LMFDB - Embedded newform 4010.2.a.d.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:	$0$
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)

$f(q)$

$=$

$q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -3.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +6.00000 q^{9} +1.00000 q^{10} +5.00000 q^{11} +3.00000 q^{12} +4.00000 q^{13} -1.00000 q^{14} -3.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -6.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +3.00000 q^{21} -5.00000 q^{22} -2.00000 q^{23} -3.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +9.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} +3.00000 q^{30} -7.00000 q^{31} -1.00000 q^{32} +15.0000 q^{33} -5.00000 q^{34} -1.00000 q^{35} +6.00000 q^{36} -8.00000 q^{37} -4.00000 q^{38} +12.0000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -3.00000 q^{42} +5.00000 q^{44} -6.00000 q^{45} +2.00000 q^{46} +3.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} +15.0000 q^{51} +4.00000 q^{52} -2.00000 q^{53} -9.00000 q^{54} -5.00000 q^{55} -1.00000 q^{56} +12.0000 q^{57} -2.00000 q^{58} -6.00000 q^{59} -3.00000 q^{60} -2.00000 q^{61} +7.00000 q^{62} +6.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -15.0000 q^{66} +7.00000 q^{67} +5.00000 q^{68} -6.00000 q^{69} +1.00000 q^{70} -13.0000 q^{71} -6.00000 q^{72} -6.00000 q^{73} +8.00000 q^{74} +3.00000 q^{75} +4.00000 q^{76} +5.00000 q^{77} -12.0000 q^{78} -1.00000 q^{80} +9.00000 q^{81} -2.00000 q^{82} +6.00000 q^{83} +3.00000 q^{84} -5.00000 q^{85} +6.00000 q^{87} -5.00000 q^{88} +6.00000 q^{89} +6.00000 q^{90} +4.00000 q^{91} -2.00000 q^{92} -21.0000 q^{93} -4.00000 q^{95} -3.00000 q^{96} -1.00000 q^{97} +6.00000 q^{98} +30.0000 q^{99} +O(q^{100})\)

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	−0.707107
$2$	−1.00000	−0.707107
$3$	3.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	3.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−3.00000	−1.22474
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	6.00000	2.00000
$10$	1.00000	0.316228
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	3.00000	0.866025
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	−1.00000	−0.267261
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	5.00000	1.21268	0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000	1.21268	0.606339	+	0.795206i	$0.292637\pi$
$18$	−6.00000	−1.41421
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	3.00000	0.654654
$22$	−5.00000	−1.06600
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	−3.00000	−0.612372
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	9.00000	1.73205
$28$	1.00000	0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	3.00000	0.547723
$31$	−7.00000	−1.25724	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	−7.00000	−1.25724	−0.628619	+	0.777714i	$0.716379\pi$
$32$	−1.00000	−0.176777
$33$	15.0000	2.61116
$34$	−5.00000	−0.857493
$35$	−1.00000	−0.169031
$36$	6.00000	1.00000
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	−4.00000	−0.648886
$39$	12.0000	1.92154
$40$	1.00000	0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	−3.00000	−0.462910
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	5.00000	0.753778
$45$	−6.00000	−0.894427
$46$	2.00000	0.294884
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	3.00000	0.433013
$49$	−6.00000	−0.857143
$50$	−1.00000	−0.141421
$51$	15.0000	2.10042
$52$	4.00000	0.554700
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−9.00000	−1.22474
$55$	−5.00000	−0.674200
$56$	−1.00000	−0.133631
$57$	12.0000	1.58944
$58$	−2.00000	−0.262613
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−3.00000	−0.387298
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	7.00000	0.889001
$63$	6.00000	0.755929
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−15.0000	−1.84637
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	5.00000	0.606339
$69$	−6.00000	−0.722315
$70$	1.00000	0.119523
$71$	−13.0000	−1.54282	−0.771408	−	0.636341i	$-0.780447\pi$
$71$	−13.0000	−1.54282	−0.771408	+	0.636341i	$0.780447\pi$
$72$	−6.00000	−0.707107
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	8.00000	0.929981
$75$	3.00000	0.346410
$76$	4.00000	0.458831
$77$	5.00000	0.569803
$78$	−12.0000	−1.35873
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	9.00000	1.00000
$82$	−2.00000	−0.220863
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	3.00000	0.327327
$85$	−5.00000	−0.542326
$86$	0	0
$87$	6.00000	0.643268
$88$	−5.00000	−0.533002
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	6.00000	0.632456
$91$	4.00000	0.419314
$92$	−2.00000	−0.208514
$93$	−21.0000	−2.17760
$94$	0	0
$95$	−4.00000	−0.410391
$96$	−3.00000	−0.306186
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	6.00000	0.606092
$99$	30.0000	3.01511
$100$	1.00000	0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	−15.0000	−1.48522
$103$	3.00000	0.295599	0.147799	−	0.989017i	$-0.452781\pi$
$103$	3.00000	0.295599	0.147799	+	0.989017i	$0.452781\pi$
$104$	−4.00000	−0.392232
$105$	−3.00000	−0.292770
$106$	2.00000	0.194257
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	9.00000	0.866025
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	5.00000	0.476731
$111$	−24.0000	−2.27798
$112$	1.00000	0.0944911
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	−12.0000	−1.12390
$115$	2.00000	0.186501
$116$	2.00000	0.185695
$117$	24.0000	2.21880
$118$	6.00000	0.552345
$119$	5.00000	0.458349
$120$	3.00000	0.273861
$121$	14.0000	1.27273
$122$	2.00000	0.181071
$123$	6.00000	0.541002
$124$	−7.00000	−0.628619
$125$	−1.00000	−0.0894427
$126$	−6.00000	−0.534522
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	4.00000	0.350823
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	15.0000	1.30558
$133$	4.00000	0.346844
$134$	−7.00000	−0.604708
$135$	−9.00000	−0.774597
$136$	−5.00000	−0.428746
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	6.00000	0.510754
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	13.0000	1.09094
$143$	20.0000	1.67248
$144$	6.00000	0.500000
$145$	−2.00000	−0.166091
$146$	6.00000	0.496564
$147$	−18.0000	−1.48461
$148$	−8.00000	−0.657596
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	−3.00000	−0.244949
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−4.00000	−0.324443
$153$	30.0000	2.42536
$154$	−5.00000	−0.402911
$155$	7.00000	0.562254
$156$	12.0000	0.960769
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	−2.00000	−0.157622
$162$	−9.00000	−0.707107
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	2.00000	0.156174
$165$	−15.0000	−1.16775
$166$	−6.00000	−0.465690
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−3.00000	−0.231455
$169$	3.00000	0.230769
$170$	5.00000	0.383482
$171$	24.0000	1.83533
$172$	0	0
$173$	−17.0000	−1.29249	−0.646243	−	0.763132i	$-0.723661\pi$
$173$	−17.0000	−1.29249	−0.646243	+	0.763132i	$0.723661\pi$
$174$	−6.00000	−0.454859
$175$	1.00000	0.0755929
$176$	5.00000	0.376889
$177$	−18.0000	−1.35296
$178$	−6.00000	−0.449719
$179$	−11.0000	−0.822179	−0.411089	−	0.911595i	$-0.634852\pi$
$179$	−11.0000	−0.822179	−0.411089	+	0.911595i	$0.634852\pi$
$180$	−6.00000	−0.447214
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	−4.00000	−0.296500
$183$	−6.00000	−0.443533
$184$	2.00000	0.147442
$185$	8.00000	0.588172
$186$	21.0000	1.53979
$187$	25.0000	1.82818
$188$	0	0
$189$	9.00000	0.654654
$190$	4.00000	0.290191
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	3.00000	0.216506
$193$	22.0000	1.58359	0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000	1.58359	0.791797	+	0.610784i	$0.209146\pi$
$194$	1.00000	0.0717958
$195$	−12.0000	−0.859338
$196$	−6.00000	−0.428571
$197$	1.00000	0.0712470	0.0356235	−	0.999365i	$-0.488658\pi$
$197$	1.00000	0.0712470	0.0356235	+	0.999365i	$0.488658\pi$
$198$	−30.0000	−2.13201
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	−1.00000	−0.0707107
$201$	21.0000	1.48123
$202$	18.0000	1.26648
$203$	2.00000	0.140372
$204$	15.0000	1.05021
$205$	−2.00000	−0.139686
$206$	−3.00000	−0.209020
$207$	−12.0000	−0.834058
$208$	4.00000	0.277350
$209$	20.0000	1.38343
$210$	3.00000	0.207020
$211$	26.0000	1.78991	0.894957	−	0.446153i	$-0.147206\pi$
$211$	26.0000	1.78991	0.894957	+	0.446153i	$0.147206\pi$
$212$	−2.00000	−0.137361
$213$	−39.0000	−2.67224
$214$	20.0000	1.36717
$215$	0	0
$216$	−9.00000	−0.612372
$217$	−7.00000	−0.475191
$218$	−6.00000	−0.406371
$219$	−18.0000	−1.21633
$220$	−5.00000	−0.337100
$221$	20.0000	1.34535
$222$	24.0000	1.61077
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	−1.00000	−0.0668153
$225$	6.00000	0.400000
$226$	18.0000	1.19734
$227$	9.00000	0.597351	0.298675	−	0.954355i	$-0.403455\pi$
$227$	9.00000	0.597351	0.298675	+	0.954355i	$0.403455\pi$
$228$	12.0000	0.794719
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	−2.00000	−0.131876
$231$	15.0000	0.986928
$232$	−2.00000	−0.131306
$233$	15.0000	0.982683	0.491341	−	0.870967i	$-0.336507\pi$
$233$	15.0000	0.982683	0.491341	+	0.870967i	$0.336507\pi$
$234$	−24.0000	−1.56893
$235$	0	0
$236$	−6.00000	−0.390567
$237$	0	0
$238$	−5.00000	−0.324102
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	−3.00000	−0.193649
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	−14.0000	−0.899954
$243$	0	0
$244$	−2.00000	−0.128037
$245$	6.00000	0.383326
$246$	−6.00000	−0.382546
$247$	16.0000	1.01806
$248$	7.00000	0.444500
$249$	18.0000	1.14070
$250$	1.00000	0.0632456
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	6.00000	0.377964
$253$	−10.0000	−0.628695
$254$	−12.0000	−0.752947
$255$	−15.0000	−0.939336
$256$	1.00000	0.0625000
$257$	−8.00000	−0.499026	−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000	−0.499026	−0.249513	+	0.968371i	$0.580271\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	−4.00000	−0.248069
$261$	12.0000	0.742781
$262$	6.00000	0.370681
$263$	−31.0000	−1.91154	−0.955771	−	0.294112i	$-0.904976\pi$
$263$	−31.0000	−1.91154	−0.955771	+	0.294112i	$0.904976\pi$
$264$	−15.0000	−0.923186
$265$	2.00000	0.122859
$266$	−4.00000	−0.245256
$267$	18.0000	1.10158
$268$	7.00000	0.427593
$269$	−1.00000	−0.0609711	−0.0304855	−	0.999535i	$-0.509705\pi$
$269$	−1.00000	−0.0609711	−0.0304855	+	0.999535i	$0.509705\pi$
$270$	9.00000	0.547723
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	5.00000	0.303170
$273$	12.0000	0.726273
$274$	−9.00000	−0.543710
$275$	5.00000	0.301511
$276$	−6.00000	−0.361158
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−10.0000	−0.599760
$279$	−42.0000	−2.51447
$280$	1.00000	0.0597614
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	−17.0000	−1.01055	−0.505273	−	0.862960i	$-0.668608\pi$
$283$	−17.0000	−1.01055	−0.505273	+	0.862960i	$0.668608\pi$
$284$	−13.0000	−0.771408
$285$	−12.0000	−0.710819
$286$	−20.0000	−1.18262
$287$	2.00000	0.118056
$288$	−6.00000	−0.353553
$289$	8.00000	0.470588
$290$	2.00000	0.117444
$291$	−3.00000	−0.175863
$292$	−6.00000	−0.351123
$293$	−8.00000	−0.467365	−0.233682	−	0.972313i	$-0.575078\pi$
$293$	−8.00000	−0.467365	−0.233682	+	0.972313i	$0.575078\pi$
$294$	18.0000	1.04978
$295$	6.00000	0.349334
$296$	8.00000	0.464991
$297$	45.0000	2.61116
$298$	−4.00000	−0.231714
$299$	−8.00000	−0.462652
$300$	3.00000	0.173205
$301$	0	0
$302$	−16.0000	−0.920697
$303$	−54.0000	−3.10222
$304$	4.00000	0.229416
$305$	2.00000	0.114520
$306$	−30.0000	−1.71499
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	5.00000	0.284901
$309$	9.00000	0.511992
$310$	−7.00000	−0.397573
$311$	32.0000	1.81455	0.907277	−	0.420534i	$-0.138157\pi$
$311$	32.0000	1.81455	0.907277	+	0.420534i	$0.138157\pi$
$312$	−12.0000	−0.679366
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	−14.0000	−0.790066
$315$	−6.00000	−0.338062
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	6.00000	0.336463
$319$	10.0000	0.559893
$320$	−1.00000	−0.0559017
$321$	−60.0000	−3.34887
$322$	2.00000	0.111456
$323$	20.0000	1.11283
$324$	9.00000	0.500000
$325$	4.00000	0.221880
$326$	−8.00000	−0.443079
$327$	18.0000	0.995402
$328$	−2.00000	−0.110432
$329$	0	0
$330$	15.0000	0.825723
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	6.00000	0.329293
$333$	−48.0000	−2.63038
$334$	−12.0000	−0.656611
$335$	−7.00000	−0.382451
$336$	3.00000	0.163663
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	−3.00000	−0.163178
$339$	−54.0000	−2.93288
$340$	−5.00000	−0.271163
$341$	−35.0000	−1.89536
$342$	−24.0000	−1.29777
$343$	−13.0000	−0.701934
$344$	0	0
$345$	6.00000	0.323029
$346$	17.0000	0.913926
$347$	−3.00000	−0.161048	−0.0805242	−	0.996753i	$-0.525659\pi$
$347$	−3.00000	−0.161048	−0.0805242	+	0.996753i	$0.525659\pi$
$348$	6.00000	0.321634
$349$	5.00000	0.267644	0.133822	−	0.991005i	$-0.457275\pi$
$349$	5.00000	0.267644	0.133822	+	0.991005i	$0.457275\pi$
$350$	−1.00000	−0.0534522
$351$	36.0000	1.92154
$352$	−5.00000	−0.266501
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	18.0000	0.956689
$355$	13.0000	0.689968
$356$	6.00000	0.317999
$357$	15.0000	0.793884
$358$	11.0000	0.581368
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	6.00000	0.316228
$361$	−3.00000	−0.157895
$362$	−6.00000	−0.315353
$363$	42.0000	2.20443
$364$	4.00000	0.209657
$365$	6.00000	0.314054
$366$	6.00000	0.313625
$367$	−14.0000	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$367$	−14.0000	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$368$	−2.00000	−0.104257
$369$	12.0000	0.624695
$370$	−8.00000	−0.415900
$371$	−2.00000	−0.103835
$372$	−21.0000	−1.08880
$373$	−29.0000	−1.50156	−0.750782	−	0.660551i	$-0.770323\pi$
$373$	−29.0000	−1.50156	−0.750782	+	0.660551i	$0.770323\pi$
$374$	−25.0000	−1.29272
$375$	−3.00000	−0.154919
$376$	0	0
$377$	8.00000	0.412021
$378$	−9.00000	−0.462910
$379$	−17.0000	−0.873231	−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000	−0.873231	−0.436616	+	0.899648i	$0.643823\pi$
$380$	−4.00000	−0.205196
$381$	36.0000	1.84434
$382$	12.0000	0.613973
$383$	−15.0000	−0.766464	−0.383232	−	0.923652i	$-0.625189\pi$
$383$	−15.0000	−0.766464	−0.383232	+	0.923652i	$0.625189\pi$
$384$	−3.00000	−0.153093
$385$	−5.00000	−0.254824
$386$	−22.0000	−1.11977
$387$	0	0
$388$	−1.00000	−0.0507673
$389$	−21.0000	−1.06474	−0.532371	−	0.846511i	$-0.678699\pi$
$389$	−21.0000	−1.06474	−0.532371	+	0.846511i	$0.678699\pi$
$390$	12.0000	0.607644
$391$	−10.0000	−0.505722
$392$	6.00000	0.303046
$393$	−18.0000	−0.907980
$394$	−1.00000	−0.0503793
$395$	0	0
$396$	30.0000	1.50756
$397$	7.00000	0.351320	0.175660	−	0.984451i	$-0.443794\pi$
$397$	7.00000	0.351320	0.175660	+	0.984451i	$0.443794\pi$
$398$	−3.00000	−0.150376
$399$	12.0000	0.600751
$400$	1.00000	0.0500000
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	−21.0000	−1.04738
$403$	−28.0000	−1.39478
$404$	−18.0000	−0.895533
$405$	−9.00000	−0.447214
$406$	−2.00000	−0.0992583
$407$	−40.0000	−1.98273
$408$	−15.0000	−0.742611
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	2.00000	0.0987730
$411$	27.0000	1.33181
$412$	3.00000	0.147799
$413$	−6.00000	−0.295241
$414$	12.0000	0.589768
$415$	−6.00000	−0.294528
$416$	−4.00000	−0.196116
$417$	30.0000	1.46911
$418$	−20.0000	−0.978232
$419$	3.00000	0.146560	0.0732798	−	0.997311i	$-0.476653\pi$
$419$	3.00000	0.146560	0.0732798	+	0.997311i	$0.476653\pi$
$420$	−3.00000	−0.146385
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−26.0000	−1.26566
$423$	0	0
$424$	2.00000	0.0971286
$425$	5.00000	0.242536
$426$	39.0000	1.88956
$427$	−2.00000	−0.0967868
$428$	−20.0000	−0.966736
$429$	60.0000	2.89683
$430$	0	0
$431$	31.0000	1.49322	0.746609	−	0.665263i	$-0.231681\pi$
$431$	31.0000	1.49322	0.746609	+	0.665263i	$0.231681\pi$
$432$	9.00000	0.433013
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	7.00000	0.336011
$435$	−6.00000	−0.287678
$436$	6.00000	0.287348
$437$	−8.00000	−0.382692
$438$	18.0000	0.860073
$439$	−13.0000	−0.620456	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000	−0.620456	−0.310228	+	0.950662i	$0.600405\pi$
$440$	5.00000	0.238366
$441$	−36.0000	−1.71429
$442$	−20.0000	−0.951303
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−24.0000	−1.13899
$445$	−6.00000	−0.284427
$446$	19.0000	0.899676
$447$	12.0000	0.567581
$448$	1.00000	0.0472456
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	−6.00000	−0.282843
$451$	10.0000	0.470882
$452$	−18.0000	−0.846649
$453$	48.0000	2.25524
$454$	−9.00000	−0.422391
$455$	−4.00000	−0.187523
$456$	−12.0000	−0.561951
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	4.00000	0.186908
$459$	45.0000	2.10042
$460$	2.00000	0.0932505
$461$	−27.0000	−1.25752	−0.628758	−	0.777601i	$-0.716436\pi$
$461$	−27.0000	−1.25752	−0.628758	+	0.777601i	$0.716436\pi$
$462$	−15.0000	−0.697863
$463$	18.0000	0.836531	0.418265	−	0.908325i	$-0.362638\pi$
$463$	18.0000	0.836531	0.418265	+	0.908325i	$0.362638\pi$
$464$	2.00000	0.0928477
$465$	21.0000	0.973852
$466$	−15.0000	−0.694862
$467$	−39.0000	−1.80470	−0.902352	−	0.430999i	$-0.858161\pi$
$467$	−39.0000	−1.80470	−0.902352	+	0.430999i	$0.858161\pi$
$468$	24.0000	1.10940
$469$	7.00000	0.323230
$470$	0	0
$471$	42.0000	1.93526
$472$	6.00000	0.276172
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	5.00000	0.229175
$477$	−12.0000	−0.549442
$478$	6.00000	0.274434
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	3.00000	0.136931
$481$	−32.0000	−1.45907
$482$	5.00000	0.227744
$483$	−6.00000	−0.273009
$484$	14.0000	0.636364
$485$	1.00000	0.0454077
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	2.00000	0.0905357
$489$	24.0000	1.08532
$490$	−6.00000	−0.271052
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	6.00000	0.270501
$493$	10.0000	0.450377
$494$	−16.0000	−0.719874
$495$	−30.0000	−1.34840
$496$	−7.00000	−0.314309
$497$	−13.0000	−0.583130
$498$	−18.0000	−0.806599
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	−1.00000	−0.0447214
$501$	36.0000	1.60836
$502$	−2.00000	−0.0892644
$503$	17.0000	0.757993	0.378996	−	0.925398i	$-0.376269\pi$
$503$	17.0000	0.757993	0.378996	+	0.925398i	$0.376269\pi$
$504$	−6.00000	−0.267261
$505$	18.0000	0.800989
$506$	10.0000	0.444554
$507$	9.00000	0.399704
$508$	12.0000	0.532414
$509$	11.0000	0.487566	0.243783	−	0.969830i	$-0.421611\pi$
$509$	11.0000	0.487566	0.243783	+	0.969830i	$0.421611\pi$
$510$	15.0000	0.664211
$511$	−6.00000	−0.265424
$512$	−1.00000	−0.0441942
$513$	36.0000	1.58944
$514$	8.00000	0.352865
$515$	−3.00000	−0.132196
$516$	0	0
$517$	0	0
$518$	8.00000	0.351500
$519$	−51.0000	−2.23865
$520$	4.00000	0.175412
$521$	38.0000	1.66481	0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000	1.66481	0.832405	+	0.554168i	$0.186963\pi$
$522$	−12.0000	−0.525226
$523$	−23.0000	−1.00572	−0.502860	−	0.864368i	$-0.667719\pi$
$523$	−23.0000	−1.00572	−0.502860	+	0.864368i	$0.667719\pi$
$524$	−6.00000	−0.262111
$525$	3.00000	0.130931
$526$	31.0000	1.35166
$527$	−35.0000	−1.52462
$528$	15.0000	0.652791
$529$	−19.0000	−0.826087
$530$	−2.00000	−0.0868744
$531$	−36.0000	−1.56227
$532$	4.00000	0.173422
$533$	8.00000	0.346518
$534$	−18.0000	−0.778936
$535$	20.0000	0.864675
$536$	−7.00000	−0.302354
$537$	−33.0000	−1.42406
$538$	1.00000	0.0431131
$539$	−30.0000	−1.29219
$540$	−9.00000	−0.387298
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	20.0000	0.859074
$543$	18.0000	0.772454
$544$	−5.00000	−0.214373
$545$	−6.00000	−0.257012
$546$	−12.0000	−0.513553
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	9.00000	0.384461
$549$	−12.0000	−0.512148
$550$	−5.00000	−0.213201
$551$	8.00000	0.340811
$552$	6.00000	0.255377
$553$	0	0
$554$	−10.0000	−0.424859
$555$	24.0000	1.01874
$556$	10.0000	0.424094
$557$	−37.0000	−1.56774	−0.783870	−	0.620925i	$-0.786757\pi$
$557$	−37.0000	−1.56774	−0.783870	+	0.620925i	$0.786757\pi$
$558$	42.0000	1.77800
$559$	0	0
$560$	−1.00000	−0.0422577
$561$	75.0000	3.16650
$562$	14.0000	0.590554
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	17.0000	0.714563
$567$	9.00000	0.377964
$568$	13.0000	0.545468
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	12.0000	0.502625
$571$	−30.0000	−1.25546	−0.627730	−	0.778431i	$-0.716016\pi$
$571$	−30.0000	−1.25546	−0.627730	+	0.778431i	$0.716016\pi$
$572$	20.0000	0.836242
$573$	−36.0000	−1.50392
$574$	−2.00000	−0.0834784
$575$	−2.00000	−0.0834058
$576$	6.00000	0.250000
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−8.00000	−0.332756
$579$	66.0000	2.74287
$580$	−2.00000	−0.0830455
$581$	6.00000	0.248922
$582$	3.00000	0.124354
$583$	−10.0000	−0.414158
$584$	6.00000	0.248282
$585$	−24.0000	−0.992278
$586$	8.00000	0.330477
$587$	26.0000	1.07313	0.536567	−	0.843857i	$-0.319721\pi$
$587$	26.0000	1.07313	0.536567	+	0.843857i	$0.319721\pi$
$588$	−18.0000	−0.742307
$589$	−28.0000	−1.15372
$590$	−6.00000	−0.247016
$591$	3.00000	0.123404
$592$	−8.00000	−0.328798
$593$	3.00000	0.123195	0.0615976	−	0.998101i	$-0.480380\pi$
$593$	3.00000	0.123195	0.0615976	+	0.998101i	$0.480380\pi$
$594$	−45.0000	−1.84637
$595$	−5.00000	−0.204980
$596$	4.00000	0.163846
$597$	9.00000	0.368345
$598$	8.00000	0.327144
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	−3.00000	−0.122474
$601$	9.00000	0.367118	0.183559	−	0.983009i	$-0.441238\pi$
$601$	9.00000	0.367118	0.183559	+	0.983009i	$0.441238\pi$
$602$	0	0
$603$	42.0000	1.71037
$604$	16.0000	0.651031
$605$	−14.0000	−0.569181
$606$	54.0000	2.19360
$607$	39.0000	1.58296	0.791481	−	0.611194i	$-0.209311\pi$
$607$	39.0000	1.58296	0.791481	+	0.611194i	$0.209311\pi$
$608$	−4.00000	−0.162221
$609$	6.00000	0.243132
$610$	−2.00000	−0.0809776
$611$	0	0
$612$	30.0000	1.21268
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	−18.0000	−0.726421
$615$	−6.00000	−0.241943
$616$	−5.00000	−0.201456
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−9.00000	−0.362033
$619$	−15.0000	−0.602901	−0.301450	−	0.953482i	$-0.597471\pi$
$619$	−15.0000	−0.602901	−0.301450	+	0.953482i	$0.597471\pi$
$620$	7.00000	0.281127
$621$	−18.0000	−0.722315
$622$	−32.0000	−1.28308
$623$	6.00000	0.240385
$624$	12.0000	0.480384
$625$	1.00000	0.0400000
$626$	8.00000	0.319744
$627$	60.0000	2.39617
$628$	14.0000	0.558661
$629$	−40.0000	−1.59490
$630$	6.00000	0.239046
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	78.0000	3.10022
$634$	−2.00000	−0.0794301
$635$	−12.0000	−0.476205
$636$	−6.00000	−0.237915
$637$	−24.0000	−0.950915
$638$	−10.0000	−0.395904
$639$	−78.0000	−3.08563
$640$	1.00000	0.0395285
$641$	−32.0000	−1.26392	−0.631962	−	0.774999i	$-0.717750\pi$
$641$	−32.0000	−1.26392	−0.631962	+	0.774999i	$0.717750\pi$
$642$	60.0000	2.36801
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	−2.00000	−0.0788110
$645$	0	0
$646$	−20.0000	−0.786889
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	−9.00000	−0.353553
$649$	−30.0000	−1.17760
$650$	−4.00000	−0.156893
$651$	−21.0000	−0.823055
$652$	8.00000	0.313304
$653$	33.0000	1.29139	0.645695	−	0.763596i	$-0.276568\pi$
$653$	33.0000	1.29139	0.645695	+	0.763596i	$0.276568\pi$
$654$	−18.0000	−0.703856
$655$	6.00000	0.234439
$656$	2.00000	0.0780869
$657$	−36.0000	−1.40449
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	−15.0000	−0.583874
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	13.0000	0.505259
$663$	60.0000	2.33021
$664$	−6.00000	−0.232845
$665$	−4.00000	−0.155113
$666$	48.0000	1.85996
$667$	−4.00000	−0.154881
$668$	12.0000	0.464294
$669$	−57.0000	−2.20375
$670$	7.00000	0.270434
$671$	−10.0000	−0.386046
$672$	−3.00000	−0.115728
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−34.0000	−1.30963
$675$	9.00000	0.346410
$676$	3.00000	0.115385
$677$	49.0000	1.88322	0.941611	−	0.336701i	$-0.109311\pi$
$677$	49.0000	1.88322	0.941611	+	0.336701i	$0.109311\pi$
$678$	54.0000	2.07386
$679$	−1.00000	−0.0383765
$680$	5.00000	0.191741
$681$	27.0000	1.03464
$682$	35.0000	1.34022
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	24.0000	0.917663
$685$	−9.00000	−0.343872
$686$	13.0000	0.496342
$687$	−12.0000	−0.457829
$688$	0	0
$689$	−8.00000	−0.304776
$690$	−6.00000	−0.228416
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	−17.0000	−0.646243
$693$	30.0000	1.13961
$694$	3.00000	0.113878
$695$	−10.0000	−0.379322
$696$	−6.00000	−0.227429
$697$	10.0000	0.378777
$698$	−5.00000	−0.189253
$699$	45.0000	1.70206
$700$	1.00000	0.0377964
$701$	−11.0000	−0.415464	−0.207732	−	0.978186i	$-0.566608\pi$
$701$	−11.0000	−0.415464	−0.207732	+	0.978186i	$0.566608\pi$
$702$	−36.0000	−1.35873
$703$	−32.0000	−1.20690
$704$	5.00000	0.188445
$705$	0	0
$706$	−26.0000	−0.978523
$707$	−18.0000	−0.676960
$708$	−18.0000	−0.676481
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	−13.0000	−0.487881
$711$	0	0
$712$	−6.00000	−0.224860
$713$	14.0000	0.524304
$714$	−15.0000	−0.561361
$715$	−20.0000	−0.747958
$716$	−11.0000	−0.411089
$717$	−18.0000	−0.672222
$718$	0	0
$719$	44.0000	1.64092	0.820462	−	0.571702i	$-0.193717\pi$
$719$	44.0000	1.64092	0.820462	+	0.571702i	$0.193717\pi$
$720$	−6.00000	−0.223607
$721$	3.00000	0.111726
$722$	3.00000	0.111648
$723$	−15.0000	−0.557856
$724$	6.00000	0.222988
$725$	2.00000	0.0742781
$726$	−42.0000	−1.55877
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	−4.00000	−0.148250
$729$	−27.0000	−1.00000
$730$	−6.00000	−0.222070
$731$	0	0
$732$	−6.00000	−0.221766
$733$	35.0000	1.29275	0.646377	−	0.763018i	$-0.276283\pi$
$733$	35.0000	1.29275	0.646377	+	0.763018i	$0.276283\pi$
$734$	14.0000	0.516749
$735$	18.0000	0.663940
$736$	2.00000	0.0737210
$737$	35.0000	1.28924
$738$	−12.0000	−0.441726
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	8.00000	0.294086
$741$	48.0000	1.76332
$742$	2.00000	0.0734223
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	21.0000	0.769897
$745$	−4.00000	−0.146549
$746$	29.0000	1.06177
$747$	36.0000	1.31717
$748$	25.0000	0.914091
$749$	−20.0000	−0.730784
$750$	3.00000	0.109545
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	−8.00000	−0.291343
$755$	−16.0000	−0.582300
$756$	9.00000	0.327327
$757$	−9.00000	−0.327111	−0.163555	−	0.986534i	$-0.552296\pi$
$757$	−9.00000	−0.327111	−0.163555	+	0.986534i	$0.552296\pi$
$758$	17.0000	0.617468
$759$	−30.0000	−1.08893
$760$	4.00000	0.145095
$761$	−38.0000	−1.37750	−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000	−1.37750	−0.688749	+	0.724999i	$0.741840\pi$
$762$	−36.0000	−1.30414
$763$	6.00000	0.217215
$764$	−12.0000	−0.434145
$765$	−30.0000	−1.08465
$766$	15.0000	0.541972
$767$	−24.0000	−0.866590
$768$	3.00000	0.108253
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	5.00000	0.180187
$771$	−24.0000	−0.864339
$772$	22.0000	0.791797
$773$	−39.0000	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$774$	0	0
$775$	−7.00000	−0.251447
$776$	1.00000	0.0358979
$777$	−24.0000	−0.860995
$778$	21.0000	0.752886
$779$	8.00000	0.286630
$780$	−12.0000	−0.429669
$781$	−65.0000	−2.32588
$782$	10.0000	0.357599
$783$	18.0000	0.643268
$784$	−6.00000	−0.214286
$785$	−14.0000	−0.499681
$786$	18.0000	0.642039
$787$	5.00000	0.178231	0.0891154	−	0.996021i	$-0.471596\pi$
$787$	5.00000	0.178231	0.0891154	+	0.996021i	$0.471596\pi$
$788$	1.00000	0.0356235
$789$	−93.0000	−3.31089
$790$	0	0
$791$	−18.0000	−0.640006
$792$	−30.0000	−1.06600
$793$	−8.00000	−0.284088
$794$	−7.00000	−0.248421
$795$	6.00000	0.212798
$796$	3.00000	0.106332
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	−12.0000	−0.424795
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	36.0000	1.27200
$802$	−1.00000	−0.0353112
$803$	−30.0000	−1.05868
$804$	21.0000	0.740613
$805$	2.00000	0.0704907
$806$	28.0000	0.986258
$807$	−3.00000	−0.105605
$808$	18.0000	0.633238
$809$	−15.0000	−0.527372	−0.263686	−	0.964609i	$-0.584938\pi$
$809$	−15.0000	−0.527372	−0.263686	+	0.964609i	$0.584938\pi$
$810$	9.00000	0.316228
$811$	−20.0000	−0.702295	−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000	−0.702295	−0.351147	+	0.936320i	$0.614208\pi$
$812$	2.00000	0.0701862
$813$	−60.0000	−2.10429
$814$	40.0000	1.40200
$815$	−8.00000	−0.280228
$816$	15.0000	0.525105
$817$	0	0
$818$	26.0000	0.909069
$819$	24.0000	0.838628
$820$	−2.00000	−0.0698430
$821$	−27.0000	−0.942306	−0.471153	−	0.882051i	$-0.656162\pi$
$821$	−27.0000	−0.942306	−0.471153	+	0.882051i	$0.656162\pi$
$822$	−27.0000	−0.941733
$823$	−30.0000	−1.04573	−0.522867	−	0.852414i	$-0.675138\pi$
$823$	−30.0000	−1.04573	−0.522867	+	0.852414i	$0.675138\pi$
$824$	−3.00000	−0.104510
$825$	15.0000	0.522233
$826$	6.00000	0.208767
$827$	−2.00000	−0.0695468	−0.0347734	−	0.999395i	$-0.511071\pi$
$827$	−2.00000	−0.0695468	−0.0347734	+	0.999395i	$0.511071\pi$
$828$	−12.0000	−0.417029
$829$	27.0000	0.937749	0.468874	−	0.883265i	$-0.344660\pi$
$829$	27.0000	0.937749	0.468874	+	0.883265i	$0.344660\pi$
$830$	6.00000	0.208263
$831$	30.0000	1.04069
$832$	4.00000	0.138675
$833$	−30.0000	−1.03944
$834$	−30.0000	−1.03882
$835$	−12.0000	−0.415277
$836$	20.0000	0.691714
$837$	−63.0000	−2.17760
$838$	−3.00000	−0.103633
$839$	−53.0000	−1.82976	−0.914882	−	0.403722i	$-0.867716\pi$
$839$	−53.0000	−1.82976	−0.914882	+	0.403722i	$0.867716\pi$
$840$	3.00000	0.103510
$841$	−25.0000	−0.862069
$842$	−22.0000	−0.758170
$843$	−42.0000	−1.44656
$844$	26.0000	0.894957
$845$	−3.00000	−0.103203
$846$	0	0
$847$	14.0000	0.481046
$848$	−2.00000	−0.0686803
$849$	−51.0000	−1.75032
$850$	−5.00000	−0.171499
$851$	16.0000	0.548473
$852$	−39.0000	−1.33612
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	2.00000	0.0684386
$855$	−24.0000	−0.820783
$856$	20.0000	0.683586
$857$	24.0000	0.819824	0.409912	−	0.912125i	$-0.365559\pi$
$857$	24.0000	0.819824	0.409912	+	0.912125i	$0.365559\pi$
$858$	−60.0000	−2.04837
$859$	7.00000	0.238837	0.119418	−	0.992844i	$-0.461897\pi$
$859$	7.00000	0.238837	0.119418	+	0.992844i	$0.461897\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	−31.0000	−1.05586
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	−9.00000	−0.306186
$865$	17.0000	0.578017
$866$	−8.00000	−0.271851
$867$	24.0000	0.815083
$868$	−7.00000	−0.237595
$869$	0	0
$870$	6.00000	0.203419
$871$	28.0000	0.948744
$872$	−6.00000	−0.203186
$873$	−6.00000	−0.203069
$874$	8.00000	0.270604
$875$	−1.00000	−0.0338062
$876$	−18.0000	−0.608164
$877$	36.0000	1.21563	0.607817	−	0.794077i	$-0.292045\pi$
$877$	36.0000	1.21563	0.607817	+	0.794077i	$0.292045\pi$
$878$	13.0000	0.438729
$879$	−24.0000	−0.809500
$880$	−5.00000	−0.168550
$881$	−42.0000	−1.41502	−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000	−1.41502	−0.707508	+	0.706705i	$0.750181\pi$
$882$	36.0000	1.21218
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	20.0000	0.672673
$885$	18.0000	0.605063
$886$	36.0000	1.20944
$887$	−48.0000	−1.61168	−0.805841	−	0.592132i	$-0.798286\pi$
$887$	−48.0000	−1.61168	−0.805841	+	0.592132i	$0.798286\pi$
$888$	24.0000	0.805387
$889$	12.0000	0.402467
$890$	6.00000	0.201120
$891$	45.0000	1.50756
$892$	−19.0000	−0.636167
$893$	0	0
$894$	−12.0000	−0.401340
$895$	11.0000	0.367689
$896$	−1.00000	−0.0334077
$897$	−24.0000	−0.801337
$898$	−12.0000	−0.400445
$899$	−14.0000	−0.466926
$900$	6.00000	0.200000
$901$	−10.0000	−0.333148
$902$	−10.0000	−0.332964
$903$	0	0
$904$	18.0000	0.598671
$905$	−6.00000	−0.199447
$906$	−48.0000	−1.59469
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	9.00000	0.298675
$909$	−108.000	−3.58213
$910$	4.00000	0.132599
$911$	−52.0000	−1.72284	−0.861418	−	0.507896i	$-0.830423\pi$
$911$	−52.0000	−1.72284	−0.861418	+	0.507896i	$0.830423\pi$
$912$	12.0000	0.397360
$913$	30.0000	0.992855
$914$	2.00000	0.0661541
$915$	6.00000	0.198354
$916$	−4.00000	−0.132164
$917$	−6.00000	−0.198137
$918$	−45.0000	−1.48522
$919$	37.0000	1.22052	0.610259	−	0.792202i	$-0.291065\pi$
$919$	37.0000	1.22052	0.610259	+	0.792202i	$0.291065\pi$
$920$	−2.00000	−0.0659380
$921$	54.0000	1.77936
$922$	27.0000	0.889198
$923$	−52.0000	−1.71160
$924$	15.0000	0.493464
$925$	−8.00000	−0.263038
$926$	−18.0000	−0.591517
$927$	18.0000	0.591198
$928$	−2.00000	−0.0656532
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	−21.0000	−0.688617
$931$	−24.0000	−0.786568
$932$	15.0000	0.491341
$933$	96.0000	3.14290
$934$	39.0000	1.27612
$935$	−25.0000	−0.817587
$936$	−24.0000	−0.784465
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	−7.00000	−0.228558
$939$	−24.0000	−0.783210
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−42.0000	−1.36843
$943$	−4.00000	−0.130258
$944$	−6.00000	−0.195283
$945$	−9.00000	−0.292770
$946$	0	0
$947$	14.0000	0.454939	0.227469	−	0.973785i	$-0.426955\pi$
$947$	14.0000	0.454939	0.227469	+	0.973785i	$0.426955\pi$
$948$	0	0
$949$	−24.0000	−0.779073
$950$	−4.00000	−0.129777
$951$	6.00000	0.194563
$952$	−5.00000	−0.162051
$953$	8.00000	0.259145	0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000	0.259145	0.129573	+	0.991570i	$0.458639\pi$
$954$	12.0000	0.388514
$955$	12.0000	0.388311
$956$	−6.00000	−0.194054
$957$	30.0000	0.969762
$958$	−6.00000	−0.193851
$959$	9.00000	0.290625
$960$	−3.00000	−0.0968246
$961$	18.0000	0.580645
$962$	32.0000	1.03172
$963$	−120.000	−3.86695
$964$	−5.00000	−0.161039
$965$	−22.0000	−0.708205
$966$	6.00000	0.193047
$967$	−52.0000	−1.67221	−0.836104	−	0.548572i	$-0.815172\pi$
$967$	−52.0000	−1.67221	−0.836104	+	0.548572i	$0.815172\pi$
$968$	−14.0000	−0.449977
$969$	60.0000	1.92748
$970$	−1.00000	−0.0321081
$971$	−40.0000	−1.28366	−0.641831	−	0.766846i	$-0.721825\pi$
$971$	−40.0000	−1.28366	−0.641831	+	0.766846i	$0.721825\pi$
$972$	0	0
$973$	10.0000	0.320585
$974$	−24.0000	−0.769010
$975$	12.0000	0.384308
$976$	−2.00000	−0.0640184
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−24.0000	−0.767435
$979$	30.0000	0.958804
$980$	6.00000	0.191663
$981$	36.0000	1.14939
$982$	12.0000	0.382935
$983$	−9.00000	−0.287055	−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000	−0.287055	−0.143528	+	0.989646i	$0.545845\pi$
$984$	−6.00000	−0.191273
$985$	−1.00000	−0.0318626
$986$	−10.0000	−0.318465
$987$	0	0
$988$	16.0000	0.509028
$989$	0	0
$990$	30.0000	0.953463
$991$	−47.0000	−1.49300	−0.746502	−	0.665383i	$-0.768268\pi$
$991$	−47.0000	−1.49300	−0.746502	+	0.665383i	$0.768268\pi$
$992$	7.00000	0.222250
$993$	−39.0000	−1.23763
$994$	13.0000	0.412335
$995$	−3.00000	−0.0951064
$996$	18.0000	0.570352
$997$	58.0000	1.83688	0.918439	−	0.395562i	$-0.129450\pi$
$997$	58.0000	1.83688	0.918439	+	0.395562i	$0.129450\pi$
$998$	−12.0000	−0.379853
$999$	−72.0000	−2.27798

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4010.2.a.d.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