0.986403
$790$	0	0
$791$	4354.00	0.195715
$792$	−792.000	−0.0355335
$793$	−11914.0	−0.533516
$794$	−17988.0	−0.803993
$795$	22176.0	0.989310
$796$	4000.00	0.178111
$797$	−1711.00	−0.0760436	−0.0380218	−	0.999277i	$-0.512106\pi$
$797$	−1711.00	−0.0760436	−0.0380218	+	0.999277i	$0.512106\pi$
$798$	−630.000	−0.0279471
$799$	−6854.00	−0.303476
$800$	128.000	0.00565685
$801$	−7830.00	−0.345393
$802$	−17304.0	−0.761877
$803$	−2497.00	−0.109735
$804$	4932.00	0.216341
$805$	7084.00	0.310159
$806$	10508.0	0.459217
$807$	−19170.0	−0.836203
$808$	11184.0	0.486945
$809$	15165.0	0.659052	0.329526	−	0.944147i	$-0.393111\pi$
$809$	15165.0	0.659052	0.329526	+	0.944147i	$0.393111\pi$
$810$	−1782.00	−0.0773001
$811$	4097.00	0.177392	0.0886961	−	0.996059i	$-0.471730\pi$
$811$	4097.00	0.177392	0.0886961	+	0.996059i	$0.471730\pi$
$812$	−5740.00	−0.248072
$813$	−26481.0	−1.14235
$814$	9482.00	0.408285
$815$	−23617.0	−1.01505
$816$	2208.00	0.0947248
$817$	6720.00	0.287764
$818$	−19940.0	−0.852305
$819$	2331.00	0.0994527
$820$	−352.000	−0.0149907
$821$	−46713.0	−1.98574	−0.992871	−	0.119190i	$-0.961970\pi$
$821$	−46713.0	−1.98574	−0.992871	+	0.119190i	$0.961970\pi$
$822$	3924.00	0.166503
$823$	−27907.0	−1.18199	−0.590994	−	0.806676i	$-0.701264\pi$
$823$	−27907.0	−1.18199	−0.590994	+	0.806676i	$0.701264\pi$
$824$	−10704.0	−0.452538
$825$	132.000	0.00557048
$826$	−8610.00	−0.362688
$827$	−211.000	−0.00887205	−0.00443603	−	0.999990i	$-0.501412\pi$
$827$	−211.000	−0.00887205	−0.00443603	+	0.999990i	$0.501412\pi$
$828$	−3312.00	−0.139010
$829$	34720.0	1.45461	0.727307	−	0.686312i	$-0.240772\pi$
$829$	34720.0	1.45461	0.727307	+	0.686312i	$0.240772\pi$
$830$	28644.0	1.19789
$831$	5148.00	0.214900
$832$	−2368.00	−0.0986726
$833$	−2254.00	−0.0937533
$834$	9840.00	0.408551
$835$	−19096.0	−0.791430
$836$	660.000	0.0273045
$837$	−3834.00	−0.158330
$838$	−25070.0	−1.03345
$839$	−10155.0	−0.417866	−0.208933	−	0.977930i	$-0.566999\pi$
$839$	−10155.0	−0.417866	−0.208933	+	0.977930i	$0.566999\pi$
$840$	−1848.00	−0.0759072
$841$	17636.0	0.723113
$842$	26386.0	1.07995
$843$	15069.0	0.615663
$844$	−23912.0	−0.975219
$845$	−9108.00	−0.370798
$846$	−2682.00	−0.108994
$847$	−847.000	−0.0343604
$848$	−10752.0	−0.435407
$849$	6651.00	0.268860
$850$	−368.000	−0.0148498
$851$	39652.0	1.59724
$852$	11616.0	0.467086
$853$	−33922.0	−1.36163	−0.680813	−	0.732457i	$-0.738373\pi$
$853$	−33922.0	−1.36163	−0.680813	+	0.732457i	$0.738373\pi$
$854$	4508.00	0.180633
$855$	1485.00	0.0593987
$856$	−8872.00	−0.354251
$857$	12254.0	0.488435	0.244217	−	0.969721i	$-0.421469\pi$
$857$	12254.0	0.488435	0.244217	+	0.969721i	$0.421469\pi$
$858$	−2442.00	−0.0971661
$859$	13120.0	0.521128	0.260564	−	0.965457i	$-0.416092\pi$
$859$	13120.0	0.521128	0.260564	+	0.965457i	$0.416092\pi$
$860$	19712.0	0.781597
$861$	−168.000	−0.00664974
$862$	−19634.0	−0.775797
$863$	−35492.0	−1.39996	−0.699978	−	0.714165i	$-0.746807\pi$
$863$	−35492.0	−1.39996	−0.699978	+	0.714165i	$0.746807\pi$
$864$	864.000	0.0340207
$865$	−12232.0	−0.480810
$866$	−5896.00	−0.231356
$867$	8391.00	0.328689
$868$	−3976.00	−0.155477
$869$	0	0
$870$	13530.0	0.527253
$871$	15207.0	0.591584
$872$	4720.00	0.183302
$873$	−15624.0	−0.605719
$874$	2760.00	0.106817
$875$	9933.00	0.383768
$876$	2724.00	0.105063
$877$	49504.0	1.90608	0.953040	−	0.302846i	$-0.0979368\pi$
$877$	49504.0	1.90608	0.953040	+	0.302846i	$0.0979368\pi$
$878$	−230.000	−0.00884069
$879$	−18144.0	−0.696225
$880$	1936.00	0.0741620
$881$	−6453.00	−0.246773	−0.123387	−	0.992359i	$-0.539376\pi$
$881$	−6453.00	−0.246773	−0.123387	+	0.992359i	$0.539376\pi$
$882$	−882.000	−0.0336718
$883$	−17717.0	−0.675226	−0.337613	−	0.941285i	$-0.609620\pi$
$883$	−17717.0	−0.675226	−0.337613	+	0.941285i	$0.609620\pi$
$884$	6808.00	0.259025
$885$	20295.0	0.770858
$886$	−16176.0	−0.613367
$887$	1354.00	0.0512546	0.0256273	−	0.999672i	$-0.491842\pi$
$887$	1354.00	0.0512546	0.0256273	+	0.999672i	$0.491842\pi$
$888$	−10344.0	−0.390903
$889$	−4718.00	−0.177994
$890$	19140.0	0.720870
$891$	891.000	0.0335013
$892$	15112.0	0.567250
$893$	2235.00	0.0837530
$894$	−11850.0	−0.443315
$895$	−4180.00	−0.156114
$896$	896.000	0.0334077
$897$	−10212.0	−0.380121
$898$	3600.00	0.133779
$899$	29110.0	1.07995
$900$	−144.000	−0.00533333
$901$	30912.0	1.14298
$902$	176.000	0.00649685
$903$	9408.00	0.346709
$904$	4976.00	0.183074
$905$	−28028.0	−1.02948
$906$	7812.00	0.286464
$907$	−35596.0	−1.30314	−0.651569	−	0.758590i	$-0.725889\pi$
$907$	−35596.0	−1.30314	−0.651569	+	0.758590i	$0.725889\pi$
$908$	10696.0	0.390924
$909$	−12582.0	−0.459096
$910$	−5698.00	−0.207568
$911$	9462.00	0.344116	0.172058	−	0.985087i	$-0.444958\pi$
$911$	9462.00	0.344116	0.172058	+	0.985087i	$0.444958\pi$
$912$	−720.000	−0.0261421
$913$	−14322.0	−0.519156
$914$	512.000	0.0185289
$915$	−10626.0	−0.383918
$916$	8960.00	0.323195
$917$	−4704.00	−0.169400
$918$	−2484.00	−0.0893074
$919$	42780.0	1.53556	0.767781	−	0.640712i	$-0.221361\pi$
$919$	42780.0	1.53556	0.767781	+	0.640712i	$0.221361\pi$
$920$	8096.00	0.290127
$921$	8868.00	0.317275
$922$	−144.000	−0.00514359
$923$	35816.0	1.27725
$924$	924.000	0.0328976
$925$	1724.00	0.0612808
$926$	27094.0	0.961516
$927$	12042.0	0.426657
$928$	−6560.00	−0.232050
$929$	−11175.0	−0.394661	−0.197330	−	0.980337i	$-0.563227\pi$
$929$	−11175.0	−0.394661	−0.197330	+	0.980337i	$0.563227\pi$
$930$	9372.00	0.330452
$931$	735.000	0.0258740
$932$	23112.0	0.812295
$933$	5304.00	0.186115
$934$	−9258.00	−0.324337
$935$	−5566.00	−0.194682
$936$	2664.00	0.0930294
$937$	−21546.0	−0.751203	−0.375601	−	0.926781i	$-0.622564\pi$
$937$	−21546.0	−0.751203	−0.375601	+	0.926781i	$0.622564\pi$
$938$	−5754.00	−0.200293
$939$	−22134.0	−0.769239
$940$	6556.00	0.227482
$941$	−7268.00	−0.251785	−0.125893	−	0.992044i	$-0.540179\pi$
$941$	−7268.00	−0.251785	−0.125893	+	0.992044i	$0.540179\pi$
$942$	16584.0	0.573605
$943$	736.000	0.0254162
$944$	−9840.00	−0.339263
$945$	2079.00	0.0715660
$946$	−9856.00	−0.338738
$947$	914.000	0.0313633	0.0156816	−	0.999877i	$-0.495008\pi$
$947$	914.000	0.0313633	0.0156816	+	0.999877i	$0.495008\pi$
$948$	0	0
$949$	8399.00	0.287295
$950$	120.000	0.00409823
$951$	−3672.00	−0.125208
$952$	−2576.00	−0.0876982
$953$	2693.00	0.0915371	0.0457685	−	0.998952i	$-0.485426\pi$
$953$	2693.00	0.0915371	0.0457685	+	0.998952i	$0.485426\pi$
$954$	12096.0	0.410506
$955$	7942.00	0.269107
$956$	−11300.0	−0.382289
$957$	−6765.00	−0.228507
$958$	38420.0	1.29571
$959$	−4578.00	−0.154152
$960$	−2112.00	−0.0710047
$961$	−9627.00	−0.323151
$962$	−31894.0	−1.06892
$963$	9981.00	0.333991
$964$	−8292.00	−0.277041
$965$	17908.0	0.597387
$966$	3864.00	0.128698
$967$	18074.0	0.601055	0.300528	−	0.953773i	$-0.402837\pi$
$967$	18074.0	0.601055	0.300528	+	0.953773i	$0.402837\pi$
$968$	−968.000	−0.0321412
$969$	2070.00	0.0686254
$970$	38192.0	1.26420
$971$	−40093.0	−1.32507	−0.662536	−	0.749030i	$-0.730520\pi$
$971$	−40093.0	−1.32507	−0.662536	+	0.749030i	$0.730520\pi$
$972$	−972.000	−0.0320750
$973$	−11480.0	−0.378245
$974$	−37408.0	−1.23063
$975$	−444.000	−0.0145840
$976$	5152.00	0.168967
$977$	−49096.0	−1.60770	−0.803849	−	0.594834i	$-0.797218\pi$
$977$	−49096.0	−1.60770	−0.803849	+	0.594834i	$0.797218\pi$
$978$	−12882.0	−0.421187
$979$	−9570.00	−0.312419
$980$	2156.00	0.0702764
$981$	−5310.00	−0.172819
$982$	−10794.0	−0.350764
$983$	−4632.00	−0.150293	−0.0751464	−	0.997173i	$-0.523942\pi$
$983$	−4632.00	−0.150293	−0.0751464	+	0.997173i	$0.523942\pi$
$984$	−192.000	−0.00622026
$985$	−66.0000	−0.00213496
$986$	18860.0	0.609153
$987$	3129.00	0.100909
$988$	−2220.00	−0.0714854
$989$	−41216.0	−1.32517
$990$	−2178.00	−0.0699206
$991$	−51803.0	−1.66052	−0.830261	−	0.557375i	$-0.811808\pi$
$991$	−51803.0	−1.66052	−0.830261	+	0.557375i	$0.811808\pi$
$992$	−4544.00	−0.145436
$993$	−16176.0	−0.516948
$994$	−13552.0	−0.432438
$995$	11000.0	0.350476
$996$	15624.0	0.497054
$997$	11334.0	0.360031	0.180016	−	0.983664i	$-0.442385\pi$
$997$	11334.0	0.360031	0.180016	+	0.983664i	$0.442385\pi$
$998$	−31030.0	−0.984206
$999$	11637.0	0.368547

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	462.4.a.c.1.1	✓	1
3.2	odd	2		1386.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.c.1.1	✓	1	1.1	even	1	trivial
1386.4.a.i.1.1		1	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:	\( N \)	\(=\)	\( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	462.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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