LMFDB - Embedded newform 294.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [294,2,Mod(1,294)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(294, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("294.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	294.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,1,1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.34760181943$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	294.1

$q$ -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} -3.00000 q^{10} +3.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -4.00000 q^{19} +3.00000 q^{20} -3.00000 q^{22} -1.00000 q^{24} +4.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} +9.00000 q^{29} -3.00000 q^{30} -1.00000 q^{31} -1.00000 q^{32} +3.00000 q^{33} +1.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} -4.00000 q^{39} -3.00000 q^{40} -10.0000 q^{43} +3.00000 q^{44} +3.00000 q^{45} -6.00000 q^{47} +1.00000 q^{48} -4.00000 q^{50} -4.00000 q^{52} -3.00000 q^{53} -1.00000 q^{54} +9.00000 q^{55} -4.00000 q^{57} -9.00000 q^{58} +3.00000 q^{59} +3.00000 q^{60} -10.0000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} -3.00000 q^{66} -10.0000 q^{67} -6.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -8.00000 q^{74} +4.00000 q^{75} -4.00000 q^{76} +4.00000 q^{78} -1.00000 q^{79} +3.00000 q^{80} +1.00000 q^{81} -9.00000 q^{83} +10.0000 q^{86} +9.00000 q^{87} -3.00000 q^{88} +6.00000 q^{89} -3.00000 q^{90} -1.00000 q^{93} +6.00000 q^{94} -12.0000 q^{95} -1.00000 q^{96} -1.00000 q^{97} +3.00000 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
$5$	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	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−3.00000	−0.639602
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	4.00000	0.800000
$26$	4.00000	0.784465
$27$	1.00000	0.192450
$28$	0	0
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	−3.00000	−0.547723
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	−1.00000	−0.176777
$33$	3.00000	0.522233
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	4.00000	0.648886
$39$	−4.00000	−0.640513
$40$	−3.00000	−0.474342
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	3.00000	0.452267
$45$	3.00000	0.447214
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−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
$57$	−4.00000	−0.529813
$58$	−9.00000	−1.18176
$59$	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
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	−3.00000	−0.369274
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−8.00000	−0.929981
$75$	4.00000	0.461880
$76$	−4.00000	−0.458831
$77$	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
$82$	0	0
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	0	0
$85$	0	0
$86$	10.0000	1.07833
$87$	9.00000	0.964901
$88$	−3.00000	−0.319801
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−3.00000	−0.316228
$91$	0	0
$92$	0	0
$93$	−1.00000	−0.103695
$94$	6.00000	0.618853
$95$	−12.0000	−1.23117
$96$	−1.00000	−0.102062
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	4.00000	0.400000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	3.00000	0.291386
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	1.00000	0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−9.00000	−0.858116
$111$	8.00000	0.759326
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	9.00000	0.835629
$117$	−4.00000	−0.369800
$118$	−3.00000	−0.276172
$119$	0	0
$120$	−3.00000	−0.273861
$121$	−2.00000	−0.181818
$122$	10.0000	0.905357
$123$	0	0
$124$	−1.00000	−0.0898027
$125$	−3.00000	−0.268328
$126$	0	0
$127$	5.00000	0.443678	0.221839	−	0.975083i	$-0.428794\pi$
$127$	5.00000	0.443678	0.221839	+	0.975083i	$0.428794\pi$
$128$	−1.00000	−0.0883883
$129$	−10.0000	−0.880451
$130$	12.0000	1.05247
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	3.00000	0.261116
$133$	0	0
$134$	10.0000	0.863868
$135$	3.00000	0.258199
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	6.00000	0.503509
$143$	−12.0000	−1.00349
$144$	1.00000	0.0833333
$145$	27.0000	2.24223
$146$	−2.00000	−0.165521
$147$	0	0
$148$	8.00000	0.657596
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	−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	$0.512955\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	0	0
$155$	−3.00000	−0.240966
$156$	−4.00000	−0.320256
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	1.00000	0.0795557
$159$	−3.00000	−0.237915
$160$	−3.00000	−0.237171
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	0	0
$165$	9.00000	0.700649
$166$	9.00000	0.698535
$167$	6.00000	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$167$	6.00000	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−10.0000	−0.762493
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	−9.00000	−0.682288
$175$	0	0
$176$	3.00000	0.226134
$177$	3.00000	0.225494
$178$	−6.00000	−0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	3.00000	0.223607
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	24.0000	1.76452
$186$	1.00000	0.0733236
$187$	0	0
$188$	−6.00000	−0.437595
$189$	0	0
$190$	12.0000	0.870572
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000	0.0721688
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	1.00000	0.0717958
$195$	−12.0000	−0.859338
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−3.00000	−0.213201
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	−4.00000	−0.282843
$201$	−10.0000	−0.705346
$202$	18.0000	1.26648
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−12.0000	−0.830057
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	−3.00000	−0.206041
$213$	−6.00000	−0.411113
$214$	3.00000	0.205076
$215$	−30.0000	−2.04598
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−14.0000	−0.948200
$219$	2.00000	0.135147
$220$	9.00000	0.606780
$221$	0	0
$222$	−8.00000	−0.536925
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	−27.0000	−1.79205	−0.896026	−	0.444001i	$-0.853559\pi$
$227$	−27.0000	−1.79205	−0.896026	+	0.444001i	$0.853559\pi$
$228$	−4.00000	−0.264906
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	0	0
$231$	0	0
$232$	−9.00000	−0.590879
$233$	−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
$241$	−1.00000	−0.0644157	−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000	−0.0644157	−0.0322078	+	0.999481i	$0.510254\pi$
$242$	2.00000	0.128565
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	16.0000	1.01806
$248$	1.00000	0.0635001
$249$	−9.00000	−0.570352
$250$	3.00000	0.189737
$251$	27.0000	1.70422	0.852112	−	0.523359i	$-0.175321\pi$
$251$	27.0000	1.70422	0.852112	+	0.523359i	$0.175321\pi$
$252$	0	0
$253$	0	0
$254$	−5.00000	−0.313728
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	10.0000	0.622573
$259$	0	0
$260$	−12.0000	−0.744208
$261$	9.00000	0.557086
$262$	9.00000	0.556022
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	−3.00000	−0.184637
$265$	−9.00000	−0.552866
$266$	0	0
$267$	6.00000	0.367194
$268$	−10.0000	−0.610847
$269$	21.0000	1.28039	0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000	1.28039	0.640196	+	0.768211i	$0.278853\pi$
$270$	−3.00000	−0.182574
$271$	11.0000	0.668202	0.334101	−	0.942537i	$-0.391567\pi$
$271$	11.0000	0.668202	0.334101	+	0.942537i	$0.391567\pi$
$272$	0	0
$273$	0	0
$274$	−18.0000	−1.08742
$275$	12.0000	0.723627
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−2.00000	−0.119952
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	6.00000	0.357295
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−6.00000	−0.356034
$285$	−12.0000	−0.710819
$286$	12.0000	0.709575
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−17.0000	−1.00000
$290$	−27.0000	−1.58549
$291$	−1.00000	−0.0586210
$292$	2.00000	0.117041
$293$	33.0000	1.92788	0.963940	−	0.266119i	$-0.0857413\pi$
$293$	33.0000	1.92788	0.963940	+	0.266119i	$0.0857413\pi$
$294$	0	0
$295$	9.00000	0.524000
$296$	−8.00000	−0.464991
$297$	3.00000	0.174078
$298$	−18.0000	−1.04271
$299$	0	0
$300$	4.00000	0.230940
$301$	0	0
$302$	1.00000	0.0575435
$303$	−18.0000	−1.03407
$304$	−4.00000	−0.229416
$305$	−30.0000	−1.71780
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	3.00000	0.170389
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	4.00000	0.226455
$313$	−31.0000	−1.75222	−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000	−1.75222	−0.876112	+	0.482108i	$0.839871\pi$
$314$	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
$319$	27.0000	1.51171
$320$	3.00000	0.167705
$321$	−3.00000	−0.167444
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−16.0000	−0.887520
$326$	16.0000	0.886158
$327$	14.0000	0.774202
$328$	0	0
$329$	0	0
$330$	−9.00000	−0.495434
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	−9.00000	−0.493939
$333$	8.00000	0.438397
$334$	−6.00000	−0.328305
$335$	−30.0000	−1.63908
$336$	0	0
$337$	−7.00000	−0.381314	−0.190657	−	0.981657i	$-0.561062\pi$
$337$	−7.00000	−0.381314	−0.190657	+	0.981657i	$0.561062\pi$
$338$	−3.00000	−0.163178
$339$	0	0
$340$	0	0
$341$	−3.00000	−0.162459
$342$	4.00000	0.216295
$343$	0	0
$344$	10.0000	0.539164
$345$	0	0
$346$	−18.0000	−0.967686
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	9.00000	0.482451
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	−3.00000	−0.159901
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	−3.00000	−0.159448
$355$	−18.0000	−0.955341
$356$	6.00000	0.317999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	−3.00000	−0.158114
$361$	−3.00000	−0.157895
$362$	−8.00000	−0.420471
$363$	−2.00000	−0.104973
$364$	0	0
$365$	6.00000	0.314054
$366$	10.0000	0.522708
$367$	−19.0000	−0.991792	−0.495896	−	0.868382i	$-0.665160\pi$
$367$	−19.0000	−0.991792	−0.495896	+	0.868382i	$0.665160\pi$
$368$	0	0
$369$	0	0
$370$	−24.0000	−1.24770
$371$	0	0
$372$	−1.00000	−0.0518476
$373$	8.00000	0.414224	0.207112	−	0.978317i	$-0.433593\pi$
$373$	8.00000	0.414224	0.207112	+	0.978317i	$0.433593\pi$
$374$	0	0
$375$	−3.00000	−0.154919
$376$	6.00000	0.309426
$377$	−36.0000	−1.85409
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	−12.0000	−0.615587
$381$	5.00000	0.256158
$382$	0	0
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	19.0000	0.967075
$387$	−10.0000	−0.508329
$388$	−1.00000	−0.0507673
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	12.0000	0.607644
$391$	0	0
$392$	0	0
$393$	−9.00000	−0.453990
$394$	−6.00000	−0.302276
$395$	−3.00000	−0.150946
$396$	3.00000	0.150756
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	−20.0000	−1.00251
$399$	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	$0.295465\pi$
$402$	10.0000	0.498755
$403$	4.00000	0.199254
$404$	−18.0000	−0.895533
$405$	3.00000	0.149071
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	8.00000	0.394132
$413$	0	0
$414$	0	0
$415$	−27.0000	−1.32538
$416$	4.00000	0.196116
$417$	2.00000	0.0979404
$418$	12.0000	0.586939
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−14.0000	−0.681509
$423$	−6.00000	−0.291730
$424$	3.00000	0.145693
$425$	0	0
$426$	6.00000	0.290701
$427$	0	0
$428$	−3.00000	−0.145010
$429$	−12.0000	−0.579365
$430$	30.0000	1.44673
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	1.00000	0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	27.0000	1.29455
$436$	14.0000	0.670478
$437$	0	0
$438$	−2.00000	−0.0955637
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	−9.00000	−0.429058
$441$	0	0
$442$	0	0
$443$	−33.0000	−1.56788	−0.783939	−	0.620838i	$-0.786792\pi$
$443$	−33.0000	−1.56788	−0.783939	+	0.620838i	$0.786792\pi$
$444$	8.00000	0.379663
$445$	18.0000	0.853282
$446$	19.0000	0.899676
$447$	18.0000	0.851371
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	−4.00000	−0.188562
$451$	0	0
$452$	0	0
$453$	−1.00000	−0.0469841
$454$	27.0000	1.26717
$455$	0	0
$456$	4.00000	0.187317
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	4.00000	0.186908
$459$	0	0
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	9.00000	0.417815
$465$	−3.00000	−0.139122
$466$	24.0000	1.11178
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	−4.00000	−0.184900
$469$	0	0
$470$	18.0000	0.830278
$471$	−4.00000	−0.184310
$472$	−3.00000	−0.138086
$473$	−30.0000	−1.37940
$474$	1.00000	0.0459315
$475$	−16.0000	−0.734130
$476$	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
$481$	−32.0000	−1.45907
$482$	1.00000	0.0455488
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	−3.00000	−0.136223
$486$	−1.00000	−0.0453609
$487$	41.0000	1.85789	0.928944	−	0.370221i	$-0.120718\pi$
$487$	41.0000	1.85789	0.928944	+	0.370221i	$0.120718\pi$
$488$	10.0000	0.452679
$489$	−16.0000	−0.723545
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	0	0
$493$	0	0
$494$	−16.0000	−0.719874
$495$	9.00000	0.404520
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	9.00000	0.403300
$499$	2.00000	0.0895323	0.0447661	−	0.998997i	$-0.485746\pi$
$499$	2.00000	0.0895323	0.0447661	+	0.998997i	$0.485746\pi$
$500$	−3.00000	−0.134164
$501$	6.00000	0.268060
$502$	−27.0000	−1.20507
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	−54.0000	−2.40297
$506$	0	0
$507$	3.00000	0.133235
$508$	5.00000	0.221839
$509$	−3.00000	−0.132973	−0.0664863	−	0.997787i	$-0.521179\pi$
$509$	−3.00000	−0.132973	−0.0664863	+	0.997787i	$0.521179\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	−6.00000	−0.264649
$515$	24.0000	1.05757
$516$	−10.0000	−0.440225
$517$	−18.0000	−0.791639
$518$	0	0
$519$	18.0000	0.790112
$520$	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	$0.629012\pi$
$522$	−9.00000	−0.393919
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−9.00000	−0.393167
$525$	0	0
$526$	6.00000	0.261612
$527$	0	0
$528$	3.00000	0.130558
$529$	−23.0000	−1.00000
$530$	9.00000	0.390935
$531$	3.00000	0.130189
$532$	0	0
$533$	0	0
$534$	−6.00000	−0.259645
$535$	−9.00000	−0.389104
$536$	10.0000	0.431934
$537$	12.0000	0.517838
$538$	−21.0000	−0.905374
$539$	0	0
$540$	3.00000	0.129099
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	−11.0000	−0.472490
$543$	8.00000	0.343313
$544$	0	0
$545$	42.0000	1.79908
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	18.0000	0.768922
$549$	−10.0000	−0.426790
$550$	−12.0000	−0.511682
$551$	−36.0000	−1.53365
$552$	0	0
$553$	0	0
$554$	−8.00000	−0.339887
$555$	24.0000	1.01874
$556$	2.00000	0.0848189
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	1.00000	0.0423334
$559$	40.0000	1.69182
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	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
$565$	0	0
$566$	−14.0000	−0.588464
$567$	0	0
$568$	6.00000	0.251754
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	12.0000	0.502625
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	−12.0000	−0.501745
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	17.0000	0.707107
$579$	−19.0000	−0.789613
$580$	27.0000	1.12111
$581$	0	0
$582$	1.00000	0.0414513
$583$	−9.00000	−0.372742
$584$	−2.00000	−0.0827606
$585$	−12.0000	−0.496139
$586$	−33.0000	−1.36322
$587$	21.0000	0.866763	0.433381	−	0.901211i	$-0.357320\pi$
$587$	21.0000	0.866763	0.433381	+	0.901211i	$0.357320\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	−9.00000	−0.370524
$591$	6.00000	0.246807
$592$	8.00000	0.328798
$593$	24.0000	0.985562	0.492781	−	0.870153i	$-0.335980\pi$
$593$	24.0000	0.985562	0.492781	+	0.870153i	$0.335980\pi$
$594$	−3.00000	−0.123091
$595$	0	0
$596$	18.0000	0.737309
$597$	20.0000	0.818546
$598$	0	0
$599$	−18.0000	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$599$	−18.0000	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$600$	−4.00000	−0.163299
$601$	11.0000	0.448699	0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000	0.448699	0.224350	+	0.974509i	$0.427974\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	−1.00000	−0.0406894
$605$	−6.00000	−0.243935
$606$	18.0000	0.731200
$607$	−7.00000	−0.284121	−0.142061	−	0.989858i	$-0.545373\pi$
$607$	−7.00000	−0.284121	−0.142061	+	0.989858i	$0.545373\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	30.0000	1.21466
$611$	24.0000	0.970936
$612$	0	0
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−8.00000	−0.321807
$619$	−34.0000	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$619$	−34.0000	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$620$	−3.00000	−0.120483
$621$	0	0
$622$	−24.0000	−0.962312
$623$	0	0
$624$	−4.00000	−0.160128
$625$	−29.0000	−1.16000
$626$	31.0000	1.23901
$627$	−12.0000	−0.479234
$628$	−4.00000	−0.159617
$629$	0	0
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	1.00000	0.0397779
$633$	14.0000	0.556450
$634$	−9.00000	−0.357436
$635$	15.0000	0.595257
$636$	−3.00000	−0.118958
$637$	0	0
$638$	−27.0000	−1.06894
$639$	−6.00000	−0.237356
$640$	−3.00000	−0.118585
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	3.00000	0.118401
$643$	−34.0000	−1.34083	−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000	−1.34083	−0.670415	+	0.741987i	$0.733884\pi$
$644$	0	0
$645$	−30.0000	−1.18125
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	−1.00000	−0.0392837
$649$	9.00000	0.353281
$650$	16.0000	0.627572
$651$	0	0
$652$	−16.0000	−0.626608
$653$	3.00000	0.117399	0.0586995	−	0.998276i	$-0.481305\pi$
$653$	3.00000	0.117399	0.0586995	+	0.998276i	$0.481305\pi$
$654$	−14.0000	−0.547443
$655$	−27.0000	−1.05498
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	9.00000	0.350325
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	9.00000	0.349268
$665$	0	0
$666$	−8.00000	−0.309994
$667$	0	0
$668$	6.00000	0.232147
$669$	−19.0000	−0.734582
$670$	30.0000	1.15900
$671$	−30.0000	−1.15814
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	7.00000	0.269630
$675$	4.00000	0.153960
$676$	3.00000	0.115385
$677$	−33.0000	−1.26829	−0.634147	−	0.773213i	$-0.718648\pi$
$677$	−33.0000	−1.26829	−0.634147	+	0.773213i	$0.718648\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−27.0000	−1.03464
$682$	3.00000	0.114876
$683$	33.0000	1.26271	0.631355	−	0.775494i	$-0.282499\pi$
$683$	33.0000	1.26271	0.631355	+	0.775494i	$0.282499\pi$
$684$	−4.00000	−0.152944
$685$	54.0000	2.06323
$686$	0	0
$687$	−4.00000	−0.152610
$688$	−10.0000	−0.381246
$689$	12.0000	0.457164
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−12.0000	−0.455514
$695$	6.00000	0.227593
$696$	−9.00000	−0.341144
$697$	0	0
$698$	−26.0000	−0.984115
$699$	−24.0000	−0.907763
$700$	0	0
$701$	−15.0000	−0.566542	−0.283271	−	0.959040i	$-0.591420\pi$
$701$	−15.0000	−0.566542	−0.283271	+	0.959040i	$0.591420\pi$
$702$	4.00000	0.150970
$703$	−32.0000	−1.20690
$704$	3.00000	0.113067
$705$	−18.0000	−0.677919
$706$	−24.0000	−0.903252
$707$	0	0
$708$	3.00000	0.112747
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	18.0000	0.675528
$711$	−1.00000	−0.0375029
$712$	−6.00000	−0.224860
$713$	0	0
$714$	0	0
$715$	−36.0000	−1.34632
$716$	12.0000	0.448461
$717$	−24.0000	−0.896296
$718$	−30.0000	−1.11959
$719$	−18.0000	−0.671287	−0.335643	−	0.941989i	$-0.608954\pi$
$719$	−18.0000	−0.671287	−0.335643	+	0.941989i	$0.608954\pi$
$720$	3.00000	0.111803
$721$	0	0
$722$	3.00000	0.111648
$723$	−1.00000	−0.0371904
$724$	8.00000	0.297318
$725$	36.0000	1.33701
$726$	2.00000	0.0742270
$727$	−13.0000	−0.482143	−0.241072	−	0.970507i	$-0.577499\pi$
$727$	−13.0000	−0.482143	−0.241072	+	0.970507i	$0.577499\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	0	0
$732$	−10.0000	−0.369611
$733$	−10.0000	−0.369358	−0.184679	−	0.982799i	$-0.559125\pi$
$733$	−10.0000	−0.369358	−0.184679	+	0.982799i	$0.559125\pi$
$734$	19.0000	0.701303
$735$	0	0
$736$	0	0
$737$	−30.0000	−1.10506
$738$	0	0
$739$	50.0000	1.83928	0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000	1.83928	0.919640	+	0.392763i	$0.128481\pi$
$740$	24.0000	0.882258
$741$	16.0000	0.587775
$742$	0	0
$743$	42.0000	1.54083	0.770415	−	0.637542i	$-0.220049\pi$
$743$	42.0000	1.54083	0.770415	+	0.637542i	$0.220049\pi$
$744$	1.00000	0.0366618
$745$	54.0000	1.97841
$746$	−8.00000	−0.292901
$747$	−9.00000	−0.329293
$748$	0	0
$749$	0	0
$750$	3.00000	0.109545
$751$	−7.00000	−0.255434	−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000	−0.255434	−0.127717	+	0.991811i	$0.540765\pi$
$752$	−6.00000	−0.218797
$753$	27.0000	0.983935
$754$	36.0000	1.31104
$755$	−3.00000	−0.109181
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−8.00000	−0.290573
$759$	0	0
$760$	12.0000	0.435286
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−5.00000	−0.181131
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−18.0000	−0.650366
$767$	−12.0000	−0.433295
$768$	1.00000	0.0360844
$769$	−19.0000	−0.685158	−0.342579	−	0.939489i	$-0.611300\pi$
$769$	−19.0000	−0.685158	−0.342579	+	0.939489i	$0.611300\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−19.0000	−0.683825
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	10.0000	0.359443
$775$	−4.00000	−0.143684
$776$	1.00000	0.0358979
$777$	0	0
$778$	6.00000	0.215110
$779$	0	0
$780$	−12.0000	−0.429669
$781$	−18.0000	−0.644091
$782$	0	0
$783$	9.00000	0.321634
$784$	0	0
$785$	−12.0000	−0.428298
$786$	9.00000	0.321019
$787$	50.0000	1.78231	0.891154	−	0.453701i	$-0.149897\pi$
$787$	50.0000	1.78231	0.891154	+	0.453701i	$0.149897\pi$
$788$	6.00000	0.213741
$789$	−6.00000	−0.213606
$790$	3.00000	0.106735
$791$	0	0
$792$	−3.00000	−0.106600
$793$	40.0000	1.42044
$794$	4.00000	0.141955
$795$	−9.00000	−0.319197
$796$	20.0000	0.708881
$797$	−33.0000	−1.16892	−0.584460	−	0.811423i	$-0.698694\pi$
$797$	−33.0000	−1.16892	−0.584460	+	0.811423i	$0.698694\pi$
$798$	0	0
$799$	0	0
$800$	−4.00000	−0.141421
$801$	6.00000	0.212000
$802$	−24.0000	−0.847469
$803$	6.00000	0.211735
$804$	−10.0000	−0.352673
$805$	0	0
$806$	−4.00000	−0.140894
$807$	21.0000	0.739235
$808$	18.0000	0.633238
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	−3.00000	−0.105409
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	11.0000	0.385787
$814$	−24.0000	−0.841200
$815$	−48.0000	−1.68137
$816$	0	0
$817$	40.0000	1.39942
$818$	25.0000	0.874105
$819$	0	0
$820$	0	0
$821$	−3.00000	−0.104701	−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000	−0.104701	−0.0523504	+	0.998629i	$0.516671\pi$
$822$	−18.0000	−0.627822
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	−8.00000	−0.278693
$825$	12.0000	0.417786
$826$	0	0
$827$	15.0000	0.521601	0.260801	−	0.965393i	$-0.416014\pi$
$827$	15.0000	0.521601	0.260801	+	0.965393i	$0.416014\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	27.0000	0.937184
$831$	8.00000	0.277517
$832$	−4.00000	−0.138675
$833$	0	0
$834$	−2.00000	−0.0692543
$835$	18.0000	0.622916
$836$	−12.0000	−0.415029
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	22.0000	0.758170
$843$	6.00000	0.206651
$844$	14.0000	0.481900
$845$	9.00000	0.309609
$846$	6.00000	0.206284
$847$	0	0
$848$	−3.00000	−0.103020
$849$	14.0000	0.480479
$850$	0	0
$851$	0	0
$852$	−6.00000	−0.205557
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	0	0
$855$	−12.0000	−0.410391
$856$	3.00000	0.102538
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	12.0000	0.409673
$859$	50.0000	1.70598	0.852989	−	0.521929i	$-0.174787\pi$
$859$	50.0000	1.70598	0.852989	+	0.521929i	$0.174787\pi$
$860$	−30.0000	−1.02299
$861$	0	0
$862$	−12.0000	−0.408722
$863$	6.00000	0.204242	0.102121	−	0.994772i	$-0.467437\pi$
$863$	6.00000	0.204242	0.102121	+	0.994772i	$0.467437\pi$
$864$	−1.00000	−0.0340207
$865$	54.0000	1.83606
$866$	34.0000	1.15537
$867$	−17.0000	−0.577350
$868$	0	0
$869$	−3.00000	−0.101768
$870$	−27.0000	−0.915386
$871$	40.0000	1.35535
$872$	−14.0000	−0.474100
$873$	−1.00000	−0.0338449
$874$	0	0
$875$	0	0
$876$	2.00000	0.0675737
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−35.0000	−1.18119
$879$	33.0000	1.11306
$880$	9.00000	0.303390
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	0	0
$883$	32.0000	1.07689	0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000	1.07689	0.538443	+	0.842662i	$0.319013\pi$
$884$	0	0
$885$	9.00000	0.302532
$886$	33.0000	1.10866
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	−8.00000	−0.268462
$889$	0	0
$890$	−18.0000	−0.603361
$891$	3.00000	0.100504
$892$	−19.0000	−0.636167
$893$	24.0000	0.803129
$894$	−18.0000	−0.602010
$895$	36.0000	1.20335
$896$	0	0
$897$	0	0
$898$	−12.0000	−0.400445
$899$	−9.00000	−0.300167
$900$	4.00000	0.133333
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	24.0000	0.797787
$906$	1.00000	0.0332228
$907$	8.00000	0.265636	0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000	0.265636	0.132818	+	0.991140i	$0.457597\pi$
$908$	−27.0000	−0.896026
$909$	−18.0000	−0.597022
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	−4.00000	−0.132453
$913$	−27.0000	−0.893570
$914$	1.00000	0.0330771
$915$	−30.0000	−0.991769
$916$	−4.00000	−0.132164
$917$	0	0
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	−30.0000	−0.987997
$923$	24.0000	0.789970
$924$	0	0
$925$	32.0000	1.05215
$926$	−8.00000	−0.262896
$927$	8.00000	0.262754
$928$	−9.00000	−0.295439
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	3.00000	0.0983739
$931$	0	0
$932$	−24.0000	−0.786146
$933$	24.0000	0.785725
$934$	−36.0000	−1.17796
$935$	0	0
$936$	4.00000	0.130744
$937$	35.0000	1.14340	0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000	1.14340	0.571700	+	0.820463i	$0.306284\pi$
$938$	0	0
$939$	−31.0000	−1.01165
$940$	−18.0000	−0.587095
$941$	−9.00000	−0.293392	−0.146696	−	0.989182i	$-0.546864\pi$
$941$	−9.00000	−0.293392	−0.146696	+	0.989182i	$0.546864\pi$
$942$	4.00000	0.130327
$943$	0	0
$944$	3.00000	0.0976417
$945$	0	0
$946$	30.0000	0.975384
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	−1.00000	−0.0324785
$949$	−8.00000	−0.259691
$950$	16.0000	0.519109
$951$	9.00000	0.291845
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	−24.0000	−0.776215
$957$	27.0000	0.872786
$958$	18.0000	0.581554
$959$	0	0
$960$	3.00000	0.0968246
$961$	−30.0000	−0.967742
$962$	32.0000	1.03172
$963$	−3.00000	−0.0966736
$964$	−1.00000	−0.0322078
$965$	−57.0000	−1.83489
$966$	0	0
$967$	−1.00000	−0.0321578	−0.0160789	−	0.999871i	$-0.505118\pi$
$967$	−1.00000	−0.0321578	−0.0160789	+	0.999871i	$0.505118\pi$
$968$	2.00000	0.0642824
$969$	0	0
$970$	3.00000	0.0963242
$971$	−39.0000	−1.25157	−0.625785	−	0.779996i	$-0.715221\pi$
$971$	−39.0000	−1.25157	−0.625785	+	0.779996i	$0.715221\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−41.0000	−1.31372
$975$	−16.0000	−0.512410
$976$	−10.0000	−0.320092
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	16.0000	0.511624
$979$	18.0000	0.575282
$980$	0	0
$981$	14.0000	0.446986
$982$	33.0000	1.05307
$983$	36.0000	1.14822	0.574111	−	0.818778i	$-0.305348\pi$
$983$	36.0000	1.14822	0.574111	+	0.818778i	$0.305348\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	0	0
$988$	16.0000	0.509028
$989$	0	0
$990$	−9.00000	−0.286039
$991$	−13.0000	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$991$	−13.0000	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$992$	1.00000	0.0317500
$993$	20.0000	0.634681
$994$	0	0
$995$	60.0000	1.90213
$996$	−9.00000	−0.285176
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	−2.00000	−0.0633089
$999$	8.00000	0.253109

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.2.a.d.1.1		1
3.2	odd	2		882.2.a.g.1.1		1
4.3	odd	2		2352.2.a.m.1.1		1
5.4	even	2		7350.2.a.ce.1.1		1
7.2	even	3		42.2.e.b.25.1	✓	2
7.3	odd	6		294.2.e.f.79.1		2
7.4	even	3		42.2.e.b.37.1	yes	2
7.5	odd	6		294.2.e.f.67.1		2
7.6	odd	2		294.2.a.a.1.1		1
8.3	odd	2		9408.2.a.bu.1.1		1
8.5	even	2		9408.2.a.d.1.1		1
12.11	even	2		7056.2.a.g.1.1		1
21.2	odd	6		126.2.g.b.109.1		2
21.5	even	6		882.2.g.b.361.1		2
21.11	odd	6		126.2.g.b.37.1		2
21.17	even	6		882.2.g.b.667.1		2
21.20	even	2		882.2.a.k.1.1		1
28.3	even	6		2352.2.q.m.961.1		2
28.11	odd	6		336.2.q.d.289.1		2
28.19	even	6		2352.2.q.m.1537.1		2
28.23	odd	6		336.2.q.d.193.1		2
28.27	even	2		2352.2.a.n.1.1		1
35.2	odd	12		1050.2.o.b.949.1		4
35.4	even	6		1050.2.i.e.751.1		2
35.9	even	6		1050.2.i.e.151.1		2
35.18	odd	12		1050.2.o.b.499.1		4
35.23	odd	12		1050.2.o.b.949.2		4
35.32	odd	12		1050.2.o.b.499.2		4
35.34	odd	2		7350.2.a.cw.1.1		1
56.11	odd	6		1344.2.q.j.961.1		2
56.13	odd	2		9408.2.a.db.1.1		1
56.27	even	2		9408.2.a.bm.1.1		1
56.37	even	6		1344.2.q.v.193.1		2
56.51	odd	6		1344.2.q.j.193.1		2
56.53	even	6		1344.2.q.v.961.1		2
63.2	odd	6		1134.2.h.a.109.1		2
63.4	even	3		1134.2.h.p.541.1		2
63.11	odd	6		1134.2.e.p.919.1		2
63.16	even	3		1134.2.h.p.109.1		2
63.23	odd	6		1134.2.e.p.865.1		2
63.25	even	3		1134.2.e.a.919.1		2
63.32	odd	6		1134.2.h.a.541.1		2
63.58	even	3		1134.2.e.a.865.1		2
84.11	even	6		1008.2.s.n.289.1		2
84.23	even	6		1008.2.s.n.865.1		2
84.83	odd	2		7056.2.a.bz.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.e.b.25.1	✓	2	7.2	even	3
42.2.e.b.37.1	yes	2	7.4	even	3
126.2.g.b.37.1		2	21.11	odd	6
126.2.g.b.109.1		2	21.2	odd	6
294.2.a.a.1.1		1	7.6	odd	2
294.2.a.d.1.1		1	1.1	even	1	trivial
294.2.e.f.67.1		2	7.5	odd	6
294.2.e.f.79.1		2	7.3	odd	6
336.2.q.d.193.1		2	28.23	odd	6
336.2.q.d.289.1		2	28.11	odd	6
882.2.a.g.1.1		1	3.2	odd	2
882.2.a.k.1.1		1	21.20	even	2
882.2.g.b.361.1		2	21.5	even	6
882.2.g.b.667.1		2	21.17	even	6
1008.2.s.n.289.1		2	84.11	even	6
1008.2.s.n.865.1		2	84.23	even	6
1050.2.i.e.151.1		2	35.9	even	6
1050.2.i.e.751.1		2	35.4	even	6
1050.2.o.b.499.1		4	35.18	odd	12
1050.2.o.b.499.2		4	35.32	odd	12
1050.2.o.b.949.1		4	35.2	odd	12
1050.2.o.b.949.2		4	35.23	odd	12
1134.2.e.a.865.1		2	63.58	even	3
1134.2.e.a.919.1		2	63.25	even	3
1134.2.e.p.865.1		2	63.23	odd	6
1134.2.e.p.919.1		2	63.11	odd	6
1134.2.h.a.109.1		2	63.2	odd	6
1134.2.h.a.541.1		2	63.32	odd	6
1134.2.h.p.109.1		2	63.16	even	3
1134.2.h.p.541.1		2	63.4	even	3
1344.2.q.j.193.1		2	56.51	odd	6
1344.2.q.j.961.1		2	56.11	odd	6
1344.2.q.v.193.1		2	56.37	even	6
1344.2.q.v.961.1		2	56.53	even	6
2352.2.a.m.1.1		1	4.3	odd	2
2352.2.a.n.1.1		1	28.27	even	2
2352.2.q.m.961.1		2	28.3	even	6
2352.2.q.m.1537.1		2	28.19	even	6
7056.2.a.g.1.1		1	12.11	even	2
7056.2.a.bz.1.1		1	84.83	odd	2
7350.2.a.ce.1.1		1	5.4	even	2
7350.2.a.cw.1.1		1	35.34	odd	2
9408.2.a.d.1.1		1	8.5	even	2
9408.2.a.bm.1.1		1	56.27	even	2
9408.2.a.bu.1.1		1	8.3	odd	2
9408.2.a.db.1.1		1	56.13	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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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	294.2.a.d.1.1

Level	$294$
Weight	$2$
Character	294.1
Self dual	yes
Analytic conductor	$2.348$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$