LMFDB - Embedded newform 33.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,4,Mod(1,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("33.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	33.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:	$1.94706303019$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	33.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -4.00000 q^{5} +3.00000 q^{6} -26.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} -4.00000 q^{5} +3.00000 q^{6} -26.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +4.00000 q^{10} +11.0000 q^{11} +21.0000 q^{12} -32.0000 q^{13} +26.0000 q^{14} +12.0000 q^{15} +41.0000 q^{16} +74.0000 q^{17} -9.00000 q^{18} -60.0000 q^{19} +28.0000 q^{20} +78.0000 q^{21} -11.0000 q^{22} -182.000 q^{23} -45.0000 q^{24} -109.000 q^{25} +32.0000 q^{26} -27.0000 q^{27} +182.000 q^{28} -90.0000 q^{29} -12.0000 q^{30} -8.00000 q^{31} -161.000 q^{32} -33.0000 q^{33} -74.0000 q^{34} +104.000 q^{35} -63.0000 q^{36} -66.0000 q^{37} +60.0000 q^{38} +96.0000 q^{39} -60.0000 q^{40} +422.000 q^{41} -78.0000 q^{42} +408.000 q^{43} -77.0000 q^{44} -36.0000 q^{45} +182.000 q^{46} -506.000 q^{47} -123.000 q^{48} +333.000 q^{49} +109.000 q^{50} -222.000 q^{51} +224.000 q^{52} +348.000 q^{53} +27.0000 q^{54} -44.0000 q^{55} -390.000 q^{56} +180.000 q^{57} +90.0000 q^{58} -200.000 q^{59} -84.0000 q^{60} +132.000 q^{61} +8.00000 q^{62} -234.000 q^{63} -167.000 q^{64} +128.000 q^{65} +33.0000 q^{66} -1036.00 q^{67} -518.000 q^{68} +546.000 q^{69} -104.000 q^{70} +762.000 q^{71} +135.000 q^{72} -542.000 q^{73} +66.0000 q^{74} +327.000 q^{75} +420.000 q^{76} -286.000 q^{77} -96.0000 q^{78} -550.000 q^{79} -164.000 q^{80} +81.0000 q^{81} -422.000 q^{82} -132.000 q^{83} -546.000 q^{84} -296.000 q^{85} -408.000 q^{86} +270.000 q^{87} +165.000 q^{88} +570.000 q^{89} +36.0000 q^{90} +832.000 q^{91} +1274.00 q^{92} +24.0000 q^{93} +506.000 q^{94} +240.000 q^{95} +483.000 q^{96} +14.0000 q^{97} -333.000 q^{98} +99.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.353553	−0.176777	−	0.984251i	$-0.556567\pi$
$2$	−1.00000	−0.353553	−0.176777	+	0.984251i	$0.556567\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.00000	−0.875000
$5$	−4.00000	−0.357771	−0.178885	−	0.983870i	$-0.557249\pi$
$5$	−4.00000	−0.357771	−0.178885	+	0.983870i	$0.557249\pi$
$6$	3.00000	0.204124
$7$	−26.0000	−1.40387	−0.701934	−	0.712242i	$-0.747680\pi$
$7$	−26.0000	−1.40387	−0.701934	+	0.712242i	$0.747680\pi$
$8$	15.0000	0.662913
$9$	9.00000	0.333333
$10$	4.00000	0.126491
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	21.0000	0.505181
$13$	−32.0000	−0.682708	−0.341354	−	0.939935i	$-0.610885\pi$
$13$	−32.0000	−0.682708	−0.341354	+	0.939935i	$0.610885\pi$
$14$	26.0000	0.496342
$15$	12.0000	0.206559
$16$	41.0000	0.640625
$17$	74.0000	1.05574	0.527872	−	0.849324i	$-0.322990\pi$
$17$	74.0000	1.05574	0.527872	+	0.849324i	$0.322990\pi$
$18$	−9.00000	−0.117851
$19$	−60.0000	−0.724471	−0.362235	−	0.932087i	$-0.617986\pi$
$19$	−60.0000	−0.724471	−0.362235	+	0.932087i	$0.617986\pi$
$20$	28.0000	0.313050
$21$	78.0000	0.810524
$22$	−11.0000	−0.106600
$23$	−182.000	−1.64998	−0.824992	−	0.565145i	$-0.808820\pi$
$23$	−182.000	−1.64998	−0.824992	+	0.565145i	$0.808820\pi$
$24$	−45.0000	−0.382733
$25$	−109.000	−0.872000
$26$	32.0000	0.241374
$27$	−27.0000	−0.192450
$28$	182.000	1.22838
$29$	−90.0000	−0.576296	−0.288148	−	0.957586i	$-0.593039\pi$
$29$	−90.0000	−0.576296	−0.288148	+	0.957586i	$0.593039\pi$
$30$	−12.0000	−0.0730297
$31$	−8.00000	−0.0463498	−0.0231749	−	0.999731i	$-0.507377\pi$
$31$	−8.00000	−0.0463498	−0.0231749	+	0.999731i	$0.507377\pi$
$32$	−161.000	−0.889408
$33$	−33.0000	−0.174078
$34$	−74.0000	−0.373262
$35$	104.000	0.502263
$36$	−63.0000	−0.291667
$37$	−66.0000	−0.293252	−0.146626	−	0.989192i	$-0.546841\pi$
$37$	−66.0000	−0.293252	−0.146626	+	0.989192i	$0.546841\pi$
$38$	60.0000	0.256139
$39$	96.0000	0.394162
$40$	−60.0000	−0.237171
$41$	422.000	1.60745	0.803724	−	0.595003i	$-0.202849\pi$
$41$	422.000	1.60745	0.803724	+	0.595003i	$0.202849\pi$
$42$	−78.0000	−0.286563
$43$	408.000	1.44696	0.723482	−	0.690344i	$-0.242541\pi$
$43$	408.000	1.44696	0.723482	+	0.690344i	$0.242541\pi$
$44$	−77.0000	−0.263822
$45$	−36.0000	−0.119257
$46$	182.000	0.583357
$47$	−506.000	−1.57038	−0.785188	−	0.619257i	$-0.787434\pi$
$47$	−506.000	−1.57038	−0.785188	+	0.619257i	$0.787434\pi$
$48$	−123.000	−0.369865
$49$	333.000	0.970845
$50$	109.000	0.308299
$51$	−222.000	−0.609534
$52$	224.000	0.597369
$53$	348.000	0.901915	0.450957	−	0.892546i	$-0.351083\pi$
$53$	348.000	0.901915	0.450957	+	0.892546i	$0.351083\pi$
$54$	27.0000	0.0680414
$55$	−44.0000	−0.107872
$56$	−390.000	−0.930642
$57$	180.000	0.418273
$58$	90.0000	0.203751
$59$	−200.000	−0.441318	−0.220659	−	0.975351i	$-0.570821\pi$
$59$	−200.000	−0.441318	−0.220659	+	0.975351i	$0.570821\pi$
$60$	−84.0000	−0.180739
$61$	132.000	0.277063	0.138532	−	0.990358i	$-0.455762\pi$
$61$	132.000	0.277063	0.138532	+	0.990358i	$0.455762\pi$
$62$	8.00000	0.0163871
$63$	−234.000	−0.467956
$64$	−167.000	−0.326172
$65$	128.000	0.244253
$66$	33.0000	0.0615457
$67$	−1036.00	−1.88907	−0.944534	−	0.328414i	$-0.893486\pi$
$67$	−1036.00	−1.88907	−0.944534	+	0.328414i	$0.893486\pi$
$68$	−518.000	−0.923775
$69$	546.000	0.952618
$70$	−104.000	−0.177577
$71$	762.000	1.27370	0.636850	−	0.770987i	$-0.280237\pi$
$71$	762.000	1.27370	0.636850	+	0.770987i	$0.280237\pi$
$72$	135.000	0.220971
$73$	−542.000	−0.868990	−0.434495	−	0.900674i	$-0.643073\pi$
$73$	−542.000	−0.868990	−0.434495	+	0.900674i	$0.643073\pi$
$74$	66.0000	0.103680
$75$	327.000	0.503449
$76$	420.000	0.633912
$77$	−286.000	−0.423282
$78$	−96.0000	−0.139357
$79$	−550.000	−0.783289	−0.391645	−	0.920117i	$-0.628094\pi$
$79$	−550.000	−0.783289	−0.391645	+	0.920117i	$0.628094\pi$
$80$	−164.000	−0.229197
$81$	81.0000	0.111111
$82$	−422.000	−0.568318
$83$	−132.000	−0.174565	−0.0872824	−	0.996184i	$-0.527818\pi$
$83$	−132.000	−0.174565	−0.0872824	+	0.996184i	$0.527818\pi$
$84$	−546.000	−0.709208
$85$	−296.000	−0.377714
$86$	−408.000	−0.511579
$87$	270.000	0.332725
$88$	165.000	0.199876
$89$	570.000	0.678875	0.339438	−	0.940629i	$-0.389763\pi$
$89$	570.000	0.678875	0.339438	+	0.940629i	$0.389763\pi$
$90$	36.0000	0.0421637
$91$	832.000	0.958432
$92$	1274.00	1.44374
$93$	24.0000	0.0267600
$94$	506.000	0.555212
$95$	240.000	0.259195
$96$	483.000	0.513500
$97$	14.0000	0.0146545	0.00732724	−	0.999973i	$-0.497668\pi$
$97$	14.0000	0.0146545	0.00732724	+	0.999973i	$0.497668\pi$
$98$	−333.000	−0.343246
$99$	99.0000	0.100504
$100$	763.000	0.763000
$101$	1702.00	1.67679	0.838393	−	0.545067i	$-0.183496\pi$
$101$	1702.00	1.67679	0.838393	+	0.545067i	$0.183496\pi$
$102$	222.000	0.215503
$103$	−1132.00	−1.08291	−0.541453	−	0.840731i	$-0.682126\pi$
$103$	−1132.00	−1.08291	−0.541453	+	0.840731i	$0.682126\pi$
$104$	−480.000	−0.452576
$105$	−312.000	−0.289982
$106$	−348.000	−0.318875
$107$	564.000	0.509570	0.254785	−	0.966998i	$-0.417995\pi$
$107$	564.000	0.509570	0.254785	+	0.966998i	$0.417995\pi$
$108$	189.000	0.168394
$109$	−320.000	−0.281197	−0.140598	−	0.990067i	$-0.544903\pi$
$109$	−320.000	−0.281197	−0.140598	+	0.990067i	$0.544903\pi$
$110$	44.0000	0.0381385
$111$	198.000	0.169309
$112$	−1066.00	−0.899353
$113$	−2142.00	−1.78321	−0.891604	−	0.452817i	$-0.850419\pi$
$113$	−2142.00	−1.78321	−0.891604	+	0.452817i	$0.850419\pi$
$114$	−180.000	−0.147882
$115$	728.000	0.590316
$116$	630.000	0.504259
$117$	−288.000	−0.227569
$118$	200.000	0.156030
$119$	−1924.00	−1.48212
$120$	180.000	0.136931
$121$	121.000	0.0909091
$122$	−132.000	−0.0979567
$123$	−1266.00	−0.928060
$124$	56.0000	0.0405560
$125$	936.000	0.669747
$126$	234.000	0.165447
$127$	−1606.00	−1.12212	−0.561061	−	0.827775i	$-0.689607\pi$
$127$	−1606.00	−1.12212	−0.561061	+	0.827775i	$0.689607\pi$
$128$	1455.00	1.00473
$129$	−1224.00	−0.835405
$130$	−128.000	−0.0863565
$131$	−1908.00	−1.27254	−0.636270	−	0.771466i	$-0.719524\pi$
$131$	−1908.00	−1.27254	−0.636270	+	0.771466i	$0.719524\pi$
$132$	231.000	0.152318
$133$	1560.00	1.01706
$134$	1036.00	0.667886
$135$	108.000	0.0688530
$136$	1110.00	0.699866
$137$	−2186.00	−1.36323	−0.681615	−	0.731711i	$-0.738722\pi$
$137$	−2186.00	−1.36323	−0.681615	+	0.731711i	$0.738722\pi$
$138$	−546.000	−0.336801
$139$	2740.00	1.67197	0.835985	−	0.548753i	$-0.184897\pi$
$139$	2740.00	1.67197	0.835985	+	0.548753i	$0.184897\pi$
$140$	−728.000	−0.439480
$141$	1518.00	0.906657
$142$	−762.000	−0.450321
$143$	−352.000	−0.205844
$144$	369.000	0.213542
$145$	360.000	0.206182
$146$	542.000	0.307235
$147$	−999.000	−0.560518
$148$	462.000	0.256596
$149$	−1310.00	−0.720264	−0.360132	−	0.932901i	$-0.617268\pi$
$149$	−1310.00	−0.720264	−0.360132	+	0.932901i	$0.617268\pi$
$150$	−327.000	−0.177996
$151$	−1198.00	−0.645641	−0.322821	−	0.946460i	$-0.604631\pi$
$151$	−1198.00	−0.645641	−0.322821	+	0.946460i	$0.604631\pi$
$152$	−900.000	−0.480261
$153$	666.000	0.351914
$154$	286.000	0.149653
$155$	32.0000	0.0165826
$156$	−672.000	−0.344891
$157$	2114.00	1.07462	0.537311	−	0.843384i	$-0.319440\pi$
$157$	2114.00	1.07462	0.537311	+	0.843384i	$0.319440\pi$
$158$	550.000	0.276934
$159$	−1044.00	−0.520721
$160$	644.000	0.318204
$161$	4732.00	2.31636
$162$	−81.0000	−0.0392837
$163$	3868.00	1.85868	0.929341	−	0.369223i	$-0.120376\pi$
$163$	3868.00	1.85868	0.929341	+	0.369223i	$0.120376\pi$
$164$	−2954.00	−1.40652
$165$	132.000	0.0622799
$166$	132.000	0.0617180
$167$	2004.00	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	2004.00	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	1170.00	0.537306
$169$	−1173.00	−0.533910
$170$	296.000	0.133542
$171$	−540.000	−0.241490
$172$	−2856.00	−1.26609
$173$	678.000	0.297962	0.148981	−	0.988840i	$-0.452401\pi$
$173$	678.000	0.297962	0.148981	+	0.988840i	$0.452401\pi$
$174$	−270.000	−0.117636
$175$	2834.00	1.22417
$176$	451.000	0.193156
$177$	600.000	0.254795
$178$	−570.000	−0.240019
$179$	−1680.00	−0.701503	−0.350752	−	0.936469i	$-0.614074\pi$
$179$	−1680.00	−0.701503	−0.350752	+	0.936469i	$0.614074\pi$
$180$	252.000	0.104350
$181$	−4358.00	−1.78966	−0.894828	−	0.446412i	$-0.852702\pi$
$181$	−4358.00	−1.78966	−0.894828	+	0.446412i	$0.852702\pi$
$182$	−832.000	−0.338857
$183$	−396.000	−0.159963
$184$	−2730.00	−1.09379
$185$	264.000	0.104917
$186$	−24.0000	−0.00946110
$187$	814.000	0.318319
$188$	3542.00	1.37408
$189$	702.000	0.270175
$190$	−240.000	−0.0916391
$191$	−1778.00	−0.673568	−0.336784	−	0.941582i	$-0.609339\pi$
$191$	−1778.00	−0.673568	−0.336784	+	0.941582i	$0.609339\pi$
$192$	501.000	0.188315
$193$	−3962.00	−1.47767	−0.738837	−	0.673884i	$-0.764625\pi$
$193$	−3962.00	−1.47767	−0.738837	+	0.673884i	$0.764625\pi$
$194$	−14.0000	−0.00518114
$195$	−384.000	−0.141020
$196$	−2331.00	−0.849490
$197$	374.000	0.135261	0.0676304	−	0.997710i	$-0.478456\pi$
$197$	374.000	0.135261	0.0676304	+	0.997710i	$0.478456\pi$
$198$	−99.0000	−0.0355335
$199$	2100.00	0.748066	0.374033	−	0.927415i	$-0.377975\pi$
$199$	2100.00	0.748066	0.374033	+	0.927415i	$0.377975\pi$
$200$	−1635.00	−0.578060
$201$	3108.00	1.09065
$202$	−1702.00	−0.592833
$203$	2340.00	0.809043
$204$	1554.00	0.533342
$205$	−1688.00	−0.575098
$206$	1132.00	0.382865
$207$	−1638.00	−0.549995
$208$	−1312.00	−0.437360
$209$	−660.000	−0.218436
$210$	312.000	0.102524
$211$	2232.00	0.728233	0.364117	−	0.931353i	$-0.381371\pi$
$211$	2232.00	0.728233	0.364117	+	0.931353i	$0.381371\pi$
$212$	−2436.00	−0.789175
$213$	−2286.00	−0.735372
$214$	−564.000	−0.180160
$215$	−1632.00	−0.517681
$216$	−405.000	−0.127578
$217$	208.000	0.0650689
$218$	320.000	0.0994180
$219$	1626.00	0.501712
$220$	308.000	0.0943880
$221$	−2368.00	−0.720764
$222$	−198.000	−0.0598599
$223$	2128.00	0.639020	0.319510	−	0.947583i	$-0.396482\pi$
$223$	2128.00	0.639020	0.319510	+	0.947583i	$0.396482\pi$
$224$	4186.00	1.24861
$225$	−981.000	−0.290667
$226$	2142.00	0.630459
$227$	2964.00	0.866641	0.433321	−	0.901240i	$-0.357342\pi$
$227$	2964.00	0.866641	0.433321	+	0.901240i	$0.357342\pi$
$228$	−1260.00	−0.365989
$229$	−2550.00	−0.735846	−0.367923	−	0.929856i	$-0.619931\pi$
$229$	−2550.00	−0.735846	−0.367923	+	0.929856i	$0.619931\pi$
$230$	−728.000	−0.208708
$231$	858.000	0.244382
$232$	−1350.00	−0.382034
$233$	−3042.00	−0.855314	−0.427657	−	0.903941i	$-0.640661\pi$
$233$	−3042.00	−0.855314	−0.427657	+	0.903941i	$0.640661\pi$
$234$	288.000	0.0804579
$235$	2024.00	0.561835
$236$	1400.00	0.386154
$237$	1650.00	0.452232
$238$	1924.00	0.524010
$239$	2700.00	0.730747	0.365373	−	0.930861i	$-0.380941\pi$
$239$	2700.00	0.730747	0.365373	+	0.930861i	$0.380941\pi$
$240$	492.000	0.132327
$241$	−578.000	−0.154491	−0.0772453	−	0.997012i	$-0.524612\pi$
$241$	−578.000	−0.154491	−0.0772453	+	0.997012i	$0.524612\pi$
$242$	−121.000	−0.0321412
$243$	−243.000	−0.0641500
$244$	−924.000	−0.242430
$245$	−1332.00	−0.347340
$246$	1266.00	0.328119
$247$	1920.00	0.494602
$248$	−120.000	−0.0307258
$249$	396.000	0.100785
$250$	−936.000	−0.236791
$251$	3752.00	0.943522	0.471761	−	0.881726i	$-0.343618\pi$
$251$	3752.00	0.943522	0.471761	+	0.881726i	$0.343618\pi$
$252$	1638.00	0.409462
$253$	−2002.00	−0.497489
$254$	1606.00	0.396730
$255$	888.000	0.218073
$256$	−119.000	−0.0290527
$257$	674.000	0.163591	0.0817957	−	0.996649i	$-0.473935\pi$
$257$	674.000	0.163591	0.0817957	+	0.996649i	$0.473935\pi$
$258$	1224.00	0.295360
$259$	1716.00	0.411687
$260$	−896.000	−0.213721
$261$	−810.000	−0.192099
$262$	1908.00	0.449911
$263$	−4352.00	−1.02036	−0.510182	−	0.860066i	$-0.670422\pi$
$263$	−4352.00	−1.02036	−0.510182	+	0.860066i	$0.670422\pi$
$264$	−495.000	−0.115398
$265$	−1392.00	−0.322679
$266$	−1560.00	−0.359585
$267$	−1710.00	−0.391949
$268$	7252.00	1.65293
$269$	500.000	0.113329	0.0566646	−	0.998393i	$-0.481953\pi$
$269$	500.000	0.113329	0.0566646	+	0.998393i	$0.481953\pi$
$270$	−108.000	−0.0243432
$271$	−6538.00	−1.46552	−0.732759	−	0.680489i	$-0.761768\pi$
$271$	−6538.00	−1.46552	−0.732759	+	0.680489i	$0.761768\pi$
$272$	3034.00	0.676336
$273$	−2496.00	−0.553351
$274$	2186.00	0.481975
$275$	−1199.00	−0.262918
$276$	−3822.00	−0.833541
$277$	124.000	0.0268969	0.0134484	−	0.999910i	$-0.495719\pi$
$277$	124.000	0.0268969	0.0134484	+	0.999910i	$0.495719\pi$
$278$	−2740.00	−0.591131
$279$	−72.0000	−0.0154499
$280$	1560.00	0.332957
$281$	3642.00	0.773180	0.386590	−	0.922252i	$-0.373653\pi$
$281$	3642.00	0.773180	0.386590	+	0.922252i	$0.373653\pi$
$282$	−1518.00	−0.320552
$283$	4648.00	0.976307	0.488154	−	0.872758i	$-0.337671\pi$
$283$	4648.00	0.976307	0.488154	+	0.872758i	$0.337671\pi$
$284$	−5334.00	−1.11449
$285$	−720.000	−0.149646
$286$	352.000	0.0727769
$287$	−10972.0	−2.25664
$288$	−1449.00	−0.296469
$289$	563.000	0.114594
$290$	−360.000	−0.0728963
$291$	−42.0000	−0.00846077
$292$	3794.00	0.760367
$293$	−3102.00	−0.618501	−0.309250	−	0.950981i	$-0.600078\pi$
$293$	−3102.00	−0.618501	−0.309250	+	0.950981i	$0.600078\pi$
$294$	999.000	0.198173
$295$	800.000	0.157891
$296$	−990.000	−0.194401
$297$	−297.000	−0.0580259
$298$	1310.00	0.254652
$299$	5824.00	1.12646
$300$	−2289.00	−0.440518
$301$	−10608.0	−2.03135
$302$	1198.00	0.228269
$303$	−5106.00	−0.968093
$304$	−2460.00	−0.464114
$305$	−528.000	−0.0991252
$306$	−666.000	−0.124421
$307$	1244.00	0.231267	0.115633	−	0.993292i	$-0.463110\pi$
$307$	1244.00	0.231267	0.115633	+	0.993292i	$0.463110\pi$
$308$	2002.00	0.370372
$309$	3396.00	0.625216
$310$	−32.0000	−0.00586283
$311$	2082.00	0.379612	0.189806	−	0.981822i	$-0.439214\pi$
$311$	2082.00	0.379612	0.189806	+	0.981822i	$0.439214\pi$
$312$	1440.00	0.261295
$313$	2378.00	0.429433	0.214716	−	0.976676i	$-0.431117\pi$
$313$	2378.00	0.429433	0.214716	+	0.976676i	$0.431117\pi$
$314$	−2114.00	−0.379936
$315$	936.000	0.167421
$316$	3850.00	0.685378
$317$	−496.000	−0.0878806	−0.0439403	−	0.999034i	$-0.513991\pi$
$317$	−496.000	−0.0878806	−0.0439403	+	0.999034i	$0.513991\pi$
$318$	1044.00	0.184103
$319$	−990.000	−0.173760
$320$	668.000	0.116695
$321$	−1692.00	−0.294200
$322$	−4732.00	−0.818957
$323$	−4440.00	−0.764855
$324$	−567.000	−0.0972222
$325$	3488.00	0.595321
$326$	−3868.00	−0.657143
$327$	960.000	0.162349
$328$	6330.00	1.06560
$329$	13156.0	2.20460
$330$	−132.000	−0.0220193
$331$	−2708.00	−0.449683	−0.224842	−	0.974395i	$-0.572186\pi$
$331$	−2708.00	−0.449683	−0.224842	+	0.974395i	$0.572186\pi$
$332$	924.000	0.152744
$333$	−594.000	−0.0977507
$334$	−2004.00	−0.328305
$335$	4144.00	0.675853
$336$	3198.00	0.519242
$337$	4034.00	0.652065	0.326033	−	0.945359i	$-0.394288\pi$
$337$	4034.00	0.652065	0.326033	+	0.945359i	$0.394288\pi$
$338$	1173.00	0.188766
$339$	6426.00	1.02954
$340$	2072.00	0.330500
$341$	−88.0000	−0.0139750
$342$	540.000	0.0853797
$343$	260.000	0.0409291
$344$	6120.00	0.959210
$345$	−2184.00	−0.340819
$346$	−678.000	−0.105345
$347$	11084.0	1.71476	0.857378	−	0.514687i	$-0.172092\pi$
$347$	11084.0	1.71476	0.857378	+	0.514687i	$0.172092\pi$
$348$	−1890.00	−0.291134
$349$	−3120.00	−0.478538	−0.239269	−	0.970953i	$-0.576908\pi$
$349$	−3120.00	−0.478538	−0.239269	+	0.970953i	$0.576908\pi$
$350$	−2834.00	−0.432810
$351$	864.000	0.131387
$352$	−1771.00	−0.268167
$353$	−5622.00	−0.847674	−0.423837	−	0.905739i	$-0.639317\pi$
$353$	−5622.00	−0.847674	−0.423837	+	0.905739i	$0.639317\pi$
$354$	−600.000	−0.0900837
$355$	−3048.00	−0.455693
$356$	−3990.00	−0.594016
$357$	5772.00	0.855705
$358$	1680.00	0.248019
$359$	−8500.00	−1.24962	−0.624809	−	0.780778i	$-0.714823\pi$
$359$	−8500.00	−1.24962	−0.624809	+	0.780778i	$0.714823\pi$
$360$	−540.000	−0.0790569
$361$	−3259.00	−0.475142
$362$	4358.00	0.632739
$363$	−363.000	−0.0524864
$364$	−5824.00	−0.838628
$365$	2168.00	0.310899
$366$	396.000	0.0565553
$367$	7144.00	1.01611	0.508057	−	0.861324i	$-0.330364\pi$
$367$	7144.00	1.01611	0.508057	+	0.861324i	$0.330364\pi$
$368$	−7462.00	−1.05702
$369$	3798.00	0.535816
$370$	−264.000	−0.0370938
$371$	−9048.00	−1.26617
$372$	−168.000	−0.0234150
$373$	−632.000	−0.0877312	−0.0438656	−	0.999037i	$-0.513967\pi$
$373$	−632.000	−0.0877312	−0.0438656	+	0.999037i	$0.513967\pi$
$374$	−814.000	−0.112543
$375$	−2808.00	−0.386679
$376$	−7590.00	−1.04102
$377$	2880.00	0.393442
$378$	−702.000	−0.0955211
$379$	−4220.00	−0.571944	−0.285972	−	0.958238i	$-0.592316\pi$
$379$	−4220.00	−0.571944	−0.285972	+	0.958238i	$0.592316\pi$
$380$	−1680.00	−0.226795
$381$	4818.00	0.647857
$382$	1778.00	0.238142
$383$	8458.00	1.12842	0.564208	−	0.825632i	$-0.309181\pi$
$383$	8458.00	1.12842	0.564208	+	0.825632i	$0.309181\pi$
$384$	−4365.00	−0.580079
$385$	1144.00	0.151438
$386$	3962.00	0.522437
$387$	3672.00	0.482321
$388$	−98.0000	−0.0128227
$389$	1740.00	0.226790	0.113395	−	0.993550i	$-0.463827\pi$
$389$	1740.00	0.226790	0.113395	+	0.993550i	$0.463827\pi$
$390$	384.000	0.0498579
$391$	−13468.0	−1.74196
$392$	4995.00	0.643586
$393$	5724.00	0.734701
$394$	−374.000	−0.0478219
$395$	2200.00	0.280238
$396$	−693.000	−0.0879408
$397$	−5126.00	−0.648027	−0.324013	−	0.946053i	$-0.605032\pi$
$397$	−5126.00	−0.648027	−0.324013	+	0.946053i	$0.605032\pi$
$398$	−2100.00	−0.264481
$399$	−4680.00	−0.587201
$400$	−4469.00	−0.558625
$401$	−3098.00	−0.385802	−0.192901	−	0.981218i	$-0.561790\pi$
$401$	−3098.00	−0.385802	−0.192901	+	0.981218i	$0.561790\pi$
$402$	−3108.00	−0.385604
$403$	256.000	0.0316433
$404$	−11914.0	−1.46719
$405$	−324.000	−0.0397523
$406$	−2340.00	−0.286040
$407$	−726.000	−0.0884189
$408$	−3330.00	−0.404068
$409$	6390.00	0.772531	0.386265	−	0.922388i	$-0.373765\pi$
$409$	6390.00	0.772531	0.386265	+	0.922388i	$0.373765\pi$
$410$	1688.00	0.203328
$411$	6558.00	0.787062
$412$	7924.00	0.947542
$413$	5200.00	0.619553
$414$	1638.00	0.194452
$415$	528.000	0.0624542
$416$	5152.00	0.607206
$417$	−8220.00	−0.965312
$418$	660.000	0.0772288
$419$	9760.00	1.13796	0.568982	−	0.822350i	$-0.307337\pi$
$419$	9760.00	1.13796	0.568982	+	0.822350i	$0.307337\pi$
$420$	2184.00	0.253734
$421$	−5138.00	−0.594800	−0.297400	−	0.954753i	$-0.596119\pi$
$421$	−5138.00	−0.594800	−0.297400	+	0.954753i	$0.596119\pi$
$422$	−2232.00	−0.257469
$423$	−4554.00	−0.523459
$424$	5220.00	0.597891
$425$	−8066.00	−0.920608
$426$	2286.00	0.259993
$427$	−3432.00	−0.388960
$428$	−3948.00	−0.445873
$429$	1056.00	0.118844
$430$	1632.00	0.183028
$431$	−7008.00	−0.783210	−0.391605	−	0.920133i	$-0.628080\pi$
$431$	−7008.00	−0.783210	−0.391605	+	0.920133i	$0.628080\pi$
$432$	−1107.00	−0.123288
$433$	5578.00	0.619080	0.309540	−	0.950886i	$-0.399825\pi$
$433$	5578.00	0.619080	0.309540	+	0.950886i	$0.399825\pi$
$434$	−208.000	−0.0230053
$435$	−1080.00	−0.119039
$436$	2240.00	0.246047
$437$	10920.0	1.19536
$438$	−1626.00	−0.177382
$439$	−10430.0	−1.13393	−0.566967	−	0.823741i	$-0.691883\pi$
$439$	−10430.0	−1.13393	−0.566967	+	0.823741i	$0.691883\pi$
$440$	−660.000	−0.0715097
$441$	2997.00	0.323615
$442$	2368.00	0.254829
$443$	−4432.00	−0.475329	−0.237664	−	0.971347i	$-0.576382\pi$
$443$	−4432.00	−0.475329	−0.237664	+	0.971347i	$0.576382\pi$
$444$	−1386.00	−0.148146
$445$	−2280.00	−0.242882
$446$	−2128.00	−0.225928
$447$	3930.00	0.415845
$448$	4342.00	0.457902
$449$	−6290.00	−0.661121	−0.330561	−	0.943785i	$-0.607238\pi$
$449$	−6290.00	−0.661121	−0.330561	+	0.943785i	$0.607238\pi$
$450$	981.000	0.102766
$451$	4642.00	0.484664
$452$	14994.0	1.56031
$453$	3594.00	0.372761
$454$	−2964.00	−0.306404
$455$	−3328.00	−0.342899
$456$	2700.00	0.277279
$457$	3054.00	0.312604	0.156302	−	0.987709i	$-0.450043\pi$
$457$	3054.00	0.312604	0.156302	+	0.987709i	$0.450043\pi$
$458$	2550.00	0.260161
$459$	−1998.00	−0.203178
$460$	−5096.00	−0.516527
$461$	12882.0	1.30146	0.650732	−	0.759308i	$-0.274462\pi$
$461$	12882.0	1.30146	0.650732	+	0.759308i	$0.274462\pi$
$462$	−858.000	−0.0864021
$463$	6148.00	0.617110	0.308555	−	0.951207i	$-0.400155\pi$
$463$	6148.00	0.617110	0.308555	+	0.951207i	$0.400155\pi$
$464$	−3690.00	−0.369190
$465$	−96.0000	−0.00957396
$466$	3042.00	0.302399
$467$	5124.00	0.507731	0.253866	−	0.967240i	$-0.418298\pi$
$467$	5124.00	0.507731	0.253866	+	0.967240i	$0.418298\pi$
$468$	2016.00	0.199123
$469$	26936.0	2.65200
$470$	−2024.00	−0.198639
$471$	−6342.00	−0.620433
$472$	−3000.00	−0.292555
$473$	4488.00	0.436276
$474$	−1650.00	−0.159888
$475$	6540.00	0.631738
$476$	13468.0	1.29686
$477$	3132.00	0.300638
$478$	−2700.00	−0.258358
$479$	−16520.0	−1.57582	−0.787910	−	0.615790i	$-0.788837\pi$
$479$	−16520.0	−1.57582	−0.787910	+	0.615790i	$0.788837\pi$
$480$	−1932.00	−0.183715
$481$	2112.00	0.200206
$482$	578.000	0.0546207
$483$	−14196.0	−1.33735
$484$	−847.000	−0.0795455
$485$	−56.0000	−0.00524295
$486$	243.000	0.0226805
$487$	524.000	0.0487571	0.0243785	−	0.999703i	$-0.492239\pi$
$487$	524.000	0.0487571	0.0243785	+	0.999703i	$0.492239\pi$
$488$	1980.00	0.183669
$489$	−11604.0	−1.07311
$490$	1332.00	0.122803
$491$	−15028.0	−1.38127	−0.690636	−	0.723203i	$-0.742669\pi$
$491$	−15028.0	−1.38127	−0.690636	+	0.723203i	$0.742669\pi$
$492$	8862.00	0.812052
$493$	−6660.00	−0.608421
$494$	−1920.00	−0.174868
$495$	−396.000	−0.0359573
$496$	−328.000	−0.0296928
$497$	−19812.0	−1.78811
$498$	−396.000	−0.0356329
$499$	9020.00	0.809200	0.404600	−	0.914494i	$-0.367411\pi$
$499$	9020.00	0.809200	0.404600	+	0.914494i	$0.367411\pi$
$500$	−6552.00	−0.586029
$501$	−6012.00	−0.536120
$502$	−3752.00	−0.333586
$503$	−14812.0	−1.31299	−0.656495	−	0.754330i	$-0.727962\pi$
$503$	−14812.0	−1.31299	−0.656495	+	0.754330i	$0.727962\pi$
$504$	−3510.00	−0.310214
$505$	−6808.00	−0.599905
$506$	2002.00	0.175889
$507$	3519.00	0.308253
$508$	11242.0	0.981856
$509$	12660.0	1.10245	0.551223	−	0.834358i	$-0.314161\pi$
$509$	12660.0	1.10245	0.551223	+	0.834358i	$0.314161\pi$
$510$	−888.000	−0.0771006
$511$	14092.0	1.21995
$512$	−11521.0	−0.994455
$513$	1620.00	0.139424
$514$	−674.000	−0.0578383
$515$	4528.00	0.387432
$516$	8568.00	0.730979
$517$	−5566.00	−0.473486
$518$	−1716.00	−0.145553
$519$	−2034.00	−0.172028
$520$	1920.00	0.161918
$521$	−3738.00	−0.314328	−0.157164	−	0.987573i	$-0.550235\pi$
$521$	−3738.00	−0.314328	−0.157164	+	0.987573i	$0.550235\pi$
$522$	810.000	0.0679171
$523$	−6352.00	−0.531078	−0.265539	−	0.964100i	$-0.585550\pi$
$523$	−6352.00	−0.531078	−0.265539	+	0.964100i	$0.585550\pi$
$524$	13356.0	1.11347
$525$	−8502.00	−0.706777
$526$	4352.00	0.360753
$527$	−592.000	−0.0489334
$528$	−1353.00	−0.111518
$529$	20957.0	1.72245
$530$	1392.00	0.114084
$531$	−1800.00	−0.147106
$532$	−10920.0	−0.889929
$533$	−13504.0	−1.09742
$534$	1710.00	0.138575
$535$	−2256.00	−0.182309
$536$	−15540.0	−1.25229
$537$	5040.00	0.405013
$538$	−500.000	−0.0400679
$539$	3663.00	0.292721
$540$	−756.000	−0.0602464
$541$	−24728.0	−1.96514	−0.982569	−	0.185898i	$-0.940481\pi$
$541$	−24728.0	−1.96514	−0.982569	+	0.185898i	$0.940481\pi$
$542$	6538.00	0.518139
$543$	13074.0	1.03326
$544$	−11914.0	−0.938986
$545$	1280.00	0.100604
$546$	2496.00	0.195639
$547$	−22756.0	−1.77875	−0.889375	−	0.457178i	$-0.848860\pi$
$547$	−22756.0	−1.77875	−0.889375	+	0.457178i	$0.848860\pi$
$548$	15302.0	1.19283
$549$	1188.00	0.0923545
$550$	1199.00	0.0929555
$551$	5400.00	0.417509
$552$	8190.00	0.631503
$553$	14300.0	1.09963
$554$	−124.000	−0.00950949
$555$	−792.000	−0.0605739
$556$	−19180.0	−1.46297
$557$	−9526.00	−0.724649	−0.362325	−	0.932052i	$-0.618017\pi$
$557$	−9526.00	−0.724649	−0.362325	+	0.932052i	$0.618017\pi$
$558$	72.0000	0.00546237
$559$	−13056.0	−0.987853
$560$	4264.00	0.321762
$561$	−2442.00	−0.183781
$562$	−3642.00	−0.273360
$563$	12068.0	0.903385	0.451692	−	0.892174i	$-0.350820\pi$
$563$	12068.0	0.903385	0.451692	+	0.892174i	$0.350820\pi$
$564$	−10626.0	−0.793325
$565$	8568.00	0.637980
$566$	−4648.00	−0.345177
$567$	−2106.00	−0.155985
$568$	11430.0	0.844352
$569$	15090.0	1.11179	0.555893	−	0.831254i	$-0.312377\pi$
$569$	15090.0	1.11179	0.555893	+	0.831254i	$0.312377\pi$
$570$	720.000	0.0529079
$571$	4412.00	0.323356	0.161678	−	0.986844i	$-0.448309\pi$
$571$	4412.00	0.323356	0.161678	+	0.986844i	$0.448309\pi$
$572$	2464.00	0.180114
$573$	5334.00	0.388885
$574$	10972.0	0.797844
$575$	19838.0	1.43879
$576$	−1503.00	−0.108724
$577$	−3906.00	−0.281818	−0.140909	−	0.990023i	$-0.545002\pi$
$577$	−3906.00	−0.281818	−0.140909	+	0.990023i	$0.545002\pi$
$578$	−563.000	−0.0405151
$579$	11886.0	0.853135
$580$	−2520.00	−0.180409
$581$	3432.00	0.245066
$582$	42.0000	0.00299133
$583$	3828.00	0.271937
$584$	−8130.00	−0.576065
$585$	1152.00	0.0814177
$586$	3102.00	0.218673
$587$	−12016.0	−0.844895	−0.422448	−	0.906387i	$-0.638829\pi$
$587$	−12016.0	−0.844895	−0.422448	+	0.906387i	$0.638829\pi$
$588$	6993.00	0.490453
$589$	480.000	0.0335790
$590$	−800.000	−0.0558228
$591$	−1122.00	−0.0780929
$592$	−2706.00	−0.187865
$593$	−11342.0	−0.785430	−0.392715	−	0.919660i	$-0.628464\pi$
$593$	−11342.0	−0.785430	−0.392715	+	0.919660i	$0.628464\pi$
$594$	297.000	0.0205152
$595$	7696.00	0.530261
$596$	9170.00	0.630231
$597$	−6300.00	−0.431896
$598$	−5824.00	−0.398263
$599$	20690.0	1.41130	0.705651	−	0.708559i	$-0.250654\pi$
$599$	20690.0	1.41130	0.705651	+	0.708559i	$0.250654\pi$
$600$	4905.00	0.333743
$601$	−598.000	−0.0405872	−0.0202936	−	0.999794i	$-0.506460\pi$
$601$	−598.000	−0.0405872	−0.0202936	+	0.999794i	$0.506460\pi$
$602$	10608.0	0.718189
$603$	−9324.00	−0.629689
$604$	8386.00	0.564936
$605$	−484.000	−0.0325246
$606$	5106.00	0.342272
$607$	−166.000	−0.0111001	−0.00555003	−	0.999985i	$-0.501767\pi$
$607$	−166.000	−0.0111001	−0.00555003	+	0.999985i	$0.501767\pi$
$608$	9660.00	0.644350
$609$	−7020.00	−0.467101
$610$	528.000	0.0350461
$611$	16192.0	1.07211
$612$	−4662.00	−0.307925
$613$	20108.0	1.32488	0.662442	−	0.749113i	$-0.269520\pi$
$613$	20108.0	1.32488	0.662442	+	0.749113i	$0.269520\pi$
$614$	−1244.00	−0.0817651
$615$	5064.00	0.332033
$616$	−4290.00	−0.280599
$617$	−2286.00	−0.149159	−0.0745793	−	0.997215i	$-0.523761\pi$
$617$	−2286.00	−0.149159	−0.0745793	+	0.997215i	$0.523761\pi$
$618$	−3396.00	−0.221047
$619$	−25660.0	−1.66618	−0.833088	−	0.553141i	$-0.813429\pi$
$619$	−25660.0	−1.66618	−0.833088	+	0.553141i	$0.813429\pi$
$620$	−224.000	−0.0145098
$621$	4914.00	0.317539
$622$	−2082.00	−0.134213
$623$	−14820.0	−0.953051
$624$	3936.00	0.252510
$625$	9881.00	0.632384
$626$	−2378.00	−0.151827
$627$	1980.00	0.126114
$628$	−14798.0	−0.940294
$629$	−4884.00	−0.309599
$630$	−936.000	−0.0591923
$631$	−11408.0	−0.719723	−0.359862	−	0.933006i	$-0.617176\pi$
$631$	−11408.0	−0.719723	−0.359862	+	0.933006i	$0.617176\pi$
$632$	−8250.00	−0.519252
$633$	−6696.00	−0.420446
$634$	496.000	0.0310705
$635$	6424.00	0.401462
$636$	7308.00	0.455631
$637$	−10656.0	−0.662804
$638$	990.000	0.0614333
$639$	6858.00	0.424567
$640$	−5820.00	−0.359462
$641$	−3378.00	−0.208148	−0.104074	−	0.994570i	$-0.533188\pi$
$641$	−3378.00	−0.208148	−0.104074	+	0.994570i	$0.533188\pi$
$642$	1692.00	0.104015
$643$	−11212.0	−0.687649	−0.343824	−	0.939034i	$-0.611722\pi$
$643$	−11212.0	−0.687649	−0.343824	+	0.939034i	$0.611722\pi$
$644$	−33124.0	−2.02681
$645$	4896.00	0.298883
$646$	4440.00	0.270417
$647$	−86.0000	−0.00522567	−0.00261284	−	0.999997i	$-0.500832\pi$
$647$	−86.0000	−0.00522567	−0.00261284	+	0.999997i	$0.500832\pi$
$648$	1215.00	0.0736570
$649$	−2200.00	−0.133062
$650$	−3488.00	−0.210478
$651$	−624.000	−0.0375676
$652$	−27076.0	−1.62635
$653$	−4432.00	−0.265601	−0.132801	−	0.991143i	$-0.542397\pi$
$653$	−4432.00	−0.265601	−0.132801	+	0.991143i	$0.542397\pi$
$654$	−960.000	−0.0573990
$655$	7632.00	0.455278
$656$	17302.0	1.02977
$657$	−4878.00	−0.289663
$658$	−13156.0	−0.779444
$659$	4580.00	0.270731	0.135365	−	0.990796i	$-0.456779\pi$
$659$	4580.00	0.270731	0.135365	+	0.990796i	$0.456779\pi$
$660$	−924.000	−0.0544949
$661$	4282.00	0.251967	0.125984	−	0.992032i	$-0.459791\pi$
$661$	4282.00	0.251967	0.125984	+	0.992032i	$0.459791\pi$
$662$	2708.00	0.158987
$663$	7104.00	0.416133
$664$	−1980.00	−0.115721
$665$	−6240.00	−0.363875
$666$	594.000	0.0345601
$667$	16380.0	0.950879
$668$	−14028.0	−0.812514
$669$	−6384.00	−0.368938
$670$	−4144.00	−0.238950
$671$	1452.00	0.0835378
$672$	−12558.0	−0.720886
$673$	8438.00	0.483300	0.241650	−	0.970363i	$-0.422311\pi$
$673$	8438.00	0.483300	0.241650	+	0.970363i	$0.422311\pi$
$674$	−4034.00	−0.230540
$675$	2943.00	0.167816
$676$	8211.00	0.467171
$677$	34494.0	1.95822	0.979108	−	0.203341i	$-0.0651801\pi$
$677$	34494.0	1.95822	0.979108	+	0.203341i	$0.0651801\pi$
$678$	−6426.00	−0.363996
$679$	−364.000	−0.0205730
$680$	−4440.00	−0.250392
$681$	−8892.00	−0.500356
$682$	88.0000	0.00494090
$683$	−13712.0	−0.768192	−0.384096	−	0.923293i	$-0.625487\pi$
$683$	−13712.0	−0.768192	−0.384096	+	0.923293i	$0.625487\pi$
$684$	3780.00	0.211304
$685$	8744.00	0.487724
$686$	−260.000	−0.0144706
$687$	7650.00	0.424841
$688$	16728.0	0.926961
$689$	−11136.0	−0.615744
$690$	2184.00	0.120498
$691$	11372.0	0.626066	0.313033	−	0.949742i	$-0.398655\pi$
$691$	11372.0	0.626066	0.313033	+	0.949742i	$0.398655\pi$
$692$	−4746.00	−0.260717
$693$	−2574.00	−0.141094
$694$	−11084.0	−0.606258
$695$	−10960.0	−0.598182
$696$	4050.00	0.220567
$697$	31228.0	1.69705
$698$	3120.00	0.169189
$699$	9126.00	0.493815
$700$	−19838.0	−1.07115
$701$	−6398.00	−0.344721	−0.172360	−	0.985034i	$-0.555139\pi$
$701$	−6398.00	−0.344721	−0.172360	+	0.985034i	$0.555139\pi$
$702$	−864.000	−0.0464524
$703$	3960.00	0.212453
$704$	−1837.00	−0.0983445
$705$	−6072.00	−0.324376
$706$	5622.00	0.299698
$707$	−44252.0	−2.35399
$708$	−4200.00	−0.222946
$709$	−5830.00	−0.308816	−0.154408	−	0.988007i	$-0.549347\pi$
$709$	−5830.00	−0.308816	−0.154408	+	0.988007i	$0.549347\pi$
$710$	3048.00	0.161112
$711$	−4950.00	−0.261096
$712$	8550.00	0.450035
$713$	1456.00	0.0764763
$714$	−5772.00	−0.302537
$715$	1408.00	0.0736451
$716$	11760.0	0.613815
$717$	−8100.00	−0.421897
$718$	8500.00	0.441807
$719$	34530.0	1.79103	0.895516	−	0.445030i	$-0.146807\pi$
$719$	34530.0	1.79103	0.895516	+	0.445030i	$0.146807\pi$
$720$	−1476.00	−0.0763990
$721$	29432.0	1.52026
$722$	3259.00	0.167988
$723$	1734.00	0.0891952
$724$	30506.0	1.56595
$725$	9810.00	0.502530
$726$	363.000	0.0185567
$727$	−17316.0	−0.883377	−0.441688	−	0.897169i	$-0.645620\pi$
$727$	−17316.0	−0.883377	−0.441688	+	0.897169i	$0.645620\pi$
$728$	12480.0	0.635357
$729$	729.000	0.0370370
$730$	−2168.00	−0.109920
$731$	30192.0	1.52762
$732$	2772.00	0.139967
$733$	−27072.0	−1.36416	−0.682079	−	0.731279i	$-0.738924\pi$
$733$	−27072.0	−1.36416	−0.682079	+	0.731279i	$0.738924\pi$
$734$	−7144.00	−0.359250
$735$	3996.00	0.200537
$736$	29302.0	1.46751
$737$	−11396.0	−0.569575
$738$	−3798.00	−0.189439
$739$	−17320.0	−0.862147	−0.431073	−	0.902317i	$-0.641865\pi$
$739$	−17320.0	−0.862147	−0.431073	+	0.902317i	$0.641865\pi$
$740$	−1848.00	−0.0918025
$741$	−5760.00	−0.285559
$742$	9048.00	0.447658
$743$	14588.0	0.720299	0.360149	−	0.932895i	$-0.382726\pi$
$743$	14588.0	0.720299	0.360149	+	0.932895i	$0.382726\pi$
$744$	360.000	0.0177396
$745$	5240.00	0.257690
$746$	632.000	0.0310176
$747$	−1188.00	−0.0581883
$748$	−5698.00	−0.278529
$749$	−14664.0	−0.715368
$750$	2808.00	0.136712
$751$	26152.0	1.27071	0.635353	−	0.772222i	$-0.280855\pi$
$751$	26152.0	1.27071	0.635353	+	0.772222i	$0.280855\pi$
$752$	−20746.0	−1.00602
$753$	−11256.0	−0.544743
$754$	−2880.00	−0.139103
$755$	4792.00	0.230992
$756$	−4914.00	−0.236403
$757$	−1066.00	−0.0511815	−0.0255908	−	0.999673i	$-0.508147\pi$
$757$	−1066.00	−0.0511815	−0.0255908	+	0.999673i	$0.508147\pi$
$758$	4220.00	0.202213
$759$	6006.00	0.287225
$760$	3600.00	0.171823
$761$	−37518.0	−1.78716	−0.893578	−	0.448907i	$-0.851813\pi$
$761$	−37518.0	−1.78716	−0.893578	+	0.448907i	$0.851813\pi$
$762$	−4818.00	−0.229052
$763$	8320.00	0.394763
$764$	12446.0	0.589372
$765$	−2664.00	−0.125905
$766$	−8458.00	−0.398956
$767$	6400.00	0.301292
$768$	357.000	0.0167736
$769$	−17290.0	−0.810785	−0.405392	−	0.914143i	$-0.632865\pi$
$769$	−17290.0	−0.810785	−0.405392	+	0.914143i	$0.632865\pi$
$770$	−1144.00	−0.0535414
$771$	−2022.00	−0.0944495
$772$	27734.0	1.29296
$773$	−17172.0	−0.799009	−0.399504	−	0.916731i	$-0.630818\pi$
$773$	−17172.0	−0.799009	−0.399504	+	0.916731i	$0.630818\pi$
$774$	−3672.00	−0.170526
$775$	872.000	0.0404170
$776$	210.000	0.00971464
$777$	−5148.00	−0.237688
$778$	−1740.00	−0.0801825
$779$	−25320.0	−1.16455
$780$	2688.00	0.123392
$781$	8382.00	0.384035
$782$	13468.0	0.615876
$783$	2430.00	0.110908
$784$	13653.0	0.621948
$785$	−8456.00	−0.384468
$786$	−5724.00	−0.259756
$787$	−9536.00	−0.431921	−0.215960	−	0.976402i	$-0.569288\pi$
$787$	−9536.00	−0.431921	−0.215960	+	0.976402i	$0.569288\pi$
$788$	−2618.00	−0.118353
$789$	13056.0	0.589108
$790$	−2200.00	−0.0990791
$791$	55692.0	2.50339
$792$	1485.00	0.0666252
$793$	−4224.00	−0.189153
$794$	5126.00	0.229112
$795$	4176.00	0.186299
$796$	−14700.0	−0.654557
$797$	−20516.0	−0.911812	−0.455906	−	0.890028i	$-0.650685\pi$
$797$	−20516.0	−0.911812	−0.455906	+	0.890028i	$0.650685\pi$
$798$	4680.00	0.207607
$799$	−37444.0	−1.65791
$800$	17549.0	0.775564
$801$	5130.00	0.226292
$802$	3098.00	0.136402
$803$	−5962.00	−0.262010
$804$	−21756.0	−0.954322
$805$	−18928.0	−0.828726
$806$	−256.000	−0.0111876
$807$	−1500.00	−0.0654306
$808$	25530.0	1.11156
$809$	22470.0	0.976518	0.488259	−	0.872699i	$-0.337632\pi$
$809$	22470.0	0.976518	0.488259	+	0.872699i	$0.337632\pi$
$810$	324.000	0.0140546
$811$	−3368.00	−0.145828	−0.0729140	−	0.997338i	$-0.523230\pi$
$811$	−3368.00	−0.145828	−0.0729140	+	0.997338i	$0.523230\pi$
$812$	−16380.0	−0.707913
$813$	19614.0	0.846117
$814$	726.000	0.0312608
$815$	−15472.0	−0.664982
$816$	−9102.00	−0.390483
$817$	−24480.0	−1.04828
$818$	−6390.00	−0.273131
$819$	7488.00	0.319477
$820$	11816.0	0.503211
$821$	−10738.0	−0.456466	−0.228233	−	0.973607i	$-0.573295\pi$
$821$	−10738.0	−0.456466	−0.228233	+	0.973607i	$0.573295\pi$
$822$	−6558.00	−0.278268
$823$	−15912.0	−0.673946	−0.336973	−	0.941514i	$-0.609403\pi$
$823$	−15912.0	−0.673946	−0.336973	+	0.941514i	$0.609403\pi$
$824$	−16980.0	−0.717872
$825$	3597.00	0.151796
$826$	−5200.00	−0.219045
$827$	22924.0	0.963900	0.481950	−	0.876199i	$-0.339929\pi$
$827$	22924.0	0.963900	0.481950	+	0.876199i	$0.339929\pi$
$828$	11466.0	0.481245
$829$	−41690.0	−1.74663	−0.873313	−	0.487159i	$-0.838033\pi$
$829$	−41690.0	−1.74663	−0.873313	+	0.487159i	$0.838033\pi$
$830$	−528.000	−0.0220809
$831$	−372.000	−0.0155289
$832$	5344.00	0.222680
$833$	24642.0	1.02496
$834$	8220.00	0.341289
$835$	−8016.00	−0.332222
$836$	4620.00	0.191132
$837$	216.000	0.00892001
$838$	−9760.00	−0.402331
$839$	−16450.0	−0.676898	−0.338449	−	0.940985i	$-0.609902\pi$
$839$	−16450.0	−0.676898	−0.338449	+	0.940985i	$0.609902\pi$
$840$	−4680.00	−0.192233
$841$	−16289.0	−0.667883
$842$	5138.00	0.210294
$843$	−10926.0	−0.446396
$844$	−15624.0	−0.637204
$845$	4692.00	0.191017
$846$	4554.00	0.185071
$847$	−3146.00	−0.127624
$848$	14268.0	0.577789
$849$	−13944.0	−0.563671
$850$	8066.00	0.325484
$851$	12012.0	0.483861
$852$	16002.0	0.643450
$853$	−30892.0	−1.24000	−0.620001	−	0.784601i	$-0.712868\pi$
$853$	−30892.0	−1.24000	−0.620001	+	0.784601i	$0.712868\pi$
$854$	3432.00	0.137518
$855$	2160.00	0.0863982
$856$	8460.00	0.337800
$857$	−38906.0	−1.55076	−0.775381	−	0.631493i	$-0.782442\pi$
$857$	−38906.0	−1.55076	−0.775381	+	0.631493i	$0.782442\pi$
$858$	−1056.00	−0.0420178
$859$	−1020.00	−0.0405145	−0.0202572	−	0.999795i	$-0.506449\pi$
$859$	−1020.00	−0.0405145	−0.0202572	+	0.999795i	$0.506449\pi$
$860$	11424.0	0.452971
$861$	32916.0	1.30287
$862$	7008.00	0.276907
$863$	15078.0	0.594741	0.297370	−	0.954762i	$-0.403890\pi$
$863$	15078.0	0.594741	0.297370	+	0.954762i	$0.403890\pi$
$864$	4347.00	0.171167
$865$	−2712.00	−0.106602
$866$	−5578.00	−0.218878
$867$	−1689.00	−0.0661608
$868$	−1456.00	−0.0569353
$869$	−6050.00	−0.236171
$870$	1080.00	0.0420867
$871$	33152.0	1.28968
$872$	−4800.00	−0.186409
$873$	126.000	0.00488483
$874$	−10920.0	−0.422625
$875$	−24336.0	−0.940237
$876$	−11382.0	−0.438998
$877$	22704.0	0.874184	0.437092	−	0.899417i	$-0.356008\pi$
$877$	22704.0	0.874184	0.437092	+	0.899417i	$0.356008\pi$
$878$	10430.0	0.400906
$879$	9306.00	0.357092
$880$	−1804.00	−0.0691055
$881$	−19358.0	−0.740281	−0.370141	−	0.928976i	$-0.620690\pi$
$881$	−19358.0	−0.740281	−0.370141	+	0.928976i	$0.620690\pi$
$882$	−2997.00	−0.114415
$883$	−11252.0	−0.428833	−0.214417	−	0.976742i	$-0.568785\pi$
$883$	−11252.0	−0.428833	−0.214417	+	0.976742i	$0.568785\pi$
$884$	16576.0	0.630669
$885$	−2400.00	−0.0911583
$886$	4432.00	0.168054
$887$	43684.0	1.65362	0.826812	−	0.562478i	$-0.190152\pi$
$887$	43684.0	1.65362	0.826812	+	0.562478i	$0.190152\pi$
$888$	2970.00	0.112237
$889$	41756.0	1.57531
$890$	2280.00	0.0858717
$891$	891.000	0.0335013
$892$	−14896.0	−0.559142
$893$	30360.0	1.13769
$894$	−3930.00	−0.147023
$895$	6720.00	0.250977
$896$	−37830.0	−1.41050
$897$	−17472.0	−0.650360
$898$	6290.00	0.233742
$899$	720.000	0.0267112
$900$	6867.00	0.254333
$901$	25752.0	0.952190
$902$	−4642.00	−0.171354
$903$	31824.0	1.17280
$904$	−32130.0	−1.18211
$905$	17432.0	0.640287
$906$	−3594.00	−0.131791
$907$	45804.0	1.67684	0.838422	−	0.545022i	$-0.183479\pi$
$907$	45804.0	1.67684	0.838422	+	0.545022i	$0.183479\pi$
$908$	−20748.0	−0.758311
$909$	15318.0	0.558928
$910$	3328.00	0.121233
$911$	−15318.0	−0.557089	−0.278544	−	0.960423i	$-0.589852\pi$
$911$	−15318.0	−0.557089	−0.278544	+	0.960423i	$0.589852\pi$
$912$	7380.00	0.267956
$913$	−1452.00	−0.0526333
$914$	−3054.00	−0.110522
$915$	1584.00	0.0572300
$916$	17850.0	0.643865
$917$	49608.0	1.78648
$918$	1998.00	0.0718342
$919$	11350.0	0.407401	0.203701	−	0.979033i	$-0.434703\pi$
$919$	11350.0	0.407401	0.203701	+	0.979033i	$0.434703\pi$
$920$	10920.0	0.391328
$921$	−3732.00	−0.133522
$922$	−12882.0	−0.460137
$923$	−24384.0	−0.869566
$924$	−6006.00	−0.213834
$925$	7194.00	0.255716
$926$	−6148.00	−0.218181
$927$	−10188.0	−0.360969
$928$	14490.0	0.512562
$929$	33030.0	1.16650	0.583250	−	0.812292i	$-0.301781\pi$
$929$	33030.0	1.16650	0.583250	+	0.812292i	$0.301781\pi$
$930$	96.0000	0.00338491
$931$	−19980.0	−0.703349
$932$	21294.0	0.748399
$933$	−6246.00	−0.219169
$934$	−5124.00	−0.179510
$935$	−3256.00	−0.113885
$936$	−4320.00	−0.150859
$937$	−10006.0	−0.348860	−0.174430	−	0.984670i	$-0.555808\pi$
$937$	−10006.0	−0.348860	−0.174430	+	0.984670i	$0.555808\pi$
$938$	−26936.0	−0.937624
$939$	−7134.00	−0.247933
$940$	−14168.0	−0.491606
$941$	2622.00	0.0908340	0.0454170	−	0.998968i	$-0.485538\pi$
$941$	2622.00	0.0908340	0.0454170	+	0.998968i	$0.485538\pi$
$942$	6342.00	0.219356
$943$	−76804.0	−2.65226
$944$	−8200.00	−0.282720
$945$	−2808.00	−0.0966606
$946$	−4488.00	−0.154247
$947$	−39876.0	−1.36832	−0.684158	−	0.729334i	$-0.739830\pi$
$947$	−39876.0	−1.36832	−0.684158	+	0.729334i	$0.739830\pi$
$948$	−11550.0	−0.395703
$949$	17344.0	0.593267
$950$	−6540.00	−0.223353
$951$	1488.00	0.0507379
$952$	−28860.0	−0.982519
$953$	38918.0	1.32285	0.661426	−	0.750011i	$-0.269952\pi$
$953$	38918.0	1.32285	0.661426	+	0.750011i	$0.269952\pi$
$954$	−3132.00	−0.106292
$955$	7112.00	0.240983
$956$	−18900.0	−0.639403
$957$	2970.00	0.100320
$958$	16520.0	0.557137
$959$	56836.0	1.91380
$960$	−2004.00	−0.0673738
$961$	−29727.0	−0.997852
$962$	−2112.00	−0.0707834
$963$	5076.00	0.169857
$964$	4046.00	0.135179
$965$	15848.0	0.528669
$966$	14196.0	0.472825
$967$	1114.00	0.0370464	0.0185232	−	0.999828i	$-0.494104\pi$
$967$	1114.00	0.0370464	0.0185232	+	0.999828i	$0.494104\pi$
$968$	1815.00	0.0602648
$969$	13320.0	0.441589
$970$	56.0000	0.00185366
$971$	−1688.00	−0.0557884	−0.0278942	−	0.999611i	$-0.508880\pi$
$971$	−1688.00	−0.0557884	−0.0278942	+	0.999611i	$0.508880\pi$
$972$	1701.00	0.0561313
$973$	−71240.0	−2.34722
$974$	−524.000	−0.0172382
$975$	−10464.0	−0.343709
$976$	5412.00	0.177494
$977$	−41826.0	−1.36963	−0.684817	−	0.728715i	$-0.740118\pi$
$977$	−41826.0	−1.36963	−0.684817	+	0.728715i	$0.740118\pi$
$978$	11604.0	0.379402
$979$	6270.00	0.204689
$980$	9324.00	0.303923
$981$	−2880.00	−0.0937322
$982$	15028.0	0.488353
$983$	978.000	0.0317328	0.0158664	−	0.999874i	$-0.494949\pi$
$983$	978.000	0.0317328	0.0158664	+	0.999874i	$0.494949\pi$
$984$	−18990.0	−0.615223
$985$	−1496.00	−0.0483924
$986$	6660.00	0.215109
$987$	−39468.0	−1.27283
$988$	−13440.0	−0.432777
$989$	−74256.0	−2.38747
$990$	396.000	0.0127128
$991$	47272.0	1.51528	0.757641	−	0.652671i	$-0.226352\pi$
$991$	47272.0	1.51528	0.757641	+	0.652671i	$0.226352\pi$
$992$	1288.00	0.0412238
$993$	8124.00	0.259625
$994$	19812.0	0.632192
$995$	−8400.00	−0.267636
$996$	−2772.00	−0.0881869
$997$	51104.0	1.62335	0.811675	−	0.584109i	$-0.198556\pi$
$997$	51104.0	1.62335	0.811675	+	0.584109i	$0.198556\pi$
$998$	−9020.00	−0.286095
$999$	1782.00	0.0564364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.4.a.b.1.1	✓	1
3.2	odd	2		99.4.a.a.1.1		1
4.3	odd	2		528.4.a.h.1.1		1
5.2	odd	4		825.4.c.f.199.1		2
5.3	odd	4		825.4.c.f.199.2		2
5.4	even	2		825.4.a.f.1.1		1
7.6	odd	2		1617.4.a.d.1.1		1
8.3	odd	2		2112.4.a.h.1.1		1
8.5	even	2		2112.4.a.u.1.1		1
11.10	odd	2		363.4.a.d.1.1		1
12.11	even	2		1584.4.a.l.1.1		1
15.14	odd	2		2475.4.a.e.1.1		1
33.32	even	2		1089.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.b.1.1	✓	1	1.1	even	1	trivial
99.4.a.a.1.1		1	3.2	odd	2
363.4.a.d.1.1		1	11.10	odd	2
528.4.a.h.1.1		1	4.3	odd	2
825.4.a.f.1.1		1	5.4	even	2
825.4.c.f.199.1		2	5.2	odd	4
825.4.c.f.199.2		2	5.3	odd	4
1089.4.a.e.1.1		1	33.32	even	2
1584.4.a.l.1.1		1	12.11	even	2
1617.4.a.d.1.1		1	7.6	odd	2
2112.4.a.h.1.1		1	8.3	odd	2
2112.4.a.u.1.1		1	8.5	even	2
2475.4.a.e.1.1		1	15.14	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:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	33.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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