LMFDB - Embedded newform 8281.2.a.cx.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.565505 q^{2} -3.09111 q^{3} -1.68020 q^{4} -1.80514 q^{5} -1.74804 q^{6} -2.08118 q^{8} +6.55499 q^{9} +O(q^{10})$	\(q+0.565505 q^{2} -3.09111 q^{3} -1.68020 q^{4} -1.80514 q^{5} -1.74804 q^{6} -2.08118 q^{8} +6.55499 q^{9} -1.02082 q^{10} -3.58433 q^{11} +5.19370 q^{12} +5.57989 q^{15} +2.18349 q^{16} -6.14919 q^{17} +3.70688 q^{18} +2.51075 q^{19} +3.03300 q^{20} -2.02696 q^{22} +4.13185 q^{23} +6.43315 q^{24} -1.74147 q^{25} -10.9889 q^{27} +5.11133 q^{29} +3.15546 q^{30} -4.55318 q^{31} +5.39713 q^{32} +11.0796 q^{33} -3.47740 q^{34} -11.0137 q^{36} -11.7215 q^{37} +1.41984 q^{38} +3.75681 q^{40} +1.86640 q^{41} -1.42314 q^{43} +6.02240 q^{44} -11.8327 q^{45} +2.33658 q^{46} -8.86595 q^{47} -6.74942 q^{48} -0.984810 q^{50} +19.0078 q^{51} +2.02920 q^{53} -6.21427 q^{54} +6.47021 q^{55} -7.76102 q^{57} +2.89049 q^{58} -9.10752 q^{59} -9.37536 q^{60} -5.47841 q^{61} -2.57485 q^{62} -1.31488 q^{64} +6.26555 q^{66} +7.02055 q^{67} +10.3319 q^{68} -12.7720 q^{69} -4.49410 q^{71} -13.6421 q^{72} -9.35564 q^{73} -6.62856 q^{74} +5.38308 q^{75} -4.21858 q^{76} +5.20970 q^{79} -3.94151 q^{80} +14.3029 q^{81} +1.05546 q^{82} -9.42491 q^{83} +11.1001 q^{85} -0.804791 q^{86} -15.7997 q^{87} +7.45961 q^{88} -0.355648 q^{89} -6.69144 q^{90} -6.94235 q^{92} +14.0744 q^{93} -5.01374 q^{94} -4.53226 q^{95} -16.6831 q^{96} -10.6035 q^{97} -23.4952 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 40 q^{4} + 56 q^{9}+O(q^{10}) $ 32 * q + 40 * q^4 + 56 * q^9	$ 32 q + 40 q^{4} + 56 q^{9} + 56 q^{16} - 16 q^{22} + 48 q^{23} + 40 q^{25} + 48 q^{29} + 48 q^{30} + 184 q^{36} + 24 q^{43} + 72 q^{53} - 32 q^{64} - 48 q^{74} + 96 q^{79} + 128 q^{81} + 112 q^{88} + 168 q^{92} + 168 q^{95}+O(q^{100}) $ 32 * q + 40 * q^4 + 56 * q^9 + 56 * q^16 - 16 * q^22 + 48 * q^23 + 40 * q^25 + 48 * q^29 + 48 * q^30 + 184 * q^36 + 24 * q^43 + 72 * q^53 - 32 * q^64 - 48 * q^74 + 96 * q^79 + 128 * q^81 + 112 * q^88 + 168 * q^92 + 168 * q^95

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 $

$2$	0.565505	0.399873	0.199936	−	0.979809i	$-0.435926\pi$
$2$	0.565505	0.399873	0.199936	+	0.979809i	$0.435926\pi$
$3$	−3.09111	−1.78466	−0.892328	−	0.451388i	$-0.850929\pi$
$3$	−3.09111	−1.78466	−0.892328	+	0.451388i	$0.850929\pi$
$4$	−1.68020	−0.840102
$5$	−1.80514	−0.807283	−0.403642	−	0.914917i	$-0.632256\pi$
$5$	−1.80514	−0.807283	−0.403642	+	0.914917i	$0.632256\pi$
$6$	−1.74804	−0.713635
$7$	0	0
$7$	0	0
$8$	−2.08118	−0.735806
$9$	6.55499	2.18500
$10$	−1.02082	−0.322811
$11$	−3.58433	−1.08072	−0.540358	−	0.841436i	$-0.681711\pi$
$11$	−3.58433	−1.08072	−0.540358	+	0.841436i	$0.681711\pi$
$12$	5.19370	1.49929
$13$	0	0
$13$	0	0
$14$	0	0
$15$	5.57989	1.44072
$16$	2.18349	0.545873
$17$	−6.14919	−1.49140	−0.745699	−	0.666283i	$-0.767884\pi$
$17$	−6.14919	−1.49140	−0.745699	+	0.666283i	$0.767884\pi$
$18$	3.70688	0.873720
$19$	2.51075	0.576006	0.288003	−	0.957629i	$-0.407009\pi$
$19$	2.51075	0.576006	0.288003	+	0.957629i	$0.407009\pi$
$20$	3.03300	0.678200
$21$	0	0
$22$	−2.02696	−0.432148
$23$	4.13185	0.861550	0.430775	−	0.902459i	$-0.358240\pi$
$23$	4.13185	0.861550	0.430775	+	0.902459i	$0.358240\pi$
$24$	6.43315	1.31316
$25$	−1.74147	−0.348294
$26$	0	0
$27$	−10.9889	−2.11481
$28$	0	0
$29$	5.11133	0.949151	0.474576	−	0.880215i	$-0.342602\pi$
$29$	5.11133	0.949151	0.474576	+	0.880215i	$0.342602\pi$
$30$	3.15546	0.576106
$31$	−4.55318	−0.817775	−0.408888	−	0.912585i	$-0.634083\pi$
$31$	−4.55318	−0.817775	−0.408888	+	0.912585i	$0.634083\pi$
$32$	5.39713	0.954086
$33$	11.0796	1.92870
$34$	−3.47740	−0.596369
$35$	0	0
$36$	−11.0137	−1.83562
$37$	−11.7215	−1.92700	−0.963500	−	0.267708i	$-0.913734\pi$
$37$	−11.7215	−1.92700	−0.963500	+	0.267708i	$0.913734\pi$
$38$	1.41984	0.230329
$39$	0	0
$40$	3.75681	0.594004
$41$	1.86640	0.291483	0.145742	−	0.989323i	$-0.453443\pi$
$41$	1.86640	0.291483	0.145742	+	0.989323i	$0.453443\pi$
$42$	0	0
$43$	−1.42314	−0.217026	−0.108513	−	0.994095i	$-0.534609\pi$
$43$	−1.42314	−0.217026	−0.108513	+	0.994095i	$0.534609\pi$
$44$	6.02240	0.907911
$45$	−11.8327	−1.76391
$46$	2.33658	0.344510
$47$	−8.86595	−1.29323	−0.646616	−	0.762816i	$-0.723816\pi$
$47$	−8.86595	−1.29323	−0.646616	+	0.762816i	$0.723816\pi$
$48$	−6.74942	−0.974195
$49$	0	0
$50$	−0.984810	−0.139273
$51$	19.0078	2.66163
$52$	0	0
$53$	2.02920	0.278732	0.139366	−	0.990241i	$-0.455494\pi$
$53$	2.02920	0.278732	0.139366	+	0.990241i	$0.455494\pi$
$54$	−6.21427	−0.845655
$55$	6.47021	0.872443
$56$	0	0
$57$	−7.76102	−1.02797
$58$	2.89049	0.379540
$59$	−9.10752	−1.18570	−0.592849	−	0.805314i	$-0.701997\pi$
$59$	−9.10752	−1.18570	−0.592849	+	0.805314i	$0.701997\pi$
$60$	−9.37536	−1.21035
$61$	−5.47841	−0.701438	−0.350719	−	0.936481i	$-0.614063\pi$
$61$	−5.47841	−0.701438	−0.350719	+	0.936481i	$0.614063\pi$
$62$	−2.57485	−0.327006
$63$	0	0
$64$	−1.31488	−0.164360
$65$	0	0
$66$	6.26555	0.771236
$67$	7.02055	0.857697	0.428849	−	0.903376i	$-0.358919\pi$
$67$	7.02055	0.857697	0.428849	+	0.903376i	$0.358919\pi$
$68$	10.3319	1.25293
$69$	−12.7720	−1.53757
$70$	0	0
$71$	−4.49410	−0.533351	−0.266676	−	0.963786i	$-0.585925\pi$
$71$	−4.49410	−0.533351	−0.266676	+	0.963786i	$0.585925\pi$
$72$	−13.6421	−1.60773
$73$	−9.35564	−1.09499	−0.547497	−	0.836807i	$-0.684419\pi$
$73$	−9.35564	−1.09499	−0.547497	+	0.836807i	$0.684419\pi$
$74$	−6.62856	−0.770555
$75$	5.38308	0.621584
$76$	−4.21858	−0.483904
$77$	0	0
$78$	0	0
$79$	5.20970	0.586137	0.293069	−	0.956091i	$-0.405324\pi$
$79$	5.20970	0.586137	0.293069	+	0.956091i	$0.405324\pi$
$80$	−3.94151	−0.440674
$81$	14.3029	1.58921
$82$	1.05546	0.116556
$83$	−9.42491	−1.03452	−0.517259	−	0.855829i	$-0.673048\pi$
$83$	−9.42491	−1.03452	−0.517259	+	0.855829i	$0.673048\pi$
$84$	0	0
$85$	11.1001	1.20398
$86$	−0.804791	−0.0867829
$87$	−15.7997	−1.69391
$88$	7.45961	0.795197
$89$	−0.355648	−0.0376986	−0.0188493	−	0.999822i	$-0.506000\pi$
$89$	−0.355648	−0.0376986	−0.0188493	+	0.999822i	$0.506000\pi$
$90$	−6.69144	−0.705340
$91$	0	0
$92$	−6.94235	−0.723790
$93$	14.0744	1.45945
$94$	−5.01374	−0.517128
$95$	−4.53226	−0.465000
$96$	−16.6831	−1.70272
$97$	−10.6035	−1.07663	−0.538313	−	0.842745i	$-0.680938\pi$
$97$	−10.6035	−1.07663	−0.538313	+	0.842745i	$0.680938\pi$
$98$	0	0
$99$	−23.4952	−2.36136
$100$	2.92602	0.292602
$101$	3.42508	0.340808	0.170404	−	0.985374i	$-0.445493\pi$
$101$	3.42508	0.340808	0.170404	+	0.985374i	$0.445493\pi$
$102$	10.7490	1.06431
$103$	−12.9558	−1.27658	−0.638288	−	0.769798i	$-0.720357\pi$
$103$	−12.9558	−1.27658	−0.638288	+	0.769798i	$0.720357\pi$
$104$	0	0
$105$	0	0
$106$	1.14752	0.111457
$107$	−18.1696	−1.75653	−0.878263	−	0.478179i	$-0.841297\pi$
$107$	−18.1696	−1.75653	−0.878263	+	0.478179i	$0.841297\pi$
$108$	18.4635	1.77666
$109$	−13.9294	−1.33420	−0.667098	−	0.744970i	$-0.732464\pi$
$109$	−13.9294	−1.33420	−0.667098	+	0.744970i	$0.732464\pi$
$110$	3.65894	0.348866
$111$	36.2324	3.43903
$112$	0	0
$113$	−10.1281	−0.952770	−0.476385	−	0.879237i	$-0.658053\pi$
$113$	−10.1281	−0.952770	−0.476385	+	0.879237i	$0.658053\pi$
$114$	−4.38890	−0.411058
$115$	−7.45857	−0.695515
$116$	−8.58808	−0.797383
$117$	0	0
$118$	−5.15035	−0.474128
$119$	0	0
$120$	−11.6127	−1.06009
$121$	1.84739	0.167945
$122$	−3.09807	−0.280486
$123$	−5.76927	−0.520197
$124$	7.65027	0.687014
$125$	12.1693	1.08845
$126$	0	0
$127$	6.10219	0.541482	0.270741	−	0.962652i	$-0.412731\pi$
$127$	6.10219	0.541482	0.270741	+	0.962652i	$0.412731\pi$
$128$	−11.5378	−1.01981
$129$	4.39908	0.387317
$130$	0	0
$131$	−5.97266	−0.521833	−0.260917	−	0.965361i	$-0.584025\pi$
$131$	−5.97266	−0.521833	−0.260917	+	0.965361i	$0.584025\pi$
$132$	−18.6159	−1.62031
$133$	0	0
$134$	3.97016	0.342970
$135$	19.8365	1.70725
$136$	12.7975	1.09738
$137$	−3.60737	−0.308198	−0.154099	−	0.988055i	$-0.549247\pi$
$137$	−3.60737	−0.308198	−0.154099	+	0.988055i	$0.549247\pi$
$138$	−7.22265	−0.614833
$139$	−12.9167	−1.09558	−0.547789	−	0.836617i	$-0.684530\pi$
$139$	−12.9167	−1.09558	−0.547789	+	0.836617i	$0.684530\pi$
$140$	0	0
$141$	27.4057	2.30797
$142$	−2.54144	−0.213273
$143$	0	0
$144$	14.3128	1.19273
$145$	−9.22668	−0.766234
$146$	−5.29067	−0.437859
$147$	0	0
$148$	19.6945	1.61888
$149$	−2.19416	−0.179753	−0.0898763	−	0.995953i	$-0.528647\pi$
$149$	−2.19416	−0.179753	−0.0898763	+	0.995953i	$0.528647\pi$
$150$	3.04416	0.248555
$151$	−1.94893	−0.158602	−0.0793009	−	0.996851i	$-0.525269\pi$
$151$	−1.94893	−0.158602	−0.0793009	+	0.996851i	$0.525269\pi$
$152$	−5.22532	−0.423829
$153$	−40.3078	−3.25870
$154$	0	0
$155$	8.21913	0.660176
$156$	0	0
$157$	−18.3508	−1.46455	−0.732275	−	0.681009i	$-0.761541\pi$
$157$	−18.3508	−1.46455	−0.732275	+	0.681009i	$0.761541\pi$
$158$	2.94611	0.234380
$159$	−6.27248	−0.497440
$160$	−9.74257	−0.770218
$161$	0	0
$162$	8.08836	0.635482
$163$	−22.0599	−1.72786	−0.863931	−	0.503611i	$-0.832004\pi$
$163$	−22.0599	−1.72786	−0.863931	+	0.503611i	$0.832004\pi$
$164$	−3.13594	−0.244876
$165$	−20.0002	−1.55701
$166$	−5.32984	−0.413676
$167$	2.83878	0.219672	0.109836	−	0.993950i	$-0.464967\pi$
$167$	2.83878	0.219672	0.109836	+	0.993950i	$0.464967\pi$
$168$	0	0
$169$	0	0
$170$	6.27719	0.481439
$171$	16.4580	1.25857
$172$	2.39116	0.182324
$173$	3.52822	0.268246	0.134123	−	0.990965i	$-0.457178\pi$
$173$	3.52822	0.268246	0.134123	+	0.990965i	$0.457178\pi$
$174$	−8.93483	−0.677347
$175$	0	0
$176$	−7.82635	−0.589933
$177$	28.1524	2.11606
$178$	−0.201121	−0.0150746
$179$	−6.69470	−0.500385	−0.250193	−	0.968196i	$-0.580494\pi$
$179$	−6.69470	−0.500385	−0.250193	+	0.968196i	$0.580494\pi$
$180$	19.8813	1.48186
$181$	−12.5317	−0.931476	−0.465738	−	0.884923i	$-0.654211\pi$
$181$	−12.5317	−0.931476	−0.465738	+	0.884923i	$0.654211\pi$
$182$	0	0
$183$	16.9344	1.25183
$184$	−8.59910	−0.633934
$185$	21.1589	1.55563
$186$	7.95915	0.583593
$187$	22.0407	1.61178
$188$	14.8966	1.08645
$189$	0	0
$190$	−2.56302	−0.185941
$191$	3.03641	0.219707	0.109854	−	0.993948i	$-0.464962\pi$
$191$	3.03641	0.219707	0.109854	+	0.993948i	$0.464962\pi$
$192$	4.06444	0.293326
$193$	26.9449	1.93954	0.969769	−	0.244023i	$-0.0784673\pi$
$193$	26.9449	1.93954	0.969769	+	0.244023i	$0.0784673\pi$
$194$	−5.99635	−0.430513
$195$	0	0
$196$	0	0
$197$	−9.29898	−0.662525	−0.331262	−	0.943539i	$-0.607475\pi$
$197$	−9.29898	−0.662525	−0.331262	+	0.943539i	$0.607475\pi$
$198$	−13.2867	−0.944242
$199$	17.3848	1.23238	0.616190	−	0.787598i	$-0.288676\pi$
$199$	17.3848	1.23238	0.616190	+	0.787598i	$0.288676\pi$
$200$	3.62430	0.256277
$201$	−21.7013	−1.53069
$202$	1.93690	0.136280
$203$	0	0
$204$	−31.9370	−2.23604
$205$	−3.36912	−0.235310
$206$	−7.32659	−0.510468
$207$	27.0842	1.88248
$208$	0	0
$209$	−8.99936	−0.622499
$210$	0	0
$211$	21.8236	1.50240	0.751199	−	0.660075i	$-0.229476\pi$
$211$	21.8236	1.50240	0.751199	+	0.660075i	$0.229476\pi$
$212$	−3.40947	−0.234163
$213$	13.8918	0.951849
$214$	−10.2750	−0.702386
$215$	2.56896	0.175202
$216$	22.8698	1.55609
$217$	0	0
$218$	−7.87716	−0.533508
$219$	28.9194	1.95419
$220$	−10.8713	−0.732941
$221$	0	0
$222$	20.4896	1.37517
$223$	1.91187	0.128029	0.0640143	−	0.997949i	$-0.479610\pi$
$223$	1.91187	0.128029	0.0640143	+	0.997949i	$0.479610\pi$
$224$	0	0
$225$	−11.4153	−0.761020
$226$	−5.72749	−0.380987
$227$	10.3347	0.685939	0.342969	−	0.939347i	$-0.388567\pi$
$227$	10.3347	0.685939	0.342969	+	0.939347i	$0.388567\pi$
$228$	13.0401	0.863602
$229$	−6.10394	−0.403360	−0.201680	−	0.979451i	$-0.564640\pi$
$229$	−6.10394	−0.403360	−0.201680	+	0.979451i	$0.564640\pi$
$230$	−4.21786	−0.278118
$231$	0	0
$232$	−10.6376	−0.698391
$233$	7.12776	0.466955	0.233478	−	0.972362i	$-0.424989\pi$
$233$	7.12776	0.466955	0.233478	+	0.972362i	$0.424989\pi$
$234$	0	0
$235$	16.0043	1.04400
$236$	15.3025	0.996107
$237$	−16.1038	−1.04605
$238$	0	0
$239$	−7.32349	−0.473717	−0.236859	−	0.971544i	$-0.576118\pi$
$239$	−7.32349	−0.473717	−0.236859	+	0.971544i	$0.576118\pi$
$240$	12.1837	0.786451
$241$	0.263098	0.0169477	0.00847383	−	0.999964i	$-0.497303\pi$
$241$	0.263098	0.0169477	0.00847383	+	0.999964i	$0.497303\pi$
$242$	1.04471	0.0671566
$243$	−11.2453	−0.721384
$244$	9.20484	0.589280
$245$	0	0
$246$	−3.26255	−0.208013
$247$	0	0
$248$	9.47596	0.601724
$249$	29.1335	1.84626
$250$	6.88180	0.435243
$251$	−13.2819	−0.838347	−0.419173	−	0.907906i	$-0.637680\pi$
$251$	−13.2819	−0.838347	−0.419173	+	0.907906i	$0.637680\pi$
$252$	0	0
$253$	−14.8099	−0.931090
$254$	3.45082	0.216524
$255$	−34.3118	−2.14869
$256$	−3.89494	−0.243434
$257$	−17.4144	−1.08628	−0.543139	−	0.839643i	$-0.682764\pi$
$257$	−17.4144	−1.08628	−0.543139	+	0.839643i	$0.682764\pi$
$258$	2.48770	0.154878
$259$	0	0
$260$	0	0
$261$	33.5047	2.07389
$262$	−3.37757	−0.208667
$263$	−12.7276	−0.784820	−0.392410	−	0.919790i	$-0.628359\pi$
$263$	−12.7276	−0.784820	−0.392410	+	0.919790i	$0.628359\pi$
$264$	−23.0585	−1.41915
$265$	−3.66299	−0.225015
$266$	0	0
$267$	1.09935	0.0672790
$268$	−11.7960	−0.720553
$269$	−2.16163	−0.131797	−0.0658985	−	0.997826i	$-0.520991\pi$
$269$	−2.16163	−0.131797	−0.0658985	+	0.997826i	$0.520991\pi$
$270$	11.2176	0.682683
$271$	5.82966	0.354127	0.177063	−	0.984199i	$-0.443340\pi$
$271$	5.82966	0.354127	0.177063	+	0.984199i	$0.443340\pi$
$272$	−13.4267	−0.814113
$273$	0	0
$274$	−2.03999	−0.123240
$275$	6.24199	0.376406
$276$	21.4596	1.29172
$277$	−11.6473	−0.699820	−0.349910	−	0.936783i	$-0.613788\pi$
$277$	−11.6473	−0.699820	−0.349910	+	0.936783i	$0.613788\pi$
$278$	−7.30445	−0.438092
$279$	−29.8460	−1.78684
$280$	0	0
$281$	24.7940	1.47908	0.739542	−	0.673110i	$-0.235042\pi$
$281$	24.7940	1.47908	0.739542	+	0.673110i	$0.235042\pi$
$282$	15.4981	0.922896
$283$	15.1432	0.900172	0.450086	−	0.892985i	$-0.351393\pi$
$283$	15.1432	0.900172	0.450086	+	0.892985i	$0.351393\pi$
$284$	7.55100	0.448069
$285$	14.0097	0.829865
$286$	0	0
$287$	0	0
$288$	35.3781	2.08467
$289$	20.8125	1.22427
$290$	−5.21773	−0.306396
$291$	32.7767	1.92141
$292$	15.7194	0.919907
$293$	−24.9203	−1.45586	−0.727929	−	0.685652i	$-0.759517\pi$
$293$	−24.9203	−1.45586	−0.727929	+	0.685652i	$0.759517\pi$
$294$	0	0
$295$	16.4404	0.957194
$296$	24.3945	1.41790
$297$	39.3877	2.28551
$298$	−1.24081	−0.0718782
$299$	0	0
$300$	−9.04467	−0.522194
$301$	0	0
$302$	−1.10213	−0.0634205
$303$	−10.5873	−0.608225
$304$	5.48221	0.314426
$305$	9.88930	0.566259
$306$	−22.7943	−1.30306
$307$	−3.90158	−0.222675	−0.111337	−	0.993783i	$-0.535513\pi$
$307$	−3.90158	−0.222675	−0.111337	+	0.993783i	$0.535513\pi$
$308$	0	0
$309$	40.0479	2.27825
$310$	4.64796	0.263986
$311$	19.7180	1.11810	0.559052	−	0.829132i	$-0.311165\pi$
$311$	19.7180	1.11810	0.559052	+	0.829132i	$0.311165\pi$
$312$	0	0
$313$	2.89235	0.163485	0.0817427	−	0.996653i	$-0.473951\pi$
$313$	2.89235	0.163485	0.0817427	+	0.996653i	$0.473951\pi$
$314$	−10.3775	−0.585633
$315$	0	0
$316$	−8.75336	−0.492415
$317$	15.9537	0.896046	0.448023	−	0.894022i	$-0.352128\pi$
$317$	15.9537	0.896046	0.448023	+	0.894022i	$0.352128\pi$
$318$	−3.54712	−0.198913
$319$	−18.3207	−1.02576
$320$	2.37354	0.132685
$321$	56.1644	3.13479
$322$	0	0
$323$	−15.4391	−0.859054
$324$	−24.0318	−1.33510
$325$	0	0
$326$	−12.4750	−0.690924
$327$	43.0574	2.38108
$328$	−3.88431	−0.214475
$329$	0	0
$330$	−11.3102	−0.622606
$331$	−3.82678	−0.210339	−0.105169	−	0.994454i	$-0.533538\pi$
$331$	−3.82678	−0.210339	−0.105169	+	0.994454i	$0.533538\pi$
$332$	15.8358	0.869101
$333$	−76.8342	−4.21049
$334$	1.60535	0.0878407
$335$	−12.6731	−0.692405
$336$	0	0
$337$	−9.78172	−0.532844	−0.266422	−	0.963856i	$-0.585842\pi$
$337$	−9.78172	−0.532844	−0.266422	+	0.963856i	$0.585842\pi$
$338$	0	0
$339$	31.3071	1.70037
$340$	−18.6505	−1.01147
$341$	16.3201	0.883782
$342$	9.30706	0.503268
$343$	0	0
$344$	2.96180	0.159689
$345$	23.0553	1.24126
$346$	1.99523	0.107264
$347$	5.33403	0.286346	0.143173	−	0.989698i	$-0.454269\pi$
$347$	5.33403	0.286346	0.143173	+	0.989698i	$0.454269\pi$
$348$	26.5467	1.42305
$349$	−14.7030	−0.787032	−0.393516	−	0.919318i	$-0.628741\pi$
$349$	−14.7030	−0.787032	−0.393516	+	0.919318i	$0.628741\pi$
$350$	0	0
$351$	0	0
$352$	−19.3451	−1.03110
$353$	35.5173	1.89039	0.945196	−	0.326502i	$-0.105870\pi$
$353$	35.5173	1.89039	0.945196	+	0.326502i	$0.105870\pi$
$354$	15.9203	0.846156
$355$	8.11248	0.430566
$356$	0.597560	0.0316706
$357$	0	0
$358$	−3.78589	−0.200090
$359$	−15.9147	−0.839945	−0.419972	−	0.907537i	$-0.637960\pi$
$359$	−15.9147	−0.839945	−0.419972	+	0.907537i	$0.637960\pi$
$360$	24.6259	1.29790
$361$	−12.6961	−0.668217
$362$	−7.08676	−0.372472
$363$	−5.71051	−0.299724
$364$	0	0
$365$	16.8882	0.883971
$366$	9.57649	0.500571
$367$	−18.2446	−0.952361	−0.476181	−	0.879348i	$-0.657979\pi$
$367$	−18.2446	−0.952361	−0.476181	+	0.879348i	$0.657979\pi$
$368$	9.02186	0.470297
$369$	12.2343	0.636890
$370$	11.9655	0.622056
$371$	0	0
$372$	−23.6479	−1.22608
$373$	−13.5744	−0.702856	−0.351428	−	0.936215i	$-0.614304\pi$
$373$	−13.5744	−0.702856	−0.351428	+	0.936215i	$0.614304\pi$
$374$	12.4641	0.644505
$375$	−37.6167	−1.94252
$376$	18.4516	0.951568
$377$	0	0
$378$	0	0
$379$	−7.23572	−0.371674	−0.185837	−	0.982581i	$-0.559500\pi$
$379$	−7.23572	−0.371674	−0.185837	+	0.982581i	$0.559500\pi$
$380$	7.61512	0.390648
$381$	−18.8626	−0.966359
$382$	1.71711	0.0878549
$383$	20.3464	1.03965	0.519827	−	0.854272i	$-0.325996\pi$
$383$	20.3464	1.03965	0.519827	+	0.854272i	$0.325996\pi$
$384$	35.6647	1.82001
$385$	0	0
$386$	15.2375	0.775569
$387$	−9.32864	−0.474202
$388$	17.8161	0.904475
$389$	2.48306	0.125896	0.0629481	−	0.998017i	$-0.479950\pi$
$389$	2.48306	0.125896	0.0629481	+	0.998017i	$0.479950\pi$
$390$	0	0
$391$	−25.4075	−1.28491
$392$	0	0
$393$	18.4622	0.931293
$394$	−5.25862	−0.264926
$395$	−9.40424	−0.473179
$396$	39.4767	1.98378
$397$	−10.7391	−0.538980	−0.269490	−	0.963003i	$-0.586855\pi$
$397$	−10.7391	−0.538980	−0.269490	+	0.963003i	$0.586855\pi$
$398$	9.83122	0.492795
$399$	0	0
$400$	−3.80248	−0.190124
$401$	−14.7540	−0.736778	−0.368389	−	0.929672i	$-0.620091\pi$
$401$	−14.7540	−0.736778	−0.368389	+	0.929672i	$0.620091\pi$
$402$	−12.2722	−0.612083
$403$	0	0
$404$	−5.75483	−0.286313
$405$	−25.8187	−1.28294
$406$	0	0
$407$	42.0136	2.08254
$408$	−39.5586	−1.95844
$409$	−6.34448	−0.313714	−0.156857	−	0.987621i	$-0.550136\pi$
$409$	−6.34448	−0.313714	−0.156857	+	0.987621i	$0.550136\pi$
$410$	−1.90526	−0.0940939
$411$	11.1508	0.550028
$412$	21.7684	1.07245
$413$	0	0
$414$	15.3163	0.752754
$415$	17.0133	0.835150
$416$	0	0
$417$	39.9269	1.95523
$418$	−5.08919	−0.248920
$419$	18.0851	0.883513	0.441756	−	0.897135i	$-0.354356\pi$
$419$	18.0851	0.883513	0.441756	+	0.897135i	$0.354356\pi$
$420$	0	0
$421$	−19.7043	−0.960327	−0.480164	−	0.877179i	$-0.659423\pi$
$421$	−19.7043	−0.960327	−0.480164	+	0.877179i	$0.659423\pi$
$422$	12.3414	0.600768
$423$	−58.1162	−2.82571
$424$	−4.22312	−0.205093
$425$	10.7086	0.519444
$426$	7.85587	0.380618
$427$	0	0
$428$	30.5287	1.47566
$429$	0	0
$430$	1.45276	0.0700584
$431$	−10.5039	−0.505953	−0.252977	−	0.967472i	$-0.581410\pi$
$431$	−10.5039	−0.505953	−0.252977	+	0.967472i	$0.581410\pi$
$432$	−23.9941	−1.15442
$433$	−9.11805	−0.438186	−0.219093	−	0.975704i	$-0.570310\pi$
$433$	−9.11805	−0.438186	−0.219093	+	0.975704i	$0.570310\pi$
$434$	0	0
$435$	28.5207	1.36746
$436$	23.4043	1.12086
$437$	10.3741	0.496258
$438$	16.3541	0.781427
$439$	29.2801	1.39746	0.698731	−	0.715384i	$-0.253748\pi$
$439$	29.2801	1.39746	0.698731	+	0.715384i	$0.253748\pi$
$440$	−13.4656	−0.641949
$441$	0	0
$442$	0	0
$443$	−26.7353	−1.27023	−0.635117	−	0.772416i	$-0.719048\pi$
$443$	−26.7353	−1.27023	−0.635117	+	0.772416i	$0.719048\pi$
$444$	−60.8779	−2.88914
$445$	0.641994	0.0304334
$446$	1.08118	0.0511951
$447$	6.78240	0.320797
$448$	0	0
$449$	−21.2155	−1.00122	−0.500611	−	0.865673i	$-0.666891\pi$
$449$	−21.2155	−1.00122	−0.500611	+	0.865673i	$0.666891\pi$
$450$	−6.45542	−0.304311
$451$	−6.68980	−0.315010
$452$	17.0172	0.800424
$453$	6.02437	0.283050
$454$	5.84433	0.274288
$455$	0	0
$456$	16.1521	0.756389
$457$	−14.6446	−0.685046	−0.342523	−	0.939509i	$-0.611282\pi$
$457$	−14.6446	−0.685046	−0.342523	+	0.939509i	$0.611282\pi$
$458$	−3.45181	−0.161293
$459$	67.5726	3.15402
$460$	12.5319	0.584304
$461$	24.8946	1.15946	0.579729	−	0.814810i	$-0.303159\pi$
$461$	24.8946	1.15946	0.579729	+	0.814810i	$0.303159\pi$
$462$	0	0
$463$	−10.1069	−0.469706	−0.234853	−	0.972031i	$-0.575461\pi$
$463$	−10.1069	−0.469706	−0.234853	+	0.972031i	$0.575461\pi$
$464$	11.1606	0.518116
$465$	−25.4063	−1.17819
$466$	4.03079	0.186723
$467$	11.4825	0.531346	0.265673	−	0.964063i	$-0.414406\pi$
$467$	11.4825	0.531346	0.265673	+	0.964063i	$0.414406\pi$
$468$	0	0
$469$	0	0
$470$	9.05051	0.417469
$471$	56.7243	2.61372
$472$	18.9543	0.872444
$473$	5.10099	0.234544
$474$	−9.10678	−0.418288
$475$	−4.37240	−0.200619
$476$	0	0
$477$	13.3014	0.609028
$478$	−4.14147	−0.189427
$479$	−1.08116	−0.0493997	−0.0246998	−	0.999695i	$-0.507863\pi$
$479$	−1.08116	−0.0493997	−0.0246998	+	0.999695i	$0.507863\pi$
$480$	30.1154	1.37457
$481$	0	0
$482$	0.148784	0.00677691
$483$	0	0
$484$	−3.10400	−0.141091
$485$	19.1409	0.869142
$486$	−6.35926	−0.288462
$487$	−25.2677	−1.14499	−0.572493	−	0.819909i	$-0.694024\pi$
$487$	−25.2677	−1.14499	−0.572493	+	0.819909i	$0.694024\pi$
$488$	11.4015	0.516123
$489$	68.1895	3.08364
$490$	0	0
$491$	41.7180	1.88271	0.941355	−	0.337419i	$-0.109554\pi$
$491$	41.7180	1.88271	0.941355	+	0.337419i	$0.109554\pi$
$492$	9.69354	0.437019
$493$	−31.4306	−1.41556
$494$	0	0
$495$	42.4122	1.90628
$496$	−9.94183	−0.446401
$497$	0	0
$498$	16.4751	0.738269
$499$	−10.4684	−0.468628	−0.234314	−	0.972161i	$-0.575284\pi$
$499$	−10.4684	−0.468628	−0.234314	+	0.972161i	$0.575284\pi$
$500$	−20.4469	−0.914413
$501$	−8.77501	−0.392038
$502$	−7.51099	−0.335232
$503$	2.57470	0.114800	0.0574001	−	0.998351i	$-0.481719\pi$
$503$	2.57470	0.114800	0.0574001	+	0.998351i	$0.481719\pi$
$504$	0	0
$505$	−6.18275	−0.275129
$506$	−8.37508	−0.372318
$507$	0	0
$508$	−10.2529	−0.454900
$509$	7.52681	0.333620	0.166810	−	0.985989i	$-0.446653\pi$
$509$	7.52681	0.333620	0.166810	+	0.985989i	$0.446653\pi$
$510$	−19.4035	−0.859202
$511$	0	0
$512$	20.8730	0.922467
$513$	−27.5903	−1.21814
$514$	−9.84792	−0.434373
$515$	23.3871	1.03056
$516$	−7.39135	−0.325386
$517$	31.7785	1.39761
$518$	0	0
$519$	−10.9061	−0.478726
$520$	0	0
$521$	0.0156677	0.000686413 0	0.000343207	−	1.00000i	$-0.499891\pi$
$521$	0.0156677	0.000686413 0	0.000343207		1.00000i	$0.499891\pi$
$522$	18.9471	0.829292
$523$	24.0715	1.05257	0.526286	−	0.850308i	$-0.323584\pi$
$523$	24.0715	1.05257	0.526286	+	0.850308i	$0.323584\pi$
$524$	10.0353	0.438393
$525$	0	0
$526$	−7.19755	−0.313828
$527$	27.9984	1.21963
$528$	24.1921	1.05283
$529$	−5.92781	−0.257731
$530$	−2.07144	−0.0899775
$531$	−59.6997	−2.59075
$532$	0	0
$533$	0	0
$534$	0.621687	0.0269030
$535$	32.7987	1.41801
$536$	−14.6110	−0.631099
$537$	20.6941	0.893015
$538$	−1.22241	−0.0527020
$539$	0	0
$540$	−33.3293	−1.43426
$541$	−37.7763	−1.62413	−0.812065	−	0.583567i	$-0.801656\pi$
$541$	−37.7763	−1.62413	−0.812065	+	0.583567i	$0.801656\pi$
$542$	3.29671	0.141606
$543$	38.7370	1.66236
$544$	−33.1879	−1.42292
$545$	25.1445	1.07707
$546$	0	0
$547$	28.4925	1.21825	0.609126	−	0.793074i	$-0.291521\pi$
$547$	28.4925	1.21825	0.609126	+	0.793074i	$0.291521\pi$
$548$	6.06111	0.258918
$549$	−35.9109	−1.53264
$550$	3.52988	0.150515
$551$	12.8333	0.546717
$552$	26.5808	1.13135
$553$	0	0
$554$	−6.58663	−0.279839
$555$	−65.4046	−2.77627
$556$	21.7026	0.920397
$557$	19.8646	0.841692	0.420846	−	0.907132i	$-0.361733\pi$
$557$	19.8646	0.841692	0.420846	+	0.907132i	$0.361733\pi$
$558$	−16.8781	−0.714507
$559$	0	0
$560$	0	0
$561$	−68.1303	−2.87646
$562$	14.0211	0.591445
$563$	−15.5397	−0.654919	−0.327459	−	0.944865i	$-0.606192\pi$
$563$	−15.5397	−0.654919	−0.327459	+	0.944865i	$0.606192\pi$
$564$	−46.0471	−1.93893
$565$	18.2826	0.769155
$566$	8.56358	0.359954
$567$	0	0
$568$	9.35301	0.392443
$569$	−31.1088	−1.30415	−0.652075	−	0.758154i	$-0.726101\pi$
$569$	−31.1088	−1.30415	−0.652075	+	0.758154i	$0.726101\pi$
$570$	7.92258	0.331840
$571$	−0.602653	−0.0252202	−0.0126101	−	0.999920i	$-0.504014\pi$
$571$	−0.602653	−0.0252202	−0.0126101	+	0.999920i	$0.504014\pi$
$572$	0	0
$573$	−9.38590	−0.392102
$574$	0	0
$575$	−7.19549	−0.300073
$576$	−8.61902	−0.359126
$577$	−5.21634	−0.217159	−0.108580	−	0.994088i	$-0.534630\pi$
$577$	−5.21634	−0.217159	−0.108580	+	0.994088i	$0.534630\pi$
$578$	11.7696	0.489550
$579$	−83.2899	−3.46141
$580$	15.5027	0.643714
$581$	0	0
$582$	18.5354	0.768318
$583$	−7.27331	−0.301230
$584$	19.4707	0.805704
$585$	0	0
$586$	−14.0926	−0.582158
$587$	17.1895	0.709487	0.354743	−	0.934964i	$-0.384568\pi$
$587$	17.1895	0.709487	0.354743	+	0.934964i	$0.384568\pi$
$588$	0	0
$589$	−11.4319	−0.471044
$590$	9.29711	0.382756
$591$	28.7442	1.18238
$592$	−25.5938	−1.05190
$593$	−10.6452	−0.437147	−0.218574	−	0.975820i	$-0.570140\pi$
$593$	−10.6452	−0.437147	−0.218574	+	0.975820i	$0.570140\pi$
$594$	22.2740	0.913912
$595$	0	0
$596$	3.68664	0.151011
$597$	−53.7385	−2.19937
$598$	0	0
$599$	25.1472	1.02749	0.513744	−	0.857944i	$-0.328258\pi$
$599$	25.1472	1.02749	0.513744	+	0.857944i	$0.328258\pi$
$600$	−11.2031	−0.457366
$601$	−3.24667	−0.132434	−0.0662172	−	0.997805i	$-0.521093\pi$
$601$	−3.24667	−0.132434	−0.0662172	+	0.997805i	$0.521093\pi$
$602$	0	0
$603$	46.0196	1.87406
$604$	3.27460	0.133242
$605$	−3.33481	−0.135579
$606$	−5.98718	−0.243213
$607$	17.1124	0.694572	0.347286	−	0.937759i	$-0.387103\pi$
$607$	17.1124	0.694572	0.347286	+	0.937759i	$0.387103\pi$
$608$	13.5509	0.549560
$609$	0	0
$610$	5.59245	0.226432
$611$	0	0
$612$	67.7254	2.73764
$613$	28.0427	1.13264	0.566318	−	0.824187i	$-0.308368\pi$
$613$	28.0427	1.13264	0.566318	+	0.824187i	$0.308368\pi$
$614$	−2.20636	−0.0890415
$615$	10.4143	0.419947
$616$	0	0
$617$	42.0926	1.69459	0.847293	−	0.531126i	$-0.178231\pi$
$617$	42.0926	1.69459	0.847293	+	0.531126i	$0.178231\pi$
$618$	22.6473	0.911009
$619$	−27.1308	−1.09048	−0.545239	−	0.838280i	$-0.683561\pi$
$619$	−27.1308	−1.09048	−0.545239	+	0.838280i	$0.683561\pi$
$620$	−13.8098	−0.554615
$621$	−45.4044	−1.82201
$622$	11.1506	0.447099
$623$	0	0
$624$	0	0
$625$	−13.2599	−0.530398
$626$	1.63564	0.0653733
$627$	27.8180	1.11095
$628$	30.8330	1.23037
$629$	72.0776	2.87392
$630$	0	0
$631$	16.0095	0.637329	0.318665	−	0.947868i	$-0.396766\pi$
$631$	16.0095	0.637329	0.318665	+	0.947868i	$0.396766\pi$
$632$	−10.8423	−0.431284
$633$	−67.4593	−2.68126
$634$	9.02188	0.358304
$635$	−11.0153	−0.437129
$636$	10.5390	0.417900
$637$	0	0
$638$	−10.3604	−0.410174
$639$	−29.4588	−1.16537
$640$	20.8274	0.823275
$641$	−14.6814	−0.579880	−0.289940	−	0.957045i	$-0.593635\pi$
$641$	−14.6814	−0.579880	−0.289940	+	0.957045i	$0.593635\pi$
$642$	31.7613	1.25352
$643$	−37.1435	−1.46480	−0.732399	−	0.680875i	$-0.761600\pi$
$643$	−37.1435	−1.46480	−0.732399	+	0.680875i	$0.761600\pi$
$644$	0	0
$645$	−7.94095	−0.312675
$646$	−8.73089	−0.343512
$647$	14.2284	0.559377	0.279689	−	0.960091i	$-0.409769\pi$
$647$	14.2284	0.559377	0.279689	+	0.960091i	$0.409769\pi$
$648$	−29.7668	−1.16935
$649$	32.6443	1.28140
$650$	0	0
$651$	0	0
$652$	37.0650	1.45158
$653$	−21.8455	−0.854880	−0.427440	−	0.904044i	$-0.640585\pi$
$653$	−21.8455	−0.854880	−0.427440	+	0.904044i	$0.640585\pi$
$654$	24.3492	0.952129
$655$	10.7815	0.421267
$656$	4.07528	0.159113
$657$	−61.3261	−2.39256
$658$	0	0
$659$	33.1051	1.28959	0.644796	−	0.764355i	$-0.276943\pi$
$659$	33.1051	1.28959	0.644796	+	0.764355i	$0.276943\pi$
$660$	33.6043	1.30805
$661$	−23.3428	−0.907930	−0.453965	−	0.891020i	$-0.649991\pi$
$661$	−23.3428	−0.907930	−0.453965	+	0.891020i	$0.649991\pi$
$662$	−2.16406	−0.0841087
$663$	0	0
$664$	19.6149	0.761206
$665$	0	0
$666$	−43.4501	−1.68366
$667$	21.1193	0.817741
$668$	−4.76974	−0.184547
$669$	−5.90982	−0.228487
$670$	−7.16670	−0.276874
$671$	19.6364	0.758055
$672$	0	0
$673$	−48.4847	−1.86895	−0.934474	−	0.356030i	$-0.884130\pi$
$673$	−48.4847	−1.86895	−0.934474	+	0.356030i	$0.884130\pi$
$674$	−5.53162	−0.213070
$675$	19.1368	0.736575
$676$	0	0
$677$	−43.8976	−1.68712	−0.843560	−	0.537034i	$-0.819545\pi$
$677$	−43.8976	−1.68712	−0.843560	+	0.537034i	$0.819545\pi$
$678$	17.7043	0.679930
$679$	0	0
$680$	−23.1013	−0.885896
$681$	−31.9458	−1.22416
$682$	9.22909	0.353400
$683$	−44.9278	−1.71912	−0.859558	−	0.511039i	$-0.829261\pi$
$683$	−44.9278	−1.71912	−0.859558	+	0.511039i	$0.829261\pi$
$684$	−27.6527	−1.05733
$685$	6.51180	0.248803
$686$	0	0
$687$	18.8680	0.719859
$688$	−3.10741	−0.118469
$689$	0	0
$690$	13.0379	0.496344
$691$	45.4493	1.72897	0.864486	−	0.502656i	$-0.167644\pi$
$691$	45.4493	1.72897	0.864486	+	0.502656i	$0.167644\pi$
$692$	−5.92813	−0.225354
$693$	0	0
$694$	3.01643	0.114502
$695$	23.3164	0.884441
$696$	32.8820	1.24639
$697$	−11.4769	−0.434717
$698$	−8.31461	−0.314713
$699$	−22.0327	−0.833354
$700$	0	0
$701$	−28.7914	−1.08744	−0.543718	−	0.839268i	$-0.682984\pi$
$701$	−28.7914	−1.08744	−0.543718	+	0.839268i	$0.682984\pi$
$702$	0	0
$703$	−29.4298	−1.10996
$704$	4.71296	0.177626
$705$	−49.4711	−1.86319
$706$	20.0852	0.755916
$707$	0	0
$708$	−47.3017	−1.77771
$709$	17.3626	0.652068	0.326034	−	0.945358i	$-0.394288\pi$
$709$	17.3626	0.652068	0.326034	+	0.945358i	$0.394288\pi$
$710$	4.58765	0.172171
$711$	34.1495	1.28071
$712$	0.740165	0.0277389
$713$	−18.8131	−0.704554
$714$	0	0
$715$	0	0
$716$	11.2485	0.420374
$717$	22.6377	0.845422
$718$	−8.99984	−0.335871
$719$	9.98708	0.372455	0.186228	−	0.982507i	$-0.440374\pi$
$719$	9.98708	0.372455	0.186228	+	0.982507i	$0.440374\pi$
$720$	−25.8365	−0.962871
$721$	0	0
$722$	−7.17972	−0.267202
$723$	−0.813267	−0.0302457
$724$	21.0558	0.782534
$725$	−8.90123	−0.330583
$726$	−3.22932	−0.119851
$727$	−8.00409	−0.296855	−0.148428	−	0.988923i	$-0.547421\pi$
$727$	−8.00409	−0.296855	−0.148428	+	0.988923i	$0.547421\pi$
$728$	0	0
$729$	−8.14827	−0.301788
$730$	9.55039	0.353476
$731$	8.75114	0.323672
$732$	−28.4532	−1.05166
$733$	−44.8461	−1.65643	−0.828214	−	0.560412i	$-0.810643\pi$
$733$	−44.8461	−1.65643	−0.828214	+	0.560412i	$0.810643\pi$
$734$	−10.3174	−0.380823
$735$	0	0
$736$	22.3001	0.821993
$737$	−25.1640	−0.926926
$738$	6.91854	0.254675
$739$	25.0756	0.922421	0.461210	−	0.887291i	$-0.347415\pi$
$739$	25.0756	0.922421	0.461210	+	0.887291i	$0.347415\pi$
$740$	−35.5513	−1.30689
$741$	0	0
$742$	0	0
$743$	18.4191	0.675733	0.337866	−	0.941194i	$-0.390295\pi$
$743$	18.4191	0.675733	0.337866	+	0.941194i	$0.390295\pi$
$744$	−29.2913	−1.07387
$745$	3.96077	0.145111
$746$	−7.67639	−0.281053
$747$	−61.7802	−2.26042
$748$	−37.0329	−1.35406
$749$	0	0
$750$	−21.2724	−0.776760
$751$	11.6918	0.426638	0.213319	−	0.976983i	$-0.431573\pi$
$751$	11.6918	0.426638	0.213319	+	0.976983i	$0.431573\pi$
$752$	−19.3587	−0.705940
$753$	41.0559	1.49616
$754$	0	0
$755$	3.51810	0.128037
$756$	0	0
$757$	15.5490	0.565137	0.282569	−	0.959247i	$-0.408814\pi$
$757$	15.5490	0.565137	0.282569	+	0.959247i	$0.408814\pi$
$758$	−4.09184	−0.148622
$759$	45.7791	1.66168
$760$	9.43243	0.342150
$761$	31.9505	1.15821	0.579103	−	0.815254i	$-0.303403\pi$
$761$	31.9505	1.15821	0.579103	+	0.815254i	$0.303403\pi$
$762$	−10.6669	−0.386420
$763$	0	0
$764$	−5.10179	−0.184576
$765$	72.7613	2.63069
$766$	11.5060	0.415729
$767$	0	0
$768$	12.0397	0.434446
$769$	−14.5302	−0.523972	−0.261986	−	0.965072i	$-0.584377\pi$
$769$	−14.5302	−0.523972	−0.261986	+	0.965072i	$0.584377\pi$
$770$	0	0
$771$	53.8298	1.93863
$772$	−45.2730	−1.62941
$773$	42.1103	1.51460	0.757301	−	0.653066i	$-0.226518\pi$
$773$	42.1103	1.51460	0.757301	+	0.653066i	$0.226518\pi$
$774$	−5.27540	−0.189620
$775$	7.92922	0.284826
$776$	22.0678	0.792188
$777$	0	0
$778$	1.40418	0.0503424
$779$	4.68608	0.167896
$780$	0	0
$781$	16.1083	0.576401
$782$	−14.3681	−0.513802
$783$	−56.1678	−2.00727
$784$	0	0
$785$	33.1257	1.18231
$786$	10.4405	0.372399
$787$	6.09474	0.217254	0.108627	−	0.994083i	$-0.465355\pi$
$787$	6.09474	0.217254	0.108627	+	0.994083i	$0.465355\pi$
$788$	15.6242	0.556588
$789$	39.3426	1.40063
$790$	−5.31815	−0.189211
$791$	0	0
$792$	48.8976	1.73750
$793$	0	0
$794$	−6.07302	−0.215523
$795$	11.3227	0.401575
$796$	−29.2101	−1.03532
$797$	36.8038	1.30366	0.651830	−	0.758366i	$-0.274002\pi$
$797$	36.8038	1.30366	0.651830	+	0.758366i	$0.274002\pi$
$798$	0	0
$799$	54.5184	1.92872
$800$	−9.39893	−0.332302
$801$	−2.33127	−0.0823712
$802$	−8.34345	−0.294617
$803$	33.5337	1.18338
$804$	36.4627	1.28594
$805$	0	0
$806$	0	0
$807$	6.68185	0.235212
$808$	−7.12819	−0.250769
$809$	−8.73333	−0.307048	−0.153524	−	0.988145i	$-0.549062\pi$
$809$	−8.73333	−0.307048	−0.153524	+	0.988145i	$0.549062\pi$
$810$	−14.6006	−0.513014
$811$	−37.1024	−1.30284	−0.651421	−	0.758716i	$-0.725827\pi$
$811$	−37.1024	−1.30284	−0.651421	+	0.758716i	$0.725827\pi$
$812$	0	0
$813$	−18.0202	−0.631995
$814$	23.7589	0.832750
$815$	39.8211	1.39487
$816$	41.5035	1.45291
$817$	−3.57314	−0.125009
$818$	−3.58784	−0.125446
$819$	0	0
$820$	5.66081	0.197684
$821$	−42.1578	−1.47132	−0.735658	−	0.677353i	$-0.763127\pi$
$821$	−42.1578	−1.47132	−0.735658	+	0.677353i	$0.763127\pi$
$822$	6.30583	0.219941
$823$	−18.5452	−0.646445	−0.323223	−	0.946323i	$-0.604766\pi$
$823$	−18.5452	−0.646445	−0.323223	+	0.946323i	$0.604766\pi$
$824$	26.9633	0.939312
$825$	−19.2947	−0.671756
$826$	0	0
$827$	−12.5050	−0.434841	−0.217420	−	0.976078i	$-0.569764\pi$
$827$	−12.5050	−0.434841	−0.217420	+	0.976078i	$0.569764\pi$
$828$	−45.5070	−1.58148
$829$	−45.7666	−1.58954	−0.794770	−	0.606911i	$-0.792409\pi$
$829$	−45.7666	−1.58954	−0.794770	+	0.606911i	$0.792409\pi$
$830$	9.62111	0.333954
$831$	36.0032	1.24894
$832$	0	0
$833$	0	0
$834$	22.5789	0.781843
$835$	−5.12440	−0.177337
$836$	15.1208	0.522962
$837$	50.0343	1.72944
$838$	10.2272	0.353293
$839$	−12.8196	−0.442580	−0.221290	−	0.975208i	$-0.571027\pi$
$839$	−12.8196	−0.442580	−0.221290	+	0.975208i	$0.571027\pi$
$840$	0	0
$841$	−2.87426	−0.0991124
$842$	−11.1429	−0.384009
$843$	−76.6410	−2.63966
$844$	−36.6681	−1.26217
$845$	0	0
$846$	−32.8650	−1.12992
$847$	0	0
$848$	4.43074	0.152152
$849$	−46.8095	−1.60650
$850$	6.05578	0.207712
$851$	−48.4314	−1.66021
$852$	−23.3410	−0.799650
$853$	4.51954	0.154746	0.0773730	−	0.997002i	$-0.475347\pi$
$853$	4.51954	0.154746	0.0773730	+	0.997002i	$0.475347\pi$
$854$	0	0
$855$	−29.7089	−1.01602
$856$	37.8142	1.29246
$857$	47.2746	1.61487	0.807435	−	0.589957i	$-0.200855\pi$
$857$	47.2746	1.61487	0.807435	+	0.589957i	$0.200855\pi$
$858$	0	0
$859$	48.0543	1.63959	0.819795	−	0.572657i	$-0.194087\pi$
$859$	48.0543	1.63959	0.819795	+	0.572657i	$0.194087\pi$
$860$	−4.31638	−0.147187
$861$	0	0
$862$	−5.93999	−0.202317
$863$	27.5799	0.938831	0.469415	−	0.882977i	$-0.344465\pi$
$863$	27.5799	0.938831	0.469415	+	0.882977i	$0.344465\pi$
$864$	−59.3083	−2.01771
$865$	−6.36893	−0.216550
$866$	−5.15631	−0.175218
$867$	−64.3339	−2.18489
$868$	0	0
$869$	−18.6733	−0.633447
$870$	16.1286	0.546811
$871$	0	0
$872$	28.9895	0.981710
$873$	−69.5060	−2.35242
$874$	5.86659	0.198440
$875$	0	0
$876$	−48.5904	−1.64172
$877$	−49.4487	−1.66976	−0.834882	−	0.550430i	$-0.814464\pi$
$877$	−49.4487	−1.66976	−0.834882	+	0.550430i	$0.814464\pi$
$878$	16.5581	0.558807
$879$	77.0315	2.59821
$880$	14.1277	0.476243
$881$	−1.23791	−0.0417061	−0.0208530	−	0.999783i	$-0.506638\pi$
$881$	−1.23791	−0.0417061	−0.0208530	+	0.999783i	$0.506638\pi$
$882$	0	0
$883$	41.4537	1.39503	0.697514	−	0.716571i	$-0.254289\pi$
$883$	41.4537	1.39503	0.697514	+	0.716571i	$0.254289\pi$
$884$	0	0
$885$	−50.8190	−1.70826
$886$	−15.1190	−0.507932
$887$	−1.17172	−0.0393426	−0.0196713	−	0.999807i	$-0.506262\pi$
$887$	−1.17172	−0.0393426	−0.0196713	+	0.999807i	$0.506262\pi$
$888$	−75.4061	−2.53046
$889$	0	0
$890$	0.363051	0.0121695
$891$	−51.2662	−1.71748
$892$	−3.21234	−0.107557
$893$	−22.2602	−0.744910
$894$	3.83548	0.128278
$895$	12.0849	0.403952
$896$	0	0
$897$	0	0
$898$	−11.9975	−0.400361
$899$	−23.2728	−0.776192
$900$	19.1800	0.639335
$901$	−12.4779	−0.415700
$902$	−3.78312	−0.125964
$903$	0	0
$904$	21.0783	0.701054
$905$	22.6215	0.751965
$906$	3.40681	0.113184
$907$	−42.9537	−1.42625	−0.713127	−	0.701035i	$-0.752722\pi$
$907$	−42.9537	−1.42625	−0.713127	+	0.701035i	$0.752722\pi$
$908$	−17.3644	−0.576258
$909$	22.4513	0.744664
$910$	0	0
$911$	24.5412	0.813085	0.406543	−	0.913632i	$-0.366734\pi$
$911$	24.5412	0.813085	0.406543	+	0.913632i	$0.366734\pi$
$912$	−16.9461	−0.561142
$913$	33.7820	1.11802
$914$	−8.28161	−0.273931
$915$	−30.5689	−1.01058
$916$	10.2559	0.338863
$917$	0	0
$918$	38.2127	1.26121
$919$	36.6615	1.20935	0.604675	−	0.796472i	$-0.293303\pi$
$919$	36.6615	1.20935	0.604675	+	0.796472i	$0.293303\pi$
$920$	15.5226	0.511765
$921$	12.0602	0.397398
$922$	14.0780	0.463635
$923$	0	0
$924$	0	0
$925$	20.4126	0.671162
$926$	−5.71549	−0.187823
$927$	−84.9253	−2.78931
$928$	27.5865	0.905572
$929$	−25.9653	−0.851894	−0.425947	−	0.904748i	$-0.640059\pi$
$929$	−25.9653	−0.851894	−0.425947	+	0.904748i	$0.640059\pi$
$930$	−14.3674	−0.471125
$931$	0	0
$932$	−11.9761	−0.392290
$933$	−60.9506	−1.99543
$934$	6.49340	0.212471
$935$	−39.7865	−1.30116
$936$	0	0
$937$	11.8632	0.387554	0.193777	−	0.981046i	$-0.437926\pi$
$937$	11.8632	0.387554	0.193777	+	0.981046i	$0.437926\pi$
$938$	0	0
$939$	−8.94059	−0.291765
$940$	−26.8905	−0.877070
$941$	9.51954	0.310328	0.155164	−	0.987889i	$-0.450409\pi$
$941$	9.51954	0.310328	0.155164	+	0.987889i	$0.450409\pi$
$942$	32.0779	1.04515
$943$	7.71170	0.251128
$944$	−19.8862	−0.647241
$945$	0	0
$946$	2.88464	0.0937876
$947$	−42.5868	−1.38388	−0.691942	−	0.721953i	$-0.743245\pi$
$947$	−42.5868	−1.38388	−0.691942	+	0.721953i	$0.743245\pi$
$948$	27.0576	0.878791
$949$	0	0
$950$	−2.47261	−0.0802222
$951$	−49.3146	−1.59913
$952$	0	0
$953$	31.4390	1.01841	0.509204	−	0.860646i	$-0.329940\pi$
$953$	31.4390	1.01841	0.509204	+	0.860646i	$0.329940\pi$
$954$	7.52199	0.243534
$955$	−5.48115	−0.177366
$956$	12.3050	0.397971
$957$	56.6313	1.83063
$958$	−0.611404	−0.0197536
$959$	0	0
$960$	−7.33689	−0.236797
$961$	−10.2686	−0.331244
$962$	0	0
$963$	−119.102	−3.83800
$964$	−0.442059	−0.0142378
$965$	−48.6394	−1.56576
$966$	0	0
$967$	6.05894	0.194842	0.0974212	−	0.995243i	$-0.468941\pi$
$967$	6.05894	0.194842	0.0974212	+	0.995243i	$0.468941\pi$
$968$	−3.84475	−0.123575
$969$	47.7240	1.53312
$970$	10.8243	0.347546
$971$	−43.4692	−1.39499	−0.697497	−	0.716588i	$-0.745703\pi$
$971$	−43.4692	−1.39499	−0.697497	+	0.716588i	$0.745703\pi$
$972$	18.8943	0.606036
$973$	0	0
$974$	−14.2890	−0.457849
$975$	0	0
$976$	−11.9621	−0.382896
$977$	−28.8178	−0.921964	−0.460982	−	0.887409i	$-0.652503\pi$
$977$	−28.8178	−0.921964	−0.460982	+	0.887409i	$0.652503\pi$
$978$	38.5615	1.23306
$979$	1.27476	0.0407414
$980$	0	0
$981$	−91.3071	−2.91521
$982$	23.5918	0.752844
$983$	−31.1850	−0.994647	−0.497323	−	0.867565i	$-0.665684\pi$
$983$	−31.1850	−0.994647	−0.497323	+	0.867565i	$0.665684\pi$
$984$	12.0069	0.382765
$985$	16.7860	0.534845
$986$	−17.7742	−0.566044
$987$	0	0
$988$	0	0
$989$	−5.88019	−0.186979
$990$	23.9843	0.762271
$991$	−62.1395	−1.97393	−0.986963	−	0.160946i	$-0.948546\pi$
$991$	−62.1395	−1.97393	−0.986963	+	0.160946i	$0.948546\pi$
$992$	−24.5741	−0.780228
$993$	11.8290	0.375382
$994$	0	0
$995$	−31.3821	−0.994879
$996$	−48.9502	−1.55105
$997$	17.3144	0.548351	0.274176	−	0.961680i	$-0.411595\pi$
$997$	17.3144	0.548351	0.274176	+	0.961680i	$0.411595\pi$
$998$	−5.91991	−0.187392
$999$	128.806	4.07524

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cx.1.19		32
7.6	odd	2	inner	8281.2.a.cx.1.20		32
13.2	odd	12		637.2.q.j.589.10	yes	32
13.7	odd	12		637.2.q.j.491.10	yes	32
13.12	even	2	inner	8281.2.a.cx.1.13		32
91.2	odd	12		637.2.k.j.459.8		32
91.20	even	12		637.2.q.j.491.9	✓	32
91.33	even	12		637.2.u.j.361.8		32
91.41	even	12		637.2.q.j.589.9	yes	32
91.46	odd	12		637.2.k.j.569.10		32
91.54	even	12		637.2.k.j.459.7		32
91.59	even	12		637.2.k.j.569.9		32
91.67	odd	12		637.2.u.j.30.7		32
91.72	odd	12		637.2.u.j.361.7		32
91.80	even	12		637.2.u.j.30.8		32
91.90	odd	2	inner	8281.2.a.cx.1.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.k.j.459.7		32	91.54	even	12
637.2.k.j.459.8		32	91.2	odd	12
637.2.k.j.569.9		32	91.59	even	12
637.2.k.j.569.10		32	91.46	odd	12
637.2.q.j.491.9	✓	32	91.20	even	12
637.2.q.j.491.10	yes	32	13.7	odd	12
637.2.q.j.589.9	yes	32	91.41	even	12
637.2.q.j.589.10	yes	32	13.2	odd	12
637.2.u.j.30.7		32	91.67	odd	12
637.2.u.j.30.8		32	91.80	even	12
637.2.u.j.361.7		32	91.72	odd	12
637.2.u.j.361.8		32	91.33	even	12
8281.2.a.cx.1.13		32	13.12	even	2	inner
8281.2.a.cx.1.14		32	91.90	odd	2	inner
8281.2.a.cx.1.19		32	1.1	even	1	trivial
8281.2.a.cx.1.20		32	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+0.565505 q^{2} -3.09111 q^{3} -1.68020 q^{4} -1.80514 q^{5} -1.74804 q^{6} -2.08118 q^{8} +6.55499 q^{9} +O(q^{10})\)	\(q+0.565505 q^{2} -3.09111 q^{3} -1.68020 q^{4} -1.80514 q^{5} -1.74804 q^{6} -2.08118 q^{8} +6.55499 q^{9} -1.02082 q^{10} -3.58433 q^{11} +5.19370 q^{12} +5.57989 q^{15} +2.18349 q^{16} -6.14919 q^{17} +3.70688 q^{18} +2.51075 q^{19} +3.03300 q^{20} -2.02696 q^{22} +4.13185 q^{23} +6.43315 q^{24} -1.74147 q^{25} -10.9889 q^{27} +5.11133 q^{29} +3.15546 q^{30} -4.55318 q^{31} +5.39713 q^{32} +11.0796 q^{33} -3.47740 q^{34} -11.0137 q^{36} -11.7215 q^{37} +1.41984 q^{38} +3.75681 q^{40} +1.86640 q^{41} -1.42314 q^{43} +6.02240 q^{44} -11.8327 q^{45} +2.33658 q^{46} -8.86595 q^{47} -6.74942 q^{48} -0.984810 q^{50} +19.0078 q^{51} +2.02920 q^{53} -6.21427 q^{54} +6.47021 q^{55} -7.76102 q^{57} +2.89049 q^{58} -9.10752 q^{59} -9.37536 q^{60} -5.47841 q^{61} -2.57485 q^{62} -1.31488 q^{64} +6.26555 q^{66} +7.02055 q^{67} +10.3319 q^{68} -12.7720 q^{69} -4.49410 q^{71} -13.6421 q^{72} -9.35564 q^{73} -6.62856 q^{74} +5.38308 q^{75} -4.21858 q^{76} +5.20970 q^{79} -3.94151 q^{80} +14.3029 q^{81} +1.05546 q^{82} -9.42491 q^{83} +11.1001 q^{85} -0.804791 q^{86} -15.7997 q^{87} +7.45961 q^{88} -0.355648 q^{89} -6.69144 q^{90} -6.94235 q^{92} +14.0744 q^{93} -5.01374 q^{94} -4.53226 q^{95} -16.6831 q^{96} -10.6035 q^{97} -23.4952 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 40 q^{4} + 56 q^{9}+O(q^{10}) \) 32 * q + 40 * q^4 + 56 * q^9	\( 32 q + 40 q^{4} + 56 q^{9} + 56 q^{16} - 16 q^{22} + 48 q^{23} + 40 q^{25} + 48 q^{29} + 48 q^{30} + 184 q^{36} + 24 q^{43} + 72 q^{53} - 32 q^{64} - 48 q^{74} + 96 q^{79} + 128 q^{81} + 112 q^{88} + 168 q^{92} + 168 q^{95}+O(q^{100}) \) 32 * q + 40 * q^4 + 56 * q^9 + 56 * q^16 - 16 * q^22 + 48 * q^23 + 40 * q^25 + 48 * q^29 + 48 * q^30 + 184 * q^36 + 24 * q^43 + 72 * q^53 - 32 * q^64 - 48 * q^74 + 96 * q^79 + 128 * q^81 + 112 * q^88 + 168 * q^92 + 168 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more