LMFDB - Embedded newform 4010.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4010,2,Mod(1,4010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4010, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4010 = 2 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4010.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:	$32.0200112105$
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	$\chi$	$=$	4010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +3.00000 q^{11} +2.00000 q^{12} +1.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} -8.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} -3.00000 q^{22} -2.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} -4.00000 q^{27} -4.00000 q^{29} +2.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} +8.00000 q^{34} +1.00000 q^{36} -10.0000 q^{37} +2.00000 q^{39} +1.00000 q^{40} +7.00000 q^{41} -11.0000 q^{43} +3.00000 q^{44} -1.00000 q^{45} +4.00000 q^{47} +2.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -16.0000 q^{51} +1.00000 q^{52} -5.00000 q^{53} +4.00000 q^{54} -3.00000 q^{55} +4.00000 q^{58} +4.00000 q^{59} -2.00000 q^{60} -13.0000 q^{61} -1.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -6.00000 q^{66} -8.00000 q^{68} +9.00000 q^{71} -1.00000 q^{72} +9.00000 q^{73} +10.0000 q^{74} +2.00000 q^{75} -2.00000 q^{78} +3.00000 q^{79} -1.00000 q^{80} -11.0000 q^{81} -7.00000 q^{82} -5.00000 q^{83} +8.00000 q^{85} +11.0000 q^{86} -8.00000 q^{87} -3.00000 q^{88} +7.00000 q^{89} +1.00000 q^{90} +2.00000 q^{93} -4.00000 q^{94} -2.00000 q^{96} -18.0000 q^{97} +7.00000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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	−0.707107
$2$	−1.00000	−0.707107
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.00000	−0.816497
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	2.00000	0.577350
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	1.00000	0.250000
$17$	−8.00000	−1.94029	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−8.00000	−1.94029	−0.970143	+	0.242536i	$0.922021\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−3.00000	−0.639602
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−2.00000	−0.408248
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	2.00000	0.365148
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	−1.00000	−0.176777
$33$	6.00000	1.04447
$34$	8.00000	1.37199
$35$	0	0
$36$	1.00000	0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	1.00000	0.158114
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
$43$	−11.0000	−1.67748	−0.838742	+	0.544529i	$0.816708\pi$
$44$	3.00000	0.452267
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	2.00000	0.288675
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	−16.0000	−2.24045
$52$	1.00000	0.138675
$53$	−5.00000	−0.686803	−0.343401	−	0.939189i	$-0.611579\pi$
$53$	−5.00000	−0.686803	−0.343401	+	0.939189i	$0.611579\pi$
$54$	4.00000	0.544331
$55$	−3.00000	−0.404520
$56$	0	0
$57$	0	0
$58$	4.00000	0.525226
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	−2.00000	−0.258199
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−6.00000	−0.738549
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−8.00000	−0.970143
$69$	0	0
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	−1.00000	−0.117851
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	10.0000	1.16248
$75$	2.00000	0.230940
$76$	0	0
$77$	0	0
$78$	−2.00000	−0.226455
$79$	3.00000	0.337526	0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000	0.337526	0.168763	+	0.985657i	$0.446023\pi$
$80$	−1.00000	−0.111803
$81$	−11.0000	−1.22222
$82$	−7.00000	−0.773021
$83$	−5.00000	−0.548821	−0.274411	−	0.961613i	$-0.588483\pi$
$83$	−5.00000	−0.548821	−0.274411	+	0.961613i	$0.588483\pi$
$84$	0	0
$85$	8.00000	0.867722
$86$	11.0000	1.18616
$87$	−8.00000	−0.857690
$88$	−3.00000	−0.319801
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	0	0
$93$	2.00000	0.207390
$94$	−4.00000	−0.412568
$95$	0	0
$96$	−2.00000	−0.204124
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	7.00000	0.707107
$99$	3.00000	0.301511
$100$	1.00000	0.100000
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	16.0000	1.58424
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	5.00000	0.485643
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−4.00000	−0.384900
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	3.00000	0.286039
$111$	−20.0000	−1.89832
$112$	0	0
$113$	15.0000	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$114$	0	0
$115$	0	0
$116$	−4.00000	−0.371391
$117$	1.00000	0.0924500
$118$	−4.00000	−0.368230
$119$	0	0
$120$	2.00000	0.182574
$121$	−2.00000	−0.181818
$122$	13.0000	1.17696
$123$	14.0000	1.26234
$124$	1.00000	0.0898027
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.00000	−0.266207	−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000	−0.266207	−0.133103	+	0.991102i	$0.542494\pi$
$128$	−1.00000	−0.0883883
$129$	−22.0000	−1.93699
$130$	1.00000	0.0877058
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	6.00000	0.522233
$133$	0	0
$134$	0	0
$135$	4.00000	0.344265
$136$	8.00000	0.685994
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−9.00000	−0.755263
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−9.00000	−0.744845
$147$	−14.0000	−1.15470
$148$	−10.0000	−0.821995
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−2.00000	−0.163299
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	−8.00000	−0.646762
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	2.00000	0.160128
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	−3.00000	−0.238667
$159$	−10.0000	−0.793052
$160$	1.00000	0.0790569
$161$	0	0
$162$	11.0000	0.864242
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	7.00000	0.546608
$165$	−6.00000	−0.467099
$166$	5.00000	0.388075
$167$	19.0000	1.47026	0.735132	−	0.677924i	$-0.237120\pi$
$167$	19.0000	1.47026	0.735132	+	0.677924i	$0.237120\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−8.00000	−0.613572
$171$	0	0
$172$	−11.0000	−0.838742
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	8.00000	0.606478
$175$	0	0
$176$	3.00000	0.226134
$177$	8.00000	0.601317
$178$	−7.00000	−0.524672
$179$	−23.0000	−1.71910	−0.859550	−	0.511051i	$-0.829256\pi$
$179$	−23.0000	−1.71910	−0.859550	+	0.511051i	$0.829256\pi$
$180$	−1.00000	−0.0745356
$181$	−4.00000	−0.297318	−0.148659	−	0.988889i	$-0.547496\pi$
$181$	−4.00000	−0.297318	−0.148659	+	0.988889i	$0.547496\pi$
$182$	0	0
$183$	−26.0000	−1.92198
$184$	0	0
$185$	10.0000	0.735215
$186$	−2.00000	−0.146647
$187$	−24.0000	−1.75505
$188$	4.00000	0.291730
$189$	0	0
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	2.00000	0.144338
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	18.0000	1.29232
$195$	−2.00000	−0.143223
$196$	−7.00000	−0.500000
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−3.00000	−0.213201
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−10.0000	−0.703598
$203$	0	0
$204$	−16.0000	−1.12022
$205$	−7.00000	−0.488901
$206$	−6.00000	−0.418040
$207$	0	0
$208$	1.00000	0.0693375
$209$	0	0
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	−5.00000	−0.343401
$213$	18.0000	1.23334
$214$	−12.0000	−0.820303
$215$	11.0000	0.750194
$216$	4.00000	0.272166
$217$	0	0
$218$	14.0000	0.948200
$219$	18.0000	1.21633
$220$	−3.00000	−0.202260
$221$	−8.00000	−0.538138
$222$	20.0000	1.34231
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−15.0000	−0.997785
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	4.00000	0.262613
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	−1.00000	−0.0653720
$235$	−4.00000	−0.260931
$236$	4.00000	0.260378
$237$	6.00000	0.389742
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−2.00000	−0.129099
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	2.00000	0.128565
$243$	−10.0000	−0.641500
$244$	−13.0000	−0.832240
$245$	7.00000	0.447214
$246$	−14.0000	−0.892607
$247$	0	0
$248$	−1.00000	−0.0635001
$249$	−10.0000	−0.633724
$250$	1.00000	0.0632456
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	0	0
$254$	3.00000	0.188237
$255$	16.0000	1.00196
$256$	1.00000	0.0625000
$257$	−17.0000	−1.06043	−0.530215	−	0.847863i	$-0.677889\pi$
$257$	−17.0000	−1.06043	−0.530215	+	0.847863i	$0.677889\pi$
$258$	22.0000	1.36966
$259$	0	0
$260$	−1.00000	−0.0620174
$261$	−4.00000	−0.247594
$262$	8.00000	0.494242
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	−6.00000	−0.369274
$265$	5.00000	0.307148
$266$	0	0
$267$	14.0000	0.856786
$268$	0	0
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	−4.00000	−0.243432
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−8.00000	−0.485071
$273$	0	0
$274$	22.0000	1.32907
$275$	3.00000	0.180907
$276$	0	0
$277$	1.00000	0.0600842	0.0300421	−	0.999549i	$-0.490436\pi$
$277$	1.00000	0.0600842	0.0300421	+	0.999549i	$0.490436\pi$
$278$	−4.00000	−0.239904
$279$	1.00000	0.0598684
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	−8.00000	−0.476393
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	9.00000	0.534052
$285$	0	0
$286$	−3.00000	−0.177394
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	47.0000	2.76471
$290$	−4.00000	−0.234888
$291$	−36.0000	−2.11036
$292$	9.00000	0.526685
$293$	−1.00000	−0.0584206	−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000	−0.0584206	−0.0292103	+	0.999573i	$0.509299\pi$
$294$	14.0000	0.816497
$295$	−4.00000	−0.232889
$296$	10.0000	0.581238
$297$	−12.0000	−0.696311
$298$	10.0000	0.579284
$299$	0	0
$300$	2.00000	0.115470
$301$	0	0
$302$	−12.0000	−0.690522
$303$	20.0000	1.14897
$304$	0	0
$305$	13.0000	0.744378
$306$	8.00000	0.457330
$307$	23.0000	1.31268	0.656340	−	0.754466i	$-0.272104\pi$
$307$	23.0000	1.31268	0.656340	+	0.754466i	$0.272104\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	1.00000	0.0567962
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	−2.00000	−0.113228
$313$	−31.0000	−1.75222	−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000	−1.75222	−0.876112	+	0.482108i	$0.839871\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	3.00000	0.168763
$317$	−9.00000	−0.505490	−0.252745	−	0.967533i	$-0.581333\pi$
$317$	−9.00000	−0.505490	−0.252745	+	0.967533i	$0.581333\pi$
$318$	10.0000	0.560772
$319$	−12.0000	−0.671871
$320$	−1.00000	−0.0559017
$321$	24.0000	1.33955
$322$	0	0
$323$	0	0
$324$	−11.0000	−0.611111
$325$	1.00000	0.0554700
$326$	10.0000	0.553849
$327$	−28.0000	−1.54840
$328$	−7.00000	−0.386510
$329$	0	0
$330$	6.00000	0.330289
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	−5.00000	−0.274411
$333$	−10.0000	−0.547997
$334$	−19.0000	−1.03963
$335$	0	0
$336$	0	0
$337$	−3.00000	−0.163420	−0.0817102	−	0.996656i	$-0.526038\pi$
$337$	−3.00000	−0.163420	−0.0817102	+	0.996656i	$0.526038\pi$
$338$	12.0000	0.652714
$339$	30.0000	1.62938
$340$	8.00000	0.433861
$341$	3.00000	0.162459
$342$	0	0
$343$	0	0
$344$	11.0000	0.593080
$345$	0	0
$346$	16.0000	0.860165
$347$	10.0000	0.536828	0.268414	−	0.963304i	$-0.413500\pi$
$347$	10.0000	0.536828	0.268414	+	0.963304i	$0.413500\pi$
$348$	−8.00000	−0.428845
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	−3.00000	−0.159901
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	−8.00000	−0.425195
$355$	−9.00000	−0.477670
$356$	7.00000	0.370999
$357$	0	0
$358$	23.0000	1.21559
$359$	27.0000	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$360$	1.00000	0.0527046
$361$	−19.0000	−1.00000
$362$	4.00000	0.210235
$363$	−4.00000	−0.209946
$364$	0	0
$365$	−9.00000	−0.471082
$366$	26.0000	1.35904
$367$	−35.0000	−1.82699	−0.913493	−	0.406855i	$-0.866625\pi$
$367$	−35.0000	−1.82699	−0.913493	+	0.406855i	$0.866625\pi$
$368$	0	0
$369$	7.00000	0.364405
$370$	−10.0000	−0.519875
$371$	0	0
$372$	2.00000	0.103695
$373$	8.00000	0.414224	0.207112	−	0.978317i	$-0.433593\pi$
$373$	8.00000	0.414224	0.207112	+	0.978317i	$0.433593\pi$
$374$	24.0000	1.24101
$375$	−2.00000	−0.103280
$376$	−4.00000	−0.206284
$377$	−4.00000	−0.206010
$378$	0	0
$379$	35.0000	1.79783	0.898915	−	0.438124i	$-0.144357\pi$
$379$	35.0000	1.79783	0.898915	+	0.438124i	$0.144357\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	−3.00000	−0.153493
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	4.00000	0.203595
$387$	−11.0000	−0.559161
$388$	−18.0000	−0.913812
$389$	37.0000	1.87597	0.937987	−	0.346670i	$-0.112688\pi$
$389$	37.0000	1.87597	0.937987	+	0.346670i	$0.112688\pi$
$390$	2.00000	0.101274
$391$	0	0
$392$	7.00000	0.353553
$393$	−16.0000	−0.807093
$394$	−6.00000	−0.302276
$395$	−3.00000	−0.150946
$396$	3.00000	0.150756
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	1.00000	0.0500000
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	1.00000	0.0498135
$404$	10.0000	0.497519
$405$	11.0000	0.546594
$406$	0	0
$407$	−30.0000	−1.48704
$408$	16.0000	0.792118
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	7.00000	0.345705
$411$	−44.0000	−2.17036
$412$	6.00000	0.295599
$413$	0	0
$414$	0	0
$415$	5.00000	0.245440
$416$	−1.00000	−0.0490290
$417$	8.00000	0.391762
$418$	0	0
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	−16.0000	−0.778868
$423$	4.00000	0.194487
$424$	5.00000	0.242821
$425$	−8.00000	−0.388057
$426$	−18.0000	−0.872103
$427$	0	0
$428$	12.0000	0.580042
$429$	6.00000	0.289683
$430$	−11.0000	−0.530467
$431$	1.00000	0.0481683	0.0240842	−	0.999710i	$-0.492333\pi$
$431$	1.00000	0.0481683	0.0240842	+	0.999710i	$0.492333\pi$
$432$	−4.00000	−0.192450
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	0	0
$435$	8.00000	0.383571
$436$	−14.0000	−0.670478
$437$	0	0
$438$	−18.0000	−0.860073
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	3.00000	0.143019
$441$	−7.00000	−0.333333
$442$	8.00000	0.380521
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	−20.0000	−0.949158
$445$	−7.00000	−0.331832
$446$	−20.0000	−0.947027
$447$	−20.0000	−0.945968
$448$	0	0
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	−1.00000	−0.0471405
$451$	21.0000	0.988851
$452$	15.0000	0.705541
$453$	24.0000	1.12762
$454$	4.00000	0.187729
$455$	0	0
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	10.0000	0.467269
$459$	32.0000	1.49363
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	5.00000	0.232370	0.116185	−	0.993228i	$-0.462933\pi$
$463$	5.00000	0.232370	0.116185	+	0.993228i	$0.462933\pi$
$464$	−4.00000	−0.185695
$465$	−2.00000	−0.0927478
$466$	−12.0000	−0.555889
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	1.00000	0.0462250
$469$	0	0
$470$	4.00000	0.184506
$471$	−36.0000	−1.65879
$472$	−4.00000	−0.184115
$473$	−33.0000	−1.51734
$474$	−6.00000	−0.275589
$475$	0	0
$476$	0	0
$477$	−5.00000	−0.228934
$478$	6.00000	0.274434
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	2.00000	0.0912871
$481$	−10.0000	−0.455961
$482$	−17.0000	−0.774329
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	18.0000	0.817338
$486$	10.0000	0.453609
$487$	14.0000	0.634401	0.317200	−	0.948359i	$-0.397257\pi$
$487$	14.0000	0.634401	0.317200	+	0.948359i	$0.397257\pi$
$488$	13.0000	0.588482
$489$	−20.0000	−0.904431
$490$	−7.00000	−0.316228
$491$	−15.0000	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$492$	14.0000	0.631169
$493$	32.0000	1.44121
$494$	0	0
$495$	−3.00000	−0.134840
$496$	1.00000	0.0449013
$497$	0	0
$498$	10.0000	0.448111
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−1.00000	−0.0447214
$501$	38.0000	1.69771
$502$	20.0000	0.892644
$503$	−42.0000	−1.87269	−0.936344	−	0.351085i	$-0.885813\pi$
$503$	−42.0000	−1.87269	−0.936344	+	0.351085i	$0.885813\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	−24.0000	−1.06588
$508$	−3.00000	−0.133103
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	−16.0000	−0.708492
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	17.0000	0.749838
$515$	−6.00000	−0.264392
$516$	−22.0000	−0.968496
$517$	12.0000	0.527759
$518$	0	0
$519$	−32.0000	−1.40464
$520$	1.00000	0.0438529
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	4.00000	0.175075
$523$	−2.00000	−0.0874539	−0.0437269	−	0.999044i	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	−18.0000	−0.784837
$527$	−8.00000	−0.348485
$528$	6.00000	0.261116
$529$	−23.0000	−1.00000
$530$	−5.00000	−0.217186
$531$	4.00000	0.173585
$532$	0	0
$533$	7.00000	0.303204
$534$	−14.0000	−0.605839
$535$	−12.0000	−0.518805
$536$	0	0
$537$	−46.0000	−1.98505
$538$	3.00000	0.129339
$539$	−21.0000	−0.904534
$540$	4.00000	0.172133
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−8.00000	−0.343629
$543$	−8.00000	−0.343313
$544$	8.00000	0.342997
$545$	14.0000	0.599694
$546$	0	0
$547$	15.0000	0.641354	0.320677	−	0.947189i	$-0.396090\pi$
$547$	15.0000	0.641354	0.320677	+	0.947189i	$0.396090\pi$
$548$	−22.0000	−0.939793
$549$	−13.0000	−0.554826
$550$	−3.00000	−0.127920
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−1.00000	−0.0424859
$555$	20.0000	0.848953
$556$	4.00000	0.169638
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	−1.00000	−0.0423334
$559$	−11.0000	−0.465250
$560$	0	0
$561$	−48.0000	−2.02656
$562$	22.0000	0.928014
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	8.00000	0.336861
$565$	−15.0000	−0.631055
$566$	12.0000	0.504398
$567$	0	0
$568$	−9.00000	−0.377632
$569$	40.0000	1.67689	0.838444	−	0.544988i	$-0.183466\pi$
$569$	40.0000	1.67689	0.838444	+	0.544988i	$0.183466\pi$
$570$	0	0
$571$	−38.0000	−1.59025	−0.795125	−	0.606445i	$-0.792595\pi$
$571$	−38.0000	−1.59025	−0.795125	+	0.606445i	$0.792595\pi$
$572$	3.00000	0.125436
$573$	6.00000	0.250654
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−33.0000	−1.37381	−0.686904	−	0.726748i	$-0.741031\pi$
$577$	−33.0000	−1.37381	−0.686904	+	0.726748i	$0.741031\pi$
$578$	−47.0000	−1.95494
$579$	−8.00000	−0.332469
$580$	4.00000	0.166091
$581$	0	0
$582$	36.0000	1.49225
$583$	−15.0000	−0.621237
$584$	−9.00000	−0.372423
$585$	−1.00000	−0.0413449
$586$	1.00000	0.0413096
$587$	−5.00000	−0.206372	−0.103186	−	0.994662i	$-0.532904\pi$
$587$	−5.00000	−0.206372	−0.103186	+	0.994662i	$0.532904\pi$
$588$	−14.0000	−0.577350
$589$	0	0
$590$	4.00000	0.164677
$591$	12.0000	0.493614
$592$	−10.0000	−0.410997
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	12.0000	0.492366
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−8.00000	−0.327418
$598$	0	0
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	−2.00000	−0.0816497
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	0	0
$604$	12.0000	0.488273
$605$	2.00000	0.0813116
$606$	−20.0000	−0.812444
$607$	36.0000	1.46119	0.730597	−	0.682808i	$-0.239242\pi$
$607$	36.0000	1.46119	0.730597	+	0.682808i	$0.239242\pi$
$608$	0	0
$609$	0	0
$610$	−13.0000	−0.526355
$611$	4.00000	0.161823
$612$	−8.00000	−0.323381
$613$	−41.0000	−1.65597	−0.827987	−	0.560747i	$-0.810514\pi$
$613$	−41.0000	−1.65597	−0.827987	+	0.560747i	$0.810514\pi$
$614$	−23.0000	−0.928204
$615$	−14.0000	−0.564534
$616$	0	0
$617$	28.0000	1.12724	0.563619	−	0.826035i	$-0.309409\pi$
$617$	28.0000	1.12724	0.563619	+	0.826035i	$0.309409\pi$
$618$	−12.0000	−0.482711
$619$	−17.0000	−0.683288	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−17.0000	−0.683288	−0.341644	+	0.939829i	$0.610984\pi$
$620$	−1.00000	−0.0401610
$621$	0	0
$622$	0	0
$623$	0	0
$624$	2.00000	0.0800641
$625$	1.00000	0.0400000
$626$	31.0000	1.23901
$627$	0	0
$628$	−18.0000	−0.718278
$629$	80.0000	3.18981
$630$	0	0
$631$	28.0000	1.11466	0.557331	−	0.830290i	$-0.311825\pi$
$631$	28.0000	1.11466	0.557331	+	0.830290i	$0.311825\pi$
$632$	−3.00000	−0.119334
$633$	32.0000	1.27189
$634$	9.00000	0.357436
$635$	3.00000	0.119051
$636$	−10.0000	−0.396526
$637$	−7.00000	−0.277350
$638$	12.0000	0.475085
$639$	9.00000	0.356034
$640$	1.00000	0.0395285
$641$	−4.00000	−0.157991	−0.0789953	−	0.996875i	$-0.525171\pi$
$641$	−4.00000	−0.157991	−0.0789953	+	0.996875i	$0.525171\pi$
$642$	−24.0000	−0.947204
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	22.0000	0.866249
$646$	0	0
$647$	−3.00000	−0.117942	−0.0589711	−	0.998260i	$-0.518782\pi$
$647$	−3.00000	−0.117942	−0.0589711	+	0.998260i	$0.518782\pi$
$648$	11.0000	0.432121
$649$	12.0000	0.471041
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	−10.0000	−0.391630
$653$	−22.0000	−0.860927	−0.430463	−	0.902608i	$-0.641650\pi$
$653$	−22.0000	−0.860927	−0.430463	+	0.902608i	$0.641650\pi$
$654$	28.0000	1.09489
$655$	8.00000	0.312586
$656$	7.00000	0.273304
$657$	9.00000	0.351123
$658$	0	0
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	−6.00000	−0.233550
$661$	−41.0000	−1.59472	−0.797358	−	0.603507i	$-0.793769\pi$
$661$	−41.0000	−1.59472	−0.797358	+	0.603507i	$0.793769\pi$
$662$	−11.0000	−0.427527
$663$	−16.0000	−0.621389
$664$	5.00000	0.194038
$665$	0	0
$666$	10.0000	0.387492
$667$	0	0
$668$	19.0000	0.735132
$669$	40.0000	1.54649
$670$	0	0
$671$	−39.0000	−1.50558
$672$	0	0
$673$	−42.0000	−1.61898	−0.809491	−	0.587133i	$-0.800257\pi$
$673$	−42.0000	−1.61898	−0.809491	+	0.587133i	$0.800257\pi$
$674$	3.00000	0.115556
$675$	−4.00000	−0.153960
$676$	−12.0000	−0.461538
$677$	12.0000	0.461197	0.230599	−	0.973049i	$-0.425932\pi$
$677$	12.0000	0.461197	0.230599	+	0.973049i	$0.425932\pi$
$678$	−30.0000	−1.15214
$679$	0	0
$680$	−8.00000	−0.306786
$681$	−8.00000	−0.306561
$682$	−3.00000	−0.114876
$683$	6.00000	0.229584	0.114792	−	0.993390i	$-0.463380\pi$
$683$	6.00000	0.229584	0.114792	+	0.993390i	$0.463380\pi$
$684$	0	0
$685$	22.0000	0.840577
$686$	0	0
$687$	−20.0000	−0.763048
$688$	−11.0000	−0.419371
$689$	−5.00000	−0.190485
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	−16.0000	−0.608229
$693$	0	0
$694$	−10.0000	−0.379595
$695$	−4.00000	−0.151729
$696$	8.00000	0.303239
$697$	−56.0000	−2.12115
$698$	10.0000	0.378506
$699$	24.0000	0.907763
$700$	0	0
$701$	45.0000	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$702$	4.00000	0.150970
$703$	0	0
$704$	3.00000	0.113067
$705$	−8.00000	−0.301297
$706$	−6.00000	−0.225813
$707$	0	0
$708$	8.00000	0.300658
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	9.00000	0.337764
$711$	3.00000	0.112509
$712$	−7.00000	−0.262336
$713$	0	0
$714$	0	0
$715$	−3.00000	−0.112194
$716$	−23.0000	−0.859550
$717$	−12.0000	−0.448148
$718$	−27.0000	−1.00763
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	19.0000	0.707107
$723$	34.0000	1.26447
$724$	−4.00000	−0.148659
$725$	−4.00000	−0.148556
$726$	4.00000	0.148454
$727$	−29.0000	−1.07555	−0.537775	−	0.843088i	$-0.680735\pi$
$727$	−29.0000	−1.07555	−0.537775	+	0.843088i	$0.680735\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	9.00000	0.333105
$731$	88.0000	3.25480
$732$	−26.0000	−0.960988
$733$	8.00000	0.295487	0.147743	−	0.989026i	$-0.452799\pi$
$733$	8.00000	0.295487	0.147743	+	0.989026i	$0.452799\pi$
$734$	35.0000	1.29187
$735$	14.0000	0.516398
$736$	0	0
$737$	0	0
$738$	−7.00000	−0.257674
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	10.0000	0.367607
$741$	0	0
$742$	0	0
$743$	37.0000	1.35740	0.678699	−	0.734416i	$-0.262544\pi$
$743$	37.0000	1.35740	0.678699	+	0.734416i	$0.262544\pi$
$744$	−2.00000	−0.0733236
$745$	10.0000	0.366372
$746$	−8.00000	−0.292901
$747$	−5.00000	−0.182940
$748$	−24.0000	−0.877527
$749$	0	0
$750$	2.00000	0.0730297
$751$	−12.0000	−0.437886	−0.218943	−	0.975738i	$-0.570261\pi$
$751$	−12.0000	−0.437886	−0.218943	+	0.975738i	$0.570261\pi$
$752$	4.00000	0.145865
$753$	−40.0000	−1.45768
$754$	4.00000	0.145671
$755$	−12.0000	−0.436725
$756$	0	0
$757$	46.0000	1.67190	0.835949	−	0.548807i	$-0.184918\pi$
$757$	46.0000	1.67190	0.835949	+	0.548807i	$0.184918\pi$
$758$	−35.0000	−1.27126
$759$	0	0
$760$	0	0
$761$	33.0000	1.19625	0.598125	−	0.801403i	$-0.295913\pi$
$761$	33.0000	1.19625	0.598125	+	0.801403i	$0.295913\pi$
$762$	6.00000	0.217357
$763$	0	0
$764$	3.00000	0.108536
$765$	8.00000	0.289241
$766$	−12.0000	−0.433578
$767$	4.00000	0.144432
$768$	2.00000	0.0721688
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	−34.0000	−1.22448
$772$	−4.00000	−0.143963
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	11.0000	0.395387
$775$	1.00000	0.0359211
$776$	18.0000	0.646162
$777$	0	0
$778$	−37.0000	−1.32651
$779$	0	0
$780$	−2.00000	−0.0716115
$781$	27.0000	0.966136
$782$	0	0
$783$	16.0000	0.571793
$784$	−7.00000	−0.250000
$785$	18.0000	0.642448
$786$	16.0000	0.570701
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	6.00000	0.213741
$789$	36.0000	1.28163
$790$	3.00000	0.106735
$791$	0	0
$792$	−3.00000	−0.106600
$793$	−13.0000	−0.461644
$794$	14.0000	0.496841
$795$	10.0000	0.354663
$796$	−4.00000	−0.141776
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	−32.0000	−1.13208
$800$	−1.00000	−0.0353553
$801$	7.00000	0.247333
$802$	1.00000	0.0353112
$803$	27.0000	0.952809
$804$	0	0
$805$	0	0
$806$	−1.00000	−0.0352235
$807$	−6.00000	−0.211210
$808$	−10.0000	−0.351799
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	−11.0000	−0.386501
$811$	−24.0000	−0.842754	−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000	−0.842754	−0.421377	+	0.906886i	$0.638453\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	30.0000	1.05150
$815$	10.0000	0.350285
$816$	−16.0000	−0.560112
$817$	0	0
$818$	18.0000	0.629355
$819$	0	0
$820$	−7.00000	−0.244451
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	44.0000	1.53468
$823$	19.0000	0.662298	0.331149	−	0.943578i	$-0.392564\pi$
$823$	19.0000	0.662298	0.331149	+	0.943578i	$0.392564\pi$
$824$	−6.00000	−0.209020
$825$	6.00000	0.208893
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	33.0000	1.14614	0.573069	−	0.819507i	$-0.305753\pi$
$829$	33.0000	1.14614	0.573069	+	0.819507i	$0.305753\pi$
$830$	−5.00000	−0.173553
$831$	2.00000	0.0693792
$832$	1.00000	0.0346688
$833$	56.0000	1.94029
$834$	−8.00000	−0.277017
$835$	−19.0000	−0.657522
$836$	0	0
$837$	−4.00000	−0.138260
$838$	28.0000	0.967244
$839$	27.0000	0.932144	0.466072	−	0.884747i	$-0.345669\pi$
$839$	27.0000	0.932144	0.466072	+	0.884747i	$0.345669\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	2.00000	0.0689246
$843$	−44.0000	−1.51544
$844$	16.0000	0.550743
$845$	12.0000	0.412813
$846$	−4.00000	−0.137523
$847$	0	0
$848$	−5.00000	−0.171701
$849$	−24.0000	−0.823678
$850$	8.00000	0.274398
$851$	0	0
$852$	18.0000	0.616670
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−7.00000	−0.239115	−0.119558	−	0.992827i	$-0.538148\pi$
$857$	−7.00000	−0.239115	−0.119558	+	0.992827i	$0.538148\pi$
$858$	−6.00000	−0.204837
$859$	35.0000	1.19418	0.597092	−	0.802173i	$-0.296323\pi$
$859$	35.0000	1.19418	0.597092	+	0.802173i	$0.296323\pi$
$860$	11.0000	0.375097
$861$	0	0
$862$	−1.00000	−0.0340601
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	4.00000	0.136083
$865$	16.0000	0.544016
$866$	−10.0000	−0.339814
$867$	94.0000	3.19241
$868$	0	0
$869$	9.00000	0.305304
$870$	−8.00000	−0.271225
$871$	0	0
$872$	14.0000	0.474100
$873$	−18.0000	−0.609208
$874$	0	0
$875$	0	0
$876$	18.0000	0.608164
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	−28.0000	−0.944954
$879$	−2.00000	−0.0674583
$880$	−3.00000	−0.101130
$881$	38.0000	1.28025	0.640126	−	0.768270i	$-0.278882\pi$
$881$	38.0000	1.28025	0.640126	+	0.768270i	$0.278882\pi$
$882$	7.00000	0.235702
$883$	−44.0000	−1.48072	−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000	−1.48072	−0.740359	+	0.672212i	$0.765344\pi$
$884$	−8.00000	−0.269069
$885$	−8.00000	−0.268917
$886$	−20.0000	−0.671913
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	20.0000	0.671156
$889$	0	0
$890$	7.00000	0.234641
$891$	−33.0000	−1.10554
$892$	20.0000	0.669650
$893$	0	0
$894$	20.0000	0.668900
$895$	23.0000	0.768805
$896$	0	0
$897$	0	0
$898$	−14.0000	−0.467186
$899$	−4.00000	−0.133407
$900$	1.00000	0.0333333
$901$	40.0000	1.33259
$902$	−21.0000	−0.699224
$903$	0	0
$904$	−15.0000	−0.498893
$905$	4.00000	0.132964
$906$	−24.0000	−0.797347
$907$	−38.0000	−1.26177	−0.630885	−	0.775877i	$-0.717308\pi$
$907$	−38.0000	−1.26177	−0.630885	+	0.775877i	$0.717308\pi$
$908$	−4.00000	−0.132745
$909$	10.0000	0.331679
$910$	0	0
$911$	−58.0000	−1.92163	−0.960813	−	0.277198i	$-0.910594\pi$
$911$	−58.0000	−1.92163	−0.960813	+	0.277198i	$0.910594\pi$
$912$	0	0
$913$	−15.0000	−0.496428
$914$	−1.00000	−0.0330771
$915$	26.0000	0.859533
$916$	−10.0000	−0.330409
$917$	0	0
$918$	−32.0000	−1.05616
$919$	−53.0000	−1.74831	−0.874154	−	0.485648i	$-0.838584\pi$
$919$	−53.0000	−1.74831	−0.874154	+	0.485648i	$0.838584\pi$
$920$	0	0
$921$	46.0000	1.51575
$922$	−10.0000	−0.329332
$923$	9.00000	0.296239
$924$	0	0
$925$	−10.0000	−0.328798
$926$	−5.00000	−0.164310
$927$	6.00000	0.197066
$928$	4.00000	0.131306
$929$	8.00000	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$930$	2.00000	0.0655826
$931$	0	0
$932$	12.0000	0.393073
$933$	0	0
$934$	6.00000	0.196326
$935$	24.0000	0.784884
$936$	−1.00000	−0.0326860
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	−62.0000	−2.02329
$940$	−4.00000	−0.130466
$941$	1.00000	0.0325991	0.0162995	−	0.999867i	$-0.494811\pi$
$941$	1.00000	0.0325991	0.0162995	+	0.999867i	$0.494811\pi$
$942$	36.0000	1.17294
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	33.0000	1.07292
$947$	−47.0000	−1.52729	−0.763647	−	0.645634i	$-0.776593\pi$
$947$	−47.0000	−1.52729	−0.763647	+	0.645634i	$0.776593\pi$
$948$	6.00000	0.194871
$949$	9.00000	0.292152
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	9.00000	0.291539	0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000	0.291539	0.145769	+	0.989319i	$0.453434\pi$
$954$	5.00000	0.161881
$955$	−3.00000	−0.0970777
$956$	−6.00000	−0.194054
$957$	−24.0000	−0.775810
$958$	40.0000	1.29234
$959$	0	0
$960$	−2.00000	−0.0645497
$961$	−30.0000	−0.967742
$962$	10.0000	0.322413
$963$	12.0000	0.386695
$964$	17.0000	0.547533
$965$	4.00000	0.128765
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	−18.0000	−0.577945
$971$	32.0000	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$972$	−10.0000	−0.320750
$973$	0	0
$974$	−14.0000	−0.448589
$975$	2.00000	0.0640513
$976$	−13.0000	−0.416120
$977$	−11.0000	−0.351921	−0.175961	−	0.984397i	$-0.556303\pi$
$977$	−11.0000	−0.351921	−0.175961	+	0.984397i	$0.556303\pi$
$978$	20.0000	0.639529
$979$	21.0000	0.671163
$980$	7.00000	0.223607
$981$	−14.0000	−0.446986
$982$	15.0000	0.478669
$983$	38.0000	1.21201	0.606006	−	0.795460i	$-0.292771\pi$
$983$	38.0000	1.21201	0.606006	+	0.795460i	$0.292771\pi$
$984$	−14.0000	−0.446304
$985$	−6.00000	−0.191176
$986$	−32.0000	−1.01909
$987$	0	0
$988$	0	0
$989$	0	0
$990$	3.00000	0.0953463
$991$	61.0000	1.93773	0.968864	−	0.247592i	$-0.0796392\pi$
$991$	61.0000	1.93773	0.968864	+	0.247592i	$0.0796392\pi$
$992$	−1.00000	−0.0317500
$993$	22.0000	0.698149
$994$	0	0
$995$	4.00000	0.126809
$996$	−10.0000	−0.316862
$997$	−22.0000	−0.696747	−0.348373	−	0.937356i	$-0.613266\pi$
$997$	−22.0000	−0.696747	−0.348373	+	0.937356i	$0.613266\pi$
$998$	4.00000	0.126618
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4010.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.b.1.1	✓	1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4010 = 2 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4010.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more