LMFDB - Embedded newform 6223.2.a.t.1.53

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6223,2,Mod(1,6223)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, 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("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6223 = 7^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6223.a (trivial)

Newform invariants

comment:select newform

sage:traces = [74,18,0,86,0] 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:	$49.6909051778$
Analytic rank:	$0$
Dimension:	$74$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.53
Character	$\chi$	$=$	6223.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.75327 q^{2} -3.28812 q^{3} +1.07397 q^{4} -1.87758 q^{5} -5.76498 q^{6} -1.62359 q^{8} +7.81176 q^{9} -3.29190 q^{10} -2.70975 q^{11} -3.53134 q^{12} -3.71676 q^{13} +6.17370 q^{15} -4.99453 q^{16} +2.05065 q^{17} +13.6962 q^{18} -7.60959 q^{19} -2.01645 q^{20} -4.75092 q^{22} +4.81105 q^{23} +5.33856 q^{24} -1.47471 q^{25} -6.51650 q^{26} -15.8217 q^{27} -8.01094 q^{29} +10.8242 q^{30} +0.938179 q^{31} -5.50959 q^{32} +8.90998 q^{33} +3.59535 q^{34} +8.38958 q^{36} +4.23700 q^{37} -13.3417 q^{38} +12.2212 q^{39} +3.04841 q^{40} -7.94113 q^{41} -5.32646 q^{43} -2.91018 q^{44} -14.6672 q^{45} +8.43508 q^{46} -7.32875 q^{47} +16.4226 q^{48} -2.58557 q^{50} -6.74279 q^{51} -3.99168 q^{52} -8.15367 q^{53} -27.7397 q^{54} +5.08775 q^{55} +25.0213 q^{57} -14.0454 q^{58} +13.7515 q^{59} +6.63035 q^{60} -8.43911 q^{61} +1.64488 q^{62} +0.329233 q^{64} +6.97851 q^{65} +15.6216 q^{66} -10.8139 q^{67} +2.20233 q^{68} -15.8193 q^{69} -3.70175 q^{71} -12.6831 q^{72} -2.98643 q^{73} +7.42861 q^{74} +4.84902 q^{75} -8.17245 q^{76} +21.4271 q^{78} -0.365977 q^{79} +9.37761 q^{80} +28.5884 q^{81} -13.9230 q^{82} -15.4825 q^{83} -3.85025 q^{85} -9.33874 q^{86} +26.3410 q^{87} +4.39951 q^{88} -5.66403 q^{89} -25.7156 q^{90} +5.16690 q^{92} -3.08485 q^{93} -12.8493 q^{94} +14.2876 q^{95} +18.1162 q^{96} +2.08128 q^{97} -21.1679 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 74 q + 18 q^{2} + 86 q^{4} + 54 q^{8} + 114 q^{9} + 28 q^{11} + 32 q^{15} + 118 q^{16} + 54 q^{18} + 20 q^{22} + 64 q^{23} + 130 q^{25} + 36 q^{29} + 68 q^{30} + 146 q^{32} + 162 q^{36} + 48 q^{37} + 24 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) $ 74 * q + 18 * q^2 + 86 * q^4 + 54 * q^8 + 114 * q^9 + 28 * q^11 + 32 * q^15 + 118 * q^16 + 54 * q^18 + 20 * q^22 + 64 * q^23 + 130 * q^25 + 36 * q^29 + 68 * q^30 + 146 * q^32 + 162 * q^36 + 48 * q^37 + 24 * q^39 + 36 * q^43 + 76 * q^44 + 40 * q^46 + 94 * q^50 - 40 * q^51 + 136 * q^53 + 112 * q^57 - 44 * q^58 + 116 * q^60 + 150 * q^64 + 128 * q^65 + 56 * q^67 + 60 * q^71 + 58 * q^72 + 116 * q^74 + 76 * q^78 + 24 * q^79 + 274 * q^81 + 48 * q^85 + 76 * q^88 + 192 * q^92 + 104 * q^93 + 64 * q^95 + 72 * q^99

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 $

$2$	1.75327	1.23975	0.619876	−	0.784700i	$-0.287183\pi$
$2$	1.75327	1.23975	0.619876	+	0.784700i	$0.287183\pi$
$3$	−3.28812	−1.89840	−0.949200	−	0.314674i	$-0.898105\pi$
$3$	−3.28812	−1.89840	−0.949200	+	0.314674i	$0.898105\pi$
$4$	1.07397	0.536983
$5$	−1.87758	−0.839678	−0.419839	−	0.907599i	$-0.637913\pi$
$5$	−1.87758	−0.839678	−0.419839	+	0.907599i	$0.637913\pi$
$6$	−5.76498	−2.35354
$7$	0	0
$7$	0	0
$8$	−1.62359	−0.574025
$9$	7.81176	2.60392
$10$	−3.29190	−1.04099
$11$	−2.70975	−0.817019	−0.408509	−	0.912754i	$-0.633951\pi$
$11$	−2.70975	−0.817019	−0.408509	+	0.912754i	$0.633951\pi$
$12$	−3.53134	−1.01941
$13$	−3.71676	−1.03084	−0.515422	−	0.856936i	$-0.672365\pi$
$13$	−3.71676	−1.03084	−0.515422	+	0.856936i	$0.672365\pi$
$14$	0	0
$15$	6.17370	1.59404
$16$	−4.99453	−1.24863
$17$	2.05065	0.497356	0.248678	−	0.968586i	$-0.420004\pi$
$17$	2.05065	0.497356	0.248678	+	0.968586i	$0.420004\pi$
$18$	13.6962	3.22822
$19$	−7.60959	−1.74576	−0.872880	−	0.487935i	$-0.837750\pi$
$19$	−7.60959	−1.74576	−0.872880	+	0.487935i	$0.837750\pi$
$20$	−2.01645	−0.450893
$21$	0	0
$22$	−4.75092	−1.01290
$23$	4.81105	1.00317	0.501586	−	0.865108i	$-0.332750\pi$
$23$	4.81105	1.00317	0.501586	+	0.865108i	$0.332750\pi$
$24$	5.33856	1.08973
$25$	−1.47471	−0.294942
$26$	−6.51650	−1.27799
$27$	−15.8217	−3.04488
$28$	0	0
$29$	−8.01094	−1.48759	−0.743797	−	0.668405i	$-0.766977\pi$
$29$	−8.01094	−1.48759	−0.743797	+	0.668405i	$0.766977\pi$
$30$	10.8242	1.97622
$31$	0.938179	0.168502	0.0842510	−	0.996445i	$-0.473150\pi$
$31$	0.938179	0.168502	0.0842510	+	0.996445i	$0.473150\pi$
$32$	−5.50959	−0.973968
$33$	8.90998	1.55103
$34$	3.59535	0.616597
$35$	0	0
$36$	8.38958	1.39826
$37$	4.23700	0.696558	0.348279	−	0.937391i	$-0.386766\pi$
$37$	4.23700	0.696558	0.348279	+	0.937391i	$0.386766\pi$
$38$	−13.3417	−2.16431
$39$	12.2212	1.95696
$40$	3.04841	0.481996
$41$	−7.94113	−1.24020	−0.620098	−	0.784524i	$-0.712907\pi$
$41$	−7.94113	−1.24020	−0.620098	+	0.784524i	$0.712907\pi$
$42$	0	0
$43$	−5.32646	−0.812278	−0.406139	−	0.913811i	$-0.633125\pi$
$43$	−5.32646	−0.812278	−0.406139	+	0.913811i	$0.633125\pi$
$44$	−2.91018	−0.438726
$45$	−14.6672	−2.18645
$46$	8.43508	1.24368
$47$	−7.32875	−1.06901	−0.534504	−	0.845166i	$-0.679502\pi$
$47$	−7.32875	−1.06901	−0.534504	+	0.845166i	$0.679502\pi$
$48$	16.4226	2.37040
$49$	0	0
$50$	−2.58557	−0.365654
$51$	−6.74279	−0.944180
$52$	−3.99168	−0.553546
$53$	−8.15367	−1.11999	−0.559996	−	0.828495i	$-0.689197\pi$
$53$	−8.15367	−1.11999	−0.559996	+	0.828495i	$0.689197\pi$
$54$	−27.7397	−3.77490
$55$	5.08775	0.686032
$56$	0	0
$57$	25.0213	3.31415
$58$	−14.0454	−1.84425
$59$	13.7515	1.79029	0.895143	−	0.445779i	$-0.147073\pi$
$59$	13.7515	1.79029	0.895143	+	0.445779i	$0.147073\pi$
$60$	6.63035	0.855975
$61$	−8.43911	−1.08052	−0.540259	−	0.841499i	$-0.681674\pi$
$61$	−8.43911	−1.08052	−0.540259	+	0.841499i	$0.681674\pi$
$62$	1.64488	0.208901
$63$	0	0
$64$	0.329233	0.0411541
$65$	6.97851	0.865577
$66$	15.6216	1.92289
$67$	−10.8139	−1.32113	−0.660565	−	0.750769i	$-0.729683\pi$
$67$	−10.8139	−1.32113	−0.660565	+	0.750769i	$0.729683\pi$
$68$	2.20233	0.267072
$69$	−15.8193	−1.90442
$70$	0	0
$71$	−3.70175	−0.439317	−0.219658	−	0.975577i	$-0.570494\pi$
$71$	−3.70175	−0.439317	−0.219658	+	0.975577i	$0.570494\pi$
$72$	−12.6831	−1.49472
$73$	−2.98643	−0.349535	−0.174768	−	0.984610i	$-0.555917\pi$
$73$	−2.98643	−0.349535	−0.174768	+	0.984610i	$0.555917\pi$
$74$	7.42861	0.863559
$75$	4.84902	0.559917
$76$	−8.17245	−0.937444
$77$	0	0
$78$	21.4271	2.42614
$79$	−0.365977	−0.0411756	−0.0205878	−	0.999788i	$-0.506554\pi$
$79$	−0.365977	−0.0411756	−0.0205878	+	0.999788i	$0.506554\pi$
$80$	9.37761	1.04845
$81$	28.5884	3.17649
$82$	−13.9230	−1.53754
$83$	−15.4825	−1.69942	−0.849711	−	0.527248i	$-0.823224\pi$
$83$	−15.4825	−1.69942	−0.849711	+	0.527248i	$0.823224\pi$
$84$	0	0
$85$	−3.85025	−0.417618
$86$	−9.33874	−1.00702
$87$	26.3410	2.82405
$88$	4.39951	0.468990
$89$	−5.66403	−0.600386	−0.300193	−	0.953878i	$-0.597051\pi$
$89$	−5.66403	−0.600386	−0.300193	+	0.953878i	$0.597051\pi$
$90$	−25.7156	−2.71066
$91$	0	0
$92$	5.16690	0.538687
$93$	−3.08485	−0.319884
$94$	−12.8493	−1.32530
$95$	14.2876	1.46588
$96$	18.1162	1.84898
$97$	2.08128	0.211322	0.105661	−	0.994402i	$-0.466304\pi$
$97$	2.08128	0.211322	0.105661	+	0.994402i	$0.466304\pi$
$98$	0	0
$99$	−21.1679	−2.12745
$100$	−1.58379	−0.158379
$101$	6.21607	0.618522	0.309261	−	0.950977i	$-0.399918\pi$
$101$	6.21607	0.618522	0.309261	+	0.950977i	$0.399918\pi$
$102$	−11.8220	−1.17055
$103$	14.0549	1.38487	0.692434	−	0.721481i	$-0.256538\pi$
$103$	14.0549	1.38487	0.692434	+	0.721481i	$0.256538\pi$
$104$	6.03450	0.591731
$105$	0	0
$106$	−14.2956	−1.38851
$107$	−18.3549	−1.77443	−0.887216	−	0.461355i	$-0.847364\pi$
$107$	−18.3549	−1.77443	−0.887216	+	0.461355i	$0.847364\pi$
$108$	−16.9920	−1.63505
$109$	13.7838	1.32025	0.660123	−	0.751158i	$-0.270504\pi$
$109$	13.7838	1.32025	0.660123	+	0.751158i	$0.270504\pi$
$110$	8.92022	0.850510
$111$	−13.9318	−1.32235
$112$	0	0
$113$	8.19499	0.770919	0.385460	−	0.922725i	$-0.374043\pi$
$113$	8.19499	0.770919	0.385460	+	0.922725i	$0.374043\pi$
$114$	43.8691	4.10872
$115$	−9.03311	−0.842341
$116$	−8.60348	−0.798813
$117$	−29.0345	−2.68424
$118$	24.1101	2.21951
$119$	0	0
$120$	−10.0236	−0.915022
$121$	−3.65728	−0.332480
$122$	−14.7961	−1.33957
$123$	26.1114	2.35439
$124$	1.00757	0.0904827
$125$	12.1568	1.08733
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	11.5964	1.02499
$129$	17.5141	1.54203
$130$	12.2352	1.07310
$131$	15.8959	1.38883	0.694417	−	0.719573i	$-0.255663\pi$
$131$	15.8959	1.38883	0.694417	+	0.719573i	$0.255663\pi$
$132$	9.56902	0.832876
$133$	0	0
$134$	−18.9597	−1.63787
$135$	29.7064	2.55672
$136$	−3.32941	−0.285495
$137$	−0.634025	−0.0541684	−0.0270842	−	0.999633i	$-0.508622\pi$
$137$	−0.634025	−0.0541684	−0.0270842	+	0.999633i	$0.508622\pi$
$138$	−27.7356	−2.36101
$139$	−1.89489	−0.160723	−0.0803613	−	0.996766i	$-0.525607\pi$
$139$	−1.89489	−0.160723	−0.0803613	+	0.996766i	$0.525607\pi$
$140$	0	0
$141$	24.0979	2.02941
$142$	−6.49018	−0.544644
$143$	10.0715	0.842220
$144$	−39.0161	−3.25134
$145$	15.0412	1.24910
$146$	−5.23603	−0.433337
$147$	0	0
$148$	4.55039	0.374040
$149$	0.813532	0.0666471	0.0333236	−	0.999445i	$-0.489391\pi$
$149$	0.813532	0.0666471	0.0333236	+	0.999445i	$0.489391\pi$
$150$	8.50166	0.694158
$151$	−15.1759	−1.23500	−0.617500	−	0.786571i	$-0.711855\pi$
$151$	−15.1759	−1.23500	−0.617500	+	0.786571i	$0.711855\pi$
$152$	12.3548	1.00211
$153$	16.0192	1.29508
$154$	0	0
$155$	−1.76150	−0.141487
$156$	13.1251	1.05085
$157$	13.7481	1.09722	0.548608	−	0.836080i	$-0.315158\pi$
$157$	13.7481	1.09722	0.548608	+	0.836080i	$0.315158\pi$
$158$	−0.641657	−0.0510475
$159$	26.8103	2.12619
$160$	10.3447	0.817819
$161$	0	0
$162$	50.1232	3.93805
$163$	−6.04408	−0.473409	−0.236704	−	0.971582i	$-0.576067\pi$
$163$	−6.04408	−0.473409	−0.236704	+	0.971582i	$0.576067\pi$
$164$	−8.52851	−0.665965
$165$	−16.7292	−1.30236
$166$	−27.1450	−2.10686
$167$	−13.1442	−1.01713	−0.508566	−	0.861023i	$-0.669824\pi$
$167$	−13.1442	−1.01713	−0.508566	+	0.861023i	$0.669824\pi$
$168$	0	0
$169$	0.814327	0.0626405
$170$	−6.75054	−0.517743
$171$	−59.4443	−4.54582
$172$	−5.72044	−0.436180
$173$	−13.9027	−1.05700	−0.528502	−	0.848932i	$-0.677246\pi$
$173$	−13.9027	−1.05700	−0.528502	+	0.848932i	$0.677246\pi$
$174$	46.1829	3.50112
$175$	0	0
$176$	13.5339	1.02016
$177$	−45.2165	−3.39868
$178$	−9.93060	−0.744330
$179$	−0.526339	−0.0393404	−0.0196702	−	0.999807i	$-0.506262\pi$
$179$	−0.526339	−0.0393404	−0.0196702	+	0.999807i	$0.506262\pi$
$180$	−15.7521	−1.17409
$181$	3.81393	0.283487	0.141743	−	0.989903i	$-0.454729\pi$
$181$	3.81393	0.283487	0.141743	+	0.989903i	$0.454729\pi$
$182$	0	0
$183$	27.7489	2.05125
$184$	−7.81116	−0.575847
$185$	−7.95528	−0.584884
$186$	−5.40858	−0.396577
$187$	−5.55674	−0.406349
$188$	−7.87084	−0.574040
$189$	0	0
$190$	25.0500	1.81732
$191$	17.7280	1.28275	0.641376	−	0.767227i	$-0.278364\pi$
$191$	17.7280	1.28275	0.641376	+	0.767227i	$0.278364\pi$
$192$	−1.08256	−0.0781269
$193$	19.1342	1.37731	0.688656	−	0.725088i	$-0.258201\pi$
$193$	19.1342	1.37731	0.688656	+	0.725088i	$0.258201\pi$
$194$	3.64906	0.261987
$195$	−22.9462	−1.64321
$196$	0	0
$197$	9.96740	0.710148	0.355074	−	0.934838i	$-0.384456\pi$
$197$	9.96740	0.710148	0.355074	+	0.934838i	$0.384456\pi$
$198$	−37.1131	−2.63751
$199$	−3.94001	−0.279300	−0.139650	−	0.990201i	$-0.544598\pi$
$199$	−3.94001	−0.279300	−0.139650	+	0.990201i	$0.544598\pi$
$200$	2.39432	0.169304
$201$	35.5575	2.50803
$202$	10.8985	0.766813
$203$	0	0
$204$	−7.24154	−0.507009
$205$	14.9101	1.04137
$206$	24.6420	1.71689
$207$	37.5828	2.61218
$208$	18.5635	1.28715
$209$	20.6201	1.42632
$210$	0	0
$211$	−12.6786	−0.872834	−0.436417	−	0.899745i	$-0.643753\pi$
$211$	−12.6786	−0.872834	−0.436417	+	0.899745i	$0.643753\pi$
$212$	−8.75677	−0.601417
$213$	12.1718	0.833999
$214$	−32.1811	−2.19985
$215$	10.0008	0.682051
$216$	25.6879	1.74784
$217$	0	0
$218$	24.1667	1.63678
$219$	9.81976	0.663558
$220$	5.46408	0.368388
$221$	−7.62178	−0.512696
$222$	−24.4262	−1.63938
$223$	14.0427	0.940368	0.470184	−	0.882568i	$-0.344188\pi$
$223$	14.0427	0.940368	0.470184	+	0.882568i	$0.344188\pi$
$224$	0	0
$225$	−11.5201	−0.768004
$226$	14.3680	0.955748
$227$	9.37755	0.622410	0.311205	−	0.950343i	$-0.399267\pi$
$227$	9.37755	0.622410	0.311205	+	0.950343i	$0.399267\pi$
$228$	26.8720	1.77964
$229$	29.2694	1.93418	0.967089	−	0.254440i	$-0.0818912\pi$
$229$	29.2694	1.93418	0.967089	+	0.254440i	$0.0818912\pi$
$230$	−15.8375	−1.04429
$231$	0	0
$232$	13.0065	0.853917
$233$	18.7377	1.22755	0.613775	−	0.789481i	$-0.289650\pi$
$233$	18.7377	1.22755	0.613775	+	0.789481i	$0.289650\pi$
$234$	−50.9054	−3.32779
$235$	13.7603	0.897622
$236$	14.7686	0.961354
$237$	1.20338	0.0781677
$238$	0	0
$239$	−0.286738	−0.0185476	−0.00927378	−	0.999957i	$-0.502952\pi$
$239$	−0.286738	−0.0185476	−0.00927378	+	0.999957i	$0.502952\pi$
$240$	−30.8347	−1.99037
$241$	0.0989937	0.00637674	0.00318837	−	0.999995i	$-0.498985\pi$
$241$	0.0989937	0.00637674	0.00318837	+	0.999995i	$0.498985\pi$
$242$	−6.41221	−0.412193
$243$	−46.5371	−2.98535
$244$	−9.06333	−0.580220
$245$	0	0
$246$	45.7805	2.91886
$247$	28.2830	1.79961
$248$	−1.52322	−0.0967244
$249$	50.9083	3.22618
$250$	21.3141	1.34802
$251$	−21.4397	−1.35326	−0.676632	−	0.736321i	$-0.736561\pi$
$251$	−21.4397	−1.35326	−0.676632	+	0.736321i	$0.736561\pi$
$252$	0	0
$253$	−13.0367	−0.819611
$254$	1.75327	0.110010
$255$	12.6601	0.792807
$256$	19.6732	1.22958
$257$	−10.1539	−0.633386	−0.316693	−	0.948528i	$-0.602572\pi$
$257$	−10.1539	−0.633386	−0.316693	+	0.948528i	$0.602572\pi$
$258$	30.7069	1.91173
$259$	0	0
$260$	7.49468	0.464801
$261$	−62.5796	−3.87358
$262$	27.8699	1.72181
$263$	−0.170410	−0.0105079	−0.00525397	−	0.999986i	$-0.501672\pi$
$263$	−0.170410	−0.0105079	−0.00525397	+	0.999986i	$0.501672\pi$
$264$	−14.4661	−0.890330
$265$	15.3091	0.940432
$266$	0	0
$267$	18.6240	1.13977
$268$	−11.6138	−0.709424
$269$	23.4143	1.42759	0.713797	−	0.700352i	$-0.246974\pi$
$269$	23.4143	1.42759	0.713797	+	0.700352i	$0.246974\pi$
$270$	52.0835	3.16970
$271$	−10.7567	−0.653424	−0.326712	−	0.945124i	$-0.605941\pi$
$271$	−10.7567	−0.653424	−0.326712	+	0.945124i	$0.605941\pi$
$272$	−10.2420	−0.621014
$273$	0	0
$274$	−1.11162	−0.0671554
$275$	3.99608	0.240973
$276$	−16.9894	−1.02264
$277$	−25.3150	−1.52103	−0.760516	−	0.649319i	$-0.775054\pi$
$277$	−25.3150	−1.52103	−0.760516	+	0.649319i	$0.775054\pi$
$278$	−3.32226	−0.199256
$279$	7.32883	0.438766
$280$	0	0
$281$	−12.4718	−0.744004	−0.372002	−	0.928232i	$-0.621328\pi$
$281$	−12.4718	−0.744004	−0.372002	+	0.928232i	$0.621328\pi$
$282$	42.2501	2.51596
$283$	20.5940	1.22418	0.612092	−	0.790787i	$-0.290328\pi$
$283$	20.5940	1.22418	0.612092	+	0.790787i	$0.290328\pi$
$284$	−3.97555	−0.235906
$285$	−46.9794	−2.78282
$286$	17.6581	1.04414
$287$	0	0
$288$	−43.0397	−2.53614
$289$	−12.7948	−0.752637
$290$	26.3712	1.54857
$291$	−6.84352	−0.401174
$292$	−3.20733	−0.187695
$293$	9.80199	0.572638	0.286319	−	0.958134i	$-0.407568\pi$
$293$	9.80199	0.572638	0.286319	+	0.958134i	$0.407568\pi$
$294$	0	0
$295$	−25.8194	−1.50326
$296$	−6.87914	−0.399842
$297$	42.8727	2.48773
$298$	1.42634	0.0826259
$299$	−17.8815	−1.03411
$300$	5.20769	0.300666
$301$	0	0
$302$	−26.6076	−1.53109
$303$	−20.4392	−1.17420
$304$	38.0063	2.17981
$305$	15.8451	0.907287
$306$	28.0860	1.60557
$307$	14.7492	0.841781	0.420890	−	0.907112i	$-0.361718\pi$
$307$	14.7492	0.841781	0.420890	+	0.907112i	$0.361718\pi$
$308$	0	0
$309$	−46.2142	−2.62903
$310$	−3.08840	−0.175409
$311$	−16.1745	−0.917174	−0.458587	−	0.888649i	$-0.651644\pi$
$311$	−16.1745	−0.917174	−0.458587	+	0.888649i	$0.651644\pi$
$312$	−19.8422	−1.12334
$313$	0.701554	0.0396542	0.0198271	−	0.999803i	$-0.493688\pi$
$313$	0.701554	0.0396542	0.0198271	+	0.999803i	$0.493688\pi$
$314$	24.1041	1.36028
$315$	0	0
$316$	−0.393047	−0.0221106
$317$	21.6294	1.21483	0.607415	−	0.794385i	$-0.292207\pi$
$317$	21.6294	1.21483	0.607415	+	0.794385i	$0.292207\pi$
$318$	47.0057	2.63595
$319$	21.7076	1.21539
$320$	−0.618160	−0.0345562
$321$	60.3531	3.36858
$322$	0	0
$323$	−15.6046	−0.868264
$324$	30.7030	1.70572
$325$	5.48114	0.304039
$326$	−10.5969	−0.586909
$327$	−45.3227	−2.50635
$328$	12.8931	0.711904
$329$	0	0
$330$	−29.3308	−1.61461
$331$	19.5706	1.07570	0.537850	−	0.843041i	$-0.319237\pi$
$331$	19.5706	1.07570	0.537850	+	0.843041i	$0.319237\pi$
$332$	−16.6277	−0.912562
$333$	33.0984	1.81378
$334$	−23.0454	−1.26099
$335$	20.3039	1.10932
$336$	0	0
$337$	−18.5193	−1.00881	−0.504404	−	0.863468i	$-0.668288\pi$
$337$	−18.5193	−1.00881	−0.504404	+	0.863468i	$0.668288\pi$
$338$	1.42774	0.0776587
$339$	−26.9461	−1.46351
$340$	−4.13504	−0.224254
$341$	−2.54223	−0.137669
$342$	−104.222	−5.63569
$343$	0	0
$344$	8.64799	0.466268
$345$	29.7020	1.59910
$346$	−24.3752	−1.31042
$347$	−35.5600	−1.90896	−0.954481	−	0.298272i	$-0.903590\pi$
$347$	−35.5600	−1.90896	−0.954481	+	0.298272i	$0.903590\pi$
$348$	28.2893	1.51647
$349$	9.31611	0.498680	0.249340	−	0.968416i	$-0.419786\pi$
$349$	9.31611	0.498680	0.249340	+	0.968416i	$0.419786\pi$
$350$	0	0
$351$	58.8054	3.13880
$352$	14.9296	0.795750
$353$	−7.76581	−0.413332	−0.206666	−	0.978412i	$-0.566261\pi$
$353$	−7.76581	−0.413332	−0.206666	+	0.978412i	$0.566261\pi$
$354$	−79.2769	−4.21352
$355$	6.95032	0.368884
$356$	−6.08298	−0.322397
$357$	0	0
$358$	−0.922816	−0.0487723
$359$	−9.11633	−0.481142	−0.240571	−	0.970632i	$-0.577335\pi$
$359$	−9.11633	−0.481142	−0.240571	+	0.970632i	$0.577335\pi$
$360$	23.8135	1.25508
$361$	38.9059	2.04768
$362$	6.68686	0.351453
$363$	12.0256	0.631180
$364$	0	0
$365$	5.60725	0.293497
$366$	48.6513	2.54305
$367$	6.61489	0.345294	0.172647	−	0.984984i	$-0.444768\pi$
$367$	6.61489	0.345294	0.172647	+	0.984984i	$0.444768\pi$
$368$	−24.0289	−1.25259
$369$	−62.0343	−3.22937
$370$	−13.9478	−0.725111
$371$	0	0
$372$	−3.31303	−0.171772
$373$	−19.4162	−1.00533	−0.502666	−	0.864481i	$-0.667647\pi$
$373$	−19.4162	−1.00533	−0.502666	+	0.864481i	$0.667647\pi$
$374$	−9.74248	−0.503772
$375$	−39.9729	−2.06419
$376$	11.8989	0.613638
$377$	29.7748	1.53348
$378$	0	0
$379$	−6.00616	−0.308516	−0.154258	−	0.988031i	$-0.549299\pi$
$379$	−6.00616	−0.308516	−0.154258	+	0.988031i	$0.549299\pi$
$380$	15.3444	0.787151
$381$	−3.28812	−0.168456
$382$	31.0820	1.59029
$383$	−27.5205	−1.40623	−0.703117	−	0.711074i	$-0.748209\pi$
$383$	−27.5205	−1.40623	−0.703117	+	0.711074i	$0.748209\pi$
$384$	−38.1305	−1.94584
$385$	0	0
$386$	33.5475	1.70752
$387$	−41.6091	−2.11511
$388$	2.23523	0.113477
$389$	14.1529	0.717578	0.358789	−	0.933419i	$-0.383190\pi$
$389$	14.1529	0.717578	0.358789	+	0.933419i	$0.383190\pi$
$390$	−40.2310	−2.03717
$391$	9.86577	0.498933
$392$	0	0
$393$	−52.2678	−2.63656
$394$	17.4756	0.880407
$395$	0.687149	0.0345742
$396$	−22.7336	−1.14241
$397$	34.8688	1.75001	0.875006	−	0.484112i	$-0.160857\pi$
$397$	34.8688	1.75001	0.875006	+	0.484112i	$0.160857\pi$
$398$	−6.90792	−0.346263
$399$	0	0
$400$	7.36547	0.368273
$401$	17.4112	0.869474	0.434737	−	0.900558i	$-0.356841\pi$
$401$	17.4112	0.869474	0.434737	+	0.900558i	$0.356841\pi$
$402$	62.3420	3.10933
$403$	−3.48699	−0.173699
$404$	6.67585	0.332136
$405$	−53.6768	−2.66722
$406$	0	0
$407$	−11.4812	−0.569101
$408$	10.9475	0.541983
$409$	−12.9475	−0.640212	−0.320106	−	0.947382i	$-0.603719\pi$
$409$	−12.9475	−0.640212	−0.320106	+	0.947382i	$0.603719\pi$
$410$	26.1414	1.29103
$411$	2.08475	0.102833
$412$	15.0945	0.743651
$413$	0	0
$414$	65.8928	3.23846
$415$	29.0695	1.42697
$416$	20.4779	1.00401
$417$	6.23064	0.305116
$418$	36.1526	1.76828
$419$	−28.3893	−1.38691	−0.693455	−	0.720500i	$-0.743912\pi$
$419$	−28.3893	−1.38691	−0.693455	+	0.720500i	$0.743912\pi$
$420$	0	0
$421$	−34.9959	−1.70560	−0.852798	−	0.522241i	$-0.825096\pi$
$421$	−34.9959	−1.70560	−0.852798	+	0.522241i	$0.825096\pi$
$422$	−22.2291	−1.08210
$423$	−57.2505	−2.78361
$424$	13.2382	0.642904
$425$	−3.02411	−0.146691
$426$	21.3405	1.03395
$427$	0	0
$428$	−19.7125	−0.952840
$429$	−33.1163	−1.59887
$430$	17.5342	0.845574
$431$	−19.3699	−0.933017	−0.466508	−	0.884517i	$-0.654488\pi$
$431$	−19.3699	−0.933017	−0.466508	+	0.884517i	$0.654488\pi$
$432$	79.0218	3.80194
$433$	11.7846	0.566330	0.283165	−	0.959071i	$-0.408616\pi$
$433$	11.7846	0.566330	0.283165	+	0.959071i	$0.408616\pi$
$434$	0	0
$435$	−49.4572	−2.37129
$436$	14.8033	0.708950
$437$	−36.6101	−1.75130
$438$	17.2167	0.822647
$439$	−18.7778	−0.896214	−0.448107	−	0.893980i	$-0.647902\pi$
$439$	−18.7778	−0.896214	−0.448107	+	0.893980i	$0.647902\pi$
$440$	−8.26042	−0.393800
$441$	0	0
$442$	−13.3631	−0.635616
$443$	32.3556	1.53726	0.768630	−	0.639694i	$-0.220939\pi$
$443$	32.3556	1.53726	0.768630	+	0.639694i	$0.220939\pi$
$444$	−14.9623	−0.710077
$445$	10.6347	0.504131
$446$	24.6207	1.16582
$447$	−2.67499	−0.126523
$448$	0	0
$449$	10.5288	0.496886	0.248443	−	0.968647i	$-0.420081\pi$
$449$	10.5288	0.496886	0.248443	+	0.968647i	$0.420081\pi$
$450$	−20.1978	−0.952135
$451$	21.5184	1.01326
$452$	8.80114	0.413971
$453$	49.9004	2.34452
$454$	16.4414	0.771633
$455$	0	0
$456$	−40.6243	−1.90241
$457$	−0.338624	−0.0158402	−0.00792008	−	0.999969i	$-0.502521\pi$
$457$	−0.338624	−0.0158402	−0.00792008	+	0.999969i	$0.502521\pi$
$458$	51.3173	2.39790
$459$	−32.4447	−1.51439
$460$	−9.70125	−0.452323
$461$	10.1873	0.474472	0.237236	−	0.971452i	$-0.423759\pi$
$461$	10.1873	0.474472	0.237236	+	0.971452i	$0.423759\pi$
$462$	0	0
$463$	34.5047	1.60357	0.801785	−	0.597612i	$-0.203884\pi$
$463$	34.5047	1.60357	0.801785	+	0.597612i	$0.203884\pi$
$464$	40.0109	1.85746
$465$	5.79204	0.268599
$466$	32.8524	1.52186
$467$	−1.87037	−0.0865505	−0.0432752	−	0.999063i	$-0.513779\pi$
$467$	−1.87037	−0.0865505	−0.0432752	+	0.999063i	$0.513779\pi$
$468$	−31.1821	−1.44139
$469$	0	0
$470$	24.1255	1.11283
$471$	−45.2054	−2.08296
$472$	−22.3267	−1.02767
$473$	14.4334	0.663646
$474$	2.10985	0.0969086
$475$	11.2219	0.514897
$476$	0	0
$477$	−63.6945	−2.91637
$478$	−0.502731	−0.0229944
$479$	−29.5238	−1.34898	−0.674489	−	0.738285i	$-0.735636\pi$
$479$	−29.5238	−1.34898	−0.674489	+	0.738285i	$0.735636\pi$
$480$	−34.0146	−1.55255
$481$	−15.7479	−0.718043
$482$	0.173563	0.00790558
$483$	0	0
$484$	−3.92780	−0.178536
$485$	−3.90777	−0.177443
$486$	−81.5922	−3.70110
$487$	−19.8864	−0.901137	−0.450569	−	0.892742i	$-0.648779\pi$
$487$	−19.8864	−0.901137	−0.450569	+	0.892742i	$0.648779\pi$
$488$	13.7017	0.620245
$489$	19.8737	0.898719
$490$	0	0
$491$	−22.8686	−1.03204	−0.516022	−	0.856576i	$-0.672588\pi$
$491$	−22.8686	−1.03204	−0.516022	+	0.856576i	$0.672588\pi$
$492$	28.0428	1.26427
$493$	−16.4276	−0.739863
$494$	49.5879	2.23107
$495$	39.7443	1.78637
$496$	−4.68576	−0.210397
$497$	0	0
$498$	89.2562	3.99967
$499$	−26.5633	−1.18914	−0.594569	−	0.804044i	$-0.702677\pi$
$499$	−26.5633	−1.18914	−0.594569	+	0.804044i	$0.702677\pi$
$500$	13.0560	0.583880
$501$	43.2199	1.93092
$502$	−37.5897	−1.67771
$503$	−32.0499	−1.42904	−0.714518	−	0.699617i	$-0.753354\pi$
$503$	−32.0499	−1.42904	−0.714518	+	0.699617i	$0.753354\pi$
$504$	0	0
$505$	−11.6711	−0.519359
$506$	−22.8569	−1.01611
$507$	−2.67761	−0.118917
$508$	1.07397	0.0476496
$509$	12.7522	0.565230	0.282615	−	0.959233i	$-0.408798\pi$
$509$	12.7522	0.565230	0.282615	+	0.959233i	$0.408798\pi$
$510$	22.1966	0.982883
$511$	0	0
$512$	11.2997	0.499381
$513$	120.397	5.31564
$514$	−17.8026	−0.785241
$515$	−26.3891	−1.16284
$516$	18.8095	0.828043
$517$	19.8591	0.873400
$518$	0	0
$519$	45.7138	2.00661
$520$	−11.3302	−0.496863
$521$	3.09515	0.135601	0.0678005	−	0.997699i	$-0.478402\pi$
$521$	3.09515	0.135601	0.0678005	+	0.997699i	$0.478402\pi$
$522$	−109.719	−4.80227
$523$	40.9064	1.78871	0.894356	−	0.447357i	$-0.147634\pi$
$523$	40.9064	1.78871	0.894356	+	0.447357i	$0.147634\pi$
$524$	17.0717	0.745780
$525$	0	0
$526$	−0.298776	−0.0130272
$527$	1.92388	0.0838054
$528$	−44.5012	−1.93666
$529$	0.146161	0.00635482
$530$	26.8411	1.16590
$531$	107.423	4.66177
$532$	0	0
$533$	29.5153	1.27845
$534$	32.6530	1.41304
$535$	34.4627	1.48995
$536$	17.5573	0.758362
$537$	1.73067	0.0746838
$538$	41.0517	1.76986
$539$	0	0
$540$	31.9037	1.37292
$541$	−5.45110	−0.234361	−0.117181	−	0.993111i	$-0.537386\pi$
$541$	−5.45110	−0.234361	−0.117181	+	0.993111i	$0.537386\pi$
$542$	−18.8595	−0.810083
$543$	−12.5407	−0.538172
$544$	−11.2983	−0.484408
$545$	−25.8801	−1.10858
$546$	0	0
$547$	−21.9805	−0.939820	−0.469910	−	0.882714i	$-0.655714\pi$
$547$	−21.9805	−0.939820	−0.469910	+	0.882714i	$0.655714\pi$
$548$	−0.680922	−0.0290876
$549$	−65.9244	−2.81358
$550$	7.00622	0.298746
$551$	60.9600	2.59698
$552$	25.6841	1.09319
$553$	0	0
$554$	−44.3841	−1.88570
$555$	26.1580	1.11034
$556$	−2.03505	−0.0863054
$557$	15.4277	0.653692	0.326846	−	0.945078i	$-0.394014\pi$
$557$	15.4277	0.653692	0.326846	+	0.945078i	$0.394014\pi$
$558$	12.8494	0.543961
$559$	19.7972	0.837332
$560$	0	0
$561$	18.2712	0.771413
$562$	−21.8664	−0.922379
$563$	6.21949	0.262120	0.131060	−	0.991374i	$-0.458162\pi$
$563$	6.21949	0.262120	0.131060	+	0.991374i	$0.458162\pi$
$564$	25.8803	1.08976
$565$	−15.3867	−0.647324
$566$	36.1068	1.51768
$567$	0	0
$568$	6.01012	0.252179
$569$	5.05757	0.212024	0.106012	−	0.994365i	$-0.466192\pi$
$569$	5.05757	0.212024	0.106012	+	0.994365i	$0.466192\pi$
$570$	−82.3677	−3.45000
$571$	27.9872	1.17123	0.585614	−	0.810590i	$-0.300853\pi$
$571$	27.9872	1.17123	0.585614	+	0.810590i	$0.300853\pi$
$572$	10.8164	0.452258
$573$	−58.2918	−2.43518
$574$	0	0
$575$	−7.09489	−0.295877
$576$	2.57189	0.107162
$577$	−30.1732	−1.25613	−0.628063	−	0.778162i	$-0.716152\pi$
$577$	−30.1732	−1.25613	−0.628063	+	0.778162i	$0.716152\pi$
$578$	−22.4328	−0.933083
$579$	−62.9157	−2.61469
$580$	16.1537	0.670746
$581$	0	0
$582$	−11.9986	−0.497356
$583$	22.0944	0.915055
$584$	4.84874	0.200642
$585$	54.5144	2.25389
$586$	17.1856	0.709929
$587$	−23.8119	−0.982821	−0.491411	−	0.870928i	$-0.663519\pi$
$587$	−23.8119	−0.982821	−0.491411	+	0.870928i	$0.663519\pi$
$588$	0	0
$589$	−7.13916	−0.294164
$590$	−45.2685	−1.86367
$591$	−32.7741	−1.34814
$592$	−21.1618	−0.869745
$593$	−8.57484	−0.352126	−0.176063	−	0.984379i	$-0.556336\pi$
$593$	−8.57484	−0.352126	−0.176063	+	0.984379i	$0.556336\pi$
$594$	75.1676	3.08416
$595$	0	0
$596$	0.873706	0.0357884
$597$	12.9553	0.530223
$598$	−31.3512	−1.28205
$599$	−14.6458	−0.598412	−0.299206	−	0.954189i	$-0.596722\pi$
$599$	−14.6458	−0.598412	−0.299206	+	0.954189i	$0.596722\pi$
$600$	−7.87282	−0.321407
$601$	−8.46937	−0.345473	−0.172736	−	0.984968i	$-0.555261\pi$
$601$	−8.46937	−0.345473	−0.172736	+	0.984968i	$0.555261\pi$
$602$	0	0
$603$	−84.4757	−3.44012
$604$	−16.2985	−0.663175
$605$	6.86682	0.279176
$606$	−35.8355	−1.45572
$607$	−39.7008	−1.61140	−0.805702	−	0.592321i	$-0.798212\pi$
$607$	−39.7008	−1.61140	−0.805702	+	0.592321i	$0.798212\pi$
$608$	41.9258	1.70031
$609$	0	0
$610$	27.7808	1.12481
$611$	27.2392	1.10198
$612$	17.2041	0.695434
$613$	−22.4593	−0.907122	−0.453561	−	0.891225i	$-0.649847\pi$
$613$	−22.4593	−0.907122	−0.453561	+	0.891225i	$0.649847\pi$
$614$	25.8594	1.04360
$615$	−49.0262	−1.97693
$616$	0	0
$617$	27.2386	1.09658	0.548291	−	0.836287i	$-0.315278\pi$
$617$	27.2386	1.09658	0.548291	+	0.836287i	$0.315278\pi$
$618$	−81.0261	−3.25935
$619$	−41.2808	−1.65921	−0.829607	−	0.558347i	$-0.811436\pi$
$619$	−41.2808	−1.65921	−0.829607	+	0.558347i	$0.811436\pi$
$620$	−1.89180	−0.0759763
$621$	−76.1188	−3.05454
$622$	−28.3584	−1.13707
$623$	0	0
$624$	−61.0390	−2.44352
$625$	−15.4517	−0.618068
$626$	1.23002	0.0491613
$627$	−67.8013	−2.70772
$628$	14.7650	0.589187
$629$	8.68860	0.346437
$630$	0	0
$631$	20.4482	0.814029	0.407014	−	0.913422i	$-0.366570\pi$
$631$	20.4482	0.814029	0.407014	+	0.913422i	$0.366570\pi$
$632$	0.594196	0.0236358
$633$	41.6890	1.65699
$634$	37.9223	1.50609
$635$	−1.87758	−0.0745093
$636$	28.7933	1.14173
$637$	0	0
$638$	38.0594	1.50678
$639$	−28.9172	−1.14395
$640$	−21.7732	−0.860660
$641$	−9.51793	−0.375936	−0.187968	−	0.982175i	$-0.560190\pi$
$641$	−9.51793	−0.375936	−0.187968	+	0.982175i	$0.560190\pi$
$642$	105.815	4.17620
$643$	−22.0482	−0.869498	−0.434749	−	0.900552i	$-0.643163\pi$
$643$	−22.0482	−0.869498	−0.434749	+	0.900552i	$0.643163\pi$
$644$	0	0
$645$	−32.8840	−1.29481
$646$	−27.3591	−1.07643
$647$	46.6585	1.83433	0.917167	−	0.398503i	$-0.130470\pi$
$647$	46.6585	1.83433	0.917167	+	0.398503i	$0.130470\pi$
$648$	−46.4158	−1.82338
$649$	−37.2629	−1.46270
$650$	9.60993	0.376933
$651$	0	0
$652$	−6.49114	−0.254213
$653$	14.3345	0.560951	0.280475	−	0.959861i	$-0.409508\pi$
$653$	14.3345	0.560951	0.280475	+	0.959861i	$0.409508\pi$
$654$	−79.4631	−3.10725
$655$	−29.8458	−1.16617
$656$	39.6622	1.54855
$657$	−23.3293	−0.910163
$658$	0	0
$659$	−37.1916	−1.44878	−0.724389	−	0.689391i	$-0.757878\pi$
$659$	−37.1916	−1.44878	−0.724389	+	0.689391i	$0.757878\pi$
$660$	−17.9666	−0.699348
$661$	−0.139390	−0.00542165	−0.00271082	−	0.999996i	$-0.500863\pi$
$661$	−0.139390	−0.00542165	−0.00271082	+	0.999996i	$0.500863\pi$
$662$	34.3127	1.33360
$663$	25.0614	0.973303
$664$	25.1372	0.975512
$665$	0	0
$666$	58.0306	2.24864
$667$	−38.5410	−1.49231
$668$	−14.1165	−0.546183
$669$	−46.1741	−1.78519
$670$	35.5983	1.37528
$671$	22.8679	0.882804
$672$	0	0
$673$	−48.9422	−1.88658	−0.943291	−	0.331966i	$-0.892288\pi$
$673$	−48.9422	−1.88658	−0.943291	+	0.331966i	$0.892288\pi$
$674$	−32.4693	−1.25067
$675$	23.3324	0.898063
$676$	0.874560	0.0336369
$677$	26.0416	1.00086	0.500430	−	0.865777i	$-0.333175\pi$
$677$	26.0416	1.00086	0.500430	+	0.865777i	$0.333175\pi$
$678$	−47.2439	−1.81439
$679$	0	0
$680$	6.25123	0.239724
$681$	−30.8346	−1.18158
$682$	−4.45722	−0.170676
$683$	40.6990	1.55730	0.778651	−	0.627457i	$-0.215904\pi$
$683$	40.6990	1.55730	0.778651	+	0.627457i	$0.215904\pi$
$684$	−63.8412	−2.44103
$685$	1.19043	0.0454840
$686$	0	0
$687$	−96.2415	−3.67184
$688$	26.6032	1.01424
$689$	30.3052	1.15454
$690$	52.0757	1.98249
$691$	34.5133	1.31295	0.656473	−	0.754350i	$-0.272048\pi$
$691$	34.5133	1.31295	0.656473	+	0.754350i	$0.272048\pi$
$692$	−14.9310	−0.567593
$693$	0	0
$694$	−62.3464	−2.36664
$695$	3.55780	0.134955
$696$	−42.7669	−1.62108
$697$	−16.2845	−0.616819
$698$	16.3337	0.618239
$699$	−61.6120	−2.33038
$700$	0	0
$701$	−6.47321	−0.244490	−0.122245	−	0.992500i	$-0.539009\pi$
$701$	−6.47321	−0.244490	−0.122245	+	0.992500i	$0.539009\pi$
$702$	103.102	3.89133
$703$	−32.2418	−1.21602
$704$	−0.892137	−0.0336237
$705$	−45.2456	−1.70405
$706$	−13.6156	−0.512429
$707$	0	0
$708$	−48.5610	−1.82503
$709$	−3.40874	−0.128018	−0.0640090	−	0.997949i	$-0.520389\pi$
$709$	−3.40874	−0.128018	−0.0640090	+	0.997949i	$0.520389\pi$
$710$	12.1858	0.457325
$711$	−2.85892	−0.107218
$712$	9.19606	0.344637
$713$	4.51362	0.169037
$714$	0	0
$715$	−18.9100	−0.707193
$716$	−0.565270	−0.0211251
$717$	0.942832	0.0352107
$718$	−15.9834	−0.596496
$719$	−30.9697	−1.15498	−0.577488	−	0.816399i	$-0.695967\pi$
$719$	−30.9697	−1.15498	−0.577488	+	0.816399i	$0.695967\pi$
$720$	73.2557	2.73008
$721$	0	0
$722$	68.2126	2.53861
$723$	−0.325504	−0.0121056
$724$	4.09603	0.152228
$725$	11.8138	0.438753
$726$	21.0842	0.782506
$727$	−19.3050	−0.715985	−0.357992	−	0.933725i	$-0.616539\pi$
$727$	−19.3050	−0.715985	−0.357992	+	0.933725i	$0.616539\pi$
$728$	0	0
$729$	67.2546	2.49091
$730$	9.83105	0.363863
$731$	−10.9227	−0.403991
$732$	29.8014	1.10149
$733$	−8.16857	−0.301713	−0.150856	−	0.988556i	$-0.548203\pi$
$733$	−8.16857	−0.301713	−0.150856	+	0.988556i	$0.548203\pi$
$734$	11.5977	0.428079
$735$	0	0
$736$	−26.5069	−0.977058
$737$	29.3029	1.07939
$738$	−108.763	−4.00362
$739$	−20.0774	−0.738561	−0.369280	−	0.929318i	$-0.620396\pi$
$739$	−20.0774	−0.738561	−0.369280	+	0.929318i	$0.620396\pi$
$740$	−8.54371	−0.314073
$741$	−92.9982	−3.41637
$742$	0	0
$743$	12.8439	0.471198	0.235599	−	0.971850i	$-0.424295\pi$
$743$	12.8439	0.471198	0.235599	+	0.971850i	$0.424295\pi$
$744$	5.00853	0.183622
$745$	−1.52747	−0.0559621
$746$	−34.0419	−1.24636
$747$	−120.945	−4.42516
$748$	−5.96775	−0.218203
$749$	0	0
$750$	−70.0835	−2.55909
$751$	−28.8402	−1.05239	−0.526197	−	0.850363i	$-0.676383\pi$
$751$	−28.8402	−1.05239	−0.526197	+	0.850363i	$0.676383\pi$
$752$	36.6037	1.33480
$753$	70.4965	2.56904
$754$	52.2033	1.90113
$755$	28.4940	1.03700
$756$	0	0
$757$	14.1748	0.515192	0.257596	−	0.966253i	$-0.417070\pi$
$757$	14.1748	0.515192	0.257596	+	0.966253i	$0.417070\pi$
$758$	−10.5304	−0.382483
$759$	42.8663	1.55595
$760$	−23.1972	−0.841450
$761$	23.1893	0.840612	0.420306	−	0.907382i	$-0.361923\pi$
$761$	23.1893	0.840612	0.420306	+	0.907382i	$0.361923\pi$
$762$	−5.76498	−0.208843
$763$	0	0
$764$	19.0393	0.688816
$765$	−30.0773	−1.08745
$766$	−48.2510	−1.74338
$767$	−51.1109	−1.84551
$768$	−64.6881	−2.33423
$769$	−8.59536	−0.309957	−0.154978	−	0.987918i	$-0.549531\pi$
$769$	−8.59536	−0.309957	−0.154978	+	0.987918i	$0.549531\pi$
$770$	0	0
$771$	33.3874	1.20242
$772$	20.5495	0.739594
$773$	−18.8578	−0.678269	−0.339135	−	0.940738i	$-0.610134\pi$
$773$	−18.8578	−0.678269	−0.339135	+	0.940738i	$0.610134\pi$
$774$	−72.9520	−2.62221
$775$	−1.38354	−0.0496982
$776$	−3.37915	−0.121304
$777$	0	0
$778$	24.8138	0.889618
$779$	60.4288	2.16509
$780$	−24.6435	−0.882377
$781$	10.0308	0.358930
$782$	17.2974	0.618553
$783$	126.747	4.52955
$784$	0	0
$785$	−25.8131	−0.921308
$786$	−91.6397	−3.26868
$787$	−25.2512	−0.900108	−0.450054	−	0.893001i	$-0.648595\pi$
$787$	−25.2512	−0.900108	−0.450054	+	0.893001i	$0.648595\pi$
$788$	10.7047	0.381338
$789$	0.560330	0.0199483
$790$	1.20476	0.0428634
$791$	0	0
$792$	34.3680	1.22121
$793$	31.3662	1.11385
$794$	61.1344	2.16958
$795$	−50.3383	−1.78532
$796$	−4.23144	−0.149980
$797$	36.4067	1.28959	0.644796	−	0.764355i	$-0.276943\pi$
$797$	36.4067	1.28959	0.644796	+	0.764355i	$0.276943\pi$
$798$	0	0
$799$	−15.0287	−0.531677
$800$	8.12504	0.287264
$801$	−44.2461	−1.56336
$802$	30.5266	1.07793
$803$	8.09247	0.285577
$804$	38.1875	1.34677
$805$	0	0
$806$	−6.11365	−0.215344
$807$	−76.9891	−2.71015
$808$	−10.0923	−0.355047
$809$	28.2036	0.991586	0.495793	−	0.868441i	$-0.334877\pi$
$809$	28.2036	0.991586	0.495793	+	0.868441i	$0.334877\pi$
$810$	−94.1102	−3.30669
$811$	−46.5628	−1.63504	−0.817520	−	0.575900i	$-0.804652\pi$
$811$	−46.5628	−1.63504	−0.817520	+	0.575900i	$0.804652\pi$
$812$	0	0
$813$	35.3694	1.24046
$814$	−20.1296	−0.705544
$815$	11.3482	0.397511
$816$	33.6771	1.17893
$817$	40.5322	1.41804
$818$	−22.7005	−0.793704
$819$	0	0
$820$	16.0129	0.559196
$821$	9.20580	0.321285	0.160642	−	0.987013i	$-0.448643\pi$
$821$	9.20580	0.321285	0.160642	+	0.987013i	$0.448643\pi$
$822$	3.65514	0.127488
$823$	−48.6812	−1.69692	−0.848460	−	0.529259i	$-0.822470\pi$
$823$	−48.6812	−1.69692	−0.848460	+	0.529259i	$0.822470\pi$
$824$	−22.8194	−0.794950
$825$	−13.1396	−0.457463
$826$	0	0
$827$	−41.6871	−1.44960	−0.724801	−	0.688959i	$-0.758068\pi$
$827$	−41.6871	−1.44960	−0.724801	+	0.688959i	$0.758068\pi$
$828$	40.3626	1.40270
$829$	−16.8859	−0.586472	−0.293236	−	0.956040i	$-0.594732\pi$
$829$	−16.8859	−0.586472	−0.293236	+	0.956040i	$0.594732\pi$
$830$	50.9668	1.76908
$831$	83.2389	2.88753
$832$	−1.22368	−0.0424235
$833$	0	0
$834$	10.9240	0.378268
$835$	24.6793	0.854063
$836$	22.1453	0.765910
$837$	−14.8436	−0.513069
$838$	−49.7742	−1.71942
$839$	−20.5719	−0.710220	−0.355110	−	0.934824i	$-0.615557\pi$
$839$	−20.5719	−0.710220	−0.355110	+	0.934824i	$0.615557\pi$
$840$	0	0
$841$	35.1752	1.21294
$842$	−61.3574	−2.11451
$843$	41.0087	1.41242
$844$	−13.6164	−0.468697
$845$	−1.52896	−0.0525978
$846$	−100.376	−3.45099
$847$	0	0
$848$	40.7237	1.39846
$849$	−67.7155	−2.32399
$850$	−5.30209	−0.181860
$851$	20.3844	0.698768
$852$	13.0721	0.447843
$853$	−16.4923	−0.564685	−0.282342	−	0.959314i	$-0.591111\pi$
$853$	−16.4923	−0.564685	−0.282342	+	0.959314i	$0.591111\pi$
$854$	0	0
$855$	111.611	3.81702
$856$	29.8008	1.01857
$857$	−48.0788	−1.64234	−0.821169	−	0.570684i	$-0.806678\pi$
$857$	−48.0788	−1.64234	−0.821169	+	0.570684i	$0.806678\pi$
$858$	−58.0619	−1.98220
$859$	−17.7113	−0.604303	−0.302151	−	0.953260i	$-0.597705\pi$
$859$	−17.7113	−0.604303	−0.302151	+	0.953260i	$0.597705\pi$
$860$	10.7406	0.366250
$861$	0	0
$862$	−33.9608	−1.15671
$863$	22.1323	0.753393	0.376697	−	0.926337i	$-0.377060\pi$
$863$	22.1323	0.753393	0.376697	+	0.926337i	$0.377060\pi$
$864$	87.1711	2.96562
$865$	26.1034	0.887542
$866$	20.6616	0.702109
$867$	42.0710	1.42881
$868$	0	0
$869$	0.991704	0.0336412
$870$	−86.7119	−2.93981
$871$	40.1927	1.36188
$872$	−22.3792	−0.757854
$873$	16.2585	0.550267
$874$	−64.1875	−2.17117
$875$	0	0
$876$	10.5461	0.356320
$877$	−10.0761	−0.340244	−0.170122	−	0.985423i	$-0.554416\pi$
$877$	−10.0761	−0.340244	−0.170122	+	0.985423i	$0.554416\pi$
$878$	−32.9226	−1.11108
$879$	−32.2302	−1.08710
$880$	−25.4109	−0.856602
$881$	−38.0263	−1.28114	−0.640569	−	0.767901i	$-0.721301\pi$
$881$	−38.0263	−1.28114	−0.640569	+	0.767901i	$0.721301\pi$
$882$	0	0
$883$	46.8196	1.57561	0.787803	−	0.615928i	$-0.211219\pi$
$883$	46.8196	1.57561	0.787803	+	0.615928i	$0.211219\pi$
$884$	−8.18554	−0.275309
$885$	84.8974	2.85380
$886$	56.7282	1.90582
$887$	44.7914	1.50395	0.751975	−	0.659192i	$-0.229102\pi$
$887$	44.7914	1.50395	0.751975	+	0.659192i	$0.229102\pi$
$888$	22.6195	0.759060
$889$	0	0
$890$	18.6455	0.624997
$891$	−77.4672	−2.59525
$892$	15.0814	0.504962
$893$	55.7688	1.86623
$894$	−4.69000	−0.156857
$895$	0.988241	0.0330333
$896$	0	0
$897$	58.7967	1.96316
$898$	18.4599	0.616015
$899$	−7.51570	−0.250663
$900$	−12.3722	−0.412406
$901$	−16.7203	−0.557035
$902$	37.7277	1.25620
$903$	0	0
$904$	−13.3053	−0.442527
$905$	−7.16094	−0.238038
$906$	87.4890	2.90663
$907$	−11.2243	−0.372698	−0.186349	−	0.982484i	$-0.559666\pi$
$907$	−11.2243	−0.372698	−0.186349	+	0.982484i	$0.559666\pi$
$908$	10.0712	0.334224
$909$	48.5584	1.61058
$910$	0	0
$911$	−6.23132	−0.206453	−0.103226	−	0.994658i	$-0.532917\pi$
$911$	−6.23132	−0.206453	−0.103226	+	0.994658i	$0.532917\pi$
$912$	−124.970	−4.13815
$913$	41.9536	1.38846
$914$	−0.593700	−0.0196379
$915$	−52.1006	−1.72239
$916$	31.4344	1.03862
$917$	0	0
$918$	−56.8845	−1.87747
$919$	46.4827	1.53332	0.766661	−	0.642052i	$-0.221917\pi$
$919$	46.4827	1.53332	0.766661	+	0.642052i	$0.221917\pi$
$920$	14.6661	0.483525
$921$	−48.4972	−1.59804
$922$	17.8612	0.588227
$923$	13.7585	0.452867
$924$	0	0
$925$	−6.24833	−0.205444
$926$	60.4962	1.98803
$927$	109.793	3.60609
$928$	44.1370	1.44887
$929$	0.578340	0.0189747	0.00948736	−	0.999955i	$-0.496980\pi$
$929$	0.578340	0.0189747	0.00948736	+	0.999955i	$0.496980\pi$
$930$	10.1550	0.332997
$931$	0	0
$932$	20.1237	0.659174
$933$	53.1839	1.74116
$934$	−3.27927	−0.107301
$935$	10.4332	0.341202
$936$	47.1401	1.54082
$937$	8.44113	0.275760	0.137880	−	0.990449i	$-0.455971\pi$
$937$	8.44113	0.275760	0.137880	+	0.990449i	$0.455971\pi$
$938$	0	0
$939$	−2.30680	−0.0752795
$940$	14.7781	0.482008
$941$	−30.0364	−0.979160	−0.489580	−	0.871958i	$-0.662850\pi$
$941$	−30.0364	−0.979160	−0.489580	+	0.871958i	$0.662850\pi$
$942$	−79.2574	−2.58235
$943$	−38.2052	−1.24413
$944$	−68.6820	−2.23541
$945$	0	0
$946$	25.3056	0.822756
$947$	−8.40307	−0.273063	−0.136531	−	0.990636i	$-0.543595\pi$
$947$	−8.40307	−0.273063	−0.136531	+	0.990636i	$0.543595\pi$
$948$	1.29239	0.0419748
$949$	11.0999	0.360317
$950$	19.6751	0.638344
$951$	−71.1202	−2.30623
$952$	0	0
$953$	12.3922	0.401421	0.200711	−	0.979651i	$-0.435675\pi$
$953$	12.3922	0.401421	0.200711	+	0.979651i	$0.435675\pi$
$954$	−111.674	−3.61558
$955$	−33.2856	−1.07710
$956$	−0.307948	−0.00995974
$957$	−71.3773	−2.30730
$958$	−51.7633	−1.67240
$959$	0	0
$960$	2.03259	0.0656014
$961$	−30.1198	−0.971607
$962$	−27.6104	−0.890195
$963$	−143.384	−4.62048
$964$	0.106316	0.00342421
$965$	−35.9260	−1.15650
$966$	0	0
$967$	−48.1863	−1.54957	−0.774783	−	0.632227i	$-0.782141\pi$
$967$	−48.1863	−1.54957	−0.774783	+	0.632227i	$0.782141\pi$
$968$	5.93792	0.190852
$969$	51.3099	1.64831
$970$	−6.85138	−0.219985
$971$	24.7386	0.793898	0.396949	−	0.917841i	$-0.370069\pi$
$971$	24.7386	0.793898	0.396949	+	0.917841i	$0.370069\pi$
$972$	−49.9793	−1.60309
$973$	0	0
$974$	−34.8662	−1.11719
$975$	−18.0227	−0.577187
$976$	42.1494	1.34917
$977$	50.2894	1.60890	0.804451	−	0.594020i	$-0.202460\pi$
$977$	50.2894	1.60890	0.804451	+	0.594020i	$0.202460\pi$
$978$	34.8440	1.11419
$979$	15.3481	0.490527
$980$	0	0
$981$	107.676	3.43781
$982$	−40.0948	−1.27948
$983$	46.4876	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$983$	46.4876	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$984$	−42.3942	−1.35148
$985$	−18.7146	−0.596295
$986$	−28.8021	−0.917247
$987$	0	0
$988$	30.3751	0.966359
$989$	−25.6258	−0.814855
$990$	69.6827	2.21466
$991$	59.7987	1.89957	0.949784	−	0.312907i	$-0.101303\pi$
$991$	59.7987	1.89957	0.949784	+	0.312907i	$0.101303\pi$
$992$	−5.16899	−0.164116
$993$	−64.3507	−2.04211
$994$	0	0
$995$	7.39768	0.234522
$996$	54.6738	1.73241
$997$	−8.35771	−0.264691	−0.132346	−	0.991204i	$-0.542251\pi$
$997$	−8.35771	−0.264691	−0.132346	+	0.991204i	$0.542251\pi$
$998$	−46.5728	−1.47424
$999$	−67.0364	−2.12094

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.t.1.53	✓	74
7.6	odd	2	inner	6223.2.a.t.1.54	yes	74

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6223.2.a.t.1.53	✓	74	1.1	even	1	trivial
6223.2.a.t.1.54	yes	74	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q+1.75327 q^{2} -3.28812 q^{3} +1.07397 q^{4} -1.87758 q^{5} -5.76498 q^{6} -1.62359 q^{8} +7.81176 q^{9} -3.29190 q^{10} -2.70975 q^{11} -3.53134 q^{12} -3.71676 q^{13} +6.17370 q^{15} -4.99453 q^{16} +2.05065 q^{17} +13.6962 q^{18} -7.60959 q^{19} -2.01645 q^{20} -4.75092 q^{22} +4.81105 q^{23} +5.33856 q^{24} -1.47471 q^{25} -6.51650 q^{26} -15.8217 q^{27} -8.01094 q^{29} +10.8242 q^{30} +0.938179 q^{31} -5.50959 q^{32} +8.90998 q^{33} +3.59535 q^{34} +8.38958 q^{36} +4.23700 q^{37} -13.3417 q^{38} +12.2212 q^{39} +3.04841 q^{40} -7.94113 q^{41} -5.32646 q^{43} -2.91018 q^{44} -14.6672 q^{45} +8.43508 q^{46} -7.32875 q^{47} +16.4226 q^{48} -2.58557 q^{50} -6.74279 q^{51} -3.99168 q^{52} -8.15367 q^{53} -27.7397 q^{54} +5.08775 q^{55} +25.0213 q^{57} -14.0454 q^{58} +13.7515 q^{59} +6.63035 q^{60} -8.43911 q^{61} +1.64488 q^{62} +0.329233 q^{64} +6.97851 q^{65} +15.6216 q^{66} -10.8139 q^{67} +2.20233 q^{68} -15.8193 q^{69} -3.70175 q^{71} -12.6831 q^{72} -2.98643 q^{73} +7.42861 q^{74} +4.84902 q^{75} -8.17245 q^{76} +21.4271 q^{78} -0.365977 q^{79} +9.37761 q^{80} +28.5884 q^{81} -13.9230 q^{82} -15.4825 q^{83} -3.85025 q^{85} -9.33874 q^{86} +26.3410 q^{87} +4.39951 q^{88} -5.66403 q^{89} -25.7156 q^{90} +5.16690 q^{92} -3.08485 q^{93} -12.8493 q^{94} +14.2876 q^{95} +18.1162 q^{96} +2.08128 q^{97} -21.1679 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 74 q + 18 q^{2} + 86 q^{4} + 54 q^{8} + 114 q^{9} + 28 q^{11} + 32 q^{15} + 118 q^{16} + 54 q^{18} + 20 q^{22} + 64 q^{23} + 130 q^{25} + 36 q^{29} + 68 q^{30} + 146 q^{32} + 162 q^{36} + 48 q^{37} + 24 q^{39}+ \cdots + 72 q^{99}+O(q^{100}) \) 74 * q + 18 * q^2 + 86 * q^4 + 54 * q^8 + 114 * q^9 + 28 * q^11 + 32 * q^15 + 118 * q^16 + 54 * q^18 + 20 * q^22 + 64 * q^23 + 130 * q^25 + 36 * q^29 + 68 * q^30 + 146 * q^32 + 162 * q^36 + 48 * q^37 + 24 * q^39 + 36 * q^43 + 76 * q^44 + 40 * q^46 + 94 * q^50 - 40 * q^51 + 136 * q^53 + 112 * q^57 - 44 * q^58 + 116 * q^60 + 150 * q^64 + 128 * q^65 + 56 * q^67 + 60 * q^71 + 58 * q^72 + 116 * q^74 + 76 * q^78 + 24 * q^79 + 274 * q^81 + 48 * q^85 + 76 * q^88 + 192 * q^92 + 104 * q^93 + 64 * q^95 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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