LMFDB - Embedded newform 1386.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,4,Mod(1,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, 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("1386.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1386.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:	$81.7766472680$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1386.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +21.0000 q^{5} +7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} +21.0000 q^{5} +7.00000 q^{7} +8.00000 q^{8} +42.0000 q^{10} +11.0000 q^{11} +65.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} +54.0000 q^{17} +65.0000 q^{19} +84.0000 q^{20} +22.0000 q^{22} -132.000 q^{23} +316.000 q^{25} +130.000 q^{26} +28.0000 q^{28} -39.0000 q^{29} -178.000 q^{31} +32.0000 q^{32} +108.000 q^{34} +147.000 q^{35} -439.000 q^{37} +130.000 q^{38} +168.000 q^{40} -96.0000 q^{41} +272.000 q^{43} +44.0000 q^{44} -264.000 q^{46} +375.000 q^{47} +49.0000 q^{49} +632.000 q^{50} +260.000 q^{52} -612.000 q^{53} +231.000 q^{55} +56.0000 q^{56} -78.0000 q^{58} +507.000 q^{59} +758.000 q^{61} -356.000 q^{62} +64.0000 q^{64} +1365.00 q^{65} -1087.00 q^{67} +216.000 q^{68} +294.000 q^{70} -673.000 q^{73} -878.000 q^{74} +260.000 q^{76} +77.0000 q^{77} -700.000 q^{79} +336.000 q^{80} -192.000 q^{82} -1218.00 q^{83} +1134.00 q^{85} +544.000 q^{86} +88.0000 q^{88} +1350.00 q^{89} +455.000 q^{91} -528.000 q^{92} +750.000 q^{94} +1365.00 q^{95} -808.000 q^{97} +98.0000 q^{98} +O(q^{100})\)

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
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	21.0000	1.87830	0.939149	−	0.343511i	$-0.111616\pi$
$5$	21.0000	1.87830	0.939149	+	0.343511i	$0.111616\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	42.0000	1.32816
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	65.0000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	65.0000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	54.0000	0.770407	0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000	0.770407	0.385204	+	0.922832i	$0.374131\pi$
$18$	0	0
$19$	65.0000	0.784843	0.392422	−	0.919785i	$-0.371637\pi$
$19$	65.0000	0.784843	0.392422	+	0.919785i	$0.371637\pi$
$20$	84.0000	0.939149
$21$	0	0
$22$	22.0000	0.213201
$23$	−132.000	−1.19669	−0.598346	−	0.801238i	$-0.704175\pi$
$23$	−132.000	−1.19669	−0.598346	+	0.801238i	$0.704175\pi$
$24$	0	0
$25$	316.000	2.52800
$26$	130.000	0.980581
$27$	0	0
$28$	28.0000	0.188982
$29$	−39.0000	−0.249728	−0.124864	−	0.992174i	$-0.539849\pi$
$29$	−39.0000	−0.249728	−0.124864	+	0.992174i	$0.539849\pi$
$30$	0	0
$31$	−178.000	−1.03128	−0.515641	−	0.856805i	$-0.672446\pi$
$31$	−178.000	−1.03128	−0.515641	+	0.856805i	$0.672446\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	108.000	0.544760
$35$	147.000	0.709930
$36$	0	0
$37$	−439.000	−1.95057	−0.975286	−	0.220947i	$-0.929085\pi$
$37$	−439.000	−1.95057	−0.975286	+	0.220947i	$0.929085\pi$
$38$	130.000	0.554968
$39$	0	0
$40$	168.000	0.664078
$41$	−96.0000	−0.365675	−0.182838	−	0.983143i	$-0.558528\pi$
$41$	−96.0000	−0.365675	−0.182838	+	0.983143i	$0.558528\pi$
$42$	0	0
$43$	272.000	0.964642	0.482321	−	0.875995i	$-0.339794\pi$
$43$	272.000	0.964642	0.482321	+	0.875995i	$0.339794\pi$
$44$	44.0000	0.150756
$45$	0	0
$46$	−264.000	−0.846189
$47$	375.000	1.16382	0.581908	−	0.813254i	$-0.302306\pi$
$47$	375.000	1.16382	0.581908	+	0.813254i	$0.302306\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	632.000	1.78757
$51$	0	0
$52$	260.000	0.693375
$53$	−612.000	−1.58613	−0.793063	−	0.609140i	$-0.791515\pi$
$53$	−612.000	−1.58613	−0.793063	+	0.609140i	$0.791515\pi$
$54$	0	0
$55$	231.000	0.566328
$56$	56.0000	0.133631
$57$	0	0
$58$	−78.0000	−0.176585
$59$	507.000	1.11874	0.559371	−	0.828917i	$-0.311043\pi$
$59$	507.000	1.11874	0.559371	+	0.828917i	$0.311043\pi$
$60$	0	0
$61$	758.000	1.59102	0.795508	−	0.605943i	$-0.207204\pi$
$61$	758.000	1.59102	0.795508	+	0.605943i	$0.207204\pi$
$62$	−356.000	−0.729227
$63$	0	0
$64$	64.0000	0.125000
$65$	1365.00	2.60473
$66$	0	0
$67$	−1087.00	−1.98206	−0.991031	−	0.133630i	$-0.957337\pi$
$67$	−1087.00	−1.98206	−0.991031	+	0.133630i	$0.957337\pi$
$68$	216.000	0.385204
$69$	0	0
$70$	294.000	0.501996
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−673.000	−1.07902	−0.539512	−	0.841978i	$-0.681391\pi$
$73$	−673.000	−1.07902	−0.539512	+	0.841978i	$0.681391\pi$
$74$	−878.000	−1.37926
$75$	0	0
$76$	260.000	0.392422
$77$	77.0000	0.113961
$78$	0	0
$79$	−700.000	−0.996913	−0.498457	−	0.866915i	$-0.666100\pi$
$79$	−700.000	−0.996913	−0.498457	+	0.866915i	$0.666100\pi$
$80$	336.000	0.469574
$81$	0	0
$82$	−192.000	−0.258571
$83$	−1218.00	−1.61076	−0.805379	−	0.592761i	$-0.798038\pi$
$83$	−1218.00	−1.61076	−0.805379	+	0.592761i	$0.798038\pi$
$84$	0	0
$85$	1134.00	1.44705
$86$	544.000	0.682105
$87$	0	0
$88$	88.0000	0.106600
$89$	1350.00	1.60786	0.803931	−	0.594723i	$-0.202738\pi$
$89$	1350.00	1.60786	0.803931	+	0.594723i	$0.202738\pi$
$90$	0	0
$91$	455.000	0.524142
$92$	−528.000	−0.598346
$93$	0	0
$94$	750.000	0.822942
$95$	1365.00	1.47417
$96$	0	0
$97$	−808.000	−0.845773	−0.422886	−	0.906183i	$-0.638983\pi$
$97$	−808.000	−0.845773	−0.422886	+	0.906183i	$0.638983\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	1264.00	1.26400
$101$	1194.00	1.17631	0.588156	−	0.808748i	$-0.299854\pi$
$101$	1194.00	1.17631	0.588156	+	0.808748i	$0.299854\pi$
$102$	0	0
$103$	−1546.00	−1.47895	−0.739475	−	0.673184i	$-0.764926\pi$
$103$	−1546.00	−1.47895	−0.739475	+	0.673184i	$0.764926\pi$
$104$	520.000	0.490290
$105$	0	0
$106$	−1224.00	−1.12156
$107$	−1395.00	−1.26037	−0.630186	−	0.776444i	$-0.717021\pi$
$107$	−1395.00	−1.26037	−0.630186	+	0.776444i	$0.717021\pi$
$108$	0	0
$109$	−250.000	−0.219685	−0.109842	−	0.993949i	$-0.535035\pi$
$109$	−250.000	−0.219685	−0.109842	+	0.993949i	$0.535035\pi$
$110$	462.000	0.400454
$111$	0	0
$112$	112.000	0.0944911
$113$	−222.000	−0.184814	−0.0924071	−	0.995721i	$-0.529456\pi$
$113$	−222.000	−0.184814	−0.0924071	+	0.995721i	$0.529456\pi$
$114$	0	0
$115$	−2772.00	−2.24774
$116$	−156.000	−0.124864
$117$	0	0
$118$	1014.00	0.791070
$119$	378.000	0.291187
$120$	0	0
$121$	121.000	0.0909091
$122$	1516.00	1.12502
$123$	0	0
$124$	−712.000	−0.515641
$125$	4011.00	2.87004
$126$	0	0
$127$	−142.000	−0.0992162	−0.0496081	−	0.998769i	$-0.515797\pi$
$127$	−142.000	−0.0992162	−0.0496081	+	0.998769i	$0.515797\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	2730.00	1.84182
$131$	552.000	0.368156	0.184078	−	0.982912i	$-0.441070\pi$
$131$	552.000	0.368156	0.184078	+	0.982912i	$0.441070\pi$
$132$	0	0
$133$	455.000	0.296643
$134$	−2174.00	−1.40153
$135$	0	0
$136$	432.000	0.272380
$137$	726.000	0.452747	0.226374	−	0.974041i	$-0.427313\pi$
$137$	726.000	0.452747	0.226374	+	0.974041i	$0.427313\pi$
$138$	0	0
$139$	56.0000	0.0341716	0.0170858	−	0.999854i	$-0.494561\pi$
$139$	56.0000	0.0341716	0.0170858	+	0.999854i	$0.494561\pi$
$140$	588.000	0.354965
$141$	0	0
$142$	0	0
$143$	715.000	0.418121
$144$	0	0
$145$	−819.000	−0.469064
$146$	−1346.00	−0.762985
$147$	0	0
$148$	−1756.00	−0.975286
$149$	−3219.00	−1.76987	−0.884935	−	0.465714i	$-0.845798\pi$
$149$	−3219.00	−1.76987	−0.884935	+	0.465714i	$0.845798\pi$
$150$	0	0
$151$	−106.000	−0.0571269	−0.0285634	−	0.999592i	$-0.509093\pi$
$151$	−106.000	−0.0571269	−0.0285634	+	0.999592i	$0.509093\pi$
$152$	520.000	0.277484
$153$	0	0
$154$	154.000	0.0805823
$155$	−3738.00	−1.93705
$156$	0	0
$157$	2864.00	1.45587	0.727937	−	0.685644i	$-0.240480\pi$
$157$	2864.00	1.45587	0.727937	+	0.685644i	$0.240480\pi$
$158$	−1400.00	−0.704924
$159$	0	0
$160$	672.000	0.332039
$161$	−924.000	−0.452307
$162$	0	0
$163$	−1159.00	−0.556932	−0.278466	−	0.960446i	$-0.589826\pi$
$163$	−1159.00	−0.556932	−0.278466	+	0.960446i	$0.589826\pi$
$164$	−384.000	−0.182838
$165$	0	0
$166$	−2436.00	−1.13898
$167$	2280.00	1.05648	0.528239	−	0.849096i	$-0.322853\pi$
$167$	2280.00	1.05648	0.528239	+	0.849096i	$0.322853\pi$
$168$	0	0
$169$	2028.00	0.923077
$170$	2268.00	1.02322
$171$	0	0
$172$	1088.00	0.482321
$173$	3156.00	1.38697	0.693486	−	0.720470i	$-0.256074\pi$
$173$	3156.00	1.38697	0.693486	+	0.720470i	$0.256074\pi$
$174$	0	0
$175$	2212.00	0.955494
$176$	176.000	0.0753778
$177$	0	0
$178$	2700.00	1.13693
$179$	−864.000	−0.360773	−0.180387	−	0.983596i	$-0.557735\pi$
$179$	−864.000	−0.360773	−0.180387	+	0.983596i	$0.557735\pi$
$180$	0	0
$181$	−160.000	−0.0657056	−0.0328528	−	0.999460i	$-0.510459\pi$
$181$	−160.000	−0.0657056	−0.0328528	+	0.999460i	$0.510459\pi$
$182$	910.000	0.370625
$183$	0	0
$184$	−1056.00	−0.423094
$185$	−9219.00	−3.66375
$186$	0	0
$187$	594.000	0.232287
$188$	1500.00	0.581908
$189$	0	0
$190$	2730.00	1.04239
$191$	1410.00	0.534157	0.267079	−	0.963675i	$-0.413942\pi$
$191$	1410.00	0.534157	0.267079	+	0.963675i	$0.413942\pi$
$192$	0	0
$193$	2936.00	1.09502	0.547508	−	0.836801i	$-0.315577\pi$
$193$	2936.00	1.09502	0.547508	+	0.836801i	$0.315577\pi$
$194$	−1616.00	−0.598052
$195$	0	0
$196$	196.000	0.0714286
$197$	1170.00	0.423142	0.211571	−	0.977363i	$-0.432142\pi$
$197$	1170.00	0.423142	0.211571	+	0.977363i	$0.432142\pi$
$198$	0	0
$199$	5168.00	1.84095	0.920477	−	0.390797i	$-0.127801\pi$
$199$	5168.00	1.84095	0.920477	+	0.390797i	$0.127801\pi$
$200$	2528.00	0.893783
$201$	0	0
$202$	2388.00	0.831778
$203$	−273.000	−0.0943884
$204$	0	0
$205$	−2016.00	−0.686847
$206$	−3092.00	−1.04578
$207$	0	0
$208$	1040.00	0.346688
$209$	715.000	0.236639
$210$	0	0
$211$	2162.00	0.705394	0.352697	−	0.935738i	$-0.385265\pi$
$211$	2162.00	0.705394	0.352697	+	0.935738i	$0.385265\pi$
$212$	−2448.00	−0.793063
$213$	0	0
$214$	−2790.00	−0.891217
$215$	5712.00	1.81188
$216$	0	0
$217$	−1246.00	−0.389788
$218$	−500.000	−0.155341
$219$	0	0
$220$	924.000	0.283164
$221$	3510.00	1.06836
$222$	0	0
$223$	−2518.00	−0.756133	−0.378067	−	0.925778i	$-0.623411\pi$
$223$	−2518.00	−0.756133	−0.378067	+	0.925778i	$0.623411\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−444.000	−0.130683
$227$	2058.00	0.601737	0.300868	−	0.953666i	$-0.402724\pi$
$227$	2058.00	0.601737	0.300868	+	0.953666i	$0.402724\pi$
$228$	0	0
$229$	−1168.00	−0.337046	−0.168523	−	0.985698i	$-0.553900\pi$
$229$	−1168.00	−0.337046	−0.168523	+	0.985698i	$0.553900\pi$
$230$	−5544.00	−1.58939
$231$	0	0
$232$	−312.000	−0.0882923
$233$	2250.00	0.632628	0.316314	−	0.948654i	$-0.397555\pi$
$233$	2250.00	0.632628	0.316314	+	0.948654i	$0.397555\pi$
$234$	0	0
$235$	7875.00	2.18599
$236$	2028.00	0.559371
$237$	0	0
$238$	756.000	0.205900
$239$	−3849.00	−1.04172	−0.520860	−	0.853642i	$-0.674389\pi$
$239$	−3849.00	−1.04172	−0.520860	+	0.853642i	$0.674389\pi$
$240$	0	0
$241$	−1771.00	−0.473362	−0.236681	−	0.971587i	$-0.576060\pi$
$241$	−1771.00	−0.473362	−0.236681	+	0.971587i	$0.576060\pi$
$242$	242.000	0.0642824
$243$	0	0
$244$	3032.00	0.795508
$245$	1029.00	0.268328
$246$	0	0
$247$	4225.00	1.08838
$248$	−1424.00	−0.364613
$249$	0	0
$250$	8022.00	2.02942
$251$	−3555.00	−0.893983	−0.446991	−	0.894538i	$-0.647504\pi$
$251$	−3555.00	−0.893983	−0.446991	+	0.894538i	$0.647504\pi$
$252$	0	0
$253$	−1452.00	−0.360816
$254$	−284.000	−0.0701565
$255$	0	0
$256$	256.000	0.0625000
$257$	3621.00	0.878878	0.439439	−	0.898272i	$-0.355177\pi$
$257$	3621.00	0.878878	0.439439	+	0.898272i	$0.355177\pi$
$258$	0	0
$259$	−3073.00	−0.737247
$260$	5460.00	1.30236
$261$	0	0
$262$	1104.00	0.260326
$263$	321.000	0.0752612	0.0376306	−	0.999292i	$-0.488019\pi$
$263$	321.000	0.0752612	0.0376306	+	0.999292i	$0.488019\pi$
$264$	0	0
$265$	−12852.0	−2.97922
$266$	910.000	0.209758
$267$	0	0
$268$	−4348.00	−0.991031
$269$	8370.00	1.89713	0.948565	−	0.316583i	$-0.102536\pi$
$269$	8370.00	1.89713	0.948565	+	0.316583i	$0.102536\pi$
$270$	0	0
$271$	1541.00	0.345421	0.172710	−	0.984973i	$-0.444747\pi$
$271$	1541.00	0.345421	0.172710	+	0.984973i	$0.444747\pi$
$272$	864.000	0.192602
$273$	0	0
$274$	1452.00	0.320141
$275$	3476.00	0.762221
$276$	0	0
$277$	−3112.00	−0.675025	−0.337513	−	0.941321i	$-0.609586\pi$
$277$	−3112.00	−0.675025	−0.337513	+	0.941321i	$0.609586\pi$
$278$	112.000	0.0241630
$279$	0	0
$280$	1176.00	0.250998
$281$	4821.00	1.02348	0.511738	−	0.859142i	$-0.329002\pi$
$281$	4821.00	1.02348	0.511738	+	0.859142i	$0.329002\pi$
$282$	0	0
$283$	3737.00	0.784953	0.392476	−	0.919762i	$-0.371618\pi$
$283$	3737.00	0.784953	0.392476	+	0.919762i	$0.371618\pi$
$284$	0	0
$285$	0	0
$286$	1430.00	0.295656
$287$	−672.000	−0.138212
$288$	0	0
$289$	−1997.00	−0.406473
$290$	−1638.00	−0.331678
$291$	0	0
$292$	−2692.00	−0.539512
$293$	4080.00	0.813502	0.406751	−	0.913539i	$-0.366662\pi$
$293$	4080.00	0.813502	0.406751	+	0.913539i	$0.366662\pi$
$294$	0	0
$295$	10647.0	2.10133
$296$	−3512.00	−0.689631
$297$	0	0
$298$	−6438.00	−1.25149
$299$	−8580.00	−1.65951
$300$	0	0
$301$	1904.00	0.364600
$302$	−212.000	−0.0403948
$303$	0	0
$304$	1040.00	0.196211
$305$	15918.0	2.98840
$306$	0	0
$307$	−9772.00	−1.81667	−0.908335	−	0.418244i	$-0.862646\pi$
$307$	−9772.00	−1.81667	−0.908335	+	0.418244i	$0.862646\pi$
$308$	308.000	0.0569803
$309$	0	0
$310$	−7476.00	−1.36970
$311$	−5136.00	−0.936450	−0.468225	−	0.883609i	$-0.655106\pi$
$311$	−5136.00	−0.936450	−0.468225	+	0.883609i	$0.655106\pi$
$312$	0	0
$313$	−4282.00	−0.773268	−0.386634	−	0.922233i	$-0.626362\pi$
$313$	−4282.00	−0.773268	−0.386634	+	0.922233i	$0.626362\pi$
$314$	5728.00	1.02946
$315$	0	0
$316$	−2800.00	−0.498457
$317$	708.000	0.125442	0.0627212	−	0.998031i	$-0.480022\pi$
$317$	708.000	0.125442	0.0627212	+	0.998031i	$0.480022\pi$
$318$	0	0
$319$	−429.000	−0.0752959
$320$	1344.00	0.234787
$321$	0	0
$322$	−1848.00	−0.319829
$323$	3510.00	0.604649
$324$	0	0
$325$	20540.0	3.50571
$326$	−2318.00	−0.393810
$327$	0	0
$328$	−768.000	−0.129286
$329$	2625.00	0.439881
$330$	0	0
$331$	2648.00	0.439720	0.219860	−	0.975531i	$-0.429440\pi$
$331$	2648.00	0.439720	0.219860	+	0.975531i	$0.429440\pi$
$332$	−4872.00	−0.805379
$333$	0	0
$334$	4560.00	0.747042
$335$	−22827.0	−3.72290
$336$	0	0
$337$	−4444.00	−0.718339	−0.359169	−	0.933272i	$-0.616940\pi$
$337$	−4444.00	−0.718339	−0.359169	+	0.933272i	$0.616940\pi$
$338$	4056.00	0.652714
$339$	0	0
$340$	4536.00	0.723527
$341$	−1958.00	−0.310943
$342$	0	0
$343$	343.000	0.0539949
$344$	2176.00	0.341052
$345$	0	0
$346$	6312.00	0.980738
$347$	3072.00	0.475255	0.237628	−	0.971356i	$-0.423630\pi$
$347$	3072.00	0.475255	0.237628	+	0.971356i	$0.423630\pi$
$348$	0	0
$349$	4205.00	0.644953	0.322476	−	0.946578i	$-0.395485\pi$
$349$	4205.00	0.644953	0.322476	+	0.946578i	$0.395485\pi$
$350$	4424.00	0.675636
$351$	0	0
$352$	352.000	0.0533002
$353$	−849.000	−0.128011	−0.0640053	−	0.997950i	$-0.520387\pi$
$353$	−849.000	−0.128011	−0.0640053	+	0.997950i	$0.520387\pi$
$354$	0	0
$355$	0	0
$356$	5400.00	0.803931
$357$	0	0
$358$	−1728.00	−0.255105
$359$	−6876.00	−1.01087	−0.505434	−	0.862865i	$-0.668667\pi$
$359$	−6876.00	−1.01087	−0.505434	+	0.862865i	$0.668667\pi$
$360$	0	0
$361$	−2634.00	−0.384021
$362$	−320.000	−0.0464608
$363$	0	0
$364$	1820.00	0.262071
$365$	−14133.0	−2.02673
$366$	0	0
$367$	−8206.00	−1.16717	−0.583583	−	0.812054i	$-0.698350\pi$
$367$	−8206.00	−1.16717	−0.583583	+	0.812054i	$0.698350\pi$
$368$	−2112.00	−0.299173
$369$	0	0
$370$	−18438.0	−2.59066
$371$	−4284.00	−0.599499
$372$	0	0
$373$	−9322.00	−1.29403	−0.647017	−	0.762475i	$-0.723984\pi$
$373$	−9322.00	−1.29403	−0.647017	+	0.762475i	$0.723984\pi$
$374$	1188.00	0.164251
$375$	0	0
$376$	3000.00	0.411471
$377$	−2535.00	−0.346311
$378$	0	0
$379$	−9601.00	−1.30124	−0.650620	−	0.759403i	$-0.725491\pi$
$379$	−9601.00	−1.30124	−0.650620	+	0.759403i	$0.725491\pi$
$380$	5460.00	0.737084
$381$	0	0
$382$	2820.00	0.377706
$383$	−888.000	−0.118472	−0.0592359	−	0.998244i	$-0.518866\pi$
$383$	−888.000	−0.118472	−0.0592359	+	0.998244i	$0.518866\pi$
$384$	0	0
$385$	1617.00	0.214052
$386$	5872.00	0.774293
$387$	0	0
$388$	−3232.00	−0.422886
$389$	10212.0	1.33103	0.665513	−	0.746386i	$-0.268213\pi$
$389$	10212.0	1.33103	0.665513	+	0.746386i	$0.268213\pi$
$390$	0	0
$391$	−7128.00	−0.921940
$392$	392.000	0.0505076
$393$	0	0
$394$	2340.00	0.299207
$395$	−14700.0	−1.87250
$396$	0	0
$397$	−2554.00	−0.322876	−0.161438	−	0.986883i	$-0.551613\pi$
$397$	−2554.00	−0.322876	−0.161438	+	0.986883i	$0.551613\pi$
$398$	10336.0	1.30175
$399$	0	0
$400$	5056.00	0.632000
$401$	−492.000	−0.0612701	−0.0306350	−	0.999531i	$-0.509753\pi$
$401$	−492.000	−0.0612701	−0.0306350	+	0.999531i	$0.509753\pi$
$402$	0	0
$403$	−11570.0	−1.43013
$404$	4776.00	0.588156
$405$	0	0
$406$	−546.000	−0.0667427
$407$	−4829.00	−0.588120
$408$	0	0
$409$	−13354.0	−1.61446	−0.807228	−	0.590239i	$-0.799033\pi$
$409$	−13354.0	−1.61446	−0.807228	+	0.590239i	$0.799033\pi$
$410$	−4032.00	−0.485674
$411$	0	0
$412$	−6184.00	−0.739475
$413$	3549.00	0.422845
$414$	0	0
$415$	−25578.0	−3.02548
$416$	2080.00	0.245145
$417$	0	0
$418$	1430.00	0.167329
$419$	−4515.00	−0.526425	−0.263213	−	0.964738i	$-0.584782\pi$
$419$	−4515.00	−0.526425	−0.263213	+	0.964738i	$0.584782\pi$
$420$	0	0
$421$	−1177.00	−0.136255	−0.0681276	−	0.997677i	$-0.521703\pi$
$421$	−1177.00	−0.136255	−0.0681276	+	0.997677i	$0.521703\pi$
$422$	4324.00	0.498789
$423$	0	0
$424$	−4896.00	−0.560780
$425$	17064.0	1.94759
$426$	0	0
$427$	5306.00	0.601347
$428$	−5580.00	−0.630186
$429$	0	0
$430$	11424.0	1.28120
$431$	−1215.00	−0.135788	−0.0678938	−	0.997693i	$-0.521628\pi$
$431$	−1215.00	−0.135788	−0.0678938	+	0.997693i	$0.521628\pi$
$432$	0	0
$433$	−6712.00	−0.744938	−0.372469	−	0.928045i	$-0.621489\pi$
$433$	−6712.00	−0.744938	−0.372469	+	0.928045i	$0.621489\pi$
$434$	−2492.00	−0.275622
$435$	0	0
$436$	−1000.00	−0.109842
$437$	−8580.00	−0.939215
$438$	0	0
$439$	−11851.0	−1.28842	−0.644211	−	0.764848i	$-0.722814\pi$
$439$	−11851.0	−1.28842	−0.644211	+	0.764848i	$0.722814\pi$
$440$	1848.00	0.200227
$441$	0	0
$442$	7020.00	0.755446
$443$	7620.00	0.817240	0.408620	−	0.912705i	$-0.366010\pi$
$443$	7620.00	0.817240	0.408620	+	0.912705i	$0.366010\pi$
$444$	0	0
$445$	28350.0	3.02004
$446$	−5036.00	−0.534667
$447$	0	0
$448$	448.000	0.0472456
$449$	2340.00	0.245950	0.122975	−	0.992410i	$-0.460757\pi$
$449$	2340.00	0.245950	0.122975	+	0.992410i	$0.460757\pi$
$450$	0	0
$451$	−1056.00	−0.110255
$452$	−888.000	−0.0924071
$453$	0	0
$454$	4116.00	0.425492
$455$	9555.00	0.984495
$456$	0	0
$457$	−13984.0	−1.43139	−0.715694	−	0.698414i	$-0.753889\pi$
$457$	−13984.0	−1.43139	−0.715694	+	0.698414i	$0.753889\pi$
$458$	−2336.00	−0.238328
$459$	0	0
$460$	−11088.0	−1.12387
$461$	−5844.00	−0.590417	−0.295208	−	0.955433i	$-0.595389\pi$
$461$	−5844.00	−0.590417	−0.295208	+	0.955433i	$0.595389\pi$
$462$	0	0
$463$	7625.00	0.765365	0.382682	−	0.923880i	$-0.375000\pi$
$463$	7625.00	0.765365	0.382682	+	0.923880i	$0.375000\pi$
$464$	−624.000	−0.0624321
$465$	0	0
$466$	4500.00	0.447336
$467$	13959.0	1.38318	0.691590	−	0.722290i	$-0.256910\pi$
$467$	13959.0	1.38318	0.691590	+	0.722290i	$0.256910\pi$
$468$	0	0
$469$	−7609.00	−0.749149
$470$	15750.0	1.54573
$471$	0	0
$472$	4056.00	0.395535
$473$	2992.00	0.290851
$474$	0	0
$475$	20540.0	1.98408
$476$	1512.00	0.145593
$477$	0	0
$478$	−7698.00	−0.736607
$479$	2922.00	0.278726	0.139363	−	0.990241i	$-0.455495\pi$
$479$	2922.00	0.278726	0.139363	+	0.990241i	$0.455495\pi$
$480$	0	0
$481$	−28535.0	−2.70496
$482$	−3542.00	−0.334717
$483$	0	0
$484$	484.000	0.0454545
$485$	−16968.0	−1.58861
$486$	0	0
$487$	−2464.00	−0.229270	−0.114635	−	0.993408i	$-0.536570\pi$
$487$	−2464.00	−0.229270	−0.114635	+	0.993408i	$0.536570\pi$
$488$	6064.00	0.562509
$489$	0	0
$490$	2058.00	0.189737
$491$	−18555.0	−1.70545	−0.852724	−	0.522361i	$-0.825051\pi$
$491$	−18555.0	−1.70545	−0.852724	+	0.522361i	$0.825051\pi$
$492$	0	0
$493$	−2106.00	−0.192392
$494$	8450.00	0.769602
$495$	0	0
$496$	−2848.00	−0.257821
$497$	0	0
$498$	0	0
$499$	14519.0	1.30252	0.651262	−	0.758853i	$-0.274240\pi$
$499$	14519.0	1.30252	0.651262	+	0.758853i	$0.274240\pi$
$500$	16044.0	1.43502
$501$	0	0
$502$	−7110.00	−0.632141
$503$	4644.00	0.411661	0.205831	−	0.978588i	$-0.434010\pi$
$503$	4644.00	0.411661	0.205831	+	0.978588i	$0.434010\pi$
$504$	0	0
$505$	25074.0	2.20946
$506$	−2904.00	−0.255135
$507$	0	0
$508$	−568.000	−0.0496081
$509$	−17214.0	−1.49901	−0.749506	−	0.661998i	$-0.769709\pi$
$509$	−17214.0	−1.49901	−0.749506	+	0.661998i	$0.769709\pi$
$510$	0	0
$511$	−4711.00	−0.407832
$512$	512.000	0.0441942
$513$	0	0
$514$	7242.00	0.621461
$515$	−32466.0	−2.77791
$516$	0	0
$517$	4125.00	0.350904
$518$	−6146.00	−0.521312
$519$	0	0
$520$	10920.0	0.920911
$521$	6957.00	0.585013	0.292506	−	0.956264i	$-0.405511\pi$
$521$	6957.00	0.585013	0.292506	+	0.956264i	$0.405511\pi$
$522$	0	0
$523$	15311.0	1.28012	0.640061	−	0.768325i	$-0.278909\pi$
$523$	15311.0	1.28012	0.640061	+	0.768325i	$0.278909\pi$
$524$	2208.00	0.184078
$525$	0	0
$526$	642.000	0.0532177
$527$	−9612.00	−0.794507
$528$	0	0
$529$	5257.00	0.432070
$530$	−25704.0	−2.10662
$531$	0	0
$532$	1820.00	0.148321
$533$	−6240.00	−0.507100
$534$	0	0
$535$	−29295.0	−2.36735
$536$	−8696.00	−0.700765
$537$	0	0
$538$	16740.0	1.34147
$539$	539.000	0.0430730
$540$	0	0
$541$	−6172.00	−0.490490	−0.245245	−	0.969461i	$-0.578868\pi$
$541$	−6172.00	−0.490490	−0.245245	+	0.969461i	$0.578868\pi$
$542$	3082.00	0.244250
$543$	0	0
$544$	1728.00	0.136190
$545$	−5250.00	−0.412634
$546$	0	0
$547$	−15064.0	−1.17750	−0.588748	−	0.808317i	$-0.700379\pi$
$547$	−15064.0	−1.17750	−0.588748	+	0.808317i	$0.700379\pi$
$548$	2904.00	0.226374
$549$	0	0
$550$	6952.00	0.538971
$551$	−2535.00	−0.195998
$552$	0	0
$553$	−4900.00	−0.376798
$554$	−6224.00	−0.477315
$555$	0	0
$556$	224.000	0.0170858
$557$	13509.0	1.02764	0.513819	−	0.857898i	$-0.328230\pi$
$557$	13509.0	1.02764	0.513819	+	0.857898i	$0.328230\pi$
$558$	0	0
$559$	17680.0	1.33772
$560$	2352.00	0.177482
$561$	0	0
$562$	9642.00	0.723707
$563$	6780.00	0.507536	0.253768	−	0.967265i	$-0.418330\pi$
$563$	6780.00	0.507536	0.253768	+	0.967265i	$0.418330\pi$
$564$	0	0
$565$	−4662.00	−0.347136
$566$	7474.00	0.555045
$567$	0	0
$568$	0	0
$569$	−6258.00	−0.461070	−0.230535	−	0.973064i	$-0.574048\pi$
$569$	−6258.00	−0.461070	−0.230535	+	0.973064i	$0.574048\pi$
$570$	0	0
$571$	21944.0	1.60828	0.804140	−	0.594440i	$-0.202626\pi$
$571$	21944.0	1.60828	0.804140	+	0.594440i	$0.202626\pi$
$572$	2860.00	0.209061
$573$	0	0
$574$	−1344.00	−0.0977308
$575$	−41712.0	−3.02524
$576$	0	0
$577$	26894.0	1.94040	0.970201	−	0.242302i	$-0.0779024\pi$
$577$	26894.0	1.94040	0.970201	+	0.242302i	$0.0779024\pi$
$578$	−3994.00	−0.287420
$579$	0	0
$580$	−3276.00	−0.234532
$581$	−8526.00	−0.608809
$582$	0	0
$583$	−6732.00	−0.478235
$584$	−5384.00	−0.381492
$585$	0	0
$586$	8160.00	0.575233
$587$	−1551.00	−0.109057	−0.0545286	−	0.998512i	$-0.517366\pi$
$587$	−1551.00	−0.109057	−0.0545286	+	0.998512i	$0.517366\pi$
$588$	0	0
$589$	−11570.0	−0.809395
$590$	21294.0	1.48586
$591$	0	0
$592$	−7024.00	−0.487643
$593$	−2808.00	−0.194453	−0.0972266	−	0.995262i	$-0.530997\pi$
$593$	−2808.00	−0.194453	−0.0972266	+	0.995262i	$0.530997\pi$
$594$	0	0
$595$	7938.00	0.546935
$596$	−12876.0	−0.884935
$597$	0	0
$598$	−17160.0	−1.17345
$599$	−9708.00	−0.662201	−0.331100	−	0.943596i	$-0.607420\pi$
$599$	−9708.00	−0.662201	−0.331100	+	0.943596i	$0.607420\pi$
$600$	0	0
$601$	−15973.0	−1.08411	−0.542057	−	0.840342i	$-0.682354\pi$
$601$	−15973.0	−1.08411	−0.542057	+	0.840342i	$0.682354\pi$
$602$	3808.00	0.257811
$603$	0	0
$604$	−424.000	−0.0285634
$605$	2541.00	0.170754
$606$	0	0
$607$	−10555.0	−0.705790	−0.352895	−	0.935663i	$-0.614803\pi$
$607$	−10555.0	−0.705790	−0.352895	+	0.935663i	$0.614803\pi$
$608$	2080.00	0.138742
$609$	0	0
$610$	31836.0	2.11312
$611$	24375.0	1.61392
$612$	0	0
$613$	6374.00	0.419973	0.209986	−	0.977704i	$-0.432658\pi$
$613$	6374.00	0.419973	0.209986	+	0.977704i	$0.432658\pi$
$614$	−19544.0	−1.28458
$615$	0	0
$616$	616.000	0.0402911
$617$	21306.0	1.39019	0.695095	−	0.718918i	$-0.255362\pi$
$617$	21306.0	1.39019	0.695095	+	0.718918i	$0.255362\pi$
$618$	0	0
$619$	13826.0	0.897761	0.448880	−	0.893592i	$-0.351823\pi$
$619$	13826.0	0.897761	0.448880	+	0.893592i	$0.351823\pi$
$620$	−14952.0	−0.968527
$621$	0	0
$622$	−10272.0	−0.662170
$623$	9450.00	0.607715
$624$	0	0
$625$	44731.0	2.86278
$626$	−8564.00	−0.546783
$627$	0	0
$628$	11456.0	0.727937
$629$	−23706.0	−1.50273
$630$	0	0
$631$	23420.0	1.47755	0.738776	−	0.673951i	$-0.235404\pi$
$631$	23420.0	1.47755	0.738776	+	0.673951i	$0.235404\pi$
$632$	−5600.00	−0.352462
$633$	0	0
$634$	1416.00	0.0887012
$635$	−2982.00	−0.186358
$636$	0	0
$637$	3185.00	0.198107
$638$	−858.000	−0.0532422
$639$	0	0
$640$	2688.00	0.166020
$641$	6522.00	0.401878	0.200939	−	0.979604i	$-0.435601\pi$
$641$	6522.00	0.401878	0.200939	+	0.979604i	$0.435601\pi$
$642$	0	0
$643$	−11968.0	−0.734015	−0.367008	−	0.930218i	$-0.619618\pi$
$643$	−11968.0	−0.734015	−0.367008	+	0.930218i	$0.619618\pi$
$644$	−3696.00	−0.226153
$645$	0	0
$646$	7020.00	0.427551
$647$	1269.00	0.0771090	0.0385545	−	0.999256i	$-0.487725\pi$
$647$	1269.00	0.0771090	0.0385545	+	0.999256i	$0.487725\pi$
$648$	0	0
$649$	5577.00	0.337313
$650$	41080.0	2.47891
$651$	0	0
$652$	−4636.00	−0.278466
$653$	12864.0	0.770915	0.385457	−	0.922726i	$-0.374044\pi$
$653$	12864.0	0.770915	0.385457	+	0.922726i	$0.374044\pi$
$654$	0	0
$655$	11592.0	0.691507
$656$	−1536.00	−0.0914188
$657$	0	0
$658$	5250.00	0.311043
$659$	10761.0	0.636099	0.318049	−	0.948074i	$-0.396972\pi$
$659$	10761.0	0.636099	0.318049	+	0.948074i	$0.396972\pi$
$660$	0	0
$661$	7688.00	0.452388	0.226194	−	0.974082i	$-0.427372\pi$
$661$	7688.00	0.452388	0.226194	+	0.974082i	$0.427372\pi$
$662$	5296.00	0.310929
$663$	0	0
$664$	−9744.00	−0.569489
$665$	9555.00	0.557183
$666$	0	0
$667$	5148.00	0.298848
$668$	9120.00	0.528239
$669$	0	0
$670$	−45654.0	−2.63249
$671$	8338.00	0.479709
$672$	0	0
$673$	−19708.0	−1.12881	−0.564404	−	0.825499i	$-0.690894\pi$
$673$	−19708.0	−1.12881	−0.564404	+	0.825499i	$0.690894\pi$
$674$	−8888.00	−0.507942
$675$	0	0
$676$	8112.00	0.461538
$677$	−4566.00	−0.259211	−0.129605	−	0.991566i	$-0.541371\pi$
$677$	−4566.00	−0.259211	−0.129605	+	0.991566i	$0.541371\pi$
$678$	0	0
$679$	−5656.00	−0.319672
$680$	9072.00	0.511611
$681$	0	0
$682$	−3916.00	−0.219870
$683$	−8478.00	−0.474966	−0.237483	−	0.971392i	$-0.576322\pi$
$683$	−8478.00	−0.474966	−0.237483	+	0.971392i	$0.576322\pi$
$684$	0	0
$685$	15246.0	0.850394
$686$	686.000	0.0381802
$687$	0	0
$688$	4352.00	0.241161
$689$	−39780.0	−2.19956
$690$	0	0
$691$	−6514.00	−0.358617	−0.179308	−	0.983793i	$-0.557386\pi$
$691$	−6514.00	−0.358617	−0.179308	+	0.983793i	$0.557386\pi$
$692$	12624.0	0.693486
$693$	0	0
$694$	6144.00	0.336056
$695$	1176.00	0.0641845
$696$	0	0
$697$	−5184.00	−0.281719
$698$	8410.00	0.456050
$699$	0	0
$700$	8848.00	0.477747
$701$	90.0000	0.00484915	0.00242457	−	0.999997i	$-0.499228\pi$
$701$	90.0000	0.00484915	0.00242457	+	0.999997i	$0.499228\pi$
$702$	0	0
$703$	−28535.0	−1.53089
$704$	704.000	0.0376889
$705$	0	0
$706$	−1698.00	−0.0905171
$707$	8358.00	0.444604
$708$	0	0
$709$	−3085.00	−0.163413	−0.0817064	−	0.996656i	$-0.526037\pi$
$709$	−3085.00	−0.163413	−0.0817064	+	0.996656i	$0.526037\pi$
$710$	0	0
$711$	0	0
$712$	10800.0	0.568465
$713$	23496.0	1.23413
$714$	0	0
$715$	15015.0	0.785355
$716$	−3456.00	−0.180387
$717$	0	0
$718$	−13752.0	−0.714791
$719$	−8415.00	−0.436476	−0.218238	−	0.975896i	$-0.570031\pi$
$719$	−8415.00	−0.436476	−0.218238	+	0.975896i	$0.570031\pi$
$720$	0	0
$721$	−10822.0	−0.558991
$722$	−5268.00	−0.271544
$723$	0	0
$724$	−640.000	−0.0328528
$725$	−12324.0	−0.631313
$726$	0	0
$727$	−16360.0	−0.834606	−0.417303	−	0.908767i	$-0.637025\pi$
$727$	−16360.0	−0.834606	−0.417303	+	0.908767i	$0.637025\pi$
$728$	3640.00	0.185312
$729$	0	0
$730$	−28266.0	−1.43311
$731$	14688.0	0.743167
$732$	0	0
$733$	3206.00	0.161550	0.0807751	−	0.996732i	$-0.474260\pi$
$733$	3206.00	0.161550	0.0807751	+	0.996732i	$0.474260\pi$
$734$	−16412.0	−0.825311
$735$	0	0
$736$	−4224.00	−0.211547
$737$	−11957.0	−0.597614
$738$	0	0
$739$	−23128.0	−1.15125	−0.575627	−	0.817712i	$-0.695242\pi$
$739$	−23128.0	−1.15125	−0.575627	+	0.817712i	$0.695242\pi$
$740$	−36876.0	−1.83188
$741$	0	0
$742$	−8568.00	−0.423910
$743$	−38031.0	−1.87782	−0.938911	−	0.344159i	$-0.888164\pi$
$743$	−38031.0	−1.87782	−0.938911	+	0.344159i	$0.888164\pi$
$744$	0	0
$745$	−67599.0	−3.32434
$746$	−18644.0	−0.915021
$747$	0	0
$748$	2376.00	0.116143
$749$	−9765.00	−0.476376
$750$	0	0
$751$	641.000	0.0311457	0.0155729	−	0.999879i	$-0.495043\pi$
$751$	641.000	0.0311457	0.0155729	+	0.999879i	$0.495043\pi$
$752$	6000.00	0.290954
$753$	0	0
$754$	−5070.00	−0.244879
$755$	−2226.00	−0.107301
$756$	0	0
$757$	3809.00	0.182880	0.0914402	−	0.995811i	$-0.470853\pi$
$757$	3809.00	0.182880	0.0914402	+	0.995811i	$0.470853\pi$
$758$	−19202.0	−0.920116
$759$	0	0
$760$	10920.0	0.521197
$761$	2676.00	0.127470	0.0637352	−	0.997967i	$-0.479699\pi$
$761$	2676.00	0.127470	0.0637352	+	0.997967i	$0.479699\pi$
$762$	0	0
$763$	−1750.00	−0.0830331
$764$	5640.00	0.267079
$765$	0	0
$766$	−1776.00	−0.0837722
$767$	32955.0	1.55142
$768$	0	0
$769$	26885.0	1.26073	0.630363	−	0.776301i	$-0.282906\pi$
$769$	26885.0	1.26073	0.630363	+	0.776301i	$0.282906\pi$
$770$	3234.00	0.151357
$771$	0	0
$772$	11744.0	0.547508
$773$	−32319.0	−1.50380	−0.751898	−	0.659280i	$-0.770861\pi$
$773$	−32319.0	−1.50380	−0.751898	+	0.659280i	$0.770861\pi$
$774$	0	0
$775$	−56248.0	−2.60708
$776$	−6464.00	−0.299026
$777$	0	0
$778$	20424.0	0.941177
$779$	−6240.00	−0.286998
$780$	0	0
$781$	0	0
$782$	−14256.0	−0.651910
$783$	0	0
$784$	784.000	0.0357143
$785$	60144.0	2.73456
$786$	0	0
$787$	−5569.00	−0.252241	−0.126120	−	0.992015i	$-0.540253\pi$
$787$	−5569.00	−0.252241	−0.126120	+	0.992015i	$0.540253\pi$
$788$	4680.00	0.211571
$789$	0	0
$790$	−29400.0	−1.32406
$791$	−1554.00	−0.0698532
$792$	0	0
$793$	49270.0	2.20634
$794$	−5108.00	−0.228307
$795$	0	0
$796$	20672.0	0.920477
$797$	13431.0	0.596927	0.298463	−	0.954421i	$-0.403526\pi$
$797$	13431.0	0.596927	0.298463	+	0.954421i	$0.403526\pi$
$798$	0	0
$799$	20250.0	0.896613
$800$	10112.0	0.446891
$801$	0	0
$802$	−984.000	−0.0433245
$803$	−7403.00	−0.325338
$804$	0	0
$805$	−19404.0	−0.849567
$806$	−23140.0	−1.01126
$807$	0	0
$808$	9552.00	0.415889
$809$	−22959.0	−0.997769	−0.498885	−	0.866668i	$-0.666257\pi$
$809$	−22959.0	−0.997769	−0.498885	+	0.866668i	$0.666257\pi$
$810$	0	0
$811$	11135.0	0.482124	0.241062	−	0.970510i	$-0.422504\pi$
$811$	11135.0	0.482124	0.241062	+	0.970510i	$0.422504\pi$
$812$	−1092.00	−0.0471942
$813$	0	0
$814$	−9658.00	−0.415863
$815$	−24339.0	−1.04608
$816$	0	0
$817$	17680.0	0.757093
$818$	−26708.0	−1.14159
$819$	0	0
$820$	−8064.00	−0.343423
$821$	18411.0	0.782641	0.391321	−	0.920254i	$-0.372018\pi$
$821$	18411.0	0.782641	0.391321	+	0.920254i	$0.372018\pi$
$822$	0	0
$823$	8345.00	0.353449	0.176724	−	0.984260i	$-0.443450\pi$
$823$	8345.00	0.353449	0.176724	+	0.984260i	$0.443450\pi$
$824$	−12368.0	−0.522888
$825$	0	0
$826$	7098.00	0.298996
$827$	2133.00	0.0896876	0.0448438	−	0.998994i	$-0.485721\pi$
$827$	2133.00	0.0896876	0.0448438	+	0.998994i	$0.485721\pi$
$828$	0	0
$829$	26768.0	1.12146	0.560730	−	0.827998i	$-0.310520\pi$
$829$	26768.0	1.12146	0.560730	+	0.827998i	$0.310520\pi$
$830$	−51156.0	−2.13934
$831$	0	0
$832$	4160.00	0.173344
$833$	2646.00	0.110058
$834$	0	0
$835$	47880.0	1.98438
$836$	2860.00	0.118320
$837$	0	0
$838$	−9030.00	−0.372239
$839$	21399.0	0.880543	0.440271	−	0.897865i	$-0.354882\pi$
$839$	21399.0	0.880543	0.440271	+	0.897865i	$0.354882\pi$
$840$	0	0
$841$	−22868.0	−0.937636
$842$	−2354.00	−0.0963470
$843$	0	0
$844$	8648.00	0.352697
$845$	42588.0	1.73381
$846$	0	0
$847$	847.000	0.0343604
$848$	−9792.00	−0.396531
$849$	0	0
$850$	34128.0	1.37715
$851$	57948.0	2.33423
$852$	0	0
$853$	−11014.0	−0.442101	−0.221051	−	0.975262i	$-0.570949\pi$
$853$	−11014.0	−0.442101	−0.221051	+	0.975262i	$0.570949\pi$
$854$	10612.0	0.425217
$855$	0	0
$856$	−11160.0	−0.445609
$857$	−38514.0	−1.53514	−0.767569	−	0.640966i	$-0.778534\pi$
$857$	−38514.0	−1.53514	−0.767569	+	0.640966i	$0.778534\pi$
$858$	0	0
$859$	−43108.0	−1.71225	−0.856127	−	0.516766i	$-0.827136\pi$
$859$	−43108.0	−1.71225	−0.856127	+	0.516766i	$0.827136\pi$
$860$	22848.0	0.905942
$861$	0	0
$862$	−2430.00	−0.0960164
$863$	−25056.0	−0.988315	−0.494158	−	0.869372i	$-0.664523\pi$
$863$	−25056.0	−0.988315	−0.494158	+	0.869372i	$0.664523\pi$
$864$	0	0
$865$	66276.0	2.60515
$866$	−13424.0	−0.526751
$867$	0	0
$868$	−4984.00	−0.194894
$869$	−7700.00	−0.300581
$870$	0	0
$871$	−70655.0	−2.74863
$872$	−2000.00	−0.0776704
$873$	0	0
$874$	−17160.0	−0.664125
$875$	28077.0	1.08477
$876$	0	0
$877$	38432.0	1.47977	0.739884	−	0.672735i	$-0.234880\pi$
$877$	38432.0	1.47977	0.739884	+	0.672735i	$0.234880\pi$
$878$	−23702.0	−0.911052
$879$	0	0
$880$	3696.00	0.141582
$881$	9621.00	0.367923	0.183961	−	0.982933i	$-0.441108\pi$
$881$	9621.00	0.367923	0.183961	+	0.982933i	$0.441108\pi$
$882$	0	0
$883$	17255.0	0.657618	0.328809	−	0.944396i	$-0.393353\pi$
$883$	17255.0	0.657618	0.328809	+	0.944396i	$0.393353\pi$
$884$	14040.0	0.534181
$885$	0	0
$886$	15240.0	0.577876
$887$	15906.0	0.602110	0.301055	−	0.953607i	$-0.402661\pi$
$887$	15906.0	0.602110	0.301055	+	0.953607i	$0.402661\pi$
$888$	0	0
$889$	−994.000	−0.0375002
$890$	56700.0	2.13549
$891$	0	0
$892$	−10072.0	−0.378067
$893$	24375.0	0.913414
$894$	0	0
$895$	−18144.0	−0.677639
$896$	896.000	0.0334077
$897$	0	0
$898$	4680.00	0.173913
$899$	6942.00	0.257540
$900$	0	0
$901$	−33048.0	−1.22196
$902$	−2112.00	−0.0779622
$903$	0	0
$904$	−1776.00	−0.0653417
$905$	−3360.00	−0.123415
$906$	0	0
$907$	−49372.0	−1.80746	−0.903732	−	0.428098i	$-0.859184\pi$
$907$	−49372.0	−1.80746	−0.903732	+	0.428098i	$0.859184\pi$
$908$	8232.00	0.300868
$909$	0	0
$910$	19110.0	0.696143
$911$	−42618.0	−1.54994	−0.774971	−	0.631997i	$-0.782236\pi$
$911$	−42618.0	−1.54994	−0.774971	+	0.631997i	$0.782236\pi$
$912$	0	0
$913$	−13398.0	−0.485662
$914$	−27968.0	−1.01214
$915$	0	0
$916$	−4672.00	−0.168523
$917$	3864.00	0.139150
$918$	0	0
$919$	−28456.0	−1.02141	−0.510706	−	0.859756i	$-0.670616\pi$
$919$	−28456.0	−1.02141	−0.510706	+	0.859756i	$0.670616\pi$
$920$	−22176.0	−0.794697
$921$	0	0
$922$	−11688.0	−0.417488
$923$	0	0
$924$	0	0
$925$	−138724.	−4.93105
$926$	15250.0	0.541194
$927$	0	0
$928$	−1248.00	−0.0441461
$929$	12831.0	0.453145	0.226572	−	0.973994i	$-0.427248\pi$
$929$	12831.0	0.453145	0.226572	+	0.973994i	$0.427248\pi$
$930$	0	0
$931$	3185.00	0.112120
$932$	9000.00	0.316314
$933$	0	0
$934$	27918.0	0.978057
$935$	12474.0	0.436303
$936$	0	0
$937$	−35206.0	−1.22746	−0.613730	−	0.789516i	$-0.710332\pi$
$937$	−35206.0	−1.22746	−0.613730	+	0.789516i	$0.710332\pi$
$938$	−15218.0	−0.529728
$939$	0	0
$940$	31500.0	1.09300
$941$	−46680.0	−1.61714	−0.808568	−	0.588403i	$-0.799757\pi$
$941$	−46680.0	−1.61714	−0.808568	+	0.588403i	$0.799757\pi$
$942$	0	0
$943$	12672.0	0.437600
$944$	8112.00	0.279685
$945$	0	0
$946$	5984.00	0.205662
$947$	−37110.0	−1.27340	−0.636702	−	0.771110i	$-0.719702\pi$
$947$	−37110.0	−1.27340	−0.636702	+	0.771110i	$0.719702\pi$
$948$	0	0
$949$	−43745.0	−1.49634
$950$	41080.0	1.40296
$951$	0	0
$952$	3024.00	0.102950
$953$	11241.0	0.382090	0.191045	−	0.981581i	$-0.438812\pi$
$953$	11241.0	0.382090	0.191045	+	0.981581i	$0.438812\pi$
$954$	0	0
$955$	29610.0	1.00331
$956$	−15396.0	−0.520860
$957$	0	0
$958$	5844.00	0.197089
$959$	5082.00	0.171122
$960$	0	0
$961$	1893.00	0.0635427
$962$	−57070.0	−1.91269
$963$	0	0
$964$	−7084.00	−0.236681
$965$	61656.0	2.05676
$966$	0	0
$967$	10010.0	0.332885	0.166443	−	0.986051i	$-0.446772\pi$
$967$	10010.0	0.332885	0.166443	+	0.986051i	$0.446772\pi$
$968$	968.000	0.0321412
$969$	0	0
$970$	−33936.0	−1.12332
$971$	−29367.0	−0.970579	−0.485289	−	0.874354i	$-0.661286\pi$
$971$	−29367.0	−0.970579	−0.485289	+	0.874354i	$0.661286\pi$
$972$	0	0
$973$	392.000	0.0129157
$974$	−4928.00	−0.162118
$975$	0	0
$976$	12128.0	0.397754
$977$	32016.0	1.04840	0.524198	−	0.851597i	$-0.324365\pi$
$977$	32016.0	1.04840	0.524198	+	0.851597i	$0.324365\pi$
$978$	0	0
$979$	14850.0	0.484789
$980$	4116.00	0.134164
$981$	0	0
$982$	−37110.0	−1.20593
$983$	−6888.00	−0.223492	−0.111746	−	0.993737i	$-0.535644\pi$
$983$	−6888.00	−0.223492	−0.111746	+	0.993737i	$0.535644\pi$
$984$	0	0
$985$	24570.0	0.794787
$986$	−4212.00	−0.136042
$987$	0	0
$988$	16900.0	0.544191
$989$	−35904.0	−1.15438
$990$	0	0
$991$	12233.0	0.392123	0.196062	−	0.980592i	$-0.437185\pi$
$991$	12233.0	0.392123	0.196062	+	0.980592i	$0.437185\pi$
$992$	−5696.00	−0.182307
$993$	0	0
$994$	0	0
$995$	108528.	3.45786
$996$	0	0
$997$	7562.00	0.240212	0.120106	−	0.992761i	$-0.461677\pi$
$997$	7562.00	0.240212	0.120106	+	0.992761i	$0.461677\pi$
$998$	29038.0	0.921024
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.a.1.1	✓	1	3.2	odd	2
1386.4.a.n.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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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