LMFDB - Embedded newform 1254.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1254,2,Mod(1,1254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1254, 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("1254.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1254 = 2 \cdot 3 \cdot 11 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1254.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:	$10.0132404135$
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	$\chi$	$=$	1254.1

$q$ -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} +1.00000 q^{10} +1.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +1.00000 q^{20} -2.00000 q^{21} +1.00000 q^{22} +4.00000 q^{23} +1.00000 q^{24} -4.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -2.00000 q^{28} -5.00000 q^{29} +1.00000 q^{30} +2.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +3.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} +3.00000 q^{37} -1.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} +2.00000 q^{41} -2.00000 q^{42} +9.00000 q^{43} +1.00000 q^{44} +1.00000 q^{45} +4.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} -4.00000 q^{50} +3.00000 q^{51} +4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +1.00000 q^{55} -2.00000 q^{56} -1.00000 q^{57} -5.00000 q^{58} +1.00000 q^{60} -13.0000 q^{61} +2.00000 q^{62} -2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +1.00000 q^{66} +13.0000 q^{67} +3.00000 q^{68} +4.00000 q^{69} -2.00000 q^{70} -8.00000 q^{71} +1.00000 q^{72} -16.0000 q^{73} +3.00000 q^{74} -4.00000 q^{75} -1.00000 q^{76} -2.00000 q^{77} +4.00000 q^{78} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -6.00000 q^{83} -2.00000 q^{84} +3.00000 q^{85} +9.00000 q^{86} -5.00000 q^{87} +1.00000 q^{88} -5.00000 q^{89} +1.00000 q^{90} -8.00000 q^{91} +4.00000 q^{92} +2.00000 q^{93} +8.00000 q^{94} -1.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} +1.00000 q^{99} +O(q^{100})

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
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	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	$0.312833\pi$
$14$	−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
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.00000	0.223607
$21$	−2.00000	−0.436436
$22$	1.00000	0.213201
$23$	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
$26$	4.00000	0.784465
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	1.00000	0.182574
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	3.00000	0.514496
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	−1.00000	−0.162221
$39$	4.00000	0.640513
$40$	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
$43$	9.00000	1.37249	0.686244	−	0.727372i	$-0.259258\pi$
$43$	9.00000	1.37249	0.686244	+	0.727372i	$0.259258\pi$
$44$	1.00000	0.150756
$45$	1.00000	0.149071
$46$	4.00000	0.589768
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	3.00000	0.420084
$52$	4.00000	0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	1.00000	0.134840
$56$	−2.00000	−0.267261
$57$	−1.00000	−0.132453
$58$	−5.00000	−0.656532
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000	0.129099
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	2.00000	0.254000
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	1.00000	0.123091
$67$	13.0000	1.58820	0.794101	−	0.607785i	$-0.207942\pi$
$67$	13.0000	1.58820	0.794101	+	0.607785i	$0.207942\pi$
$68$	3.00000	0.363803
$69$	4.00000	0.481543
$70$	−2.00000	−0.239046
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	1.00000	0.117851
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	3.00000	0.348743
$75$	−4.00000	−0.461880
$76$	−1.00000	−0.114708
$77$	−2.00000	−0.227921
$78$	4.00000	0.452911
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−2.00000	−0.218218
$85$	3.00000	0.325396
$86$	9.00000	0.970495
$87$	−5.00000	−0.536056
$88$	1.00000	0.106600
$89$	−5.00000	−0.529999	−0.264999	−	0.964249i	$-0.585372\pi$
$89$	−5.00000	−0.529999	−0.264999	+	0.964249i	$0.585372\pi$
$90$	1.00000	0.105409
$91$	−8.00000	−0.838628
$92$	4.00000	0.417029
$93$	2.00000	0.207390
$94$	8.00000	0.825137
$95$	−1.00000	−0.102598
$96$	1.00000	0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−3.00000	−0.303046
$99$	1.00000	0.100504
$100$	−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
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	4.00000	0.392232
$105$	−2.00000	−0.195180
$106$	−6.00000	−0.582772
$107$	−7.00000	−0.676716	−0.338358	−	0.941018i	$-0.609871\pi$
$107$	−7.00000	−0.676716	−0.338358	+	0.941018i	$0.609871\pi$
$108$	1.00000	0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	1.00000	0.0953463
$111$	3.00000	0.284747
$112$	−2.00000	−0.188982
$113$	−11.0000	−1.03479	−0.517396	−	0.855746i	$-0.673099\pi$
$113$	−11.0000	−1.03479	−0.517396	+	0.855746i	$0.673099\pi$
$114$	−1.00000	−0.0936586
$115$	4.00000	0.373002
$116$	−5.00000	−0.464238
$117$	4.00000	0.369800
$118$	0	0
$119$	−6.00000	−0.550019
$120$	1.00000	0.0912871
$121$	1.00000	0.0909091
$122$	−13.0000	−1.17696
$123$	2.00000	0.180334
$124$	2.00000	0.179605
$125$	−9.00000	−0.804984
$126$	−2.00000	−0.178174
$127$	13.0000	1.15356	0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000	1.15356	0.576782	+	0.816898i	$0.304308\pi$
$128$	1.00000	0.0883883
$129$	9.00000	0.792406
$130$	4.00000	0.350823
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	1.00000	0.0870388
$133$	2.00000	0.173422
$134$	13.0000	1.12303
$135$	1.00000	0.0860663
$136$	3.00000	0.257248
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	4.00000	0.340503
$139$	15.0000	1.27228	0.636142	−	0.771572i	$-0.280529\pi$
$139$	15.0000	1.27228	0.636142	+	0.771572i	$0.280529\pi$
$140$	−2.00000	−0.169031
$141$	8.00000	0.673722
$142$	−8.00000	−0.671345
$143$	4.00000	0.334497
$144$	1.00000	0.0833333
$145$	−5.00000	−0.415227
$146$	−16.0000	−1.32417
$147$	−3.00000	−0.247436
$148$	3.00000	0.246598
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−4.00000	−0.326599
$151$	7.00000	0.569652	0.284826	−	0.958579i	$-0.408064\pi$
$151$	7.00000	0.569652	0.284826	+	0.958579i	$0.408064\pi$
$152$	−1.00000	−0.0811107
$153$	3.00000	0.242536
$154$	−2.00000	−0.161165
$155$	2.00000	0.160644
$156$	4.00000	0.320256
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	−8.00000	−0.630488
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	2.00000	0.156174
$165$	1.00000	0.0778499
$166$	−6.00000	−0.465690
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−2.00000	−0.154303
$169$	3.00000	0.230769
$170$	3.00000	0.230089
$171$	−1.00000	−0.0764719
$172$	9.00000	0.686244
$173$	−1.00000	−0.0760286	−0.0380143	−	0.999277i	$-0.512103\pi$
$173$	−1.00000	−0.0760286	−0.0380143	+	0.999277i	$0.512103\pi$
$174$	−5.00000	−0.379049
$175$	8.00000	0.604743
$176$	1.00000	0.0753778
$177$	0	0
$178$	−5.00000	−0.374766
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	1.00000	0.0745356
$181$	−3.00000	−0.222988	−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000	−0.222988	−0.111494	+	0.993765i	$0.535564\pi$
$182$	−8.00000	−0.592999
$183$	−13.0000	−0.960988
$184$	4.00000	0.294884
$185$	3.00000	0.220564
$186$	2.00000	0.146647
$187$	3.00000	0.219382
$188$	8.00000	0.583460
$189$	−2.00000	−0.145479
$190$	−1.00000	−0.0725476
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	1.00000	0.0721688
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	8.00000	0.574367
$195$	4.00000	0.286446
$196$	−3.00000	−0.214286
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	1.00000	0.0710669
$199$	−15.0000	−1.06332	−0.531661	−	0.846957i	$-0.678432\pi$
$199$	−15.0000	−1.06332	−0.531661	+	0.846957i	$0.678432\pi$
$200$	−4.00000	−0.282843
$201$	13.0000	0.916949
$202$	2.00000	0.140720
$203$	10.0000	0.701862
$204$	3.00000	0.210042
$205$	2.00000	0.139686
$206$	−6.00000	−0.418040
$207$	4.00000	0.278019
$208$	4.00000	0.277350
$209$	−1.00000	−0.0691714
$210$	−2.00000	−0.138013
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−6.00000	−0.412082
$213$	−8.00000	−0.548151
$214$	−7.00000	−0.478510
$215$	9.00000	0.613795
$216$	1.00000	0.0680414
$217$	−4.00000	−0.271538
$218$	−10.0000	−0.677285
$219$	−16.0000	−1.08118
$220$	1.00000	0.0674200
$221$	12.0000	0.807207
$222$	3.00000	0.201347
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−2.00000	−0.133631
$225$	−4.00000	−0.266667
$226$	−11.0000	−0.731709
$227$	−27.0000	−1.79205	−0.896026	−	0.444001i	$-0.853559\pi$
$227$	−27.0000	−1.79205	−0.896026	+	0.444001i	$0.853559\pi$
$228$	−1.00000	−0.0662266
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	4.00000	0.263752
$231$	−2.00000	−0.131590
$232$	−5.00000	−0.328266
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	4.00000	0.261488
$235$	8.00000	0.521862
$236$	0	0
$237$	0	0
$238$	−6.00000	−0.388922
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	1.00000	0.0645497
$241$	−3.00000	−0.193247	−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000	−0.193247	−0.0966235	+	0.995321i	$0.530804\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	−13.0000	−0.832240
$245$	−3.00000	−0.191663
$246$	2.00000	0.127515
$247$	−4.00000	−0.254514
$248$	2.00000	0.127000
$249$	−6.00000	−0.380235
$250$	−9.00000	−0.569210
$251$	17.0000	1.07303	0.536515	−	0.843891i	$-0.319740\pi$
$251$	17.0000	1.07303	0.536515	+	0.843891i	$0.319740\pi$
$252$	−2.00000	−0.125988
$253$	4.00000	0.251478
$254$	13.0000	0.815693
$255$	3.00000	0.187867
$256$	1.00000	0.0625000
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	9.00000	0.560316
$259$	−6.00000	−0.372822
$260$	4.00000	0.248069
$261$	−5.00000	−0.309492
$262$	−18.0000	−1.11204
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	1.00000	0.0615457
$265$	−6.00000	−0.368577
$266$	2.00000	0.122628
$267$	−5.00000	−0.305995
$268$	13.0000	0.794101
$269$	20.0000	1.21942	0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000	1.21942	0.609711	+	0.792624i	$0.291286\pi$
$270$	1.00000	0.0608581
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	3.00000	0.181902
$273$	−8.00000	−0.484182
$274$	8.00000	0.483298
$275$	−4.00000	−0.241209
$276$	4.00000	0.240772
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	15.0000	0.899640
$279$	2.00000	0.119737
$280$	−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	$0.680403\pi$
$282$	8.00000	0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−8.00000	−0.474713
$285$	−1.00000	−0.0592349
$286$	4.00000	0.236525
$287$	−4.00000	−0.236113
$288$	1.00000	0.0589256
$289$	−8.00000	−0.470588
$290$	−5.00000	−0.293610
$291$	8.00000	0.468968
$292$	−16.0000	−0.936329
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	3.00000	0.174371
$297$	1.00000	0.0580259
$298$	−10.0000	−0.579284
$299$	16.0000	0.925304
$300$	−4.00000	−0.230940
$301$	−18.0000	−1.03750
$302$	7.00000	0.402805
$303$	2.00000	0.114897
$304$	−1.00000	−0.0573539
$305$	−13.0000	−0.744378
$306$	3.00000	0.171499
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	−2.00000	−0.113961
$309$	−6.00000	−0.341328
$310$	2.00000	0.113592
$311$	−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
$313$	29.0000	1.63918	0.819588	−	0.572953i	$-0.194202\pi$
$313$	29.0000	1.63918	0.819588	+	0.572953i	$0.194202\pi$
$314$	8.00000	0.451466
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	−6.00000	−0.336463
$319$	−5.00000	−0.279946
$320$	1.00000	0.0559017
$321$	−7.00000	−0.390702
$322$	−8.00000	−0.445823
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	−16.0000	−0.887520
$326$	4.00000	0.221540
$327$	−10.0000	−0.553001
$328$	2.00000	0.110432
$329$	−16.0000	−0.882109
$330$	1.00000	0.0550482
$331$	27.0000	1.48405	0.742027	−	0.670370i	$-0.233865\pi$
$331$	27.0000	1.48405	0.742027	+	0.670370i	$0.233865\pi$
$332$	−6.00000	−0.329293
$333$	3.00000	0.164399
$334$	−12.0000	−0.656611
$335$	13.0000	0.710266
$336$	−2.00000	−0.109109
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	3.00000	0.163178
$339$	−11.0000	−0.597438
$340$	3.00000	0.162698
$341$	2.00000	0.108306
$342$	−1.00000	−0.0540738
$343$	20.0000	1.07990
$344$	9.00000	0.485247
$345$	4.00000	0.215353
$346$	−1.00000	−0.0537603
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	−5.00000	−0.268028
$349$	15.0000	0.802932	0.401466	−	0.915874i	$-0.368501\pi$
$349$	15.0000	0.802932	0.401466	+	0.915874i	$0.368501\pi$
$350$	8.00000	0.427618
$351$	4.00000	0.213504
$352$	1.00000	0.0533002
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	−5.00000	−0.264999
$357$	−6.00000	−0.317554
$358$	0	0
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	1.00000	0.0527046
$361$	1.00000	0.0526316
$362$	−3.00000	−0.157676
$363$	1.00000	0.0524864
$364$	−8.00000	−0.419314
$365$	−16.0000	−0.837478
$366$	−13.0000	−0.679521
$367$	3.00000	0.156599	0.0782994	−	0.996930i	$-0.475051\pi$
$367$	3.00000	0.156599	0.0782994	+	0.996930i	$0.475051\pi$
$368$	4.00000	0.208514
$369$	2.00000	0.104116
$370$	3.00000	0.155963
$371$	12.0000	0.623009
$372$	2.00000	0.103695
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	3.00000	0.155126
$375$	−9.00000	−0.464758
$376$	8.00000	0.412568
$377$	−20.0000	−1.03005
$378$	−2.00000	−0.102869
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	−1.00000	−0.0512989
$381$	13.0000	0.666010
$382$	12.0000	0.613973
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	1.00000	0.0510310
$385$	−2.00000	−0.101929
$386$	−11.0000	−0.559885
$387$	9.00000	0.457496
$388$	8.00000	0.406138
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	4.00000	0.202548
$391$	12.0000	0.606866
$392$	−3.00000	−0.151523
$393$	−18.0000	−0.907980
$394$	−12.0000	−0.604551
$395$	0	0
$396$	1.00000	0.0502519
$397$	−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
$399$	2.00000	0.100125
$400$	−4.00000	−0.200000
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	13.0000	0.648381
$403$	8.00000	0.398508
$404$	2.00000	0.0995037
$405$	1.00000	0.0496904
$406$	10.0000	0.496292
$407$	3.00000	0.148704
$408$	3.00000	0.148522
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	2.00000	0.0987730
$411$	8.00000	0.394611
$412$	−6.00000	−0.295599
$413$	0	0
$414$	4.00000	0.196589
$415$	−6.00000	−0.294528
$416$	4.00000	0.196116
$417$	15.0000	0.734553
$418$	−1.00000	−0.0489116
$419$	35.0000	1.70986	0.854931	−	0.518742i	$-0.173599\pi$
$419$	35.0000	1.70986	0.854931	+	0.518742i	$0.173599\pi$
$420$	−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	$0.319894\pi$
$422$	12.0000	0.584151
$423$	8.00000	0.388973
$424$	−6.00000	−0.291386
$425$	−12.0000	−0.582086
$426$	−8.00000	−0.387601
$427$	26.0000	1.25823
$428$	−7.00000	−0.338358
$429$	4.00000	0.193122
$430$	9.00000	0.434019
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	1.00000	0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−4.00000	−0.192006
$435$	−5.00000	−0.239732
$436$	−10.0000	−0.478913
$437$	−4.00000	−0.191346
$438$	−16.0000	−0.764510
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	1.00000	0.0476731
$441$	−3.00000	−0.142857
$442$	12.0000	0.570782
$443$	−41.0000	−1.94797	−0.973984	−	0.226615i	$-0.927234\pi$
$443$	−41.0000	−1.94797	−0.973984	+	0.226615i	$0.927234\pi$
$444$	3.00000	0.142374
$445$	−5.00000	−0.237023
$446$	−16.0000	−0.757622
$447$	−10.0000	−0.472984
$448$	−2.00000	−0.0944911
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	−4.00000	−0.188562
$451$	2.00000	0.0941763
$452$	−11.0000	−0.517396
$453$	7.00000	0.328889
$454$	−27.0000	−1.26717
$455$	−8.00000	−0.375046
$456$	−1.00000	−0.0468293
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	20.0000	0.934539
$459$	3.00000	0.140028
$460$	4.00000	0.186501
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	−2.00000	−0.0930484
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	−5.00000	−0.232119
$465$	2.00000	0.0927478
$466$	−6.00000	−0.277945
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	4.00000	0.184900
$469$	−26.0000	−1.20057
$470$	8.00000	0.369012
$471$	8.00000	0.368621
$472$	0	0
$473$	9.00000	0.413820
$474$	0	0
$475$	4.00000	0.183533
$476$	−6.00000	−0.275010
$477$	−6.00000	−0.274721
$478$	−15.0000	−0.686084
$479$	−5.00000	−0.228456	−0.114228	−	0.993455i	$-0.536439\pi$
$479$	−5.00000	−0.228456	−0.114228	+	0.993455i	$0.536439\pi$
$480$	1.00000	0.0456435
$481$	12.0000	0.547153
$482$	−3.00000	−0.136646
$483$	−8.00000	−0.364013
$484$	1.00000	0.0454545
$485$	8.00000	0.363261
$486$	1.00000	0.0453609
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	−13.0000	−0.588482
$489$	4.00000	0.180886
$490$	−3.00000	−0.135526
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	2.00000	0.0901670
$493$	−15.0000	−0.675566
$494$	−4.00000	−0.179969
$495$	1.00000	0.0449467
$496$	2.00000	0.0898027
$497$	16.0000	0.717698
$498$	−6.00000	−0.268866
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	−9.00000	−0.402492
$501$	−12.0000	−0.536120
$502$	17.0000	0.758747
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	−2.00000	−0.0890871
$505$	2.00000	0.0889988
$506$	4.00000	0.177822
$507$	3.00000	0.133235
$508$	13.0000	0.576782
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	3.00000	0.132842
$511$	32.0000	1.41560
$512$	1.00000	0.0441942
$513$	−1.00000	−0.0441511
$514$	13.0000	0.573405
$515$	−6.00000	−0.264392
$516$	9.00000	0.396203
$517$	8.00000	0.351840
$518$	−6.00000	−0.263625
$519$	−1.00000	−0.0438951
$520$	4.00000	0.175412
$521$	7.00000	0.306676	0.153338	−	0.988174i	$-0.450998\pi$
$521$	7.00000	0.306676	0.153338	+	0.988174i	$0.450998\pi$
$522$	−5.00000	−0.218844
$523$	14.0000	0.612177	0.306089	−	0.952003i	$-0.400980\pi$
$523$	14.0000	0.612177	0.306089	+	0.952003i	$0.400980\pi$
$524$	−18.0000	−0.786334
$525$	8.00000	0.349149
$526$	−16.0000	−0.697633
$527$	6.00000	0.261364
$528$	1.00000	0.0435194
$529$	−7.00000	−0.304348
$530$	−6.00000	−0.260623
$531$	0	0
$532$	2.00000	0.0867110
$533$	8.00000	0.346518
$534$	−5.00000	−0.216371
$535$	−7.00000	−0.302636
$536$	13.0000	0.561514
$537$	0	0
$538$	20.0000	0.862261
$539$	−3.00000	−0.129219
$540$	1.00000	0.0430331
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−28.0000	−1.20270
$543$	−3.00000	−0.128742
$544$	3.00000	0.128624
$545$	−10.0000	−0.428353
$546$	−8.00000	−0.342368
$547$	−42.0000	−1.79579	−0.897895	−	0.440209i	$-0.854904\pi$
$547$	−42.0000	−1.79579	−0.897895	+	0.440209i	$0.854904\pi$
$548$	8.00000	0.341743
$549$	−13.0000	−0.554826
$550$	−4.00000	−0.170561
$551$	5.00000	0.213007
$552$	4.00000	0.170251
$553$	0	0
$554$	13.0000	0.552317
$555$	3.00000	0.127343
$556$	15.0000	0.636142
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	2.00000	0.0846668
$559$	36.0000	1.52264
$560$	−2.00000	−0.0845154
$561$	3.00000	0.126660
$562$	−18.0000	−0.759284
$563$	19.0000	0.800755	0.400377	−	0.916350i	$-0.368879\pi$
$563$	19.0000	0.800755	0.400377	+	0.916350i	$0.368879\pi$
$564$	8.00000	0.336861
$565$	−11.0000	−0.462773
$566$	4.00000	0.168133
$567$	−2.00000	−0.0839921
$568$	−8.00000	−0.335673
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	−1.00000	−0.0418854
$571$	−23.0000	−0.962520	−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000	−0.962520	−0.481260	+	0.876578i	$0.659821\pi$
$572$	4.00000	0.167248
$573$	12.0000	0.501307
$574$	−4.00000	−0.166957
$575$	−16.0000	−0.667246
$576$	1.00000	0.0416667
$577$	33.0000	1.37381	0.686904	−	0.726748i	$-0.258969\pi$
$577$	33.0000	1.37381	0.686904	+	0.726748i	$0.258969\pi$
$578$	−8.00000	−0.332756
$579$	−11.0000	−0.457144
$580$	−5.00000	−0.207614
$581$	12.0000	0.497844
$582$	8.00000	0.331611
$583$	−6.00000	−0.248495
$584$	−16.0000	−0.662085
$585$	4.00000	0.165380
$586$	−21.0000	−0.867502
$587$	3.00000	0.123823	0.0619116	−	0.998082i	$-0.480280\pi$
$587$	3.00000	0.123823	0.0619116	+	0.998082i	$0.480280\pi$
$588$	−3.00000	−0.123718
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−12.0000	−0.493614
$592$	3.00000	0.123299
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	1.00000	0.0410305
$595$	−6.00000	−0.245976
$596$	−10.0000	−0.409616
$597$	−15.0000	−0.613909
$598$	16.0000	0.654289
$599$	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	$0.487012\pi$
$602$	−18.0000	−0.733625
$603$	13.0000	0.529401
$604$	7.00000	0.284826
$605$	1.00000	0.0406558
$606$	2.00000	0.0812444
$607$	48.0000	1.94826	0.974130	−	0.225989i	$-0.0725612\pi$
$607$	48.0000	1.94826	0.974130	+	0.225989i	$0.0725612\pi$
$608$	−1.00000	−0.0405554
$609$	10.0000	0.405220
$610$	−13.0000	−0.526355
$611$	32.0000	1.29458
$612$	3.00000	0.121268
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	28.0000	1.12999
$615$	2.00000	0.0806478
$616$	−2.00000	−0.0805823
$617$	38.0000	1.52982	0.764911	−	0.644136i	$-0.222783\pi$
$617$	38.0000	1.52982	0.764911	+	0.644136i	$0.222783\pi$
$618$	−6.00000	−0.241355
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	2.00000	0.0803219
$621$	4.00000	0.160514
$622$	−18.0000	−0.721734
$623$	10.0000	0.400642
$624$	4.00000	0.160128
$625$	11.0000	0.440000
$626$	29.0000	1.15907
$627$	−1.00000	−0.0399362
$628$	8.00000	0.319235
$629$	9.00000	0.358854
$630$	−2.00000	−0.0796819
$631$	−23.0000	−0.915616	−0.457808	−	0.889051i	$-0.651365\pi$
$631$	−23.0000	−0.915616	−0.457808	+	0.889051i	$0.651365\pi$
$632$	0	0
$633$	12.0000	0.476957
$634$	−22.0000	−0.873732
$635$	13.0000	0.515889
$636$	−6.00000	−0.237915
$637$	−12.0000	−0.475457
$638$	−5.00000	−0.197952
$639$	−8.00000	−0.316475
$640$	1.00000	0.0395285
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	−7.00000	−0.276268
$643$	−6.00000	−0.236617	−0.118308	−	0.992977i	$-0.537747\pi$
$643$	−6.00000	−0.236617	−0.118308	+	0.992977i	$0.537747\pi$
$644$	−8.00000	−0.315244
$645$	9.00000	0.354375
$646$	−3.00000	−0.118033
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−16.0000	−0.627572
$651$	−4.00000	−0.156772
$652$	4.00000	0.156652
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	−10.0000	−0.391031
$655$	−18.0000	−0.703318
$656$	2.00000	0.0780869
$657$	−16.0000	−0.624219
$658$	−16.0000	−0.623745
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	1.00000	0.0389249
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	27.0000	1.04938
$663$	12.0000	0.466041
$664$	−6.00000	−0.232845
$665$	2.00000	0.0775567
$666$	3.00000	0.116248
$667$	−20.0000	−0.774403
$668$	−12.0000	−0.464294
$669$	−16.0000	−0.618596
$670$	13.0000	0.502234
$671$	−13.0000	−0.501859
$672$	−2.00000	−0.0771517
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	13.0000	0.500741
$675$	−4.00000	−0.153960
$676$	3.00000	0.115385
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	−11.0000	−0.422452
$679$	−16.0000	−0.614024
$680$	3.00000	0.115045
$681$	−27.0000	−1.03464
$682$	2.00000	0.0765840
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	−1.00000	−0.0382360
$685$	8.00000	0.305664
$686$	20.0000	0.763604
$687$	20.0000	0.763048
$688$	9.00000	0.343122
$689$	−24.0000	−0.914327
$690$	4.00000	0.152277
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	−1.00000	−0.0380143
$693$	−2.00000	−0.0759737
$694$	−12.0000	−0.455514
$695$	15.0000	0.568982
$696$	−5.00000	−0.189525
$697$	6.00000	0.227266
$698$	15.0000	0.567758
$699$	−6.00000	−0.226941
$700$	8.00000	0.302372
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	4.00000	0.150970
$703$	−3.00000	−0.113147
$704$	1.00000	0.0376889
$705$	8.00000	0.301297
$706$	24.0000	0.903252
$707$	−4.00000	−0.150435
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	−8.00000	−0.300235
$711$	0	0
$712$	−5.00000	−0.187383
$713$	8.00000	0.299602
$714$	−6.00000	−0.224544
$715$	4.00000	0.149592
$716$	0	0
$717$	−15.0000	−0.560185
$718$	20.0000	0.746393
$719$	10.0000	0.372937	0.186469	−	0.982461i	$-0.440296\pi$
$719$	10.0000	0.372937	0.186469	+	0.982461i	$0.440296\pi$
$720$	1.00000	0.0372678
$721$	12.0000	0.446903
$722$	1.00000	0.0372161
$723$	−3.00000	−0.111571
$724$	−3.00000	−0.111494
$725$	20.0000	0.742781
$726$	1.00000	0.0371135
$727$	−7.00000	−0.259616	−0.129808	−	0.991539i	$-0.541436\pi$
$727$	−7.00000	−0.259616	−0.129808	+	0.991539i	$0.541436\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	−16.0000	−0.592187
$731$	27.0000	0.998631
$732$	−13.0000	−0.480494
$733$	29.0000	1.07114	0.535570	−	0.844491i	$-0.320097\pi$
$733$	29.0000	1.07114	0.535570	+	0.844491i	$0.320097\pi$
$734$	3.00000	0.110732
$735$	−3.00000	−0.110657
$736$	4.00000	0.147442
$737$	13.0000	0.478861
$738$	2.00000	0.0736210
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	3.00000	0.110282
$741$	−4.00000	−0.146944
$742$	12.0000	0.440534
$743$	14.0000	0.513610	0.256805	−	0.966463i	$-0.417330\pi$
$743$	14.0000	0.513610	0.256805	+	0.966463i	$0.417330\pi$
$744$	2.00000	0.0733236
$745$	−10.0000	−0.366372
$746$	−16.0000	−0.585802
$747$	−6.00000	−0.219529
$748$	3.00000	0.109691
$749$	14.0000	0.511549
$750$	−9.00000	−0.328634
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	8.00000	0.291730
$753$	17.0000	0.619514
$754$	−20.0000	−0.728357
$755$	7.00000	0.254756
$756$	−2.00000	−0.0727393
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	25.0000	0.908041
$759$	4.00000	0.145191
$760$	−1.00000	−0.0362738
$761$	27.0000	0.978749	0.489375	−	0.872074i	$-0.337225\pi$
$761$	27.0000	0.978749	0.489375	+	0.872074i	$0.337225\pi$
$762$	13.0000	0.470940
$763$	20.0000	0.724049
$764$	12.0000	0.434145
$765$	3.00000	0.108465
$766$	−21.0000	−0.758761
$767$	0	0
$768$	1.00000	0.0360844
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	−2.00000	−0.0720750
$771$	13.0000	0.468184
$772$	−11.0000	−0.395899
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	9.00000	0.323498
$775$	−8.00000	−0.287368
$776$	8.00000	0.287183
$777$	−6.00000	−0.215249
$778$	−10.0000	−0.358517
$779$	−2.00000	−0.0716574
$780$	4.00000	0.143223
$781$	−8.00000	−0.286263
$782$	12.0000	0.429119
$783$	−5.00000	−0.178685
$784$	−3.00000	−0.107143
$785$	8.00000	0.285532
$786$	−18.0000	−0.642039
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	−12.0000	−0.427482
$789$	−16.0000	−0.569615
$790$	0	0
$791$	22.0000	0.782230
$792$	1.00000	0.0355335
$793$	−52.0000	−1.84657
$794$	−22.0000	−0.780751
$795$	−6.00000	−0.212798
$796$	−15.0000	−0.531661
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	2.00000	0.0707992
$799$	24.0000	0.849059
$800$	−4.00000	−0.141421
$801$	−5.00000	−0.176666
$802$	27.0000	0.953403
$803$	−16.0000	−0.564628
$804$	13.0000	0.458475
$805$	−8.00000	−0.281963
$806$	8.00000	0.281788
$807$	20.0000	0.704033
$808$	2.00000	0.0703598
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	1.00000	0.0351364
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	10.0000	0.350931
$813$	−28.0000	−0.982003
$814$	3.00000	0.105150
$815$	4.00000	0.140114
$816$	3.00000	0.105021
$817$	−9.00000	−0.314870
$818$	10.0000	0.349642
$819$	−8.00000	−0.279543
$820$	2.00000	0.0698430
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	8.00000	0.279032
$823$	19.0000	0.662298	0.331149	−	0.943578i	$-0.392564\pi$
$823$	19.0000	0.662298	0.331149	+	0.943578i	$0.392564\pi$
$824$	−6.00000	−0.209020
$825$	−4.00000	−0.139262
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	4.00000	0.139010
$829$	−45.0000	−1.56291	−0.781457	−	0.623959i	$-0.785523\pi$
$829$	−45.0000	−1.56291	−0.781457	+	0.623959i	$0.785523\pi$
$830$	−6.00000	−0.208263
$831$	13.0000	0.450965
$832$	4.00000	0.138675
$833$	−9.00000	−0.311832
$834$	15.0000	0.519408
$835$	−12.0000	−0.415277
$836$	−1.00000	−0.0345857
$837$	2.00000	0.0691301
$838$	35.0000	1.20905
$839$	−15.0000	−0.517858	−0.258929	−	0.965896i	$-0.583369\pi$
$839$	−15.0000	−0.517858	−0.258929	+	0.965896i	$0.583369\pi$
$840$	−2.00000	−0.0690066
$841$	−4.00000	−0.137931
$842$	22.0000	0.758170
$843$	−18.0000	−0.619953
$844$	12.0000	0.413057
$845$	3.00000	0.103203
$846$	8.00000	0.275046
$847$	−2.00000	−0.0687208
$848$	−6.00000	−0.206041
$849$	4.00000	0.137280
$850$	−12.0000	−0.411597
$851$	12.0000	0.411355
$852$	−8.00000	−0.274075
$853$	19.0000	0.650548	0.325274	−	0.945620i	$-0.394544\pi$
$853$	19.0000	0.650548	0.325274	+	0.945620i	$0.394544\pi$
$854$	26.0000	0.889702
$855$	−1.00000	−0.0341993
$856$	−7.00000	−0.239255
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	4.00000	0.136558
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	9.00000	0.306897
$861$	−4.00000	−0.136320
$862$	−18.0000	−0.613082
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	1.00000	0.0340207
$865$	−1.00000	−0.0340010
$866$	−26.0000	−0.883516
$867$	−8.00000	−0.271694
$868$	−4.00000	−0.135769
$869$	0	0
$870$	−5.00000	−0.169516
$871$	52.0000	1.76195
$872$	−10.0000	−0.338643
$873$	8.00000	0.270759
$874$	−4.00000	−0.135302
$875$	18.0000	0.608511
$876$	−16.0000	−0.540590
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	35.0000	1.18119
$879$	−21.0000	−0.708312
$880$	1.00000	0.0337100
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	−3.00000	−0.101015
$883$	−6.00000	−0.201916	−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000	−0.201916	−0.100958	+	0.994891i	$0.532191\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−41.0000	−1.37742
$887$	−52.0000	−1.74599	−0.872995	−	0.487730i	$-0.837825\pi$
$887$	−52.0000	−1.74599	−0.872995	+	0.487730i	$0.837825\pi$
$888$	3.00000	0.100673
$889$	−26.0000	−0.872012
$890$	−5.00000	−0.167600
$891$	1.00000	0.0335013
$892$	−16.0000	−0.535720
$893$	−8.00000	−0.267710
$894$	−10.0000	−0.334450
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	16.0000	0.534224
$898$	−10.0000	−0.333704
$899$	−10.0000	−0.333519
$900$	−4.00000	−0.133333
$901$	−18.0000	−0.599667
$902$	2.00000	0.0665927
$903$	−18.0000	−0.599002
$904$	−11.0000	−0.365855
$905$	−3.00000	−0.0997234
$906$	7.00000	0.232559
$907$	23.0000	0.763702	0.381851	−	0.924224i	$-0.375287\pi$
$907$	23.0000	0.763702	0.381851	+	0.924224i	$0.375287\pi$
$908$	−27.0000	−0.896026
$909$	2.00000	0.0663358
$910$	−8.00000	−0.265197
$911$	57.0000	1.88849	0.944247	−	0.329238i	$-0.106792\pi$
$911$	57.0000	1.88849	0.944247	+	0.329238i	$0.106792\pi$
$912$	−1.00000	−0.0331133
$913$	−6.00000	−0.198571
$914$	−2.00000	−0.0661541
$915$	−13.0000	−0.429767
$916$	20.0000	0.660819
$917$	36.0000	1.18882
$918$	3.00000	0.0990148
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	4.00000	0.131876
$921$	28.0000	0.922631
$922$	12.0000	0.395199
$923$	−32.0000	−1.05329
$924$	−2.00000	−0.0657952
$925$	−12.0000	−0.394558
$926$	−31.0000	−1.01872
$927$	−6.00000	−0.197066
$928$	−5.00000	−0.164133
$929$	20.0000	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$929$	20.0000	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$930$	2.00000	0.0655826
$931$	3.00000	0.0983210
$932$	−6.00000	−0.196537
$933$	−18.0000	−0.589294
$934$	8.00000	0.261768
$935$	3.00000	0.0981105
$936$	4.00000	0.130744
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	−26.0000	−0.848930
$939$	29.0000	0.946379
$940$	8.00000	0.260931
$941$	−33.0000	−1.07577	−0.537885	−	0.843018i	$-0.680776\pi$
$941$	−33.0000	−1.07577	−0.537885	+	0.843018i	$0.680776\pi$
$942$	8.00000	0.260654
$943$	8.00000	0.260516
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	9.00000	0.292615
$947$	−7.00000	−0.227469	−0.113735	−	0.993511i	$-0.536281\pi$
$947$	−7.00000	−0.227469	−0.113735	+	0.993511i	$0.536281\pi$
$948$	0	0
$949$	−64.0000	−2.07753
$950$	4.00000	0.129777
$951$	−22.0000	−0.713399
$952$	−6.00000	−0.194461
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	−6.00000	−0.194257
$955$	12.0000	0.388311
$956$	−15.0000	−0.485135
$957$	−5.00000	−0.161627
$958$	−5.00000	−0.161543
$959$	−16.0000	−0.516667
$960$	1.00000	0.0322749
$961$	−27.0000	−0.870968
$962$	12.0000	0.386896
$963$	−7.00000	−0.225572
$964$	−3.00000	−0.0966235
$965$	−11.0000	−0.354103
$966$	−8.00000	−0.257396
$967$	38.0000	1.22200	0.610999	−	0.791632i	$-0.290768\pi$
$967$	38.0000	1.22200	0.610999	+	0.791632i	$0.290768\pi$
$968$	1.00000	0.0321412
$969$	−3.00000	−0.0963739
$970$	8.00000	0.256865
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	−30.0000	−0.961756
$974$	18.0000	0.576757
$975$	−16.0000	−0.512410
$976$	−13.0000	−0.416120
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	4.00000	0.127906
$979$	−5.00000	−0.159801
$980$	−3.00000	−0.0958315
$981$	−10.0000	−0.319275
$982$	22.0000	0.702048
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	2.00000	0.0637577
$985$	−12.0000	−0.382352
$986$	−15.0000	−0.477697
$987$	−16.0000	−0.509286
$988$	−4.00000	−0.127257
$989$	36.0000	1.14473
$990$	1.00000	0.0317821
$991$	42.0000	1.33417	0.667087	−	0.744980i	$-0.267541\pi$
$991$	42.0000	1.33417	0.667087	+	0.744980i	$0.267541\pi$
$992$	2.00000	0.0635001
$993$	27.0000	0.856819
$994$	16.0000	0.507489
$995$	−15.0000	−0.475532
$996$	−6.00000	−0.190117
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	0	0
$999$	3.00000	0.0949158

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a_n

instead) (See

a_n

instead) Display

a_n

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1254.2.a.j.1.1	✓	1
3.2	odd	2		3762.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1254.2.a.j.1.1	✓	1	1.1	even	1	trivial
3762.2.a.d.1.1		1	3.2	odd	2

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1254.2.a.j.1.1

Level	$1254$
Weight	$2$
Character	1254.1
Self dual	yes
Analytic conductor	$10.013$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$