LMFDB - Embedded newform 8450.2.a.by.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8450,2,Mod(1,8450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, 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("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,-1,3,0,-1,-4,3,-4,0,2,-1,0,-4,0,3,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.4735897080$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} -0.445042 q^{6} +0.246980 q^{7} +1.00000 q^{8} -2.80194 q^{9} -0.801938 q^{11} -0.445042 q^{12} +0.246980 q^{14} +1.00000 q^{16} +1.55496 q^{17} -2.80194 q^{18} -1.55496 q^{19} -0.109916 q^{21} -0.801938 q^{22} +0.692021 q^{23} -0.445042 q^{24} +2.58211 q^{27} +0.246980 q^{28} -0.286208 q^{29} +4.15883 q^{31} +1.00000 q^{32} +0.356896 q^{33} +1.55496 q^{34} -2.80194 q^{36} -8.28382 q^{37} -1.55496 q^{38} +8.54288 q^{41} -0.109916 q^{42} -5.62565 q^{43} -0.801938 q^{44} +0.692021 q^{46} -6.02715 q^{47} -0.445042 q^{48} -6.93900 q^{49} -0.692021 q^{51} +0.868313 q^{53} +2.58211 q^{54} +0.246980 q^{56} +0.692021 q^{57} -0.286208 q^{58} +4.30798 q^{59} -4.30798 q^{61} +4.15883 q^{62} -0.692021 q^{63} +1.00000 q^{64} +0.356896 q^{66} -6.49396 q^{67} +1.55496 q^{68} -0.307979 q^{69} -6.58211 q^{71} -2.80194 q^{72} +7.47219 q^{73} -8.28382 q^{74} -1.55496 q^{76} -0.198062 q^{77} +4.72886 q^{79} +7.25667 q^{81} +8.54288 q^{82} -7.93362 q^{83} -0.109916 q^{84} -5.62565 q^{86} +0.127375 q^{87} -0.801938 q^{88} -4.34721 q^{89} +0.692021 q^{92} -1.85086 q^{93} -6.02715 q^{94} -0.445042 q^{96} +7.83877 q^{97} -6.93900 q^{98} +2.24698 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 5 q^{17} - 4 q^{18} - 5 q^{19} - q^{21} + 2 q^{22} - 3 q^{23} - q^{24} + 2 q^{27} - 4 q^{28}+ \cdots + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9 + 2 * q^11 - q^12 - 4 * q^14 + 3 * q^16 + 5 * q^17 - 4 * q^18 - 5 * q^19 - q^21 + 2 * q^22 - 3 * q^23 - q^24 + 2 * q^27 - 4 * q^28 - 9 * q^29 + 4 * q^31 + 3 * q^32 - 3 * q^33 + 5 * q^34 - 4 * q^36 + 8 * q^37 - 5 * q^38 + 7 * q^41 - q^42 - 5 * q^43 + 2 * q^44 - 3 * q^46 - 12 * q^47 - q^48 - 11 * q^49 + 3 * q^51 + 5 * q^53 + 2 * q^54 - 4 * q^56 - 3 * q^57 - 9 * q^58 + 18 * q^59 - 18 * q^61 + 4 * q^62 + 3 * q^63 + 3 * q^64 - 3 * q^66 - 10 * q^67 + 5 * q^68 - 6 * q^69 - 14 * q^71 - 4 * q^72 + 16 * q^73 + 8 * q^74 - 5 * q^76 - 5 * q^77 - 19 * q^79 - 5 * q^81 + 7 * q^82 - 17 * q^83 - q^84 - 5 * q^86 + 17 * q^87 + 2 * q^88 - 31 * q^89 - 3 * q^92 + 8 * q^93 - 12 * q^94 - q^96 - 9 * q^97 - 11 * q^98 + 2 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−0.445042	−0.256945	−0.128473	−	0.991713i	$-0.541007\pi$
$3$	−0.445042	−0.256945	−0.128473	+	0.991713i	$0.541007\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−0.445042	−0.181688
$7$	0.246980	0.0933495	0.0466748	−	0.998910i	$-0.485138\pi$
$7$	0.246980	0.0933495	0.0466748	+	0.998910i	$0.485138\pi$
$8$	1.00000	0.353553
$9$	−2.80194	−0.933979
$10$	0	0
$11$	−0.801938	−0.241793	−0.120897	−	0.992665i	$-0.538577\pi$
$11$	−0.801938	−0.241793	−0.120897	+	0.992665i	$0.538577\pi$
$12$	−0.445042	−0.128473
$13$	0	0
$13$	0	0
$14$	0.246980	0.0660081
$15$	0	0
$16$	1.00000	0.250000
$17$	1.55496	0.377133	0.188566	−	0.982060i	$-0.439616\pi$
$17$	1.55496	0.377133	0.188566	+	0.982060i	$0.439616\pi$
$18$	−2.80194	−0.660423
$19$	−1.55496	−0.356732	−0.178366	−	0.983964i	$-0.557081\pi$
$19$	−1.55496	−0.356732	−0.178366	+	0.983964i	$0.557081\pi$
$20$	0	0
$21$	−0.109916	−0.0239857
$22$	−0.801938	−0.170974
$23$	0.692021	0.144296	0.0721482	−	0.997394i	$-0.477015\pi$
$23$	0.692021	0.144296	0.0721482	+	0.997394i	$0.477015\pi$
$24$	−0.445042	−0.0908438
$25$	0	0
$26$	0	0
$27$	2.58211	0.496926
$28$	0.246980	0.0466748
$29$	−0.286208	−0.0531475	−0.0265738	−	0.999647i	$-0.508460\pi$
$29$	−0.286208	−0.0531475	−0.0265738	+	0.999647i	$0.508460\pi$
$30$	0	0
$31$	4.15883	0.746949	0.373474	−	0.927641i	$-0.378166\pi$
$31$	4.15883	0.746949	0.373474	+	0.927641i	$0.378166\pi$
$32$	1.00000	0.176777
$33$	0.356896	0.0621276
$34$	1.55496	0.266673
$35$	0	0
$36$	−2.80194	−0.466990
$37$	−8.28382	−1.36185	−0.680925	−	0.732353i	$-0.738422\pi$
$37$	−8.28382	−1.36185	−0.680925	+	0.732353i	$0.738422\pi$
$38$	−1.55496	−0.252248
$39$	0	0
$40$	0	0
$41$	8.54288	1.33417	0.667087	−	0.744980i	$-0.267541\pi$
$41$	8.54288	1.33417	0.667087	+	0.744980i	$0.267541\pi$
$42$	−0.109916	−0.0169604
$43$	−5.62565	−0.857903	−0.428951	−	0.903328i	$-0.641117\pi$
$43$	−5.62565	−0.857903	−0.428951	+	0.903328i	$0.641117\pi$
$44$	−0.801938	−0.120897
$45$	0	0
$46$	0.692021	0.102033
$47$	−6.02715	−0.879150	−0.439575	−	0.898206i	$-0.644871\pi$
$47$	−6.02715	−0.879150	−0.439575	+	0.898206i	$0.644871\pi$
$48$	−0.445042	−0.0642363
$49$	−6.93900	−0.991286
$50$	0	0
$51$	−0.692021	−0.0969024
$52$	0	0
$53$	0.868313	0.119272	0.0596360	−	0.998220i	$-0.481006\pi$
$53$	0.868313	0.119272	0.0596360	+	0.998220i	$0.481006\pi$
$54$	2.58211	0.351380
$55$	0	0
$56$	0.246980	0.0330040
$57$	0.692021	0.0916605
$58$	−0.286208	−0.0375810
$59$	4.30798	0.560851	0.280426	−	0.959876i	$-0.409524\pi$
$59$	4.30798	0.560851	0.280426	+	0.959876i	$0.409524\pi$
$60$	0	0
$61$	−4.30798	−0.551580	−0.275790	−	0.961218i	$-0.588939\pi$
$61$	−4.30798	−0.551580	−0.275790	+	0.961218i	$0.588939\pi$
$62$	4.15883	0.528172
$63$	−0.692021	−0.0871865
$64$	1.00000	0.125000
$65$	0	0
$66$	0.356896	0.0439308
$67$	−6.49396	−0.793363	−0.396682	−	0.917956i	$-0.629838\pi$
$67$	−6.49396	−0.793363	−0.396682	+	0.917956i	$0.629838\pi$
$68$	1.55496	0.188566
$69$	−0.307979	−0.0370763
$70$	0	0
$71$	−6.58211	−0.781152	−0.390576	−	0.920571i	$-0.627724\pi$
$71$	−6.58211	−0.781152	−0.390576	+	0.920571i	$0.627724\pi$
$72$	−2.80194	−0.330212
$73$	7.47219	0.874554	0.437277	−	0.899327i	$-0.355943\pi$
$73$	7.47219	0.874554	0.437277	+	0.899327i	$0.355943\pi$
$74$	−8.28382	−0.962974
$75$	0	0
$76$	−1.55496	−0.178366
$77$	−0.198062	−0.0225713
$78$	0	0
$79$	4.72886	0.532038	0.266019	−	0.963968i	$-0.414292\pi$
$79$	4.72886	0.532038	0.266019	+	0.963968i	$0.414292\pi$
$80$	0	0
$81$	7.25667	0.806296
$82$	8.54288	0.943403
$83$	−7.93362	−0.870828	−0.435414	−	0.900230i	$-0.643398\pi$
$83$	−7.93362	−0.870828	−0.435414	+	0.900230i	$0.643398\pi$
$84$	−0.109916	−0.0119928
$85$	0	0
$86$	−5.62565	−0.606629
$87$	0.127375	0.0136560
$88$	−0.801938	−0.0854868
$89$	−4.34721	−0.460803	−0.230402	−	0.973096i	$-0.574004\pi$
$89$	−4.34721	−0.460803	−0.230402	+	0.973096i	$0.574004\pi$
$90$	0	0
$91$	0	0
$92$	0.692021	0.0721482
$93$	−1.85086	−0.191925
$94$	−6.02715	−0.621653
$95$	0	0
$96$	−0.445042	−0.0454219
$97$	7.83877	0.795907	0.397953	−	0.917406i	$-0.369721\pi$
$97$	7.83877	0.795907	0.397953	+	0.917406i	$0.369721\pi$
$98$	−6.93900	−0.700945
$99$	2.24698	0.225830
$100$	0	0
$101$	−0.192685	−0.0191729	−0.00958646	−	0.999954i	$-0.503052\pi$
$101$	−0.192685	−0.0191729	−0.00958646	+	0.999954i	$0.503052\pi$
$102$	−0.692021	−0.0685203
$103$	14.8485	1.46306	0.731531	−	0.681808i	$-0.238806\pi$
$103$	14.8485	1.46306	0.731531	+	0.681808i	$0.238806\pi$
$104$	0	0
$105$	0	0
$106$	0.868313	0.0843381
$107$	−8.03684	−0.776950	−0.388475	−	0.921459i	$-0.626998\pi$
$107$	−8.03684	−0.776950	−0.388475	+	0.921459i	$0.626998\pi$
$108$	2.58211	0.248463
$109$	−17.9584	−1.72010	−0.860050	−	0.510209i	$-0.829568\pi$
$109$	−17.9584	−1.72010	−0.860050	+	0.510209i	$0.829568\pi$
$110$	0	0
$111$	3.68664	0.349921
$112$	0.246980	0.0233374
$113$	−12.3937	−1.16590	−0.582952	−	0.812507i	$-0.698102\pi$
$113$	−12.3937	−1.16590	−0.582952	+	0.812507i	$0.698102\pi$
$114$	0.692021	0.0648137
$115$	0	0
$116$	−0.286208	−0.0265738
$117$	0	0
$118$	4.30798	0.396582
$119$	0.384043	0.0352052
$120$	0	0
$121$	−10.3569	−0.941536
$122$	−4.30798	−0.390026
$123$	−3.80194	−0.342809
$124$	4.15883	0.373474
$125$	0	0
$126$	−0.692021	−0.0616502
$127$	14.2784	1.26701	0.633503	−	0.773740i	$-0.281616\pi$
$127$	14.2784	1.26701	0.633503	+	0.773740i	$0.281616\pi$
$128$	1.00000	0.0883883
$129$	2.50365	0.220434
$130$	0	0
$131$	−13.9922	−1.22251	−0.611253	−	0.791435i	$-0.709334\pi$
$131$	−13.9922	−1.22251	−0.611253	+	0.791435i	$0.709334\pi$
$132$	0.356896	0.0310638
$133$	−0.384043	−0.0333007
$134$	−6.49396	−0.560993
$135$	0	0
$136$	1.55496	0.133337
$137$	−2.87502	−0.245629	−0.122815	−	0.992430i	$-0.539192\pi$
$137$	−2.87502	−0.245629	−0.122815	+	0.992430i	$0.539192\pi$
$138$	−0.307979	−0.0262169
$139$	−21.3991	−1.81505	−0.907524	−	0.419999i	$-0.862030\pi$
$139$	−21.3991	−1.81505	−0.907524	+	0.419999i	$0.862030\pi$
$140$	0	0
$141$	2.68233	0.225893
$142$	−6.58211	−0.552358
$143$	0	0
$144$	−2.80194	−0.233495
$145$	0	0
$146$	7.47219	0.618403
$147$	3.08815	0.254706
$148$	−8.28382	−0.680925
$149$	−14.3080	−1.17216	−0.586078	−	0.810255i	$-0.699329\pi$
$149$	−14.3080	−1.17216	−0.586078	+	0.810255i	$0.699329\pi$
$150$	0	0
$151$	4.48427	0.364925	0.182462	−	0.983213i	$-0.441593\pi$
$151$	4.48427	0.364925	0.182462	+	0.983213i	$0.441593\pi$
$152$	−1.55496	−0.126124
$153$	−4.35690	−0.352234
$154$	−0.198062	−0.0159603
$155$	0	0
$156$	0	0
$157$	0.457123	0.0364824	0.0182412	−	0.999834i	$-0.494193\pi$
$157$	0.457123	0.0364824	0.0182412	+	0.999834i	$0.494193\pi$
$158$	4.72886	0.376208
$159$	−0.386436	−0.0306464
$160$	0	0
$161$	0.170915	0.0134700
$162$	7.25667	0.570138
$163$	8.78448	0.688054	0.344027	−	0.938960i	$-0.388209\pi$
$163$	8.78448	0.688054	0.344027	+	0.938960i	$0.388209\pi$
$164$	8.54288	0.667087
$165$	0	0
$166$	−7.93362	−0.615769
$167$	−16.9366	−1.31059	−0.655297	−	0.755371i	$-0.727457\pi$
$167$	−16.9366	−1.31059	−0.655297	+	0.755371i	$0.727457\pi$
$168$	−0.109916	−0.00848022
$169$	0	0
$170$	0	0
$171$	4.35690	0.333180
$172$	−5.62565	−0.428951
$173$	−6.26875	−0.476604	−0.238302	−	0.971191i	$-0.576591\pi$
$173$	−6.26875	−0.476604	−0.238302	+	0.971191i	$0.576591\pi$
$174$	0.127375	0.00965625
$175$	0	0
$176$	−0.801938	−0.0604483
$177$	−1.91723	−0.144108
$178$	−4.34721	−0.325837
$179$	−19.0344	−1.42270	−0.711351	−	0.702837i	$-0.751916\pi$
$179$	−19.0344	−1.42270	−0.711351	+	0.702837i	$0.751916\pi$
$180$	0	0
$181$	−4.83579	−0.359441	−0.179721	−	0.983718i	$-0.557519\pi$
$181$	−4.83579	−0.359441	−0.179721	+	0.983718i	$0.557519\pi$
$182$	0	0
$183$	1.91723	0.141726
$184$	0.692021	0.0510165
$185$	0	0
$186$	−1.85086	−0.135711
$187$	−1.24698	−0.0911882
$188$	−6.02715	−0.439575
$189$	0.637727	0.0463878
$190$	0	0
$191$	−3.19806	−0.231404	−0.115702	−	0.993284i	$-0.536912\pi$
$191$	−3.19806	−0.231404	−0.115702	+	0.993284i	$0.536912\pi$
$192$	−0.445042	−0.0321181
$193$	−7.65817	−0.551247	−0.275624	−	0.961266i	$-0.588884\pi$
$193$	−7.65817	−0.551247	−0.275624	+	0.961266i	$0.588884\pi$
$194$	7.83877	0.562791
$195$	0	0
$196$	−6.93900	−0.495643
$197$	−18.4547	−1.31485	−0.657423	−	0.753522i	$-0.728353\pi$
$197$	−18.4547	−1.31485	−0.657423	+	0.753522i	$0.728353\pi$
$198$	2.24698	0.159686
$199$	−16.1709	−1.14633	−0.573163	−	0.819441i	$-0.694284\pi$
$199$	−16.1709	−1.14633	−0.573163	+	0.819441i	$0.694284\pi$
$200$	0	0
$201$	2.89008	0.203851
$202$	−0.192685	−0.0135573
$203$	−0.0706876	−0.00496130
$204$	−0.692021	−0.0484512
$205$	0	0
$206$	14.8485	1.03454
$207$	−1.93900	−0.134770
$208$	0	0
$209$	1.24698	0.0862554
$210$	0	0
$211$	0.104539	0.00719679	0.00359840	−	0.999994i	$-0.498855\pi$
$211$	0.104539	0.00719679	0.00359840	+	0.999994i	$0.498855\pi$
$212$	0.868313	0.0596360
$213$	2.92931	0.200713
$214$	−8.03684	−0.549387
$215$	0	0
$216$	2.58211	0.175690
$217$	1.02715	0.0697273
$218$	−17.9584	−1.21629
$219$	−3.32544	−0.224712
$220$	0	0
$221$	0	0
$222$	3.68664	0.247431
$223$	17.9541	1.20229	0.601147	−	0.799139i	$-0.294711\pi$
$223$	17.9541	1.20229	0.601147	+	0.799139i	$0.294711\pi$
$224$	0.246980	0.0165020
$225$	0	0
$226$	−12.3937	−0.824419
$227$	23.5623	1.56388	0.781941	−	0.623353i	$-0.214230\pi$
$227$	23.5623	1.56388	0.781941	+	0.623353i	$0.214230\pi$
$228$	0.692021	0.0458302
$229$	3.44265	0.227497	0.113748	−	0.993510i	$-0.463714\pi$
$229$	3.44265	0.227497	0.113748	+	0.993510i	$0.463714\pi$
$230$	0	0
$231$	0.0881460	0.00579958
$232$	−0.286208	−0.0187905
$233$	19.4131	1.27180	0.635898	−	0.771773i	$-0.280630\pi$
$233$	19.4131	1.27180	0.635898	+	0.771773i	$0.280630\pi$
$234$	0	0
$235$	0	0
$236$	4.30798	0.280426
$237$	−2.10454	−0.136705
$238$	0.384043	0.0248938
$239$	−29.0495	−1.87906	−0.939528	−	0.342471i	$-0.888736\pi$
$239$	−29.0495	−1.87906	−0.939528	+	0.342471i	$0.888736\pi$
$240$	0	0
$241$	12.3394	0.794853	0.397427	−	0.917634i	$-0.369903\pi$
$241$	12.3394	0.794853	0.397427	+	0.917634i	$0.369903\pi$
$242$	−10.3569	−0.665766
$243$	−10.9758	−0.704100
$244$	−4.30798	−0.275790
$245$	0	0
$246$	−3.80194	−0.242403
$247$	0	0
$248$	4.15883	0.264086
$249$	3.53079	0.223755
$250$	0	0
$251$	−17.9051	−1.13016	−0.565081	−	0.825035i	$-0.691155\pi$
$251$	−17.9051	−1.13016	−0.565081	+	0.825035i	$0.691155\pi$
$252$	−0.692021	−0.0435933
$253$	−0.554958	−0.0348899
$254$	14.2784	0.895909
$255$	0	0
$256$	1.00000	0.0625000
$257$	21.9041	1.36634	0.683170	−	0.730260i	$-0.260601\pi$
$257$	21.9041	1.36634	0.683170	+	0.730260i	$0.260601\pi$
$258$	2.50365	0.155870
$259$	−2.04593	−0.127128
$260$	0	0
$261$	0.801938	0.0496387
$262$	−13.9922	−0.864443
$263$	−18.3720	−1.13286	−0.566432	−	0.824109i	$-0.691677\pi$
$263$	−18.3720	−1.13286	−0.566432	+	0.824109i	$0.691677\pi$
$264$	0.356896	0.0219654
$265$	0	0
$266$	−0.384043	−0.0235472
$267$	1.93469	0.118401
$268$	−6.49396	−0.396682
$269$	−29.4969	−1.79846	−0.899230	−	0.437476i	$-0.855873\pi$
$269$	−29.4969	−1.79846	−0.899230	+	0.437476i	$0.855873\pi$
$270$	0	0
$271$	11.0435	0.670847	0.335424	−	0.942067i	$-0.391121\pi$
$271$	11.0435	0.670847	0.335424	+	0.942067i	$0.391121\pi$
$272$	1.55496	0.0942832
$273$	0	0
$274$	−2.87502	−0.173686
$275$	0	0
$276$	−0.307979	−0.0185381
$277$	15.9390	0.957682	0.478841	−	0.877902i	$-0.341057\pi$
$277$	15.9390	0.957682	0.478841	+	0.877902i	$0.341057\pi$
$278$	−21.3991	−1.28343
$279$	−11.6528	−0.697634
$280$	0	0
$281$	14.4886	0.864316	0.432158	−	0.901798i	$-0.357752\pi$
$281$	14.4886	0.864316	0.432158	+	0.901798i	$0.357752\pi$
$282$	2.68233	0.159731
$283$	−0.445042	−0.0264550	−0.0132275	−	0.999913i	$-0.504211\pi$
$283$	−0.445042	−0.0264550	−0.0132275	+	0.999913i	$0.504211\pi$
$284$	−6.58211	−0.390576
$285$	0	0
$286$	0	0
$287$	2.10992	0.124544
$288$	−2.80194	−0.165106
$289$	−14.5821	−0.857771
$290$	0	0
$291$	−3.48858	−0.204504
$292$	7.47219	0.437277
$293$	23.3099	1.36178	0.680889	−	0.732386i	$-0.261593\pi$
$293$	23.3099	1.36178	0.680889	+	0.732386i	$0.261593\pi$
$294$	3.08815	0.180104
$295$	0	0
$296$	−8.28382	−0.481487
$297$	−2.07069	−0.120153
$298$	−14.3080	−0.828839
$299$	0	0
$300$	0	0
$301$	−1.38942	−0.0800848
$302$	4.48427	0.258041
$303$	0.0857531	0.00492639
$304$	−1.55496	−0.0891830
$305$	0	0
$306$	−4.35690	−0.249067
$307$	8.45904	0.482783	0.241392	−	0.970428i	$-0.422396\pi$
$307$	8.45904	0.482783	0.241392	+	0.970428i	$0.422396\pi$
$308$	−0.198062	−0.0112856
$309$	−6.60819	−0.375927
$310$	0	0
$311$	24.2838	1.37701	0.688504	−	0.725232i	$-0.258268\pi$
$311$	24.2838	1.37701	0.688504	+	0.725232i	$0.258268\pi$
$312$	0	0
$313$	−2.04593	−0.115643	−0.0578215	−	0.998327i	$-0.518415\pi$
$313$	−2.04593	−0.115643	−0.0578215	+	0.998327i	$0.518415\pi$
$314$	0.457123	0.0257970
$315$	0	0
$316$	4.72886	0.266019
$317$	16.1444	0.906758	0.453379	−	0.891318i	$-0.350218\pi$
$317$	16.1444	0.906758	0.453379	+	0.891318i	$0.350218\pi$
$318$	−0.386436	−0.0216702
$319$	0.229521	0.0128507
$320$	0	0
$321$	3.57673	0.199634
$322$	0.170915	0.00952473
$323$	−2.41789	−0.134535
$324$	7.25667	0.403148
$325$	0	0
$326$	8.78448	0.486527
$327$	7.99223	0.441971
$328$	8.54288	0.471701
$329$	−1.48858	−0.0820682
$330$	0	0
$331$	−25.2760	−1.38930	−0.694649	−	0.719349i	$-0.744440\pi$
$331$	−25.2760	−1.38930	−0.694649	+	0.719349i	$0.744440\pi$
$332$	−7.93362	−0.435414
$333$	23.2107	1.27194
$334$	−16.9366	−0.926730
$335$	0	0
$336$	−0.109916	−0.00599642
$337$	−7.82132	−0.426054	−0.213027	−	0.977046i	$-0.568332\pi$
$337$	−7.82132	−0.426054	−0.213027	+	0.977046i	$0.568332\pi$
$338$	0	0
$339$	5.51573	0.299573
$340$	0	0
$341$	−3.33513	−0.180607
$342$	4.35690	0.235594
$343$	−3.44265	−0.185886
$344$	−5.62565	−0.303314
$345$	0	0
$346$	−6.26875	−0.337010
$347$	20.7439	1.11359	0.556796	−	0.830649i	$-0.312030\pi$
$347$	20.7439	1.11359	0.556796	+	0.830649i	$0.312030\pi$
$348$	0.127375	0.00682800
$349$	11.9463	0.639471	0.319735	−	0.947507i	$-0.396406\pi$
$349$	11.9463	0.639471	0.319735	+	0.947507i	$0.396406\pi$
$350$	0	0
$351$	0	0
$352$	−0.801938	−0.0427434
$353$	1.60388	0.0853657	0.0426828	−	0.999089i	$-0.486410\pi$
$353$	1.60388	0.0853657	0.0426828	+	0.999089i	$0.486410\pi$
$354$	−1.91723	−0.101900
$355$	0	0
$356$	−4.34721	−0.230402
$357$	−0.170915	−0.00904579
$358$	−19.0344	−1.00600
$359$	22.1183	1.16736	0.583679	−	0.811984i	$-0.301613\pi$
$359$	22.1183	1.16736	0.583679	+	0.811984i	$0.301613\pi$
$360$	0	0
$361$	−16.5821	−0.872742
$362$	−4.83579	−0.254163
$363$	4.60925	0.241923
$364$	0	0
$365$	0	0
$366$	1.91723	0.100215
$367$	−16.1129	−0.841087	−0.420543	−	0.907272i	$-0.638161\pi$
$367$	−16.1129	−0.841087	−0.420543	+	0.907272i	$0.638161\pi$
$368$	0.692021	0.0360741
$369$	−23.9366	−1.24609
$370$	0	0
$371$	0.214456	0.0111340
$372$	−1.85086	−0.0959624
$373$	−36.7851	−1.90466	−0.952329	−	0.305072i	$-0.901320\pi$
$373$	−36.7851	−1.90466	−0.952329	+	0.305072i	$0.901320\pi$
$374$	−1.24698	−0.0644798
$375$	0	0
$376$	−6.02715	−0.310826
$377$	0	0
$378$	0.637727	0.0328012
$379$	27.7415	1.42499	0.712493	−	0.701679i	$-0.247566\pi$
$379$	27.7415	1.42499	0.712493	+	0.701679i	$0.247566\pi$
$380$	0	0
$381$	−6.35450	−0.325551
$382$	−3.19806	−0.163627
$383$	−27.0116	−1.38023	−0.690114	−	0.723700i	$-0.742440\pi$
$383$	−27.0116	−1.38023	−0.690114	+	0.723700i	$0.742440\pi$
$384$	−0.445042	−0.0227109
$385$	0	0
$386$	−7.65817	−0.389791
$387$	15.7627	0.801264
$388$	7.83877	0.397953
$389$	12.5284	0.635215	0.317608	−	0.948222i	$-0.397121\pi$
$389$	12.5284	0.635215	0.317608	+	0.948222i	$0.397121\pi$
$390$	0	0
$391$	1.07606	0.0544189
$392$	−6.93900	−0.350472
$393$	6.22713	0.314117
$394$	−18.4547	−0.929736
$395$	0	0
$396$	2.24698	0.112915
$397$	−13.5918	−0.682153	−0.341076	−	0.940036i	$-0.610791\pi$
$397$	−13.5918	−0.682153	−0.341076	+	0.940036i	$0.610791\pi$
$398$	−16.1709	−0.810575
$399$	0.170915	0.00855646
$400$	0	0
$401$	−9.39075	−0.468952	−0.234476	−	0.972122i	$-0.575337\pi$
$401$	−9.39075	−0.468952	−0.234476	+	0.972122i	$0.575337\pi$
$402$	2.89008	0.144144
$403$	0	0
$404$	−0.192685	−0.00958646
$405$	0	0
$406$	−0.0706876	−0.00350817
$407$	6.64310	0.329286
$408$	−0.692021	−0.0342602
$409$	−29.7493	−1.47101	−0.735504	−	0.677520i	$-0.763055\pi$
$409$	−29.7493	−1.47101	−0.735504	+	0.677520i	$0.763055\pi$
$410$	0	0
$411$	1.27950	0.0631133
$412$	14.8485	0.731531
$413$	1.06398	0.0523552
$414$	−1.93900	−0.0952967
$415$	0	0
$416$	0	0
$417$	9.52350	0.466368
$418$	1.24698	0.0609918
$419$	17.0043	0.830715	0.415357	−	0.909658i	$-0.363656\pi$
$419$	17.0043	0.830715	0.415357	+	0.909658i	$0.363656\pi$
$420$	0	0
$421$	15.1763	0.739647	0.369824	−	0.929102i	$-0.379418\pi$
$421$	15.1763	0.739647	0.369824	+	0.929102i	$0.379418\pi$
$422$	0.104539	0.00508890
$423$	16.8877	0.821108
$424$	0.868313	0.0421690
$425$	0	0
$426$	2.92931	0.141926
$427$	−1.06398	−0.0514897
$428$	−8.03684	−0.388475
$429$	0	0
$430$	0	0
$431$	10.4306	0.502423	0.251211	−	0.967932i	$-0.419171\pi$
$431$	10.4306	0.502423	0.251211	+	0.967932i	$0.419171\pi$
$432$	2.58211	0.124232
$433$	−25.4021	−1.22075	−0.610373	−	0.792114i	$-0.708981\pi$
$433$	−25.4021	−1.22075	−0.610373	+	0.792114i	$0.708981\pi$
$434$	1.02715	0.0493046
$435$	0	0
$436$	−17.9584	−0.860050
$437$	−1.07606	−0.0514751
$438$	−3.32544	−0.158896
$439$	−28.4838	−1.35946	−0.679729	−	0.733464i	$-0.737902\pi$
$439$	−28.4838	−1.35946	−0.679729	+	0.733464i	$0.737902\pi$
$440$	0	0
$441$	19.4426	0.925840
$442$	0	0
$443$	−35.8509	−1.70333	−0.851663	−	0.524090i	$-0.824405\pi$
$443$	−35.8509	−1.70333	−0.851663	+	0.524090i	$0.824405\pi$
$444$	3.68664	0.174960
$445$	0	0
$446$	17.9541	0.850150
$447$	6.36765	0.301179
$448$	0.246980	0.0116687
$449$	−5.48294	−0.258756	−0.129378	−	0.991595i	$-0.541298\pi$
$449$	−5.48294	−0.258756	−0.129378	+	0.991595i	$0.541298\pi$
$450$	0	0
$451$	−6.85086	−0.322594
$452$	−12.3937	−0.582952
$453$	−1.99569	−0.0937656
$454$	23.5623	1.10583
$455$	0	0
$456$	0.692021	0.0324069
$457$	−20.6383	−0.965420	−0.482710	−	0.875780i	$-0.660347\pi$
$457$	−20.6383	−0.965420	−0.482710	+	0.875780i	$0.660347\pi$
$458$	3.44265	0.160864
$459$	4.01507	0.187407
$460$	0	0
$461$	−15.4547	−0.719799	−0.359899	−	0.932991i	$-0.617189\pi$
$461$	−15.4547	−0.719799	−0.359899	+	0.932991i	$0.617189\pi$
$462$	0.0881460	0.00410092
$463$	17.1830	0.798562	0.399281	−	0.916829i	$-0.369260\pi$
$463$	17.1830	0.798562	0.399281	+	0.916829i	$0.369260\pi$
$464$	−0.286208	−0.0132869
$465$	0	0
$466$	19.4131	0.899295
$467$	12.1655	0.562954	0.281477	−	0.959568i	$-0.409176\pi$
$467$	12.1655	0.562954	0.281477	+	0.959568i	$0.409176\pi$
$468$	0	0
$469$	−1.60388	−0.0740601
$470$	0	0
$471$	−0.203439	−0.00937398
$472$	4.30798	0.198291
$473$	4.51142	0.207435
$474$	−2.10454	−0.0966647
$475$	0	0
$476$	0.384043	0.0176026
$477$	−2.43296	−0.111398
$478$	−29.0495	−1.32869
$479$	15.8998	0.726479	0.363240	−	0.931696i	$-0.381671\pi$
$479$	15.8998	0.726479	0.363240	+	0.931696i	$0.381671\pi$
$480$	0	0
$481$	0	0
$482$	12.3394	0.562046
$483$	−0.0760644	−0.00346105
$484$	−10.3569	−0.470768
$485$	0	0
$486$	−10.9758	−0.497874
$487$	−3.85815	−0.174830	−0.0874148	−	0.996172i	$-0.527861\pi$
$487$	−3.85815	−0.174830	−0.0874148	+	0.996172i	$0.527861\pi$
$488$	−4.30798	−0.195013
$489$	−3.90946	−0.176792
$490$	0	0
$491$	12.3351	0.556676	0.278338	−	0.960483i	$-0.410216\pi$
$491$	12.3351	0.556676	0.278338	+	0.960483i	$0.410216\pi$
$492$	−3.80194	−0.171405
$493$	−0.445042	−0.0200437
$494$	0	0
$495$	0	0
$496$	4.15883	0.186737
$497$	−1.62565	−0.0729202
$498$	3.53079	0.158219
$499$	−16.6843	−0.746890	−0.373445	−	0.927652i	$-0.621823\pi$
$499$	−16.6843	−0.746890	−0.373445	+	0.927652i	$0.621823\pi$
$500$	0	0
$501$	7.53750	0.336751
$502$	−17.9051	−0.799146
$503$	0.781495	0.0348452	0.0174226	−	0.999848i	$-0.494454\pi$
$503$	0.781495	0.0348452	0.0174226	+	0.999848i	$0.494454\pi$
$504$	−0.692021	−0.0308251
$505$	0	0
$506$	−0.554958	−0.0246709
$507$	0	0
$508$	14.2784	0.633503
$509$	13.1328	0.582099	0.291049	−	0.956708i	$-0.405996\pi$
$509$	13.1328	0.582099	0.291049	+	0.956708i	$0.405996\pi$
$510$	0	0
$511$	1.84548	0.0816392
$512$	1.00000	0.0441942
$513$	−4.01507	−0.177269
$514$	21.9041	0.966148
$515$	0	0
$516$	2.50365	0.110217
$517$	4.83340	0.212573
$518$	−2.04593	−0.0898932
$519$	2.78986	0.122461
$520$	0	0
$521$	16.9148	0.741053	0.370526	−	0.928822i	$-0.379177\pi$
$521$	16.9148	0.741053	0.370526	+	0.928822i	$0.379177\pi$
$522$	0.801938	0.0350999
$523$	−5.66056	−0.247519	−0.123760	−	0.992312i	$-0.539495\pi$
$523$	−5.66056	−0.247519	−0.123760	+	0.992312i	$0.539495\pi$
$524$	−13.9922	−0.611253
$525$	0	0
$526$	−18.3720	−0.801056
$527$	6.46681	0.281699
$528$	0.356896	0.0155319
$529$	−22.5211	−0.979179
$530$	0	0
$531$	−12.0707	−0.523823
$532$	−0.384043	−0.0166504
$533$	0	0
$534$	1.93469	0.0837222
$535$	0	0
$536$	−6.49396	−0.280496
$537$	8.47112	0.365556
$538$	−29.4969	−1.27170
$539$	5.56465	0.239686
$540$	0	0
$541$	20.2338	0.869920	0.434960	−	0.900450i	$-0.356762\pi$
$541$	20.2338	0.869920	0.434960	+	0.900450i	$0.356762\pi$
$542$	11.0435	0.474361
$543$	2.15213	0.0923567
$544$	1.55496	0.0666683
$545$	0	0
$546$	0	0
$547$	6.73317	0.287890	0.143945	−	0.989586i	$-0.454021\pi$
$547$	6.73317	0.287890	0.143945	+	0.989586i	$0.454021\pi$
$548$	−2.87502	−0.122815
$549$	12.0707	0.515164
$550$	0	0
$551$	0.445042	0.0189594
$552$	−0.307979	−0.0131084
$553$	1.16793	0.0496655
$554$	15.9390	0.677183
$555$	0	0
$556$	−21.3991	−0.907524
$557$	16.4547	0.697209	0.348605	−	0.937270i	$-0.386656\pi$
$557$	16.4547	0.697209	0.348605	+	0.937270i	$0.386656\pi$
$558$	−11.6528	−0.493302
$559$	0	0
$560$	0	0
$561$	0.554958	0.0234304
$562$	14.4886	0.611164
$563$	−22.8834	−0.964419	−0.482210	−	0.876056i	$-0.660166\pi$
$563$	−22.8834	−0.964419	−0.482210	+	0.876056i	$0.660166\pi$
$564$	2.68233	0.112947
$565$	0	0
$566$	−0.445042	−0.0187065
$567$	1.79225	0.0752674
$568$	−6.58211	−0.276179
$569$	−30.3787	−1.27354	−0.636770	−	0.771054i	$-0.719730\pi$
$569$	−30.3787	−1.27354	−0.636770	+	0.771054i	$0.719730\pi$
$570$	0	0
$571$	5.79092	0.242343	0.121171	−	0.992632i	$-0.461335\pi$
$571$	5.79092	0.242343	0.121171	+	0.992632i	$0.461335\pi$
$572$	0	0
$573$	1.42327	0.0594580
$574$	2.10992	0.0880662
$575$	0	0
$576$	−2.80194	−0.116747
$577$	44.3279	1.84540	0.922698	−	0.385523i	$-0.125979\pi$
$577$	44.3279	1.84540	0.922698	+	0.385523i	$0.125979\pi$
$578$	−14.5821	−0.606536
$579$	3.40821	0.141640
$580$	0	0
$581$	−1.95944	−0.0812914
$582$	−3.48858	−0.144606
$583$	−0.696333	−0.0288392
$584$	7.47219	0.309201
$585$	0	0
$586$	23.3099	0.962923
$587$	11.1263	0.459232	0.229616	−	0.973281i	$-0.426253\pi$
$587$	11.1263	0.459232	0.229616	+	0.973281i	$0.426253\pi$
$588$	3.08815	0.127353
$589$	−6.46681	−0.266460
$590$	0	0
$591$	8.21313	0.337843
$592$	−8.28382	−0.340463
$593$	10.9748	0.450680	0.225340	−	0.974280i	$-0.427651\pi$
$593$	10.9748	0.450680	0.225340	+	0.974280i	$0.427651\pi$
$594$	−2.07069	−0.0849613
$595$	0	0
$596$	−14.3080	−0.586078
$597$	7.19673	0.294543
$598$	0	0
$599$	17.8756	0.730378	0.365189	−	0.930933i	$-0.381004\pi$
$599$	17.8756	0.730378	0.365189	+	0.930933i	$0.381004\pi$
$600$	0	0
$601$	−27.9105	−1.13849	−0.569247	−	0.822167i	$-0.692765\pi$
$601$	−27.9105	−1.13849	−0.569247	+	0.822167i	$0.692765\pi$
$602$	−1.38942	−0.0566285
$603$	18.1957	0.740985
$604$	4.48427	0.182462
$605$	0	0
$606$	0.0857531	0.00348348
$607$	36.2446	1.47112	0.735561	−	0.677458i	$-0.236919\pi$
$607$	36.2446	1.47112	0.735561	+	0.677458i	$0.236919\pi$
$608$	−1.55496	−0.0630619
$609$	0.0314589	0.00127478
$610$	0	0
$611$	0	0
$612$	−4.35690	−0.176117
$613$	4.39612	0.177558	0.0887789	−	0.996051i	$-0.471704\pi$
$613$	4.39612	0.177558	0.0887789	+	0.996051i	$0.471704\pi$
$614$	8.45904	0.341379
$615$	0	0
$616$	−0.198062	−0.00798016
$617$	−3.84979	−0.154987	−0.0774934	−	0.996993i	$-0.524692\pi$
$617$	−3.84979	−0.154987	−0.0774934	+	0.996993i	$0.524692\pi$
$618$	−6.60819	−0.265820
$619$	−4.56896	−0.183642	−0.0918210	−	0.995776i	$-0.529269\pi$
$619$	−4.56896	−0.183642	−0.0918210	+	0.995776i	$0.529269\pi$
$620$	0	0
$621$	1.78687	0.0717047
$622$	24.2838	0.973692
$623$	−1.07367	−0.0430157
$624$	0	0
$625$	0	0
$626$	−2.04593	−0.0817719
$627$	−0.554958	−0.0221629
$628$	0.457123	0.0182412
$629$	−12.8810	−0.513599
$630$	0	0
$631$	5.00777	0.199356	0.0996781	−	0.995020i	$-0.468219\pi$
$631$	5.00777	0.199356	0.0996781	+	0.995020i	$0.468219\pi$
$632$	4.72886	0.188104
$633$	−0.0465244	−0.00184918
$634$	16.1444	0.641174
$635$	0	0
$636$	−0.386436	−0.0153232
$637$	0	0
$638$	0.229521	0.00908683
$639$	18.4426	0.729580
$640$	0	0
$641$	−7.09485	−0.280230	−0.140115	−	0.990135i	$-0.544747\pi$
$641$	−7.09485	−0.280230	−0.140115	+	0.990135i	$0.544747\pi$
$642$	3.57673	0.141162
$643$	1.30665	0.0515293	0.0257646	−	0.999668i	$-0.491798\pi$
$643$	1.30665	0.0515293	0.0257646	+	0.999668i	$0.491798\pi$
$644$	0.170915	0.00673500
$645$	0	0
$646$	−2.41789	−0.0951308
$647$	12.6015	0.495415	0.247708	−	0.968835i	$-0.420323\pi$
$647$	12.6015	0.495415	0.247708	+	0.968835i	$0.420323\pi$
$648$	7.25667	0.285069
$649$	−3.45473	−0.135610
$650$	0	0
$651$	−0.457123	−0.0179161
$652$	8.78448	0.344027
$653$	−20.7885	−0.813518	−0.406759	−	0.913536i	$-0.633341\pi$
$653$	−20.7885	−0.813518	−0.406759	+	0.913536i	$0.633341\pi$
$654$	7.99223	0.312521
$655$	0	0
$656$	8.54288	0.333543
$657$	−20.9366	−0.816815
$658$	−1.48858	−0.0580310
$659$	−3.58987	−0.139842	−0.0699208	−	0.997553i	$-0.522275\pi$
$659$	−3.58987	−0.139842	−0.0699208	+	0.997553i	$0.522275\pi$
$660$	0	0
$661$	−3.19913	−0.124432	−0.0622158	−	0.998063i	$-0.519817\pi$
$661$	−3.19913	−0.124432	−0.0622158	+	0.998063i	$0.519817\pi$
$662$	−25.2760	−0.982381
$663$	0	0
$664$	−7.93362	−0.307884
$665$	0	0
$666$	23.2107	0.899398
$667$	−0.198062	−0.00766900
$668$	−16.9366	−0.655297
$669$	−7.99031	−0.308923
$670$	0	0
$671$	3.45473	0.133368
$672$	−0.109916	−0.00424011
$673$	5.65710	0.218065	0.109033	−	0.994038i	$-0.465225\pi$
$673$	5.65710	0.218065	0.109033	+	0.994038i	$0.465225\pi$
$674$	−7.82132	−0.301266
$675$	0	0
$676$	0	0
$677$	−35.2905	−1.35632	−0.678162	−	0.734912i	$-0.737223\pi$
$677$	−35.2905	−1.35632	−0.678162	+	0.734912i	$0.737223\pi$
$678$	5.51573	0.211830
$679$	1.93602	0.0742975
$680$	0	0
$681$	−10.4862	−0.401832
$682$	−3.33513	−0.127709
$683$	38.6112	1.47742	0.738708	−	0.674026i	$-0.235436\pi$
$683$	38.6112	1.47742	0.738708	+	0.674026i	$0.235436\pi$
$684$	4.35690	0.166590
$685$	0	0
$686$	−3.44265	−0.131441
$687$	−1.53212	−0.0584541
$688$	−5.62565	−0.214476
$689$	0	0
$690$	0	0
$691$	−32.6939	−1.24374	−0.621868	−	0.783122i	$-0.713626\pi$
$691$	−32.6939	−1.24374	−0.621868	+	0.783122i	$0.713626\pi$
$692$	−6.26875	−0.238302
$693$	0.554958	0.0210811
$694$	20.7439	0.787429
$695$	0	0
$696$	0.127375	0.00482812
$697$	13.2838	0.503160
$698$	11.9463	0.452174
$699$	−8.63965	−0.326781
$700$	0	0
$701$	10.1360	0.382831	0.191416	−	0.981509i	$-0.438692\pi$
$701$	10.1360	0.382831	0.191416	+	0.981509i	$0.438692\pi$
$702$	0	0
$703$	12.8810	0.485816
$704$	−0.801938	−0.0302242
$705$	0	0
$706$	1.60388	0.0603626
$707$	−0.0475894	−0.00178978
$708$	−1.91723	−0.0720539
$709$	−30.3086	−1.13826	−0.569131	−	0.822247i	$-0.692720\pi$
$709$	−30.3086	−1.13826	−0.569131	+	0.822247i	$0.692720\pi$
$710$	0	0
$711$	−13.2500	−0.496912
$712$	−4.34721	−0.162918
$713$	2.87800	0.107782
$714$	−0.170915	−0.00639634
$715$	0	0
$716$	−19.0344	−0.711351
$717$	12.9282	0.482814
$718$	22.1183	0.825447
$719$	−22.1293	−0.825283	−0.412642	−	0.910893i	$-0.635394\pi$
$719$	−22.1293	−0.825283	−0.412642	+	0.910893i	$0.635394\pi$
$720$	0	0
$721$	3.66727	0.136576
$722$	−16.5821	−0.617122
$723$	−5.49157	−0.204234
$724$	−4.83579	−0.179721
$725$	0	0
$726$	4.60925	0.171065
$727$	−41.6195	−1.54358	−0.771791	−	0.635876i	$-0.780639\pi$
$727$	−41.6195	−1.54358	−0.771791	+	0.635876i	$0.780639\pi$
$728$	0	0
$729$	−16.8853	−0.625381
$730$	0	0
$731$	−8.74764	−0.323543
$732$	1.91723	0.0708629
$733$	28.4993	1.05265	0.526323	−	0.850284i	$-0.323570\pi$
$733$	28.4993	1.05265	0.526323	+	0.850284i	$0.323570\pi$
$734$	−16.1129	−0.594738
$735$	0	0
$736$	0.692021	0.0255082
$737$	5.20775	0.191830
$738$	−23.9366	−0.881119
$739$	33.5362	1.23365	0.616824	−	0.787101i	$-0.288419\pi$
$739$	33.5362	1.23365	0.616824	+	0.787101i	$0.288419\pi$
$740$	0	0
$741$	0	0
$742$	0.214456	0.00787292
$743$	21.5918	0.792126	0.396063	−	0.918223i	$-0.370376\pi$
$743$	21.5918	0.792126	0.396063	+	0.918223i	$0.370376\pi$
$744$	−1.85086	−0.0678556
$745$	0	0
$746$	−36.7851	−1.34680
$747$	22.2295	0.813336
$748$	−1.24698	−0.0455941
$749$	−1.98493	−0.0725279
$750$	0	0
$751$	−9.09352	−0.331827	−0.165914	−	0.986140i	$-0.553057\pi$
$751$	−9.09352	−0.331827	−0.165914	+	0.986140i	$0.553057\pi$
$752$	−6.02715	−0.219787
$753$	7.96854	0.290390
$754$	0	0
$755$	0	0
$756$	0.637727	0.0231939
$757$	22.4155	0.814705	0.407353	−	0.913271i	$-0.366452\pi$
$757$	22.4155	0.814705	0.407353	+	0.913271i	$0.366452\pi$
$758$	27.7415	1.00762
$759$	0.246980	0.00896479
$760$	0	0
$761$	21.3086	0.772435	0.386218	−	0.922408i	$-0.373781\pi$
$761$	21.3086	0.772435	0.386218	+	0.922408i	$0.373781\pi$
$762$	−6.35450	−0.230199
$763$	−4.43535	−0.160571
$764$	−3.19806	−0.115702
$765$	0	0
$766$	−27.0116	−0.975969
$767$	0	0
$768$	−0.445042	−0.0160591
$769$	−24.4276	−0.880881	−0.440441	−	0.897782i	$-0.645178\pi$
$769$	−24.4276	−0.880881	−0.440441	+	0.897782i	$0.645178\pi$
$770$	0	0
$771$	−9.74823	−0.351074
$772$	−7.65817	−0.275624
$773$	−24.1433	−0.868374	−0.434187	−	0.900823i	$-0.642964\pi$
$773$	−24.1433	−0.868374	−0.434187	+	0.900823i	$0.642964\pi$
$774$	15.7627	0.566579
$775$	0	0
$776$	7.83877	0.281396
$777$	0.910526	0.0326649
$778$	12.5284	0.449165
$779$	−13.2838	−0.475942
$780$	0	0
$781$	5.27844	0.188877
$782$	1.07606	0.0384800
$783$	−0.739020	−0.0264104
$784$	−6.93900	−0.247821
$785$	0	0
$786$	6.22713	0.222114
$787$	52.6969	1.87844	0.939221	−	0.343312i	$-0.111549\pi$
$787$	52.6969	1.87844	0.939221	+	0.343312i	$0.111549\pi$
$788$	−18.4547	−0.657423
$789$	8.17629	0.291084
$790$	0	0
$791$	−3.06100	−0.108837
$792$	2.24698	0.0798429
$793$	0	0
$794$	−13.5918	−0.482355
$795$	0	0
$796$	−16.1709	−0.573163
$797$	−30.9135	−1.09501	−0.547506	−	0.836802i	$-0.684423\pi$
$797$	−30.9135	−1.09501	−0.547506	+	0.836802i	$0.684423\pi$
$798$	0.170915	0.00605033
$799$	−9.37196	−0.331556
$800$	0	0
$801$	12.1806	0.430380
$802$	−9.39075	−0.331599
$803$	−5.99223	−0.211461
$804$	2.89008	0.101925
$805$	0	0
$806$	0	0
$807$	13.1274	0.462105
$808$	−0.192685	−0.00677865
$809$	42.5706	1.49670	0.748352	−	0.663302i	$-0.230846\pi$
$809$	42.5706	1.49670	0.748352	+	0.663302i	$0.230846\pi$
$810$	0	0
$811$	24.6655	0.866122	0.433061	−	0.901365i	$-0.357434\pi$
$811$	24.6655	0.866122	0.433061	+	0.901365i	$0.357434\pi$
$812$	−0.0706876	−0.00248065
$813$	−4.91484	−0.172371
$814$	6.64310	0.232841
$815$	0	0
$816$	−0.692021	−0.0242256
$817$	8.74764	0.306041
$818$	−29.7493	−1.04016
$819$	0	0
$820$	0	0
$821$	14.9849	0.522978	0.261489	−	0.965206i	$-0.415787\pi$
$821$	14.9849	0.522978	0.261489	+	0.965206i	$0.415787\pi$
$822$	1.27950	0.0446278
$823$	−33.1377	−1.15511	−0.577553	−	0.816353i	$-0.695992\pi$
$823$	−33.1377	−1.15511	−0.577553	+	0.816353i	$0.695992\pi$
$824$	14.8485	0.517271
$825$	0	0
$826$	1.06398	0.0370207
$827$	−40.9711	−1.42470	−0.712352	−	0.701823i	$-0.752370\pi$
$827$	−40.9711	−1.42470	−0.712352	+	0.701823i	$0.752370\pi$
$828$	−1.93900	−0.0673849
$829$	−13.0362	−0.452767	−0.226384	−	0.974038i	$-0.572690\pi$
$829$	−13.0362	−0.452767	−0.226384	+	0.974038i	$0.572690\pi$
$830$	0	0
$831$	−7.09352	−0.246072
$832$	0	0
$833$	−10.7899	−0.373846
$834$	9.52350	0.329772
$835$	0	0
$836$	1.24698	0.0431277
$837$	10.7385	0.371178
$838$	17.0043	0.587404
$839$	18.4588	0.637268	0.318634	−	0.947878i	$-0.396776\pi$
$839$	18.4588	0.637268	0.318634	+	0.947878i	$0.396776\pi$
$840$	0	0
$841$	−28.9181	−0.997175
$842$	15.1763	0.523010
$843$	−6.44803	−0.222082
$844$	0.104539	0.00359840
$845$	0	0
$846$	16.8877	0.580611
$847$	−2.55794	−0.0878919
$848$	0.868313	0.0298180
$849$	0.198062	0.00679748
$850$	0	0
$851$	−5.73258	−0.196510
$852$	2.92931	0.100357
$853$	12.5700	0.430389	0.215195	−	0.976571i	$-0.430961\pi$
$853$	12.5700	0.430389	0.215195	+	0.976571i	$0.430961\pi$
$854$	−1.06398	−0.0364087
$855$	0	0
$856$	−8.03684	−0.274693
$857$	40.2174	1.37380	0.686901	−	0.726751i	$-0.258971\pi$
$857$	40.2174	1.37380	0.686901	+	0.726751i	$0.258971\pi$
$858$	0	0
$859$	−12.0271	−0.410361	−0.205180	−	0.978724i	$-0.565778\pi$
$859$	−12.0271	−0.410361	−0.205180	+	0.978724i	$0.565778\pi$
$860$	0	0
$861$	−0.939001	−0.0320011
$862$	10.4306	0.355267
$863$	22.8810	0.778878	0.389439	−	0.921052i	$-0.372669\pi$
$863$	22.8810	0.778878	0.389439	+	0.921052i	$0.372669\pi$
$864$	2.58211	0.0878450
$865$	0	0
$866$	−25.4021	−0.863198
$867$	6.48965	0.220400
$868$	1.02715	0.0348636
$869$	−3.79225	−0.128643
$870$	0	0
$871$	0	0
$872$	−17.9584	−0.608147
$873$	−21.9638	−0.743360
$874$	−1.07606	−0.0363984
$875$	0	0
$876$	−3.32544	−0.112356
$877$	−27.7265	−0.936256	−0.468128	−	0.883661i	$-0.655071\pi$
$877$	−27.7265	−0.936256	−0.468128	+	0.883661i	$0.655071\pi$
$878$	−28.4838	−0.961282
$879$	−10.3739	−0.349902
$880$	0	0
$881$	46.9590	1.58209	0.791044	−	0.611759i	$-0.209538\pi$
$881$	46.9590	1.58209	0.791044	+	0.611759i	$0.209538\pi$
$882$	19.4426	0.654668
$883$	47.7904	1.60828	0.804138	−	0.594442i	$-0.202627\pi$
$883$	47.7904	1.60828	0.804138	+	0.594442i	$0.202627\pi$
$884$	0	0
$885$	0	0
$886$	−35.8509	−1.20443
$887$	11.2838	0.378873	0.189437	−	0.981893i	$-0.439334\pi$
$887$	11.2838	0.378873	0.189437	+	0.981893i	$0.439334\pi$
$888$	3.68664	0.123716
$889$	3.52648	0.118274
$890$	0	0
$891$	−5.81940	−0.194957
$892$	17.9541	0.601147
$893$	9.37196	0.313621
$894$	6.36765	0.212966
$895$	0	0
$896$	0.246980	0.00825101
$897$	0	0
$898$	−5.48294	−0.182968
$899$	−1.19029	−0.0396985
$900$	0	0
$901$	1.35019	0.0449814
$902$	−6.85086	−0.228109
$903$	0.618350	0.0205774
$904$	−12.3937	−0.412209
$905$	0	0
$906$	−1.99569	−0.0663023
$907$	−35.5545	−1.18057	−0.590284	−	0.807196i	$-0.700984\pi$
$907$	−35.5545	−1.18057	−0.590284	+	0.807196i	$0.700984\pi$
$908$	23.5623	0.781941
$909$	0.539893	0.0179071
$910$	0	0
$911$	−49.2683	−1.63233	−0.816165	−	0.577818i	$-0.803904\pi$
$911$	−49.2683	−1.63233	−0.816165	+	0.577818i	$0.803904\pi$
$912$	0.692021	0.0229151
$913$	6.36227	0.210560
$914$	−20.6383	−0.682655
$915$	0	0
$916$	3.44265	0.113748
$917$	−3.45580	−0.114120
$918$	4.01507	0.132517
$919$	35.6577	1.17624	0.588119	−	0.808774i	$-0.299868\pi$
$919$	35.6577	1.17624	0.588119	+	0.808774i	$0.299868\pi$
$920$	0	0
$921$	−3.76463	−0.124049
$922$	−15.4547	−0.508974
$923$	0	0
$924$	0.0881460	0.00289979
$925$	0	0
$926$	17.1830	0.564668
$927$	−41.6045	−1.36647
$928$	−0.286208	−0.00939525
$929$	23.8944	0.783950	0.391975	−	0.919976i	$-0.371792\pi$
$929$	23.8944	0.783950	0.391975	+	0.919976i	$0.371792\pi$
$930$	0	0
$931$	10.7899	0.353623
$932$	19.4131	0.635898
$933$	−10.8073	−0.353816
$934$	12.1655	0.398069
$935$	0	0
$936$	0	0
$937$	−56.0796	−1.83204	−0.916021	−	0.401130i	$-0.868617\pi$
$937$	−56.0796	−1.83204	−0.916021	+	0.401130i	$0.868617\pi$
$938$	−1.60388	−0.0523684
$939$	0.910526	0.0297139
$940$	0	0
$941$	−30.7942	−1.00386	−0.501931	−	0.864908i	$-0.667377\pi$
$941$	−30.7942	−1.00386	−0.501931	+	0.864908i	$0.667377\pi$
$942$	−0.203439	−0.00662840
$943$	5.91185	0.192516
$944$	4.30798	0.140213
$945$	0	0
$946$	4.51142	0.146679
$947$	39.6152	1.28732	0.643661	−	0.765311i	$-0.277415\pi$
$947$	39.6152	1.28732	0.643661	+	0.765311i	$0.277415\pi$
$948$	−2.10454	−0.0683523
$949$	0	0
$950$	0	0
$951$	−7.18492	−0.232987
$952$	0.384043	0.0124469
$953$	29.1661	0.944784	0.472392	−	0.881389i	$-0.343391\pi$
$953$	29.1661	0.944784	0.472392	+	0.881389i	$0.343391\pi$
$954$	−2.43296	−0.0787700
$955$	0	0
$956$	−29.0495	−0.939528
$957$	−0.102147	−0.00330193
$958$	15.8998	0.513698
$959$	−0.710071	−0.0229294
$960$	0	0
$961$	−13.7041	−0.442068
$962$	0	0
$963$	22.5187	0.725655
$964$	12.3394	0.397427
$965$	0	0
$966$	−0.0760644	−0.00244733
$967$	16.2808	0.523556	0.261778	−	0.965128i	$-0.415691\pi$
$967$	16.2808	0.523556	0.261778	+	0.965128i	$0.415691\pi$
$968$	−10.3569	−0.332883
$969$	1.07606	0.0345682
$970$	0	0
$971$	−14.5545	−0.467076	−0.233538	−	0.972348i	$-0.575030\pi$
$971$	−14.5545	−0.467076	−0.233538	+	0.972348i	$0.575030\pi$
$972$	−10.9758	−0.352050
$973$	−5.28514	−0.169434
$974$	−3.85815	−0.123623
$975$	0	0
$976$	−4.30798	−0.137895
$977$	−0.932559	−0.0298352	−0.0149176	−	0.999889i	$-0.504749\pi$
$977$	−0.932559	−0.0298352	−0.0149176	+	0.999889i	$0.504749\pi$
$978$	−3.90946	−0.125011
$979$	3.48619	0.111419
$980$	0	0
$981$	50.3183	1.60654
$982$	12.3351	0.393630
$983$	14.2784	0.455411	0.227706	−	0.973730i	$-0.426878\pi$
$983$	14.2784	0.455411	0.227706	+	0.973730i	$0.426878\pi$
$984$	−3.80194	−0.121201
$985$	0	0
$986$	−0.445042	−0.0141730
$987$	0.662481	0.0210870
$988$	0	0
$989$	−3.89307	−0.123792
$990$	0	0
$991$	−32.8528	−1.04360	−0.521801	−	0.853067i	$-0.674740\pi$
$991$	−32.8528	−1.04360	−0.521801	+	0.853067i	$0.674740\pi$
$992$	4.15883	0.132043
$993$	11.2489	0.356973
$994$	−1.62565	−0.0515624
$995$	0	0
$996$	3.53079	0.111878
$997$	−5.88816	−0.186480	−0.0932400	−	0.995644i	$-0.529722\pi$
$997$	−5.88816	−0.186480	−0.0932400	+	0.995644i	$0.529722\pi$
$998$	−16.6843	−0.528131
$999$	−21.3897	−0.676740

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.by.1.2	yes	3
5.4	even	2		8450.2.a.bw.1.2	yes	3
13.12	even	2		8450.2.a.bp.1.2	✓	3
65.64	even	2		8450.2.a.cf.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8450.2.a.bp.1.2	✓	3	13.12	even	2
8450.2.a.bw.1.2	yes	3	5.4	even	2
8450.2.a.by.1.2	yes	3	1.1	even	1	trivial
8450.2.a.cf.1.2	yes	3	65.64	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.445042 q^{3} +1.00000 q^{4} -0.445042 q^{6} +0.246980 q^{7} +1.00000 q^{8} -2.80194 q^{9} -0.801938 q^{11} -0.445042 q^{12} +0.246980 q^{14} +1.00000 q^{16} +1.55496 q^{17} -2.80194 q^{18} -1.55496 q^{19} -0.109916 q^{21} -0.801938 q^{22} +0.692021 q^{23} -0.445042 q^{24} +2.58211 q^{27} +0.246980 q^{28} -0.286208 q^{29} +4.15883 q^{31} +1.00000 q^{32} +0.356896 q^{33} +1.55496 q^{34} -2.80194 q^{36} -8.28382 q^{37} -1.55496 q^{38} +8.54288 q^{41} -0.109916 q^{42} -5.62565 q^{43} -0.801938 q^{44} +0.692021 q^{46} -6.02715 q^{47} -0.445042 q^{48} -6.93900 q^{49} -0.692021 q^{51} +0.868313 q^{53} +2.58211 q^{54} +0.246980 q^{56} +0.692021 q^{57} -0.286208 q^{58} +4.30798 q^{59} -4.30798 q^{61} +4.15883 q^{62} -0.692021 q^{63} +1.00000 q^{64} +0.356896 q^{66} -6.49396 q^{67} +1.55496 q^{68} -0.307979 q^{69} -6.58211 q^{71} -2.80194 q^{72} +7.47219 q^{73} -8.28382 q^{74} -1.55496 q^{76} -0.198062 q^{77} +4.72886 q^{79} +7.25667 q^{81} +8.54288 q^{82} -7.93362 q^{83} -0.109916 q^{84} -5.62565 q^{86} +0.127375 q^{87} -0.801938 q^{88} -4.34721 q^{89} +0.692021 q^{92} -1.85086 q^{93} -6.02715 q^{94} -0.445042 q^{96} +7.83877 q^{97} -6.93900 q^{98} +2.24698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 5 q^{17} - 4 q^{18} - 5 q^{19} - q^{21} + 2 q^{22} - 3 q^{23} - q^{24} + 2 q^{27} - 4 q^{28}+ \cdots + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9 + 2 * q^11 - q^12 - 4 * q^14 + 3 * q^16 + 5 * q^17 - 4 * q^18 - 5 * q^19 - q^21 + 2 * q^22 - 3 * q^23 - q^24 + 2 * q^27 - 4 * q^28 - 9 * q^29 + 4 * q^31 + 3 * q^32 - 3 * q^33 + 5 * q^34 - 4 * q^36 + 8 * q^37 - 5 * q^38 + 7 * q^41 - q^42 - 5 * q^43 + 2 * q^44 - 3 * q^46 - 12 * q^47 - q^48 - 11 * q^49 + 3 * q^51 + 5 * q^53 + 2 * q^54 - 4 * q^56 - 3 * q^57 - 9 * q^58 + 18 * q^59 - 18 * q^61 + 4 * q^62 + 3 * q^63 + 3 * q^64 - 3 * q^66 - 10 * q^67 + 5 * q^68 - 6 * q^69 - 14 * q^71 - 4 * q^72 + 16 * q^73 + 8 * q^74 - 5 * q^76 - 5 * q^77 - 19 * q^79 - 5 * q^81 + 7 * q^82 - 17 * q^83 - q^84 - 5 * q^86 + 17 * q^87 + 2 * q^88 - 31 * q^89 - 3 * q^92 + 8 * q^93 - 12 * q^94 - q^96 - 9 * q^97 - 11 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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