LMFDB - Embedded newform 186.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [186,2,Mod(1,186)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(186, 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("186.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 186 = 2 \cdot 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	186.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.48521747760$
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	$\chi$	$=$	186.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -5.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} -3.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} +1.00000 q^{30} +1.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} +3.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} -5.00000 q^{38} -1.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{42} -6.00000 q^{43} -3.00000 q^{44} +1.00000 q^{45} +4.00000 q^{46} -7.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} +3.00000 q^{51} -1.00000 q^{52} +14.0000 q^{53} +1.00000 q^{54} -3.00000 q^{55} -2.00000 q^{56} -5.00000 q^{57} +10.0000 q^{59} +1.00000 q^{60} +7.00000 q^{61} +1.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} -3.00000 q^{66} -7.00000 q^{67} +3.00000 q^{68} +4.00000 q^{69} -2.00000 q^{70} -3.00000 q^{71} +1.00000 q^{72} -6.00000 q^{73} -2.00000 q^{74} -4.00000 q^{75} -5.00000 q^{76} +6.00000 q^{77} -1.00000 q^{78} +15.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -1.00000 q^{83} -2.00000 q^{84} +3.00000 q^{85} -6.00000 q^{86} -3.00000 q^{88} +10.0000 q^{89} +1.00000 q^{90} +2.00000 q^{91} +4.00000 q^{92} +1.00000 q^{93} -7.00000 q^{94} -5.00000 q^{95} +1.00000 q^{96} +13.0000 q^{97} -3.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	−2.00000	−0.534522
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	1.00000	0.235702
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	−3.00000	−0.639602
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	−4.00000	−0.800000
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	1.00000	0.182574
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	−3.00000	−0.522233
$34$	3.00000	0.514496
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−5.00000	−0.811107
$39$	−1.00000	−0.160128
$40$	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	$0.450084\pi$
$42$	−2.00000	−0.308607
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−3.00000	−0.452267
$45$	1.00000	0.149071
$46$	4.00000	0.589768
$47$	−7.00000	−1.02105	−0.510527	−	0.859861i	$-0.670550\pi$
$47$	−7.00000	−1.02105	−0.510527	+	0.859861i	$0.670550\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	3.00000	0.420084
$52$	−1.00000	−0.138675
$53$	14.0000	1.92305	0.961524	−	0.274721i	$-0.0885855\pi$
$53$	14.0000	1.92305	0.961524	+	0.274721i	$0.0885855\pi$
$54$	1.00000	0.136083
$55$	−3.00000	−0.404520
$56$	−2.00000	−0.267261
$57$	−5.00000	−0.662266
$58$	0	0
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	1.00000	0.129099
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	1.00000	0.127000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	−3.00000	−0.369274
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	3.00000	0.363803
$69$	4.00000	0.481543
$70$	−2.00000	−0.239046
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	1.00000	0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.00000	−0.232495
$75$	−4.00000	−0.461880
$76$	−5.00000	−0.573539
$77$	6.00000	0.683763
$78$	−1.00000	−0.113228
$79$	15.0000	1.68763	0.843816	−	0.536633i	$-0.180304\pi$
$79$	15.0000	1.68763	0.843816	+	0.536633i	$0.180304\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−1.00000	−0.109764	−0.0548821	−	0.998493i	$-0.517478\pi$
$83$	−1.00000	−0.109764	−0.0548821	+	0.998493i	$0.517478\pi$
$84$	−2.00000	−0.218218
$85$	3.00000	0.325396
$86$	−6.00000	−0.646997
$87$	0	0
$88$	−3.00000	−0.319801
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	1.00000	0.105409
$91$	2.00000	0.209657
$92$	4.00000	0.417029
$93$	1.00000	0.103695
$94$	−7.00000	−0.721995
$95$	−5.00000	−0.512989
$96$	1.00000	0.102062
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	−3.00000	−0.303046
$99$	−3.00000	−0.301511
$100$	−4.00000	−0.400000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	3.00000	0.297044
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	−1.00000	−0.0980581
$105$	−2.00000	−0.195180
$106$	14.0000	1.35980
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	1.00000	0.0962250
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	−3.00000	−0.286039
$111$	−2.00000	−0.189832
$112$	−2.00000	−0.188982
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	−5.00000	−0.468293
$115$	4.00000	0.373002
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	10.0000	0.920575
$119$	−6.00000	−0.550019
$120$	1.00000	0.0912871
$121$	−2.00000	−0.181818
$122$	7.00000	0.633750
$123$	2.00000	0.180334
$124$	1.00000	0.0898027
$125$	−9.00000	−0.804984
$126$	−2.00000	−0.178174
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	1.00000	0.0883883
$129$	−6.00000	−0.528271
$130$	−1.00000	−0.0877058
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	−3.00000	−0.261116
$133$	10.0000	0.867110
$134$	−7.00000	−0.604708
$135$	1.00000	0.0860663
$136$	3.00000	0.257248
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	4.00000	0.340503
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	−2.00000	−0.169031
$141$	−7.00000	−0.589506
$142$	−3.00000	−0.251754
$143$	3.00000	0.250873
$144$	1.00000	0.0833333
$145$	0	0
$146$	−6.00000	−0.496564
$147$	−3.00000	−0.247436
$148$	−2.00000	−0.164399
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	−4.00000	−0.326599
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	−5.00000	−0.405554
$153$	3.00000	0.242536
$154$	6.00000	0.483494
$155$	1.00000	0.0803219
$156$	−1.00000	−0.0800641
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	15.0000	1.19334
$159$	14.0000	1.11027
$160$	1.00000	0.0790569
$161$	−8.00000	−0.630488
$162$	1.00000	0.0785674
$163$	−21.0000	−1.64485	−0.822423	−	0.568876i	$-0.807379\pi$
$163$	−21.0000	−1.64485	−0.822423	+	0.568876i	$0.807379\pi$
$164$	2.00000	0.156174
$165$	−3.00000	−0.233550
$166$	−1.00000	−0.0776151
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	−2.00000	−0.154303
$169$	−12.0000	−0.923077
$170$	3.00000	0.230089
$171$	−5.00000	−0.382360
$172$	−6.00000	−0.457496
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	0	0
$175$	8.00000	0.604743
$176$	−3.00000	−0.226134
$177$	10.0000	0.751646
$178$	10.0000	0.749532
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	1.00000	0.0745356
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	2.00000	0.148250
$183$	7.00000	0.517455
$184$	4.00000	0.294884
$185$	−2.00000	−0.147043
$186$	1.00000	0.0733236
$187$	−9.00000	−0.658145
$188$	−7.00000	−0.510527
$189$	−2.00000	−0.145479
$190$	−5.00000	−0.362738
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000	0.0721688
$193$	−21.0000	−1.51161	−0.755807	−	0.654795i	$-0.772755\pi$
$193$	−21.0000	−1.51161	−0.755807	+	0.654795i	$0.772755\pi$
$194$	13.0000	0.933346
$195$	−1.00000	−0.0716115
$196$	−3.00000	−0.214286
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	−3.00000	−0.213201
$199$	15.0000	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$199$	15.0000	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$200$	−4.00000	−0.282843
$201$	−7.00000	−0.493742
$202$	2.00000	0.140720
$203$	0	0
$204$	3.00000	0.210042
$205$	2.00000	0.139686
$206$	14.0000	0.975426
$207$	4.00000	0.278019
$208$	−1.00000	−0.0693375
$209$	15.0000	1.03757
$210$	−2.00000	−0.138013
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	14.0000	0.961524
$213$	−3.00000	−0.205557
$214$	−12.0000	−0.820303
$215$	−6.00000	−0.409197
$216$	1.00000	0.0680414
$217$	−2.00000	−0.135769
$218$	10.0000	0.677285
$219$	−6.00000	−0.405442
$220$	−3.00000	−0.202260
$221$	−3.00000	−0.201802
$222$	−2.00000	−0.134231
$223$	9.00000	0.602685	0.301342	−	0.953516i	$-0.402565\pi$
$223$	9.00000	0.602685	0.301342	+	0.953516i	$0.402565\pi$
$224$	−2.00000	−0.133631
$225$	−4.00000	−0.266667
$226$	14.0000	0.931266
$227$	−22.0000	−1.46019	−0.730096	−	0.683345i	$-0.760525\pi$
$227$	−22.0000	−1.46019	−0.730096	+	0.683345i	$0.760525\pi$
$228$	−5.00000	−0.331133
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	4.00000	0.263752
$231$	6.00000	0.394771
$232$	0	0
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	−1.00000	−0.0653720
$235$	−7.00000	−0.456630
$236$	10.0000	0.650945
$237$	15.0000	0.974355
$238$	−6.00000	−0.388922
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	1.00000	0.0645497
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−2.00000	−0.128565
$243$	1.00000	0.0641500
$244$	7.00000	0.448129
$245$	−3.00000	−0.191663
$246$	2.00000	0.127515
$247$	5.00000	0.318142
$248$	1.00000	0.0635001
$249$	−1.00000	−0.0633724
$250$	−9.00000	−0.569210
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	−2.00000	−0.125988
$253$	−12.0000	−0.754434
$254$	−12.0000	−0.752947
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	28.0000	1.74659	0.873296	−	0.487190i	$-0.161978\pi$
$257$	28.0000	1.74659	0.873296	+	0.487190i	$0.161978\pi$
$258$	−6.00000	−0.373544
$259$	4.00000	0.248548
$260$	−1.00000	−0.0620174
$261$	0	0
$262$	−18.0000	−1.11204
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	−3.00000	−0.184637
$265$	14.0000	0.860013
$266$	10.0000	0.613139
$267$	10.0000	0.611990
$268$	−7.00000	−0.427593
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	1.00000	0.0608581
$271$	7.00000	0.425220	0.212610	−	0.977137i	$-0.431804\pi$
$271$	7.00000	0.425220	0.212610	+	0.977137i	$0.431804\pi$
$272$	3.00000	0.181902
$273$	2.00000	0.121046
$274$	−2.00000	−0.120824
$275$	12.0000	0.723627
$276$	4.00000	0.240772
$277$	−17.0000	−1.02143	−0.510716	−	0.859750i	$-0.670619\pi$
$277$	−17.0000	−1.02143	−0.510716	+	0.859750i	$0.670619\pi$
$278$	20.0000	1.19952
$279$	1.00000	0.0598684
$280$	−2.00000	−0.119523
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−7.00000	−0.416844
$283$	9.00000	0.534994	0.267497	−	0.963559i	$-0.413803\pi$
$283$	9.00000	0.534994	0.267497	+	0.963559i	$0.413803\pi$
$284$	−3.00000	−0.178017
$285$	−5.00000	−0.296174
$286$	3.00000	0.177394
$287$	−4.00000	−0.236113
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	0	0
$291$	13.0000	0.762073
$292$	−6.00000	−0.351123
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	−3.00000	−0.174964
$295$	10.0000	0.582223
$296$	−2.00000	−0.116248
$297$	−3.00000	−0.174078
$298$	5.00000	0.289642
$299$	−4.00000	−0.231326
$300$	−4.00000	−0.230940
$301$	12.0000	0.691669
$302$	−3.00000	−0.172631
$303$	2.00000	0.114897
$304$	−5.00000	−0.286770
$305$	7.00000	0.400819
$306$	3.00000	0.171499
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	6.00000	0.341882
$309$	14.0000	0.796432
$310$	1.00000	0.0567962
$311$	27.0000	1.53103	0.765515	−	0.643418i	$-0.222484\pi$
$311$	27.0000	1.53103	0.765515	+	0.643418i	$0.222484\pi$
$312$	−1.00000	−0.0566139
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	8.00000	0.451466
$315$	−2.00000	−0.112687
$316$	15.0000	0.843816
$317$	−17.0000	−0.954815	−0.477408	−	0.878682i	$-0.658423\pi$
$317$	−17.0000	−0.954815	−0.477408	+	0.878682i	$0.658423\pi$
$318$	14.0000	0.785081
$319$	0	0
$320$	1.00000	0.0559017
$321$	−12.0000	−0.669775
$322$	−8.00000	−0.445823
$323$	−15.0000	−0.834622
$324$	1.00000	0.0555556
$325$	4.00000	0.221880
$326$	−21.0000	−1.16308
$327$	10.0000	0.553001
$328$	2.00000	0.110432
$329$	14.0000	0.771845
$330$	−3.00000	−0.165145
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	−1.00000	−0.0548821
$333$	−2.00000	−0.109599
$334$	−2.00000	−0.109435
$335$	−7.00000	−0.382451
$336$	−2.00000	−0.109109
$337$	−32.0000	−1.74315	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000	−1.74315	−0.871576	+	0.490261i	$0.836901\pi$
$338$	−12.0000	−0.652714
$339$	14.0000	0.760376
$340$	3.00000	0.162698
$341$	−3.00000	−0.162459
$342$	−5.00000	−0.270369
$343$	20.0000	1.07990
$344$	−6.00000	−0.323498
$345$	4.00000	0.215353
$346$	−21.0000	−1.12897
$347$	−7.00000	−0.375780	−0.187890	−	0.982190i	$-0.560165\pi$
$347$	−7.00000	−0.375780	−0.187890	+	0.982190i	$0.560165\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	8.00000	0.427618
$351$	−1.00000	−0.0533761
$352$	−3.00000	−0.159901
$353$	9.00000	0.479022	0.239511	−	0.970894i	$-0.423013\pi$
$353$	9.00000	0.479022	0.239511	+	0.970894i	$0.423013\pi$
$354$	10.0000	0.531494
$355$	−3.00000	−0.159223
$356$	10.0000	0.529999
$357$	−6.00000	−0.317554
$358$	−5.00000	−0.264258
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	1.00000	0.0527046
$361$	6.00000	0.315789
$362$	−18.0000	−0.946059
$363$	−2.00000	−0.104973
$364$	2.00000	0.104828
$365$	−6.00000	−0.314054
$366$	7.00000	0.365896
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	4.00000	0.208514
$369$	2.00000	0.104116
$370$	−2.00000	−0.103975
$371$	−28.0000	−1.45369
$372$	1.00000	0.0518476
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	−9.00000	−0.465379
$375$	−9.00000	−0.464758
$376$	−7.00000	−0.360997
$377$	0	0
$378$	−2.00000	−0.102869
$379$	−35.0000	−1.79783	−0.898915	−	0.438124i	$-0.855643\pi$
$379$	−35.0000	−1.79783	−0.898915	+	0.438124i	$0.855643\pi$
$380$	−5.00000	−0.256495
$381$	−12.0000	−0.614779
$382$	−8.00000	−0.409316
$383$	−26.0000	−1.32854	−0.664269	−	0.747494i	$-0.731257\pi$
$383$	−26.0000	−1.32854	−0.664269	+	0.747494i	$0.731257\pi$
$384$	1.00000	0.0510310
$385$	6.00000	0.305788
$386$	−21.0000	−1.06887
$387$	−6.00000	−0.304997
$388$	13.0000	0.659975
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	−1.00000	−0.0506370
$391$	12.0000	0.606866
$392$	−3.00000	−0.151523
$393$	−18.0000	−0.907980
$394$	−22.0000	−1.10834
$395$	15.0000	0.754732
$396$	−3.00000	−0.150756
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	15.0000	0.751882
$399$	10.0000	0.500626
$400$	−4.00000	−0.200000
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	−7.00000	−0.349128
$403$	−1.00000	−0.0498135
$404$	2.00000	0.0995037
$405$	1.00000	0.0496904
$406$	0	0
$407$	6.00000	0.297409
$408$	3.00000	0.148522
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	2.00000	0.0987730
$411$	−2.00000	−0.0986527
$412$	14.0000	0.689730
$413$	−20.0000	−0.984136
$414$	4.00000	0.196589
$415$	−1.00000	−0.0490881
$416$	−1.00000	−0.0490290
$417$	20.0000	0.979404
$418$	15.0000	0.733674
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	−2.00000	−0.0975900
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−8.00000	−0.389434
$423$	−7.00000	−0.340352
$424$	14.0000	0.679900
$425$	−12.0000	−0.582086
$426$	−3.00000	−0.145350
$427$	−14.0000	−0.677507
$428$	−12.0000	−0.580042
$429$	3.00000	0.144841
$430$	−6.00000	−0.289346
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	1.00000	0.0481125
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	10.0000	0.478913
$437$	−20.0000	−0.956730
$438$	−6.00000	−0.286691
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	−3.00000	−0.143019
$441$	−3.00000	−0.142857
$442$	−3.00000	−0.142695
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	−2.00000	−0.0949158
$445$	10.0000	0.474045
$446$	9.00000	0.426162
$447$	5.00000	0.236492
$448$	−2.00000	−0.0944911
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	−4.00000	−0.188562
$451$	−6.00000	−0.282529
$452$	14.0000	0.658505
$453$	−3.00000	−0.140952
$454$	−22.0000	−1.03251
$455$	2.00000	0.0937614
$456$	−5.00000	−0.234146
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−5.00000	−0.233635
$459$	3.00000	0.140028
$460$	4.00000	0.186501
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	6.00000	0.279145
$463$	19.0000	0.883005	0.441502	−	0.897260i	$-0.354446\pi$
$463$	19.0000	0.883005	0.441502	+	0.897260i	$0.354446\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	4.00000	0.185296
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−1.00000	−0.0462250
$469$	14.0000	0.646460
$470$	−7.00000	−0.322886
$471$	8.00000	0.368621
$472$	10.0000	0.460287
$473$	18.0000	0.827641
$474$	15.0000	0.688973
$475$	20.0000	0.917663
$476$	−6.00000	−0.275010
$477$	14.0000	0.641016
$478$	20.0000	0.914779
$479$	35.0000	1.59919	0.799595	−	0.600539i	$-0.205047\pi$
$479$	35.0000	1.59919	0.799595	+	0.600539i	$0.205047\pi$
$480$	1.00000	0.0456435
$481$	2.00000	0.0911922
$482$	2.00000	0.0910975
$483$	−8.00000	−0.364013
$484$	−2.00000	−0.0909091
$485$	13.0000	0.590300
$486$	1.00000	0.0453609
$487$	−17.0000	−0.770344	−0.385172	−	0.922845i	$-0.625858\pi$
$487$	−17.0000	−0.770344	−0.385172	+	0.922845i	$0.625858\pi$
$488$	7.00000	0.316875
$489$	−21.0000	−0.949653
$490$	−3.00000	−0.135526
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	2.00000	0.0901670
$493$	0	0
$494$	5.00000	0.224961
$495$	−3.00000	−0.134840
$496$	1.00000	0.0449013
$497$	6.00000	0.269137
$498$	−1.00000	−0.0448111
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	−9.00000	−0.402492
$501$	−2.00000	−0.0893534
$502$	12.0000	0.535586
$503$	9.00000	0.401290	0.200645	−	0.979664i	$-0.435696\pi$
$503$	9.00000	0.401290	0.200645	+	0.979664i	$0.435696\pi$
$504$	−2.00000	−0.0890871
$505$	2.00000	0.0889988
$506$	−12.0000	−0.533465
$507$	−12.0000	−0.532939
$508$	−12.0000	−0.532414
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	3.00000	0.132842
$511$	12.0000	0.530849
$512$	1.00000	0.0441942
$513$	−5.00000	−0.220755
$514$	28.0000	1.23503
$515$	14.0000	0.616914
$516$	−6.00000	−0.264135
$517$	21.0000	0.923579
$518$	4.00000	0.175750
$519$	−21.0000	−0.921798
$520$	−1.00000	−0.0438529
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	−18.0000	−0.786334
$525$	8.00000	0.349149
$526$	24.0000	1.04645
$527$	3.00000	0.130682
$528$	−3.00000	−0.130558
$529$	−7.00000	−0.304348
$530$	14.0000	0.608121
$531$	10.0000	0.433963
$532$	10.0000	0.433555
$533$	−2.00000	−0.0866296
$534$	10.0000	0.432742
$535$	−12.0000	−0.518805
$536$	−7.00000	−0.302354
$537$	−5.00000	−0.215766
$538$	10.0000	0.431131
$539$	9.00000	0.387657
$540$	1.00000	0.0430331
$541$	−8.00000	−0.343947	−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000	−0.343947	−0.171973	+	0.985102i	$0.555014\pi$
$542$	7.00000	0.300676
$543$	−18.0000	−0.772454
$544$	3.00000	0.128624
$545$	10.0000	0.428353
$546$	2.00000	0.0855921
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−2.00000	−0.0854358
$549$	7.00000	0.298753
$550$	12.0000	0.511682
$551$	0	0
$552$	4.00000	0.170251
$553$	−30.0000	−1.27573
$554$	−17.0000	−0.722261
$555$	−2.00000	−0.0848953
$556$	20.0000	0.848189
$557$	−32.0000	−1.35588	−0.677942	−	0.735116i	$-0.737128\pi$
$557$	−32.0000	−1.35588	−0.677942	+	0.735116i	$0.737128\pi$
$558$	1.00000	0.0423334
$559$	6.00000	0.253773
$560$	−2.00000	−0.0845154
$561$	−9.00000	−0.379980
$562$	−18.0000	−0.759284
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	−7.00000	−0.294753
$565$	14.0000	0.588984
$566$	9.00000	0.378298
$567$	−2.00000	−0.0839921
$568$	−3.00000	−0.125877
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	−5.00000	−0.209427
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	3.00000	0.125436
$573$	−8.00000	−0.334205
$574$	−4.00000	−0.166957
$575$	−16.0000	−0.667246
$576$	1.00000	0.0416667
$577$	43.0000	1.79011	0.895057	−	0.445952i	$-0.147135\pi$
$577$	43.0000	1.79011	0.895057	+	0.445952i	$0.147135\pi$
$578$	−8.00000	−0.332756
$579$	−21.0000	−0.872730
$580$	0	0
$581$	2.00000	0.0829740
$582$	13.0000	0.538867
$583$	−42.0000	−1.73946
$584$	−6.00000	−0.248282
$585$	−1.00000	−0.0413449
$586$	14.0000	0.578335
$587$	−17.0000	−0.701665	−0.350833	−	0.936438i	$-0.614101\pi$
$587$	−17.0000	−0.701665	−0.350833	+	0.936438i	$0.614101\pi$
$588$	−3.00000	−0.123718
$589$	−5.00000	−0.206021
$590$	10.0000	0.411693
$591$	−22.0000	−0.904959
$592$	−2.00000	−0.0821995
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	−3.00000	−0.123091
$595$	−6.00000	−0.245976
$596$	5.00000	0.204808
$597$	15.0000	0.613909
$598$	−4.00000	−0.163572
$599$	15.0000	0.612883	0.306442	−	0.951889i	$-0.400862\pi$
$599$	15.0000	0.612883	0.306442	+	0.951889i	$0.400862\pi$
$600$	−4.00000	−0.163299
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	12.0000	0.489083
$603$	−7.00000	−0.285062
$604$	−3.00000	−0.122068
$605$	−2.00000	−0.0813116
$606$	2.00000	0.0812444
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	−5.00000	−0.202777
$609$	0	0
$610$	7.00000	0.283422
$611$	7.00000	0.283190
$612$	3.00000	0.121268
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	8.00000	0.322854
$615$	2.00000	0.0806478
$616$	6.00000	0.241747
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	14.0000	0.563163
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	1.00000	0.0401610
$621$	4.00000	0.160514
$622$	27.0000	1.08260
$623$	−20.0000	−0.801283
$624$	−1.00000	−0.0400320
$625$	11.0000	0.440000
$626$	−26.0000	−1.03917
$627$	15.0000	0.599042
$628$	8.00000	0.319235
$629$	−6.00000	−0.239236
$630$	−2.00000	−0.0796819
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	15.0000	0.596668
$633$	−8.00000	−0.317971
$634$	−17.0000	−0.675156
$635$	−12.0000	−0.476205
$636$	14.0000	0.555136
$637$	3.00000	0.118864
$638$	0	0
$639$	−3.00000	−0.118678
$640$	1.00000	0.0395285
$641$	27.0000	1.06644	0.533218	−	0.845978i	$-0.320983\pi$
$641$	27.0000	1.06644	0.533218	+	0.845978i	$0.320983\pi$
$642$	−12.0000	−0.473602
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	−8.00000	−0.315244
$645$	−6.00000	−0.236250
$646$	−15.0000	−0.590167
$647$	38.0000	1.49393	0.746967	−	0.664861i	$-0.231509\pi$
$647$	38.0000	1.49393	0.746967	+	0.664861i	$0.231509\pi$
$648$	1.00000	0.0392837
$649$	−30.0000	−1.17760
$650$	4.00000	0.156893
$651$	−2.00000	−0.0783862
$652$	−21.0000	−0.822423
$653$	−31.0000	−1.21312	−0.606562	−	0.795036i	$-0.707452\pi$
$653$	−31.0000	−1.21312	−0.606562	+	0.795036i	$0.707452\pi$
$654$	10.0000	0.391031
$655$	−18.0000	−0.703318
$656$	2.00000	0.0780869
$657$	−6.00000	−0.234082
$658$	14.0000	0.545777
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	−3.00000	−0.116775
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	32.0000	1.24372
$663$	−3.00000	−0.116510
$664$	−1.00000	−0.0388075
$665$	10.0000	0.387783
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	−2.00000	−0.0773823
$669$	9.00000	0.347960
$670$	−7.00000	−0.270434
$671$	−21.0000	−0.810696
$672$	−2.00000	−0.0771517
$673$	24.0000	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$674$	−32.0000	−1.23259
$675$	−4.00000	−0.153960
$676$	−12.0000	−0.461538
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	14.0000	0.537667
$679$	−26.0000	−0.997788
$680$	3.00000	0.115045
$681$	−22.0000	−0.843042
$682$	−3.00000	−0.114876
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	−5.00000	−0.191180
$685$	−2.00000	−0.0764161
$686$	20.0000	0.763604
$687$	−5.00000	−0.190762
$688$	−6.00000	−0.228748
$689$	−14.0000	−0.533358
$690$	4.00000	0.152277
$691$	−23.0000	−0.874961	−0.437481	−	0.899228i	$-0.644129\pi$
$691$	−23.0000	−0.874961	−0.437481	+	0.899228i	$0.644129\pi$
$692$	−21.0000	−0.798300
$693$	6.00000	0.227921
$694$	−7.00000	−0.265716
$695$	20.0000	0.758643
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	4.00000	0.151294
$700$	8.00000	0.302372
$701$	−23.0000	−0.868698	−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000	−0.868698	−0.434349	+	0.900745i	$0.643022\pi$
$702$	−1.00000	−0.0377426
$703$	10.0000	0.377157
$704$	−3.00000	−0.113067
$705$	−7.00000	−0.263635
$706$	9.00000	0.338719
$707$	−4.00000	−0.150435
$708$	10.0000	0.375823
$709$	−35.0000	−1.31445	−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000	−1.31445	−0.657226	+	0.753693i	$0.728270\pi$
$710$	−3.00000	−0.112588
$711$	15.0000	0.562544
$712$	10.0000	0.374766
$713$	4.00000	0.149801
$714$	−6.00000	−0.224544
$715$	3.00000	0.112194
$716$	−5.00000	−0.186859
$717$	20.0000	0.746914
$718$	−15.0000	−0.559795
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	1.00000	0.0372678
$721$	−28.0000	−1.04277
$722$	6.00000	0.223297
$723$	2.00000	0.0743808
$724$	−18.0000	−0.668965
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	−18.0000	−0.665754
$732$	7.00000	0.258727
$733$	−46.0000	−1.69905	−0.849524	−	0.527549i	$-0.823111\pi$
$733$	−46.0000	−1.69905	−0.849524	+	0.527549i	$0.823111\pi$
$734$	−17.0000	−0.627481
$735$	−3.00000	−0.110657
$736$	4.00000	0.147442
$737$	21.0000	0.773545
$738$	2.00000	0.0736210
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−2.00000	−0.0735215
$741$	5.00000	0.183680
$742$	−28.0000	−1.02791
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	1.00000	0.0366618
$745$	5.00000	0.183186
$746$	−16.0000	−0.585802
$747$	−1.00000	−0.0365881
$748$	−9.00000	−0.329073
$749$	24.0000	0.876941
$750$	−9.00000	−0.328634
$751$	22.0000	0.802791	0.401396	−	0.915905i	$-0.368525\pi$
$751$	22.0000	0.802791	0.401396	+	0.915905i	$0.368525\pi$
$752$	−7.00000	−0.255264
$753$	12.0000	0.437304
$754$	0	0
$755$	−3.00000	−0.109181
$756$	−2.00000	−0.0727393
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−35.0000	−1.27126
$759$	−12.0000	−0.435572
$760$	−5.00000	−0.181369
$761$	−3.00000	−0.108750	−0.0543750	−	0.998521i	$-0.517317\pi$
$761$	−3.00000	−0.108750	−0.0543750	+	0.998521i	$0.517317\pi$
$762$	−12.0000	−0.434714
$763$	−20.0000	−0.724049
$764$	−8.00000	−0.289430
$765$	3.00000	0.108465
$766$	−26.0000	−0.939418
$767$	−10.0000	−0.361079
$768$	1.00000	0.0360844
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	6.00000	0.216225
$771$	28.0000	1.00840
$772$	−21.0000	−0.755807
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	−6.00000	−0.215666
$775$	−4.00000	−0.143684
$776$	13.0000	0.466673
$777$	4.00000	0.143499
$778$	30.0000	1.07555
$779$	−10.0000	−0.358287
$780$	−1.00000	−0.0358057
$781$	9.00000	0.322045
$782$	12.0000	0.429119
$783$	0	0
$784$	−3.00000	−0.107143
$785$	8.00000	0.285532
$786$	−18.0000	−0.642039
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	−22.0000	−0.783718
$789$	24.0000	0.854423
$790$	15.0000	0.533676
$791$	−28.0000	−0.995565
$792$	−3.00000	−0.106600
$793$	−7.00000	−0.248577
$794$	−22.0000	−0.780751
$795$	14.0000	0.496529
$796$	15.0000	0.531661
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	10.0000	0.353996
$799$	−21.0000	−0.742927
$800$	−4.00000	−0.141421
$801$	10.0000	0.353333
$802$	−3.00000	−0.105934
$803$	18.0000	0.635206
$804$	−7.00000	−0.246871
$805$	−8.00000	−0.281963
$806$	−1.00000	−0.0352235
$807$	10.0000	0.352017
$808$	2.00000	0.0703598
$809$	15.0000	0.527372	0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000	0.527372	0.263686	+	0.964609i	$0.415062\pi$
$810$	1.00000	0.0351364
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	0	0
$813$	7.00000	0.245501
$814$	6.00000	0.210300
$815$	−21.0000	−0.735598
$816$	3.00000	0.105021
$817$	30.0000	1.04957
$818$	30.0000	1.04893
$819$	2.00000	0.0698857
$820$	2.00000	0.0698430
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	−2.00000	−0.0697580
$823$	9.00000	0.313720	0.156860	−	0.987621i	$-0.449863\pi$
$823$	9.00000	0.313720	0.156860	+	0.987621i	$0.449863\pi$
$824$	14.0000	0.487713
$825$	12.0000	0.417786
$826$	−20.0000	−0.695889
$827$	43.0000	1.49526	0.747628	−	0.664117i	$-0.231193\pi$
$827$	43.0000	1.49526	0.747628	+	0.664117i	$0.231193\pi$
$828$	4.00000	0.139010
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	−1.00000	−0.0347105
$831$	−17.0000	−0.589723
$832$	−1.00000	−0.0346688
$833$	−9.00000	−0.311832
$834$	20.0000	0.692543
$835$	−2.00000	−0.0692129
$836$	15.0000	0.518786
$837$	1.00000	0.0345651
$838$	0	0
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	−2.00000	−0.0690066
$841$	−29.0000	−1.00000
$842$	22.0000	0.758170
$843$	−18.0000	−0.619953
$844$	−8.00000	−0.275371
$845$	−12.0000	−0.412813
$846$	−7.00000	−0.240665
$847$	4.00000	0.137442
$848$	14.0000	0.480762
$849$	9.00000	0.308879
$850$	−12.0000	−0.411597
$851$	−8.00000	−0.274236
$852$	−3.00000	−0.102778
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	−14.0000	−0.479070
$855$	−5.00000	−0.170996
$856$	−12.0000	−0.410152
$857$	28.0000	0.956462	0.478231	−	0.878234i	$-0.341278\pi$
$857$	28.0000	0.956462	0.478231	+	0.878234i	$0.341278\pi$
$858$	3.00000	0.102418
$859$	10.0000	0.341196	0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000	0.341196	0.170598	+	0.985341i	$0.445430\pi$
$860$	−6.00000	−0.204598
$861$	−4.00000	−0.136320
$862$	32.0000	1.08992
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	1.00000	0.0340207
$865$	−21.0000	−0.714021
$866$	24.0000	0.815553
$867$	−8.00000	−0.271694
$868$	−2.00000	−0.0678844
$869$	−45.0000	−1.52652
$870$	0	0
$871$	7.00000	0.237186
$872$	10.0000	0.338643
$873$	13.0000	0.439983
$874$	−20.0000	−0.676510
$875$	18.0000	0.608511
$876$	−6.00000	−0.202721
$877$	8.00000	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$877$	8.00000	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$878$	−20.0000	−0.674967
$879$	14.0000	0.472208
$880$	−3.00000	−0.101130
$881$	7.00000	0.235836	0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000	0.235836	0.117918	+	0.993023i	$0.462378\pi$
$882$	−3.00000	−0.101015
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	−3.00000	−0.100901
$885$	10.0000	0.336146
$886$	−6.00000	−0.201574
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	−2.00000	−0.0671156
$889$	24.0000	0.804934
$890$	10.0000	0.335201
$891$	−3.00000	−0.100504
$892$	9.00000	0.301342
$893$	35.0000	1.17123
$894$	5.00000	0.167225
$895$	−5.00000	−0.167132
$896$	−2.00000	−0.0668153
$897$	−4.00000	−0.133556
$898$	−15.0000	−0.500556
$899$	0	0
$900$	−4.00000	−0.133333
$901$	42.0000	1.39922
$902$	−6.00000	−0.199778
$903$	12.0000	0.399335
$904$	14.0000	0.465633
$905$	−18.0000	−0.598340
$906$	−3.00000	−0.0996683
$907$	−37.0000	−1.22856	−0.614282	−	0.789086i	$-0.710554\pi$
$907$	−37.0000	−1.22856	−0.614282	+	0.789086i	$0.710554\pi$
$908$	−22.0000	−0.730096
$909$	2.00000	0.0663358
$910$	2.00000	0.0662994
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−5.00000	−0.165567
$913$	3.00000	0.0992855
$914$	−22.0000	−0.727695
$915$	7.00000	0.231413
$916$	−5.00000	−0.165205
$917$	36.0000	1.18882
$918$	3.00000	0.0990148
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	4.00000	0.131876
$921$	8.00000	0.263609
$922$	−18.0000	−0.592798
$923$	3.00000	0.0987462
$924$	6.00000	0.197386
$925$	8.00000	0.263038
$926$	19.0000	0.624379
$927$	14.0000	0.459820
$928$	0	0
$929$	−10.0000	−0.328089	−0.164045	−	0.986453i	$-0.552454\pi$
$929$	−10.0000	−0.328089	−0.164045	+	0.986453i	$0.552454\pi$
$930$	1.00000	0.0327913
$931$	15.0000	0.491605
$932$	4.00000	0.131024
$933$	27.0000	0.883940
$934$	−12.0000	−0.392652
$935$	−9.00000	−0.294331
$936$	−1.00000	−0.0326860
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	14.0000	0.457116
$939$	−26.0000	−0.848478
$940$	−7.00000	−0.228315
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	8.00000	0.260654
$943$	8.00000	0.260516
$944$	10.0000	0.325472
$945$	−2.00000	−0.0650600
$946$	18.0000	0.585230
$947$	3.00000	0.0974869	0.0487435	−	0.998811i	$-0.484478\pi$
$947$	3.00000	0.0974869	0.0487435	+	0.998811i	$0.484478\pi$
$948$	15.0000	0.487177
$949$	6.00000	0.194768
$950$	20.0000	0.648886
$951$	−17.0000	−0.551263
$952$	−6.00000	−0.194461
$953$	39.0000	1.26333	0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000	1.26333	0.631667	+	0.775240i	$0.282371\pi$
$954$	14.0000	0.453267
$955$	−8.00000	−0.258874
$956$	20.0000	0.646846
$957$	0	0
$958$	35.0000	1.13080
$959$	4.00000	0.129167
$960$	1.00000	0.0322749
$961$	1.00000	0.0322581
$962$	2.00000	0.0644826
$963$	−12.0000	−0.386695
$964$	2.00000	0.0644157
$965$	−21.0000	−0.676014
$966$	−8.00000	−0.257396
$967$	43.0000	1.38279	0.691393	−	0.722478i	$-0.256997\pi$
$967$	43.0000	1.38279	0.691393	+	0.722478i	$0.256997\pi$
$968$	−2.00000	−0.0642824
$969$	−15.0000	−0.481869
$970$	13.0000	0.417405
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	1.00000	0.0320750
$973$	−40.0000	−1.28234
$974$	−17.0000	−0.544715
$975$	4.00000	0.128103
$976$	7.00000	0.224065
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−21.0000	−0.671506
$979$	−30.0000	−0.958804
$980$	−3.00000	−0.0958315
$981$	10.0000	0.319275
$982$	−28.0000	−0.893516
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	2.00000	0.0637577
$985$	−22.0000	−0.700978
$986$	0	0
$987$	14.0000	0.445625
$988$	5.00000	0.159071
$989$	−24.0000	−0.763156
$990$	−3.00000	−0.0953463
$991$	12.0000	0.381193	0.190596	−	0.981669i	$-0.438958\pi$
$991$	12.0000	0.381193	0.190596	+	0.981669i	$0.438958\pi$
$992$	1.00000	0.0317500
$993$	32.0000	1.01549
$994$	6.00000	0.190308
$995$	15.0000	0.475532
$996$	−1.00000	−0.0316862
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	40.0000	1.26618
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	186.2.a.c.1.1	✓	1
3.2	odd	2		558.2.a.a.1.1		1
4.3	odd	2		1488.2.a.f.1.1		1
5.2	odd	4		4650.2.d.r.3349.2		2
5.3	odd	4		4650.2.d.r.3349.1		2
5.4	even	2		4650.2.a.i.1.1		1
7.6	odd	2		9114.2.a.t.1.1		1
8.3	odd	2		5952.2.a.v.1.1		1
8.5	even	2		5952.2.a.d.1.1		1
12.11	even	2		4464.2.a.h.1.1		1
31.30	odd	2		5766.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
186.2.a.c.1.1	✓	1	1.1	even	1	trivial
558.2.a.a.1.1		1	3.2	odd	2
1488.2.a.f.1.1		1	4.3	odd	2
4464.2.a.h.1.1		1	12.11	even	2
4650.2.a.i.1.1		1	5.4	even	2
4650.2.d.r.3349.1		2	5.3	odd	4
4650.2.d.r.3349.2		2	5.2	odd	4
5766.2.a.h.1.1		1	31.30	odd	2
5952.2.a.d.1.1		1	8.5	even	2
5952.2.a.v.1.1		1	8.3	odd	2
9114.2.a.t.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:	\( N \)	\(=\)	\( 186 = 2 \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	186.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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