LMFDB - Embedded newform 8450.2.a.cy.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 = [9,9,0,9,0,0,-4,9,15,0,-1,0,0,-4,0,9,2] 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} - 21x^{7} - 3x^{6} + 133x^{5} + 28x^{4} - 249x^{3} + 21x^{2} + 126x - 43 $ x^9 - 21x^7 - 3x^6 + 133x^5 + 28x^4 - 249x^3 + 21x^2 + 126*x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.943657$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.943657 q^{3} +1.00000 q^{4} +0.943657 q^{6} +4.61826 q^{7} +1.00000 q^{8} -2.10951 q^{9} +2.88138 q^{11} +0.943657 q^{12} +4.61826 q^{14} +1.00000 q^{16} +3.15453 q^{17} -2.10951 q^{18} -0.0635301 q^{19} +4.35806 q^{21} +2.88138 q^{22} -1.56571 q^{23} +0.943657 q^{24} -4.82163 q^{27} +4.61826 q^{28} -8.04979 q^{29} -7.30744 q^{31} +1.00000 q^{32} +2.71904 q^{33} +3.15453 q^{34} -2.10951 q^{36} +7.82786 q^{37} -0.0635301 q^{38} +11.7352 q^{41} +4.35806 q^{42} +4.77211 q^{43} +2.88138 q^{44} -1.56571 q^{46} +5.61549 q^{47} +0.943657 q^{48} +14.3284 q^{49} +2.97680 q^{51} -1.26236 q^{53} -4.82163 q^{54} +4.61826 q^{56} -0.0599506 q^{57} -8.04979 q^{58} -10.3344 q^{59} +4.48215 q^{61} -7.30744 q^{62} -9.74228 q^{63} +1.00000 q^{64} +2.71904 q^{66} +14.9547 q^{67} +3.15453 q^{68} -1.47750 q^{69} +5.03136 q^{71} -2.10951 q^{72} +9.09682 q^{73} +7.82786 q^{74} -0.0635301 q^{76} +13.3070 q^{77} +9.16845 q^{79} +1.77857 q^{81} +11.7352 q^{82} +5.55586 q^{83} +4.35806 q^{84} +4.77211 q^{86} -7.59624 q^{87} +2.88138 q^{88} -6.75492 q^{89} -1.56571 q^{92} -6.89571 q^{93} +5.61549 q^{94} +0.943657 q^{96} +6.91035 q^{97} +14.3284 q^{98} -6.07831 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + 9 q^{4} - 4 q^{7} + 9 q^{8} + 15 q^{9} - q^{11} - 4 q^{14} + 9 q^{16} + 2 q^{17} + 15 q^{18} + 10 q^{19} + 3 q^{21} - q^{22} - q^{23} - 9 q^{27} - 4 q^{28} - q^{29} - 4 q^{31} + 9 q^{32}+ \cdots - 62 q^{99}+O(q^{100}) $ 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^7 + 9 * q^8 + 15 * q^9 - q^11 - 4 * q^14 + 9 * q^16 + 2 * q^17 + 15 * q^18 + 10 * q^19 + 3 * q^21 - q^22 - q^23 - 9 * q^27 - 4 * q^28 - q^29 - 4 * q^31 + 9 * q^32 + 30 * q^33 + 2 * q^34 + 15 * q^36 + 18 * q^37 + 10 * q^38 - 2 * q^41 + 3 * q^42 - 5 * q^43 - q^44 - q^46 - 10 * q^47 + 39 * q^49 + 38 * q^51 + 3 * q^53 - 9 * q^54 - 4 * q^56 + 14 * q^57 - q^58 - 20 * q^59 + 40 * q^61 - 4 * q^62 - 45 * q^63 + 9 * q^64 + 30 * q^66 + 15 * q^67 + 2 * q^68 + 68 * q^69 + 46 * q^71 + 15 * q^72 + 13 * q^73 + 18 * q^74 + 10 * q^76 + 13 * q^77 + 37 * q^79 + 53 * q^81 - 2 * q^82 + 2 * q^83 + 3 * q^84 - 5 * q^86 + q^87 - q^88 + 14 * q^89 - q^92 + 2 * q^93 - 10 * q^94 + 23 * q^97 + 39 * q^98 - 62 * 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
$3$	0.943657	0.544821	0.272410	−	0.962181i	$-0.412179\pi$
$3$	0.943657	0.544821	0.272410	+	0.962181i	$0.412179\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.943657	0.385246
$7$	4.61826	1.74554	0.872770	−	0.488132i	$-0.162322\pi$
$7$	4.61826	1.74554	0.872770	+	0.488132i	$0.162322\pi$
$8$	1.00000	0.353553
$9$	−2.10951	−0.703170
$10$	0	0
$11$	2.88138	0.868770	0.434385	−	0.900727i	$-0.356966\pi$
$11$	2.88138	0.868770	0.434385	+	0.900727i	$0.356966\pi$
$12$	0.943657	0.272410
$13$	0	0
$13$	0	0
$14$	4.61826	1.23428
$15$	0	0
$16$	1.00000	0.250000
$17$	3.15453	0.765087	0.382543	−	0.923938i	$-0.375048\pi$
$17$	3.15453	0.765087	0.382543	+	0.923938i	$0.375048\pi$
$18$	−2.10951	−0.497217
$19$	−0.0635301	−0.0145748	−0.00728740	−	0.999973i	$-0.502320\pi$
$19$	−0.0635301	−0.0145748	−0.00728740	+	0.999973i	$0.502320\pi$
$20$	0	0
$21$	4.35806	0.951006
$22$	2.88138	0.614313
$23$	−1.56571	−0.326474	−0.163237	−	0.986587i	$-0.552193\pi$
$23$	−1.56571	−0.326474	−0.163237	+	0.986587i	$0.552193\pi$
$24$	0.943657	0.192623
$25$	0	0
$26$	0	0
$27$	−4.82163	−0.927923
$28$	4.61826	0.872770
$29$	−8.04979	−1.49481	−0.747404	−	0.664370i	$-0.768700\pi$
$29$	−8.04979	−1.49481	−0.747404	+	0.664370i	$0.768700\pi$
$30$	0	0
$31$	−7.30744	−1.31245	−0.656227	−	0.754563i	$-0.727849\pi$
$31$	−7.30744	−1.31245	−0.656227	+	0.754563i	$0.727849\pi$
$32$	1.00000	0.176777
$33$	2.71904	0.473324
$34$	3.15453	0.540998
$35$	0	0
$36$	−2.10951	−0.351585
$37$	7.82786	1.28689	0.643446	−	0.765491i	$-0.277504\pi$
$37$	7.82786	1.28689	0.643446	+	0.765491i	$0.277504\pi$
$38$	−0.0635301	−0.0103059
$39$	0	0
$40$	0	0
$41$	11.7352	1.83274	0.916368	−	0.400336i	$-0.131107\pi$
$41$	11.7352	1.83274	0.916368	+	0.400336i	$0.131107\pi$
$42$	4.35806	0.672463
$43$	4.77211	0.727740	0.363870	−	0.931450i	$-0.381455\pi$
$43$	4.77211	0.727740	0.363870	+	0.931450i	$0.381455\pi$
$44$	2.88138	0.434385
$45$	0	0
$46$	−1.56571	−0.230852
$47$	5.61549	0.819104	0.409552	−	0.912287i	$-0.365685\pi$
$47$	5.61549	0.819104	0.409552	+	0.912287i	$0.365685\pi$
$48$	0.943657	0.136205
$49$	14.3284	2.04691
$50$	0	0
$51$	2.97680	0.416835
$52$	0	0
$53$	−1.26236	−0.173399	−0.0866993	−	0.996235i	$-0.527632\pi$
$53$	−1.26236	−0.173399	−0.0866993	+	0.996235i	$0.527632\pi$
$54$	−4.82163	−0.656140
$55$	0	0
$56$	4.61826	0.617141
$57$	−0.0599506	−0.00794065
$58$	−8.04979	−1.05699
$59$	−10.3344	−1.34543	−0.672715	−	0.739902i	$-0.734872\pi$
$59$	−10.3344	−1.34543	−0.672715	+	0.739902i	$0.734872\pi$
$60$	0	0
$61$	4.48215	0.573880	0.286940	−	0.957949i	$-0.407362\pi$
$61$	4.48215	0.573880	0.286940	+	0.957949i	$0.407362\pi$
$62$	−7.30744	−0.928045
$63$	−9.74228	−1.22741
$64$	1.00000	0.125000
$65$	0	0
$66$	2.71904	0.334690
$67$	14.9547	1.82701	0.913506	−	0.406826i	$-0.133364\pi$
$67$	14.9547	1.82701	0.913506	+	0.406826i	$0.133364\pi$
$68$	3.15453	0.382543
$69$	−1.47750	−0.177870
$70$	0	0
$71$	5.03136	0.597113	0.298556	−	0.954392i	$-0.403495\pi$
$71$	5.03136	0.597113	0.298556	+	0.954392i	$0.403495\pi$
$72$	−2.10951	−0.248608
$73$	9.09682	1.06470	0.532351	−	0.846523i	$-0.321309\pi$
$73$	9.09682	1.06470	0.532351	+	0.846523i	$0.321309\pi$
$74$	7.82786	0.909971
$75$	0	0
$76$	−0.0635301	−0.00728740
$77$	13.3070	1.51647
$78$	0	0
$79$	9.16845	1.03153	0.515765	−	0.856730i	$-0.327508\pi$
$79$	9.16845	1.03153	0.515765	+	0.856730i	$0.327508\pi$
$80$	0	0
$81$	1.77857	0.197619
$82$	11.7352	1.29594
$83$	5.55586	0.609835	0.304917	−	0.952379i	$-0.401371\pi$
$83$	5.55586	0.609835	0.304917	+	0.952379i	$0.401371\pi$
$84$	4.35806	0.475503
$85$	0	0
$86$	4.77211	0.514590
$87$	−7.59624	−0.814402
$88$	2.88138	0.307157
$89$	−6.75492	−0.716020	−0.358010	−	0.933718i	$-0.616545\pi$
$89$	−6.75492	−0.716020	−0.358010	+	0.933718i	$0.616545\pi$
$90$	0	0
$91$	0	0
$92$	−1.56571	−0.163237
$93$	−6.89571	−0.715052
$94$	5.61549	0.579194
$95$	0	0
$96$	0.943657	0.0963116
$97$	6.91035	0.701640	0.350820	−	0.936443i	$-0.385903\pi$
$97$	6.91035	0.701640	0.350820	+	0.936443i	$0.385903\pi$
$98$	14.3284	1.44738
$99$	−6.07831	−0.610893
$100$	0	0
$101$	−11.5103	−1.14532	−0.572658	−	0.819795i	$-0.694088\pi$
$101$	−11.5103	−1.14532	−0.572658	+	0.819795i	$0.694088\pi$
$102$	2.97680	0.294747
$103$	−5.83062	−0.574508	−0.287254	−	0.957854i	$-0.592742\pi$
$103$	−5.83062	−0.574508	−0.287254	+	0.957854i	$0.592742\pi$
$104$	0	0
$105$	0	0
$106$	−1.26236	−0.122611
$107$	−13.5760	−1.31244	−0.656220	−	0.754569i	$-0.727846\pi$
$107$	−13.5760	−1.31244	−0.656220	+	0.754569i	$0.727846\pi$
$108$	−4.82163	−0.463961
$109$	−9.22925	−0.884002	−0.442001	−	0.897015i	$-0.645731\pi$
$109$	−9.22925	−0.884002	−0.442001	+	0.897015i	$0.645731\pi$
$110$	0	0
$111$	7.38682	0.701126
$112$	4.61826	0.436385
$113$	14.6232	1.37564	0.687819	−	0.725882i	$-0.258568\pi$
$113$	14.6232	1.37564	0.687819	+	0.725882i	$0.258568\pi$
$114$	−0.0599506	−0.00561489
$115$	0	0
$116$	−8.04979	−0.747404
$117$	0	0
$118$	−10.3344	−0.951362
$119$	14.5685	1.33549
$120$	0	0
$121$	−2.69763	−0.245239
$122$	4.48215	0.405794
$123$	11.0740	0.998513
$124$	−7.30744	−0.656227
$125$	0	0
$126$	−9.74228	−0.867911
$127$	−7.21042	−0.639821	−0.319911	−	0.947448i	$-0.603653\pi$
$127$	−7.21042	−0.639821	−0.319911	+	0.947448i	$0.603653\pi$
$128$	1.00000	0.0883883
$129$	4.50323	0.396488
$130$	0	0
$131$	−1.54747	−0.135203	−0.0676015	−	0.997712i	$-0.521535\pi$
$131$	−1.54747	−0.135203	−0.0676015	+	0.997712i	$0.521535\pi$
$132$	2.71904	0.236662
$133$	−0.293399	−0.0254409
$134$	14.9547	1.29189
$135$	0	0
$136$	3.15453	0.270499
$137$	−7.07504	−0.604462	−0.302231	−	0.953235i	$-0.597731\pi$
$137$	−7.07504	−0.604462	−0.302231	+	0.953235i	$0.597731\pi$
$138$	−1.47750	−0.125773
$139$	−13.6714	−1.15959	−0.579795	−	0.814762i	$-0.696868\pi$
$139$	−13.6714	−1.15959	−0.579795	+	0.814762i	$0.696868\pi$
$140$	0	0
$141$	5.29910	0.446265
$142$	5.03136	0.422223
$143$	0	0
$144$	−2.10951	−0.175793
$145$	0	0
$146$	9.09682	0.752859
$147$	13.5211	1.11520
$148$	7.82786	0.643446
$149$	−12.5436	−1.02761	−0.513805	−	0.857907i	$-0.671765\pi$
$149$	−12.5436	−1.02761	−0.513805	+	0.857907i	$0.671765\pi$
$150$	0	0
$151$	−19.6644	−1.60026	−0.800132	−	0.599824i	$-0.795237\pi$
$151$	−19.6644	−1.60026	−0.800132	+	0.599824i	$0.795237\pi$
$152$	−0.0635301	−0.00515297
$153$	−6.65452	−0.537986
$154$	13.3070	1.07231
$155$	0	0
$156$	0	0
$157$	20.3758	1.62616	0.813081	−	0.582150i	$-0.197788\pi$
$157$	20.3758	1.62616	0.813081	+	0.582150i	$0.197788\pi$
$158$	9.16845	0.729403
$159$	−1.19124	−0.0944711
$160$	0	0
$161$	−7.23087	−0.569873
$162$	1.77857	0.139738
$163$	20.9608	1.64177	0.820887	−	0.571091i	$-0.193480\pi$
$163$	20.9608	1.64177	0.820887	+	0.571091i	$0.193480\pi$
$164$	11.7352	0.916368
$165$	0	0
$166$	5.55586	0.431218
$167$	−21.7851	−1.68578	−0.842890	−	0.538086i	$-0.819147\pi$
$167$	−21.7851	−1.68578	−0.842890	+	0.538086i	$0.819147\pi$
$168$	4.35806	0.336231
$169$	0	0
$170$	0	0
$171$	0.134017	0.0102486
$172$	4.77211	0.363870
$173$	−12.3298	−0.937415	−0.468708	−	0.883353i	$-0.655280\pi$
$173$	−12.3298	−0.937415	−0.468708	+	0.883353i	$0.655280\pi$
$174$	−7.59624	−0.575869
$175$	0	0
$176$	2.88138	0.217192
$177$	−9.75217	−0.733018
$178$	−6.75492	−0.506303
$179$	19.9819	1.49352	0.746760	−	0.665094i	$-0.231608\pi$
$179$	19.9819	1.49352	0.746760	+	0.665094i	$0.231608\pi$
$180$	0	0
$181$	4.90650	0.364697	0.182349	−	0.983234i	$-0.441630\pi$
$181$	4.90650	0.364697	0.182349	+	0.983234i	$0.441630\pi$
$182$	0	0
$183$	4.22961	0.312662
$184$	−1.56571	−0.115426
$185$	0	0
$186$	−6.89571	−0.505618
$187$	9.08942	0.664684
$188$	5.61549	0.409552
$189$	−22.2675	−1.61973
$190$	0	0
$191$	−14.8526	−1.07470	−0.537349	−	0.843360i	$-0.680574\pi$
$191$	−14.8526	−1.07470	−0.537349	+	0.843360i	$0.680574\pi$
$192$	0.943657	0.0681026
$193$	−26.4730	−1.90557	−0.952784	−	0.303649i	$-0.901795\pi$
$193$	−26.4730	−1.90557	−0.952784	+	0.303649i	$0.901795\pi$
$194$	6.91035	0.496135
$195$	0	0
$196$	14.3284	1.02345
$197$	4.97462	0.354427	0.177213	−	0.984172i	$-0.443292\pi$
$197$	4.97462	0.354427	0.177213	+	0.984172i	$0.443292\pi$
$198$	−6.07831	−0.431967
$199$	14.6426	1.03799	0.518994	−	0.854778i	$-0.326307\pi$
$199$	14.6426	1.03799	0.518994	+	0.854778i	$0.326307\pi$
$200$	0	0
$201$	14.1121	0.995394
$202$	−11.5103	−0.809860
$203$	−37.1760	−2.60925
$204$	2.97680	0.208417
$205$	0	0
$206$	−5.83062	−0.406239
$207$	3.30289	0.229567
$208$	0	0
$209$	−0.183055	−0.0126621
$210$	0	0
$211$	6.86084	0.472319	0.236160	−	0.971714i	$-0.424111\pi$
$211$	6.86084	0.472319	0.236160	+	0.971714i	$0.424111\pi$
$212$	−1.26236	−0.0866993
$213$	4.74788	0.325319
$214$	−13.5760	−0.928035
$215$	0	0
$216$	−4.82163	−0.328070
$217$	−33.7477	−2.29094
$218$	−9.22925	−0.625084
$219$	8.58428	0.580072
$220$	0	0
$221$	0	0
$222$	7.38682	0.495771
$223$	−21.0169	−1.40739	−0.703697	−	0.710500i	$-0.748469\pi$
$223$	−21.0169	−1.40739	−0.703697	+	0.710500i	$0.748469\pi$
$224$	4.61826	0.308571
$225$	0	0
$226$	14.6232	0.972723
$227$	6.56269	0.435581	0.217791	−	0.975996i	$-0.430115\pi$
$227$	6.56269	0.435581	0.217791	+	0.975996i	$0.430115\pi$
$228$	−0.0599506	−0.00397033
$229$	16.0898	1.06325	0.531623	−	0.846981i	$-0.321583\pi$
$229$	16.0898	1.06325	0.531623	+	0.846981i	$0.321583\pi$
$230$	0	0
$231$	12.5572	0.826205
$232$	−8.04979	−0.528494
$233$	−6.58375	−0.431316	−0.215658	−	0.976469i	$-0.569190\pi$
$233$	−6.58375	−0.431316	−0.215658	+	0.976469i	$0.569190\pi$
$234$	0	0
$235$	0	0
$236$	−10.3344	−0.672715
$237$	8.65187	0.561999
$238$	14.5685	0.944333
$239$	−11.5368	−0.746256	−0.373128	−	0.927780i	$-0.621715\pi$
$239$	−11.5368	−0.746256	−0.373128	+	0.927780i	$0.621715\pi$
$240$	0	0
$241$	−21.7563	−1.40144	−0.700722	−	0.713434i	$-0.747139\pi$
$241$	−21.7563	−1.40144	−0.700722	+	0.713434i	$0.747139\pi$
$242$	−2.69763	−0.173410
$243$	16.1432	1.03559
$244$	4.48215	0.286940
$245$	0	0
$246$	11.0740	0.706055
$247$	0	0
$248$	−7.30744	−0.464023
$249$	5.24283	0.332251
$250$	0	0
$251$	22.7936	1.43872	0.719361	−	0.694637i	$-0.244435\pi$
$251$	22.7936	1.43872	0.719361	+	0.694637i	$0.244435\pi$
$252$	−9.74228	−0.613706
$253$	−4.51142	−0.283630
$254$	−7.21042	−0.452422
$255$	0	0
$256$	1.00000	0.0625000
$257$	16.4152	1.02395	0.511976	−	0.859000i	$-0.328914\pi$
$257$	16.4152	1.02395	0.511976	+	0.859000i	$0.328914\pi$
$258$	4.50323	0.280359
$259$	36.1511	2.24632
$260$	0	0
$261$	16.9811	1.05110
$262$	−1.54747	−0.0956029
$263$	−6.81715	−0.420364	−0.210182	−	0.977662i	$-0.567406\pi$
$263$	−6.81715	−0.420364	−0.210182	+	0.977662i	$0.567406\pi$
$264$	2.71904	0.167345
$265$	0	0
$266$	−0.293399	−0.0179894
$267$	−6.37433	−0.390103
$268$	14.9547	0.913506
$269$	4.49766	0.274227	0.137114	−	0.990555i	$-0.456217\pi$
$269$	4.49766	0.274227	0.137114	+	0.990555i	$0.456217\pi$
$270$	0	0
$271$	−15.7937	−0.959401	−0.479700	−	0.877432i	$-0.659255\pi$
$271$	−15.7937	−0.959401	−0.479700	+	0.877432i	$0.659255\pi$
$272$	3.15453	0.191272
$273$	0	0
$274$	−7.07504	−0.427419
$275$	0	0
$276$	−1.47750	−0.0889348
$277$	−11.9344	−0.717068	−0.358534	−	0.933517i	$-0.616723\pi$
$277$	−11.9344	−0.717068	−0.358534	+	0.933517i	$0.616723\pi$
$278$	−13.6714	−0.819955
$279$	15.4151	0.922879
$280$	0	0
$281$	24.2556	1.44697	0.723484	−	0.690341i	$-0.242540\pi$
$281$	24.2556	1.44697	0.723484	+	0.690341i	$0.242540\pi$
$282$	5.29910	0.315557
$283$	22.1023	1.31385	0.656924	−	0.753957i	$-0.271857\pi$
$283$	22.1023	1.31385	0.656924	+	0.753957i	$0.271857\pi$
$284$	5.03136	0.298556
$285$	0	0
$286$	0	0
$287$	54.1964	3.19912
$288$	−2.10951	−0.124304
$289$	−7.04892	−0.414643
$290$	0	0
$291$	6.52101	0.382268
$292$	9.09682	0.532351
$293$	−10.5760	−0.617854	−0.308927	−	0.951086i	$-0.599970\pi$
$293$	−10.5760	−0.617854	−0.308927	+	0.951086i	$0.599970\pi$
$294$	13.5211	0.788564
$295$	0	0
$296$	7.82786	0.454985
$297$	−13.8930	−0.806151
$298$	−12.5436	−0.726630
$299$	0	0
$300$	0	0
$301$	22.0388	1.27030
$302$	−19.6644	−1.13156
$303$	−10.8618	−0.623992
$304$	−0.0635301	−0.00364370
$305$	0	0
$306$	−6.65452	−0.380414
$307$	4.38566	0.250303	0.125151	−	0.992138i	$-0.460058\pi$
$307$	4.38566	0.250303	0.125151	+	0.992138i	$0.460058\pi$
$308$	13.3070	0.758236
$309$	−5.50211	−0.313004
$310$	0	0
$311$	35.0919	1.98988	0.994941	−	0.100462i	$-0.0320321\pi$
$311$	35.0919	1.98988	0.994941	+	0.100462i	$0.0320321\pi$
$312$	0	0
$313$	19.0745	1.07815	0.539076	−	0.842257i	$-0.318774\pi$
$313$	19.0745	1.07815	0.539076	+	0.842257i	$0.318774\pi$
$314$	20.3758	1.14987
$315$	0	0
$316$	9.16845	0.515765
$317$	6.85928	0.385256	0.192628	−	0.981272i	$-0.438299\pi$
$317$	6.85928	0.385256	0.192628	+	0.981272i	$0.438299\pi$
$318$	−1.19124	−0.0668012
$319$	−23.1945	−1.29864
$320$	0	0
$321$	−12.8111	−0.715045
$322$	−7.23087	−0.402961
$323$	−0.200408	−0.0111510
$324$	1.77857	0.0988095
$325$	0	0
$326$	20.9608	1.16091
$327$	−8.70925	−0.481623
$328$	11.7352	0.647970
$329$	25.9338	1.42978
$330$	0	0
$331$	6.83284	0.375567	0.187783	−	0.982210i	$-0.439870\pi$
$331$	6.83284	0.375567	0.187783	+	0.982210i	$0.439870\pi$
$332$	5.55586	0.304917
$333$	−16.5130	−0.904905
$334$	−21.7851	−1.19203
$335$	0	0
$336$	4.35806	0.237752
$337$	15.1320	0.824293	0.412146	−	0.911118i	$-0.364779\pi$
$337$	15.1320	0.824293	0.412146	+	0.911118i	$0.364779\pi$
$338$	0	0
$339$	13.7993	0.749476
$340$	0	0
$341$	−21.0555	−1.14022
$342$	0.134017	0.00724683
$343$	33.8443	1.82742
$344$	4.77211	0.257295
$345$	0	0
$346$	−12.3298	−0.662853
$347$	8.17548	0.438883	0.219441	−	0.975626i	$-0.429576\pi$
$347$	8.17548	0.438883	0.219441	+	0.975626i	$0.429576\pi$
$348$	−7.59624	−0.407201
$349$	22.5745	1.20839	0.604193	−	0.796838i	$-0.293496\pi$
$349$	22.5745	1.20839	0.604193	+	0.796838i	$0.293496\pi$
$350$	0	0
$351$	0	0
$352$	2.88138	0.153578
$353$	8.26337	0.439815	0.219907	−	0.975521i	$-0.429424\pi$
$353$	8.26337	0.439815	0.219907	+	0.975521i	$0.429424\pi$
$354$	−9.75217	−0.518322
$355$	0	0
$356$	−6.75492	−0.358010
$357$	13.7476	0.727602
$358$	19.9819	1.05608
$359$	0.714798	0.0377256	0.0188628	−	0.999822i	$-0.493995\pi$
$359$	0.714798	0.0377256	0.0188628	+	0.999822i	$0.493995\pi$
$360$	0	0
$361$	−18.9960	−0.999788
$362$	4.90650	0.257880
$363$	−2.54564	−0.133611
$364$	0	0
$365$	0	0
$366$	4.22961	0.221085
$367$	−2.67301	−0.139530	−0.0697649	−	0.997563i	$-0.522225\pi$
$367$	−2.67301	−0.139530	−0.0697649	+	0.997563i	$0.522225\pi$
$368$	−1.56571	−0.0816184
$369$	−24.7556	−1.28873
$370$	0	0
$371$	−5.82991	−0.302674
$372$	−6.89571	−0.357526
$373$	−38.5507	−1.99608	−0.998039	−	0.0625886i	$-0.980064\pi$
$373$	−38.5507	−1.99608	−0.998039	+	0.0625886i	$0.980064\pi$
$374$	9.08942	0.470003
$375$	0	0
$376$	5.61549	0.289597
$377$	0	0
$378$	−22.2675	−1.14532
$379$	−11.8965	−0.611082	−0.305541	−	0.952179i	$-0.598837\pi$
$379$	−11.8965	−0.611082	−0.305541	+	0.952179i	$0.598837\pi$
$380$	0	0
$381$	−6.80416	−0.348588
$382$	−14.8526	−0.759926
$383$	−8.52308	−0.435509	−0.217755	−	0.976004i	$-0.569873\pi$
$383$	−8.52308	−0.435509	−0.217755	+	0.976004i	$0.569873\pi$
$384$	0.943657	0.0481558
$385$	0	0
$386$	−26.4730	−1.34744
$387$	−10.0668	−0.511725
$388$	6.91035	0.350820
$389$	26.5619	1.34674	0.673370	−	0.739306i	$-0.264846\pi$
$389$	26.5619	1.34674	0.673370	+	0.739306i	$0.264846\pi$
$390$	0	0
$391$	−4.93909	−0.249781
$392$	14.3284	0.723691
$393$	−1.46028	−0.0736614
$394$	4.97462	0.250618
$395$	0	0
$396$	−6.07831	−0.305447
$397$	−19.1267	−0.959940	−0.479970	−	0.877285i	$-0.659353\pi$
$397$	−19.1267	−0.959940	−0.479970	+	0.877285i	$0.659353\pi$
$398$	14.6426	0.733969
$399$	−0.276868	−0.0138607
$400$	0	0
$401$	20.4347	1.02046	0.510230	−	0.860038i	$-0.329560\pi$
$401$	20.4347	1.02046	0.510230	+	0.860038i	$0.329560\pi$
$402$	14.1121	0.703850
$403$	0	0
$404$	−11.5103	−0.572658
$405$	0	0
$406$	−37.1760	−1.84502
$407$	22.5551	1.11801
$408$	2.97680	0.147373
$409$	−6.39485	−0.316205	−0.158102	−	0.987423i	$-0.550538\pi$
$409$	−6.39485	−0.316205	−0.158102	+	0.987423i	$0.550538\pi$
$410$	0	0
$411$	−6.67641	−0.329323
$412$	−5.83062	−0.287254
$413$	−47.7272	−2.34850
$414$	3.30289	0.162328
$415$	0	0
$416$	0	0
$417$	−12.9011	−0.631769
$418$	−0.183055	−0.00895349
$419$	−7.16820	−0.350190	−0.175095	−	0.984552i	$-0.556023\pi$
$419$	−7.16820	−0.350190	−0.175095	+	0.984552i	$0.556023\pi$
$420$	0	0
$421$	−13.2519	−0.645860	−0.322930	−	0.946423i	$-0.604668\pi$
$421$	−13.2519	−0.645860	−0.322930	+	0.946423i	$0.604668\pi$
$422$	6.86084	0.333980
$423$	−11.8459	−0.575970
$424$	−1.26236	−0.0613056
$425$	0	0
$426$	4.74788	0.230036
$427$	20.6997	1.00173
$428$	−13.5760	−0.656220
$429$	0	0
$430$	0	0
$431$	−15.3450	−0.739142	−0.369571	−	0.929202i	$-0.620495\pi$
$431$	−15.3450	−0.739142	−0.369571	+	0.929202i	$0.620495\pi$
$432$	−4.82163	−0.231981
$433$	−37.2884	−1.79197	−0.895983	−	0.444088i	$-0.853528\pi$
$433$	−37.2884	−1.79197	−0.895983	+	0.444088i	$0.853528\pi$
$434$	−33.7477	−1.61994
$435$	0	0
$436$	−9.22925	−0.442001
$437$	0.0994698	0.00475829
$438$	8.58428	0.410173
$439$	7.54167	0.359944	0.179972	−	0.983672i	$-0.442399\pi$
$439$	7.54167	0.359944	0.179972	+	0.983672i	$0.442399\pi$
$440$	0	0
$441$	−30.2258	−1.43933
$442$	0	0
$443$	−14.2813	−0.678526	−0.339263	−	0.940692i	$-0.610178\pi$
$443$	−14.2813	−0.678526	−0.339263	+	0.940692i	$0.610178\pi$
$444$	7.38682	0.350563
$445$	0	0
$446$	−21.0169	−0.995178
$447$	−11.8368	−0.559864
$448$	4.61826	0.218192
$449$	−40.5951	−1.91580	−0.957900	−	0.287103i	$-0.907308\pi$
$449$	−40.5951	−1.91580	−0.957900	+	0.287103i	$0.907308\pi$
$450$	0	0
$451$	33.8137	1.59223
$452$	14.6232	0.687819
$453$	−18.5564	−0.871857
$454$	6.56269	0.308002
$455$	0	0
$456$	−0.0599506	−0.00280745
$457$	−6.07253	−0.284061	−0.142031	−	0.989862i	$-0.545363\pi$
$457$	−6.07253	−0.284061	−0.142031	+	0.989862i	$0.545363\pi$
$458$	16.0898	0.751828
$459$	−15.2100	−0.709941
$460$	0	0
$461$	−24.0995	−1.12243	−0.561214	−	0.827671i	$-0.689665\pi$
$461$	−24.0995	−1.12243	−0.561214	+	0.827671i	$0.689665\pi$
$462$	12.5572	0.584216
$463$	−9.72355	−0.451892	−0.225946	−	0.974140i	$-0.572547\pi$
$463$	−9.72355	−0.451892	−0.225946	+	0.974140i	$0.572547\pi$
$464$	−8.04979	−0.373702
$465$	0	0
$466$	−6.58375	−0.304986
$467$	2.62475	0.121459	0.0607295	−	0.998154i	$-0.480657\pi$
$467$	2.62475	0.121459	0.0607295	+	0.998154i	$0.480657\pi$
$468$	0	0
$469$	69.0649	3.18912
$470$	0	0
$471$	19.2277	0.885967
$472$	−10.3344	−0.475681
$473$	13.7503	0.632238
$474$	8.65187	0.397394
$475$	0	0
$476$	14.5685	0.667744
$477$	2.66296	0.121929
$478$	−11.5368	−0.527682
$479$	16.7841	0.766886	0.383443	−	0.923565i	$-0.374738\pi$
$479$	16.7841	0.766886	0.383443	+	0.923565i	$0.374738\pi$
$480$	0	0
$481$	0	0
$482$	−21.7563	−0.990971
$483$	−6.82346	−0.310478
$484$	−2.69763	−0.122619
$485$	0	0
$486$	16.1432	0.732272
$487$	−31.1107	−1.40976	−0.704881	−	0.709325i	$-0.749000\pi$
$487$	−31.1107	−1.40976	−0.704881	+	0.709325i	$0.749000\pi$
$488$	4.48215	0.202897
$489$	19.7798	0.894472
$490$	0	0
$491$	−18.8663	−0.851425	−0.425712	−	0.904858i	$-0.639976\pi$
$491$	−18.8663	−0.851425	−0.425712	+	0.904858i	$0.639976\pi$
$492$	11.0740	0.499257
$493$	−25.3933	−1.14366
$494$	0	0
$495$	0	0
$496$	−7.30744	−0.328114
$497$	23.2362	1.04228
$498$	5.24283	0.234937
$499$	38.1163	1.70632	0.853159	−	0.521650i	$-0.174683\pi$
$499$	38.1163	1.70632	0.853159	+	0.521650i	$0.174683\pi$
$500$	0	0
$501$	−20.5576	−0.918448
$502$	22.7936	1.01733
$503$	4.47130	0.199365	0.0996827	−	0.995019i	$-0.468217\pi$
$503$	4.47130	0.199365	0.0996827	+	0.995019i	$0.468217\pi$
$504$	−9.74228	−0.433956
$505$	0	0
$506$	−4.51142	−0.200557
$507$	0	0
$508$	−7.21042	−0.319911
$509$	−28.7307	−1.27347	−0.636734	−	0.771084i	$-0.719715\pi$
$509$	−28.7307	−1.27347	−0.636734	+	0.771084i	$0.719715\pi$
$510$	0	0
$511$	42.0115	1.85848
$512$	1.00000	0.0441942
$513$	0.306318	0.0135243
$514$	16.4152	0.724043
$515$	0	0
$516$	4.50323	0.198244
$517$	16.1804	0.711613
$518$	36.1511	1.58839
$519$	−11.6351	−0.510723
$520$	0	0
$521$	−41.4647	−1.81660	−0.908301	−	0.418318i	$-0.862620\pi$
$521$	−41.4647	−1.81660	−0.908301	+	0.418318i	$0.862620\pi$
$522$	16.9811	0.743243
$523$	−6.73304	−0.294415	−0.147208	−	0.989106i	$-0.547029\pi$
$523$	−6.73304	−0.294415	−0.147208	+	0.989106i	$0.547029\pi$
$524$	−1.54747	−0.0676015
$525$	0	0
$526$	−6.81715	−0.297242
$527$	−23.0515	−1.00414
$528$	2.71904	0.118331
$529$	−20.5485	−0.893415
$530$	0	0
$531$	21.8006	0.946066
$532$	−0.293399	−0.0127204
$533$	0	0
$534$	−6.37433	−0.275844
$535$	0	0
$536$	14.9547	0.645946
$537$	18.8561	0.813701
$538$	4.49766	0.193908
$539$	41.2855	1.77829
$540$	0	0
$541$	34.4004	1.47899	0.739494	−	0.673163i	$-0.235065\pi$
$541$	34.4004	1.47899	0.739494	+	0.673163i	$0.235065\pi$
$542$	−15.7937	−0.678399
$543$	4.63006	0.198695
$544$	3.15453	0.135249
$545$	0	0
$546$	0	0
$547$	8.24645	0.352593	0.176296	−	0.984337i	$-0.443588\pi$
$547$	8.24645	0.352593	0.176296	+	0.984337i	$0.443588\pi$
$548$	−7.07504	−0.302231
$549$	−9.45514	−0.403535
$550$	0	0
$551$	0.511404	0.0217865
$552$	−1.47750	−0.0628864
$553$	42.3423	1.80058
$554$	−11.9344	−0.507043
$555$	0	0
$556$	−13.6714	−0.579795
$557$	31.3070	1.32652	0.663259	−	0.748390i	$-0.269173\pi$
$557$	31.3070	1.32652	0.663259	+	0.748390i	$0.269173\pi$
$558$	15.4151	0.652574
$559$	0	0
$560$	0	0
$561$	8.57729	0.362134
$562$	24.2556	1.02316
$563$	−20.8507	−0.878752	−0.439376	−	0.898303i	$-0.644800\pi$
$563$	−20.8507	−0.878752	−0.439376	+	0.898303i	$0.644800\pi$
$564$	5.29910	0.223132
$565$	0	0
$566$	22.1023	0.929030
$567$	8.21391	0.344952
$568$	5.03136	0.211111
$569$	4.15326	0.174114	0.0870568	−	0.996203i	$-0.472254\pi$
$569$	4.15326	0.174114	0.0870568	+	0.996203i	$0.472254\pi$
$570$	0	0
$571$	37.3681	1.56381	0.781904	−	0.623399i	$-0.214249\pi$
$571$	37.3681	1.56381	0.781904	+	0.623399i	$0.214249\pi$
$572$	0	0
$573$	−14.0158	−0.585518
$574$	54.1964	2.26212
$575$	0	0
$576$	−2.10951	−0.0878963
$577$	−9.10013	−0.378843	−0.189422	−	0.981896i	$-0.560661\pi$
$577$	−9.10013	−0.378843	−0.189422	+	0.981896i	$0.560661\pi$
$578$	−7.04892	−0.293197
$579$	−24.9814	−1.03819
$580$	0	0
$581$	25.6584	1.06449
$582$	6.52101	0.270304
$583$	−3.63734	−0.150643
$584$	9.09682	0.376429
$585$	0	0
$586$	−10.5760	−0.436889
$587$	17.7062	0.730814	0.365407	−	0.930848i	$-0.380930\pi$
$587$	17.7062	0.730814	0.365407	+	0.930848i	$0.380930\pi$
$588$	13.5211	0.557599
$589$	0.464242	0.0191288
$590$	0	0
$591$	4.69433	0.193099
$592$	7.82786	0.321723
$593$	−12.3192	−0.505888	−0.252944	−	0.967481i	$-0.581399\pi$
$593$	−12.3192	−0.505888	−0.252944	+	0.967481i	$0.581399\pi$
$594$	−13.8930	−0.570035
$595$	0	0
$596$	−12.5436	−0.513805
$597$	13.8176	0.565518
$598$	0	0
$599$	25.0676	1.02423	0.512116	−	0.858916i	$-0.328862\pi$
$599$	25.0676	1.02423	0.512116	+	0.858916i	$0.328862\pi$
$600$	0	0
$601$	−21.7771	−0.888307	−0.444154	−	0.895951i	$-0.646496\pi$
$601$	−21.7771	−0.888307	−0.444154	+	0.895951i	$0.646496\pi$
$602$	22.0388	0.898236
$603$	−31.5472	−1.28470
$604$	−19.6644	−0.800132
$605$	0	0
$606$	−10.8618	−0.441229
$607$	−16.0040	−0.649582	−0.324791	−	0.945786i	$-0.605294\pi$
$607$	−16.0040	−0.649582	−0.324791	+	0.945786i	$0.605294\pi$
$608$	−0.0635301	−0.00257649
$609$	−35.0814	−1.42157
$610$	0	0
$611$	0	0
$612$	−6.65452	−0.268993
$613$	−22.9014	−0.924980	−0.462490	−	0.886625i	$-0.653044\pi$
$613$	−22.9014	−0.924980	−0.462490	+	0.886625i	$0.653044\pi$
$614$	4.38566	0.176991
$615$	0	0
$616$	13.3070	0.536154
$617$	38.6712	1.55684	0.778422	−	0.627741i	$-0.216020\pi$
$617$	38.6712	1.55684	0.778422	+	0.627741i	$0.216020\pi$
$618$	−5.50211	−0.221327
$619$	−5.21721	−0.209698	−0.104849	−	0.994488i	$-0.533436\pi$
$619$	−5.21721	−0.209698	−0.104849	+	0.994488i	$0.533436\pi$
$620$	0	0
$621$	7.54928	0.302942
$622$	35.0919	1.40706
$623$	−31.1960	−1.24984
$624$	0	0
$625$	0	0
$626$	19.0745	0.762369
$627$	−0.172741	−0.00689860
$628$	20.3758	0.813081
$629$	24.6933	0.984584
$630$	0	0
$631$	27.2996	1.08678	0.543389	−	0.839481i	$-0.317141\pi$
$631$	27.2996	1.08678	0.543389	+	0.839481i	$0.317141\pi$
$632$	9.16845	0.364701
$633$	6.47428	0.257329
$634$	6.85928	0.272417
$635$	0	0
$636$	−1.19124	−0.0472356
$637$	0	0
$638$	−23.1945	−0.918280
$639$	−10.6137	−0.419872
$640$	0	0
$641$	0.580201	0.0229166	0.0114583	−	0.999934i	$-0.496353\pi$
$641$	0.580201	0.0229166	0.0114583	+	0.999934i	$0.496353\pi$
$642$	−12.8111	−0.505613
$643$	−26.8715	−1.05971	−0.529855	−	0.848088i	$-0.677754\pi$
$643$	−26.8715	−1.05971	−0.529855	+	0.848088i	$0.677754\pi$
$644$	−7.23087	−0.284936
$645$	0	0
$646$	−0.200408	−0.00788494
$647$	5.20846	0.204766	0.102383	−	0.994745i	$-0.467353\pi$
$647$	5.20846	0.204766	0.102383	+	0.994745i	$0.467353\pi$
$648$	1.77857	0.0698689
$649$	−29.7775	−1.16887
$650$	0	0
$651$	−31.8462	−1.24815
$652$	20.9608	0.820887
$653$	−30.2356	−1.18321	−0.591604	−	0.806228i	$-0.701505\pi$
$653$	−30.2356	−1.18321	−0.591604	+	0.806228i	$0.701505\pi$
$654$	−8.70925	−0.340559
$655$	0	0
$656$	11.7352	0.458184
$657$	−19.1899	−0.748667
$658$	25.9338	1.01101
$659$	−32.6906	−1.27345	−0.636723	−	0.771092i	$-0.719711\pi$
$659$	−32.6906	−1.27345	−0.636723	+	0.771092i	$0.719711\pi$
$660$	0	0
$661$	−22.2015	−0.863538	−0.431769	−	0.901984i	$-0.642111\pi$
$661$	−22.2015	−0.863538	−0.431769	+	0.901984i	$0.642111\pi$
$662$	6.83284	0.265566
$663$	0	0
$664$	5.55586	0.215609
$665$	0	0
$666$	−16.5130	−0.639864
$667$	12.6037	0.488015
$668$	−21.7851	−0.842890
$669$	−19.8327	−0.766778
$670$	0	0
$671$	12.9148	0.498570
$672$	4.35806	0.168116
$673$	11.5383	0.444769	0.222384	−	0.974959i	$-0.428616\pi$
$673$	11.5383	0.444769	0.222384	+	0.974959i	$0.428616\pi$
$674$	15.1320	0.582863
$675$	0	0
$676$	0	0
$677$	−33.4315	−1.28488	−0.642439	−	0.766337i	$-0.722077\pi$
$677$	−33.4315	−1.28488	−0.642439	+	0.766337i	$0.722077\pi$
$678$	13.7993	0.529960
$679$	31.9138	1.22474
$680$	0	0
$681$	6.19293	0.237314
$682$	−21.0555	−0.806258
$683$	−6.25127	−0.239198	−0.119599	−	0.992822i	$-0.538161\pi$
$683$	−6.25127	−0.239198	−0.119599	+	0.992822i	$0.538161\pi$
$684$	0.134017	0.00512428
$685$	0	0
$686$	33.8443	1.29218
$687$	15.1833	0.579278
$688$	4.77211	0.181935
$689$	0	0
$690$	0	0
$691$	35.3665	1.34540	0.672702	−	0.739914i	$-0.265134\pi$
$691$	35.3665	1.34540	0.672702	+	0.739914i	$0.265134\pi$
$692$	−12.3298	−0.468708
$693$	−28.0712	−1.06634
$694$	8.17548	0.310337
$695$	0	0
$696$	−7.59624	−0.287935
$697$	37.0192	1.40220
$698$	22.5745	0.854458
$699$	−6.21280	−0.234990
$700$	0	0
$701$	−10.6110	−0.400773	−0.200387	−	0.979717i	$-0.564220\pi$
$701$	−10.6110	−0.400773	−0.200387	+	0.979717i	$0.564220\pi$
$702$	0	0
$703$	−0.497305	−0.0187562
$704$	2.88138	0.108596
$705$	0	0
$706$	8.26337	0.310996
$707$	−53.1575	−1.99919
$708$	−9.75217	−0.366509
$709$	−1.18072	−0.0443428	−0.0221714	−	0.999754i	$-0.507058\pi$
$709$	−1.18072	−0.0443428	−0.0221714	+	0.999754i	$0.507058\pi$
$710$	0	0
$711$	−19.3409	−0.725342
$712$	−6.75492	−0.253151
$713$	11.4413	0.428482
$714$	13.7476	0.514492
$715$	0	0
$716$	19.9819	0.746760
$717$	−10.8868	−0.406576
$718$	0.714798	0.0266760
$719$	27.4219	1.02266	0.511332	−	0.859383i	$-0.329152\pi$
$719$	27.4219	1.02266	0.511332	+	0.859383i	$0.329152\pi$
$720$	0	0
$721$	−26.9273	−1.00283
$722$	−18.9960	−0.706957
$723$	−20.5305	−0.763536
$724$	4.90650	0.182349
$725$	0	0
$726$	−2.54564	−0.0944774
$727$	−33.3438	−1.23665	−0.618326	−	0.785922i	$-0.712189\pi$
$727$	−33.3438	−1.23665	−0.618326	+	0.785922i	$0.712189\pi$
$728$	0	0
$729$	9.89797	0.366592
$730$	0	0
$731$	15.0538	0.556784
$732$	4.22961	0.156331
$733$	0.370479	0.0136840	0.00684198	−	0.999977i	$-0.497822\pi$
$733$	0.370479	0.0136840	0.00684198	+	0.999977i	$0.497822\pi$
$734$	−2.67301	−0.0986624
$735$	0	0
$736$	−1.56571	−0.0577129
$737$	43.0903	1.58725
$738$	−24.7556	−0.911267
$739$	−25.8069	−0.949323	−0.474661	−	0.880168i	$-0.657429\pi$
$739$	−25.8069	−0.949323	−0.474661	+	0.880168i	$0.657429\pi$
$740$	0	0
$741$	0	0
$742$	−5.82991	−0.214023
$743$	−18.3669	−0.673815	−0.336907	−	0.941538i	$-0.609381\pi$
$743$	−18.3669	−0.673815	−0.336907	+	0.941538i	$0.609381\pi$
$744$	−6.89571	−0.252809
$745$	0	0
$746$	−38.5507	−1.41144
$747$	−11.7201	−0.428818
$748$	9.08942	0.332342
$749$	−62.6975	−2.29092
$750$	0	0
$751$	37.1618	1.35605	0.678026	−	0.735038i	$-0.262836\pi$
$751$	37.1618	1.35605	0.678026	+	0.735038i	$0.262836\pi$
$752$	5.61549	0.204776
$753$	21.5094	0.783845
$754$	0	0
$755$	0	0
$756$	−22.2675	−0.809863
$757$	21.7326	0.789886	0.394943	−	0.918706i	$-0.370764\pi$
$757$	21.7326	0.789886	0.394943	+	0.918706i	$0.370764\pi$
$758$	−11.8965	−0.432100
$759$	−4.25723	−0.154528
$760$	0	0
$761$	−16.5794	−0.601004	−0.300502	−	0.953781i	$-0.597154\pi$
$761$	−16.5794	−0.601004	−0.300502	+	0.953781i	$0.597154\pi$
$762$	−6.80416	−0.246489
$763$	−42.6231	−1.54306
$764$	−14.8526	−0.537349
$765$	0	0
$766$	−8.52308	−0.307951
$767$	0	0
$768$	0.943657	0.0340513
$769$	−6.12291	−0.220798	−0.110399	−	0.993887i	$-0.535213\pi$
$769$	−6.12291	−0.220798	−0.110399	+	0.993887i	$0.535213\pi$
$770$	0	0
$771$	15.4903	0.557870
$772$	−26.4730	−0.952784
$773$	50.2815	1.80850	0.904250	−	0.427004i	$-0.140431\pi$
$773$	50.2815	1.80850	0.904250	+	0.427004i	$0.140431\pi$
$774$	−10.0668	−0.361844
$775$	0	0
$776$	6.91035	0.248067
$777$	34.1143	1.22384
$778$	26.5619	0.952289
$779$	−0.745541	−0.0267118
$780$	0	0
$781$	14.4973	0.518754
$782$	−4.93909	−0.176622
$783$	38.8131	1.38707
$784$	14.3284	0.511727
$785$	0	0
$786$	−1.46028	−0.0520864
$787$	−6.12082	−0.218184	−0.109092	−	0.994032i	$-0.534794\pi$
$787$	−6.12082	−0.218184	−0.109092	+	0.994032i	$0.534794\pi$
$788$	4.97462	0.177213
$789$	−6.43306	−0.229023
$790$	0	0
$791$	67.5340	2.40123
$792$	−6.07831	−0.215983
$793$	0	0
$794$	−19.1267	−0.678780
$795$	0	0
$796$	14.6426	0.518994
$797$	−33.3362	−1.18083	−0.590414	−	0.807100i	$-0.701036\pi$
$797$	−33.3362	−1.18083	−0.590414	+	0.807100i	$0.701036\pi$
$798$	−0.276868	−0.00980101
$799$	17.7143	0.626685
$800$	0	0
$801$	14.2496	0.503484
$802$	20.4347	0.721574
$803$	26.2114	0.924982
$804$	14.1121	0.497697
$805$	0	0
$806$	0	0
$807$	4.24425	0.149405
$808$	−11.5103	−0.404930
$809$	−20.2237	−0.711029	−0.355514	−	0.934671i	$-0.615694\pi$
$809$	−20.2237	−0.711029	−0.355514	+	0.934671i	$0.615694\pi$
$810$	0	0
$811$	−35.9625	−1.26281	−0.631407	−	0.775452i	$-0.717522\pi$
$811$	−35.9625	−1.26281	−0.631407	+	0.775452i	$0.717522\pi$
$812$	−37.1760	−1.30462
$813$	−14.9039	−0.522701
$814$	22.5551	0.790555
$815$	0	0
$816$	2.97680	0.104209
$817$	−0.303172	−0.0106067
$818$	−6.39485	−0.223591
$819$	0	0
$820$	0	0
$821$	−19.5926	−0.683786	−0.341893	−	0.939739i	$-0.611068\pi$
$821$	−19.5926	−0.683786	−0.341893	+	0.939739i	$0.611068\pi$
$822$	−6.67641	−0.232867
$823$	16.7688	0.584524	0.292262	−	0.956338i	$-0.405592\pi$
$823$	16.7688	0.584524	0.292262	+	0.956338i	$0.405592\pi$
$824$	−5.83062	−0.203119
$825$	0	0
$826$	−47.7272	−1.66064
$827$	−8.40977	−0.292436	−0.146218	−	0.989252i	$-0.546710\pi$
$827$	−8.40977	−0.292436	−0.146218	+	0.989252i	$0.546710\pi$
$828$	3.30289	0.114783
$829$	7.31026	0.253896	0.126948	−	0.991909i	$-0.459482\pi$
$829$	7.31026	0.253896	0.126948	+	0.991909i	$0.459482\pi$
$830$	0	0
$831$	−11.2620	−0.390673
$832$	0	0
$833$	45.1993	1.56606
$834$	−12.9011	−0.446728
$835$	0	0
$836$	−0.183055	−0.00633107
$837$	35.2337	1.21786
$838$	−7.16820	−0.247621
$839$	−6.10748	−0.210853	−0.105427	−	0.994427i	$-0.533621\pi$
$839$	−6.10748	−0.210853	−0.105427	+	0.994427i	$0.533621\pi$
$840$	0	0
$841$	35.7991	1.23445
$842$	−13.2519	−0.456692
$843$	22.8890	0.788338
$844$	6.86084	0.236160
$845$	0	0
$846$	−11.8459	−0.407272
$847$	−12.4584	−0.428074
$848$	−1.26236	−0.0433496
$849$	20.8570	0.715811
$850$	0	0
$851$	−12.2562	−0.420137
$852$	4.74788	0.162660
$853$	32.4711	1.11179	0.555894	−	0.831253i	$-0.312376\pi$
$853$	32.4711	1.11179	0.555894	+	0.831253i	$0.312376\pi$
$854$	20.6997	0.708330
$855$	0	0
$856$	−13.5760	−0.464018
$857$	−38.1166	−1.30204	−0.651020	−	0.759061i	$-0.725659\pi$
$857$	−38.1166	−1.30204	−0.651020	+	0.759061i	$0.725659\pi$
$858$	0	0
$859$	8.61622	0.293982	0.146991	−	0.989138i	$-0.453041\pi$
$859$	8.61622	0.293982	0.146991	+	0.989138i	$0.453041\pi$
$860$	0	0
$861$	51.1429	1.74294
$862$	−15.3450	−0.522652
$863$	−0.0887238	−0.00302019	−0.00151010	−	0.999999i	$-0.500481\pi$
$863$	−0.0887238	−0.00302019	−0.00151010	+	0.999999i	$0.500481\pi$
$864$	−4.82163	−0.164035
$865$	0	0
$866$	−37.2884	−1.26711
$867$	−6.65177	−0.225906
$868$	−33.7477	−1.14547
$869$	26.4178	0.896163
$870$	0	0
$871$	0	0
$872$	−9.22925	−0.312542
$873$	−14.5775	−0.493373
$874$	0.0994698	0.00336462
$875$	0	0
$876$	8.58428	0.290036
$877$	−37.9504	−1.28149	−0.640747	−	0.767752i	$-0.721375\pi$
$877$	−37.9504	−1.28149	−0.640747	+	0.767752i	$0.721375\pi$
$878$	7.54167	0.254519
$879$	−9.98008	−0.336620
$880$	0	0
$881$	−3.49722	−0.117824	−0.0589121	−	0.998263i	$-0.518763\pi$
$881$	−3.49722	−0.117824	−0.0589121	+	0.998263i	$0.518763\pi$
$882$	−30.2258	−1.01776
$883$	4.73473	0.159336	0.0796682	−	0.996821i	$-0.474614\pi$
$883$	4.73473	0.159336	0.0796682	+	0.996821i	$0.474614\pi$
$884$	0	0
$885$	0	0
$886$	−14.2813	−0.479790
$887$	−51.7330	−1.73702	−0.868512	−	0.495668i	$-0.834923\pi$
$887$	−51.7330	−1.73702	−0.868512	+	0.495668i	$0.834923\pi$
$888$	7.38682	0.247885
$889$	−33.2996	−1.11683
$890$	0	0
$891$	5.12474	0.171685
$892$	−21.0169	−0.703697
$893$	−0.356753	−0.0119383
$894$	−11.8368	−0.395883
$895$	0	0
$896$	4.61826	0.154285
$897$	0	0
$898$	−40.5951	−1.35467
$899$	58.8233	1.96187
$900$	0	0
$901$	−3.98216	−0.132665
$902$	33.8137	1.12587
$903$	20.7971	0.692085
$904$	14.6232	0.486362
$905$	0	0
$906$	−18.5564	−0.616496
$907$	−17.2248	−0.571941	−0.285971	−	0.958238i	$-0.592316\pi$
$907$	−17.2248	−0.571941	−0.285971	+	0.958238i	$0.592316\pi$
$908$	6.56269	0.217791
$909$	24.2811	0.805352
$910$	0	0
$911$	9.77031	0.323705	0.161852	−	0.986815i	$-0.448253\pi$
$911$	9.77031	0.323705	0.161852	+	0.986815i	$0.448253\pi$
$912$	−0.0599506	−0.00198516
$913$	16.0086	0.529806
$914$	−6.07253	−0.200862
$915$	0	0
$916$	16.0898	0.531623
$917$	−7.14662	−0.236002
$918$	−15.2100	−0.502004
$919$	22.0711	0.728057	0.364028	−	0.931388i	$-0.381401\pi$
$919$	22.0711	0.728057	0.364028	+	0.931388i	$0.381401\pi$
$920$	0	0
$921$	4.13856	0.136370
$922$	−24.0995	−0.793676
$923$	0	0
$924$	12.5572	0.413103
$925$	0	0
$926$	−9.72355	−0.319536
$927$	12.2998	0.403977
$928$	−8.04979	−0.264247
$929$	54.9763	1.80372	0.901858	−	0.432033i	$-0.142204\pi$
$929$	54.9763	1.80372	0.901858	+	0.432033i	$0.142204\pi$
$930$	0	0
$931$	−0.910282	−0.0298333
$932$	−6.58375	−0.215658
$933$	33.1148	1.08413
$934$	2.62475	0.0858845
$935$	0	0
$936$	0	0
$937$	7.46577	0.243896	0.121948	−	0.992536i	$-0.461086\pi$
$937$	7.46577	0.243896	0.121948	+	0.992536i	$0.461086\pi$
$938$	69.0649	2.25505
$939$	17.9998	0.587400
$940$	0	0
$941$	22.2620	0.725721	0.362860	−	0.931844i	$-0.381800\pi$
$941$	22.2620	0.725721	0.362860	+	0.931844i	$0.381800\pi$
$942$	19.2277	0.626473
$943$	−18.3740	−0.598340
$944$	−10.3344	−0.336357
$945$	0	0
$946$	13.7503	0.447060
$947$	−37.0892	−1.20524	−0.602619	−	0.798029i	$-0.705876\pi$
$947$	−37.0892	−1.20524	−0.602619	+	0.798029i	$0.705876\pi$
$948$	8.65187	0.281000
$949$	0	0
$950$	0	0
$951$	6.47281	0.209895
$952$	14.5685	0.472167
$953$	5.14282	0.166592	0.0832962	−	0.996525i	$-0.473455\pi$
$953$	5.14282	0.166592	0.0832962	+	0.996525i	$0.473455\pi$
$954$	2.66296	0.0862166
$955$	0	0
$956$	−11.5368	−0.373128
$957$	−21.8877	−0.707528
$958$	16.7841	0.542270
$959$	−32.6744	−1.05511
$960$	0	0
$961$	22.3986	0.722536
$962$	0	0
$963$	28.6387	0.922869
$964$	−21.7563	−0.700722
$965$	0	0
$966$	−6.82346	−0.219541
$967$	20.1037	0.646490	0.323245	−	0.946315i	$-0.395226\pi$
$967$	20.1037	0.646490	0.323245	+	0.946315i	$0.395226\pi$
$968$	−2.69763	−0.0867050
$969$	−0.189116	−0.00607529
$970$	0	0
$971$	−45.5778	−1.46266	−0.731330	−	0.682024i	$-0.761100\pi$
$971$	−45.5778	−1.46266	−0.731330	+	0.682024i	$0.761100\pi$
$972$	16.1432	0.517795
$973$	−63.1380	−2.02411
$974$	−31.1107	−0.996853
$975$	0	0
$976$	4.48215	0.143470
$977$	−29.9732	−0.958929	−0.479464	−	0.877561i	$-0.659169\pi$
$977$	−29.9732	−0.958929	−0.479464	+	0.877561i	$0.659169\pi$
$978$	19.7798	0.632487
$979$	−19.4635	−0.622057
$980$	0	0
$981$	19.4692	0.621604
$982$	−18.8663	−0.602048
$983$	−41.2661	−1.31618	−0.658092	−	0.752938i	$-0.728636\pi$
$983$	−41.2661	−1.31618	−0.658092	+	0.752938i	$0.728636\pi$
$984$	11.0740	0.353028
$985$	0	0
$986$	−25.3933	−0.808688
$987$	24.4726	0.778973
$988$	0	0
$989$	−7.47175	−0.237588
$990$	0	0
$991$	−48.4215	−1.53816	−0.769080	−	0.639152i	$-0.779285\pi$
$991$	−48.4215	−1.53816	−0.769080	+	0.639152i	$0.779285\pi$
$992$	−7.30744	−0.232011
$993$	6.44786	0.204617
$994$	23.2362	0.737006
$995$	0	0
$996$	5.24283	0.166125
$997$	2.66944	0.0845421	0.0422711	−	0.999106i	$-0.486541\pi$
$997$	2.66944	0.0845421	0.0422711	+	0.999106i	$0.486541\pi$
$998$	38.1163	1.20655
$999$	−37.7430	−1.19414

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cy.1.6	yes	9
5.4	even	2		8450.2.a.cu.1.4	✓	9
13.12	even	2		8450.2.a.cv.1.6	yes	9
65.64	even	2		8450.2.a.cz.1.4	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8450.2.a.cu.1.4	✓	9	5.4	even	2
8450.2.a.cv.1.6	yes	9	13.12	even	2
8450.2.a.cy.1.6	yes	9	1.1	even	1	trivial
8450.2.a.cz.1.4	yes	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.943657 q^{3} +1.00000 q^{4} +0.943657 q^{6} +4.61826 q^{7} +1.00000 q^{8} -2.10951 q^{9} +2.88138 q^{11} +0.943657 q^{12} +4.61826 q^{14} +1.00000 q^{16} +3.15453 q^{17} -2.10951 q^{18} -0.0635301 q^{19} +4.35806 q^{21} +2.88138 q^{22} -1.56571 q^{23} +0.943657 q^{24} -4.82163 q^{27} +4.61826 q^{28} -8.04979 q^{29} -7.30744 q^{31} +1.00000 q^{32} +2.71904 q^{33} +3.15453 q^{34} -2.10951 q^{36} +7.82786 q^{37} -0.0635301 q^{38} +11.7352 q^{41} +4.35806 q^{42} +4.77211 q^{43} +2.88138 q^{44} -1.56571 q^{46} +5.61549 q^{47} +0.943657 q^{48} +14.3284 q^{49} +2.97680 q^{51} -1.26236 q^{53} -4.82163 q^{54} +4.61826 q^{56} -0.0599506 q^{57} -8.04979 q^{58} -10.3344 q^{59} +4.48215 q^{61} -7.30744 q^{62} -9.74228 q^{63} +1.00000 q^{64} +2.71904 q^{66} +14.9547 q^{67} +3.15453 q^{68} -1.47750 q^{69} +5.03136 q^{71} -2.10951 q^{72} +9.09682 q^{73} +7.82786 q^{74} -0.0635301 q^{76} +13.3070 q^{77} +9.16845 q^{79} +1.77857 q^{81} +11.7352 q^{82} +5.55586 q^{83} +4.35806 q^{84} +4.77211 q^{86} -7.59624 q^{87} +2.88138 q^{88} -6.75492 q^{89} -1.56571 q^{92} -6.89571 q^{93} +5.61549 q^{94} +0.943657 q^{96} +6.91035 q^{97} +14.3284 q^{98} -6.07831 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{7} + 9 q^{8} + 15 q^{9} - q^{11} - 4 q^{14} + 9 q^{16} + 2 q^{17} + 15 q^{18} + 10 q^{19} + 3 q^{21} - q^{22} - q^{23} - 9 q^{27} - 4 q^{28} - q^{29} - 4 q^{31} + 9 q^{32}+ \cdots - 62 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^7 + 9 * q^8 + 15 * q^9 - q^11 - 4 * q^14 + 9 * q^16 + 2 * q^17 + 15 * q^18 + 10 * q^19 + 3 * q^21 - q^22 - q^23 - 9 * q^27 - 4 * q^28 - q^29 - 4 * q^31 + 9 * q^32 + 30 * q^33 + 2 * q^34 + 15 * q^36 + 18 * q^37 + 10 * q^38 - 2 * q^41 + 3 * q^42 - 5 * q^43 - q^44 - q^46 - 10 * q^47 + 39 * q^49 + 38 * q^51 + 3 * q^53 - 9 * q^54 - 4 * q^56 + 14 * q^57 - q^58 - 20 * q^59 + 40 * q^61 - 4 * q^62 - 45 * q^63 + 9 * q^64 + 30 * q^66 + 15 * q^67 + 2 * q^68 + 68 * q^69 + 46 * q^71 + 15 * q^72 + 13 * q^73 + 18 * q^74 + 10 * q^76 + 13 * q^77 + 37 * q^79 + 53 * q^81 - 2 * q^82 + 2 * q^83 + 3 * q^84 - 5 * q^86 + q^87 - q^88 + 14 * q^89 - q^92 + 2 * q^93 - 10 * q^94 + 23 * q^97 + 39 * q^98 - 62 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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