LMFDB - Embedded newform 6223.2.a.k.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6223, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6223.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [16,-2,4,12,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.6909051778$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.102239$ of defining polynomial
Character	$\chi$	$=$	6223.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.102239 q^{2} +3.12743 q^{3} -1.98955 q^{4} -0.280585 q^{5} -0.319747 q^{6} +0.407889 q^{8} +6.78082 q^{9} +0.0286868 q^{10} +0.974733 q^{11} -6.22217 q^{12} +1.41349 q^{13} -0.877509 q^{15} +3.93739 q^{16} -6.28548 q^{17} -0.693267 q^{18} -1.00698 q^{19} +0.558236 q^{20} -0.0996561 q^{22} +1.39508 q^{23} +1.27564 q^{24} -4.92127 q^{25} -0.144515 q^{26} +11.8242 q^{27} -2.24230 q^{29} +0.0897160 q^{30} +9.08093 q^{31} -1.21833 q^{32} +3.04841 q^{33} +0.642624 q^{34} -13.4908 q^{36} +8.73158 q^{37} +0.102954 q^{38} +4.42059 q^{39} -0.114447 q^{40} +3.37544 q^{41} -4.53376 q^{43} -1.93928 q^{44} -1.90259 q^{45} -0.142632 q^{46} +3.98353 q^{47} +12.3139 q^{48} +0.503148 q^{50} -19.6574 q^{51} -2.81221 q^{52} +12.4108 q^{53} -1.20890 q^{54} -0.273495 q^{55} -3.14927 q^{57} +0.229251 q^{58} +2.70787 q^{59} +1.74584 q^{60} +6.02128 q^{61} -0.928429 q^{62} -7.75022 q^{64} -0.396604 q^{65} -0.311668 q^{66} -1.29868 q^{67} +12.5053 q^{68} +4.36302 q^{69} +1.77583 q^{71} +2.76582 q^{72} +14.2675 q^{73} -0.892711 q^{74} -15.3909 q^{75} +2.00344 q^{76} -0.451959 q^{78} -9.13520 q^{79} -1.10477 q^{80} +16.6370 q^{81} -0.345103 q^{82} -2.94615 q^{83} +1.76361 q^{85} +0.463529 q^{86} -7.01263 q^{87} +0.397583 q^{88} +8.92572 q^{89} +0.194520 q^{90} -2.77558 q^{92} +28.4000 q^{93} -0.407274 q^{94} +0.282544 q^{95} -3.81026 q^{96} +1.25797 q^{97} +6.60948 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) $ 16 * q - 2 * q^2 + 4 * q^3 + 12 * q^4 + 9 * q^5 + 12 * q^6 - 6 * q^8 + 14 * q^9 + 2 * q^10 - 22 * q^11 + 10 * q^12 + 4 * q^13 - 14 * q^15 + 12 * q^16 + 18 * q^17 - 5 * q^18 + 15 * q^19 + 40 * q^20 - 11 * q^22 - 5 * q^23 + 26 * q^24 + 15 * q^25 + 24 * q^26 + 10 * q^27 - 12 * q^29 + 3 * q^30 + 32 * q^31 - 9 * q^32 + 10 * q^33 + 14 * q^34 + 10 * q^36 - 2 * q^37 - 3 * q^38 - 41 * q^39 + 14 * q^40 + 45 * q^41 - 3 * q^43 - 54 * q^44 + 22 * q^45 + 49 * q^47 - 10 * q^48 - 6 * q^50 - 12 * q^51 - 38 * q^52 + 16 * q^53 + 67 * q^54 - 7 * q^55 + 8 * q^57 + 16 * q^58 + 35 * q^59 + 56 * q^60 + 11 * q^61 - 17 * q^62 - 2 * q^64 + 14 * q^65 - 86 * q^66 + 17 * q^67 + 71 * q^68 + 17 * q^69 - 81 * q^71 + 13 * q^72 + 15 * q^73 + 13 * q^74 + 48 * q^75 - 14 * q^76 - 10 * q^78 - 34 * q^79 + 33 * q^80 + 40 * q^81 + 14 * q^82 + 39 * q^83 - 17 * q^85 + 36 * q^86 - 7 * q^87 + 61 * q^88 + 32 * q^89 - 79 * q^90 + 37 * q^92 + 27 * q^93 - 13 * q^94 - 33 * q^95 + 55 * q^96 + 4 * q^97 - 73 * 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 $

$2$	−0.102239	−0.0722942	−0.0361471	−	0.999346i	$-0.511508\pi$
$2$	−0.102239	−0.0722942	−0.0361471	+	0.999346i	$0.511508\pi$
$3$	3.12743	1.80562	0.902811	−	0.430037i	$-0.141500\pi$
$3$	3.12743	1.80562	0.902811	+	0.430037i	$0.141500\pi$
$4$	−1.98955	−0.994774
$5$	−0.280585	−0.125481	−0.0627406	−	0.998030i	$-0.519984\pi$
$5$	−0.280585	−0.125481	−0.0627406	+	0.998030i	$0.519984\pi$
$6$	−0.319747	−0.130536
$7$	0	0
$7$	0	0
$8$	0.407889	0.144211
$9$	6.78082	2.26027
$10$	0.0286868	0.00907157
$11$	0.974733	0.293893	0.146947	−	0.989144i	$-0.453055\pi$
$11$	0.974733	0.293893	0.146947	+	0.989144i	$0.453055\pi$
$12$	−6.22217	−1.79619
$13$	1.41349	0.392032	0.196016	−	0.980601i	$-0.437200\pi$
$13$	1.41349	0.392032	0.196016	+	0.980601i	$0.437200\pi$
$14$	0	0
$15$	−0.877509	−0.226572
$16$	3.93739	0.984348
$17$	−6.28548	−1.52445	−0.762227	−	0.647310i	$-0.775894\pi$
$17$	−6.28548	−1.52445	−0.762227	+	0.647310i	$0.775894\pi$
$18$	−0.693267	−0.163405
$19$	−1.00698	−0.231018	−0.115509	−	0.993306i	$-0.536850\pi$
$19$	−1.00698	−0.231018	−0.115509	+	0.993306i	$0.536850\pi$
$20$	0.558236	0.124825
$21$	0	0
$22$	−0.0996561	−0.0212468
$23$	1.39508	0.290895	0.145447	−	0.989366i	$-0.453538\pi$
$23$	1.39508	0.290895	0.145447	+	0.989366i	$0.453538\pi$
$24$	1.27564	0.260390
$25$	−4.92127	−0.984254
$26$	−0.144515	−0.0283416
$27$	11.8242	2.27557
$28$	0	0
$29$	−2.24230	−0.416384	−0.208192	−	0.978088i	$-0.566758\pi$
$29$	−2.24230	−0.416384	−0.208192	+	0.978088i	$0.566758\pi$
$30$	0.0897160	0.0163798
$31$	9.08093	1.63098	0.815492	−	0.578769i	$-0.196467\pi$
$31$	9.08093	1.63098	0.815492	+	0.578769i	$0.196467\pi$
$32$	−1.21833	−0.215373
$33$	3.04841	0.530660
$34$	0.642624	0.110209
$35$	0	0
$36$	−13.4908	−2.24846
$37$	8.73158	1.43546	0.717731	−	0.696320i	$-0.245181\pi$
$37$	8.73158	1.43546	0.717731	+	0.696320i	$0.245181\pi$
$38$	0.102954	0.0167013
$39$	4.42059	0.707862
$40$	−0.114447	−0.0180957
$41$	3.37544	0.527155	0.263577	−	0.964638i	$-0.415098\pi$
$41$	3.37544	0.527155	0.263577	+	0.964638i	$0.415098\pi$
$42$	0	0
$43$	−4.53376	−0.691392	−0.345696	−	0.938346i	$-0.612357\pi$
$43$	−4.53376	−0.691392	−0.345696	+	0.938346i	$0.612357\pi$
$44$	−1.93928	−0.292357
$45$	−1.90259	−0.283622
$46$	−0.142632	−0.0210300
$47$	3.98353	0.581057	0.290529	−	0.956866i	$-0.406169\pi$
$47$	3.98353	0.581057	0.290529	+	0.956866i	$0.406169\pi$
$48$	12.3139	1.77736
$49$	0	0
$50$	0.503148	0.0711559
$51$	−19.6574	−2.75259
$52$	−2.81221	−0.389983
$53$	12.4108	1.70475	0.852375	−	0.522930i	$-0.175161\pi$
$53$	12.4108	1.70475	0.852375	+	0.522930i	$0.175161\pi$
$54$	−1.20890	−0.164511
$55$	−0.273495	−0.0368781
$56$	0	0
$57$	−3.14927	−0.417132
$58$	0.229251	0.0301021
$59$	2.70787	0.352534	0.176267	−	0.984342i	$-0.443598\pi$
$59$	2.70787	0.352534	0.176267	+	0.984342i	$0.443598\pi$
$60$	1.74584	0.225388
$61$	6.02128	0.770946	0.385473	−	0.922719i	$-0.374038\pi$
$61$	6.02128	0.770946	0.385473	+	0.922719i	$0.374038\pi$
$62$	−0.928429	−0.117911
$63$	0	0
$64$	−7.75022	−0.968778
$65$	−0.396604	−0.0491927
$66$	−0.311668	−0.0383636
$67$	−1.29868	−0.158658	−0.0793292	−	0.996848i	$-0.525278\pi$
$67$	−1.29868	−0.158658	−0.0793292	+	0.996848i	$0.525278\pi$
$68$	12.5053	1.51649
$69$	4.36302	0.525246
$70$	0	0
$71$	1.77583	0.210752	0.105376	−	0.994432i	$-0.466395\pi$
$71$	1.77583	0.210752	0.105376	+	0.994432i	$0.466395\pi$
$72$	2.76582	0.325955
$73$	14.2675	1.66989	0.834943	−	0.550336i	$-0.185500\pi$
$73$	14.2675	1.66989	0.834943	+	0.550336i	$0.185500\pi$
$74$	−0.892711	−0.103776
$75$	−15.3909	−1.77719
$76$	2.00344	0.229811
$77$	0	0
$78$	−0.451959	−0.0511743
$79$	−9.13520	−1.02779	−0.513895	−	0.857853i	$-0.671798\pi$
$79$	−9.13520	−1.02779	−0.513895	+	0.857853i	$0.671798\pi$
$80$	−1.10477	−0.123517
$81$	16.6370	1.84856
$82$	−0.345103	−0.0381102
$83$	−2.94615	−0.323382	−0.161691	−	0.986841i	$-0.551695\pi$
$83$	−2.94615	−0.323382	−0.161691	+	0.986841i	$0.551695\pi$
$84$	0	0
$85$	1.76361	0.191290
$86$	0.463529	0.0499837
$87$	−7.01263	−0.751832
$88$	0.397583	0.0423825
$89$	8.92572	0.946124	0.473062	−	0.881029i	$-0.343149\pi$
$89$	8.92572	0.946124	0.473062	+	0.881029i	$0.343149\pi$
$90$	0.194520	0.0205042
$91$	0	0
$92$	−2.77558	−0.289374
$93$	28.4000	2.94494
$94$	−0.407274	−0.0420071
$95$	0.282544	0.0289884
$96$	−3.81026	−0.388883
$97$	1.25797	0.127727	0.0638636	−	0.997959i	$-0.479658\pi$
$97$	1.25797	0.127727	0.0638636	+	0.997959i	$0.479658\pi$
$98$	0	0
$99$	6.60948	0.664278
$100$	9.79110	0.979110
$101$	9.32514	0.927886	0.463943	−	0.885865i	$-0.346434\pi$
$101$	9.32514	0.927886	0.463943	+	0.885865i	$0.346434\pi$
$102$	2.00976	0.198996
$103$	−11.5869	−1.14169	−0.570846	−	0.821057i	$-0.693385\pi$
$103$	−11.5869	−1.14169	−0.570846	+	0.821057i	$0.693385\pi$
$104$	0.576548	0.0565351
$105$	0	0
$106$	−1.26887	−0.123244
$107$	−0.203921	−0.0197138	−0.00985689	−	0.999951i	$-0.503138\pi$
$107$	−0.203921	−0.0197138	−0.00985689	+	0.999951i	$0.503138\pi$
$108$	−23.5249	−2.26368
$109$	17.5817	1.68402	0.842012	−	0.539459i	$-0.181371\pi$
$109$	17.5817	1.68402	0.842012	+	0.539459i	$0.181371\pi$
$110$	0.0279620	0.00266607
$111$	27.3074	2.59190
$112$	0	0
$113$	−16.0844	−1.51309	−0.756546	−	0.653941i	$-0.773114\pi$
$113$	−16.0844	−1.51309	−0.756546	+	0.653941i	$0.773114\pi$
$114$	0.321980	0.0301562
$115$	−0.391438	−0.0365018
$116$	4.46116	0.414208
$117$	9.58462	0.886099
$118$	−0.276851	−0.0254862
$119$	0	0
$120$	−0.357926	−0.0326740
$121$	−10.0499	−0.913627
$122$	−0.615612	−0.0557349
$123$	10.5564	0.951842
$124$	−18.0669	−1.62246
$125$	2.78376	0.248987
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	3.22905	0.285410
$129$	−14.1790	−1.24839
$130$	0.0405486	0.00355634
$131$	16.7647	1.46474	0.732370	−	0.680907i	$-0.238414\pi$
$131$	16.7647	1.46474	0.732370	+	0.680907i	$0.238414\pi$
$132$	−6.06495	−0.527886
$133$	0	0
$134$	0.132776	0.0114701
$135$	−3.31770	−0.285542
$136$	−2.56378	−0.219842
$137$	16.3690	1.39849	0.699247	−	0.714880i	$-0.253519\pi$
$137$	16.3690	1.39849	0.699247	+	0.714880i	$0.253519\pi$
$138$	−0.446072	−0.0379722
$139$	−1.38049	−0.117092	−0.0585459	−	0.998285i	$-0.518646\pi$
$139$	−1.38049	−0.117092	−0.0585459	+	0.998285i	$0.518646\pi$
$140$	0	0
$141$	12.4582	1.04917
$142$	−0.181560	−0.0152362
$143$	1.37778	0.115215
$144$	26.6987	2.22489
$145$	0.629154	0.0522484
$146$	−1.45870	−0.120723
$147$	0	0
$148$	−17.3719	−1.42796
$149$	16.3718	1.34123	0.670616	−	0.741804i	$-0.266030\pi$
$149$	16.3718	1.34123	0.670616	+	0.741804i	$0.266030\pi$
$150$	1.57356	0.128481
$151$	−22.0724	−1.79623	−0.898113	−	0.439765i	$-0.855062\pi$
$151$	−22.0724	−1.79623	−0.898113	+	0.439765i	$0.855062\pi$
$152$	−0.410738	−0.0333153
$153$	−42.6207	−3.44568
$154$	0	0
$155$	−2.54797	−0.204658
$156$	−8.79498	−0.704162
$157$	−5.31714	−0.424354	−0.212177	−	0.977231i	$-0.568055\pi$
$157$	−5.31714	−0.424354	−0.212177	+	0.977231i	$0.568055\pi$
$158$	0.933978	0.0743033
$159$	38.8138	3.07814
$160$	0.341846	0.0270253
$161$	0	0
$162$	−1.70096	−0.133640
$163$	14.8072	1.15979	0.579896	−	0.814690i	$-0.303093\pi$
$163$	14.8072	1.15979	0.579896	+	0.814690i	$0.303093\pi$
$164$	−6.71559	−0.524399
$165$	−0.855337	−0.0665879
$166$	0.301213	0.0233787
$167$	−6.35017	−0.491391	−0.245696	−	0.969347i	$-0.579016\pi$
$167$	−6.35017	−0.491391	−0.245696	+	0.969347i	$0.579016\pi$
$168$	0	0
$169$	−11.0020	−0.846311
$170$	−0.180310	−0.0138292
$171$	−6.82818	−0.522164
$172$	9.02014	0.687779
$173$	−2.22737	−0.169344	−0.0846719	−	0.996409i	$-0.526984\pi$
$173$	−2.22737	−0.169344	−0.0846719	+	0.996409i	$0.526984\pi$
$174$	0.716967	0.0543531
$175$	0	0
$176$	3.83791	0.289293
$177$	8.46866	0.636543
$178$	−0.912560	−0.0683993
$179$	3.65043	0.272846	0.136423	−	0.990651i	$-0.456439\pi$
$179$	3.65043	0.272846	0.136423	+	0.990651i	$0.456439\pi$
$180$	3.78530	0.282139
$181$	−8.51775	−0.633119	−0.316560	−	0.948573i	$-0.602528\pi$
$181$	−8.51775	−0.633119	−0.316560	+	0.948573i	$0.602528\pi$
$182$	0	0
$183$	18.8311	1.39204
$184$	0.569038	0.0419501
$185$	−2.44995	−0.180124
$186$	−2.90360	−0.212902
$187$	−6.12667	−0.448026
$188$	−7.92542	−0.578021
$189$	0	0
$190$	−0.0288872	−0.00209570
$191$	−17.7340	−1.28319	−0.641593	−	0.767045i	$-0.721726\pi$
$191$	−17.7340	−1.28319	−0.641593	+	0.767045i	$0.721726\pi$
$192$	−24.2383	−1.74925
$193$	−2.13995	−0.154037	−0.0770186	−	0.997030i	$-0.524540\pi$
$193$	−2.13995	−0.154037	−0.0770186	+	0.997030i	$0.524540\pi$
$194$	−0.128614	−0.00923394
$195$	−1.24035	−0.0888234
$196$	0	0
$197$	−17.0094	−1.21187	−0.605936	−	0.795513i	$-0.707201\pi$
$197$	−17.0094	−1.21187	−0.605936	+	0.795513i	$0.707201\pi$
$198$	−0.675750	−0.0480235
$199$	7.03395	0.498624	0.249312	−	0.968423i	$-0.419796\pi$
$199$	7.03395	0.498624	0.249312	+	0.968423i	$0.419796\pi$
$200$	−2.00733	−0.141940
$201$	−4.06152	−0.286477
$202$	−0.953397	−0.0670808
$203$	0	0
$204$	39.1093	2.73820
$205$	−0.947095	−0.0661480
$206$	1.18464	0.0825378
$207$	9.45979	0.657501
$208$	5.56547	0.385896
$209$	−0.981541	−0.0678946
$210$	0	0
$211$	2.25672	0.155359	0.0776797	−	0.996978i	$-0.475249\pi$
$211$	2.25672	0.155359	0.0776797	+	0.996978i	$0.475249\pi$
$212$	−24.6918	−1.69584
$213$	5.55378	0.380539
$214$	0.0208488	0.00142519
$215$	1.27210	0.0867568
$216$	4.82298	0.328162
$217$	0	0
$218$	−1.79754	−0.121745
$219$	44.6206	3.01518
$220$	0.544131	0.0366853
$221$	−8.88447	−0.597634
$222$	−2.79189	−0.187379
$223$	16.8223	1.12650	0.563252	−	0.826285i	$-0.309550\pi$
$223$	16.8223	1.12650	0.563252	+	0.826285i	$0.309550\pi$
$224$	0	0
$225$	−33.3702	−2.22468
$226$	1.64446	0.109388
$227$	26.9785	1.79063	0.895314	−	0.445435i	$-0.146951\pi$
$227$	26.9785	1.79063	0.895314	+	0.445435i	$0.146951\pi$
$228$	6.26563	0.414951
$229$	11.5853	0.765576	0.382788	−	0.923836i	$-0.374964\pi$
$229$	11.5853	0.765576	0.382788	+	0.923836i	$0.374964\pi$
$230$	0.0400204	0.00263887
$231$	0	0
$232$	−0.914608	−0.0600470
$233$	−10.3197	−0.676064	−0.338032	−	0.941135i	$-0.609761\pi$
$233$	−10.3197	−0.676064	−0.338032	+	0.941135i	$0.609761\pi$
$234$	−0.979926	−0.0640598
$235$	−1.11772	−0.0729118
$236$	−5.38743	−0.350692
$237$	−28.5697	−1.85580
$238$	0	0
$239$	−15.1850	−0.982233	−0.491116	−	0.871094i	$-0.663411\pi$
$239$	−15.1850	−0.982233	−0.491116	+	0.871094i	$0.663411\pi$
$240$	−3.45510	−0.223025
$241$	−5.19032	−0.334338	−0.167169	−	0.985928i	$-0.553463\pi$
$241$	−5.19032	−0.334338	−0.167169	+	0.985928i	$0.553463\pi$
$242$	1.02750	0.0660499
$243$	16.5584	1.06222
$244$	−11.9796	−0.766917
$245$	0	0
$246$	−1.07928	−0.0688126
$247$	−1.42336	−0.0905665
$248$	3.70401	0.235205
$249$	−9.21389	−0.583907
$250$	−0.284610	−0.0180003
$251$	21.6309	1.36533	0.682664	−	0.730732i	$-0.260821\pi$
$251$	21.6309	1.36533	0.682664	+	0.730732i	$0.260821\pi$
$252$	0	0
$253$	1.35983	0.0854919
$254$	0.102239	0.00641507
$255$	5.51557	0.345398
$256$	15.1703	0.948144
$257$	−10.2922	−0.642010	−0.321005	−	0.947077i	$-0.604021\pi$
$257$	−10.2922	−0.642010	−0.321005	+	0.947077i	$0.604021\pi$
$258$	1.44966	0.0902516
$259$	0	0
$260$	0.789062	0.0489356
$261$	−15.2046	−0.941141
$262$	−1.71401	−0.105892
$263$	−14.2342	−0.877719	−0.438859	−	0.898556i	$-0.644617\pi$
$263$	−14.2342	−0.877719	−0.438859	+	0.898556i	$0.644617\pi$
$264$	1.24341	0.0765267
$265$	−3.48227	−0.213914
$266$	0	0
$267$	27.9145	1.70834
$268$	2.58378	0.157829
$269$	24.6333	1.50192	0.750958	−	0.660350i	$-0.229592\pi$
$269$	24.6333	1.50192	0.750958	+	0.660350i	$0.229592\pi$
$270$	0.339200	0.0206430
$271$	4.57306	0.277794	0.138897	−	0.990307i	$-0.455644\pi$
$271$	4.57306	0.277794	0.138897	+	0.990307i	$0.455644\pi$
$272$	−24.7484	−1.50059
$273$	0	0
$274$	−1.67355	−0.101103
$275$	−4.79693	−0.289266
$276$	−8.68043	−0.522500
$277$	−24.5754	−1.47659	−0.738296	−	0.674477i	$-0.764369\pi$
$277$	−24.5754	−1.47659	−0.738296	+	0.674477i	$0.764369\pi$
$278$	0.141141	0.00846505
$279$	61.5761	3.68647
$280$	0	0
$281$	13.4450	0.802061	0.401030	−	0.916065i	$-0.368652\pi$
$281$	13.4450	0.802061	0.401030	+	0.916065i	$0.368652\pi$
$282$	−1.27372	−0.0758489
$283$	25.9931	1.54513	0.772564	−	0.634937i	$-0.218974\pi$
$283$	25.9931	1.54513	0.772564	+	0.634937i	$0.218974\pi$
$284$	−3.53310	−0.209651
$285$	0.883638	0.0523422
$286$	−0.140863	−0.00832941
$287$	0	0
$288$	−8.26130	−0.486802
$289$	22.5073	1.32396
$290$	−0.0643243	−0.00377726
$291$	3.93420	0.230627
$292$	−28.3859	−1.66116
$293$	15.0055	0.876628	0.438314	−	0.898822i	$-0.355576\pi$
$293$	15.0055	0.876628	0.438314	+	0.898822i	$0.355576\pi$
$294$	0	0
$295$	−0.759786	−0.0442364
$296$	3.56151	0.207009
$297$	11.5255	0.668776
$298$	−1.67385	−0.0969634
$299$	1.97193	0.114040
$300$	30.6210	1.76790
$301$	0	0
$302$	2.25667	0.129857
$303$	29.1637	1.67541
$304$	−3.96489	−0.227402
$305$	−1.68948	−0.0967393
$306$	4.35752	0.249103
$307$	−15.2999	−0.873209	−0.436605	−	0.899653i	$-0.643819\pi$
$307$	−15.2999	−0.873209	−0.436605	+	0.899653i	$0.643819\pi$
$308$	0	0
$309$	−36.2373	−2.06147
$310$	0.260503	0.0147956
$311$	7.23014	0.409984	0.204992	−	0.978764i	$-0.434283\pi$
$311$	7.23014	0.409984	0.204992	+	0.978764i	$0.434283\pi$
$312$	1.80311	0.102081
$313$	−1.04458	−0.0590433	−0.0295217	−	0.999564i	$-0.509398\pi$
$313$	−1.04458	−0.0590433	−0.0295217	+	0.999564i	$0.509398\pi$
$314$	0.543622	0.0306784
$315$	0	0
$316$	18.1749	1.02242
$317$	4.47560	0.251375	0.125687	−	0.992070i	$-0.459886\pi$
$317$	4.47560	0.251375	0.125687	+	0.992070i	$0.459886\pi$
$318$	−3.96830	−0.222531
$319$	−2.18564	−0.122372
$320$	2.17459	0.121563
$321$	−0.637748	−0.0355956
$322$	0	0
$323$	6.32938	0.352176
$324$	−33.1001	−1.83890
$325$	−6.95617	−0.385859
$326$	−1.51388	−0.0838463
$327$	54.9856	3.04071
$328$	1.37680	0.0760212
$329$	0	0
$330$	0.0874491	0.00481392
$331$	17.2234	0.946683	0.473342	−	0.880879i	$-0.343048\pi$
$331$	17.2234	0.946683	0.473342	+	0.880879i	$0.343048\pi$
$332$	5.86151	0.321692
$333$	59.2072	3.24453
$334$	0.649238	0.0355247
$335$	0.364388	0.0199087
$336$	0	0
$337$	6.92934	0.377465	0.188733	−	0.982029i	$-0.439562\pi$
$337$	6.92934	0.377465	0.188733	+	0.982029i	$0.439562\pi$
$338$	1.12484	0.0611834
$339$	−50.3028	−2.73207
$340$	−3.50878	−0.190291
$341$	8.85148	0.479335
$342$	0.698109	0.0377494
$343$	0	0
$344$	−1.84927	−0.0997061
$345$	−1.22420	−0.0659085
$346$	0.227725	0.0122426
$347$	−10.8206	−0.580881	−0.290441	−	0.956893i	$-0.593802\pi$
$347$	−10.8206	−0.580881	−0.290441	+	0.956893i	$0.593802\pi$
$348$	13.9519	0.747903
$349$	−17.2151	−0.921502	−0.460751	−	0.887529i	$-0.652420\pi$
$349$	−17.2151	−0.921502	−0.460751	+	0.887529i	$0.652420\pi$
$350$	0	0
$351$	16.7135	0.892098
$352$	−1.18755	−0.0632967
$353$	35.8111	1.90603	0.953016	−	0.302921i	$-0.0979619\pi$
$353$	35.8111	1.90603	0.953016	+	0.302921i	$0.0979619\pi$
$354$	−0.865831	−0.0460184
$355$	−0.498270	−0.0264454
$356$	−17.7581	−0.941179
$357$	0	0
$358$	−0.373218	−0.0197252
$359$	12.5791	0.663899	0.331950	−	0.943297i	$-0.392294\pi$
$359$	12.5791	0.663899	0.331950	+	0.943297i	$0.392294\pi$
$360$	−0.776047	−0.0409012
$361$	−17.9860	−0.946631
$362$	0.870850	0.0457709
$363$	−31.4303	−1.64967
$364$	0	0
$365$	−4.00325	−0.209539
$366$	−1.92528	−0.100636
$367$	3.95956	0.206688	0.103344	−	0.994646i	$-0.467046\pi$
$367$	3.95956	0.206688	0.103344	+	0.994646i	$0.467046\pi$
$368$	5.49298	0.286341
$369$	22.8882	1.19151
$370$	0.250481	0.0130219
$371$	0	0
$372$	−56.5031	−2.92955
$373$	30.1501	1.56112	0.780558	−	0.625084i	$-0.214935\pi$
$373$	30.1501	1.56112	0.780558	+	0.625084i	$0.214935\pi$
$374$	0.626387	0.0323897
$375$	8.70600	0.449576
$376$	1.62484	0.0837946
$377$	−3.16947	−0.163236
$378$	0	0
$379$	22.0540	1.13284	0.566419	−	0.824117i	$-0.308328\pi$
$379$	22.0540	1.13284	0.566419	+	0.824117i	$0.308328\pi$
$380$	−0.562135	−0.0288369
$381$	−3.12743	−0.160223
$382$	1.81311	0.0927669
$383$	−9.13278	−0.466663	−0.233332	−	0.972397i	$-0.574963\pi$
$383$	−9.13278	−0.466663	−0.233332	+	0.972397i	$0.574963\pi$
$384$	10.0986	0.515343
$385$	0	0
$386$	0.218788	0.0111360
$387$	−30.7426	−1.56273
$388$	−2.50279	−0.127060
$389$	−18.5278	−0.939397	−0.469699	−	0.882827i	$-0.655637\pi$
$389$	−18.5278	−0.939397	−0.469699	+	0.882827i	$0.655637\pi$
$390$	0.126813	0.00642141
$391$	−8.76876	−0.443455
$392$	0	0
$393$	52.4305	2.64477
$394$	1.73904	0.0876114
$395$	2.56320	0.128968
$396$	−13.1499	−0.660806
$397$	17.2658	0.866545	0.433272	−	0.901263i	$-0.357359\pi$
$397$	17.2658	0.866545	0.433272	+	0.901263i	$0.357359\pi$
$398$	−0.719147	−0.0360476
$399$	0	0
$400$	−19.3770	−0.968849
$401$	29.4145	1.46889	0.734445	−	0.678668i	$-0.237442\pi$
$401$	29.4145	1.46889	0.734445	+	0.678668i	$0.237442\pi$
$402$	0.415247	0.0207106
$403$	12.8358	0.639397
$404$	−18.5528	−0.923037
$405$	−4.66809	−0.231959
$406$	0	0
$407$	8.51095	0.421872
$408$	−8.01804	−0.396952
$409$	−36.8255	−1.82090	−0.910452	−	0.413615i	$-0.864266\pi$
$409$	−36.8255	−1.82090	−0.910452	+	0.413615i	$0.864266\pi$
$410$	0.0968305	0.00478212
$411$	51.1928	2.52515
$412$	23.0527	1.13573
$413$	0	0
$414$	−0.967163	−0.0475335
$415$	0.826646	0.0405784
$416$	−1.72211	−0.0844332
$417$	−4.31739	−0.211423
$418$	0.100352	0.00490839
$419$	−29.4840	−1.44039	−0.720193	−	0.693774i	$-0.755947\pi$
$419$	−29.4840	−1.44039	−0.720193	+	0.693774i	$0.755947\pi$
$420$	0	0
$421$	−4.30802	−0.209960	−0.104980	−	0.994474i	$-0.533478\pi$
$421$	−4.30802	−0.209960	−0.104980	+	0.994474i	$0.533478\pi$
$422$	−0.230726	−0.0112316
$423$	27.0116	1.31335
$424$	5.06222	0.245843
$425$	30.9326	1.50045
$426$	−0.567815	−0.0275107
$427$	0	0
$428$	0.405710	0.0196108
$429$	4.30890	0.208036
$430$	−0.130059	−0.00627201
$431$	−22.0058	−1.05998	−0.529992	−	0.848003i	$-0.677805\pi$
$431$	−22.0058	−1.05998	−0.529992	+	0.848003i	$0.677805\pi$
$432$	46.5566	2.23996
$433$	−15.7052	−0.754744	−0.377372	−	0.926062i	$-0.623172\pi$
$433$	−15.7052	−0.754744	−0.377372	+	0.926062i	$0.623172\pi$
$434$	0	0
$435$	1.96763	0.0943409
$436$	−34.9797	−1.67522
$437$	−1.40483	−0.0672019
$438$	−4.56199	−0.217980
$439$	−25.5991	−1.22178	−0.610889	−	0.791716i	$-0.709188\pi$
$439$	−25.5991	−1.22178	−0.610889	+	0.791716i	$0.709188\pi$
$440$	−0.111556	−0.00531821
$441$	0	0
$442$	0.908344	0.0432055
$443$	17.3757	0.825545	0.412772	−	0.910834i	$-0.364561\pi$
$443$	17.3757	0.825545	0.412772	+	0.910834i	$0.364561\pi$
$444$	−54.3293	−2.57836
$445$	−2.50442	−0.118721
$446$	−1.71990	−0.0814397
$447$	51.2018	2.42176
$448$	0	0
$449$	−26.0563	−1.22967	−0.614836	−	0.788655i	$-0.710778\pi$
$449$	−26.0563	−1.22967	−0.614836	+	0.788655i	$0.710778\pi$
$450$	3.41175	0.160832
$451$	3.29015	0.154927
$452$	32.0006	1.50518
$453$	−69.0298	−3.24330
$454$	−2.75827	−0.129452
$455$	0	0
$456$	−1.28455	−0.0601548
$457$	4.60151	0.215250	0.107625	−	0.994192i	$-0.465675\pi$
$457$	4.60151	0.215250	0.107625	+	0.994192i	$0.465675\pi$
$458$	−1.18447	−0.0553467
$459$	−74.3210	−3.46901
$460$	0.778785	0.0363110
$461$	24.8341	1.15664	0.578320	−	0.815810i	$-0.303708\pi$
$461$	24.8341	1.15664	0.578320	+	0.815810i	$0.303708\pi$
$462$	0	0
$463$	25.8050	1.19926	0.599630	−	0.800277i	$-0.295314\pi$
$463$	25.8050	1.19926	0.599630	+	0.800277i	$0.295314\pi$
$464$	−8.82880	−0.409867
$465$	−7.96859	−0.369535
$466$	1.05508	0.0488755
$467$	−13.0335	−0.603119	−0.301559	−	0.953447i	$-0.597507\pi$
$467$	−13.0335	−0.603119	−0.301559	+	0.953447i	$0.597507\pi$
$468$	−19.0691	−0.881468
$469$	0	0
$470$	0.114275	0.00527110
$471$	−16.6290	−0.766224
$472$	1.10451	0.0508391
$473$	−4.41921	−0.203195
$474$	2.92095	0.134164
$475$	4.95565	0.227381
$476$	0	0
$477$	84.1552	3.85320
$478$	1.55250	0.0710097
$479$	2.00396	0.0915634	0.0457817	−	0.998951i	$-0.485422\pi$
$479$	2.00396	0.0915634	0.0457817	+	0.998951i	$0.485422\pi$
$480$	1.06910	0.0487975
$481$	12.3420	0.562747
$482$	0.530656	0.0241707
$483$	0	0
$484$	19.9947	0.908852
$485$	−0.352966	−0.0160274
$486$	−1.69292	−0.0767924
$487$	−16.6173	−0.753003	−0.376502	−	0.926416i	$-0.622873\pi$
$487$	−16.6173	−0.753003	−0.376502	+	0.926416i	$0.622873\pi$
$488$	2.45601	0.111179
$489$	46.3086	2.09415
$490$	0	0
$491$	−20.9773	−0.946692	−0.473346	−	0.880876i	$-0.656954\pi$
$491$	−20.9773	−0.946692	−0.473346	+	0.880876i	$0.656954\pi$
$492$	−21.0025	−0.946867
$493$	14.0939	0.634758
$494$	0.145524	0.00654743
$495$	−1.85452	−0.0833545
$496$	35.7552	1.60545
$497$	0	0
$498$	0.942023	0.0422131
$499$	18.9857	0.849916	0.424958	−	0.905213i	$-0.360289\pi$
$499$	18.9857	0.849916	0.424958	+	0.905213i	$0.360289\pi$
$500$	−5.53841	−0.247685
$501$	−19.8597	−0.887267
$502$	−2.21153	−0.0987053
$503$	3.24409	0.144647	0.0723234	−	0.997381i	$-0.476959\pi$
$503$	3.24409	0.144647	0.0723234	+	0.997381i	$0.476959\pi$
$504$	0	0
$505$	−2.61649	−0.116432
$506$	−0.139028	−0.00618057
$507$	−34.4081	−1.52812
$508$	1.98955	0.0882719
$509$	31.1652	1.38137	0.690686	−	0.723155i	$-0.257309\pi$
$509$	31.1652	1.38137	0.690686	+	0.723155i	$0.257309\pi$
$510$	−0.563908	−0.0249703
$511$	0	0
$512$	−8.00910	−0.353956
$513$	−11.9068	−0.525699
$514$	1.05227	0.0464136
$515$	3.25111	0.143261
$516$	28.2098	1.24187
$517$	3.88288	0.170769
$518$	0	0
$519$	−6.96594	−0.305771
$520$	−0.161770	−0.00709410
$521$	−37.3656	−1.63701	−0.818507	−	0.574496i	$-0.805198\pi$
$521$	−37.3656	−1.63701	−0.818507	+	0.574496i	$0.805198\pi$
$522$	1.55451	0.0680390
$523$	−8.19492	−0.358339	−0.179170	−	0.983818i	$-0.557341\pi$
$523$	−8.19492	−0.358339	−0.179170	+	0.983818i	$0.557341\pi$
$524$	−33.3542	−1.45708
$525$	0	0
$526$	1.45530	0.0634540
$527$	−57.0780	−2.48636
$528$	12.0028	0.522354
$529$	−21.0537	−0.915380
$530$	0.356025	0.0154648
$531$	18.3615	0.796823
$532$	0	0
$533$	4.77115	0.206661
$534$	−2.85397	−0.123503
$535$	0.0572171	0.00247371
$536$	−0.529715	−0.0228802
$537$	11.4165	0.492657
$538$	−2.51849	−0.108580
$539$	0	0
$540$	6.60072	0.284050
$541$	−39.2494	−1.68747	−0.843733	−	0.536764i	$-0.819647\pi$
$541$	−39.2494	−1.68747	−0.843733	+	0.536764i	$0.819647\pi$
$542$	−0.467547	−0.0200829
$543$	−26.6387	−1.14317
$544$	7.65782	0.328326
$545$	−4.93316	−0.211313
$546$	0	0
$547$	4.58146	0.195889	0.0979446	−	0.995192i	$-0.468773\pi$
$547$	4.58146	0.195889	0.0979446	+	0.995192i	$0.468773\pi$
$548$	−32.5668	−1.39119
$549$	40.8292	1.74255
$550$	0.490435	0.0209122
$551$	2.25796	0.0961923
$552$	1.77963	0.0757460
$553$	0	0
$554$	2.51257	0.106749
$555$	−7.66203	−0.325235
$556$	2.74655	0.116480
$557$	−15.9166	−0.674410	−0.337205	−	0.941431i	$-0.609482\pi$
$557$	−15.9166	−0.674410	−0.337205	+	0.941431i	$0.609482\pi$
$558$	−6.29551	−0.266510
$559$	−6.40843	−0.271048
$560$	0	0
$561$	−19.1607	−0.808966
$562$	−1.37461	−0.0579843
$563$	−7.61666	−0.321004	−0.160502	−	0.987036i	$-0.551311\pi$
$563$	−7.61666	−0.321004	−0.160502	+	0.987036i	$0.551311\pi$
$564$	−24.7862	−1.04369
$565$	4.51303	0.189865
$566$	−2.65752	−0.111704
$567$	0	0
$568$	0.724341	0.0303927
$569$	3.85842	0.161754	0.0808768	−	0.996724i	$-0.474228\pi$
$569$	3.85842	0.161754	0.0808768	+	0.996724i	$0.474228\pi$
$570$	−0.0903426	−0.00378404
$571$	32.1197	1.34417	0.672085	−	0.740474i	$-0.265399\pi$
$571$	32.1197	1.34417	0.672085	+	0.740474i	$0.265399\pi$
$572$	−2.74115	−0.114613
$573$	−55.4618	−2.31695
$574$	0	0
$575$	−6.86557	−0.286314
$576$	−52.5528	−2.18970
$577$	−17.3278	−0.721364	−0.360682	−	0.932689i	$-0.617456\pi$
$577$	−17.3278	−0.721364	−0.360682	+	0.932689i	$0.617456\pi$
$578$	−2.30113	−0.0957145
$579$	−6.69255	−0.278133
$580$	−1.25173	−0.0519753
$581$	0	0
$582$	−0.402231	−0.0166730
$583$	12.0972	0.501014
$584$	5.81956	0.240815
$585$	−2.68930	−0.111189
$586$	−1.53415	−0.0633751
$587$	−43.2794	−1.78633	−0.893166	−	0.449728i	$-0.851521\pi$
$587$	−43.2794	−1.78633	−0.893166	+	0.449728i	$0.851521\pi$
$588$	0	0
$589$	−9.14436	−0.376787
$590$	0.0776800	0.00319804
$591$	−53.1958	−2.18818
$592$	34.3796	1.41299
$593$	17.9037	0.735216	0.367608	−	0.929981i	$-0.380177\pi$
$593$	17.9037	0.735216	0.367608	+	0.929981i	$0.380177\pi$
$594$	−1.17836	−0.0483486
$595$	0	0
$596$	−32.5725	−1.33422
$597$	21.9982	0.900326
$598$	−0.201609	−0.00824442
$599$	28.8831	1.18013	0.590066	−	0.807355i	$-0.299102\pi$
$599$	28.8831	1.18013	0.590066	+	0.807355i	$0.299102\pi$
$600$	−6.27779	−0.256290
$601$	−12.3997	−0.505795	−0.252898	−	0.967493i	$-0.581384\pi$
$601$	−12.3997	−0.505795	−0.252898	+	0.967493i	$0.581384\pi$
$602$	0	0
$603$	−8.80608	−0.358611
$604$	43.9141	1.78684
$605$	2.81985	0.114643
$606$	−2.98168	−0.121123
$607$	12.6815	0.514726	0.257363	−	0.966315i	$-0.417146\pi$
$607$	12.6815	0.514726	0.257363	+	0.966315i	$0.417146\pi$
$608$	1.22684	0.0497551
$609$	0	0
$610$	0.172731	0.00699369
$611$	5.63068	0.227793
$612$	84.7959	3.42767
$613$	4.45146	0.179793	0.0898965	−	0.995951i	$-0.471346\pi$
$613$	4.45146	0.179793	0.0898965	+	0.995951i	$0.471346\pi$
$614$	1.56425	0.0631280
$615$	−2.96197	−0.119438
$616$	0	0
$617$	−24.0389	−0.967769	−0.483885	−	0.875132i	$-0.660775\pi$
$617$	−24.0389	−0.967769	−0.483885	+	0.875132i	$0.660775\pi$
$618$	3.70488	0.149032
$619$	−17.6987	−0.711371	−0.355686	−	0.934606i	$-0.615753\pi$
$619$	−17.6987	−0.711371	−0.355686	+	0.934606i	$0.615753\pi$
$620$	5.06930	0.203588
$621$	16.4958	0.661952
$622$	−0.739205	−0.0296394
$623$	0	0
$624$	17.4056	0.696782
$625$	23.8253	0.953011
$626$	0.106798	0.00426849
$627$	−3.06970	−0.122592
$628$	10.5787	0.422136
$629$	−54.8822	−2.18830
$630$	0	0
$631$	4.02635	0.160286	0.0801432	−	0.996783i	$-0.474462\pi$
$631$	4.02635	0.160286	0.0801432	+	0.996783i	$0.474462\pi$
$632$	−3.72615	−0.148218
$633$	7.05775	0.280520
$634$	−0.457583	−0.0181729
$635$	0.280585	0.0111347
$636$	−77.2219	−3.06205
$637$	0	0
$638$	0.223459	0.00884681
$639$	12.0416	0.476357
$640$	−0.906021	−0.0358136
$641$	−28.9089	−1.14183	−0.570917	−	0.821008i	$-0.693412\pi$
$641$	−28.9089	−1.14183	−0.570917	+	0.821008i	$0.693412\pi$
$642$	0.0652030	0.00257336
$643$	4.68023	0.184570	0.0922851	−	0.995733i	$-0.470583\pi$
$643$	4.68023	0.184570	0.0922851	+	0.995733i	$0.470583\pi$
$644$	0	0
$645$	3.97842	0.156650
$646$	−0.647113	−0.0254603
$647$	−11.1375	−0.437860	−0.218930	−	0.975741i	$-0.570257\pi$
$647$	−11.1375	−0.437860	−0.218930	+	0.975741i	$0.570257\pi$
$648$	6.78605	0.266581
$649$	2.63945	0.103607
$650$	0.711195	0.0278954
$651$	0	0
$652$	−29.4597	−1.15373
$653$	36.8949	1.44381	0.721904	−	0.691993i	$-0.243267\pi$
$653$	36.8949	1.44381	0.721904	+	0.691993i	$0.243267\pi$
$654$	−5.62169	−0.219826
$655$	−4.70392	−0.183797
$656$	13.2904	0.518903
$657$	96.7454	3.77440
$658$	0	0
$659$	−22.7628	−0.886713	−0.443357	−	0.896345i	$-0.646212\pi$
$659$	−22.7628	−0.886713	−0.443357	+	0.896345i	$0.646212\pi$
$660$	1.70173	0.0662398
$661$	−33.3559	−1.29740	−0.648698	−	0.761046i	$-0.724686\pi$
$661$	−33.3559	−1.29740	−0.648698	+	0.761046i	$0.724686\pi$
$662$	−1.76091	−0.0684397
$663$	−27.7856	−1.07910
$664$	−1.20170	−0.0466352
$665$	0	0
$666$	−6.05331	−0.234561
$667$	−3.12819	−0.121124
$668$	12.6340	0.488823
$669$	52.6105	2.03404
$670$	−0.0372548	−0.00143928
$671$	5.86914	0.226576
$672$	0	0
$673$	−5.38199	−0.207461	−0.103730	−	0.994605i	$-0.533078\pi$
$673$	−5.38199	−0.207461	−0.103730	+	0.994605i	$0.533078\pi$
$674$	−0.708452	−0.0272885
$675$	−58.1903	−2.23974
$676$	21.8891	0.841888
$677$	−35.4290	−1.36165	−0.680823	−	0.732448i	$-0.738378\pi$
$677$	−35.4290	−1.36165	−0.680823	+	0.732448i	$0.738378\pi$
$678$	5.14293	0.197513
$679$	0	0
$680$	0.719357	0.0275861
$681$	84.3735	3.23320
$682$	−0.904970	−0.0346531
$683$	−8.04397	−0.307794	−0.153897	−	0.988087i	$-0.549182\pi$
$683$	−8.04397	−0.307794	−0.153897	+	0.988087i	$0.549182\pi$
$684$	13.5850	0.519435
$685$	−4.59288	−0.175485
$686$	0	0
$687$	36.2321	1.38234
$688$	−17.8512	−0.680571
$689$	17.5425	0.668317
$690$	0.125161	0.00476480
$691$	16.1470	0.614261	0.307131	−	0.951667i	$-0.400631\pi$
$691$	16.1470	0.614261	0.307131	+	0.951667i	$0.400631\pi$
$692$	4.43146	0.168459
$693$	0	0
$694$	1.10629	0.0419943
$695$	0.387345	0.0146928
$696$	−2.86037	−0.108422
$697$	−21.2162	−0.803622
$698$	1.76006	0.0666192
$699$	−32.2741	−1.22072
$700$	0	0
$701$	−31.0098	−1.17122	−0.585611	−	0.810592i	$-0.699145\pi$
$701$	−31.0098	−1.17122	−0.585611	+	0.810592i	$0.699145\pi$
$702$	−1.70877	−0.0644935
$703$	−8.79256	−0.331618
$704$	−7.55440	−0.284717
$705$	−3.49558	−0.131651
$706$	−3.66130	−0.137795
$707$	0	0
$708$	−16.8488	−0.633217
$709$	−13.3540	−0.501520	−0.250760	−	0.968049i	$-0.580681\pi$
$709$	−13.3540	−0.501520	−0.250760	+	0.968049i	$0.580681\pi$
$710$	0.0509429	0.00191185
$711$	−61.9441	−2.32309
$712$	3.64070	0.136441
$713$	12.6686	0.474444
$714$	0	0
$715$	−0.386583	−0.0144574
$716$	−7.26270	−0.271420
$717$	−47.4899	−1.77354
$718$	−1.28608	−0.0479961
$719$	−6.07809	−0.226674	−0.113337	−	0.993557i	$-0.536154\pi$
$719$	−6.07809	−0.226674	−0.113337	+	0.993557i	$0.536154\pi$
$720$	−7.49125	−0.279182
$721$	0	0
$722$	1.83888	0.0684359
$723$	−16.2324	−0.603688
$724$	16.9465	0.629811
$725$	11.0350	0.409828
$726$	3.21342	0.119261
$727$	18.5668	0.688605	0.344302	−	0.938859i	$-0.388115\pi$
$727$	18.5668	0.688605	0.344302	+	0.938859i	$0.388115\pi$
$728$	0	0
$729$	1.87414	0.0694126
$730$	0.409289	0.0151485
$731$	28.4969	1.05400
$732$	−37.4654	−1.38476
$733$	−31.9202	−1.17900	−0.589499	−	0.807769i	$-0.700675\pi$
$733$	−31.9202	−1.17900	−0.589499	+	0.807769i	$0.700675\pi$
$734$	−0.404823	−0.0149423
$735$	0	0
$736$	−1.69968	−0.0626509
$737$	−1.26586	−0.0466286
$738$	−2.34008	−0.0861394
$739$	−47.3446	−1.74160	−0.870799	−	0.491639i	$-0.836398\pi$
$739$	−47.3446	−1.74160	−0.870799	+	0.491639i	$0.836398\pi$
$740$	4.87428	0.179182
$741$	−4.45147	−0.163529
$742$	0	0
$743$	−1.56541	−0.0574293	−0.0287147	−	0.999588i	$-0.509141\pi$
$743$	−1.56541	−0.0574293	−0.0287147	+	0.999588i	$0.509141\pi$
$744$	11.5840	0.424691
$745$	−4.59369	−0.168300
$746$	−3.08253	−0.112860
$747$	−19.9773	−0.730932
$748$	12.1893	0.445685
$749$	0	0
$750$	−0.890097	−0.0325017
$751$	−54.4033	−1.98520	−0.992601	−	0.121418i	$-0.961256\pi$
$751$	−54.4033	−1.98520	−0.992601	+	0.121418i	$0.961256\pi$
$752$	15.6847	0.571963
$753$	67.6490	2.46527
$754$	0.324044	0.0118010
$755$	6.19317	0.225393
$756$	0	0
$757$	−18.1815	−0.660817	−0.330409	−	0.943838i	$-0.607187\pi$
$757$	−18.1815	−0.660817	−0.330409	+	0.943838i	$0.607187\pi$
$758$	−2.25479	−0.0818976
$759$	4.25278	0.154366
$760$	0.115247	0.00418044
$761$	−26.0404	−0.943962	−0.471981	−	0.881609i	$-0.656461\pi$
$761$	−26.0404	−0.943962	−0.471981	+	0.881609i	$0.656461\pi$
$762$	0.319747	0.0115832
$763$	0	0
$764$	35.2826	1.27648
$765$	11.9587	0.432368
$766$	0.933730	0.0337370
$767$	3.82754	0.138205
$768$	47.4441	1.71199
$769$	34.5026	1.24419	0.622097	−	0.782940i	$-0.286281\pi$
$769$	34.5026	1.24419	0.622097	+	0.782940i	$0.286281\pi$
$770$	0	0
$771$	−32.1881	−1.15923
$772$	4.25754	0.153232
$773$	23.2583	0.836542	0.418271	−	0.908322i	$-0.362636\pi$
$773$	23.2583	0.836542	0.418271	+	0.908322i	$0.362636\pi$
$774$	3.14311	0.112977
$775$	−44.6897	−1.60530
$776$	0.513111	0.0184196
$777$	0	0
$778$	1.89427	0.0679130
$779$	−3.39901	−0.121782
$780$	2.46774	0.0883591
$781$	1.73096	0.0619386
$782$	0.896513	0.0320592
$783$	−26.5134	−0.947513
$784$	0	0
$785$	1.49191	0.0532485
$786$	−5.36046	−0.191201
$787$	26.2539	0.935850	0.467925	−	0.883768i	$-0.345002\pi$
$787$	26.2539	0.935850	0.467925	+	0.883768i	$0.345002\pi$
$788$	33.8411	1.20554
$789$	−44.5165	−1.58483
$790$	−0.262060	−0.00932367
$791$	0	0
$792$	2.69594	0.0957959
$793$	8.51103	0.302235
$794$	−1.76524	−0.0626462
$795$	−10.8906	−0.386248
$796$	−13.9944	−0.496018
$797$	−34.9004	−1.23623	−0.618117	−	0.786086i	$-0.712104\pi$
$797$	−34.9004	−1.23623	−0.618117	+	0.786086i	$0.712104\pi$
$798$	0	0
$799$	−25.0384	−0.885795
$800$	5.99576	0.211982
$801$	60.5236	2.13850
$802$	−3.00732	−0.106192
$803$	13.9070	0.490768
$804$	8.08058	0.284980
$805$	0	0
$806$	−1.31233	−0.0462247
$807$	77.0388	2.71189
$808$	3.80362	0.133811
$809$	29.4086	1.03395	0.516975	−	0.856001i	$-0.327058\pi$
$809$	29.4086	1.03395	0.516975	+	0.856001i	$0.327058\pi$
$810$	0.477263	0.0167693
$811$	−41.8547	−1.46972	−0.734859	−	0.678220i	$-0.762751\pi$
$811$	−41.8547	−1.46972	−0.734859	+	0.678220i	$0.762751\pi$
$812$	0	0
$813$	14.3019	0.501591
$814$	−0.870155	−0.0304989
$815$	−4.15468	−0.145532
$816$	−77.3989	−2.70950
$817$	4.56543	0.159724
$818$	3.76502	0.131641
$819$	0	0
$820$	1.88429	0.0658023
$821$	−46.6870	−1.62939	−0.814694	−	0.579891i	$-0.803095\pi$
$821$	−46.6870	−1.62939	−0.814694	+	0.579891i	$0.803095\pi$
$822$	−5.23392	−0.182554
$823$	15.4362	0.538071	0.269035	−	0.963130i	$-0.413295\pi$
$823$	15.4362	0.538071	0.269035	+	0.963130i	$0.413295\pi$
$824$	−4.72618	−0.164644
$825$	−15.0020	−0.522304
$826$	0	0
$827$	−22.0580	−0.767030	−0.383515	−	0.923535i	$-0.625287\pi$
$827$	−22.0580	−0.767030	−0.383515	+	0.923535i	$0.625287\pi$
$828$	−18.8207	−0.654064
$829$	−47.3457	−1.64438	−0.822191	−	0.569211i	$-0.807249\pi$
$829$	−47.3457	−1.64438	−0.822191	+	0.569211i	$0.807249\pi$
$830$	−0.0845158	−0.00293359
$831$	−76.8578	−2.66617
$832$	−10.9549	−0.379792
$833$	0	0
$834$	0.441407	0.0152847
$835$	1.78176	0.0616604
$836$	1.95282	0.0675398
$837$	107.375	3.71142
$838$	3.01442	0.104132
$839$	−33.7490	−1.16514	−0.582572	−	0.812779i	$-0.697954\pi$
$839$	−33.7490	−1.16514	−0.582572	+	0.812779i	$0.697954\pi$
$840$	0	0
$841$	−23.9721	−0.826624
$842$	0.440450	0.0151789
$843$	42.0482	1.44822
$844$	−4.48986	−0.154547
$845$	3.08700	0.106196
$846$	−2.76165	−0.0949474
$847$	0	0
$848$	48.8661	1.67807
$849$	81.2915	2.78992
$850$	−3.16253	−0.108474
$851$	12.1813	0.417568
$852$	−11.0495	−0.378550
$853$	−6.51623	−0.223111	−0.111556	−	0.993758i	$-0.535583\pi$
$853$	−6.51623	−0.223111	−0.111556	+	0.993758i	$0.535583\pi$
$854$	0	0
$855$	1.91588	0.0655218
$856$	−0.0831771	−0.00284294
$857$	−18.3967	−0.628419	−0.314209	−	0.949354i	$-0.601739\pi$
$857$	−18.3967	−0.628419	−0.314209	+	0.949354i	$0.601739\pi$
$858$	−0.440539	−0.0150398
$859$	−43.9167	−1.49842	−0.749209	−	0.662333i	$-0.769566\pi$
$859$	−43.9167	−1.49842	−0.749209	+	0.662333i	$0.769566\pi$
$860$	−2.53091	−0.0863034
$861$	0	0
$862$	2.24986	0.0766307
$863$	−46.1967	−1.57255	−0.786277	−	0.617875i	$-0.787994\pi$
$863$	−46.1967	−1.57255	−0.786277	+	0.617875i	$0.787994\pi$
$864$	−14.4059	−0.490098
$865$	0.624966	0.0212495
$866$	1.60569	0.0545636
$867$	70.3900	2.39057
$868$	0	0
$869$	−8.90438	−0.302060
$870$	−0.201170	−0.00682030
$871$	−1.83567	−0.0621992
$872$	7.17139	0.242854
$873$	8.53005	0.288698
$874$	0.143629	0.00485831
$875$	0	0
$876$	−88.7749	−2.99943
$877$	−41.8033	−1.41160	−0.705799	−	0.708412i	$-0.749412\pi$
$877$	−41.8033	−1.41160	−0.705799	+	0.708412i	$0.749412\pi$
$878$	2.61724	0.0883275
$879$	46.9285	1.58286
$880$	−1.07686	−0.0363009
$881$	17.9451	0.604587	0.302293	−	0.953215i	$-0.402248\pi$
$881$	17.9451	0.604587	0.302293	+	0.953215i	$0.402248\pi$
$882$	0	0
$883$	46.4060	1.56169	0.780843	−	0.624728i	$-0.214790\pi$
$883$	46.4060	1.56169	0.780843	+	0.624728i	$0.214790\pi$
$884$	17.6761	0.594511
$885$	−2.37618	−0.0798743
$886$	−1.77648	−0.0596821
$887$	−34.5487	−1.16003	−0.580016	−	0.814605i	$-0.696954\pi$
$887$	−34.5487	−1.16003	−0.580016	+	0.814605i	$0.696954\pi$
$888$	11.1384	0.373780
$889$	0	0
$890$	0.256050	0.00858283
$891$	16.2166	0.543278
$892$	−33.4687	−1.12062
$893$	−4.01135	−0.134235
$894$	−5.23484	−0.175079
$895$	−1.02425	−0.0342370
$896$	0	0
$897$	6.16709	0.205913
$898$	2.66398	0.0888982
$899$	−20.3621	−0.679115
$900$	66.3917	2.21306
$901$	−78.0077	−2.59881
$902$	−0.336383	−0.0112003
$903$	0	0
$904$	−6.56064	−0.218204
$905$	2.38995	0.0794446
$906$	7.05757	0.234472
$907$	−15.3750	−0.510518	−0.255259	−	0.966873i	$-0.582161\pi$
$907$	−15.3750	−0.510518	−0.255259	+	0.966873i	$0.582161\pi$
$908$	−53.6751	−1.78127
$909$	63.2321	2.09728
$910$	0	0
$911$	25.2602	0.836908	0.418454	−	0.908238i	$-0.362572\pi$
$911$	25.2602	0.836908	0.418454	+	0.908238i	$0.362572\pi$
$912$	−12.3999	−0.410603
$913$	−2.87171	−0.0950399
$914$	−0.470456	−0.0155613
$915$	−5.28373	−0.174675
$916$	−23.0494	−0.761575
$917$	0	0
$918$	7.59854	0.250789
$919$	25.4760	0.840376	0.420188	−	0.907437i	$-0.361964\pi$
$919$	25.4760	0.840376	0.420188	+	0.907437i	$0.361964\pi$
$920$	−0.159663	−0.00526395
$921$	−47.8492	−1.57669
$922$	−2.53903	−0.0836184
$923$	2.51012	0.0826216
$924$	0	0
$925$	−42.9705	−1.41286
$926$	−2.63829	−0.0866996
$927$	−78.5687	−2.58054
$928$	2.73187	0.0896780
$929$	−23.7913	−0.780566	−0.390283	−	0.920695i	$-0.627623\pi$
$929$	−23.7913	−0.780566	−0.390283	+	0.920695i	$0.627623\pi$
$930$	0.814704	0.0267152
$931$	0	0
$932$	20.5315	0.672531
$933$	22.6118	0.740276
$934$	1.33254	0.0436020
$935$	1.71905	0.0562189
$936$	3.90946	0.127785
$937$	29.6899	0.969927	0.484964	−	0.874534i	$-0.338833\pi$
$937$	29.6899	0.969927	0.484964	+	0.874534i	$0.338833\pi$
$938$	0	0
$939$	−3.26686	−0.106610
$940$	2.22375	0.0725308
$941$	40.2348	1.31162	0.655809	−	0.754927i	$-0.272328\pi$
$941$	40.2348	1.31162	0.655809	+	0.754927i	$0.272328\pi$
$942$	1.70014	0.0553935
$943$	4.70901	0.153346
$944$	10.6619	0.347016
$945$	0	0
$946$	0.451817	0.0146899
$947$	48.3106	1.56988	0.784942	−	0.619569i	$-0.212693\pi$
$947$	48.3106	1.56988	0.784942	+	0.619569i	$0.212693\pi$
$948$	56.8408	1.84610
$949$	20.1670	0.654649
$950$	−0.506662	−0.0164383
$951$	13.9971	0.453888
$952$	0	0
$953$	46.8476	1.51754	0.758772	−	0.651357i	$-0.225800\pi$
$953$	46.8476	1.51754	0.758772	+	0.651357i	$0.225800\pi$
$954$	−8.60398	−0.278564
$955$	4.97588	0.161016
$956$	30.2112	0.977099
$957$	−6.83544	−0.220958
$958$	−0.204884	−0.00661950
$959$	0	0
$960$	6.80089	0.219498
$961$	51.4633	1.66011
$962$	−1.26184	−0.0406833
$963$	−1.38275	−0.0445585
$964$	10.3264	0.332591
$965$	0.600438	0.0193288
$966$	0	0
$967$	−11.3136	−0.363822	−0.181911	−	0.983315i	$-0.558228\pi$
$967$	−11.3136	−0.363822	−0.181911	+	0.983315i	$0.558228\pi$
$968$	−4.09924	−0.131755
$969$	19.7947	0.635898
$970$	0.0360871	0.00115869
$971$	48.0486	1.54195	0.770976	−	0.636864i	$-0.219769\pi$
$971$	48.0486	1.54195	0.770976	+	0.636864i	$0.219769\pi$
$972$	−32.9437	−1.05667
$973$	0	0
$974$	1.69895	0.0544378
$975$	−21.7549	−0.696716
$976$	23.7081	0.758879
$977$	−1.06017	−0.0339177	−0.0169588	−	0.999856i	$-0.505398\pi$
$977$	−1.06017	−0.0339177	−0.0169588	+	0.999856i	$0.505398\pi$
$978$	−4.73456	−0.151395
$979$	8.70019	0.278059
$980$	0	0
$981$	119.218	3.80635
$982$	2.14471	0.0684404
$983$	−2.35200	−0.0750172	−0.0375086	−	0.999296i	$-0.511942\pi$
$983$	−2.35200	−0.0750172	−0.0375086	+	0.999296i	$0.511942\pi$
$984$	4.30585	0.137266
$985$	4.77259	0.152067
$986$	−1.44095	−0.0458893
$987$	0	0
$988$	2.83185	0.0900931
$989$	−6.32497	−0.201122
$990$	0.189605	0.00602604
$991$	−26.3037	−0.835564	−0.417782	−	0.908547i	$-0.637193\pi$
$991$	−26.3037	−0.835564	−0.417782	+	0.908547i	$0.637193\pi$
$992$	−11.0636	−0.351270
$993$	53.8649	1.70935
$994$	0	0
$995$	−1.97362	−0.0625679
$996$	18.3315	0.580855
$997$	24.0943	0.763076	0.381538	−	0.924353i	$-0.375395\pi$
$997$	24.0943	0.763076	0.381538	+	0.924353i	$0.375395\pi$
$998$	−1.94109	−0.0614440
$999$	103.244	3.26650

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.k.1.8		16
7.6	odd	2		889.2.a.c.1.8	✓	16
21.20	even	2		8001.2.a.t.1.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.8	✓	16	7.6	odd	2
6223.2.a.k.1.8		16	1.1	even	1	trivial
8001.2.a.t.1.9		16	21.20	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:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q-0.102239 q^{2} +3.12743 q^{3} -1.98955 q^{4} -0.280585 q^{5} -0.319747 q^{6} +0.407889 q^{8} +6.78082 q^{9} +0.0286868 q^{10} +0.974733 q^{11} -6.22217 q^{12} +1.41349 q^{13} -0.877509 q^{15} +3.93739 q^{16} -6.28548 q^{17} -0.693267 q^{18} -1.00698 q^{19} +0.558236 q^{20} -0.0996561 q^{22} +1.39508 q^{23} +1.27564 q^{24} -4.92127 q^{25} -0.144515 q^{26} +11.8242 q^{27} -2.24230 q^{29} +0.0897160 q^{30} +9.08093 q^{31} -1.21833 q^{32} +3.04841 q^{33} +0.642624 q^{34} -13.4908 q^{36} +8.73158 q^{37} +0.102954 q^{38} +4.42059 q^{39} -0.114447 q^{40} +3.37544 q^{41} -4.53376 q^{43} -1.93928 q^{44} -1.90259 q^{45} -0.142632 q^{46} +3.98353 q^{47} +12.3139 q^{48} +0.503148 q^{50} -19.6574 q^{51} -2.81221 q^{52} +12.4108 q^{53} -1.20890 q^{54} -0.273495 q^{55} -3.14927 q^{57} +0.229251 q^{58} +2.70787 q^{59} +1.74584 q^{60} +6.02128 q^{61} -0.928429 q^{62} -7.75022 q^{64} -0.396604 q^{65} -0.311668 q^{66} -1.29868 q^{67} +12.5053 q^{68} +4.36302 q^{69} +1.77583 q^{71} +2.76582 q^{72} +14.2675 q^{73} -0.892711 q^{74} -15.3909 q^{75} +2.00344 q^{76} -0.451959 q^{78} -9.13520 q^{79} -1.10477 q^{80} +16.6370 q^{81} -0.345103 q^{82} -2.94615 q^{83} +1.76361 q^{85} +0.463529 q^{86} -7.01263 q^{87} +0.397583 q^{88} +8.92572 q^{89} +0.194520 q^{90} -2.77558 q^{92} +28.4000 q^{93} -0.407274 q^{94} +0.282544 q^{95} -3.81026 q^{96} +1.25797 q^{97} +6.60948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 4 q^{3} + 12 q^{4} + 9 q^{5} + 12 q^{6} - 6 q^{8} + 14 q^{9} + 2 q^{10} - 22 q^{11} + 10 q^{12} + 4 q^{13} - 14 q^{15} + 12 q^{16} + 18 q^{17} - 5 q^{18} + 15 q^{19} + 40 q^{20} - 11 q^{22}+ \cdots - 73 q^{99}+O(q^{100}) \) 16 * q - 2 * q^2 + 4 * q^3 + 12 * q^4 + 9 * q^5 + 12 * q^6 - 6 * q^8 + 14 * q^9 + 2 * q^10 - 22 * q^11 + 10 * q^12 + 4 * q^13 - 14 * q^15 + 12 * q^16 + 18 * q^17 - 5 * q^18 + 15 * q^19 + 40 * q^20 - 11 * q^22 - 5 * q^23 + 26 * q^24 + 15 * q^25 + 24 * q^26 + 10 * q^27 - 12 * q^29 + 3 * q^30 + 32 * q^31 - 9 * q^32 + 10 * q^33 + 14 * q^34 + 10 * q^36 - 2 * q^37 - 3 * q^38 - 41 * q^39 + 14 * q^40 + 45 * q^41 - 3 * q^43 - 54 * q^44 + 22 * q^45 + 49 * q^47 - 10 * q^48 - 6 * q^50 - 12 * q^51 - 38 * q^52 + 16 * q^53 + 67 * q^54 - 7 * q^55 + 8 * q^57 + 16 * q^58 + 35 * q^59 + 56 * q^60 + 11 * q^61 - 17 * q^62 - 2 * q^64 + 14 * q^65 - 86 * q^66 + 17 * q^67 + 71 * q^68 + 17 * q^69 - 81 * q^71 + 13 * q^72 + 15 * q^73 + 13 * q^74 + 48 * q^75 - 14 * q^76 - 10 * q^78 - 34 * q^79 + 33 * q^80 + 40 * q^81 + 14 * q^82 + 39 * q^83 - 17 * q^85 + 36 * q^86 - 7 * q^87 + 61 * q^88 + 32 * q^89 - 79 * q^90 + 37 * q^92 + 27 * q^93 - 13 * q^94 - 33 * q^95 + 55 * q^96 + 4 * q^97 - 73 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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