LMFDB - Embedded newform 147.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [147,6,Mod(1,147)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(147, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("147.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$147 = 3 \cdot 7^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	147.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:	$23.5764215125$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	147.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-6.00000 q^{2} +9.00000 q^{3} +4.00000 q^{4} -78.0000 q^{5} -54.0000 q^{6} +168.000 q^{8} +81.0000 q^{9} +468.000 q^{10} +444.000 q^{11} +36.0000 q^{12} +442.000 q^{13} -702.000 q^{15} -1136.00 q^{16} +126.000 q^{17} -486.000 q^{18} -2684.00 q^{19} -312.000 q^{20} -2664.00 q^{22} +4200.00 q^{23} +1512.00 q^{24} +2959.00 q^{25} -2652.00 q^{26} +729.000 q^{27} -5442.00 q^{29} +4212.00 q^{30} -80.0000 q^{31} +1440.00 q^{32} +3996.00 q^{33} -756.000 q^{34} +324.000 q^{36} -5434.00 q^{37} +16104.0 q^{38} +3978.00 q^{39} -13104.0 q^{40} -7962.00 q^{41} -11524.0 q^{43} +1776.00 q^{44} -6318.00 q^{45} -25200.0 q^{46} +13920.0 q^{47} -10224.0 q^{48} -17754.0 q^{50} +1134.00 q^{51} +1768.00 q^{52} -9594.00 q^{53} -4374.00 q^{54} -34632.0 q^{55} -24156.0 q^{57} +32652.0 q^{58} -27492.0 q^{59} -2808.00 q^{60} -49478.0 q^{61} +480.000 q^{62} +27712.0 q^{64} -34476.0 q^{65} -23976.0 q^{66} -59356.0 q^{67} +504.000 q^{68} +37800.0 q^{69} +32040.0 q^{71} +13608.0 q^{72} +61846.0 q^{73} +32604.0 q^{74} +26631.0 q^{75} -10736.0 q^{76} -23868.0 q^{78} -65776.0 q^{79} +88608.0 q^{80} +6561.00 q^{81} +47772.0 q^{82} -40188.0 q^{83} -9828.00 q^{85} +69144.0 q^{86} -48978.0 q^{87} +74592.0 q^{88} +7974.00 q^{89} +37908.0 q^{90} +16800.0 q^{92} -720.000 q^{93} -83520.0 q^{94} +209352. q^{95} +12960.0 q^{96} +143662. q^{97} +35964.0 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−6.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−6.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	4.00000	0.125000
$5$	−78.0000	−1.39531	−0.697653	−	0.716436i	$-0.745772\pi$
$5$	−78.0000	−1.39531	−0.697653	+	0.716436i	$0.745772\pi$
$6$	−54.0000	−0.612372
$7$	0	0
$7$	0	0
$8$	168.000	0.928078
$9$	81.0000	0.333333
$10$	468.000	1.47995
$11$	444.000	1.10637	0.553186	−	0.833058i	$-0.313412\pi$
$11$	444.000	1.10637	0.553186	+	0.833058i	$0.313412\pi$
$12$	36.0000	0.0721688
$13$	442.000	0.725377	0.362689	−	0.931910i	$-0.381859\pi$
$13$	442.000	0.725377	0.362689	+	0.931910i	$0.381859\pi$
$14$	0	0
$15$	−702.000	−0.805581
$16$	−1136.00	−1.10938
$17$	126.000	0.105742	0.0528711	−	0.998601i	$-0.483163\pi$
$17$	126.000	0.105742	0.0528711	+	0.998601i	$0.483163\pi$
$18$	−486.000	−0.353553
$19$	−2684.00	−1.70568	−0.852842	−	0.522169i	$-0.825123\pi$
$19$	−2684.00	−1.70568	−0.852842	+	0.522169i	$0.825123\pi$
$20$	−312.000	−0.174413
$21$	0	0
$22$	−2664.00	−1.17348
$23$	4200.00	1.65550	0.827751	−	0.561096i	$-0.189620\pi$
$23$	4200.00	1.65550	0.827751	+	0.561096i	$0.189620\pi$
$24$	1512.00	0.535826
$25$	2959.00	0.946880
$26$	−2652.00	−0.769379
$27$	729.000	0.192450
$28$	0	0
$29$	−5442.00	−1.20161	−0.600805	−	0.799396i	$-0.705153\pi$
$29$	−5442.00	−1.20161	−0.600805	+	0.799396i	$0.705153\pi$
$30$	4212.00	0.854447
$31$	−80.0000	−0.0149515	−0.00747577	−	0.999972i	$-0.502380\pi$
$31$	−80.0000	−0.0149515	−0.00747577	+	0.999972i	$0.502380\pi$
$32$	1440.00	0.248592
$33$	3996.00	0.638764
$34$	−756.000	−0.112157
$35$	0	0
$36$	324.000	0.0416667
$37$	−5434.00	−0.652552	−0.326276	−	0.945274i	$-0.605794\pi$
$37$	−5434.00	−0.652552	−0.326276	+	0.945274i	$0.605794\pi$
$38$	16104.0	1.80915
$39$	3978.00	0.418797
$40$	−13104.0	−1.29495
$41$	−7962.00	−0.739712	−0.369856	−	0.929089i	$-0.620593\pi$
$41$	−7962.00	−0.739712	−0.369856	+	0.929089i	$0.620593\pi$
$42$	0	0
$43$	−11524.0	−0.950456	−0.475228	−	0.879863i	$-0.657634\pi$
$43$	−11524.0	−0.950456	−0.475228	+	0.879863i	$0.657634\pi$
$44$	1776.00	0.138297
$45$	−6318.00	−0.465102
$46$	−25200.0	−1.75592
$47$	13920.0	0.919167	0.459584	−	0.888134i	$-0.347999\pi$
$47$	13920.0	0.919167	0.459584	+	0.888134i	$0.347999\pi$
$48$	−10224.0	−0.640498
$49$	0	0
$50$	−17754.0	−1.00432
$51$	1134.00	0.0610503
$52$	1768.00	0.0906721
$53$	−9594.00	−0.469148	−0.234574	−	0.972098i	$-0.575370\pi$
$53$	−9594.00	−0.469148	−0.234574	+	0.972098i	$0.575370\pi$
$54$	−4374.00	−0.204124
$55$	−34632.0	−1.54373
$56$	0	0
$57$	−24156.0	−0.984777
$58$	32652.0	1.27450
$59$	−27492.0	−1.02820	−0.514098	−	0.857731i	$-0.671873\pi$
$59$	−27492.0	−1.02820	−0.514098	+	0.857731i	$0.671873\pi$
$60$	−2808.00	−0.100698
$61$	−49478.0	−1.70250	−0.851251	−	0.524759i	$-0.824155\pi$
$61$	−49478.0	−1.70250	−0.851251	+	0.524759i	$0.824155\pi$
$62$	480.000	0.0158585
$63$	0	0
$64$	27712.0	0.845703
$65$	−34476.0	−1.01212
$66$	−23976.0	−0.677512
$67$	−59356.0	−1.61539	−0.807695	−	0.589600i	$-0.799285\pi$
$67$	−59356.0	−1.61539	−0.807695	+	0.589600i	$0.799285\pi$
$68$	504.000	0.0132178
$69$	37800.0	0.955805
$70$	0	0
$71$	32040.0	0.754304	0.377152	−	0.926151i	$-0.376903\pi$
$71$	32040.0	0.754304	0.377152	+	0.926151i	$0.376903\pi$
$72$	13608.0	0.309359
$73$	61846.0	1.35833	0.679164	−	0.733987i	$-0.262343\pi$
$73$	61846.0	1.35833	0.679164	+	0.733987i	$0.262343\pi$
$74$	32604.0	0.692136
$75$	26631.0	0.546681
$76$	−10736.0	−0.213210
$77$	0	0
$78$	−23868.0	−0.444201
$79$	−65776.0	−1.18577	−0.592884	−	0.805288i	$-0.702011\pi$
$79$	−65776.0	−1.18577	−0.592884	+	0.805288i	$0.702011\pi$
$80$	88608.0	1.54792
$81$	6561.00	0.111111
$82$	47772.0	0.784583
$83$	−40188.0	−0.640326	−0.320163	−	0.947362i	$-0.603738\pi$
$83$	−40188.0	−0.640326	−0.320163	+	0.947362i	$0.603738\pi$
$84$	0	0
$85$	−9828.00	−0.147543
$86$	69144.0	1.00811
$87$	−48978.0	−0.693750
$88$	74592.0	1.02680
$89$	7974.00	0.106709	0.0533545	−	0.998576i	$-0.483009\pi$
$89$	7974.00	0.106709	0.0533545	+	0.998576i	$0.483009\pi$
$90$	37908.0	0.493315
$91$	0	0
$92$	16800.0	0.206938
$93$	−720.000	−0.00863227
$94$	−83520.0	−0.974924
$95$	209352.	2.37995
$96$	12960.0	0.143525
$97$	143662.	1.55029	0.775144	−	0.631784i	$-0.217677\pi$
$97$	143662.	1.55029	0.775144	+	0.631784i	$0.217677\pi$
$98$	0	0
$99$	35964.0	0.368791
$100$	11836.0	0.118360
$101$	2706.00	0.0263952	0.0131976	−	0.999913i	$-0.495799\pi$
$101$	2706.00	0.0263952	0.0131976	+	0.999913i	$0.495799\pi$
$102$	−6804.00	−0.0647536
$103$	−131768.	−1.22382	−0.611909	−	0.790928i	$-0.709598\pi$
$103$	−131768.	−1.22382	−0.611909	+	0.790928i	$0.709598\pi$
$104$	74256.0	0.673206
$105$	0	0
$106$	57564.0	0.497607
$107$	−128916.	−1.08855	−0.544274	−	0.838908i	$-0.683195\pi$
$107$	−128916.	−1.08855	−0.544274	+	0.838908i	$0.683195\pi$
$108$	2916.00	0.0240563
$109$	−100978.	−0.814068	−0.407034	−	0.913413i	$-0.633437\pi$
$109$	−100978.	−0.814068	−0.407034	+	0.913413i	$0.633437\pi$
$110$	207792.	1.63737
$111$	−48906.0	−0.376751
$112$	0	0
$113$	220146.	1.62186	0.810932	−	0.585140i	$-0.198960\pi$
$113$	220146.	1.62186	0.810932	+	0.585140i	$0.198960\pi$
$114$	144936.	1.04451
$115$	−327600.	−2.30993
$116$	−21768.0	−0.150201
$117$	35802.0	0.241792
$118$	164952.	1.09057
$119$	0	0
$120$	−117936.	−0.747641
$121$	36085.0	0.224059
$122$	296868.	1.80578
$123$	−71658.0	−0.427073
$124$	−320.000	−0.00186894
$125$	12948.0	0.0741187
$126$	0	0
$127$	−74320.0	−0.408880	−0.204440	−	0.978879i	$-0.565537\pi$
$127$	−74320.0	−0.408880	−0.204440	+	0.978879i	$0.565537\pi$
$128$	−212352.	−1.14560
$129$	−103716.	−0.548746
$130$	206856.	1.07352
$131$	155316.	0.790748	0.395374	−	0.918520i	$-0.370615\pi$
$131$	155316.	0.790748	0.395374	+	0.918520i	$0.370615\pi$
$132$	15984.0	0.0798455
$133$	0	0
$134$	356136.	1.71338
$135$	−56862.0	−0.268527
$136$	21168.0	0.0981369
$137$	−264246.	−1.20284	−0.601419	−	0.798934i	$-0.705398\pi$
$137$	−264246.	−1.20284	−0.601419	+	0.798934i	$0.705398\pi$
$138$	−226800.	−1.01378
$139$	−224612.	−0.986043	−0.493022	−	0.870017i	$-0.664108\pi$
$139$	−224612.	−0.986043	−0.493022	+	0.870017i	$0.664108\pi$
$140$	0	0
$141$	125280.	0.530682
$142$	−192240.	−0.800061
$143$	196248.	0.802537
$144$	−92016.0	−0.369792
$145$	424476.	1.67661
$146$	−371076.	−1.44072
$147$	0	0
$148$	−21736.0	−0.0815690
$149$	−82074.0	−0.302859	−0.151429	−	0.988468i	$-0.548388\pi$
$149$	−82074.0	−0.302859	−0.151429	+	0.988468i	$0.548388\pi$
$150$	−159786.	−0.579843
$151$	−287032.	−1.02444	−0.512222	−	0.858853i	$-0.671177\pi$
$151$	−287032.	−1.02444	−0.512222	+	0.858853i	$0.671177\pi$
$152$	−450912.	−1.58301
$153$	10206.0	0.0352474
$154$	0	0
$155$	6240.00	0.0208620
$156$	15912.0	0.0523496
$157$	−129878.	−0.420520	−0.210260	−	0.977646i	$-0.567431\pi$
$157$	−129878.	−0.420520	−0.210260	+	0.977646i	$0.567431\pi$
$158$	394656.	1.25770
$159$	−86346.0	−0.270863
$160$	−112320.	−0.346862
$161$	0	0
$162$	−39366.0	−0.117851
$163$	555284.	1.63699	0.818495	−	0.574513i	$-0.194809\pi$
$163$	555284.	1.63699	0.818495	+	0.574513i	$0.194809\pi$
$164$	−31848.0	−0.0924640
$165$	−311688.	−0.891272
$166$	241128.	0.679168
$167$	−43512.0	−0.120731	−0.0603654	−	0.998176i	$-0.519227\pi$
$167$	−43512.0	−0.120731	−0.0603654	+	0.998176i	$0.519227\pi$
$168$	0	0
$169$	−175929.	−0.473828
$170$	58968.0	0.156493
$171$	−217404.	−0.568561
$172$	−46096.0	−0.118807
$173$	18330.0	0.0465637	0.0232818	−	0.999729i	$-0.492588\pi$
$173$	18330.0	0.0465637	0.0232818	+	0.999729i	$0.492588\pi$
$174$	293868.	0.735833
$175$	0	0
$176$	−504384.	−1.22738
$177$	−247428.	−0.593630
$178$	−47844.0	−0.113182
$179$	−153324.	−0.357666	−0.178833	−	0.983879i	$-0.557232\pi$
$179$	−153324.	−0.357666	−0.178833	+	0.983879i	$0.557232\pi$
$180$	−25272.0	−0.0581378
$181$	382066.	0.866846	0.433423	−	0.901191i	$-0.357306\pi$
$181$	382066.	0.866846	0.433423	+	0.901191i	$0.357306\pi$
$182$	0	0
$183$	−445302.	−0.982940
$184$	705600.	1.53643
$185$	423852.	0.910510
$186$	4320.00	0.00915591
$187$	55944.0	0.116990
$188$	55680.0	0.114896
$189$	0	0
$190$	−1.25611e6	−2.52432
$191$	−273408.	−0.542285	−0.271143	−	0.962539i	$-0.587402\pi$
$191$	−273408.	−0.542285	−0.271143	+	0.962539i	$0.587402\pi$
$192$	249408.	0.488267
$193$	153602.	0.296827	0.148414	−	0.988925i	$-0.452583\pi$
$193$	153602.	0.296827	0.148414	+	0.988925i	$0.452583\pi$
$194$	−861972.	−1.64433
$195$	−310284.	−0.584350
$196$	0	0
$197$	154422.	0.283494	0.141747	−	0.989903i	$-0.454728\pi$
$197$	154422.	0.283494	0.141747	+	0.989903i	$0.454728\pi$
$198$	−215784.	−0.391162
$199$	366856.	0.656694	0.328347	−	0.944557i	$-0.393508\pi$
$199$	366856.	0.656694	0.328347	+	0.944557i	$0.393508\pi$
$200$	497112.	0.878778
$201$	−534204.	−0.932646
$202$	−16236.0	−0.0279963
$203$	0	0
$204$	4536.00	0.00763128
$205$	621036.	1.03212
$206$	790608.	1.29806
$207$	340200.	0.551834
$208$	−502112.	−0.804715
$209$	−1.19170e6	−1.88712
$210$	0	0
$211$	520244.	0.804453	0.402227	−	0.915540i	$-0.368236\pi$
$211$	520244.	0.804453	0.402227	+	0.915540i	$0.368236\pi$
$212$	−38376.0	−0.0586435
$213$	288360.	0.435498
$214$	773496.	1.15458
$215$	898872.	1.32618
$216$	122472.	0.178609
$217$	0	0
$218$	605868.	0.863449
$219$	556614.	0.784231
$220$	−138528.	−0.192966
$221$	55692.0	0.0767030
$222$	293436.	0.399605
$223$	−304736.	−0.410357	−0.205178	−	0.978725i	$-0.565777\pi$
$223$	−304736.	−0.410357	−0.205178	+	0.978725i	$0.565777\pi$
$224$	0	0
$225$	239679.	0.315627
$226$	−1.32088e6	−1.72025
$227$	−288588.	−0.371718	−0.185859	−	0.982576i	$-0.559507\pi$
$227$	−288588.	−0.371718	−0.185859	+	0.982576i	$0.559507\pi$
$228$	−96624.0	−0.123097
$229$	−772190.	−0.973051	−0.486525	−	0.873666i	$-0.661736\pi$
$229$	−772190.	−0.973051	−0.486525	+	0.873666i	$0.661736\pi$
$230$	1.96560e6	2.45005
$231$	0	0
$232$	−914256.	−1.11519
$233$	252234.	0.304378	0.152189	−	0.988351i	$-0.451368\pi$
$233$	252234.	0.304378	0.152189	+	0.988351i	$0.451368\pi$
$234$	−214812.	−0.256460
$235$	−1.08576e6	−1.28252
$236$	−109968.	−0.128525
$237$	−591984.	−0.684603
$238$	0	0
$239$	−1.45114e6	−1.64329	−0.821643	−	0.570002i	$-0.806942\pi$
$239$	−1.45114e6	−1.64329	−0.821643	+	0.570002i	$0.806942\pi$
$240$	797472.	0.893691
$241$	146398.	0.162365	0.0811825	−	0.996699i	$-0.474130\pi$
$241$	146398.	0.162365	0.0811825	+	0.996699i	$0.474130\pi$
$242$	−216510.	−0.237651
$243$	59049.0	0.0641500
$244$	−197912.	−0.212813
$245$	0	0
$246$	429948.	0.452979
$247$	−1.18633e6	−1.23726
$248$	−13440.0	−0.0138762
$249$	−361692.	−0.369692
$250$	−77688.0	−0.0786147
$251$	−607860.	−0.609003	−0.304501	−	0.952512i	$-0.598490\pi$
$251$	−607860.	−0.609003	−0.304501	+	0.952512i	$0.598490\pi$
$252$	0	0
$253$	1.86480e6	1.83160
$254$	445920.	0.433683
$255$	−88452.0	−0.0851838
$256$	387328.	0.369385
$257$	−95586.0	−0.0902737	−0.0451369	−	0.998981i	$-0.514372\pi$
$257$	−95586.0	−0.0902737	−0.0451369	+	0.998981i	$0.514372\pi$
$258$	622296.	0.582033
$259$	0	0
$260$	−137904.	−0.126515
$261$	−440802.	−0.400537
$262$	−931896.	−0.838715
$263$	−2.20034e6	−1.96156	−0.980779	−	0.195121i	$-0.937490\pi$
$263$	−2.20034e6	−1.96156	−0.980779	+	0.195121i	$0.937490\pi$
$264$	671328.	0.592823
$265$	748332.	0.654605
$266$	0	0
$267$	71766.0	0.0616085
$268$	−237424.	−0.201924
$269$	−1.77025e6	−1.49160	−0.745801	−	0.666169i	$-0.767933\pi$
$269$	−1.77025e6	−1.49160	−0.745801	+	0.666169i	$0.767933\pi$
$270$	341172.	0.284816
$271$	223504.	0.184868	0.0924341	−	0.995719i	$-0.470535\pi$
$271$	223504.	0.184868	0.0924341	+	0.995719i	$0.470535\pi$
$272$	−143136.	−0.117308
$273$	0	0
$274$	1.58548e6	1.27580
$275$	1.31380e6	1.04760
$276$	151200.	0.119476
$277$	−342778.	−0.268419	−0.134210	−	0.990953i	$-0.542850\pi$
$277$	−342778.	−0.268419	−0.134210	+	0.990953i	$0.542850\pi$
$278$	1.34767e6	1.04586
$279$	−6480.00	−0.00498384
$280$	0	0
$281$	480378.	0.362925	0.181463	−	0.983398i	$-0.441917\pi$
$281$	480378.	0.362925	0.181463	+	0.983398i	$0.441917\pi$
$282$	−751680.	−0.562873
$283$	29980.0	0.0222518	0.0111259	−	0.999938i	$-0.496458\pi$
$283$	29980.0	0.0222518	0.0111259	+	0.999938i	$0.496458\pi$
$284$	128160.	0.0942880
$285$	1.88417e6	1.37407
$286$	−1.17749e6	−0.851219
$287$	0	0
$288$	116640.	0.0828641
$289$	−1.40398e6	−0.988819
$290$	−2.54686e6	−1.77832
$291$	1.29296e6	0.895060
$292$	247384.	0.169791
$293$	198066.	0.134785	0.0673924	−	0.997727i	$-0.478532\pi$
$293$	198066.	0.134785	0.0673924	+	0.997727i	$0.478532\pi$
$294$	0	0
$295$	2.14438e6	1.43465
$296$	−912912.	−0.605619
$297$	323676.	0.212921
$298$	492444.	0.321230
$299$	1.85640e6	1.20086
$300$	106524.	0.0683352
$301$	0	0
$302$	1.72219e6	1.08659
$303$	24354.0	0.0152393
$304$	3.04902e6	1.89224
$305$	3.85928e6	2.37551
$306$	−61236.0	−0.0373855
$307$	1.04564e6	0.633191	0.316595	−	0.948561i	$-0.397460\pi$
$307$	1.04564e6	0.633191	0.316595	+	0.948561i	$0.397460\pi$
$308$	0	0
$309$	−1.18591e6	−0.706572
$310$	−37440.0	−0.0221275
$311$	−1.83718e6	−1.07708	−0.538542	−	0.842598i	$-0.681025\pi$
$311$	−1.83718e6	−1.07708	−0.538542	+	0.842598i	$0.681025\pi$
$312$	668304.	0.388676
$313$	365494.	0.210872	0.105436	−	0.994426i	$-0.466376\pi$
$313$	365494.	0.210872	0.105436	+	0.994426i	$0.466376\pi$
$314$	779268.	0.446029
$315$	0	0
$316$	−263104.	−0.148221
$317$	−28338.0	−0.0158388	−0.00791938	−	0.999969i	$-0.502521\pi$
$317$	−28338.0	−0.0158388	−0.00791938	+	0.999969i	$0.502521\pi$
$318$	518076.	0.287293
$319$	−2.41625e6	−1.32943
$320$	−2.16154e6	−1.18001
$321$	−1.16024e6	−0.628473
$322$	0	0
$323$	−338184.	−0.180363
$324$	26244.0	0.0138889
$325$	1.30788e6	0.686845
$326$	−3.33170e6	−1.73629
$327$	−908802.	−0.470002
$328$	−1.33762e6	−0.686510
$329$	0	0
$330$	1.87013e6	0.945337
$331$	1.93392e6	0.970214	0.485107	−	0.874455i	$-0.338781\pi$
$331$	1.93392e6	0.970214	0.485107	+	0.874455i	$0.338781\pi$
$332$	−160752.	−0.0800408
$333$	−440154.	−0.217517
$334$	261072.	0.128054
$335$	4.62977e6	2.25397
$336$	0	0
$337$	−1.88817e6	−0.905664	−0.452832	−	0.891596i	$-0.649586\pi$
$337$	−1.88817e6	−0.905664	−0.452832	+	0.891596i	$0.649586\pi$
$338$	1.05557e6	0.502570
$339$	1.98131e6	0.936384
$340$	−39312.0	−0.0184428
$341$	−35520.0	−0.0165420
$342$	1.30442e6	0.603050
$343$	0	0
$344$	−1.93603e6	−0.882097
$345$	−2.94840e6	−1.33364
$346$	−109980.	−0.0493882
$347$	2.91937e6	1.30156	0.650782	−	0.759264i	$-0.274441\pi$
$347$	2.91937e6	1.30156	0.650782	+	0.759264i	$0.274441\pi$
$348$	−195912.	−0.0867187
$349$	780682.	0.343092	0.171546	−	0.985176i	$-0.445124\pi$
$349$	780682.	0.343092	0.171546	+	0.985176i	$0.445124\pi$
$350$	0	0
$351$	322218.	0.139599
$352$	639360.	0.275036
$353$	−1.33437e6	−0.569954	−0.284977	−	0.958534i	$-0.591986\pi$
$353$	−1.33437e6	−0.569954	−0.284977	+	0.958534i	$0.591986\pi$
$354$	1.48457e6	0.629639
$355$	−2.49912e6	−1.05249
$356$	31896.0	0.0133386
$357$	0	0
$358$	919944.	0.379362
$359$	1.01743e6	0.416648	0.208324	−	0.978060i	$-0.433199\pi$
$359$	1.01743e6	0.416648	0.208324	+	0.978060i	$0.433199\pi$
$360$	−1.06142e6	−0.431651
$361$	4.72776e6	1.90936
$362$	−2.29240e6	−0.919429
$363$	324765.	0.129361
$364$	0	0
$365$	−4.82399e6	−1.89528
$366$	2.67181e6	1.04257
$367$	−837680.	−0.324648	−0.162324	−	0.986737i	$-0.551899\pi$
$367$	−837680.	−0.324648	−0.162324	+	0.986737i	$0.551899\pi$
$368$	−4.77120e6	−1.83657
$369$	−644922.	−0.246571
$370$	−2.54311e6	−0.965742
$371$	0	0
$372$	−2880.00	−0.00107903
$373$	−1.51993e6	−0.565655	−0.282827	−	0.959171i	$-0.591272\pi$
$373$	−1.51993e6	−0.565655	−0.282827	+	0.959171i	$0.591272\pi$
$374$	−335664.	−0.124087
$375$	116532.	0.0427924
$376$	2.33856e6	0.853059
$377$	−2.40536e6	−0.871620
$378$	0	0
$379$	2.64465e6	0.945737	0.472869	−	0.881133i	$-0.343219\pi$
$379$	2.64465e6	0.945737	0.472869	+	0.881133i	$0.343219\pi$
$380$	837408.	0.297494
$381$	−668880.	−0.236067
$382$	1.64045e6	0.575180
$383$	−2.01336e6	−0.701333	−0.350667	−	0.936500i	$-0.614045\pi$
$383$	−2.01336e6	−0.701333	−0.350667	+	0.936500i	$0.614045\pi$
$384$	−1.91117e6	−0.661410
$385$	0	0
$386$	−921612.	−0.314833
$387$	−933444.	−0.316819
$388$	574648.	0.193786
$389$	−726234.	−0.243334	−0.121667	−	0.992571i	$-0.538824\pi$
$389$	−726234.	−0.243334	−0.121667	+	0.992571i	$0.538824\pi$
$390$	1.86170e6	0.619796
$391$	529200.	0.175056
$392$	0	0
$393$	1.39784e6	0.456538
$394$	−926532.	−0.300691
$395$	5.13053e6	1.65451
$396$	143856.	0.0460988
$397$	−4.57578e6	−1.45710	−0.728549	−	0.684993i	$-0.759805\pi$
$397$	−4.57578e6	−1.45710	−0.728549	+	0.684993i	$0.759805\pi$
$398$	−2.20114e6	−0.696529
$399$	0	0
$400$	−3.36142e6	−1.05045
$401$	−33870.0	−0.0105185	−0.00525926	−	0.999986i	$-0.501674\pi$
$401$	−33870.0	−0.0105185	−0.00525926	+	0.999986i	$0.501674\pi$
$402$	3.20522e6	0.989221
$403$	−35360.0	−0.0108455
$404$	10824.0	0.00329940
$405$	−511758.	−0.155034
$406$	0	0
$407$	−2.41270e6	−0.721966
$408$	190512.	0.0566594
$409$	5.86178e6	1.73269	0.866346	−	0.499444i	$-0.166462\pi$
$409$	5.86178e6	1.73269	0.866346	+	0.499444i	$0.166462\pi$
$410$	−3.72622e6	−1.09473
$411$	−2.37821e6	−0.694459
$412$	−527072.	−0.152977
$413$	0	0
$414$	−2.04120e6	−0.585308
$415$	3.13466e6	0.893451
$416$	636480.	0.180323
$417$	−2.02151e6	−0.569292
$418$	7.15018e6	2.00159
$419$	−302748.	−0.0842454	−0.0421227	−	0.999112i	$-0.513412\pi$
$419$	−302748.	−0.0842454	−0.0421227	+	0.999112i	$0.513412\pi$
$420$	0	0
$421$	−5.36708e6	−1.47582	−0.737909	−	0.674900i	$-0.764187\pi$
$421$	−5.36708e6	−1.47582	−0.737909	+	0.674900i	$0.764187\pi$
$422$	−3.12146e6	−0.853252
$423$	1.12752e6	0.306389
$424$	−1.61179e6	−0.435406
$425$	372834.	0.100125
$426$	−1.73016e6	−0.461915
$427$	0	0
$428$	−515664.	−0.136068
$429$	1.76623e6	0.463345
$430$	−5.39323e6	−1.40662
$431$	1.17706e6	0.305214	0.152607	−	0.988287i	$-0.451233\pi$
$431$	1.17706e6	0.305214	0.152607	+	0.988287i	$0.451233\pi$
$432$	−828144.	−0.213499
$433$	3.66249e6	0.938766	0.469383	−	0.882995i	$-0.344476\pi$
$433$	3.66249e6	0.938766	0.469383	+	0.882995i	$0.344476\pi$
$434$	0	0
$435$	3.82028e6	0.967994
$436$	−403912.	−0.101758
$437$	−1.12728e7	−2.82376
$438$	−3.33968e6	−0.831802
$439$	2.53674e6	0.628225	0.314113	−	0.949386i	$-0.398293\pi$
$439$	2.53674e6	0.628225	0.314113	+	0.949386i	$0.398293\pi$
$440$	−5.81818e6	−1.43270
$441$	0	0
$442$	−334152.	−0.0813558
$443$	6.01504e6	1.45623	0.728113	−	0.685457i	$-0.240397\pi$
$443$	6.01504e6	1.45623	0.728113	+	0.685457i	$0.240397\pi$
$444$	−195624.	−0.0470939
$445$	−621972.	−0.148892
$446$	1.82842e6	0.435249
$447$	−738666.	−0.174856
$448$	0	0
$449$	5.65965e6	1.32487	0.662436	−	0.749119i	$-0.269523\pi$
$449$	5.65965e6	1.32487	0.662436	+	0.749119i	$0.269523\pi$
$450$	−1.43807e6	−0.334773
$451$	−3.53513e6	−0.818397
$452$	880584.	0.202733
$453$	−2.58329e6	−0.591463
$454$	1.73153e6	0.394267
$455$	0	0
$456$	−4.05821e6	−0.913949
$457$	−6.46159e6	−1.44727	−0.723634	−	0.690184i	$-0.757530\pi$
$457$	−6.46159e6	−1.44727	−0.723634	+	0.690184i	$0.757530\pi$
$458$	4.63314e6	1.03208
$459$	91854.0	0.0203501
$460$	−1.31040e6	−0.288742
$461$	3.37353e6	0.739320	0.369660	−	0.929167i	$-0.379474\pi$
$461$	3.37353e6	0.739320	0.369660	+	0.929167i	$0.379474\pi$
$462$	0	0
$463$	−4.54974e6	−0.986358	−0.493179	−	0.869928i	$-0.664165\pi$
$463$	−4.54974e6	−0.986358	−0.493179	+	0.869928i	$0.664165\pi$
$464$	6.18211e6	1.33304
$465$	56160.0	0.0120447
$466$	−1.51340e6	−0.322842
$467$	−2.01136e6	−0.426773	−0.213386	−	0.976968i	$-0.568449\pi$
$467$	−2.01136e6	−0.426773	−0.213386	+	0.976968i	$0.568449\pi$
$468$	143208.	0.0302240
$469$	0	0
$470$	6.51456e6	1.36032
$471$	−1.16890e6	−0.242787
$472$	−4.61866e6	−0.954247
$473$	−5.11666e6	−1.05156
$474$	3.55190e6	0.726132
$475$	−7.94196e6	−1.61508
$476$	0	0
$477$	−777114.	−0.156383
$478$	8.70682e6	1.74297
$479$	7.60402e6	1.51427	0.757137	−	0.653257i	$-0.226598\pi$
$479$	7.60402e6	1.51427	0.757137	+	0.653257i	$0.226598\pi$
$480$	−1.01088e6	−0.200261
$481$	−2.40183e6	−0.473347
$482$	−878388.	−0.172214
$483$	0	0
$484$	144340.	0.0280074
$485$	−1.12056e7	−2.16313
$486$	−354294.	−0.0680414
$487$	673112.	0.128607	0.0643035	−	0.997930i	$-0.479517\pi$
$487$	673112.	0.128607	0.0643035	+	0.997930i	$0.479517\pi$
$488$	−8.31230e6	−1.58005
$489$	4.99756e6	0.945117
$490$	0	0
$491$	−2.47170e6	−0.462692	−0.231346	−	0.972872i	$-0.574313\pi$
$491$	−2.47170e6	−0.462692	−0.231346	+	0.972872i	$0.574313\pi$
$492$	−286632.	−0.0533841
$493$	−685692.	−0.127061
$494$	7.11797e6	1.31232
$495$	−2.80519e6	−0.514576
$496$	90880.0	0.0165869
$497$	0	0
$498$	2.17015e6	0.392118
$499$	6.08152e6	1.09335	0.546677	−	0.837343i	$-0.315892\pi$
$499$	6.08152e6	1.09335	0.546677	+	0.837343i	$0.315892\pi$
$500$	51792.0	0.00926483
$501$	−391608.	−0.0697039
$502$	3.64716e6	0.645945
$503$	846216.	0.149129	0.0745644	−	0.997216i	$-0.476243\pi$
$503$	846216.	0.149129	0.0745644	+	0.997216i	$0.476243\pi$
$504$	0	0
$505$	−211068.	−0.0368293
$506$	−1.11888e7	−1.94271
$507$	−1.58336e6	−0.273565
$508$	−297280.	−0.0511101
$509$	7.66785e6	1.31183	0.655917	−	0.754833i	$-0.272282\pi$
$509$	7.66785e6	1.31183	0.655917	+	0.754833i	$0.272282\pi$
$510$	530712.	0.0903511
$511$	0	0
$512$	4.47130e6	0.753804
$513$	−1.95664e6	−0.328259
$514$	573516.	0.0957498
$515$	1.02779e7	1.70760
$516$	−414864.	−0.0685933
$517$	6.18048e6	1.01694
$518$	0	0
$519$	164970.	0.0268835
$520$	−5.79197e6	−0.939329
$521$	9.68938e6	1.56387	0.781937	−	0.623357i	$-0.214232\pi$
$521$	9.68938e6	1.56387	0.781937	+	0.623357i	$0.214232\pi$
$522$	2.64481e6	0.424833
$523$	7.51678e6	1.20165	0.600824	−	0.799381i	$-0.294839\pi$
$523$	7.51678e6	1.20165	0.600824	+	0.799381i	$0.294839\pi$
$524$	621264.	0.0988435
$525$	0	0
$526$	1.32021e7	2.08055
$527$	−10080.0	−0.00158101
$528$	−4.53946e6	−0.708629
$529$	1.12037e7	1.74069
$530$	−4.48999e6	−0.694314
$531$	−2.22685e6	−0.342732
$532$	0	0
$533$	−3.51920e6	−0.536570
$534$	−430596.	−0.0653457
$535$	1.00554e7	1.51886
$536$	−9.97181e6	−1.49921
$537$	−1.37992e6	−0.206499
$538$	1.06215e7	1.58208
$539$	0	0
$540$	−227448.	−0.0335659
$541$	7.34325e6	1.07869	0.539343	−	0.842086i	$-0.318673\pi$
$541$	7.34325e6	1.07869	0.539343	+	0.842086i	$0.318673\pi$
$542$	−1.34102e6	−0.196082
$543$	3.43859e6	0.500474
$544$	181440.	0.0262867
$545$	7.87628e6	1.13587
$546$	0	0
$547$	2.18296e6	0.311945	0.155973	−	0.987761i	$-0.450149\pi$
$547$	2.18296e6	0.311945	0.155973	+	0.987761i	$0.450149\pi$
$548$	−1.05698e6	−0.150355
$549$	−4.00772e6	−0.567501
$550$	−7.88278e6	−1.11115
$551$	1.46063e7	2.04957
$552$	6.35040e6	0.887061
$553$	0	0
$554$	2.05667e6	0.284702
$555$	3.81467e6	0.525683
$556$	−898448.	−0.123255
$557$	1.25466e7	1.71351	0.856755	−	0.515724i	$-0.172477\pi$
$557$	1.25466e7	1.71351	0.856755	+	0.515724i	$0.172477\pi$
$558$	38880.0	0.00528617
$559$	−5.09361e6	−0.689439
$560$	0	0
$561$	503496.	0.0675443
$562$	−2.88227e6	−0.384940
$563$	−5.15972e6	−0.686050	−0.343025	−	0.939326i	$-0.611451\pi$
$563$	−5.15972e6	−0.686050	−0.343025	+	0.939326i	$0.611451\pi$
$564$	501120.	0.0663352
$565$	−1.71714e7	−2.26300
$566$	−179880.	−0.0236016
$567$	0	0
$568$	5.38272e6	0.700053
$569$	1.17452e7	1.52083	0.760414	−	0.649439i	$-0.224996\pi$
$569$	1.17452e7	1.52083	0.760414	+	0.649439i	$0.224996\pi$
$570$	−1.13050e7	−1.45742
$571$	−7.54728e6	−0.968725	−0.484362	−	0.874867i	$-0.660948\pi$
$571$	−7.54728e6	−0.968725	−0.484362	+	0.874867i	$0.660948\pi$
$572$	784992.	0.100317
$573$	−2.46067e6	−0.313089
$574$	0	0
$575$	1.24278e7	1.56756
$576$	2.24467e6	0.281901
$577$	−9.28483e6	−1.16101	−0.580503	−	0.814258i	$-0.697144\pi$
$577$	−9.28483e6	−1.16101	−0.580503	+	0.814258i	$0.697144\pi$
$578$	8.42389e6	1.04880
$579$	1.38242e6	0.171373
$580$	1.69790e6	0.209577
$581$	0	0
$582$	−7.75775e6	−0.949354
$583$	−4.25974e6	−0.519053
$584$	1.03901e7	1.26063
$585$	−2.79256e6	−0.337374
$586$	−1.18840e6	−0.142961
$587$	−1.47623e6	−0.176831	−0.0884155	−	0.996084i	$-0.528180\pi$
$587$	−1.47623e6	−0.176831	−0.0884155	+	0.996084i	$0.528180\pi$
$588$	0	0
$589$	214720.	0.0255026
$590$	−1.28663e7	−1.52168
$591$	1.38980e6	0.163675
$592$	6.17302e6	0.723925
$593$	1.24007e7	1.44813	0.724067	−	0.689729i	$-0.242270\pi$
$593$	1.24007e7	1.44813	0.724067	+	0.689729i	$0.242270\pi$
$594$	−1.94206e6	−0.225837
$595$	0	0
$596$	−328296.	−0.0378573
$597$	3.30170e6	0.379142
$598$	−1.11384e7	−1.27371
$599$	−3.69127e6	−0.420348	−0.210174	−	0.977664i	$-0.567403\pi$
$599$	−3.69127e6	−0.420348	−0.210174	+	0.977664i	$0.567403\pi$
$600$	4.47401e6	0.507363
$601$	−9.12223e6	−1.03018	−0.515092	−	0.857135i	$-0.672242\pi$
$601$	−9.12223e6	−1.03018	−0.515092	+	0.857135i	$0.672242\pi$
$602$	0	0
$603$	−4.80784e6	−0.538464
$604$	−1.14813e6	−0.128055
$605$	−2.81463e6	−0.312632
$606$	−146124.	−0.0161637
$607$	5.67914e6	0.625620	0.312810	−	0.949816i	$-0.398730\pi$
$607$	5.67914e6	0.625620	0.312810	+	0.949816i	$0.398730\pi$
$608$	−3.86496e6	−0.424020
$609$	0	0
$610$	−2.31557e7	−2.51961
$611$	6.15264e6	0.666743
$612$	40824.0	0.00440592
$613$	−1.40106e7	−1.50593	−0.752966	−	0.658060i	$-0.771377\pi$
$613$	−1.40106e7	−1.50593	−0.752966	+	0.658060i	$0.771377\pi$
$614$	−6.27382e6	−0.671600
$615$	5.58932e6	0.595897
$616$	0	0
$617$	−253686.	−0.0268277	−0.0134139	−	0.999910i	$-0.504270\pi$
$617$	−253686.	−0.0268277	−0.0134139	+	0.999910i	$0.504270\pi$
$618$	7.11547e6	0.749433
$619$	−4.30034e6	−0.451103	−0.225552	−	0.974231i	$-0.572418\pi$
$619$	−4.30034e6	−0.451103	−0.225552	+	0.974231i	$0.572418\pi$
$620$	24960.0	0.00260775
$621$	3.06180e6	0.318602
$622$	1.10231e7	1.14242
$623$	0	0
$624$	−4.51901e6	−0.464603
$625$	−1.02568e7	−1.05030
$626$	−2.19296e6	−0.223664
$627$	−1.07253e7	−1.08953
$628$	−519512.	−0.0525650
$629$	−684684.	−0.0690023
$630$	0	0
$631$	1.04150e7	1.04132	0.520662	−	0.853763i	$-0.325685\pi$
$631$	1.04150e7	1.04132	0.520662	+	0.853763i	$0.325685\pi$
$632$	−1.10504e7	−1.10048
$633$	4.68220e6	0.464451
$634$	170028.	0.0167995
$635$	5.79696e6	0.570514
$636$	−345384.	−0.0338579
$637$	0	0
$638$	1.44975e7	1.41007
$639$	2.59524e6	0.251435
$640$	1.65635e7	1.59846
$641$	4.52714e6	0.435190	0.217595	−	0.976039i	$-0.430179\pi$
$641$	4.52714e6	0.435190	0.217595	+	0.976039i	$0.430179\pi$
$642$	6.96146e6	0.666596
$643$	−1.49687e7	−1.42776	−0.713882	−	0.700266i	$-0.753065\pi$
$643$	−1.49687e7	−1.42776	−0.713882	+	0.700266i	$0.753065\pi$
$644$	0	0
$645$	8.08985e6	0.765669
$646$	2.02910e6	0.191304
$647$	1.73020e7	1.62493	0.812465	−	0.583010i	$-0.198125\pi$
$647$	1.73020e7	1.62493	0.812465	+	0.583010i	$0.198125\pi$
$648$	1.10225e6	0.103120
$649$	−1.22064e7	−1.13757
$650$	−7.84727e6	−0.728509
$651$	0	0
$652$	2.22114e6	0.204624
$653$	4.07470e6	0.373949	0.186975	−	0.982365i	$-0.440132\pi$
$653$	4.07470e6	0.373949	0.186975	+	0.982365i	$0.440132\pi$
$654$	5.45281e6	0.498513
$655$	−1.21146e7	−1.10334
$656$	9.04483e6	0.820618
$657$	5.00953e6	0.452776
$658$	0	0
$659$	−3.79475e6	−0.340384	−0.170192	−	0.985411i	$-0.554439\pi$
$659$	−3.79475e6	−0.340384	−0.170192	+	0.985411i	$0.554439\pi$
$660$	−1.24675e6	−0.111409
$661$	−1.64261e7	−1.46228	−0.731142	−	0.682225i	$-0.761012\pi$
$661$	−1.64261e7	−1.46228	−0.731142	+	0.682225i	$0.761012\pi$
$662$	−1.16035e7	−1.02907
$663$	501228.	0.0442845
$664$	−6.75158e6	−0.594272
$665$	0	0
$666$	2.64092e6	0.230712
$667$	−2.28564e7	−1.98927
$668$	−174048.	−0.0150913
$669$	−2.74262e6	−0.236920
$670$	−2.77786e7	−2.39069
$671$	−2.19682e7	−1.88360
$672$	0	0
$673$	5.50675e6	0.468660	0.234330	−	0.972157i	$-0.424710\pi$
$673$	5.50675e6	0.468660	0.234330	+	0.972157i	$0.424710\pi$
$674$	1.13290e7	0.960602
$675$	2.15711e6	0.182227
$676$	−703716.	−0.0592285
$677$	−1.83957e7	−1.54257	−0.771286	−	0.636488i	$-0.780386\pi$
$677$	−1.83957e7	−1.54257	−0.771286	+	0.636488i	$0.780386\pi$
$678$	−1.18879e7	−0.993185
$679$	0	0
$680$	−1.65110e6	−0.136931
$681$	−2.59729e6	−0.214612
$682$	213120.	0.0175454
$683$	1.75835e6	0.144229	0.0721146	−	0.997396i	$-0.477025\pi$
$683$	1.75835e6	0.144229	0.0721146	+	0.997396i	$0.477025\pi$
$684$	−869616.	−0.0710702
$685$	2.06112e7	1.67833
$686$	0	0
$687$	−6.94971e6	−0.561791
$688$	1.30913e7	1.05441
$689$	−4.24055e6	−0.340309
$690$	1.76904e7	1.41454
$691$	5.36314e6	0.427291	0.213646	−	0.976911i	$-0.431466\pi$
$691$	5.36314e6	0.427291	0.213646	+	0.976911i	$0.431466\pi$
$692$	73320.0	0.00582046
$693$	0	0
$694$	−1.75162e7	−1.38052
$695$	1.75197e7	1.37583
$696$	−8.22830e6	−0.643854
$697$	−1.00321e6	−0.0782187
$698$	−4.68409e6	−0.363904
$699$	2.27011e6	0.175733
$700$	0	0
$701$	−2.12606e7	−1.63411	−0.817054	−	0.576561i	$-0.804394\pi$
$701$	−2.12606e7	−1.63411	−0.817054	+	0.576561i	$0.804394\pi$
$702$	−1.93331e6	−0.148067
$703$	1.45849e7	1.11305
$704$	1.23041e7	0.935662
$705$	−9.77184e6	−0.740463
$706$	8.00622e6	0.604527
$707$	0	0
$708$	−989712.	−0.0742037
$709$	2.07729e6	0.155196	0.0775980	−	0.996985i	$-0.475275\pi$
$709$	2.07729e6	0.155196	0.0775980	+	0.996985i	$0.475275\pi$
$710$	1.49947e7	1.11633
$711$	−5.32786e6	−0.395256
$712$	1.33963e6	0.0990343
$713$	−336000.	−0.0247523
$714$	0	0
$715$	−1.53073e7	−1.11979
$716$	−613296.	−0.0447082
$717$	−1.30602e7	−0.948752
$718$	−6.10459e6	−0.441922
$719$	−4.23619e6	−0.305600	−0.152800	−	0.988257i	$-0.548829\pi$
$719$	−4.23619e6	−0.305600	−0.152800	+	0.988257i	$0.548829\pi$
$720$	7.17725e6	0.515973
$721$	0	0
$722$	−2.83665e7	−2.02518
$723$	1.31758e6	0.0937415
$724$	1.52826e6	0.108356
$725$	−1.61029e7	−1.13778
$726$	−1.94859e6	−0.137208
$727$	−2.14524e7	−1.50536	−0.752678	−	0.658389i	$-0.771238\pi$
$727$	−2.14524e7	−1.50536	−0.752678	+	0.658389i	$0.771238\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	2.89439e7	2.01025
$731$	−1.45202e6	−0.100503
$732$	−1.78121e6	−0.122867
$733$	1.48892e7	1.02355	0.511777	−	0.859118i	$-0.328987\pi$
$733$	1.48892e7	1.02355	0.511777	+	0.859118i	$0.328987\pi$
$734$	5.02608e6	0.344341
$735$	0	0
$736$	6.04800e6	0.411545
$737$	−2.63541e7	−1.78722
$738$	3.86953e6	0.261528
$739$	6.99324e6	0.471050	0.235525	−	0.971868i	$-0.424319\pi$
$739$	6.99324e6	0.471050	0.235525	+	0.971868i	$0.424319\pi$
$740$	1.69541e6	0.113814
$741$	−1.06770e7	−0.714335
$742$	0	0
$743$	1.90428e6	0.126549	0.0632745	−	0.997996i	$-0.479846\pi$
$743$	1.90428e6	0.126549	0.0632745	+	0.997996i	$0.479846\pi$
$744$	−120960.	−0.00801142
$745$	6.40177e6	0.422581
$746$	9.11958e6	0.599968
$747$	−3.25523e6	−0.213442
$748$	223776.	0.0146238
$749$	0	0
$750$	−699192.	−0.0453882
$751$	1.95361e7	1.26398	0.631988	−	0.774978i	$-0.282239\pi$
$751$	1.95361e7	1.26398	0.631988	+	0.774978i	$0.282239\pi$
$752$	−1.58131e7	−1.01970
$753$	−5.47074e6	−0.351608
$754$	1.44322e7	0.924493
$755$	2.23885e7	1.42941
$756$	0	0
$757$	1.25183e6	0.0793973	0.0396986	−	0.999212i	$-0.487360\pi$
$757$	1.25183e6	0.0793973	0.0396986	+	0.999212i	$0.487360\pi$
$758$	−1.58679e7	−1.00311
$759$	1.67832e7	1.05748
$760$	3.51711e7	2.20878
$761$	−2.04472e7	−1.27989	−0.639944	−	0.768422i	$-0.721042\pi$
$761$	−2.04472e7	−1.27989	−0.639944	+	0.768422i	$0.721042\pi$
$762$	4.01328e6	0.250387
$763$	0	0
$764$	−1.09363e6	−0.0677857
$765$	−796068.	−0.0491809
$766$	1.20802e7	0.743876
$767$	−1.21515e7	−0.745831
$768$	3.48595e6	0.213264
$769$	−2.21064e6	−0.134804	−0.0674020	−	0.997726i	$-0.521471\pi$
$769$	−2.21064e6	−0.134804	−0.0674020	+	0.997726i	$0.521471\pi$
$770$	0	0
$771$	−860274.	−0.0521196
$772$	614408.	0.0371034
$773$	−1.29151e7	−0.777405	−0.388703	−	0.921363i	$-0.627077\pi$
$773$	−1.29151e7	−0.777405	−0.388703	+	0.921363i	$0.627077\pi$
$774$	5.60066e6	0.336037
$775$	−236720.	−0.0141573
$776$	2.41352e7	1.43879
$777$	0	0
$778$	4.35740e6	0.258095
$779$	2.13700e7	1.26171
$780$	−1.24114e6	−0.0730437
$781$	1.42258e7	0.834541
$782$	−3.17520e6	−0.185675
$783$	−3.96722e6	−0.231250
$784$	0	0
$785$	1.01305e7	0.586754
$786$	−8.38706e6	−0.484232
$787$	1.35499e7	0.779830	0.389915	−	0.920851i	$-0.372504\pi$
$787$	1.35499e7	0.779830	0.389915	+	0.920851i	$0.372504\pi$
$788$	617688.	0.0354367
$789$	−1.98031e7	−1.13251
$790$	−3.07832e7	−1.75487
$791$	0	0
$792$	6.04195e6	0.342266
$793$	−2.18693e7	−1.23496
$794$	2.74547e7	1.54549
$795$	6.73499e6	0.377937
$796$	1.46742e6	0.0820867
$797$	2.45956e7	1.37155	0.685776	−	0.727813i	$-0.259463\pi$
$797$	2.45956e7	1.37155	0.685776	+	0.727813i	$0.259463\pi$
$798$	0	0
$799$	1.75392e6	0.0971948
$800$	4.26096e6	0.235387
$801$	645894.	0.0355697
$802$	203220.	0.0111566
$803$	2.74596e7	1.50282
$804$	−2.13682e6	−0.116581
$805$	0	0
$806$	212160.	0.0115034
$807$	−1.59322e7	−0.861177
$808$	454608.	0.0244968
$809$	1.55237e7	0.833920	0.416960	−	0.908925i	$-0.363095\pi$
$809$	1.55237e7	0.833920	0.416960	+	0.908925i	$0.363095\pi$
$810$	3.07055e6	0.164438
$811$	2.66262e7	1.42153	0.710766	−	0.703429i	$-0.248349\pi$
$811$	2.66262e7	1.42153	0.710766	+	0.703429i	$0.248349\pi$
$812$	0	0
$813$	2.01154e6	0.106734
$814$	1.44762e7	0.765760
$815$	−4.33122e7	−2.28410
$816$	−1.28822e6	−0.0677276
$817$	3.09304e7	1.62118
$818$	−3.51707e7	−1.83780
$819$	0	0
$820$	2.48414e6	0.129016
$821$	−1.23891e7	−0.641477	−0.320739	−	0.947168i	$-0.603931\pi$
$821$	−1.23891e7	−0.641477	−0.320739	+	0.947168i	$0.603931\pi$
$822$	1.42693e7	0.736585
$823$	−3.65630e6	−0.188166	−0.0940831	−	0.995564i	$-0.529992\pi$
$823$	−3.65630e6	−0.188166	−0.0940831	+	0.995564i	$0.529992\pi$
$824$	−2.21370e7	−1.13580
$825$	1.18242e7	0.604833
$826$	0	0
$827$	2.80463e7	1.42597	0.712987	−	0.701178i	$-0.247342\pi$
$827$	2.80463e7	1.42597	0.712987	+	0.701178i	$0.247342\pi$
$828$	1.36080e6	0.0689792
$829$	−2.11153e7	−1.06712	−0.533558	−	0.845763i	$-0.679145\pi$
$829$	−2.11153e7	−1.06712	−0.533558	+	0.845763i	$0.679145\pi$
$830$	−1.88080e7	−0.947648
$831$	−3.08500e6	−0.154972
$832$	1.22487e7	0.613454
$833$	0	0
$834$	1.21290e7	0.603826
$835$	3.39394e6	0.168456
$836$	−4.76678e6	−0.235890
$837$	−58320.0	−0.00287742
$838$	1.81649e6	0.0893557
$839$	−1.33947e7	−0.656944	−0.328472	−	0.944514i	$-0.606534\pi$
$839$	−1.33947e7	−0.656944	−0.328472	+	0.944514i	$0.606534\pi$
$840$	0	0
$841$	9.10422e6	0.443867
$842$	3.22025e7	1.56534
$843$	4.32340e6	0.209535
$844$	2.08098e6	0.100557
$845$	1.37225e7	0.661135
$846$	−6.76512e6	−0.324975
$847$	0	0
$848$	1.08988e7	0.520461
$849$	269820.	0.0128471
$850$	−2.23700e6	−0.106199
$851$	−2.28228e7	−1.08030
$852$	1.15344e6	0.0544372
$853$	−3.01513e7	−1.41884	−0.709420	−	0.704786i	$-0.751043\pi$
$853$	−3.01513e7	−1.41884	−0.709420	+	0.704786i	$0.751043\pi$
$854$	0	0
$855$	1.69575e7	0.793317
$856$	−2.16579e7	−1.01026
$857$	−2.39894e7	−1.11575	−0.557875	−	0.829925i	$-0.688383\pi$
$857$	−2.39894e7	−1.11575	−0.557875	+	0.829925i	$0.688383\pi$
$858$	−1.05974e7	−0.491452
$859$	8.87576e6	0.410414	0.205207	−	0.978719i	$-0.434213\pi$
$859$	8.87576e6	0.410414	0.205207	+	0.978719i	$0.434213\pi$
$860$	3.59549e6	0.165772
$861$	0	0
$862$	−7.06234e6	−0.323728
$863$	−8.71286e6	−0.398230	−0.199115	−	0.979976i	$-0.563807\pi$
$863$	−8.71286e6	−0.398230	−0.199115	+	0.979976i	$0.563807\pi$
$864$	1.04976e6	0.0478416
$865$	−1.42974e6	−0.0649706
$866$	−2.19750e7	−0.995711
$867$	−1.26358e7	−0.570895
$868$	0	0
$869$	−2.92045e7	−1.31190
$870$	−2.29217e7	−1.02671
$871$	−2.62354e7	−1.17177
$872$	−1.69643e7	−0.755518
$873$	1.16366e7	0.516763
$874$	6.76368e7	2.99505
$875$	0	0
$876$	2.22646e6	0.0980288
$877$	−2.95788e7	−1.29862	−0.649310	−	0.760524i	$-0.724942\pi$
$877$	−2.95788e7	−1.29862	−0.649310	+	0.760524i	$0.724942\pi$
$878$	−1.52205e7	−0.666333
$879$	1.78259e6	0.0778180
$880$	3.93420e7	1.71257
$881$	−2.45670e7	−1.06638	−0.533190	−	0.845995i	$-0.679007\pi$
$881$	−2.45670e7	−1.06638	−0.533190	+	0.845995i	$0.679007\pi$
$882$	0	0
$883$	1.45682e7	0.628788	0.314394	−	0.949293i	$-0.398199\pi$
$883$	1.45682e7	0.628788	0.314394	+	0.949293i	$0.398199\pi$
$884$	222768.	0.00958787
$885$	1.92994e7	0.828295
$886$	−3.60902e7	−1.54456
$887$	−1.61714e7	−0.690141	−0.345070	−	0.938577i	$-0.612145\pi$
$887$	−1.61714e7	−0.690141	−0.345070	+	0.938577i	$0.612145\pi$
$888$	−8.21621e6	−0.349654
$889$	0	0
$890$	3.73183e6	0.157924
$891$	2.91308e6	0.122930
$892$	−1.21894e6	−0.0512946
$893$	−3.73613e7	−1.56781
$894$	4.43200e6	0.185462
$895$	1.19593e7	0.499054
$896$	0	0
$897$	1.67076e7	0.693319
$898$	−3.39579e7	−1.40524
$899$	435360.	0.0179659
$900$	958716.	0.0394533
$901$	−1.20884e6	−0.0496087
$902$	2.12108e7	0.868041
$903$	0	0
$904$	3.69845e7	1.50522
$905$	−2.98011e7	−1.20952
$906$	1.54997e7	0.627341
$907$	3.14446e7	1.26919	0.634596	−	0.772844i	$-0.281167\pi$
$907$	3.14446e7	1.26919	0.634596	+	0.772844i	$0.281167\pi$
$908$	−1.15435e6	−0.0464648
$909$	219186.	0.00879839
$910$	0	0
$911$	1.51427e7	0.604514	0.302257	−	0.953227i	$-0.402260\pi$
$911$	1.51427e7	0.604514	0.302257	+	0.953227i	$0.402260\pi$
$912$	2.74412e7	1.09249
$913$	−1.78435e7	−0.708439
$914$	3.87695e7	1.53506
$915$	3.47336e7	1.37150
$916$	−3.08876e6	−0.121631
$917$	0	0
$918$	−551124.	−0.0215845
$919$	4.14876e7	1.62043	0.810214	−	0.586134i	$-0.199351\pi$
$919$	4.14876e7	1.62043	0.810214	+	0.586134i	$0.199351\pi$
$920$	−5.50368e7	−2.14380
$921$	9.41072e6	0.365573
$922$	−2.02412e7	−0.784167
$923$	1.41617e7	0.547155
$924$	0	0
$925$	−1.60792e7	−0.617889
$926$	2.72985e7	1.04619
$927$	−1.06732e7	−0.407939
$928$	−7.83648e6	−0.298711
$929$	1.78495e7	0.678556	0.339278	−	0.940686i	$-0.389817\pi$
$929$	1.78495e7	0.678556	0.339278	+	0.940686i	$0.389817\pi$
$930$	−336960.	−0.0127753
$931$	0	0
$932$	1.00894e6	0.0380473
$933$	−1.65346e7	−0.621855
$934$	1.20681e7	0.452661
$935$	−4.36363e6	−0.163237
$936$	6.01474e6	0.224402
$937$	−2.96399e7	−1.10288	−0.551439	−	0.834215i	$-0.685921\pi$
$937$	−2.96399e7	−1.10288	−0.551439	+	0.834215i	$0.685921\pi$
$938$	0	0
$939$	3.28945e6	0.121747
$940$	−4.34304e6	−0.160315
$941$	3.22282e7	1.18648	0.593242	−	0.805024i	$-0.297848\pi$
$941$	3.22282e7	1.18648	0.593242	+	0.805024i	$0.297848\pi$
$942$	7.01341e6	0.257515
$943$	−3.34404e7	−1.22459
$944$	3.12309e7	1.14066
$945$	0	0
$946$	3.06999e7	1.11535
$947$	4.84885e7	1.75697	0.878484	−	0.477772i	$-0.158556\pi$
$947$	4.84885e7	1.75697	0.878484	+	0.477772i	$0.158556\pi$
$948$	−2.36794e6	−0.0855754
$949$	2.73359e7	0.985300
$950$	4.76517e7	1.71305
$951$	−255042.	−0.00914451
$952$	0	0
$953$	−2.03264e7	−0.724983	−0.362491	−	0.931987i	$-0.618074\pi$
$953$	−2.03264e7	−0.724983	−0.362491	+	0.931987i	$0.618074\pi$
$954$	4.66268e6	0.165869
$955$	2.13258e7	0.756654
$956$	−5.80454e6	−0.205411
$957$	−2.17462e7	−0.767546
$958$	−4.56241e7	−1.60613
$959$	0	0
$960$	−1.94538e7	−0.681282
$961$	−2.86228e7	−0.999776
$962$	1.44110e7	0.502060
$963$	−1.04422e7	−0.362849
$964$	585592.	0.0202956
$965$	−1.19810e7	−0.414165
$966$	0	0
$967$	−3.66292e6	−0.125968	−0.0629841	−	0.998015i	$-0.520062\pi$
$967$	−3.66292e6	−0.125968	−0.0629841	+	0.998015i	$0.520062\pi$
$968$	6.06228e6	0.207945
$969$	−3.04366e6	−0.104132
$970$	6.72338e7	2.29434
$971$	−1.48741e6	−0.0506271	−0.0253136	−	0.999680i	$-0.508058\pi$
$971$	−1.48741e6	−0.0506271	−0.0253136	+	0.999680i	$0.508058\pi$
$972$	236196.	0.00801875
$973$	0	0
$974$	−4.03867e6	−0.136408
$975$	1.17709e7	0.396550
$976$	5.62070e7	1.88871
$977$	4.07930e7	1.36725	0.683627	−	0.729831i	$-0.260401\pi$
$977$	4.07930e7	1.36725	0.683627	+	0.729831i	$0.260401\pi$
$978$	−2.99853e7	−1.00245
$979$	3.54046e6	0.118060
$980$	0	0
$981$	−8.17922e6	−0.271356
$982$	1.48302e7	0.490759
$983$	9.26326e6	0.305759	0.152880	−	0.988245i	$-0.451145\pi$
$983$	9.26326e6	0.305759	0.152880	+	0.988245i	$0.451145\pi$
$984$	−1.20385e7	−0.396357
$985$	−1.20449e7	−0.395561
$986$	4.11415e6	0.134768
$987$	0	0
$988$	−4.74531e6	−0.154658
$989$	−4.84008e7	−1.57348
$990$	1.68312e7	0.545790
$991$	−5.22051e7	−1.68861	−0.844303	−	0.535866i	$-0.819985\pi$
$991$	−5.22051e7	−1.68861	−0.844303	+	0.535866i	$0.819985\pi$
$992$	−115200.	−0.00371684
$993$	1.74052e7	0.560153
$994$	0	0
$995$	−2.86148e7	−0.916289
$996$	−1.44677e6	−0.0462116
$997$	1.86609e7	0.594560	0.297280	−	0.954790i	$-0.403921\pi$
$997$	1.86609e7	0.594560	0.297280	+	0.954790i	$0.403921\pi$
$998$	−3.64891e7	−1.15968
$999$	−3.96139e6	−0.125584

Display

a_p

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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	147.6.a.b.1.1		1
3.2	odd	2		441.6.a.j.1.1		1
7.2	even	3		147.6.e.i.67.1		2
7.3	odd	6		147.6.e.j.79.1		2
7.4	even	3		147.6.e.i.79.1		2
7.5	odd	6		147.6.e.j.67.1		2
7.6	odd	2		21.6.a.a.1.1	✓	1
21.20	even	2		63.6.a.d.1.1		1
28.27	even	2		336.6.a.r.1.1		1
35.13	even	4		525.6.d.b.274.2		2
35.27	even	4		525.6.d.b.274.1		2
35.34	odd	2		525.6.a.d.1.1		1
84.83	odd	2		1008.6.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.a.1.1	✓	1	7.6	odd	2
63.6.a.d.1.1		1	21.20	even	2
147.6.a.b.1.1		1	1.1	even	1	trivial
147.6.e.i.67.1		2	7.2	even	3
147.6.e.i.79.1		2	7.4	even	3
147.6.e.j.67.1		2	7.5	odd	6
147.6.e.j.79.1		2	7.3	odd	6
336.6.a.r.1.1		1	28.27	even	2
441.6.a.j.1.1		1	3.2	odd	2
525.6.a.d.1.1		1	35.34	odd	2
525.6.d.b.274.1		2	35.27	even	4
525.6.d.b.274.2		2	35.13	even	4
1008.6.a.c.1.1		1	84.83	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

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	147.6.a.b.1.1

Level	$147$
Weight	$6$
Character	147.1
Self dual	yes
Analytic conductor	$23.576$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$