LMFDB - Embedded newform 5586.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5586,2,Mod(1,5586)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5586, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5586.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5586.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:	$44.6044345691$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 798)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5586.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -3.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} +3.00000 q^{20} -3.00000 q^{22} -6.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} -3.00000 q^{29} +3.00000 q^{30} +1.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +1.00000 q^{38} -4.00000 q^{39} -3.00000 q^{40} +6.00000 q^{41} +8.00000 q^{43} +3.00000 q^{44} +3.00000 q^{45} +6.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} +6.00000 q^{51} +4.00000 q^{52} -3.00000 q^{53} +1.00000 q^{54} +9.00000 q^{55} +1.00000 q^{57} +3.00000 q^{58} +3.00000 q^{59} -3.00000 q^{60} -2.00000 q^{61} -1.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} +3.00000 q^{66} +2.00000 q^{67} -6.00000 q^{68} +6.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} -2.00000 q^{73} -2.00000 q^{74} -4.00000 q^{75} -1.00000 q^{76} +4.00000 q^{78} -1.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +9.00000 q^{83} -18.0000 q^{85} -8.00000 q^{86} +3.00000 q^{87} -3.00000 q^{88} +12.0000 q^{89} -3.00000 q^{90} -6.00000 q^{92} -1.00000 q^{93} -6.00000 q^{94} -3.00000 q^{95} +1.00000 q^{96} -17.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.00000	−0.948683
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−3.00000	−0.774597
$16$	1.00000	0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	3.00000	0.670820
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	−4.00000	−0.784465
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	3.00000	0.547723
$31$	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
$33$	−3.00000	−0.522233
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	1.00000	0.162221
$39$	−4.00000	−0.640513
$40$	−3.00000	−0.474342
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	3.00000	0.452267
$45$	3.00000	0.447214
$46$	6.00000	0.884652
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	6.00000	0.840168
$52$	4.00000	0.554700
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	1.00000	0.136083
$55$	9.00000	1.21356
$56$	0	0
$57$	1.00000	0.132453
$58$	3.00000	0.393919
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	−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	$0.540868\pi$
$62$	−1.00000	−0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	3.00000	0.369274
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−6.00000	−0.727607
$69$	6.00000	0.722315
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−2.00000	−0.232495
$75$	−4.00000	−0.461880
$76$	−1.00000	−0.114708
$77$	0	0
$78$	4.00000	0.452911
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	−18.0000	−1.95237
$86$	−8.00000	−0.862662
$87$	3.00000	0.321634
$88$	−3.00000	−0.319801
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−1.00000	−0.103695
$94$	−6.00000	−0.618853
$95$	−3.00000	−0.307794
$96$	1.00000	0.102062
$97$	−17.0000	−1.72609	−0.863044	−	0.505128i	$-0.831445\pi$
$97$	−17.0000	−1.72609	−0.863044	+	0.505128i	$0.831445\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	4.00000	0.400000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−6.00000	−0.594089
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	3.00000	0.291386
$107$	15.0000	1.45010	0.725052	−	0.688694i	$-0.241816\pi$
$107$	15.0000	1.45010	0.725052	+	0.688694i	$0.241816\pi$
$108$	−1.00000	−0.0962250
$109$	8.00000	0.766261	0.383131	−	0.923694i	$-0.374846\pi$
$109$	8.00000	0.766261	0.383131	+	0.923694i	$0.374846\pi$
$110$	−9.00000	−0.858116
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	−1.00000	−0.0936586
$115$	−18.0000	−1.67851
$116$	−3.00000	−0.278543
$117$	4.00000	0.369800
$118$	−3.00000	−0.276172
$119$	0	0
$120$	3.00000	0.273861
$121$	−2.00000	−0.181818
$122$	2.00000	0.181071
$123$	−6.00000	−0.541002
$124$	1.00000	0.0898027
$125$	−3.00000	−0.268328
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	−12.0000	−1.05247
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	−3.00000	−0.261116
$133$	0	0
$134$	−2.00000	−0.172774
$135$	−3.00000	−0.258199
$136$	6.00000	0.514496
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	−6.00000	−0.510754
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−12.0000	−1.00702
$143$	12.0000	1.00349
$144$	1.00000	0.0833333
$145$	−9.00000	−0.747409
$146$	2.00000	0.165521
$147$	0	0
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	4.00000	0.326599
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	1.00000	0.0811107
$153$	−6.00000	−0.485071
$154$	0	0
$155$	3.00000	0.240966
$156$	−4.00000	−0.320256
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	1.00000	0.0795557
$159$	3.00000	0.237915
$160$	−3.00000	−0.237171
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	6.00000	0.468521
$165$	−9.00000	−0.700649
$166$	−9.00000	−0.698535
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	18.0000	1.38054
$171$	−1.00000	−0.0764719
$172$	8.00000	0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−3.00000	−0.227429
$175$	0	0
$176$	3.00000	0.226134
$177$	−3.00000	−0.225494
$178$	−12.0000	−0.899438
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	3.00000	0.223607
$181$	16.0000	1.18927	0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000	1.18927	0.594635	+	0.803996i	$0.297296\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	6.00000	0.442326
$185$	6.00000	0.441129
$186$	1.00000	0.0733236
$187$	−18.0000	−1.31629
$188$	6.00000	0.437595
$189$	0	0
$190$	3.00000	0.217643
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	17.0000	1.22053
$195$	−12.0000	−0.859338
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−3.00000	−0.213201
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−4.00000	−0.282843
$201$	−2.00000	−0.141069
$202$	6.00000	0.422159
$203$	0	0
$204$	6.00000	0.420084
$205$	18.0000	1.25717
$206$	−16.0000	−1.11477
$207$	−6.00000	−0.417029
$208$	4.00000	0.277350
$209$	−3.00000	−0.207514
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−3.00000	−0.206041
$213$	−12.0000	−0.822226
$214$	−15.0000	−1.02538
$215$	24.0000	1.63679
$216$	1.00000	0.0680414
$217$	0	0
$218$	−8.00000	−0.541828
$219$	2.00000	0.135147
$220$	9.00000	0.606780
$221$	−24.0000	−1.61441
$222$	2.00000	0.134231
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	6.00000	0.399114
$227$	−15.0000	−0.995585	−0.497792	−	0.867296i	$-0.665856\pi$
$227$	−15.0000	−0.995585	−0.497792	+	0.867296i	$0.665856\pi$
$228$	1.00000	0.0662266
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	18.0000	1.18688
$231$	0	0
$232$	3.00000	0.196960
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	−4.00000	−0.261488
$235$	18.0000	1.17419
$236$	3.00000	0.195283
$237$	1.00000	0.0649570
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−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	$0.551485\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	6.00000	0.382546
$247$	−4.00000	−0.254514
$248$	−1.00000	−0.0635001
$249$	−9.00000	−0.570352
$250$	3.00000	0.189737
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	0	0
$253$	−18.0000	−1.13165
$254$	−17.0000	−1.06667
$255$	18.0000	1.12720
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	8.00000	0.498058
$259$	0	0
$260$	12.0000	0.744208
$261$	−3.00000	−0.185695
$262$	−21.0000	−1.29738
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	3.00000	0.184637
$265$	−9.00000	−0.552866
$266$	0	0
$267$	−12.0000	−0.734388
$268$	2.00000	0.122169
$269$	27.0000	1.64622	0.823110	−	0.567883i	$-0.192237\pi$
$269$	27.0000	1.64622	0.823110	+	0.567883i	$0.192237\pi$
$270$	3.00000	0.182574
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	18.0000	1.08742
$275$	12.0000	0.723627
$276$	6.00000	0.361158
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−16.0000	−0.959616
$279$	1.00000	0.0598684
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	6.00000	0.357295
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	12.0000	0.712069
$285$	3.00000	0.177705
$286$	−12.0000	−0.709575
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	19.0000	1.11765
$290$	9.00000	0.528498
$291$	17.0000	0.996558
$292$	−2.00000	−0.117041
$293$	15.0000	0.876309	0.438155	−	0.898900i	$-0.355632\pi$
$293$	15.0000	0.876309	0.438155	+	0.898900i	$0.355632\pi$
$294$	0	0
$295$	9.00000	0.524000
$296$	−2.00000	−0.116248
$297$	−3.00000	−0.174078
$298$	−6.00000	−0.347571
$299$	−24.0000	−1.38796
$300$	−4.00000	−0.230940
$301$	0	0
$302$	1.00000	0.0575435
$303$	6.00000	0.344691
$304$	−1.00000	−0.0573539
$305$	−6.00000	−0.343559
$306$	6.00000	0.342997
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	−3.00000	−0.170389
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	4.00000	0.226455
$313$	−17.0000	−0.960897	−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000	−0.960897	−0.480448	+	0.877023i	$0.659526\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−1.00000	−0.0562544
$317$	9.00000	0.505490	0.252745	−	0.967533i	$-0.418667\pi$
$317$	9.00000	0.505490	0.252745	+	0.967533i	$0.418667\pi$
$318$	−3.00000	−0.168232
$319$	−9.00000	−0.503903
$320$	3.00000	0.167705
$321$	−15.0000	−0.837218
$322$	0	0
$323$	6.00000	0.333849
$324$	1.00000	0.0555556
$325$	16.0000	0.887520
$326$	4.00000	0.221540
$327$	−8.00000	−0.442401
$328$	−6.00000	−0.331295
$329$	0	0
$330$	9.00000	0.495434
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	9.00000	0.493939
$333$	2.00000	0.109599
$334$	12.0000	0.656611
$335$	6.00000	0.327815
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	−3.00000	−0.163178
$339$	6.00000	0.325875
$340$	−18.0000	−0.976187
$341$	3.00000	0.162459
$342$	1.00000	0.0540738
$343$	0	0
$344$	−8.00000	−0.431331
$345$	18.0000	0.969087
$346$	−6.00000	−0.322562
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	3.00000	0.160817
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	−3.00000	−0.159901
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	3.00000	0.159448
$355$	36.0000	1.91068
$356$	12.0000	0.635999
$357$	0	0
$358$	24.0000	1.26844
$359$	−18.0000	−0.950004	−0.475002	−	0.879985i	$-0.657553\pi$
$359$	−18.0000	−0.950004	−0.475002	+	0.879985i	$0.657553\pi$
$360$	−3.00000	−0.158114
$361$	1.00000	0.0526316
$362$	−16.0000	−0.840941
$363$	2.00000	0.104973
$364$	0	0
$365$	−6.00000	−0.314054
$366$	−2.00000	−0.104542
$367$	37.0000	1.93138	0.965692	−	0.259690i	$-0.0836203\pi$
$367$	37.0000	1.93138	0.965692	+	0.259690i	$0.0836203\pi$
$368$	−6.00000	−0.312772
$369$	6.00000	0.312348
$370$	−6.00000	−0.311925
$371$	0	0
$372$	−1.00000	−0.0518476
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	18.0000	0.930758
$375$	3.00000	0.154919
$376$	−6.00000	−0.309426
$377$	−12.0000	−0.618031
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	−3.00000	−0.153897
$381$	−17.0000	−0.870936
$382$	12.0000	0.613973
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−11.0000	−0.559885
$387$	8.00000	0.406663
$388$	−17.0000	−0.863044
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	12.0000	0.607644
$391$	36.0000	1.82060
$392$	0	0
$393$	−21.0000	−1.05931
$394$	6.00000	0.302276
$395$	−3.00000	−0.150946
$396$	3.00000	0.150756
$397$	4.00000	0.200754	0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000	0.200754	0.100377	+	0.994949i	$0.467995\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	4.00000	0.200000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	2.00000	0.0997509
$403$	4.00000	0.199254
$404$	−6.00000	−0.298511
$405$	3.00000	0.149071
$406$	0	0
$407$	6.00000	0.297409
$408$	−6.00000	−0.297044
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	−18.0000	−0.888957
$411$	18.0000	0.887875
$412$	16.0000	0.788263
$413$	0	0
$414$	6.00000	0.294884
$415$	27.0000	1.32538
$416$	−4.00000	−0.196116
$417$	−16.0000	−0.783523
$418$	3.00000	0.146735
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−20.0000	−0.973585
$423$	6.00000	0.291730
$424$	3.00000	0.145693
$425$	−24.0000	−1.16417
$426$	12.0000	0.581402
$427$	0	0
$428$	15.0000	0.725052
$429$	−12.0000	−0.579365
$430$	−24.0000	−1.15738
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	−1.00000	−0.0481125
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	8.00000	0.383131
$437$	6.00000	0.287019
$438$	−2.00000	−0.0955637
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	−9.00000	−0.429058
$441$	0	0
$442$	24.0000	1.14156
$443$	15.0000	0.712672	0.356336	−	0.934358i	$-0.384026\pi$
$443$	15.0000	0.712672	0.356336	+	0.934358i	$0.384026\pi$
$444$	−2.00000	−0.0949158
$445$	36.0000	1.70656
$446$	−19.0000	−0.899676
$447$	−6.00000	−0.283790
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	−4.00000	−0.188562
$451$	18.0000	0.847587
$452$	−6.00000	−0.282216
$453$	1.00000	0.0469841
$454$	15.0000	0.703985
$455$	0	0
$456$	−1.00000	−0.0468293
$457$	−25.0000	−1.16945	−0.584725	−	0.811231i	$-0.698798\pi$
$457$	−25.0000	−1.16945	−0.584725	+	0.811231i	$0.698798\pi$
$458$	−28.0000	−1.30835
$459$	6.00000	0.280056
$460$	−18.0000	−0.839254
$461$	42.0000	1.95614	0.978068	−	0.208288i	$-0.0667892\pi$
$461$	42.0000	1.95614	0.978068	+	0.208288i	$0.0667892\pi$
$462$	0	0
$463$	32.0000	1.48717	0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000	1.48717	0.743583	+	0.668644i	$0.233125\pi$
$464$	−3.00000	−0.139272
$465$	−3.00000	−0.139122
$466$	24.0000	1.11178
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	4.00000	0.184900
$469$	0	0
$470$	−18.0000	−0.830278
$471$	−4.00000	−0.184310
$472$	−3.00000	−0.138086
$473$	24.0000	1.10352
$474$	−1.00000	−0.0459315
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−3.00000	−0.137361
$478$	24.0000	1.09773
$479$	−18.0000	−0.822441	−0.411220	−	0.911536i	$-0.634897\pi$
$479$	−18.0000	−0.822441	−0.411220	+	0.911536i	$0.634897\pi$
$480$	3.00000	0.136931
$481$	8.00000	0.364769
$482$	5.00000	0.227744
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−51.0000	−2.31579
$486$	1.00000	0.0453609
$487$	−43.0000	−1.94852	−0.974258	−	0.225436i	$-0.927619\pi$
$487$	−43.0000	−1.94852	−0.974258	+	0.225436i	$0.927619\pi$
$488$	2.00000	0.0905357
$489$	4.00000	0.180886
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	−6.00000	−0.270501
$493$	18.0000	0.810679
$494$	4.00000	0.179969
$495$	9.00000	0.404520
$496$	1.00000	0.0449013
$497$	0	0
$498$	9.00000	0.403300
$499$	26.0000	1.16392	0.581960	−	0.813217i	$-0.302286\pi$
$499$	26.0000	1.16392	0.581960	+	0.813217i	$0.302286\pi$
$500$	−3.00000	−0.134164
$501$	12.0000	0.536120
$502$	−21.0000	−0.937276
$503$	−30.0000	−1.33763	−0.668817	−	0.743427i	$-0.733199\pi$
$503$	−30.0000	−1.33763	−0.668817	+	0.743427i	$0.733199\pi$
$504$	0	0
$505$	−18.0000	−0.800989
$506$	18.0000	0.800198
$507$	−3.00000	−0.133235
$508$	17.0000	0.754253
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	−18.0000	−0.797053
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	−18.0000	−0.793946
$515$	48.0000	2.11513
$516$	−8.00000	−0.352180
$517$	18.0000	0.791639
$518$	0	0
$519$	−6.00000	−0.263371
$520$	−12.0000	−0.526235
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	3.00000	0.131306
$523$	−2.00000	−0.0874539	−0.0437269	−	0.999044i	$-0.513923\pi$
$523$	−2.00000	−0.0874539	−0.0437269	+	0.999044i	$0.513923\pi$
$524$	21.0000	0.917389
$525$	0	0
$526$	12.0000	0.523225
$527$	−6.00000	−0.261364
$528$	−3.00000	−0.130558
$529$	13.0000	0.565217
$530$	9.00000	0.390935
$531$	3.00000	0.130189
$532$	0	0
$533$	24.0000	1.03956
$534$	12.0000	0.519291
$535$	45.0000	1.94552
$536$	−2.00000	−0.0863868
$537$	24.0000	1.03568
$538$	−27.0000	−1.16405
$539$	0	0
$540$	−3.00000	−0.129099
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−7.00000	−0.300676
$543$	−16.0000	−0.686626
$544$	6.00000	0.257248
$545$	24.0000	1.02805
$546$	0	0
$547$	38.0000	1.62476	0.812381	−	0.583127i	$-0.198171\pi$
$547$	38.0000	1.62476	0.812381	+	0.583127i	$0.198171\pi$
$548$	−18.0000	−0.768922
$549$	−2.00000	−0.0853579
$550$	−12.0000	−0.511682
$551$	3.00000	0.127804
$552$	−6.00000	−0.255377
$553$	0	0
$554$	22.0000	0.934690
$555$	−6.00000	−0.254686
$556$	16.0000	0.678551
$557$	45.0000	1.90671	0.953356	−	0.301849i	$-0.0976040\pi$
$557$	45.0000	1.90671	0.953356	+	0.301849i	$0.0976040\pi$
$558$	−1.00000	−0.0423334
$559$	32.0000	1.35346
$560$	0	0
$561$	18.0000	0.759961
$562$	18.0000	0.759284
$563$	39.0000	1.64365	0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000	1.64365	0.821827	+	0.569737i	$0.192955\pi$
$564$	−6.00000	−0.252646
$565$	−18.0000	−0.757266
$566$	−16.0000	−0.672530
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	−3.00000	−0.125656
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	12.0000	0.501745
$573$	12.0000	0.501307
$574$	0	0
$575$	−24.0000	−1.00087
$576$	1.00000	0.0416667
$577$	13.0000	0.541197	0.270599	−	0.962692i	$-0.412778\pi$
$577$	13.0000	0.541197	0.270599	+	0.962692i	$0.412778\pi$
$578$	−19.0000	−0.790296
$579$	−11.0000	−0.457144
$580$	−9.00000	−0.373705
$581$	0	0
$582$	−17.0000	−0.704673
$583$	−9.00000	−0.372742
$584$	2.00000	0.0827606
$585$	12.0000	0.496139
$586$	−15.0000	−0.619644
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	0	0
$589$	−1.00000	−0.0412043
$590$	−9.00000	−0.370524
$591$	6.00000	0.246807
$592$	2.00000	0.0821995
$593$	−12.0000	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$593$	−12.0000	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	6.00000	0.245770
$597$	8.00000	0.327418
$598$	24.0000	0.981433
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	4.00000	0.163299
$601$	43.0000	1.75401	0.877003	−	0.480484i	$-0.159539\pi$
$601$	43.0000	1.75401	0.877003	+	0.480484i	$0.159539\pi$
$602$	0	0
$603$	2.00000	0.0814463
$604$	−1.00000	−0.0406894
$605$	−6.00000	−0.243935
$606$	−6.00000	−0.243733
$607$	−41.0000	−1.66414	−0.832069	−	0.554672i	$-0.812844\pi$
$607$	−41.0000	−1.66414	−0.832069	+	0.554672i	$0.812844\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	6.00000	0.242933
$611$	24.0000	0.970936
$612$	−6.00000	−0.242536
$613$	38.0000	1.53481	0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000	1.53481	0.767403	+	0.641165i	$0.221549\pi$
$614$	26.0000	1.04927
$615$	−18.0000	−0.725830
$616$	0	0
$617$	12.0000	0.483102	0.241551	−	0.970388i	$-0.422344\pi$
$617$	12.0000	0.483102	0.241551	+	0.970388i	$0.422344\pi$
$618$	16.0000	0.643614
$619$	46.0000	1.84890	0.924448	−	0.381308i	$-0.124526\pi$
$619$	46.0000	1.84890	0.924448	+	0.381308i	$0.124526\pi$
$620$	3.00000	0.120483
$621$	6.00000	0.240772
$622$	18.0000	0.721734
$623$	0	0
$624$	−4.00000	−0.160128
$625$	−29.0000	−1.16000
$626$	17.0000	0.679457
$627$	3.00000	0.119808
$628$	4.00000	0.159617
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−13.0000	−0.517522	−0.258761	−	0.965941i	$-0.583314\pi$
$631$	−13.0000	−0.517522	−0.258761	+	0.965941i	$0.583314\pi$
$632$	1.00000	0.0397779
$633$	−20.0000	−0.794929
$634$	−9.00000	−0.357436
$635$	51.0000	2.02387
$636$	3.00000	0.118958
$637$	0	0
$638$	9.00000	0.356313
$639$	12.0000	0.474713
$640$	−3.00000	−0.118585
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	15.0000	0.592003
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	0	0
$645$	−24.0000	−0.944999
$646$	−6.00000	−0.236067
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−1.00000	−0.0392837
$649$	9.00000	0.353281
$650$	−16.0000	−0.627572
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	8.00000	0.312825
$655$	63.0000	2.46161
$656$	6.00000	0.234261
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	−9.00000	−0.350325
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	−26.0000	−1.01052
$663$	24.0000	0.932083
$664$	−9.00000	−0.349268
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	18.0000	0.696963
$668$	−12.0000	−0.464294
$669$	−19.0000	−0.734582
$670$	−6.00000	−0.231800
$671$	−6.00000	−0.231627
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	13.0000	0.500741
$675$	−4.00000	−0.153960
$676$	3.00000	0.115385
$677$	−3.00000	−0.115299	−0.0576497	−	0.998337i	$-0.518361\pi$
$677$	−3.00000	−0.115299	−0.0576497	+	0.998337i	$0.518361\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	18.0000	0.690268
$681$	15.0000	0.574801
$682$	−3.00000	−0.114876
$683$	39.0000	1.49229	0.746147	−	0.665782i	$-0.231902\pi$
$683$	39.0000	1.49229	0.746147	+	0.665782i	$0.231902\pi$
$684$	−1.00000	−0.0382360
$685$	−54.0000	−2.06323
$686$	0	0
$687$	−28.0000	−1.06827
$688$	8.00000	0.304997
$689$	−12.0000	−0.457164
$690$	−18.0000	−0.685248
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	0	0
$695$	48.0000	1.82074
$696$	−3.00000	−0.113715
$697$	−36.0000	−1.36360
$698$	−16.0000	−0.605609
$699$	24.0000	0.907763
$700$	0	0
$701$	51.0000	1.92624	0.963122	−	0.269066i	$-0.0867150\pi$
$701$	51.0000	1.92624	0.963122	+	0.269066i	$0.0867150\pi$
$702$	4.00000	0.150970
$703$	−2.00000	−0.0754314
$704$	3.00000	0.113067
$705$	−18.0000	−0.677919
$706$	6.00000	0.225813
$707$	0	0
$708$	−3.00000	−0.112747
$709$	−40.0000	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$709$	−40.0000	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$710$	−36.0000	−1.35106
$711$	−1.00000	−0.0375029
$712$	−12.0000	−0.449719
$713$	−6.00000	−0.224702
$714$	0	0
$715$	36.0000	1.34632
$716$	−24.0000	−0.896922
$717$	24.0000	0.896296
$718$	18.0000	0.671754
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	−1.00000	−0.0372161
$723$	5.00000	0.185952
$724$	16.0000	0.594635
$725$	−12.0000	−0.445669
$726$	−2.00000	−0.0742270
$727$	19.0000	0.704671	0.352335	−	0.935874i	$-0.385388\pi$
$727$	19.0000	0.704671	0.352335	+	0.935874i	$0.385388\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	6.00000	0.222070
$731$	−48.0000	−1.77534
$732$	2.00000	0.0739221
$733$	16.0000	0.590973	0.295487	−	0.955347i	$-0.404518\pi$
$733$	16.0000	0.590973	0.295487	+	0.955347i	$0.404518\pi$
$734$	−37.0000	−1.36569
$735$	0	0
$736$	6.00000	0.221163
$737$	6.00000	0.221013
$738$	−6.00000	−0.220863
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	6.00000	0.220564
$741$	4.00000	0.146944
$742$	0	0
$743$	6.00000	0.220119	0.110059	−	0.993925i	$-0.464896\pi$
$743$	6.00000	0.220119	0.110059	+	0.993925i	$0.464896\pi$
$744$	1.00000	0.0366618
$745$	18.0000	0.659469
$746$	4.00000	0.146450
$747$	9.00000	0.329293
$748$	−18.0000	−0.658145
$749$	0	0
$750$	−3.00000	−0.109545
$751$	−31.0000	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$751$	−31.0000	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$752$	6.00000	0.218797
$753$	−21.0000	−0.765283
$754$	12.0000	0.437014
$755$	−3.00000	−0.109181
$756$	0	0
$757$	−40.0000	−1.45382	−0.726912	−	0.686730i	$-0.759045\pi$
$757$	−40.0000	−1.45382	−0.726912	+	0.686730i	$0.759045\pi$
$758$	−8.00000	−0.290573
$759$	18.0000	0.653359
$760$	3.00000	0.108821
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	17.0000	0.615845
$763$	0	0
$764$	−12.0000	−0.434145
$765$	−18.0000	−0.650791
$766$	6.00000	0.216789
$767$	12.0000	0.433295
$768$	−1.00000	−0.0360844
$769$	31.0000	1.11789	0.558944	−	0.829205i	$-0.311207\pi$
$769$	31.0000	1.11789	0.558944	+	0.829205i	$0.311207\pi$
$770$	0	0
$771$	−18.0000	−0.648254
$772$	11.0000	0.395899
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−8.00000	−0.287554
$775$	4.00000	0.143684
$776$	17.0000	0.610264
$777$	0	0
$778$	18.0000	0.645331
$779$	−6.00000	−0.214972
$780$	−12.0000	−0.429669
$781$	36.0000	1.28818
$782$	−36.0000	−1.28736
$783$	3.00000	0.107211
$784$	0	0
$785$	12.0000	0.428298
$786$	21.0000	0.749045
$787$	46.0000	1.63972	0.819861	−	0.572562i	$-0.194050\pi$
$787$	46.0000	1.63972	0.819861	+	0.572562i	$0.194050\pi$
$788$	−6.00000	−0.213741
$789$	12.0000	0.427211
$790$	3.00000	0.106735
$791$	0	0
$792$	−3.00000	−0.106600
$793$	−8.00000	−0.284088
$794$	−4.00000	−0.141955
$795$	9.00000	0.319197
$796$	−8.00000	−0.283552
$797$	−39.0000	−1.38145	−0.690725	−	0.723117i	$-0.742709\pi$
$797$	−39.0000	−1.38145	−0.690725	+	0.723117i	$0.742709\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	−4.00000	−0.141421
$801$	12.0000	0.423999
$802$	−24.0000	−0.847469
$803$	−6.00000	−0.211735
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	−4.00000	−0.140894
$807$	−27.0000	−0.950445
$808$	6.00000	0.211079
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	−3.00000	−0.105409
$811$	16.0000	0.561836	0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000	0.561836	0.280918	+	0.959732i	$0.409361\pi$
$812$	0	0
$813$	−7.00000	−0.245501
$814$	−6.00000	−0.210300
$815$	−12.0000	−0.420342
$816$	6.00000	0.210042
$817$	−8.00000	−0.279885
$818$	−19.0000	−0.664319
$819$	0	0
$820$	18.0000	0.628587
$821$	27.0000	0.942306	0.471153	−	0.882051i	$-0.343838\pi$
$821$	27.0000	0.942306	0.471153	+	0.882051i	$0.343838\pi$
$822$	−18.0000	−0.627822
$823$	32.0000	1.11545	0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000	1.11545	0.557725	+	0.830026i	$0.311674\pi$
$824$	−16.0000	−0.557386
$825$	−12.0000	−0.417786
$826$	0	0
$827$	33.0000	1.14752	0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000	1.14752	0.573761	+	0.819023i	$0.305484\pi$
$828$	−6.00000	−0.208514
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	−27.0000	−0.937184
$831$	22.0000	0.763172
$832$	4.00000	0.138675
$833$	0	0
$834$	16.0000	0.554035
$835$	−36.0000	−1.24583
$836$	−3.00000	−0.103757
$837$	−1.00000	−0.0345651
$838$	12.0000	0.414533
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	10.0000	0.344623
$843$	18.0000	0.619953
$844$	20.0000	0.688428
$845$	9.00000	0.309609
$846$	−6.00000	−0.206284
$847$	0	0
$848$	−3.00000	−0.103020
$849$	−16.0000	−0.549119
$850$	24.0000	0.823193
$851$	−12.0000	−0.411355
$852$	−12.0000	−0.411113
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	−15.0000	−0.512689
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	12.0000	0.409673
$859$	−26.0000	−0.887109	−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000	−0.887109	−0.443554	+	0.896248i	$0.646283\pi$
$860$	24.0000	0.818393
$861$	0	0
$862$	−6.00000	−0.204361
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	1.00000	0.0340207
$865$	18.0000	0.612018
$866$	2.00000	0.0679628
$867$	−19.0000	−0.645274
$868$	0	0
$869$	−3.00000	−0.101768
$870$	−9.00000	−0.305129
$871$	8.00000	0.271070
$872$	−8.00000	−0.270914
$873$	−17.0000	−0.575363
$874$	−6.00000	−0.202953
$875$	0	0
$876$	2.00000	0.0675737
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	35.0000	1.18119
$879$	−15.0000	−0.505937
$880$	9.00000	0.303390
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	2.00000	0.0673054	0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000	0.0673054	0.0336527	+	0.999434i	$0.489286\pi$
$884$	−24.0000	−0.807207
$885$	−9.00000	−0.302532
$886$	−15.0000	−0.503935
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	−36.0000	−1.20672
$891$	3.00000	0.100504
$892$	19.0000	0.636167
$893$	−6.00000	−0.200782
$894$	6.00000	0.200670
$895$	−72.0000	−2.40669
$896$	0	0
$897$	24.0000	0.801337
$898$	−30.0000	−1.00111
$899$	−3.00000	−0.100056
$900$	4.00000	0.133333
$901$	18.0000	0.599667
$902$	−18.0000	−0.599334
$903$	0	0
$904$	6.00000	0.199557
$905$	48.0000	1.59557
$906$	−1.00000	−0.0332228
$907$	−34.0000	−1.12895	−0.564476	−	0.825450i	$-0.690922\pi$
$907$	−34.0000	−1.12895	−0.564476	+	0.825450i	$0.690922\pi$
$908$	−15.0000	−0.497792
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−30.0000	−0.993944	−0.496972	−	0.867766i	$-0.665555\pi$
$911$	−30.0000	−0.993944	−0.496972	+	0.867766i	$0.665555\pi$
$912$	1.00000	0.0331133
$913$	27.0000	0.893570
$914$	25.0000	0.826927
$915$	6.00000	0.198354
$916$	28.0000	0.925146
$917$	0	0
$918$	−6.00000	−0.198030
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	18.0000	0.593442
$921$	26.0000	0.856729
$922$	−42.0000	−1.38320
$923$	48.0000	1.57994
$924$	0	0
$925$	8.00000	0.263038
$926$	−32.0000	−1.05159
$927$	16.0000	0.525509
$928$	3.00000	0.0984798
$929$	−24.0000	−0.787414	−0.393707	−	0.919236i	$-0.628808\pi$
$929$	−24.0000	−0.787414	−0.393707	+	0.919236i	$0.628808\pi$
$930$	3.00000	0.0983739
$931$	0	0
$932$	−24.0000	−0.786146
$933$	18.0000	0.589294
$934$	12.0000	0.392652
$935$	−54.0000	−1.76599
$936$	−4.00000	−0.130744
$937$	−47.0000	−1.53542	−0.767712	−	0.640796i	$-0.778605\pi$
$937$	−47.0000	−1.53542	−0.767712	+	0.640796i	$0.778605\pi$
$938$	0	0
$939$	17.0000	0.554774
$940$	18.0000	0.587095
$941$	45.0000	1.46696	0.733479	−	0.679712i	$-0.237895\pi$
$941$	45.0000	1.46696	0.733479	+	0.679712i	$0.237895\pi$
$942$	4.00000	0.130327
$943$	−36.0000	−1.17232
$944$	3.00000	0.0976417
$945$	0	0
$946$	−24.0000	−0.780307
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	1.00000	0.0324785
$949$	−8.00000	−0.259691
$950$	4.00000	0.129777
$951$	−9.00000	−0.291845
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	3.00000	0.0971286
$955$	−36.0000	−1.16493
$956$	−24.0000	−0.776215
$957$	9.00000	0.290929
$958$	18.0000	0.581554
$959$	0	0
$960$	−3.00000	−0.0968246
$961$	−30.0000	−0.967742
$962$	−8.00000	−0.257930
$963$	15.0000	0.483368
$964$	−5.00000	−0.161039
$965$	33.0000	1.06231
$966$	0	0
$967$	−7.00000	−0.225105	−0.112552	−	0.993646i	$-0.535903\pi$
$967$	−7.00000	−0.225105	−0.112552	+	0.993646i	$0.535903\pi$
$968$	2.00000	0.0642824
$969$	−6.00000	−0.192748
$970$	51.0000	1.63751
$971$	−3.00000	−0.0962746	−0.0481373	−	0.998841i	$-0.515328\pi$
$971$	−3.00000	−0.0962746	−0.0481373	+	0.998841i	$0.515328\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	43.0000	1.37781
$975$	−16.0000	−0.512410
$976$	−2.00000	−0.0640184
$977$	−54.0000	−1.72761	−0.863807	−	0.503824i	$-0.831926\pi$
$977$	−54.0000	−1.72761	−0.863807	+	0.503824i	$0.831926\pi$
$978$	−4.00000	−0.127906
$979$	36.0000	1.15056
$980$	0	0
$981$	8.00000	0.255420
$982$	33.0000	1.05307
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	6.00000	0.191273
$985$	−18.0000	−0.573528
$986$	−18.0000	−0.573237
$987$	0	0
$988$	−4.00000	−0.127257
$989$	−48.0000	−1.52631
$990$	−9.00000	−0.286039
$991$	11.0000	0.349427	0.174713	−	0.984619i	$-0.444100\pi$
$991$	11.0000	0.349427	0.174713	+	0.984619i	$0.444100\pi$
$992$	−1.00000	−0.0317500
$993$	−26.0000	−0.825085
$994$	0	0
$995$	−24.0000	−0.760851
$996$	−9.00000	−0.285176
$997$	−62.0000	−1.96356	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000	−1.96356	−0.981780	+	0.190022i	$0.939144\pi$
$998$	−26.0000	−0.823016
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5586.2.a.k.1.1		1
7.3	odd	6		798.2.j.f.457.1	✓	2
7.5	odd	6		798.2.j.f.571.1	yes	2
7.6	odd	2		5586.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.j.f.457.1	✓	2	7.3	odd	6
798.2.j.f.571.1	yes	2	7.5	odd	6
5586.2.a.k.1.1		1	1.1	even	1	trivial
5586.2.a.l.1.1		1	7.6	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:	\( N \)	\(=\)	\( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5586.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more