LMFDB - Embedded newform 8019.2.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8019 = 3^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8019.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.0320373809$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53209 q^{2} +0.347296 q^{4} -3.22668 q^{5} -0.652704 q^{7} -2.53209 q^{8} +O(q^{10})$	\(q+1.53209 q^{2} +0.347296 q^{4} -3.22668 q^{5} -0.652704 q^{7} -2.53209 q^{8} -4.94356 q^{10} -1.00000 q^{11} +2.29086 q^{13} -1.00000 q^{14} -4.57398 q^{16} +2.46791 q^{17} +6.71688 q^{19} -1.12061 q^{20} -1.53209 q^{22} +0.694593 q^{23} +5.41147 q^{25} +3.50980 q^{26} -0.226682 q^{28} +7.43376 q^{29} -1.36959 q^{31} -1.94356 q^{32} +3.78106 q^{34} +2.10607 q^{35} -4.45336 q^{37} +10.2909 q^{38} +8.17024 q^{40} +0.509800 q^{41} -11.5817 q^{43} -0.347296 q^{44} +1.06418 q^{46} +7.71688 q^{47} -6.57398 q^{49} +8.29086 q^{50} +0.795607 q^{52} -6.30541 q^{53} +3.22668 q^{55} +1.65270 q^{56} +11.3892 q^{58} -0.758770 q^{59} +1.75877 q^{61} -2.09833 q^{62} +6.17024 q^{64} -7.39187 q^{65} -5.31046 q^{67} +0.857097 q^{68} +3.22668 q^{70} +10.6159 q^{71} -10.4192 q^{73} -6.82295 q^{74} +2.33275 q^{76} +0.652704 q^{77} -10.2686 q^{79} +14.7588 q^{80} +0.781059 q^{82} +10.2540 q^{83} -7.96316 q^{85} -17.7442 q^{86} +2.53209 q^{88} +18.3182 q^{89} -1.49525 q^{91} +0.241230 q^{92} +11.8229 q^{94} -21.6732 q^{95} +10.6750 q^{97} -10.0719 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{5} - 3 q^{7} - 3 q^{8} - 3 q^{11} - 9 q^{13} - 3 q^{14} - 6 q^{16} + 12 q^{17} + 12 q^{19} - 9 q^{20} + 6 q^{25} + 12 q^{26} + 6 q^{28} + 6 q^{29} + 3 q^{31} + 9 q^{32} - 6 q^{34} - 6 q^{35} + 15 q^{38} + 3 q^{40} + 3 q^{41} - 3 q^{43} - 6 q^{46} + 15 q^{47} - 12 q^{49} + 9 q^{50} + 3 q^{52} - 21 q^{53} + 3 q^{55} + 6 q^{56} + 30 q^{58} + 9 q^{59} - 6 q^{61} - 18 q^{62} - 3 q^{64} - 9 q^{65} - 3 q^{67} + 3 q^{68} + 3 q^{70} + 21 q^{71} + 3 q^{73} - 12 q^{76} + 3 q^{77} - 21 q^{79} + 33 q^{80} - 15 q^{82} + 3 q^{83} - 12 q^{85} - 24 q^{86} + 3 q^{88} + 18 q^{89} + 12 q^{91} + 12 q^{92} + 15 q^{94} - 3 q^{95} + 27 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^5 - 3 * q^7 - 3 * q^8 - 3 * q^11 - 9 * q^13 - 3 * q^14 - 6 * q^16 + 12 * q^17 + 12 * q^19 - 9 * q^20 + 6 * q^25 + 12 * q^26 + 6 * q^28 + 6 * q^29 + 3 * q^31 + 9 * q^32 - 6 * q^34 - 6 * q^35 + 15 * q^38 + 3 * q^40 + 3 * q^41 - 3 * q^43 - 6 * q^46 + 15 * q^47 - 12 * q^49 + 9 * q^50 + 3 * q^52 - 21 * q^53 + 3 * q^55 + 6 * q^56 + 30 * q^58 + 9 * q^59 - 6 * q^61 - 18 * q^62 - 3 * q^64 - 9 * q^65 - 3 * q^67 + 3 * q^68 + 3 * q^70 + 21 * q^71 + 3 * q^73 - 12 * q^76 + 3 * q^77 - 21 * q^79 + 33 * q^80 - 15 * q^82 + 3 * q^83 - 12 * q^85 - 24 * q^86 + 3 * q^88 + 18 * q^89 + 12 * q^91 + 12 * q^92 + 15 * q^94 - 3 * q^95 + 27 * q^97 + 3 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.53209	1.08335	0.541675	−	0.840588i	$-0.317790\pi$
$2$	1.53209	1.08335	0.541675	+	0.840588i	$0.317790\pi$
$3$	0	0
$3$	0	0
$4$	0.347296	0.173648
$5$	−3.22668	−1.44302	−0.721508	−	0.692406i	$-0.756551\pi$
$5$	−3.22668	−1.44302	−0.721508	+	0.692406i	$0.756551\pi$
$6$	0	0
$7$	−0.652704	−0.246699	−0.123349	−	0.992363i	$-0.539364\pi$
$7$	−0.652704	−0.246699	−0.123349	+	0.992363i	$0.539364\pi$
$8$	−2.53209	−0.895229
$9$	0	0
$10$	−4.94356	−1.56329
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.29086	0.635370	0.317685	−	0.948196i	$-0.397095\pi$
$13$	2.29086	0.635370	0.317685	+	0.948196i	$0.397095\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	−4.57398	−1.14349
$17$	2.46791	0.598556	0.299278	−	0.954166i	$-0.403254\pi$
$17$	2.46791	0.598556	0.299278	+	0.954166i	$0.403254\pi$
$18$	0	0
$19$	6.71688	1.54096	0.770479	−	0.637465i	$-0.220017\pi$
$19$	6.71688	1.54096	0.770479	+	0.637465i	$0.220017\pi$
$20$	−1.12061	−0.250577
$21$	0	0
$22$	−1.53209	−0.326642
$23$	0.694593	0.144833	0.0724163	−	0.997374i	$-0.476929\pi$
$23$	0.694593	0.144833	0.0724163	+	0.997374i	$0.476929\pi$
$24$	0	0
$25$	5.41147	1.08229
$26$	3.50980	0.688328
$27$	0	0
$28$	−0.226682	−0.0428388
$29$	7.43376	1.38042	0.690208	−	0.723611i	$-0.257519\pi$
$29$	7.43376	1.38042	0.690208	+	0.723611i	$0.257519\pi$
$30$	0	0
$31$	−1.36959	−0.245985	−0.122992	−	0.992408i	$-0.539249\pi$
$31$	−1.36959	−0.245985	−0.122992	+	0.992408i	$0.539249\pi$
$32$	−1.94356	−0.343577
$33$	0	0
$34$	3.78106	0.648446
$35$	2.10607	0.355990
$36$	0	0
$37$	−4.45336	−0.732128	−0.366064	−	0.930590i	$-0.619295\pi$
$37$	−4.45336	−0.732128	−0.366064	+	0.930590i	$0.619295\pi$
$38$	10.2909	1.66940
$39$	0	0
$40$	8.17024	1.29183
$41$	0.509800	0.0796174	0.0398087	−	0.999207i	$-0.487325\pi$
$41$	0.509800	0.0796174	0.0398087	+	0.999207i	$0.487325\pi$
$42$	0	0
$43$	−11.5817	−1.76620	−0.883098	−	0.469189i	$-0.844546\pi$
$43$	−11.5817	−1.76620	−0.883098	+	0.469189i	$0.844546\pi$
$44$	−0.347296	−0.0523569
$45$	0	0
$46$	1.06418	0.156904
$47$	7.71688	1.12562	0.562811	−	0.826585i	$-0.309720\pi$
$47$	7.71688	1.12562	0.562811	+	0.826585i	$0.309720\pi$
$48$	0	0
$49$	−6.57398	−0.939140
$50$	8.29086	1.17250
$51$	0	0
$52$	0.795607	0.110331
$53$	−6.30541	−0.866114	−0.433057	−	0.901366i	$-0.642565\pi$
$53$	−6.30541	−0.866114	−0.433057	+	0.901366i	$0.642565\pi$
$54$	0	0
$55$	3.22668	0.435086
$56$	1.65270	0.220852
$57$	0	0
$58$	11.3892	1.49547
$59$	−0.758770	−0.0987835	−0.0493918	−	0.998779i	$-0.515728\pi$
$59$	−0.758770	−0.0987835	−0.0493918	+	0.998779i	$0.515728\pi$
$60$	0	0
$61$	1.75877	0.225187	0.112594	−	0.993641i	$-0.464084\pi$
$61$	1.75877	0.225187	0.112594	+	0.993641i	$0.464084\pi$
$62$	−2.09833	−0.266488
$63$	0	0
$64$	6.17024	0.771281
$65$	−7.39187	−0.916849
$66$	0	0
$67$	−5.31046	−0.648776	−0.324388	−	0.945924i	$-0.605158\pi$
$67$	−5.31046	−0.648776	−0.324388	+	0.945924i	$0.605158\pi$
$68$	0.857097	0.103938
$69$	0	0
$70$	3.22668	0.385662
$71$	10.6159	1.25987	0.629936	−	0.776647i	$-0.283081\pi$
$71$	10.6159	1.25987	0.629936	+	0.776647i	$0.283081\pi$
$72$	0	0
$73$	−10.4192	−1.21948	−0.609738	−	0.792603i	$-0.708726\pi$
$73$	−10.4192	−1.21948	−0.609738	+	0.792603i	$0.708726\pi$
$74$	−6.82295	−0.793152
$75$	0	0
$76$	2.33275	0.267585
$77$	0.652704	0.0743825
$78$	0	0
$79$	−10.2686	−1.15530	−0.577652	−	0.816283i	$-0.696031\pi$
$79$	−10.2686	−1.15530	−0.577652	+	0.816283i	$0.696031\pi$
$80$	14.7588	1.65008
$81$	0	0
$82$	0.781059	0.0862536
$83$	10.2540	1.12553	0.562763	−	0.826619i	$-0.309739\pi$
$83$	10.2540	1.12553	0.562763	+	0.826619i	$0.309739\pi$
$84$	0	0
$85$	−7.96316	−0.863726
$86$	−17.7442	−1.91341
$87$	0	0
$88$	2.53209	0.269922
$89$	18.3182	1.94173	0.970863	−	0.239636i	$-0.0770283\pi$
$89$	18.3182	1.94173	0.970863	+	0.239636i	$0.0770283\pi$
$90$	0	0
$91$	−1.49525	−0.156745
$92$	0.241230	0.0251499
$93$	0	0
$94$	11.8229	1.21944
$95$	−21.6732	−2.22363
$96$	0	0
$97$	10.6750	1.08388	0.541941	−	0.840417i	$-0.317690\pi$
$97$	10.6750	1.08388	0.541941	+	0.840417i	$0.317690\pi$
$98$	−10.0719	−1.01742
$99$	0	0
$100$	1.87939	0.187939
$101$	1.01960	0.101454	0.0507270	−	0.998713i	$-0.483846\pi$
$101$	1.01960	0.101454	0.0507270	+	0.998713i	$0.483846\pi$
$102$	0	0
$103$	−5.07873	−0.500422	−0.250211	−	0.968191i	$-0.580500\pi$
$103$	−5.07873	−0.500422	−0.250211	+	0.968191i	$0.580500\pi$
$104$	−5.80066	−0.568801
$105$	0	0
$106$	−9.66044	−0.938305
$107$	−11.6382	−1.12510	−0.562551	−	0.826762i	$-0.690180\pi$
$107$	−11.6382	−1.12510	−0.562551	+	0.826762i	$0.690180\pi$
$108$	0	0
$109$	−13.8871	−1.33015	−0.665073	−	0.746779i	$-0.731599\pi$
$109$	−13.8871	−1.33015	−0.665073	+	0.746779i	$0.731599\pi$
$110$	4.94356	0.471350
$111$	0	0
$112$	2.98545	0.282099
$113$	−18.2986	−1.72139	−0.860694	−	0.509123i	$-0.829970\pi$
$113$	−18.2986	−1.72139	−0.860694	+	0.509123i	$0.829970\pi$
$114$	0	0
$115$	−2.24123	−0.208996
$116$	2.58172	0.239707
$117$	0	0
$118$	−1.16250	−0.107017
$119$	−1.61081	−0.147663
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.69459	0.243957
$123$	0	0
$124$	−0.475652	−0.0427148
$125$	−1.32770	−0.118753
$126$	0	0
$127$	8.92902	0.792322	0.396161	−	0.918181i	$-0.370342\pi$
$127$	8.92902	0.792322	0.396161	+	0.918181i	$0.370342\pi$
$128$	13.3405	1.17914
$129$	0	0
$130$	−11.3250	−0.993269
$131$	0.985452	0.0860993	0.0430497	−	0.999073i	$-0.486293\pi$
$131$	0.985452	0.0860993	0.0430497	+	0.999073i	$0.486293\pi$
$132$	0	0
$133$	−4.38413	−0.380153
$134$	−8.13610	−0.702852
$135$	0	0
$136$	−6.24897	−0.535845
$137$	19.5476	1.67006	0.835031	−	0.550203i	$-0.185450\pi$
$137$	19.5476	1.67006	0.835031	+	0.550203i	$0.185450\pi$
$138$	0	0
$139$	−15.9213	−1.35042	−0.675212	−	0.737623i	$-0.735948\pi$
$139$	−15.9213	−1.35042	−0.675212	+	0.737623i	$0.735948\pi$
$140$	0.731429	0.0618171
$141$	0	0
$142$	16.2645	1.36488
$143$	−2.29086	−0.191571
$144$	0	0
$145$	−23.9864	−1.99196
$146$	−15.9632	−1.32112
$147$	0	0
$148$	−1.54664	−0.127133
$149$	−7.33956	−0.601280	−0.300640	−	0.953738i	$-0.597200\pi$
$149$	−7.33956	−0.601280	−0.300640	+	0.953738i	$0.597200\pi$
$150$	0	0
$151$	−1.96585	−0.159979	−0.0799894	−	0.996796i	$-0.525489\pi$
$151$	−1.96585	−0.159979	−0.0799894	+	0.996796i	$0.525489\pi$
$152$	−17.0077	−1.37951
$153$	0	0
$154$	1.00000	0.0805823
$155$	4.41921	0.354960
$156$	0	0
$157$	−18.7050	−1.49282	−0.746412	−	0.665485i	$-0.768225\pi$
$157$	−18.7050	−1.49282	−0.746412	+	0.665485i	$0.768225\pi$
$158$	−15.7324	−1.25160
$159$	0	0
$160$	6.27126	0.495787
$161$	−0.453363	−0.0357300
$162$	0	0
$163$	−16.8007	−1.31593	−0.657965	−	0.753049i	$-0.728582\pi$
$163$	−16.8007	−1.31593	−0.657965	+	0.753049i	$0.728582\pi$
$164$	0.177052	0.0138254
$165$	0	0
$166$	15.7101	1.21934
$167$	−14.5321	−1.12453	−0.562263	−	0.826958i	$-0.690069\pi$
$167$	−14.5321	−1.12453	−0.562263	+	0.826958i	$0.690069\pi$
$168$	0	0
$169$	−7.75196	−0.596305
$170$	−12.2003	−0.935718
$171$	0	0
$172$	−4.02229	−0.306697
$173$	0.199340	0.0151556	0.00757779	−	0.999971i	$-0.497588\pi$
$173$	0.199340	0.0151556	0.00757779	+	0.999971i	$0.497588\pi$
$174$	0	0
$175$	−3.53209	−0.267001
$176$	4.57398	0.344777
$177$	0	0
$178$	28.0651	2.10357
$179$	−13.4466	−1.00504	−0.502521	−	0.864565i	$-0.667594\pi$
$179$	−13.4466	−1.00504	−0.502521	+	0.864565i	$0.667594\pi$
$180$	0	0
$181$	−18.3773	−1.36598	−0.682988	−	0.730430i	$-0.739320\pi$
$181$	−18.3773	−1.36598	−0.682988	+	0.730430i	$0.739320\pi$
$182$	−2.29086	−0.169810
$183$	0	0
$184$	−1.75877	−0.129658
$185$	14.3696	1.05647
$186$	0	0
$187$	−2.46791	−0.180472
$188$	2.68004	0.195462
$189$	0	0
$190$	−33.2053	−2.40897
$191$	−26.0428	−1.88439	−0.942196	−	0.335062i	$-0.891243\pi$
$191$	−26.0428	−1.88439	−0.942196	+	0.335062i	$0.891243\pi$
$192$	0	0
$193$	−8.82295	−0.635090	−0.317545	−	0.948243i	$-0.602858\pi$
$193$	−8.82295	−0.635090	−0.317545	+	0.948243i	$0.602858\pi$
$194$	16.3550	1.17422
$195$	0	0
$196$	−2.28312	−0.163080
$197$	1.95037	0.138958	0.0694791	−	0.997583i	$-0.477866\pi$
$197$	1.95037	0.138958	0.0694791	+	0.997583i	$0.477866\pi$
$198$	0	0
$199$	−4.87164	−0.345342	−0.172671	−	0.984980i	$-0.555240\pi$
$199$	−4.87164	−0.345342	−0.172671	+	0.984980i	$0.555240\pi$
$200$	−13.7023	−0.968901
$201$	0	0
$202$	1.56212	0.109910
$203$	−4.85204	−0.340547
$204$	0	0
$205$	−1.64496	−0.114889
$206$	−7.78106	−0.542132
$207$	0	0
$208$	−10.4783	−0.726542
$209$	−6.71688	−0.464616
$210$	0	0
$211$	23.3327	1.60629	0.803146	−	0.595782i	$-0.203158\pi$
$211$	23.3327	1.60629	0.803146	+	0.595782i	$0.203158\pi$
$212$	−2.18984	−0.150399
$213$	0	0
$214$	−17.8307	−1.21888
$215$	37.3705	2.54865
$216$	0	0
$217$	0.893933	0.0606841
$218$	−21.2763	−1.44101
$219$	0	0
$220$	1.12061	0.0755518
$221$	5.65364	0.380305
$222$	0	0
$223$	8.11112	0.543161	0.271580	−	0.962416i	$-0.412454\pi$
$223$	8.11112	0.543161	0.271580	+	0.962416i	$0.412454\pi$
$224$	1.26857	0.0847599
$225$	0	0
$226$	−28.0351	−1.86487
$227$	21.8726	1.45173	0.725867	−	0.687835i	$-0.241439\pi$
$227$	21.8726	1.45173	0.725867	+	0.687835i	$0.241439\pi$
$228$	0	0
$229$	−19.4338	−1.28422	−0.642110	−	0.766613i	$-0.721941\pi$
$229$	−19.4338	−1.28422	−0.642110	+	0.766613i	$0.721941\pi$
$230$	−3.43376	−0.226416
$231$	0	0
$232$	−18.8229	−1.23579
$233$	3.19253	0.209150	0.104575	−	0.994517i	$-0.466652\pi$
$233$	3.19253	0.209150	0.104575	+	0.994517i	$0.466652\pi$
$234$	0	0
$235$	−24.8999	−1.62429
$236$	−0.263518	−0.0171536
$237$	0	0
$238$	−2.46791	−0.159971
$239$	−20.7050	−1.33930	−0.669648	−	0.742678i	$-0.733555\pi$
$239$	−20.7050	−1.33930	−0.669648	+	0.742678i	$0.733555\pi$
$240$	0	0
$241$	4.48751	0.289066	0.144533	−	0.989500i	$-0.453832\pi$
$241$	4.48751	0.289066	0.144533	+	0.989500i	$0.453832\pi$
$242$	1.53209	0.0984864
$243$	0	0
$244$	0.610815	0.0391034
$245$	21.2121	1.35519
$246$	0	0
$247$	15.3874	0.979079
$248$	3.46791	0.220213
$249$	0	0
$250$	−2.03415	−0.128651
$251$	−2.34730	−0.148160	−0.0740800	−	0.997252i	$-0.523602\pi$
$251$	−2.34730	−0.148160	−0.0740800	+	0.997252i	$0.523602\pi$
$252$	0	0
$253$	−0.694593	−0.0436687
$254$	13.6800	0.858362
$255$	0	0
$256$	8.09833	0.506145
$257$	−23.0351	−1.43689	−0.718444	−	0.695584i	$-0.755146\pi$
$257$	−23.0351	−1.43689	−0.718444	+	0.695584i	$0.755146\pi$
$258$	0	0
$259$	2.90673	0.180615
$260$	−2.56717	−0.159209
$261$	0	0
$262$	1.50980	0.0932758
$263$	−19.6313	−1.21052	−0.605260	−	0.796028i	$-0.706931\pi$
$263$	−19.6313	−1.21052	−0.605260	+	0.796028i	$0.706931\pi$
$264$	0	0
$265$	20.3455	1.24982
$266$	−6.71688	−0.411838
$267$	0	0
$268$	−1.84430	−0.112659
$269$	−24.7374	−1.50827	−0.754133	−	0.656721i	$-0.771943\pi$
$269$	−24.7374	−1.50827	−0.754133	+	0.656721i	$0.771943\pi$
$270$	0	0
$271$	1.94356	0.118063	0.0590315	−	0.998256i	$-0.481199\pi$
$271$	1.94356	0.118063	0.0590315	+	0.998256i	$0.481199\pi$
$272$	−11.2882	−0.684446
$273$	0	0
$274$	29.9486	1.80926
$275$	−5.41147	−0.326324
$276$	0	0
$277$	21.3259	1.28135	0.640676	−	0.767812i	$-0.278654\pi$
$277$	21.3259	1.28135	0.640676	+	0.767812i	$0.278654\pi$
$278$	−24.3928	−1.46298
$279$	0	0
$280$	−5.33275	−0.318693
$281$	3.86215	0.230396	0.115198	−	0.993343i	$-0.463250\pi$
$281$	3.86215	0.230396	0.115198	+	0.993343i	$0.463250\pi$
$282$	0	0
$283$	−7.88713	−0.468841	−0.234420	−	0.972135i	$-0.575319\pi$
$283$	−7.88713	−0.468841	−0.234420	+	0.972135i	$0.575319\pi$
$284$	3.68685	0.218774
$285$	0	0
$286$	−3.50980	−0.207539
$287$	−0.332748	−0.0196415
$288$	0	0
$289$	−10.9094	−0.641730
$290$	−36.7493	−2.15799
$291$	0	0
$292$	−3.61856	−0.211760
$293$	11.8699	0.693446	0.346723	−	0.937968i	$-0.387294\pi$
$293$	11.8699	0.693446	0.346723	+	0.937968i	$0.387294\pi$
$294$	0	0
$295$	2.44831	0.142546
$296$	11.2763	0.655422
$297$	0	0
$298$	−11.2449	−0.651397
$299$	1.59121	0.0920223
$300$	0	0
$301$	7.55943	0.435718
$302$	−3.01186	−0.173313
$303$	0	0
$304$	−30.7229	−1.76208
$305$	−5.67499	−0.324949
$306$	0	0
$307$	33.8384	1.93126	0.965631	−	0.259918i	$-0.0836957\pi$
$307$	33.8384	1.93126	0.965631	+	0.259918i	$0.0836957\pi$
$308$	0.226682	0.0129164
$309$	0	0
$310$	6.77063	0.384546
$311$	12.0104	0.681049	0.340524	−	0.940236i	$-0.389395\pi$
$311$	12.0104	0.681049	0.340524	+	0.940236i	$0.389395\pi$
$312$	0	0
$313$	−3.18984	−0.180301	−0.0901503	−	0.995928i	$-0.528735\pi$
$313$	−3.18984	−0.180301	−0.0901503	+	0.995928i	$0.528735\pi$
$314$	−28.6578	−1.61725
$315$	0	0
$316$	−3.56624	−0.200617
$317$	−25.4115	−1.42725	−0.713625	−	0.700528i	$-0.752948\pi$
$317$	−25.4115	−1.42725	−0.713625	+	0.700528i	$0.752948\pi$
$318$	0	0
$319$	−7.43376	−0.416211
$320$	−19.9094	−1.11297
$321$	0	0
$322$	−0.694593	−0.0387081
$323$	16.5767	0.922350
$324$	0	0
$325$	12.3969	0.687658
$326$	−25.7401	−1.42561
$327$	0	0
$328$	−1.29086	−0.0712758
$329$	−5.03684	−0.277690
$330$	0	0
$331$	8.34049	0.458435	0.229217	−	0.973375i	$-0.426383\pi$
$331$	8.34049	0.458435	0.229217	+	0.973375i	$0.426383\pi$
$332$	3.56118	0.195445
$333$	0	0
$334$	−22.2645	−1.21826
$335$	17.1352	0.936194
$336$	0	0
$337$	18.8648	1.02763	0.513817	−	0.857900i	$-0.328231\pi$
$337$	18.8648	1.02763	0.513817	+	0.857900i	$0.328231\pi$
$338$	−11.8767	−0.646007
$339$	0	0
$340$	−2.76558	−0.149985
$341$	1.36959	0.0741672
$342$	0	0
$343$	8.85978	0.478383
$344$	29.3259	1.58115
$345$	0	0
$346$	0.305407	0.0164188
$347$	−25.6928	−1.37926	−0.689632	−	0.724160i	$-0.742228\pi$
$347$	−25.6928	−1.37926	−0.689632	+	0.724160i	$0.742228\pi$
$348$	0	0
$349$	3.50475	0.187605	0.0938024	−	0.995591i	$-0.470098\pi$
$349$	3.50475	0.187605	0.0938024	+	0.995591i	$0.470098\pi$
$350$	−5.41147	−0.289255
$351$	0	0
$352$	1.94356	0.103592
$353$	11.3719	0.605268	0.302634	−	0.953107i	$-0.402134\pi$
$353$	11.3719	0.605268	0.302634	+	0.953107i	$0.402134\pi$
$354$	0	0
$355$	−34.2540	−1.81801
$356$	6.36184	0.337177
$357$	0	0
$358$	−20.6013	−1.08881
$359$	22.5425	1.18975	0.594874	−	0.803819i	$-0.297202\pi$
$359$	22.5425	1.18975	0.594874	+	0.803819i	$0.297202\pi$
$360$	0	0
$361$	26.1165	1.37455
$362$	−28.1557	−1.47983
$363$	0	0
$364$	−0.519296	−0.0272185
$365$	33.6195	1.75972
$366$	0	0
$367$	17.0087	0.887846	0.443923	−	0.896065i	$-0.353586\pi$
$367$	17.0087	0.887846	0.443923	+	0.896065i	$0.353586\pi$
$368$	−3.17705	−0.165615
$369$	0	0
$370$	22.0155	1.14453
$371$	4.11556	0.213669
$372$	0	0
$373$	31.3851	1.62506	0.812529	−	0.582921i	$-0.198090\pi$
$373$	31.3851	1.62506	0.812529	+	0.582921i	$0.198090\pi$
$374$	−3.78106	−0.195514
$375$	0	0
$376$	−19.5398	−1.00769
$377$	17.0297	0.877074
$378$	0	0
$379$	−7.53209	−0.386897	−0.193449	−	0.981110i	$-0.561967\pi$
$379$	−7.53209	−0.386897	−0.193449	+	0.981110i	$0.561967\pi$
$380$	−7.52704	−0.386129
$381$	0	0
$382$	−39.8999	−2.04146
$383$	29.5398	1.50941	0.754707	−	0.656062i	$-0.227779\pi$
$383$	29.5398	1.50941	0.754707	+	0.656062i	$0.227779\pi$
$384$	0	0
$385$	−2.10607	−0.107335
$386$	−13.5175	−0.688025
$387$	0	0
$388$	3.70739	0.188214
$389$	−1.55943	−0.0790662	−0.0395331	−	0.999218i	$-0.512587\pi$
$389$	−1.55943	−0.0790662	−0.0395331	+	0.999218i	$0.512587\pi$
$390$	0	0
$391$	1.71419	0.0866905
$392$	16.6459	0.840745
$393$	0	0
$394$	2.98814	0.150540
$395$	33.1334	1.66712
$396$	0	0
$397$	−10.7178	−0.537912	−0.268956	−	0.963153i	$-0.586679\pi$
$397$	−10.7178	−0.537912	−0.268956	+	0.963153i	$0.586679\pi$
$398$	−7.46379	−0.374126
$399$	0	0
$400$	−24.7520	−1.23760
$401$	−11.1453	−0.556568	−0.278284	−	0.960499i	$-0.589766\pi$
$401$	−11.1453	−0.556568	−0.278284	+	0.960499i	$0.589766\pi$
$402$	0	0
$403$	−3.13753	−0.156291
$404$	0.354103	0.0176173
$405$	0	0
$406$	−7.43376	−0.368931
$407$	4.45336	0.220745
$408$	0	0
$409$	−19.2216	−0.950448	−0.475224	−	0.879865i	$-0.657633\pi$
$409$	−19.2216	−0.950448	−0.475224	+	0.879865i	$0.657633\pi$
$410$	−2.52023	−0.124465
$411$	0	0
$412$	−1.76382	−0.0868973
$413$	0.495252	0.0243698
$414$	0	0
$415$	−33.0865	−1.62415
$416$	−4.45243	−0.218298
$417$	0	0
$418$	−10.2909	−0.503342
$419$	−26.9026	−1.31428	−0.657139	−	0.753769i	$-0.728234\pi$
$419$	−26.9026	−1.31428	−0.657139	+	0.753769i	$0.728234\pi$
$420$	0	0
$421$	−8.26857	−0.402985	−0.201493	−	0.979490i	$-0.564579\pi$
$421$	−8.26857	−0.402985	−0.201493	+	0.979490i	$0.564579\pi$
$422$	35.7478	1.74018
$423$	0	0
$424$	15.9659	0.775370
$425$	13.3550	0.647814
$426$	0	0
$427$	−1.14796	−0.0555535
$428$	−4.04189	−0.195372
$429$	0	0
$430$	57.2550	2.76108
$431$	−7.80241	−0.375829	−0.187915	−	0.982185i	$-0.560173\pi$
$431$	−7.80241	−0.375829	−0.187915	+	0.982185i	$0.560173\pi$
$432$	0	0
$433$	15.6810	0.753580	0.376790	−	0.926299i	$-0.377028\pi$
$433$	15.6810	0.753580	0.376790	+	0.926299i	$0.377028\pi$
$434$	1.36959	0.0657422
$435$	0	0
$436$	−4.82295	−0.230977
$437$	4.66550	0.223181
$438$	0	0
$439$	−24.3277	−1.16110	−0.580549	−	0.814225i	$-0.697162\pi$
$439$	−24.3277	−1.16110	−0.580549	+	0.814225i	$0.697162\pi$
$440$	−8.17024	−0.389501
$441$	0	0
$442$	8.66187	0.412003
$443$	0.808400	0.0384083	0.0192041	−	0.999816i	$-0.493887\pi$
$443$	0.808400	0.0384083	0.0192041	+	0.999816i	$0.493887\pi$
$444$	0	0
$445$	−59.1070	−2.80194
$446$	12.4270	0.588433
$447$	0	0
$448$	−4.02734	−0.190274
$449$	14.3669	0.678016	0.339008	−	0.940784i	$-0.389909\pi$
$449$	14.3669	0.678016	0.339008	+	0.940784i	$0.389909\pi$
$450$	0	0
$451$	−0.509800	−0.0240056
$452$	−6.35504	−0.298916
$453$	0	0
$454$	33.5107	1.57274
$455$	4.82470	0.226186
$456$	0	0
$457$	4.17799	0.195438	0.0977190	−	0.995214i	$-0.468845\pi$
$457$	4.17799	0.195438	0.0977190	+	0.995214i	$0.468845\pi$
$458$	−29.7743	−1.39126
$459$	0	0
$460$	−0.778371	−0.0362917
$461$	−36.0300	−1.67809	−0.839043	−	0.544065i	$-0.816884\pi$
$461$	−36.0300	−1.67809	−0.839043	+	0.544065i	$0.816884\pi$
$462$	0	0
$463$	−1.41241	−0.0656402	−0.0328201	−	0.999461i	$-0.510449\pi$
$463$	−1.41241	−0.0656402	−0.0328201	+	0.999461i	$0.510449\pi$
$464$	−34.0019	−1.57850
$465$	0	0
$466$	4.89124	0.226583
$467$	26.2003	1.21240	0.606202	−	0.795311i	$-0.292692\pi$
$467$	26.2003	1.21240	0.606202	+	0.795311i	$0.292692\pi$
$468$	0	0
$469$	3.46616	0.160052
$470$	−38.1489	−1.75968
$471$	0	0
$472$	1.92127	0.0884338
$473$	11.5817	0.532528
$474$	0	0
$475$	36.3482	1.66777
$476$	−0.559430	−0.0256414
$477$	0	0
$478$	−31.7219	−1.45093
$479$	11.8256	0.540327	0.270164	−	0.962814i	$-0.412922\pi$
$479$	11.8256	0.540327	0.270164	+	0.962814i	$0.412922\pi$
$480$	0	0
$481$	−10.2020	−0.465172
$482$	6.87527	0.313160
$483$	0	0
$484$	0.347296	0.0157862
$485$	−34.4448	−1.56406
$486$	0	0
$487$	−18.0273	−0.816897	−0.408448	−	0.912781i	$-0.633930\pi$
$487$	−18.0273	−0.816897	−0.408448	+	0.912781i	$0.633930\pi$
$488$	−4.45336	−0.201594
$489$	0	0
$490$	32.4989	1.46815
$491$	−25.7793	−1.16340	−0.581702	−	0.813402i	$-0.697613\pi$
$491$	−25.7793	−1.16340	−0.581702	+	0.813402i	$0.697613\pi$
$492$	0	0
$493$	18.3459	0.826256
$494$	23.5749	1.06069
$495$	0	0
$496$	6.26445	0.281282
$497$	−6.92902	−0.310809
$498$	0	0
$499$	29.9982	1.34291	0.671453	−	0.741047i	$-0.265671\pi$
$499$	29.9982	1.34291	0.671453	+	0.741047i	$0.265671\pi$
$500$	−0.461104	−0.0206212
$501$	0	0
$502$	−3.59627	−0.160509
$503$	−6.88713	−0.307082	−0.153541	−	0.988142i	$-0.549068\pi$
$503$	−6.88713	−0.307082	−0.153541	+	0.988142i	$0.549068\pi$
$504$	0	0
$505$	−3.28993	−0.146400
$506$	−1.06418	−0.0473085
$507$	0	0
$508$	3.10101	0.137585
$509$	19.5827	0.867986	0.433993	−	0.900916i	$-0.357104\pi$
$509$	19.5827	0.867986	0.433993	+	0.900916i	$0.357104\pi$
$510$	0	0
$511$	6.80066	0.300843
$512$	−14.2736	−0.630811
$513$	0	0
$514$	−35.2918	−1.55665
$515$	16.3874	0.722116
$516$	0	0
$517$	−7.71688	−0.339388
$518$	4.45336	0.195670
$519$	0	0
$520$	18.7169	0.820790
$521$	−11.8803	−0.520486	−0.260243	−	0.965543i	$-0.583803\pi$
$521$	−11.8803	−0.520486	−0.260243	+	0.965543i	$0.583803\pi$
$522$	0	0
$523$	37.2591	1.62923	0.814613	−	0.580005i	$-0.196949\pi$
$523$	37.2591	1.62923	0.814613	+	0.580005i	$0.196949\pi$
$524$	0.342244	0.0149510
$525$	0	0
$526$	−30.0770	−1.31142
$527$	−3.38001	−0.147236
$528$	0	0
$529$	−22.5175	−0.979024
$530$	31.1712	1.35399
$531$	0	0
$532$	−1.52259	−0.0660128
$533$	1.16788	0.0505865
$534$	0	0
$535$	37.5526	1.62354
$536$	13.4466	0.580803
$537$	0	0
$538$	−37.8999	−1.63398
$539$	6.57398	0.283161
$540$	0	0
$541$	36.0310	1.54909	0.774546	−	0.632518i	$-0.217978\pi$
$541$	36.0310	1.54909	0.774546	+	0.632518i	$0.217978\pi$
$542$	2.97771	0.127904
$543$	0	0
$544$	−4.79654	−0.205650
$545$	44.8093	1.91942
$546$	0	0
$547$	−8.80159	−0.376329	−0.188164	−	0.982138i	$-0.560254\pi$
$547$	−8.80159	−0.376329	−0.188164	+	0.982138i	$0.560254\pi$
$548$	6.78880	0.290003
$549$	0	0
$550$	−8.29086	−0.353523
$551$	49.9317	2.12716
$552$	0	0
$553$	6.70233	0.285012
$554$	32.6732	1.38815
$555$	0	0
$556$	−5.52940	−0.234499
$557$	15.5503	0.658886	0.329443	−	0.944176i	$-0.393139\pi$
$557$	15.5503	0.658886	0.329443	+	0.944176i	$0.393139\pi$
$558$	0	0
$559$	−26.5321	−1.12219
$560$	−9.63310	−0.407073
$561$	0	0
$562$	5.91716	0.249600
$563$	17.2918	0.728762	0.364381	−	0.931250i	$-0.381281\pi$
$563$	17.2918	0.728762	0.364381	+	0.931250i	$0.381281\pi$
$564$	0	0
$565$	59.0438	2.48399
$566$	−12.0838	−0.507919
$567$	0	0
$568$	−26.8803	−1.12787
$569$	−37.4570	−1.57028	−0.785139	−	0.619319i	$-0.787409\pi$
$569$	−37.4570	−1.57028	−0.785139	+	0.619319i	$0.787409\pi$
$570$	0	0
$571$	24.0087	1.00473	0.502366	−	0.864655i	$-0.332463\pi$
$571$	24.0087	1.00473	0.502366	+	0.864655i	$0.332463\pi$
$572$	−0.795607	−0.0332660
$573$	0	0
$574$	−0.509800	−0.0212786
$575$	3.75877	0.156752
$576$	0	0
$577$	−27.8726	−1.16035	−0.580175	−	0.814492i	$-0.697016\pi$
$577$	−27.8726	−1.16035	−0.580175	+	0.814492i	$0.697016\pi$
$578$	−16.7142	−0.695219
$579$	0	0
$580$	−8.33038	−0.345900
$581$	−6.69284	−0.277666
$582$	0	0
$583$	6.30541	0.261143
$584$	26.3824	1.09171
$585$	0	0
$586$	18.1857	0.751245
$587$	−22.5408	−0.930357	−0.465178	−	0.885217i	$-0.654010\pi$
$587$	−22.5408	−0.930357	−0.465178	+	0.885217i	$0.654010\pi$
$588$	0	0
$589$	−9.19934	−0.379052
$590$	3.75103	0.154427
$591$	0	0
$592$	20.3696	0.837185
$593$	−7.12061	−0.292409	−0.146204	−	0.989254i	$-0.546706\pi$
$593$	−7.12061	−0.292409	−0.146204	+	0.989254i	$0.546706\pi$
$594$	0	0
$595$	5.19759	0.213080
$596$	−2.54900	−0.104411
$597$	0	0
$598$	2.43788	0.0996924
$599$	15.4766	0.632356	0.316178	−	0.948700i	$-0.397600\pi$
$599$	15.4766	0.632356	0.316178	+	0.948700i	$0.397600\pi$
$600$	0	0
$601$	−6.75784	−0.275658	−0.137829	−	0.990456i	$-0.544012\pi$
$601$	−6.75784	−0.275658	−0.137829	+	0.990456i	$0.544012\pi$
$602$	11.5817	0.472036
$603$	0	0
$604$	−0.682733	−0.0277800
$605$	−3.22668	−0.131183
$606$	0	0
$607$	24.8280	1.00774	0.503869	−	0.863780i	$-0.331910\pi$
$607$	24.8280	1.00774	0.503869	+	0.863780i	$0.331910\pi$
$608$	−13.0547	−0.529437
$609$	0	0
$610$	−8.69459	−0.352034
$611$	17.6783	0.715187
$612$	0	0
$613$	−35.0915	−1.41733	−0.708667	−	0.705544i	$-0.750703\pi$
$613$	−35.0915	−1.41733	−0.708667	+	0.705544i	$0.750703\pi$
$614$	51.8435	2.09223
$615$	0	0
$616$	−1.65270	−0.0665893
$617$	15.3550	0.618171	0.309085	−	0.951034i	$-0.399977\pi$
$617$	15.3550	0.618171	0.309085	+	0.951034i	$0.399977\pi$
$618$	0	0
$619$	−18.6878	−0.751126	−0.375563	−	0.926797i	$-0.622551\pi$
$619$	−18.6878	−0.751126	−0.375563	+	0.926797i	$0.622551\pi$
$620$	1.53478	0.0616381
$621$	0	0
$622$	18.4010	0.737815
$623$	−11.9564	−0.479021
$624$	0	0
$625$	−22.7733	−0.910933
$626$	−4.88713	−0.195329
$627$	0	0
$628$	−6.49619	−0.259226
$629$	−10.9905	−0.438220
$630$	0	0
$631$	−27.6159	−1.09937	−0.549685	−	0.835372i	$-0.685252\pi$
$631$	−27.6159	−1.09937	−0.549685	+	0.835372i	$0.685252\pi$
$632$	26.0009	1.03426
$633$	0	0
$634$	−38.9326	−1.54621
$635$	−28.8111	−1.14333
$636$	0	0
$637$	−15.0601	−0.596701
$638$	−11.3892	−0.450902
$639$	0	0
$640$	−43.0455	−1.70152
$641$	21.0196	0.830224	0.415112	−	0.909770i	$-0.363742\pi$
$641$	21.0196	0.830224	0.415112	+	0.909770i	$0.363742\pi$
$642$	0	0
$643$	−11.2172	−0.442363	−0.221181	−	0.975233i	$-0.570991\pi$
$643$	−11.2172	−0.442363	−0.221181	+	0.975233i	$0.570991\pi$
$644$	−0.157451	−0.00620445
$645$	0	0
$646$	25.3969	0.999229
$647$	−29.5175	−1.16045	−0.580227	−	0.814455i	$-0.697036\pi$
$647$	−29.5175	−1.16045	−0.580227	+	0.814455i	$0.697036\pi$
$648$	0	0
$649$	0.758770	0.0297843
$650$	18.9932	0.744974
$651$	0	0
$652$	−5.83481	−0.228509
$653$	27.1908	1.06406	0.532029	−	0.846726i	$-0.321430\pi$
$653$	27.1908	1.06406	0.532029	+	0.846726i	$0.321430\pi$
$654$	0	0
$655$	−3.17974	−0.124243
$656$	−2.33181	−0.0910421
$657$	0	0
$658$	−7.71688	−0.300835
$659$	13.9368	0.542899	0.271449	−	0.962453i	$-0.412497\pi$
$659$	13.9368	0.542899	0.271449	+	0.962453i	$0.412497\pi$
$660$	0	0
$661$	13.7769	0.535861	0.267930	−	0.963438i	$-0.413660\pi$
$661$	13.7769	0.535861	0.267930	+	0.963438i	$0.413660\pi$
$662$	12.7784	0.496645
$663$	0	0
$664$	−25.9641	−1.00760
$665$	14.1462	0.548566
$666$	0	0
$667$	5.16344	0.199929
$668$	−5.04694	−0.195272
$669$	0	0
$670$	26.2526	1.01423
$671$	−1.75877	−0.0678966
$672$	0	0
$673$	−26.6459	−1.02712	−0.513562	−	0.858053i	$-0.671674\pi$
$673$	−26.6459	−1.02712	−0.513562	+	0.858053i	$0.671674\pi$
$674$	28.9026	1.11329
$675$	0	0
$676$	−2.69223	−0.103547
$677$	7.26445	0.279196	0.139598	−	0.990208i	$-0.455419\pi$
$677$	7.26445	0.279196	0.139598	+	0.990208i	$0.455419\pi$
$678$	0	0
$679$	−6.96761	−0.267392
$680$	20.1634	0.773233
$681$	0	0
$682$	2.09833	0.0803491
$683$	−23.5996	−0.903012	−0.451506	−	0.892268i	$-0.649113\pi$
$683$	−23.5996	−0.903012	−0.451506	+	0.892268i	$0.649113\pi$
$684$	0	0
$685$	−63.0738	−2.40993
$686$	13.5740	0.518257
$687$	0	0
$688$	52.9745	2.01963
$689$	−14.4448	−0.550303
$690$	0	0
$691$	−41.0915	−1.56320	−0.781598	−	0.623783i	$-0.785595\pi$
$691$	−41.0915	−1.56320	−0.781598	+	0.623783i	$0.785595\pi$
$692$	0.0692302	0.00263174
$693$	0	0
$694$	−39.3637	−1.49423
$695$	51.3729	1.94868
$696$	0	0
$697$	1.25814	0.0476555
$698$	5.36959	0.203242
$699$	0	0
$700$	−1.22668	−0.0463642
$701$	19.9222	0.752451	0.376226	−	0.926528i	$-0.377222\pi$
$701$	19.9222	0.752451	0.376226	+	0.926528i	$0.377222\pi$
$702$	0	0
$703$	−29.9127	−1.12818
$704$	−6.17024	−0.232550
$705$	0	0
$706$	17.4228	0.655717
$707$	−0.665497	−0.0250286
$708$	0	0
$709$	−14.8316	−0.557013	−0.278507	−	0.960434i	$-0.589839\pi$
$709$	−14.8316	−0.557013	−0.278507	+	0.960434i	$0.589839\pi$
$710$	−52.4802	−1.96955
$711$	0	0
$712$	−46.3833	−1.73829
$713$	−0.951304	−0.0356266
$714$	0	0
$715$	7.39187	0.276440
$716$	−4.66994	−0.174524
$717$	0	0
$718$	34.5371	1.28891
$719$	−42.6236	−1.58959	−0.794796	−	0.606876i	$-0.792422\pi$
$719$	−42.6236	−1.58959	−0.794796	+	0.606876i	$0.792422\pi$
$720$	0	0
$721$	3.31490	0.123453
$722$	40.0128	1.48912
$723$	0	0
$724$	−6.38238	−0.237199
$725$	40.2276	1.49402
$726$	0	0
$727$	16.9641	0.629164	0.314582	−	0.949230i	$-0.398136\pi$
$727$	16.9641	0.629164	0.314582	+	0.949230i	$0.398136\pi$
$728$	3.78611	0.140323
$729$	0	0
$730$	51.5080	1.90640
$731$	−28.5827	−1.05717
$732$	0	0
$733$	−13.8753	−0.512495	−0.256247	−	0.966611i	$-0.582486\pi$
$733$	−13.8753	−0.512495	−0.256247	+	0.966611i	$0.582486\pi$
$734$	26.0588	0.961848
$735$	0	0
$736$	−1.34998	−0.0497611
$737$	5.31046	0.195613
$738$	0	0
$739$	13.4652	0.495326	0.247663	−	0.968846i	$-0.420337\pi$
$739$	13.4652	0.495326	0.247663	+	0.968846i	$0.420337\pi$
$740$	4.99050	0.183455
$741$	0	0
$742$	6.30541	0.231479
$743$	−0.0169120	−0.000620442 0	−0.000310221	−	1.00000i	$-0.500099\pi$
$743$	−0.0169120	−0.000620442 0	−0.000310221		1.00000i	$0.500099\pi$
$744$	0	0
$745$	23.6824	0.867656
$746$	48.0847	1.76051
$747$	0	0
$748$	−0.857097	−0.0313386
$749$	7.59627	0.277562
$750$	0	0
$751$	−37.9932	−1.38639	−0.693196	−	0.720749i	$-0.743798\pi$
$751$	−37.9932	−1.38639	−0.693196	+	0.720749i	$0.743798\pi$
$752$	−35.2968	−1.28714
$753$	0	0
$754$	26.0910	0.950179
$755$	6.34318	0.230852
$756$	0	0
$757$	38.3919	1.39538	0.697688	−	0.716402i	$-0.254212\pi$
$757$	38.3919	1.39538	0.697688	+	0.716402i	$0.254212\pi$
$758$	−11.5398	−0.419145
$759$	0	0
$760$	54.8786	1.99065
$761$	−17.2635	−0.625802	−0.312901	−	0.949786i	$-0.601301\pi$
$761$	−17.2635	−0.625802	−0.312901	+	0.949786i	$0.601301\pi$
$762$	0	0
$763$	9.06418	0.328145
$764$	−9.04458	−0.327221
$765$	0	0
$766$	45.2576	1.63523
$767$	−1.73824	−0.0627641
$768$	0	0
$769$	9.04788	0.326275	0.163137	−	0.986603i	$-0.447839\pi$
$769$	9.04788	0.326275	0.163137	+	0.986603i	$0.447839\pi$
$770$	−3.22668	−0.116282
$771$	0	0
$772$	−3.06418	−0.110282
$773$	−32.8675	−1.18216	−0.591081	−	0.806612i	$-0.701299\pi$
$773$	−32.8675	−1.18216	−0.591081	+	0.806612i	$0.701299\pi$
$774$	0	0
$775$	−7.41147	−0.266228
$776$	−27.0300	−0.970322
$777$	0	0
$778$	−2.38919	−0.0856564
$779$	3.42427	0.122687
$780$	0	0
$781$	−10.6159	−0.379866
$782$	2.62630	0.0939162
$783$	0	0
$784$	30.0692	1.07390
$785$	60.3551	2.15417
$786$	0	0
$787$	−2.92221	−0.104165	−0.0520827	−	0.998643i	$-0.516586\pi$
$787$	−2.92221	−0.104165	−0.0520827	+	0.998643i	$0.516586\pi$
$788$	0.677356	0.0241298
$789$	0	0
$790$	50.7633	1.80608
$791$	11.9436	0.424664
$792$	0	0
$793$	4.02910	0.143077
$794$	−16.4206	−0.582747
$795$	0	0
$796$	−1.69190	−0.0599680
$797$	34.1121	1.20831	0.604155	−	0.796866i	$-0.293511\pi$
$797$	34.1121	1.20831	0.604155	+	0.796866i	$0.293511\pi$
$798$	0	0
$799$	19.0446	0.673749
$800$	−10.5175	−0.371851
$801$	0	0
$802$	−17.0755	−0.602958
$803$	10.4192	0.367686
$804$	0	0
$805$	1.46286	0.0515590
$806$	−4.80697	−0.169318
$807$	0	0
$808$	−2.58172	−0.0908245
$809$	37.4986	1.31838	0.659189	−	0.751977i	$-0.270900\pi$
$809$	37.4986	1.31838	0.659189	+	0.751977i	$0.270900\pi$
$810$	0	0
$811$	−26.0888	−0.916103	−0.458051	−	0.888926i	$-0.651452\pi$
$811$	−26.0888	−0.916103	−0.458051	+	0.888926i	$0.651452\pi$
$812$	−1.68510	−0.0591353
$813$	0	0
$814$	6.82295	0.239144
$815$	54.2104	1.89891
$816$	0	0
$817$	−77.7930	−2.72163
$818$	−29.4492	−1.02967
$819$	0	0
$820$	−0.571290	−0.0199503
$821$	−24.6774	−0.861246	−0.430623	−	0.902532i	$-0.641706\pi$
$821$	−24.6774	−0.861246	−0.430623	+	0.902532i	$0.641706\pi$
$822$	0	0
$823$	−55.8180	−1.94569	−0.972847	−	0.231450i	$-0.925653\pi$
$823$	−55.8180	−1.94569	−0.972847	+	0.231450i	$0.925653\pi$
$824$	12.8598	0.447992
$825$	0	0
$826$	0.758770	0.0264010
$827$	−5.61999	−0.195426	−0.0977130	−	0.995215i	$-0.531153\pi$
$827$	−5.61999	−0.195426	−0.0977130	+	0.995215i	$0.531153\pi$
$828$	0	0
$829$	−21.4037	−0.743382	−0.371691	−	0.928356i	$-0.621222\pi$
$829$	−21.4037	−0.743382	−0.371691	+	0.928356i	$0.621222\pi$
$830$	−50.6914	−1.75952
$831$	0	0
$832$	14.1352	0.490049
$833$	−16.2240	−0.562128
$834$	0	0
$835$	46.8904	1.62271
$836$	−2.33275	−0.0806798
$837$	0	0
$838$	−41.2172	−1.42382
$839$	36.7383	1.26835	0.634174	−	0.773190i	$-0.281340\pi$
$839$	36.7383	1.26835	0.634174	+	0.773190i	$0.281340\pi$
$840$	0	0
$841$	26.2608	0.905546
$842$	−12.6682	−0.436574
$843$	0	0
$844$	8.10338	0.278930
$845$	25.0131	0.860477
$846$	0	0
$847$	−0.652704	−0.0224272
$848$	28.8408	0.990397
$849$	0	0
$850$	20.4611	0.701810
$851$	−3.09327	−0.106036
$852$	0	0
$853$	−38.5098	−1.31855	−0.659275	−	0.751902i	$-0.729137\pi$
$853$	−38.5098	−1.31855	−0.659275	+	0.751902i	$0.729137\pi$
$854$	−1.75877	−0.0601839
$855$	0	0
$856$	29.4688	1.00722
$857$	14.9573	0.510931	0.255466	−	0.966818i	$-0.417771\pi$
$857$	14.9573	0.510931	0.255466	+	0.966818i	$0.417771\pi$
$858$	0	0
$859$	−45.2354	−1.54341	−0.771705	−	0.635980i	$-0.780596\pi$
$859$	−45.2354	−1.54341	−0.771705	+	0.635980i	$0.780596\pi$
$860$	12.9786	0.442568
$861$	0	0
$862$	−11.9540	−0.407155
$863$	−5.85111	−0.199174	−0.0995871	−	0.995029i	$-0.531752\pi$
$863$	−5.85111	−0.199174	−0.0995871	+	0.995029i	$0.531752\pi$
$864$	0	0
$865$	−0.643208	−0.0218697
$866$	24.0247	0.816391
$867$	0	0
$868$	0.310460	0.0105377
$869$	10.2686	0.348337
$870$	0	0
$871$	−12.1655	−0.412213
$872$	35.1634	1.19078
$873$	0	0
$874$	7.14796	0.241783
$875$	0.866592	0.0292962
$876$	0	0
$877$	11.8871	0.401400	0.200700	−	0.979653i	$-0.435678\pi$
$877$	11.8871	0.401400	0.200700	+	0.979653i	$0.435678\pi$
$878$	−37.2722	−1.25788
$879$	0	0
$880$	−14.7588	−0.497518
$881$	−43.0634	−1.45084	−0.725421	−	0.688306i	$-0.758355\pi$
$881$	−43.0634	−1.45084	−0.725421	+	0.688306i	$0.758355\pi$
$882$	0	0
$883$	−31.6533	−1.06522	−0.532609	−	0.846361i	$-0.678789\pi$
$883$	−31.6533	−1.06522	−0.532609	+	0.846361i	$0.678789\pi$
$884$	1.96349	0.0660392
$885$	0	0
$886$	1.23854	0.0416096
$887$	−4.55169	−0.152831	−0.0764154	−	0.997076i	$-0.524348\pi$
$887$	−4.55169	−0.152831	−0.0764154	+	0.997076i	$0.524348\pi$
$888$	0	0
$889$	−5.82800	−0.195465
$890$	−90.5572	−3.03548
$891$	0	0
$892$	2.81696	0.0943189
$893$	51.8334	1.73454
$894$	0	0
$895$	43.3878	1.45029
$896$	−8.70739	−0.290893
$897$	0	0
$898$	22.0114	0.734529
$899$	−10.1812	−0.339561
$900$	0	0
$901$	−15.5612	−0.518418
$902$	−0.781059	−0.0260064
$903$	0	0
$904$	46.3337	1.54104
$905$	59.2978	1.97112
$906$	0	0
$907$	6.36421	0.211320	0.105660	−	0.994402i	$-0.466304\pi$
$907$	6.36421	0.211320	0.105660	+	0.994402i	$0.466304\pi$
$908$	7.59627	0.252091
$909$	0	0
$910$	7.39187	0.245038
$911$	16.4165	0.543904	0.271952	−	0.962311i	$-0.412331\pi$
$911$	16.4165	0.543904	0.271952	+	0.962311i	$0.412331\pi$
$912$	0	0
$913$	−10.2540	−0.339359
$914$	6.40104	0.211728
$915$	0	0
$916$	−6.74928	−0.223002
$917$	−0.643208	−0.0212406
$918$	0	0
$919$	39.0830	1.28923	0.644614	−	0.764508i	$-0.277018\pi$
$919$	39.0830	1.28923	0.644614	+	0.764508i	$0.277018\pi$
$920$	5.67499	0.187099
$921$	0	0
$922$	−55.2012	−1.81795
$923$	24.3195	0.800485
$924$	0	0
$925$	−24.0993	−0.792379
$926$	−2.16393	−0.0711113
$927$	0	0
$928$	−14.4480	−0.474278
$929$	−8.98721	−0.294861	−0.147430	−	0.989072i	$-0.547100\pi$
$929$	−8.98721	−0.294861	−0.147430	+	0.989072i	$0.547100\pi$
$930$	0	0
$931$	−44.1566	−1.44718
$932$	1.10876	0.0363185
$933$	0	0
$934$	40.1411	1.31346
$935$	7.96316	0.260423
$936$	0	0
$937$	35.8982	1.17274	0.586371	−	0.810043i	$-0.300556\pi$
$937$	35.8982	1.17274	0.586371	+	0.810043i	$0.300556\pi$
$938$	5.31046	0.173393
$939$	0	0
$940$	−8.64765	−0.282055
$941$	8.21389	0.267765	0.133883	−	0.990997i	$-0.457256\pi$
$941$	8.21389	0.267765	0.133883	+	0.990997i	$0.457256\pi$
$942$	0	0
$943$	0.354103	0.0115312
$944$	3.47060	0.112958
$945$	0	0
$946$	17.7442	0.576914
$947$	6.78013	0.220324	0.110162	−	0.993914i	$-0.464863\pi$
$947$	6.78013	0.220324	0.110162	+	0.993914i	$0.464863\pi$
$948$	0	0
$949$	−23.8690	−0.774819
$950$	55.6887	1.80678
$951$	0	0
$952$	4.07873	0.132192
$953$	44.3286	1.43595	0.717973	−	0.696071i	$-0.245070\pi$
$953$	44.3286	1.43595	0.717973	+	0.696071i	$0.245070\pi$
$954$	0	0
$955$	84.0319	2.71921
$956$	−7.19078	−0.232566
$957$	0	0
$958$	18.1179	0.585364
$959$	−12.7588	−0.412002
$960$	0	0
$961$	−29.1242	−0.939492
$962$	−15.6304	−0.503945
$963$	0	0
$964$	1.55850	0.0501958
$965$	28.4688	0.916445
$966$	0	0
$967$	11.1088	0.357233	0.178617	−	0.983919i	$-0.442838\pi$
$967$	11.1088	0.357233	0.178617	+	0.983919i	$0.442838\pi$
$968$	−2.53209	−0.0813844
$969$	0	0
$970$	−52.7725	−1.69442
$971$	−49.0369	−1.57367	−0.786835	−	0.617163i	$-0.788282\pi$
$971$	−49.0369	−1.57367	−0.786835	+	0.617163i	$0.788282\pi$
$972$	0	0
$973$	10.3919	0.333148
$974$	−27.6195	−0.884986
$975$	0	0
$976$	−8.04458	−0.257501
$977$	−23.2003	−0.742243	−0.371121	−	0.928584i	$-0.621027\pi$
$977$	−23.2003	−0.742243	−0.371121	+	0.928584i	$0.621027\pi$
$978$	0	0
$979$	−18.3182	−0.585452
$980$	7.36690	0.235327
$981$	0	0
$982$	−39.4962	−1.26037
$983$	27.3122	0.871124	0.435562	−	0.900159i	$-0.356550\pi$
$983$	27.3122	0.871124	0.435562	+	0.900159i	$0.356550\pi$
$984$	0	0
$985$	−6.29322	−0.200519
$986$	28.1075	0.895125
$987$	0	0
$988$	5.34400	0.170015
$989$	−8.04458	−0.255803
$990$	0	0
$991$	−6.29322	−0.199911	−0.0999554	−	0.994992i	$-0.531870\pi$
$991$	−6.29322	−0.199911	−0.0999554	+	0.994992i	$0.531870\pi$
$992$	2.66187	0.0845146
$993$	0	0
$994$	−10.6159	−0.336715
$995$	15.7192	0.498334
$996$	0	0
$997$	−20.7820	−0.658172	−0.329086	−	0.944300i	$-0.606741\pi$
$997$	−20.7820	−0.658172	−0.329086	+	0.944300i	$0.606741\pi$
$998$	45.9600	1.45484
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.a.1.3		3
3.2	odd	2		8019.2.a.b.1.1		3
27.2	odd	18		891.2.j.a.793.1		6
27.13	even	9		297.2.j.a.34.1	✓	6
27.14	odd	18		891.2.j.a.100.1		6
27.25	even	9		297.2.j.a.166.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.a.34.1	✓	6	27.13	even	9
297.2.j.a.166.1	yes	6	27.25	even	9
891.2.j.a.100.1		6	27.14	odd	18
891.2.j.a.793.1		6	27.2	odd	18
8019.2.a.a.1.3		3	1.1	even	1	trivial
8019.2.a.b.1.1		3	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53209 q^{2} +0.347296 q^{4} -3.22668 q^{5} -0.652704 q^{7} -2.53209 q^{8} +O(q^{10})\)	\(q+1.53209 q^{2} +0.347296 q^{4} -3.22668 q^{5} -0.652704 q^{7} -2.53209 q^{8} -4.94356 q^{10} -1.00000 q^{11} +2.29086 q^{13} -1.00000 q^{14} -4.57398 q^{16} +2.46791 q^{17} +6.71688 q^{19} -1.12061 q^{20} -1.53209 q^{22} +0.694593 q^{23} +5.41147 q^{25} +3.50980 q^{26} -0.226682 q^{28} +7.43376 q^{29} -1.36959 q^{31} -1.94356 q^{32} +3.78106 q^{34} +2.10607 q^{35} -4.45336 q^{37} +10.2909 q^{38} +8.17024 q^{40} +0.509800 q^{41} -11.5817 q^{43} -0.347296 q^{44} +1.06418 q^{46} +7.71688 q^{47} -6.57398 q^{49} +8.29086 q^{50} +0.795607 q^{52} -6.30541 q^{53} +3.22668 q^{55} +1.65270 q^{56} +11.3892 q^{58} -0.758770 q^{59} +1.75877 q^{61} -2.09833 q^{62} +6.17024 q^{64} -7.39187 q^{65} -5.31046 q^{67} +0.857097 q^{68} +3.22668 q^{70} +10.6159 q^{71} -10.4192 q^{73} -6.82295 q^{74} +2.33275 q^{76} +0.652704 q^{77} -10.2686 q^{79} +14.7588 q^{80} +0.781059 q^{82} +10.2540 q^{83} -7.96316 q^{85} -17.7442 q^{86} +2.53209 q^{88} +18.3182 q^{89} -1.49525 q^{91} +0.241230 q^{92} +11.8229 q^{94} -21.6732 q^{95} +10.6750 q^{97} -10.0719 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{5} - 3 q^{7} - 3 q^{8} - 3 q^{11} - 9 q^{13} - 3 q^{14} - 6 q^{16} + 12 q^{17} + 12 q^{19} - 9 q^{20} + 6 q^{25} + 12 q^{26} + 6 q^{28} + 6 q^{29} + 3 q^{31} + 9 q^{32} - 6 q^{34} - 6 q^{35} + 15 q^{38} + 3 q^{40} + 3 q^{41} - 3 q^{43} - 6 q^{46} + 15 q^{47} - 12 q^{49} + 9 q^{50} + 3 q^{52} - 21 q^{53} + 3 q^{55} + 6 q^{56} + 30 q^{58} + 9 q^{59} - 6 q^{61} - 18 q^{62} - 3 q^{64} - 9 q^{65} - 3 q^{67} + 3 q^{68} + 3 q^{70} + 21 q^{71} + 3 q^{73} - 12 q^{76} + 3 q^{77} - 21 q^{79} + 33 q^{80} - 15 q^{82} + 3 q^{83} - 12 q^{85} - 24 q^{86} + 3 q^{88} + 18 q^{89} + 12 q^{91} + 12 q^{92} + 15 q^{94} - 3 q^{95} + 27 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^5 - 3 * q^7 - 3 * q^8 - 3 * q^11 - 9 * q^13 - 3 * q^14 - 6 * q^16 + 12 * q^17 + 12 * q^19 - 9 * q^20 + 6 * q^25 + 12 * q^26 + 6 * q^28 + 6 * q^29 + 3 * q^31 + 9 * q^32 - 6 * q^34 - 6 * q^35 + 15 * q^38 + 3 * q^40 + 3 * q^41 - 3 * q^43 - 6 * q^46 + 15 * q^47 - 12 * q^49 + 9 * q^50 + 3 * q^52 - 21 * q^53 + 3 * q^55 + 6 * q^56 + 30 * q^58 + 9 * q^59 - 6 * q^61 - 18 * q^62 - 3 * q^64 - 9 * q^65 - 3 * q^67 + 3 * q^68 + 3 * q^70 + 21 * q^71 + 3 * q^73 - 12 * q^76 + 3 * q^77 - 21 * q^79 + 33 * q^80 - 15 * q^82 + 3 * q^83 - 12 * q^85 - 24 * q^86 + 3 * q^88 + 18 * q^89 + 12 * q^91 + 12 * q^92 + 15 * q^94 - 3 * q^95 + 27 * q^97 + 3 * q^98

Introduction

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