LMFDB - Embedded newform 110.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [110,4,Mod(1,110)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(110, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("110.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$110 = 2 \cdot 5 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	110.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:	$6.49021010063$
Analytic rank:	$0$
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	$\chi$	$=$	110.1

$q$ -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} +7.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +14.0000 q^{6} +11.0000 q^{7} +8.00000 q^{8} +22.0000 q^{9} -10.0000 q^{10} +11.0000 q^{11} +28.0000 q^{12} +2.00000 q^{13} +22.0000 q^{14} -35.0000 q^{15} +16.0000 q^{16} -9.00000 q^{17} +44.0000 q^{18} -85.0000 q^{19} -20.0000 q^{20} +77.0000 q^{21} +22.0000 q^{22} -138.000 q^{23} +56.0000 q^{24} +25.0000 q^{25} +4.00000 q^{26} -35.0000 q^{27} +44.0000 q^{28} +45.0000 q^{29} -70.0000 q^{30} +227.000 q^{31} +32.0000 q^{32} +77.0000 q^{33} -18.0000 q^{34} -55.0000 q^{35} +88.0000 q^{36} -19.0000 q^{37} -170.000 q^{38} +14.0000 q^{39} -40.0000 q^{40} -138.000 q^{41} +154.000 q^{42} -88.0000 q^{43} +44.0000 q^{44} -110.000 q^{45} -276.000 q^{46} -534.000 q^{47} +112.000 q^{48} -222.000 q^{49} +50.0000 q^{50} -63.0000 q^{51} +8.00000 q^{52} +297.000 q^{53} -70.0000 q^{54} -55.0000 q^{55} +88.0000 q^{56} -595.000 q^{57} +90.0000 q^{58} -450.000 q^{59} -140.000 q^{60} +287.000 q^{61} +454.000 q^{62} +242.000 q^{63} +64.0000 q^{64} -10.0000 q^{65} +154.000 q^{66} -304.000 q^{67} -36.0000 q^{68} -966.000 q^{69} -110.000 q^{70} +777.000 q^{71} +176.000 q^{72} +962.000 q^{73} -38.0000 q^{74} +175.000 q^{75} -340.000 q^{76} +121.000 q^{77} +28.0000 q^{78} +290.000 q^{79} -80.0000 q^{80} -839.000 q^{81} -276.000 q^{82} +1422.00 q^{83} +308.000 q^{84} +45.0000 q^{85} -176.000 q^{86} +315.000 q^{87} +88.0000 q^{88} -1455.00 q^{89} -220.000 q^{90} +22.0000 q^{91} -552.000 q^{92} +1589.00 q^{93} -1068.00 q^{94} +425.000 q^{95} +224.000 q^{96} +116.000 q^{97} -444.000 q^{98} +242.000 q^{99} +O(q^{100})

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
$3$	7.00000	1.34715	0.673575	−	0.739119i	$-0.264758\pi$
$3$	7.00000	1.34715	0.673575	+	0.739119i	$0.264758\pi$
$4$	4.00000	0.500000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	14.0000	0.952579
$7$	11.0000	0.593944	0.296972	−	0.954886i	$-0.404023\pi$
$7$	11.0000	0.593944	0.296972	+	0.954886i	$0.404023\pi$
$8$	8.00000	0.353553
$9$	22.0000	0.814815
$10$	−10.0000	−0.316228
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	28.0000	0.673575
$13$	2.00000	0.0426692	0.0213346	−	0.999772i	$-0.493208\pi$
$13$	2.00000	0.0426692	0.0213346	+	0.999772i	$0.493208\pi$
$14$	22.0000	0.419982
$15$	−35.0000	−0.602464
$16$	16.0000	0.250000
$17$	−9.00000	−0.128401	−0.0642006	−	0.997937i	$-0.520450\pi$
$17$	−9.00000	−0.128401	−0.0642006	+	0.997937i	$0.520450\pi$
$18$	44.0000	0.576161
$19$	−85.0000	−1.02633	−0.513167	−	0.858289i	$-0.671528\pi$
$19$	−85.0000	−1.02633	−0.513167	+	0.858289i	$0.671528\pi$
$20$	−20.0000	−0.223607
$21$	77.0000	0.800132
$22$	22.0000	0.213201
$23$	−138.000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−138.000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	56.0000	0.476290
$25$	25.0000	0.200000
$26$	4.00000	0.0301717
$27$	−35.0000	−0.249472
$28$	44.0000	0.296972
$29$	45.0000	0.288148	0.144074	−	0.989567i	$-0.453980\pi$
$29$	45.0000	0.288148	0.144074	+	0.989567i	$0.453980\pi$
$30$	−70.0000	−0.426006
$31$	227.000	1.31517	0.657587	−	0.753378i	$-0.271577\pi$
$31$	227.000	1.31517	0.657587	+	0.753378i	$0.271577\pi$
$32$	32.0000	0.176777
$33$	77.0000	0.406181
$34$	−18.0000	−0.0907934
$35$	−55.0000	−0.265620
$36$	88.0000	0.407407
$37$	−19.0000	−0.0844211	−0.0422106	−	0.999109i	$-0.513440\pi$
$37$	−19.0000	−0.0844211	−0.0422106	+	0.999109i	$0.513440\pi$
$38$	−170.000	−0.725727
$39$	14.0000	0.0574819
$40$	−40.0000	−0.158114
$41$	−138.000	−0.525658	−0.262829	−	0.964842i	$-0.584656\pi$
$41$	−138.000	−0.525658	−0.262829	+	0.964842i	$0.584656\pi$
$42$	154.000	0.565779
$43$	−88.0000	−0.312090	−0.156045	−	0.987750i	$-0.549875\pi$
$43$	−88.0000	−0.312090	−0.156045	+	0.987750i	$0.549875\pi$
$44$	44.0000	0.150756
$45$	−110.000	−0.364396
$46$	−276.000	−0.884652
$47$	−534.000	−1.65727	−0.828637	−	0.559786i	$-0.810883\pi$
$47$	−534.000	−1.65727	−0.828637	+	0.559786i	$0.810883\pi$
$48$	112.000	0.336788
$49$	−222.000	−0.647230
$50$	50.0000	0.141421
$51$	−63.0000	−0.172976
$52$	8.00000	0.0213346
$53$	297.000	0.769737	0.384869	−	0.922971i	$-0.374247\pi$
$53$	297.000	0.769737	0.384869	+	0.922971i	$0.374247\pi$
$54$	−70.0000	−0.176404
$55$	−55.0000	−0.134840
$56$	88.0000	0.209991
$57$	−595.000	−1.38263
$58$	90.0000	0.203751
$59$	−450.000	−0.992966	−0.496483	−	0.868046i	$-0.665376\pi$
$59$	−450.000	−0.992966	−0.496483	+	0.868046i	$0.665376\pi$
$60$	−140.000	−0.301232
$61$	287.000	0.602403	0.301202	−	0.953561i	$-0.402612\pi$
$61$	287.000	0.602403	0.301202	+	0.953561i	$0.402612\pi$
$62$	454.000	0.929969
$63$	242.000	0.483955
$64$	64.0000	0.125000
$65$	−10.0000	−0.0190823
$66$	154.000	0.287213
$67$	−304.000	−0.554321	−0.277161	−	0.960824i	$-0.589393\pi$
$67$	−304.000	−0.554321	−0.277161	+	0.960824i	$0.589393\pi$
$68$	−36.0000	−0.0642006
$69$	−966.000	−1.68540
$70$	−110.000	−0.187822
$71$	777.000	1.29877	0.649387	−	0.760458i	$-0.275026\pi$
$71$	777.000	1.29877	0.649387	+	0.760458i	$0.275026\pi$
$72$	176.000	0.288081
$73$	962.000	1.54238	0.771189	−	0.636606i	$-0.219662\pi$
$73$	962.000	1.54238	0.771189	+	0.636606i	$0.219662\pi$
$74$	−38.0000	−0.0596947
$75$	175.000	0.269430
$76$	−340.000	−0.513167
$77$	121.000	0.179081
$78$	28.0000	0.0406458
$79$	290.000	0.413007	0.206503	−	0.978446i	$-0.433792\pi$
$79$	290.000	0.413007	0.206503	+	0.978446i	$0.433792\pi$
$80$	−80.0000	−0.111803
$81$	−839.000	−1.15089
$82$	−276.000	−0.371696
$83$	1422.00	1.88054	0.940270	−	0.340430i	$-0.110573\pi$
$83$	1422.00	1.88054	0.940270	+	0.340430i	$0.110573\pi$
$84$	308.000	0.400066
$85$	45.0000	0.0574228
$86$	−176.000	−0.220681
$87$	315.000	0.388179
$88$	88.0000	0.106600
$89$	−1455.00	−1.73292	−0.866459	−	0.499248i	$-0.833610\pi$
$89$	−1455.00	−1.73292	−0.866459	+	0.499248i	$0.833610\pi$
$90$	−220.000	−0.257667
$91$	22.0000	0.0253431
$92$	−552.000	−0.625543
$93$	1589.00	1.77174
$94$	−1068.00	−1.17187
$95$	425.000	0.458990
$96$	224.000	0.238145
$97$	116.000	0.121423	0.0607114	−	0.998155i	$-0.480663\pi$
$97$	116.000	0.121423	0.0607114	+	0.998155i	$0.480663\pi$
$98$	−444.000	−0.457661
$99$	242.000	0.245676
$100$	100.000	0.100000
$101$	402.000	0.396045	0.198022	−	0.980198i	$-0.436548\pi$
$101$	402.000	0.396045	0.198022	+	0.980198i	$0.436548\pi$
$102$	−126.000	−0.122312
$103$	1412.00	1.35076	0.675381	−	0.737469i	$-0.263979\pi$
$103$	1412.00	1.35076	0.675381	+	0.737469i	$0.263979\pi$
$104$	16.0000	0.0150859
$105$	−385.000	−0.357830
$106$	594.000	0.544287
$107$	2136.00	1.92986	0.964930	−	0.262509i	$-0.0845500\pi$
$107$	2136.00	1.92986	0.964930	+	0.262509i	$0.0845500\pi$
$108$	−140.000	−0.124736
$109$	230.000	0.202110	0.101055	−	0.994881i	$-0.467778\pi$
$109$	230.000	0.202110	0.101055	+	0.994881i	$0.467778\pi$
$110$	−110.000	−0.0953463
$111$	−133.000	−0.113728
$112$	176.000	0.148486
$113$	−108.000	−0.0899096	−0.0449548	−	0.998989i	$-0.514314\pi$
$113$	−108.000	−0.0899096	−0.0449548	+	0.998989i	$0.514314\pi$
$114$	−1190.00	−0.977664
$115$	690.000	0.559503
$116$	180.000	0.144074
$117$	44.0000	0.0347675
$118$	−900.000	−0.702133
$119$	−99.0000	−0.0762632
$120$	−280.000	−0.213003
$121$	121.000	0.0909091
$122$	574.000	0.425963
$123$	−966.000	−0.708141
$124$	908.000	0.657587
$125$	−125.000	−0.0894427
$126$	484.000	0.342208
$127$	1376.00	0.961419	0.480710	−	0.876880i	$-0.340379\pi$
$127$	1376.00	0.961419	0.480710	+	0.876880i	$0.340379\pi$
$128$	128.000	0.0883883
$129$	−616.000	−0.420432
$130$	−20.0000	−0.0134932
$131$	−1593.00	−1.06245	−0.531225	−	0.847231i	$-0.678268\pi$
$131$	−1593.00	−1.06245	−0.531225	+	0.847231i	$0.678268\pi$
$132$	308.000	0.203091
$133$	−935.000	−0.609585
$134$	−608.000	−0.391964
$135$	175.000	0.111567
$136$	−72.0000	−0.0453967
$137$	756.000	0.471456	0.235728	−	0.971819i	$-0.424253\pi$
$137$	756.000	0.471456	0.235728	+	0.971819i	$0.424253\pi$
$138$	−1932.00	−1.19176
$139$	1460.00	0.890903	0.445452	−	0.895306i	$-0.353043\pi$
$139$	1460.00	0.890903	0.445452	+	0.895306i	$0.353043\pi$
$140$	−220.000	−0.132810
$141$	−3738.00	−2.23260
$142$	1554.00	0.918372
$143$	22.0000	0.0128653
$144$	352.000	0.203704
$145$	−225.000	−0.128864
$146$	1924.00	1.09063
$147$	−1554.00	−0.871917
$148$	−76.0000	−0.0422106
$149$	−2145.00	−1.17936	−0.589682	−	0.807635i	$-0.700747\pi$
$149$	−2145.00	−1.17936	−0.589682	+	0.807635i	$0.700747\pi$
$150$	350.000	0.190516
$151$	2702.00	1.45620	0.728098	−	0.685473i	$-0.240404\pi$
$151$	2702.00	1.45620	0.728098	+	0.685473i	$0.240404\pi$
$152$	−680.000	−0.362864
$153$	−198.000	−0.104623
$154$	242.000	0.126629
$155$	−1135.00	−0.588164
$156$	56.0000	0.0287410
$157$	1421.00	0.722345	0.361172	−	0.932499i	$-0.382377\pi$
$157$	1421.00	0.722345	0.361172	+	0.932499i	$0.382377\pi$
$158$	580.000	0.292040
$159$	2079.00	1.03695
$160$	−160.000	−0.0790569
$161$	−1518.00	−0.743076
$162$	−1678.00	−0.813803
$163$	−2773.00	−1.33250	−0.666252	−	0.745727i	$-0.732102\pi$
$163$	−2773.00	−1.33250	−0.666252	+	0.745727i	$0.732102\pi$
$164$	−552.000	−0.262829
$165$	−385.000	−0.181650
$166$	2844.00	1.32974
$167$	−2109.00	−0.977241	−0.488621	−	0.872496i	$-0.662500\pi$
$167$	−2109.00	−0.977241	−0.488621	+	0.872496i	$0.662500\pi$
$168$	616.000	0.282889
$169$	−2193.00	−0.998179
$170$	90.0000	0.0406040
$171$	−1870.00	−0.836272
$172$	−352.000	−0.156045
$173$	−3558.00	−1.56364	−0.781820	−	0.623504i	$-0.785708\pi$
$173$	−3558.00	−1.56364	−0.781820	+	0.623504i	$0.785708\pi$
$174$	630.000	0.274484
$175$	275.000	0.118789
$176$	176.000	0.0753778
$177$	−3150.00	−1.33768
$178$	−2910.00	−1.22536
$179$	300.000	0.125268	0.0626342	−	0.998037i	$-0.480050\pi$
$179$	300.000	0.125268	0.0626342	+	0.998037i	$0.480050\pi$
$180$	−440.000	−0.182198
$181$	3122.00	1.28208	0.641040	−	0.767508i	$-0.278503\pi$
$181$	3122.00	1.28208	0.641040	+	0.767508i	$0.278503\pi$
$182$	44.0000	0.0179203
$183$	2009.00	0.811528
$184$	−1104.00	−0.442326
$185$	95.0000	0.0377543
$186$	3178.00	1.25281
$187$	−99.0000	−0.0387144
$188$	−2136.00	−0.828637
$189$	−385.000	−0.148173
$190$	850.000	0.324555
$191$	−1548.00	−0.586436	−0.293218	−	0.956046i	$-0.594726\pi$
$191$	−1548.00	−0.586436	−0.293218	+	0.956046i	$0.594726\pi$
$192$	448.000	0.168394
$193$	827.000	0.308439	0.154220	−	0.988037i	$-0.450714\pi$
$193$	827.000	0.308439	0.154220	+	0.988037i	$0.450714\pi$
$194$	232.000	0.0858589
$195$	−70.0000	−0.0257067
$196$	−888.000	−0.323615
$197$	456.000	0.164917	0.0824585	−	0.996594i	$-0.473723\pi$
$197$	456.000	0.164917	0.0824585	+	0.996594i	$0.473723\pi$
$198$	484.000	0.173719
$199$	2825.00	1.00633	0.503163	−	0.864191i	$-0.332169\pi$
$199$	2825.00	1.00633	0.503163	+	0.864191i	$0.332169\pi$
$200$	200.000	0.0707107
$201$	−2128.00	−0.746754
$202$	804.000	0.280046
$203$	495.000	0.171144
$204$	−252.000	−0.0864879
$205$	690.000	0.235081
$206$	2824.00	0.955133
$207$	−3036.00	−1.01940
$208$	32.0000	0.0106673
$209$	−935.000	−0.309451
$210$	−770.000	−0.253024
$211$	−2233.00	−0.728560	−0.364280	−	0.931290i	$-0.618685\pi$
$211$	−2233.00	−0.728560	−0.364280	+	0.931290i	$0.618685\pi$
$212$	1188.00	0.384869
$213$	5439.00	1.74964
$214$	4272.00	1.36462
$215$	440.000	0.139571
$216$	−280.000	−0.0882018
$217$	2497.00	0.781140
$218$	460.000	0.142913
$219$	6734.00	2.07782
$220$	−220.000	−0.0674200
$221$	−18.0000	−0.00547878
$222$	−266.000	−0.0804178
$223$	5402.00	1.62217	0.811087	−	0.584926i	$-0.198876\pi$
$223$	5402.00	1.62217	0.811087	+	0.584926i	$0.198876\pi$
$224$	352.000	0.104995
$225$	550.000	0.162963
$226$	−216.000	−0.0635757
$227$	2826.00	0.826292	0.413146	−	0.910665i	$-0.364430\pi$
$227$	2826.00	0.826292	0.413146	+	0.910665i	$0.364430\pi$
$228$	−2380.00	−0.691313
$229$	−2410.00	−0.695447	−0.347723	−	0.937597i	$-0.613045\pi$
$229$	−2410.00	−0.695447	−0.347723	+	0.937597i	$0.613045\pi$
$230$	1380.00	0.395628
$231$	847.000	0.241249
$232$	360.000	0.101876
$233$	687.000	0.193163	0.0965813	−	0.995325i	$-0.469209\pi$
$233$	687.000	0.193163	0.0965813	+	0.995325i	$0.469209\pi$
$234$	88.0000	0.0245844
$235$	2670.00	0.741156
$236$	−1800.00	−0.496483
$237$	2030.00	0.556383
$238$	−198.000	−0.0539262
$239$	1650.00	0.446567	0.223284	−	0.974753i	$-0.428322\pi$
$239$	1650.00	0.446567	0.223284	+	0.974753i	$0.428322\pi$
$240$	−560.000	−0.150616
$241$	2882.00	0.770315	0.385158	−	0.922851i	$-0.374147\pi$
$241$	2882.00	0.770315	0.385158	+	0.922851i	$0.374147\pi$
$242$	242.000	0.0642824
$243$	−4928.00	−1.30095
$244$	1148.00	0.301202
$245$	1110.00	0.289450
$246$	−1932.00	−0.500731
$247$	−170.000	−0.0437929
$248$	1816.00	0.464984
$249$	9954.00	2.53337
$250$	−250.000	−0.0632456
$251$	−7698.00	−1.93583	−0.967915	−	0.251277i	$-0.919150\pi$
$251$	−7698.00	−1.93583	−0.967915	+	0.251277i	$0.919150\pi$
$252$	968.000	0.241977
$253$	−1518.00	−0.377217
$254$	2752.00	0.679826
$255$	315.000	0.0773571
$256$	256.000	0.0625000
$257$	−6354.00	−1.54222	−0.771112	−	0.636699i	$-0.780299\pi$
$257$	−6354.00	−1.54222	−0.771112	+	0.636699i	$0.780299\pi$
$258$	−1232.00	−0.297291
$259$	−209.000	−0.0501414
$260$	−40.0000	−0.00954113
$261$	990.000	0.234787
$262$	−3186.00	−0.751266
$263$	−5223.00	−1.22458	−0.612289	−	0.790634i	$-0.709751\pi$
$263$	−5223.00	−1.22458	−0.612289	+	0.790634i	$0.709751\pi$
$264$	616.000	0.143607
$265$	−1485.00	−0.344237
$266$	−1870.00	−0.431042
$267$	−10185.0	−2.33450
$268$	−1216.00	−0.277161
$269$	−4500.00	−1.01996	−0.509981	−	0.860186i	$-0.670348\pi$
$269$	−4500.00	−1.01996	−0.509981	+	0.860186i	$0.670348\pi$
$270$	350.000	0.0788901
$271$	−1708.00	−0.382855	−0.191427	−	0.981507i	$-0.561312\pi$
$271$	−1708.00	−0.382855	−0.191427	+	0.981507i	$0.561312\pi$
$272$	−144.000	−0.0321003
$273$	154.000	0.0341410
$274$	1512.00	0.333370
$275$	275.000	0.0603023
$276$	−3864.00	−0.842701
$277$	5636.00	1.22251	0.611253	−	0.791435i	$-0.290666\pi$
$277$	5636.00	1.22251	0.611253	+	0.791435i	$0.290666\pi$
$278$	2920.00	0.629964
$279$	4994.00	1.07162
$280$	−440.000	−0.0939108
$281$	−5058.00	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−5058.00	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−7476.00	−1.57869
$283$	−4858.00	−1.02042	−0.510209	−	0.860051i	$-0.670432\pi$
$283$	−4858.00	−1.02042	−0.510209	+	0.860051i	$0.670432\pi$
$284$	3108.00	0.649387
$285$	2975.00	0.618329
$286$	44.0000	0.00909711
$287$	−1518.00	−0.312212
$288$	704.000	0.144040
$289$	−4832.00	−0.983513
$290$	−450.000	−0.0911204
$291$	812.000	0.163575
$292$	3848.00	0.771189
$293$	9402.00	1.87464	0.937322	−	0.348464i	$-0.113297\pi$
$293$	9402.00	1.87464	0.937322	+	0.348464i	$0.113297\pi$
$294$	−3108.00	−0.616538
$295$	2250.00	0.444068
$296$	−152.000	−0.0298474
$297$	−385.000	−0.0752187
$298$	−4290.00	−0.833936
$299$	−276.000	−0.0533829
$300$	700.000	0.134715
$301$	−968.000	−0.185364
$302$	5404.00	1.02969
$303$	2814.00	0.533532
$304$	−1360.00	−0.256583
$305$	−1435.00	−0.269403
$306$	−396.000	−0.0739798
$307$	−6574.00	−1.22214	−0.611072	−	0.791575i	$-0.709261\pi$
$307$	−6574.00	−1.22214	−0.611072	+	0.791575i	$0.709261\pi$
$308$	484.000	0.0895405
$309$	9884.00	1.81968
$310$	−2270.00	−0.415895
$311$	−8253.00	−1.50477	−0.752387	−	0.658721i	$-0.771098\pi$
$311$	−8253.00	−1.50477	−0.752387	+	0.658721i	$0.771098\pi$
$312$	112.000	0.0203229
$313$	−4858.00	−0.877286	−0.438643	−	0.898661i	$-0.644541\pi$
$313$	−4858.00	−0.877286	−0.438643	+	0.898661i	$0.644541\pi$
$314$	2842.00	0.510775
$315$	−1210.00	−0.216431
$316$	1160.00	0.206503
$317$	711.000	0.125974	0.0629870	−	0.998014i	$-0.479937\pi$
$317$	711.000	0.125974	0.0629870	+	0.998014i	$0.479937\pi$
$318$	4158.00	0.733236
$319$	495.000	0.0868799
$320$	−320.000	−0.0559017
$321$	14952.0	2.59981
$322$	−3036.00	−0.525434
$323$	765.000	0.131782
$324$	−3356.00	−0.575446
$325$	50.0000	0.00853385
$326$	−5546.00	−0.942222
$327$	1610.00	0.272273
$328$	−1104.00	−0.185848
$329$	−5874.00	−0.984329
$330$	−770.000	−0.128446
$331$	8552.00	1.42012	0.710061	−	0.704140i	$-0.248667\pi$
$331$	8552.00	1.42012	0.710061	+	0.704140i	$0.248667\pi$
$332$	5688.00	0.940270
$333$	−418.000	−0.0687876
$334$	−4218.00	−0.691014
$335$	1520.00	0.247900
$336$	1232.00	0.200033
$337$	1121.00	0.181201	0.0906005	−	0.995887i	$-0.471121\pi$
$337$	1121.00	0.181201	0.0906005	+	0.995887i	$0.471121\pi$
$338$	−4386.00	−0.705819
$339$	−756.000	−0.121122
$340$	180.000	0.0287114
$341$	2497.00	0.396540
$342$	−3740.00	−0.591333
$343$	−6215.00	−0.978363
$344$	−704.000	−0.110341
$345$	4830.00	0.753735
$346$	−7116.00	−1.10566
$347$	10986.0	1.69959	0.849797	−	0.527109i	$-0.176724\pi$
$347$	10986.0	1.69959	0.849797	+	0.527109i	$0.176724\pi$
$348$	1260.00	0.194089
$349$	−1090.00	−0.167182	−0.0835908	−	0.996500i	$-0.526639\pi$
$349$	−1090.00	−0.167182	−0.0835908	+	0.996500i	$0.526639\pi$
$350$	550.000	0.0839964
$351$	−70.0000	−0.0106448
$352$	352.000	0.0533002
$353$	462.000	0.0696594	0.0348297	−	0.999393i	$-0.488911\pi$
$353$	462.000	0.0696594	0.0348297	+	0.999393i	$0.488911\pi$
$354$	−6300.00	−0.945879
$355$	−3885.00	−0.580829
$356$	−5820.00	−0.866459
$357$	−693.000	−0.102738
$358$	600.000	0.0885782
$359$	−60.0000	−0.00882083	−0.00441042	−	0.999990i	$-0.501404\pi$
$359$	−60.0000	−0.00882083	−0.00441042	+	0.999990i	$0.501404\pi$
$360$	−880.000	−0.128834
$361$	366.000	0.0533605
$362$	6244.00	0.906567
$363$	847.000	0.122468
$364$	88.0000	0.0126716
$365$	−4810.00	−0.689772
$366$	4018.00	0.573837
$367$	−1924.00	−0.273657	−0.136828	−	0.990595i	$-0.543691\pi$
$367$	−1924.00	−0.273657	−0.136828	+	0.990595i	$0.543691\pi$
$368$	−2208.00	−0.312772
$369$	−3036.00	−0.428314
$370$	190.000	0.0266963
$371$	3267.00	0.457181
$372$	6356.00	0.885869
$373$	−6238.00	−0.865929	−0.432964	−	0.901411i	$-0.642532\pi$
$373$	−6238.00	−0.865929	−0.432964	+	0.901411i	$0.642532\pi$
$374$	−198.000	−0.0273752
$375$	−875.000	−0.120493
$376$	−4272.00	−0.585935
$377$	90.0000	0.0122951
$378$	−770.000	−0.104774
$379$	830.000	0.112491	0.0562457	−	0.998417i	$-0.482087\pi$
$379$	830.000	0.112491	0.0562457	+	0.998417i	$0.482087\pi$
$380$	1700.00	0.229495
$381$	9632.00	1.29518
$382$	−3096.00	−0.414673
$383$	3762.00	0.501904	0.250952	−	0.968000i	$-0.419256\pi$
$383$	3762.00	0.501904	0.250952	+	0.968000i	$0.419256\pi$
$384$	896.000	0.119072
$385$	−605.000	−0.0800874
$386$	1654.00	0.218099
$387$	−1936.00	−0.254296
$388$	464.000	0.0607114
$389$	−5550.00	−0.723383	−0.361692	−	0.932298i	$-0.617801\pi$
$389$	−5550.00	−0.723383	−0.361692	+	0.932298i	$0.617801\pi$
$390$	−140.000	−0.0181774
$391$	1242.00	0.160641
$392$	−1776.00	−0.228830
$393$	−11151.0	−1.43128
$394$	912.000	0.116614
$395$	−1450.00	−0.184702
$396$	968.000	0.122838
$397$	8786.00	1.11072	0.555361	−	0.831609i	$-0.312580\pi$
$397$	8786.00	1.11072	0.555361	+	0.831609i	$0.312580\pi$
$398$	5650.00	0.711580
$399$	−6545.00	−0.821203
$400$	400.000	0.0500000
$401$	−3333.00	−0.415068	−0.207534	−	0.978228i	$-0.566544\pi$
$401$	−3333.00	−0.415068	−0.207534	+	0.978228i	$0.566544\pi$
$402$	−4256.00	−0.528035
$403$	454.000	0.0561175
$404$	1608.00	0.198022
$405$	4195.00	0.514694
$406$	990.000	0.121017
$407$	−209.000	−0.0254539
$408$	−504.000	−0.0611562
$409$	−12820.0	−1.54990	−0.774949	−	0.632024i	$-0.782224\pi$
$409$	−12820.0	−1.54990	−0.774949	+	0.632024i	$0.782224\pi$
$410$	1380.00	0.166228
$411$	5292.00	0.635122
$412$	5648.00	0.675381
$413$	−4950.00	−0.589767
$414$	−6072.00	−0.720827
$415$	−7110.00	−0.841003
$416$	64.0000	0.00754293
$417$	10220.0	1.20018
$418$	−1870.00	−0.218815
$419$	13680.0	1.59502	0.797508	−	0.603308i	$-0.206151\pi$
$419$	13680.0	1.59502	0.797508	+	0.603308i	$0.206151\pi$
$420$	−1540.00	−0.178915
$421$	4952.00	0.573268	0.286634	−	0.958040i	$-0.407464\pi$
$421$	4952.00	0.573268	0.286634	+	0.958040i	$0.407464\pi$
$422$	−4466.00	−0.515169
$423$	−11748.0	−1.35037
$424$	2376.00	0.272143
$425$	−225.000	−0.0256802
$426$	10878.0	1.23719
$427$	3157.00	0.357794
$428$	8544.00	0.964930
$429$	154.000	0.0173314
$430$	880.000	0.0986916
$431$	16032.0	1.79173	0.895863	−	0.444330i	$-0.146558\pi$
$431$	16032.0	1.79173	0.895863	+	0.444330i	$0.146558\pi$
$432$	−560.000	−0.0623681
$433$	692.000	0.0768023	0.0384012	−	0.999262i	$-0.487774\pi$
$433$	692.000	0.0768023	0.0384012	+	0.999262i	$0.487774\pi$
$434$	4994.00	0.552349
$435$	−1575.00	−0.173599
$436$	920.000	0.101055
$437$	11730.0	1.28403
$438$	13468.0	1.46924
$439$	−17920.0	−1.94823	−0.974117	−	0.226043i	$-0.927421\pi$
$439$	−17920.0	−1.94823	−0.974117	+	0.226043i	$0.927421\pi$
$440$	−440.000	−0.0476731
$441$	−4884.00	−0.527373
$442$	−36.0000	−0.00387408
$443$	15852.0	1.70012	0.850058	−	0.526689i	$-0.176567\pi$
$443$	15852.0	1.70012	0.850058	+	0.526689i	$0.176567\pi$
$444$	−532.000	−0.0568640
$445$	7275.00	0.774984
$446$	10804.0	1.14705
$447$	−15015.0	−1.58878
$448$	704.000	0.0742430
$449$	6930.00	0.728390	0.364195	−	0.931323i	$-0.381344\pi$
$449$	6930.00	0.728390	0.364195	+	0.931323i	$0.381344\pi$
$450$	1100.00	0.115232
$451$	−1518.00	−0.158492
$452$	−432.000	−0.0449548
$453$	18914.0	1.96172
$454$	5652.00	0.584276
$455$	−110.000	−0.0113338
$456$	−4760.00	−0.488832
$457$	−619.000	−0.0633602	−0.0316801	−	0.999498i	$-0.510086\pi$
$457$	−619.000	−0.0633602	−0.0316801	+	0.999498i	$0.510086\pi$
$458$	−4820.00	−0.491755
$459$	315.000	0.0320326
$460$	2760.00	0.279751
$461$	−10533.0	−1.06414	−0.532072	−	0.846699i	$-0.678587\pi$
$461$	−10533.0	−1.06414	−0.532072	+	0.846699i	$0.678587\pi$
$462$	1694.00	0.170589
$463$	−11698.0	−1.17419	−0.587097	−	0.809516i	$-0.699729\pi$
$463$	−11698.0	−1.17419	−0.587097	+	0.809516i	$0.699729\pi$
$464$	720.000	0.0720370
$465$	−7945.00	−0.792345
$466$	1374.00	0.136587
$467$	1431.00	0.141796	0.0708981	−	0.997484i	$-0.477413\pi$
$467$	1431.00	0.141796	0.0708981	+	0.997484i	$0.477413\pi$
$468$	176.000	0.0173838
$469$	−3344.00	−0.329236
$470$	5340.00	0.524076
$471$	9947.00	0.973107
$472$	−3600.00	−0.351067
$473$	−968.000	−0.0940987
$474$	4060.00	0.393422
$475$	−2125.00	−0.205267
$476$	−396.000	−0.0381316
$477$	6534.00	0.627194
$478$	3300.00	0.315771
$479$	−4740.00	−0.452142	−0.226071	−	0.974111i	$-0.572588\pi$
$479$	−4740.00	−0.452142	−0.226071	+	0.974111i	$0.572588\pi$
$480$	−1120.00	−0.106502
$481$	−38.0000	−0.00360218
$482$	5764.00	0.544695
$483$	−10626.0	−1.00103
$484$	484.000	0.0454545
$485$	−580.000	−0.0543019
$486$	−9856.00	−0.919912
$487$	−2104.00	−0.195773	−0.0978864	−	0.995198i	$-0.531208\pi$
$487$	−2104.00	−0.195773	−0.0978864	+	0.995198i	$0.531208\pi$
$488$	2296.00	0.212982
$489$	−19411.0	−1.79508
$490$	2220.00	0.204672
$491$	−12933.0	−1.18871	−0.594357	−	0.804202i	$-0.702593\pi$
$491$	−12933.0	−1.18871	−0.594357	+	0.804202i	$0.702593\pi$
$492$	−3864.00	−0.354070
$493$	−405.000	−0.0369985
$494$	−340.000	−0.0309662
$495$	−1210.00	−0.109870
$496$	3632.00	0.328794
$497$	8547.00	0.771399
$498$	19908.0	1.79136
$499$	17540.0	1.57354	0.786772	−	0.617244i	$-0.211751\pi$
$499$	17540.0	1.57354	0.786772	+	0.617244i	$0.211751\pi$
$500$	−500.000	−0.0447214
$501$	−14763.0	−1.31649
$502$	−15396.0	−1.36884
$503$	7632.00	0.676529	0.338264	−	0.941051i	$-0.390160\pi$
$503$	7632.00	0.676529	0.338264	+	0.941051i	$0.390160\pi$
$504$	1936.00	0.171104
$505$	−2010.00	−0.177116
$506$	−3036.00	−0.266733
$507$	−15351.0	−1.34470
$508$	5504.00	0.480710
$509$	2340.00	0.203770	0.101885	−	0.994796i	$-0.467513\pi$
$509$	2340.00	0.203770	0.101885	+	0.994796i	$0.467513\pi$
$510$	630.000	0.0546997
$511$	10582.0	0.916086
$512$	512.000	0.0441942
$513$	2975.00	0.256042
$514$	−12708.0	−1.09052
$515$	−7060.00	−0.604079
$516$	−2464.00	−0.210216
$517$	−5874.00	−0.499687
$518$	−418.000	−0.0354553
$519$	−24906.0	−2.10646
$520$	−80.0000	−0.00674660
$521$	−16338.0	−1.37386	−0.686930	−	0.726724i	$-0.741042\pi$
$521$	−16338.0	−1.37386	−0.686930	+	0.726724i	$0.741042\pi$
$522$	1980.00	0.166020
$523$	7712.00	0.644784	0.322392	−	0.946606i	$-0.395513\pi$
$523$	7712.00	0.644784	0.322392	+	0.946606i	$0.395513\pi$
$524$	−6372.00	−0.531225
$525$	1925.00	0.160026
$526$	−10446.0	−0.865907
$527$	−2043.00	−0.168870
$528$	1232.00	0.101545
$529$	6877.00	0.565217
$530$	−2970.00	−0.243412
$531$	−9900.00	−0.809084
$532$	−3740.00	−0.304792
$533$	−276.000	−0.0224294
$534$	−20370.0	−1.65074
$535$	−10680.0	−0.863059
$536$	−2432.00	−0.195982
$537$	2100.00	0.168755
$538$	−9000.00	−0.721222
$539$	−2442.00	−0.195147
$540$	700.000	0.0557837
$541$	14537.0	1.15526	0.577629	−	0.816300i	$-0.303978\pi$
$541$	14537.0	1.15526	0.577629	+	0.816300i	$0.303978\pi$
$542$	−3416.00	−0.270719
$543$	21854.0	1.72715
$544$	−288.000	−0.0226983
$545$	−1150.00	−0.0903864
$546$	308.000	0.0241414
$547$	14096.0	1.10183	0.550915	−	0.834561i	$-0.314279\pi$
$547$	14096.0	1.10183	0.550915	+	0.834561i	$0.314279\pi$
$548$	3024.00	0.235728
$549$	6314.00	0.490847
$550$	550.000	0.0426401
$551$	−3825.00	−0.295736
$552$	−7728.00	−0.595880
$553$	3190.00	0.245303
$554$	11272.0	0.864443
$555$	665.000	0.0508607
$556$	5840.00	0.445452
$557$	2706.00	0.205847	0.102924	−	0.994689i	$-0.467180\pi$
$557$	2706.00	0.205847	0.102924	+	0.994689i	$0.467180\pi$
$558$	9988.00	0.757752
$559$	−176.000	−0.0133166
$560$	−880.000	−0.0664050
$561$	−693.000	−0.0521542
$562$	−10116.0	−0.759284
$563$	20502.0	1.53474	0.767368	−	0.641207i	$-0.221566\pi$
$563$	20502.0	1.53474	0.767368	+	0.641207i	$0.221566\pi$
$564$	−14952.0	−1.11630
$565$	540.000	0.0402088
$566$	−9716.00	−0.721544
$567$	−9229.00	−0.683565
$568$	6216.00	0.459186
$569$	12960.0	0.954853	0.477427	−	0.878672i	$-0.341570\pi$
$569$	12960.0	0.954853	0.477427	+	0.878672i	$0.341570\pi$
$570$	5950.00	0.437225
$571$	−14443.0	−1.05853	−0.529265	−	0.848457i	$-0.677532\pi$
$571$	−14443.0	−1.05853	−0.529265	+	0.848457i	$0.677532\pi$
$572$	88.0000	0.00643263
$573$	−10836.0	−0.790018
$574$	−3036.00	−0.220767
$575$	−3450.00	−0.250217
$576$	1408.00	0.101852
$577$	−10234.0	−0.738383	−0.369192	−	0.929353i	$-0.620365\pi$
$577$	−10234.0	−0.738383	−0.369192	+	0.929353i	$0.620365\pi$
$578$	−9664.00	−0.695449
$579$	5789.00	0.415514
$580$	−900.000	−0.0644318
$581$	15642.0	1.11694
$582$	1624.00	0.115665
$583$	3267.00	0.232085
$584$	7696.00	0.545313
$585$	−220.000	−0.0155485
$586$	18804.0	1.32557
$587$	9741.00	0.684930	0.342465	−	0.939531i	$-0.388738\pi$
$587$	9741.00	0.684930	0.342465	+	0.939531i	$0.388738\pi$
$588$	−6216.00	−0.435958
$589$	−19295.0	−1.34981
$590$	4500.00	0.314004
$591$	3192.00	0.222168
$592$	−304.000	−0.0211053
$593$	−16518.0	−1.14387	−0.571933	−	0.820300i	$-0.693806\pi$
$593$	−16518.0	−1.14387	−0.571933	+	0.820300i	$0.693806\pi$
$594$	−770.000	−0.0531877
$595$	495.000	0.0341059
$596$	−8580.00	−0.589682
$597$	19775.0	1.35567
$598$	−552.000	−0.0377474
$599$	13635.0	0.930068	0.465034	−	0.885293i	$-0.346042\pi$
$599$	13635.0	0.930068	0.465034	+	0.885293i	$0.346042\pi$
$600$	1400.00	0.0952579
$601$	−13438.0	−0.912059	−0.456029	−	0.889965i	$-0.650729\pi$
$601$	−13438.0	−0.912059	−0.456029	+	0.889965i	$0.650729\pi$
$602$	−1936.00	−0.131072
$603$	−6688.00	−0.451669
$604$	10808.0	0.728098
$605$	−605.000	−0.0406558
$606$	5628.00	0.377264
$607$	−15199.0	−1.01632	−0.508162	−	0.861262i	$-0.669675\pi$
$607$	−15199.0	−1.01632	−0.508162	+	0.861262i	$0.669675\pi$
$608$	−2720.00	−0.181432
$609$	3465.00	0.230556
$610$	−2870.00	−0.190497
$611$	−1068.00	−0.0707147
$612$	−792.000	−0.0523116
$613$	21122.0	1.39170	0.695848	−	0.718189i	$-0.255029\pi$
$613$	21122.0	1.39170	0.695848	+	0.718189i	$0.255029\pi$
$614$	−13148.0	−0.864186
$615$	4830.00	0.316690
$616$	968.000	0.0633147
$617$	−2304.00	−0.150333	−0.0751666	−	0.997171i	$-0.523949\pi$
$617$	−2304.00	−0.150333	−0.0751666	+	0.997171i	$0.523949\pi$
$618$	19768.0	1.28671
$619$	−14020.0	−0.910358	−0.455179	−	0.890400i	$-0.650425\pi$
$619$	−14020.0	−0.910358	−0.455179	+	0.890400i	$0.650425\pi$
$620$	−4540.00	−0.294082
$621$	4830.00	0.312111
$622$	−16506.0	−1.06404
$623$	−16005.0	−1.02926
$624$	224.000	0.0143705
$625$	625.000	0.0400000
$626$	−9716.00	−0.620335
$627$	−6545.00	−0.416877
$628$	5684.00	0.361172
$629$	171.000	0.0108398
$630$	−2420.00	−0.153040
$631$	22457.0	1.41680	0.708399	−	0.705813i	$-0.249418\pi$
$631$	22457.0	1.41680	0.708399	+	0.705813i	$0.249418\pi$
$632$	2320.00	0.146020
$633$	−15631.0	−0.981479
$634$	1422.00	0.0890770
$635$	−6880.00	−0.429960
$636$	8316.00	0.518476
$637$	−444.000	−0.0276168
$638$	990.000	0.0614333
$639$	17094.0	1.05826
$640$	−640.000	−0.0395285
$641$	8847.00	0.545141	0.272571	−	0.962136i	$-0.412126\pi$
$641$	8847.00	0.545141	0.272571	+	0.962136i	$0.412126\pi$
$642$	29904.0	1.83834
$643$	−1213.00	−0.0743951	−0.0371976	−	0.999308i	$-0.511843\pi$
$643$	−1213.00	−0.0743951	−0.0371976	+	0.999308i	$0.511843\pi$
$644$	−6072.00	−0.371538
$645$	3080.00	0.188023
$646$	1530.00	0.0931843
$647$	−9774.00	−0.593904	−0.296952	−	0.954892i	$-0.595970\pi$
$647$	−9774.00	−0.593904	−0.296952	+	0.954892i	$0.595970\pi$
$648$	−6712.00	−0.406902
$649$	−4950.00	−0.299391
$650$	100.000	0.00603434
$651$	17479.0	1.05231
$652$	−11092.0	−0.666252
$653$	−14433.0	−0.864942	−0.432471	−	0.901648i	$-0.642358\pi$
$653$	−14433.0	−0.864942	−0.432471	+	0.901648i	$0.642358\pi$
$654$	3220.00	0.192526
$655$	7965.00	0.475142
$656$	−2208.00	−0.131415
$657$	21164.0	1.25675
$658$	−11748.0	−0.696025
$659$	22905.0	1.35395	0.676974	−	0.736007i	$-0.263291\pi$
$659$	22905.0	1.35395	0.676974	+	0.736007i	$0.263291\pi$
$660$	−1540.00	−0.0908249
$661$	2942.00	0.173117	0.0865587	−	0.996247i	$-0.472413\pi$
$661$	2942.00	0.173117	0.0865587	+	0.996247i	$0.472413\pi$
$662$	17104.0	1.00418
$663$	−126.000	−0.00738075
$664$	11376.0	0.664871
$665$	4675.00	0.272615
$666$	−836.000	−0.0486402
$667$	−6210.00	−0.360498
$668$	−8436.00	−0.488621
$669$	37814.0	2.18531
$670$	3040.00	0.175292
$671$	3157.00	0.181631
$672$	2464.00	0.141445
$673$	−13813.0	−0.791162	−0.395581	−	0.918431i	$-0.629457\pi$
$673$	−13813.0	−0.791162	−0.395581	+	0.918431i	$0.629457\pi$
$674$	2242.00	0.128129
$675$	−875.000	−0.0498945
$676$	−8772.00	−0.499090
$677$	8436.00	0.478910	0.239455	−	0.970908i	$-0.423031\pi$
$677$	8436.00	0.478910	0.239455	+	0.970908i	$0.423031\pi$
$678$	−1512.00	−0.0856460
$679$	1276.00	0.0721184
$680$	360.000	0.0203020
$681$	19782.0	1.11314
$682$	4994.00	0.280396
$683$	10377.0	0.581354	0.290677	−	0.956821i	$-0.406119\pi$
$683$	10377.0	0.581354	0.290677	+	0.956821i	$0.406119\pi$
$684$	−7480.00	−0.418136
$685$	−3780.00	−0.210841
$686$	−12430.0	−0.691807
$687$	−16870.0	−0.936871
$688$	−1408.00	−0.0780225
$689$	594.000	0.0328441
$690$	9660.00	0.532971
$691$	18602.0	1.02410	0.512050	−	0.858956i	$-0.328886\pi$
$691$	18602.0	1.02410	0.512050	+	0.858956i	$0.328886\pi$
$692$	−14232.0	−0.781820
$693$	2662.00	0.145918
$694$	21972.0	1.20179
$695$	−7300.00	−0.398424
$696$	2520.00	0.137242
$697$	1242.00	0.0674951
$698$	−2180.00	−0.118215
$699$	4809.00	0.260219
$700$	1100.00	0.0593944
$701$	−31353.0	−1.68928	−0.844641	−	0.535333i	$-0.820186\pi$
$701$	−31353.0	−1.68928	−0.844641	+	0.535333i	$0.820186\pi$
$702$	−140.000	−0.00752701
$703$	1615.00	0.0866442
$704$	704.000	0.0376889
$705$	18690.0	0.998448
$706$	924.000	0.0492567
$707$	4422.00	0.235228
$708$	−12600.0	−0.668838
$709$	27470.0	1.45509	0.727544	−	0.686061i	$-0.240662\pi$
$709$	27470.0	1.45509	0.727544	+	0.686061i	$0.240662\pi$
$710$	−7770.00	−0.410708
$711$	6380.00	0.336524
$712$	−11640.0	−0.612679
$713$	−31326.0	−1.64540
$714$	−1386.00	−0.0726467
$715$	−110.000	−0.00575352
$716$	1200.00	0.0626342
$717$	11550.0	0.601594
$718$	−120.000	−0.00623727
$719$	−10425.0	−0.540733	−0.270366	−	0.962758i	$-0.587145\pi$
$719$	−10425.0	−0.540733	−0.270366	+	0.962758i	$0.587145\pi$
$720$	−1760.00	−0.0910991
$721$	15532.0	0.802277
$722$	732.000	0.0377316
$723$	20174.0	1.03773
$724$	12488.0	0.641040
$725$	1125.00	0.0576296
$726$	1694.00	0.0865981
$727$	−26854.0	−1.36996	−0.684979	−	0.728563i	$-0.740189\pi$
$727$	−26854.0	−1.36996	−0.684979	+	0.728563i	$0.740189\pi$
$728$	176.000	0.00896016
$729$	−11843.0	−0.601687
$730$	−9620.00	−0.487743
$731$	792.000	0.0400727
$732$	8036.00	0.405764
$733$	−22228.0	−1.12007	−0.560034	−	0.828470i	$-0.689212\pi$
$733$	−22228.0	−1.12007	−0.560034	+	0.828470i	$0.689212\pi$
$734$	−3848.00	−0.193504
$735$	7770.00	0.389933
$736$	−4416.00	−0.221163
$737$	−3344.00	−0.167134
$738$	−6072.00	−0.302864
$739$	−12640.0	−0.629188	−0.314594	−	0.949226i	$-0.601868\pi$
$739$	−12640.0	−0.629188	−0.314594	+	0.949226i	$0.601868\pi$
$740$	380.000	0.0188771
$741$	−1190.00	−0.0589956
$742$	6534.00	0.323276
$743$	24507.0	1.21006	0.605030	−	0.796203i	$-0.293161\pi$
$743$	24507.0	1.21006	0.605030	+	0.796203i	$0.293161\pi$
$744$	12712.0	0.626404
$745$	10725.0	0.527428
$746$	−12476.0	−0.612304
$747$	31284.0	1.53229
$748$	−396.000	−0.0193572
$749$	23496.0	1.14623
$750$	−1750.00	−0.0852013
$751$	17087.0	0.830244	0.415122	−	0.909766i	$-0.363739\pi$
$751$	17087.0	0.830244	0.415122	+	0.909766i	$0.363739\pi$
$752$	−8544.00	−0.414319
$753$	−53886.0	−2.60786
$754$	180.000	0.00869392
$755$	−13510.0	−0.651231
$756$	−1540.00	−0.0740863
$757$	7106.00	0.341178	0.170589	−	0.985342i	$-0.445433\pi$
$757$	7106.00	0.341178	0.170589	+	0.985342i	$0.445433\pi$
$758$	1660.00	0.0795434
$759$	−10626.0	−0.508168
$760$	3400.00	0.162278
$761$	−2238.00	−0.106606	−0.0533032	−	0.998578i	$-0.516975\pi$
$761$	−2238.00	−0.106606	−0.0533032	+	0.998578i	$0.516975\pi$
$762$	19264.0	0.915828
$763$	2530.00	0.120042
$764$	−6192.00	−0.293218
$765$	990.000	0.0467889
$766$	7524.00	0.354900
$767$	−900.000	−0.0423691
$768$	1792.00	0.0841969
$769$	−12730.0	−0.596951	−0.298476	−	0.954417i	$-0.596478\pi$
$769$	−12730.0	−0.596951	−0.298476	+	0.954417i	$0.596478\pi$
$770$	−1210.00	−0.0566304
$771$	−44478.0	−2.07761
$772$	3308.00	0.154220
$773$	−22323.0	−1.03868	−0.519342	−	0.854567i	$-0.673823\pi$
$773$	−22323.0	−1.03868	−0.519342	+	0.854567i	$0.673823\pi$
$774$	−3872.00	−0.179814
$775$	5675.00	0.263035
$776$	928.000	0.0429295
$777$	−1463.00	−0.0675480
$778$	−11100.0	−0.511509
$779$	11730.0	0.539500
$780$	−280.000	−0.0128533
$781$	8547.00	0.391595
$782$	2484.00	0.113590
$783$	−1575.00	−0.0718849
$784$	−3552.00	−0.161808
$785$	−7105.00	−0.323042
$786$	−22302.0	−1.01207
$787$	40376.0	1.82878	0.914389	−	0.404836i	$-0.132671\pi$
$787$	40376.0	1.82878	0.914389	+	0.404836i	$0.132671\pi$
$788$	1824.00	0.0824585
$789$	−36561.0	−1.64969
$790$	−2900.00	−0.130604
$791$	−1188.00	−0.0534013
$792$	1936.00	0.0868596
$793$	574.000	0.0257041
$794$	17572.0	0.785399
$795$	−10395.0	−0.463739
$796$	11300.0	0.503163
$797$	−21114.0	−0.938389	−0.469195	−	0.883095i	$-0.655456\pi$
$797$	−21114.0	−0.938389	−0.469195	+	0.883095i	$0.655456\pi$
$798$	−13090.0	−0.580678
$799$	4806.00	0.212796
$800$	800.000	0.0353553
$801$	−32010.0	−1.41201
$802$	−6666.00	−0.293497
$803$	10582.0	0.465044
$804$	−8512.00	−0.373377
$805$	7590.00	0.332313
$806$	908.000	0.0396811
$807$	−31500.0	−1.37404
$808$	3216.00	0.140023
$809$	−38550.0	−1.67533	−0.837667	−	0.546181i	$-0.816081\pi$
$809$	−38550.0	−1.67533	−0.837667	+	0.546181i	$0.816081\pi$
$810$	8390.00	0.363944
$811$	−43063.0	−1.86455	−0.932273	−	0.361756i	$-0.882177\pi$
$811$	−43063.0	−1.86455	−0.932273	+	0.361756i	$0.882177\pi$
$812$	1980.00	0.0855719
$813$	−11956.0	−0.515763
$814$	−418.000	−0.0179986
$815$	13865.0	0.595914
$816$	−1008.00	−0.0432439
$817$	7480.00	0.320309
$818$	−25640.0	−1.09594
$819$	484.000	0.0206500
$820$	2760.00	0.117541
$821$	−2298.00	−0.0976867	−0.0488433	−	0.998806i	$-0.515554\pi$
$821$	−2298.00	−0.0976867	−0.0488433	+	0.998806i	$0.515554\pi$
$822$	10584.0	0.449099
$823$	−5428.00	−0.229901	−0.114950	−	0.993371i	$-0.536671\pi$
$823$	−5428.00	−0.229901	−0.114950	+	0.993371i	$0.536671\pi$
$824$	11296.0	0.477567
$825$	1925.00	0.0812362
$826$	−9900.00	−0.417028
$827$	−5004.00	−0.210406	−0.105203	−	0.994451i	$-0.533549\pi$
$827$	−5004.00	−0.210406	−0.105203	+	0.994451i	$0.533549\pi$
$828$	−12144.0	−0.509702
$829$	−25180.0	−1.05493	−0.527465	−	0.849577i	$-0.676858\pi$
$829$	−25180.0	−1.05493	−0.527465	+	0.849577i	$0.676858\pi$
$830$	−14220.0	−0.594679
$831$	39452.0	1.64690
$832$	128.000	0.00533366
$833$	1998.00	0.0831052
$834$	20440.0	0.848656
$835$	10545.0	0.437036
$836$	−3740.00	−0.154726
$837$	−7945.00	−0.328100
$838$	27360.0	1.12785
$839$	−8760.00	−0.360463	−0.180232	−	0.983624i	$-0.557685\pi$
$839$	−8760.00	−0.360463	−0.180232	+	0.983624i	$0.557685\pi$
$840$	−3080.00	−0.126512
$841$	−22364.0	−0.916971
$842$	9904.00	0.405361
$843$	−35406.0	−1.44656
$844$	−8932.00	−0.364280
$845$	10965.0	0.446399
$846$	−23496.0	−0.954857
$847$	1331.00	0.0539949
$848$	4752.00	0.192434
$849$	−34006.0	−1.37466
$850$	−450.000	−0.0181587
$851$	2622.00	0.105618
$852$	21756.0	0.874822
$853$	35462.0	1.42344	0.711721	−	0.702462i	$-0.247916\pi$
$853$	35462.0	1.42344	0.711721	+	0.702462i	$0.247916\pi$
$854$	6314.00	0.252998
$855$	9350.00	0.373992
$856$	17088.0	0.682308
$857$	−12489.0	−0.497802	−0.248901	−	0.968529i	$-0.580069\pi$
$857$	−12489.0	−0.497802	−0.248901	+	0.968529i	$0.580069\pi$
$858$	308.000	0.0122552
$859$	10970.0	0.435729	0.217865	−	0.975979i	$-0.430091\pi$
$859$	10970.0	0.435729	0.217865	+	0.975979i	$0.430091\pi$
$860$	1760.00	0.0697855
$861$	−10626.0	−0.420596
$862$	32064.0	1.26694
$863$	36942.0	1.45715	0.728575	−	0.684966i	$-0.240183\pi$
$863$	36942.0	1.45715	0.728575	+	0.684966i	$0.240183\pi$
$864$	−1120.00	−0.0441009
$865$	17790.0	0.699281
$866$	1384.00	0.0543074
$867$	−33824.0	−1.32494
$868$	9988.00	0.390570
$869$	3190.00	0.124526
$870$	−3150.00	−0.122753
$871$	−608.000	−0.0236525
$872$	1840.00	0.0714567
$873$	2552.00	0.0989371
$874$	23460.0	0.907948
$875$	−1375.00	−0.0531240
$876$	26936.0	1.03891
$877$	−48094.0	−1.85179	−0.925895	−	0.377782i	$-0.876687\pi$
$877$	−48094.0	−1.85179	−0.925895	+	0.377782i	$0.876687\pi$
$878$	−35840.0	−1.37761
$879$	65814.0	2.52543
$880$	−880.000	−0.0337100
$881$	−28878.0	−1.10434	−0.552171	−	0.833731i	$-0.686200\pi$
$881$	−28878.0	−1.10434	−0.552171	+	0.833731i	$0.686200\pi$
$882$	−9768.00	−0.372909
$883$	6497.00	0.247612	0.123806	−	0.992306i	$-0.460490\pi$
$883$	6497.00	0.247612	0.123806	+	0.992306i	$0.460490\pi$
$884$	−72.0000	−0.00273939
$885$	15750.0	0.598227
$886$	31704.0	1.20216
$887$	15696.0	0.594160	0.297080	−	0.954853i	$-0.403987\pi$
$887$	15696.0	0.594160	0.297080	+	0.954853i	$0.403987\pi$
$888$	−1064.00	−0.0402089
$889$	15136.0	0.571029
$890$	14550.0	0.547997
$891$	−9229.00	−0.347007
$892$	21608.0	0.811087
$893$	45390.0	1.70092
$894$	−30030.0	−1.12344
$895$	−1500.00	−0.0560218
$896$	1408.00	0.0524977
$897$	−1932.00	−0.0719148
$898$	13860.0	0.515049
$899$	10215.0	0.378965
$900$	2200.00	0.0814815
$901$	−2673.00	−0.0988352
$902$	−3036.00	−0.112071
$903$	−6776.00	−0.249713
$904$	−864.000	−0.0317878
$905$	−15610.0	−0.573363
$906$	37828.0	1.38714
$907$	9551.00	0.349654	0.174827	−	0.984599i	$-0.444063\pi$
$907$	9551.00	0.349654	0.174827	+	0.984599i	$0.444063\pi$
$908$	11304.0	0.413146
$909$	8844.00	0.322703
$910$	−220.000	−0.00801421
$911$	1317.00	0.0478970	0.0239485	−	0.999713i	$-0.492376\pi$
$911$	1317.00	0.0478970	0.0239485	+	0.999713i	$0.492376\pi$
$912$	−9520.00	−0.345656
$913$	15642.0	0.567004
$914$	−1238.00	−0.0448024
$915$	−10045.0	−0.362926
$916$	−9640.00	−0.347723
$917$	−17523.0	−0.631036
$918$	630.000	0.0226504
$919$	43460.0	1.55997	0.779985	−	0.625798i	$-0.215226\pi$
$919$	43460.0	1.55997	0.779985	+	0.625798i	$0.215226\pi$
$920$	5520.00	0.197814
$921$	−46018.0	−1.64641
$922$	−21066.0	−0.752464
$923$	1554.00	0.0554177
$924$	3388.00	0.120624
$925$	−475.000	−0.0168842
$926$	−23396.0	−0.830281
$927$	31064.0	1.10062
$928$	1440.00	0.0509378
$929$	40995.0	1.44780	0.723898	−	0.689907i	$-0.242349\pi$
$929$	40995.0	1.44780	0.723898	+	0.689907i	$0.242349\pi$
$930$	−15890.0	−0.560273
$931$	18870.0	0.664274
$932$	2748.00	0.0965813
$933$	−57771.0	−2.02716
$934$	2862.00	0.100265
$935$	495.000	0.0173136
$936$	352.000	0.0122922
$937$	−31174.0	−1.08688	−0.543442	−	0.839447i	$-0.682879\pi$
$937$	−31174.0	−1.08688	−0.543442	+	0.839447i	$0.682879\pi$
$938$	−6688.00	−0.232805
$939$	−34006.0	−1.18184
$940$	10680.0	0.370578
$941$	−41103.0	−1.42393	−0.711966	−	0.702214i	$-0.752195\pi$
$941$	−41103.0	−1.42393	−0.711966	+	0.702214i	$0.752195\pi$
$942$	19894.0	0.688091
$943$	19044.0	0.657644
$944$	−7200.00	−0.248242
$945$	1925.00	0.0662648
$946$	−1936.00	−0.0665378
$947$	18861.0	0.647202	0.323601	−	0.946194i	$-0.395106\pi$
$947$	18861.0	0.647202	0.323601	+	0.946194i	$0.395106\pi$
$948$	8120.00	0.278191
$949$	1924.00	0.0658121
$950$	−4250.00	−0.145145
$951$	4977.00	0.169706
$952$	−792.000	−0.0269631
$953$	4857.00	0.165093	0.0825465	−	0.996587i	$-0.473695\pi$
$953$	4857.00	0.165093	0.0825465	+	0.996587i	$0.473695\pi$
$954$	13068.0	0.443493
$955$	7740.00	0.262262
$956$	6600.00	0.223284
$957$	3465.00	0.117040
$958$	−9480.00	−0.319713
$959$	8316.00	0.280018
$960$	−2240.00	−0.0753080
$961$	21738.0	0.729683
$962$	−76.0000	−0.00254713
$963$	46992.0	1.57248
$964$	11528.0	0.385158
$965$	−4135.00	−0.137938
$966$	−21252.0	−0.707838
$967$	−18859.0	−0.627161	−0.313580	−	0.949562i	$-0.601529\pi$
$967$	−18859.0	−0.627161	−0.313580	+	0.949562i	$0.601529\pi$
$968$	968.000	0.0321412
$969$	5355.00	0.177531
$970$	−1160.00	−0.0383973
$971$	−20568.0	−0.679772	−0.339886	−	0.940467i	$-0.610388\pi$
$971$	−20568.0	−0.679772	−0.339886	+	0.940467i	$0.610388\pi$
$972$	−19712.0	−0.650476
$973$	16060.0	0.529147
$974$	−4208.00	−0.138432
$975$	350.000	0.0114964
$976$	4592.00	0.150601
$977$	45636.0	1.49440	0.747198	−	0.664601i	$-0.231399\pi$
$977$	45636.0	1.49440	0.747198	+	0.664601i	$0.231399\pi$
$978$	−38822.0	−1.26932
$979$	−16005.0	−0.522494
$980$	4440.00	0.144725
$981$	5060.00	0.164682
$982$	−25866.0	−0.840547
$983$	−33618.0	−1.09079	−0.545396	−	0.838179i	$-0.683621\pi$
$983$	−33618.0	−1.09079	−0.545396	+	0.838179i	$0.683621\pi$
$984$	−7728.00	−0.250365
$985$	−2280.00	−0.0737531
$986$	−810.000	−0.0261619
$987$	−41118.0	−1.32604
$988$	−680.000	−0.0218964
$989$	12144.0	0.390452
$990$	−2420.00	−0.0776895
$991$	−42928.0	−1.37604	−0.688019	−	0.725693i	$-0.741519\pi$
$991$	−42928.0	−1.37604	−0.688019	+	0.725693i	$0.741519\pi$
$992$	7264.00	0.232492
$993$	59864.0	1.91312
$994$	17094.0	0.545462
$995$	−14125.0	−0.450043
$996$	39816.0	1.26668
$997$	55316.0	1.75715	0.878573	−	0.477607i	$-0.158496\pi$
$997$	55316.0	1.75715	0.878573	+	0.477607i	$0.158496\pi$
$998$	35080.0	1.11266
$999$	665.000	0.0210607

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	110.4.a.g.1.1	✓	1
3.2	odd	2		990.4.a.i.1.1		1
4.3	odd	2		880.4.a.c.1.1		1
5.2	odd	4		550.4.b.j.199.2		2
5.3	odd	4		550.4.b.j.199.1		2
5.4	even	2		550.4.a.b.1.1		1
11.10	odd	2		1210.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.4.a.g.1.1	✓	1	1.1	even	1	trivial
550.4.a.b.1.1		1	5.4	even	2
550.4.b.j.199.1		2	5.3	odd	4
550.4.b.j.199.2		2	5.2	odd	4
880.4.a.c.1.1		1	4.3	odd	2
990.4.a.i.1.1		1	3.2	odd	2
1210.4.a.g.1.1		1	11.10	odd	2

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

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Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	110.4.a.g.1.1

Level	$110$
Weight	$4$
Character	110.1
Self dual	yes
Analytic conductor	$6.490$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$