LMFDB - Embedded newform 1470.4.a.bc.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)

$f(q)$

$=$

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

\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +5.00000 q^{5} +6.00000 q^{6} +8.00000 q^{8} +9.00000 q^{9} +10.0000 q^{10} -4.00000 q^{11} +12.0000 q^{12} +42.0000 q^{13} +15.0000 q^{15} +16.0000 q^{16} +86.0000 q^{17} +18.0000 q^{18} +96.0000 q^{19} +20.0000 q^{20} -8.00000 q^{22} -96.0000 q^{23} +24.0000 q^{24} +25.0000 q^{25} +84.0000 q^{26} +27.0000 q^{27} -78.0000 q^{29} +30.0000 q^{30} -80.0000 q^{31} +32.0000 q^{32} -12.0000 q^{33} +172.000 q^{34} +36.0000 q^{36} +50.0000 q^{37} +192.000 q^{38} +126.000 q^{39} +40.0000 q^{40} +26.0000 q^{41} -32.0000 q^{43} -16.0000 q^{44} +45.0000 q^{45} -192.000 q^{46} +20.0000 q^{47} +48.0000 q^{48} +50.0000 q^{50} +258.000 q^{51} +168.000 q^{52} -382.000 q^{53} +54.0000 q^{54} -20.0000 q^{55} +288.000 q^{57} -156.000 q^{58} -356.000 q^{59} +60.0000 q^{60} +134.000 q^{61} -160.000 q^{62} +64.0000 q^{64} +210.000 q^{65} -24.0000 q^{66} +888.000 q^{67} +344.000 q^{68} -288.000 q^{69} +868.000 q^{71} +72.0000 q^{72} +70.0000 q^{73} +100.000 q^{74} +75.0000 q^{75} +384.000 q^{76} +252.000 q^{78} +400.000 q^{79} +80.0000 q^{80} +81.0000 q^{81} +52.0000 q^{82} +1052.00 q^{83} +430.000 q^{85} -64.0000 q^{86} -234.000 q^{87} -32.0000 q^{88} +634.000 q^{89} +90.0000 q^{90} -384.000 q^{92} -240.000 q^{93} +40.0000 q^{94} +480.000 q^{95} +96.0000 q^{96} -1202.00 q^{97} -36.0000 q^{99} +O(q^{100})\)

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
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	10.0000	0.316228
$11$	−4.00000	−0.109640	−0.0548202	−	0.998496i	$-0.517459\pi$
$11$	−4.00000	−0.109640	−0.0548202	+	0.998496i	$0.517459\pi$
$12$	12.0000	0.288675
$13$	42.0000	0.896054	0.448027	−	0.894020i	$-0.352127\pi$
$13$	42.0000	0.896054	0.448027	+	0.894020i	$0.352127\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	16.0000	0.250000
$17$	86.0000	1.22694	0.613472	−	0.789716i	$-0.289772\pi$
$17$	86.0000	1.22694	0.613472	+	0.789716i	$0.289772\pi$
$18$	18.0000	0.235702
$19$	96.0000	1.15915	0.579577	−	0.814918i	$-0.303218\pi$
$19$	96.0000	1.15915	0.579577	+	0.814918i	$0.303218\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	−8.00000	−0.0775275
$23$	−96.0000	−0.870321	−0.435161	−	0.900353i	$-0.643308\pi$
$23$	−96.0000	−0.870321	−0.435161	+	0.900353i	$0.643308\pi$
$24$	24.0000	0.204124
$25$	25.0000	0.200000
$26$	84.0000	0.633606
$27$	27.0000	0.192450
$28$	0	0
$29$	−78.0000	−0.499456	−0.249728	−	0.968316i	$-0.580341\pi$
$29$	−78.0000	−0.499456	−0.249728	+	0.968316i	$0.580341\pi$
$30$	30.0000	0.182574
$31$	−80.0000	−0.463498	−0.231749	−	0.972776i	$-0.574445\pi$
$31$	−80.0000	−0.463498	−0.231749	+	0.972776i	$0.574445\pi$
$32$	32.0000	0.176777
$33$	−12.0000	−0.0633010
$34$	172.000	0.867581
$35$	0	0
$36$	36.0000	0.166667
$37$	50.0000	0.222161	0.111080	−	0.993811i	$-0.464569\pi$
$37$	50.0000	0.222161	0.111080	+	0.993811i	$0.464569\pi$
$38$	192.000	0.819645
$39$	126.000	0.517337
$40$	40.0000	0.158114
$41$	26.0000	0.0990370	0.0495185	−	0.998773i	$-0.484231\pi$
$41$	26.0000	0.0990370	0.0495185	+	0.998773i	$0.484231\pi$
$42$	0	0
$43$	−32.0000	−0.113487	−0.0567437	−	0.998389i	$-0.518072\pi$
$43$	−32.0000	−0.113487	−0.0567437	+	0.998389i	$0.518072\pi$
$44$	−16.0000	−0.0548202
$45$	45.0000	0.149071
$46$	−192.000	−0.615410
$47$	20.0000	0.0620702	0.0310351	−	0.999518i	$-0.490120\pi$
$47$	20.0000	0.0620702	0.0310351	+	0.999518i	$0.490120\pi$
$48$	48.0000	0.144338
$49$	0	0
$50$	50.0000	0.141421
$51$	258.000	0.708377
$52$	168.000	0.448027
$53$	−382.000	−0.990033	−0.495016	−	0.868884i	$-0.664838\pi$
$53$	−382.000	−0.990033	−0.495016	+	0.868884i	$0.664838\pi$
$54$	54.0000	0.136083
$55$	−20.0000	−0.0490327
$56$	0	0
$57$	288.000	0.669237
$58$	−156.000	−0.353169
$59$	−356.000	−0.785547	−0.392773	−	0.919635i	$-0.628484\pi$
$59$	−356.000	−0.785547	−0.392773	+	0.919635i	$0.628484\pi$
$60$	60.0000	0.129099
$61$	134.000	0.281261	0.140631	−	0.990062i	$-0.455087\pi$
$61$	134.000	0.281261	0.140631	+	0.990062i	$0.455087\pi$
$62$	−160.000	−0.327742
$63$	0	0
$64$	64.0000	0.125000
$65$	210.000	0.400728
$66$	−24.0000	−0.0447605
$67$	888.000	1.61920	0.809600	−	0.586981i	$-0.199684\pi$
$67$	888.000	1.61920	0.809600	+	0.586981i	$0.199684\pi$
$68$	344.000	0.613472
$69$	−288.000	−0.502480
$70$	0	0
$71$	868.000	1.45088	0.725441	−	0.688284i	$-0.241636\pi$
$71$	868.000	1.45088	0.725441	+	0.688284i	$0.241636\pi$
$72$	72.0000	0.117851
$73$	70.0000	0.112231	0.0561156	−	0.998424i	$-0.482128\pi$
$73$	70.0000	0.112231	0.0561156	+	0.998424i	$0.482128\pi$
$74$	100.000	0.157091
$75$	75.0000	0.115470
$76$	384.000	0.579577
$77$	0	0
$78$	252.000	0.365813
$79$	400.000	0.569665	0.284832	−	0.958577i	$-0.408062\pi$
$79$	400.000	0.569665	0.284832	+	0.958577i	$0.408062\pi$
$80$	80.0000	0.111803
$81$	81.0000	0.111111
$82$	52.0000	0.0700297
$83$	1052.00	1.39123	0.695614	−	0.718415i	$-0.255132\pi$
$83$	1052.00	1.39123	0.695614	+	0.718415i	$0.255132\pi$
$84$	0	0
$85$	430.000	0.548706
$86$	−64.0000	−0.0802476
$87$	−234.000	−0.288361
$88$	−32.0000	−0.0387638
$89$	634.000	0.755100	0.377550	−	0.925989i	$-0.376767\pi$
$89$	634.000	0.755100	0.377550	+	0.925989i	$0.376767\pi$
$90$	90.0000	0.105409
$91$	0	0
$92$	−384.000	−0.435161
$93$	−240.000	−0.267600
$94$	40.0000	0.0438903
$95$	480.000	0.518389
$96$	96.0000	0.102062
$97$	−1202.00	−1.25819	−0.629096	−	0.777328i	$-0.716575\pi$
$97$	−1202.00	−1.25819	−0.629096	+	0.777328i	$0.716575\pi$
$98$	0	0
$99$	−36.0000	−0.0365468
$100$	100.000	0.100000
$101$	462.000	0.455156	0.227578	−	0.973760i	$-0.426919\pi$
$101$	462.000	0.455156	0.227578	+	0.973760i	$0.426919\pi$
$102$	516.000	0.500898
$103$	416.000	0.397958	0.198979	−	0.980004i	$-0.436237\pi$
$103$	416.000	0.397958	0.198979	+	0.980004i	$0.436237\pi$
$104$	336.000	0.316803
$105$	0	0
$106$	−764.000	−0.700059
$107$	−876.000	−0.791459	−0.395730	−	0.918367i	$-0.629508\pi$
$107$	−876.000	−0.791459	−0.395730	+	0.918367i	$0.629508\pi$
$108$	108.000	0.0962250
$109$	1542.00	1.35502	0.677508	−	0.735515i	$-0.263060\pi$
$109$	1542.00	1.35502	0.677508	+	0.735515i	$0.263060\pi$
$110$	−40.0000	−0.0346714
$111$	150.000	0.128265
$112$	0	0
$113$	−482.000	−0.401263	−0.200632	−	0.979667i	$-0.564299\pi$
$113$	−482.000	−0.401263	−0.200632	+	0.979667i	$0.564299\pi$
$114$	576.000	0.473222
$115$	−480.000	−0.389219
$116$	−312.000	−0.249728
$117$	378.000	0.298685
$118$	−712.000	−0.555465
$119$	0	0
$120$	120.000	0.0912871
$121$	−1315.00	−0.987979
$122$	268.000	0.198882
$123$	78.0000	0.0571791
$124$	−320.000	−0.231749
$125$	125.000	0.0894427
$126$	0	0
$127$	880.000	0.614861	0.307431	−	0.951571i	$-0.400531\pi$
$127$	880.000	0.614861	0.307431	+	0.951571i	$0.400531\pi$
$128$	128.000	0.0883883
$129$	−96.0000	−0.0655219
$130$	420.000	0.283357
$131$	1580.00	1.05378	0.526890	−	0.849933i	$-0.323358\pi$
$131$	1580.00	1.05378	0.526890	+	0.849933i	$0.323358\pi$
$132$	−48.0000	−0.0316505
$133$	0	0
$134$	1776.00	1.14495
$135$	135.000	0.0860663
$136$	688.000	0.433791
$137$	−1170.00	−0.729634	−0.364817	−	0.931079i	$-0.618868\pi$
$137$	−1170.00	−0.729634	−0.364817	+	0.931079i	$0.618868\pi$
$138$	−576.000	−0.355307
$139$	−2656.00	−1.62071	−0.810356	−	0.585938i	$-0.800726\pi$
$139$	−2656.00	−1.62071	−0.810356	+	0.585938i	$0.800726\pi$
$140$	0	0
$141$	60.0000	0.0358363
$142$	1736.00	1.02593
$143$	−168.000	−0.0982438
$144$	144.000	0.0833333
$145$	−390.000	−0.223364
$146$	140.000	0.0793595
$147$	0	0
$148$	200.000	0.111080
$149$	−2766.00	−1.52080	−0.760401	−	0.649454i	$-0.774998\pi$
$149$	−2766.00	−1.52080	−0.760401	+	0.649454i	$0.774998\pi$
$150$	150.000	0.0816497
$151$	2792.00	1.50470	0.752350	−	0.658763i	$-0.228920\pi$
$151$	2792.00	1.50470	0.752350	+	0.658763i	$0.228920\pi$
$152$	768.000	0.409823
$153$	774.000	0.408982
$154$	0	0
$155$	−400.000	−0.207282
$156$	504.000	0.258669
$157$	1970.00	1.00142	0.500711	−	0.865615i	$-0.333072\pi$
$157$	1970.00	1.00142	0.500711	+	0.865615i	$0.333072\pi$
$158$	800.000	0.402814
$159$	−1146.00	−0.571596
$160$	160.000	0.0790569
$161$	0	0
$162$	162.000	0.0785674
$163$	−1720.00	−0.826508	−0.413254	−	0.910616i	$-0.635608\pi$
$163$	−1720.00	−0.826508	−0.413254	+	0.910616i	$0.635608\pi$
$164$	104.000	0.0495185
$165$	−60.0000	−0.0283091
$166$	2104.00	0.983747
$167$	−1164.00	−0.539359	−0.269680	−	0.962950i	$-0.586918\pi$
$167$	−1164.00	−0.539359	−0.269680	+	0.962950i	$0.586918\pi$
$168$	0	0
$169$	−433.000	−0.197087
$170$	860.000	0.387994
$171$	864.000	0.386384
$172$	−128.000	−0.0567437
$173$	4214.00	1.85193	0.925967	−	0.377605i	$-0.123252\pi$
$173$	4214.00	1.85193	0.925967	+	0.377605i	$0.123252\pi$
$174$	−468.000	−0.203902
$175$	0	0
$176$	−64.0000	−0.0274101
$177$	−1068.00	−0.453536
$178$	1268.00	0.533936
$179$	3132.00	1.30780	0.653901	−	0.756580i	$-0.273131\pi$
$179$	3132.00	1.30780	0.653901	+	0.756580i	$0.273131\pi$
$180$	180.000	0.0745356
$181$	−562.000	−0.230791	−0.115395	−	0.993320i	$-0.536814\pi$
$181$	−562.000	−0.230791	−0.115395	+	0.993320i	$0.536814\pi$
$182$	0	0
$183$	402.000	0.162386
$184$	−768.000	−0.307705
$185$	250.000	0.0993533
$186$	−480.000	−0.189222
$187$	−344.000	−0.134523
$188$	80.0000	0.0310351
$189$	0	0
$190$	960.000	0.366556
$191$	−1140.00	−0.431872	−0.215936	−	0.976408i	$-0.569280\pi$
$191$	−1140.00	−0.431872	−0.215936	+	0.976408i	$0.569280\pi$
$192$	192.000	0.0721688
$193$	−2374.00	−0.885411	−0.442705	−	0.896667i	$-0.645981\pi$
$193$	−2374.00	−0.885411	−0.442705	+	0.896667i	$0.645981\pi$
$194$	−2404.00	−0.889676
$195$	630.000	0.231360
$196$	0	0
$197$	−3230.00	−1.16816	−0.584081	−	0.811695i	$-0.698545\pi$
$197$	−3230.00	−1.16816	−0.584081	+	0.811695i	$0.698545\pi$
$198$	−72.0000	−0.0258425
$199$	4528.00	1.61297	0.806486	−	0.591253i	$-0.201367\pi$
$199$	4528.00	1.61297	0.806486	+	0.591253i	$0.201367\pi$
$200$	200.000	0.0707107
$201$	2664.00	0.934846
$202$	924.000	0.321844
$203$	0	0
$204$	1032.00	0.354188
$205$	130.000	0.0442907
$206$	832.000	0.281399
$207$	−864.000	−0.290107
$208$	672.000	0.224014
$209$	−384.000	−0.127090
$210$	0	0
$211$	−5300.00	−1.72923	−0.864614	−	0.502437i	$-0.832437\pi$
$211$	−5300.00	−1.72923	−0.864614	+	0.502437i	$0.832437\pi$
$212$	−1528.00	−0.495016
$213$	2604.00	0.837667
$214$	−1752.00	−0.559646
$215$	−160.000	−0.0507531
$216$	216.000	0.0680414
$217$	0	0
$218$	3084.00	0.958141
$219$	210.000	0.0647967
$220$	−80.0000	−0.0245164
$221$	3612.00	1.09941
$222$	300.000	0.0906968
$223$	−472.000	−0.141737	−0.0708687	−	0.997486i	$-0.522577\pi$
$223$	−472.000	−0.141737	−0.0708687	+	0.997486i	$0.522577\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	−964.000	−0.283736
$227$	5876.00	1.71808	0.859039	−	0.511910i	$-0.171062\pi$
$227$	5876.00	1.71808	0.859039	+	0.511910i	$0.171062\pi$
$228$	1152.00	0.334619
$229$	−4282.00	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$229$	−4282.00	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$230$	−960.000	−0.275220
$231$	0	0
$232$	−624.000	−0.176585
$233$	−5226.00	−1.46938	−0.734692	−	0.678400i	$-0.762673\pi$
$233$	−5226.00	−1.46938	−0.734692	+	0.678400i	$0.762673\pi$
$234$	756.000	0.211202
$235$	100.000	0.0277586
$236$	−1424.00	−0.392773
$237$	1200.00	0.328896
$238$	0	0
$239$	3620.00	0.979742	0.489871	−	0.871795i	$-0.337044\pi$
$239$	3620.00	0.979742	0.489871	+	0.871795i	$0.337044\pi$
$240$	240.000	0.0645497
$241$	1358.00	0.362973	0.181486	−	0.983393i	$-0.441909\pi$
$241$	1358.00	0.362973	0.181486	+	0.983393i	$0.441909\pi$
$242$	−2630.00	−0.698607
$243$	243.000	0.0641500
$244$	536.000	0.140631
$245$	0	0
$246$	156.000	0.0404317
$247$	4032.00	1.03866
$248$	−640.000	−0.163871
$249$	3156.00	0.803226
$250$	250.000	0.0632456
$251$	−7788.00	−1.95846	−0.979231	−	0.202745i	$-0.935014\pi$
$251$	−7788.00	−1.95846	−0.979231	+	0.202745i	$0.935014\pi$
$252$	0	0
$253$	384.000	0.0954224
$254$	1760.00	0.434773
$255$	1290.00	0.316796
$256$	256.000	0.0625000
$257$	−4450.00	−1.08009	−0.540045	−	0.841636i	$-0.681593\pi$
$257$	−4450.00	−1.08009	−0.540045	+	0.841636i	$0.681593\pi$
$258$	−192.000	−0.0463310
$259$	0	0
$260$	840.000	0.200364
$261$	−702.000	−0.166485
$262$	3160.00	0.745135
$263$	2008.00	0.470793	0.235397	−	0.971899i	$-0.424361\pi$
$263$	2008.00	0.470793	0.235397	+	0.971899i	$0.424361\pi$
$264$	−96.0000	−0.0223803
$265$	−1910.00	−0.442756
$266$	0	0
$267$	1902.00	0.435957
$268$	3552.00	0.809600
$269$	−2058.00	−0.466463	−0.233231	−	0.972421i	$-0.574930\pi$
$269$	−2058.00	−0.466463	−0.233231	+	0.972421i	$0.574930\pi$
$270$	270.000	0.0608581
$271$	−2160.00	−0.484172	−0.242086	−	0.970255i	$-0.577832\pi$
$271$	−2160.00	−0.484172	−0.242086	+	0.970255i	$0.577832\pi$
$272$	1376.00	0.306736
$273$	0	0
$274$	−2340.00	−0.515929
$275$	−100.000	−0.0219281
$276$	−1152.00	−0.251240
$277$	90.0000	0.0195219	0.00976097	−	0.999952i	$-0.496893\pi$
$277$	90.0000	0.0195219	0.00976097	+	0.999952i	$0.496893\pi$
$278$	−5312.00	−1.14602
$279$	−720.000	−0.154499
$280$	0	0
$281$	3706.00	0.786767	0.393383	−	0.919375i	$-0.371305\pi$
$281$	3706.00	0.786767	0.393383	+	0.919375i	$0.371305\pi$
$282$	120.000	0.0253401
$283$	6028.00	1.26617	0.633087	−	0.774080i	$-0.281787\pi$
$283$	6028.00	1.26617	0.633087	+	0.774080i	$0.281787\pi$
$284$	3472.00	0.725441
$285$	1440.00	0.299292
$286$	−336.000	−0.0694689
$287$	0	0
$288$	288.000	0.0589256
$289$	2483.00	0.505394
$290$	−780.000	−0.157942
$291$	−3606.00	−0.726417
$292$	280.000	0.0561156
$293$	−170.000	−0.0338959	−0.0169480	−	0.999856i	$-0.505395\pi$
$293$	−170.000	−0.0338959	−0.0169480	+	0.999856i	$0.505395\pi$
$294$	0	0
$295$	−1780.00	−0.351307
$296$	400.000	0.0785457
$297$	−108.000	−0.0211003
$298$	−5532.00	−1.07537
$299$	−4032.00	−0.779855
$300$	300.000	0.0577350
$301$	0	0
$302$	5584.00	1.06398
$303$	1386.00	0.262784
$304$	1536.00	0.289788
$305$	670.000	0.125784
$306$	1548.00	0.289194
$307$	−10396.0	−1.93267	−0.966337	−	0.257279i	$-0.917174\pi$
$307$	−10396.0	−1.93267	−0.966337	+	0.257279i	$0.917174\pi$
$308$	0	0
$309$	1248.00	0.229761
$310$	−800.000	−0.146571
$311$	−6672.00	−1.21651	−0.608255	−	0.793742i	$-0.708130\pi$
$311$	−6672.00	−1.21651	−0.608255	+	0.793742i	$0.708130\pi$
$312$	1008.00	0.182906
$313$	5830.00	1.05281	0.526407	−	0.850232i	$-0.323539\pi$
$313$	5830.00	1.05281	0.526407	+	0.850232i	$0.323539\pi$
$314$	3940.00	0.708112
$315$	0	0
$316$	1600.00	0.284832
$317$	−4726.00	−0.837346	−0.418673	−	0.908137i	$-0.637505\pi$
$317$	−4726.00	−0.837346	−0.418673	+	0.908137i	$0.637505\pi$
$318$	−2292.00	−0.404179
$319$	312.000	0.0547606
$320$	320.000	0.0559017
$321$	−2628.00	−0.456949
$322$	0	0
$323$	8256.00	1.42222
$324$	324.000	0.0555556
$325$	1050.00	0.179211
$326$	−3440.00	−0.584429
$327$	4626.00	0.782319
$328$	208.000	0.0350149
$329$	0	0
$330$	−120.000	−0.0200175
$331$	3076.00	0.510792	0.255396	−	0.966836i	$-0.417794\pi$
$331$	3076.00	0.510792	0.255396	+	0.966836i	$0.417794\pi$
$332$	4208.00	0.695614
$333$	450.000	0.0740536
$334$	−2328.00	−0.381385
$335$	4440.00	0.724129
$336$	0	0
$337$	9226.00	1.49131	0.745656	−	0.666331i	$-0.232136\pi$
$337$	9226.00	1.49131	0.745656	+	0.666331i	$0.232136\pi$
$338$	−866.000	−0.139362
$339$	−1446.00	−0.231669
$340$	1720.00	0.274353
$341$	320.000	0.0508181
$342$	1728.00	0.273215
$343$	0	0
$344$	−256.000	−0.0401238
$345$	−1440.00	−0.224716
$346$	8428.00	1.30951
$347$	284.000	0.0439364	0.0219682	−	0.999759i	$-0.493007\pi$
$347$	284.000	0.0439364	0.0219682	+	0.999759i	$0.493007\pi$
$348$	−936.000	−0.144181
$349$	7742.00	1.18745	0.593725	−	0.804668i	$-0.297657\pi$
$349$	7742.00	1.18745	0.593725	+	0.804668i	$0.297657\pi$
$350$	0	0
$351$	1134.00	0.172446
$352$	−128.000	−0.0193819
$353$	−8130.00	−1.22583	−0.612913	−	0.790151i	$-0.710002\pi$
$353$	−8130.00	−1.22583	−0.612913	+	0.790151i	$0.710002\pi$
$354$	−2136.00	−0.320698
$355$	4340.00	0.648854
$356$	2536.00	0.377550
$357$	0	0
$358$	6264.00	0.924756
$359$	−6452.00	−0.948534	−0.474267	−	0.880381i	$-0.657287\pi$
$359$	−6452.00	−0.948534	−0.474267	+	0.880381i	$0.657287\pi$
$360$	360.000	0.0527046
$361$	2357.00	0.343636
$362$	−1124.00	−0.163194
$363$	−3945.00	−0.570410
$364$	0	0
$365$	350.000	0.0501913
$366$	804.000	0.114824
$367$	−1504.00	−0.213919	−0.106959	−	0.994263i	$-0.534111\pi$
$367$	−1504.00	−0.213919	−0.106959	+	0.994263i	$0.534111\pi$
$368$	−1536.00	−0.217580
$369$	234.000	0.0330123
$370$	500.000	0.0702534
$371$	0	0
$372$	−960.000	−0.133800
$373$	−12814.0	−1.77878	−0.889388	−	0.457152i	$-0.848869\pi$
$373$	−12814.0	−1.77878	−0.889388	+	0.457152i	$0.848869\pi$
$374$	−688.000	−0.0951220
$375$	375.000	0.0516398
$376$	160.000	0.0219451
$377$	−3276.00	−0.447540
$378$	0	0
$379$	−7020.00	−0.951433	−0.475717	−	0.879599i	$-0.657811\pi$
$379$	−7020.00	−0.951433	−0.475717	+	0.879599i	$0.657811\pi$
$380$	1920.00	0.259195
$381$	2640.00	0.354990
$382$	−2280.00	−0.305379
$383$	−6732.00	−0.898144	−0.449072	−	0.893496i	$-0.648245\pi$
$383$	−6732.00	−0.898144	−0.449072	+	0.893496i	$0.648245\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−4748.00	−0.626080
$387$	−288.000	−0.0378291
$388$	−4808.00	−0.629096
$389$	5026.00	0.655086	0.327543	−	0.944836i	$-0.393779\pi$
$389$	5026.00	0.655086	0.327543	+	0.944836i	$0.393779\pi$
$390$	1260.00	0.163596
$391$	−8256.00	−1.06784
$392$	0	0
$393$	4740.00	0.608400
$394$	−6460.00	−0.826015
$395$	2000.00	0.254762
$396$	−144.000	−0.0182734
$397$	−13462.0	−1.70186	−0.850930	−	0.525279i	$-0.823961\pi$
$397$	−13462.0	−1.70186	−0.850930	+	0.525279i	$0.823961\pi$
$398$	9056.00	1.14054
$399$	0	0
$400$	400.000	0.0500000
$401$	−3982.00	−0.495889	−0.247945	−	0.968774i	$-0.579755\pi$
$401$	−3982.00	−0.495889	−0.247945	+	0.968774i	$0.579755\pi$
$402$	5328.00	0.661036
$403$	−3360.00	−0.415319
$404$	1848.00	0.227578
$405$	405.000	0.0496904
$406$	0	0
$407$	−200.000	−0.0243578
$408$	2064.00	0.250449
$409$	11958.0	1.44568	0.722842	−	0.691013i	$-0.242835\pi$
$409$	11958.0	1.44568	0.722842	+	0.691013i	$0.242835\pi$
$410$	260.000	0.0313183
$411$	−3510.00	−0.421254
$412$	1664.00	0.198979
$413$	0	0
$414$	−1728.00	−0.205137
$415$	5260.00	0.622176
$416$	1344.00	0.158401
$417$	−7968.00	−0.935719
$418$	−768.000	−0.0898663
$419$	2964.00	0.345587	0.172793	−	0.984958i	$-0.444721\pi$
$419$	2964.00	0.345587	0.172793	+	0.984958i	$0.444721\pi$
$420$	0	0
$421$	−9394.00	−1.08750	−0.543748	−	0.839249i	$-0.682995\pi$
$421$	−9394.00	−1.08750	−0.543748	+	0.839249i	$0.682995\pi$
$422$	−10600.0	−1.22275
$423$	180.000	0.0206901
$424$	−3056.00	−0.350029
$425$	2150.00	0.245389
$426$	5208.00	0.592320
$427$	0	0
$428$	−3504.00	−0.395730
$429$	−504.000	−0.0567211
$430$	−320.000	−0.0358878
$431$	5580.00	0.623617	0.311809	−	0.950145i	$-0.399065\pi$
$431$	5580.00	0.623617	0.311809	+	0.950145i	$0.399065\pi$
$432$	432.000	0.0481125
$433$	−5906.00	−0.655483	−0.327742	−	0.944767i	$-0.606288\pi$
$433$	−5906.00	−0.655483	−0.327742	+	0.944767i	$0.606288\pi$
$434$	0	0
$435$	−1170.00	−0.128959
$436$	6168.00	0.677508
$437$	−9216.00	−1.00884
$438$	420.000	0.0458182
$439$	4616.00	0.501844	0.250922	−	0.968007i	$-0.419266\pi$
$439$	4616.00	0.501844	0.250922	+	0.968007i	$0.419266\pi$
$440$	−160.000	−0.0173357
$441$	0	0
$442$	7224.00	0.777400
$443$	−7492.00	−0.803512	−0.401756	−	0.915747i	$-0.631600\pi$
$443$	−7492.00	−0.803512	−0.401756	+	0.915747i	$0.631600\pi$
$444$	600.000	0.0641323
$445$	3170.00	0.337691
$446$	−944.000	−0.100224
$447$	−8298.00	−0.878036
$448$	0	0
$449$	−14214.0	−1.49399	−0.746993	−	0.664831i	$-0.768503\pi$
$449$	−14214.0	−1.49399	−0.746993	+	0.664831i	$0.768503\pi$
$450$	450.000	0.0471405
$451$	−104.000	−0.0108585
$452$	−1928.00	−0.200632
$453$	8376.00	0.868739
$454$	11752.0	1.21486
$455$	0	0
$456$	2304.00	0.236611
$457$	12970.0	1.32760	0.663798	−	0.747912i	$-0.268944\pi$
$457$	12970.0	1.32760	0.663798	+	0.747912i	$0.268944\pi$
$458$	−8564.00	−0.873732
$459$	2322.00	0.236126
$460$	−1920.00	−0.194610
$461$	11158.0	1.12729	0.563644	−	0.826018i	$-0.309399\pi$
$461$	11158.0	1.12729	0.563644	+	0.826018i	$0.309399\pi$
$462$	0	0
$463$	7544.00	0.757234	0.378617	−	0.925553i	$-0.376400\pi$
$463$	7544.00	0.757234	0.378617	+	0.925553i	$0.376400\pi$
$464$	−1248.00	−0.124864
$465$	−1200.00	−0.119675
$466$	−10452.0	−1.03901
$467$	7004.00	0.694018	0.347009	−	0.937862i	$-0.387197\pi$
$467$	7004.00	0.694018	0.347009	+	0.937862i	$0.387197\pi$
$468$	1512.00	0.149342
$469$	0	0
$470$	200.000	0.0196283
$471$	5910.00	0.578171
$472$	−2848.00	−0.277733
$473$	128.000	0.0124428
$474$	2400.00	0.232565
$475$	2400.00	0.231831
$476$	0	0
$477$	−3438.00	−0.330011
$478$	7240.00	0.692782
$479$	11544.0	1.10117	0.550583	−	0.834780i	$-0.314405\pi$
$479$	11544.0	1.10117	0.550583	+	0.834780i	$0.314405\pi$
$480$	480.000	0.0456435
$481$	2100.00	0.199068
$482$	2716.00	0.256661
$483$	0	0
$484$	−5260.00	−0.493989
$485$	−6010.00	−0.562680
$486$	486.000	0.0453609
$487$	−424.000	−0.0394523	−0.0197262	−	0.999805i	$-0.506279\pi$
$487$	−424.000	−0.0394523	−0.0197262	+	0.999805i	$0.506279\pi$
$488$	1072.00	0.0994409
$489$	−5160.00	−0.477185
$490$	0	0
$491$	−10428.0	−0.958471	−0.479235	−	0.877686i	$-0.659086\pi$
$491$	−10428.0	−0.958471	−0.479235	+	0.877686i	$0.659086\pi$
$492$	312.000	0.0285895
$493$	−6708.00	−0.612806
$494$	8064.00	0.734446
$495$	−180.000	−0.0163442
$496$	−1280.00	−0.115874
$497$	0	0
$498$	6312.00	0.567967
$499$	−6756.00	−0.606092	−0.303046	−	0.952976i	$-0.598004\pi$
$499$	−6756.00	−0.606092	−0.303046	+	0.952976i	$0.598004\pi$
$500$	500.000	0.0447214
$501$	−3492.00	−0.311399
$502$	−15576.0	−1.38484
$503$	−12012.0	−1.06479	−0.532394	−	0.846497i	$-0.678708\pi$
$503$	−12012.0	−1.06479	−0.532394	+	0.846497i	$0.678708\pi$
$504$	0	0
$505$	2310.00	0.203552
$506$	768.000	0.0674738
$507$	−1299.00	−0.113788
$508$	3520.00	0.307431
$509$	6646.00	0.578740	0.289370	−	0.957217i	$-0.406554\pi$
$509$	6646.00	0.578740	0.289370	+	0.957217i	$0.406554\pi$
$510$	2580.00	0.224008
$511$	0	0
$512$	512.000	0.0441942
$513$	2592.00	0.223079
$514$	−8900.00	−0.763740
$515$	2080.00	0.177972
$516$	−384.000	−0.0327610
$517$	−80.0000	−0.00680541
$518$	0	0
$519$	12642.0	1.06921
$520$	1680.00	0.141679
$521$	−9814.00	−0.825257	−0.412629	−	0.910899i	$-0.635389\pi$
$521$	−9814.00	−0.825257	−0.412629	+	0.910899i	$0.635389\pi$
$522$	−1404.00	−0.117723
$523$	7700.00	0.643781	0.321891	−	0.946777i	$-0.395682\pi$
$523$	7700.00	0.643781	0.321891	+	0.946777i	$0.395682\pi$
$524$	6320.00	0.526890
$525$	0	0
$526$	4016.00	0.332901
$527$	−6880.00	−0.568686
$528$	−192.000	−0.0158252
$529$	−2951.00	−0.242541
$530$	−3820.00	−0.313076
$531$	−3204.00	−0.261849
$532$	0	0
$533$	1092.00	0.0887425
$534$	3804.00	0.308268
$535$	−4380.00	−0.353951
$536$	7104.00	0.572474
$537$	9396.00	0.755060
$538$	−4116.00	−0.329839
$539$	0	0
$540$	540.000	0.0430331
$541$	15390.0	1.22305	0.611523	−	0.791227i	$-0.290557\pi$
$541$	15390.0	1.22305	0.611523	+	0.791227i	$0.290557\pi$
$542$	−4320.00	−0.342361
$543$	−1686.00	−0.133247
$544$	2752.00	0.216895
$545$	7710.00	0.605982
$546$	0	0
$547$	−23064.0	−1.80283	−0.901413	−	0.432961i	$-0.857469\pi$
$547$	−23064.0	−1.80283	−0.901413	+	0.432961i	$0.857469\pi$
$548$	−4680.00	−0.364817
$549$	1206.00	0.0937538
$550$	−200.000	−0.0155055
$551$	−7488.00	−0.578947
$552$	−2304.00	−0.177654
$553$	0	0
$554$	180.000	0.0138041
$555$	750.000	0.0573617
$556$	−10624.0	−0.810356
$557$	−1654.00	−0.125821	−0.0629104	−	0.998019i	$-0.520038\pi$
$557$	−1654.00	−0.125821	−0.0629104	+	0.998019i	$0.520038\pi$
$558$	−1440.00	−0.109247
$559$	−1344.00	−0.101691
$560$	0	0
$561$	−1032.00	−0.0776668
$562$	7412.00	0.556328
$563$	−23212.0	−1.73760	−0.868800	−	0.495163i	$-0.835108\pi$
$563$	−23212.0	−1.73760	−0.868800	+	0.495163i	$0.835108\pi$
$564$	240.000	0.0179181
$565$	−2410.00	−0.179450
$566$	12056.0	0.895321
$567$	0	0
$568$	6944.00	0.512964
$569$	−7494.00	−0.552135	−0.276068	−	0.961138i	$-0.589031\pi$
$569$	−7494.00	−0.552135	−0.276068	+	0.961138i	$0.589031\pi$
$570$	2880.00	0.211631
$571$	−13668.0	−1.00173	−0.500865	−	0.865525i	$-0.666985\pi$
$571$	−13668.0	−1.00173	−0.500865	+	0.865525i	$0.666985\pi$
$572$	−672.000	−0.0491219
$573$	−3420.00	−0.249341
$574$	0	0
$575$	−2400.00	−0.174064
$576$	576.000	0.0416667
$577$	−19138.0	−1.38081	−0.690403	−	0.723425i	$-0.742567\pi$
$577$	−19138.0	−1.38081	−0.690403	+	0.723425i	$0.742567\pi$
$578$	4966.00	0.357367
$579$	−7122.00	−0.511192
$580$	−1560.00	−0.111682
$581$	0	0
$582$	−7212.00	−0.513655
$583$	1528.00	0.108548
$584$	560.000	0.0396797
$585$	1890.00	0.133576
$586$	−340.000	−0.0239680
$587$	5516.00	0.387853	0.193926	−	0.981016i	$-0.437878\pi$
$587$	5516.00	0.387853	0.193926	+	0.981016i	$0.437878\pi$
$588$	0	0
$589$	−7680.00	−0.537265
$590$	−3560.00	−0.248412
$591$	−9690.00	−0.674439
$592$	800.000	0.0555402
$593$	14886.0	1.03085	0.515426	−	0.856934i	$-0.327634\pi$
$593$	14886.0	1.03085	0.515426	+	0.856934i	$0.327634\pi$
$594$	−216.000	−0.0149202
$595$	0	0
$596$	−11064.0	−0.760401
$597$	13584.0	0.931250
$598$	−8064.00	−0.551441
$599$	3636.00	0.248018	0.124009	−	0.992281i	$-0.460425\pi$
$599$	3636.00	0.248018	0.124009	+	0.992281i	$0.460425\pi$
$600$	600.000	0.0408248
$601$	−20826.0	−1.41349	−0.706747	−	0.707466i	$-0.749838\pi$
$601$	−20826.0	−1.41349	−0.706747	+	0.707466i	$0.749838\pi$
$602$	0	0
$603$	7992.00	0.539734
$604$	11168.0	0.752350
$605$	−6575.00	−0.441838
$606$	2772.00	0.185817
$607$	10184.0	0.680982	0.340491	−	0.940248i	$-0.389407\pi$
$607$	10184.0	0.680982	0.340491	+	0.940248i	$0.389407\pi$
$608$	3072.00	0.204911
$609$	0	0
$610$	1340.00	0.0889426
$611$	840.000	0.0556183
$612$	3096.00	0.204491
$613$	6114.00	0.402842	0.201421	−	0.979505i	$-0.435444\pi$
$613$	6114.00	0.402842	0.201421	+	0.979505i	$0.435444\pi$
$614$	−20792.0	−1.36661
$615$	390.000	0.0255712
$616$	0	0
$617$	−13754.0	−0.897431	−0.448716	−	0.893675i	$-0.648118\pi$
$617$	−13754.0	−0.897431	−0.448716	+	0.893675i	$0.648118\pi$
$618$	2496.00	0.162466
$619$	17416.0	1.13087	0.565435	−	0.824793i	$-0.308708\pi$
$619$	17416.0	1.13087	0.565435	+	0.824793i	$0.308708\pi$
$620$	−1600.00	−0.103641
$621$	−2592.00	−0.167493
$622$	−13344.0	−0.860202
$623$	0	0
$624$	2016.00	0.129334
$625$	625.000	0.0400000
$626$	11660.0	0.744453
$627$	−1152.00	−0.0733755
$628$	7880.00	0.500711
$629$	4300.00	0.272579
$630$	0	0
$631$	−18592.0	−1.17296	−0.586478	−	0.809965i	$-0.699486\pi$
$631$	−18592.0	−1.17296	−0.586478	+	0.809965i	$0.699486\pi$
$632$	3200.00	0.201407
$633$	−15900.0	−0.998370
$634$	−9452.00	−0.592093
$635$	4400.00	0.274974
$636$	−4584.00	−0.285798
$637$	0	0
$638$	624.000	0.0387216
$639$	7812.00	0.483627
$640$	640.000	0.0395285
$641$	−8198.00	−0.505151	−0.252575	−	0.967577i	$-0.581278\pi$
$641$	−8198.00	−0.505151	−0.252575	+	0.967577i	$0.581278\pi$
$642$	−5256.00	−0.323112
$643$	17852.0	1.09489	0.547445	−	0.836842i	$-0.315601\pi$
$643$	17852.0	1.09489	0.547445	+	0.836842i	$0.315601\pi$
$644$	0	0
$645$	−480.000	−0.0293023
$646$	16512.0	1.00566
$647$	−8228.00	−0.499963	−0.249981	−	0.968251i	$-0.580425\pi$
$647$	−8228.00	−0.499963	−0.249981	+	0.968251i	$0.580425\pi$
$648$	648.000	0.0392837
$649$	1424.00	0.0861277
$650$	2100.00	0.126721
$651$	0	0
$652$	−6880.00	−0.413254
$653$	4426.00	0.265242	0.132621	−	0.991167i	$-0.457661\pi$
$653$	4426.00	0.265242	0.132621	+	0.991167i	$0.457661\pi$
$654$	9252.00	0.553183
$655$	7900.00	0.471265
$656$	416.000	0.0247593
$657$	630.000	0.0374104
$658$	0	0
$659$	−21292.0	−1.25860	−0.629301	−	0.777162i	$-0.716659\pi$
$659$	−21292.0	−1.25860	−0.629301	+	0.777162i	$0.716659\pi$
$660$	−240.000	−0.0141545
$661$	−1138.00	−0.0669638	−0.0334819	−	0.999439i	$-0.510660\pi$
$661$	−1138.00	−0.0669638	−0.0334819	+	0.999439i	$0.510660\pi$
$662$	6152.00	0.361185
$663$	10836.0	0.634744
$664$	8416.00	0.491874
$665$	0	0
$666$	900.000	0.0523638
$667$	7488.00	0.434687
$668$	−4656.00	−0.269680
$669$	−1416.00	−0.0818322
$670$	8880.00	0.512036
$671$	−536.000	−0.0308376
$672$	0	0
$673$	2746.00	0.157282	0.0786408	−	0.996903i	$-0.474942\pi$
$673$	2746.00	0.157282	0.0786408	+	0.996903i	$0.474942\pi$
$674$	18452.0	1.05452
$675$	675.000	0.0384900
$676$	−1732.00	−0.0985435
$677$	−30354.0	−1.72319	−0.861595	−	0.507597i	$-0.830534\pi$
$677$	−30354.0	−1.72319	−0.861595	+	0.507597i	$0.830534\pi$
$678$	−2892.00	−0.163815
$679$	0	0
$680$	3440.00	0.193997
$681$	17628.0	0.991933
$682$	640.000	0.0359338
$683$	−5804.00	−0.325159	−0.162580	−	0.986695i	$-0.551981\pi$
$683$	−5804.00	−0.325159	−0.162580	+	0.986695i	$0.551981\pi$
$684$	3456.00	0.193192
$685$	−5850.00	−0.326302
$686$	0	0
$687$	−12846.0	−0.713400
$688$	−512.000	−0.0283718
$689$	−16044.0	−0.887123
$690$	−2880.00	−0.158898
$691$	9104.00	0.501205	0.250602	−	0.968090i	$-0.419371\pi$
$691$	9104.00	0.501205	0.250602	+	0.968090i	$0.419371\pi$
$692$	16856.0	0.925967
$693$	0	0
$694$	568.000	0.0310677
$695$	−13280.0	−0.724804
$696$	−1872.00	−0.101951
$697$	2236.00	0.121513
$698$	15484.0	0.839653
$699$	−15678.0	−0.848350
$700$	0	0
$701$	−12622.0	−0.680066	−0.340033	−	0.940413i	$-0.610438\pi$
$701$	−12622.0	−0.680066	−0.340033	+	0.940413i	$0.610438\pi$
$702$	2268.00	0.121938
$703$	4800.00	0.257518
$704$	−256.000	−0.0137051
$705$	300.000	0.0160265
$706$	−16260.0	−0.866789
$707$	0	0
$708$	−4272.00	−0.226768
$709$	21190.0	1.12244	0.561218	−	0.827668i	$-0.310333\pi$
$709$	21190.0	1.12244	0.561218	+	0.827668i	$0.310333\pi$
$710$	8680.00	0.458809
$711$	3600.00	0.189888
$712$	5072.00	0.266968
$713$	7680.00	0.403392
$714$	0	0
$715$	−840.000	−0.0439360
$716$	12528.0	0.653901
$717$	10860.0	0.565654
$718$	−12904.0	−0.670714
$719$	−25560.0	−1.32577	−0.662884	−	0.748722i	$-0.730668\pi$
$719$	−25560.0	−1.32577	−0.662884	+	0.748722i	$0.730668\pi$
$720$	720.000	0.0372678
$721$	0	0
$722$	4714.00	0.242987
$723$	4074.00	0.209563
$724$	−2248.00	−0.115395
$725$	−1950.00	−0.0998913
$726$	−7890.00	−0.403341
$727$	22656.0	1.15580	0.577899	−	0.816109i	$-0.303873\pi$
$727$	22656.0	1.15580	0.577899	+	0.816109i	$0.303873\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	700.000	0.0354906
$731$	−2752.00	−0.139243
$732$	1608.00	0.0811932
$733$	16922.0	0.852699	0.426349	−	0.904559i	$-0.359799\pi$
$733$	16922.0	0.852699	0.426349	+	0.904559i	$0.359799\pi$
$734$	−3008.00	−0.151263
$735$	0	0
$736$	−3072.00	−0.153852
$737$	−3552.00	−0.177530
$738$	468.000	0.0233432
$739$	10972.0	0.546159	0.273080	−	0.961991i	$-0.411958\pi$
$739$	10972.0	0.546159	0.273080	+	0.961991i	$0.411958\pi$
$740$	1000.00	0.0496767
$741$	12096.0	0.599673
$742$	0	0
$743$	10512.0	0.519042	0.259521	−	0.965738i	$-0.416435\pi$
$743$	10512.0	0.519042	0.259521	+	0.965738i	$0.416435\pi$
$744$	−1920.00	−0.0946110
$745$	−13830.0	−0.680123
$746$	−25628.0	−1.25779
$747$	9468.00	0.463743
$748$	−1376.00	−0.0672614
$749$	0	0
$750$	750.000	0.0365148
$751$	−22312.0	−1.08412	−0.542062	−	0.840339i	$-0.682356\pi$
$751$	−22312.0	−1.08412	−0.542062	+	0.840339i	$0.682356\pi$
$752$	320.000	0.0155176
$753$	−23364.0	−1.13072
$754$	−6552.00	−0.316459
$755$	13960.0	0.672922
$756$	0	0
$757$	1762.00	0.0845984	0.0422992	−	0.999105i	$-0.486532\pi$
$757$	1762.00	0.0845984	0.0422992	+	0.999105i	$0.486532\pi$
$758$	−14040.0	−0.672765
$759$	1152.00	0.0550922
$760$	3840.00	0.183278
$761$	−20238.0	−0.964030	−0.482015	−	0.876163i	$-0.660095\pi$
$761$	−20238.0	−0.964030	−0.482015	+	0.876163i	$0.660095\pi$
$762$	5280.00	0.251016
$763$	0	0
$764$	−4560.00	−0.215936
$765$	3870.00	0.182902
$766$	−13464.0	−0.635084
$767$	−14952.0	−0.703892
$768$	768.000	0.0360844
$769$	14246.0	0.668041	0.334021	−	0.942566i	$-0.391594\pi$
$769$	14246.0	0.668041	0.334021	+	0.942566i	$0.391594\pi$
$770$	0	0
$771$	−13350.0	−0.623591
$772$	−9496.00	−0.442705
$773$	−39834.0	−1.85347	−0.926733	−	0.375720i	$-0.877395\pi$
$773$	−39834.0	−1.85347	−0.926733	+	0.375720i	$0.877395\pi$
$774$	−576.000	−0.0267492
$775$	−2000.00	−0.0926995
$776$	−9616.00	−0.444838
$777$	0	0
$778$	10052.0	0.463216
$779$	2496.00	0.114799
$780$	2520.00	0.115680
$781$	−3472.00	−0.159075
$782$	−16512.0	−0.755074
$783$	−2106.00	−0.0961204
$784$	0	0
$785$	9850.00	0.447849
$786$	9480.00	0.430204
$787$	34052.0	1.54234	0.771171	−	0.636628i	$-0.219671\pi$
$787$	34052.0	1.54234	0.771171	+	0.636628i	$0.219671\pi$
$788$	−12920.0	−0.584081
$789$	6024.00	0.271813
$790$	4000.00	0.180144
$791$	0	0
$792$	−288.000	−0.0129213
$793$	5628.00	0.252025
$794$	−26924.0	−1.20340
$795$	−5730.00	−0.255625
$796$	18112.0	0.806486
$797$	−14442.0	−0.641859	−0.320930	−	0.947103i	$-0.603995\pi$
$797$	−14442.0	−0.641859	−0.320930	+	0.947103i	$0.603995\pi$
$798$	0	0
$799$	1720.00	0.0761567
$800$	800.000	0.0353553
$801$	5706.00	0.251700
$802$	−7964.00	−0.350647
$803$	−280.000	−0.0123051
$804$	10656.0	0.467423
$805$	0	0
$806$	−6720.00	−0.293675
$807$	−6174.00	−0.269312
$808$	3696.00	0.160922
$809$	21450.0	0.932190	0.466095	−	0.884735i	$-0.345660\pi$
$809$	21450.0	0.932190	0.466095	+	0.884735i	$0.345660\pi$
$810$	810.000	0.0351364
$811$	−4496.00	−0.194668	−0.0973341	−	0.995252i	$-0.531032\pi$
$811$	−4496.00	−0.194668	−0.0973341	+	0.995252i	$0.531032\pi$
$812$	0	0
$813$	−6480.00	−0.279537
$814$	−400.000	−0.0172236
$815$	−8600.00	−0.369626
$816$	4128.00	0.177094
$817$	−3072.00	−0.131549
$818$	23916.0	1.02225
$819$	0	0
$820$	520.000	0.0221454
$821$	−40182.0	−1.70811	−0.854057	−	0.520180i	$-0.825865\pi$
$821$	−40182.0	−1.70811	−0.854057	+	0.520180i	$0.825865\pi$
$822$	−7020.00	−0.297872
$823$	−2048.00	−0.0867422	−0.0433711	−	0.999059i	$-0.513810\pi$
$823$	−2048.00	−0.0867422	−0.0433711	+	0.999059i	$0.513810\pi$
$824$	3328.00	0.140699
$825$	−300.000	−0.0126602
$826$	0	0
$827$	−20612.0	−0.866686	−0.433343	−	0.901229i	$-0.642666\pi$
$827$	−20612.0	−0.866686	−0.433343	+	0.901229i	$0.642666\pi$
$828$	−3456.00	−0.145054
$829$	−38786.0	−1.62496	−0.812481	−	0.582988i	$-0.801884\pi$
$829$	−38786.0	−1.62496	−0.812481	+	0.582988i	$0.801884\pi$
$830$	10520.0	0.439945
$831$	270.000	0.0112710
$832$	2688.00	0.112007
$833$	0	0
$834$	−15936.0	−0.661653
$835$	−5820.00	−0.241209
$836$	−1536.00	−0.0635451
$837$	−2160.00	−0.0892001
$838$	5928.00	0.244367
$839$	−13624.0	−0.560611	−0.280306	−	0.959911i	$-0.590436\pi$
$839$	−13624.0	−0.560611	−0.280306	+	0.959911i	$0.590436\pi$
$840$	0	0
$841$	−18305.0	−0.750543
$842$	−18788.0	−0.768975
$843$	11118.0	0.454240
$844$	−21200.0	−0.864614
$845$	−2165.00	−0.0881400
$846$	360.000	0.0146301
$847$	0	0
$848$	−6112.00	−0.247508
$849$	18084.0	0.731026
$850$	4300.00	0.173516
$851$	−4800.00	−0.193351
$852$	10416.0	0.418834
$853$	9778.00	0.392488	0.196244	−	0.980555i	$-0.437126\pi$
$853$	9778.00	0.392488	0.196244	+	0.980555i	$0.437126\pi$
$854$	0	0
$855$	4320.00	0.172796
$856$	−7008.00	−0.279823
$857$	−10314.0	−0.411108	−0.205554	−	0.978646i	$-0.565900\pi$
$857$	−10314.0	−0.411108	−0.205554	+	0.978646i	$0.565900\pi$
$858$	−1008.00	−0.0401079
$859$	2280.00	0.0905618	0.0452809	−	0.998974i	$-0.485582\pi$
$859$	2280.00	0.0905618	0.0452809	+	0.998974i	$0.485582\pi$
$860$	−640.000	−0.0253765
$861$	0	0
$862$	11160.0	0.440964
$863$	464.000	0.0183021	0.00915107	−	0.999958i	$-0.497087\pi$
$863$	464.000	0.0183021	0.00915107	+	0.999958i	$0.497087\pi$
$864$	864.000	0.0340207
$865$	21070.0	0.828210
$866$	−11812.0	−0.463497
$867$	7449.00	0.291789
$868$	0	0
$869$	−1600.00	−0.0624583
$870$	−2340.00	−0.0911879
$871$	37296.0	1.45089
$872$	12336.0	0.479071
$873$	−10818.0	−0.419397
$874$	−18432.0	−0.713354
$875$	0	0
$876$	840.000	0.0323984
$877$	38546.0	1.48416	0.742079	−	0.670313i	$-0.233840\pi$
$877$	38546.0	1.48416	0.742079	+	0.670313i	$0.233840\pi$
$878$	9232.00	0.354858
$879$	−510.000	−0.0195698
$880$	−320.000	−0.0122582
$881$	−7318.00	−0.279852	−0.139926	−	0.990162i	$-0.544686\pi$
$881$	−7318.00	−0.279852	−0.139926	+	0.990162i	$0.544686\pi$
$882$	0	0
$883$	42600.0	1.62356	0.811780	−	0.583963i	$-0.198499\pi$
$883$	42600.0	1.62356	0.811780	+	0.583963i	$0.198499\pi$
$884$	14448.0	0.549705
$885$	−5340.00	−0.202827
$886$	−14984.0	−0.568169
$887$	−31372.0	−1.18756	−0.593782	−	0.804626i	$-0.702366\pi$
$887$	−31372.0	−1.18756	−0.593782	+	0.804626i	$0.702366\pi$
$888$	1200.00	0.0453484
$889$	0	0
$890$	6340.00	0.238783
$891$	−324.000	−0.0121823
$892$	−1888.00	−0.0708687
$893$	1920.00	0.0719489
$894$	−16596.0	−0.620865
$895$	15660.0	0.584867
$896$	0	0
$897$	−12096.0	−0.450249
$898$	−28428.0	−1.05641
$899$	6240.00	0.231497
$900$	900.000	0.0333333
$901$	−32852.0	−1.21472
$902$	−208.000	−0.00767810
$903$	0	0
$904$	−3856.00	−0.141868
$905$	−2810.00	−0.103213
$906$	16752.0	0.614291
$907$	−16336.0	−0.598046	−0.299023	−	0.954246i	$-0.596661\pi$
$907$	−16336.0	−0.598046	−0.299023	+	0.954246i	$0.596661\pi$
$908$	23504.0	0.859039
$909$	4158.00	0.151719
$910$	0	0
$911$	−34260.0	−1.24598	−0.622988	−	0.782231i	$-0.714082\pi$
$911$	−34260.0	−1.24598	−0.622988	+	0.782231i	$0.714082\pi$
$912$	4608.00	0.167309
$913$	−4208.00	−0.152535
$914$	25940.0	0.938752
$915$	2010.00	0.0726214
$916$	−17128.0	−0.617822
$917$	0	0
$918$	4644.00	0.166966
$919$	52160.0	1.87225	0.936126	−	0.351665i	$-0.114384\pi$
$919$	52160.0	1.87225	0.936126	+	0.351665i	$0.114384\pi$
$920$	−3840.00	−0.137610
$921$	−31188.0	−1.11583
$922$	22316.0	0.797113
$923$	36456.0	1.30007
$924$	0	0
$925$	1250.00	0.0444322
$926$	15088.0	0.535445
$927$	3744.00	0.132653
$928$	−2496.00	−0.0882923
$929$	31018.0	1.09544	0.547722	−	0.836660i	$-0.315495\pi$
$929$	31018.0	1.09544	0.547722	+	0.836660i	$0.315495\pi$
$930$	−2400.00	−0.0846227
$931$	0	0
$932$	−20904.0	−0.734692
$933$	−20016.0	−0.702352
$934$	14008.0	0.490745
$935$	−1720.00	−0.0601604
$936$	3024.00	0.105601
$937$	14614.0	0.509518	0.254759	−	0.967005i	$-0.418004\pi$
$937$	14614.0	0.509518	0.254759	+	0.967005i	$0.418004\pi$
$938$	0	0
$939$	17490.0	0.607843
$940$	400.000	0.0138793
$941$	−4234.00	−0.146678	−0.0733392	−	0.997307i	$-0.523366\pi$
$941$	−4234.00	−0.146678	−0.0733392	+	0.997307i	$0.523366\pi$
$942$	11820.0	0.408828
$943$	−2496.00	−0.0861940
$944$	−5696.00	−0.196387
$945$	0	0
$946$	256.000	0.00879839
$947$	−34292.0	−1.17671	−0.588353	−	0.808604i	$-0.700223\pi$
$947$	−34292.0	−1.17671	−0.588353	+	0.808604i	$0.700223\pi$
$948$	4800.00	0.164448
$949$	2940.00	0.100565
$950$	4800.00	0.163929
$951$	−14178.0	−0.483442
$952$	0	0
$953$	53958.0	1.83407	0.917036	−	0.398804i	$-0.130575\pi$
$953$	53958.0	1.83407	0.917036	+	0.398804i	$0.130575\pi$
$954$	−6876.00	−0.233353
$955$	−5700.00	−0.193139
$956$	14480.0	0.489871
$957$	936.000	0.0316161
$958$	23088.0	0.778642
$959$	0	0
$960$	960.000	0.0322749
$961$	−23391.0	−0.785170
$962$	4200.00	0.140762
$963$	−7884.00	−0.263820
$964$	5432.00	0.181486
$965$	−11870.0	−0.395968
$966$	0	0
$967$	−16904.0	−0.562147	−0.281073	−	0.959686i	$-0.590690\pi$
$967$	−16904.0	−0.562147	−0.281073	+	0.959686i	$0.590690\pi$
$968$	−10520.0	−0.349303
$969$	24768.0	0.821117
$970$	−12020.0	−0.397875
$971$	−34428.0	−1.13784	−0.568922	−	0.822391i	$-0.692639\pi$
$971$	−34428.0	−1.13784	−0.568922	+	0.822391i	$0.692639\pi$
$972$	972.000	0.0320750
$973$	0	0
$974$	−848.000	−0.0278970
$975$	3150.00	0.103467
$976$	2144.00	0.0703153
$977$	−16434.0	−0.538148	−0.269074	−	0.963120i	$-0.586718\pi$
$977$	−16434.0	−0.538148	−0.269074	+	0.963120i	$0.586718\pi$
$978$	−10320.0	−0.337420
$979$	−2536.00	−0.0827895
$980$	0	0
$981$	13878.0	0.451672
$982$	−20856.0	−0.677741
$983$	−25348.0	−0.822457	−0.411229	−	0.911532i	$-0.634900\pi$
$983$	−25348.0	−0.822457	−0.411229	+	0.911532i	$0.634900\pi$
$984$	624.000	0.0202158
$985$	−16150.0	−0.522418
$986$	−13416.0	−0.433319
$987$	0	0
$988$	16128.0	0.519332
$989$	3072.00	0.0987704
$990$	−360.000	−0.0115571
$991$	46744.0	1.49836	0.749179	−	0.662368i	$-0.230448\pi$
$991$	46744.0	1.49836	0.749179	+	0.662368i	$0.230448\pi$
$992$	−2560.00	−0.0819356
$993$	9228.00	0.294906
$994$	0	0
$995$	22640.0	0.721343
$996$	12624.0	0.401613
$997$	−29678.0	−0.942740	−0.471370	−	0.881936i	$-0.656240\pi$
$997$	−29678.0	−0.942740	−0.471370	+	0.881936i	$0.656240\pi$
$998$	−13512.0	−0.428572
$999$	1350.00	0.0427549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.4.a.bc.1.1		1
7.6	odd	2		210.4.a.g.1.1	✓	1
21.20	even	2		630.4.a.e.1.1		1
28.27	even	2		1680.4.a.q.1.1		1
35.13	even	4		1050.4.g.c.799.1		2
35.27	even	4		1050.4.g.c.799.2		2
35.34	odd	2		1050.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.a.g.1.1	✓	1	7.6	odd	2
630.4.a.e.1.1		1	21.20	even	2
1050.4.a.k.1.1		1	35.34	odd	2
1050.4.g.c.799.1		2	35.13	even	4
1050.4.g.c.799.2		2	35.27	even	4
1470.4.a.bc.1.1		1	1.1	even	1	trivial
1680.4.a.q.1.1		1	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1470.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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