LMFDB - Embedded newform 2310.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2310,4,Mod(1,2310)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2310, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2310.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2310.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:	$136.294412113$
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	$\chi$	$=$	2310.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -11.0000 q^{11} -12.0000 q^{12} +14.0000 q^{13} +14.0000 q^{14} -15.0000 q^{15} +16.0000 q^{16} -78.0000 q^{17} -18.0000 q^{18} +4.00000 q^{19} +20.0000 q^{20} +21.0000 q^{21} +22.0000 q^{22} +128.000 q^{23} +24.0000 q^{24} +25.0000 q^{25} -28.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +110.000 q^{29} +30.0000 q^{30} -184.000 q^{31} -32.0000 q^{32} +33.0000 q^{33} +156.000 q^{34} -35.0000 q^{35} +36.0000 q^{36} +158.000 q^{37} -8.00000 q^{38} -42.0000 q^{39} -40.0000 q^{40} -342.000 q^{41} -42.0000 q^{42} -84.0000 q^{43} -44.0000 q^{44} +45.0000 q^{45} -256.000 q^{46} +264.000 q^{47} -48.0000 q^{48} +49.0000 q^{49} -50.0000 q^{50} +234.000 q^{51} +56.0000 q^{52} +78.0000 q^{53} +54.0000 q^{54} -55.0000 q^{55} +56.0000 q^{56} -12.0000 q^{57} -220.000 q^{58} +828.000 q^{59} -60.0000 q^{60} +366.000 q^{61} +368.000 q^{62} -63.0000 q^{63} +64.0000 q^{64} +70.0000 q^{65} -66.0000 q^{66} +692.000 q^{67} -312.000 q^{68} -384.000 q^{69} +70.0000 q^{70} -880.000 q^{71} -72.0000 q^{72} -886.000 q^{73} -316.000 q^{74} -75.0000 q^{75} +16.0000 q^{76} +77.0000 q^{77} +84.0000 q^{78} -1200.00 q^{79} +80.0000 q^{80} +81.0000 q^{81} +684.000 q^{82} -1228.00 q^{83} +84.0000 q^{84} -390.000 q^{85} +168.000 q^{86} -330.000 q^{87} +88.0000 q^{88} -630.000 q^{89} -90.0000 q^{90} -98.0000 q^{91} +512.000 q^{92} +552.000 q^{93} -528.000 q^{94} +20.0000 q^{95} +96.0000 q^{96} -1310.00 q^{97} -98.0000 q^{98} -99.0000 q^{99} +O(q^{100})\)

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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−12.0000	−0.288675
$13$	14.0000	0.298685	0.149342	−	0.988786i	$-0.452284\pi$
$13$	14.0000	0.298685	0.149342	+	0.988786i	$0.452284\pi$
$14$	14.0000	0.267261
$15$	−15.0000	−0.258199
$16$	16.0000	0.250000
$17$	−78.0000	−1.11281	−0.556405	−	0.830911i	$-0.687820\pi$
$17$	−78.0000	−1.11281	−0.556405	+	0.830911i	$0.687820\pi$
$18$	−18.0000	−0.235702
$19$	4.00000	0.0482980	0.0241490	−	0.999708i	$-0.492312\pi$
$19$	4.00000	0.0482980	0.0241490	+	0.999708i	$0.492312\pi$
$20$	20.0000	0.223607
$21$	21.0000	0.218218
$22$	22.0000	0.213201
$23$	128.000	1.16043	0.580214	−	0.814464i	$-0.302969\pi$
$23$	128.000	1.16043	0.580214	+	0.814464i	$0.302969\pi$
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	−28.0000	−0.211202
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	110.000	0.704362	0.352181	−	0.935932i	$-0.385440\pi$
$29$	110.000	0.704362	0.352181	+	0.935932i	$0.385440\pi$
$30$	30.0000	0.182574
$31$	−184.000	−1.06604	−0.533022	−	0.846101i	$-0.678944\pi$
$31$	−184.000	−1.06604	−0.533022	+	0.846101i	$0.678944\pi$
$32$	−32.0000	−0.176777
$33$	33.0000	0.174078
$34$	156.000	0.786876
$35$	−35.0000	−0.169031
$36$	36.0000	0.166667
$37$	158.000	0.702028	0.351014	−	0.936370i	$-0.385837\pi$
$37$	158.000	0.702028	0.351014	+	0.936370i	$0.385837\pi$
$38$	−8.00000	−0.0341519
$39$	−42.0000	−0.172446
$40$	−40.0000	−0.158114
$41$	−342.000	−1.30272	−0.651359	−	0.758770i	$-0.725801\pi$
$41$	−342.000	−1.30272	−0.651359	+	0.758770i	$0.725801\pi$
$42$	−42.0000	−0.154303
$43$	−84.0000	−0.297904	−0.148952	−	0.988844i	$-0.547590\pi$
$43$	−84.0000	−0.297904	−0.148952	+	0.988844i	$0.547590\pi$
$44$	−44.0000	−0.150756
$45$	45.0000	0.149071
$46$	−256.000	−0.820547
$47$	264.000	0.819327	0.409663	−	0.912237i	$-0.365646\pi$
$47$	264.000	0.819327	0.409663	+	0.912237i	$0.365646\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−50.0000	−0.141421
$51$	234.000	0.642481
$52$	56.0000	0.149342
$53$	78.0000	0.202153	0.101077	−	0.994879i	$-0.467771\pi$
$53$	78.0000	0.202153	0.101077	+	0.994879i	$0.467771\pi$
$54$	54.0000	0.136083
$55$	−55.0000	−0.134840
$56$	56.0000	0.133631
$57$	−12.0000	−0.0278849
$58$	−220.000	−0.498059
$59$	828.000	1.82706	0.913529	−	0.406774i	$-0.133346\pi$
$59$	828.000	1.82706	0.913529	+	0.406774i	$0.133346\pi$
$60$	−60.0000	−0.129099
$61$	366.000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	366.000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	368.000	0.753807
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	70.0000	0.133576
$66$	−66.0000	−0.123091
$67$	692.000	1.26181	0.630905	−	0.775860i	$-0.282684\pi$
$67$	692.000	1.26181	0.630905	+	0.775860i	$0.282684\pi$
$68$	−312.000	−0.556405
$69$	−384.000	−0.669973
$70$	70.0000	0.119523
$71$	−880.000	−1.47094	−0.735470	−	0.677557i	$-0.763039\pi$
$71$	−880.000	−1.47094	−0.735470	+	0.677557i	$0.763039\pi$
$72$	−72.0000	−0.117851
$73$	−886.000	−1.42053	−0.710263	−	0.703936i	$-0.751424\pi$
$73$	−886.000	−1.42053	−0.710263	+	0.703936i	$0.751424\pi$
$74$	−316.000	−0.496409
$75$	−75.0000	−0.115470
$76$	16.0000	0.0241490
$77$	77.0000	0.113961
$78$	84.0000	0.121938
$79$	−1200.00	−1.70899	−0.854497	−	0.519456i	$-0.826135\pi$
$79$	−1200.00	−1.70899	−0.854497	+	0.519456i	$0.826135\pi$
$80$	80.0000	0.111803
$81$	81.0000	0.111111
$82$	684.000	0.921161
$83$	−1228.00	−1.62398	−0.811991	−	0.583670i	$-0.801616\pi$
$83$	−1228.00	−1.62398	−0.811991	+	0.583670i	$0.801616\pi$
$84$	84.0000	0.109109
$85$	−390.000	−0.497664
$86$	168.000	0.210650
$87$	−330.000	−0.406663
$88$	88.0000	0.106600
$89$	−630.000	−0.750336	−0.375168	−	0.926957i	$-0.622415\pi$
$89$	−630.000	−0.750336	−0.375168	+	0.926957i	$0.622415\pi$
$90$	−90.0000	−0.105409
$91$	−98.0000	−0.112892
$92$	512.000	0.580214
$93$	552.000	0.615481
$94$	−528.000	−0.579352
$95$	20.0000	0.0215995
$96$	96.0000	0.102062
$97$	−1310.00	−1.37124	−0.685620	−	0.727959i	$-0.740469\pi$
$97$	−1310.00	−1.37124	−0.685620	+	0.727959i	$0.740469\pi$
$98$	−98.0000	−0.101015
$99$	−99.0000	−0.100504
$100$	100.000	0.100000
$101$	1014.00	0.998978	0.499489	−	0.866320i	$-0.333521\pi$
$101$	1014.00	0.998978	0.499489	+	0.866320i	$0.333521\pi$
$102$	−468.000	−0.454303
$103$	1200.00	1.14796	0.573978	−	0.818871i	$-0.305399\pi$
$103$	1200.00	1.14796	0.573978	+	0.818871i	$0.305399\pi$
$104$	−112.000	−0.105601
$105$	105.000	0.0975900
$106$	−156.000	−0.142944
$107$	1916.00	1.73109	0.865545	−	0.500831i	$-0.166972\pi$
$107$	1916.00	1.73109	0.865545	+	0.500831i	$0.166972\pi$
$108$	−108.000	−0.0962250
$109$	942.000	0.827773	0.413886	−	0.910329i	$-0.364171\pi$
$109$	942.000	0.827773	0.413886	+	0.910329i	$0.364171\pi$
$110$	110.000	0.0953463
$111$	−474.000	−0.405316
$112$	−112.000	−0.0944911
$113$	1074.00	0.894101	0.447051	−	0.894509i	$-0.352474\pi$
$113$	1074.00	0.894101	0.447051	+	0.894509i	$0.352474\pi$
$114$	24.0000	0.0197176
$115$	640.000	0.518959
$116$	440.000	0.352181
$117$	126.000	0.0995616
$118$	−1656.00	−1.29193
$119$	546.000	0.420603
$120$	120.000	0.0912871
$121$	121.000	0.0909091
$122$	−732.000	−0.543214
$123$	1026.00	0.752124
$124$	−736.000	−0.533022
$125$	125.000	0.0894427
$126$	126.000	0.0890871
$127$	2528.00	1.76633	0.883164	−	0.469064i	$-0.155409\pi$
$127$	2528.00	1.76633	0.883164	+	0.469064i	$0.155409\pi$
$128$	−128.000	−0.0883883
$129$	252.000	0.171995
$130$	−140.000	−0.0944524
$131$	−1820.00	−1.21385	−0.606924	−	0.794760i	$-0.707597\pi$
$131$	−1820.00	−1.21385	−0.606924	+	0.794760i	$0.707597\pi$
$132$	132.000	0.0870388
$133$	−28.0000	−0.0182549
$134$	−1384.00	−0.892234
$135$	−135.000	−0.0860663
$136$	624.000	0.393438
$137$	−550.000	−0.342990	−0.171495	−	0.985185i	$-0.554860\pi$
$137$	−550.000	−0.342990	−0.171495	+	0.985185i	$0.554860\pi$
$138$	768.000	0.473743
$139$	−1556.00	−0.949483	−0.474742	−	0.880125i	$-0.657459\pi$
$139$	−1556.00	−0.949483	−0.474742	+	0.880125i	$0.657459\pi$
$140$	−140.000	−0.0845154
$141$	−792.000	−0.473039
$142$	1760.00	1.04011
$143$	−154.000	−0.0900568
$144$	144.000	0.0833333
$145$	550.000	0.315000
$146$	1772.00	1.00446
$147$	−147.000	−0.0824786
$148$	632.000	0.351014
$149$	−810.000	−0.445354	−0.222677	−	0.974892i	$-0.571480\pi$
$149$	−810.000	−0.445354	−0.222677	+	0.974892i	$0.571480\pi$
$150$	150.000	0.0816497
$151$	−776.000	−0.418212	−0.209106	−	0.977893i	$-0.567055\pi$
$151$	−776.000	−0.418212	−0.209106	+	0.977893i	$0.567055\pi$
$152$	−32.0000	−0.0170759
$153$	−702.000	−0.370937
$154$	−154.000	−0.0805823
$155$	−920.000	−0.476750
$156$	−168.000	−0.0862229
$157$	−2106.00	−1.07055	−0.535277	−	0.844676i	$-0.679793\pi$
$157$	−2106.00	−1.07055	−0.535277	+	0.844676i	$0.679793\pi$
$158$	2400.00	1.20844
$159$	−234.000	−0.116713
$160$	−160.000	−0.0790569
$161$	−896.000	−0.438601
$162$	−162.000	−0.0785674
$163$	756.000	0.363279	0.181640	−	0.983365i	$-0.441860\pi$
$163$	756.000	0.363279	0.181640	+	0.983365i	$0.441860\pi$
$164$	−1368.00	−0.651359
$165$	165.000	0.0778499
$166$	2456.00	1.14833
$167$	904.000	0.418884	0.209442	−	0.977821i	$-0.432835\pi$
$167$	904.000	0.418884	0.209442	+	0.977821i	$0.432835\pi$
$168$	−168.000	−0.0771517
$169$	−2001.00	−0.910787
$170$	780.000	0.351902
$171$	36.0000	0.0160993
$172$	−336.000	−0.148952
$173$	1486.00	0.653055	0.326527	−	0.945188i	$-0.394121\pi$
$173$	1486.00	0.653055	0.326527	+	0.945188i	$0.394121\pi$
$174$	660.000	0.287554
$175$	−175.000	−0.0755929
$176$	−176.000	−0.0753778
$177$	−2484.00	−1.05485
$178$	1260.00	0.530567
$179$	4580.00	1.91243	0.956216	−	0.292662i	$-0.0945412\pi$
$179$	4580.00	1.91243	0.956216	+	0.292662i	$0.0945412\pi$
$180$	180.000	0.0745356
$181$	62.0000	0.0254609	0.0127305	−	0.999919i	$-0.495948\pi$
$181$	62.0000	0.0254609	0.0127305	+	0.999919i	$0.495948\pi$
$182$	196.000	0.0798268
$183$	−1098.00	−0.443533
$184$	−1024.00	−0.410273
$185$	790.000	0.313957
$186$	−1104.00	−0.435211
$187$	858.000	0.335525
$188$	1056.00	0.409663
$189$	189.000	0.0727393
$190$	−40.0000	−0.0152732
$191$	−2712.00	−1.02740	−0.513700	−	0.857970i	$-0.671726\pi$
$191$	−2712.00	−1.02740	−0.513700	+	0.857970i	$0.671726\pi$
$192$	−192.000	−0.0721688
$193$	514.000	0.191702	0.0958511	−	0.995396i	$-0.469443\pi$
$193$	514.000	0.191702	0.0958511	+	0.995396i	$0.469443\pi$
$194$	2620.00	0.969614
$195$	−210.000	−0.0771201
$196$	196.000	0.0714286
$197$	4502.00	1.62819	0.814097	−	0.580729i	$-0.197232\pi$
$197$	4502.00	1.62819	0.814097	+	0.580729i	$0.197232\pi$
$198$	198.000	0.0710669
$199$	−560.000	−0.199484	−0.0997421	−	0.995013i	$-0.531802\pi$
$199$	−560.000	−0.199484	−0.0997421	+	0.995013i	$0.531802\pi$
$200$	−200.000	−0.0707107
$201$	−2076.00	−0.728506
$202$	−2028.00	−0.706384
$203$	−770.000	−0.266224
$204$	936.000	0.321241
$205$	−1710.00	−0.582593
$206$	−2400.00	−0.811728
$207$	1152.00	0.386809
$208$	224.000	0.0746712
$209$	−44.0000	−0.0145624
$210$	−210.000	−0.0690066
$211$	2612.00	0.852216	0.426108	−	0.904672i	$-0.359884\pi$
$211$	2612.00	0.852216	0.426108	+	0.904672i	$0.359884\pi$
$212$	312.000	0.101077
$213$	2640.00	0.849248
$214$	−3832.00	−1.22407
$215$	−420.000	−0.133227
$216$	216.000	0.0680414
$217$	1288.00	0.402927
$218$	−1884.00	−0.585324
$219$	2658.00	0.820142
$220$	−220.000	−0.0674200
$221$	−1092.00	−0.332379
$222$	948.000	0.286602
$223$	2904.00	0.872046	0.436023	−	0.899936i	$-0.356387\pi$
$223$	2904.00	0.872046	0.436023	+	0.899936i	$0.356387\pi$
$224$	224.000	0.0668153
$225$	225.000	0.0666667
$226$	−2148.00	−0.632225
$227$	−1068.00	−0.312272	−0.156136	−	0.987736i	$-0.549904\pi$
$227$	−1068.00	−0.312272	−0.156136	+	0.987736i	$0.549904\pi$
$228$	−48.0000	−0.0139424
$229$	686.000	0.197957	0.0989785	−	0.995090i	$-0.468442\pi$
$229$	686.000	0.197957	0.0989785	+	0.995090i	$0.468442\pi$
$230$	−1280.00	−0.366960
$231$	−231.000	−0.0657952
$232$	−880.000	−0.249029
$233$	−5558.00	−1.56273	−0.781366	−	0.624073i	$-0.785477\pi$
$233$	−5558.00	−1.56273	−0.781366	+	0.624073i	$0.785477\pi$
$234$	−252.000	−0.0704007
$235$	1320.00	0.366414
$236$	3312.00	0.913529
$237$	3600.00	0.986688
$238$	−1092.00	−0.297411
$239$	−6112.00	−1.65419	−0.827097	−	0.562059i	$-0.810009\pi$
$239$	−6112.00	−1.65419	−0.827097	+	0.562059i	$0.810009\pi$
$240$	−240.000	−0.0645497
$241$	−3358.00	−0.897543	−0.448771	−	0.893647i	$-0.648138\pi$
$241$	−3358.00	−0.897543	−0.448771	+	0.893647i	$0.648138\pi$
$242$	−242.000	−0.0642824
$243$	−243.000	−0.0641500
$244$	1464.00	0.384111
$245$	245.000	0.0638877
$246$	−2052.00	−0.531832
$247$	56.0000	0.0144259
$248$	1472.00	0.376904
$249$	3684.00	0.937606
$250$	−250.000	−0.0632456
$251$	−1332.00	−0.334961	−0.167480	−	0.985875i	$-0.553563\pi$
$251$	−1332.00	−0.334961	−0.167480	+	0.985875i	$0.553563\pi$
$252$	−252.000	−0.0629941
$253$	−1408.00	−0.349882
$254$	−5056.00	−1.24898
$255$	1170.00	0.287326
$256$	256.000	0.0625000
$257$	−1182.00	−0.286892	−0.143446	−	0.989658i	$-0.545818\pi$
$257$	−1182.00	−0.286892	−0.143446	+	0.989658i	$0.545818\pi$
$258$	−504.000	−0.121619
$259$	−1106.00	−0.265342
$260$	280.000	0.0667879
$261$	990.000	0.234787
$262$	3640.00	0.858320
$263$	−3912.00	−0.917202	−0.458601	−	0.888642i	$-0.651649\pi$
$263$	−3912.00	−0.917202	−0.458601	+	0.888642i	$0.651649\pi$
$264$	−264.000	−0.0615457
$265$	390.000	0.0904057
$266$	56.0000	0.0129082
$267$	1890.00	0.433206
$268$	2768.00	0.630905
$269$	1926.00	0.436544	0.218272	−	0.975888i	$-0.429958\pi$
$269$	1926.00	0.436544	0.218272	+	0.975888i	$0.429958\pi$
$270$	270.000	0.0608581
$271$	−144.000	−0.0322781	−0.0161391	−	0.999870i	$-0.505137\pi$
$271$	−144.000	−0.0322781	−0.0161391	+	0.999870i	$0.505137\pi$
$272$	−1248.00	−0.278203
$273$	294.000	0.0651783
$274$	1100.00	0.242531
$275$	−275.000	−0.0603023
$276$	−1536.00	−0.334987
$277$	−7994.00	−1.73398	−0.866991	−	0.498324i	$-0.833949\pi$
$277$	−7994.00	−1.73398	−0.866991	+	0.498324i	$0.833949\pi$
$278$	3112.00	0.671386
$279$	−1656.00	−0.355348
$280$	280.000	0.0597614
$281$	5610.00	1.19098	0.595489	−	0.803364i	$-0.296959\pi$
$281$	5610.00	1.19098	0.595489	+	0.803364i	$0.296959\pi$
$282$	1584.00	0.334489
$283$	3836.00	0.805747	0.402874	−	0.915256i	$-0.368011\pi$
$283$	3836.00	0.805747	0.402874	+	0.915256i	$0.368011\pi$
$284$	−3520.00	−0.735470
$285$	−60.0000	−0.0124705
$286$	308.000	0.0636798
$287$	2394.00	0.492381
$288$	−288.000	−0.0589256
$289$	1171.00	0.238347
$290$	−1100.00	−0.222739
$291$	3930.00	0.791686
$292$	−3544.00	−0.710263
$293$	2998.00	0.597765	0.298882	−	0.954290i	$-0.403386\pi$
$293$	2998.00	0.597765	0.298882	+	0.954290i	$0.403386\pi$
$294$	294.000	0.0583212
$295$	4140.00	0.817085
$296$	−1264.00	−0.248204
$297$	297.000	0.0580259
$298$	1620.00	0.314913
$299$	1792.00	0.346602
$300$	−300.000	−0.0577350
$301$	588.000	0.112597
$302$	1552.00	0.295720
$303$	−3042.00	−0.576760
$304$	64.0000	0.0120745
$305$	1830.00	0.343559
$306$	1404.00	0.262292
$307$	−5468.00	−1.01653	−0.508266	−	0.861200i	$-0.669713\pi$
$307$	−5468.00	−1.01653	−0.508266	+	0.861200i	$0.669713\pi$
$308$	308.000	0.0569803
$309$	−3600.00	−0.662773
$310$	1840.00	0.337113
$311$	−6000.00	−1.09398	−0.546992	−	0.837138i	$-0.684227\pi$
$311$	−6000.00	−1.09398	−0.546992	+	0.837138i	$0.684227\pi$
$312$	336.000	0.0609688
$313$	−8022.00	−1.44866	−0.724329	−	0.689454i	$-0.757851\pi$
$313$	−8022.00	−1.44866	−0.724329	+	0.689454i	$0.757851\pi$
$314$	4212.00	0.756997
$315$	−315.000	−0.0563436
$316$	−4800.00	−0.854497
$317$	4854.00	0.860025	0.430012	−	0.902823i	$-0.358509\pi$
$317$	4854.00	0.860025	0.430012	+	0.902823i	$0.358509\pi$
$318$	468.000	0.0825287
$319$	−1210.00	−0.212373
$320$	320.000	0.0559017
$321$	−5748.00	−0.999446
$322$	1792.00	0.310137
$323$	−312.000	−0.0537466
$324$	324.000	0.0555556
$325$	350.000	0.0597369
$326$	−1512.00	−0.256877
$327$	−2826.00	−0.477915
$328$	2736.00	0.460580
$329$	−1848.00	−0.309676
$330$	−330.000	−0.0550482
$331$	−2740.00	−0.454997	−0.227499	−	0.973778i	$-0.573055\pi$
$331$	−2740.00	−0.454997	−0.227499	+	0.973778i	$0.573055\pi$
$332$	−4912.00	−0.811991
$333$	1422.00	0.234009
$334$	−1808.00	−0.296196
$335$	3460.00	0.564298
$336$	336.000	0.0545545
$337$	−11966.0	−1.93421	−0.967106	−	0.254373i	$-0.918131\pi$
$337$	−11966.0	−1.93421	−0.967106	+	0.254373i	$0.918131\pi$
$338$	4002.00	0.644024
$339$	−3222.00	−0.516209
$340$	−1560.00	−0.248832
$341$	2024.00	0.321424
$342$	−72.0000	−0.0113840
$343$	−343.000	−0.0539949
$344$	672.000	0.105325
$345$	−1920.00	−0.299621
$346$	−2972.00	−0.461780
$347$	−6228.00	−0.963506	−0.481753	−	0.876307i	$-0.660000\pi$
$347$	−6228.00	−0.963506	−0.481753	+	0.876307i	$0.660000\pi$
$348$	−1320.00	−0.203332
$349$	2078.00	0.318719	0.159359	−	0.987221i	$-0.449057\pi$
$349$	2078.00	0.318719	0.159359	+	0.987221i	$0.449057\pi$
$350$	350.000	0.0534522
$351$	−378.000	−0.0574819
$352$	352.000	0.0533002
$353$	7170.00	1.08108	0.540539	−	0.841319i	$-0.318220\pi$
$353$	7170.00	1.08108	0.540539	+	0.841319i	$0.318220\pi$
$354$	4968.00	0.745893
$355$	−4400.00	−0.657825
$356$	−2520.00	−0.375168
$357$	−1638.00	−0.242835
$358$	−9160.00	−1.35229
$359$	7752.00	1.13965	0.569826	−	0.821766i	$-0.307011\pi$
$359$	7752.00	1.13965	0.569826	+	0.821766i	$0.307011\pi$
$360$	−360.000	−0.0527046
$361$	−6843.00	−0.997667
$362$	−124.000	−0.0180036
$363$	−363.000	−0.0524864
$364$	−392.000	−0.0564461
$365$	−4430.00	−0.635279
$366$	2196.00	0.313625
$367$	−1800.00	−0.256020	−0.128010	−	0.991773i	$-0.540859\pi$
$367$	−1800.00	−0.256020	−0.128010	+	0.991773i	$0.540859\pi$
$368$	2048.00	0.290107
$369$	−3078.00	−0.434239
$370$	−1580.00	−0.222001
$371$	−546.000	−0.0764068
$372$	2208.00	0.307741
$373$	−9114.00	−1.26516	−0.632580	−	0.774495i	$-0.718004\pi$
$373$	−9114.00	−1.26516	−0.632580	+	0.774495i	$0.718004\pi$
$374$	−1716.00	−0.237252
$375$	−375.000	−0.0516398
$376$	−2112.00	−0.289676
$377$	1540.00	0.210382
$378$	−378.000	−0.0514344
$379$	−2996.00	−0.406053	−0.203027	−	0.979173i	$-0.565078\pi$
$379$	−2996.00	−0.406053	−0.203027	+	0.979173i	$0.565078\pi$
$380$	80.0000	0.0107998
$381$	−7584.00	−1.01979
$382$	5424.00	0.726482
$383$	5240.00	0.699090	0.349545	−	0.936920i	$-0.386336\pi$
$383$	5240.00	0.699090	0.349545	+	0.936920i	$0.386336\pi$
$384$	384.000	0.0510310
$385$	385.000	0.0509647
$386$	−1028.00	−0.135554
$387$	−756.000	−0.0993014
$388$	−5240.00	−0.685620
$389$	−11762.0	−1.53305	−0.766526	−	0.642214i	$-0.778016\pi$
$389$	−11762.0	−1.53305	−0.766526	+	0.642214i	$0.778016\pi$
$390$	420.000	0.0545321
$391$	−9984.00	−1.29134
$392$	−392.000	−0.0505076
$393$	5460.00	0.700816
$394$	−9004.00	−1.15131
$395$	−6000.00	−0.764285
$396$	−396.000	−0.0502519
$397$	10262.0	1.29732	0.648659	−	0.761079i	$-0.275330\pi$
$397$	10262.0	1.29732	0.648659	+	0.761079i	$0.275330\pi$
$398$	1120.00	0.141057
$399$	84.0000	0.0105395
$400$	400.000	0.0500000
$401$	−10094.0	−1.25703	−0.628517	−	0.777796i	$-0.716338\pi$
$401$	−10094.0	−1.25703	−0.628517	+	0.777796i	$0.716338\pi$
$402$	4152.00	0.515132
$403$	−2576.00	−0.318411
$404$	4056.00	0.499489
$405$	405.000	0.0496904
$406$	1540.00	0.188249
$407$	−1738.00	−0.211669
$408$	−1872.00	−0.227151
$409$	−16454.0	−1.98924	−0.994619	−	0.103605i	$-0.966962\pi$
$409$	−16454.0	−1.98924	−0.994619	+	0.103605i	$0.966962\pi$
$410$	3420.00	0.411956
$411$	1650.00	0.198026
$412$	4800.00	0.573978
$413$	−5796.00	−0.690563
$414$	−2304.00	−0.273516
$415$	−6140.00	−0.726267
$416$	−448.000	−0.0528005
$417$	4668.00	0.548185
$418$	88.0000	0.0102972
$419$	−3948.00	−0.460316	−0.230158	−	0.973153i	$-0.573924\pi$
$419$	−3948.00	−0.460316	−0.230158	+	0.973153i	$0.573924\pi$
$420$	420.000	0.0487950
$421$	−14674.0	−1.69873	−0.849367	−	0.527803i	$-0.823016\pi$
$421$	−14674.0	−1.69873	−0.849367	+	0.527803i	$0.823016\pi$
$422$	−5224.00	−0.602607
$423$	2376.00	0.273109
$424$	−624.000	−0.0714720
$425$	−1950.00	−0.222562
$426$	−5280.00	−0.600509
$427$	−2562.00	−0.290360
$428$	7664.00	0.865545
$429$	462.000	0.0519943
$430$	840.000	0.0942056
$431$	12128.0	1.35542	0.677709	−	0.735330i	$-0.262973\pi$
$431$	12128.0	1.35542	0.677709	+	0.735330i	$0.262973\pi$
$432$	−432.000	−0.0481125
$433$	−16366.0	−1.81640	−0.908198	−	0.418540i	$-0.862542\pi$
$433$	−16366.0	−1.81640	−0.908198	+	0.418540i	$0.862542\pi$
$434$	−2576.00	−0.284912
$435$	−1650.00	−0.181865
$436$	3768.00	0.413886
$437$	512.000	0.0560464
$438$	−5316.00	−0.579928
$439$	−13528.0	−1.47074	−0.735372	−	0.677664i	$-0.762992\pi$
$439$	−13528.0	−1.47074	−0.735372	+	0.677664i	$0.762992\pi$
$440$	440.000	0.0476731
$441$	441.000	0.0476190
$442$	2184.00	0.235028
$443$	3196.00	0.342769	0.171384	−	0.985204i	$-0.445176\pi$
$443$	3196.00	0.342769	0.171384	+	0.985204i	$0.445176\pi$
$444$	−1896.00	−0.202658
$445$	−3150.00	−0.335560
$446$	−5808.00	−0.616630
$447$	2430.00	0.257125
$448$	−448.000	−0.0472456
$449$	−8318.00	−0.874278	−0.437139	−	0.899394i	$-0.644008\pi$
$449$	−8318.00	−0.874278	−0.437139	+	0.899394i	$0.644008\pi$
$450$	−450.000	−0.0471405
$451$	3762.00	0.392784
$452$	4296.00	0.447051
$453$	2328.00	0.241455
$454$	2136.00	0.220809
$455$	−490.000	−0.0504869
$456$	96.0000	0.00985880
$457$	−13382.0	−1.36977	−0.684884	−	0.728653i	$-0.740147\pi$
$457$	−13382.0	−1.36977	−0.684884	+	0.728653i	$0.740147\pi$
$458$	−1372.00	−0.139977
$459$	2106.00	0.214160
$460$	2560.00	0.259480
$461$	3534.00	0.357039	0.178519	−	0.983936i	$-0.442869\pi$
$461$	3534.00	0.357039	0.178519	+	0.983936i	$0.442869\pi$
$462$	462.000	0.0465242
$463$	3928.00	0.394276	0.197138	−	0.980376i	$-0.436835\pi$
$463$	3928.00	0.394276	0.197138	+	0.980376i	$0.436835\pi$
$464$	1760.00	0.176090
$465$	2760.00	0.275251
$466$	11116.0	1.10502
$467$	−10572.0	−1.04757	−0.523784	−	0.851851i	$-0.675480\pi$
$467$	−10572.0	−1.04757	−0.523784	+	0.851851i	$0.675480\pi$
$468$	504.000	0.0497808
$469$	−4844.00	−0.476919
$470$	−2640.00	−0.259094
$471$	6318.00	0.618085
$472$	−6624.00	−0.645963
$473$	924.000	0.0898215
$474$	−7200.00	−0.697694
$475$	100.000	0.00965961
$476$	2184.00	0.210301
$477$	702.000	0.0673844
$478$	12224.0	1.16969
$479$	10608.0	1.01188	0.505941	−	0.862568i	$-0.331145\pi$
$479$	10608.0	1.01188	0.505941	+	0.862568i	$0.331145\pi$
$480$	480.000	0.0456435
$481$	2212.00	0.209685
$482$	6716.00	0.634659
$483$	2688.00	0.253226
$484$	484.000	0.0454545
$485$	−6550.00	−0.613237
$486$	486.000	0.0453609
$487$	16224.0	1.50961	0.754805	−	0.655950i	$-0.227732\pi$
$487$	16224.0	1.50961	0.754805	+	0.655950i	$0.227732\pi$
$488$	−2928.00	−0.271607
$489$	−2268.00	−0.209739
$490$	−490.000	−0.0451754
$491$	1772.00	0.162870	0.0814351	−	0.996679i	$-0.474050\pi$
$491$	1772.00	0.162870	0.0814351	+	0.996679i	$0.474050\pi$
$492$	4104.00	0.376062
$493$	−8580.00	−0.783821
$494$	−112.000	−0.0102006
$495$	−495.000	−0.0449467
$496$	−2944.00	−0.266511
$497$	6160.00	0.555963
$498$	−7368.00	−0.662988
$499$	−15084.0	−1.35321	−0.676606	−	0.736345i	$-0.736550\pi$
$499$	−15084.0	−1.35321	−0.676606	+	0.736345i	$0.736550\pi$
$500$	500.000	0.0447214
$501$	−2712.00	−0.241843
$502$	2664.00	0.236853
$503$	16488.0	1.46156	0.730779	−	0.682614i	$-0.239157\pi$
$503$	16488.0	1.46156	0.730779	+	0.682614i	$0.239157\pi$
$504$	504.000	0.0445435
$505$	5070.00	0.446757
$506$	2816.00	0.247404
$507$	6003.00	0.525843
$508$	10112.0	0.883164
$509$	−7818.00	−0.680799	−0.340400	−	0.940281i	$-0.610562\pi$
$509$	−7818.00	−0.680799	−0.340400	+	0.940281i	$0.610562\pi$
$510$	−2340.00	−0.203170
$511$	6202.00	0.536909
$512$	−512.000	−0.0441942
$513$	−108.000	−0.00929496
$514$	2364.00	0.202863
$515$	6000.00	0.513382
$516$	1008.00	0.0859975
$517$	−2904.00	−0.247036
$518$	2212.00	0.187625
$519$	−4458.00	−0.377041
$520$	−560.000	−0.0472262
$521$	−8230.00	−0.692059	−0.346030	−	0.938224i	$-0.612470\pi$
$521$	−8230.00	−0.692059	−0.346030	+	0.938224i	$0.612470\pi$
$522$	−1980.00	−0.166020
$523$	−8036.00	−0.671873	−0.335937	−	0.941885i	$-0.609053\pi$
$523$	−8036.00	−0.671873	−0.335937	+	0.941885i	$0.609053\pi$
$524$	−7280.00	−0.606924
$525$	525.000	0.0436436
$526$	7824.00	0.648560
$527$	14352.0	1.18631
$528$	528.000	0.0435194
$529$	4217.00	0.346593
$530$	−780.000	−0.0639265
$531$	7452.00	0.609019
$532$	−112.000	−0.00912747
$533$	−4788.00	−0.389102
$534$	−3780.00	−0.306323
$535$	9580.00	0.774167
$536$	−5536.00	−0.446117
$537$	−13740.0	−1.10414
$538$	−3852.00	−0.308683
$539$	−539.000	−0.0430730
$540$	−540.000	−0.0430331
$541$	−7874.00	−0.625748	−0.312874	−	0.949795i	$-0.601292\pi$
$541$	−7874.00	−0.625748	−0.312874	+	0.949795i	$0.601292\pi$
$542$	288.000	0.0228241
$543$	−186.000	−0.0146999
$544$	2496.00	0.196719
$545$	4710.00	0.370191
$546$	−588.000	−0.0460881
$547$	−18444.0	−1.44170	−0.720849	−	0.693092i	$-0.756248\pi$
$547$	−18444.0	−1.44170	−0.720849	+	0.693092i	$0.756248\pi$
$548$	−2200.00	−0.171495
$549$	3294.00	0.256074
$550$	550.000	0.0426401
$551$	440.000	0.0340193
$552$	3072.00	0.236871
$553$	8400.00	0.645939
$554$	15988.0	1.22611
$555$	−2370.00	−0.181263
$556$	−6224.00	−0.474742
$557$	−22770.0	−1.73213	−0.866065	−	0.499932i	$-0.833358\pi$
$557$	−22770.0	−1.73213	−0.866065	+	0.499932i	$0.833358\pi$
$558$	3312.00	0.251269
$559$	−1176.00	−0.0889794
$560$	−560.000	−0.0422577
$561$	−2574.00	−0.193715
$562$	−11220.0	−0.842148
$563$	−16508.0	−1.23575	−0.617877	−	0.786275i	$-0.712007\pi$
$563$	−16508.0	−1.23575	−0.617877	+	0.786275i	$0.712007\pi$
$564$	−3168.00	−0.236519
$565$	5370.00	0.399854
$566$	−7672.00	−0.569749
$567$	−567.000	−0.0419961
$568$	7040.00	0.520056
$569$	−19238.0	−1.41740	−0.708699	−	0.705511i	$-0.750717\pi$
$569$	−19238.0	−1.41740	−0.708699	+	0.705511i	$0.750717\pi$
$570$	120.000	0.00881798
$571$	1356.00	0.0993815	0.0496907	−	0.998765i	$-0.484176\pi$
$571$	1356.00	0.0993815	0.0496907	+	0.998765i	$0.484176\pi$
$572$	−616.000	−0.0450284
$573$	8136.00	0.593170
$574$	−4788.00	−0.348166
$575$	3200.00	0.232086
$576$	576.000	0.0416667
$577$	−17662.0	−1.27431	−0.637157	−	0.770734i	$-0.719890\pi$
$577$	−17662.0	−1.27431	−0.637157	+	0.770734i	$0.719890\pi$
$578$	−2342.00	−0.168537
$579$	−1542.00	−0.110679
$580$	2200.00	0.157500
$581$	8596.00	0.613808
$582$	−7860.00	−0.559807
$583$	−858.000	−0.0609515
$584$	7088.00	0.502232
$585$	630.000	0.0445253
$586$	−5996.00	−0.422683
$587$	−404.000	−0.0284069	−0.0142035	−	0.999899i	$-0.504521\pi$
$587$	−404.000	−0.0284069	−0.0142035	+	0.999899i	$0.504521\pi$
$588$	−588.000	−0.0412393
$589$	−736.000	−0.0514879
$590$	−8280.00	−0.577766
$591$	−13506.0	−0.940038
$592$	2528.00	0.175507
$593$	4578.00	0.317025	0.158513	−	0.987357i	$-0.449330\pi$
$593$	4578.00	0.317025	0.158513	+	0.987357i	$0.449330\pi$
$594$	−594.000	−0.0410305
$595$	2730.00	0.188099
$596$	−3240.00	−0.222677
$597$	1680.00	0.115172
$598$	−3584.00	−0.245085
$599$	5856.00	0.399449	0.199724	−	0.979852i	$-0.435995\pi$
$599$	5856.00	0.399449	0.199724	+	0.979852i	$0.435995\pi$
$600$	600.000	0.0408248
$601$	−6358.00	−0.431528	−0.215764	−	0.976446i	$-0.569224\pi$
$601$	−6358.00	−0.431528	−0.215764	+	0.976446i	$0.569224\pi$
$602$	−1176.00	−0.0796182
$603$	6228.00	0.420603
$604$	−3104.00	−0.209106
$605$	605.000	0.0406558
$606$	6084.00	0.407831
$607$	15488.0	1.03565	0.517824	−	0.855487i	$-0.326742\pi$
$607$	15488.0	1.03565	0.517824	+	0.855487i	$0.326742\pi$
$608$	−128.000	−0.00853797
$609$	2310.00	0.153704
$610$	−3660.00	−0.242933
$611$	3696.00	0.244720
$612$	−2808.00	−0.185468
$613$	−8474.00	−0.558339	−0.279169	−	0.960242i	$-0.590059\pi$
$613$	−8474.00	−0.558339	−0.279169	+	0.960242i	$0.590059\pi$
$614$	10936.0	0.718796
$615$	5130.00	0.336360
$616$	−616.000	−0.0402911
$617$	16794.0	1.09579	0.547894	−	0.836548i	$-0.315430\pi$
$617$	16794.0	1.09579	0.547894	+	0.836548i	$0.315430\pi$
$618$	7200.00	0.468651
$619$	−12692.0	−0.824127	−0.412063	−	0.911155i	$-0.635192\pi$
$619$	−12692.0	−0.824127	−0.412063	+	0.911155i	$0.635192\pi$
$620$	−3680.00	−0.238375
$621$	−3456.00	−0.223324
$622$	12000.0	0.773563
$623$	4410.00	0.283600
$624$	−672.000	−0.0431114
$625$	625.000	0.0400000
$626$	16044.0	1.02436
$627$	132.000	0.00840761
$628$	−8424.00	−0.535277
$629$	−12324.0	−0.781224
$630$	630.000	0.0398410
$631$	−27904.0	−1.76044	−0.880222	−	0.474561i	$-0.842607\pi$
$631$	−27904.0	−1.76044	−0.880222	+	0.474561i	$0.842607\pi$
$632$	9600.00	0.604221
$633$	−7836.00	−0.492027
$634$	−9708.00	−0.608129
$635$	12640.0	0.789926
$636$	−936.000	−0.0583566
$637$	686.000	0.0426692
$638$	2420.00	0.150170
$639$	−7920.00	−0.490314
$640$	−640.000	−0.0395285
$641$	24962.0	1.53813	0.769064	−	0.639172i	$-0.220723\pi$
$641$	24962.0	1.53813	0.769064	+	0.639172i	$0.220723\pi$
$642$	11496.0	0.706715
$643$	9332.00	0.572345	0.286173	−	0.958178i	$-0.407617\pi$
$643$	9332.00	0.572345	0.286173	+	0.958178i	$0.407617\pi$
$644$	−3584.00	−0.219300
$645$	1260.00	0.0769185
$646$	624.000	0.0380046
$647$	−6320.00	−0.384026	−0.192013	−	0.981392i	$-0.561502\pi$
$647$	−6320.00	−0.384026	−0.192013	+	0.981392i	$0.561502\pi$
$648$	−648.000	−0.0392837
$649$	−9108.00	−0.550879
$650$	−700.000	−0.0422404
$651$	−3864.00	−0.232630
$652$	3024.00	0.181640
$653$	13782.0	0.825929	0.412964	−	0.910747i	$-0.364493\pi$
$653$	13782.0	0.825929	0.412964	+	0.910747i	$0.364493\pi$
$654$	5652.00	0.337937
$655$	−9100.00	−0.542850
$656$	−5472.00	−0.325679
$657$	−7974.00	−0.473509
$658$	3696.00	0.218974
$659$	−25004.0	−1.47802	−0.739012	−	0.673692i	$-0.764707\pi$
$659$	−25004.0	−1.47802	−0.739012	+	0.673692i	$0.764707\pi$
$660$	660.000	0.0389249
$661$	30158.0	1.77460	0.887300	−	0.461193i	$-0.152579\pi$
$661$	30158.0	1.77460	0.887300	+	0.461193i	$0.152579\pi$
$662$	5480.00	0.321731
$663$	3276.00	0.191899
$664$	9824.00	0.574164
$665$	−140.000	−0.00816386
$666$	−2844.00	−0.165470
$667$	14080.0	0.817361
$668$	3616.00	0.209442
$669$	−8712.00	−0.503476
$670$	−6920.00	−0.399019
$671$	−4026.00	−0.231627
$672$	−672.000	−0.0385758
$673$	−20030.0	−1.14725	−0.573625	−	0.819118i	$-0.694463\pi$
$673$	−20030.0	−1.14725	−0.573625	+	0.819118i	$0.694463\pi$
$674$	23932.0	1.36769
$675$	−675.000	−0.0384900
$676$	−8004.00	−0.455394
$677$	−8618.00	−0.489242	−0.244621	−	0.969619i	$-0.578663\pi$
$677$	−8618.00	−0.489242	−0.244621	+	0.969619i	$0.578663\pi$
$678$	6444.00	0.365015
$679$	9170.00	0.518280
$680$	3120.00	0.175951
$681$	3204.00	0.180290
$682$	−4048.00	−0.227281
$683$	26604.0	1.49044	0.745222	−	0.666816i	$-0.232343\pi$
$683$	26604.0	1.49044	0.745222	+	0.666816i	$0.232343\pi$
$684$	144.000	0.00804967
$685$	−2750.00	−0.153390
$686$	686.000	0.0381802
$687$	−2058.00	−0.114291
$688$	−1344.00	−0.0744760
$689$	1092.00	0.0603801
$690$	3840.00	0.211864
$691$	−21980.0	−1.21007	−0.605035	−	0.796199i	$-0.706841\pi$
$691$	−21980.0	−1.21007	−0.605035	+	0.796199i	$0.706841\pi$
$692$	5944.00	0.326527
$693$	693.000	0.0379869
$694$	12456.0	0.681302
$695$	−7780.00	−0.424622
$696$	2640.00	0.143777
$697$	26676.0	1.44968
$698$	−4156.00	−0.225368
$699$	16674.0	0.902244
$700$	−700.000	−0.0377964
$701$	23550.0	1.26886	0.634430	−	0.772980i	$-0.281235\pi$
$701$	23550.0	1.26886	0.634430	+	0.772980i	$0.281235\pi$
$702$	756.000	0.0406458
$703$	632.000	0.0339066
$704$	−704.000	−0.0376889
$705$	−3960.00	−0.211549
$706$	−14340.0	−0.764438
$707$	−7098.00	−0.377578
$708$	−9936.00	−0.527426
$709$	20126.0	1.06608	0.533038	−	0.846091i	$-0.321050\pi$
$709$	20126.0	1.06608	0.533038	+	0.846091i	$0.321050\pi$
$710$	8800.00	0.465152
$711$	−10800.0	−0.569665
$712$	5040.00	0.265284
$713$	−23552.0	−1.23707
$714$	3276.00	0.171710
$715$	−770.000	−0.0402746
$716$	18320.0	0.956216
$717$	18336.0	0.955049
$718$	−15504.0	−0.805855
$719$	−20648.0	−1.07099	−0.535494	−	0.844539i	$-0.679875\pi$
$719$	−20648.0	−1.07099	−0.535494	+	0.844539i	$0.679875\pi$
$720$	720.000	0.0372678
$721$	−8400.00	−0.433887
$722$	13686.0	0.705457
$723$	10074.0	0.518197
$724$	248.000	0.0127305
$725$	2750.00	0.140872
$726$	726.000	0.0371135
$727$	21712.0	1.10764	0.553819	−	0.832637i	$-0.313170\pi$
$727$	21712.0	1.10764	0.553819	+	0.832637i	$0.313170\pi$
$728$	784.000	0.0399134
$729$	729.000	0.0370370
$730$	8860.00	0.449210
$731$	6552.00	0.331511
$732$	−4392.00	−0.221766
$733$	28862.0	1.45436	0.727178	−	0.686449i	$-0.240832\pi$
$733$	28862.0	1.45436	0.727178	+	0.686449i	$0.240832\pi$
$734$	3600.00	0.181033
$735$	−735.000	−0.0368856
$736$	−4096.00	−0.205137
$737$	−7612.00	−0.380450
$738$	6156.00	0.307054
$739$	−19548.0	−0.973051	−0.486526	−	0.873666i	$-0.661736\pi$
$739$	−19548.0	−0.973051	−0.486526	+	0.873666i	$0.661736\pi$
$740$	3160.00	0.156978
$741$	−168.000	−0.00832879
$742$	1092.00	0.0540277
$743$	20360.0	1.00530	0.502649	−	0.864491i	$-0.332359\pi$
$743$	20360.0	1.00530	0.502649	+	0.864491i	$0.332359\pi$
$744$	−4416.00	−0.217605
$745$	−4050.00	−0.199168
$746$	18228.0	0.894604
$747$	−11052.0	−0.541327
$748$	3432.00	0.167762
$749$	−13412.0	−0.654291
$750$	750.000	0.0365148
$751$	−32456.0	−1.57701	−0.788506	−	0.615027i	$-0.789145\pi$
$751$	−32456.0	−1.57701	−0.788506	+	0.615027i	$0.789145\pi$
$752$	4224.00	0.204832
$753$	3996.00	0.193390
$754$	−3080.00	−0.148763
$755$	−3880.00	−0.187030
$756$	756.000	0.0363696
$757$	−31490.0	−1.51192	−0.755960	−	0.654618i	$-0.772830\pi$
$757$	−31490.0	−1.51192	−0.755960	+	0.654618i	$0.772830\pi$
$758$	5992.00	0.287123
$759$	4224.00	0.202005
$760$	−160.000	−0.00763659
$761$	−12998.0	−0.619155	−0.309578	−	0.950874i	$-0.600188\pi$
$761$	−12998.0	−0.619155	−0.309578	+	0.950874i	$0.600188\pi$
$762$	15168.0	0.721101
$763$	−6594.00	−0.312869
$764$	−10848.0	−0.513700
$765$	−3510.00	−0.165888
$766$	−10480.0	−0.494331
$767$	11592.0	0.545714
$768$	−768.000	−0.0360844
$769$	16210.0	0.760140	0.380070	−	0.924958i	$-0.375900\pi$
$769$	16210.0	0.760140	0.380070	+	0.924958i	$0.375900\pi$
$770$	−770.000	−0.0360375
$771$	3546.00	0.165637
$772$	2056.00	0.0958511
$773$	−25554.0	−1.18902	−0.594511	−	0.804088i	$-0.702654\pi$
$773$	−25554.0	−1.18902	−0.594511	+	0.804088i	$0.702654\pi$
$774$	1512.00	0.0702167
$775$	−4600.00	−0.213209
$776$	10480.0	0.484807
$777$	3318.00	0.153195
$778$	23524.0	1.08403
$779$	−1368.00	−0.0629187
$780$	−840.000	−0.0385600
$781$	9680.00	0.443505
$782$	19968.0	0.913113
$783$	−2970.00	−0.135554
$784$	784.000	0.0357143
$785$	−10530.0	−0.478767
$786$	−10920.0	−0.495552
$787$	9604.00	0.435001	0.217500	−	0.976060i	$-0.430210\pi$
$787$	9604.00	0.435001	0.217500	+	0.976060i	$0.430210\pi$
$788$	18008.0	0.814097
$789$	11736.0	0.529547
$790$	12000.0	0.540431
$791$	−7518.00	−0.337938
$792$	792.000	0.0355335
$793$	5124.00	0.229456
$794$	−20524.0	−0.917342
$795$	−1170.00	−0.0521958
$796$	−2240.00	−0.0997421
$797$	30694.0	1.36416	0.682081	−	0.731277i	$-0.261075\pi$
$797$	30694.0	1.36416	0.682081	+	0.731277i	$0.261075\pi$
$798$	−168.000	−0.00745255
$799$	−20592.0	−0.911755
$800$	−800.000	−0.0353553
$801$	−5670.00	−0.250112
$802$	20188.0	0.888857
$803$	9746.00	0.428305
$804$	−8304.00	−0.364253
$805$	−4480.00	−0.196148
$806$	5152.00	0.225151
$807$	−5778.00	−0.252039
$808$	−8112.00	−0.353192
$809$	35370.0	1.53714	0.768568	−	0.639768i	$-0.220970\pi$
$809$	35370.0	1.53714	0.768568	+	0.639768i	$0.220970\pi$
$810$	−810.000	−0.0351364
$811$	19372.0	0.838771	0.419385	−	0.907808i	$-0.362246\pi$
$811$	19372.0	0.838771	0.419385	+	0.907808i	$0.362246\pi$
$812$	−3080.00	−0.133112
$813$	432.000	0.0186358
$814$	3476.00	0.149673
$815$	3780.00	0.162463
$816$	3744.00	0.160620
$817$	−336.000	−0.0143882
$818$	32908.0	1.40660
$819$	−882.000	−0.0376307
$820$	−6840.00	−0.291297
$821$	21270.0	0.904176	0.452088	−	0.891973i	$-0.350679\pi$
$821$	21270.0	0.904176	0.452088	+	0.891973i	$0.350679\pi$
$822$	−3300.00	−0.140025
$823$	41792.0	1.77008	0.885041	−	0.465513i	$-0.154130\pi$
$823$	41792.0	1.77008	0.885041	+	0.465513i	$0.154130\pi$
$824$	−9600.00	−0.405864
$825$	825.000	0.0348155
$826$	11592.0	0.488302
$827$	35356.0	1.48664	0.743318	−	0.668938i	$-0.233251\pi$
$827$	35356.0	1.48664	0.743318	+	0.668938i	$0.233251\pi$
$828$	4608.00	0.193405
$829$	−5466.00	−0.229001	−0.114501	−	0.993423i	$-0.536527\pi$
$829$	−5466.00	−0.229001	−0.114501	+	0.993423i	$0.536527\pi$
$830$	12280.0	0.513548
$831$	23982.0	1.00111
$832$	896.000	0.0373356
$833$	−3822.00	−0.158973
$834$	−9336.00	−0.387625
$835$	4520.00	0.187331
$836$	−176.000	−0.00728120
$837$	4968.00	0.205160
$838$	7896.00	0.325493
$839$	−8064.00	−0.331824	−0.165912	−	0.986141i	$-0.553057\pi$
$839$	−8064.00	−0.331824	−0.165912	+	0.986141i	$0.553057\pi$
$840$	−840.000	−0.0345033
$841$	−12289.0	−0.503875
$842$	29348.0	1.20119
$843$	−16830.0	−0.687611
$844$	10448.0	0.426108
$845$	−10005.0	−0.407317
$846$	−4752.00	−0.193117
$847$	−847.000	−0.0343604
$848$	1248.00	0.0505383
$849$	−11508.0	−0.465199
$850$	3900.00	0.157375
$851$	20224.0	0.814653
$852$	10560.0	0.424624
$853$	−29914.0	−1.20075	−0.600373	−	0.799720i	$-0.704981\pi$
$853$	−29914.0	−1.20075	−0.600373	+	0.799720i	$0.704981\pi$
$854$	5124.00	0.205316
$855$	180.000	0.00719985
$856$	−15328.0	−0.612033
$857$	1770.00	0.0705508	0.0352754	−	0.999378i	$-0.488769\pi$
$857$	1770.00	0.0705508	0.0352754	+	0.999378i	$0.488769\pi$
$858$	−924.000	−0.0367655
$859$	12572.0	0.499361	0.249681	−	0.968328i	$-0.419674\pi$
$859$	12572.0	0.499361	0.249681	+	0.968328i	$0.419674\pi$
$860$	−1680.00	−0.0666134
$861$	−7182.00	−0.284276
$862$	−24256.0	−0.958425
$863$	−37944.0	−1.49667	−0.748336	−	0.663319i	$-0.769147\pi$
$863$	−37944.0	−1.49667	−0.748336	+	0.663319i	$0.769147\pi$
$864$	864.000	0.0340207
$865$	7430.00	0.292055
$866$	32732.0	1.28439
$867$	−3513.00	−0.137610
$868$	5152.00	0.201463
$869$	13200.0	0.515281
$870$	3300.00	0.128598
$871$	9688.00	0.376883
$872$	−7536.00	−0.292662
$873$	−11790.0	−0.457080
$874$	−1024.00	−0.0396308
$875$	−875.000	−0.0338062
$876$	10632.0	0.410071
$877$	4590.00	0.176731	0.0883656	−	0.996088i	$-0.471836\pi$
$877$	4590.00	0.176731	0.0883656	+	0.996088i	$0.471836\pi$
$878$	27056.0	1.03997
$879$	−8994.00	−0.345120
$880$	−880.000	−0.0337100
$881$	−10478.0	−0.400696	−0.200348	−	0.979725i	$-0.564207\pi$
$881$	−10478.0	−0.400696	−0.200348	+	0.979725i	$0.564207\pi$
$882$	−882.000	−0.0336718
$883$	9876.00	0.376392	0.188196	−	0.982132i	$-0.439736\pi$
$883$	9876.00	0.376392	0.188196	+	0.982132i	$0.439736\pi$
$884$	−4368.00	−0.166190
$885$	−12420.0	−0.471744
$886$	−6392.00	−0.242374
$887$	−14344.0	−0.542981	−0.271491	−	0.962441i	$-0.587517\pi$
$887$	−14344.0	−0.542981	−0.271491	+	0.962441i	$0.587517\pi$
$888$	3792.00	0.143301
$889$	−17696.0	−0.667609
$890$	6300.00	0.237277
$891$	−891.000	−0.0335013
$892$	11616.0	0.436023
$893$	1056.00	0.0395719
$894$	−4860.00	−0.181815
$895$	22900.0	0.855265
$896$	896.000	0.0334077
$897$	−5376.00	−0.200111
$898$	16636.0	0.618208
$899$	−20240.0	−0.750881
$900$	900.000	0.0333333
$901$	−6084.00	−0.224958
$902$	−7524.00	−0.277740
$903$	−1764.00	−0.0650080
$904$	−8592.00	−0.316112
$905$	310.000	0.0113865
$906$	−4656.00	−0.170734
$907$	−26708.0	−0.977756	−0.488878	−	0.872352i	$-0.662594\pi$
$907$	−26708.0	−0.977756	−0.488878	+	0.872352i	$0.662594\pi$
$908$	−4272.00	−0.156136
$909$	9126.00	0.332993
$910$	980.000	0.0356997
$911$	−31032.0	−1.12858	−0.564290	−	0.825577i	$-0.690850\pi$
$911$	−31032.0	−1.12858	−0.564290	+	0.825577i	$0.690850\pi$
$912$	−192.000	−0.00697122
$913$	13508.0	0.489649
$914$	26764.0	0.968572
$915$	−5490.00	−0.198354
$916$	2744.00	0.0989785
$917$	12740.0	0.458792
$918$	−4212.00	−0.151434
$919$	35336.0	1.26836	0.634182	−	0.773184i	$-0.281337\pi$
$919$	35336.0	1.26836	0.634182	+	0.773184i	$0.281337\pi$
$920$	−5120.00	−0.183480
$921$	16404.0	0.586895
$922$	−7068.00	−0.252464
$923$	−12320.0	−0.439347
$924$	−924.000	−0.0328976
$925$	3950.00	0.140406
$926$	−7856.00	−0.278795
$927$	10800.0	0.382652
$928$	−3520.00	−0.124515
$929$	6498.00	0.229486	0.114743	−	0.993395i	$-0.463396\pi$
$929$	6498.00	0.229486	0.114743	+	0.993395i	$0.463396\pi$
$930$	−5520.00	−0.194632
$931$	196.000	0.00689972
$932$	−22232.0	−0.781366
$933$	18000.0	0.631612
$934$	21144.0	0.740742
$935$	4290.00	0.150051
$936$	−1008.00	−0.0352003
$937$	−43958.0	−1.53260	−0.766300	−	0.642484i	$-0.777904\pi$
$937$	−43958.0	−1.53260	−0.766300	+	0.642484i	$0.777904\pi$
$938$	9688.00	0.337233
$939$	24066.0	0.836384
$940$	5280.00	0.183207
$941$	−9154.00	−0.317122	−0.158561	−	0.987349i	$-0.550685\pi$
$941$	−9154.00	−0.317122	−0.158561	+	0.987349i	$0.550685\pi$
$942$	−12636.0	−0.437052
$943$	−43776.0	−1.51171
$944$	13248.0	0.456764
$945$	945.000	0.0325300
$946$	−1848.00	−0.0635134
$947$	16404.0	0.562892	0.281446	−	0.959577i	$-0.409186\pi$
$947$	16404.0	0.562892	0.281446	+	0.959577i	$0.409186\pi$
$948$	14400.0	0.493344
$949$	−12404.0	−0.424290
$950$	−200.000	−0.00683038
$951$	−14562.0	−0.496535
$952$	−4368.00	−0.148706
$953$	21130.0	0.718224	0.359112	−	0.933294i	$-0.383080\pi$
$953$	21130.0	0.718224	0.359112	+	0.933294i	$0.383080\pi$
$954$	−1404.00	−0.0476480
$955$	−13560.0	−0.459467
$956$	−24448.0	−0.827097
$957$	3630.00	0.122614
$958$	−21216.0	−0.715509
$959$	3850.00	0.129638
$960$	−960.000	−0.0322749
$961$	4065.00	0.136451
$962$	−4424.00	−0.148270
$963$	17244.0	0.577030
$964$	−13432.0	−0.448771
$965$	2570.00	0.0857318
$966$	−5376.00	−0.179058
$967$	10088.0	0.335479	0.167740	−	0.985831i	$-0.446353\pi$
$967$	10088.0	0.335479	0.167740	+	0.985831i	$0.446353\pi$
$968$	−968.000	−0.0321412
$969$	936.000	0.0310306
$970$	13100.0	0.433624
$971$	42684.0	1.41071	0.705353	−	0.708857i	$-0.250789\pi$
$971$	42684.0	1.41071	0.705353	+	0.708857i	$0.250789\pi$
$972$	−972.000	−0.0320750
$973$	10892.0	0.358871
$974$	−32448.0	−1.06745
$975$	−1050.00	−0.0344891
$976$	5856.00	0.192055
$977$	23346.0	0.764488	0.382244	−	0.924061i	$-0.375151\pi$
$977$	23346.0	0.764488	0.382244	+	0.924061i	$0.375151\pi$
$978$	4536.00	0.148308
$979$	6930.00	0.226235
$980$	980.000	0.0319438
$981$	8478.00	0.275924
$982$	−3544.00	−0.115167
$983$	−23952.0	−0.777162	−0.388581	−	0.921415i	$-0.627035\pi$
$983$	−23952.0	−0.777162	−0.388581	+	0.921415i	$0.627035\pi$
$984$	−8208.00	−0.265916
$985$	22510.0	0.728150
$986$	17160.0	0.554245
$987$	5544.00	0.178792
$988$	224.000	0.00721294
$989$	−10752.0	−0.345696
$990$	990.000	0.0317821
$991$	42312.0	1.35629	0.678146	−	0.734927i	$-0.262784\pi$
$991$	42312.0	1.35629	0.678146	+	0.734927i	$0.262784\pi$
$992$	5888.00	0.188452
$993$	8220.00	0.262693
$994$	−12320.0	−0.393125
$995$	−2800.00	−0.0892120
$996$	14736.0	0.468803
$997$	27270.0	0.866248	0.433124	−	0.901334i	$-0.357411\pi$
$997$	27270.0	0.866248	0.433124	+	0.901334i	$0.357411\pi$
$998$	30168.0	0.956865
$999$	−4266.00	−0.135105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2310.4.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2310.4.a.b.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2310.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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