LMFDB - Embedded newform 2175.2.a.u.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,2,Mod(1,2175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,3,5,0,-1,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.16425$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.16425 q^{2} +1.00000 q^{3} +2.68397 q^{4} +2.16425 q^{6} +4.84822 q^{7} +1.48028 q^{8} +1.00000 q^{9} +3.00000 q^{11} +2.68397 q^{12} -0.519721 q^{13} +10.4927 q^{14} -2.16425 q^{16} +2.84822 q^{17} +2.16425 q^{18} -3.36794 q^{19} +4.84822 q^{21} +6.49274 q^{22} -6.80877 q^{23} +1.48028 q^{24} -1.12481 q^{26} +1.00000 q^{27} +13.0125 q^{28} -1.00000 q^{29} -7.64453 q^{32} +3.00000 q^{33} +6.16425 q^{34} +2.68397 q^{36} -10.8088 q^{37} -7.28905 q^{38} -0.519721 q^{39} +7.84822 q^{41} +10.4927 q^{42} -4.17671 q^{43} +8.05191 q^{44} -14.7359 q^{46} -6.21616 q^{47} -2.16425 q^{48} +16.5052 q^{49} +2.84822 q^{51} -1.39492 q^{52} -3.44084 q^{53} +2.16425 q^{54} +7.17671 q^{56} -3.36794 q^{57} -2.16425 q^{58} +12.3285 q^{59} +5.36794 q^{61} +4.84822 q^{63} -12.2162 q^{64} +6.49274 q^{66} +15.8088 q^{67} +7.64453 q^{68} -6.80877 q^{69} -14.9855 q^{71} +1.48028 q^{72} -8.58409 q^{73} -23.3929 q^{74} -9.03944 q^{76} +14.5447 q^{77} -1.12481 q^{78} -1.59262 q^{79} +1.00000 q^{81} +16.9855 q^{82} +6.48028 q^{83} +13.0125 q^{84} -9.03944 q^{86} -1.00000 q^{87} +4.44084 q^{88} +14.4408 q^{89} -2.51972 q^{91} -18.2745 q^{92} -13.4533 q^{94} -7.64453 q^{96} -11.1373 q^{97} +35.7214 q^{98} +3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 4 q^{7} + 3 q^{9} + 9 q^{11} + 5 q^{12} - 6 q^{13} + 9 q^{14} + q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + 4 q^{21} - 3 q^{22} - q^{23} + 13 q^{26} + 3 q^{27}+ \cdots + 9 q^{99}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 - q^6 + 4 * q^7 + 3 * q^9 + 9 * q^11 + 5 * q^12 - 6 * q^13 + 9 * q^14 + q^16 - 2 * q^17 - q^18 - 4 * q^19 + 4 * q^21 - 3 * q^22 - q^23 + 13 * q^26 + 3 * q^27 + 21 * q^28 - 3 * q^29 - 11 * q^32 + 9 * q^33 + 11 * q^34 + 5 * q^36 - 13 * q^37 + 2 * q^38 - 6 * q^39 + 13 * q^41 + 9 * q^42 + 13 * q^43 + 15 * q^44 - 32 * q^46 - 2 * q^47 + q^48 + 9 * q^49 - 2 * q^51 - 25 * q^52 + 3 * q^53 - q^54 - 4 * q^56 - 4 * q^57 + q^58 + 22 * q^59 + 10 * q^61 + 4 * q^63 - 20 * q^64 - 3 * q^66 + 28 * q^67 + 11 * q^68 - q^69 - 3 * q^73 - 28 * q^74 - 36 * q^76 + 12 * q^77 + 13 * q^78 - 2 * q^79 + 3 * q^81 + 6 * q^82 + 15 * q^83 + 21 * q^84 - 36 * q^86 - 3 * q^87 + 30 * q^89 - 12 * q^91 + 14 * q^92 - 9 * q^94 - 11 * q^96 + q^97 + 50 * q^98 + 9 * 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 $

$2$	2.16425	1.53035	0.765177	−	0.643820i	$-0.222651\pi$
$2$	2.16425	1.53035	0.765177	+	0.643820i	$0.222651\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.68397	1.34198
$5$	0	0
$5$	0	0
$6$	2.16425	0.883551
$7$	4.84822	1.83245	0.916227	−	0.400660i	$-0.131219\pi$
$7$	4.84822	1.83245	0.916227	+	0.400660i	$0.131219\pi$
$8$	1.48028	0.523358
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	2.68397	0.774795
$13$	−0.519721	−0.144145	−0.0720724	−	0.997399i	$-0.522961\pi$
$13$	−0.519721	−0.144145	−0.0720724	+	0.997399i	$0.522961\pi$
$14$	10.4927	2.80430
$15$	0	0
$16$	−2.16425	−0.541062
$17$	2.84822	0.690794	0.345397	−	0.938457i	$-0.387744\pi$
$17$	2.84822	0.690794	0.345397	+	0.938457i	$0.387744\pi$
$18$	2.16425	0.510118
$19$	−3.36794	−0.772658	−0.386329	−	0.922361i	$-0.626257\pi$
$19$	−3.36794	−0.772658	−0.386329	+	0.922361i	$0.626257\pi$
$20$	0	0
$21$	4.84822	1.05797
$22$	6.49274	1.38426
$23$	−6.80877	−1.41973	−0.709864	−	0.704339i	$-0.751244\pi$
$23$	−6.80877	−1.41973	−0.709864	+	0.704339i	$0.751244\pi$
$24$	1.48028	0.302161
$25$	0	0
$26$	−1.12481	−0.220593
$27$	1.00000	0.192450
$28$	13.0125	2.45912
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−7.64453	−1.35137
$33$	3.00000	0.522233
$34$	6.16425	1.05716
$35$	0	0
$36$	2.68397	0.447328
$37$	−10.8088	−1.77695	−0.888476	−	0.458923i	$-0.848235\pi$
$37$	−10.8088	−1.77695	−0.888476	+	0.458923i	$0.848235\pi$
$38$	−7.28905	−1.18244
$39$	−0.519721	−0.0832220
$40$	0	0
$41$	7.84822	1.22569	0.612843	−	0.790205i	$-0.290026\pi$
$41$	7.84822	1.22569	0.612843	+	0.790205i	$0.290026\pi$
$42$	10.4927	1.61907
$43$	−4.17671	−0.636943	−0.318471	−	0.947932i	$-0.603169\pi$
$43$	−4.17671	−0.636943	−0.318471	+	0.947932i	$0.603169\pi$
$44$	8.05191	1.21387
$45$	0	0
$46$	−14.7359	−2.17269
$47$	−6.21616	−0.906719	−0.453360	−	0.891328i	$-0.649775\pi$
$47$	−6.21616	−0.906719	−0.453360	+	0.891328i	$0.649775\pi$
$48$	−2.16425	−0.312382
$49$	16.5052	2.35789
$50$	0	0
$51$	2.84822	0.398830
$52$	−1.39492	−0.193440
$53$	−3.44084	−0.472635	−0.236318	−	0.971676i	$-0.575941\pi$
$53$	−3.44084	−0.472635	−0.236318	+	0.971676i	$0.575941\pi$
$54$	2.16425	0.294517
$55$	0	0
$56$	7.17671	0.959029
$57$	−3.36794	−0.446094
$58$	−2.16425	−0.284180
$59$	12.3285	1.60503	0.802517	−	0.596630i	$-0.203494\pi$
$59$	12.3285	1.60503	0.802517	+	0.596630i	$0.203494\pi$
$60$	0	0
$61$	5.36794	0.687294	0.343647	−	0.939099i	$-0.388338\pi$
$61$	5.36794	0.687294	0.343647	+	0.939099i	$0.388338\pi$
$62$	0	0
$63$	4.84822	0.610818
$64$	−12.2162	−1.52702
$65$	0	0
$66$	6.49274	0.799202
$67$	15.8088	1.93135	0.965675	−	0.259755i	$-0.0836417\pi$
$67$	15.8088	1.93135	0.965675	+	0.259755i	$0.0836417\pi$
$68$	7.64453	0.927035
$69$	−6.80877	−0.819680
$70$	0	0
$71$	−14.9855	−1.77845	−0.889225	−	0.457470i	$-0.848756\pi$
$71$	−14.9855	−1.77845	−0.889225	+	0.457470i	$0.848756\pi$
$72$	1.48028	0.174453
$73$	−8.58409	−1.00469	−0.502346	−	0.864667i	$-0.667530\pi$
$73$	−8.58409	−1.00469	−0.502346	+	0.864667i	$0.667530\pi$
$74$	−23.3929	−2.71937
$75$	0	0
$76$	−9.03944	−1.03690
$77$	14.5447	1.65752
$78$	−1.12481	−0.127359
$79$	−1.59262	−0.179184	−0.0895918	−	0.995979i	$-0.528556\pi$
$79$	−1.59262	−0.179184	−0.0895918	+	0.995979i	$0.528556\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	16.9855	1.87573
$83$	6.48028	0.711303	0.355652	−	0.934619i	$-0.384259\pi$
$83$	6.48028	0.711303	0.355652	+	0.934619i	$0.384259\pi$
$84$	13.0125	1.41978
$85$	0	0
$86$	−9.03944	−0.974748
$87$	−1.00000	−0.107211
$88$	4.44084	0.473395
$89$	14.4408	1.53073	0.765363	−	0.643599i	$-0.222560\pi$
$89$	14.4408	1.53073	0.765363	+	0.643599i	$0.222560\pi$
$90$	0	0
$91$	−2.51972	−0.264139
$92$	−18.2745	−1.90525
$93$	0	0
$94$	−13.4533	−1.38760
$95$	0	0
$96$	−7.64453	−0.780216
$97$	−11.1373	−1.13082	−0.565409	−	0.824811i	$-0.691282\pi$
$97$	−11.1373	−1.13082	−0.565409	+	0.824811i	$0.691282\pi$
$98$	35.7214	3.60840
$99$	3.00000	0.301511
$100$	0	0
$101$	5.63206	0.560411	0.280206	−	0.959940i	$-0.409597\pi$
$101$	5.63206	0.560411	0.280206	+	0.959940i	$0.409597\pi$
$102$	6.16425	0.610351
$103$	−12.3534	−1.21722	−0.608610	−	0.793470i	$-0.708272\pi$
$103$	−12.3534	−1.21722	−0.608610	+	0.793470i	$0.708272\pi$
$104$	−0.769332	−0.0754392
$105$	0	0
$106$	−7.44682	−0.723299
$107$	6.96056	0.672903	0.336451	−	0.941701i	$-0.390773\pi$
$107$	6.96056	0.672903	0.336451	+	0.941701i	$0.390773\pi$
$108$	2.68397	0.258265
$109$	−7.73588	−0.740963	−0.370481	−	0.928840i	$-0.620807\pi$
$109$	−7.73588	−0.740963	−0.370481	+	0.928840i	$0.620807\pi$
$110$	0	0
$111$	−10.8088	−1.02592
$112$	−10.4927	−0.991471
$113$	7.50521	0.706031	0.353015	−	0.935618i	$-0.385156\pi$
$113$	7.50521	0.706031	0.353015	+	0.935618i	$0.385156\pi$
$114$	−7.28905	−0.682682
$115$	0	0
$116$	−2.68397	−0.249200
$117$	−0.519721	−0.0480482
$118$	26.6819	2.45627
$119$	13.8088	1.26585
$120$	0	0
$121$	−2.00000	−0.181818
$122$	11.6175	1.05180
$123$	7.84822	0.707650
$124$	0	0
$125$	0	0
$126$	10.4927	0.934768
$127$	6.58409	0.584244	0.292122	−	0.956381i	$-0.405639\pi$
$127$	6.58409	0.584244	0.292122	+	0.956381i	$0.405639\pi$
$128$	−11.1497	−0.985507
$129$	−4.17671	−0.367739
$130$	0	0
$131$	−7.50521	−0.655733	−0.327867	−	0.944724i	$-0.606330\pi$
$131$	−7.50521	−0.655733	−0.327867	+	0.944724i	$0.606330\pi$
$132$	8.05191	0.700829
$133$	−16.3285	−1.41586
$134$	34.2141	2.95565
$135$	0	0
$136$	4.21616	0.361532
$137$	0.224681	0.0191958	0.00959790	−	0.999954i	$-0.496945\pi$
$137$	0.224681	0.0191958	0.00959790	+	0.999954i	$0.496945\pi$
$138$	−14.7359	−1.25440
$139$	−4.36794	−0.370484	−0.185242	−	0.982693i	$-0.559307\pi$
$139$	−4.36794	−0.370484	−0.185242	+	0.982693i	$0.559307\pi$
$140$	0	0
$141$	−6.21616	−0.523495
$142$	−32.4323	−2.72166
$143$	−1.55916	−0.130384
$144$	−2.16425	−0.180354
$145$	0	0
$146$	−18.5781	−1.53754
$147$	16.5052	1.36133
$148$	−29.0104	−2.38464
$149$	10.3285	0.846143	0.423072	−	0.906096i	$-0.360952\pi$
$149$	10.3285	0.846143	0.423072	+	0.906096i	$0.360952\pi$
$150$	0	0
$151$	−5.84822	−0.475921	−0.237961	−	0.971275i	$-0.576479\pi$
$151$	−5.84822	−0.475921	−0.237961	+	0.971275i	$0.576479\pi$
$152$	−4.98549	−0.404376
$153$	2.84822	0.230265
$154$	31.4782	2.53659
$155$	0	0
$156$	−1.39492	−0.111683
$157$	−7.36794	−0.588025	−0.294013	−	0.955801i	$-0.594991\pi$
$157$	−7.36794	−0.588025	−0.294013	+	0.955801i	$0.594991\pi$
$158$	−3.44682	−0.274215
$159$	−3.44084	−0.272876
$160$	0	0
$161$	−33.0104	−2.60159
$162$	2.16425	0.170039
$163$	4.55916	0.357101	0.178551	−	0.983931i	$-0.442859\pi$
$163$	4.55916	0.357101	0.178551	+	0.983931i	$0.442859\pi$
$164$	21.0644	1.64485
$165$	0	0
$166$	14.0249	1.08855
$167$	−19.3929	−1.50067	−0.750333	−	0.661061i	$-0.770107\pi$
$167$	−19.3929	−1.50067	−0.750333	+	0.661061i	$0.770107\pi$
$168$	7.17671	0.553695
$169$	−12.7299	−0.979222
$170$	0	0
$171$	−3.36794	−0.257553
$172$	−11.2102	−0.854767
$173$	−3.92710	−0.298572	−0.149286	−	0.988794i	$-0.547698\pi$
$173$	−3.92710	−0.298572	−0.149286	+	0.988794i	$0.547698\pi$
$174$	−2.16425	−0.164071
$175$	0	0
$176$	−6.49274	−0.489409
$177$	12.3285	0.926666
$178$	31.2536	2.34255
$179$	4.40738	0.329423	0.164712	−	0.986342i	$-0.447331\pi$
$179$	4.40738	0.329423	0.164712	+	0.986342i	$0.447331\pi$
$180$	0	0
$181$	−7.65699	−0.569140	−0.284570	−	0.958655i	$-0.591851\pi$
$181$	−7.65699	−0.569140	−0.284570	+	0.958655i	$0.591851\pi$
$182$	−5.45330	−0.404226
$183$	5.36794	0.396809
$184$	−10.0789	−0.743025
$185$	0	0
$186$	0	0
$187$	8.54465	0.624847
$188$	−16.6840	−1.21680
$189$	4.84822	0.352656
$190$	0	0
$191$	18.5052	1.33899	0.669495	−	0.742817i	$-0.266511\pi$
$191$	18.5052	1.33899	0.669495	+	0.742817i	$0.266511\pi$
$192$	−12.2162	−0.881625
$193$	17.3929	1.25197	0.625983	−	0.779837i	$-0.284698\pi$
$193$	17.3929	1.25197	0.625983	+	0.779837i	$0.284698\pi$
$194$	−24.1038	−1.73055
$195$	0	0
$196$	44.2995	3.16425
$197$	−12.8088	−0.912587	−0.456294	−	0.889829i	$-0.650823\pi$
$197$	−12.8088	−0.912587	−0.456294	+	0.889829i	$0.650823\pi$
$198$	6.49274	0.461419
$199$	−7.63206	−0.541023	−0.270511	−	0.962717i	$-0.587193\pi$
$199$	−7.63206	−0.541023	−0.270511	+	0.962717i	$0.587193\pi$
$200$	0	0
$201$	15.8088	1.11506
$202$	12.1892	0.857628
$203$	−4.84822	−0.340278
$204$	7.64453	0.535224
$205$	0	0
$206$	−26.7359	−1.86278
$207$	−6.80877	−0.473243
$208$	1.12481	0.0779912
$209$	−10.1038	−0.698895
$210$	0	0
$211$	7.47175	0.514377	0.257188	−	0.966361i	$-0.417204\pi$
$211$	7.47175	0.514377	0.257188	+	0.966361i	$0.417204\pi$
$212$	−9.23510	−0.634269
$213$	−14.9855	−1.02679
$214$	15.0644	1.02978
$215$	0	0
$216$	1.48028	0.100720
$217$	0	0
$218$	−16.7424	−1.13394
$219$	−8.58409	−0.580059
$220$	0	0
$221$	−1.48028	−0.0995743
$222$	−23.3929	−1.57003
$223$	−8.54465	−0.572192	−0.286096	−	0.958201i	$-0.592358\pi$
$223$	−8.54465	−0.572192	−0.286096	+	0.958201i	$0.592358\pi$
$224$	−37.0623	−2.47633
$225$	0	0
$226$	16.2431	1.08048
$227$	−3.13727	−0.208228	−0.104114	−	0.994565i	$-0.533201\pi$
$227$	−3.13727	−0.208228	−0.104114	+	0.994565i	$0.533201\pi$
$228$	−9.03944	−0.598652
$229$	−18.0499	−1.19277	−0.596384	−	0.802699i	$-0.703396\pi$
$229$	−18.0499	−1.19277	−0.596384	+	0.802699i	$0.703396\pi$
$230$	0	0
$231$	14.5447	0.956968
$232$	−1.48028	−0.0971851
$233$	−1.92710	−0.126249	−0.0631243	−	0.998006i	$-0.520106\pi$
$233$	−1.92710	−0.126249	−0.0631243	+	0.998006i	$0.520106\pi$
$234$	−1.12481	−0.0735308
$235$	0	0
$236$	33.0893	2.15393
$237$	−1.59262	−0.103452
$238$	29.8856	1.93720
$239$	17.3929	1.12505	0.562526	−	0.826780i	$-0.309830\pi$
$239$	17.3929	1.12505	0.562526	+	0.826780i	$0.309830\pi$
$240$	0	0
$241$	20.6175	1.32809	0.664047	−	0.747691i	$-0.268838\pi$
$241$	20.6175	1.32809	0.664047	+	0.747691i	$0.268838\pi$
$242$	−4.32850	−0.278246
$243$	1.00000	0.0641500
$244$	14.4074	0.922338
$245$	0	0
$246$	16.9855	1.08296
$247$	1.75039	0.111375
$248$	0	0
$249$	6.48028	0.410671
$250$	0	0
$251$	−16.1622	−1.02015	−0.510075	−	0.860130i	$-0.670382\pi$
$251$	−16.1622	−1.02015	−0.510075	+	0.860130i	$0.670382\pi$
$252$	13.0125	0.819708
$253$	−20.4263	−1.28419
$254$	14.2496	0.894100
$255$	0	0
$256$	0.301518	0.0188449
$257$	10.5592	0.658663	0.329331	−	0.944214i	$-0.393177\pi$
$257$	10.5592	0.658663	0.329331	+	0.944214i	$0.393177\pi$
$258$	−9.03944	−0.562771
$259$	−52.4033	−3.25618
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−16.2431	−1.00350
$263$	13.4718	0.830704	0.415352	−	0.909661i	$-0.363658\pi$
$263$	13.4718	0.830704	0.415352	+	0.909661i	$0.363658\pi$
$264$	4.44084	0.273315
$265$	0	0
$266$	−35.3389	−2.16677
$267$	14.4408	0.883765
$268$	42.4303	2.59184
$269$	−6.44084	−0.392705	−0.196352	−	0.980533i	$-0.562910\pi$
$269$	−6.44084	−0.392705	−0.196352	+	0.980533i	$0.562910\pi$
$270$	0	0
$271$	26.0499	1.58242	0.791208	−	0.611547i	$-0.209452\pi$
$271$	26.0499	1.58242	0.791208	+	0.611547i	$0.209452\pi$
$272$	−6.16425	−0.373762
$273$	−2.51972	−0.152500
$274$	0.486265	0.0293764
$275$	0	0
$276$	−18.2745	−1.10000
$277$	32.7154	1.96568	0.982838	−	0.184469i	$-0.0590567\pi$
$277$	32.7154	1.96568	0.982838	+	0.184469i	$0.0590567\pi$
$278$	−9.45330	−0.566971
$279$	0	0
$280$	0	0
$281$	−17.6425	−1.05246	−0.526231	−	0.850342i	$-0.676395\pi$
$281$	−17.6425	−1.05246	−0.526231	+	0.850342i	$0.676395\pi$
$282$	−13.4533	−0.801132
$283$	−26.4323	−1.57124	−0.785619	−	0.618711i	$-0.787655\pi$
$283$	−26.4323	−1.57124	−0.785619	+	0.618711i	$0.787655\pi$
$284$	−40.2206	−2.38665
$285$	0	0
$286$	−3.37442	−0.199533
$287$	38.0499	2.24601
$288$	−7.64453	−0.450458
$289$	−8.88766	−0.522803
$290$	0	0
$291$	−11.1373	−0.652878
$292$	−23.0394	−1.34828
$293$	−15.5052	−0.905824	−0.452912	−	0.891555i	$-0.649615\pi$
$293$	−15.5052	−0.905824	−0.452912	+	0.891555i	$0.649615\pi$
$294$	35.7214	2.08331
$295$	0	0
$296$	−16.0000	−0.929981
$297$	3.00000	0.174078
$298$	22.3534	1.29490
$299$	3.53866	0.204646
$300$	0	0
$301$	−20.2496	−1.16717
$302$	−12.6570	−0.728328
$303$	5.63206	0.323553
$304$	7.28905	0.418056
$305$	0	0
$306$	6.16425	0.352387
$307$	−11.1123	−0.634215	−0.317107	−	0.948390i	$-0.602712\pi$
$307$	−11.1123	−0.634215	−0.317107	+	0.948390i	$0.602712\pi$
$308$	39.0374	2.22436
$309$	−12.3534	−0.702762
$310$	0	0
$311$	11.0000	0.623753	0.311876	−	0.950123i	$-0.399043\pi$
$311$	11.0000	0.623753	0.311876	+	0.950123i	$0.399043\pi$
$312$	−0.769332	−0.0435549
$313$	−17.3225	−0.979126	−0.489563	−	0.871968i	$-0.662844\pi$
$313$	−17.3225	−0.979126	−0.489563	+	0.871968i	$0.662844\pi$
$314$	−15.9460	−0.899887
$315$	0	0
$316$	−4.27454	−0.240462
$317$	−22.2411	−1.24918	−0.624592	−	0.780951i	$-0.714735\pi$
$317$	−22.2411	−1.24918	−0.624592	+	0.780951i	$0.714735\pi$
$318$	−7.44682	−0.417597
$319$	−3.00000	−0.167968
$320$	0	0
$321$	6.96056	0.388500
$322$	−71.4427	−3.98135
$323$	−9.59262	−0.533748
$324$	2.68397	0.149109
$325$	0	0
$326$	9.86716	0.546491
$327$	−7.73588	−0.427795
$328$	11.6175	0.641472
$329$	−30.1373	−1.66152
$330$	0	0
$331$	29.5387	1.62359	0.811796	−	0.583941i	$-0.198490\pi$
$331$	29.5387	1.62359	0.811796	+	0.583941i	$0.198490\pi$
$332$	17.3929	0.954558
$333$	−10.8088	−0.592317
$334$	−41.9710	−2.29655
$335$	0	0
$336$	−10.4927	−0.572426
$337$	9.34301	0.508946	0.254473	−	0.967080i	$-0.418098\pi$
$337$	9.34301	0.508946	0.254473	+	0.967080i	$0.418098\pi$
$338$	−27.5506	−1.49856
$339$	7.50521	0.407627
$340$	0	0
$341$	0	0
$342$	−7.28905	−0.394147
$343$	46.0833	2.48827
$344$	−6.18270	−0.333349
$345$	0	0
$346$	−8.49922	−0.456921
$347$	25.8482	1.38760	0.693802	−	0.720165i	$-0.255934\pi$
$347$	25.8482	1.38760	0.693802	+	0.720165i	$0.255934\pi$
$348$	−2.68397	−0.143876
$349$	−33.9770	−1.81875	−0.909373	−	0.415983i	$-0.863438\pi$
$349$	−33.9770	−1.81875	−0.909373	+	0.415983i	$0.863438\pi$
$350$	0	0
$351$	−0.519721	−0.0277407
$352$	−22.9336	−1.22236
$353$	24.7359	1.31656	0.658279	−	0.752774i	$-0.271285\pi$
$353$	24.7359	1.31656	0.658279	+	0.752774i	$0.271285\pi$
$354$	26.6819	1.41813
$355$	0	0
$356$	38.7588	2.05421
$357$	13.8088	0.730838
$358$	9.53866	0.504134
$359$	−16.0729	−0.848295	−0.424148	−	0.905593i	$-0.639426\pi$
$359$	−16.0729	−0.848295	−0.424148	+	0.905593i	$0.639426\pi$
$360$	0	0
$361$	−7.65699	−0.403000
$362$	−16.5716	−0.870985
$363$	−2.00000	−0.104973
$364$	−6.76285	−0.354470
$365$	0	0
$366$	11.6175	0.607259
$367$	7.46577	0.389710	0.194855	−	0.980832i	$-0.437576\pi$
$367$	7.46577	0.389710	0.194855	+	0.980832i	$0.437576\pi$
$368$	14.7359	0.768161
$369$	7.84822	0.408562
$370$	0	0
$371$	−16.6819	−0.866082
$372$	0	0
$373$	−17.0893	−0.884851	−0.442425	−	0.896805i	$-0.645882\pi$
$373$	−17.0893	−0.884851	−0.442425	+	0.896805i	$0.645882\pi$
$374$	18.4927	0.956237
$375$	0	0
$376$	−9.20164	−0.474539
$377$	0.519721	0.0267670
$378$	10.4927	0.539689
$379$	19.1682	0.984604	0.492302	−	0.870425i	$-0.336156\pi$
$379$	19.1682	0.984604	0.492302	+	0.870425i	$0.336156\pi$
$380$	0	0
$381$	6.58409	0.337313
$382$	40.0499	2.04913
$383$	−13.8731	−0.708885	−0.354442	−	0.935078i	$-0.615329\pi$
$383$	−13.8731	−0.708885	−0.354442	+	0.935078i	$0.615329\pi$
$384$	−11.1497	−0.568983
$385$	0	0
$386$	37.6425	1.91595
$387$	−4.17671	−0.212314
$388$	−29.8921	−1.51754
$389$	20.3389	1.03122	0.515612	−	0.856822i	$-0.327565\pi$
$389$	20.3389	1.03122	0.515612	+	0.856822i	$0.327565\pi$
$390$	0	0
$391$	−19.3929	−0.980740
$392$	24.4323	1.23402
$393$	−7.50521	−0.378588
$394$	−27.7214	−1.39658
$395$	0	0
$396$	8.05191	0.404624
$397$	−32.1287	−1.61250	−0.806248	−	0.591578i	$-0.798505\pi$
$397$	−32.1287	−1.61250	−0.806248	+	0.591578i	$0.798505\pi$
$398$	−16.5177	−0.827956
$399$	−16.3285	−0.817447
$400$	0	0
$401$	21.0353	1.05046	0.525228	−	0.850962i	$-0.323980\pi$
$401$	21.0353	1.05046	0.525228	+	0.850962i	$0.323980\pi$
$402$	34.2141	1.70644
$403$	0	0
$404$	15.1163	0.752063
$405$	0	0
$406$	−10.4927	−0.520746
$407$	−32.4263	−1.60731
$408$	4.21616	0.208731
$409$	−7.51373	−0.371530	−0.185765	−	0.982594i	$-0.559476\pi$
$409$	−7.51373	−0.371530	−0.185765	+	0.982594i	$0.559476\pi$
$410$	0	0
$411$	0.224681	0.0110827
$412$	−33.1562	−1.63349
$413$	59.7712	2.94115
$414$	−14.7359	−0.724229
$415$	0	0
$416$	3.97302	0.194793
$417$	−4.36794	−0.213899
$418$	−21.8672	−1.06956
$419$	−14.6570	−0.716041	−0.358020	−	0.933714i	$-0.616548\pi$
$419$	−14.6570	−0.716041	−0.358020	+	0.933714i	$0.616548\pi$
$420$	0	0
$421$	−25.4967	−1.24263	−0.621316	−	0.783560i	$-0.713402\pi$
$421$	−25.4967	−1.24263	−0.621316	+	0.783560i	$0.713402\pi$
$422$	16.1707	0.787179
$423$	−6.21616	−0.302240
$424$	−5.09340	−0.247357
$425$	0	0
$426$	−32.4323	−1.57135
$427$	26.0249	1.25943
$428$	18.6819	0.903025
$429$	−1.55916	−0.0752771
$430$	0	0
$431$	36.1957	1.74348	0.871742	−	0.489966i	$-0.162991\pi$
$431$	36.1957	1.74348	0.871742	+	0.489966i	$0.162991\pi$
$432$	−2.16425	−0.104127
$433$	14.5301	0.698274	0.349137	−	0.937072i	$-0.386475\pi$
$433$	14.5301	0.698274	0.349137	+	0.937072i	$0.386475\pi$
$434$	0	0
$435$	0	0
$436$	−20.7629	−0.994360
$437$	22.9315	1.09696
$438$	−18.5781	−0.887696
$439$	10.5696	0.504459	0.252229	−	0.967667i	$-0.418836\pi$
$439$	10.5696	0.504459	0.252229	+	0.967667i	$0.418836\pi$
$440$	0	0
$441$	16.5052	0.785962
$442$	−3.20369	−0.152384
$443$	−39.0019	−1.85304	−0.926518	−	0.376251i	$-0.877213\pi$
$443$	−39.0019	−1.85304	−0.926518	+	0.376251i	$0.877213\pi$
$444$	−29.0104	−1.37677
$445$	0	0
$446$	−18.4927	−0.875657
$447$	10.3285	0.488521
$448$	−59.2266	−2.79819
$449$	−20.9460	−0.988505	−0.494252	−	0.869318i	$-0.664558\pi$
$449$	−20.9460	−0.988505	−0.494252	+	0.869318i	$0.664558\pi$
$450$	0	0
$451$	23.5447	1.10867
$452$	20.1437	0.947482
$453$	−5.84822	−0.274773
$454$	−6.78983	−0.318663
$455$	0	0
$456$	−4.98549	−0.233467
$457$	25.9126	1.21214	0.606070	−	0.795411i	$-0.292745\pi$
$457$	25.9126	1.21214	0.606070	+	0.795411i	$0.292745\pi$
$458$	−39.0644	−1.82536
$459$	2.84822	0.132943
$460$	0	0
$461$	25.7693	1.20020	0.600099	−	0.799926i	$-0.295128\pi$
$461$	25.7693	1.20020	0.600099	+	0.799926i	$0.295128\pi$
$462$	31.4782	1.46450
$463$	33.2805	1.54668	0.773339	−	0.633993i	$-0.218585\pi$
$463$	33.2805	1.54668	0.773339	+	0.633993i	$0.218585\pi$
$464$	2.16425	0.100473
$465$	0	0
$466$	−4.17073	−0.193205
$467$	34.4822	1.59564	0.797822	−	0.602893i	$-0.205985\pi$
$467$	34.4822	1.59564	0.797822	+	0.602893i	$0.205985\pi$
$468$	−1.39492	−0.0644800
$469$	76.6444	3.53911
$470$	0	0
$471$	−7.36794	−0.339497
$472$	18.2496	0.840006
$473$	−12.5301	−0.576136
$474$	−3.44682	−0.158318
$475$	0	0
$476$	37.0623	1.69875
$477$	−3.44084	−0.157545
$478$	37.6425	1.72173
$479$	−13.2641	−0.606053	−0.303027	−	0.952982i	$-0.597997\pi$
$479$	−13.2641	−0.606053	−0.303027	+	0.952982i	$0.597997\pi$
$480$	0	0
$481$	5.61755	0.256138
$482$	44.6215	2.03245
$483$	−33.0104	−1.50203
$484$	−5.36794	−0.243997
$485$	0	0
$486$	2.16425	0.0981723
$487$	23.2351	1.05288	0.526441	−	0.850211i	$-0.323526\pi$
$487$	23.2351	1.05288	0.526441	+	0.850211i	$0.323526\pi$
$488$	7.94605	0.359701
$489$	4.55916	0.206173
$490$	0	0
$491$	17.9211	0.808769	0.404384	−	0.914589i	$-0.367486\pi$
$491$	17.9211	0.808769	0.404384	+	0.914589i	$0.367486\pi$
$492$	21.0644	0.949655
$493$	−2.84822	−0.128277
$494$	3.78828	0.170443
$495$	0	0
$496$	0	0
$497$	−72.6529	−3.25893
$498$	14.0249	0.628472
$499$	−2.46986	−0.110566	−0.0552831	−	0.998471i	$-0.517606\pi$
$499$	−2.46986	−0.110566	−0.0552831	+	0.998471i	$0.517606\pi$
$500$	0	0
$501$	−19.3929	−0.866409
$502$	−34.9790	−1.56119
$503$	21.2266	0.946446	0.473223	−	0.880943i	$-0.343090\pi$
$503$	21.2266	0.946446	0.473223	+	0.880943i	$0.343090\pi$
$504$	7.17671	0.319676
$505$	0	0
$506$	−44.2076	−1.96527
$507$	−12.7299	−0.565354
$508$	17.6715	0.784046
$509$	37.9460	1.68193	0.840964	−	0.541090i	$-0.181988\pi$
$509$	37.9460	1.68193	0.840964	+	0.541090i	$0.181988\pi$
$510$	0	0
$511$	−41.6175	−1.84105
$512$	22.9520	1.01435
$513$	−3.36794	−0.148698
$514$	22.8526	1.00799
$515$	0	0
$516$	−11.2102	−0.493500
$517$	−18.6485	−0.820159
$518$	−113.414	−4.98311
$519$	−3.92710	−0.172381
$520$	0	0
$521$	8.40738	0.368334	0.184167	−	0.982895i	$-0.441041\pi$
$521$	8.40738	0.368334	0.184167	+	0.982895i	$0.441041\pi$
$522$	−2.16425	−0.0947266
$523$	−3.45535	−0.151092	−0.0755459	−	0.997142i	$-0.524070\pi$
$523$	−3.45535	−0.151092	−0.0755459	+	0.997142i	$0.524070\pi$
$524$	−20.1437	−0.879984
$525$	0	0
$526$	29.1562	1.27127
$527$	0	0
$528$	−6.49274	−0.282560
$529$	23.3594	1.01563
$530$	0	0
$531$	12.3285	0.535011
$532$	−43.8252	−1.90006
$533$	−4.07888	−0.176676
$534$	31.2536	1.35247
$535$	0	0
$536$	23.4014	1.01079
$537$	4.40738	0.190193
$538$	−13.9396	−0.600977
$539$	49.5156	2.13279
$540$	0	0
$541$	16.7359	0.719532	0.359766	−	0.933043i	$-0.382857\pi$
$541$	16.7359	0.719532	0.359766	+	0.933043i	$0.382857\pi$
$542$	56.3784	2.42166
$543$	−7.65699	−0.328593
$544$	−21.7733	−0.933521
$545$	0	0
$546$	−5.45330	−0.233380
$547$	−12.1123	−0.517886	−0.258943	−	0.965893i	$-0.583374\pi$
$547$	−12.1123	−0.517886	−0.258943	+	0.965893i	$0.583374\pi$
$548$	0.603037	0.0257605
$549$	5.36794	0.229098
$550$	0	0
$551$	3.36794	0.143479
$552$	−10.0789	−0.428986
$553$	−7.72136	−0.328346
$554$	70.8042	3.00818
$555$	0	0
$556$	−11.7234	−0.497183
$557$	−2.33448	−0.0989152	−0.0494576	−	0.998776i	$-0.515749\pi$
$557$	−2.33448	−0.0989152	−0.0494576	+	0.998776i	$0.515749\pi$
$558$	0	0
$559$	2.17073	0.0918119
$560$	0	0
$561$	8.54465	0.360755
$562$	−38.1827	−1.61064
$563$	4.13727	0.174365	0.0871826	−	0.996192i	$-0.472214\pi$
$563$	4.13727	0.174365	0.0871826	+	0.996192i	$0.472214\pi$
$564$	−16.6840	−0.702522
$565$	0	0
$566$	−57.2061	−2.40455
$567$	4.84822	0.203606
$568$	−22.1827	−0.930765
$569$	30.6485	1.28485	0.642425	−	0.766348i	$-0.277928\pi$
$569$	30.6485	1.28485	0.642425	+	0.766348i	$0.277928\pi$
$570$	0	0
$571$	−35.2121	−1.47358	−0.736789	−	0.676122i	$-0.763659\pi$
$571$	−35.2121	−1.47358	−0.736789	+	0.676122i	$0.763659\pi$
$572$	−4.18475	−0.174973
$573$	18.5052	0.773066
$574$	82.3493	3.43719
$575$	0	0
$576$	−12.2162	−0.509006
$577$	3.03944	0.126534	0.0632668	−	0.997997i	$-0.479848\pi$
$577$	3.03944	0.126534	0.0632668	+	0.997997i	$0.479848\pi$
$578$	−19.2351	−0.800075
$579$	17.3929	0.722823
$580$	0	0
$581$	31.4178	1.30343
$582$	−24.1038	−0.999135
$583$	−10.3225	−0.427515
$584$	−12.7069	−0.525813
$585$	0	0
$586$	−33.5571	−1.38623
$587$	3.51373	0.145027	0.0725137	−	0.997367i	$-0.476898\pi$
$587$	3.51373	0.145027	0.0725137	+	0.997367i	$0.476898\pi$
$588$	44.2995	1.82688
$589$	0	0
$590$	0	0
$591$	−12.8088	−0.526883
$592$	23.3929	0.961441
$593$	16.8397	0.691523	0.345762	−	0.938322i	$-0.387621\pi$
$593$	16.8397	0.691523	0.345762	+	0.938322i	$0.387621\pi$
$594$	6.49274	0.266401
$595$	0	0
$596$	27.7214	1.13551
$597$	−7.63206	−0.312359
$598$	7.65855	0.313181
$599$	−8.16220	−0.333498	−0.166749	−	0.985999i	$-0.553327\pi$
$599$	−8.16220	−0.333498	−0.166749	+	0.985999i	$0.553327\pi$
$600$	0	0
$601$	−6.30357	−0.257128	−0.128564	−	0.991701i	$-0.541037\pi$
$601$	−6.30357	−0.257128	−0.128564	+	0.991701i	$0.541037\pi$
$602$	−43.8252	−1.78618
$603$	15.8088	0.643783
$604$	−15.6964	−0.638679
$605$	0	0
$606$	12.1892	0.495152
$607$	−16.1707	−0.656350	−0.328175	−	0.944617i	$-0.606433\pi$
$607$	−16.1707	−0.656350	−0.328175	+	0.944617i	$0.606433\pi$
$608$	25.7463	1.04415
$609$	−4.84822	−0.196460
$610$	0	0
$611$	3.23067	0.130699
$612$	7.64453	0.309012
$613$	−0.469861	−0.0189775	−0.00948876	−	0.999955i	$-0.503020\pi$
$613$	−0.469861	−0.0189775	−0.00948876	+	0.999955i	$0.503020\pi$
$614$	−24.0499	−0.970573
$615$	0	0
$616$	21.5301	0.867474
$617$	22.2076	0.894046	0.447023	−	0.894523i	$-0.352484\pi$
$617$	22.2076	0.894046	0.447023	+	0.894523i	$0.352484\pi$
$618$	−26.7359	−1.07547
$619$	−40.5781	−1.63097	−0.815486	−	0.578777i	$-0.803530\pi$
$619$	−40.5781	−1.63097	−0.815486	+	0.578777i	$0.803530\pi$
$620$	0	0
$621$	−6.80877	−0.273227
$622$	23.8067	0.954563
$623$	70.0123	2.80498
$624$	1.12481	0.0450283
$625$	0	0
$626$	−37.4902	−1.49841
$627$	−10.1038	−0.403507
$628$	−19.7753	−0.789121
$629$	−30.7857	−1.22751
$630$	0	0
$631$	6.07036	0.241657	0.120829	−	0.992673i	$-0.461445\pi$
$631$	6.07036	0.241657	0.120829	+	0.992673i	$0.461445\pi$
$632$	−2.35752	−0.0937771
$633$	7.47175	0.296976
$634$	−48.1352	−1.91169
$635$	0	0
$636$	−9.23510	−0.366195
$637$	−8.57811	−0.339877
$638$	−6.49274	−0.257050
$639$	−14.9855	−0.592817
$640$	0	0
$641$	−40.7712	−1.61037	−0.805183	−	0.593026i	$-0.797933\pi$
$641$	−40.7712	−1.61037	−0.805183	+	0.593026i	$0.797933\pi$
$642$	15.0644	0.594543
$643$	−48.5447	−1.91441	−0.957207	−	0.289404i	$-0.906543\pi$
$643$	−48.5447	−1.91441	−0.957207	+	0.289404i	$0.906543\pi$
$644$	−88.5989	−3.49129
$645$	0	0
$646$	−20.7608	−0.816823
$647$	−24.6879	−0.970582	−0.485291	−	0.874353i	$-0.661286\pi$
$647$	−24.6879	−0.970582	−0.485291	+	0.874353i	$0.661286\pi$
$648$	1.48028	0.0581508
$649$	36.9855	1.45181
$650$	0	0
$651$	0	0
$652$	12.2367	0.479224
$653$	24.1912	0.946676	0.473338	−	0.880881i	$-0.343049\pi$
$653$	24.1912	0.946676	0.473338	+	0.880881i	$0.343049\pi$
$654$	−16.7424	−0.654678
$655$	0	0
$656$	−16.9855	−0.663172
$657$	−8.58409	−0.334897
$658$	−65.2245	−2.54272
$659$	−26.9041	−1.04803	−0.524017	−	0.851708i	$-0.675567\pi$
$659$	−26.9041	−1.04803	−0.524017	+	0.851708i	$0.675567\pi$
$660$	0	0
$661$	−6.24707	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$661$	−6.24707	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$662$	63.9290	2.48467
$663$	−1.48028	−0.0574893
$664$	9.59262	0.372266
$665$	0	0
$666$	−23.3929	−0.906455
$667$	6.80877	0.263637
$668$	−52.0499	−2.01387
$669$	−8.54465	−0.330355
$670$	0	0
$671$	16.1038	0.621681
$672$	−37.0623	−1.42971
$673$	17.4014	0.670774	0.335387	−	0.942080i	$-0.391133\pi$
$673$	17.4014	0.670774	0.335387	+	0.942080i	$0.391133\pi$
$674$	20.2206	0.778868
$675$	0	0
$676$	−34.1666	−1.31410
$677$	32.2121	1.23801	0.619005	−	0.785387i	$-0.287536\pi$
$677$	32.2121	1.23801	0.619005	+	0.785387i	$0.287536\pi$
$678$	16.2431	0.623814
$679$	−53.9959	−2.07217
$680$	0	0
$681$	−3.13727	−0.120220
$682$	0	0
$683$	−48.9375	−1.87254	−0.936271	−	0.351278i	$-0.885747\pi$
$683$	−48.9375	−1.87254	−0.936271	+	0.351278i	$0.885747\pi$
$684$	−9.03944	−0.345632
$685$	0	0
$686$	99.7357	3.80793
$687$	−18.0499	−0.688645
$688$	9.03944	0.344626
$689$	1.78828	0.0681279
$690$	0	0
$691$	31.6759	1.20501	0.602505	−	0.798115i	$-0.294169\pi$
$691$	31.6759	1.20501	0.602505	+	0.798115i	$0.294169\pi$
$692$	−10.5402	−0.400679
$693$	14.5447	0.552506
$694$	55.9420	2.12353
$695$	0	0
$696$	−1.48028	−0.0561098
$697$	22.3534	0.846696
$698$	−73.5346	−2.78332
$699$	−1.92710	−0.0728897
$700$	0	0
$701$	46.2206	1.74573	0.872864	−	0.487964i	$-0.162260\pi$
$701$	46.2206	1.74573	0.872864	+	0.487964i	$0.162260\pi$
$702$	−1.12481	−0.0424530
$703$	36.4033	1.37298
$704$	−36.6485	−1.38124
$705$	0	0
$706$	53.5346	2.01480
$707$	27.3055	1.02693
$708$	33.0893	1.24357
$709$	−1.97696	−0.0742464	−0.0371232	−	0.999311i	$-0.511819\pi$
$709$	−1.97696	−0.0742464	−0.0371232	+	0.999311i	$0.511819\pi$
$710$	0	0
$711$	−1.59262	−0.0597279
$712$	21.3765	0.801117
$713$	0	0
$714$	29.8856	1.11844
$715$	0	0
$716$	11.8293	0.442081
$717$	17.3929	0.649549
$718$	−34.7857	−1.29819
$719$	19.2891	0.719360	0.359680	−	0.933076i	$-0.382886\pi$
$719$	19.2891	0.719360	0.359680	+	0.933076i	$0.382886\pi$
$720$	0	0
$721$	−59.8921	−2.23050
$722$	−16.5716	−0.616732
$723$	20.6175	0.766775
$724$	−20.5511	−0.763777
$725$	0	0
$726$	−4.32850	−0.160646
$727$	−1.92112	−0.0712502	−0.0356251	−	0.999365i	$-0.511342\pi$
$727$	−1.92112	−0.0712502	−0.0356251	+	0.999365i	$0.511342\pi$
$728$	−3.72989	−0.138239
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.8962	−0.439996
$732$	14.4074	0.532512
$733$	−31.8002	−1.17457	−0.587284	−	0.809381i	$-0.699803\pi$
$733$	−31.8002	−1.17457	−0.587284	+	0.809381i	$0.699803\pi$
$734$	16.1578	0.596394
$735$	0	0
$736$	52.0499	1.91858
$737$	47.4263	1.74697
$738$	16.9855	0.625244
$739$	11.4967	0.422912	0.211456	−	0.977387i	$-0.432179\pi$
$739$	11.4967	0.422912	0.211456	+	0.977387i	$0.432179\pi$
$740$	0	0
$741$	1.75039	0.0643021
$742$	−36.1038	−1.32541
$743$	25.3843	0.931261	0.465631	−	0.884979i	$-0.345828\pi$
$743$	25.3843	0.931261	0.465631	+	0.884979i	$0.345828\pi$
$744$	0	0
$745$	0	0
$746$	−36.9855	−1.35413
$747$	6.48028	0.237101
$748$	22.9336	0.838535
$749$	33.7463	1.23306
$750$	0	0
$751$	50.8356	1.85502	0.927509	−	0.373802i	$-0.121946\pi$
$751$	50.8356	1.85502	0.927509	+	0.373802i	$0.121946\pi$
$752$	13.4533	0.490591
$753$	−16.1622	−0.588983
$754$	1.12481	0.0409630
$755$	0	0
$756$	13.0125	0.473259
$757$	−34.9546	−1.27045	−0.635223	−	0.772329i	$-0.719092\pi$
$757$	−34.9546	−1.27045	−0.635223	+	0.772329i	$0.719092\pi$
$758$	41.4847	1.50679
$759$	−20.4263	−0.741429
$760$	0	0
$761$	17.1931	0.623250	0.311625	−	0.950205i	$-0.399127\pi$
$761$	17.1931	0.623250	0.311625	+	0.950205i	$0.399127\pi$
$762$	14.2496	0.516209
$763$	−37.5052	−1.35778
$764$	49.6674	1.79690
$765$	0	0
$766$	−30.0249	−1.08484
$767$	−6.40738	−0.231357
$768$	0.301518	0.0108801
$769$	−13.2102	−0.476371	−0.238185	−	0.971220i	$-0.576553\pi$
$769$	−13.2102	−0.476371	−0.238185	+	0.971220i	$0.576553\pi$
$770$	0	0
$771$	10.5592	0.380279
$772$	46.6819	1.68012
$773$	18.0499	0.649208	0.324604	−	0.945850i	$-0.394769\pi$
$773$	18.0499	0.649208	0.324604	+	0.945850i	$0.394769\pi$
$774$	−9.03944	−0.324916
$775$	0	0
$776$	−16.4863	−0.591822
$777$	−52.4033	−1.87996
$778$	44.0185	1.57814
$779$	−26.4323	−0.947036
$780$	0	0
$781$	−44.9565	−1.60867
$782$	−41.9710	−1.50088
$783$	−1.00000	−0.0357371
$784$	−35.7214	−1.27576
$785$	0	0
$786$	−16.2431	−0.579373
$787$	−9.31398	−0.332008	−0.166004	−	0.986125i	$-0.553086\pi$
$787$	−9.31398	−0.332008	−0.166004	+	0.986125i	$0.553086\pi$
$788$	−34.3784	−1.22468
$789$	13.4718	0.479607
$790$	0	0
$791$	36.3869	1.29377
$792$	4.44084	0.157798
$793$	−2.78983	−0.0990698
$794$	−69.5346	−2.46769
$795$	0	0
$796$	−20.4842	−0.726044
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	−35.3389	−1.25098
$799$	−17.7050	−0.626356
$800$	0	0
$801$	14.4408	0.510242
$802$	45.5257	1.60757
$803$	−25.7523	−0.908778
$804$	42.4303	1.49640
$805$	0	0
$806$	0	0
$807$	−6.44084	−0.226728
$808$	8.33702	0.293295
$809$	3.85674	0.135596	0.0677979	−	0.997699i	$-0.478403\pi$
$809$	3.85674	0.135596	0.0677979	+	0.997699i	$0.478403\pi$
$810$	0	0
$811$	42.7883	1.50250	0.751250	−	0.660018i	$-0.229451\pi$
$811$	42.7883	1.50250	0.751250	+	0.660018i	$0.229451\pi$
$812$	−13.0125	−0.456648
$813$	26.0499	0.913608
$814$	−70.1786	−2.45976
$815$	0	0
$816$	−6.16425	−0.215792
$817$	14.0669	0.492139
$818$	−16.2616	−0.568573
$819$	−2.51972	−0.0880462
$820$	0	0
$821$	17.7463	0.619350	0.309675	−	0.950842i	$-0.399780\pi$
$821$	17.7463	0.619350	0.309675	+	0.950842i	$0.399780\pi$
$822$	0.486265	0.0169605
$823$	49.2600	1.71710	0.858548	−	0.512733i	$-0.171367\pi$
$823$	49.2600	1.71710	0.858548	+	0.512733i	$0.171367\pi$
$824$	−18.2865	−0.637041
$825$	0	0
$826$	129.360	4.50100
$827$	−19.7463	−0.686646	−0.343323	−	0.939217i	$-0.611553\pi$
$827$	−19.7463	−0.686646	−0.343323	+	0.939217i	$0.611553\pi$
$828$	−18.2745	−0.635084
$829$	47.5387	1.65109	0.825543	−	0.564339i	$-0.190869\pi$
$829$	47.5387	1.65109	0.825543	+	0.564339i	$0.190869\pi$
$830$	0	0
$831$	32.7154	1.13488
$832$	6.34899	0.220112
$833$	47.0104	1.62881
$834$	−9.45330	−0.327341
$835$	0	0
$836$	−27.1183	−0.937907
$837$	0	0
$838$	−31.7214	−1.09580
$839$	30.4658	1.05180	0.525898	−	0.850548i	$-0.323730\pi$
$839$	30.4658	1.05180	0.525898	+	0.850548i	$0.323730\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−55.1811	−1.90167
$843$	−17.6425	−0.607639
$844$	20.0540	0.690286
$845$	0	0
$846$	−13.4533	−0.462534
$847$	−9.69643	−0.333173
$848$	7.44682	0.255725
$849$	−26.4323	−0.907155
$850$	0	0
$851$	73.5945	2.52279
$852$	−40.2206	−1.37793
$853$	41.5406	1.42232	0.711161	−	0.703029i	$-0.248170\pi$
$853$	41.5406	1.42232	0.711161	+	0.703029i	$0.248170\pi$
$854$	56.3244	1.92738
$855$	0	0
$856$	10.3036	0.352169
$857$	8.05585	0.275182	0.137591	−	0.990489i	$-0.456064\pi$
$857$	8.05585	0.275182	0.137591	+	0.990489i	$0.456064\pi$
$858$	−3.37442	−0.115201
$859$	−50.6529	−1.72825	−0.864127	−	0.503273i	$-0.832129\pi$
$859$	−50.6529	−1.72825	−0.864127	+	0.503273i	$0.832129\pi$
$860$	0	0
$861$	38.0499	1.29674
$862$	78.3364	2.66815
$863$	25.2600	0.859861	0.429931	−	0.902862i	$-0.358538\pi$
$863$	25.2600	0.859861	0.429931	+	0.902862i	$0.358538\pi$
$864$	−7.64453	−0.260072
$865$	0	0
$866$	31.4468	1.06861
$867$	−8.88766	−0.301841
$868$	0	0
$869$	−4.77786	−0.162078
$870$	0	0
$871$	−8.21616	−0.278394
$872$	−11.4513	−0.387788
$873$	−11.1373	−0.376939
$874$	49.6295	1.67874
$875$	0	0
$876$	−23.0394	−0.778431
$877$	−24.9315	−0.841878	−0.420939	−	0.907089i	$-0.638299\pi$
$877$	−24.9315	−0.841878	−0.420939	+	0.907089i	$0.638299\pi$
$878$	22.8752	0.772000
$879$	−15.5052	−0.522978
$880$	0	0
$881$	−21.9356	−0.739030	−0.369515	−	0.929225i	$-0.620476\pi$
$881$	−21.9356	−0.739030	−0.369515	+	0.929225i	$0.620476\pi$
$882$	35.7214	1.20280
$883$	−7.39287	−0.248790	−0.124395	−	0.992233i	$-0.539699\pi$
$883$	−7.39287	−0.248790	−0.124395	+	0.992233i	$0.539699\pi$
$884$	−3.97302	−0.133627
$885$	0	0
$886$	−84.4098	−2.83580
$887$	−19.5472	−0.656330	−0.328165	−	0.944620i	$-0.606430\pi$
$887$	−19.5472	−0.656330	−0.328165	+	0.944620i	$0.606430\pi$
$888$	−16.0000	−0.536925
$889$	31.9211	1.07060
$890$	0	0
$891$	3.00000	0.100504
$892$	−22.9336	−0.767873
$893$	20.9356	0.700584
$894$	22.3534	0.747610
$895$	0	0
$896$	−54.0563	−1.80590
$897$	3.53866	0.118153
$898$	−45.3324	−1.51276
$899$	0	0
$900$	0	0
$901$	−9.80025	−0.326494
$902$	50.9565	1.69666
$903$	−20.2496	−0.673865
$904$	11.1098	0.369507
$905$	0	0
$906$	−12.6570	−0.420500
$907$	−26.2266	−0.870839	−0.435420	−	0.900228i	$-0.643400\pi$
$907$	−26.2266	−0.870839	−0.435420	+	0.900228i	$0.643400\pi$
$908$	−8.42034	−0.279439
$909$	5.63206	0.186804
$910$	0	0
$911$	−28.1392	−0.932292	−0.466146	−	0.884708i	$-0.654358\pi$
$911$	−28.1392	−0.932292	−0.466146	+	0.884708i	$0.654358\pi$
$912$	7.28905	0.241365
$913$	19.4408	0.643398
$914$	56.0813	1.85500
$915$	0	0
$916$	−48.4453	−1.60068
$917$	−36.3869	−1.20160
$918$	6.16425	0.203450
$919$	−42.9520	−1.41686	−0.708428	−	0.705783i	$-0.750595\pi$
$919$	−42.9520	−1.41686	−0.708428	+	0.705783i	$0.750595\pi$
$920$	0	0
$921$	−11.1123	−0.366164
$922$	55.7712	1.83673
$923$	7.78828	0.256354
$924$	39.0374	1.28424
$925$	0	0
$926$	72.0273	2.36696
$927$	−12.3534	−0.405740
$928$	7.64453	0.250944
$929$	−56.4822	−1.85312	−0.926560	−	0.376147i	$-0.877249\pi$
$929$	−56.4822	−1.85312	−0.926560	+	0.376147i	$0.877249\pi$
$930$	0	0
$931$	−55.5885	−1.82184
$932$	−5.17228	−0.169424
$933$	11.0000	0.360124
$934$	74.6280	2.44190
$935$	0	0
$936$	−0.769332	−0.0251464
$937$	58.3449	1.90604	0.953022	−	0.302900i	$-0.0979548\pi$
$937$	58.3449	1.90604	0.953022	+	0.302900i	$0.0979548\pi$
$938$	165.877	5.41609
$939$	−17.3225	−0.565299
$940$	0	0
$941$	0.457241	0.0149056	0.00745281	−	0.999972i	$-0.497628\pi$
$941$	0.457241	0.0149056	0.00745281	+	0.999972i	$0.497628\pi$
$942$	−15.9460	−0.519550
$943$	−53.4367	−1.74014
$944$	−26.6819	−0.868423
$945$	0	0
$946$	−27.1183	−0.881693
$947$	−15.0309	−0.488439	−0.244220	−	0.969720i	$-0.578532\pi$
$947$	−15.0309	−0.488439	−0.244220	+	0.969720i	$0.578532\pi$
$948$	−4.27454	−0.138831
$949$	4.46134	0.144821
$950$	0	0
$951$	−22.2411	−0.721217
$952$	20.4408	0.662491
$953$	4.19975	0.136043	0.0680216	−	0.997684i	$-0.478331\pi$
$953$	4.19975	0.136043	0.0680216	+	0.997684i	$0.478331\pi$
$954$	−7.44682	−0.241100
$955$	0	0
$956$	46.6819	1.50980
$957$	−3.00000	−0.0969762
$958$	−28.7069	−0.927476
$959$	1.08930	0.0351754
$960$	0	0
$961$	−31.0000	−1.00000
$962$	12.1578	0.391982
$963$	6.96056	0.224301
$964$	55.3369	1.78228
$965$	0	0
$966$	−71.4427	−2.29863
$967$	−24.3055	−0.781611	−0.390805	−	0.920473i	$-0.627803\pi$
$967$	−24.3055	−0.781611	−0.390805	+	0.920473i	$0.627803\pi$
$968$	−2.96056	−0.0951559
$969$	−9.59262	−0.308159
$970$	0	0
$971$	−30.8088	−0.988701	−0.494350	−	0.869263i	$-0.664594\pi$
$971$	−30.8088	−0.988701	−0.494350	+	0.869263i	$0.664594\pi$
$972$	2.68397	0.0860884
$973$	−21.1767	−0.678894
$974$	50.2865	1.61128
$975$	0	0
$976$	−11.6175	−0.371869
$977$	32.0729	1.02610	0.513051	−	0.858358i	$-0.328515\pi$
$977$	32.0729	1.02610	0.513051	+	0.858358i	$0.328515\pi$
$978$	9.86716	0.315517
$979$	43.3225	1.38459
$980$	0	0
$981$	−7.73588	−0.246988
$982$	38.7857	1.23770
$983$	−28.3534	−0.904334	−0.452167	−	0.891933i	$-0.649349\pi$
$983$	−28.3534	−0.904334	−0.452167	+	0.891933i	$0.649349\pi$
$984$	11.6175	0.370354
$985$	0	0
$986$	−6.16425	−0.196310
$987$	−30.1373	−0.959280
$988$	4.69799	0.149463
$989$	28.4383	0.904285
$990$	0	0
$991$	−14.6425	−0.465134	−0.232567	−	0.972580i	$-0.574712\pi$
$991$	−14.6425	−0.465134	−0.232567	+	0.972580i	$0.574712\pi$
$992$	0	0
$993$	29.5387	0.937381
$994$	−157.239	−4.98731
$995$	0	0
$996$	17.3929	0.551114
$997$	34.7128	1.09937	0.549683	−	0.835373i	$-0.314748\pi$
$997$	34.7128	1.09937	0.549683	+	0.835373i	$0.314748\pi$
$998$	−5.34539	−0.169205
$999$	−10.8088	−0.341974

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.u.1.3		3
3.2	odd	2		6525.2.a.bf.1.1		3
5.2	odd	4		2175.2.c.m.349.5		6
5.3	odd	4		2175.2.c.m.349.2		6
5.4	even	2		435.2.a.i.1.1	✓	3
15.14	odd	2		1305.2.a.q.1.3		3
20.19	odd	2		6960.2.a.cl.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.i.1.1	✓	3	5.4	even	2
1305.2.a.q.1.3		3	15.14	odd	2
2175.2.a.u.1.3		3	1.1	even	1	trivial
2175.2.c.m.349.2		6	5.3	odd	4
2175.2.c.m.349.5		6	5.2	odd	4
6525.2.a.bf.1.1		3	3.2	odd	2
6960.2.a.cl.1.3		3	20.19	odd	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q+2.16425 q^{2} +1.00000 q^{3} +2.68397 q^{4} +2.16425 q^{6} +4.84822 q^{7} +1.48028 q^{8} +1.00000 q^{9} +3.00000 q^{11} +2.68397 q^{12} -0.519721 q^{13} +10.4927 q^{14} -2.16425 q^{16} +2.84822 q^{17} +2.16425 q^{18} -3.36794 q^{19} +4.84822 q^{21} +6.49274 q^{22} -6.80877 q^{23} +1.48028 q^{24} -1.12481 q^{26} +1.00000 q^{27} +13.0125 q^{28} -1.00000 q^{29} -7.64453 q^{32} +3.00000 q^{33} +6.16425 q^{34} +2.68397 q^{36} -10.8088 q^{37} -7.28905 q^{38} -0.519721 q^{39} +7.84822 q^{41} +10.4927 q^{42} -4.17671 q^{43} +8.05191 q^{44} -14.7359 q^{46} -6.21616 q^{47} -2.16425 q^{48} +16.5052 q^{49} +2.84822 q^{51} -1.39492 q^{52} -3.44084 q^{53} +2.16425 q^{54} +7.17671 q^{56} -3.36794 q^{57} -2.16425 q^{58} +12.3285 q^{59} +5.36794 q^{61} +4.84822 q^{63} -12.2162 q^{64} +6.49274 q^{66} +15.8088 q^{67} +7.64453 q^{68} -6.80877 q^{69} -14.9855 q^{71} +1.48028 q^{72} -8.58409 q^{73} -23.3929 q^{74} -9.03944 q^{76} +14.5447 q^{77} -1.12481 q^{78} -1.59262 q^{79} +1.00000 q^{81} +16.9855 q^{82} +6.48028 q^{83} +13.0125 q^{84} -9.03944 q^{86} -1.00000 q^{87} +4.44084 q^{88} +14.4408 q^{89} -2.51972 q^{91} -18.2745 q^{92} -13.4533 q^{94} -7.64453 q^{96} -11.1373 q^{97} +35.7214 q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - q^{6} + 4 q^{7} + 3 q^{9} + 9 q^{11} + 5 q^{12} - 6 q^{13} + 9 q^{14} + q^{16} - 2 q^{17} - q^{18} - 4 q^{19} + 4 q^{21} - 3 q^{22} - q^{23} + 13 q^{26} + 3 q^{27}+ \cdots + 9 q^{99}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 - q^6 + 4 * q^7 + 3 * q^9 + 9 * q^11 + 5 * q^12 - 6 * q^13 + 9 * q^14 + q^16 - 2 * q^17 - q^18 - 4 * q^19 + 4 * q^21 - 3 * q^22 - q^23 + 13 * q^26 + 3 * q^27 + 21 * q^28 - 3 * q^29 - 11 * q^32 + 9 * q^33 + 11 * q^34 + 5 * q^36 - 13 * q^37 + 2 * q^38 - 6 * q^39 + 13 * q^41 + 9 * q^42 + 13 * q^43 + 15 * q^44 - 32 * q^46 - 2 * q^47 + q^48 + 9 * q^49 - 2 * q^51 - 25 * q^52 + 3 * q^53 - q^54 - 4 * q^56 - 4 * q^57 + q^58 + 22 * q^59 + 10 * q^61 + 4 * q^63 - 20 * q^64 - 3 * q^66 + 28 * q^67 + 11 * q^68 - q^69 - 3 * q^73 - 28 * q^74 - 36 * q^76 + 12 * q^77 + 13 * q^78 - 2 * q^79 + 3 * q^81 + 6 * q^82 + 15 * q^83 + 21 * q^84 - 36 * q^86 - 3 * q^87 + 30 * q^89 - 12 * q^91 + 14 * q^92 - 9 * q^94 - 11 * q^96 + q^97 + 50 * q^98 + 9 * q^99

Introduction

L-functions

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Database

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