LMFDB - Embedded newform 503.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [503,2,Mod(1,503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(503, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("503.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	503.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:	$4.01647522167$
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	$\chi$	$=$	503.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +3.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +3.00000 q^{6} +3.00000 q^{7} -3.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +3.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +3.00000 q^{6} +3.00000 q^{7} -3.00000 q^{8} +6.00000 q^{9} -2.00000 q^{10} +3.00000 q^{11} -3.00000 q^{12} +5.00000 q^{13} +3.00000 q^{14} -6.00000 q^{15} -1.00000 q^{16} -8.00000 q^{17} +6.00000 q^{18} +4.00000 q^{19} +2.00000 q^{20} +9.00000 q^{21} +3.00000 q^{22} -5.00000 q^{23} -9.00000 q^{24} -1.00000 q^{25} +5.00000 q^{26} +9.00000 q^{27} -3.00000 q^{28} -6.00000 q^{30} -2.00000 q^{31} +5.00000 q^{32} +9.00000 q^{33} -8.00000 q^{34} -6.00000 q^{35} -6.00000 q^{36} +4.00000 q^{37} +4.00000 q^{38} +15.0000 q^{39} +6.00000 q^{40} -10.0000 q^{41} +9.00000 q^{42} -1.00000 q^{43} -3.00000 q^{44} -12.0000 q^{45} -5.00000 q^{46} -3.00000 q^{47} -3.00000 q^{48} +2.00000 q^{49} -1.00000 q^{50} -24.0000 q^{51} -5.00000 q^{52} -12.0000 q^{53} +9.00000 q^{54} -6.00000 q^{55} -9.00000 q^{56} +12.0000 q^{57} +12.0000 q^{59} +6.00000 q^{60} -11.0000 q^{61} -2.00000 q^{62} +18.0000 q^{63} +7.00000 q^{64} -10.0000 q^{65} +9.00000 q^{66} +7.00000 q^{67} +8.00000 q^{68} -15.0000 q^{69} -6.00000 q^{70} -8.00000 q^{71} -18.0000 q^{72} -6.00000 q^{73} +4.00000 q^{74} -3.00000 q^{75} -4.00000 q^{76} +9.00000 q^{77} +15.0000 q^{78} -4.00000 q^{79} +2.00000 q^{80} +9.00000 q^{81} -10.0000 q^{82} +15.0000 q^{83} -9.00000 q^{84} +16.0000 q^{85} -1.00000 q^{86} -9.00000 q^{88} -12.0000 q^{90} +15.0000 q^{91} +5.00000 q^{92} -6.00000 q^{93} -3.00000 q^{94} -8.00000 q^{95} +15.0000 q^{96} -6.00000 q^{97} +2.00000 q^{98} +18.0000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	3.00000	1.22474
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−3.00000	−1.06066
$9$	6.00000	2.00000
$10$	−2.00000	−0.632456
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−3.00000	−0.866025
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	3.00000	0.801784
$15$	−6.00000	−1.54919
$16$	−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	$0.922021\pi$
$18$	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	$0.348268\pi$
$20$	2.00000	0.447214
$21$	9.00000	1.96396
$22$	3.00000	0.639602
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	−9.00000	−1.83712
$25$	−1.00000	−0.200000
$26$	5.00000	0.980581
$27$	9.00000	1.73205
$28$	−3.00000	−0.566947
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−6.00000	−1.09545
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	5.00000	0.883883
$33$	9.00000	1.56670
$34$	−8.00000	−1.37199
$35$	−6.00000	−1.01419
$36$	−6.00000	−1.00000
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	4.00000	0.648886
$39$	15.0000	2.40192
$40$	6.00000	0.948683
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	9.00000	1.38873
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.00000	−0.452267
$45$	−12.0000	−1.78885
$46$	−5.00000	−0.737210
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	−3.00000	−0.433013
$49$	2.00000	0.285714
$50$	−1.00000	−0.141421
$51$	−24.0000	−3.36067
$52$	−5.00000	−0.693375
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	9.00000	1.22474
$55$	−6.00000	−0.809040
$56$	−9.00000	−1.20268
$57$	12.0000	1.58944
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	6.00000	0.774597
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	−2.00000	−0.254000
$63$	18.0000	2.26779
$64$	7.00000	0.875000
$65$	−10.0000	−1.24035
$66$	9.00000	1.10782
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	8.00000	0.970143
$69$	−15.0000	−1.80579
$70$	−6.00000	−0.717137
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−18.0000	−2.12132
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	4.00000	0.464991
$75$	−3.00000	−0.346410
$76$	−4.00000	−0.458831
$77$	9.00000	1.02565
$78$	15.0000	1.69842
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	2.00000	0.223607
$81$	9.00000	1.00000
$82$	−10.0000	−1.10432
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	−9.00000	−0.981981
$85$	16.0000	1.73544
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−9.00000	−0.959403
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−12.0000	−1.26491
$91$	15.0000	1.57243
$92$	5.00000	0.521286
$93$	−6.00000	−0.622171
$94$	−3.00000	−0.309426
$95$	−8.00000	−0.820783
$96$	15.0000	1.53093
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	2.00000	0.202031
$99$	18.0000	1.80907
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−24.0000	−2.37635
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	−15.0000	−1.47087
$105$	−18.0000	−1.75662
$106$	−12.0000	−1.16554
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	−9.00000	−0.866025
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−6.00000	−0.572078
$111$	12.0000	1.13899
$112$	−3.00000	−0.283473
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	12.0000	1.12390
$115$	10.0000	0.932505
$116$	0	0
$117$	30.0000	2.77350
$118$	12.0000	1.10469
$119$	−24.0000	−2.20008
$120$	18.0000	1.64317
$121$	−2.00000	−0.181818
$122$	−11.0000	−0.995893
$123$	−30.0000	−2.70501
$124$	2.00000	0.179605
$125$	12.0000	1.07331
$126$	18.0000	1.60357
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	−3.00000	−0.265165
$129$	−3.00000	−0.264135
$130$	−10.0000	−0.877058
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	−9.00000	−0.783349
$133$	12.0000	1.04053
$134$	7.00000	0.604708
$135$	−18.0000	−1.54919
$136$	24.0000	2.05798
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	−15.0000	−1.27688
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	6.00000	0.507093
$141$	−9.00000	−0.757937
$142$	−8.00000	−0.671345
$143$	15.0000	1.25436
$144$	−6.00000	−0.500000
$145$	0	0
$146$	−6.00000	−0.496564
$147$	6.00000	0.494872
$148$	−4.00000	−0.328798
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−3.00000	−0.244949
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	−12.0000	−0.973329
$153$	−48.0000	−3.88057
$154$	9.00000	0.725241
$155$	4.00000	0.321288
$156$	−15.0000	−1.20096
$157$	−24.0000	−1.91541	−0.957704	−	0.287754i	$-0.907091\pi$
$157$	−24.0000	−1.91541	−0.957704	+	0.287754i	$0.907091\pi$
$158$	−4.00000	−0.318223
$159$	−36.0000	−2.85499
$160$	−10.0000	−0.790569
$161$	−15.0000	−1.18217
$162$	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	$0.398564\pi$
$164$	10.0000	0.780869
$165$	−18.0000	−1.40130
$166$	15.0000	1.16423
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	−27.0000	−2.08310
$169$	12.0000	0.923077
$170$	16.0000	1.22714
$171$	24.0000	1.83533
$172$	1.00000	0.0762493
$173$	−5.00000	−0.380143	−0.190071	−	0.981770i	$-0.560872\pi$
$173$	−5.00000	−0.380143	−0.190071	+	0.981770i	$0.560872\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	−3.00000	−0.226134
$177$	36.0000	2.70593
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	12.0000	0.894427
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	15.0000	1.11187
$183$	−33.0000	−2.43943
$184$	15.0000	1.10581
$185$	−8.00000	−0.588172
$186$	−6.00000	−0.439941
$187$	−24.0000	−1.75505
$188$	3.00000	0.218797
$189$	27.0000	1.96396
$190$	−8.00000	−0.580381
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	21.0000	1.51554
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	−6.00000	−0.430775
$195$	−30.0000	−2.14834
$196$	−2.00000	−0.142857
$197$	3.00000	0.213741	0.106871	−	0.994273i	$-0.465917\pi$
$197$	3.00000	0.213741	0.106871	+	0.994273i	$0.465917\pi$
$198$	18.0000	1.27920
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	3.00000	0.212132
$201$	21.0000	1.48123
$202$	0	0
$203$	0	0
$204$	24.0000	1.68034
$205$	20.0000	1.39686
$206$	2.00000	0.139347
$207$	−30.0000	−2.08514
$208$	−5.00000	−0.346688
$209$	12.0000	0.830057
$210$	−18.0000	−1.24212
$211$	2.00000	0.137686	0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000	0.137686	0.0688428	+	0.997628i	$0.478069\pi$
$212$	12.0000	0.824163
$213$	−24.0000	−1.64445
$214$	18.0000	1.23045
$215$	2.00000	0.136399
$216$	−27.0000	−1.83712
$217$	−6.00000	−0.407307
$218$	10.0000	0.677285
$219$	−18.0000	−1.21633
$220$	6.00000	0.404520
$221$	−40.0000	−2.69069
$222$	12.0000	0.805387
$223$	−17.0000	−1.13840	−0.569202	−	0.822198i	$-0.692748\pi$
$223$	−17.0000	−1.13840	−0.569202	+	0.822198i	$0.692748\pi$
$224$	15.0000	1.00223
$225$	−6.00000	−0.400000
$226$	13.0000	0.864747
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−12.0000	−0.794719
$229$	19.0000	1.25556	0.627778	−	0.778393i	$-0.283965\pi$
$229$	19.0000	1.25556	0.627778	+	0.778393i	$0.283965\pi$
$230$	10.0000	0.659380
$231$	27.0000	1.77647
$232$	0	0
$233$	3.00000	0.196537	0.0982683	−	0.995160i	$-0.468670\pi$
$233$	3.00000	0.196537	0.0982683	+	0.995160i	$0.468670\pi$
$234$	30.0000	1.96116
$235$	6.00000	0.391397
$236$	−12.0000	−0.781133
$237$	−12.0000	−0.779484
$238$	−24.0000	−1.55569
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	6.00000	0.387298
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	11.0000	0.704203
$245$	−4.00000	−0.255551
$246$	−30.0000	−1.91273
$247$	20.0000	1.27257
$248$	6.00000	0.381000
$249$	45.0000	2.85176
$250$	12.0000	0.758947
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	−18.0000	−1.13389
$253$	−15.0000	−0.943042
$254$	2.00000	0.125491
$255$	48.0000	3.00588
$256$	−17.0000	−1.06250
$257$	27.0000	1.68421	0.842107	−	0.539311i	$-0.181315\pi$
$257$	27.0000	1.68421	0.842107	+	0.539311i	$0.181315\pi$
$258$	−3.00000	−0.186772
$259$	12.0000	0.745644
$260$	10.0000	0.620174
$261$	0	0
$262$	15.0000	0.926703
$263$	−5.00000	−0.308313	−0.154157	−	0.988046i	$-0.549266\pi$
$263$	−5.00000	−0.308313	−0.154157	+	0.988046i	$0.549266\pi$
$264$	−27.0000	−1.66174
$265$	24.0000	1.47431
$266$	12.0000	0.735767
$267$	0	0
$268$	−7.00000	−0.427593
$269$	22.0000	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$270$	−18.0000	−1.09545
$271$	25.0000	1.51864	0.759321	−	0.650716i	$-0.225531\pi$
$271$	25.0000	1.51864	0.759321	+	0.650716i	$0.225531\pi$
$272$	8.00000	0.485071
$273$	45.0000	2.72352
$274$	−12.0000	−0.724947
$275$	−3.00000	−0.180907
$276$	15.0000	0.902894
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	2.00000	0.119952
$279$	−12.0000	−0.718421
$280$	18.0000	1.07571
$281$	−33.0000	−1.96861	−0.984307	−	0.176462i	$-0.943535\pi$
$281$	−33.0000	−1.96861	−0.984307	+	0.176462i	$0.943535\pi$
$282$	−9.00000	−0.535942
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	8.00000	0.474713
$285$	−24.0000	−1.42164
$286$	15.0000	0.886969
$287$	−30.0000	−1.77084
$288$	30.0000	1.76777
$289$	47.0000	2.76471
$290$	0	0
$291$	−18.0000	−1.05518
$292$	6.00000	0.351123
$293$	1.00000	0.0584206	0.0292103	−	0.999573i	$-0.490701\pi$
$293$	1.00000	0.0584206	0.0292103	+	0.999573i	$0.490701\pi$
$294$	6.00000	0.349927
$295$	−24.0000	−1.39733
$296$	−12.0000	−0.697486
$297$	27.0000	1.56670
$298$	−18.0000	−1.04271
$299$	−25.0000	−1.44579
$300$	3.00000	0.173205
$301$	−3.00000	−0.172917
$302$	6.00000	0.345261
$303$	0	0
$304$	−4.00000	−0.229416
$305$	22.0000	1.25972
$306$	−48.0000	−2.74398
$307$	−14.0000	−0.799022	−0.399511	−	0.916728i	$-0.630820\pi$
$307$	−14.0000	−0.799022	−0.399511	+	0.916728i	$0.630820\pi$
$308$	−9.00000	−0.512823
$309$	6.00000	0.341328
$310$	4.00000	0.227185
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	−45.0000	−2.54762
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	−24.0000	−1.35440
$315$	−36.0000	−2.02837
$316$	4.00000	0.225018
$317$	−17.0000	−0.954815	−0.477408	−	0.878682i	$-0.658423\pi$
$317$	−17.0000	−0.954815	−0.477408	+	0.878682i	$0.658423\pi$
$318$	−36.0000	−2.01878
$319$	0	0
$320$	−14.0000	−0.782624
$321$	54.0000	3.01399
$322$	−15.0000	−0.835917
$323$	−32.0000	−1.78053
$324$	−9.00000	−0.500000
$325$	−5.00000	−0.277350
$326$	8.00000	0.443079
$327$	30.0000	1.65900
$328$	30.0000	1.65647
$329$	−9.00000	−0.496186
$330$	−18.0000	−0.990867
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	−15.0000	−0.823232
$333$	24.0000	1.31519
$334$	6.00000	0.328305
$335$	−14.0000	−0.764902
$336$	−9.00000	−0.490990
$337$	32.0000	1.74315	0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000	1.74315	0.871576	+	0.490261i	$0.163099\pi$
$338$	12.0000	0.652714
$339$	39.0000	2.11819
$340$	−16.0000	−0.867722
$341$	−6.00000	−0.324918
$342$	24.0000	1.29777
$343$	−15.0000	−0.809924
$344$	3.00000	0.161749
$345$	30.0000	1.61515
$346$	−5.00000	−0.268802
$347$	−26.0000	−1.39575	−0.697877	−	0.716218i	$-0.745872\pi$
$347$	−26.0000	−1.39575	−0.697877	+	0.716218i	$0.745872\pi$
$348$	0	0
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	−3.00000	−0.160357
$351$	45.0000	2.40192
$352$	15.0000	0.799503
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	36.0000	1.91338
$355$	16.0000	0.849192
$356$	0	0
$357$	−72.0000	−3.81064
$358$	12.0000	0.634220
$359$	−34.0000	−1.79445	−0.897226	−	0.441572i	$-0.854421\pi$
$359$	−34.0000	−1.79445	−0.897226	+	0.441572i	$0.854421\pi$
$360$	36.0000	1.89737
$361$	−3.00000	−0.157895
$362$	−16.0000	−0.840941
$363$	−6.00000	−0.314918
$364$	−15.0000	−0.786214
$365$	12.0000	0.628109
$366$	−33.0000	−1.72494
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	5.00000	0.260643
$369$	−60.0000	−3.12348
$370$	−8.00000	−0.415900
$371$	−36.0000	−1.86903
$372$	6.00000	0.311086
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	−24.0000	−1.24101
$375$	36.0000	1.85903
$376$	9.00000	0.464140
$377$	0	0
$378$	27.0000	1.38873
$379$	−21.0000	−1.07870	−0.539349	−	0.842082i	$-0.681330\pi$
$379$	−21.0000	−1.07870	−0.539349	+	0.842082i	$0.681330\pi$
$380$	8.00000	0.410391
$381$	6.00000	0.307389
$382$	8.00000	0.409316
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−9.00000	−0.459279
$385$	−18.0000	−0.917365
$386$	12.0000	0.610784
$387$	−6.00000	−0.304997
$388$	6.00000	0.304604
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	−30.0000	−1.51911
$391$	40.0000	2.02289
$392$	−6.00000	−0.303046
$393$	45.0000	2.26995
$394$	3.00000	0.151138
$395$	8.00000	0.402524
$396$	−18.0000	−0.904534
$397$	29.0000	1.45547	0.727734	−	0.685859i	$-0.240573\pi$
$397$	29.0000	1.45547	0.727734	+	0.685859i	$0.240573\pi$
$398$	−8.00000	−0.401004
$399$	36.0000	1.80225
$400$	1.00000	0.0500000
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	21.0000	1.04738
$403$	−10.0000	−0.498135
$404$	0	0
$405$	−18.0000	−0.894427
$406$	0	0
$407$	12.0000	0.594818
$408$	72.0000	3.56453
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	20.0000	0.987730
$411$	−36.0000	−1.77575
$412$	−2.00000	−0.0985329
$413$	36.0000	1.77144
$414$	−30.0000	−1.47442
$415$	−30.0000	−1.47264
$416$	25.0000	1.22573
$417$	6.00000	0.293821
$418$	12.0000	0.586939
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	18.0000	0.878310
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	2.00000	0.0973585
$423$	−18.0000	−0.875190
$424$	36.0000	1.74831
$425$	8.00000	0.388057
$426$	−24.0000	−1.16280
$427$	−33.0000	−1.59698
$428$	−18.0000	−0.870063
$429$	45.0000	2.17262
$430$	2.00000	0.0964486
$431$	10.0000	0.481683	0.240842	−	0.970564i	$-0.422577\pi$
$431$	10.0000	0.481683	0.240842	+	0.970564i	$0.422577\pi$
$432$	−9.00000	−0.433013
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−20.0000	−0.956730
$438$	−18.0000	−0.860073
$439$	36.0000	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$440$	18.0000	0.858116
$441$	12.0000	0.571429
$442$	−40.0000	−1.90261
$443$	35.0000	1.66290	0.831450	−	0.555599i	$-0.187511\pi$
$443$	35.0000	1.66290	0.831450	+	0.555599i	$0.187511\pi$
$444$	−12.0000	−0.569495
$445$	0	0
$446$	−17.0000	−0.804973
$447$	−54.0000	−2.55411
$448$	21.0000	0.992157
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−6.00000	−0.282843
$451$	−30.0000	−1.41264
$452$	−13.0000	−0.611469
$453$	18.0000	0.845714
$454$	18.0000	0.844782
$455$	−30.0000	−1.40642
$456$	−36.0000	−1.68585
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	19.0000	0.887812
$459$	−72.0000	−3.36067
$460$	−10.0000	−0.466252
$461$	8.00000	0.372597	0.186299	−	0.982493i	$-0.440351\pi$
$461$	8.00000	0.372597	0.186299	+	0.982493i	$0.440351\pi$
$462$	27.0000	1.25615
$463$	−35.0000	−1.62659	−0.813294	−	0.581853i	$-0.802328\pi$
$463$	−35.0000	−1.62659	−0.813294	+	0.581853i	$0.802328\pi$
$464$	0	0
$465$	12.0000	0.556487
$466$	3.00000	0.138972
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	−30.0000	−1.38675
$469$	21.0000	0.969690
$470$	6.00000	0.276759
$471$	−72.0000	−3.31758
$472$	−36.0000	−1.65703
$473$	−3.00000	−0.137940
$474$	−12.0000	−0.551178
$475$	−4.00000	−0.183533
$476$	24.0000	1.10004
$477$	−72.0000	−3.29665
$478$	24.0000	1.09773
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	−30.0000	−1.36931
$481$	20.0000	0.911922
$482$	−10.0000	−0.455488
$483$	−45.0000	−2.04757
$484$	2.00000	0.0909091
$485$	12.0000	0.544892
$486$	0	0
$487$	−8.00000	−0.362515	−0.181257	−	0.983436i	$-0.558017\pi$
$487$	−8.00000	−0.362515	−0.181257	+	0.983436i	$0.558017\pi$
$488$	33.0000	1.49384
$489$	24.0000	1.08532
$490$	−4.00000	−0.180702
$491$	2.00000	0.0902587	0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000	0.0902587	0.0451294	+	0.998981i	$0.485630\pi$
$492$	30.0000	1.35250
$493$	0	0
$494$	20.0000	0.899843
$495$	−36.0000	−1.61808
$496$	2.00000	0.0898027
$497$	−24.0000	−1.07655
$498$	45.0000	2.01650
$499$	42.0000	1.88018	0.940089	−	0.340929i	$-0.110742\pi$
$499$	42.0000	1.88018	0.940089	+	0.340929i	$0.110742\pi$
$500$	−12.0000	−0.536656
$501$	18.0000	0.804181
$502$	2.00000	0.0892644
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	−54.0000	−2.40535
$505$	0	0
$506$	−15.0000	−0.666831
$507$	36.0000	1.59882
$508$	−2.00000	−0.0887357
$509$	13.0000	0.576215	0.288107	−	0.957598i	$-0.406974\pi$
$509$	13.0000	0.576215	0.288107	+	0.957598i	$0.406974\pi$
$510$	48.0000	2.12548
$511$	−18.0000	−0.796273
$512$	−11.0000	−0.486136
$513$	36.0000	1.58944
$514$	27.0000	1.19092
$515$	−4.00000	−0.176261
$516$	3.00000	0.132068
$517$	−9.00000	−0.395820
$518$	12.0000	0.527250
$519$	−15.0000	−0.658427
$520$	30.0000	1.31559
$521$	−17.0000	−0.744784	−0.372392	−	0.928076i	$-0.621462\pi$
$521$	−17.0000	−0.744784	−0.372392	+	0.928076i	$0.621462\pi$
$522$	0	0
$523$	−6.00000	−0.262362	−0.131181	−	0.991358i	$-0.541877\pi$
$523$	−6.00000	−0.262362	−0.131181	+	0.991358i	$0.541877\pi$
$524$	−15.0000	−0.655278
$525$	−9.00000	−0.392792
$526$	−5.00000	−0.218010
$527$	16.0000	0.696971
$528$	−9.00000	−0.391675
$529$	2.00000	0.0869565
$530$	24.0000	1.04249
$531$	72.0000	3.12453
$532$	−12.0000	−0.520266
$533$	−50.0000	−2.16574
$534$	0	0
$535$	−36.0000	−1.55642
$536$	−21.0000	−0.907062
$537$	36.0000	1.55351
$538$	22.0000	0.948487
$539$	6.00000	0.258438
$540$	18.0000	0.774597
$541$	42.0000	1.80572	0.902861	−	0.429934i	$-0.141463\pi$
$541$	42.0000	1.80572	0.902861	+	0.429934i	$0.141463\pi$
$542$	25.0000	1.07384
$543$	−48.0000	−2.05988
$544$	−40.0000	−1.71499
$545$	−20.0000	−0.856706
$546$	45.0000	1.92582
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	12.0000	0.512615
$549$	−66.0000	−2.81681
$550$	−3.00000	−0.127920
$551$	0	0
$552$	45.0000	1.91533
$553$	−12.0000	−0.510292
$554$	−8.00000	−0.339887
$555$	−24.0000	−1.01874
$556$	−2.00000	−0.0848189
$557$	−23.0000	−0.974541	−0.487271	−	0.873251i	$-0.662007\pi$
$557$	−23.0000	−0.974541	−0.487271	+	0.873251i	$0.662007\pi$
$558$	−12.0000	−0.508001
$559$	−5.00000	−0.211477
$560$	6.00000	0.253546
$561$	−72.0000	−3.03984
$562$	−33.0000	−1.39202
$563$	−34.0000	−1.43293	−0.716465	−	0.697623i	$-0.754241\pi$
$563$	−34.0000	−1.43293	−0.716465	+	0.697623i	$0.754241\pi$
$564$	9.00000	0.378968
$565$	−26.0000	−1.09383
$566$	20.0000	0.840663
$567$	27.0000	1.13389
$568$	24.0000	1.00702
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	−24.0000	−1.00525
$571$	−6.00000	−0.251092	−0.125546	−	0.992088i	$-0.540068\pi$
$571$	−6.00000	−0.251092	−0.125546	+	0.992088i	$0.540068\pi$
$572$	−15.0000	−0.627182
$573$	24.0000	1.00261
$574$	−30.0000	−1.25218
$575$	5.00000	0.208514
$576$	42.0000	1.75000
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	47.0000	1.95494
$579$	36.0000	1.49611
$580$	0	0
$581$	45.0000	1.86691
$582$	−18.0000	−0.746124
$583$	−36.0000	−1.49097
$584$	18.0000	0.744845
$585$	−60.0000	−2.48069
$586$	1.00000	0.0413096
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	−6.00000	−0.247436
$589$	−8.00000	−0.329634
$590$	−24.0000	−0.988064
$591$	9.00000	0.370211
$592$	−4.00000	−0.164399
$593$	−32.0000	−1.31408	−0.657041	−	0.753855i	$-0.728192\pi$
$593$	−32.0000	−1.31408	−0.657041	+	0.753855i	$0.728192\pi$
$594$	27.0000	1.10782
$595$	48.0000	1.96781
$596$	18.0000	0.737309
$597$	−24.0000	−0.982255
$598$	−25.0000	−1.02233
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	9.00000	0.367423
$601$	−23.0000	−0.938190	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	−23.0000	−0.938190	−0.469095	+	0.883148i	$0.655420\pi$
$602$	−3.00000	−0.122271
$603$	42.0000	1.71037
$604$	−6.00000	−0.244137
$605$	4.00000	0.162623
$606$	0	0
$607$	11.0000	0.446476	0.223238	−	0.974764i	$-0.428337\pi$
$607$	11.0000	0.446476	0.223238	+	0.974764i	$0.428337\pi$
$608$	20.0000	0.811107
$609$	0	0
$610$	22.0000	0.890754
$611$	−15.0000	−0.606835
$612$	48.0000	1.94029
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−14.0000	−0.564994
$615$	60.0000	2.41943
$616$	−27.0000	−1.08786
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	6.00000	0.241355
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	−4.00000	−0.160644
$621$	−45.0000	−1.80579
$622$	−20.0000	−0.801927
$623$	0	0
$624$	−15.0000	−0.600481
$625$	−19.0000	−0.760000
$626$	−8.00000	−0.319744
$627$	36.0000	1.43770
$628$	24.0000	0.957704
$629$	−32.0000	−1.27592
$630$	−36.0000	−1.43427
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	12.0000	0.477334
$633$	6.00000	0.238479
$634$	−17.0000	−0.675156
$635$	−4.00000	−0.158735
$636$	36.0000	1.42749
$637$	10.0000	0.396214
$638$	0	0
$639$	−48.0000	−1.89885
$640$	6.00000	0.237171
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	54.0000	2.13121
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	15.0000	0.591083
$645$	6.00000	0.236250
$646$	−32.0000	−1.25902
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−27.0000	−1.06066
$649$	36.0000	1.41312
$650$	−5.00000	−0.196116
$651$	−18.0000	−0.705476
$652$	−8.00000	−0.313304
$653$	45.0000	1.76099	0.880493	−	0.474059i	$-0.157212\pi$
$653$	45.0000	1.76099	0.880493	+	0.474059i	$0.157212\pi$
$654$	30.0000	1.17309
$655$	−30.0000	−1.17220
$656$	10.0000	0.390434
$657$	−36.0000	−1.40449
$658$	−9.00000	−0.350857
$659$	35.0000	1.36341	0.681703	−	0.731629i	$-0.261240\pi$
$659$	35.0000	1.36341	0.681703	+	0.731629i	$0.261240\pi$
$660$	18.0000	0.700649
$661$	−31.0000	−1.20576	−0.602880	−	0.797832i	$-0.705980\pi$
$661$	−31.0000	−1.20576	−0.602880	+	0.797832i	$0.705980\pi$
$662$	0	0
$663$	−120.000	−4.66041
$664$	−45.0000	−1.74634
$665$	−24.0000	−0.930680
$666$	24.0000	0.929981
$667$	0	0
$668$	−6.00000	−0.232147
$669$	−51.0000	−1.97177
$670$	−14.0000	−0.540867
$671$	−33.0000	−1.27395
$672$	45.0000	1.73591
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	32.0000	1.23259
$675$	−9.00000	−0.346410
$676$	−12.0000	−0.461538
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	39.0000	1.49779
$679$	−18.0000	−0.690777
$680$	−48.0000	−1.84072
$681$	54.0000	2.06928
$682$	−6.00000	−0.229752
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	−24.0000	−0.917663
$685$	24.0000	0.916993
$686$	−15.0000	−0.572703
$687$	57.0000	2.17469
$688$	1.00000	0.0381246
$689$	−60.0000	−2.28582
$690$	30.0000	1.14208
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	5.00000	0.190071
$693$	54.0000	2.05129
$694$	−26.0000	−0.986947
$695$	−4.00000	−0.151729
$696$	0	0
$697$	80.0000	3.03022
$698$	4.00000	0.151402
$699$	9.00000	0.340411
$700$	3.00000	0.113389
$701$	1.00000	0.0377695	0.0188847	−	0.999822i	$-0.493988\pi$
$701$	1.00000	0.0377695	0.0188847	+	0.999822i	$0.493988\pi$
$702$	45.0000	1.69842
$703$	16.0000	0.603451
$704$	21.0000	0.791467
$705$	18.0000	0.677919
$706$	6.00000	0.225813
$707$	0	0
$708$	−36.0000	−1.35296
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	16.0000	0.600469
$711$	−24.0000	−0.900070
$712$	0	0
$713$	10.0000	0.374503
$714$	−72.0000	−2.69453
$715$	−30.0000	−1.12194
$716$	−12.0000	−0.448461
$717$	72.0000	2.68889
$718$	−34.0000	−1.26887
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	12.0000	0.447214
$721$	6.00000	0.223452
$722$	−3.00000	−0.111648
$723$	−30.0000	−1.11571
$724$	16.0000	0.594635
$725$	0	0
$726$	−6.00000	−0.222681
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	−45.0000	−1.66781
$729$	−27.0000	−1.00000
$730$	12.0000	0.444140
$731$	8.00000	0.295891
$732$	33.0000	1.21972
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	17.0000	0.627481
$735$	−12.0000	−0.442627
$736$	−25.0000	−0.921512
$737$	21.0000	0.773545
$738$	−60.0000	−2.20863
$739$	−19.0000	−0.698926	−0.349463	−	0.936950i	$-0.613636\pi$
$739$	−19.0000	−0.698926	−0.349463	+	0.936950i	$0.613636\pi$
$740$	8.00000	0.294086
$741$	60.0000	2.20416
$742$	−36.0000	−1.32160
$743$	14.0000	0.513610	0.256805	−	0.966463i	$-0.417330\pi$
$743$	14.0000	0.513610	0.256805	+	0.966463i	$0.417330\pi$
$744$	18.0000	0.659912
$745$	36.0000	1.31894
$746$	−11.0000	−0.402739
$747$	90.0000	3.29293
$748$	24.0000	0.877527
$749$	54.0000	1.97312
$750$	36.0000	1.31453
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	3.00000	0.109399
$753$	6.00000	0.218652
$754$	0	0
$755$	−12.0000	−0.436725
$756$	−27.0000	−0.981981
$757$	28.0000	1.01768	0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000	1.01768	0.508839	+	0.860862i	$0.330075\pi$
$758$	−21.0000	−0.762754
$759$	−45.0000	−1.63340
$760$	24.0000	0.870572
$761$	−33.0000	−1.19625	−0.598125	−	0.801403i	$-0.704087\pi$
$761$	−33.0000	−1.19625	−0.598125	+	0.801403i	$0.704087\pi$
$762$	6.00000	0.217357
$763$	30.0000	1.08607
$764$	−8.00000	−0.289430
$765$	96.0000	3.47089
$766$	−16.0000	−0.578103
$767$	60.0000	2.16647
$768$	−51.0000	−1.84030
$769$	−24.0000	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$770$	−18.0000	−0.648675
$771$	81.0000	2.91714
$772$	−12.0000	−0.431889
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	−6.00000	−0.215666
$775$	2.00000	0.0718421
$776$	18.0000	0.646162
$777$	36.0000	1.29149
$778$	−26.0000	−0.932145
$779$	−40.0000	−1.43315
$780$	30.0000	1.07417
$781$	−24.0000	−0.858788
$782$	40.0000	1.43040
$783$	0	0
$784$	−2.00000	−0.0714286
$785$	48.0000	1.71319
$786$	45.0000	1.60510
$787$	−14.0000	−0.499046	−0.249523	−	0.968369i	$-0.580274\pi$
$787$	−14.0000	−0.499046	−0.249523	+	0.968369i	$0.580274\pi$
$788$	−3.00000	−0.106871
$789$	−15.0000	−0.534014
$790$	8.00000	0.284627
$791$	39.0000	1.38668
$792$	−54.0000	−1.91881
$793$	−55.0000	−1.95311
$794$	29.0000	1.02917
$795$	72.0000	2.55358
$796$	8.00000	0.283552
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	36.0000	1.27439
$799$	24.0000	0.849059
$800$	−5.00000	−0.176777
$801$	0	0
$802$	−3.00000	−0.105934
$803$	−18.0000	−0.635206
$804$	−21.0000	−0.740613
$805$	30.0000	1.05736
$806$	−10.0000	−0.352235
$807$	66.0000	2.32331
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	−18.0000	−0.632456
$811$	−13.0000	−0.456492	−0.228246	−	0.973604i	$-0.573299\pi$
$811$	−13.0000	−0.456492	−0.228246	+	0.973604i	$0.573299\pi$
$812$	0	0
$813$	75.0000	2.63036
$814$	12.0000	0.420600
$815$	−16.0000	−0.560456
$816$	24.0000	0.840168
$817$	−4.00000	−0.139942
$818$	−18.0000	−0.629355
$819$	90.0000	3.14485
$820$	−20.0000	−0.698430
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	−36.0000	−1.25564
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−6.00000	−0.209020
$825$	−9.00000	−0.313340
$826$	36.0000	1.25260
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	30.0000	1.04257
$829$	−36.0000	−1.25033	−0.625166	−	0.780492i	$-0.714969\pi$
$829$	−36.0000	−1.25033	−0.625166	+	0.780492i	$0.714969\pi$
$830$	−30.0000	−1.04132
$831$	−24.0000	−0.832551
$832$	35.0000	1.21341
$833$	−16.0000	−0.554367
$834$	6.00000	0.207763
$835$	−12.0000	−0.415277
$836$	−12.0000	−0.415029
$837$	−18.0000	−0.622171
$838$	0	0
$839$	−41.0000	−1.41548	−0.707739	−	0.706474i	$-0.750285\pi$
$839$	−41.0000	−1.41548	−0.707739	+	0.706474i	$0.750285\pi$
$840$	54.0000	1.86318
$841$	−29.0000	−1.00000
$842$	14.0000	0.482472
$843$	−99.0000	−3.40974
$844$	−2.00000	−0.0688428
$845$	−24.0000	−0.825625
$846$	−18.0000	−0.618853
$847$	−6.00000	−0.206162
$848$	12.0000	0.412082
$849$	60.0000	2.05919
$850$	8.00000	0.274398
$851$	−20.0000	−0.685591
$852$	24.0000	0.822226
$853$	−7.00000	−0.239675	−0.119838	−	0.992793i	$-0.538237\pi$
$853$	−7.00000	−0.239675	−0.119838	+	0.992793i	$0.538237\pi$
$854$	−33.0000	−1.12924
$855$	−48.0000	−1.64157
$856$	−54.0000	−1.84568
$857$	−7.00000	−0.239115	−0.119558	−	0.992827i	$-0.538148\pi$
$857$	−7.00000	−0.239115	−0.119558	+	0.992827i	$0.538148\pi$
$858$	45.0000	1.53627
$859$	−18.0000	−0.614152	−0.307076	−	0.951685i	$-0.599351\pi$
$859$	−18.0000	−0.614152	−0.307076	+	0.951685i	$0.599351\pi$
$860$	−2.00000	−0.0681994
$861$	−90.0000	−3.06719
$862$	10.0000	0.340601
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	45.0000	1.53093
$865$	10.0000	0.340010
$866$	2.00000	0.0679628
$867$	141.000	4.78861
$868$	6.00000	0.203653
$869$	−12.0000	−0.407072
$870$	0	0
$871$	35.0000	1.18593
$872$	−30.0000	−1.01593
$873$	−36.0000	−1.21842
$874$	−20.0000	−0.676510
$875$	36.0000	1.21702
$876$	18.0000	0.608164
$877$	46.0000	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$877$	46.0000	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$878$	36.0000	1.21494
$879$	3.00000	0.101187
$880$	6.00000	0.202260
$881$	10.0000	0.336909	0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000	0.336909	0.168454	+	0.985709i	$0.446122\pi$
$882$	12.0000	0.404061
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	40.0000	1.34535
$885$	−72.0000	−2.42025
$886$	35.0000	1.17585
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−36.0000	−1.20808
$889$	6.00000	0.201234
$890$	0	0
$891$	27.0000	0.904534
$892$	17.0000	0.569202
$893$	−12.0000	−0.401565
$894$	−54.0000	−1.80603
$895$	−24.0000	−0.802232
$896$	−9.00000	−0.300669
$897$	−75.0000	−2.50418
$898$	6.00000	0.200223
$899$	0	0
$900$	6.00000	0.200000
$901$	96.0000	3.19822
$902$	−30.0000	−0.998891
$903$	−9.00000	−0.299501
$904$	−39.0000	−1.29712
$905$	32.0000	1.06372
$906$	18.0000	0.598010
$907$	−46.0000	−1.52740	−0.763702	−	0.645568i	$-0.776621\pi$
$907$	−46.0000	−1.52740	−0.763702	+	0.645568i	$0.776621\pi$
$908$	−18.0000	−0.597351
$909$	0	0
$910$	−30.0000	−0.994490
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	−12.0000	−0.397360
$913$	45.0000	1.48928
$914$	26.0000	0.860004
$915$	66.0000	2.18189
$916$	−19.0000	−0.627778
$917$	45.0000	1.48603
$918$	−72.0000	−2.37635
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	−30.0000	−0.989071
$921$	−42.0000	−1.38395
$922$	8.00000	0.263466
$923$	−40.0000	−1.31662
$924$	−27.0000	−0.888235
$925$	−4.00000	−0.131519
$926$	−35.0000	−1.15017
$927$	12.0000	0.394132
$928$	0	0
$929$	24.0000	0.787414	0.393707	−	0.919236i	$-0.371192\pi$
$929$	24.0000	0.787414	0.393707	+	0.919236i	$0.371192\pi$
$930$	12.0000	0.393496
$931$	8.00000	0.262189
$932$	−3.00000	−0.0982683
$933$	−60.0000	−1.96431
$934$	12.0000	0.392652
$935$	48.0000	1.56977
$936$	−90.0000	−2.94174
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	21.0000	0.685674
$939$	−24.0000	−0.783210
$940$	−6.00000	−0.195698
$941$	−31.0000	−1.01057	−0.505286	−	0.862952i	$-0.668613\pi$
$941$	−31.0000	−1.01057	−0.505286	+	0.862952i	$0.668613\pi$
$942$	−72.0000	−2.34589
$943$	50.0000	1.62822
$944$	−12.0000	−0.390567
$945$	−54.0000	−1.75662
$946$	−3.00000	−0.0975384
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	12.0000	0.389742
$949$	−30.0000	−0.973841
$950$	−4.00000	−0.129777
$951$	−51.0000	−1.65379
$952$	72.0000	2.33353
$953$	61.0000	1.97598	0.987992	−	0.154506i	$-0.0493785\pi$
$953$	61.0000	1.97598	0.987992	+	0.154506i	$0.0493785\pi$
$954$	−72.0000	−2.33109
$955$	−16.0000	−0.517748
$956$	−24.0000	−0.776215
$957$	0	0
$958$	10.0000	0.323085
$959$	−36.0000	−1.16250
$960$	−42.0000	−1.35554
$961$	−27.0000	−0.870968
$962$	20.0000	0.644826
$963$	108.000	3.48025
$964$	10.0000	0.322078
$965$	−24.0000	−0.772587
$966$	−45.0000	−1.44785
$967$	20.0000	0.643157	0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000	0.643157	0.321578	+	0.946883i	$0.395787\pi$
$968$	6.00000	0.192847
$969$	−96.0000	−3.08396
$970$	12.0000	0.385297
$971$	−27.0000	−0.866471	−0.433236	−	0.901281i	$-0.642628\pi$
$971$	−27.0000	−0.866471	−0.433236	+	0.901281i	$0.642628\pi$
$972$	0	0
$973$	6.00000	0.192351
$974$	−8.00000	−0.256337
$975$	−15.0000	−0.480384
$976$	11.0000	0.352101
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	24.0000	0.767435
$979$	0	0
$980$	4.00000	0.127775
$981$	60.0000	1.91565
$982$	2.00000	0.0638226
$983$	20.0000	0.637901	0.318950	−	0.947771i	$-0.396670\pi$
$983$	20.0000	0.637901	0.318950	+	0.947771i	$0.396670\pi$
$984$	90.0000	2.86910
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−27.0000	−0.859419
$988$	−20.0000	−0.636285
$989$	5.00000	0.158991
$990$	−36.0000	−1.14416
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	−10.0000	−0.317500
$993$	0	0
$994$	−24.0000	−0.761234
$995$	16.0000	0.507234
$996$	−45.0000	−1.42588
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	42.0000	1.32949
$999$	36.0000	1.13899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	503.2.a.c.1.1	✓	1
3.2	odd	2		4527.2.a.d.1.1		1
4.3	odd	2		8048.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.c.1.1	✓	1	1.1	even	1	trivial
4527.2.a.d.1.1		1	3.2	odd	2
8048.2.a.a.1.1		1	4.3	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:	\( N \)	\(=\)	\( 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	503.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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