LMFDB - Embedded newform 7203.2.a.k.1.21

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7203,2,Mod(1,7203)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7203, 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("7203.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7203 = 3 \cdot 7^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7203.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,-6,24,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$57.5162445759$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	7203.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.88124 q^{2} +1.00000 q^{3} +1.53907 q^{4} -2.44079 q^{5} +1.88124 q^{6} -0.867123 q^{8} +1.00000 q^{9} -4.59171 q^{10} +3.12497 q^{11} +1.53907 q^{12} -3.06454 q^{13} -2.44079 q^{15} -4.70940 q^{16} +4.97470 q^{17} +1.88124 q^{18} +3.78182 q^{19} -3.75654 q^{20} +5.87882 q^{22} -8.35677 q^{23} -0.867123 q^{24} +0.957455 q^{25} -5.76514 q^{26} +1.00000 q^{27} -8.89375 q^{29} -4.59171 q^{30} +3.12101 q^{31} -7.12528 q^{32} +3.12497 q^{33} +9.35861 q^{34} +1.53907 q^{36} -7.30956 q^{37} +7.11452 q^{38} -3.06454 q^{39} +2.11646 q^{40} +4.34364 q^{41} +4.14344 q^{43} +4.80954 q^{44} -2.44079 q^{45} -15.7211 q^{46} +1.75234 q^{47} -4.70940 q^{48} +1.80120 q^{50} +4.97470 q^{51} -4.71654 q^{52} -0.616802 q^{53} +1.88124 q^{54} -7.62739 q^{55} +3.78182 q^{57} -16.7313 q^{58} +2.96521 q^{59} -3.75654 q^{60} -8.12485 q^{61} +5.87137 q^{62} -3.98556 q^{64} +7.47990 q^{65} +5.87882 q^{66} -7.14360 q^{67} +7.65640 q^{68} -8.35677 q^{69} -10.0195 q^{71} -0.867123 q^{72} -2.05129 q^{73} -13.7511 q^{74} +0.957455 q^{75} +5.82048 q^{76} -5.76514 q^{78} +4.68957 q^{79} +11.4947 q^{80} +1.00000 q^{81} +8.17144 q^{82} -12.2277 q^{83} -12.1422 q^{85} +7.79482 q^{86} -8.89375 q^{87} -2.70973 q^{88} -2.12414 q^{89} -4.59171 q^{90} -12.8616 q^{92} +3.12101 q^{93} +3.29658 q^{94} -9.23063 q^{95} -7.12528 q^{96} -19.4268 q^{97} +3.12497 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 6 q^{2} + 24 q^{3} + 18 q^{4} - 6 q^{6} - 18 q^{8} + 24 q^{9} - 12 q^{10} - 12 q^{11} + 18 q^{12} - 14 q^{13} + 6 q^{16} + 2 q^{17} - 6 q^{18} - 26 q^{19} + 6 q^{20} - 24 q^{22} - 24 q^{23} - 18 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) $ 24 * q - 6 * q^2 + 24 * q^3 + 18 * q^4 - 6 * q^6 - 18 * q^8 + 24 * q^9 - 12 * q^10 - 12 * q^11 + 18 * q^12 - 14 * q^13 + 6 * q^16 + 2 * q^17 - 6 * q^18 - 26 * q^19 + 6 * q^20 - 24 * q^22 - 24 * q^23 - 18 * q^24 - 13 * q^26 + 24 * q^27 - 36 * q^29 - 12 * q^30 - 36 * q^31 - 42 * q^32 - 12 * q^33 - 4 * q^34 + 18 * q^36 - 50 * q^37 + 38 * q^38 - 14 * q^39 - 28 * q^40 + 25 * q^41 - 36 * q^43 - 30 * q^44 - 30 * q^46 + 6 * q^47 + 6 * q^48 - 6 * q^50 + 2 * q^51 - 41 * q^52 - 63 * q^53 - 6 * q^54 - 44 * q^55 - 26 * q^57 - 3 * q^58 - 41 * q^59 + 6 * q^60 - 38 * q^61 + 22 * q^62 - 6 * q^64 - 14 * q^65 - 24 * q^66 - 34 * q^67 + 22 * q^68 - 24 * q^69 - 32 * q^71 - 18 * q^72 - 13 * q^73 - 43 * q^74 - 105 * q^76 - 13 * q^78 - 34 * q^79 + 55 * q^80 + 24 * q^81 + 28 * q^82 + 42 * q^83 - 92 * q^85 + 40 * q^86 - 36 * q^87 - 10 * q^88 + 32 * q^89 - 12 * q^90 - 14 * q^92 - 36 * q^93 - 64 * q^94 - 38 * q^95 - 42 * q^96 - 62 * q^97 - 12 * 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 $

$2$	1.88124	1.33024	0.665119	−	0.746737i	$-0.268381\pi$
$2$	1.88124	1.33024	0.665119	+	0.746737i	$0.268381\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.53907	0.769534
$5$	−2.44079	−1.09155	−0.545777	−	0.837930i	$-0.683765\pi$
$5$	−2.44079	−1.09155	−0.545777	+	0.837930i	$0.683765\pi$
$6$	1.88124	0.768014
$7$	0	0
$7$	0	0
$8$	−0.867123	−0.306574
$9$	1.00000	0.333333
$10$	−4.59171	−1.45203
$11$	3.12497	0.942213	0.471107	−	0.882076i	$-0.343855\pi$
$11$	3.12497	0.942213	0.471107	+	0.882076i	$0.343855\pi$
$12$	1.53907	0.444291
$13$	−3.06454	−0.849950	−0.424975	−	0.905205i	$-0.639717\pi$
$13$	−3.06454	−0.849950	−0.424975	+	0.905205i	$0.639717\pi$
$14$	0	0
$15$	−2.44079	−0.630209
$16$	−4.70940	−1.17735
$17$	4.97470	1.20654	0.603271	−	0.797536i	$-0.293864\pi$
$17$	4.97470	1.20654	0.603271	+	0.797536i	$0.293864\pi$
$18$	1.88124	0.443413
$19$	3.78182	0.867609	0.433804	−	0.901007i	$-0.357171\pi$
$19$	3.78182	0.867609	0.433804	+	0.901007i	$0.357171\pi$
$20$	−3.75654	−0.839989
$21$	0	0
$22$	5.87882	1.25337
$23$	−8.35677	−1.74251	−0.871254	−	0.490833i	$-0.836693\pi$
$23$	−8.35677	−1.74251	−0.871254	+	0.490833i	$0.836693\pi$
$24$	−0.867123	−0.177001
$25$	0.957455	0.191491
$26$	−5.76514	−1.13064
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.89375	−1.65153	−0.825764	−	0.564015i	$-0.809256\pi$
$29$	−8.89375	−1.65153	−0.825764	+	0.564015i	$0.809256\pi$
$30$	−4.59171	−0.838329
$31$	3.12101	0.560550	0.280275	−	0.959920i	$-0.409574\pi$
$31$	3.12101	0.560550	0.280275	+	0.959920i	$0.409574\pi$
$32$	−7.12528	−1.25958
$33$	3.12497	0.543987
$34$	9.35861	1.60499
$35$	0	0
$36$	1.53907	0.256511
$37$	−7.30956	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$37$	−7.30956	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$38$	7.11452	1.15413
$39$	−3.06454	−0.490719
$40$	2.11646	0.334642
$41$	4.34364	0.678363	0.339182	−	0.940721i	$-0.389850\pi$
$41$	4.34364	0.678363	0.339182	+	0.940721i	$0.389850\pi$
$42$	0	0
$43$	4.14344	0.631869	0.315935	−	0.948781i	$-0.397682\pi$
$43$	4.14344	0.631869	0.315935	+	0.948781i	$0.397682\pi$
$44$	4.80954	0.725065
$45$	−2.44079	−0.363851
$46$	−15.7211	−2.31795
$47$	1.75234	0.255606	0.127803	−	0.991800i	$-0.459208\pi$
$47$	1.75234	0.255606	0.127803	+	0.991800i	$0.459208\pi$
$48$	−4.70940	−0.679744
$49$	0	0
$50$	1.80120	0.254729
$51$	4.97470	0.696597
$52$	−4.71654	−0.654066
$53$	−0.616802	−0.0847243	−0.0423621	−	0.999102i	$-0.513488\pi$
$53$	−0.616802	−0.0847243	−0.0423621	+	0.999102i	$0.513488\pi$
$54$	1.88124	0.256005
$55$	−7.62739	−1.02848
$56$	0	0
$57$	3.78182	0.500914
$58$	−16.7313	−2.19693
$59$	2.96521	0.386038	0.193019	−	0.981195i	$-0.438172\pi$
$59$	2.96521	0.386038	0.193019	+	0.981195i	$0.438172\pi$
$60$	−3.75654	−0.484968
$61$	−8.12485	−1.04028	−0.520140	−	0.854081i	$-0.674120\pi$
$61$	−8.12485	−1.04028	−0.520140	+	0.854081i	$0.674120\pi$
$62$	5.87137	0.745665
$63$	0	0
$64$	−3.98556	−0.498195
$65$	7.47990	0.927767
$66$	5.87882	0.723632
$67$	−7.14360	−0.872729	−0.436365	−	0.899770i	$-0.643734\pi$
$67$	−7.14360	−0.872729	−0.436365	+	0.899770i	$0.643734\pi$
$68$	7.65640	0.928475
$69$	−8.35677	−1.00604
$70$	0	0
$71$	−10.0195	−1.18910	−0.594548	−	0.804060i	$-0.702669\pi$
$71$	−10.0195	−1.18910	−0.594548	+	0.804060i	$0.702669\pi$
$72$	−0.867123	−0.102191
$73$	−2.05129	−0.240086	−0.120043	−	0.992769i	$-0.538303\pi$
$73$	−2.05129	−0.240086	−0.120043	+	0.992769i	$0.538303\pi$
$74$	−13.7511	−1.59853
$75$	0.957455	0.110557
$76$	5.82048	0.667655
$77$	0	0
$78$	−5.76514	−0.652773
$79$	4.68957	0.527618	0.263809	−	0.964575i	$-0.415021\pi$
$79$	4.68957	0.527618	0.263809	+	0.964575i	$0.415021\pi$
$80$	11.4947	1.28514
$81$	1.00000	0.111111
$82$	8.17144	0.902385
$83$	−12.2277	−1.34216	−0.671082	−	0.741383i	$-0.734170\pi$
$83$	−12.2277	−1.34216	−0.671082	+	0.741383i	$0.734170\pi$
$84$	0	0
$85$	−12.1422	−1.31701
$86$	7.79482	0.840537
$87$	−8.89375	−0.953510
$88$	−2.70973	−0.288858
$89$	−2.12414	−0.225158	−0.112579	−	0.993643i	$-0.535911\pi$
$89$	−2.12414	−0.225158	−0.112579	+	0.993643i	$0.535911\pi$
$90$	−4.59171	−0.484009
$91$	0	0
$92$	−12.8616	−1.34092
$93$	3.12101	0.323634
$94$	3.29658	0.340016
$95$	−9.23063	−0.947042
$96$	−7.12528	−0.727221
$97$	−19.4268	−1.97249	−0.986247	−	0.165280i	$-0.947147\pi$
$97$	−19.4268	−1.97249	−0.986247	+	0.165280i	$0.947147\pi$
$98$	0	0
$99$	3.12497	0.314071
$100$	1.47359	0.147359
$101$	13.3576	1.32913	0.664563	−	0.747232i	$-0.268618\pi$
$101$	13.3576	1.32913	0.664563	+	0.747232i	$0.268618\pi$
$102$	9.35861	0.926640
$103$	−11.8996	−1.17250	−0.586249	−	0.810131i	$-0.699396\pi$
$103$	−11.8996	−1.17250	−0.586249	+	0.810131i	$0.699396\pi$
$104$	2.65733	0.260573
$105$	0	0
$106$	−1.16035	−0.112704
$107$	−9.20277	−0.889666	−0.444833	−	0.895614i	$-0.646737\pi$
$107$	−9.20277	−0.889666	−0.444833	+	0.895614i	$0.646737\pi$
$108$	1.53907	0.148097
$109$	−14.9121	−1.42832	−0.714159	−	0.699983i	$-0.753191\pi$
$109$	−14.9121	−1.42832	−0.714159	+	0.699983i	$0.753191\pi$
$110$	−14.3490	−1.36812
$111$	−7.30956	−0.693793
$112$	0	0
$113$	6.04421	0.568591	0.284296	−	0.958737i	$-0.408240\pi$
$113$	6.04421	0.568591	0.284296	+	0.958737i	$0.408240\pi$
$114$	7.11452	0.666335
$115$	20.3971	1.90204
$116$	−13.6881	−1.27091
$117$	−3.06454	−0.283317
$118$	5.57828	0.513522
$119$	0	0
$120$	2.11646	0.193206
$121$	−1.23458	−0.112234
$122$	−15.2848	−1.38382
$123$	4.34364	0.391653
$124$	4.80345	0.431362
$125$	9.86700	0.882532
$126$	0	0
$127$	14.8274	1.31572	0.657858	−	0.753142i	$-0.271463\pi$
$127$	14.8274	1.31572	0.657858	+	0.753142i	$0.271463\pi$
$128$	6.75276	0.596865
$129$	4.14344	0.364810
$130$	14.0715	1.23415
$131$	−9.42020	−0.823047	−0.411524	−	0.911399i	$-0.635003\pi$
$131$	−9.42020	−0.823047	−0.411524	+	0.911399i	$0.635003\pi$
$132$	4.80954	0.418617
$133$	0	0
$134$	−13.4388	−1.16094
$135$	−2.44079	−0.210070
$136$	−4.31367	−0.369894
$137$	−8.94971	−0.764625	−0.382313	−	0.924033i	$-0.624872\pi$
$137$	−8.94971	−0.764625	−0.382313	+	0.924033i	$0.624872\pi$
$138$	−15.7211	−1.33827
$139$	−17.1971	−1.45864	−0.729320	−	0.684173i	$-0.760163\pi$
$139$	−17.1971	−1.45864	−0.729320	+	0.684173i	$0.760163\pi$
$140$	0	0
$141$	1.75234	0.147574
$142$	−18.8491	−1.58178
$143$	−9.57659	−0.800834
$144$	−4.70940	−0.392450
$145$	21.7078	1.80273
$146$	−3.85898	−0.319371
$147$	0	0
$148$	−11.2499	−0.924738
$149$	10.7119	0.877552	0.438776	−	0.898596i	$-0.355412\pi$
$149$	10.7119	0.877552	0.438776	+	0.898596i	$0.355412\pi$
$150$	1.80120	0.147068
$151$	−2.49200	−0.202796	−0.101398	−	0.994846i	$-0.532332\pi$
$151$	−2.49200	−0.202796	−0.101398	+	0.994846i	$0.532332\pi$
$152$	−3.27930	−0.265987
$153$	4.97470	0.402181
$154$	0	0
$155$	−7.61773	−0.611870
$156$	−4.71654	−0.377625
$157$	−10.6777	−0.852170	−0.426085	−	0.904683i	$-0.640108\pi$
$157$	−10.6777	−0.852170	−0.426085	+	0.904683i	$0.640108\pi$
$158$	8.82221	0.701857
$159$	−0.616802	−0.0489156
$160$	17.3913	1.37490
$161$	0	0
$162$	1.88124	0.147804
$163$	20.4082	1.59849	0.799246	−	0.601003i	$-0.205232\pi$
$163$	20.4082	1.59849	0.799246	+	0.601003i	$0.205232\pi$
$164$	6.68517	0.522024
$165$	−7.62739	−0.593791
$166$	−23.0032	−1.78540
$167$	−6.08401	−0.470795	−0.235398	−	0.971899i	$-0.575639\pi$
$167$	−6.08401	−0.470795	−0.235398	+	0.971899i	$0.575639\pi$
$168$	0	0
$169$	−3.60860	−0.277585
$170$	−22.8424	−1.75193
$171$	3.78182	0.289203
$172$	6.37705	0.486245
$173$	4.69180	0.356711	0.178355	−	0.983966i	$-0.442922\pi$
$173$	4.69180	0.356711	0.178355	+	0.983966i	$0.442922\pi$
$174$	−16.7313	−1.26840
$175$	0	0
$176$	−14.7167	−1.10932
$177$	2.96521	0.222879
$178$	−3.99602	−0.299514
$179$	5.55986	0.415563	0.207782	−	0.978175i	$-0.433376\pi$
$179$	5.55986	0.415563	0.207782	+	0.978175i	$0.433376\pi$
$180$	−3.75654	−0.279996
$181$	20.2113	1.50229	0.751145	−	0.660137i	$-0.229502\pi$
$181$	20.2113	1.50229	0.751145	+	0.660137i	$0.229502\pi$
$182$	0	0
$183$	−8.12485	−0.600606
$184$	7.24635	0.534208
$185$	17.8411	1.31170
$186$	5.87137	0.430510
$187$	15.5458	1.13682
$188$	2.69698	0.196697
$189$	0	0
$190$	−17.3650	−1.25979
$191$	−11.5657	−0.836867	−0.418434	−	0.908247i	$-0.637421\pi$
$191$	−11.5657	−0.836867	−0.418434	+	0.908247i	$0.637421\pi$
$192$	−3.98556	−0.287633
$193$	3.21953	0.231747	0.115873	−	0.993264i	$-0.463033\pi$
$193$	3.21953	0.231747	0.115873	+	0.993264i	$0.463033\pi$
$194$	−36.5465	−2.62389
$195$	7.47990	0.535646
$196$	0	0
$197$	−17.0222	−1.21278	−0.606389	−	0.795168i	$-0.707383\pi$
$197$	−17.0222	−1.21278	−0.606389	+	0.795168i	$0.707383\pi$
$198$	5.87882	0.417789
$199$	23.3830	1.65758	0.828788	−	0.559563i	$-0.189031\pi$
$199$	23.3830	1.65758	0.828788	+	0.559563i	$0.189031\pi$
$200$	−0.830231	−0.0587062
$201$	−7.14360	−0.503871
$202$	25.1288	1.76806
$203$	0	0
$204$	7.65640	0.536055
$205$	−10.6019	−0.740470
$206$	−22.3859	−1.55970
$207$	−8.35677	−0.580836
$208$	14.4322	1.00069
$209$	11.8181	0.817472
$210$	0	0
$211$	−18.9230	−1.30271	−0.651357	−	0.758772i	$-0.725800\pi$
$211$	−18.9230	−1.30271	−0.651357	+	0.758772i	$0.725800\pi$
$212$	−0.949301	−0.0651983
$213$	−10.0195	−0.686525
$214$	−17.3126	−1.18347
$215$	−10.1133	−0.689720
$216$	−0.867123	−0.0590002
$217$	0	0
$218$	−28.0532	−1.90000
$219$	−2.05129	−0.138613
$220$	−11.7391	−0.791448
$221$	−15.2452	−1.02550
$222$	−13.7511	−0.922910
$223$	−1.79025	−0.119884	−0.0599420	−	0.998202i	$-0.519092\pi$
$223$	−1.79025	−0.119884	−0.0599420	+	0.998202i	$0.519092\pi$
$224$	0	0
$225$	0.957455	0.0638303
$226$	11.3706	0.756362
$227$	−5.39445	−0.358042	−0.179021	−	0.983845i	$-0.557293\pi$
$227$	−5.39445	−0.358042	−0.179021	+	0.983845i	$0.557293\pi$
$228$	5.82048	0.385471
$229$	26.9928	1.78374	0.891869	−	0.452294i	$-0.149394\pi$
$229$	26.9928	1.78374	0.891869	+	0.452294i	$0.149394\pi$
$230$	38.3719	2.53017
$231$	0	0
$232$	7.71198	0.506316
$233$	−4.72429	−0.309498	−0.154749	−	0.987954i	$-0.549457\pi$
$233$	−4.72429	−0.309498	−0.154749	+	0.987954i	$0.549457\pi$
$234$	−5.76514	−0.376879
$235$	−4.27710	−0.279007
$236$	4.56367	0.297069
$237$	4.68957	0.304620
$238$	0	0
$239$	11.0565	0.715182	0.357591	−	0.933878i	$-0.383598\pi$
$239$	11.0565	0.715182	0.357591	+	0.933878i	$0.383598\pi$
$240$	11.4947	0.741978
$241$	−11.0464	−0.711564	−0.355782	−	0.934569i	$-0.615785\pi$
$241$	−11.0464	−0.711564	−0.355782	+	0.934569i	$0.615785\pi$
$242$	−2.32254	−0.149299
$243$	1.00000	0.0641500
$244$	−12.5047	−0.800532
$245$	0	0
$246$	8.17144	0.520992
$247$	−11.5895	−0.737424
$248$	−2.70630	−0.171850
$249$	−12.2277	−0.774898
$250$	18.5622	1.17398
$251$	0.781279	0.0493139	0.0246569	−	0.999696i	$-0.492151\pi$
$251$	0.781279	0.0493139	0.0246569	+	0.999696i	$0.492151\pi$
$252$	0	0
$253$	−26.1146	−1.64181
$254$	27.8938	1.75022
$255$	−12.1422	−0.760374
$256$	20.6747	1.29217
$257$	−0.850342	−0.0530429	−0.0265214	−	0.999648i	$-0.508443\pi$
$257$	−0.850342	−0.0530429	−0.0265214	+	0.999648i	$0.508443\pi$
$258$	7.79482	0.485284
$259$	0	0
$260$	11.5121	0.713949
$261$	−8.89375	−0.550509
$262$	−17.7217	−1.09485
$263$	−25.8474	−1.59382	−0.796909	−	0.604100i	$-0.793533\pi$
$263$	−25.8474	−1.59382	−0.796909	+	0.604100i	$0.793533\pi$
$264$	−2.70973	−0.166772
$265$	1.50548	0.0924812
$266$	0	0
$267$	−2.12414	−0.129995
$268$	−10.9945	−0.671595
$269$	−15.5617	−0.948813	−0.474407	−	0.880306i	$-0.657337\pi$
$269$	−15.5617	−0.948813	−0.474407	+	0.880306i	$0.657337\pi$
$270$	−4.59171	−0.279443
$271$	2.33777	0.142009	0.0710046	−	0.997476i	$-0.477379\pi$
$271$	2.33777	0.142009	0.0710046	+	0.997476i	$0.477379\pi$
$272$	−23.4279	−1.42052
$273$	0	0
$274$	−16.8366	−1.01713
$275$	2.99201	0.180425
$276$	−12.8616	−0.774180
$277$	−7.74059	−0.465087	−0.232543	−	0.972586i	$-0.574705\pi$
$277$	−7.74059	−0.465087	−0.232543	+	0.972586i	$0.574705\pi$
$278$	−32.3519	−1.94034
$279$	3.12101	0.186850
$280$	0	0
$281$	26.4752	1.57938	0.789689	−	0.613507i	$-0.210242\pi$
$281$	26.4752	1.57938	0.789689	+	0.613507i	$0.210242\pi$
$282$	3.29658	0.196309
$283$	−20.1078	−1.19528	−0.597642	−	0.801763i	$-0.703896\pi$
$283$	−20.1078	−1.19528	−0.597642	+	0.801763i	$0.703896\pi$
$284$	−15.4207	−0.915050
$285$	−9.23063	−0.546775
$286$	−18.0159	−1.06530
$287$	0	0
$288$	−7.12528	−0.419861
$289$	7.74762	0.455742
$290$	40.8376	2.39806
$291$	−19.4268	−1.13882
$292$	−3.15708	−0.184754
$293$	9.06933	0.529836	0.264918	−	0.964271i	$-0.414655\pi$
$293$	9.06933	0.529836	0.264918	+	0.964271i	$0.414655\pi$
$294$	0	0
$295$	−7.23746	−0.421381
$296$	6.33829	0.368406
$297$	3.12497	0.181329
$298$	20.1516	1.16735
$299$	25.6097	1.48104
$300$	1.47359	0.0850777
$301$	0	0
$302$	−4.68806	−0.269767
$303$	13.3576	0.767372
$304$	−17.8101	−1.02148
$305$	19.8310	1.13552
$306$	9.35861	0.534996
$307$	−5.34057	−0.304803	−0.152401	−	0.988319i	$-0.548701\pi$
$307$	−5.34057	−0.304803	−0.152401	+	0.988319i	$0.548701\pi$
$308$	0	0
$309$	−11.8996	−0.676942
$310$	−14.3308	−0.813934
$311$	−12.7374	−0.722270	−0.361135	−	0.932514i	$-0.617611\pi$
$311$	−12.7374	−0.722270	−0.361135	+	0.932514i	$0.617611\pi$
$312$	2.65733	0.150442
$313$	−0.185603	−0.0104909	−0.00524545	−	0.999986i	$-0.501670\pi$
$313$	−0.185603	−0.0104909	−0.00524545	+	0.999986i	$0.501670\pi$
$314$	−20.0872	−1.13359
$315$	0	0
$316$	7.21757	0.406020
$317$	−2.09076	−0.117429	−0.0587144	−	0.998275i	$-0.518700\pi$
$317$	−2.09076	−0.117429	−0.0587144	+	0.998275i	$0.518700\pi$
$318$	−1.16035	−0.0650694
$319$	−27.7927	−1.55609
$320$	9.72792	0.543807
$321$	−9.20277	−0.513649
$322$	0	0
$323$	18.8134	1.04681
$324$	1.53907	0.0855038
$325$	−2.93416	−0.162758
$326$	38.3927	2.12638
$327$	−14.9121	−0.824640
$328$	−3.76647	−0.207969
$329$	0	0
$330$	−14.3490	−0.789884
$331$	5.86784	0.322526	0.161263	−	0.986912i	$-0.448443\pi$
$331$	5.86784	0.322526	0.161263	+	0.986912i	$0.448443\pi$
$332$	−18.8193	−1.03284
$333$	−7.30956	−0.400562
$334$	−11.4455	−0.626270
$335$	17.4360	0.952632
$336$	0	0
$337$	−14.8966	−0.811469	−0.405735	−	0.913991i	$-0.632984\pi$
$337$	−14.8966	−0.811469	−0.405735	+	0.913991i	$0.632984\pi$
$338$	−6.78865	−0.369254
$339$	6.04421	0.328276
$340$	−18.6877	−1.01348
$341$	9.75305	0.528157
$342$	7.11452	0.384709
$343$	0	0
$344$	−3.59288	−0.193715
$345$	20.3971	1.09814
$346$	8.82640	0.474510
$347$	−9.70753	−0.521128	−0.260564	−	0.965457i	$-0.583908\pi$
$347$	−9.70753	−0.521128	−0.260564	+	0.965457i	$0.583908\pi$
$348$	−13.6881	−0.733759
$349$	−28.1924	−1.50910	−0.754552	−	0.656240i	$-0.772146\pi$
$349$	−28.1924	−1.50910	−0.754552	+	0.656240i	$0.772146\pi$
$350$	0	0
$351$	−3.06454	−0.163573
$352$	−22.2663	−1.18680
$353$	32.7538	1.74331	0.871655	−	0.490120i	$-0.163047\pi$
$353$	32.7538	1.74331	0.871655	+	0.490120i	$0.163047\pi$
$354$	5.57828	0.296482
$355$	24.4555	1.29796
$356$	−3.26920	−0.173267
$357$	0	0
$358$	10.4594	0.552799
$359$	31.1370	1.64335	0.821674	−	0.569958i	$-0.193040\pi$
$359$	31.1370	1.64335	0.821674	+	0.569958i	$0.193040\pi$
$360$	2.11646	0.111547
$361$	−4.69784	−0.247255
$362$	38.0222	1.99840
$363$	−1.23458	−0.0647986
$364$	0	0
$365$	5.00677	0.262066
$366$	−15.2848	−0.798949
$367$	3.96940	0.207201	0.103601	−	0.994619i	$-0.466964\pi$
$367$	3.96940	0.207201	0.103601	+	0.994619i	$0.466964\pi$
$368$	39.3554	2.05154
$369$	4.34364	0.226121
$370$	33.5634	1.74488
$371$	0	0
$372$	4.80345	0.249047
$373$	20.4178	1.05719	0.528596	−	0.848873i	$-0.322719\pi$
$373$	20.4178	1.05719	0.528596	+	0.848873i	$0.322719\pi$
$374$	29.2453	1.51224
$375$	9.86700	0.509530
$376$	−1.51950	−0.0783621
$377$	27.2553	1.40372
$378$	0	0
$379$	22.5419	1.15790	0.578951	−	0.815363i	$-0.303462\pi$
$379$	22.5419	1.15790	0.578951	+	0.815363i	$0.303462\pi$
$380$	−14.2066	−0.728782
$381$	14.8274	0.759629
$382$	−21.7579	−1.11323
$383$	19.1727	0.979679	0.489839	−	0.871813i	$-0.337055\pi$
$383$	19.1727	0.979679	0.489839	+	0.871813i	$0.337055\pi$
$384$	6.75276	0.344600
$385$	0	0
$386$	6.05671	0.308279
$387$	4.14344	0.210623
$388$	−29.8992	−1.51790
$389$	29.7518	1.50848	0.754239	−	0.656600i	$-0.228006\pi$
$389$	29.7518	1.50848	0.754239	+	0.656600i	$0.228006\pi$
$390$	14.0715	0.712538
$391$	−41.5724	−2.10241
$392$	0	0
$393$	−9.42020	−0.475186
$394$	−32.0228	−1.61328
$395$	−11.4463	−0.575923
$396$	4.80954	0.241688
$397$	32.7072	1.64153	0.820763	−	0.571269i	$-0.193549\pi$
$397$	32.7072	1.64153	0.820763	+	0.571269i	$0.193549\pi$
$398$	43.9890	2.20497
$399$	0	0
$400$	−4.50904	−0.225452
$401$	−1.69001	−0.0843950	−0.0421975	−	0.999109i	$-0.513436\pi$
$401$	−1.69001	−0.0843950	−0.0421975	+	0.999109i	$0.513436\pi$
$402$	−13.4388	−0.670268
$403$	−9.56445	−0.476439
$404$	20.5582	1.02281
$405$	−2.44079	−0.121284
$406$	0	0
$407$	−22.8421	−1.13224
$408$	−4.31367	−0.213559
$409$	0.438800	0.0216972	0.0108486	−	0.999941i	$-0.496547\pi$
$409$	0.438800	0.0216972	0.0108486	+	0.999941i	$0.496547\pi$
$410$	−19.9448	−0.985002
$411$	−8.94971	−0.441457
$412$	−18.3142	−0.902277
$413$	0	0
$414$	−15.7211	−0.772650
$415$	29.8452	1.46504
$416$	21.8357	1.07058
$417$	−17.1971	−0.842146
$418$	22.2326	1.08743
$419$	17.7290	0.866117	0.433059	−	0.901366i	$-0.357434\pi$
$419$	17.7290	0.866117	0.433059	+	0.901366i	$0.357434\pi$
$420$	0	0
$421$	−12.8327	−0.625427	−0.312713	−	0.949848i	$-0.601238\pi$
$421$	−12.8327	−0.625427	−0.312713	+	0.949848i	$0.601238\pi$
$422$	−35.5987	−1.73292
$423$	1.75234	0.0852018
$424$	0.534843	0.0259743
$425$	4.76305	0.231042
$426$	−18.8491	−0.913242
$427$	0	0
$428$	−14.1637	−0.684628
$429$	−9.57659	−0.462362
$430$	−19.0255	−0.917492
$431$	−28.7998	−1.38724	−0.693620	−	0.720341i	$-0.743985\pi$
$431$	−28.7998	−1.38724	−0.693620	+	0.720341i	$0.743985\pi$
$432$	−4.70940	−0.226581
$433$	3.39719	0.163259	0.0816293	−	0.996663i	$-0.473988\pi$
$433$	3.39719	0.163259	0.0816293	+	0.996663i	$0.473988\pi$
$434$	0	0
$435$	21.7078	1.04081
$436$	−22.9507	−1.09914
$437$	−31.6038	−1.51182
$438$	−3.85898	−0.184389
$439$	13.6035	0.649259	0.324629	−	0.945841i	$-0.394760\pi$
$439$	13.6035	0.649259	0.324629	+	0.945841i	$0.394760\pi$
$440$	6.61388	0.315304
$441$	0	0
$442$	−28.6798	−1.36416
$443$	33.1359	1.57433	0.787166	−	0.616741i	$-0.211548\pi$
$443$	33.1359	1.57433	0.787166	+	0.616741i	$0.211548\pi$
$444$	−11.2499	−0.533898
$445$	5.18458	0.245773
$446$	−3.36789	−0.159474
$447$	10.7119	0.506655
$448$	0	0
$449$	−0.154675	−0.00729955	−0.00364977	−	0.999993i	$-0.501162\pi$
$449$	−0.154675	−0.00729955	−0.00364977	+	0.999993i	$0.501162\pi$
$450$	1.80120	0.0849095
$451$	13.5737	0.639163
$452$	9.30245	0.437551
$453$	−2.49200	−0.117085
$454$	−10.1483	−0.476281
$455$	0	0
$456$	−3.27930	−0.153567
$457$	24.2464	1.13420	0.567098	−	0.823650i	$-0.308066\pi$
$457$	24.2464	1.13420	0.567098	+	0.823650i	$0.308066\pi$
$458$	50.7801	2.37280
$459$	4.97470	0.232199
$460$	31.3926	1.46369
$461$	−17.5384	−0.816843	−0.408421	−	0.912794i	$-0.633921\pi$
$461$	−17.5384	−0.816843	−0.408421	+	0.912794i	$0.633921\pi$
$462$	0	0
$463$	−33.7820	−1.56998	−0.784991	−	0.619508i	$-0.787332\pi$
$463$	−33.7820	−1.56998	−0.784991	+	0.619508i	$0.787332\pi$
$464$	41.8843	1.94443
$465$	−7.61773	−0.353264
$466$	−8.88752	−0.411707
$467$	−12.4132	−0.574416	−0.287208	−	0.957868i	$-0.592727\pi$
$467$	−12.4132	−0.574416	−0.287208	+	0.957868i	$0.592727\pi$
$468$	−4.71654	−0.218022
$469$	0	0
$470$	−8.04626	−0.371146
$471$	−10.6777	−0.492000
$472$	−2.57120	−0.118349
$473$	12.9481	0.595356
$474$	8.82221	0.405218
$475$	3.62092	0.166139
$476$	0	0
$477$	−0.616802	−0.0282414
$478$	20.7999	0.951363
$479$	24.2228	1.10677	0.553384	−	0.832926i	$-0.313336\pi$
$479$	24.2228	1.10677	0.553384	+	0.832926i	$0.313336\pi$
$480$	17.3913	0.793801
$481$	22.4004	1.02137
$482$	−20.7810	−0.946549
$483$	0	0
$484$	−1.90010	−0.0863683
$485$	47.4167	2.15308
$486$	1.88124	0.0853348
$487$	−16.5843	−0.751507	−0.375753	−	0.926720i	$-0.622616\pi$
$487$	−16.5843	−0.751507	−0.375753	+	0.926720i	$0.622616\pi$
$488$	7.04524	0.318923
$489$	20.4082	0.922890
$490$	0	0
$491$	19.6133	0.885137	0.442569	−	0.896735i	$-0.354067\pi$
$491$	19.6133	0.885137	0.442569	+	0.896735i	$0.354067\pi$
$492$	6.68517	0.301391
$493$	−44.2437	−1.99264
$494$	−21.8027	−0.980950
$495$	−7.62739	−0.342826
$496$	−14.6981	−0.659964
$497$	0	0
$498$	−23.0032	−1.03080
$499$	−4.62050	−0.206842	−0.103421	−	0.994638i	$-0.532979\pi$
$499$	−4.62050	−0.206842	−0.103421	+	0.994638i	$0.532979\pi$
$500$	15.1860	0.679138
$501$	−6.08401	−0.271814
$502$	1.46977	0.0655992
$503$	18.4737	0.823703	0.411851	−	0.911251i	$-0.364882\pi$
$503$	18.4737	0.823703	0.411851	+	0.911251i	$0.364882\pi$
$504$	0	0
$505$	−32.6030	−1.45081
$506$	−49.1279	−2.18400
$507$	−3.60860	−0.160264
$508$	22.8203	1.01249
$509$	−33.3903	−1.48000	−0.739999	−	0.672608i	$-0.765174\pi$
$509$	−33.3903	−1.48000	−0.739999	+	0.672608i	$0.765174\pi$
$510$	−22.8424	−1.01148
$511$	0	0
$512$	25.3886	1.12203
$513$	3.78182	0.166971
$514$	−1.59970	−0.0705597
$515$	29.0443	1.27984
$516$	6.37705	0.280734
$517$	5.47602	0.240835
$518$	0	0
$519$	4.69180	0.205947
$520$	−6.48599	−0.284429
$521$	−0.932535	−0.0408551	−0.0204275	−	0.999791i	$-0.506503\pi$
$521$	−0.932535	−0.0408551	−0.0204275	+	0.999791i	$0.506503\pi$
$522$	−16.7313	−0.732309
$523$	−19.3199	−0.844802	−0.422401	−	0.906409i	$-0.638813\pi$
$523$	−19.3199	−0.844802	−0.422401	+	0.906409i	$0.638813\pi$
$524$	−14.4983	−0.633363
$525$	0	0
$526$	−48.6251	−2.12016
$527$	15.5261	0.676327
$528$	−14.7167	−0.640464
$529$	46.8357	2.03633
$530$	2.83218	0.123022
$531$	2.96521	0.128679
$532$	0	0
$533$	−13.3113	−0.576575
$534$	−3.99602	−0.172925
$535$	22.4620	0.971119
$536$	6.19438	0.267556
$537$	5.55986	0.239926
$538$	−29.2753	−1.26215
$539$	0	0
$540$	−3.75654	−0.161656
$541$	−37.3193	−1.60448	−0.802241	−	0.597000i	$-0.796359\pi$
$541$	−37.3193	−1.60448	−0.802241	+	0.597000i	$0.796359\pi$
$542$	4.39790	0.188906
$543$	20.2113	0.867348
$544$	−35.4461	−1.51974
$545$	36.3973	1.55909
$546$	0	0
$547$	12.1711	0.520398	0.260199	−	0.965555i	$-0.416212\pi$
$547$	12.1711	0.520398	0.260199	+	0.965555i	$0.416212\pi$
$548$	−13.7742	−0.588405
$549$	−8.12485	−0.346760
$550$	5.62870	0.240009
$551$	−33.6346	−1.43288
$552$	7.24635	0.308425
$553$	0	0
$554$	−14.5619	−0.618677
$555$	17.8411	0.757313
$556$	−26.4675	−1.12247
$557$	−12.9566	−0.548987	−0.274494	−	0.961589i	$-0.588510\pi$
$557$	−12.9566	−0.548987	−0.274494	+	0.961589i	$0.588510\pi$
$558$	5.87137	0.248555
$559$	−12.6977	−0.537058
$560$	0	0
$561$	15.5458	0.656343
$562$	49.8062	2.10095
$563$	23.6463	0.996574	0.498287	−	0.867012i	$-0.333963\pi$
$563$	23.6463	0.996574	0.498287	+	0.867012i	$0.333963\pi$
$564$	2.69698	0.113563
$565$	−14.7526	−0.620648
$566$	−37.8276	−1.59001
$567$	0	0
$568$	8.68814	0.364546
$569$	−15.0554	−0.631154	−0.315577	−	0.948900i	$-0.602198\pi$
$569$	−15.0554	−0.631154	−0.315577	+	0.948900i	$0.602198\pi$
$570$	−17.3650	−0.727341
$571$	6.29054	0.263251	0.131626	−	0.991300i	$-0.457980\pi$
$571$	6.29054	0.263251	0.131626	+	0.991300i	$0.457980\pi$
$572$	−14.7390	−0.616270
$573$	−11.5657	−0.483165
$574$	0	0
$575$	−8.00123	−0.333674
$576$	−3.98556	−0.166065
$577$	33.5603	1.39713	0.698566	−	0.715546i	$-0.253822\pi$
$577$	33.5603	1.39713	0.698566	+	0.715546i	$0.253822\pi$
$578$	14.5751	0.606246
$579$	3.21953	0.133799
$580$	33.4098	1.38727
$581$	0	0
$582$	−36.5465	−1.51490
$583$	−1.92749	−0.0798283
$584$	1.77872	0.0736040
$585$	7.47990	0.309256
$586$	17.0616	0.704808
$587$	−19.8347	−0.818667	−0.409334	−	0.912385i	$-0.634239\pi$
$587$	−19.8347	−0.818667	−0.409334	+	0.912385i	$0.634239\pi$
$588$	0	0
$589$	11.8031	0.486338
$590$	−13.6154	−0.560537
$591$	−17.0222	−0.700198
$592$	34.4237	1.41480
$593$	26.3481	1.08198	0.540992	−	0.841027i	$-0.318049\pi$
$593$	26.3481	1.08198	0.540992	+	0.841027i	$0.318049\pi$
$594$	5.87882	0.241211
$595$	0	0
$596$	16.4863	0.675306
$597$	23.3830	0.957002
$598$	48.1779	1.97014
$599$	11.0077	0.449762	0.224881	−	0.974386i	$-0.427801\pi$
$599$	11.0077	0.449762	0.224881	+	0.974386i	$0.427801\pi$
$600$	−0.830231	−0.0338940
$601$	25.8420	1.05412	0.527059	−	0.849828i	$-0.323295\pi$
$601$	25.8420	1.05412	0.527059	+	0.849828i	$0.323295\pi$
$602$	0	0
$603$	−7.14360	−0.290910
$604$	−3.83536	−0.156059
$605$	3.01335	0.122510
$606$	25.1288	1.02079
$607$	12.1297	0.492330	0.246165	−	0.969228i	$-0.420830\pi$
$607$	12.1297	0.492330	0.246165	+	0.969228i	$0.420830\pi$
$608$	−26.9465	−1.09283
$609$	0	0
$610$	37.3070	1.51052
$611$	−5.37012	−0.217252
$612$	7.65640	0.309492
$613$	−0.287193	−0.0115996	−0.00579982	−	0.999983i	$-0.501846\pi$
$613$	−0.287193	−0.0115996	−0.00579982	+	0.999983i	$0.501846\pi$
$614$	−10.0469	−0.405460
$615$	−10.6019	−0.427511
$616$	0	0
$617$	−10.4448	−0.420491	−0.210246	−	0.977649i	$-0.567426\pi$
$617$	−10.4448	−0.420491	−0.210246	+	0.977649i	$0.567426\pi$
$618$	−22.3859	−0.900494
$619$	−18.3218	−0.736416	−0.368208	−	0.929743i	$-0.620029\pi$
$619$	−18.3218	−0.736416	−0.368208	+	0.929743i	$0.620029\pi$
$620$	−11.7242	−0.470855
$621$	−8.35677	−0.335346
$622$	−23.9621	−0.960792
$623$	0	0
$624$	14.4322	0.577749
$625$	−28.8706	−1.15482
$626$	−0.349164	−0.0139554
$627$	11.8181	0.471968
$628$	−16.4336	−0.655774
$629$	−36.3629	−1.44988
$630$	0	0
$631$	−11.7491	−0.467725	−0.233863	−	0.972270i	$-0.575137\pi$
$631$	−11.7491	−0.467725	−0.233863	+	0.972270i	$0.575137\pi$
$632$	−4.06643	−0.161754
$633$	−18.9230	−0.752122
$634$	−3.93323	−0.156208
$635$	−36.1905	−1.43618
$636$	−0.949301	−0.0376422
$637$	0	0
$638$	−52.2848	−2.06997
$639$	−10.0195	−0.396365
$640$	−16.4821	−0.651510
$641$	32.3072	1.27606	0.638029	−	0.770012i	$-0.279750\pi$
$641$	32.3072	1.27606	0.638029	+	0.770012i	$0.279750\pi$
$642$	−17.3126	−0.683275
$643$	−8.60507	−0.339351	−0.169675	−	0.985500i	$-0.554272\pi$
$643$	−8.60507	−0.339351	−0.169675	+	0.985500i	$0.554272\pi$
$644$	0	0
$645$	−10.1133	−0.398210
$646$	35.3926	1.39250
$647$	5.21343	0.204961	0.102481	−	0.994735i	$-0.467322\pi$
$647$	5.21343	0.204961	0.102481	+	0.994735i	$0.467322\pi$
$648$	−0.867123	−0.0340638
$649$	9.26619	0.363730
$650$	−5.51986	−0.216507
$651$	0	0
$652$	31.4096	1.23010
$653$	−2.74874	−0.107567	−0.0537834	−	0.998553i	$-0.517128\pi$
$653$	−2.74874	−0.107567	−0.0537834	+	0.998553i	$0.517128\pi$
$654$	−28.0532	−1.09697
$655$	22.9927	0.898401
$656$	−20.4560	−0.798672
$657$	−2.05129	−0.0800285
$658$	0	0
$659$	40.0883	1.56162	0.780809	−	0.624770i	$-0.214807\pi$
$659$	40.0883	1.56162	0.780809	+	0.624770i	$0.214807\pi$
$660$	−11.7391	−0.456943
$661$	21.5867	0.839625	0.419813	−	0.907611i	$-0.362096\pi$
$661$	21.5867	0.839625	0.419813	+	0.907611i	$0.362096\pi$
$662$	11.0388	0.429036
$663$	−15.2452	−0.592073
$664$	10.6029	0.411473
$665$	0	0
$666$	−13.7511	−0.532842
$667$	74.3231	2.87780
$668$	−9.36372	−0.362293
$669$	−1.79025	−0.0692150
$670$	32.8014	1.26723
$671$	−25.3899	−0.980166
$672$	0	0
$673$	31.6950	1.22175	0.610876	−	0.791726i	$-0.290817\pi$
$673$	31.6950	1.22175	0.610876	+	0.791726i	$0.290817\pi$
$674$	−28.0241	−1.07945
$675$	0.957455	0.0368524
$676$	−5.55388	−0.213611
$677$	−9.83778	−0.378097	−0.189048	−	0.981968i	$-0.560540\pi$
$677$	−9.83778	−0.378097	−0.189048	+	0.981968i	$0.560540\pi$
$678$	11.3706	0.436686
$679$	0	0
$680$	10.5288	0.403760
$681$	−5.39445	−0.206716
$682$	18.3478	0.702575
$683$	−34.8927	−1.33513	−0.667565	−	0.744551i	$-0.732663\pi$
$683$	−34.8927	−1.33513	−0.667565	+	0.744551i	$0.732663\pi$
$684$	5.82048	0.222552
$685$	21.8444	0.834630
$686$	0	0
$687$	26.9928	1.02984
$688$	−19.5132	−0.743932
$689$	1.89021	0.0720114
$690$	38.3719	1.46079
$691$	−46.9324	−1.78539	−0.892696	−	0.450659i	$-0.851189\pi$
$691$	−46.9324	−1.78539	−0.892696	+	0.450659i	$0.851189\pi$
$692$	7.22100	0.274501
$693$	0	0
$694$	−18.2622	−0.693224
$695$	41.9745	1.59218
$696$	7.71198	0.292322
$697$	21.6083	0.818473
$698$	−53.0367	−2.00747
$699$	−4.72429	−0.178689
$700$	0	0
$701$	8.03395	0.303438	0.151719	−	0.988424i	$-0.451519\pi$
$701$	8.03395	0.303438	0.151719	+	0.988424i	$0.451519\pi$
$702$	−5.76514	−0.217591
$703$	−27.6434	−1.04259
$704$	−12.4548	−0.469406
$705$	−4.27710	−0.161085
$706$	61.6178	2.31902
$707$	0	0
$708$	4.56367	0.171513
$709$	18.4946	0.694579	0.347289	−	0.937758i	$-0.387102\pi$
$709$	18.4946	0.694579	0.347289	+	0.937758i	$0.387102\pi$
$710$	46.0067	1.72660
$711$	4.68957	0.175873
$712$	1.84189	0.0690278
$713$	−26.0816	−0.976762
$714$	0	0
$715$	23.3744	0.874154
$716$	8.55701	0.319790
$717$	11.0565	0.412911
$718$	58.5762	2.18604
$719$	34.0782	1.27090	0.635451	−	0.772141i	$-0.280814\pi$
$719$	34.0782	1.27090	0.635451	+	0.772141i	$0.280814\pi$
$720$	11.4947	0.428381
$721$	0	0
$722$	−8.83777	−0.328908
$723$	−11.0464	−0.410822
$724$	31.1065	1.15606
$725$	−8.51536	−0.316253
$726$	−2.32254	−0.0861976
$727$	−34.4379	−1.27723	−0.638616	−	0.769525i	$-0.720493\pi$
$727$	−34.4379	−1.27723	−0.638616	+	0.769525i	$0.720493\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	9.41895	0.348611
$731$	20.6124	0.762377
$732$	−12.5047	−0.462187
$733$	34.5125	1.27475	0.637374	−	0.770554i	$-0.280021\pi$
$733$	34.5125	1.27475	0.637374	+	0.770554i	$0.280021\pi$
$734$	7.46740	0.275627
$735$	0	0
$736$	59.5444	2.19483
$737$	−22.3235	−0.822297
$738$	8.17144	0.300795
$739$	−5.01415	−0.184449	−0.0922243	−	0.995738i	$-0.529398\pi$
$739$	−5.01415	−0.184449	−0.0922243	+	0.995738i	$0.529398\pi$
$740$	27.4587	1.00940
$741$	−11.5895	−0.425752
$742$	0	0
$743$	0.462539	0.0169689	0.00848445	−	0.999964i	$-0.497299\pi$
$743$	0.462539	0.0169689	0.00848445	+	0.999964i	$0.497299\pi$
$744$	−2.70630	−0.0992177
$745$	−26.1455	−0.957896
$746$	38.4108	1.40632
$747$	−12.2277	−0.447388
$748$	23.9260	0.874821
$749$	0	0
$750$	18.5622	0.677796
$751$	−33.8518	−1.23527	−0.617635	−	0.786465i	$-0.711909\pi$
$751$	−33.8518	−1.23527	−0.617635	+	0.786465i	$0.711909\pi$
$752$	−8.25249	−0.300937
$753$	0.781279	0.0284714
$754$	51.2737	1.86728
$755$	6.08246	0.221363
$756$	0	0
$757$	−48.1596	−1.75039	−0.875196	−	0.483769i	$-0.839268\pi$
$757$	−48.1596	−1.75039	−0.875196	+	0.483769i	$0.839268\pi$
$758$	42.4068	1.54028
$759$	−26.1146	−0.947901
$760$	8.00409	0.290339
$761$	−13.6833	−0.496020	−0.248010	−	0.968758i	$-0.579777\pi$
$761$	−13.6833	−0.496020	−0.248010	+	0.968758i	$0.579777\pi$
$762$	27.8938	1.01049
$763$	0	0
$764$	−17.8005	−0.643998
$765$	−12.1422	−0.439002
$766$	36.0685	1.30321
$767$	−9.08701	−0.328113
$768$	20.6747	0.746034
$769$	29.3762	1.05933	0.529666	−	0.848206i	$-0.322317\pi$
$769$	29.3762	1.05933	0.529666	+	0.848206i	$0.322317\pi$
$770$	0	0
$771$	−0.850342	−0.0306243
$772$	4.95508	0.178337
$773$	29.7784	1.07105	0.535527	−	0.844518i	$-0.320113\pi$
$773$	29.7784	1.07105	0.535527	+	0.844518i	$0.320113\pi$
$774$	7.79482	0.280179
$775$	2.98822	0.107340
$776$	16.8454	0.604716
$777$	0	0
$778$	55.9704	2.00663
$779$	16.4269	0.588554
$780$	11.5121	0.412198
$781$	−31.3106	−1.12038
$782$	−78.2078	−2.79670
$783$	−8.89375	−0.317837
$784$	0	0
$785$	26.0619	0.930190
$786$	−17.7217	−0.632111
$787$	12.3603	0.440597	0.220298	−	0.975433i	$-0.429297\pi$
$787$	12.3603	0.440597	0.220298	+	0.975433i	$0.429297\pi$
$788$	−26.1983	−0.933275
$789$	−25.8474	−0.920191
$790$	−21.5332	−0.766115
$791$	0	0
$792$	−2.70973	−0.0962861
$793$	24.8989	0.884187
$794$	61.5301	2.18362
$795$	1.50548	0.0533940
$796$	35.9880	1.27556
$797$	−25.9322	−0.918567	−0.459284	−	0.888290i	$-0.651894\pi$
$797$	−25.9322	−0.918567	−0.459284	+	0.888290i	$0.651894\pi$
$798$	0	0
$799$	8.71738	0.308399
$800$	−6.82213	−0.241199
$801$	−2.12414	−0.0750528
$802$	−3.17931	−0.112266
$803$	−6.41022	−0.226212
$804$	−10.9945	−0.387746
$805$	0	0
$806$	−17.9930	−0.633778
$807$	−15.5617	−0.547797
$808$	−11.5826	−0.407476
$809$	4.63400	0.162923	0.0814613	−	0.996677i	$-0.474041\pi$
$809$	4.63400	0.162923	0.0814613	+	0.996677i	$0.474041\pi$
$810$	−4.59171	−0.161336
$811$	−40.4196	−1.41932	−0.709662	−	0.704542i	$-0.751152\pi$
$811$	−40.4196	−1.41932	−0.709662	+	0.704542i	$0.751152\pi$
$812$	0	0
$813$	2.33777	0.0819891
$814$	−42.9716	−1.50615
$815$	−49.8121	−1.74484
$816$	−23.4279	−0.820139
$817$	15.6698	0.548216
$818$	0.825488	0.0288625
$819$	0	0
$820$	−16.3171	−0.569817
$821$	−3.52675	−0.123084	−0.0615422	−	0.998104i	$-0.519602\pi$
$821$	−3.52675	−0.123084	−0.0615422	+	0.998104i	$0.519602\pi$
$822$	−16.8366	−0.587242
$823$	−12.4054	−0.432425	−0.216212	−	0.976346i	$-0.569370\pi$
$823$	−12.4054	−0.432425	−0.216212	+	0.976346i	$0.569370\pi$
$824$	10.3184	0.359458
$825$	2.99201	0.104169
$826$	0	0
$827$	−2.04753	−0.0711995	−0.0355997	−	0.999366i	$-0.511334\pi$
$827$	−2.04753	−0.0711995	−0.0355997	+	0.999366i	$0.511334\pi$
$828$	−12.8616	−0.446973
$829$	8.55373	0.297083	0.148542	−	0.988906i	$-0.452542\pi$
$829$	8.55373	0.297083	0.148542	+	0.988906i	$0.452542\pi$
$830$	56.1461	1.94886
$831$	−7.74059	−0.268518
$832$	12.2139	0.423441
$833$	0	0
$834$	−32.3519	−1.12025
$835$	14.8498	0.513899
$836$	18.1888	0.629073
$837$	3.12101	0.107878
$838$	33.3525	1.15214
$839$	7.17783	0.247806	0.123903	−	0.992294i	$-0.460459\pi$
$839$	7.17783	0.247806	0.123903	+	0.992294i	$0.460459\pi$
$840$	0	0
$841$	50.0988	1.72755
$842$	−24.1414	−0.831967
$843$	26.4752	0.911854
$844$	−29.1238	−1.00248
$845$	8.80783	0.302999
$846$	3.29658	0.113339
$847$	0	0
$848$	2.90477	0.0997502
$849$	−20.1078	−0.690098
$850$	8.96044	0.307341
$851$	61.0844	2.09394
$852$	−15.4207	−0.528304
$853$	35.6724	1.22140	0.610701	−	0.791862i	$-0.290888\pi$
$853$	35.6724	1.22140	0.610701	+	0.791862i	$0.290888\pi$
$854$	0	0
$855$	−9.23063	−0.315681
$856$	7.97994	0.272749
$857$	32.3150	1.10386	0.551930	−	0.833890i	$-0.313891\pi$
$857$	32.3150	1.10386	0.551930	+	0.833890i	$0.313891\pi$
$858$	−18.0159	−0.615052
$859$	3.70986	0.126579	0.0632894	−	0.997995i	$-0.479841\pi$
$859$	3.70986	0.126579	0.0632894	+	0.997995i	$0.479841\pi$
$860$	−15.5650	−0.530763
$861$	0	0
$862$	−54.1795	−1.84536
$863$	16.0976	0.547970	0.273985	−	0.961734i	$-0.411658\pi$
$863$	16.0976	0.547970	0.273985	+	0.961734i	$0.411658\pi$
$864$	−7.12528	−0.242407
$865$	−11.4517	−0.389369
$866$	6.39093	0.217173
$867$	7.74762	0.263123
$868$	0	0
$869$	14.6547	0.497128
$870$	40.8376	1.38452
$871$	21.8918	0.741777
$872$	12.9306	0.437886
$873$	−19.4268	−0.657498
$874$	−59.4544	−2.01107
$875$	0	0
$876$	−3.15708	−0.106668
$877$	−34.1440	−1.15296	−0.576480	−	0.817111i	$-0.695574\pi$
$877$	−34.1440	−1.15296	−0.576480	+	0.817111i	$0.695574\pi$
$878$	25.5914	0.863669
$879$	9.06933	0.305901
$880$	35.9205	1.21088
$881$	25.0315	0.843334	0.421667	−	0.906751i	$-0.361445\pi$
$881$	25.0315	0.843334	0.421667	+	0.906751i	$0.361445\pi$
$882$	0	0
$883$	−20.9210	−0.704047	−0.352024	−	0.935991i	$-0.614506\pi$
$883$	−20.9210	−0.704047	−0.352024	+	0.935991i	$0.614506\pi$
$884$	−23.4633	−0.789158
$885$	−7.23746	−0.243285
$886$	62.3365	2.09424
$887$	10.7698	0.361616	0.180808	−	0.983518i	$-0.442129\pi$
$887$	10.7698	0.361616	0.180808	+	0.983518i	$0.442129\pi$
$888$	6.33829	0.212699
$889$	0	0
$890$	9.75345	0.326936
$891$	3.12497	0.104690
$892$	−2.75532	−0.0922548
$893$	6.62705	0.221766
$894$	20.1516	0.673972
$895$	−13.5705	−0.453610
$896$	0	0
$897$	25.6097	0.855082
$898$	−0.290980	−0.00971014
$899$	−27.7575	−0.925764
$900$	1.47359	0.0491196
$901$	−3.06840	−0.102223
$902$	25.5355	0.850239
$903$	0	0
$904$	−5.24107	−0.174315
$905$	−49.3314	−1.63983
$906$	−4.68806	−0.155750
$907$	17.2959	0.574300	0.287150	−	0.957886i	$-0.407292\pi$
$907$	17.2959	0.574300	0.287150	+	0.957886i	$0.407292\pi$
$908$	−8.30242	−0.275526
$909$	13.3576	0.443042
$910$	0	0
$911$	−14.7576	−0.488942	−0.244471	−	0.969657i	$-0.578614\pi$
$911$	−14.7576	−0.488942	−0.244471	+	0.969657i	$0.578614\pi$
$912$	−17.8101	−0.589752
$913$	−38.2111	−1.26460
$914$	45.6133	1.50875
$915$	19.8310	0.655594
$916$	41.5438	1.37265
$917$	0	0
$918$	9.35861	0.308880
$919$	57.3976	1.89337	0.946687	−	0.322156i	$-0.104407\pi$
$919$	57.3976	1.89337	0.946687	+	0.322156i	$0.104407\pi$
$920$	−17.6868	−0.583117
$921$	−5.34057	−0.175978
$922$	−32.9939	−1.08660
$923$	30.7051	1.01067
$924$	0	0
$925$	−6.99857	−0.230112
$926$	−63.5520	−2.08845
$927$	−11.8996	−0.390833
$928$	63.3705	2.08024
$929$	−7.63146	−0.250380	−0.125190	−	0.992133i	$-0.539954\pi$
$929$	−7.63146	−0.250380	−0.125190	+	0.992133i	$0.539954\pi$
$930$	−14.3308	−0.469925
$931$	0	0
$932$	−7.27100	−0.238170
$933$	−12.7374	−0.417003
$934$	−23.3523	−0.764110
$935$	−37.9440	−1.24090
$936$	2.65733	0.0868576
$937$	16.7284	0.546491	0.273246	−	0.961944i	$-0.411903\pi$
$937$	16.7284	0.546491	0.273246	+	0.961944i	$0.411903\pi$
$938$	0	0
$939$	−0.185603	−0.00605693
$940$	−6.58275	−0.214706
$941$	15.2258	0.496347	0.248173	−	0.968716i	$-0.420170\pi$
$941$	15.2258	0.496347	0.248173	+	0.968716i	$0.420170\pi$
$942$	−20.0872	−0.654478
$943$	−36.2988	−1.18205
$944$	−13.9644	−0.454502
$945$	0	0
$946$	24.3586	0.791965
$947$	−41.1178	−1.33615	−0.668074	−	0.744095i	$-0.732881\pi$
$947$	−41.1178	−1.33615	−0.668074	+	0.744095i	$0.732881\pi$
$948$	7.21757	0.234416
$949$	6.28626	0.204061
$950$	6.81182	0.221005
$951$	−2.09076	−0.0677976
$952$	0	0
$953$	−15.7632	−0.510619	−0.255309	−	0.966859i	$-0.582177\pi$
$953$	−15.7632	−0.510619	−0.255309	+	0.966859i	$0.582177\pi$
$954$	−1.16035	−0.0375678
$955$	28.2295	0.913486
$956$	17.0166	0.550357
$957$	−27.7927	−0.898410
$958$	45.5690	1.47227
$959$	0	0
$960$	9.72792	0.313967
$961$	−21.2593	−0.685784
$962$	42.1406	1.35867
$963$	−9.20277	−0.296555
$964$	−17.0012	−0.547573
$965$	−7.85819	−0.252964
$966$	0	0
$967$	−40.4050	−1.29934	−0.649669	−	0.760217i	$-0.725093\pi$
$967$	−40.4050	−1.29934	−0.649669	+	0.760217i	$0.725093\pi$
$968$	1.07053	0.0344082
$969$	18.8134	0.604374
$970$	89.2023	2.86411
$971$	54.8703	1.76087	0.880435	−	0.474166i	$-0.157250\pi$
$971$	54.8703	1.76087	0.880435	+	0.474166i	$0.157250\pi$
$972$	1.53907	0.0493657
$973$	0	0
$974$	−31.1991	−0.999683
$975$	−2.93416	−0.0939682
$976$	38.2632	1.22478
$977$	25.6340	0.820103	0.410052	−	0.912062i	$-0.365511\pi$
$977$	25.6340	0.820103	0.410052	+	0.912062i	$0.365511\pi$
$978$	38.3927	1.22766
$979$	−6.63787	−0.212147
$980$	0	0
$981$	−14.9121	−0.476106
$982$	36.8974	1.17744
$983$	31.7721	1.01337	0.506687	−	0.862130i	$-0.330870\pi$
$983$	31.7721	1.01337	0.506687	+	0.862130i	$0.330870\pi$
$984$	−3.76647	−0.120071
$985$	41.5475	1.32381
$986$	−83.2331	−2.65068
$987$	0	0
$988$	−17.8371	−0.567473
$989$	−34.6258	−1.10104
$990$	−14.3490	−0.456040
$991$	−34.1030	−1.08332	−0.541659	−	0.840598i	$-0.682204\pi$
$991$	−34.1030	−1.08332	−0.541659	+	0.840598i	$0.682204\pi$
$992$	−22.2381	−0.706059
$993$	5.86784	0.186210
$994$	0	0
$995$	−57.0729	−1.80933
$996$	−18.8193	−0.596311
$997$	−30.5373	−0.967127	−0.483564	−	0.875309i	$-0.660658\pi$
$997$	−30.5373	−0.967127	−0.483564	+	0.875309i	$0.660658\pi$
$998$	−8.69228	−0.275149
$999$	−7.30956	−0.231264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7203.2.a.k.1.21		24
7.6	odd	2		7203.2.a.i.1.21		24
49.12	odd	42		147.2.m.a.46.4	yes	48
49.45	odd	42		147.2.m.a.16.4	✓	48
147.110	even	42		441.2.bb.c.46.1		48
147.143	even	42		441.2.bb.c.163.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.2.m.a.16.4	✓	48	49.45	odd	42
147.2.m.a.46.4	yes	48	49.12	odd	42
441.2.bb.c.46.1		48	147.110	even	42
441.2.bb.c.163.1		48	147.143	even	42
7203.2.a.i.1.21		24	7.6	odd	2
7203.2.a.k.1.21		24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7203 = 3 \cdot 7^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7203.a (trivial)

\(f(q)\)	\(=\)	\(q+1.88124 q^{2} +1.00000 q^{3} +1.53907 q^{4} -2.44079 q^{5} +1.88124 q^{6} -0.867123 q^{8} +1.00000 q^{9} -4.59171 q^{10} +3.12497 q^{11} +1.53907 q^{12} -3.06454 q^{13} -2.44079 q^{15} -4.70940 q^{16} +4.97470 q^{17} +1.88124 q^{18} +3.78182 q^{19} -3.75654 q^{20} +5.87882 q^{22} -8.35677 q^{23} -0.867123 q^{24} +0.957455 q^{25} -5.76514 q^{26} +1.00000 q^{27} -8.89375 q^{29} -4.59171 q^{30} +3.12101 q^{31} -7.12528 q^{32} +3.12497 q^{33} +9.35861 q^{34} +1.53907 q^{36} -7.30956 q^{37} +7.11452 q^{38} -3.06454 q^{39} +2.11646 q^{40} +4.34364 q^{41} +4.14344 q^{43} +4.80954 q^{44} -2.44079 q^{45} -15.7211 q^{46} +1.75234 q^{47} -4.70940 q^{48} +1.80120 q^{50} +4.97470 q^{51} -4.71654 q^{52} -0.616802 q^{53} +1.88124 q^{54} -7.62739 q^{55} +3.78182 q^{57} -16.7313 q^{58} +2.96521 q^{59} -3.75654 q^{60} -8.12485 q^{61} +5.87137 q^{62} -3.98556 q^{64} +7.47990 q^{65} +5.87882 q^{66} -7.14360 q^{67} +7.65640 q^{68} -8.35677 q^{69} -10.0195 q^{71} -0.867123 q^{72} -2.05129 q^{73} -13.7511 q^{74} +0.957455 q^{75} +5.82048 q^{76} -5.76514 q^{78} +4.68957 q^{79} +11.4947 q^{80} +1.00000 q^{81} +8.17144 q^{82} -12.2277 q^{83} -12.1422 q^{85} +7.79482 q^{86} -8.89375 q^{87} -2.70973 q^{88} -2.12414 q^{89} -4.59171 q^{90} -12.8616 q^{92} +3.12101 q^{93} +3.29658 q^{94} -9.23063 q^{95} -7.12528 q^{96} -19.4268 q^{97} +3.12497 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} + 24 q^{3} + 18 q^{4} - 6 q^{6} - 18 q^{8} + 24 q^{9} - 12 q^{10} - 12 q^{11} + 18 q^{12} - 14 q^{13} + 6 q^{16} + 2 q^{17} - 6 q^{18} - 26 q^{19} + 6 q^{20} - 24 q^{22} - 24 q^{23} - 18 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) \) 24 * q - 6 * q^2 + 24 * q^3 + 18 * q^4 - 6 * q^6 - 18 * q^8 + 24 * q^9 - 12 * q^10 - 12 * q^11 + 18 * q^12 - 14 * q^13 + 6 * q^16 + 2 * q^17 - 6 * q^18 - 26 * q^19 + 6 * q^20 - 24 * q^22 - 24 * q^23 - 18 * q^24 - 13 * q^26 + 24 * q^27 - 36 * q^29 - 12 * q^30 - 36 * q^31 - 42 * q^32 - 12 * q^33 - 4 * q^34 + 18 * q^36 - 50 * q^37 + 38 * q^38 - 14 * q^39 - 28 * q^40 + 25 * q^41 - 36 * q^43 - 30 * q^44 - 30 * q^46 + 6 * q^47 + 6 * q^48 - 6 * q^50 + 2 * q^51 - 41 * q^52 - 63 * q^53 - 6 * q^54 - 44 * q^55 - 26 * q^57 - 3 * q^58 - 41 * q^59 + 6 * q^60 - 38 * q^61 + 22 * q^62 - 6 * q^64 - 14 * q^65 - 24 * q^66 - 34 * q^67 + 22 * q^68 - 24 * q^69 - 32 * q^71 - 18 * q^72 - 13 * q^73 - 43 * q^74 - 105 * q^76 - 13 * q^78 - 34 * q^79 + 55 * q^80 + 24 * q^81 + 28 * q^82 + 42 * q^83 - 92 * q^85 + 40 * q^86 - 36 * q^87 - 10 * q^88 + 32 * q^89 - 12 * q^90 - 14 * q^92 - 36 * q^93 - 64 * q^94 - 38 * q^95 - 42 * q^96 - 62 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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