LMFDB - Embedded newform 169.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.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:	$9.97132279097$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +9.00000 q^{5} +3.00000 q^{6} -15.0000 q^{7} +21.0000 q^{8} -26.0000 q^{9} +O(q^{10})$

\(q-3.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +9.00000 q^{5} +3.00000 q^{6} -15.0000 q^{7} +21.0000 q^{8} -26.0000 q^{9} -27.0000 q^{10} +48.0000 q^{11} -1.00000 q^{12} +45.0000 q^{14} -9.00000 q^{15} -71.0000 q^{16} +45.0000 q^{17} +78.0000 q^{18} -6.00000 q^{19} +9.00000 q^{20} +15.0000 q^{21} -144.000 q^{22} -162.000 q^{23} -21.0000 q^{24} -44.0000 q^{25} +53.0000 q^{27} -15.0000 q^{28} -144.000 q^{29} +27.0000 q^{30} -264.000 q^{31} +45.0000 q^{32} -48.0000 q^{33} -135.000 q^{34} -135.000 q^{35} -26.0000 q^{36} -303.000 q^{37} +18.0000 q^{38} +189.000 q^{40} +192.000 q^{41} -45.0000 q^{42} +97.0000 q^{43} +48.0000 q^{44} -234.000 q^{45} +486.000 q^{46} -111.000 q^{47} +71.0000 q^{48} -118.000 q^{49} +132.000 q^{50} -45.0000 q^{51} -414.000 q^{53} -159.000 q^{54} +432.000 q^{55} -315.000 q^{56} +6.00000 q^{57} +432.000 q^{58} -522.000 q^{59} -9.00000 q^{60} +376.000 q^{61} +792.000 q^{62} +390.000 q^{63} +433.000 q^{64} +144.000 q^{66} +36.0000 q^{67} +45.0000 q^{68} +162.000 q^{69} +405.000 q^{70} -357.000 q^{71} -546.000 q^{72} +1098.00 q^{73} +909.000 q^{74} +44.0000 q^{75} -6.00000 q^{76} -720.000 q^{77} -830.000 q^{79} -639.000 q^{80} +649.000 q^{81} -576.000 q^{82} +438.000 q^{83} +15.0000 q^{84} +405.000 q^{85} -291.000 q^{86} +144.000 q^{87} +1008.00 q^{88} +438.000 q^{89} +702.000 q^{90} -162.000 q^{92} +264.000 q^{93} +333.000 q^{94} -54.0000 q^{95} -45.0000 q^{96} +852.000 q^{97} +354.000 q^{98} -1248.00 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	−1.00000	−0.192450	−0.0962250	−	0.995360i	$-0.530677\pi$
$3$	−1.00000	−0.192450	−0.0962250	+	0.995360i	$0.530677\pi$
$4$	1.00000	0.125000
$5$	9.00000	0.804984	0.402492	−	0.915423i	$-0.368144\pi$
$5$	9.00000	0.804984	0.402492	+	0.915423i	$0.368144\pi$
$6$	3.00000	0.204124
$7$	−15.0000	−0.809924	−0.404962	−	0.914334i	$-0.632715\pi$
$7$	−15.0000	−0.809924	−0.404962	+	0.914334i	$0.632715\pi$
$8$	21.0000	0.928078
$9$	−26.0000	−0.962963
$10$	−27.0000	−0.853815
$11$	48.0000	1.31569	0.657843	−	0.753155i	$-0.271469\pi$
$11$	48.0000	1.31569	0.657843	+	0.753155i	$0.271469\pi$
$12$	−1.00000	−0.0240563
$13$	0	0
$13$	0	0
$14$	45.0000	0.859054
$15$	−9.00000	−0.154919
$16$	−71.0000	−1.10938
$17$	45.0000	0.642006	0.321003	−	0.947078i	$-0.395980\pi$
$17$	45.0000	0.642006	0.321003	+	0.947078i	$0.395980\pi$
$18$	78.0000	1.02138
$19$	−6.00000	−0.0724471	−0.0362235	−	0.999344i	$-0.511533\pi$
$19$	−6.00000	−0.0724471	−0.0362235	+	0.999344i	$0.511533\pi$
$20$	9.00000	0.100623
$21$	15.0000	0.155870
$22$	−144.000	−1.39550
$23$	−162.000	−1.46867	−0.734333	−	0.678789i	$-0.762505\pi$
$23$	−162.000	−1.46867	−0.734333	+	0.678789i	$0.762505\pi$
$24$	−21.0000	−0.178609
$25$	−44.0000	−0.352000
$26$	0	0
$27$	53.0000	0.377772
$28$	−15.0000	−0.101240
$29$	−144.000	−0.922073	−0.461037	−	0.887381i	$-0.652522\pi$
$29$	−144.000	−0.922073	−0.461037	+	0.887381i	$0.652522\pi$
$30$	27.0000	0.164317
$31$	−264.000	−1.52954	−0.764771	−	0.644302i	$-0.777148\pi$
$31$	−264.000	−1.52954	−0.764771	+	0.644302i	$0.777148\pi$
$32$	45.0000	0.248592
$33$	−48.0000	−0.253204
$34$	−135.000	−0.680950
$35$	−135.000	−0.651976
$36$	−26.0000	−0.120370
$37$	−303.000	−1.34629	−0.673147	−	0.739509i	$-0.735058\pi$
$37$	−303.000	−1.34629	−0.673147	+	0.739509i	$0.735058\pi$
$38$	18.0000	0.0768417
$39$	0	0
$40$	189.000	0.747088
$41$	192.000	0.731350	0.365675	−	0.930743i	$-0.380838\pi$
$41$	192.000	0.731350	0.365675	+	0.930743i	$0.380838\pi$
$42$	−45.0000	−0.165325
$43$	97.0000	0.344008	0.172004	−	0.985096i	$-0.444976\pi$
$43$	97.0000	0.344008	0.172004	+	0.985096i	$0.444976\pi$
$44$	48.0000	0.164461
$45$	−234.000	−0.775170
$46$	486.000	1.55776
$47$	−111.000	−0.344490	−0.172245	−	0.985054i	$-0.555102\pi$
$47$	−111.000	−0.344490	−0.172245	+	0.985054i	$0.555102\pi$
$48$	71.0000	0.213499
$49$	−118.000	−0.344023
$50$	132.000	0.373352
$51$	−45.0000	−0.123554
$52$	0	0
$53$	−414.000	−1.07297	−0.536484	−	0.843911i	$-0.680248\pi$
$53$	−414.000	−1.07297	−0.536484	+	0.843911i	$0.680248\pi$
$54$	−159.000	−0.400688
$55$	432.000	1.05911
$56$	−315.000	−0.751672
$57$	6.00000	0.0139424
$58$	432.000	0.978007
$59$	−522.000	−1.15184	−0.575920	−	0.817506i	$-0.695356\pi$
$59$	−522.000	−1.15184	−0.575920	+	0.817506i	$0.695356\pi$
$60$	−9.00000	−0.0193649
$61$	376.000	0.789211	0.394605	−	0.918851i	$-0.370881\pi$
$61$	376.000	0.789211	0.394605	+	0.918851i	$0.370881\pi$
$62$	792.000	1.62232
$63$	390.000	0.779927
$64$	433.000	0.845703
$65$	0	0
$66$	144.000	0.268563
$67$	36.0000	0.0656433	0.0328216	−	0.999461i	$-0.489551\pi$
$67$	36.0000	0.0656433	0.0328216	+	0.999461i	$0.489551\pi$
$68$	45.0000	0.0802508
$69$	162.000	0.282645
$70$	405.000	0.691525
$71$	−357.000	−0.596734	−0.298367	−	0.954451i	$-0.596442\pi$
$71$	−357.000	−0.596734	−0.298367	+	0.954451i	$0.596442\pi$
$72$	−546.000	−0.893704
$73$	1098.00	1.76043	0.880214	−	0.474578i	$-0.157399\pi$
$73$	1098.00	1.76043	0.880214	+	0.474578i	$0.157399\pi$
$74$	909.000	1.42796
$75$	44.0000	0.0677424
$76$	−6.00000	−0.00905588
$77$	−720.000	−1.06561
$78$	0	0
$79$	−830.000	−1.18205	−0.591027	−	0.806652i	$-0.701277\pi$
$79$	−830.000	−1.18205	−0.591027	+	0.806652i	$0.701277\pi$
$80$	−639.000	−0.893030
$81$	649.000	0.890261
$82$	−576.000	−0.775714
$83$	438.000	0.579238	0.289619	−	0.957142i	$-0.406471\pi$
$83$	438.000	0.579238	0.289619	+	0.957142i	$0.406471\pi$
$84$	15.0000	0.0194837
$85$	405.000	0.516805
$86$	−291.000	−0.364876
$87$	144.000	0.177453
$88$	1008.00	1.22106
$89$	438.000	0.521662	0.260831	−	0.965384i	$-0.416003\pi$
$89$	438.000	0.521662	0.260831	+	0.965384i	$0.416003\pi$
$90$	702.000	0.822192
$91$	0	0
$92$	−162.000	−0.183583
$93$	264.000	0.294360
$94$	333.000	0.365386
$95$	−54.0000	−0.0583188
$96$	−45.0000	−0.0478416
$97$	852.000	0.891830	0.445915	−	0.895075i	$-0.352878\pi$
$97$	852.000	0.891830	0.445915	+	0.895075i	$0.352878\pi$
$98$	354.000	0.364892
$99$	−1248.00	−1.26696
$100$	−44.0000	−0.0440000
$101$	−396.000	−0.390133	−0.195067	−	0.980790i	$-0.562492\pi$
$101$	−396.000	−0.390133	−0.195067	+	0.980790i	$0.562492\pi$
$102$	135.000	0.131049
$103$	−182.000	−0.174107	−0.0870534	−	0.996204i	$-0.527745\pi$
$103$	−182.000	−0.174107	−0.0870534	+	0.996204i	$0.527745\pi$
$104$	0	0
$105$	135.000	0.125473
$106$	1242.00	1.13805
$107$	−612.000	−0.552937	−0.276469	−	0.961023i	$-0.589164\pi$
$107$	−612.000	−0.552937	−0.276469	+	0.961023i	$0.589164\pi$
$108$	53.0000	0.0472215
$109$	−1083.00	−0.951675	−0.475838	−	0.879533i	$-0.657855\pi$
$109$	−1083.00	−0.951675	−0.475838	+	0.879533i	$0.657855\pi$
$110$	−1296.00	−1.12335
$111$	303.000	0.259094
$112$	1065.00	0.898509
$113$	90.0000	0.0749247	0.0374623	−	0.999298i	$-0.488073\pi$
$113$	90.0000	0.0749247	0.0374623	+	0.999298i	$0.488073\pi$
$114$	−18.0000	−0.0147882
$115$	−1458.00	−1.18225
$116$	−144.000	−0.115259
$117$	0	0
$118$	1566.00	1.22171
$119$	−675.000	−0.519976
$120$	−189.000	−0.143777
$121$	973.000	0.731029
$122$	−1128.00	−0.837085
$123$	−192.000	−0.140748
$124$	−264.000	−0.191193
$125$	−1521.00	−1.08834
$126$	−1170.00	−0.827237
$127$	−2086.00	−1.45750	−0.728750	−	0.684780i	$-0.759898\pi$
$127$	−2086.00	−1.45750	−0.728750	+	0.684780i	$0.759898\pi$
$128$	−1659.00	−1.14560
$129$	−97.0000	−0.0662044
$130$	0	0
$131$	−1467.00	−0.978415	−0.489208	−	0.872167i	$-0.662714\pi$
$131$	−1467.00	−0.978415	−0.489208	+	0.872167i	$0.662714\pi$
$132$	−48.0000	−0.0316505
$133$	90.0000	0.0586766
$134$	−108.000	−0.0696252
$135$	477.000	0.304101
$136$	945.000	0.595831
$137$	414.000	0.258178	0.129089	−	0.991633i	$-0.458795\pi$
$137$	414.000	0.258178	0.129089	+	0.991633i	$0.458795\pi$
$138$	−486.000	−0.299790
$139$	−2419.00	−1.47609	−0.738046	−	0.674750i	$-0.764251\pi$
$139$	−2419.00	−1.47609	−0.738046	+	0.674750i	$0.764251\pi$
$140$	−135.000	−0.0814970
$141$	111.000	0.0662971
$142$	1071.00	0.632932
$143$	0	0
$144$	1846.00	1.06829
$145$	−1296.00	−0.742255
$146$	−3294.00	−1.86721
$147$	118.000	0.0662073
$148$	−303.000	−0.168287
$149$	930.000	0.511333	0.255666	−	0.966765i	$-0.417705\pi$
$149$	930.000	0.511333	0.255666	+	0.966765i	$0.417705\pi$
$150$	−132.000	−0.0718517
$151$	1683.00	0.907024	0.453512	−	0.891250i	$-0.350171\pi$
$151$	1683.00	0.907024	0.453512	+	0.891250i	$0.350171\pi$
$152$	−126.000	−0.0672365
$153$	−1170.00	−0.618228
$154$	2160.00	1.13025
$155$	−2376.00	−1.23126
$156$	0	0
$157$	1874.00	0.952621	0.476310	−	0.879277i	$-0.341974\pi$
$157$	1874.00	0.952621	0.476310	+	0.879277i	$0.341974\pi$
$158$	2490.00	1.25376
$159$	414.000	0.206493
$160$	405.000	0.200113
$161$	2430.00	1.18951
$162$	−1947.00	−0.944264
$163$	1194.00	0.573750	0.286875	−	0.957968i	$-0.407384\pi$
$163$	1194.00	0.573750	0.286875	+	0.957968i	$0.407384\pi$
$164$	192.000	0.0914188
$165$	−432.000	−0.203825
$166$	−1314.00	−0.614375
$167$	2388.00	1.10652	0.553260	−	0.833008i	$-0.313383\pi$
$167$	2388.00	1.10652	0.553260	+	0.833008i	$0.313383\pi$
$168$	315.000	0.144659
$169$	0	0
$170$	−1215.00	−0.548154
$171$	156.000	0.0697638
$172$	97.0000	0.0430011
$173$	1566.00	0.688213	0.344106	−	0.938931i	$-0.388182\pi$
$173$	1566.00	0.688213	0.344106	+	0.938931i	$0.388182\pi$
$174$	−432.000	−0.188217
$175$	660.000	0.285093
$176$	−3408.00	−1.45959
$177$	522.000	0.221672
$178$	−1314.00	−0.553306
$179$	657.000	0.274338	0.137169	−	0.990548i	$-0.456200\pi$
$179$	657.000	0.274338	0.137169	+	0.990548i	$0.456200\pi$
$180$	−234.000	−0.0968963
$181$	1222.00	0.501826	0.250913	−	0.968010i	$-0.419269\pi$
$181$	1222.00	0.501826	0.250913	+	0.968010i	$0.419269\pi$
$182$	0	0
$183$	−376.000	−0.151884
$184$	−3402.00	−1.36304
$185$	−2727.00	−1.08375
$186$	−792.000	−0.312216
$187$	2160.00	0.844678
$188$	−111.000	−0.0430612
$189$	−795.000	−0.305967
$190$	162.000	0.0618564
$191$	1260.00	0.477332	0.238666	−	0.971102i	$-0.423290\pi$
$191$	1260.00	0.477332	0.238666	+	0.971102i	$0.423290\pi$
$192$	−433.000	−0.162756
$193$	−342.000	−0.127553	−0.0637764	−	0.997964i	$-0.520314\pi$
$193$	−342.000	−0.127553	−0.0637764	+	0.997964i	$0.520314\pi$
$194$	−2556.00	−0.945928
$195$	0	0
$196$	−118.000	−0.0430029
$197$	−81.0000	−0.0292945	−0.0146472	−	0.999893i	$-0.504663\pi$
$197$	−81.0000	−0.0292945	−0.0146472	+	0.999893i	$0.504663\pi$
$198$	3744.00	1.34381
$199$	−1996.00	−0.711019	−0.355509	−	0.934673i	$-0.615693\pi$
$199$	−1996.00	−0.711019	−0.355509	+	0.934673i	$0.615693\pi$
$200$	−924.000	−0.326683
$201$	−36.0000	−0.0126331
$202$	1188.00	0.413799
$203$	2160.00	0.746809
$204$	−45.0000	−0.0154443
$205$	1728.00	0.588726
$206$	546.000	0.184668
$207$	4212.00	1.41427
$208$	0	0
$209$	−288.000	−0.0953176
$210$	−405.000	−0.133084
$211$	2833.00	0.924321	0.462161	−	0.886796i	$-0.347074\pi$
$211$	2833.00	0.924321	0.462161	+	0.886796i	$0.347074\pi$
$212$	−414.000	−0.134121
$213$	357.000	0.114841
$214$	1836.00	0.586478
$215$	873.000	0.276921
$216$	1113.00	0.350602
$217$	3960.00	1.23881
$218$	3249.00	1.00940
$219$	−1098.00	−0.338794
$220$	432.000	0.132388
$221$	0	0
$222$	−909.000	−0.274811
$223$	3507.00	1.05312	0.526561	−	0.850138i	$-0.323481\pi$
$223$	3507.00	1.05312	0.526561	+	0.850138i	$0.323481\pi$
$224$	−675.000	−0.201341
$225$	1144.00	0.338963
$226$	−270.000	−0.0794696
$227$	228.000	0.0666647	0.0333324	−	0.999444i	$-0.489388\pi$
$227$	228.000	0.0666647	0.0333324	+	0.999444i	$0.489388\pi$
$228$	6.00000	0.00174281
$229$	−5493.00	−1.58510	−0.792549	−	0.609808i	$-0.791247\pi$
$229$	−5493.00	−1.58510	−0.792549	+	0.609808i	$0.791247\pi$
$230$	4374.00	1.25397
$231$	720.000	0.205076
$232$	−3024.00	−0.855756
$233$	−3627.00	−1.01980	−0.509898	−	0.860235i	$-0.670317\pi$
$233$	−3627.00	−1.01980	−0.509898	+	0.860235i	$0.670317\pi$
$234$	0	0
$235$	−999.000	−0.277309
$236$	−522.000	−0.143980
$237$	830.000	0.227486
$238$	2025.00	0.551518
$239$	−6075.00	−1.64418	−0.822090	−	0.569357i	$-0.807192\pi$
$239$	−6075.00	−1.64418	−0.822090	+	0.569357i	$0.807192\pi$
$240$	639.000	0.171864
$241$	−210.000	−0.0561298	−0.0280649	−	0.999606i	$-0.508935\pi$
$241$	−210.000	−0.0561298	−0.0280649	+	0.999606i	$0.508935\pi$
$242$	−2919.00	−0.775374
$243$	−2080.00	−0.549103
$244$	376.000	0.0986514
$245$	−1062.00	−0.276933
$246$	576.000	0.149286
$247$	0	0
$248$	−5544.00	−1.41953
$249$	−438.000	−0.111474
$250$	4563.00	1.15436
$251$	−7092.00	−1.78344	−0.891719	−	0.452589i	$-0.850501\pi$
$251$	−7092.00	−1.78344	−0.891719	+	0.452589i	$0.850501\pi$
$252$	390.000	0.0974908
$253$	−7776.00	−1.93230
$254$	6258.00	1.54591
$255$	−405.000	−0.0994592
$256$	1513.00	0.369385
$257$	5805.00	1.40897	0.704486	−	0.709718i	$-0.251177\pi$
$257$	5805.00	1.40897	0.704486	+	0.709718i	$0.251177\pi$
$258$	291.000	0.0702204
$259$	4545.00	1.09040
$260$	0	0
$261$	3744.00	0.887923
$262$	4401.00	1.03777
$263$	792.000	0.185691	0.0928457	−	0.995681i	$-0.470404\pi$
$263$	792.000	0.185691	0.0928457	+	0.995681i	$0.470404\pi$
$264$	−1008.00	−0.234993
$265$	−3726.00	−0.863722
$266$	−270.000	−0.0622359
$267$	−438.000	−0.100394
$268$	36.0000	0.00820541
$269$	5472.00	1.24027	0.620137	−	0.784493i	$-0.287077\pi$
$269$	5472.00	1.24027	0.620137	+	0.784493i	$0.287077\pi$
$270$	−1431.00	−0.322548
$271$	−2331.00	−0.522502	−0.261251	−	0.965271i	$-0.584135\pi$
$271$	−2331.00	−0.522502	−0.261251	+	0.965271i	$0.584135\pi$
$272$	−3195.00	−0.712225
$273$	0	0
$274$	−1242.00	−0.273839
$275$	−2112.00	−0.463121
$276$	162.000	0.0353306
$277$	1384.00	0.300204	0.150102	−	0.988671i	$-0.452040\pi$
$277$	1384.00	0.300204	0.150102	+	0.988671i	$0.452040\pi$
$278$	7257.00	1.56563
$279$	6864.00	1.47289
$280$	−2835.00	−0.605084
$281$	4062.00	0.862344	0.431172	−	0.902270i	$-0.358100\pi$
$281$	4062.00	0.862344	0.431172	+	0.902270i	$0.358100\pi$
$282$	−333.000	−0.0703187
$283$	3764.00	0.790624	0.395312	−	0.918547i	$-0.370636\pi$
$283$	3764.00	0.790624	0.395312	+	0.918547i	$0.370636\pi$
$284$	−357.000	−0.0745917
$285$	54.0000	0.0112235
$286$	0	0
$287$	−2880.00	−0.592338
$288$	−1170.00	−0.239385
$289$	−2888.00	−0.587828
$290$	3888.00	0.787280
$291$	−852.000	−0.171633
$292$	1098.00	0.220053
$293$	−4227.00	−0.842812	−0.421406	−	0.906872i	$-0.638463\pi$
$293$	−4227.00	−0.842812	−0.421406	+	0.906872i	$0.638463\pi$
$294$	−354.000	−0.0702235
$295$	−4698.00	−0.927214
$296$	−6363.00	−1.24947
$297$	2544.00	0.497030
$298$	−2790.00	−0.542350
$299$	0	0
$300$	44.0000	0.00846780
$301$	−1455.00	−0.278621
$302$	−5049.00	−0.962044
$303$	396.000	0.0750812
$304$	426.000	0.0803710
$305$	3384.00	0.635303
$306$	3510.00	0.655730
$307$	−306.000	−0.0568871	−0.0284436	−	0.999595i	$-0.509055\pi$
$307$	−306.000	−0.0568871	−0.0284436	+	0.999595i	$0.509055\pi$
$308$	−720.000	−0.133201
$309$	182.000	0.0335069
$310$	7128.00	1.30595
$311$	2106.00	0.383988	0.191994	−	0.981396i	$-0.438505\pi$
$311$	2106.00	0.383988	0.191994	+	0.981396i	$0.438505\pi$
$312$	0	0
$313$	10051.0	1.81507	0.907534	−	0.419979i	$-0.137963\pi$
$313$	10051.0	1.81507	0.907534	+	0.419979i	$0.137963\pi$
$314$	−5622.00	−1.01041
$315$	3510.00	0.627829
$316$	−830.000	−0.147757
$317$	2154.00	0.381643	0.190821	−	0.981625i	$-0.438885\pi$
$317$	2154.00	0.381643	0.190821	+	0.981625i	$0.438885\pi$
$318$	−1242.00	−0.219019
$319$	−6912.00	−1.21316
$320$	3897.00	0.680778
$321$	612.000	0.106413
$322$	−7290.00	−1.26166
$323$	−270.000	−0.0465115
$324$	649.000	0.111283
$325$	0	0
$326$	−3582.00	−0.608554
$327$	1083.00	0.183150
$328$	4032.00	0.678750
$329$	1665.00	0.279010
$330$	1296.00	0.216189
$331$	−10770.0	−1.78844	−0.894219	−	0.447630i	$-0.852268\pi$
$331$	−10770.0	−1.78844	−0.894219	+	0.447630i	$0.852268\pi$
$332$	438.000	0.0724047
$333$	7878.00	1.29643
$334$	−7164.00	−1.17364
$335$	324.000	0.0528418
$336$	−1065.00	−0.172918
$337$	2171.00	0.350926	0.175463	−	0.984486i	$-0.443858\pi$
$337$	2171.00	0.350926	0.175463	+	0.984486i	$0.443858\pi$
$338$	0	0
$339$	−90.0000	−0.0144193
$340$	405.000	0.0646006
$341$	−12672.0	−2.01240
$342$	−468.000	−0.0739957
$343$	6915.00	1.08856
$344$	2037.00	0.319267
$345$	1458.00	0.227525
$346$	−4698.00	−0.729960
$347$	−7047.00	−1.09021	−0.545105	−	0.838368i	$-0.683510\pi$
$347$	−7047.00	−1.09021	−0.545105	+	0.838368i	$0.683510\pi$
$348$	144.000	0.0221816
$349$	6873.00	1.05416	0.527082	−	0.849814i	$-0.323286\pi$
$349$	6873.00	1.05416	0.527082	+	0.849814i	$0.323286\pi$
$350$	−1980.00	−0.302387
$351$	0	0
$352$	2160.00	0.327069
$353$	9318.00	1.40495	0.702475	−	0.711709i	$-0.252078\pi$
$353$	9318.00	1.40495	0.702475	+	0.711709i	$0.252078\pi$
$354$	−1566.00	−0.235119
$355$	−3213.00	−0.480362
$356$	438.000	0.0652077
$357$	675.000	0.100069
$358$	−1971.00	−0.290979
$359$	−4128.00	−0.606873	−0.303437	−	0.952852i	$-0.598134\pi$
$359$	−4128.00	−0.606873	−0.303437	+	0.952852i	$0.598134\pi$
$360$	−4914.00	−0.719418
$361$	−6823.00	−0.994751
$362$	−3666.00	−0.532267
$363$	−973.000	−0.140687
$364$	0	0
$365$	9882.00	1.41712
$366$	1128.00	0.161097
$367$	−2536.00	−0.360703	−0.180352	−	0.983602i	$-0.557724\pi$
$367$	−2536.00	−0.360703	−0.180352	+	0.983602i	$0.557724\pi$
$368$	11502.0	1.62930
$369$	−4992.00	−0.704263
$370$	8181.00	1.14949
$371$	6210.00	0.869022
$372$	264.000	0.0367951
$373$	−92.0000	−0.0127710	−0.00638550	−	0.999980i	$-0.502033\pi$
$373$	−92.0000	−0.0127710	−0.00638550	+	0.999980i	$0.502033\pi$
$374$	−6480.00	−0.895917
$375$	1521.00	0.209451
$376$	−2331.00	−0.319713
$377$	0	0
$378$	2385.00	0.324527
$379$	−10182.0	−1.37998	−0.689992	−	0.723817i	$-0.742386\pi$
$379$	−10182.0	−1.37998	−0.689992	+	0.723817i	$0.742386\pi$
$380$	−54.0000	−0.00728985
$381$	2086.00	0.280496
$382$	−3780.00	−0.506287
$383$	579.000	0.0772468	0.0386234	−	0.999254i	$-0.487703\pi$
$383$	579.000	0.0772468	0.0386234	+	0.999254i	$0.487703\pi$
$384$	1659.00	0.220470
$385$	−6480.00	−0.857796
$386$	1026.00	0.135290
$387$	−2522.00	−0.331267
$388$	852.000	0.111479
$389$	−2106.00	−0.274495	−0.137247	−	0.990537i	$-0.543826\pi$
$389$	−2106.00	−0.274495	−0.137247	+	0.990537i	$0.543826\pi$
$390$	0	0
$391$	−7290.00	−0.942893
$392$	−2478.00	−0.319280
$393$	1467.00	0.188296
$394$	243.000	0.0310715
$395$	−7470.00	−0.951535
$396$	−1248.00	−0.158370
$397$	1974.00	0.249552	0.124776	−	0.992185i	$-0.460179\pi$
$397$	1974.00	0.249552	0.124776	+	0.992185i	$0.460179\pi$
$398$	5988.00	0.754149
$399$	−90.0000	−0.0112923
$400$	3124.00	0.390500
$401$	−11886.0	−1.48020	−0.740098	−	0.672499i	$-0.765221\pi$
$401$	−11886.0	−1.48020	−0.740098	+	0.672499i	$0.765221\pi$
$402$	108.000	0.0133994
$403$	0	0
$404$	−396.000	−0.0487667
$405$	5841.00	0.716646
$406$	−6480.00	−0.792111
$407$	−14544.0	−1.77130
$408$	−945.000	−0.114668
$409$	−1254.00	−0.151605	−0.0758023	−	0.997123i	$-0.524152\pi$
$409$	−1254.00	−0.151605	−0.0758023	+	0.997123i	$0.524152\pi$
$410$	−5184.00	−0.624438
$411$	−414.000	−0.0496864
$412$	−182.000	−0.0217633
$413$	7830.00	0.932903
$414$	−12636.0	−1.50006
$415$	3942.00	0.466278
$416$	0	0
$417$	2419.00	0.284074
$418$	864.000	0.101100
$419$	5823.00	0.678931	0.339466	−	0.940618i	$-0.389754\pi$
$419$	5823.00	0.678931	0.339466	+	0.940618i	$0.389754\pi$
$420$	135.000	0.0156841
$421$	7341.00	0.849830	0.424915	−	0.905233i	$-0.360304\pi$
$421$	7341.00	0.849830	0.424915	+	0.905233i	$0.360304\pi$
$422$	−8499.00	−0.980391
$423$	2886.00	0.331731
$424$	−8694.00	−0.995797
$425$	−1980.00	−0.225986
$426$	−1071.00	−0.121808
$427$	−5640.00	−0.639201
$428$	−612.000	−0.0691171
$429$	0	0
$430$	−2619.00	−0.293720
$431$	7485.00	0.836519	0.418260	−	0.908328i	$-0.362640\pi$
$431$	7485.00	0.836519	0.418260	+	0.908328i	$0.362640\pi$
$432$	−3763.00	−0.419091
$433$	15203.0	1.68732	0.843660	−	0.536878i	$-0.180396\pi$
$433$	15203.0	1.68732	0.843660	+	0.536878i	$0.180396\pi$
$434$	−11880.0	−1.31396
$435$	1296.00	0.142847
$436$	−1083.00	−0.118959
$437$	972.000	0.106401
$438$	3294.00	0.359346
$439$	1762.00	0.191562	0.0957809	−	0.995402i	$-0.469465\pi$
$439$	1762.00	0.191562	0.0957809	+	0.995402i	$0.469465\pi$
$440$	9072.00	0.982933
$441$	3068.00	0.331282
$442$	0	0
$443$	−7317.00	−0.784743	−0.392372	−	0.919807i	$-0.628345\pi$
$443$	−7317.00	−0.784743	−0.392372	+	0.919807i	$0.628345\pi$
$444$	303.000	0.0323868
$445$	3942.00	0.419930
$446$	−10521.0	−1.11700
$447$	−930.000	−0.0984060
$448$	−6495.00	−0.684955
$449$	5016.00	0.527215	0.263608	−	0.964630i	$-0.415088\pi$
$449$	5016.00	0.527215	0.263608	+	0.964630i	$0.415088\pi$
$450$	−3432.00	−0.359525
$451$	9216.00	0.962227
$452$	90.0000	0.00936558
$453$	−1683.00	−0.174557
$454$	−684.000	−0.0707086
$455$	0	0
$456$	126.000	0.0129397
$457$	−9870.00	−1.01028	−0.505141	−	0.863037i	$-0.668560\pi$
$457$	−9870.00	−1.01028	−0.505141	+	0.863037i	$0.668560\pi$
$458$	16479.0	1.68125
$459$	2385.00	0.242532
$460$	−1458.00	−0.147782
$461$	14541.0	1.46907	0.734536	−	0.678570i	$-0.237400\pi$
$461$	14541.0	1.46907	0.734536	+	0.678570i	$0.237400\pi$
$462$	−2160.00	−0.217516
$463$	2112.00	0.211993	0.105997	−	0.994366i	$-0.466197\pi$
$463$	2112.00	0.211993	0.105997	+	0.994366i	$0.466197\pi$
$464$	10224.0	1.02293
$465$	2376.00	0.236956
$466$	10881.0	1.08166
$467$	−3276.00	−0.324615	−0.162307	−	0.986740i	$-0.551894\pi$
$467$	−3276.00	−0.324615	−0.162307	+	0.986740i	$0.551894\pi$
$468$	0	0
$469$	−540.000	−0.0531661
$470$	2997.00	0.294130
$471$	−1874.00	−0.183332
$472$	−10962.0	−1.06900
$473$	4656.00	0.452607
$474$	−2490.00	−0.241286
$475$	264.000	0.0255014
$476$	−675.000	−0.0649970
$477$	10764.0	1.03323
$478$	18225.0	1.74392
$479$	15453.0	1.47404	0.737020	−	0.675870i	$-0.236232\pi$
$479$	15453.0	1.47404	0.737020	+	0.675870i	$0.236232\pi$
$480$	−405.000	−0.0385117
$481$	0	0
$482$	630.000	0.0595347
$483$	−2430.00	−0.228921
$484$	973.000	0.0913787
$485$	7668.00	0.717909
$486$	6240.00	0.582412
$487$	3660.00	0.340555	0.170278	−	0.985396i	$-0.445534\pi$
$487$	3660.00	0.340555	0.170278	+	0.985396i	$0.445534\pi$
$488$	7896.00	0.732449
$489$	−1194.00	−0.110418
$490$	3186.00	0.293732
$491$	−747.000	−0.0686591	−0.0343296	−	0.999411i	$-0.510930\pi$
$491$	−747.000	−0.0686591	−0.0343296	+	0.999411i	$0.510930\pi$
$492$	−192.000	−0.0175936
$493$	−6480.00	−0.591977
$494$	0	0
$495$	−11232.0	−1.01988
$496$	18744.0	1.69684
$497$	5355.00	0.483309
$498$	1314.00	0.118236
$499$	15804.0	1.41780	0.708902	−	0.705307i	$-0.249191\pi$
$499$	15804.0	1.41780	0.708902	+	0.705307i	$0.249191\pi$
$500$	−1521.00	−0.136042
$501$	−2388.00	−0.212950
$502$	21276.0	1.89162
$503$	−12078.0	−1.07064	−0.535319	−	0.844650i	$-0.679809\pi$
$503$	−12078.0	−1.07064	−0.535319	+	0.844650i	$0.679809\pi$
$504$	8190.00	0.723833
$505$	−3564.00	−0.314051
$506$	23328.0	2.04952
$507$	0	0
$508$	−2086.00	−0.182188
$509$	−16110.0	−1.40287	−0.701437	−	0.712731i	$-0.747458\pi$
$509$	−16110.0	−1.40287	−0.701437	+	0.712731i	$0.747458\pi$
$510$	1215.00	0.105492
$511$	−16470.0	−1.42581
$512$	8733.00	0.753804
$513$	−318.000	−0.0273685
$514$	−17415.0	−1.49444
$515$	−1638.00	−0.140153
$516$	−97.0000	−0.00827556
$517$	−5328.00	−0.453240
$518$	−13635.0	−1.15654
$519$	−1566.00	−0.132447
$520$	0	0
$521$	3915.00	0.329212	0.164606	−	0.986359i	$-0.447365\pi$
$521$	3915.00	0.329212	0.164606	+	0.986359i	$0.447365\pi$
$522$	−11232.0	−0.941784
$523$	16184.0	1.35311	0.676555	−	0.736392i	$-0.263472\pi$
$523$	16184.0	1.35311	0.676555	+	0.736392i	$0.263472\pi$
$524$	−1467.00	−0.122302
$525$	−660.000	−0.0548662
$526$	−2376.00	−0.196955
$527$	−11880.0	−0.981975
$528$	3408.00	0.280898
$529$	14077.0	1.15698
$530$	11178.0	0.916116
$531$	13572.0	1.10918
$532$	90.0000	0.00733458
$533$	0	0
$534$	1314.00	0.106484
$535$	−5508.00	−0.445106
$536$	756.000	0.0609221
$537$	−657.000	−0.0527964
$538$	−16416.0	−1.31551
$539$	−5664.00	−0.452627
$540$	477.000	0.0380126
$541$	7923.00	0.629642	0.314821	−	0.949151i	$-0.398055\pi$
$541$	7923.00	0.629642	0.314821	+	0.949151i	$0.398055\pi$
$542$	6993.00	0.554198
$543$	−1222.00	−0.0965765
$544$	2025.00	0.159598
$545$	−9747.00	−0.766084
$546$	0	0
$547$	−14389.0	−1.12473	−0.562367	−	0.826888i	$-0.690109\pi$
$547$	−14389.0	−1.12473	−0.562367	+	0.826888i	$0.690109\pi$
$548$	414.000	0.0322723
$549$	−9776.00	−0.759981
$550$	6336.00	0.491214
$551$	864.000	0.0668015
$552$	3402.00	0.262317
$553$	12450.0	0.957374
$554$	−4152.00	−0.318414
$555$	2727.00	0.208567
$556$	−2419.00	−0.184512
$557$	10383.0	0.789842	0.394921	−	0.918715i	$-0.370772\pi$
$557$	10383.0	0.789842	0.394921	+	0.918715i	$0.370772\pi$
$558$	−20592.0	−1.56224
$559$	0	0
$560$	9585.00	0.723286
$561$	−2160.00	−0.162558
$562$	−12186.0	−0.914654
$563$	16425.0	1.22954	0.614770	−	0.788706i	$-0.289249\pi$
$563$	16425.0	1.22954	0.614770	+	0.788706i	$0.289249\pi$
$564$	111.000	0.00828713
$565$	810.000	0.0603132
$566$	−11292.0	−0.838583
$567$	−9735.00	−0.721043
$568$	−7497.00	−0.553815
$569$	−12213.0	−0.899817	−0.449908	−	0.893075i	$-0.648543\pi$
$569$	−12213.0	−0.899817	−0.449908	+	0.893075i	$0.648543\pi$
$570$	−162.000	−0.0119043
$571$	6383.00	0.467811	0.233906	−	0.972259i	$-0.424849\pi$
$571$	6383.00	0.467811	0.233906	+	0.972259i	$0.424849\pi$
$572$	0	0
$573$	−1260.00	−0.0918626
$574$	8640.00	0.628269
$575$	7128.00	0.516971
$576$	−11258.0	−0.814381
$577$	−6426.00	−0.463636	−0.231818	−	0.972759i	$-0.574467\pi$
$577$	−6426.00	−0.463636	−0.231818	+	0.972759i	$0.574467\pi$
$578$	8664.00	0.623486
$579$	342.000	0.0245476
$580$	−1296.00	−0.0927818
$581$	−6570.00	−0.469139
$582$	2556.00	0.182044
$583$	−19872.0	−1.41169
$584$	23058.0	1.63381
$585$	0	0
$586$	12681.0	0.893937
$587$	21330.0	1.49980	0.749901	−	0.661551i	$-0.230101\pi$
$587$	21330.0	1.49980	0.749901	+	0.661551i	$0.230101\pi$
$588$	118.000	0.00827591
$589$	1584.00	0.110811
$590$	14094.0	0.983459
$591$	81.0000	0.00563772
$592$	21513.0	1.49355
$593$	−12084.0	−0.836813	−0.418407	−	0.908260i	$-0.637411\pi$
$593$	−12084.0	−0.836813	−0.418407	+	0.908260i	$0.637411\pi$
$594$	−7632.00	−0.527180
$595$	−6075.00	−0.418573
$596$	930.000	0.0639166
$597$	1996.00	0.136836
$598$	0	0
$599$	2394.00	0.163299	0.0816496	−	0.996661i	$-0.473981\pi$
$599$	2394.00	0.163299	0.0816496	+	0.996661i	$0.473981\pi$
$600$	924.000	0.0628702
$601$	−21971.0	−1.49121	−0.745604	−	0.666389i	$-0.767839\pi$
$601$	−21971.0	−1.49121	−0.745604	+	0.666389i	$0.767839\pi$
$602$	4365.00	0.295522
$603$	−936.000	−0.0632121
$604$	1683.00	0.113378
$605$	8757.00	0.588467
$606$	−1188.00	−0.0796356
$607$	−15406.0	−1.03017	−0.515083	−	0.857141i	$-0.672239\pi$
$607$	−15406.0	−1.03017	−0.515083	+	0.857141i	$0.672239\pi$
$608$	−270.000	−0.0180098
$609$	−2160.00	−0.143724
$610$	−10152.0	−0.673840
$611$	0	0
$612$	−1170.00	−0.0772785
$613$	9630.00	0.634506	0.317253	−	0.948341i	$-0.397240\pi$
$613$	9630.00	0.634506	0.317253	+	0.948341i	$0.397240\pi$
$614$	918.000	0.0603379
$615$	−1728.00	−0.113300
$616$	−15120.0	−0.988965
$617$	−14748.0	−0.962289	−0.481144	−	0.876641i	$-0.659779\pi$
$617$	−14748.0	−0.962289	−0.481144	+	0.876641i	$0.659779\pi$
$618$	−546.000	−0.0355394
$619$	3672.00	0.238433	0.119217	−	0.992868i	$-0.461962\pi$
$619$	3672.00	0.238433	0.119217	+	0.992868i	$0.461962\pi$
$620$	−2376.00	−0.153907
$621$	−8586.00	−0.554822
$622$	−6318.00	−0.407281
$623$	−6570.00	−0.422506
$624$	0	0
$625$	−8189.00	−0.524096
$626$	−30153.0	−1.92517
$627$	288.000	0.0183439
$628$	1874.00	0.119078
$629$	−13635.0	−0.864329
$630$	−10530.0	−0.665913
$631$	19875.0	1.25390	0.626950	−	0.779059i	$-0.284303\pi$
$631$	19875.0	1.25390	0.626950	+	0.779059i	$0.284303\pi$
$632$	−17430.0	−1.09704
$633$	−2833.00	−0.177886
$634$	−6462.00	−0.404793
$635$	−18774.0	−1.17327
$636$	414.000	0.0258116
$637$	0	0
$638$	20736.0	1.28675
$639$	9282.00	0.574633
$640$	−14931.0	−0.922187
$641$	−1710.00	−0.105368	−0.0526840	−	0.998611i	$-0.516778\pi$
$641$	−1710.00	−0.105368	−0.0526840	+	0.998611i	$0.516778\pi$
$642$	−1836.00	−0.112868
$643$	16452.0	1.00903	0.504513	−	0.863404i	$-0.331672\pi$
$643$	16452.0	1.00903	0.504513	+	0.863404i	$0.331672\pi$
$644$	2430.00	0.148689
$645$	−873.000	−0.0532936
$646$	810.000	0.0493329
$647$	−25902.0	−1.57390	−0.786950	−	0.617017i	$-0.788341\pi$
$647$	−25902.0	−1.57390	−0.786950	+	0.617017i	$0.788341\pi$
$648$	13629.0	0.826231
$649$	−25056.0	−1.51546
$650$	0	0
$651$	−3960.00	−0.238410
$652$	1194.00	0.0717188
$653$	18108.0	1.08518	0.542589	−	0.839999i	$-0.317444\pi$
$653$	18108.0	1.08518	0.542589	+	0.839999i	$0.317444\pi$
$654$	−3249.00	−0.194260
$655$	−13203.0	−0.787609
$656$	−13632.0	−0.811342
$657$	−28548.0	−1.69523
$658$	−4995.00	−0.295935
$659$	−32904.0	−1.94500	−0.972502	−	0.232894i	$-0.925181\pi$
$659$	−32904.0	−1.94500	−0.972502	+	0.232894i	$0.925181\pi$
$660$	−432.000	−0.0254781
$661$	−15318.0	−0.901363	−0.450682	−	0.892685i	$-0.648819\pi$
$661$	−15318.0	−0.901363	−0.450682	+	0.892685i	$0.648819\pi$
$662$	32310.0	1.89692
$663$	0	0
$664$	9198.00	0.537578
$665$	810.000	0.0472338
$666$	−23634.0	−1.37507
$667$	23328.0	1.35422
$668$	2388.00	0.138315
$669$	−3507.00	−0.202673
$670$	−972.000	−0.0560472
$671$	18048.0	1.03835
$672$	675.000	0.0387481
$673$	7729.00	0.442691	0.221346	−	0.975195i	$-0.428955\pi$
$673$	7729.00	0.442691	0.221346	+	0.975195i	$0.428955\pi$
$674$	−6513.00	−0.372213
$675$	−2332.00	−0.132976
$676$	0	0
$677$	19242.0	1.09236	0.546182	−	0.837667i	$-0.316081\pi$
$677$	19242.0	1.09236	0.546182	+	0.837667i	$0.316081\pi$
$678$	270.000	0.0152939
$679$	−12780.0	−0.722314
$680$	8505.00	0.479635
$681$	−228.000	−0.0128296
$682$	38016.0	2.13447
$683$	−22518.0	−1.26153	−0.630767	−	0.775973i	$-0.717260\pi$
$683$	−22518.0	−1.26153	−0.630767	+	0.775973i	$0.717260\pi$
$684$	156.000	0.00872048
$685$	3726.00	0.207829
$686$	−20745.0	−1.15459
$687$	5493.00	0.305052
$688$	−6887.00	−0.381634
$689$	0	0
$690$	−4374.00	−0.241327
$691$	−9168.00	−0.504728	−0.252364	−	0.967632i	$-0.581208\pi$
$691$	−9168.00	−0.504728	−0.252364	+	0.967632i	$0.581208\pi$
$692$	1566.00	0.0860266
$693$	18720.0	1.02614
$694$	21141.0	1.15634
$695$	−21771.0	−1.18823
$696$	3024.00	0.164690
$697$	8640.00	0.469531
$698$	−20619.0	−1.11811
$699$	3627.00	0.196260
$700$	660.000	0.0356367
$701$	−1170.00	−0.0630389	−0.0315195	−	0.999503i	$-0.510035\pi$
$701$	−1170.00	−0.0630389	−0.0315195	+	0.999503i	$0.510035\pi$
$702$	0	0
$703$	1818.00	0.0975351
$704$	20784.0	1.11268
$705$	999.000	0.0533681
$706$	−27954.0	−1.49017
$707$	5940.00	0.315978
$708$	522.000	0.0277090
$709$	1662.00	0.0880363	0.0440181	−	0.999031i	$-0.485984\pi$
$709$	1662.00	0.0880363	0.0440181	+	0.999031i	$0.485984\pi$
$710$	9639.00	0.509500
$711$	21580.0	1.13827
$712$	9198.00	0.484143
$713$	42768.0	2.24639
$714$	−2025.00	−0.106140
$715$	0	0
$716$	657.000	0.0342922
$717$	6075.00	0.316423
$718$	12384.0	0.643686
$719$	−30960.0	−1.60586	−0.802930	−	0.596073i	$-0.796727\pi$
$719$	−30960.0	−1.60586	−0.802930	+	0.596073i	$0.796727\pi$
$720$	16614.0	0.859954
$721$	2730.00	0.141013
$722$	20469.0	1.05509
$723$	210.000	0.0108022
$724$	1222.00	0.0627283
$725$	6336.00	0.324570
$726$	2919.00	0.149221
$727$	8372.00	0.427098	0.213549	−	0.976932i	$-0.431498\pi$
$727$	8372.00	0.427098	0.213549	+	0.976932i	$0.431498\pi$
$728$	0	0
$729$	−15443.0	−0.784586
$730$	−29646.0	−1.50308
$731$	4365.00	0.220855
$732$	−376.000	−0.0189855
$733$	2739.00	0.138018	0.0690091	−	0.997616i	$-0.478016\pi$
$733$	2739.00	0.138018	0.0690091	+	0.997616i	$0.478016\pi$
$734$	7608.00	0.382584
$735$	1062.00	0.0532959
$736$	−7290.00	−0.365099
$737$	1728.00	0.0863659
$738$	14976.0	0.746984
$739$	6756.00	0.336297	0.168148	−	0.985762i	$-0.446221\pi$
$739$	6756.00	0.336297	0.168148	+	0.985762i	$0.446221\pi$
$740$	−2727.00	−0.135468
$741$	0	0
$742$	−18630.0	−0.921737
$743$	−29643.0	−1.46366	−0.731828	−	0.681490i	$-0.761332\pi$
$743$	−29643.0	−1.46366	−0.731828	+	0.681490i	$0.761332\pi$
$744$	5544.00	0.273189
$745$	8370.00	0.411615
$746$	276.000	0.0135457
$747$	−11388.0	−0.557785
$748$	2160.00	0.105585
$749$	9180.00	0.447837
$750$	−4563.00	−0.222156
$751$	−18128.0	−0.880826	−0.440413	−	0.897795i	$-0.645168\pi$
$751$	−18128.0	−0.880826	−0.440413	+	0.897795i	$0.645168\pi$
$752$	7881.00	0.382168
$753$	7092.00	0.343223
$754$	0	0
$755$	15147.0	0.730140
$756$	−795.000	−0.0382459
$757$	−6410.00	−0.307761	−0.153881	−	0.988089i	$-0.549177\pi$
$757$	−6410.00	−0.307761	−0.153881	+	0.988089i	$0.549177\pi$
$758$	30546.0	1.46369
$759$	7776.00	0.371872
$760$	−1134.00	−0.0541243
$761$	−28290.0	−1.34758	−0.673792	−	0.738921i	$-0.735336\pi$
$761$	−28290.0	−1.34758	−0.673792	+	0.738921i	$0.735336\pi$
$762$	−6258.00	−0.297511
$763$	16245.0	0.770784
$764$	1260.00	0.0596665
$765$	−10530.0	−0.497664
$766$	−1737.00	−0.0819326
$767$	0	0
$768$	−1513.00	−0.0710881
$769$	27960.0	1.31114	0.655568	−	0.755136i	$-0.272429\pi$
$769$	27960.0	1.31114	0.655568	+	0.755136i	$0.272429\pi$
$770$	19440.0	0.909830
$771$	−5805.00	−0.271157
$772$	−342.000	−0.0159441
$773$	5649.00	0.262847	0.131423	−	0.991326i	$-0.458045\pi$
$773$	5649.00	0.262847	0.131423	+	0.991326i	$0.458045\pi$
$774$	7566.00	0.351362
$775$	11616.0	0.538399
$776$	17892.0	0.827687
$777$	−4545.00	−0.209847
$778$	6318.00	0.291146
$779$	−1152.00	−0.0529842
$780$	0	0
$781$	−17136.0	−0.785114
$782$	21870.0	1.00009
$783$	−7632.00	−0.348334
$784$	8378.00	0.381651
$785$	16866.0	0.766845
$786$	−4401.00	−0.199718
$787$	−756.000	−0.0342420	−0.0171210	−	0.999853i	$-0.505450\pi$
$787$	−756.000	−0.0342420	−0.0171210	+	0.999853i	$0.505450\pi$
$788$	−81.0000	−0.00366181
$789$	−792.000	−0.0357363
$790$	22410.0	1.00926
$791$	−1350.00	−0.0606833
$792$	−26208.0	−1.17583
$793$	0	0
$794$	−5922.00	−0.264690
$795$	3726.00	0.166223
$796$	−1996.00	−0.0888773
$797$	−31194.0	−1.38638	−0.693192	−	0.720753i	$-0.743796\pi$
$797$	−31194.0	−1.38638	−0.693192	+	0.720753i	$0.743796\pi$
$798$	270.000	0.0119773
$799$	−4995.00	−0.221164
$800$	−1980.00	−0.0875045
$801$	−11388.0	−0.502341
$802$	35658.0	1.56998
$803$	52704.0	2.31617
$804$	−36.0000	−0.00157913
$805$	21870.0	0.957536
$806$	0	0
$807$	−5472.00	−0.238691
$808$	−8316.00	−0.362074
$809$	17055.0	0.741189	0.370594	−	0.928795i	$-0.379154\pi$
$809$	17055.0	0.741189	0.370594	+	0.928795i	$0.379154\pi$
$810$	−17523.0	−0.760118
$811$	−35520.0	−1.53795	−0.768974	−	0.639280i	$-0.779232\pi$
$811$	−35520.0	−1.53795	−0.768974	+	0.639280i	$0.779232\pi$
$812$	2160.00	0.0933512
$813$	2331.00	0.100556
$814$	43632.0	1.87875
$815$	10746.0	0.461860
$816$	3195.00	0.137068
$817$	−582.000	−0.0249224
$818$	3762.00	0.160801
$819$	0	0
$820$	1728.00	0.0735907
$821$	−1095.00	−0.0465478	−0.0232739	−	0.999729i	$-0.507409\pi$
$821$	−1095.00	−0.0465478	−0.0232739	+	0.999729i	$0.507409\pi$
$822$	1242.00	0.0527004
$823$	2554.00	0.108174	0.0540868	−	0.998536i	$-0.482775\pi$
$823$	2554.00	0.108174	0.0540868	+	0.998536i	$0.482775\pi$
$824$	−3822.00	−0.161585
$825$	2112.00	0.0891278
$826$	−23490.0	−0.989494
$827$	−21522.0	−0.904950	−0.452475	−	0.891777i	$-0.649459\pi$
$827$	−21522.0	−0.904950	−0.452475	+	0.891777i	$0.649459\pi$
$828$	4212.00	0.176784
$829$	13124.0	0.549838	0.274919	−	0.961467i	$-0.411349\pi$
$829$	13124.0	0.549838	0.274919	+	0.961467i	$0.411349\pi$
$830$	−11826.0	−0.494562
$831$	−1384.00	−0.0577743
$832$	0	0
$833$	−5310.00	−0.220865
$834$	−7257.00	−0.301306
$835$	21492.0	0.890732
$836$	−288.000	−0.0119147
$837$	−13992.0	−0.577819
$838$	−17469.0	−0.720115
$839$	23424.0	0.963869	0.481935	−	0.876207i	$-0.339934\pi$
$839$	23424.0	0.963869	0.481935	+	0.876207i	$0.339934\pi$
$840$	2835.00	0.116449
$841$	−3653.00	−0.149781
$842$	−22023.0	−0.901381
$843$	−4062.00	−0.165958
$844$	2833.00	0.115540
$845$	0	0
$846$	−8658.00	−0.351854
$847$	−14595.0	−0.592078
$848$	29394.0	1.19032
$849$	−3764.00	−0.152156
$850$	5940.00	0.239694
$851$	49086.0	1.97726
$852$	357.000	0.0143552
$853$	−31077.0	−1.24743	−0.623714	−	0.781653i	$-0.714377\pi$
$853$	−31077.0	−1.24743	−0.623714	+	0.781653i	$0.714377\pi$
$854$	16920.0	0.677975
$855$	1404.00	0.0561588
$856$	−12852.0	−0.513169
$857$	19422.0	0.774146	0.387073	−	0.922049i	$-0.373486\pi$
$857$	19422.0	0.774146	0.387073	+	0.922049i	$0.373486\pi$
$858$	0	0
$859$	1744.00	0.0692718	0.0346359	−	0.999400i	$-0.488973\pi$
$859$	1744.00	0.0692718	0.0346359	+	0.999400i	$0.488973\pi$
$860$	873.000	0.0346152
$861$	2880.00	0.113996
$862$	−22455.0	−0.887263
$863$	−19179.0	−0.756501	−0.378251	−	0.925703i	$-0.623474\pi$
$863$	−19179.0	−0.756501	−0.378251	+	0.925703i	$0.623474\pi$
$864$	2385.00	0.0939113
$865$	14094.0	0.554000
$866$	−45609.0	−1.78967
$867$	2888.00	0.113128
$868$	3960.00	0.154852
$869$	−39840.0	−1.55521
$870$	−3888.00	−0.151512
$871$	0	0
$872$	−22743.0	−0.883228
$873$	−22152.0	−0.858799
$874$	−2916.00	−0.112855
$875$	22815.0	0.881472
$876$	−1098.00	−0.0423493
$877$	−29217.0	−1.12496	−0.562479	−	0.826812i	$-0.690152\pi$
$877$	−29217.0	−1.12496	−0.562479	+	0.826812i	$0.690152\pi$
$878$	−5286.00	−0.203182
$879$	4227.00	0.162199
$880$	−30672.0	−1.17495
$881$	15633.0	0.597831	0.298916	−	0.954280i	$-0.403375\pi$
$881$	15633.0	0.597831	0.298916	+	0.954280i	$0.403375\pi$
$882$	−9204.00	−0.351377
$883$	−30589.0	−1.16580	−0.582900	−	0.812544i	$-0.698082\pi$
$883$	−30589.0	−1.16580	−0.582900	+	0.812544i	$0.698082\pi$
$884$	0	0
$885$	4698.00	0.178442
$886$	21951.0	0.832346
$887$	−25884.0	−0.979819	−0.489910	−	0.871773i	$-0.662970\pi$
$887$	−25884.0	−0.979819	−0.489910	+	0.871773i	$0.662970\pi$
$888$	6363.00	0.240460
$889$	31290.0	1.18046
$890$	−11826.0	−0.445403
$891$	31152.0	1.17130
$892$	3507.00	0.131640
$893$	666.000	0.0249573
$894$	2790.00	0.104375
$895$	5913.00	0.220838
$896$	24885.0	0.927845
$897$	0	0
$898$	−15048.0	−0.559196
$899$	38016.0	1.41035
$900$	1144.00	0.0423704
$901$	−18630.0	−0.688852
$902$	−27648.0	−1.02060
$903$	1455.00	0.0536206
$904$	1890.00	0.0695359
$905$	10998.0	0.403962
$906$	5049.00	0.185145
$907$	12305.0	0.450475	0.225237	−	0.974304i	$-0.427684\pi$
$907$	12305.0	0.450475	0.225237	+	0.974304i	$0.427684\pi$
$908$	228.000	0.00833309
$909$	10296.0	0.375684
$910$	0	0
$911$	29772.0	1.08276	0.541378	−	0.840779i	$-0.317903\pi$
$911$	29772.0	1.08276	0.541378	+	0.840779i	$0.317903\pi$
$912$	−426.000	−0.0154674
$913$	21024.0	0.762095
$914$	29610.0	1.07157
$915$	−3384.00	−0.122264
$916$	−5493.00	−0.198137
$917$	22005.0	0.792442
$918$	−7155.00	−0.257244
$919$	47644.0	1.71015	0.855076	−	0.518502i	$-0.173510\pi$
$919$	47644.0	1.71015	0.855076	+	0.518502i	$0.173510\pi$
$920$	−30618.0	−1.09722
$921$	306.000	0.0109479
$922$	−43623.0	−1.55819
$923$	0	0
$924$	720.000	0.0256345
$925$	13332.0	0.473896
$926$	−6336.00	−0.224853
$927$	4732.00	0.167658
$928$	−6480.00	−0.229220
$929$	−21924.0	−0.774277	−0.387138	−	0.922022i	$-0.626536\pi$
$929$	−21924.0	−0.774277	−0.387138	+	0.922022i	$0.626536\pi$
$930$	−7128.00	−0.251329
$931$	708.000	0.0249235
$932$	−3627.00	−0.127475
$933$	−2106.00	−0.0738985
$934$	9828.00	0.344306
$935$	19440.0	0.679953
$936$	0	0
$937$	32398.0	1.12956	0.564779	−	0.825242i	$-0.308961\pi$
$937$	32398.0	1.12956	0.564779	+	0.825242i	$0.308961\pi$
$938$	1620.00	0.0563911
$939$	−10051.0	−0.349310
$940$	−999.000	−0.0346636
$941$	−2097.00	−0.0726464	−0.0363232	−	0.999340i	$-0.511565\pi$
$941$	−2097.00	−0.0726464	−0.0363232	+	0.999340i	$0.511565\pi$
$942$	5622.00	0.194453
$943$	−31104.0	−1.07411
$944$	37062.0	1.27782
$945$	−7155.00	−0.246299
$946$	−13968.0	−0.480062
$947$	20016.0	0.686835	0.343417	−	0.939183i	$-0.388415\pi$
$947$	20016.0	0.686835	0.343417	+	0.939183i	$0.388415\pi$
$948$	830.000	0.0284358
$949$	0	0
$950$	−792.000	−0.0270483
$951$	−2154.00	−0.0734471
$952$	−14175.0	−0.482578
$953$	−24993.0	−0.849531	−0.424765	−	0.905304i	$-0.639643\pi$
$953$	−24993.0	−0.849531	−0.424765	+	0.905304i	$0.639643\pi$
$954$	−32292.0	−1.09590
$955$	11340.0	0.384245
$956$	−6075.00	−0.205523
$957$	6912.00	0.233473
$958$	−46359.0	−1.56346
$959$	−6210.00	−0.209105
$960$	−3897.00	−0.131016
$961$	39905.0	1.33950
$962$	0	0
$963$	15912.0	0.532458
$964$	−210.000	−0.00701623
$965$	−3078.00	−0.102678
$966$	7290.00	0.242807
$967$	40959.0	1.36210	0.681051	−	0.732236i	$-0.261523\pi$
$967$	40959.0	1.36210	0.681051	+	0.732236i	$0.261523\pi$
$968$	20433.0	0.678452
$969$	270.000	0.00895113
$970$	−23004.0	−0.761458
$971$	−48933.0	−1.61723	−0.808617	−	0.588335i	$-0.799784\pi$
$971$	−48933.0	−1.61723	−0.808617	+	0.588335i	$0.799784\pi$
$972$	−2080.00	−0.0686379
$973$	36285.0	1.19552
$974$	−10980.0	−0.361213
$975$	0	0
$976$	−26696.0	−0.875531
$977$	−47388.0	−1.55177	−0.775884	−	0.630876i	$-0.782696\pi$
$977$	−47388.0	−1.55177	−0.775884	+	0.630876i	$0.782696\pi$
$978$	3582.00	0.117116
$979$	21024.0	0.686343
$980$	−1062.00	−0.0346167
$981$	28158.0	0.916428
$982$	2241.00	0.0728240
$983$	−16803.0	−0.545201	−0.272600	−	0.962127i	$-0.587884\pi$
$983$	−16803.0	−0.545201	−0.272600	+	0.962127i	$0.587884\pi$
$984$	−4032.00	−0.130625
$985$	−729.000	−0.0235816
$986$	19440.0	0.627886
$987$	−1665.00	−0.0536956
$988$	0	0
$989$	−15714.0	−0.505234
$990$	33696.0	1.08175
$991$	−57526.0	−1.84397	−0.921985	−	0.387226i	$-0.873433\pi$
$991$	−57526.0	−1.84397	−0.921985	+	0.387226i	$0.873433\pi$
$992$	−11880.0	−0.380232
$993$	10770.0	0.344185
$994$	−16065.0	−0.512627
$995$	−17964.0	−0.572359
$996$	−438.000	−0.0139343
$997$	−25000.0	−0.794140	−0.397070	−	0.917788i	$-0.629973\pi$
$997$	−25000.0	−0.794140	−0.397070	+	0.917788i	$0.629973\pi$
$998$	−47412.0	−1.50381
$999$	−16059.0	−0.508593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.b.1.1		1
3.2	odd	2		1521.4.a.i.1.1		1
13.2	odd	12		169.4.e.d.147.1		4
13.3	even	3		169.4.c.c.22.1		2
13.4	even	6		169.4.c.b.146.1		2
13.5	odd	4		13.4.b.a.12.2	yes	2
13.6	odd	12		169.4.e.d.23.2		4
13.7	odd	12		169.4.e.d.23.1		4
13.8	odd	4		13.4.b.a.12.1	✓	2
13.9	even	3		169.4.c.c.146.1		2
13.10	even	6		169.4.c.b.22.1		2
13.11	odd	12		169.4.e.d.147.2		4
13.12	even	2		169.4.a.c.1.1		1
39.5	even	4		117.4.b.a.64.1		2
39.8	even	4		117.4.b.a.64.2		2
39.38	odd	2		1521.4.a.d.1.1		1
52.31	even	4		208.4.f.b.129.1		2
52.47	even	4		208.4.f.b.129.2		2
65.8	even	4		325.4.d.a.324.1		2
65.18	even	4		325.4.d.b.324.1		2
65.34	odd	4		325.4.c.b.51.2		2
65.44	odd	4		325.4.c.b.51.1		2
65.47	even	4		325.4.d.b.324.2		2
65.57	even	4		325.4.d.a.324.2		2
104.5	odd	4		832.4.f.e.129.2		2
104.21	odd	4		832.4.f.e.129.1		2
104.83	even	4		832.4.f.c.129.2		2
104.99	even	4		832.4.f.c.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.b.a.12.1	✓	2	13.8	odd	4
13.4.b.a.12.2	yes	2	13.5	odd	4
117.4.b.a.64.1		2	39.5	even	4
117.4.b.a.64.2		2	39.8	even	4
169.4.a.b.1.1		1	1.1	even	1	trivial
169.4.a.c.1.1		1	13.12	even	2
169.4.c.b.22.1		2	13.10	even	6
169.4.c.b.146.1		2	13.4	even	6
169.4.c.c.22.1		2	13.3	even	3
169.4.c.c.146.1		2	13.9	even	3
169.4.e.d.23.1		4	13.7	odd	12
169.4.e.d.23.2		4	13.6	odd	12
169.4.e.d.147.1		4	13.2	odd	12
169.4.e.d.147.2		4	13.11	odd	12
208.4.f.b.129.1		2	52.31	even	4
208.4.f.b.129.2		2	52.47	even	4
325.4.c.b.51.1		2	65.44	odd	4
325.4.c.b.51.2		2	65.34	odd	4
325.4.d.a.324.1		2	65.8	even	4
325.4.d.a.324.2		2	65.57	even	4
325.4.d.b.324.1		2	65.18	even	4
325.4.d.b.324.2		2	65.47	even	4
832.4.f.c.129.1		2	104.99	even	4
832.4.f.c.129.2		2	104.83	even	4
832.4.f.e.129.1		2	104.21	odd	4
832.4.f.e.129.2		2	104.5	odd	4
1521.4.a.d.1.1		1	39.38	odd	2
1521.4.a.i.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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