LMFDB - Embedded newform 8450.2.a.da.1.6

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:	$0$
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			1.6
Root			$2.17115$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.956446 q^{3} +1.00000 q^{4} +0.956446 q^{6} -2.15124 q^{7} +1.00000 q^{8} -2.08521 q^{9} +1.30608 q^{11} +0.956446 q^{12} -2.15124 q^{14} +1.00000 q^{16} -2.87416 q^{17} -2.08521 q^{18} -2.65614 q^{19} -2.05754 q^{21} +1.30608 q^{22} +5.11025 q^{23} +0.956446 q^{24} -4.86373 q^{27} -2.15124 q^{28} +7.96020 q^{29} -4.12897 q^{31} +1.00000 q^{32} +1.24919 q^{33} -2.87416 q^{34} -2.08521 q^{36} +2.98725 q^{37} -2.65614 q^{38} +8.69023 q^{41} -2.05754 q^{42} +3.35303 q^{43} +1.30608 q^{44} +5.11025 q^{46} +7.30423 q^{47} +0.956446 q^{48} -2.37217 q^{49} -2.74898 q^{51} +10.0804 q^{53} -4.86373 q^{54} -2.15124 q^{56} -2.54046 q^{57} +7.96020 q^{58} -14.6950 q^{59} +11.2009 q^{61} -4.12897 q^{62} +4.48579 q^{63} +1.00000 q^{64} +1.24919 q^{66} -11.0454 q^{67} -2.87416 q^{68} +4.88768 q^{69} +14.0739 q^{71} -2.08521 q^{72} -3.83547 q^{73} +2.98725 q^{74} -2.65614 q^{76} -2.80969 q^{77} -8.94252 q^{79} +1.60374 q^{81} +8.69023 q^{82} +13.9980 q^{83} -2.05754 q^{84} +3.35303 q^{86} +7.61350 q^{87} +1.30608 q^{88} +15.6235 q^{89} +5.11025 q^{92} -3.94914 q^{93} +7.30423 q^{94} +0.956446 q^{96} +5.05416 q^{97} -2.37217 q^{98} -2.72345 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.956446	0.552204	0.276102	−	0.961128i	$-0.410957\pi$
$3$	0.956446	0.552204	0.276102	+	0.961128i	$0.410957\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.956446	0.390467
$7$	−2.15124	−0.813092	−0.406546	−	0.913630i	$-0.633267\pi$
$7$	−2.15124	−0.813092	−0.406546	+	0.913630i	$0.633267\pi$
$8$	1.00000	0.353553
$9$	−2.08521	−0.695070
$10$	0	0
$11$	1.30608	0.393797	0.196899	−	0.980424i	$-0.436913\pi$
$11$	1.30608	0.393797	0.196899	+	0.980424i	$0.436913\pi$
$12$	0.956446	0.276102
$13$	0	0
$13$	0	0
$14$	−2.15124	−0.574943
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.87416	−0.697086	−0.348543	−	0.937293i	$-0.613323\pi$
$17$	−2.87416	−0.697086	−0.348543	+	0.937293i	$0.613323\pi$
$18$	−2.08521	−0.491489
$19$	−2.65614	−0.609360	−0.304680	−	0.952455i	$-0.598550\pi$
$19$	−2.65614	−0.609360	−0.304680	+	0.952455i	$0.598550\pi$
$20$	0	0
$21$	−2.05754	−0.448993
$22$	1.30608	0.278457
$23$	5.11025	1.06556	0.532780	−	0.846254i	$-0.321147\pi$
$23$	5.11025	1.06556	0.532780	+	0.846254i	$0.321147\pi$
$24$	0.956446	0.195234
$25$	0	0
$26$	0	0
$27$	−4.86373	−0.936025
$28$	−2.15124	−0.406546
$29$	7.96020	1.47817	0.739086	−	0.673611i	$-0.235258\pi$
$29$	7.96020	1.47817	0.739086	+	0.673611i	$0.235258\pi$
$30$	0	0
$31$	−4.12897	−0.741585	−0.370792	−	0.928716i	$-0.620914\pi$
$31$	−4.12897	−0.741585	−0.370792	+	0.928716i	$0.620914\pi$
$32$	1.00000	0.176777
$33$	1.24919	0.217457
$34$	−2.87416	−0.492914
$35$	0	0
$36$	−2.08521	−0.347535
$37$	2.98725	0.491101	0.245551	−	0.969384i	$-0.421031\pi$
$37$	2.98725	0.491101	0.245551	+	0.969384i	$0.421031\pi$
$38$	−2.65614	−0.430883
$39$	0	0
$40$	0	0
$41$	8.69023	1.35719	0.678593	−	0.734514i	$-0.262590\pi$
$41$	8.69023	1.35719	0.678593	+	0.734514i	$0.262590\pi$
$42$	−2.05754	−0.317486
$43$	3.35303	0.511332	0.255666	−	0.966765i	$-0.417705\pi$
$43$	3.35303	0.511332	0.255666	+	0.966765i	$0.417705\pi$
$44$	1.30608	0.196899
$45$	0	0
$46$	5.11025	0.753465
$47$	7.30423	1.06543	0.532716	−	0.846294i	$-0.321172\pi$
$47$	7.30423	1.06543	0.532716	+	0.846294i	$0.321172\pi$
$48$	0.956446	0.138051
$49$	−2.37217	−0.338882
$50$	0	0
$51$	−2.74898	−0.384934
$52$	0	0
$53$	10.0804	1.38465	0.692325	−	0.721586i	$-0.256587\pi$
$53$	10.0804	1.38465	0.692325	+	0.721586i	$0.256587\pi$
$54$	−4.86373	−0.661870
$55$	0	0
$56$	−2.15124	−0.287471
$57$	−2.54046	−0.336491
$58$	7.96020	1.04523
$59$	−14.6950	−1.91313	−0.956564	−	0.291522i	$-0.905838\pi$
$59$	−14.6950	−1.91313	−0.956564	+	0.291522i	$0.905838\pi$
$60$	0	0
$61$	11.2009	1.43412	0.717062	−	0.697009i	$-0.245486\pi$
$61$	11.2009	1.43412	0.717062	+	0.697009i	$0.245486\pi$
$62$	−4.12897	−0.524380
$63$	4.48579	0.565156
$64$	1.00000	0.125000
$65$	0	0
$66$	1.24919	0.153765
$67$	−11.0454	−1.34941	−0.674706	−	0.738087i	$-0.735730\pi$
$67$	−11.0454	−1.34941	−0.674706	+	0.738087i	$0.735730\pi$
$68$	−2.87416	−0.348543
$69$	4.88768	0.588407
$70$	0	0
$71$	14.0739	1.67026	0.835130	−	0.550053i	$-0.185392\pi$
$71$	14.0739	1.67026	0.835130	+	0.550053i	$0.185392\pi$
$72$	−2.08521	−0.245744
$73$	−3.83547	−0.448907	−0.224454	−	0.974485i	$-0.572060\pi$
$73$	−3.83547	−0.448907	−0.224454	+	0.974485i	$0.572060\pi$
$74$	2.98725	0.347261
$75$	0	0
$76$	−2.65614	−0.304680
$77$	−2.80969	−0.320193
$78$	0	0
$79$	−8.94252	−1.00611	−0.503056	−	0.864254i	$-0.667791\pi$
$79$	−8.94252	−1.00611	−0.503056	+	0.864254i	$0.667791\pi$
$80$	0	0
$81$	1.60374	0.178193
$82$	8.69023	0.959676
$83$	13.9980	1.53648	0.768239	−	0.640163i	$-0.221133\pi$
$83$	13.9980	1.53648	0.768239	+	0.640163i	$0.221133\pi$
$84$	−2.05754	−0.224496
$85$	0	0
$86$	3.35303	0.361567
$87$	7.61350	0.816253
$88$	1.30608	0.139228
$89$	15.6235	1.65609	0.828043	−	0.560665i	$-0.189454\pi$
$89$	15.6235	1.65609	0.828043	+	0.560665i	$0.189454\pi$
$90$	0	0
$91$	0	0
$92$	5.11025	0.532780
$93$	−3.94914	−0.409506
$94$	7.30423	0.753374
$95$	0	0
$96$	0.956446	0.0976169
$97$	5.05416	0.513172	0.256586	−	0.966521i	$-0.417402\pi$
$97$	5.05416	0.513172	0.256586	+	0.966521i	$0.417402\pi$
$98$	−2.37217	−0.239626
$99$	−2.72345	−0.273717
$100$	0	0
$101$	−2.02811	−0.201804	−0.100902	−	0.994896i	$-0.532173\pi$
$101$	−2.02811	−0.201804	−0.100902	+	0.994896i	$0.532173\pi$
$102$	−2.74898	−0.272189
$103$	−3.26212	−0.321426	−0.160713	−	0.987001i	$-0.551379\pi$
$103$	−3.26212	−0.321426	−0.160713	+	0.987001i	$0.551379\pi$
$104$	0	0
$105$	0	0
$106$	10.0804	0.979095
$107$	1.89051	0.182763	0.0913814	−	0.995816i	$-0.470872\pi$
$107$	1.89051	0.182763	0.0913814	+	0.995816i	$0.470872\pi$
$108$	−4.86373	−0.468013
$109$	4.29397	0.411287	0.205644	−	0.978627i	$-0.434071\pi$
$109$	4.29397	0.411287	0.205644	+	0.978627i	$0.434071\pi$
$110$	0	0
$111$	2.85715	0.271188
$112$	−2.15124	−0.203273
$113$	18.7154	1.76060	0.880300	−	0.474418i	$-0.157342\pi$
$113$	18.7154	1.76060	0.880300	+	0.474418i	$0.157342\pi$
$114$	−2.54046	−0.237935
$115$	0	0
$116$	7.96020	0.739086
$117$	0	0
$118$	−14.6950	−1.35279
$119$	6.18300	0.566795
$120$	0	0
$121$	−9.29416	−0.844924
$122$	11.2009	1.01408
$123$	8.31174	0.749444
$124$	−4.12897	−0.370792
$125$	0	0
$126$	4.48579	0.399626
$127$	17.6498	1.56617	0.783083	−	0.621918i	$-0.213646\pi$
$127$	17.6498	1.56617	0.783083	+	0.621918i	$0.213646\pi$
$128$	1.00000	0.0883883
$129$	3.20699	0.282360
$130$	0	0
$131$	8.59263	0.750742	0.375371	−	0.926875i	$-0.377515\pi$
$131$	8.59263	0.750742	0.375371	+	0.926875i	$0.377515\pi$
$132$	1.24919	0.108728
$133$	5.71399	0.495466
$134$	−11.0454	−0.954178
$135$	0	0
$136$	−2.87416	−0.246457
$137$	−6.43023	−0.549372	−0.274686	−	0.961534i	$-0.588574\pi$
$137$	−6.43023	−0.549372	−0.274686	+	0.961534i	$0.588574\pi$
$138$	4.88768	0.416067
$139$	1.79435	0.152195	0.0760975	−	0.997100i	$-0.475754\pi$
$139$	1.79435	0.152195	0.0760975	+	0.997100i	$0.475754\pi$
$140$	0	0
$141$	6.98610	0.588336
$142$	14.0739	1.18105
$143$	0	0
$144$	−2.08521	−0.173768
$145$	0	0
$146$	−3.83547	−0.317426
$147$	−2.26885	−0.187132
$148$	2.98725	0.245551
$149$	−8.49082	−0.695595	−0.347798	−	0.937570i	$-0.613070\pi$
$149$	−8.49082	−0.695595	−0.347798	+	0.937570i	$0.613070\pi$
$150$	0	0
$151$	−2.97802	−0.242347	−0.121174	−	0.992631i	$-0.538666\pi$
$151$	−2.97802	−0.242347	−0.121174	+	0.992631i	$0.538666\pi$
$152$	−2.65614	−0.215441
$153$	5.99323	0.484524
$154$	−2.80969	−0.226411
$155$	0	0
$156$	0	0
$157$	14.3503	1.14528	0.572639	−	0.819808i	$-0.305920\pi$
$157$	14.3503	1.14528	0.572639	+	0.819808i	$0.305920\pi$
$158$	−8.94252	−0.711428
$159$	9.64136	0.764609
$160$	0	0
$161$	−10.9934	−0.866398
$162$	1.60374	0.126002
$163$	−12.7139	−0.995833	−0.497916	−	0.867225i	$-0.665901\pi$
$163$	−12.7139	−0.995833	−0.497916	+	0.867225i	$0.665901\pi$
$164$	8.69023	0.678593
$165$	0	0
$166$	13.9980	1.08645
$167$	1.32171	0.102277	0.0511387	−	0.998692i	$-0.483715\pi$
$167$	1.32171	0.102277	0.0511387	+	0.998692i	$0.483715\pi$
$168$	−2.05754	−0.158743
$169$	0	0
$170$	0	0
$171$	5.53861	0.423548
$172$	3.35303	0.255666
$173$	23.9387	1.82002	0.910012	−	0.414581i	$-0.136072\pi$
$173$	23.9387	1.82002	0.910012	+	0.414581i	$0.136072\pi$
$174$	7.61350	0.577178
$175$	0	0
$176$	1.30608	0.0984493
$177$	−14.0550	−1.05644
$178$	15.6235	1.17103
$179$	−0.841778	−0.0629174	−0.0314587	−	0.999505i	$-0.510015\pi$
$179$	−0.841778	−0.0629174	−0.0314587	+	0.999505i	$0.510015\pi$
$180$	0	0
$181$	−6.20054	−0.460883	−0.230441	−	0.973086i	$-0.574017\pi$
$181$	−6.20054	−0.460883	−0.230441	+	0.973086i	$0.574017\pi$
$182$	0	0
$183$	10.7130	0.791930
$184$	5.11025	0.376732
$185$	0	0
$186$	−3.94914	−0.289565
$187$	−3.75387	−0.274511
$188$	7.30423	0.532716
$189$	10.4630	0.761074
$190$	0	0
$191$	−10.7222	−0.775829	−0.387915	−	0.921695i	$-0.626804\pi$
$191$	−10.7222	−0.775829	−0.387915	+	0.921695i	$0.626804\pi$
$192$	0.956446	0.0690255
$193$	−25.9564	−1.86838	−0.934191	−	0.356774i	$-0.883877\pi$
$193$	−25.9564	−1.86838	−0.934191	+	0.356774i	$0.883877\pi$
$194$	5.05416	0.362868
$195$	0	0
$196$	−2.37217	−0.169441
$197$	−5.14709	−0.366715	−0.183358	−	0.983046i	$-0.558697\pi$
$197$	−5.14709	−0.366715	−0.183358	+	0.983046i	$0.558697\pi$
$198$	−2.72345	−0.193547
$199$	−9.50447	−0.673754	−0.336877	−	0.941549i	$-0.609371\pi$
$199$	−9.50447	−0.673754	−0.336877	+	0.941549i	$0.609371\pi$
$200$	0	0
$201$	−10.5643	−0.745151
$202$	−2.02811	−0.142697
$203$	−17.1243	−1.20189
$204$	−2.74898	−0.192467
$205$	0	0
$206$	−3.26212	−0.227283
$207$	−10.6559	−0.740639
$208$	0	0
$209$	−3.46913	−0.239964
$210$	0	0
$211$	25.8770	1.78145	0.890724	−	0.454545i	$-0.150198\pi$
$211$	25.8770	1.78145	0.890724	+	0.454545i	$0.150198\pi$
$212$	10.0804	0.692325
$213$	13.4609	0.922325
$214$	1.89051	0.129233
$215$	0	0
$216$	−4.86373	−0.330935
$217$	8.88240	0.602977
$218$	4.29397	0.290824
$219$	−3.66842	−0.247889
$220$	0	0
$221$	0	0
$222$	2.85715	0.191759
$223$	−0.669364	−0.0448239	−0.0224120	−	0.999749i	$-0.507135\pi$
$223$	−0.669364	−0.0448239	−0.0224120	+	0.999749i	$0.507135\pi$
$224$	−2.15124	−0.143736
$225$	0	0
$226$	18.7154	1.24493
$227$	−5.40643	−0.358837	−0.179419	−	0.983773i	$-0.557422\pi$
$227$	−5.40643	−0.358837	−0.179419	+	0.983773i	$0.557422\pi$
$228$	−2.54046	−0.168246
$229$	−2.23462	−0.147668	−0.0738340	−	0.997271i	$-0.523524\pi$
$229$	−2.23462	−0.147668	−0.0738340	+	0.997271i	$0.523524\pi$
$230$	0	0
$231$	−2.68731	−0.176812
$232$	7.96020	0.522613
$233$	0.131688	0.00862719	0.00431360	−	0.999991i	$-0.498627\pi$
$233$	0.131688	0.00862719	0.00431360	+	0.999991i	$0.498627\pi$
$234$	0	0
$235$	0	0
$236$	−14.6950	−0.956564
$237$	−8.55303	−0.555579
$238$	6.18300	0.400784
$239$	22.7145	1.46928	0.734639	−	0.678458i	$-0.237351\pi$
$239$	22.7145	1.46928	0.734639	+	0.678458i	$0.237351\pi$
$240$	0	0
$241$	−21.9809	−1.41591	−0.707956	−	0.706256i	$-0.750383\pi$
$241$	−21.9809	−1.41591	−0.707956	+	0.706256i	$0.750383\pi$
$242$	−9.29416	−0.597451
$243$	16.1251	1.03442
$244$	11.2009	0.717062
$245$	0	0
$246$	8.31174	0.529937
$247$	0	0
$248$	−4.12897	−0.262190
$249$	13.3883	0.848450
$250$	0	0
$251$	9.38645	0.592468	0.296234	−	0.955115i	$-0.404269\pi$
$251$	9.38645	0.592468	0.296234	+	0.955115i	$0.404269\pi$
$252$	4.48579	0.282578
$253$	6.67438	0.419615
$254$	17.6498	1.10745
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.36768	0.459583	0.229792	−	0.973240i	$-0.426195\pi$
$257$	7.36768	0.459583	0.229792	+	0.973240i	$0.426195\pi$
$258$	3.20699	0.199659
$259$	−6.42630	−0.399311
$260$	0	0
$261$	−16.5987	−1.02743
$262$	8.59263	0.530855
$263$	−20.4624	−1.26177	−0.630884	−	0.775877i	$-0.717308\pi$
$263$	−20.4624	−1.26177	−0.630884	+	0.775877i	$0.717308\pi$
$264$	1.24919	0.0768825
$265$	0	0
$266$	5.71399	0.350347
$267$	14.9430	0.914498
$268$	−11.0454	−0.674706
$269$	28.7835	1.75496	0.877480	−	0.479614i	$-0.159223\pi$
$269$	28.7835	1.75496	0.877480	+	0.479614i	$0.159223\pi$
$270$	0	0
$271$	1.82589	0.110915	0.0554575	−	0.998461i	$-0.482338\pi$
$271$	1.82589	0.110915	0.0554575	+	0.998461i	$0.482338\pi$
$272$	−2.87416	−0.174271
$273$	0	0
$274$	−6.43023	−0.388465
$275$	0	0
$276$	4.88768	0.294203
$277$	−1.10372	−0.0663163	−0.0331581	−	0.999450i	$-0.510557\pi$
$277$	−1.10372	−0.0663163	−0.0331581	+	0.999450i	$0.510557\pi$
$278$	1.79435	0.107618
$279$	8.60977	0.515454
$280$	0	0
$281$	−2.66535	−0.159002	−0.0795008	−	0.996835i	$-0.525333\pi$
$281$	−2.66535	−0.159002	−0.0795008	+	0.996835i	$0.525333\pi$
$282$	6.98610	0.416016
$283$	24.5173	1.45740	0.728702	−	0.684831i	$-0.240124\pi$
$283$	24.5173	1.45740	0.728702	+	0.684831i	$0.240124\pi$
$284$	14.0739	0.835130
$285$	0	0
$286$	0	0
$287$	−18.6948	−1.10352
$288$	−2.08521	−0.122872
$289$	−8.73921	−0.514071
$290$	0	0
$291$	4.83403	0.283376
$292$	−3.83547	−0.224454
$293$	−8.74457	−0.510863	−0.255432	−	0.966827i	$-0.582218\pi$
$293$	−8.74457	−0.510863	−0.255432	+	0.966827i	$0.582218\pi$
$294$	−2.26885	−0.132322
$295$	0	0
$296$	2.98725	0.173631
$297$	−6.35241	−0.368604
$298$	−8.49082	−0.491860
$299$	0	0
$300$	0	0
$301$	−7.21317	−0.415760
$302$	−2.97802	−0.171366
$303$	−1.93977	−0.111437
$304$	−2.65614	−0.152340
$305$	0	0
$306$	5.99323	0.342610
$307$	12.4345	0.709674	0.354837	−	0.934928i	$-0.384536\pi$
$307$	12.4345	0.709674	0.354837	+	0.934928i	$0.384536\pi$
$308$	−2.80969	−0.160097
$309$	−3.12004	−0.177493
$310$	0	0
$311$	−0.269313	−0.0152713	−0.00763567	−	0.999971i	$-0.502431\pi$
$311$	−0.269313	−0.0152713	−0.00763567	+	0.999971i	$0.502431\pi$
$312$	0	0
$313$	−10.5866	−0.598389	−0.299195	−	0.954192i	$-0.596718\pi$
$313$	−10.5866	−0.598389	−0.299195	+	0.954192i	$0.596718\pi$
$314$	14.3503	0.809833
$315$	0	0
$316$	−8.94252	−0.503056
$317$	21.0117	1.18014	0.590068	−	0.807354i	$-0.299101\pi$
$317$	21.0117	1.18014	0.590068	+	0.807354i	$0.299101\pi$
$318$	9.64136	0.540661
$319$	10.3966	0.582100
$320$	0	0
$321$	1.80817	0.100922
$322$	−10.9934	−0.612636
$323$	7.63417	0.424777
$324$	1.60374	0.0890966
$325$	0	0
$326$	−12.7139	−0.704160
$327$	4.10695	0.227115
$328$	8.69023	0.479838
$329$	−15.7131	−0.866293
$330$	0	0
$331$	−26.6423	−1.46440	−0.732198	−	0.681092i	$-0.761505\pi$
$331$	−26.6423	−1.46440	−0.732198	+	0.681092i	$0.761505\pi$
$332$	13.9980	0.768239
$333$	−6.22905	−0.341350
$334$	1.32171	0.0723210
$335$	0	0
$336$	−2.05754	−0.112248
$337$	−3.43601	−0.187171	−0.0935856	−	0.995611i	$-0.529833\pi$
$337$	−3.43601	−0.187171	−0.0935856	+	0.995611i	$0.529833\pi$
$338$	0	0
$339$	17.9003	0.972211
$340$	0	0
$341$	−5.39276	−0.292034
$342$	5.53861	0.299494
$343$	20.1618	1.08863
$344$	3.35303	0.180783
$345$	0	0
$346$	23.9387	1.28695
$347$	13.6671	0.733686	0.366843	−	0.930283i	$-0.380439\pi$
$347$	13.6671	0.733686	0.366843	+	0.930283i	$0.380439\pi$
$348$	7.61350	0.408127
$349$	−8.18464	−0.438114	−0.219057	−	0.975712i	$-0.570298\pi$
$349$	−8.18464	−0.438114	−0.219057	+	0.975712i	$0.570298\pi$
$350$	0	0
$351$	0	0
$352$	1.30608	0.0696142
$353$	−21.4420	−1.14124	−0.570621	−	0.821214i	$-0.693297\pi$
$353$	−21.4420	−1.14124	−0.570621	+	0.821214i	$0.693297\pi$
$354$	−14.0550	−0.747014
$355$	0	0
$356$	15.6235	0.828043
$357$	5.91371	0.312987
$358$	−0.841778	−0.0444893
$359$	7.09493	0.374456	0.187228	−	0.982316i	$-0.440050\pi$
$359$	7.09493	0.374456	0.187228	+	0.982316i	$0.440050\pi$
$360$	0	0
$361$	−11.9449	−0.628680
$362$	−6.20054	−0.325893
$363$	−8.88936	−0.466571
$364$	0	0
$365$	0	0
$366$	10.7130	0.559979
$367$	23.9033	1.24774	0.623872	−	0.781526i	$-0.285559\pi$
$367$	23.9033	1.24774	0.623872	+	0.781526i	$0.285559\pi$
$368$	5.11025	0.266390
$369$	−18.1210	−0.943340
$370$	0	0
$371$	−21.6853	−1.12585
$372$	−3.94914	−0.204753
$373$	11.5127	0.596105	0.298053	−	0.954549i	$-0.403663\pi$
$373$	11.5127	0.596105	0.298053	+	0.954549i	$0.403663\pi$
$374$	−3.75387	−0.194108
$375$	0	0
$376$	7.30423	0.376687
$377$	0	0
$378$	10.4630	0.538161
$379$	−2.64445	−0.135836	−0.0679180	−	0.997691i	$-0.521636\pi$
$379$	−2.64445	−0.135836	−0.0679180	+	0.997691i	$0.521636\pi$
$380$	0	0
$381$	16.8811	0.864843
$382$	−10.7222	−0.548594
$383$	16.2424	0.829949	0.414975	−	0.909833i	$-0.363790\pi$
$383$	16.2424	0.829949	0.414975	+	0.909833i	$0.363790\pi$
$384$	0.956446	0.0488084
$385$	0	0
$386$	−25.9564	−1.32115
$387$	−6.99178	−0.355412
$388$	5.05416	0.256586
$389$	−23.7722	−1.20530	−0.602649	−	0.798006i	$-0.705888\pi$
$389$	−23.7722	−1.20530	−0.602649	+	0.798006i	$0.705888\pi$
$390$	0	0
$391$	−14.6877	−0.742787
$392$	−2.37217	−0.119813
$393$	8.21839	0.414563
$394$	−5.14709	−0.259307
$395$	0	0
$396$	−2.72345	−0.136858
$397$	−34.6274	−1.73790	−0.868949	−	0.494901i	$-0.835204\pi$
$397$	−34.6274	−1.73790	−0.868949	+	0.494901i	$0.835204\pi$
$398$	−9.50447	−0.476416
$399$	5.46513	0.273598
$400$	0	0
$401$	−4.37028	−0.218242	−0.109121	−	0.994028i	$-0.534804\pi$
$401$	−4.37028	−0.218242	−0.109121	+	0.994028i	$0.534804\pi$
$402$	−10.5643	−0.526901
$403$	0	0
$404$	−2.02811	−0.100902
$405$	0	0
$406$	−17.1243	−0.849865
$407$	3.90159	0.193394
$408$	−2.74898	−0.136095
$409$	11.2279	0.555186	0.277593	−	0.960699i	$-0.410463\pi$
$409$	11.2279	0.555186	0.277593	+	0.960699i	$0.410463\pi$
$410$	0	0
$411$	−6.15017	−0.303366
$412$	−3.26212	−0.160713
$413$	31.6125	1.55555
$414$	−10.6559	−0.523711
$415$	0	0
$416$	0	0
$417$	1.71620	0.0840427
$418$	−3.46913	−0.169681
$419$	18.3230	0.895137	0.447569	−	0.894250i	$-0.352290\pi$
$419$	18.3230	0.895137	0.447569	+	0.894250i	$0.352290\pi$
$420$	0	0
$421$	−8.92506	−0.434981	−0.217490	−	0.976062i	$-0.569787\pi$
$421$	−8.92506	−0.434981	−0.217490	+	0.976062i	$0.569787\pi$
$422$	25.8770	1.25967
$423$	−15.2309	−0.740550
$424$	10.0804	0.489548
$425$	0	0
$426$	13.4609	0.652182
$427$	−24.0958	−1.16608
$428$	1.89051	0.0913814
$429$	0	0
$430$	0	0
$431$	−34.8123	−1.67685	−0.838425	−	0.545017i	$-0.816523\pi$
$431$	−34.8123	−1.67685	−0.838425	+	0.545017i	$0.816523\pi$
$432$	−4.86373	−0.234006
$433$	37.9777	1.82509	0.912547	−	0.408972i	$-0.134113\pi$
$433$	37.9777	1.82509	0.912547	+	0.408972i	$0.134113\pi$
$434$	8.88240	0.426369
$435$	0	0
$436$	4.29397	0.205644
$437$	−13.5735	−0.649310
$438$	−3.66842	−0.175284
$439$	−16.6128	−0.792887	−0.396443	−	0.918059i	$-0.629756\pi$
$439$	−16.6128	−0.792887	−0.396443	+	0.918059i	$0.629756\pi$
$440$	0	0
$441$	4.94648	0.235547
$442$	0	0
$443$	−16.5892	−0.788177	−0.394089	−	0.919072i	$-0.628940\pi$
$443$	−16.5892	−0.788177	−0.394089	+	0.919072i	$0.628940\pi$
$444$	2.85715	0.135594
$445$	0	0
$446$	−0.669364	−0.0316953
$447$	−8.12101	−0.384111
$448$	−2.15124	−0.101636
$449$	19.4158	0.916287	0.458144	−	0.888878i	$-0.348515\pi$
$449$	19.4158	0.916287	0.458144	+	0.888878i	$0.348515\pi$
$450$	0	0
$451$	11.3501	0.534456
$452$	18.7154	0.880300
$453$	−2.84831	−0.133825
$454$	−5.40643	−0.253736
$455$	0	0
$456$	−2.54046	−0.118968
$457$	8.84923	0.413950	0.206975	−	0.978346i	$-0.433638\pi$
$457$	8.84923	0.413950	0.206975	+	0.978346i	$0.433638\pi$
$458$	−2.23462	−0.104417
$459$	13.9791	0.652490
$460$	0	0
$461$	17.3352	0.807380	0.403690	−	0.914896i	$-0.367727\pi$
$461$	17.3352	0.807380	0.403690	+	0.914896i	$0.367727\pi$
$462$	−2.68731	−0.125025
$463$	−17.0652	−0.793088	−0.396544	−	0.918016i	$-0.629791\pi$
$463$	−17.0652	−0.793088	−0.396544	+	0.918016i	$0.629791\pi$
$464$	7.96020	0.369543
$465$	0	0
$466$	0.131688	0.00610035
$467$	−22.5357	−1.04283	−0.521415	−	0.853303i	$-0.674596\pi$
$467$	−22.5357	−1.04283	−0.521415	+	0.853303i	$0.674596\pi$
$468$	0	0
$469$	23.7613	1.09720
$470$	0	0
$471$	13.7253	0.632427
$472$	−14.6950	−0.676393
$473$	4.37932	0.201361
$474$	−8.55303	−0.392854
$475$	0	0
$476$	6.18300	0.283397
$477$	−21.0198	−0.962429
$478$	22.7145	1.03894
$479$	33.3487	1.52374	0.761870	−	0.647731i	$-0.224282\pi$
$479$	33.3487	1.52374	0.761870	+	0.647731i	$0.224282\pi$
$480$	0	0
$481$	0	0
$482$	−21.9809	−1.00120
$483$	−10.5146	−0.478429
$484$	−9.29416	−0.422462
$485$	0	0
$486$	16.1251	0.731448
$487$	32.8658	1.48929	0.744647	−	0.667459i	$-0.232618\pi$
$487$	32.8658	1.48929	0.744647	+	0.667459i	$0.232618\pi$
$488$	11.2009	0.507040
$489$	−12.1602	−0.549903
$490$	0	0
$491$	−17.6525	−0.796645	−0.398322	−	0.917245i	$-0.630407\pi$
$491$	−17.6525	−0.796645	−0.398322	+	0.917245i	$0.630407\pi$
$492$	8.31174	0.374722
$493$	−22.8789	−1.03041
$494$	0	0
$495$	0	0
$496$	−4.12897	−0.185396
$497$	−30.2762	−1.35807
$498$	13.3883	0.599945
$499$	−37.3227	−1.67079	−0.835396	−	0.549648i	$-0.814762\pi$
$499$	−37.3227	−1.67079	−0.835396	+	0.549648i	$0.814762\pi$
$500$	0	0
$501$	1.26415	0.0564780
$502$	9.38645	0.418938
$503$	27.1489	1.21051	0.605254	−	0.796032i	$-0.293072\pi$
$503$	27.1489	1.21051	0.605254	+	0.796032i	$0.293072\pi$
$504$	4.48579	0.199813
$505$	0	0
$506$	6.67438	0.296712
$507$	0	0
$508$	17.6498	0.783083
$509$	11.3658	0.503781	0.251890	−	0.967756i	$-0.418948\pi$
$509$	11.3658	0.503781	0.251890	+	0.967756i	$0.418948\pi$
$510$	0	0
$511$	8.25101	0.365003
$512$	1.00000	0.0441942
$513$	12.9188	0.570377
$514$	7.36768	0.324975
$515$	0	0
$516$	3.20699	0.141180
$517$	9.53989	0.419564
$518$	−6.42630	−0.282355
$519$	22.8961	1.00503
$520$	0	0
$521$	31.9131	1.39814	0.699069	−	0.715054i	$-0.253598\pi$
$521$	31.9131	1.39814	0.699069	+	0.715054i	$0.253598\pi$
$522$	−16.5987	−0.726506
$523$	25.7706	1.12687	0.563435	−	0.826160i	$-0.309479\pi$
$523$	25.7706	1.12687	0.563435	+	0.826160i	$0.309479\pi$
$524$	8.59263	0.375371
$525$	0	0
$526$	−20.4624	−0.892205
$527$	11.8673	0.516948
$528$	1.24919	0.0543641
$529$	3.11463	0.135419
$530$	0	0
$531$	30.6422	1.32976
$532$	5.71399	0.247733
$533$	0	0
$534$	14.9430	0.646648
$535$	0	0
$536$	−11.0454	−0.477089
$537$	−0.805115	−0.0347433
$538$	28.7835	1.24094
$539$	−3.09824	−0.133451
$540$	0	0
$541$	−29.0524	−1.24906	−0.624531	−	0.781000i	$-0.714710\pi$
$541$	−29.0524	−1.24906	−0.624531	+	0.781000i	$0.714710\pi$
$542$	1.82589	0.0784287
$543$	−5.93048	−0.254501
$544$	−2.87416	−0.123229
$545$	0	0
$546$	0	0
$547$	−14.0848	−0.602221	−0.301111	−	0.953589i	$-0.597357\pi$
$547$	−14.0848	−0.602221	−0.301111	+	0.953589i	$0.597357\pi$
$548$	−6.43023	−0.274686
$549$	−23.3562	−0.996818
$550$	0	0
$551$	−21.1434	−0.900740
$552$	4.88768	0.208033
$553$	19.2375	0.818061
$554$	−1.10372	−0.0468927
$555$	0	0
$556$	1.79435	0.0760975
$557$	−1.65529	−0.0701367	−0.0350683	−	0.999385i	$-0.511165\pi$
$557$	−1.65529	−0.0701367	−0.0350683	+	0.999385i	$0.511165\pi$
$558$	8.60977	0.364481
$559$	0	0
$560$	0	0
$561$	−3.59038	−0.151586
$562$	−2.66535	−0.112431
$563$	−36.3020	−1.52995	−0.764974	−	0.644061i	$-0.777248\pi$
$563$	−36.3020	−1.52995	−0.764974	+	0.644061i	$0.777248\pi$
$564$	6.98610	0.294168
$565$	0	0
$566$	24.5173	1.03054
$567$	−3.45002	−0.144887
$568$	14.0739	0.590526
$569$	−15.1125	−0.633548	−0.316774	−	0.948501i	$-0.602600\pi$
$569$	−15.1125	−0.633548	−0.316774	+	0.948501i	$0.602600\pi$
$570$	0	0
$571$	−35.2350	−1.47454	−0.737269	−	0.675600i	$-0.763885\pi$
$571$	−35.2350	−1.47454	−0.737269	+	0.675600i	$0.763885\pi$
$572$	0	0
$573$	−10.2552	−0.428416
$574$	−18.6948	−0.780304
$575$	0	0
$576$	−2.08521	−0.0868838
$577$	25.4407	1.05911	0.529556	−	0.848275i	$-0.322359\pi$
$577$	25.4407	1.05911	0.529556	+	0.848275i	$0.322359\pi$
$578$	−8.73921	−0.363503
$579$	−24.8259	−1.03173
$580$	0	0
$581$	−30.1130	−1.24930
$582$	4.83403	0.200377
$583$	13.1658	0.545271
$584$	−3.83547	−0.158713
$585$	0	0
$586$	−8.74457	−0.361235
$587$	−16.4237	−0.677877	−0.338938	−	0.940809i	$-0.610068\pi$
$587$	−16.4237	−0.677877	−0.338938	+	0.940809i	$0.610068\pi$
$588$	−2.26885	−0.0935660
$589$	10.9671	0.451892
$590$	0	0
$591$	−4.92291	−0.202502
$592$	2.98725	0.122775
$593$	23.1420	0.950327	0.475163	−	0.879898i	$-0.342389\pi$
$593$	23.1420	0.950327	0.475163	+	0.879898i	$0.342389\pi$
$594$	−6.35241	−0.260643
$595$	0	0
$596$	−8.49082	−0.347798
$597$	−9.09052	−0.372050
$598$	0	0
$599$	−27.7657	−1.13448	−0.567238	−	0.823554i	$-0.691988\pi$
$599$	−27.7657	−1.13448	−0.567238	+	0.823554i	$0.691988\pi$
$600$	0	0
$601$	−13.2621	−0.540974	−0.270487	−	0.962724i	$-0.587185\pi$
$601$	−13.2621	−0.540974	−0.270487	+	0.962724i	$0.587185\pi$
$602$	−7.21317	−0.293987
$603$	23.0320	0.937936
$604$	−2.97802	−0.121174
$605$	0	0
$606$	−1.93977	−0.0787979
$607$	37.9308	1.53957	0.769783	−	0.638306i	$-0.220365\pi$
$607$	37.9308	1.53957	0.769783	+	0.638306i	$0.220365\pi$
$608$	−2.65614	−0.107721
$609$	−16.3785	−0.663689
$610$	0	0
$611$	0	0
$612$	5.99323	0.242262
$613$	−2.38979	−0.0965227	−0.0482614	−	0.998835i	$-0.515368\pi$
$613$	−2.38979	−0.0965227	−0.0482614	+	0.998835i	$0.515368\pi$
$614$	12.4345	0.501815
$615$	0	0
$616$	−2.80969	−0.113205
$617$	−19.0583	−0.767259	−0.383630	−	0.923487i	$-0.625326\pi$
$617$	−19.0583	−0.767259	−0.383630	+	0.923487i	$0.625326\pi$
$618$	−3.12004	−0.125506
$619$	−18.6920	−0.751294	−0.375647	−	0.926763i	$-0.622579\pi$
$619$	−18.6920	−0.751294	−0.375647	+	0.926763i	$0.622579\pi$
$620$	0	0
$621$	−24.8549	−0.997391
$622$	−0.269313	−0.0107985
$623$	−33.6098	−1.34655
$624$	0	0
$625$	0	0
$626$	−10.5866	−0.423125
$627$	−3.31803	−0.132509
$628$	14.3503	0.572639
$629$	−8.58584	−0.342340
$630$	0	0
$631$	−21.5119	−0.856377	−0.428188	−	0.903689i	$-0.640848\pi$
$631$	−21.5119	−0.856377	−0.428188	+	0.903689i	$0.640848\pi$
$632$	−8.94252	−0.355714
$633$	24.7500	0.983723
$634$	21.0117	0.834482
$635$	0	0
$636$	9.64136	0.382305
$637$	0	0
$638$	10.3966	0.411607
$639$	−29.3470	−1.16095
$640$	0	0
$641$	34.3509	1.35678	0.678390	−	0.734702i	$-0.262678\pi$
$641$	34.3509	1.35678	0.678390	+	0.734702i	$0.262678\pi$
$642$	1.80817	0.0713629
$643$	31.1063	1.22671	0.613356	−	0.789807i	$-0.289819\pi$
$643$	31.1063	1.22671	0.613356	+	0.789807i	$0.289819\pi$
$644$	−10.9934	−0.433199
$645$	0	0
$646$	7.63417	0.300362
$647$	−4.61263	−0.181341	−0.0906705	−	0.995881i	$-0.528901\pi$
$647$	−4.61263	−0.181341	−0.0906705	+	0.995881i	$0.528901\pi$
$648$	1.60374	0.0630008
$649$	−19.1928	−0.753385
$650$	0	0
$651$	8.49554	0.332966
$652$	−12.7139	−0.497916
$653$	−15.0219	−0.587853	−0.293926	−	0.955828i	$-0.594962\pi$
$653$	−15.0219	−0.587853	−0.293926	+	0.955828i	$0.594962\pi$
$654$	4.10695	0.160594
$655$	0	0
$656$	8.69023	0.339297
$657$	7.99776	0.312022
$658$	−15.7131	−0.612562
$659$	−37.6385	−1.46619	−0.733095	−	0.680126i	$-0.761925\pi$
$659$	−37.6385	−1.46619	−0.733095	+	0.680126i	$0.761925\pi$
$660$	0	0
$661$	14.5505	0.565949	0.282974	−	0.959127i	$-0.408679\pi$
$661$	14.5505	0.565949	0.282974	+	0.959127i	$0.408679\pi$
$662$	−26.6423	−1.03548
$663$	0	0
$664$	13.9980	0.543227
$665$	0	0
$666$	−6.22905	−0.241371
$667$	40.6786	1.57508
$668$	1.32171	0.0511387
$669$	−0.640210	−0.0247520
$670$	0	0
$671$	14.6292	0.564754
$672$	−2.05754	−0.0793715
$673$	−3.50799	−0.135223	−0.0676115	−	0.997712i	$-0.521538\pi$
$673$	−3.50799	−0.135223	−0.0676115	+	0.997712i	$0.521538\pi$
$674$	−3.43601	−0.132350
$675$	0	0
$676$	0	0
$677$	33.5349	1.28885	0.644425	−	0.764668i	$-0.277097\pi$
$677$	33.5349	1.28885	0.644425	+	0.764668i	$0.277097\pi$
$678$	17.9003	0.687457
$679$	−10.8727	−0.417256
$680$	0	0
$681$	−5.17096	−0.198152
$682$	−5.39276	−0.206499
$683$	−39.6392	−1.51675	−0.758376	−	0.651817i	$-0.774007\pi$
$683$	−39.6392	−1.51675	−0.758376	+	0.651817i	$0.774007\pi$
$684$	5.53861	0.211774
$685$	0	0
$686$	20.1618	0.769780
$687$	−2.13730	−0.0815429
$688$	3.35303	0.127833
$689$	0	0
$690$	0	0
$691$	17.6727	0.672300	0.336150	−	0.941808i	$-0.390875\pi$
$691$	17.6727	0.672300	0.336150	+	0.941808i	$0.390875\pi$
$692$	23.9387	0.910012
$693$	5.85879	0.222557
$694$	13.6671	0.518794
$695$	0	0
$696$	7.61350	0.288589
$697$	−24.9771	−0.946075
$698$	−8.18464	−0.309793
$699$	0.125953	0.00476397
$700$	0	0
$701$	−46.5635	−1.75868	−0.879340	−	0.476194i	$-0.842016\pi$
$701$	−46.5635	−1.75868	−0.879340	+	0.476194i	$0.842016\pi$
$702$	0	0
$703$	−7.93456	−0.299258
$704$	1.30608	0.0492247
$705$	0	0
$706$	−21.4420	−0.806980
$707$	4.36294	0.164085
$708$	−14.0550	−0.528219
$709$	−26.2769	−0.986849	−0.493425	−	0.869789i	$-0.664255\pi$
$709$	−26.2769	−0.986849	−0.493425	+	0.869789i	$0.664255\pi$
$710$	0	0
$711$	18.6470	0.699318
$712$	15.6235	0.585515
$713$	−21.1001	−0.790203
$714$	5.91371	0.221315
$715$	0	0
$716$	−0.841778	−0.0314587
$717$	21.7252	0.811342
$718$	7.09493	0.264780
$719$	11.3458	0.423128	0.211564	−	0.977364i	$-0.432144\pi$
$719$	11.3458	0.423128	0.211564	+	0.977364i	$0.432144\pi$
$720$	0	0
$721$	7.01759	0.261349
$722$	−11.9449	−0.444544
$723$	−21.0235	−0.781873
$724$	−6.20054	−0.230441
$725$	0	0
$726$	−8.88936	−0.329915
$727$	33.1374	1.22900	0.614499	−	0.788918i	$-0.289358\pi$
$727$	33.1374	1.22900	0.614499	+	0.788918i	$0.289358\pi$
$728$	0	0
$729$	10.6116	0.393020
$730$	0	0
$731$	−9.63714	−0.356443
$732$	10.7130	0.395965
$733$	−32.4103	−1.19710	−0.598550	−	0.801085i	$-0.704256\pi$
$733$	−32.4103	−1.19710	−0.598550	+	0.801085i	$0.704256\pi$
$734$	23.9033	0.882289
$735$	0	0
$736$	5.11025	0.188366
$737$	−14.4262	−0.531395
$738$	−18.1210	−0.667042
$739$	−19.0620	−0.701208	−0.350604	−	0.936524i	$-0.614024\pi$
$739$	−19.0620	−0.701208	−0.350604	+	0.936524i	$0.614024\pi$
$740$	0	0
$741$	0	0
$742$	−21.6853	−0.796094
$743$	−25.4983	−0.935441	−0.467721	−	0.883876i	$-0.654925\pi$
$743$	−25.4983	−0.935441	−0.467721	+	0.883876i	$0.654925\pi$
$744$	−3.94914	−0.144782
$745$	0	0
$746$	11.5127	0.421510
$747$	−29.1887	−1.06796
$748$	−3.75387	−0.137255
$749$	−4.06694	−0.148603
$750$	0	0
$751$	−19.5176	−0.712209	−0.356104	−	0.934446i	$-0.615895\pi$
$751$	−19.5176	−0.712209	−0.356104	+	0.934446i	$0.615895\pi$
$752$	7.30423	0.266358
$753$	8.97763	0.327163
$754$	0	0
$755$	0	0
$756$	10.4630	0.380537
$757$	23.6562	0.859801	0.429900	−	0.902876i	$-0.358549\pi$
$757$	23.6562	0.859801	0.429900	+	0.902876i	$0.358549\pi$
$758$	−2.64445	−0.0960506
$759$	6.38368	0.231713
$760$	0	0
$761$	16.2681	0.589717	0.294859	−	0.955541i	$-0.404727\pi$
$761$	16.2681	0.589717	0.294859	+	0.955541i	$0.404727\pi$
$762$	16.8811	0.611537
$763$	−9.23735	−0.334414
$764$	−10.7222	−0.387915
$765$	0	0
$766$	16.2424	0.586863
$767$	0	0
$768$	0.956446	0.0345128
$769$	−30.6878	−1.10663	−0.553314	−	0.832973i	$-0.686637\pi$
$769$	−30.6878	−1.10663	−0.553314	+	0.832973i	$0.686637\pi$
$770$	0	0
$771$	7.04679	0.253784
$772$	−25.9564	−0.934191
$773$	37.4115	1.34560	0.672798	−	0.739826i	$-0.265092\pi$
$773$	37.4115	1.34560	0.672798	+	0.739826i	$0.265092\pi$
$774$	−6.99178	−0.251314
$775$	0	0
$776$	5.05416	0.181434
$777$	−6.14640	−0.220501
$778$	−23.7722	−0.852275
$779$	−23.0825	−0.827016
$780$	0	0
$781$	18.3816	0.657744
$782$	−14.6877	−0.525230
$783$	−38.7163	−1.38361
$784$	−2.37217	−0.0847204
$785$	0	0
$786$	8.21839	0.293140
$787$	8.05503	0.287131	0.143565	−	0.989641i	$-0.454143\pi$
$787$	8.05503	0.287131	0.143565	+	0.989641i	$0.454143\pi$
$788$	−5.14709	−0.183358
$789$	−19.5712	−0.696754
$790$	0	0
$791$	−40.2613	−1.43153
$792$	−2.72345	−0.0967735
$793$	0	0
$794$	−34.6274	−1.22888
$795$	0	0
$796$	−9.50447	−0.336877
$797$	−4.94210	−0.175058	−0.0875290	−	0.996162i	$-0.527897\pi$
$797$	−4.94210	−0.175058	−0.0875290	+	0.996162i	$0.527897\pi$
$798$	5.46513	0.193463
$799$	−20.9935	−0.742697
$800$	0	0
$801$	−32.5783	−1.15110
$802$	−4.37028	−0.154320
$803$	−5.00942	−0.176779
$804$	−10.5643	−0.372576
$805$	0	0
$806$	0	0
$807$	27.5298	0.969096
$808$	−2.02811	−0.0713485
$809$	14.0553	0.494160	0.247080	−	0.968995i	$-0.420529\pi$
$809$	14.0553	0.494160	0.247080	+	0.968995i	$0.420529\pi$
$810$	0	0
$811$	−11.0393	−0.387644	−0.193822	−	0.981037i	$-0.562088\pi$
$811$	−11.0393	−0.387644	−0.193822	+	0.981037i	$0.562088\pi$
$812$	−17.1243	−0.600945
$813$	1.74636	0.0612477
$814$	3.90159	0.136750
$815$	0	0
$816$	−2.74898	−0.0962335
$817$	−8.90612	−0.311586
$818$	11.2279	0.392575
$819$	0	0
$820$	0	0
$821$	4.13968	0.144476	0.0722379	−	0.997387i	$-0.476986\pi$
$821$	4.13968	0.144476	0.0722379	+	0.997387i	$0.476986\pi$
$822$	−6.15017	−0.214512
$823$	−27.7827	−0.968445	−0.484222	−	0.874945i	$-0.660897\pi$
$823$	−27.7827	−0.968445	−0.484222	+	0.874945i	$0.660897\pi$
$824$	−3.26212	−0.113641
$825$	0	0
$826$	31.6125	1.09994
$827$	6.46886	0.224944	0.112472	−	0.993655i	$-0.464123\pi$
$827$	6.46886	0.224944	0.112472	+	0.993655i	$0.464123\pi$
$828$	−10.6559	−0.370320
$829$	13.9415	0.484208	0.242104	−	0.970250i	$-0.422162\pi$
$829$	13.9415	0.484208	0.242104	+	0.970250i	$0.422162\pi$
$830$	0	0
$831$	−1.05565	−0.0366201
$832$	0	0
$833$	6.81800	0.236230
$834$	1.71620	0.0594272
$835$	0	0
$836$	−3.46913	−0.119982
$837$	20.0822	0.694142
$838$	18.3230	0.632958
$839$	−41.0483	−1.41714	−0.708571	−	0.705639i	$-0.750660\pi$
$839$	−41.0483	−1.41714	−0.708571	+	0.705639i	$0.750660\pi$
$840$	0	0
$841$	34.3648	1.18499
$842$	−8.92506	−0.307578
$843$	−2.54927	−0.0878014
$844$	25.8770	0.890724
$845$	0	0
$846$	−15.2309	−0.523648
$847$	19.9940	0.687001
$848$	10.0804	0.346162
$849$	23.4495	0.804785
$850$	0	0
$851$	15.2656	0.523298
$852$	13.4609	0.461162
$853$	41.4242	1.41834	0.709168	−	0.705039i	$-0.249071\pi$
$853$	41.4242	1.41834	0.709168	+	0.705039i	$0.249071\pi$
$854$	−24.0958	−0.824540
$855$	0	0
$856$	1.89051	0.0646164
$857$	48.9642	1.67259	0.836293	−	0.548282i	$-0.184718\pi$
$857$	48.9642	1.67259	0.836293	+	0.548282i	$0.184718\pi$
$858$	0	0
$859$	−17.0890	−0.583068	−0.291534	−	0.956560i	$-0.594166\pi$
$859$	−17.0890	−0.583068	−0.291534	+	0.956560i	$0.594166\pi$
$860$	0	0
$861$	−17.8805	−0.609367
$862$	−34.8123	−1.18571
$863$	12.3756	0.421269	0.210635	−	0.977565i	$-0.432447\pi$
$863$	12.3756	0.421269	0.210635	+	0.977565i	$0.432447\pi$
$864$	−4.86373	−0.165467
$865$	0	0
$866$	37.9777	1.29054
$867$	−8.35858	−0.283872
$868$	8.88240	0.301488
$869$	−11.6796	−0.396204
$870$	0	0
$871$	0	0
$872$	4.29397	0.145412
$873$	−10.5390	−0.356691
$874$	−13.5735	−0.459132
$875$	0	0
$876$	−3.66842	−0.123944
$877$	29.3663	0.991629	0.495814	−	0.868429i	$-0.334870\pi$
$877$	29.3663	0.991629	0.495814	+	0.868429i	$0.334870\pi$
$878$	−16.6128	−0.560656
$879$	−8.36371	−0.282101
$880$	0	0
$881$	6.22016	0.209563	0.104781	−	0.994495i	$-0.466586\pi$
$881$	6.22016	0.209563	0.104781	+	0.994495i	$0.466586\pi$
$882$	4.94648	0.166557
$883$	32.4762	1.09291	0.546456	−	0.837488i	$-0.315977\pi$
$883$	32.4762	1.09291	0.546456	+	0.837488i	$0.315977\pi$
$884$	0	0
$885$	0	0
$886$	−16.5892	−0.557325
$887$	36.5930	1.22867	0.614337	−	0.789044i	$-0.289424\pi$
$887$	36.5930	1.22867	0.614337	+	0.789044i	$0.289424\pi$
$888$	2.85715	0.0958796
$889$	−37.9689	−1.27344
$890$	0	0
$891$	2.09461	0.0701720
$892$	−0.669364	−0.0224120
$893$	−19.4011	−0.649232
$894$	−8.12101	−0.271607
$895$	0	0
$896$	−2.15124	−0.0718678
$897$	0	0
$898$	19.4158	0.647913
$899$	−32.8674	−1.09619
$900$	0	0
$901$	−28.9727	−0.965220
$902$	11.3501	0.377918
$903$	−6.89901	−0.229585
$904$	18.7154	0.622466
$905$	0	0
$906$	−2.84831	−0.0946288
$907$	6.23158	0.206916	0.103458	−	0.994634i	$-0.467009\pi$
$907$	6.23158	0.206916	0.103458	+	0.994634i	$0.467009\pi$
$908$	−5.40643	−0.179419
$909$	4.22903	0.140268
$910$	0	0
$911$	47.4457	1.57195	0.785973	−	0.618261i	$-0.212163\pi$
$911$	47.4457	1.57195	0.785973	+	0.618261i	$0.212163\pi$
$912$	−2.54046	−0.0841229
$913$	18.2825	0.605061
$914$	8.84923	0.292707
$915$	0	0
$916$	−2.23462	−0.0738340
$917$	−18.4848	−0.610422
$918$	13.9791	0.461380
$919$	−52.7882	−1.74132	−0.870661	−	0.491883i	$-0.836309\pi$
$919$	−52.7882	−1.74132	−0.870661	+	0.491883i	$0.836309\pi$
$920$	0	0
$921$	11.8929	0.391885
$922$	17.3352	0.570904
$923$	0	0
$924$	−2.68731	−0.0884061
$925$	0	0
$926$	−17.0652	−0.560798
$927$	6.80220	0.223414
$928$	7.96020	0.261306
$929$	7.11857	0.233553	0.116776	−	0.993158i	$-0.462744\pi$
$929$	7.11857	0.233553	0.116776	+	0.993158i	$0.462744\pi$
$930$	0	0
$931$	6.30082	0.206501
$932$	0.131688	0.00431360
$933$	−0.257583	−0.00843290
$934$	−22.5357	−0.737392
$935$	0	0
$936$	0	0
$937$	5.13448	0.167736	0.0838680	−	0.996477i	$-0.473273\pi$
$937$	5.13448	0.167736	0.0838680	+	0.996477i	$0.473273\pi$
$938$	23.7613	0.775834
$939$	−10.1255	−0.330433
$940$	0	0
$941$	−44.3630	−1.44619	−0.723096	−	0.690748i	$-0.757281\pi$
$941$	−44.3630	−1.44619	−0.723096	+	0.690748i	$0.757281\pi$
$942$	13.7253	0.447193
$943$	44.4092	1.44616
$944$	−14.6950	−0.478282
$945$	0	0
$946$	4.37932	0.142384
$947$	54.3234	1.76527	0.882637	−	0.470056i	$-0.155766\pi$
$947$	54.3234	1.76527	0.882637	+	0.470056i	$0.155766\pi$
$948$	−8.55303	−0.277790
$949$	0	0
$950$	0	0
$951$	20.0966	0.651676
$952$	6.18300	0.200392
$953$	37.8138	1.22491	0.612454	−	0.790506i	$-0.290182\pi$
$953$	37.8138	1.22491	0.612454	+	0.790506i	$0.290182\pi$
$954$	−21.0198	−0.680540
$955$	0	0
$956$	22.7145	0.734639
$957$	9.94383	0.321438
$958$	33.3487	1.07745
$959$	13.8330	0.446690
$960$	0	0
$961$	−13.9516	−0.450052
$962$	0	0
$963$	−3.94212	−0.127033
$964$	−21.9809	−0.707956
$965$	0	0
$966$	−10.5146	−0.338300
$967$	−8.18366	−0.263169	−0.131584	−	0.991305i	$-0.542006\pi$
$967$	−8.18366	−0.263169	−0.131584	+	0.991305i	$0.542006\pi$
$968$	−9.29416	−0.298726
$969$	7.30167	0.234563
$970$	0	0
$971$	−19.0931	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$971$	−19.0931	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$972$	16.1251	0.517212
$973$	−3.86008	−0.123748
$974$	32.8658	1.05309
$975$	0	0
$976$	11.2009	0.358531
$977$	−2.67273	−0.0855082	−0.0427541	−	0.999086i	$-0.513613\pi$
$977$	−2.67273	−0.0855082	−0.0427541	+	0.999086i	$0.513613\pi$
$978$	−12.1602	−0.388840
$979$	20.4055	0.652162
$980$	0	0
$981$	−8.95383	−0.285874
$982$	−17.6525	−0.563313
$983$	5.29656	0.168934	0.0844670	−	0.996426i	$-0.473081\pi$
$983$	5.29656	0.168934	0.0844670	+	0.996426i	$0.473081\pi$
$984$	8.31174	0.264969
$985$	0	0
$986$	−22.8789	−0.728612
$987$	−15.0288	−0.478371
$988$	0	0
$989$	17.1348	0.544855
$990$	0	0
$991$	−26.1528	−0.830773	−0.415386	−	0.909645i	$-0.636354\pi$
$991$	−26.1528	−0.830773	−0.415386	+	0.909645i	$0.636354\pi$
$992$	−4.12897	−0.131095
$993$	−25.4820	−0.808646
$994$	−30.2762	−0.960304
$995$	0	0
$996$	13.3883	0.424225
$997$	−19.6903	−0.623597	−0.311799	−	0.950148i	$-0.600931\pi$
$997$	−19.6903	−0.623597	−0.311799	+	0.950148i	$0.600931\pi$
$998$	−37.3227	−1.18143
$999$	−14.5292	−0.459683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.da.1.6		9
5.2	odd	4		1690.2.b.g.339.13	yes	18
5.3	odd	4		1690.2.b.g.339.6	yes	18
5.4	even	2		8450.2.a.ct.1.4		9
13.12	even	2		8450.2.a.cw.1.6		9
65.8	even	4		1690.2.c.h.1689.13		18
65.12	odd	4		1690.2.b.f.339.4	✓	18
65.18	even	4		1690.2.c.g.1689.13		18
65.38	odd	4		1690.2.b.f.339.15	yes	18
65.47	even	4		1690.2.c.g.1689.6		18
65.57	even	4		1690.2.c.h.1689.6		18
65.64	even	2		8450.2.a.cx.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.b.f.339.4	✓	18	65.12	odd	4
1690.2.b.f.339.15	yes	18	65.38	odd	4
1690.2.b.g.339.6	yes	18	5.3	odd	4
1690.2.b.g.339.13	yes	18	5.2	odd	4
1690.2.c.g.1689.6		18	65.47	even	4
1690.2.c.g.1689.13		18	65.18	even	4
1690.2.c.h.1689.6		18	65.57	even	4
1690.2.c.h.1689.13		18	65.8	even	4
8450.2.a.ct.1.4		9	5.4	even	2
8450.2.a.cw.1.6		9	13.12	even	2
8450.2.a.cx.1.4		9	65.64	even	2
8450.2.a.da.1.6		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.956446 q^{3} +1.00000 q^{4} +0.956446 q^{6} -2.15124 q^{7} +1.00000 q^{8} -2.08521 q^{9} +1.30608 q^{11} +0.956446 q^{12} -2.15124 q^{14} +1.00000 q^{16} -2.87416 q^{17} -2.08521 q^{18} -2.65614 q^{19} -2.05754 q^{21} +1.30608 q^{22} +5.11025 q^{23} +0.956446 q^{24} -4.86373 q^{27} -2.15124 q^{28} +7.96020 q^{29} -4.12897 q^{31} +1.00000 q^{32} +1.24919 q^{33} -2.87416 q^{34} -2.08521 q^{36} +2.98725 q^{37} -2.65614 q^{38} +8.69023 q^{41} -2.05754 q^{42} +3.35303 q^{43} +1.30608 q^{44} +5.11025 q^{46} +7.30423 q^{47} +0.956446 q^{48} -2.37217 q^{49} -2.74898 q^{51} +10.0804 q^{53} -4.86373 q^{54} -2.15124 q^{56} -2.54046 q^{57} +7.96020 q^{58} -14.6950 q^{59} +11.2009 q^{61} -4.12897 q^{62} +4.48579 q^{63} +1.00000 q^{64} +1.24919 q^{66} -11.0454 q^{67} -2.87416 q^{68} +4.88768 q^{69} +14.0739 q^{71} -2.08521 q^{72} -3.83547 q^{73} +2.98725 q^{74} -2.65614 q^{76} -2.80969 q^{77} -8.94252 q^{79} +1.60374 q^{81} +8.69023 q^{82} +13.9980 q^{83} -2.05754 q^{84} +3.35303 q^{86} +7.61350 q^{87} +1.30608 q^{88} +15.6235 q^{89} +5.11025 q^{92} -3.94914 q^{93} +7.30423 q^{94} +0.956446 q^{96} +5.05416 q^{97} -2.37217 q^{98} -2.72345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 7 q^{3} + 9 q^{4} + 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} + 4 q^{11} + 7 q^{12} + q^{14} + 9 q^{16} + 12 q^{17} + 8 q^{18} + 6 q^{19} + 8 q^{21} + 4 q^{22} + 11 q^{23} + 7 q^{24} + 34 q^{27}+ \cdots + 16 q^{99}+O(q^{100}) \) 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

Introduction

L-functions

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