LMFDB - Embedded newform 1470.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,4,Mod(1,1470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, 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("1470.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1470.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:	$86.7328077084$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} -6.00000 q^{6} -8.00000 q^{8} +9.00000 q^{9} -10.0000 q^{10} -32.0000 q^{11} +12.0000 q^{12} +15.0000 q^{13} +15.0000 q^{15} +16.0000 q^{16} -70.0000 q^{17} -18.0000 q^{18} +15.0000 q^{19} +20.0000 q^{20} +64.0000 q^{22} -42.0000 q^{23} -24.0000 q^{24} +25.0000 q^{25} -30.0000 q^{26} +27.0000 q^{27} +90.0000 q^{29} -30.0000 q^{30} -85.0000 q^{31} -32.0000 q^{32} -96.0000 q^{33} +140.000 q^{34} +36.0000 q^{36} +113.000 q^{37} -30.0000 q^{38} +45.0000 q^{39} -40.0000 q^{40} +164.000 q^{41} +169.000 q^{43} -128.000 q^{44} +45.0000 q^{45} +84.0000 q^{46} +326.000 q^{47} +48.0000 q^{48} -50.0000 q^{50} -210.000 q^{51} +60.0000 q^{52} -44.0000 q^{53} -54.0000 q^{54} -160.000 q^{55} +45.0000 q^{57} -180.000 q^{58} -782.000 q^{59} +60.0000 q^{60} +658.000 q^{61} +170.000 q^{62} +64.0000 q^{64} +75.0000 q^{65} +192.000 q^{66} +1071.00 q^{67} -280.000 q^{68} -126.000 q^{69} +344.000 q^{71} -72.0000 q^{72} +431.000 q^{73} -226.000 q^{74} +75.0000 q^{75} +60.0000 q^{76} -90.0000 q^{78} +397.000 q^{79} +80.0000 q^{80} +81.0000 q^{81} -328.000 q^{82} +680.000 q^{83} -350.000 q^{85} -338.000 q^{86} +270.000 q^{87} +256.000 q^{88} +1534.00 q^{89} -90.0000 q^{90} -168.000 q^{92} -255.000 q^{93} -652.000 q^{94} +75.0000 q^{95} -96.0000 q^{96} -1234.00 q^{97} -288.000 q^{99} +O(q^{100})\)

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
$7$	0	0
$7$	0	0
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−10.0000	−0.316228
$11$	−32.0000	−0.877124	−0.438562	−	0.898701i	$-0.644512\pi$
$11$	−32.0000	−0.877124	−0.438562	+	0.898701i	$0.644512\pi$
$12$	12.0000	0.288675
$13$	15.0000	0.320019	0.160010	−	0.987115i	$-0.448847\pi$
$13$	15.0000	0.320019	0.160010	+	0.987115i	$0.448847\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	−70.0000	−0.998676	−0.499338	−	0.866407i	$-0.666423\pi$
$17$	−70.0000	−0.998676	−0.499338	+	0.866407i	$0.666423\pi$
$18$	−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	$0.471135\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	64.0000	0.620220
$23$	−42.0000	−0.380765	−0.190383	−	0.981710i	$-0.560973\pi$
$23$	−42.0000	−0.380765	−0.190383	+	0.981710i	$0.560973\pi$
$24$	−24.0000	−0.204124
$25$	25.0000	0.200000
$26$	−30.0000	−0.226288
$27$	27.0000	0.192450
$28$	0	0
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	−30.0000	−0.182574
$31$	−85.0000	−0.492466	−0.246233	−	0.969211i	$-0.579193\pi$
$31$	−85.0000	−0.492466	−0.246233	+	0.969211i	$0.579193\pi$
$32$	−32.0000	−0.176777
$33$	−96.0000	−0.506408
$34$	140.000	0.706171
$35$	0	0
$36$	36.0000	0.166667
$37$	113.000	0.502083	0.251042	−	0.967976i	$-0.419227\pi$
$37$	113.000	0.502083	0.251042	+	0.967976i	$0.419227\pi$
$38$	−30.0000	−0.128070
$39$	45.0000	0.184763
$40$	−40.0000	−0.158114
$41$	164.000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	164.000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	169.000	0.599355	0.299677	−	0.954041i	$-0.403121\pi$
$43$	169.000	0.599355	0.299677	+	0.954041i	$0.403121\pi$
$44$	−128.000	−0.438562
$45$	45.0000	0.149071
$46$	84.0000	0.269242
$47$	326.000	1.01174	0.505872	−	0.862608i	$-0.331171\pi$
$47$	326.000	1.01174	0.505872	+	0.862608i	$0.331171\pi$
$48$	48.0000	0.144338
$49$	0	0
$50$	−50.0000	−0.141421
$51$	−210.000	−0.576586
$52$	60.0000	0.160010
$53$	−44.0000	−0.114035	−0.0570176	−	0.998373i	$-0.518159\pi$
$53$	−44.0000	−0.114035	−0.0570176	+	0.998373i	$0.518159\pi$
$54$	−54.0000	−0.136083
$55$	−160.000	−0.392262
$56$	0	0
$57$	45.0000	0.104568
$58$	−180.000	−0.407503
$59$	−782.000	−1.72555	−0.862777	−	0.505584i	$-0.831277\pi$
$59$	−782.000	−1.72555	−0.862777	+	0.505584i	$0.831277\pi$
$60$	60.0000	0.129099
$61$	658.000	1.38112	0.690560	−	0.723276i	$-0.257364\pi$
$61$	658.000	1.38112	0.690560	+	0.723276i	$0.257364\pi$
$62$	170.000	0.348226
$63$	0	0
$64$	64.0000	0.125000
$65$	75.0000	0.143117
$66$	192.000	0.358084
$67$	1071.00	1.95289	0.976444	−	0.215772i	$-0.0692267\pi$
$67$	1071.00	1.95289	0.976444	+	0.215772i	$0.0692267\pi$
$68$	−280.000	−0.499338
$69$	−126.000	−0.219835
$70$	0	0
$71$	344.000	0.575004	0.287502	−	0.957780i	$-0.407175\pi$
$71$	344.000	0.575004	0.287502	+	0.957780i	$0.407175\pi$
$72$	−72.0000	−0.117851
$73$	431.000	0.691024	0.345512	−	0.938414i	$-0.387705\pi$
$73$	431.000	0.691024	0.345512	+	0.938414i	$0.387705\pi$
$74$	−226.000	−0.355027
$75$	75.0000	0.115470
$76$	60.0000	0.0905588
$77$	0	0
$78$	−90.0000	−0.130647
$79$	397.000	0.565392	0.282696	−	0.959210i	$-0.408771\pi$
$79$	397.000	0.565392	0.282696	+	0.959210i	$0.408771\pi$
$80$	80.0000	0.111803
$81$	81.0000	0.111111
$82$	−328.000	−0.441726
$83$	680.000	0.899273	0.449637	−	0.893212i	$-0.351553\pi$
$83$	680.000	0.899273	0.449637	+	0.893212i	$0.351553\pi$
$84$	0	0
$85$	−350.000	−0.446622
$86$	−338.000	−0.423808
$87$	270.000	0.332725
$88$	256.000	0.310110
$89$	1534.00	1.82701	0.913504	−	0.406830i	$-0.133366\pi$
$89$	1534.00	1.82701	0.913504	+	0.406830i	$0.133366\pi$
$90$	−90.0000	−0.105409
$91$	0	0
$92$	−168.000	−0.190383
$93$	−255.000	−0.284325
$94$	−652.000	−0.715411
$95$	75.0000	0.0809983
$96$	−96.0000	−0.102062
$97$	−1234.00	−1.29169	−0.645844	−	0.763469i	$-0.723494\pi$
$97$	−1234.00	−1.29169	−0.645844	+	0.763469i	$0.723494\pi$
$98$	0	0
$99$	−288.000	−0.292375
$100$	100.000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	420.000	0.407708
$103$	−1691.00	−1.61766	−0.808831	−	0.588041i	$-0.799899\pi$
$103$	−1691.00	−1.61766	−0.808831	+	0.588041i	$0.799899\pi$
$104$	−120.000	−0.113144
$105$	0	0
$106$	88.0000	0.0806351
$107$	162.000	0.146366	0.0731829	−	0.997319i	$-0.476684\pi$
$107$	162.000	0.146366	0.0731829	+	0.997319i	$0.476684\pi$
$108$	108.000	0.0962250
$109$	1221.00	1.07294	0.536471	−	0.843919i	$-0.319757\pi$
$109$	1221.00	1.07294	0.536471	+	0.843919i	$0.319757\pi$
$110$	320.000	0.277371
$111$	339.000	0.289878
$112$	0	0
$113$	−274.000	−0.228104	−0.114052	−	0.993475i	$-0.536383\pi$
$113$	−274.000	−0.228104	−0.114052	+	0.993475i	$0.536383\pi$
$114$	−90.0000	−0.0739410
$115$	−210.000	−0.170283
$116$	360.000	0.288148
$117$	135.000	0.106673
$118$	1564.00	1.22015
$119$	0	0
$120$	−120.000	−0.0912871
$121$	−307.000	−0.230654
$122$	−1316.00	−0.976599
$123$	492.000	0.360668
$124$	−340.000	−0.246233
$125$	125.000	0.0894427
$126$	0	0
$127$	529.000	0.369615	0.184808	−	0.982775i	$-0.440834\pi$
$127$	529.000	0.369615	0.184808	+	0.982775i	$0.440834\pi$
$128$	−128.000	−0.0883883
$129$	507.000	0.346038
$130$	−150.000	−0.101199
$131$	326.000	0.217426	0.108713	−	0.994073i	$-0.465327\pi$
$131$	326.000	0.217426	0.108713	+	0.994073i	$0.465327\pi$
$132$	−384.000	−0.253204
$133$	0	0
$134$	−2142.00	−1.38090
$135$	135.000	0.0860663
$136$	560.000	0.353085
$137$	894.000	0.557515	0.278758	−	0.960362i	$-0.410077\pi$
$137$	894.000	0.557515	0.278758	+	0.960362i	$0.410077\pi$
$138$	252.000	0.155447
$139$	895.000	0.546136	0.273068	−	0.961995i	$-0.411962\pi$
$139$	895.000	0.546136	0.273068	+	0.961995i	$0.411962\pi$
$140$	0	0
$141$	978.000	0.584131
$142$	−688.000	−0.406589
$143$	−480.000	−0.280697
$144$	144.000	0.0833333
$145$	450.000	0.257727
$146$	−862.000	−0.488628
$147$	0	0
$148$	452.000	0.251042
$149$	−840.000	−0.461849	−0.230924	−	0.972972i	$-0.574175\pi$
$149$	−840.000	−0.461849	−0.230924	+	0.972972i	$0.574175\pi$
$150$	−150.000	−0.0816497
$151$	920.000	0.495818	0.247909	−	0.968783i	$-0.420257\pi$
$151$	920.000	0.495818	0.247909	+	0.968783i	$0.420257\pi$
$152$	−120.000	−0.0640348
$153$	−630.000	−0.332892
$154$	0	0
$155$	−425.000	−0.220238
$156$	180.000	0.0923816
$157$	910.000	0.462585	0.231293	−	0.972884i	$-0.425704\pi$
$157$	910.000	0.462585	0.231293	+	0.972884i	$0.425704\pi$
$158$	−794.000	−0.399793
$159$	−132.000	−0.0658382
$160$	−160.000	−0.0790569
$161$	0	0
$162$	−162.000	−0.0785674
$163$	−2068.00	−0.993732	−0.496866	−	0.867827i	$-0.665516\pi$
$163$	−2068.00	−0.993732	−0.496866	+	0.867827i	$0.665516\pi$
$164$	656.000	0.312348
$165$	−480.000	−0.226472
$166$	−1360.00	−0.635882
$167$	3504.00	1.62364	0.811819	−	0.583909i	$-0.198477\pi$
$167$	3504.00	1.62364	0.811819	+	0.583909i	$0.198477\pi$
$168$	0	0
$169$	−1972.00	−0.897588
$170$	700.000	0.315809
$171$	135.000	0.0603726
$172$	676.000	0.299677
$173$	−1948.00	−0.856091	−0.428045	−	0.903757i	$-0.640798\pi$
$173$	−1948.00	−0.856091	−0.428045	+	0.903757i	$0.640798\pi$
$174$	−540.000	−0.235272
$175$	0	0
$176$	−512.000	−0.219281
$177$	−2346.00	−0.996249
$178$	−3068.00	−1.29189
$179$	−3030.00	−1.26521	−0.632606	−	0.774474i	$-0.718015\pi$
$179$	−3030.00	−1.26521	−0.632606	+	0.774474i	$0.718015\pi$
$180$	180.000	0.0745356
$181$	3133.00	1.28660	0.643298	−	0.765615i	$-0.277565\pi$
$181$	3133.00	1.28660	0.643298	+	0.765615i	$0.277565\pi$
$182$	0	0
$183$	1974.00	0.797390
$184$	336.000	0.134621
$185$	565.000	0.224539
$186$	510.000	0.201048
$187$	2240.00	0.875963
$188$	1304.00	0.505872
$189$	0	0
$190$	−150.000	−0.0572744
$191$	−4422.00	−1.67521	−0.837604	−	0.546278i	$-0.816044\pi$
$191$	−4422.00	−1.67521	−0.837604	+	0.546278i	$0.816044\pi$
$192$	192.000	0.0721688
$193$	2825.00	1.05362	0.526808	−	0.849984i	$-0.323389\pi$
$193$	2825.00	1.05362	0.526808	+	0.849984i	$0.323389\pi$
$194$	2468.00	0.913361
$195$	225.000	0.0826286
$196$	0	0
$197$	3170.00	1.14646	0.573231	−	0.819394i	$-0.305690\pi$
$197$	3170.00	1.14646	0.573231	+	0.819394i	$0.305690\pi$
$198$	576.000	0.206740
$199$	−3064.00	−1.09146	−0.545732	−	0.837960i	$-0.683748\pi$
$199$	−3064.00	−1.09146	−0.545732	+	0.837960i	$0.683748\pi$
$200$	−200.000	−0.0707107
$201$	3213.00	1.12750
$202$	0	0
$203$	0	0
$204$	−840.000	−0.288293
$205$	820.000	0.279372
$206$	3382.00	1.14386
$207$	−378.000	−0.126922
$208$	240.000	0.0800048
$209$	−480.000	−0.158863
$210$	0	0
$211$	460.000	0.150084	0.0750420	−	0.997180i	$-0.476091\pi$
$211$	460.000	0.150084	0.0750420	+	0.997180i	$0.476091\pi$
$212$	−176.000	−0.0570176
$213$	1032.00	0.331979
$214$	−324.000	−0.103496
$215$	845.000	0.268040
$216$	−216.000	−0.0680414
$217$	0	0
$218$	−2442.00	−0.758684
$219$	1293.00	0.398963
$220$	−640.000	−0.196131
$221$	−1050.00	−0.319596
$222$	−678.000	−0.204975
$223$	4156.00	1.24801	0.624005	−	0.781420i	$-0.285504\pi$
$223$	4156.00	1.24801	0.624005	+	0.781420i	$0.285504\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	548.000	0.161294
$227$	−2740.00	−0.801146	−0.400573	−	0.916265i	$-0.631189\pi$
$227$	−2740.00	−0.801146	−0.400573	+	0.916265i	$0.631189\pi$
$228$	180.000	0.0522842
$229$	4921.00	1.42004	0.710019	−	0.704182i	$-0.248686\pi$
$229$	4921.00	1.42004	0.710019	+	0.704182i	$0.248686\pi$
$230$	420.000	0.120409
$231$	0	0
$232$	−720.000	−0.203751
$233$	−4038.00	−1.13536	−0.567678	−	0.823250i	$-0.692158\pi$
$233$	−4038.00	−1.13536	−0.567678	+	0.823250i	$0.692158\pi$
$234$	−270.000	−0.0754293
$235$	1630.00	0.452466
$236$	−3128.00	−0.862777
$237$	1191.00	0.326429
$238$	0	0
$239$	−2930.00	−0.792996	−0.396498	−	0.918036i	$-0.629775\pi$
$239$	−2930.00	−0.792996	−0.396498	+	0.918036i	$0.629775\pi$
$240$	240.000	0.0645497
$241$	1834.00	0.490201	0.245100	−	0.969498i	$-0.421179\pi$
$241$	1834.00	0.490201	0.245100	+	0.969498i	$0.421179\pi$
$242$	614.000	0.163097
$243$	243.000	0.0641500
$244$	2632.00	0.690560
$245$	0	0
$246$	−984.000	−0.255031
$247$	225.000	0.0579612
$248$	680.000	0.174113
$249$	2040.00	0.519196
$250$	−250.000	−0.0632456
$251$	1014.00	0.254992	0.127496	−	0.991839i	$-0.459306\pi$
$251$	1014.00	0.254992	0.127496	+	0.991839i	$0.459306\pi$
$252$	0	0
$253$	1344.00	0.333978
$254$	−1058.00	−0.261358
$255$	−1050.00	−0.257857
$256$	256.000	0.0625000
$257$	2816.00	0.683491	0.341746	−	0.939792i	$-0.388982\pi$
$257$	2816.00	0.683491	0.341746	+	0.939792i	$0.388982\pi$
$258$	−1014.00	−0.244686
$259$	0	0
$260$	300.000	0.0715585
$261$	810.000	0.192099
$262$	−652.000	−0.153743
$263$	6788.00	1.59151	0.795753	−	0.605621i	$-0.207075\pi$
$263$	6788.00	1.59151	0.795753	+	0.605621i	$0.207075\pi$
$264$	768.000	0.179042
$265$	−220.000	−0.0509981
$266$	0	0
$267$	4602.00	1.05482
$268$	4284.00	0.976444
$269$	6198.00	1.40483	0.702414	−	0.711769i	$-0.252106\pi$
$269$	6198.00	1.40483	0.702414	+	0.711769i	$0.252106\pi$
$270$	−270.000	−0.0608581
$271$	3984.00	0.893029	0.446514	−	0.894776i	$-0.352665\pi$
$271$	3984.00	0.893029	0.446514	+	0.894776i	$0.352665\pi$
$272$	−1120.00	−0.249669
$273$	0	0
$274$	−1788.00	−0.394223
$275$	−800.000	−0.175425
$276$	−504.000	−0.109918
$277$	5127.00	1.11210	0.556050	−	0.831149i	$-0.312316\pi$
$277$	5127.00	1.11210	0.556050	+	0.831149i	$0.312316\pi$
$278$	−1790.00	−0.386176
$279$	−765.000	−0.164155
$280$	0	0
$281$	−6196.00	−1.31538	−0.657691	−	0.753288i	$-0.728467\pi$
$281$	−6196.00	−1.31538	−0.657691	+	0.753288i	$0.728467\pi$
$282$	−1956.00	−0.413043
$283$	5789.00	1.21597	0.607986	−	0.793947i	$-0.291977\pi$
$283$	5789.00	1.21597	0.607986	+	0.793947i	$0.291977\pi$
$284$	1376.00	0.287502
$285$	225.000	0.0467644
$286$	960.000	0.198482
$287$	0	0
$288$	−288.000	−0.0589256
$289$	−13.0000	−0.00264604
$290$	−900.000	−0.182241
$291$	−3702.00	−0.745756
$292$	1724.00	0.345512
$293$	−2108.00	−0.420309	−0.210155	−	0.977668i	$-0.567397\pi$
$293$	−2108.00	−0.420309	−0.210155	+	0.977668i	$0.567397\pi$
$294$	0	0
$295$	−3910.00	−0.771692
$296$	−904.000	−0.177513
$297$	−864.000	−0.168803
$298$	1680.00	0.326576
$299$	−630.000	−0.121852
$300$	300.000	0.0577350
$301$	0	0
$302$	−1840.00	−0.350596
$303$	0	0
$304$	240.000	0.0452794
$305$	3290.00	0.617655
$306$	1260.00	0.235390
$307$	−1121.00	−0.208400	−0.104200	−	0.994556i	$-0.533228\pi$
$307$	−1121.00	−0.208400	−0.104200	+	0.994556i	$0.533228\pi$
$308$	0	0
$309$	−5073.00	−0.933958
$310$	850.000	0.155731
$311$	8028.00	1.46375	0.731875	−	0.681439i	$-0.238646\pi$
$311$	8028.00	1.46375	0.731875	+	0.681439i	$0.238646\pi$
$312$	−360.000	−0.0653237
$313$	5093.00	0.919723	0.459862	−	0.887991i	$-0.347899\pi$
$313$	5093.00	0.919723	0.459862	+	0.887991i	$0.347899\pi$
$314$	−1820.00	−0.327097
$315$	0	0
$316$	1588.00	0.282696
$317$	−3746.00	−0.663711	−0.331855	−	0.943330i	$-0.607675\pi$
$317$	−3746.00	−0.663711	−0.331855	+	0.943330i	$0.607675\pi$
$318$	264.000	0.0465547
$319$	−2880.00	−0.505483
$320$	320.000	0.0559017
$321$	486.000	0.0845043
$322$	0	0
$323$	−1050.00	−0.180878
$324$	324.000	0.0555556
$325$	375.000	0.0640039
$326$	4136.00	0.702674
$327$	3663.00	0.619463
$328$	−1312.00	−0.220863
$329$	0	0
$330$	960.000	0.160140
$331$	7261.00	1.20574	0.602871	−	0.797839i	$-0.294023\pi$
$331$	7261.00	1.20574	0.602871	+	0.797839i	$0.294023\pi$
$332$	2720.00	0.449637
$333$	1017.00	0.167361
$334$	−7008.00	−1.14809
$335$	5355.00	0.873358
$336$	0	0
$337$	−2945.00	−0.476037	−0.238018	−	0.971261i	$-0.576498\pi$
$337$	−2945.00	−0.476037	−0.238018	+	0.971261i	$0.576498\pi$
$338$	3944.00	0.634690
$339$	−822.000	−0.131696
$340$	−1400.00	−0.223311
$341$	2720.00	0.431954
$342$	−270.000	−0.0426898
$343$	0	0
$344$	−1352.00	−0.211904
$345$	−630.000	−0.0983132
$346$	3896.00	0.605348
$347$	−11060.0	−1.71104	−0.855521	−	0.517767i	$-0.826763\pi$
$347$	−11060.0	−1.71104	−0.855521	+	0.517767i	$0.826763\pi$
$348$	1080.00	0.166362
$349$	6298.00	0.965972	0.482986	−	0.875628i	$-0.339552\pi$
$349$	6298.00	0.965972	0.482986	+	0.875628i	$0.339552\pi$
$350$	0	0
$351$	405.000	0.0615878
$352$	1024.00	0.155055
$353$	10092.0	1.52165	0.760826	−	0.648956i	$-0.224794\pi$
$353$	10092.0	1.52165	0.760826	+	0.648956i	$0.224794\pi$
$354$	4692.00	0.704455
$355$	1720.00	0.257150
$356$	6136.00	0.913504
$357$	0	0
$358$	6060.00	0.894640
$359$	−11122.0	−1.63509	−0.817544	−	0.575866i	$-0.804665\pi$
$359$	−11122.0	−1.63509	−0.817544	+	0.575866i	$0.804665\pi$
$360$	−360.000	−0.0527046
$361$	−6634.00	−0.967196
$362$	−6266.00	−0.909761
$363$	−921.000	−0.133168
$364$	0	0
$365$	2155.00	0.309035
$366$	−3948.00	−0.563840
$367$	5287.00	0.751987	0.375993	−	0.926622i	$-0.377302\pi$
$367$	5287.00	0.751987	0.375993	+	0.926622i	$0.377302\pi$
$368$	−672.000	−0.0951914
$369$	1476.00	0.208232
$370$	−1130.00	−0.158773
$371$	0	0
$372$	−1020.00	−0.142163
$373$	13025.0	1.80807	0.904033	−	0.427462i	$-0.140592\pi$
$373$	13025.0	1.80807	0.904033	+	0.427462i	$0.140592\pi$
$374$	−4480.00	−0.619399
$375$	375.000	0.0516398
$376$	−2608.00	−0.357706
$377$	1350.00	0.184426
$378$	0	0
$379$	13869.0	1.87969	0.939845	−	0.341601i	$-0.110969\pi$
$379$	13869.0	1.87969	0.939845	+	0.341601i	$0.110969\pi$
$380$	300.000	0.0404991
$381$	1587.00	0.213398
$382$	8844.00	1.18455
$383$	−1764.00	−0.235343	−0.117671	−	0.993053i	$-0.537543\pi$
$383$	−1764.00	−0.235343	−0.117671	+	0.993053i	$0.537543\pi$
$384$	−384.000	−0.0510310
$385$	0	0
$386$	−5650.00	−0.745019
$387$	1521.00	0.199785
$388$	−4936.00	−0.645844
$389$	−6064.00	−0.790378	−0.395189	−	0.918600i	$-0.629321\pi$
$389$	−6064.00	−0.790378	−0.395189	+	0.918600i	$0.629321\pi$
$390$	−450.000	−0.0584273
$391$	2940.00	0.380261
$392$	0	0
$393$	978.000	0.125531
$394$	−6340.00	−0.810672
$395$	1985.00	0.252851
$396$	−1152.00	−0.146187
$397$	9373.00	1.18493	0.592465	−	0.805596i	$-0.298155\pi$
$397$	9373.00	1.18493	0.592465	+	0.805596i	$0.298155\pi$
$398$	6128.00	0.771781
$399$	0	0
$400$	400.000	0.0500000
$401$	9580.00	1.19302	0.596512	−	0.802604i	$-0.296553\pi$
$401$	9580.00	1.19302	0.596512	+	0.802604i	$0.296553\pi$
$402$	−6426.00	−0.797263
$403$	−1275.00	−0.157599
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−3616.00	−0.440389
$408$	1680.00	0.203854
$409$	−10107.0	−1.22190	−0.610952	−	0.791667i	$-0.709213\pi$
$409$	−10107.0	−1.22190	−0.610952	+	0.791667i	$0.709213\pi$
$410$	−1640.00	−0.197546
$411$	2682.00	0.321882
$412$	−6764.00	−0.808831
$413$	0	0
$414$	756.000	0.0897473
$415$	3400.00	0.402167
$416$	−480.000	−0.0565720
$417$	2685.00	0.315312
$418$	960.000	0.112333
$419$	8934.00	1.04166	0.520829	−	0.853661i	$-0.325623\pi$
$419$	8934.00	1.04166	0.520829	+	0.853661i	$0.325623\pi$
$420$	0	0
$421$	3389.00	0.392327	0.196164	−	0.980571i	$-0.437152\pi$
$421$	3389.00	0.392327	0.196164	+	0.980571i	$0.437152\pi$
$422$	−920.000	−0.106125
$423$	2934.00	0.337248
$424$	352.000	0.0403175
$425$	−1750.00	−0.199735
$426$	−2064.00	−0.234744
$427$	0	0
$428$	648.000	0.0731829
$429$	−1440.00	−0.162060
$430$	−1690.00	−0.189533
$431$	−17514.0	−1.95735	−0.978677	−	0.205405i	$-0.934149\pi$
$431$	−17514.0	−1.95735	−0.978677	+	0.205405i	$0.934149\pi$
$432$	432.000	0.0481125
$433$	13457.0	1.49354	0.746769	−	0.665083i	$-0.231604\pi$
$433$	13457.0	1.49354	0.746769	+	0.665083i	$0.231604\pi$
$434$	0	0
$435$	1350.00	0.148799
$436$	4884.00	0.536471
$437$	−630.000	−0.0689634
$438$	−2586.00	−0.282109
$439$	−9428.00	−1.02500	−0.512499	−	0.858688i	$-0.671280\pi$
$439$	−9428.00	−1.02500	−0.512499	+	0.858688i	$0.671280\pi$
$440$	1280.00	0.138685
$441$	0	0
$442$	2100.00	0.225988
$443$	16888.0	1.81123	0.905613	−	0.424105i	$-0.139411\pi$
$443$	16888.0	1.81123	0.905613	+	0.424105i	$0.139411\pi$
$444$	1356.00	0.144939
$445$	7670.00	0.817063
$446$	−8312.00	−0.882477
$447$	−2520.00	−0.266649
$448$	0	0
$449$	−822.000	−0.0863977	−0.0431989	−	0.999066i	$-0.513755\pi$
$449$	−822.000	−0.0863977	−0.0431989	+	0.999066i	$0.513755\pi$
$450$	−450.000	−0.0471405
$451$	−5248.00	−0.547935
$452$	−1096.00	−0.114052
$453$	2760.00	0.286261
$454$	5480.00	0.566496
$455$	0	0
$456$	−360.000	−0.0369705
$457$	−10865.0	−1.11213	−0.556065	−	0.831139i	$-0.687689\pi$
$457$	−10865.0	−1.11213	−0.556065	+	0.831139i	$0.687689\pi$
$458$	−9842.00	−1.00412
$459$	−1890.00	−0.192195
$460$	−840.000	−0.0851417
$461$	−9518.00	−0.961600	−0.480800	−	0.876830i	$-0.659654\pi$
$461$	−9518.00	−0.961600	−0.480800	+	0.876830i	$0.659654\pi$
$462$	0	0
$463$	9227.00	0.926166	0.463083	−	0.886315i	$-0.346743\pi$
$463$	9227.00	0.926166	0.463083	+	0.886315i	$0.346743\pi$
$464$	1440.00	0.144074
$465$	−1275.00	−0.127154
$466$	8076.00	0.802819
$467$	−5530.00	−0.547961	−0.273981	−	0.961735i	$-0.588340\pi$
$467$	−5530.00	−0.547961	−0.273981	+	0.961735i	$0.588340\pi$
$468$	540.000	0.0533366
$469$	0	0
$470$	−3260.00	−0.319942
$471$	2730.00	0.267074
$472$	6256.00	0.610076
$473$	−5408.00	−0.525708
$474$	−2382.00	−0.230820
$475$	375.000	0.0362235
$476$	0	0
$477$	−396.000	−0.0380117
$478$	5860.00	0.560733
$479$	−4812.00	−0.459010	−0.229505	−	0.973307i	$-0.573711\pi$
$479$	−4812.00	−0.459010	−0.229505	+	0.973307i	$0.573711\pi$
$480$	−480.000	−0.0456435
$481$	1695.00	0.160676
$482$	−3668.00	−0.346624
$483$	0	0
$484$	−1228.00	−0.115327
$485$	−6170.00	−0.577660
$486$	−486.000	−0.0453609
$487$	8315.00	0.773693	0.386847	−	0.922144i	$-0.373564\pi$
$487$	8315.00	0.773693	0.386847	+	0.922144i	$0.373564\pi$
$488$	−5264.00	−0.488299
$489$	−6204.00	−0.573731
$490$	0	0
$491$	10212.0	0.938617	0.469309	−	0.883034i	$-0.344503\pi$
$491$	10212.0	0.938617	0.469309	+	0.883034i	$0.344503\pi$
$492$	1968.00	0.180334
$493$	−6300.00	−0.575533
$494$	−450.000	−0.0409847
$495$	−1440.00	−0.130754
$496$	−1360.00	−0.123117
$497$	0	0
$498$	−4080.00	−0.367127
$499$	−11229.0	−1.00737	−0.503686	−	0.863887i	$-0.668023\pi$
$499$	−11229.0	−1.00737	−0.503686	+	0.863887i	$0.668023\pi$
$500$	500.000	0.0447214
$501$	10512.0	0.937408
$502$	−2028.00	−0.180307
$503$	−3786.00	−0.335605	−0.167803	−	0.985821i	$-0.553667\pi$
$503$	−3786.00	−0.335605	−0.167803	+	0.985821i	$0.553667\pi$
$504$	0	0
$505$	0	0
$506$	−2688.00	−0.236158
$507$	−5916.00	−0.518222
$508$	2116.00	0.184808
$509$	−12206.0	−1.06291	−0.531455	−	0.847086i	$-0.678355\pi$
$509$	−12206.0	−1.06291	−0.531455	+	0.847086i	$0.678355\pi$
$510$	2100.00	0.182332
$511$	0	0
$512$	−512.000	−0.0441942
$513$	405.000	0.0348561
$514$	−5632.00	−0.483301
$515$	−8455.00	−0.723440
$516$	2028.00	0.173019
$517$	−10432.0	−0.887425
$518$	0	0
$519$	−5844.00	−0.494264
$520$	−600.000	−0.0505995
$521$	13952.0	1.17322	0.586611	−	0.809869i	$-0.300462\pi$
$521$	13952.0	1.17322	0.586611	+	0.809869i	$0.300462\pi$
$522$	−1620.00	−0.135834
$523$	−10181.0	−0.851212	−0.425606	−	0.904909i	$-0.639939\pi$
$523$	−10181.0	−0.851212	−0.425606	+	0.904909i	$0.639939\pi$
$524$	1304.00	0.108713
$525$	0	0
$526$	−13576.0	−1.12536
$527$	5950.00	0.491814
$528$	−1536.00	−0.126602
$529$	−10403.0	−0.855018
$530$	440.000	0.0360611
$531$	−7038.00	−0.575185
$532$	0	0
$533$	2460.00	0.199914
$534$	−9204.00	−0.745873
$535$	810.000	0.0654567
$536$	−8568.00	−0.690450
$537$	−9090.00	−0.730470
$538$	−12396.0	−0.993363
$539$	0	0
$540$	540.000	0.0430331
$541$	−14055.0	−1.11695	−0.558477	−	0.829520i	$-0.688614\pi$
$541$	−14055.0	−1.11695	−0.558477	+	0.829520i	$0.688614\pi$
$542$	−7968.00	−0.631467
$543$	9399.00	0.742817
$544$	2240.00	0.176543
$545$	6105.00	0.479834
$546$	0	0
$547$	−6144.00	−0.480253	−0.240127	−	0.970742i	$-0.577189\pi$
$547$	−6144.00	−0.480253	−0.240127	+	0.970742i	$0.577189\pi$
$548$	3576.00	0.278758
$549$	5922.00	0.460373
$550$	1600.00	0.124044
$551$	1350.00	0.104377
$552$	1008.00	0.0777234
$553$	0	0
$554$	−10254.0	−0.786373
$555$	1695.00	0.129637
$556$	3580.00	0.273068
$557$	−14342.0	−1.09101	−0.545503	−	0.838109i	$-0.683661\pi$
$557$	−14342.0	−1.09101	−0.545503	+	0.838109i	$0.683661\pi$
$558$	1530.00	0.116075
$559$	2535.00	0.191805
$560$	0	0
$561$	6720.00	0.505737
$562$	12392.0	0.930116
$563$	−9658.00	−0.722977	−0.361489	−	0.932377i	$-0.617731\pi$
$563$	−9658.00	−0.722977	−0.361489	+	0.932377i	$0.617731\pi$
$564$	3912.00	0.292065
$565$	−1370.00	−0.102011
$566$	−11578.0	−0.859823
$567$	0	0
$568$	−2752.00	−0.203295
$569$	−14592.0	−1.07509	−0.537547	−	0.843234i	$-0.680649\pi$
$569$	−14592.0	−1.07509	−0.537547	+	0.843234i	$0.680649\pi$
$570$	−450.000	−0.0330674
$571$	15699.0	1.15058	0.575291	−	0.817949i	$-0.304889\pi$
$571$	15699.0	1.15058	0.575291	+	0.817949i	$0.304889\pi$
$572$	−1920.00	−0.140348
$573$	−13266.0	−0.967182
$574$	0	0
$575$	−1050.00	−0.0761531
$576$	576.000	0.0416667
$577$	−9779.00	−0.705555	−0.352777	−	0.935707i	$-0.614763\pi$
$577$	−9779.00	−0.705555	−0.352777	+	0.935707i	$0.614763\pi$
$578$	26.0000	0.00187103
$579$	8475.00	0.608306
$580$	1800.00	0.128864
$581$	0	0
$582$	7404.00	0.527329
$583$	1408.00	0.100023
$584$	−3448.00	−0.244314
$585$	675.000	0.0477057
$586$	4216.00	0.297204
$587$	−9454.00	−0.664750	−0.332375	−	0.943147i	$-0.607850\pi$
$587$	−9454.00	−0.664750	−0.332375	+	0.943147i	$0.607850\pi$
$588$	0	0
$589$	−1275.00	−0.0891943
$590$	7820.00	0.545668
$591$	9510.00	0.661911
$592$	1808.00	0.125521
$593$	−7668.00	−0.531007	−0.265503	−	0.964110i	$-0.585538\pi$
$593$	−7668.00	−0.531007	−0.265503	+	0.964110i	$0.585538\pi$
$594$	1728.00	0.119361
$595$	0	0
$596$	−3360.00	−0.230924
$597$	−9192.00	−0.630157
$598$	1260.00	0.0861626
$599$	2052.00	0.139971	0.0699853	−	0.997548i	$-0.477705\pi$
$599$	2052.00	0.139971	0.0699853	+	0.997548i	$0.477705\pi$
$600$	−600.000	−0.0408248
$601$	−19857.0	−1.34773	−0.673863	−	0.738856i	$-0.735366\pi$
$601$	−19857.0	−1.34773	−0.673863	+	0.738856i	$0.735366\pi$
$602$	0	0
$603$	9639.00	0.650963
$604$	3680.00	0.247909
$605$	−1535.00	−0.103151
$606$	0	0
$607$	10615.0	0.709802	0.354901	−	0.934904i	$-0.384515\pi$
$607$	10615.0	0.709802	0.354901	+	0.934904i	$0.384515\pi$
$608$	−480.000	−0.0320174
$609$	0	0
$610$	−6580.00	−0.436748
$611$	4890.00	0.323778
$612$	−2520.00	−0.166446
$613$	−9642.00	−0.635296	−0.317648	−	0.948209i	$-0.602893\pi$
$613$	−9642.00	−0.635296	−0.317648	+	0.948209i	$0.602893\pi$
$614$	2242.00	0.147361
$615$	2460.00	0.161296
$616$	0	0
$617$	1694.00	0.110531	0.0552657	−	0.998472i	$-0.482399\pi$
$617$	1694.00	0.110531	0.0552657	+	0.998472i	$0.482399\pi$
$618$	10146.0	0.660408
$619$	−4399.00	−0.285639	−0.142820	−	0.989749i	$-0.545617\pi$
$619$	−4399.00	−0.285639	−0.142820	+	0.989749i	$0.545617\pi$
$620$	−1700.00	−0.110119
$621$	−1134.00	−0.0732783
$622$	−16056.0	−1.03503
$623$	0	0
$624$	720.000	0.0461908
$625$	625.000	0.0400000
$626$	−10186.0	−0.650343
$627$	−1440.00	−0.0917194
$628$	3640.00	0.231293
$629$	−7910.00	−0.501419
$630$	0	0
$631$	5336.00	0.336645	0.168322	−	0.985732i	$-0.446165\pi$
$631$	5336.00	0.336645	0.168322	+	0.985732i	$0.446165\pi$
$632$	−3176.00	−0.199896
$633$	1380.00	0.0866510
$634$	7492.00	0.469314
$635$	2645.00	0.165297
$636$	−528.000	−0.0329191
$637$	0	0
$638$	5760.00	0.357430
$639$	3096.00	0.191668
$640$	−640.000	−0.0395285
$641$	−18562.0	−1.14377	−0.571884	−	0.820335i	$-0.693787\pi$
$641$	−18562.0	−1.14377	−0.571884	+	0.820335i	$0.693787\pi$
$642$	−972.000	−0.0597536
$643$	−5687.00	−0.348792	−0.174396	−	0.984676i	$-0.555797\pi$
$643$	−5687.00	−0.348792	−0.174396	+	0.984676i	$0.555797\pi$
$644$	0	0
$645$	2535.00	0.154753
$646$	2100.00	0.127900
$647$	24712.0	1.50159	0.750795	−	0.660535i	$-0.229671\pi$
$647$	24712.0	1.50159	0.750795	+	0.660535i	$0.229671\pi$
$648$	−648.000	−0.0392837
$649$	25024.0	1.51353
$650$	−750.000	−0.0452576
$651$	0	0
$652$	−8272.00	−0.496866
$653$	20552.0	1.23164	0.615821	−	0.787886i	$-0.288825\pi$
$653$	20552.0	1.23164	0.615821	+	0.787886i	$0.288825\pi$
$654$	−7326.00	−0.438026
$655$	1630.00	0.0972357
$656$	2624.00	0.156174
$657$	3879.00	0.230341
$658$	0	0
$659$	−4472.00	−0.264347	−0.132173	−	0.991227i	$-0.542196\pi$
$659$	−4472.00	−0.264347	−0.132173	+	0.991227i	$0.542196\pi$
$660$	−1920.00	−0.113236
$661$	−10505.0	−0.618150	−0.309075	−	0.951038i	$-0.600019\pi$
$661$	−10505.0	−0.618150	−0.309075	+	0.951038i	$0.600019\pi$
$662$	−14522.0	−0.852588
$663$	−3150.00	−0.184519
$664$	−5440.00	−0.317941
$665$	0	0
$666$	−2034.00	−0.118342
$667$	−3780.00	−0.219434
$668$	14016.0	0.811819
$669$	12468.0	0.720539
$670$	−10710.0	−0.617557
$671$	−21056.0	−1.21141
$672$	0	0
$673$	19285.0	1.10458	0.552290	−	0.833652i	$-0.313754\pi$
$673$	19285.0	1.10458	0.552290	+	0.833652i	$0.313754\pi$
$674$	5890.00	0.336609
$675$	675.000	0.0384900
$676$	−7888.00	−0.448794
$677$	−18114.0	−1.02833	−0.514164	−	0.857692i	$-0.671898\pi$
$677$	−18114.0	−1.02833	−0.514164	+	0.857692i	$0.671898\pi$
$678$	1644.00	0.0931231
$679$	0	0
$680$	2800.00	0.157905
$681$	−8220.00	−0.462542
$682$	−5440.00	−0.305437
$683$	−13606.0	−0.762253	−0.381127	−	0.924523i	$-0.624464\pi$
$683$	−13606.0	−0.762253	−0.381127	+	0.924523i	$0.624464\pi$
$684$	540.000	0.0301863
$685$	4470.00	0.249328
$686$	0	0
$687$	14763.0	0.819860
$688$	2704.00	0.149839
$689$	−660.000	−0.0364935
$690$	1260.00	0.0695179
$691$	−6863.00	−0.377830	−0.188915	−	0.981993i	$-0.560497\pi$
$691$	−6863.00	−0.377830	−0.188915	+	0.981993i	$0.560497\pi$
$692$	−7792.00	−0.428045
$693$	0	0
$694$	22120.0	1.20989
$695$	4475.00	0.244239
$696$	−2160.00	−0.117636
$697$	−11480.0	−0.623868
$698$	−12596.0	−0.683045
$699$	−12114.0	−0.655499
$700$	0	0
$701$	14458.0	0.778989	0.389494	−	0.921029i	$-0.372650\pi$
$701$	14458.0	0.778989	0.389494	+	0.921029i	$0.372650\pi$
$702$	−810.000	−0.0435491
$703$	1695.00	0.0909362
$704$	−2048.00	−0.109640
$705$	4890.00	0.261231
$706$	−20184.0	−1.07597
$707$	0	0
$708$	−9384.00	−0.498125
$709$	8278.00	0.438486	0.219243	−	0.975670i	$-0.429641\pi$
$709$	8278.00	0.438486	0.219243	+	0.975670i	$0.429641\pi$
$710$	−3440.00	−0.181832
$711$	3573.00	0.188464
$712$	−12272.0	−0.645945
$713$	3570.00	0.187514
$714$	0	0
$715$	−2400.00	−0.125531
$716$	−12120.0	−0.632606
$717$	−8790.00	−0.457836
$718$	22244.0	1.15618
$719$	−29718.0	−1.54144	−0.770719	−	0.637175i	$-0.780103\pi$
$719$	−29718.0	−1.54144	−0.770719	+	0.637175i	$0.780103\pi$
$720$	720.000	0.0372678
$721$	0	0
$722$	13268.0	0.683911
$723$	5502.00	0.283017
$724$	12532.0	0.643298
$725$	2250.00	0.115259
$726$	1842.00	0.0941640
$727$	−24069.0	−1.22788	−0.613941	−	0.789352i	$-0.710417\pi$
$727$	−24069.0	−1.22788	−0.613941	+	0.789352i	$0.710417\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−4310.00	−0.218521
$731$	−11830.0	−0.598561
$732$	7896.00	0.398695
$733$	−1973.00	−0.0994194	−0.0497097	−	0.998764i	$-0.515830\pi$
$733$	−1973.00	−0.0994194	−0.0497097	+	0.998764i	$0.515830\pi$
$734$	−10574.0	−0.531735
$735$	0	0
$736$	1344.00	0.0673105
$737$	−34272.0	−1.71292
$738$	−2952.00	−0.147242
$739$	−37085.0	−1.84600	−0.923000	−	0.384800i	$-0.874270\pi$
$739$	−37085.0	−1.84600	−0.923000	+	0.384800i	$0.874270\pi$
$740$	2260.00	0.112269
$741$	675.000	0.0334639
$742$	0	0
$743$	4368.00	0.215675	0.107837	−	0.994169i	$-0.465607\pi$
$743$	4368.00	0.215675	0.107837	+	0.994169i	$0.465607\pi$
$744$	2040.00	0.100524
$745$	−4200.00	−0.206545
$746$	−26050.0	−1.27850
$747$	6120.00	0.299758
$748$	8960.00	0.437981
$749$	0	0
$750$	−750.000	−0.0365148
$751$	−25813.0	−1.25423	−0.627117	−	0.778925i	$-0.715765\pi$
$751$	−25813.0	−1.25423	−0.627117	+	0.778925i	$0.715765\pi$
$752$	5216.00	0.252936
$753$	3042.00	0.147220
$754$	−2700.00	−0.130409
$755$	4600.00	0.221737
$756$	0	0
$757$	18034.0	0.865861	0.432931	−	0.901427i	$-0.357480\pi$
$757$	18034.0	0.865861	0.432931	+	0.901427i	$0.357480\pi$
$758$	−27738.0	−1.32914
$759$	4032.00	0.192823
$760$	−600.000	−0.0286372
$761$	9882.00	0.470726	0.235363	−	0.971908i	$-0.424372\pi$
$761$	9882.00	0.470726	0.235363	+	0.971908i	$0.424372\pi$
$762$	−3174.00	−0.150895
$763$	0	0
$764$	−17688.0	−0.837604
$765$	−3150.00	−0.148874
$766$	3528.00	0.166412
$767$	−11730.0	−0.552211
$768$	768.000	0.0360844
$769$	−21845.0	−1.02438	−0.512192	−	0.858871i	$-0.671166\pi$
$769$	−21845.0	−1.02438	−0.512192	+	0.858871i	$0.671166\pi$
$770$	0	0
$771$	8448.00	0.394614
$772$	11300.0	0.526808
$773$	−2976.00	−0.138473	−0.0692363	−	0.997600i	$-0.522056\pi$
$773$	−2976.00	−0.138473	−0.0692363	+	0.997600i	$0.522056\pi$
$774$	−3042.00	−0.141269
$775$	−2125.00	−0.0984932
$776$	9872.00	0.456681
$777$	0	0
$778$	12128.0	0.558882
$779$	2460.00	0.113143
$780$	900.000	0.0413143
$781$	−11008.0	−0.504350
$782$	−5880.00	−0.268885
$783$	2430.00	0.110908
$784$	0	0
$785$	4550.00	0.206874
$786$	−1956.00	−0.0887636
$787$	4708.00	0.213243	0.106621	−	0.994300i	$-0.465997\pi$
$787$	4708.00	0.213243	0.106621	+	0.994300i	$0.465997\pi$
$788$	12680.0	0.573231
$789$	20364.0	0.918856
$790$	−3970.00	−0.178793
$791$	0	0
$792$	2304.00	0.103370
$793$	9870.00	0.441985
$794$	−18746.0	−0.837872
$795$	−660.000	−0.0294438
$796$	−12256.0	−0.545732
$797$	37950.0	1.68665	0.843324	−	0.537406i	$-0.180596\pi$
$797$	37950.0	1.68665	0.843324	+	0.537406i	$0.180596\pi$
$798$	0	0
$799$	−22820.0	−1.01040
$800$	−800.000	−0.0353553
$801$	13806.0	0.609003
$802$	−19160.0	−0.843595
$803$	−13792.0	−0.606113
$804$	12852.0	0.563750
$805$	0	0
$806$	2550.00	0.111439
$807$	18594.0	0.811078
$808$	0	0
$809$	1062.00	0.0461532	0.0230766	−	0.999734i	$-0.492654\pi$
$809$	1062.00	0.0461532	0.0230766	+	0.999734i	$0.492654\pi$
$810$	−810.000	−0.0351364
$811$	−28852.0	−1.24924	−0.624618	−	0.780930i	$-0.714745\pi$
$811$	−28852.0	−1.24924	−0.624618	+	0.780930i	$0.714745\pi$
$812$	0	0
$813$	11952.0	0.515590
$814$	7232.00	0.311402
$815$	−10340.0	−0.444410
$816$	−3360.00	−0.144146
$817$	2535.00	0.108554
$818$	20214.0	0.864017
$819$	0	0
$820$	3280.00	0.139686
$821$	−20580.0	−0.874844	−0.437422	−	0.899256i	$-0.644108\pi$
$821$	−20580.0	−0.874844	−0.437422	+	0.899256i	$0.644108\pi$
$822$	−5364.00	−0.227605
$823$	41176.0	1.74399	0.871996	−	0.489513i	$-0.162826\pi$
$823$	41176.0	1.74399	0.871996	+	0.489513i	$0.162826\pi$
$824$	13528.0	0.571930
$825$	−2400.00	−0.101282
$826$	0	0
$827$	−29158.0	−1.22603	−0.613013	−	0.790073i	$-0.710043\pi$
$827$	−29158.0	−1.22603	−0.613013	+	0.790073i	$0.710043\pi$
$828$	−1512.00	−0.0634609
$829$	−33415.0	−1.39994	−0.699970	−	0.714172i	$-0.746803\pi$
$829$	−33415.0	−1.39994	−0.699970	+	0.714172i	$0.746803\pi$
$830$	−6800.00	−0.284375
$831$	15381.0	0.642071
$832$	960.000	0.0400024
$833$	0	0
$834$	−5370.00	−0.222959
$835$	17520.0	0.726113
$836$	−1920.00	−0.0794313
$837$	−2295.00	−0.0947752
$838$	−17868.0	−0.736563
$839$	−11506.0	−0.473458	−0.236729	−	0.971576i	$-0.576075\pi$
$839$	−11506.0	−0.473458	−0.236729	+	0.971576i	$0.576075\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	−6778.00	−0.277417
$843$	−18588.0	−0.759436
$844$	1840.00	0.0750420
$845$	−9860.00	−0.401413
$846$	−5868.00	−0.238470
$847$	0	0
$848$	−704.000	−0.0285088
$849$	17367.0	0.702042
$850$	3500.00	0.141234
$851$	−4746.00	−0.191176
$852$	4128.00	0.165989
$853$	34889.0	1.40044	0.700221	−	0.713926i	$-0.253085\pi$
$853$	34889.0	1.40044	0.700221	+	0.713926i	$0.253085\pi$
$854$	0	0
$855$	675.000	0.0269994
$856$	−1296.00	−0.0517481
$857$	34818.0	1.38782	0.693909	−	0.720063i	$-0.255887\pi$
$857$	34818.0	1.38782	0.693909	+	0.720063i	$0.255887\pi$
$858$	2880.00	0.114594
$859$	12204.0	0.484744	0.242372	−	0.970183i	$-0.422075\pi$
$859$	12204.0	0.484744	0.242372	+	0.970183i	$0.422075\pi$
$860$	3380.00	0.134020
$861$	0	0
$862$	35028.0	1.38406
$863$	38422.0	1.51553	0.757764	−	0.652529i	$-0.226292\pi$
$863$	38422.0	1.51553	0.757764	+	0.652529i	$0.226292\pi$
$864$	−864.000	−0.0340207
$865$	−9740.00	−0.382855
$866$	−26914.0	−1.05609
$867$	−39.0000	−0.00152769
$868$	0	0
$869$	−12704.0	−0.495919
$870$	−2700.00	−0.105217
$871$	16065.0	0.624962
$872$	−9768.00	−0.379342
$873$	−11106.0	−0.430563
$874$	1260.00	0.0487645
$875$	0	0
$876$	5172.00	0.199481
$877$	−26230.0	−1.00995	−0.504974	−	0.863135i	$-0.668498\pi$
$877$	−26230.0	−1.00995	−0.504974	+	0.863135i	$0.668498\pi$
$878$	18856.0	0.724783
$879$	−6324.00	−0.242666
$880$	−2560.00	−0.0980654
$881$	−1768.00	−0.0676112	−0.0338056	−	0.999428i	$-0.510763\pi$
$881$	−1768.00	−0.0676112	−0.0338056	+	0.999428i	$0.510763\pi$
$882$	0	0
$883$	19887.0	0.757928	0.378964	−	0.925411i	$-0.376280\pi$
$883$	19887.0	0.757928	0.378964	+	0.925411i	$0.376280\pi$
$884$	−4200.00	−0.159798
$885$	−11730.0	−0.445536
$886$	−33776.0	−1.28073
$887$	−6436.00	−0.243630	−0.121815	−	0.992553i	$-0.538871\pi$
$887$	−6436.00	−0.243630	−0.121815	+	0.992553i	$0.538871\pi$
$888$	−2712.00	−0.102487
$889$	0	0
$890$	−15340.0	−0.577751
$891$	−2592.00	−0.0974582
$892$	16624.0	0.624005
$893$	4890.00	0.183245
$894$	5040.00	0.188549
$895$	−15150.0	−0.565820
$896$	0	0
$897$	−1890.00	−0.0703515
$898$	1644.00	0.0610924
$899$	−7650.00	−0.283806
$900$	900.000	0.0333333
$901$	3080.00	0.113884
$902$	10496.0	0.387449
$903$	0	0
$904$	2192.00	0.0806469
$905$	15665.0	0.575384
$906$	−5520.00	−0.202417
$907$	−32065.0	−1.17387	−0.586935	−	0.809634i	$-0.699666\pi$
$907$	−32065.0	−1.17387	−0.586935	+	0.809634i	$0.699666\pi$
$908$	−10960.0	−0.400573
$909$	0	0
$910$	0	0
$911$	−14394.0	−0.523485	−0.261742	−	0.965138i	$-0.584297\pi$
$911$	−14394.0	−0.523485	−0.261742	+	0.965138i	$0.584297\pi$
$912$	720.000	0.0261421
$913$	−21760.0	−0.788774
$914$	21730.0	0.786394
$915$	9870.00	0.356603
$916$	19684.0	0.710019
$917$	0	0
$918$	3780.00	0.135903
$919$	−15367.0	−0.551589	−0.275795	−	0.961217i	$-0.588941\pi$
$919$	−15367.0	−0.551589	−0.275795	+	0.961217i	$0.588941\pi$
$920$	1680.00	0.0602043
$921$	−3363.00	−0.120320
$922$	19036.0	0.679954
$923$	5160.00	0.184012
$924$	0	0
$925$	2825.00	0.100417
$926$	−18454.0	−0.654899
$927$	−15219.0	−0.539221
$928$	−2880.00	−0.101876
$929$	19936.0	0.704068	0.352034	−	0.935987i	$-0.385490\pi$
$929$	19936.0	0.704068	0.352034	+	0.935987i	$0.385490\pi$
$930$	2550.00	0.0899116
$931$	0	0
$932$	−16152.0	−0.567678
$933$	24084.0	0.845096
$934$	11060.0	0.387467
$935$	11200.0	0.391742
$936$	−1080.00	−0.0377146
$937$	45953.0	1.60215	0.801077	−	0.598561i	$-0.204260\pi$
$937$	45953.0	1.60215	0.801077	+	0.598561i	$0.204260\pi$
$938$	0	0
$939$	15279.0	0.531002
$940$	6520.00	0.226233
$941$	48782.0	1.68996	0.844978	−	0.534802i	$-0.179614\pi$
$941$	48782.0	1.68996	0.844978	+	0.534802i	$0.179614\pi$
$942$	−5460.00	−0.188850
$943$	−6888.00	−0.237862
$944$	−12512.0	−0.431389
$945$	0	0
$946$	10816.0	0.371732
$947$	51704.0	1.77419	0.887093	−	0.461591i	$-0.152721\pi$
$947$	51704.0	1.77419	0.887093	+	0.461591i	$0.152721\pi$
$948$	4764.00	0.163215
$949$	6465.00	0.221141
$950$	−750.000	−0.0256139
$951$	−11238.0	−0.383194
$952$	0	0
$953$	5916.00	0.201089	0.100545	−	0.994933i	$-0.467941\pi$
$953$	5916.00	0.201089	0.100545	+	0.994933i	$0.467941\pi$
$954$	792.000	0.0268784
$955$	−22110.0	−0.749176
$956$	−11720.0	−0.396498
$957$	−8640.00	−0.291841
$958$	9624.00	0.324569
$959$	0	0
$960$	960.000	0.0322749
$961$	−22566.0	−0.757477
$962$	−3390.00	−0.113615
$963$	1458.00	0.0487886
$964$	7336.00	0.245100
$965$	14125.0	0.471192
$966$	0	0
$967$	−44981.0	−1.49585	−0.747927	−	0.663781i	$-0.768951\pi$
$967$	−44981.0	−1.49585	−0.747927	+	0.663781i	$0.768951\pi$
$968$	2456.00	0.0815484
$969$	−3150.00	−0.104430
$970$	12340.0	0.408468
$971$	−1980.00	−0.0654390	−0.0327195	−	0.999465i	$-0.510417\pi$
$971$	−1980.00	−0.0654390	−0.0327195	+	0.999465i	$0.510417\pi$
$972$	972.000	0.0320750
$973$	0	0
$974$	−16630.0	−0.547084
$975$	1125.00	0.0369527
$976$	10528.0	0.345280
$977$	132.000	0.00432247	0.00216124	−	0.999998i	$-0.499312\pi$
$977$	132.000	0.00432247	0.00216124	+	0.999998i	$0.499312\pi$
$978$	12408.0	0.405689
$979$	−49088.0	−1.60251
$980$	0	0
$981$	10989.0	0.357647
$982$	−20424.0	−0.663703
$983$	−28798.0	−0.934398	−0.467199	−	0.884152i	$-0.654737\pi$
$983$	−28798.0	−0.934398	−0.467199	+	0.884152i	$0.654737\pi$
$984$	−3936.00	−0.127515
$985$	15850.0	0.512714
$986$	12600.0	0.406963
$987$	0	0
$988$	900.000	0.0289806
$989$	−7098.00	−0.228214
$990$	2880.00	0.0924570
$991$	38707.0	1.24073	0.620367	−	0.784311i	$-0.286983\pi$
$991$	38707.0	1.24073	0.620367	+	0.784311i	$0.286983\pi$
$992$	2720.00	0.0870565
$993$	21783.0	0.696136
$994$	0	0
$995$	−15320.0	−0.488117
$996$	8160.00	0.259598
$997$	22673.0	0.720222	0.360111	−	0.932910i	$-0.382739\pi$
$997$	22673.0	0.720222	0.360111	+	0.932910i	$0.382739\pi$
$998$	22458.0	0.712320
$999$	3051.00	0.0966260

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.4.a.l.1.1		1
7.2	even	3		210.4.i.e.151.1	yes	2
7.4	even	3		210.4.i.e.121.1	✓	2
7.6	odd	2		1470.4.a.b.1.1		1
21.2	odd	6		630.4.k.d.361.1		2
21.11	odd	6		630.4.k.d.541.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.i.e.121.1	✓	2	7.4	even	3
210.4.i.e.151.1	yes	2	7.2	even	3
630.4.k.d.361.1		2	21.2	odd	6
630.4.k.d.541.1		2	21.11	odd	6
1470.4.a.b.1.1		1	7.6	odd	2
1470.4.a.l.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:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1470.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more