LMFDB - Embedded newform 8450.2.a.bv.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [3,-3,1,3,0,-1,-2,-3,-4,0,-13,1,0,2,0,3,1] 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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1690)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ 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)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.80194 q^{3} +1.00000 q^{4} -1.80194 q^{6} -3.60388 q^{7} -1.00000 q^{8} +0.246980 q^{9} -4.44504 q^{11} +1.80194 q^{12} +3.60388 q^{14} +1.00000 q^{16} +3.15883 q^{17} -0.246980 q^{18} +0.356896 q^{19} -6.49396 q^{21} +4.44504 q^{22} +6.49396 q^{23} -1.80194 q^{24} -4.96077 q^{27} -3.60388 q^{28} -2.89008 q^{29} -3.82371 q^{31} -1.00000 q^{32} -8.00969 q^{33} -3.15883 q^{34} +0.246980 q^{36} -7.48188 q^{37} -0.356896 q^{38} -2.03923 q^{41} +6.49396 q^{42} -10.0151 q^{43} -4.44504 q^{44} -6.49396 q^{46} +11.7017 q^{47} +1.80194 q^{48} +5.98792 q^{49} +5.69202 q^{51} +12.3720 q^{53} +4.96077 q^{54} +3.60388 q^{56} +0.643104 q^{57} +2.89008 q^{58} +11.9976 q^{59} -12.3720 q^{61} +3.82371 q^{62} -0.890084 q^{63} +1.00000 q^{64} +8.00969 q^{66} +13.9976 q^{67} +3.15883 q^{68} +11.7017 q^{69} -12.8116 q^{71} -0.246980 q^{72} +6.96615 q^{73} +7.48188 q^{74} +0.356896 q^{76} +16.0194 q^{77} -5.87800 q^{79} -9.67994 q^{81} +2.03923 q^{82} +4.67456 q^{83} -6.49396 q^{84} +10.0151 q^{86} -5.20775 q^{87} +4.44504 q^{88} -8.02177 q^{89} +6.49396 q^{92} -6.89008 q^{93} -11.7017 q^{94} -1.80194 q^{96} +3.95108 q^{97} -5.98792 q^{98} -1.09783 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + q^{3} + 3 q^{4} - q^{6} - 2 q^{7} - 3 q^{8} - 4 q^{9} - 13 q^{11} + q^{12} + 2 q^{14} + 3 q^{16} + q^{17} + 4 q^{18} - 3 q^{19} - 10 q^{21} + 13 q^{22} + 10 q^{23} - q^{24} - 2 q^{27}+ \cdots + 15 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + q^3 + 3 * q^4 - q^6 - 2 * q^7 - 3 * q^8 - 4 * q^9 - 13 * q^11 + q^12 + 2 * q^14 + 3 * q^16 + q^17 + 4 * q^18 - 3 * q^19 - 10 * q^21 + 13 * q^22 + 10 * q^23 - q^24 - 2 * q^27 - 2 * q^28 - 8 * q^29 - 4 * q^31 - 3 * q^32 - 2 * q^33 - q^34 - 4 * q^36 + 6 * q^37 + 3 * q^38 - 19 * q^41 + 10 * q^42 - 5 * q^43 - 13 * q^44 - 10 * q^46 + 8 * q^47 + q^48 - q^49 + 12 * q^51 + 8 * q^53 + 2 * q^54 + 2 * q^56 + 6 * q^57 + 8 * q^58 - 5 * q^59 - 8 * q^61 + 4 * q^62 - 2 * q^63 + 3 * q^64 + 2 * q^66 + q^67 + q^68 + 8 * q^69 - 12 * q^71 + 4 * q^72 + 5 * q^73 - 6 * q^74 - 3 * q^76 + 4 * q^77 + 2 * q^79 - 5 * q^81 + 19 * q^82 - 7 * q^83 - 10 * q^84 + 5 * q^86 + 2 * q^87 + 13 * q^88 - 21 * q^89 + 10 * q^92 - 20 * q^93 - 8 * q^94 - q^96 + 21 * q^97 + q^98 + 15 * 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
$3$	1.80194	1.04035	0.520175	−	0.854060i	$-0.325867\pi$
$3$	1.80194	1.04035	0.520175	+	0.854060i	$0.325867\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.80194	−0.735638
$7$	−3.60388	−1.36214	−0.681068	−	0.732220i	$-0.738484\pi$
$7$	−3.60388	−1.36214	−0.681068	+	0.732220i	$0.738484\pi$
$8$	−1.00000	−0.353553
$9$	0.246980	0.0823265
$10$	0	0
$11$	−4.44504	−1.34023	−0.670115	−	0.742257i	$-0.733755\pi$
$11$	−4.44504	−1.34023	−0.670115	+	0.742257i	$0.733755\pi$
$12$	1.80194	0.520175
$13$	0	0
$13$	0	0
$14$	3.60388	0.963176
$15$	0	0
$16$	1.00000	0.250000
$17$	3.15883	0.766130	0.383065	−	0.923721i	$-0.374869\pi$
$17$	3.15883	0.766130	0.383065	+	0.923721i	$0.374869\pi$
$18$	−0.246980	−0.0582137
$19$	0.356896	0.0818775	0.0409388	−	0.999162i	$-0.486965\pi$
$19$	0.356896	0.0818775	0.0409388	+	0.999162i	$0.486965\pi$
$20$	0	0
$21$	−6.49396	−1.41710
$22$	4.44504	0.947686
$23$	6.49396	1.35408	0.677042	−	0.735944i	$-0.263262\pi$
$23$	6.49396	1.35408	0.677042	+	0.735944i	$0.263262\pi$
$24$	−1.80194	−0.367819
$25$	0	0
$26$	0	0
$27$	−4.96077	−0.954701
$28$	−3.60388	−0.681068
$29$	−2.89008	−0.536675	−0.268338	−	0.963325i	$-0.586474\pi$
$29$	−2.89008	−0.536675	−0.268338	+	0.963325i	$0.586474\pi$
$30$	0	0
$31$	−3.82371	−0.686758	−0.343379	−	0.939197i	$-0.611572\pi$
$31$	−3.82371	−0.686758	−0.343379	+	0.939197i	$0.611572\pi$
$32$	−1.00000	−0.176777
$33$	−8.00969	−1.39431
$34$	−3.15883	−0.541735
$35$	0	0
$36$	0.246980	0.0411633
$37$	−7.48188	−1.23001	−0.615007	−	0.788522i	$-0.710847\pi$
$37$	−7.48188	−1.23001	−0.615007	+	0.788522i	$0.710847\pi$
$38$	−0.356896	−0.0578962
$39$	0	0
$40$	0	0
$41$	−2.03923	−0.318474	−0.159237	−	0.987240i	$-0.550903\pi$
$41$	−2.03923	−0.318474	−0.159237	+	0.987240i	$0.550903\pi$
$42$	6.49396	1.00204
$43$	−10.0151	−1.52728	−0.763642	−	0.645640i	$-0.776591\pi$
$43$	−10.0151	−1.52728	−0.763642	+	0.645640i	$0.776591\pi$
$44$	−4.44504	−0.670115
$45$	0	0
$46$	−6.49396	−0.957482
$47$	11.7017	1.70687	0.853435	−	0.521199i	$-0.174515\pi$
$47$	11.7017	1.70687	0.853435	+	0.521199i	$0.174515\pi$
$48$	1.80194	0.260087
$49$	5.98792	0.855417
$50$	0	0
$51$	5.69202	0.797042
$52$	0	0
$53$	12.3720	1.69942	0.849710	−	0.527251i	$-0.176777\pi$
$53$	12.3720	1.69942	0.849710	+	0.527251i	$0.176777\pi$
$54$	4.96077	0.675075
$55$	0	0
$56$	3.60388	0.481588
$57$	0.643104	0.0851812
$58$	2.89008	0.379487
$59$	11.9976	1.56196	0.780978	−	0.624559i	$-0.214721\pi$
$59$	11.9976	1.56196	0.780978	+	0.624559i	$0.214721\pi$
$60$	0	0
$61$	−12.3720	−1.58407	−0.792034	−	0.610477i	$-0.790978\pi$
$61$	−12.3720	−1.58407	−0.792034	+	0.610477i	$0.790978\pi$
$62$	3.82371	0.485611
$63$	−0.890084	−0.112140
$64$	1.00000	0.125000
$65$	0	0
$66$	8.00969	0.985925
$67$	13.9976	1.71008	0.855040	−	0.518562i	$-0.173533\pi$
$67$	13.9976	1.71008	0.855040	+	0.518562i	$0.173533\pi$
$68$	3.15883	0.383065
$69$	11.7017	1.40872
$70$	0	0
$71$	−12.8116	−1.52046	−0.760230	−	0.649654i	$-0.774914\pi$
$71$	−12.8116	−1.52046	−0.760230	+	0.649654i	$0.774914\pi$
$72$	−0.246980	−0.0291068
$73$	6.96615	0.815326	0.407663	−	0.913132i	$-0.366344\pi$
$73$	6.96615	0.815326	0.407663	+	0.913132i	$0.366344\pi$
$74$	7.48188	0.869751
$75$	0	0
$76$	0.356896	0.0409388
$77$	16.0194	1.82558
$78$	0	0
$79$	−5.87800	−0.661327	−0.330663	−	0.943749i	$-0.607272\pi$
$79$	−5.87800	−0.661327	−0.330663	+	0.943749i	$0.607272\pi$
$80$	0	0
$81$	−9.67994	−1.07555
$82$	2.03923	0.225195
$83$	4.67456	0.513100	0.256550	−	0.966531i	$-0.417414\pi$
$83$	4.67456	0.513100	0.256550	+	0.966531i	$0.417414\pi$
$84$	−6.49396	−0.708549
$85$	0	0
$86$	10.0151	1.07995
$87$	−5.20775	−0.558330
$88$	4.44504	0.473843
$89$	−8.02177	−0.850306	−0.425153	−	0.905122i	$-0.639780\pi$
$89$	−8.02177	−0.850306	−0.425153	+	0.905122i	$0.639780\pi$
$90$	0	0
$91$	0	0
$92$	6.49396	0.677042
$93$	−6.89008	−0.714468
$94$	−11.7017	−1.20694
$95$	0	0
$96$	−1.80194	−0.183910
$97$	3.95108	0.401172	0.200586	−	0.979676i	$-0.435715\pi$
$97$	3.95108	0.401172	0.200586	+	0.979676i	$0.435715\pi$
$98$	−5.98792	−0.604871
$99$	−1.09783	−0.110337
$100$	0	0
$101$	15.0422	1.49676	0.748378	−	0.663272i	$-0.230833\pi$
$101$	15.0422	1.49676	0.748378	+	0.663272i	$0.230833\pi$
$102$	−5.69202	−0.563594
$103$	−7.82371	−0.770893	−0.385446	−	0.922730i	$-0.625953\pi$
$103$	−7.82371	−0.770893	−0.385446	+	0.922730i	$0.625953\pi$
$104$	0	0
$105$	0	0
$106$	−12.3720	−1.20167
$107$	−9.30127	−0.899188	−0.449594	−	0.893233i	$-0.648431\pi$
$107$	−9.30127	−0.899188	−0.449594	+	0.893233i	$0.648431\pi$
$108$	−4.96077	−0.477350
$109$	−0.811626	−0.0777397	−0.0388699	−	0.999244i	$-0.512376\pi$
$109$	−0.811626	−0.0777397	−0.0388699	+	0.999244i	$0.512376\pi$
$110$	0	0
$111$	−13.4819	−1.27964
$112$	−3.60388	−0.340534
$113$	−7.00969	−0.659416	−0.329708	−	0.944083i	$-0.606950\pi$
$113$	−7.00969	−0.659416	−0.329708	+	0.944083i	$0.606950\pi$
$114$	−0.643104	−0.0602322
$115$	0	0
$116$	−2.89008	−0.268338
$117$	0	0
$118$	−11.9976	−1.10447
$119$	−11.3840	−1.04357
$120$	0	0
$121$	8.75840	0.796218
$122$	12.3720	1.12010
$123$	−3.67456	−0.331324
$124$	−3.82371	−0.343379
$125$	0	0
$126$	0.890084	0.0792950
$127$	11.6582	1.03450	0.517248	−	0.855836i	$-0.326957\pi$
$127$	11.6582	1.03450	0.517248	+	0.855836i	$0.326957\pi$
$128$	−1.00000	−0.0883883
$129$	−18.0465	−1.58891
$130$	0	0
$131$	−1.34481	−0.117497	−0.0587485	−	0.998273i	$-0.518711\pi$
$131$	−1.34481	−0.117497	−0.0587485	+	0.998273i	$0.518711\pi$
$132$	−8.00969	−0.697154
$133$	−1.28621	−0.111528
$134$	−13.9976	−1.20921
$135$	0	0
$136$	−3.15883	−0.270868
$137$	7.80194	0.666565	0.333282	−	0.942827i	$-0.391844\pi$
$137$	7.80194	0.666565	0.333282	+	0.942827i	$0.391844\pi$
$138$	−11.7017	−0.996116
$139$	13.8726	1.17666	0.588330	−	0.808621i	$-0.299785\pi$
$139$	13.8726	1.17666	0.588330	+	0.808621i	$0.299785\pi$
$140$	0	0
$141$	21.0858	1.77574
$142$	12.8116	1.07513
$143$	0	0
$144$	0.246980	0.0205816
$145$	0	0
$146$	−6.96615	−0.576523
$147$	10.7899	0.889932
$148$	−7.48188	−0.615007
$149$	−12.6703	−1.03799	−0.518994	−	0.854778i	$-0.673693\pi$
$149$	−12.6703	−1.03799	−0.518994	+	0.854778i	$0.673693\pi$
$150$	0	0
$151$	4.86592	0.395983	0.197991	−	0.980204i	$-0.436558\pi$
$151$	4.86592	0.395983	0.197991	+	0.980204i	$0.436558\pi$
$152$	−0.356896	−0.0289481
$153$	0.780167	0.0630728
$154$	−16.0194	−1.29088
$155$	0	0
$156$	0	0
$157$	9.42758	0.752403	0.376202	−	0.926538i	$-0.377230\pi$
$157$	9.42758	0.752403	0.376202	+	0.926538i	$0.377230\pi$
$158$	5.87800	0.467629
$159$	22.2935	1.76799
$160$	0	0
$161$	−23.4034	−1.84445
$162$	9.67994	0.760528
$163$	5.47650	0.428953	0.214476	−	0.976729i	$-0.431196\pi$
$163$	5.47650	0.428953	0.214476	+	0.976729i	$0.431196\pi$
$164$	−2.03923	−0.159237
$165$	0	0
$166$	−4.67456	−0.362816
$167$	−2.68963	−0.208130	−0.104065	−	0.994571i	$-0.533185\pi$
$167$	−2.68963	−0.208130	−0.104065	+	0.994571i	$0.533185\pi$
$168$	6.49396	0.501020
$169$	0	0
$170$	0	0
$171$	0.0881460	0.00674069
$172$	−10.0151	−0.763642
$173$	−10.9095	−0.829431	−0.414715	−	0.909951i	$-0.636119\pi$
$173$	−10.9095	−0.829431	−0.414715	+	0.909951i	$0.636119\pi$
$174$	5.20775	0.394799
$175$	0	0
$176$	−4.44504	−0.335058
$177$	21.6189	1.62498
$178$	8.02177	0.601257
$179$	12.9825	0.970361	0.485180	−	0.874414i	$-0.338754\pi$
$179$	12.9825	0.970361	0.485180	+	0.874414i	$0.338754\pi$
$180$	0	0
$181$	22.7332	1.68974	0.844872	−	0.534969i	$-0.179677\pi$
$181$	22.7332	1.68974	0.844872	+	0.534969i	$0.179677\pi$
$182$	0	0
$183$	−22.2935	−1.64798
$184$	−6.49396	−0.478741
$185$	0	0
$186$	6.89008	0.505205
$187$	−14.0411	−1.02679
$188$	11.7017	0.853435
$189$	17.8780	1.30043
$190$	0	0
$191$	18.2198	1.31834	0.659170	−	0.751994i	$-0.270908\pi$
$191$	18.2198	1.31834	0.659170	+	0.751994i	$0.270908\pi$
$192$	1.80194	0.130044
$193$	18.7482	1.34953	0.674764	−	0.738034i	$-0.264246\pi$
$193$	18.7482	1.34953	0.674764	+	0.738034i	$0.264246\pi$
$194$	−3.95108	−0.283671
$195$	0	0
$196$	5.98792	0.427708
$197$	22.4155	1.59704	0.798519	−	0.601969i	$-0.205617\pi$
$197$	22.4155	1.59704	0.798519	+	0.601969i	$0.205617\pi$
$198$	1.09783	0.0780197
$199$	17.0858	1.21118	0.605588	−	0.795778i	$-0.292938\pi$
$199$	17.0858	1.21118	0.605588	+	0.795778i	$0.292938\pi$
$200$	0	0
$201$	25.2228	1.77908
$202$	−15.0422	−1.05837
$203$	10.4155	0.731025
$204$	5.69202	0.398521
$205$	0	0
$206$	7.82371	0.545104
$207$	1.60388	0.111477
$208$	0	0
$209$	−1.58642	−0.109735
$210$	0	0
$211$	−5.53079	−0.380756	−0.190378	−	0.981711i	$-0.560971\pi$
$211$	−5.53079	−0.380756	−0.190378	+	0.981711i	$0.560971\pi$
$212$	12.3720	0.849710
$213$	−23.0858	−1.58181
$214$	9.30127	0.635822
$215$	0	0
$216$	4.96077	0.337538
$217$	13.7802	0.935459
$218$	0.811626	0.0549703
$219$	12.5526	0.848224
$220$	0	0
$221$	0	0
$222$	13.4819	0.904844
$223$	4.27413	0.286217	0.143108	−	0.989707i	$-0.454290\pi$
$223$	4.27413	0.286217	0.143108	+	0.989707i	$0.454290\pi$
$224$	3.60388	0.240794
$225$	0	0
$226$	7.00969	0.466278
$227$	−9.95407	−0.660675	−0.330337	−	0.943863i	$-0.607162\pi$
$227$	−9.95407	−0.660675	−0.330337	+	0.943863i	$0.607162\pi$
$228$	0.643104	0.0425906
$229$	23.0315	1.52196	0.760981	−	0.648774i	$-0.224718\pi$
$229$	23.0315	1.52196	0.760981	+	0.648774i	$0.224718\pi$
$230$	0	0
$231$	28.8659	1.89924
$232$	2.89008	0.189743
$233$	−18.5700	−1.21656	−0.608281	−	0.793721i	$-0.708141\pi$
$233$	−18.5700	−1.21656	−0.608281	+	0.793721i	$0.708141\pi$
$234$	0	0
$235$	0	0
$236$	11.9976	0.780978
$237$	−10.5918	−0.688011
$238$	11.3840	0.737918
$239$	18.9530	1.22597	0.612984	−	0.790095i	$-0.289969\pi$
$239$	18.9530	1.22597	0.612984	+	0.790095i	$0.289969\pi$
$240$	0	0
$241$	−22.7439	−1.46506	−0.732532	−	0.680732i	$-0.761662\pi$
$241$	−22.7439	−1.46506	−0.732532	+	0.680732i	$0.761662\pi$
$242$	−8.75840	−0.563011
$243$	−2.56033	−0.164246
$244$	−12.3720	−0.792034
$245$	0	0
$246$	3.67456	0.234282
$247$	0	0
$248$	3.82371	0.242806
$249$	8.42327	0.533803
$250$	0	0
$251$	14.5670	0.919463	0.459732	−	0.888058i	$-0.347946\pi$
$251$	14.5670	0.919463	0.459732	+	0.888058i	$0.347946\pi$
$252$	−0.890084	−0.0560700
$253$	−28.8659	−1.81478
$254$	−11.6582	−0.731499
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00431	−0.125026	−0.0625128	−	0.998044i	$-0.519911\pi$
$257$	−2.00431	−0.125026	−0.0625128	+	0.998044i	$0.519911\pi$
$258$	18.0465	1.12353
$259$	26.9638	1.67545
$260$	0	0
$261$	−0.713792	−0.0441826
$262$	1.34481	0.0830829
$263$	−4.41550	−0.272272	−0.136136	−	0.990690i	$-0.543468\pi$
$263$	−4.41550	−0.272272	−0.136136	+	0.990690i	$0.543468\pi$
$264$	8.00969	0.492962
$265$	0	0
$266$	1.28621	0.0788625
$267$	−14.4547	−0.884615
$268$	13.9976	0.855040
$269$	−18.1521	−1.10675	−0.553377	−	0.832931i	$-0.686661\pi$
$269$	−18.1521	−1.10675	−0.553377	+	0.832931i	$0.686661\pi$
$270$	0	0
$271$	19.4711	1.18279	0.591393	−	0.806383i	$-0.298578\pi$
$271$	19.4711	1.18279	0.591393	+	0.806383i	$0.298578\pi$
$272$	3.15883	0.191532
$273$	0	0
$274$	−7.80194	−0.471332
$275$	0	0
$276$	11.7017	0.704360
$277$	18.3961	1.10532	0.552658	−	0.833408i	$-0.313614\pi$
$277$	18.3961	1.10532	0.552658	+	0.833408i	$0.313614\pi$
$278$	−13.8726	−0.832025
$279$	−0.944378	−0.0565384
$280$	0	0
$281$	15.9463	0.951276	0.475638	−	0.879641i	$-0.342217\pi$
$281$	15.9463	0.951276	0.475638	+	0.879641i	$0.342217\pi$
$282$	−21.0858	−1.25564
$283$	27.9909	1.66389	0.831943	−	0.554861i	$-0.187228\pi$
$283$	27.9909	1.66389	0.831943	+	0.554861i	$0.187228\pi$
$284$	−12.8116	−0.760230
$285$	0	0
$286$	0	0
$287$	7.34913	0.433805
$288$	−0.246980	−0.0145534
$289$	−7.02177	−0.413045
$290$	0	0
$291$	7.11960	0.417359
$292$	6.96615	0.407663
$293$	−13.5603	−0.792203	−0.396102	−	0.918207i	$-0.629637\pi$
$293$	−13.5603	−0.792203	−0.396102	+	0.918207i	$0.629637\pi$
$294$	−10.7899	−0.629277
$295$	0	0
$296$	7.48188	0.434875
$297$	22.0508	1.27952
$298$	12.6703	0.733968
$299$	0	0
$300$	0	0
$301$	36.0930	2.08037
$302$	−4.86592	−0.280002
$303$	27.1051	1.55715
$304$	0.356896	0.0204694
$305$	0	0
$306$	−0.780167	−0.0445992
$307$	−15.1588	−0.865160	−0.432580	−	0.901595i	$-0.642397\pi$
$307$	−15.1588	−0.865160	−0.432580	+	0.901595i	$0.642397\pi$
$308$	16.0194	0.912789
$309$	−14.0978	−0.801998
$310$	0	0
$311$	5.38404	0.305301	0.152651	−	0.988280i	$-0.451219\pi$
$311$	5.38404	0.305301	0.152651	+	0.988280i	$0.451219\pi$
$312$	0	0
$313$	16.5864	0.937520	0.468760	−	0.883326i	$-0.344701\pi$
$313$	16.5864	0.937520	0.468760	+	0.883326i	$0.344701\pi$
$314$	−9.42758	−0.532029
$315$	0	0
$316$	−5.87800	−0.330663
$317$	12.1086	0.680086	0.340043	−	0.940410i	$-0.389558\pi$
$317$	12.1086	0.680086	0.340043	+	0.940410i	$0.389558\pi$
$318$	−22.2935	−1.25016
$319$	12.8465	0.719268
$320$	0	0
$321$	−16.7603	−0.935470
$322$	23.4034	1.30422
$323$	1.12737	0.0627288
$324$	−9.67994	−0.537774
$325$	0	0
$326$	−5.47650	−0.303315
$327$	−1.46250	−0.0808764
$328$	2.03923	0.112598
$329$	−42.1715	−2.32499
$330$	0	0
$331$	−14.6112	−0.803103	−0.401551	−	0.915837i	$-0.631529\pi$
$331$	−14.6112	−0.803103	−0.401551	+	0.915837i	$0.631529\pi$
$332$	4.67456	0.256550
$333$	−1.84787	−0.101263
$334$	2.68963	0.147170
$335$	0	0
$336$	−6.49396	−0.354275
$337$	4.45952	0.242925	0.121463	−	0.992596i	$-0.461242\pi$
$337$	4.45952	0.242925	0.121463	+	0.992596i	$0.461242\pi$
$338$	0	0
$339$	−12.6310	−0.686023
$340$	0	0
$341$	16.9965	0.920414
$342$	−0.0881460	−0.00476639
$343$	3.64742	0.196942
$344$	10.0151	0.539976
$345$	0	0
$346$	10.9095	0.586496
$347$	−28.5870	−1.53463	−0.767316	−	0.641270i	$-0.778408\pi$
$347$	−28.5870	−1.53463	−0.767316	+	0.641270i	$0.778408\pi$
$348$	−5.20775	−0.279165
$349$	4.87933	0.261185	0.130592	−	0.991436i	$-0.458312\pi$
$349$	4.87933	0.261185	0.130592	+	0.991436i	$0.458312\pi$
$350$	0	0
$351$	0	0
$352$	4.44504	0.236922
$353$	−4.05967	−0.216074	−0.108037	−	0.994147i	$-0.534457\pi$
$353$	−4.05967	−0.216074	−0.108037	+	0.994147i	$0.534457\pi$
$354$	−21.6189	−1.14903
$355$	0	0
$356$	−8.02177	−0.425153
$357$	−20.5133	−1.08568
$358$	−12.9825	−0.686149
$359$	5.12067	0.270259	0.135129	−	0.990828i	$-0.456855\pi$
$359$	5.12067	0.270259	0.135129	+	0.990828i	$0.456855\pi$
$360$	0	0
$361$	−18.8726	−0.993296
$362$	−22.7332	−1.19483
$363$	15.7821	0.828345
$364$	0	0
$365$	0	0
$366$	22.2935	1.16530
$367$	9.28621	0.484736	0.242368	−	0.970184i	$-0.422076\pi$
$367$	9.28621	0.484736	0.242368	+	0.970184i	$0.422076\pi$
$368$	6.49396	0.338521
$369$	−0.503648	−0.0262189
$370$	0	0
$371$	−44.5870	−2.31484
$372$	−6.89008	−0.357234
$373$	7.30559	0.378269	0.189134	−	0.981951i	$-0.439432\pi$
$373$	7.30559	0.378269	0.189134	+	0.981951i	$0.439432\pi$
$374$	14.0411	0.726050
$375$	0	0
$376$	−11.7017	−0.603470
$377$	0	0
$378$	−17.8780	−0.919545
$379$	30.5700	1.57028	0.785138	−	0.619320i	$-0.212592\pi$
$379$	30.5700	1.57028	0.785138	+	0.619320i	$0.212592\pi$
$380$	0	0
$381$	21.0073	1.07624
$382$	−18.2198	−0.932208
$383$	7.93230	0.405321	0.202661	−	0.979249i	$-0.435041\pi$
$383$	7.93230	0.405321	0.202661	+	0.979249i	$0.435041\pi$
$384$	−1.80194	−0.0919548
$385$	0	0
$386$	−18.7482	−0.954260
$387$	−2.47352	−0.125736
$388$	3.95108	0.200586
$389$	1.35988	0.0689486	0.0344743	−	0.999406i	$-0.489024\pi$
$389$	1.35988	0.0689486	0.0344743	+	0.999406i	$0.489024\pi$
$390$	0	0
$391$	20.5133	1.03740
$392$	−5.98792	−0.302436
$393$	−2.42327	−0.122238
$394$	−22.4155	−1.12928
$395$	0	0
$396$	−1.09783	−0.0551683
$397$	−24.6112	−1.23520	−0.617600	−	0.786493i	$-0.711895\pi$
$397$	−24.6112	−1.23520	−0.617600	+	0.786493i	$0.711895\pi$
$398$	−17.0858	−0.856431
$399$	−2.31767	−0.116028
$400$	0	0
$401$	−4.89546	−0.244468	−0.122234	−	0.992501i	$-0.539006\pi$
$401$	−4.89546	−0.244468	−0.122234	+	0.992501i	$0.539006\pi$
$402$	−25.2228	−1.25800
$403$	0	0
$404$	15.0422	0.748378
$405$	0	0
$406$	−10.4155	−0.516913
$407$	33.2573	1.64850
$408$	−5.69202	−0.281797
$409$	4.37435	0.216298	0.108149	−	0.994135i	$-0.465508\pi$
$409$	4.37435	0.216298	0.108149	+	0.994135i	$0.465508\pi$
$410$	0	0
$411$	14.0586	0.693460
$412$	−7.82371	−0.385446
$413$	−43.2379	−2.12760
$414$	−1.60388	−0.0788262
$415$	0	0
$416$	0	0
$417$	24.9976	1.22414
$418$	1.58642	0.0775942
$419$	16.7439	0.817994	0.408997	−	0.912536i	$-0.365879\pi$
$419$	16.7439	0.817994	0.408997	+	0.912536i	$0.365879\pi$
$420$	0	0
$421$	−0.483206	−0.0235500	−0.0117750	−	0.999931i	$-0.503748\pi$
$421$	−0.483206	−0.0235500	−0.0117750	+	0.999931i	$0.503748\pi$
$422$	5.53079	0.269235
$423$	2.89008	0.140521
$424$	−12.3720	−0.600836
$425$	0	0
$426$	23.0858	1.11851
$427$	44.5870	2.15772
$428$	−9.30127	−0.449594
$429$	0	0
$430$	0	0
$431$	11.9758	0.576856	0.288428	−	0.957502i	$-0.406867\pi$
$431$	11.9758	0.576856	0.288428	+	0.957502i	$0.406867\pi$
$432$	−4.96077	−0.238675
$433$	4.15452	0.199654	0.0998268	−	0.995005i	$-0.468171\pi$
$433$	4.15452	0.199654	0.0998268	+	0.995005i	$0.468171\pi$
$434$	−13.7802	−0.661469
$435$	0	0
$436$	−0.811626	−0.0388699
$437$	2.31767	0.110869
$438$	−12.5526	−0.599785
$439$	−12.5133	−0.597229	−0.298614	−	0.954374i	$-0.596524\pi$
$439$	−12.5133	−0.597229	−0.298614	+	0.954374i	$0.596524\pi$
$440$	0	0
$441$	1.47889	0.0704235
$442$	0	0
$443$	2.27950	0.108302	0.0541512	−	0.998533i	$-0.482755\pi$
$443$	2.27950	0.108302	0.0541512	+	0.998533i	$0.482755\pi$
$444$	−13.4819	−0.639822
$445$	0	0
$446$	−4.27413	−0.202386
$447$	−22.8310	−1.07987
$448$	−3.60388	−0.170267
$449$	−20.4547	−0.965318	−0.482659	−	0.875808i	$-0.660329\pi$
$449$	−20.4547	−0.965318	−0.482659	+	0.875808i	$0.660329\pi$
$450$	0	0
$451$	9.06446	0.426829
$452$	−7.00969	−0.329708
$453$	8.76809	0.411961
$454$	9.95407	0.467167
$455$	0	0
$456$	−0.643104	−0.0301161
$457$	17.8431	0.834664	0.417332	−	0.908754i	$-0.362965\pi$
$457$	17.8431	0.834664	0.417332	+	0.908754i	$0.362965\pi$
$458$	−23.0315	−1.07619
$459$	−15.6703	−0.731425
$460$	0	0
$461$	−3.52542	−0.164195	−0.0820975	−	0.996624i	$-0.526162\pi$
$461$	−3.52542	−0.164195	−0.0820975	+	0.996624i	$0.526162\pi$
$462$	−28.8659	−1.34296
$463$	3.72587	0.173156	0.0865780	−	0.996245i	$-0.472407\pi$
$463$	3.72587	0.173156	0.0865780	+	0.996245i	$0.472407\pi$
$464$	−2.89008	−0.134169
$465$	0	0
$466$	18.5700	0.860240
$467$	12.8576	0.594977	0.297488	−	0.954725i	$-0.403851\pi$
$467$	12.8576	0.594977	0.297488	+	0.954725i	$0.403851\pi$
$468$	0	0
$469$	−50.4456	−2.32936
$470$	0	0
$471$	16.9879	0.782762
$472$	−11.9976	−0.552235
$473$	44.5174	2.04691
$474$	10.5918	0.486497
$475$	0	0
$476$	−11.3840	−0.521787
$477$	3.05562	0.139907
$478$	−18.9530	−0.866890
$479$	33.9952	1.55328	0.776640	−	0.629944i	$-0.216922\pi$
$479$	33.9952	1.55328	0.776640	+	0.629944i	$0.216922\pi$
$480$	0	0
$481$	0	0
$482$	22.7439	1.03596
$483$	−42.1715	−1.91887
$484$	8.75840	0.398109
$485$	0	0
$486$	2.56033	0.116139
$487$	25.4819	1.15469	0.577347	−	0.816499i	$-0.304088\pi$
$487$	25.4819	1.15469	0.577347	+	0.816499i	$0.304088\pi$
$488$	12.3720	0.560052
$489$	9.86831	0.446261
$490$	0	0
$491$	25.7778	1.16333	0.581667	−	0.813427i	$-0.302401\pi$
$491$	25.7778	1.16333	0.581667	+	0.813427i	$0.302401\pi$
$492$	−3.67456	−0.165662
$493$	−9.12929	−0.411163
$494$	0	0
$495$	0	0
$496$	−3.82371	−0.171690
$497$	46.1715	2.07108
$498$	−8.42327	−0.377456
$499$	21.6286	0.968230	0.484115	−	0.875004i	$-0.339142\pi$
$499$	21.6286	0.968230	0.484115	+	0.875004i	$0.339142\pi$
$500$	0	0
$501$	−4.84654	−0.216528
$502$	−14.5670	−0.650159
$503$	−26.7332	−1.19197	−0.595987	−	0.802994i	$-0.703239\pi$
$503$	−26.7332	−1.19197	−0.595987	+	0.802994i	$0.703239\pi$
$504$	0.890084	0.0396475
$505$	0	0
$506$	28.8659	1.28325
$507$	0	0
$508$	11.6582	0.517248
$509$	−35.2814	−1.56382	−0.781911	−	0.623390i	$-0.785755\pi$
$509$	−35.2814	−1.56382	−0.781911	+	0.623390i	$0.785755\pi$
$510$	0	0
$511$	−25.1051	−1.11059
$512$	−1.00000	−0.0441942
$513$	−1.77048	−0.0781685
$514$	2.00431	0.0884064
$515$	0	0
$516$	−18.0465	−0.794454
$517$	−52.0146	−2.28760
$518$	−26.9638	−1.18472
$519$	−19.6582	−0.862898
$520$	0	0
$521$	12.5700	0.550703	0.275351	−	0.961344i	$-0.411206\pi$
$521$	12.5700	0.550703	0.275351	+	0.961344i	$0.411206\pi$
$522$	0.713792	0.0312418
$523$	−5.38942	−0.235663	−0.117831	−	0.993034i	$-0.537594\pi$
$523$	−5.38942	−0.235663	−0.117831	+	0.993034i	$0.537594\pi$
$524$	−1.34481	−0.0587485
$525$	0	0
$526$	4.41550	0.192525
$527$	−12.0785	−0.526146
$528$	−8.00969	−0.348577
$529$	19.1715	0.833544
$530$	0	0
$531$	2.96316	0.128590
$532$	−1.28621	−0.0557642
$533$	0	0
$534$	14.4547	0.625517
$535$	0	0
$536$	−13.9976	−0.604605
$537$	23.3937	1.00951
$538$	18.1521	0.782594
$539$	−26.6165	−1.14646
$540$	0	0
$541$	−22.7332	−0.977375	−0.488688	−	0.872459i	$-0.662524\pi$
$541$	−22.7332	−0.977375	−0.488688	+	0.872459i	$0.662524\pi$
$542$	−19.4711	−0.836356
$543$	40.9638	1.75792
$544$	−3.15883	−0.135434
$545$	0	0
$546$	0	0
$547$	−31.9734	−1.36709	−0.683543	−	0.729910i	$-0.739562\pi$
$547$	−31.9734	−1.36709	−0.683543	+	0.729910i	$0.739562\pi$
$548$	7.80194	0.333282
$549$	−3.05562	−0.130411
$550$	0	0
$551$	−1.03146	−0.0439416
$552$	−11.7017	−0.498058
$553$	21.1836	0.900818
$554$	−18.3961	−0.781576
$555$	0	0
$556$	13.8726	0.588330
$557$	27.2164	1.15319	0.576597	−	0.817028i	$-0.304380\pi$
$557$	27.2164	1.15319	0.576597	+	0.817028i	$0.304380\pi$
$558$	0.944378	0.0399787
$559$	0	0
$560$	0	0
$561$	−25.3013	−1.06822
$562$	−15.9463	−0.672654
$563$	11.7560	0.495457	0.247728	−	0.968830i	$-0.420316\pi$
$563$	11.7560	0.495457	0.247728	+	0.968830i	$0.420316\pi$
$564$	21.0858	0.887870
$565$	0	0
$566$	−27.9909	−1.17655
$567$	34.8853	1.46504
$568$	12.8116	0.537564
$569$	−19.2731	−0.807969	−0.403984	−	0.914766i	$-0.632375\pi$
$569$	−19.2731	−0.807969	−0.403984	+	0.914766i	$0.632375\pi$
$570$	0	0
$571$	32.6329	1.36565	0.682823	−	0.730584i	$-0.260752\pi$
$571$	32.6329	1.36565	0.682823	+	0.730584i	$0.260752\pi$
$572$	0	0
$573$	32.8310	1.37153
$574$	−7.34913	−0.306747
$575$	0	0
$576$	0.246980	0.0102908
$577$	−31.4534	−1.30942	−0.654711	−	0.755879i	$-0.727210\pi$
$577$	−31.4534	−1.30942	−0.654711	+	0.755879i	$0.727210\pi$
$578$	7.02177	0.292067
$579$	33.7832	1.40398
$580$	0	0
$581$	−16.8465	−0.698912
$582$	−7.11960	−0.295117
$583$	−54.9939	−2.27761
$584$	−6.96615	−0.288261
$585$	0	0
$586$	13.5603	0.560172
$587$	−20.7616	−0.856925	−0.428462	−	0.903560i	$-0.640945\pi$
$587$	−20.7616	−0.856925	−0.428462	+	0.903560i	$0.640945\pi$
$588$	10.7899	0.444966
$589$	−1.36467	−0.0562301
$590$	0	0
$591$	40.3913	1.66148
$592$	−7.48188	−0.307503
$593$	−31.9028	−1.31009	−0.655045	−	0.755590i	$-0.727350\pi$
$593$	−31.9028	−1.31009	−0.655045	+	0.755590i	$0.727350\pi$
$594$	−22.0508	−0.904757
$595$	0	0
$596$	−12.6703	−0.518994
$597$	30.7875	1.26005
$598$	0	0
$599$	−8.07846	−0.330077	−0.165038	−	0.986287i	$-0.552775\pi$
$599$	−8.07846	−0.330077	−0.165038	+	0.986287i	$0.552775\pi$
$600$	0	0
$601$	−19.4969	−0.795297	−0.397648	−	0.917538i	$-0.630174\pi$
$601$	−19.4969	−0.795297	−0.397648	+	0.917538i	$0.630174\pi$
$602$	−36.0930	−1.47104
$603$	3.45712	0.140785
$604$	4.86592	0.197991
$605$	0	0
$606$	−27.1051	−1.10107
$607$	35.9845	1.46056	0.730282	−	0.683146i	$-0.239389\pi$
$607$	35.9845	1.46056	0.730282	+	0.683146i	$0.239389\pi$
$608$	−0.356896	−0.0144740
$609$	18.7681	0.760521
$610$	0	0
$611$	0	0
$612$	0.780167	0.0315364
$613$	18.1763	0.734134	0.367067	−	0.930195i	$-0.380362\pi$
$613$	18.1763	0.734134	0.367067	+	0.930195i	$0.380362\pi$
$614$	15.1588	0.611761
$615$	0	0
$616$	−16.0194	−0.645439
$617$	43.0532	1.73326	0.866629	−	0.498953i	$-0.166282\pi$
$617$	43.0532	1.73326	0.866629	+	0.498953i	$0.166282\pi$
$618$	14.0978	0.567098
$619$	−39.5120	−1.58812	−0.794061	−	0.607838i	$-0.792037\pi$
$619$	−39.5120	−1.58812	−0.794061	+	0.607838i	$0.792037\pi$
$620$	0	0
$621$	−32.2150	−1.29275
$622$	−5.38404	−0.215880
$623$	28.9095	1.15823
$624$	0	0
$625$	0	0
$626$	−16.5864	−0.662927
$627$	−2.85862	−0.114162
$628$	9.42758	0.376202
$629$	−23.6340	−0.942350
$630$	0	0
$631$	−23.0616	−0.918067	−0.459034	−	0.888419i	$-0.651804\pi$
$631$	−23.0616	−0.918067	−0.459034	+	0.888419i	$0.651804\pi$
$632$	5.87800	0.233814
$633$	−9.96615	−0.396119
$634$	−12.1086	−0.480893
$635$	0	0
$636$	22.2935	0.883995
$637$	0	0
$638$	−12.8465	−0.508600
$639$	−3.16421	−0.125174
$640$	0	0
$641$	34.7047	1.37075	0.685377	−	0.728189i	$-0.259638\pi$
$641$	34.7047	1.37075	0.685377	+	0.728189i	$0.259638\pi$
$642$	16.7603	0.661477
$643$	34.2107	1.34914	0.674570	−	0.738211i	$-0.264329\pi$
$643$	34.2107	1.34914	0.674570	+	0.738211i	$0.264329\pi$
$644$	−23.4034	−0.922224
$645$	0	0
$646$	−1.12737	−0.0443560
$647$	−4.45904	−0.175303	−0.0876515	−	0.996151i	$-0.527936\pi$
$647$	−4.45904	−0.175303	−0.0876515	+	0.996151i	$0.527936\pi$
$648$	9.67994	0.380264
$649$	−53.3299	−2.09338
$650$	0	0
$651$	24.8310	0.973204
$652$	5.47650	0.214476
$653$	−9.44696	−0.369688	−0.184844	−	0.982768i	$-0.559178\pi$
$653$	−9.44696	−0.369688	−0.184844	+	0.982768i	$0.559178\pi$
$654$	1.46250	0.0571883
$655$	0	0
$656$	−2.03923	−0.0796185
$657$	1.72050	0.0671230
$658$	42.1715	1.64402
$659$	−17.7778	−0.692524	−0.346262	−	0.938138i	$-0.612549\pi$
$659$	−17.7778	−0.692524	−0.346262	+	0.938138i	$0.612549\pi$
$660$	0	0
$661$	−31.2814	−1.21671	−0.608353	−	0.793666i	$-0.708170\pi$
$661$	−31.2814	−1.21671	−0.608353	+	0.793666i	$0.708170\pi$
$662$	14.6112	0.567879
$663$	0	0
$664$	−4.67456	−0.181408
$665$	0	0
$666$	1.84787	0.0716036
$667$	−18.7681	−0.726703
$668$	−2.68963	−0.104065
$669$	7.70171	0.297765
$670$	0	0
$671$	54.9939	2.12302
$672$	6.49396	0.250510
$673$	14.4614	0.557447	0.278724	−	0.960371i	$-0.410089\pi$
$673$	14.4614	0.557447	0.278724	+	0.960371i	$0.410089\pi$
$674$	−4.45952	−0.171774
$675$	0	0
$676$	0	0
$677$	−19.8974	−0.764718	−0.382359	−	0.924014i	$-0.624888\pi$
$677$	−19.8974	−0.764718	−0.382359	+	0.924014i	$0.624888\pi$
$678$	12.6310	0.485091
$679$	−14.2392	−0.546451
$680$	0	0
$681$	−17.9366	−0.687332
$682$	−16.9965	−0.650831
$683$	−40.3038	−1.54218	−0.771091	−	0.636725i	$-0.780289\pi$
$683$	−40.3038	−1.54218	−0.771091	+	0.636725i	$0.780289\pi$
$684$	0.0881460	0.00337035
$685$	0	0
$686$	−3.64742	−0.139259
$687$	41.5013	1.58337
$688$	−10.0151	−0.381821
$689$	0	0
$690$	0	0
$691$	−0.748235	−0.0284642	−0.0142321	−	0.999899i	$-0.504530\pi$
$691$	−0.748235	−0.0284642	−0.0142321	+	0.999899i	$0.504530\pi$
$692$	−10.9095	−0.414715
$693$	3.95646	0.150293
$694$	28.5870	1.08515
$695$	0	0
$696$	5.20775	0.197399
$697$	−6.44158	−0.243992
$698$	−4.87933	−0.184685
$699$	−33.4620	−1.26565
$700$	0	0
$701$	−27.5555	−1.04076	−0.520379	−	0.853935i	$-0.674209\pi$
$701$	−27.5555	−1.04076	−0.520379	+	0.853935i	$0.674209\pi$
$702$	0	0
$703$	−2.67025	−0.100710
$704$	−4.44504	−0.167529
$705$	0	0
$706$	4.05967	0.152788
$707$	−54.2103	−2.03879
$708$	21.6189	0.812490
$709$	−7.09054	−0.266291	−0.133145	−	0.991097i	$-0.542508\pi$
$709$	−7.09054	−0.266291	−0.133145	+	0.991097i	$0.542508\pi$
$710$	0	0
$711$	−1.45175	−0.0544448
$712$	8.02177	0.300629
$713$	−24.8310	−0.929928
$714$	20.5133	0.767692
$715$	0	0
$716$	12.9825	0.485180
$717$	34.1521	1.27543
$718$	−5.12067	−0.191102
$719$	−10.7138	−0.399557	−0.199779	−	0.979841i	$-0.564022\pi$
$719$	−10.7138	−0.399557	−0.199779	+	0.979841i	$0.564022\pi$
$720$	0	0
$721$	28.1957	1.05006
$722$	18.8726	0.702366
$723$	−40.9831	−1.52418
$724$	22.7332	0.844872
$725$	0	0
$726$	−15.7821	−0.585728
$727$	−1.37329	−0.0509325	−0.0254662	−	0.999676i	$-0.508107\pi$
$727$	−1.37329	−0.0509325	−0.0254662	+	0.999676i	$0.508107\pi$
$728$	0	0
$729$	24.4263	0.904676
$730$	0	0
$731$	−31.6359	−1.17010
$732$	−22.2935	−0.823992
$733$	−12.6160	−0.465981	−0.232991	−	0.972479i	$-0.574851\pi$
$733$	−12.6160	−0.465981	−0.232991	+	0.972479i	$0.574851\pi$
$734$	−9.28621	−0.342760
$735$	0	0
$736$	−6.49396	−0.239371
$737$	−62.2199	−2.29190
$738$	0.503648	0.0185395
$739$	−37.1377	−1.36613	−0.683065	−	0.730357i	$-0.739354\pi$
$739$	−37.1377	−1.36613	−0.683065	+	0.730357i	$0.739354\pi$
$740$	0	0
$741$	0	0
$742$	44.5870	1.63684
$743$	15.0530	0.552240	0.276120	−	0.961123i	$-0.410951\pi$
$743$	15.0530	0.552240	0.276120	+	0.961123i	$0.410951\pi$
$744$	6.89008	0.252603
$745$	0	0
$746$	−7.30559	−0.267476
$747$	1.15452	0.0422417
$748$	−14.0411	−0.513395
$749$	33.5206	1.22482
$750$	0	0
$751$	−22.3913	−0.817072	−0.408536	−	0.912742i	$-0.633960\pi$
$751$	−22.3913	−0.817072	−0.408536	+	0.912742i	$0.633960\pi$
$752$	11.7017	0.426717
$753$	26.2489	0.956563
$754$	0	0
$755$	0	0
$756$	17.8780	0.650217
$757$	15.8732	0.576922	0.288461	−	0.957492i	$-0.406856\pi$
$757$	15.8732	0.576922	0.288461	+	0.957492i	$0.406856\pi$
$758$	−30.5700	−1.11035
$759$	−52.0146	−1.88801
$760$	0	0
$761$	38.5682	1.39810	0.699048	−	0.715074i	$-0.253607\pi$
$761$	38.5682	1.39810	0.699048	+	0.715074i	$0.253607\pi$
$762$	−21.0073	−0.761014
$763$	2.92500	0.105892
$764$	18.2198	0.659170
$765$	0	0
$766$	−7.93230	−0.286606
$767$	0	0
$768$	1.80194	0.0650218
$769$	49.0320	1.76814	0.884070	−	0.467354i	$-0.154793\pi$
$769$	49.0320	1.76814	0.884070	+	0.467354i	$0.154793\pi$
$770$	0	0
$771$	−3.61165	−0.130070
$772$	18.7482	0.674764
$773$	−11.8888	−0.427609	−0.213804	−	0.976876i	$-0.568586\pi$
$773$	−11.8888	−0.427609	−0.213804	+	0.976876i	$0.568586\pi$
$774$	2.47352	0.0889087
$775$	0	0
$776$	−3.95108	−0.141836
$777$	48.5870	1.74305
$778$	−1.35988	−0.0487541
$779$	−0.727792	−0.0260759
$780$	0	0
$781$	56.9482	2.03777
$782$	−20.5133	−0.733555
$783$	14.3370	0.512364
$784$	5.98792	0.213854
$785$	0	0
$786$	2.42327	0.0864352
$787$	−3.25667	−0.116088	−0.0580438	−	0.998314i	$-0.518486\pi$
$787$	−3.25667	−0.116088	−0.0580438	+	0.998314i	$0.518486\pi$
$788$	22.4155	0.798519
$789$	−7.95646	−0.283257
$790$	0	0
$791$	25.2620	0.898215
$792$	1.09783	0.0390099
$793$	0	0
$794$	24.6112	0.873418
$795$	0	0
$796$	17.0858	0.605588
$797$	−36.9530	−1.30894	−0.654471	−	0.756087i	$-0.727109\pi$
$797$	−36.9530	−1.30894	−0.654471	+	0.756087i	$0.727109\pi$
$798$	2.31767	0.0820445
$799$	36.9638	1.30768
$800$	0	0
$801$	−1.98121	−0.0700027
$802$	4.89546	0.172865
$803$	−30.9648	−1.09272
$804$	25.2228	0.889540
$805$	0	0
$806$	0	0
$807$	−32.7090	−1.15141
$808$	−15.0422	−0.529183
$809$	−4.17198	−0.146679	−0.0733395	−	0.997307i	$-0.523366\pi$
$809$	−4.17198	−0.146679	−0.0733395	+	0.997307i	$0.523366\pi$
$810$	0	0
$811$	−17.0696	−0.599396	−0.299698	−	0.954034i	$-0.596886\pi$
$811$	−17.0696	−0.599396	−0.299698	+	0.954034i	$0.596886\pi$
$812$	10.4155	0.365512
$813$	35.0858	1.23051
$814$	−33.2573	−1.16567
$815$	0	0
$816$	5.69202	0.199261
$817$	−3.57434	−0.125050
$818$	−4.37435	−0.152946
$819$	0	0
$820$	0	0
$821$	−37.8297	−1.32026	−0.660132	−	0.751149i	$-0.729500\pi$
$821$	−37.8297	−1.32026	−0.660132	+	0.751149i	$0.729500\pi$
$822$	−14.0586	−0.490350
$823$	−41.0723	−1.43169	−0.715846	−	0.698258i	$-0.753959\pi$
$823$	−41.0723	−1.43169	−0.715846	+	0.698258i	$0.753959\pi$
$824$	7.82371	0.272552
$825$	0	0
$826$	43.2379	1.50444
$827$	26.9855	0.938379	0.469189	−	0.883098i	$-0.344546\pi$
$827$	26.9855	0.938379	0.469189	+	0.883098i	$0.344546\pi$
$828$	1.60388	0.0557385
$829$	−12.1655	−0.422527	−0.211263	−	0.977429i	$-0.567758\pi$
$829$	−12.1655	−0.422527	−0.211263	+	0.977429i	$0.567758\pi$
$830$	0	0
$831$	33.1487	1.14991
$832$	0	0
$833$	18.9148	0.655360
$834$	−24.9976	−0.865596
$835$	0	0
$836$	−1.58642	−0.0548674
$837$	18.9685	0.655649
$838$	−16.7439	−0.578409
$839$	−1.86725	−0.0644646	−0.0322323	−	0.999480i	$-0.510262\pi$
$839$	−1.86725	−0.0644646	−0.0322323	+	0.999480i	$0.510262\pi$
$840$	0	0
$841$	−20.6474	−0.711980
$842$	0.483206	0.0166524
$843$	28.7342	0.989660
$844$	−5.53079	−0.190378
$845$	0	0
$846$	−2.89008	−0.0993631
$847$	−31.5642	−1.08456
$848$	12.3720	0.424855
$849$	50.4379	1.73102
$850$	0	0
$851$	−48.5870	−1.66554
$852$	−23.0858	−0.790905
$853$	27.7453	0.949979	0.474990	−	0.879991i	$-0.342452\pi$
$853$	27.7453	0.949979	0.474990	+	0.879991i	$0.342452\pi$
$854$	−44.5870	−1.52574
$855$	0	0
$856$	9.30127	0.317911
$857$	24.2553	0.828547	0.414273	−	0.910153i	$-0.364036\pi$
$857$	24.2553	0.828547	0.414273	+	0.910153i	$0.364036\pi$
$858$	0	0
$859$	26.1089	0.890823	0.445411	−	0.895326i	$-0.353057\pi$
$859$	26.1089	0.890823	0.445411	+	0.895326i	$0.353057\pi$
$860$	0	0
$861$	13.2427	0.451309
$862$	−11.9758	−0.407899
$863$	39.5905	1.34768	0.673838	−	0.738880i	$-0.264645\pi$
$863$	39.5905	1.34768	0.673838	+	0.738880i	$0.264645\pi$
$864$	4.96077	0.168769
$865$	0	0
$866$	−4.15452	−0.141176
$867$	−12.6528	−0.429711
$868$	13.7802	0.467729
$869$	26.1280	0.886331
$870$	0	0
$871$	0	0
$872$	0.811626	0.0274851
$873$	0.975837	0.0330271
$874$	−2.31767	−0.0783963
$875$	0	0
$876$	12.5526	0.424112
$877$	−40.5086	−1.36788	−0.683938	−	0.729540i	$-0.739734\pi$
$877$	−40.5086	−1.36788	−0.683938	+	0.729540i	$0.739734\pi$
$878$	12.5133	0.422305
$879$	−24.4349	−0.824168
$880$	0	0
$881$	−36.3752	−1.22551	−0.612756	−	0.790272i	$-0.709939\pi$
$881$	−36.3752	−1.22551	−0.612756	+	0.790272i	$0.709939\pi$
$882$	−1.47889	−0.0497969
$883$	−6.51275	−0.219171	−0.109586	−	0.993977i	$-0.534952\pi$
$883$	−6.51275	−0.219171	−0.109586	+	0.993977i	$0.534952\pi$
$884$	0	0
$885$	0	0
$886$	−2.27950	−0.0765814
$887$	−3.31634	−0.111352	−0.0556759	−	0.998449i	$-0.517731\pi$
$887$	−3.31634	−0.111352	−0.0556759	+	0.998449i	$0.517731\pi$
$888$	13.4819	0.452422
$889$	−42.0146	−1.40912
$890$	0	0
$891$	43.0277	1.44148
$892$	4.27413	0.143108
$893$	4.17629	0.139754
$894$	22.8310	0.763583
$895$	0	0
$896$	3.60388	0.120397
$897$	0	0
$898$	20.4547	0.682583
$899$	11.0508	0.368566
$900$	0	0
$901$	39.0810	1.30198
$902$	−9.06446	−0.301813
$903$	65.0374	2.16431
$904$	7.00969	0.233139
$905$	0	0
$906$	−8.76809	−0.291300
$907$	48.3648	1.60593	0.802963	−	0.596029i	$-0.203256\pi$
$907$	48.3648	1.60593	0.802963	+	0.596029i	$0.203256\pi$
$908$	−9.95407	−0.330337
$909$	3.71512	0.123223
$910$	0	0
$911$	−18.2634	−0.605093	−0.302546	−	0.953135i	$-0.597837\pi$
$911$	−18.2634	−0.605093	−0.302546	+	0.953135i	$0.597837\pi$
$912$	0.643104	0.0212953
$913$	−20.7786	−0.687672
$914$	−17.8431	−0.590197
$915$	0	0
$916$	23.0315	0.760981
$917$	4.84654	0.160047
$918$	15.6703	0.517195
$919$	22.1608	0.731016	0.365508	−	0.930808i	$-0.380895\pi$
$919$	22.1608	0.731016	0.365508	+	0.930808i	$0.380895\pi$
$920$	0	0
$921$	−27.3153	−0.900069
$922$	3.52542	0.116103
$923$	0	0
$924$	28.8659	0.949619
$925$	0	0
$926$	−3.72587	−0.122440
$927$	−1.93230	−0.0634649
$928$	2.89008	0.0948716
$929$	−7.92798	−0.260109	−0.130054	−	0.991507i	$-0.541515\pi$
$929$	−7.92798	−0.260109	−0.130054	+	0.991507i	$0.541515\pi$
$930$	0	0
$931$	2.13706	0.0700394
$932$	−18.5700	−0.608281
$933$	9.70171	0.317620
$934$	−12.8576	−0.420712
$935$	0	0
$936$	0	0
$937$	−28.1758	−0.920464	−0.460232	−	0.887799i	$-0.652234\pi$
$937$	−28.1758	−0.920464	−0.460232	+	0.887799i	$0.652234\pi$
$938$	50.4456	1.64711
$939$	29.8877	0.975348
$940$	0	0
$941$	31.4228	1.02435	0.512177	−	0.858880i	$-0.328839\pi$
$941$	31.4228	1.02435	0.512177	+	0.858880i	$0.328839\pi$
$942$	−16.9879	−0.553496
$943$	−13.2427	−0.431241
$944$	11.9976	0.390489
$945$	0	0
$946$	−44.5174	−1.44739
$947$	−7.96077	−0.258690	−0.129345	−	0.991600i	$-0.541288\pi$
$947$	−7.96077	−0.258690	−0.129345	+	0.991600i	$0.541288\pi$
$948$	−10.5918	−0.344005
$949$	0	0
$950$	0	0
$951$	21.8189	0.707527
$952$	11.3840	0.368959
$953$	6.90515	0.223680	0.111840	−	0.993726i	$-0.464326\pi$
$953$	6.90515	0.223680	0.111840	+	0.993726i	$0.464326\pi$
$954$	−3.05562	−0.0989294
$955$	0	0
$956$	18.9530	0.612984
$957$	23.1487	0.748290
$958$	−33.9952	−1.09834
$959$	−28.1172	−0.907952
$960$	0	0
$961$	−16.3793	−0.528363
$962$	0	0
$963$	−2.29722	−0.0740270
$964$	−22.7439	−0.732532
$965$	0	0
$966$	42.1715	1.35685
$967$	31.3297	1.00750	0.503748	−	0.863850i	$-0.331954\pi$
$967$	31.3297	1.00750	0.503748	+	0.863850i	$0.331954\pi$
$968$	−8.75840	−0.281506
$969$	2.03146	0.0652599
$970$	0	0
$971$	−4.90515	−0.157414	−0.0787069	−	0.996898i	$-0.525079\pi$
$971$	−4.90515	−0.157414	−0.0787069	+	0.996898i	$0.525079\pi$
$972$	−2.56033	−0.0821228
$973$	−49.9952	−1.60277
$974$	−25.4819	−0.816492
$975$	0	0
$976$	−12.3720	−0.396017
$977$	13.5163	0.432425	0.216213	−	0.976346i	$-0.430630\pi$
$977$	13.5163	0.432425	0.216213	+	0.976346i	$0.430630\pi$
$978$	−9.86831	−0.315554
$979$	35.6571	1.13961
$980$	0	0
$981$	−0.200455	−0.00640004
$982$	−25.7778	−0.822602
$983$	31.7453	1.01252	0.506258	−	0.862382i	$-0.331028\pi$
$983$	31.7453	1.01252	0.506258	+	0.862382i	$0.331028\pi$
$984$	3.67456	0.117141
$985$	0	0
$986$	9.12929	0.290736
$987$	−75.9904	−2.41880
$988$	0	0
$989$	−65.0374	−2.06807
$990$	0	0
$991$	13.6098	0.432331	0.216165	−	0.976357i	$-0.430645\pi$
$991$	13.6098	0.432331	0.216165	+	0.976357i	$0.430645\pi$
$992$	3.82371	0.121403
$993$	−26.3284	−0.835507
$994$	−46.1715	−1.46447
$995$	0	0
$996$	8.42327	0.266902
$997$	39.4120	1.24819	0.624096	−	0.781348i	$-0.285467\pi$
$997$	39.4120	1.24819	0.624096	+	0.781348i	$0.285467\pi$
$998$	−21.6286	−0.684642
$999$	37.1159	1.17429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.bv.1.3		3
5.4	even	2		1690.2.a.r.1.1	yes	3
13.12	even	2		8450.2.a.cg.1.3		3
65.4	even	6		1690.2.e.r.991.3		6
65.9	even	6		1690.2.e.p.991.3		6
65.19	odd	12		1690.2.l.m.361.3		12
65.24	odd	12		1690.2.l.m.1161.3		12
65.29	even	6		1690.2.e.p.191.3		6
65.34	odd	4		1690.2.d.i.1351.4		6
65.44	odd	4		1690.2.d.i.1351.1		6
65.49	even	6		1690.2.e.r.191.3		6
65.54	odd	12		1690.2.l.m.1161.6		12
65.59	odd	12		1690.2.l.m.361.6		12
65.64	even	2		1690.2.a.p.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.p.1.1	✓	3	65.64	even	2
1690.2.a.r.1.1	yes	3	5.4	even	2
1690.2.d.i.1351.1		6	65.44	odd	4
1690.2.d.i.1351.4		6	65.34	odd	4
1690.2.e.p.191.3		6	65.29	even	6
1690.2.e.p.991.3		6	65.9	even	6
1690.2.e.r.191.3		6	65.49	even	6
1690.2.e.r.991.3		6	65.4	even	6
1690.2.l.m.361.3		12	65.19	odd	12
1690.2.l.m.361.6		12	65.59	odd	12
1690.2.l.m.1161.3		12	65.24	odd	12
1690.2.l.m.1161.6		12	65.54	odd	12
8450.2.a.bv.1.3		3	1.1	even	1	trivial
8450.2.a.cg.1.3		3	13.12	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.80194 q^{3} +1.00000 q^{4} -1.80194 q^{6} -3.60388 q^{7} -1.00000 q^{8} +0.246980 q^{9} -4.44504 q^{11} +1.80194 q^{12} +3.60388 q^{14} +1.00000 q^{16} +3.15883 q^{17} -0.246980 q^{18} +0.356896 q^{19} -6.49396 q^{21} +4.44504 q^{22} +6.49396 q^{23} -1.80194 q^{24} -4.96077 q^{27} -3.60388 q^{28} -2.89008 q^{29} -3.82371 q^{31} -1.00000 q^{32} -8.00969 q^{33} -3.15883 q^{34} +0.246980 q^{36} -7.48188 q^{37} -0.356896 q^{38} -2.03923 q^{41} +6.49396 q^{42} -10.0151 q^{43} -4.44504 q^{44} -6.49396 q^{46} +11.7017 q^{47} +1.80194 q^{48} +5.98792 q^{49} +5.69202 q^{51} +12.3720 q^{53} +4.96077 q^{54} +3.60388 q^{56} +0.643104 q^{57} +2.89008 q^{58} +11.9976 q^{59} -12.3720 q^{61} +3.82371 q^{62} -0.890084 q^{63} +1.00000 q^{64} +8.00969 q^{66} +13.9976 q^{67} +3.15883 q^{68} +11.7017 q^{69} -12.8116 q^{71} -0.246980 q^{72} +6.96615 q^{73} +7.48188 q^{74} +0.356896 q^{76} +16.0194 q^{77} -5.87800 q^{79} -9.67994 q^{81} +2.03923 q^{82} +4.67456 q^{83} -6.49396 q^{84} +10.0151 q^{86} -5.20775 q^{87} +4.44504 q^{88} -8.02177 q^{89} +6.49396 q^{92} -6.89008 q^{93} -11.7017 q^{94} -1.80194 q^{96} +3.95108 q^{97} -5.98792 q^{98} -1.09783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + q^{3} + 3 q^{4} - q^{6} - 2 q^{7} - 3 q^{8} - 4 q^{9} - 13 q^{11} + q^{12} + 2 q^{14} + 3 q^{16} + q^{17} + 4 q^{18} - 3 q^{19} - 10 q^{21} + 13 q^{22} + 10 q^{23} - q^{24} - 2 q^{27}+ \cdots + 15 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + q^3 + 3 * q^4 - q^6 - 2 * q^7 - 3 * q^8 - 4 * q^9 - 13 * q^11 + q^12 + 2 * q^14 + 3 * q^16 + q^17 + 4 * q^18 - 3 * q^19 - 10 * q^21 + 13 * q^22 + 10 * q^23 - q^24 - 2 * q^27 - 2 * q^28 - 8 * q^29 - 4 * q^31 - 3 * q^32 - 2 * q^33 - q^34 - 4 * q^36 + 6 * q^37 + 3 * q^38 - 19 * q^41 + 10 * q^42 - 5 * q^43 - 13 * q^44 - 10 * q^46 + 8 * q^47 + q^48 - q^49 + 12 * q^51 + 8 * q^53 + 2 * q^54 + 2 * q^56 + 6 * q^57 + 8 * q^58 - 5 * q^59 - 8 * q^61 + 4 * q^62 - 2 * q^63 + 3 * q^64 + 2 * q^66 + q^67 + q^68 + 8 * q^69 - 12 * q^71 + 4 * q^72 + 5 * q^73 - 6 * q^74 - 3 * q^76 + 4 * q^77 + 2 * q^79 - 5 * q^81 + 19 * q^82 - 7 * q^83 - 10 * q^84 + 5 * q^86 + 2 * q^87 + 13 * q^88 - 21 * q^89 + 10 * q^92 - 20 * q^93 - 8 * q^94 - q^96 + 21 * q^97 + q^98 + 15 * q^99

Introduction

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