LMFDB - Embedded newform 8450.2.a.cx.1.5

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 = [9,9,-7,9,0,-7,1,9,8,0,-4,-7,0,1,0,9,-12] 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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 14x^{7} + 17x^{6} + 53x^{5} - 69x^{4} - 33x^{3} + 26x^{2} + 8x - 1 $ x^9 - x^8 - 14x^7 + 17x^6 + 53x^5 - 69x^4 - 33x^3 + 26x^2 + 8*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1690)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.20982$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} -0.666723 q^{3} +1.00000 q^{4} -0.666723 q^{6} +2.41461 q^{7} +1.00000 q^{8} -2.55548 q^{9} +1.46118 q^{11} -0.666723 q^{12} +2.41461 q^{14} +1.00000 q^{16} -6.74749 q^{17} -2.55548 q^{18} -7.41659 q^{19} -1.60988 q^{21} +1.46118 q^{22} +7.48975 q^{23} -0.666723 q^{24} +3.70397 q^{27} +2.41461 q^{28} -0.696556 q^{29} -1.36837 q^{31} +1.00000 q^{32} -0.974200 q^{33} -6.74749 q^{34} -2.55548 q^{36} +9.25191 q^{37} -7.41659 q^{38} -4.24884 q^{41} -1.60988 q^{42} -2.38573 q^{43} +1.46118 q^{44} +7.48975 q^{46} -2.77456 q^{47} -0.666723 q^{48} -1.16965 q^{49} +4.49870 q^{51} -7.04744 q^{53} +3.70397 q^{54} +2.41461 q^{56} +4.94481 q^{57} -0.696556 q^{58} +0.0690364 q^{59} +5.91472 q^{61} -1.36837 q^{62} -6.17050 q^{63} +1.00000 q^{64} -0.974200 q^{66} -10.9958 q^{67} -6.74749 q^{68} -4.99359 q^{69} +3.60919 q^{71} -2.55548 q^{72} -9.30157 q^{73} +9.25191 q^{74} -7.41659 q^{76} +3.52818 q^{77} -0.766732 q^{79} +5.19692 q^{81} -4.24884 q^{82} -16.5800 q^{83} -1.60988 q^{84} -2.38573 q^{86} +0.464410 q^{87} +1.46118 q^{88} -17.7481 q^{89} +7.48975 q^{92} +0.912326 q^{93} -2.77456 q^{94} -0.666723 q^{96} +10.5209 q^{97} -1.16965 q^{98} -3.73401 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 7 q^{3} + 9 q^{4} - 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} - 4 q^{11} - 7 q^{12} + q^{14} + 9 q^{16} - 12 q^{17} + 8 q^{18} - 6 q^{19} - 8 q^{21} - 4 q^{22} - 11 q^{23} - 7 q^{24} - 34 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 - 7 * q^3 + 9 * q^4 - 7 * q^6 + q^7 + 9 * q^8 + 8 * q^9 - 4 * q^11 - 7 * q^12 + q^14 + 9 * q^16 - 12 * q^17 + 8 * q^18 - 6 * q^19 - 8 * q^21 - 4 * q^22 - 11 * q^23 - 7 * q^24 - 34 * q^27 + q^28 + 15 * q^29 + 6 * q^31 + 9 * q^32 - 14 * q^33 - 12 * q^34 + 8 * q^36 - 10 * q^37 - 6 * q^38 - 27 * q^41 - 8 * q^42 - 41 * q^43 - 4 * q^44 - 11 * q^46 + 5 * q^47 - 7 * q^48 - 10 * q^49 + 12 * q^51 - 18 * q^53 - 34 * q^54 + q^56 - 12 * q^57 + 15 * q^58 - 20 * q^59 - 19 * q^61 + 6 * q^62 + 39 * q^63 + 9 * q^64 - 14 * q^66 - q^67 - 12 * q^68 - 14 * q^69 + 10 * q^71 + 8 * q^72 - 12 * q^73 - 10 * q^74 - 6 * q^76 - 30 * q^77 - 14 * q^79 + 53 * q^81 - 27 * q^82 + q^83 - 8 * q^84 - 41 * q^86 - 33 * q^87 - 4 * q^88 - 43 * q^89 - 11 * q^92 - 28 * q^93 + 5 * q^94 - 7 * q^96 + 6 * q^97 - 10 * q^98 - 16 * 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
$3$	−0.666723	−0.384933	−0.192466	−	0.981304i	$-0.561649\pi$
$3$	−0.666723	−0.384933	−0.192466	+	0.981304i	$0.561649\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−0.666723	−0.272188
$7$	2.41461	0.912638	0.456319	−	0.889816i	$-0.349168\pi$
$7$	2.41461	0.912638	0.456319	+	0.889816i	$0.349168\pi$
$8$	1.00000	0.353553
$9$	−2.55548	−0.851827
$10$	0	0
$11$	1.46118	0.440561	0.220281	−	0.975437i	$-0.429303\pi$
$11$	1.46118	0.440561	0.220281	+	0.975437i	$0.429303\pi$
$12$	−0.666723	−0.192466
$13$	0	0
$13$	0	0
$14$	2.41461	0.645332
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.74749	−1.63651	−0.818253	−	0.574858i	$-0.805057\pi$
$17$	−6.74749	−1.63651	−0.818253	+	0.574858i	$0.805057\pi$
$18$	−2.55548	−0.602333
$19$	−7.41659	−1.70148	−0.850741	−	0.525585i	$-0.823846\pi$
$19$	−7.41659	−1.70148	−0.850741	+	0.525585i	$0.823846\pi$
$20$	0	0
$21$	−1.60988	−0.351304
$22$	1.46118	0.311524
$23$	7.48975	1.56172	0.780861	−	0.624705i	$-0.214781\pi$
$23$	7.48975	1.56172	0.780861	+	0.624705i	$0.214781\pi$
$24$	−0.666723	−0.136094
$25$	0	0
$26$	0	0
$27$	3.70397	0.712829
$28$	2.41461	0.456319
$29$	−0.696556	−0.129347	−0.0646736	−	0.997906i	$-0.520601\pi$
$29$	−0.696556	−0.129347	−0.0646736	+	0.997906i	$0.520601\pi$
$30$	0	0
$31$	−1.36837	−0.245767	−0.122884	−	0.992421i	$-0.539214\pi$
$31$	−1.36837	−0.245767	−0.122884	+	0.992421i	$0.539214\pi$
$32$	1.00000	0.176777
$33$	−0.974200	−0.169586
$34$	−6.74749	−1.15718
$35$	0	0
$36$	−2.55548	−0.425913
$37$	9.25191	1.52101	0.760503	−	0.649335i	$-0.224953\pi$
$37$	9.25191	1.52101	0.760503	+	0.649335i	$0.224953\pi$
$38$	−7.41659	−1.20313
$39$	0	0
$40$	0	0
$41$	−4.24884	−0.663558	−0.331779	−	0.943357i	$-0.607649\pi$
$41$	−4.24884	−0.663558	−0.331779	+	0.943357i	$0.607649\pi$
$42$	−1.60988	−0.248409
$43$	−2.38573	−0.363821	−0.181910	−	0.983315i	$-0.558228\pi$
$43$	−2.38573	−0.363821	−0.181910	+	0.983315i	$0.558228\pi$
$44$	1.46118	0.220281
$45$	0	0
$46$	7.48975	1.10430
$47$	−2.77456	−0.404711	−0.202356	−	0.979312i	$-0.564860\pi$
$47$	−2.77456	−0.404711	−0.202356	+	0.979312i	$0.564860\pi$
$48$	−0.666723	−0.0962331
$49$	−1.16965	−0.167092
$50$	0	0
$51$	4.49870	0.629944
$52$	0	0
$53$	−7.04744	−0.968040	−0.484020	−	0.875057i	$-0.660824\pi$
$53$	−7.04744	−0.968040	−0.484020	+	0.875057i	$0.660824\pi$
$54$	3.70397	0.504046
$55$	0	0
$56$	2.41461	0.322666
$57$	4.94481	0.654956
$58$	−0.696556	−0.0914623
$59$	0.0690364	0.00898777	0.00449389	−	0.999990i	$-0.498570\pi$
$59$	0.0690364	0.00898777	0.00449389	+	0.999990i	$0.498570\pi$
$60$	0	0
$61$	5.91472	0.757302	0.378651	−	0.925539i	$-0.376388\pi$
$61$	5.91472	0.757302	0.378651	+	0.925539i	$0.376388\pi$
$62$	−1.36837	−0.173784
$63$	−6.17050	−0.777409
$64$	1.00000	0.125000
$65$	0	0
$66$	−0.974200	−0.119916
$67$	−10.9958	−1.34335	−0.671674	−	0.740847i	$-0.734424\pi$
$67$	−10.9958	−1.34335	−0.671674	+	0.740847i	$0.734424\pi$
$68$	−6.74749	−0.818253
$69$	−4.99359	−0.601157
$70$	0	0
$71$	3.60919	0.428332	0.214166	−	0.976797i	$-0.431297\pi$
$71$	3.60919	0.428332	0.214166	+	0.976797i	$0.431297\pi$
$72$	−2.55548	−0.301166
$73$	−9.30157	−1.08867	−0.544333	−	0.838869i	$-0.683217\pi$
$73$	−9.30157	−1.08867	−0.544333	+	0.838869i	$0.683217\pi$
$74$	9.25191	1.07551
$75$	0	0
$76$	−7.41659	−0.850741
$77$	3.52818	0.402073
$78$	0	0
$79$	−0.766732	−0.0862641	−0.0431320	−	0.999069i	$-0.513734\pi$
$79$	−0.766732	−0.0862641	−0.0431320	+	0.999069i	$0.513734\pi$
$80$	0	0
$81$	5.19692	0.577436
$82$	−4.24884	−0.469206
$83$	−16.5800	−1.81989	−0.909947	−	0.414725i	$-0.863878\pi$
$83$	−16.5800	−1.81989	−0.909947	+	0.414725i	$0.863878\pi$
$84$	−1.60988	−0.175652
$85$	0	0
$86$	−2.38573	−0.257260
$87$	0.464410	0.0497899
$88$	1.46118	0.155762
$89$	−17.7481	−1.88130	−0.940648	−	0.339384i	$-0.889781\pi$
$89$	−17.7481	−1.88130	−0.940648	+	0.339384i	$0.889781\pi$
$90$	0	0
$91$	0	0
$92$	7.48975	0.780861
$93$	0.912326	0.0946038
$94$	−2.77456	−0.286174
$95$	0	0
$96$	−0.666723	−0.0680471
$97$	10.5209	1.06824	0.534120	−	0.845409i	$-0.320643\pi$
$97$	10.5209	1.06824	0.534120	+	0.845409i	$0.320643\pi$
$98$	−1.16965	−0.118152
$99$	−3.73401	−0.375282
$100$	0	0
$101$	14.2473	1.41765	0.708827	−	0.705382i	$-0.249225\pi$
$101$	14.2473	1.41765	0.708827	+	0.705382i	$0.249225\pi$
$102$	4.49870	0.445438
$103$	−9.72411	−0.958145	−0.479072	−	0.877775i	$-0.659027\pi$
$103$	−9.72411	−0.958145	−0.479072	+	0.877775i	$0.659027\pi$
$104$	0	0
$105$	0	0
$106$	−7.04744	−0.684508
$107$	−6.14900	−0.594446	−0.297223	−	0.954808i	$-0.596061\pi$
$107$	−6.14900	−0.594446	−0.297223	+	0.954808i	$0.596061\pi$
$108$	3.70397	0.356414
$109$	0.319202	0.0305741	0.0152870	−	0.999883i	$-0.495134\pi$
$109$	0.319202	0.0305741	0.0152870	+	0.999883i	$0.495134\pi$
$110$	0	0
$111$	−6.16846	−0.585485
$112$	2.41461	0.228159
$113$	1.95370	0.183789	0.0918945	−	0.995769i	$-0.470708\pi$
$113$	1.95370	0.183789	0.0918945	+	0.995769i	$0.470708\pi$
$114$	4.94481	0.463124
$115$	0	0
$116$	−0.696556	−0.0646736
$117$	0	0
$118$	0.0690364	0.00635531
$119$	−16.2926	−1.49354
$120$	0	0
$121$	−8.86496	−0.805906
$122$	5.91472	0.535494
$123$	2.83280	0.255425
$124$	−1.36837	−0.122884
$125$	0	0
$126$	−6.17050	−0.549711
$127$	−9.73867	−0.864167	−0.432084	−	0.901834i	$-0.642221\pi$
$127$	−9.73867	−0.864167	−0.432084	+	0.901834i	$0.642221\pi$
$128$	1.00000	0.0883883
$129$	1.59062	0.140046
$130$	0	0
$131$	10.6075	0.926778	0.463389	−	0.886155i	$-0.346633\pi$
$131$	10.6075	0.926778	0.463389	+	0.886155i	$0.346633\pi$
$132$	−0.974200	−0.0847932
$133$	−17.9082	−1.55284
$134$	−10.9958	−0.949890
$135$	0	0
$136$	−6.74749	−0.578592
$137$	6.40586	0.547289	0.273645	−	0.961831i	$-0.411771\pi$
$137$	6.40586	0.547289	0.273645	+	0.961831i	$0.411771\pi$
$138$	−4.99359	−0.425082
$139$	−16.7981	−1.42480	−0.712398	−	0.701775i	$-0.752391\pi$
$139$	−16.7981	−1.42480	−0.712398	+	0.701775i	$0.752391\pi$
$140$	0	0
$141$	1.84986	0.155786
$142$	3.60919	0.302877
$143$	0	0
$144$	−2.55548	−0.212957
$145$	0	0
$146$	−9.30157	−0.769804
$147$	0.779830	0.0643193
$148$	9.25191	0.760503
$149$	8.90415	0.729456	0.364728	−	0.931114i	$-0.381162\pi$
$149$	8.90415	0.729456	0.364728	+	0.931114i	$0.381162\pi$
$150$	0	0
$151$	−17.1722	−1.39745	−0.698726	−	0.715389i	$-0.746249\pi$
$151$	−17.1722	−1.39745	−0.698726	+	0.715389i	$0.746249\pi$
$152$	−7.41659	−0.601565
$153$	17.2431	1.39402
$154$	3.52818	0.284308
$155$	0	0
$156$	0	0
$157$	−11.0649	−0.883073	−0.441537	−	0.897243i	$-0.645566\pi$
$157$	−11.0649	−0.883073	−0.441537	+	0.897243i	$0.645566\pi$
$158$	−0.766732	−0.0609979
$159$	4.69869	0.372630
$160$	0	0
$161$	18.0848	1.42529
$162$	5.19692	0.408309
$163$	15.2437	1.19398	0.596988	−	0.802250i	$-0.296364\pi$
$163$	15.2437	1.19398	0.596988	+	0.802250i	$0.296364\pi$
$164$	−4.24884	−0.331779
$165$	0	0
$166$	−16.5800	−1.28686
$167$	−8.69198	−0.672605	−0.336303	−	0.941754i	$-0.609177\pi$
$167$	−8.69198	−0.672605	−0.336303	+	0.941754i	$0.609177\pi$
$168$	−1.60988	−0.124205
$169$	0	0
$170$	0	0
$171$	18.9529	1.44937
$172$	−2.38573	−0.181910
$173$	3.39226	0.257909	0.128954	−	0.991651i	$-0.458838\pi$
$173$	3.39226	0.257909	0.128954	+	0.991651i	$0.458838\pi$
$174$	0.464410	0.0352068
$175$	0	0
$176$	1.46118	0.110140
$177$	−0.0460281	−0.00345969
$178$	−17.7481	−1.33028
$179$	13.2996	0.994057	0.497029	−	0.867734i	$-0.334424\pi$
$179$	13.2996	0.994057	0.497029	+	0.867734i	$0.334424\pi$
$180$	0	0
$181$	−14.0010	−1.04068	−0.520341	−	0.853958i	$-0.674195\pi$
$181$	−14.0010	−1.04068	−0.520341	+	0.853958i	$0.674195\pi$
$182$	0	0
$183$	−3.94348	−0.291510
$184$	7.48975	0.552152
$185$	0	0
$186$	0.912326	0.0668950
$187$	−9.85927	−0.720981
$188$	−2.77456	−0.202356
$189$	8.94364	0.650554
$190$	0	0
$191$	−18.7576	−1.35725	−0.678627	−	0.734483i	$-0.737425\pi$
$191$	−18.7576	−1.35725	−0.678627	+	0.734483i	$0.737425\pi$
$192$	−0.666723	−0.0481166
$193$	24.3855	1.75530	0.877652	−	0.479299i	$-0.159109\pi$
$193$	24.3855	1.75530	0.877652	+	0.479299i	$0.159109\pi$
$194$	10.5209	0.755360
$195$	0	0
$196$	−1.16965	−0.0835462
$197$	−2.73133	−0.194599	−0.0972997	−	0.995255i	$-0.531021\pi$
$197$	−2.73133	−0.194599	−0.0972997	+	0.995255i	$0.531021\pi$
$198$	−3.73401	−0.265364
$199$	11.2116	0.794771	0.397385	−	0.917652i	$-0.369918\pi$
$199$	11.2116	0.794771	0.397385	+	0.917652i	$0.369918\pi$
$200$	0	0
$201$	7.33113	0.517098
$202$	14.2473	1.00243
$203$	−1.68191	−0.118047
$204$	4.49870	0.314972
$205$	0	0
$206$	−9.72411	−0.677511
$207$	−19.1399	−1.33032
$208$	0	0
$209$	−10.8369	−0.749607
$210$	0	0
$211$	13.7163	0.944266	0.472133	−	0.881527i	$-0.343484\pi$
$211$	13.7163	0.944266	0.472133	+	0.881527i	$0.343484\pi$
$212$	−7.04744	−0.484020
$213$	−2.40633	−0.164879
$214$	−6.14900	−0.420337
$215$	0	0
$216$	3.70397	0.252023
$217$	−3.30409	−0.224296
$218$	0.319202	0.0216191
$219$	6.20157	0.419063
$220$	0	0
$221$	0	0
$222$	−6.16846	−0.414000
$223$	−9.84353	−0.659172	−0.329586	−	0.944126i	$-0.606909\pi$
$223$	−9.84353	−0.659172	−0.329586	+	0.944126i	$0.606909\pi$
$224$	2.41461	0.161333
$225$	0	0
$226$	1.95370	0.129959
$227$	−19.4038	−1.28788	−0.643939	−	0.765077i	$-0.722701\pi$
$227$	−19.4038	−1.28788	−0.643939	+	0.765077i	$0.722701\pi$
$228$	4.94481	0.327478
$229$	−16.7503	−1.10689	−0.553447	−	0.832885i	$-0.686688\pi$
$229$	−16.7503	−1.10689	−0.553447	+	0.832885i	$0.686688\pi$
$230$	0	0
$231$	−2.35231	−0.154771
$232$	−0.696556	−0.0457311
$233$	−16.7974	−1.10043	−0.550217	−	0.835022i	$-0.685455\pi$
$233$	−16.7974	−1.10043	−0.550217	+	0.835022i	$0.685455\pi$
$234$	0	0
$235$	0	0
$236$	0.0690364	0.00449389
$237$	0.511198	0.0332059
$238$	−16.2926	−1.05609
$239$	18.2819	1.18256	0.591278	−	0.806467i	$-0.298623\pi$
$239$	18.2819	1.18256	0.591278	+	0.806467i	$0.298623\pi$
$240$	0	0
$241$	−7.86172	−0.506418	−0.253209	−	0.967412i	$-0.581486\pi$
$241$	−7.86172	−0.506418	−0.253209	+	0.967412i	$0.581486\pi$
$242$	−8.86496	−0.569861
$243$	−14.5768	−0.935102
$244$	5.91472	0.378651
$245$	0	0
$246$	2.83280	0.180613
$247$	0	0
$248$	−1.36837	−0.0868918
$249$	11.0543	0.700536
$250$	0	0
$251$	8.97087	0.566236	0.283118	−	0.959085i	$-0.408631\pi$
$251$	8.97087	0.566236	0.283118	+	0.959085i	$0.408631\pi$
$252$	−6.17050	−0.388705
$253$	10.9439	0.688034
$254$	−9.73867	−0.611059
$255$	0	0
$256$	1.00000	0.0625000
$257$	24.2725	1.51408	0.757040	−	0.653369i	$-0.226645\pi$
$257$	24.2725	1.51408	0.757040	+	0.653369i	$0.226645\pi$
$258$	1.59062	0.0990278
$259$	22.3398	1.38813
$260$	0	0
$261$	1.78004	0.110181
$262$	10.6075	0.655331
$263$	−28.7755	−1.77438	−0.887188	−	0.461408i	$-0.847344\pi$
$263$	−28.7755	−1.77438	−0.887188	+	0.461408i	$0.847344\pi$
$264$	−0.974200	−0.0599578
$265$	0	0
$266$	−17.9082	−1.09802
$267$	11.8331	0.724172
$268$	−10.9958	−0.671674
$269$	−26.2808	−1.60237	−0.801185	−	0.598417i	$-0.795797\pi$
$269$	−26.2808	−1.60237	−0.801185	+	0.598417i	$0.795797\pi$
$270$	0	0
$271$	14.3647	0.872594	0.436297	−	0.899803i	$-0.356290\pi$
$271$	14.3647	0.872594	0.436297	+	0.899803i	$0.356290\pi$
$272$	−6.74749	−0.409126
$273$	0	0
$274$	6.40586	0.386992
$275$	0	0
$276$	−4.99359	−0.300579
$277$	9.92492	0.596331	0.298165	−	0.954514i	$-0.403625\pi$
$277$	9.92492	0.596331	0.298165	+	0.954514i	$0.403625\pi$
$278$	−16.7981	−1.00748
$279$	3.49685	0.209351
$280$	0	0
$281$	11.7992	0.703880	0.351940	−	0.936023i	$-0.385522\pi$
$281$	11.7992	0.703880	0.351940	+	0.936023i	$0.385522\pi$
$282$	1.84986	0.110158
$283$	−13.0538	−0.775968	−0.387984	−	0.921666i	$-0.626828\pi$
$283$	−13.0538	−0.775968	−0.387984	+	0.921666i	$0.626828\pi$
$284$	3.60919	0.214166
$285$	0	0
$286$	0	0
$287$	−10.2593	−0.605588
$288$	−2.55548	−0.150583
$289$	28.5286	1.67815
$290$	0	0
$291$	−7.01455	−0.411200
$292$	−9.30157	−0.544333
$293$	16.2358	0.948506	0.474253	−	0.880389i	$-0.342718\pi$
$293$	16.2358	0.948506	0.474253	+	0.880389i	$0.342718\pi$
$294$	0.779830	0.0454806
$295$	0	0
$296$	9.25191	0.537757
$297$	5.41215	0.314045
$298$	8.90415	0.515803
$299$	0	0
$300$	0	0
$301$	−5.76062	−0.332036
$302$	−17.1722	−0.988148
$303$	−9.49897	−0.545701
$304$	−7.41659	−0.425370
$305$	0	0
$306$	17.2431	0.985721
$307$	5.66589	0.323370	0.161685	−	0.986842i	$-0.448307\pi$
$307$	5.66589	0.323370	0.161685	+	0.986842i	$0.448307\pi$
$308$	3.52818	0.201036
$309$	6.48328	0.368821
$310$	0	0
$311$	20.5465	1.16508	0.582542	−	0.812801i	$-0.302058\pi$
$311$	20.5465	1.16508	0.582542	+	0.812801i	$0.302058\pi$
$312$	0	0
$313$	11.0354	0.623756	0.311878	−	0.950122i	$-0.399042\pi$
$313$	11.0354	0.623756	0.311878	+	0.950122i	$0.399042\pi$
$314$	−11.0649	−0.624427
$315$	0	0
$316$	−0.766732	−0.0431320
$317$	−17.0656	−0.958501	−0.479251	−	0.877678i	$-0.659091\pi$
$317$	−17.0656	−0.958501	−0.479251	+	0.877678i	$0.659091\pi$
$318$	4.69869	0.263489
$319$	−1.01779	−0.0569854
$320$	0	0
$321$	4.09968	0.228822
$322$	18.0848	1.00783
$323$	50.0433	2.78448
$324$	5.19692	0.288718
$325$	0	0
$326$	15.2437	0.844269
$327$	−0.212820	−0.0117689
$328$	−4.24884	−0.234603
$329$	−6.69949	−0.369355
$330$	0	0
$331$	29.6977	1.63233	0.816167	−	0.577817i	$-0.196095\pi$
$331$	29.6977	1.63233	0.816167	+	0.577817i	$0.196095\pi$
$332$	−16.5800	−0.909947
$333$	−23.6431	−1.29563
$334$	−8.69198	−0.475604
$335$	0	0
$336$	−1.60988	−0.0878260
$337$	−11.1216	−0.605833	−0.302917	−	0.953017i	$-0.597960\pi$
$337$	−11.1216	−0.605833	−0.302917	+	0.953017i	$0.597960\pi$
$338$	0	0
$339$	−1.30258	−0.0707464
$340$	0	0
$341$	−1.99944	−0.108275
$342$	18.9529	1.02486
$343$	−19.7265	−1.06513
$344$	−2.38573	−0.128630
$345$	0	0
$346$	3.39226	0.182369
$347$	−19.1698	−1.02909	−0.514545	−	0.857463i	$-0.672039\pi$
$347$	−19.1698	−1.02909	−0.514545	+	0.857463i	$0.672039\pi$
$348$	0.464410	0.0248950
$349$	2.07169	0.110895	0.0554474	−	0.998462i	$-0.482341\pi$
$349$	2.07169	0.110895	0.0554474	+	0.998462i	$0.482341\pi$
$350$	0	0
$351$	0	0
$352$	1.46118	0.0778810
$353$	−14.2889	−0.760522	−0.380261	−	0.924879i	$-0.624166\pi$
$353$	−14.2889	−0.760522	−0.380261	+	0.924879i	$0.624166\pi$
$354$	−0.0460281	−0.00244637
$355$	0	0
$356$	−17.7481	−0.940648
$357$	10.8626	0.574911
$358$	13.2996	0.702905
$359$	12.4966	0.659543	0.329772	−	0.944061i	$-0.393028\pi$
$359$	12.4966	0.659543	0.329772	+	0.944061i	$0.393028\pi$
$360$	0	0
$361$	36.0058	1.89504
$362$	−14.0010	−0.735874
$363$	5.91047	0.310219
$364$	0	0
$365$	0	0
$366$	−3.94348	−0.206129
$367$	−20.7194	−1.08154	−0.540772	−	0.841170i	$-0.681868\pi$
$367$	−20.7194	−1.08154	−0.540772	+	0.841170i	$0.681868\pi$
$368$	7.48975	0.390430
$369$	10.8578	0.565236
$370$	0	0
$371$	−17.0168	−0.883470
$372$	0.912326	0.0473019
$373$	−31.4926	−1.63063	−0.815313	−	0.579020i	$-0.803435\pi$
$373$	−31.4926	−1.63063	−0.815313	+	0.579020i	$0.803435\pi$
$374$	−9.85927	−0.509811
$375$	0	0
$376$	−2.77456	−0.143087
$377$	0	0
$378$	8.94364	0.460011
$379$	−7.87442	−0.404482	−0.202241	−	0.979336i	$-0.564822\pi$
$379$	−7.87442	−0.404482	−0.202241	+	0.979336i	$0.564822\pi$
$380$	0	0
$381$	6.49299	0.332646
$382$	−18.7576	−0.959723
$383$	8.03386	0.410511	0.205256	−	0.978708i	$-0.434197\pi$
$383$	8.03386	0.410511	0.205256	+	0.978708i	$0.434197\pi$
$384$	−0.666723	−0.0340236
$385$	0	0
$386$	24.3855	1.24119
$387$	6.09669	0.309912
$388$	10.5209	0.534120
$389$	−21.5917	−1.09474	−0.547371	−	0.836890i	$-0.684371\pi$
$389$	−21.5917	−1.09474	−0.547371	+	0.836890i	$0.684371\pi$
$390$	0	0
$391$	−50.5370	−2.55577
$392$	−1.16965	−0.0590761
$393$	−7.07224	−0.356747
$394$	−2.73133	−0.137603
$395$	0	0
$396$	−3.73401	−0.187641
$397$	−36.8751	−1.85071	−0.925355	−	0.379102i	$-0.876233\pi$
$397$	−36.8751	−1.85071	−0.925355	+	0.379102i	$0.876233\pi$
$398$	11.2116	0.561988
$399$	11.9398	0.597737
$400$	0	0
$401$	−8.26837	−0.412902	−0.206451	−	0.978457i	$-0.566191\pi$
$401$	−8.26837	−0.412902	−0.206451	+	0.978457i	$0.566191\pi$
$402$	7.33113	0.365644
$403$	0	0
$404$	14.2473	0.708827
$405$	0	0
$406$	−1.68191	−0.0834719
$407$	13.5187	0.670096
$408$	4.49870	0.222719
$409$	−3.02146	−0.149402	−0.0747008	−	0.997206i	$-0.523800\pi$
$409$	−3.02146	−0.149402	−0.0747008	+	0.997206i	$0.523800\pi$
$410$	0	0
$411$	−4.27093	−0.210669
$412$	−9.72411	−0.479072
$413$	0.166696	0.00820258
$414$	−19.1399	−0.940676
$415$	0	0
$416$	0	0
$417$	11.1997	0.548451
$418$	−10.8369	−0.530052
$419$	−35.0376	−1.71170	−0.855850	−	0.517223i	$-0.826966\pi$
$419$	−35.0376	−1.71170	−0.855850	+	0.517223i	$0.826966\pi$
$420$	0	0
$421$	9.81291	0.478252	0.239126	−	0.970989i	$-0.423139\pi$
$421$	9.81291	0.478252	0.239126	+	0.970989i	$0.423139\pi$
$422$	13.7163	0.667697
$423$	7.09033	0.344744
$424$	−7.04744	−0.342254
$425$	0	0
$426$	−2.40633	−0.116587
$427$	14.2818	0.691143
$428$	−6.14900	−0.297223
$429$	0	0
$430$	0	0
$431$	−5.88552	−0.283495	−0.141748	−	0.989903i	$-0.545272\pi$
$431$	−5.88552	−0.283495	−0.141748	+	0.989903i	$0.545272\pi$
$432$	3.70397	0.178207
$433$	−31.1096	−1.49503	−0.747516	−	0.664244i	$-0.768754\pi$
$433$	−31.1096	−1.49503	−0.747516	+	0.664244i	$0.768754\pi$
$434$	−3.30409	−0.158601
$435$	0	0
$436$	0.319202	0.0152870
$437$	−55.5484	−2.65724
$438$	6.20157	0.296322
$439$	−2.69710	−0.128726	−0.0643629	−	0.997927i	$-0.520502\pi$
$439$	−2.69710	−0.128726	−0.0643629	+	0.997927i	$0.520502\pi$
$440$	0	0
$441$	2.98901	0.142334
$442$	0	0
$443$	19.0859	0.906799	0.453399	−	0.891308i	$-0.350211\pi$
$443$	19.0859	0.906799	0.453399	+	0.891308i	$0.350211\pi$
$444$	−6.16846	−0.292742
$445$	0	0
$446$	−9.84353	−0.466105
$447$	−5.93660	−0.280791
$448$	2.41461	0.114080
$449$	−18.3579	−0.866363	−0.433182	−	0.901307i	$-0.642609\pi$
$449$	−18.3579	−0.866363	−0.433182	+	0.901307i	$0.642609\pi$
$450$	0	0
$451$	−6.20831	−0.292338
$452$	1.95370	0.0918945
$453$	11.4491	0.537925
$454$	−19.4038	−0.910668
$455$	0	0
$456$	4.94481	0.231562
$457$	25.3873	1.18757	0.593785	−	0.804624i	$-0.297633\pi$
$457$	25.3873	1.18757	0.593785	+	0.804624i	$0.297633\pi$
$458$	−16.7503	−0.782692
$459$	−24.9925	−1.16655
$460$	0	0
$461$	−1.12031	−0.0521782	−0.0260891	−	0.999660i	$-0.508305\pi$
$461$	−1.12031	−0.0521782	−0.0260891	+	0.999660i	$0.508305\pi$
$462$	−2.35231	−0.109440
$463$	−30.6952	−1.42653	−0.713264	−	0.700895i	$-0.752784\pi$
$463$	−30.6952	−1.42653	−0.713264	+	0.700895i	$0.752784\pi$
$464$	−0.696556	−0.0323368
$465$	0	0
$466$	−16.7974	−0.778124
$467$	5.88670	0.272404	0.136202	−	0.990681i	$-0.456510\pi$
$467$	5.88670	0.272404	0.136202	+	0.990681i	$0.456510\pi$
$468$	0	0
$469$	−26.5505	−1.22599
$470$	0	0
$471$	7.37720	0.339924
$472$	0.0690364	0.00317766
$473$	−3.48597	−0.160285
$474$	0.511198	0.0234801
$475$	0	0
$476$	−16.2926	−0.746768
$477$	18.0096	0.824602
$478$	18.2819	0.836194
$479$	24.6962	1.12840	0.564200	−	0.825638i	$-0.309185\pi$
$479$	24.6962	1.12840	0.564200	+	0.825638i	$0.309185\pi$
$480$	0	0
$481$	0	0
$482$	−7.86172	−0.358091
$483$	−12.0576	−0.548639
$484$	−8.86496	−0.402953
$485$	0	0
$486$	−14.5768	−0.661217
$487$	−6.85817	−0.310774	−0.155387	−	0.987854i	$-0.549662\pi$
$487$	−6.85817	−0.310774	−0.155387	+	0.987854i	$0.549662\pi$
$488$	5.91472	0.267747
$489$	−10.1633	−0.459600
$490$	0	0
$491$	−32.0421	−1.44604	−0.723021	−	0.690826i	$-0.757247\pi$
$491$	−32.0421	−1.44604	−0.723021	+	0.690826i	$0.757247\pi$
$492$	2.83280	0.127712
$493$	4.70000	0.211677
$494$	0	0
$495$	0	0
$496$	−1.36837	−0.0614418
$497$	8.71480	0.390912
$498$	11.0543	0.495354
$499$	25.9861	1.16330	0.581648	−	0.813440i	$-0.302408\pi$
$499$	25.9861	1.16330	0.581648	+	0.813440i	$0.302408\pi$
$500$	0	0
$501$	5.79514	0.258908
$502$	8.97087	0.400389
$503$	−7.96895	−0.355318	−0.177659	−	0.984092i	$-0.556852\pi$
$503$	−7.96895	−0.355318	−0.177659	+	0.984092i	$0.556852\pi$
$504$	−6.17050	−0.274856
$505$	0	0
$506$	10.9439	0.486513
$507$	0	0
$508$	−9.73867	−0.432084
$509$	26.5772	1.17801	0.589007	−	0.808128i	$-0.299519\pi$
$509$	26.5772	1.17801	0.589007	+	0.808128i	$0.299519\pi$
$510$	0	0
$511$	−22.4597	−0.993558
$512$	1.00000	0.0441942
$513$	−27.4708	−1.21286
$514$	24.2725	1.07062
$515$	0	0
$516$	1.59062	0.0700232
$517$	−4.05412	−0.178300
$518$	22.3398	0.981554
$519$	−2.26170	−0.0992776
$520$	0	0
$521$	−16.3843	−0.717810	−0.358905	−	0.933374i	$-0.616850\pi$
$521$	−16.3843	−0.717810	−0.358905	+	0.933374i	$0.616850\pi$
$522$	1.78004	0.0779100
$523$	−12.9757	−0.567390	−0.283695	−	0.958915i	$-0.591560\pi$
$523$	−12.9757	−0.567390	−0.283695	+	0.958915i	$0.591560\pi$
$524$	10.6075	0.463389
$525$	0	0
$526$	−28.7755	−1.25467
$527$	9.23308	0.402199
$528$	−0.974200	−0.0423966
$529$	33.0964	1.43897
$530$	0	0
$531$	−0.176421	−0.00765602
$532$	−17.9082	−0.776418
$533$	0	0
$534$	11.8331	0.512067
$535$	0	0
$536$	−10.9958	−0.474945
$537$	−8.86713	−0.382645
$538$	−26.2808	−1.13305
$539$	−1.70906	−0.0736144
$540$	0	0
$541$	23.1606	0.995753	0.497877	−	0.867248i	$-0.334113\pi$
$541$	23.1606	0.995753	0.497877	+	0.867248i	$0.334113\pi$
$542$	14.3647	0.617017
$543$	9.33475	0.400593
$544$	−6.74749	−0.289296
$545$	0	0
$546$	0	0
$547$	−13.3168	−0.569387	−0.284693	−	0.958619i	$-0.591892\pi$
$547$	−13.3168	−0.569387	−0.284693	+	0.958619i	$0.591892\pi$
$548$	6.40586	0.273645
$549$	−15.1150	−0.645091
$550$	0	0
$551$	5.16607	0.220082
$552$	−4.99359	−0.212541
$553$	−1.85136	−0.0787279
$554$	9.92492	0.421669
$555$	0	0
$556$	−16.7981	−0.712398
$557$	−3.02677	−0.128248	−0.0641242	−	0.997942i	$-0.520425\pi$
$557$	−3.02677	−0.128248	−0.0641242	+	0.997942i	$0.520425\pi$
$558$	3.49685	0.148034
$559$	0	0
$560$	0	0
$561$	6.57340	0.277529
$562$	11.7992	0.497718
$563$	18.2744	0.770175	0.385087	−	0.922880i	$-0.374171\pi$
$563$	18.2744	0.770175	0.385087	+	0.922880i	$0.374171\pi$
$564$	1.84986	0.0778932
$565$	0	0
$566$	−13.0538	−0.548692
$567$	12.5486	0.526990
$568$	3.60919	0.151438
$569$	−25.9020	−1.08587	−0.542933	−	0.839776i	$-0.682686\pi$
$569$	−25.9020	−1.08587	−0.542933	+	0.839776i	$0.682686\pi$
$570$	0	0
$571$	−37.9443	−1.58792	−0.793960	−	0.607970i	$-0.791984\pi$
$571$	−37.9443	−1.58792	−0.793960	+	0.607970i	$0.791984\pi$
$572$	0	0
$573$	12.5061	0.522451
$574$	−10.2593	−0.428215
$575$	0	0
$576$	−2.55548	−0.106478
$577$	−18.6481	−0.776330	−0.388165	−	0.921590i	$-0.626891\pi$
$577$	−18.6481	−0.776330	−0.388165	+	0.921590i	$0.626891\pi$
$578$	28.5286	1.18663
$579$	−16.2583	−0.675674
$580$	0	0
$581$	−40.0343	−1.66090
$582$	−7.01455	−0.290763
$583$	−10.2976	−0.426481
$584$	−9.30157	−0.384902
$585$	0	0
$586$	16.2358	0.670695
$587$	6.63680	0.273930	0.136965	−	0.990576i	$-0.456265\pi$
$587$	6.63680	0.273930	0.136965	+	0.990576i	$0.456265\pi$
$588$	0.779830	0.0321596
$589$	10.1487	0.418168
$590$	0	0
$591$	1.82104	0.0749077
$592$	9.25191	0.380251
$593$	8.72953	0.358479	0.179239	−	0.983805i	$-0.442636\pi$
$593$	8.72953	0.358479	0.179239	+	0.983805i	$0.442636\pi$
$594$	5.41215	0.222063
$595$	0	0
$596$	8.90415	0.364728
$597$	−7.47504	−0.305933
$598$	0	0
$599$	38.0625	1.55519	0.777596	−	0.628765i	$-0.216439\pi$
$599$	38.0625	1.55519	0.777596	+	0.628765i	$0.216439\pi$
$600$	0	0
$601$	−2.03785	−0.0831256	−0.0415628	−	0.999136i	$-0.513234\pi$
$601$	−2.03785	−0.0831256	−0.0415628	+	0.999136i	$0.513234\pi$
$602$	−5.76062	−0.234785
$603$	28.0995	1.14430
$604$	−17.1722	−0.698726
$605$	0	0
$606$	−9.49897	−0.385869
$607$	−1.61460	−0.0655347	−0.0327674	−	0.999463i	$-0.510432\pi$
$607$	−1.61460	−0.0655347	−0.0327674	+	0.999463i	$0.510432\pi$
$608$	−7.41659	−0.300782
$609$	1.12137	0.0454402
$610$	0	0
$611$	0	0
$612$	17.2431	0.697010
$613$	17.8032	0.719066	0.359533	−	0.933132i	$-0.382936\pi$
$613$	17.8032	0.719066	0.359533	+	0.933132i	$0.382936\pi$
$614$	5.66589	0.228657
$615$	0	0
$616$	3.52818	0.142154
$617$	−36.9565	−1.48781	−0.743906	−	0.668284i	$-0.767029\pi$
$617$	−36.9565	−1.48781	−0.743906	+	0.668284i	$0.767029\pi$
$618$	6.48328	0.260796
$619$	31.6640	1.27268	0.636341	−	0.771408i	$-0.280447\pi$
$619$	31.6640	1.27268	0.636341	+	0.771408i	$0.280447\pi$
$620$	0	0
$621$	27.7418	1.11324
$622$	20.5465	0.823839
$623$	−42.8548	−1.71694
$624$	0	0
$625$	0	0
$626$	11.0354	0.441062
$627$	7.22524	0.288548
$628$	−11.0649	−0.441537
$629$	−62.4272	−2.48913
$630$	0	0
$631$	7.35141	0.292655	0.146327	−	0.989236i	$-0.453255\pi$
$631$	7.35141	0.292655	0.146327	+	0.989236i	$0.453255\pi$
$632$	−0.766732	−0.0304990
$633$	−9.14494	−0.363479
$634$	−17.0656	−0.677763
$635$	0	0
$636$	4.69869	0.186315
$637$	0	0
$638$	−1.01779	−0.0402947
$639$	−9.22322	−0.364865
$640$	0	0
$641$	38.9606	1.53885	0.769425	−	0.638737i	$-0.220543\pi$
$641$	38.9606	1.53885	0.769425	+	0.638737i	$0.220543\pi$
$642$	4.09968	0.161801
$643$	−0.528071	−0.0208251	−0.0104126	−	0.999946i	$-0.503314\pi$
$643$	−0.528071	−0.0208251	−0.0104126	+	0.999946i	$0.503314\pi$
$644$	18.0848	0.712643
$645$	0	0
$646$	50.0433	1.96893
$647$	30.5439	1.20081	0.600403	−	0.799698i	$-0.295007\pi$
$647$	30.5439	1.20081	0.600403	+	0.799698i	$0.295007\pi$
$648$	5.19692	0.204154
$649$	0.100874	0.00395966
$650$	0	0
$651$	2.20291	0.0863390
$652$	15.2437	0.596988
$653$	−11.9772	−0.468704	−0.234352	−	0.972152i	$-0.575297\pi$
$653$	−11.9772	−0.468704	−0.234352	+	0.972152i	$0.575297\pi$
$654$	−0.212820	−0.00832190
$655$	0	0
$656$	−4.24884	−0.165889
$657$	23.7700	0.927355
$658$	−6.69949	−0.261173
$659$	4.01683	0.156474	0.0782368	−	0.996935i	$-0.475071\pi$
$659$	4.01683	0.156474	0.0782368	+	0.996935i	$0.475071\pi$
$660$	0	0
$661$	40.3372	1.56893	0.784467	−	0.620170i	$-0.212936\pi$
$661$	40.3372	1.56893	0.784467	+	0.620170i	$0.212936\pi$
$662$	29.6977	1.15423
$663$	0	0
$664$	−16.5800	−0.643430
$665$	0	0
$666$	−23.6431	−0.916151
$667$	−5.21703	−0.202004
$668$	−8.69198	−0.336303
$669$	6.56291	0.253737
$670$	0	0
$671$	8.64245	0.333638
$672$	−1.60988	−0.0621024
$673$	41.3853	1.59529	0.797644	−	0.603129i	$-0.206080\pi$
$673$	41.3853	1.59529	0.797644	+	0.603129i	$0.206080\pi$
$674$	−11.1216	−0.428389
$675$	0	0
$676$	0	0
$677$	23.9115	0.918995	0.459497	−	0.888179i	$-0.348030\pi$
$677$	23.9115	0.918995	0.459497	+	0.888179i	$0.348030\pi$
$678$	−1.30258	−0.0500253
$679$	25.4040	0.974916
$680$	0	0
$681$	12.9370	0.495746
$682$	−1.99944	−0.0765623
$683$	23.5399	0.900729	0.450364	−	0.892845i	$-0.351294\pi$
$683$	23.5399	0.900729	0.450364	+	0.892845i	$0.351294\pi$
$684$	18.9529	0.724684
$685$	0	0
$686$	−19.7265	−0.753162
$687$	11.1678	0.426079
$688$	−2.38573	−0.0909552
$689$	0	0
$690$	0	0
$691$	−27.7945	−1.05735	−0.528677	−	0.848823i	$-0.677312\pi$
$691$	−27.7945	−1.05735	−0.528677	+	0.848823i	$0.677312\pi$
$692$	3.39226	0.128954
$693$	−9.01618	−0.342496
$694$	−19.1698	−0.727676
$695$	0	0
$696$	0.464410	0.0176034
$697$	28.6690	1.08592
$698$	2.07169	0.0784145
$699$	11.1992	0.423593
$700$	0	0
$701$	20.6527	0.780042	0.390021	−	0.920806i	$-0.372468\pi$
$701$	20.6527	0.780042	0.390021	+	0.920806i	$0.372468\pi$
$702$	0	0
$703$	−68.6176	−2.58796
$704$	1.46118	0.0550702
$705$	0	0
$706$	−14.2889	−0.537770
$707$	34.4016	1.29381
$708$	−0.0460281	−0.00172984
$709$	1.90959	0.0717161	0.0358581	−	0.999357i	$-0.488584\pi$
$709$	1.90959	0.0717161	0.0358581	+	0.999357i	$0.488584\pi$
$710$	0	0
$711$	1.95937	0.0734821
$712$	−17.7481	−0.665138
$713$	−10.2488	−0.383820
$714$	10.8626	0.406523
$715$	0	0
$716$	13.2996	0.497029
$717$	−12.1889	−0.455205
$718$	12.4966	0.466367
$719$	36.3672	1.35627	0.678135	−	0.734938i	$-0.262789\pi$
$719$	36.3672	1.35627	0.678135	+	0.734938i	$0.262789\pi$
$720$	0	0
$721$	−23.4799	−0.874439
$722$	36.0058	1.34000
$723$	5.24159	0.194937
$724$	−14.0010	−0.520341
$725$	0	0
$726$	5.91047	0.219358
$727$	2.25162	0.0835080	0.0417540	−	0.999128i	$-0.486705\pi$
$727$	2.25162	0.0835080	0.0417540	+	0.999128i	$0.486705\pi$
$728$	0	0
$729$	−5.87208	−0.217485
$730$	0	0
$731$	16.0977	0.595395
$732$	−3.94348	−0.145755
$733$	−27.7157	−1.02370	−0.511851	−	0.859074i	$-0.671040\pi$
$733$	−27.7157	−1.02370	−0.511851	+	0.859074i	$0.671040\pi$
$734$	−20.7194	−0.764766
$735$	0	0
$736$	7.48975	0.276076
$737$	−16.0668	−0.591827
$738$	10.8578	0.399682
$739$	17.4826	0.643108	0.321554	−	0.946891i	$-0.395795\pi$
$739$	17.4826	0.643108	0.321554	+	0.946891i	$0.395795\pi$
$740$	0	0
$741$	0	0
$742$	−17.0168	−0.624707
$743$	45.4090	1.66589	0.832947	−	0.553353i	$-0.186652\pi$
$743$	45.4090	1.66589	0.832947	+	0.553353i	$0.186652\pi$
$744$	0.912326	0.0334475
$745$	0	0
$746$	−31.4926	−1.15303
$747$	42.3699	1.55023
$748$	−9.85927	−0.360491
$749$	−14.8475	−0.542514
$750$	0	0
$751$	−23.3846	−0.853315	−0.426657	−	0.904413i	$-0.640309\pi$
$751$	−23.3846	−0.853315	−0.426657	+	0.904413i	$0.640309\pi$
$752$	−2.77456	−0.101178
$753$	−5.98108	−0.217963
$754$	0	0
$755$	0	0
$756$	8.94364	0.325277
$757$	−33.7347	−1.22611	−0.613055	−	0.790040i	$-0.710059\pi$
$757$	−33.7347	−1.22611	−0.613055	+	0.790040i	$0.710059\pi$
$758$	−7.87442	−0.286012
$759$	−7.29651	−0.264847
$760$	0	0
$761$	−13.4210	−0.486512	−0.243256	−	0.969962i	$-0.578215\pi$
$761$	−13.4210	−0.486512	−0.243256	+	0.969962i	$0.578215\pi$
$762$	6.49299	0.235216
$763$	0.770750	0.0279030
$764$	−18.7576	−0.678627
$765$	0	0
$766$	8.03386	0.290275
$767$	0	0
$768$	−0.666723	−0.0240583
$769$	18.6681	0.673188	0.336594	−	0.941650i	$-0.390725\pi$
$769$	18.6681	0.673188	0.336594	+	0.941650i	$0.390725\pi$
$770$	0	0
$771$	−16.1831	−0.582818
$772$	24.3855	0.877652
$773$	−2.12110	−0.0762905	−0.0381453	−	0.999272i	$-0.512145\pi$
$773$	−2.12110	−0.0762905	−0.0381453	+	0.999272i	$0.512145\pi$
$774$	6.09669	0.219141
$775$	0	0
$776$	10.5209	0.377680
$777$	−14.8944	−0.534335
$778$	−21.5917	−0.774099
$779$	31.5119	1.12903
$780$	0	0
$781$	5.27367	0.188707
$782$	−50.5370	−1.80720
$783$	−2.58002	−0.0922024
$784$	−1.16965	−0.0417731
$785$	0	0
$786$	−7.07224	−0.252258
$787$	1.88977	0.0673631	0.0336815	−	0.999433i	$-0.489277\pi$
$787$	1.88977	0.0673631	0.0336815	+	0.999433i	$0.489277\pi$
$788$	−2.73133	−0.0972997
$789$	19.1853	0.683015
$790$	0	0
$791$	4.71744	0.167733
$792$	−3.73401	−0.132682
$793$	0	0
$794$	−36.8751	−1.30865
$795$	0	0
$796$	11.2116	0.397385
$797$	−27.7575	−0.983222	−0.491611	−	0.870815i	$-0.663592\pi$
$797$	−27.7575	−0.983222	−0.491611	+	0.870815i	$0.663592\pi$
$798$	11.9398	0.422664
$799$	18.7213	0.662312
$800$	0	0
$801$	45.3549	1.60254
$802$	−8.26837	−0.291966
$803$	−13.5912	−0.479624
$804$	7.33113	0.258549
$805$	0	0
$806$	0	0
$807$	17.5220	0.616804
$808$	14.2473	0.501217
$809$	22.3988	0.787501	0.393751	−	0.919217i	$-0.371177\pi$
$809$	22.3988	0.787501	0.393751	+	0.919217i	$0.371177\pi$
$810$	0	0
$811$	10.0371	0.352450	0.176225	−	0.984350i	$-0.443611\pi$
$811$	10.0371	0.352450	0.176225	+	0.984350i	$0.443611\pi$
$812$	−1.68191	−0.0590236
$813$	−9.57728	−0.335890
$814$	13.5187	0.473830
$815$	0	0
$816$	4.49870	0.157486
$817$	17.6940	0.619034
$818$	−3.02146	−0.105643
$819$	0	0
$820$	0	0
$821$	12.9086	0.450513	0.225257	−	0.974299i	$-0.427678\pi$
$821$	12.9086	0.450513	0.225257	+	0.974299i	$0.427678\pi$
$822$	−4.27093	−0.148966
$823$	24.3811	0.849870	0.424935	−	0.905224i	$-0.360297\pi$
$823$	24.3811	0.849870	0.424935	+	0.905224i	$0.360297\pi$
$824$	−9.72411	−0.338755
$825$	0	0
$826$	0.166696	0.00580010
$827$	−11.1712	−0.388461	−0.194230	−	0.980956i	$-0.562221\pi$
$827$	−11.1712	−0.388461	−0.194230	+	0.980956i	$0.562221\pi$
$828$	−19.1399	−0.665158
$829$	28.9931	1.00697	0.503487	−	0.864003i	$-0.332050\pi$
$829$	28.9931	1.00697	0.503487	+	0.864003i	$0.332050\pi$
$830$	0	0
$831$	−6.61717	−0.229547
$832$	0	0
$833$	7.89217	0.273448
$834$	11.1997	0.387813
$835$	0	0
$836$	−10.8369	−0.374804
$837$	−5.06841	−0.175190
$838$	−35.0376	−1.21036
$839$	24.0650	0.830818	0.415409	−	0.909635i	$-0.363639\pi$
$839$	24.0650	0.830818	0.415409	+	0.909635i	$0.363639\pi$
$840$	0	0
$841$	−28.5148	−0.983269
$842$	9.81291	0.338175
$843$	−7.86678	−0.270946
$844$	13.7163	0.472133
$845$	0	0
$846$	7.09033	0.243771
$847$	−21.4055	−0.735500
$848$	−7.04744	−0.242010
$849$	8.70327	0.298695
$850$	0	0
$851$	69.2945	2.37539
$852$	−2.40633	−0.0824395
$853$	−13.7936	−0.472284	−0.236142	−	0.971719i	$-0.575883\pi$
$853$	−13.7936	−0.472284	−0.236142	+	0.971719i	$0.575883\pi$
$854$	14.2818	0.488712
$855$	0	0
$856$	−6.14900	−0.210168
$857$	14.1595	0.483678	0.241839	−	0.970316i	$-0.422249\pi$
$857$	14.1595	0.483678	0.241839	+	0.970316i	$0.422249\pi$
$858$	0	0
$859$	27.6665	0.943968	0.471984	−	0.881607i	$-0.343538\pi$
$859$	27.6665	0.943968	0.471984	+	0.881607i	$0.343538\pi$
$860$	0	0
$861$	6.84011	0.233110
$862$	−5.88552	−0.200462
$863$	−17.4318	−0.593386	−0.296693	−	0.954973i	$-0.595884\pi$
$863$	−17.4318	−0.593386	−0.296693	+	0.954973i	$0.595884\pi$
$864$	3.70397	0.126011
$865$	0	0
$866$	−31.1096	−1.05715
$867$	−19.0206	−0.645975
$868$	−3.30409	−0.112148
$869$	−1.12033	−0.0380046
$870$	0	0
$871$	0	0
$872$	0.319202	0.0108096
$873$	−26.8861	−0.909956
$874$	−55.5484	−1.87895
$875$	0	0
$876$	6.20157	0.209532
$877$	−2.17528	−0.0734539	−0.0367270	−	0.999325i	$-0.511693\pi$
$877$	−2.17528	−0.0734539	−0.0367270	+	0.999325i	$0.511693\pi$
$878$	−2.69710	−0.0910229
$879$	−10.8248	−0.365111
$880$	0	0
$881$	−6.32823	−0.213204	−0.106602	−	0.994302i	$-0.533997\pi$
$881$	−6.32823	−0.213204	−0.106602	+	0.994302i	$0.533997\pi$
$882$	2.98901	0.100645
$883$	−28.5533	−0.960895	−0.480447	−	0.877024i	$-0.659526\pi$
$883$	−28.5533	−0.960895	−0.480447	+	0.877024i	$0.659526\pi$
$884$	0	0
$885$	0	0
$886$	19.0859	0.641203
$887$	21.3012	0.715225	0.357613	−	0.933870i	$-0.383591\pi$
$887$	21.3012	0.715225	0.357613	+	0.933870i	$0.383591\pi$
$888$	−6.16846	−0.207000
$889$	−23.5151	−0.788672
$890$	0	0
$891$	7.59362	0.254396
$892$	−9.84353	−0.329586
$893$	20.5778	0.688609
$894$	−5.93660	−0.198550
$895$	0	0
$896$	2.41461	0.0806665
$897$	0	0
$898$	−18.3579	−0.612611
$899$	0.953149	0.0317893
$900$	0	0
$901$	47.5525	1.58420
$902$	−6.20831	−0.206714
$903$	3.84073	0.127812
$904$	1.95370	0.0649793
$905$	0	0
$906$	11.4491	0.380370
$907$	−46.5451	−1.54551	−0.772753	−	0.634707i	$-0.781121\pi$
$907$	−46.5451	−1.54551	−0.772753	+	0.634707i	$0.781121\pi$
$908$	−19.4038	−0.643939
$909$	−36.4086	−1.20760
$910$	0	0
$911$	−29.3014	−0.970800	−0.485400	−	0.874292i	$-0.661326\pi$
$911$	−29.3014	−0.970800	−0.485400	+	0.874292i	$0.661326\pi$
$912$	4.94481	0.163739
$913$	−24.2263	−0.801775
$914$	25.3873	0.839739
$915$	0	0
$916$	−16.7503	−0.553447
$917$	25.6129	0.845813
$918$	−24.9925	−0.824874
$919$	29.4843	0.972598	0.486299	−	0.873792i	$-0.338346\pi$
$919$	29.4843	0.972598	0.486299	+	0.873792i	$0.338346\pi$
$920$	0	0
$921$	−3.77758	−0.124476
$922$	−1.12031	−0.0368956
$923$	0	0
$924$	−2.35231	−0.0773855
$925$	0	0
$926$	−30.6952	−1.00871
$927$	24.8498	0.816173
$928$	−0.696556	−0.0228656
$929$	47.7459	1.56649	0.783246	−	0.621712i	$-0.213562\pi$
$929$	47.7459	1.56649	0.783246	+	0.621712i	$0.213562\pi$
$930$	0	0
$931$	8.67479	0.284305
$932$	−16.7974	−0.550217
$933$	−13.6988	−0.448479
$934$	5.88670	0.192619
$935$	0	0
$936$	0	0
$937$	−29.2998	−0.957182	−0.478591	−	0.878038i	$-0.658852\pi$
$937$	−29.2998	−0.957182	−0.478591	+	0.878038i	$0.658852\pi$
$938$	−26.5505	−0.866905
$939$	−7.35753	−0.240104
$940$	0	0
$941$	−7.48425	−0.243980	−0.121990	−	0.992531i	$-0.538928\pi$
$941$	−7.48425	−0.243980	−0.121990	+	0.992531i	$0.538928\pi$
$942$	7.37720	0.240362
$943$	−31.8228	−1.03629
$944$	0.0690364	0.00224694
$945$	0	0
$946$	−3.48597	−0.113339
$947$	15.9785	0.519230	0.259615	−	0.965712i	$-0.416404\pi$
$947$	15.9785	0.519230	0.259615	+	0.965712i	$0.416404\pi$
$948$	0.511198	0.0166029
$949$	0	0
$950$	0	0
$951$	11.3780	0.368958
$952$	−16.2926	−0.528045
$953$	44.7821	1.45063	0.725317	−	0.688415i	$-0.241693\pi$
$953$	44.7821	1.45063	0.725317	+	0.688415i	$0.241693\pi$
$954$	18.0096	0.583082
$955$	0	0
$956$	18.2819	0.591278
$957$	0.678585	0.0219355
$958$	24.6962	0.797899
$959$	15.4677	0.499477
$960$	0	0
$961$	−29.1276	−0.939599
$962$	0	0
$963$	15.7137	0.506365
$964$	−7.86172	−0.253209
$965$	0	0
$966$	−12.0576	−0.387946
$967$	5.51993	0.177509	0.0887546	−	0.996054i	$-0.471711\pi$
$967$	5.51993	0.177509	0.0887546	+	0.996054i	$0.471711\pi$
$968$	−8.86496	−0.284931
$969$	−33.3650	−1.07184
$970$	0	0
$971$	−38.6903	−1.24163	−0.620815	−	0.783957i	$-0.713198\pi$
$971$	−38.6903	−1.24163	−0.620815	+	0.783957i	$0.713198\pi$
$972$	−14.5768	−0.467551
$973$	−40.5609	−1.30032
$974$	−6.85817	−0.219750
$975$	0	0
$976$	5.91472	0.189326
$977$	−22.0275	−0.704722	−0.352361	−	0.935864i	$-0.614621\pi$
$977$	−22.0275	−0.704722	−0.352361	+	0.935864i	$0.614621\pi$
$978$	−10.1633	−0.324987
$979$	−25.9331	−0.828826
$980$	0	0
$981$	−0.815716	−0.0260438
$982$	−32.0421	−1.02251
$983$	49.1575	1.56788	0.783941	−	0.620835i	$-0.213206\pi$
$983$	49.1575	1.56788	0.783941	+	0.620835i	$0.213206\pi$
$984$	2.83280	0.0903064
$985$	0	0
$986$	4.70000	0.149679
$987$	4.46670	0.142177
$988$	0	0
$989$	−17.8685	−0.568186
$990$	0	0
$991$	−7.70733	−0.244831	−0.122416	−	0.992479i	$-0.539064\pi$
$991$	−7.70733	−0.244831	−0.122416	+	0.992479i	$0.539064\pi$
$992$	−1.36837	−0.0434459
$993$	−19.8001	−0.628338
$994$	8.71480	0.276417
$995$	0	0
$996$	11.0543	0.350268
$997$	23.9968	0.759986	0.379993	−	0.924989i	$-0.375926\pi$
$997$	23.9968	0.759986	0.379993	+	0.924989i	$0.375926\pi$
$998$	25.9861	0.822575
$999$	34.2688	1.08422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cx.1.5		9
5.2	odd	4		1690.2.b.f.339.14	yes	18
5.3	odd	4		1690.2.b.f.339.5	✓	18
5.4	even	2		8450.2.a.cw.1.5		9
13.12	even	2		8450.2.a.ct.1.5		9
65.8	even	4		1690.2.c.h.1689.7		18
65.12	odd	4		1690.2.b.g.339.5	yes	18
65.18	even	4		1690.2.c.g.1689.7		18
65.38	odd	4		1690.2.b.g.339.14	yes	18
65.47	even	4		1690.2.c.g.1689.12		18
65.57	even	4		1690.2.c.h.1689.12		18
65.64	even	2		8450.2.a.da.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.b.f.339.5	✓	18	5.3	odd	4
1690.2.b.f.339.14	yes	18	5.2	odd	4
1690.2.b.g.339.5	yes	18	65.12	odd	4
1690.2.b.g.339.14	yes	18	65.38	odd	4
1690.2.c.g.1689.7		18	65.18	even	4
1690.2.c.g.1689.12		18	65.47	even	4
1690.2.c.h.1689.7		18	65.8	even	4
1690.2.c.h.1689.12		18	65.57	even	4
8450.2.a.ct.1.5		9	13.12	even	2
8450.2.a.cw.1.5		9	5.4	even	2
8450.2.a.cx.1.5		9	1.1	even	1	trivial
8450.2.a.da.1.5		9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.666723 q^{3} +1.00000 q^{4} -0.666723 q^{6} +2.41461 q^{7} +1.00000 q^{8} -2.55548 q^{9} +1.46118 q^{11} -0.666723 q^{12} +2.41461 q^{14} +1.00000 q^{16} -6.74749 q^{17} -2.55548 q^{18} -7.41659 q^{19} -1.60988 q^{21} +1.46118 q^{22} +7.48975 q^{23} -0.666723 q^{24} +3.70397 q^{27} +2.41461 q^{28} -0.696556 q^{29} -1.36837 q^{31} +1.00000 q^{32} -0.974200 q^{33} -6.74749 q^{34} -2.55548 q^{36} +9.25191 q^{37} -7.41659 q^{38} -4.24884 q^{41} -1.60988 q^{42} -2.38573 q^{43} +1.46118 q^{44} +7.48975 q^{46} -2.77456 q^{47} -0.666723 q^{48} -1.16965 q^{49} +4.49870 q^{51} -7.04744 q^{53} +3.70397 q^{54} +2.41461 q^{56} +4.94481 q^{57} -0.696556 q^{58} +0.0690364 q^{59} +5.91472 q^{61} -1.36837 q^{62} -6.17050 q^{63} +1.00000 q^{64} -0.974200 q^{66} -10.9958 q^{67} -6.74749 q^{68} -4.99359 q^{69} +3.60919 q^{71} -2.55548 q^{72} -9.30157 q^{73} +9.25191 q^{74} -7.41659 q^{76} +3.52818 q^{77} -0.766732 q^{79} +5.19692 q^{81} -4.24884 q^{82} -16.5800 q^{83} -1.60988 q^{84} -2.38573 q^{86} +0.464410 q^{87} +1.46118 q^{88} -17.7481 q^{89} +7.48975 q^{92} +0.912326 q^{93} -2.77456 q^{94} -0.666723 q^{96} +10.5209 q^{97} -1.16965 q^{98} -3.73401 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 7 q^{3} + 9 q^{4} - 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} - 4 q^{11} - 7 q^{12} + q^{14} + 9 q^{16} - 12 q^{17} + 8 q^{18} - 6 q^{19} - 8 q^{21} - 4 q^{22} - 11 q^{23} - 7 q^{24} - 34 q^{27}+ \cdots - 16 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 7 * q^3 + 9 * q^4 - 7 * q^6 + q^7 + 9 * q^8 + 8 * q^9 - 4 * q^11 - 7 * q^12 + q^14 + 9 * q^16 - 12 * q^17 + 8 * q^18 - 6 * q^19 - 8 * q^21 - 4 * q^22 - 11 * q^23 - 7 * q^24 - 34 * q^27 + q^28 + 15 * q^29 + 6 * q^31 + 9 * q^32 - 14 * q^33 - 12 * q^34 + 8 * q^36 - 10 * q^37 - 6 * q^38 - 27 * q^41 - 8 * q^42 - 41 * q^43 - 4 * q^44 - 11 * q^46 + 5 * q^47 - 7 * q^48 - 10 * q^49 + 12 * q^51 - 18 * q^53 - 34 * q^54 + q^56 - 12 * q^57 + 15 * q^58 - 20 * q^59 - 19 * q^61 + 6 * q^62 + 39 * q^63 + 9 * q^64 - 14 * q^66 - q^67 - 12 * q^68 - 14 * q^69 + 10 * q^71 + 8 * q^72 - 12 * q^73 - 10 * q^74 - 6 * q^76 - 30 * q^77 - 14 * q^79 + 53 * q^81 - 27 * q^82 + q^83 - 8 * q^84 - 41 * q^86 - 33 * q^87 - 4 * q^88 - 43 * q^89 - 11 * q^92 - 28 * q^93 + 5 * q^94 - 7 * q^96 + 6 * q^97 - 10 * q^98 - 16 * q^99

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