LMFDB - Embedded newform 297.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [297,2,Mod(1,297)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(297, 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("297.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	297.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:	$2.37155694003$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	297.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{2} -1.46410 q^{4} -2.73205 q^{5} -0.267949 q^{7} -2.53590 q^{8} +O(q^{10})$	\(q+0.732051 q^{2} -1.46410 q^{4} -2.73205 q^{5} -0.267949 q^{7} -2.53590 q^{8} -2.00000 q^{10} -1.00000 q^{11} -3.73205 q^{13} -0.196152 q^{14} +1.07180 q^{16} -6.19615 q^{17} +5.19615 q^{19} +4.00000 q^{20} -0.732051 q^{22} -8.00000 q^{23} +2.46410 q^{25} -2.73205 q^{26} +0.392305 q^{28} +4.73205 q^{29} +7.46410 q^{31} +5.85641 q^{32} -4.53590 q^{34} +0.732051 q^{35} +0.464102 q^{37} +3.80385 q^{38} +6.92820 q^{40} +5.46410 q^{41} -3.46410 q^{43} +1.46410 q^{44} -5.85641 q^{46} +0.196152 q^{47} -6.92820 q^{49} +1.80385 q^{50} +5.46410 q^{52} -1.26795 q^{53} +2.73205 q^{55} +0.679492 q^{56} +3.46410 q^{58} +0.196152 q^{59} -7.73205 q^{61} +5.46410 q^{62} +2.14359 q^{64} +10.1962 q^{65} -7.92820 q^{67} +9.07180 q^{68} +0.535898 q^{70} -13.8564 q^{71} -7.19615 q^{73} +0.339746 q^{74} -7.60770 q^{76} +0.267949 q^{77} -2.26795 q^{79} -2.92820 q^{80} +4.00000 q^{82} +16.9282 q^{85} -2.53590 q^{86} +2.53590 q^{88} +17.6603 q^{89} +1.00000 q^{91} +11.7128 q^{92} +0.143594 q^{94} -14.1962 q^{95} +15.3923 q^{97} -5.07180 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8	$ 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8} - 4 q^{10} - 2 q^{11} - 4 q^{13} + 10 q^{14} + 16 q^{16} - 2 q^{17} + 8 q^{20} + 2 q^{22} - 16 q^{23} - 2 q^{25} - 2 q^{26} - 20 q^{28} + 6 q^{29} + 8 q^{31} - 16 q^{32} - 16 q^{34} - 2 q^{35} - 6 q^{37} + 18 q^{38} + 4 q^{41} - 4 q^{44} + 16 q^{46} - 10 q^{47} + 14 q^{50} + 4 q^{52} - 6 q^{53} + 2 q^{55} + 36 q^{56} - 10 q^{59} - 12 q^{61} + 4 q^{62} + 32 q^{64} + 10 q^{65} - 2 q^{67} + 32 q^{68} + 8 q^{70} - 4 q^{73} + 18 q^{74} - 36 q^{76} + 4 q^{77} - 8 q^{79} + 8 q^{80} + 8 q^{82} + 20 q^{85} - 12 q^{86} + 12 q^{88} + 18 q^{89} + 2 q^{91} - 32 q^{92} + 28 q^{94} - 18 q^{95} + 10 q^{97} - 24 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8 - 4 * q^10 - 2 * q^11 - 4 * q^13 + 10 * q^14 + 16 * q^16 - 2 * q^17 + 8 * q^20 + 2 * q^22 - 16 * q^23 - 2 * q^25 - 2 * q^26 - 20 * q^28 + 6 * q^29 + 8 * q^31 - 16 * q^32 - 16 * q^34 - 2 * q^35 - 6 * q^37 + 18 * q^38 + 4 * q^41 - 4 * q^44 + 16 * q^46 - 10 * q^47 + 14 * q^50 + 4 * q^52 - 6 * q^53 + 2 * q^55 + 36 * q^56 - 10 * q^59 - 12 * q^61 + 4 * q^62 + 32 * q^64 + 10 * q^65 - 2 * q^67 + 32 * q^68 + 8 * q^70 - 4 * q^73 + 18 * q^74 - 36 * q^76 + 4 * q^77 - 8 * q^79 + 8 * q^80 + 8 * q^82 + 20 * q^85 - 12 * q^86 + 12 * q^88 + 18 * q^89 + 2 * q^91 - 32 * q^92 + 28 * q^94 - 18 * q^95 + 10 * q^97 - 24 * 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 $

$2$	0.732051	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$2$	0.732051	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$3$	0	0
$3$	0	0
$4$	−1.46410	−0.732051
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	0	0
$7$	−0.267949	−0.101275	−0.0506376	−	0.998717i	$-0.516125\pi$
$7$	−0.267949	−0.101275	−0.0506376	+	0.998717i	$0.516125\pi$
$8$	−2.53590	−0.896575
$9$	0	0
$10$	−2.00000	−0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−3.73205	−1.03508	−0.517542	−	0.855658i	$-0.673153\pi$
$13$	−3.73205	−1.03508	−0.517542	+	0.855658i	$0.673153\pi$
$14$	−0.196152	−0.0524239
$15$	0	0
$16$	1.07180	0.267949
$17$	−6.19615	−1.50279	−0.751394	−	0.659854i	$-0.770618\pi$
$17$	−6.19615	−1.50279	−0.751394	+	0.659854i	$0.770618\pi$
$18$	0	0
$19$	5.19615	1.19208	0.596040	−	0.802955i	$-0.296740\pi$
$19$	5.19615	1.19208	0.596040	+	0.802955i	$0.296740\pi$
$20$	4.00000	0.894427
$21$	0	0
$22$	−0.732051	−0.156074
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	−2.73205	−0.535799
$27$	0	0
$28$	0.392305	0.0741386
$29$	4.73205	0.878720	0.439360	−	0.898311i	$-0.355205\pi$
$29$	4.73205	0.878720	0.439360	+	0.898311i	$0.355205\pi$
$30$	0	0
$31$	7.46410	1.34059	0.670296	−	0.742094i	$-0.266167\pi$
$31$	7.46410	1.34059	0.670296	+	0.742094i	$0.266167\pi$
$32$	5.85641	1.03528
$33$	0	0
$34$	−4.53590	−0.777900
$35$	0.732051	0.123739
$36$	0	0
$37$	0.464102	0.0762978	0.0381489	−	0.999272i	$-0.487854\pi$
$37$	0.464102	0.0762978	0.0381489	+	0.999272i	$0.487854\pi$
$38$	3.80385	0.617066
$39$	0	0
$40$	6.92820	1.09545
$41$	5.46410	0.853349	0.426675	−	0.904405i	$-0.359685\pi$
$41$	5.46410	0.853349	0.426675	+	0.904405i	$0.359685\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	1.46410	0.220722
$45$	0	0
$46$	−5.85641	−0.863480
$47$	0.196152	0.0286118	0.0143059	−	0.999898i	$-0.495446\pi$
$47$	0.196152	0.0286118	0.0143059	+	0.999898i	$0.495446\pi$
$48$	0	0
$49$	−6.92820	−0.989743
$50$	1.80385	0.255103
$51$	0	0
$52$	5.46410	0.757735
$53$	−1.26795	−0.174166	−0.0870831	−	0.996201i	$-0.527755\pi$
$53$	−1.26795	−0.174166	−0.0870831	+	0.996201i	$0.527755\pi$
$54$	0	0
$55$	2.73205	0.368390
$56$	0.679492	0.0908009
$57$	0	0
$58$	3.46410	0.454859
$59$	0.196152	0.0255369	0.0127684	−	0.999918i	$-0.495936\pi$
$59$	0.196152	0.0255369	0.0127684	+	0.999918i	$0.495936\pi$
$60$	0	0
$61$	−7.73205	−0.989988	−0.494994	−	0.868896i	$-0.664830\pi$
$61$	−7.73205	−0.989988	−0.494994	+	0.868896i	$0.664830\pi$
$62$	5.46410	0.693942
$63$	0	0
$64$	2.14359	0.267949
$65$	10.1962	1.26468
$66$	0	0
$67$	−7.92820	−0.968584	−0.484292	−	0.874906i	$-0.660923\pi$
$67$	−7.92820	−0.968584	−0.484292	+	0.874906i	$0.660923\pi$
$68$	9.07180	1.10012
$69$	0	0
$70$	0.535898	0.0640521
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	0	0
$73$	−7.19615	−0.842246	−0.421123	−	0.907004i	$-0.638364\pi$
$73$	−7.19615	−0.842246	−0.421123	+	0.907004i	$0.638364\pi$
$74$	0.339746	0.0394947
$75$	0	0
$76$	−7.60770	−0.872662
$77$	0.267949	0.0305356
$78$	0	0
$79$	−2.26795	−0.255164	−0.127582	−	0.991828i	$-0.540722\pi$
$79$	−2.26795	−0.255164	−0.127582	+	0.991828i	$0.540722\pi$
$80$	−2.92820	−0.327383
$81$	0	0
$82$	4.00000	0.441726
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	16.9282	1.83612
$86$	−2.53590	−0.273453
$87$	0	0
$88$	2.53590	0.270328
$89$	17.6603	1.87198	0.935992	−	0.352022i	$-0.114506\pi$
$89$	17.6603	1.87198	0.935992	+	0.352022i	$0.114506\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	11.7128	1.22115
$93$	0	0
$94$	0.143594	0.0148105
$95$	−14.1962	−1.45649
$96$	0	0
$97$	15.3923	1.56285	0.781426	−	0.623998i	$-0.214493\pi$
$97$	15.3923	1.56285	0.781426	+	0.623998i	$0.214493\pi$
$98$	−5.07180	−0.512329
$99$	0	0
$100$	−3.60770	−0.360770
$101$	−10.1962	−1.01456	−0.507278	−	0.861783i	$-0.669348\pi$
$101$	−10.1962	−1.01456	−0.507278	+	0.861783i	$0.669348\pi$
$102$	0	0
$103$	−14.8564	−1.46385	−0.731923	−	0.681388i	$-0.761377\pi$
$103$	−14.8564	−1.46385	−0.731923	+	0.681388i	$0.761377\pi$
$104$	9.46410	0.928032
$105$	0	0
$106$	−0.928203	−0.0901551
$107$	11.6603	1.12724	0.563620	−	0.826034i	$-0.309408\pi$
$107$	11.6603	1.12724	0.563620	+	0.826034i	$0.309408\pi$
$108$	0	0
$109$	−15.8564	−1.51877	−0.759384	−	0.650643i	$-0.774500\pi$
$109$	−15.8564	−1.51877	−0.759384	+	0.650643i	$0.774500\pi$
$110$	2.00000	0.190693
$111$	0	0
$112$	−0.287187	−0.0271366
$113$	−5.66025	−0.532472	−0.266236	−	0.963908i	$-0.585780\pi$
$113$	−5.66025	−0.532472	−0.266236	+	0.963908i	$0.585780\pi$
$114$	0	0
$115$	21.8564	2.03812
$116$	−6.92820	−0.643268
$117$	0	0
$118$	0.143594	0.0132189
$119$	1.66025	0.152195
$120$	0	0
$121$	1.00000	0.0909091
$122$	−5.66025	−0.512455
$123$	0	0
$124$	−10.9282	−0.981382
$125$	6.92820	0.619677
$126$	0	0
$127$	−1.46410	−0.129918	−0.0649590	−	0.997888i	$-0.520692\pi$
$127$	−1.46410	−0.129918	−0.0649590	+	0.997888i	$0.520692\pi$
$128$	−10.1436	−0.896575
$129$	0	0
$130$	7.46410	0.654645
$131$	−9.12436	−0.797199	−0.398599	−	0.917125i	$-0.630504\pi$
$131$	−9.12436	−0.797199	−0.398599	+	0.917125i	$0.630504\pi$
$132$	0	0
$133$	−1.39230	−0.120728
$134$	−5.80385	−0.501376
$135$	0	0
$136$	15.7128	1.34736
$137$	−12.3923	−1.05875	−0.529373	−	0.848389i	$-0.677573\pi$
$137$	−12.3923	−1.05875	−0.529373	+	0.848389i	$0.677573\pi$
$138$	0	0
$139$	3.19615	0.271094	0.135547	−	0.990771i	$-0.456721\pi$
$139$	3.19615	0.271094	0.135547	+	0.990771i	$0.456721\pi$
$140$	−1.07180	−0.0905834
$141$	0	0
$142$	−10.1436	−0.851231
$143$	3.73205	0.312090
$144$	0	0
$145$	−12.9282	−1.07363
$146$	−5.26795	−0.435979
$147$	0	0
$148$	−0.679492	−0.0558539
$149$	17.8564	1.46285	0.731427	−	0.681920i	$-0.238855\pi$
$149$	17.8564	1.46285	0.731427	+	0.681920i	$0.238855\pi$
$150$	0	0
$151$	−15.1962	−1.23665	−0.618323	−	0.785924i	$-0.712188\pi$
$151$	−15.1962	−1.23665	−0.618323	+	0.785924i	$0.712188\pi$
$152$	−13.1769	−1.06879
$153$	0	0
$154$	0.196152	0.0158064
$155$	−20.3923	−1.63795
$156$	0	0
$157$	17.4641	1.39379	0.696894	−	0.717175i	$-0.254565\pi$
$157$	17.4641	1.39379	0.696894	+	0.717175i	$0.254565\pi$
$158$	−1.66025	−0.132083
$159$	0	0
$160$	−16.0000	−1.26491
$161$	2.14359	0.168939
$162$	0	0
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	−8.00000	−0.624695
$165$	0	0
$166$	0	0
$167$	−19.2679	−1.49100	−0.745499	−	0.666506i	$-0.767789\pi$
$167$	−19.2679	−1.49100	−0.745499	+	0.666506i	$0.767789\pi$
$168$	0	0
$169$	0.928203	0.0714002
$170$	12.3923	0.950446
$171$	0	0
$172$	5.07180	0.386721
$173$	−16.3923	−1.24628	−0.623142	−	0.782109i	$-0.714144\pi$
$173$	−16.3923	−1.24628	−0.623142	+	0.782109i	$0.714144\pi$
$174$	0	0
$175$	−0.660254	−0.0499105
$176$	−1.07180	−0.0807897
$177$	0	0
$178$	12.9282	0.969010
$179$	3.80385	0.284313	0.142156	−	0.989844i	$-0.454596\pi$
$179$	3.80385	0.284313	0.142156	+	0.989844i	$0.454596\pi$
$180$	0	0
$181$	18.3205	1.36175	0.680876	−	0.732398i	$-0.261599\pi$
$181$	18.3205	1.36175	0.680876	+	0.732398i	$0.261599\pi$
$182$	0.732051	0.0542632
$183$	0	0
$184$	20.2872	1.49559
$185$	−1.26795	−0.0932215
$186$	0	0
$187$	6.19615	0.453108
$188$	−0.287187	−0.0209453
$189$	0	0
$190$	−10.3923	−0.753937
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	−13.1962	−0.949880	−0.474940	−	0.880018i	$-0.657530\pi$
$193$	−13.1962	−0.949880	−0.474940	+	0.880018i	$0.657530\pi$
$194$	11.2679	0.808992
$195$	0	0
$196$	10.1436	0.724542
$197$	7.66025	0.545771	0.272885	−	0.962047i	$-0.412022\pi$
$197$	7.66025	0.545771	0.272885	+	0.962047i	$0.412022\pi$
$198$	0	0
$199$	−15.0000	−1.06332	−0.531661	−	0.846957i	$-0.678432\pi$
$199$	−15.0000	−1.06332	−0.531661	+	0.846957i	$0.678432\pi$
$200$	−6.24871	−0.441851
$201$	0	0
$202$	−7.46410	−0.525172
$203$	−1.26795	−0.0889926
$204$	0	0
$205$	−14.9282	−1.04263
$206$	−10.8756	−0.757742
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−5.19615	−0.359425
$210$	0	0
$211$	−26.6603	−1.83537	−0.917684	−	0.397312i	$-0.869943\pi$
$211$	−26.6603	−1.83537	−0.917684	+	0.397312i	$0.869943\pi$
$212$	1.85641	0.127499
$213$	0	0
$214$	8.53590	0.583502
$215$	9.46410	0.645446
$216$	0	0
$217$	−2.00000	−0.135769
$218$	−11.6077	−0.786172
$219$	0	0
$220$	−4.00000	−0.269680
$221$	23.1244	1.55551
$222$	0	0
$223$	−9.85641	−0.660034	−0.330017	−	0.943975i	$-0.607054\pi$
$223$	−9.85641	−0.660034	−0.330017	+	0.943975i	$0.607054\pi$
$224$	−1.56922	−0.104848
$225$	0	0
$226$	−4.14359	−0.275628
$227$	12.3397	0.819018	0.409509	−	0.912306i	$-0.365700\pi$
$227$	12.3397	0.819018	0.409509	+	0.912306i	$0.365700\pi$
$228$	0	0
$229$	4.39230	0.290252	0.145126	−	0.989413i	$-0.453641\pi$
$229$	4.39230	0.290252	0.145126	+	0.989413i	$0.453641\pi$
$230$	16.0000	1.05501
$231$	0	0
$232$	−12.0000	−0.787839
$233$	23.6603	1.55003	0.775017	−	0.631940i	$-0.217741\pi$
$233$	23.6603	1.55003	0.775017	+	0.631940i	$0.217741\pi$
$234$	0	0
$235$	−0.535898	−0.0349582
$236$	−0.287187	−0.0186943
$237$	0	0
$238$	1.21539	0.0787821
$239$	4.39230	0.284115	0.142057	−	0.989858i	$-0.454628\pi$
$239$	4.39230	0.284115	0.142057	+	0.989858i	$0.454628\pi$
$240$	0	0
$241$	4.80385	0.309443	0.154722	−	0.987958i	$-0.450552\pi$
$241$	4.80385	0.309443	0.154722	+	0.987958i	$0.450552\pi$
$242$	0.732051	0.0470580
$243$	0	0
$244$	11.3205	0.724721
$245$	18.9282	1.20928
$246$	0	0
$247$	−19.3923	−1.23390
$248$	−18.9282	−1.20194
$249$	0	0
$250$	5.07180	0.320769
$251$	20.5885	1.29953	0.649766	−	0.760134i	$-0.274867\pi$
$251$	20.5885	1.29953	0.649766	+	0.760134i	$0.274867\pi$
$252$	0	0
$253$	8.00000	0.502956
$254$	−1.07180	−0.0672505
$255$	0	0
$256$	−11.7128	−0.732051
$257$	9.26795	0.578119	0.289059	−	0.957311i	$-0.406657\pi$
$257$	9.26795	0.578119	0.289059	+	0.957311i	$0.406657\pi$
$258$	0	0
$259$	−0.124356	−0.00772708
$260$	−14.9282	−0.925808
$261$	0	0
$262$	−6.67949	−0.412660
$263$	−29.1244	−1.79588	−0.897942	−	0.440113i	$-0.854938\pi$
$263$	−29.1244	−1.79588	−0.897942	+	0.440113i	$0.854938\pi$
$264$	0	0
$265$	3.46410	0.212798
$266$	−1.01924	−0.0624935
$267$	0	0
$268$	11.6077	0.709053
$269$	13.0718	0.797002	0.398501	−	0.917168i	$-0.369531\pi$
$269$	13.0718	0.797002	0.398501	+	0.917168i	$0.369531\pi$
$270$	0	0
$271$	−5.73205	−0.348197	−0.174099	−	0.984728i	$-0.555701\pi$
$271$	−5.73205	−0.348197	−0.174099	+	0.984728i	$0.555701\pi$
$272$	−6.64102	−0.402671
$273$	0	0
$274$	−9.07180	−0.548047
$275$	−2.46410	−0.148591
$276$	0	0
$277$	17.8564	1.07289	0.536444	−	0.843936i	$-0.319767\pi$
$277$	17.8564	1.07289	0.536444	+	0.843936i	$0.319767\pi$
$278$	2.33975	0.140329
$279$	0	0
$280$	−1.85641	−0.110942
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	22.2487	1.32255	0.661274	−	0.750144i	$-0.270016\pi$
$283$	22.2487	1.32255	0.661274	+	0.750144i	$0.270016\pi$
$284$	20.2872	1.20382
$285$	0	0
$286$	2.73205	0.161550
$287$	−1.46410	−0.0864232
$288$	0	0
$289$	21.3923	1.25837
$290$	−9.46410	−0.555751
$291$	0	0
$292$	10.5359	0.616567
$293$	2.87564	0.167997	0.0839985	−	0.996466i	$-0.473231\pi$
$293$	2.87564	0.167997	0.0839985	+	0.996466i	$0.473231\pi$
$294$	0	0
$295$	−0.535898	−0.0312012
$296$	−1.17691	−0.0684068
$297$	0	0
$298$	13.0718	0.757229
$299$	29.8564	1.72664
$300$	0	0
$301$	0.928203	0.0535007
$302$	−11.1244	−0.640135
$303$	0	0
$304$	5.56922	0.319417
$305$	21.1244	1.20958
$306$	0	0
$307$	7.32051	0.417803	0.208902	−	0.977937i	$-0.433011\pi$
$307$	7.32051	0.417803	0.208902	+	0.977937i	$0.433011\pi$
$308$	−0.392305	−0.0223536
$309$	0	0
$310$	−14.9282	−0.847865
$311$	0.588457	0.0333684	0.0166842	−	0.999861i	$-0.494689\pi$
$311$	0.588457	0.0333684	0.0166842	+	0.999861i	$0.494689\pi$
$312$	0	0
$313$	11.9282	0.674222	0.337111	−	0.941465i	$-0.390550\pi$
$313$	11.9282	0.674222	0.337111	+	0.941465i	$0.390550\pi$
$314$	12.7846	0.721477
$315$	0	0
$316$	3.32051	0.186793
$317$	20.5885	1.15636	0.578181	−	0.815908i	$-0.303763\pi$
$317$	20.5885	1.15636	0.578181	+	0.815908i	$0.303763\pi$
$318$	0	0
$319$	−4.73205	−0.264944
$320$	−5.85641	−0.327383
$321$	0	0
$322$	1.56922	0.0874492
$323$	−32.1962	−1.79144
$324$	0	0
$325$	−9.19615	−0.510111
$326$	13.9090	0.770346
$327$	0	0
$328$	−13.8564	−0.765092
$329$	−0.0525589	−0.00289767
$330$	0	0
$331$	−6.60770	−0.363192	−0.181596	−	0.983373i	$-0.558126\pi$
$331$	−6.60770	−0.363192	−0.181596	+	0.983373i	$0.558126\pi$
$332$	0	0
$333$	0	0
$334$	−14.1051	−0.771798
$335$	21.6603	1.18343
$336$	0	0
$337$	21.9808	1.19737	0.598684	−	0.800985i	$-0.295690\pi$
$337$	21.9808	1.19737	0.598684	+	0.800985i	$0.295690\pi$
$338$	0.679492	0.0369595
$339$	0	0
$340$	−24.7846	−1.34413
$341$	−7.46410	−0.404204
$342$	0	0
$343$	3.73205	0.201512
$344$	8.78461	0.473634
$345$	0	0
$346$	−12.0000	−0.645124
$347$	−27.6603	−1.48488	−0.742440	−	0.669912i	$-0.766332\pi$
$347$	−27.6603	−1.48488	−0.742440	+	0.669912i	$0.766332\pi$
$348$	0	0
$349$	−30.1244	−1.61252	−0.806260	−	0.591561i	$-0.798512\pi$
$349$	−30.1244	−1.61252	−0.806260	+	0.591561i	$0.798512\pi$
$350$	−0.483340	−0.0258356
$351$	0	0
$352$	−5.85641	−0.312148
$353$	3.12436	0.166293	0.0831463	−	0.996537i	$-0.473503\pi$
$353$	3.12436	0.166293	0.0831463	+	0.996537i	$0.473503\pi$
$354$	0	0
$355$	37.8564	2.00921
$356$	−25.8564	−1.37039
$357$	0	0
$358$	2.78461	0.147171
$359$	−13.4641	−0.710608	−0.355304	−	0.934751i	$-0.615623\pi$
$359$	−13.4641	−0.710608	−0.355304	+	0.934751i	$0.615623\pi$
$360$	0	0
$361$	8.00000	0.421053
$362$	13.4115	0.704895
$363$	0	0
$364$	−1.46410	−0.0767398
$365$	19.6603	1.02906
$366$	0	0
$367$	0.0717968	0.00374776	0.00187388	−	0.999998i	$-0.499404\pi$
$367$	0.0717968	0.00374776	0.00187388	+	0.999998i	$0.499404\pi$
$368$	−8.57437	−0.446970
$369$	0	0
$370$	−0.928203	−0.0482550
$371$	0.339746	0.0176387
$372$	0	0
$373$	4.12436	0.213551	0.106776	−	0.994283i	$-0.465947\pi$
$373$	4.12436	0.213551	0.106776	+	0.994283i	$0.465947\pi$
$374$	4.53590	0.234546
$375$	0	0
$376$	−0.497423	−0.0256526
$377$	−17.6603	−0.909549
$378$	0	0
$379$	−12.8564	−0.660389	−0.330195	−	0.943913i	$-0.607114\pi$
$379$	−12.8564	−0.660389	−0.330195	+	0.943913i	$0.607114\pi$
$380$	20.7846	1.06623
$381$	0	0
$382$	2.92820	0.149820
$383$	−21.2679	−1.08674	−0.543371	−	0.839493i	$-0.682852\pi$
$383$	−21.2679	−1.08674	−0.543371	+	0.839493i	$0.682852\pi$
$384$	0	0
$385$	−0.732051	−0.0373088
$386$	−9.66025	−0.491694
$387$	0	0
$388$	−22.5359	−1.14409
$389$	−14.5359	−0.736999	−0.368500	−	0.929628i	$-0.620128\pi$
$389$	−14.5359	−0.736999	−0.368500	+	0.929628i	$0.620128\pi$
$390$	0	0
$391$	49.5692	2.50682
$392$	17.5692	0.887380
$393$	0	0
$394$	5.60770	0.282512
$395$	6.19615	0.311762
$396$	0	0
$397$	−0.392305	−0.0196892	−0.00984461	−	0.999952i	$-0.503134\pi$
$397$	−0.392305	−0.0196892	−0.00984461	+	0.999952i	$0.503134\pi$
$398$	−10.9808	−0.550416
$399$	0	0
$400$	2.64102	0.132051
$401$	13.4641	0.672365	0.336183	−	0.941797i	$-0.390864\pi$
$401$	13.4641	0.672365	0.336183	+	0.941797i	$0.390864\pi$
$402$	0	0
$403$	−27.8564	−1.38763
$404$	14.9282	0.742706
$405$	0	0
$406$	−0.928203	−0.0460660
$407$	−0.464102	−0.0230047
$408$	0	0
$409$	−32.9090	−1.62724	−0.813622	−	0.581394i	$-0.802507\pi$
$409$	−32.9090	−1.62724	−0.813622	+	0.581394i	$0.802507\pi$
$410$	−10.9282	−0.539705
$411$	0	0
$412$	21.7513	1.07161
$413$	−0.0525589	−0.00258625
$414$	0	0
$415$	0	0
$416$	−21.8564	−1.07160
$417$	0	0
$418$	−3.80385	−0.186052
$419$	31.7128	1.54927	0.774636	−	0.632407i	$-0.217933\pi$
$419$	31.7128	1.54927	0.774636	+	0.632407i	$0.217933\pi$
$420$	0	0
$421$	−6.85641	−0.334161	−0.167080	−	0.985943i	$-0.553434\pi$
$421$	−6.85641	−0.334161	−0.167080	+	0.985943i	$0.553434\pi$
$422$	−19.5167	−0.950056
$423$	0	0
$424$	3.21539	0.156153
$425$	−15.2679	−0.740604
$426$	0	0
$427$	2.07180	0.100261
$428$	−17.0718	−0.825196
$429$	0	0
$430$	6.92820	0.334108
$431$	−21.4641	−1.03389	−0.516945	−	0.856019i	$-0.672931\pi$
$431$	−21.4641	−1.03389	−0.516945	+	0.856019i	$0.672931\pi$
$432$	0	0
$433$	−24.3923	−1.17222	−0.586110	−	0.810232i	$-0.699341\pi$
$433$	−24.3923	−1.17222	−0.586110	+	0.810232i	$0.699341\pi$
$434$	−1.46410	−0.0702791
$435$	0	0
$436$	23.2154	1.11182
$437$	−41.5692	−1.98853
$438$	0	0
$439$	37.4641	1.78806	0.894032	−	0.448003i	$-0.147865\pi$
$439$	37.4641	1.78806	0.894032	+	0.448003i	$0.147865\pi$
$440$	−6.92820	−0.330289
$441$	0	0
$442$	16.9282	0.805193
$443$	6.73205	0.319849	0.159925	−	0.987129i	$-0.448875\pi$
$443$	6.73205	0.319849	0.159925	+	0.987129i	$0.448875\pi$
$444$	0	0
$445$	−48.2487	−2.28721
$446$	−7.21539	−0.341659
$447$	0	0
$448$	−0.574374	−0.0271366
$449$	−38.2487	−1.80507	−0.902534	−	0.430618i	$-0.858296\pi$
$449$	−38.2487	−1.80507	−0.902534	+	0.430618i	$0.858296\pi$
$450$	0	0
$451$	−5.46410	−0.257294
$452$	8.28719	0.389796
$453$	0	0
$454$	9.03332	0.423955
$455$	−2.73205	−0.128081
$456$	0	0
$457$	−24.9282	−1.16609	−0.583046	−	0.812439i	$-0.698139\pi$
$457$	−24.9282	−1.16609	−0.583046	+	0.812439i	$0.698139\pi$
$458$	3.21539	0.150245
$459$	0	0
$460$	−32.0000	−1.49201
$461$	12.3397	0.574719	0.287360	−	0.957823i	$-0.407223\pi$
$461$	12.3397	0.574719	0.287360	+	0.957823i	$0.407223\pi$
$462$	0	0
$463$	15.7846	0.733573	0.366787	−	0.930305i	$-0.380458\pi$
$463$	15.7846	0.733573	0.366787	+	0.930305i	$0.380458\pi$
$464$	5.07180	0.235452
$465$	0	0
$466$	17.3205	0.802357
$467$	5.07180	0.234695	0.117347	−	0.993091i	$-0.462561\pi$
$467$	5.07180	0.234695	0.117347	+	0.993091i	$0.462561\pi$
$468$	0	0
$469$	2.12436	0.0980936
$470$	−0.392305	−0.0180957
$471$	0	0
$472$	−0.497423	−0.0228957
$473$	3.46410	0.159280
$474$	0	0
$475$	12.8038	0.587481
$476$	−2.43078	−0.111415
$477$	0	0
$478$	3.21539	0.147069
$479$	−33.5167	−1.53142	−0.765708	−	0.643189i	$-0.777611\pi$
$479$	−33.5167	−1.53142	−0.765708	+	0.643189i	$0.777611\pi$
$480$	0	0
$481$	−1.73205	−0.0789747
$482$	3.51666	0.160179
$483$	0	0
$484$	−1.46410	−0.0665501
$485$	−42.0526	−1.90951
$486$	0	0
$487$	−8.85641	−0.401322	−0.200661	−	0.979661i	$-0.564309\pi$
$487$	−8.85641	−0.401322	−0.200661	+	0.979661i	$0.564309\pi$
$488$	19.6077	0.887599
$489$	0	0
$490$	13.8564	0.625969
$491$	−8.05256	−0.363407	−0.181703	−	0.983353i	$-0.558161\pi$
$491$	−8.05256	−0.363407	−0.181703	+	0.983353i	$0.558161\pi$
$492$	0	0
$493$	−29.3205	−1.32053
$494$	−14.1962	−0.638715
$495$	0	0
$496$	8.00000	0.359211
$497$	3.71281	0.166542
$498$	0	0
$499$	−12.5359	−0.561184	−0.280592	−	0.959827i	$-0.590531\pi$
$499$	−12.5359	−0.561184	−0.280592	+	0.959827i	$0.590531\pi$
$500$	−10.1436	−0.453635
$501$	0	0
$502$	15.0718	0.672687
$503$	−9.07180	−0.404491	−0.202246	−	0.979335i	$-0.564824\pi$
$503$	−9.07180	−0.404491	−0.202246	+	0.979335i	$0.564824\pi$
$504$	0	0
$505$	27.8564	1.23959
$506$	5.85641	0.260349
$507$	0	0
$508$	2.14359	0.0951066
$509$	22.0526	0.977462	0.488731	−	0.872434i	$-0.337460\pi$
$509$	22.0526	0.977462	0.488731	+	0.872434i	$0.337460\pi$
$510$	0	0
$511$	1.92820	0.0852987
$512$	11.7128	0.517638
$513$	0	0
$514$	6.78461	0.299256
$515$	40.5885	1.78854
$516$	0	0
$517$	−0.196152	−0.00862677
$518$	−0.0910347	−0.00399983
$519$	0	0
$520$	−25.8564	−1.13388
$521$	20.7846	0.910590	0.455295	−	0.890341i	$-0.349534\pi$
$521$	20.7846	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$522$	0	0
$523$	−11.9808	−0.523882	−0.261941	−	0.965084i	$-0.584363\pi$
$523$	−11.9808	−0.523882	−0.261941	+	0.965084i	$0.584363\pi$
$524$	13.3590	0.583590
$525$	0	0
$526$	−21.3205	−0.929618
$527$	−46.2487	−2.01463
$528$	0	0
$529$	41.0000	1.78261
$530$	2.53590	0.110152
$531$	0	0
$532$	2.03848	0.0883791
$533$	−20.3923	−0.883289
$534$	0	0
$535$	−31.8564	−1.37727
$536$	20.1051	0.868409
$537$	0	0
$538$	9.56922	0.412558
$539$	6.92820	0.298419
$540$	0	0
$541$	7.58846	0.326253	0.163127	−	0.986605i	$-0.447842\pi$
$541$	7.58846	0.326253	0.163127	+	0.986605i	$0.447842\pi$
$542$	−4.19615	−0.180240
$543$	0	0
$544$	−36.2872	−1.55580
$545$	43.3205	1.85565
$546$	0	0
$547$	−27.1962	−1.16282	−0.581412	−	0.813609i	$-0.697499\pi$
$547$	−27.1962	−1.16282	−0.581412	+	0.813609i	$0.697499\pi$
$548$	18.1436	0.775056
$549$	0	0
$550$	−1.80385	−0.0769163
$551$	24.5885	1.04750
$552$	0	0
$553$	0.607695	0.0258418
$554$	13.0718	0.555367
$555$	0	0
$556$	−4.67949	−0.198455
$557$	−32.7321	−1.38690	−0.693451	−	0.720504i	$-0.743911\pi$
$557$	−32.7321	−1.38690	−0.693451	+	0.720504i	$0.743911\pi$
$558$	0	0
$559$	12.9282	0.546805
$560$	0.784610	0.0331558
$561$	0	0
$562$	0	0
$563$	−19.7128	−0.830796	−0.415398	−	0.909640i	$-0.636358\pi$
$563$	−19.7128	−0.830796	−0.415398	+	0.909640i	$0.636358\pi$
$564$	0	0
$565$	15.4641	0.650580
$566$	16.2872	0.684602
$567$	0	0
$568$	35.1384	1.47438
$569$	−0.679492	−0.0284858	−0.0142429	−	0.999899i	$-0.504534\pi$
$569$	−0.679492	−0.0284858	−0.0142429	+	0.999899i	$0.504534\pi$
$570$	0	0
$571$	7.87564	0.329585	0.164793	−	0.986328i	$-0.447304\pi$
$571$	7.87564	0.329585	0.164793	+	0.986328i	$0.447304\pi$
$572$	−5.46410	−0.228466
$573$	0	0
$574$	−1.07180	−0.0447359
$575$	−19.7128	−0.822081
$576$	0	0
$577$	−3.00000	−0.124892	−0.0624458	−	0.998048i	$-0.519890\pi$
$577$	−3.00000	−0.124892	−0.0624458	+	0.998048i	$0.519890\pi$
$578$	15.6603	0.651381
$579$	0	0
$580$	18.9282	0.785951
$581$	0	0
$582$	0	0
$583$	1.26795	0.0525131
$584$	18.2487	0.755137
$585$	0	0
$586$	2.10512	0.0869616
$587$	32.5885	1.34507	0.672535	−	0.740066i	$-0.265206\pi$
$587$	32.5885	1.34507	0.672535	+	0.740066i	$0.265206\pi$
$588$	0	0
$589$	38.7846	1.59809
$590$	−0.392305	−0.0161509
$591$	0	0
$592$	0.497423	0.0204439
$593$	7.71281	0.316727	0.158364	−	0.987381i	$-0.449378\pi$
$593$	7.71281	0.316727	0.158364	+	0.987381i	$0.449378\pi$
$594$	0	0
$595$	−4.53590	−0.185954
$596$	−26.1436	−1.07088
$597$	0	0
$598$	21.8564	0.893775
$599$	2.24871	0.0918799	0.0459399	−	0.998944i	$-0.485372\pi$
$599$	2.24871	0.0918799	0.0459399	+	0.998944i	$0.485372\pi$
$600$	0	0
$601$	−23.8564	−0.973123	−0.486562	−	0.873646i	$-0.661749\pi$
$601$	−23.8564	−0.973123	−0.486562	+	0.873646i	$0.661749\pi$
$602$	0.679492	0.0276940
$603$	0	0
$604$	22.2487	0.905287
$605$	−2.73205	−0.111074
$606$	0	0
$607$	−6.41154	−0.260236	−0.130118	−	0.991498i	$-0.541536\pi$
$607$	−6.41154	−0.260236	−0.130118	+	0.991498i	$0.541536\pi$
$608$	30.4308	1.23413
$609$	0	0
$610$	15.4641	0.626123
$611$	−0.732051	−0.0296156
$612$	0	0
$613$	−42.9090	−1.73308	−0.866538	−	0.499110i	$-0.833660\pi$
$613$	−42.9090	−1.73308	−0.866538	+	0.499110i	$0.833660\pi$
$614$	5.35898	0.216271
$615$	0	0
$616$	−0.679492	−0.0273775
$617$	−10.1436	−0.408366	−0.204183	−	0.978933i	$-0.565454\pi$
$617$	−10.1436	−0.408366	−0.204183	+	0.978933i	$0.565454\pi$
$618$	0	0
$619$	13.5359	0.544054	0.272027	−	0.962290i	$-0.412306\pi$
$619$	13.5359	0.544054	0.272027	+	0.962290i	$0.412306\pi$
$620$	29.8564	1.19906
$621$	0	0
$622$	0.430781	0.0172727
$623$	−4.73205	−0.189586
$624$	0	0
$625$	−31.2487	−1.24995
$626$	8.73205	0.349003
$627$	0	0
$628$	−25.5692	−1.02032
$629$	−2.87564	−0.114659
$630$	0	0
$631$	17.3923	0.692377	0.346188	−	0.938165i	$-0.387476\pi$
$631$	17.3923	0.692377	0.346188	+	0.938165i	$0.387476\pi$
$632$	5.75129	0.228774
$633$	0	0
$634$	15.0718	0.598578
$635$	4.00000	0.158735
$636$	0	0
$637$	25.8564	1.02447
$638$	−3.46410	−0.137145
$639$	0	0
$640$	27.7128	1.09545
$641$	−27.1244	−1.07135	−0.535674	−	0.844425i	$-0.679942\pi$
$641$	−27.1244	−1.07135	−0.535674	+	0.844425i	$0.679942\pi$
$642$	0	0
$643$	9.60770	0.378891	0.189445	−	0.981891i	$-0.439331\pi$
$643$	9.60770	0.378891	0.189445	+	0.981891i	$0.439331\pi$
$644$	−3.13844	−0.123672
$645$	0	0
$646$	−23.5692	−0.927319
$647$	22.2487	0.874687	0.437344	−	0.899295i	$-0.355919\pi$
$647$	22.2487	0.874687	0.437344	+	0.899295i	$0.355919\pi$
$648$	0	0
$649$	−0.196152	−0.00769966
$650$	−6.73205	−0.264053
$651$	0	0
$652$	−27.8179	−1.08943
$653$	22.5359	0.881898	0.440949	−	0.897532i	$-0.354642\pi$
$653$	22.5359	0.881898	0.440949	+	0.897532i	$0.354642\pi$
$654$	0	0
$655$	24.9282	0.974025
$656$	5.85641	0.228654
$657$	0	0
$658$	−0.0384758	−0.00149994
$659$	20.3397	0.792324	0.396162	−	0.918181i	$-0.370342\pi$
$659$	20.3397	0.792324	0.396162	+	0.918181i	$0.370342\pi$
$660$	0	0
$661$	23.3923	0.909855	0.454928	−	0.890528i	$-0.349665\pi$
$661$	23.3923	0.909855	0.454928	+	0.890528i	$0.349665\pi$
$662$	−4.83717	−0.188002
$663$	0	0
$664$	0	0
$665$	3.80385	0.147507
$666$	0	0
$667$	−37.8564	−1.46581
$668$	28.2102	1.09149
$669$	0	0
$670$	15.8564	0.612586
$671$	7.73205	0.298493
$672$	0	0
$673$	49.4449	1.90596	0.952980	−	0.303034i	$-0.0979996\pi$
$673$	49.4449	1.90596	0.952980	+	0.303034i	$0.0979996\pi$
$674$	16.0910	0.619803
$675$	0	0
$676$	−1.35898	−0.0522686
$677$	18.2487	0.701355	0.350677	−	0.936496i	$-0.385951\pi$
$677$	18.2487	0.701355	0.350677	+	0.936496i	$0.385951\pi$
$678$	0	0
$679$	−4.12436	−0.158278
$680$	−42.9282	−1.64622
$681$	0	0
$682$	−5.46410	−0.209231
$683$	−8.58846	−0.328628	−0.164314	−	0.986408i	$-0.552541\pi$
$683$	−8.58846	−0.328628	−0.164314	+	0.986408i	$0.552541\pi$
$684$	0	0
$685$	33.8564	1.29359
$686$	2.73205	0.104310
$687$	0	0
$688$	−3.71281	−0.141550
$689$	4.73205	0.180277
$690$	0	0
$691$	−48.7846	−1.85585	−0.927927	−	0.372762i	$-0.878411\pi$
$691$	−48.7846	−1.85585	−0.927927	+	0.372762i	$0.878411\pi$
$692$	24.0000	0.912343
$693$	0	0
$694$	−20.2487	−0.768631
$695$	−8.73205	−0.331226
$696$	0	0
$697$	−33.8564	−1.28240
$698$	−22.0526	−0.834702
$699$	0	0
$700$	0.966679	0.0365370
$701$	−48.0526	−1.81492	−0.907460	−	0.420138i	$-0.861982\pi$
$701$	−48.0526	−1.81492	−0.907460	+	0.420138i	$0.861982\pi$
$702$	0	0
$703$	2.41154	0.0909531
$704$	−2.14359	−0.0807897
$705$	0	0
$706$	2.28719	0.0860794
$707$	2.73205	0.102749
$708$	0	0
$709$	−14.6077	−0.548604	−0.274302	−	0.961644i	$-0.588447\pi$
$709$	−14.6077	−0.548604	−0.274302	+	0.961644i	$0.588447\pi$
$710$	27.7128	1.04004
$711$	0	0
$712$	−44.7846	−1.67837
$713$	−59.7128	−2.23626
$714$	0	0
$715$	−10.1962	−0.381314
$716$	−5.56922	−0.208132
$717$	0	0
$718$	−9.85641	−0.367838
$719$	28.9808	1.08080	0.540400	−	0.841408i	$-0.318273\pi$
$719$	28.9808	1.08080	0.540400	+	0.841408i	$0.318273\pi$
$720$	0	0
$721$	3.98076	0.148251
$722$	5.85641	0.217953
$723$	0	0
$724$	−26.8231	−0.996872
$725$	11.6603	0.433051
$726$	0	0
$727$	5.32051	0.197327	0.0986634	−	0.995121i	$-0.468543\pi$
$727$	5.32051	0.197327	0.0986634	+	0.995121i	$0.468543\pi$
$728$	−2.53590	−0.0939866
$729$	0	0
$730$	14.3923	0.532683
$731$	21.4641	0.793878
$732$	0	0
$733$	11.0718	0.408946	0.204473	−	0.978872i	$-0.434452\pi$
$733$	11.0718	0.408946	0.204473	+	0.978872i	$0.434452\pi$
$734$	0.0525589	0.00193998
$735$	0	0
$736$	−46.8513	−1.72696
$737$	7.92820	0.292039
$738$	0	0
$739$	14.2487	0.524147	0.262074	−	0.965048i	$-0.415594\pi$
$739$	14.2487	0.524147	0.262074	+	0.965048i	$0.415594\pi$
$740$	1.85641	0.0682429
$741$	0	0
$742$	0.248711	0.00913048
$743$	2.58846	0.0949613	0.0474806	−	0.998872i	$-0.484881\pi$
$743$	2.58846	0.0949613	0.0474806	+	0.998872i	$0.484881\pi$
$744$	0	0
$745$	−48.7846	−1.78733
$746$	3.01924	0.110542
$747$	0	0
$748$	−9.07180	−0.331698
$749$	−3.12436	−0.114161
$750$	0	0
$751$	12.3205	0.449582	0.224791	−	0.974407i	$-0.427830\pi$
$751$	12.3205	0.449582	0.224791	+	0.974407i	$0.427830\pi$
$752$	0.210236	0.00766650
$753$	0	0
$754$	−12.9282	−0.470817
$755$	41.5167	1.51095
$756$	0	0
$757$	−6.32051	−0.229723	−0.114861	−	0.993382i	$-0.536642\pi$
$757$	−6.32051	−0.229723	−0.114861	+	0.993382i	$0.536642\pi$
$758$	−9.41154	−0.341843
$759$	0	0
$760$	36.0000	1.30586
$761$	35.3205	1.28037	0.640184	−	0.768222i	$-0.278858\pi$
$761$	35.3205	1.28037	0.640184	+	0.768222i	$0.278858\pi$
$762$	0	0
$763$	4.24871	0.153814
$764$	−5.85641	−0.211877
$765$	0	0
$766$	−15.5692	−0.562539
$767$	−0.732051	−0.0264328
$768$	0	0
$769$	5.58846	0.201525	0.100762	−	0.994911i	$-0.467872\pi$
$769$	5.58846	0.201525	0.100762	+	0.994911i	$0.467872\pi$
$770$	−0.535898	−0.0193124
$771$	0	0
$772$	19.3205	0.695360
$773$	3.12436	0.112375	0.0561876	−	0.998420i	$-0.482105\pi$
$773$	3.12436	0.112375	0.0561876	+	0.998420i	$0.482105\pi$
$774$	0	0
$775$	18.3923	0.660671
$776$	−39.0333	−1.40121
$777$	0	0
$778$	−10.6410	−0.381499
$779$	28.3923	1.01726
$780$	0	0
$781$	13.8564	0.495821
$782$	36.2872	1.29763
$783$	0	0
$784$	−7.42563	−0.265201
$785$	−47.7128	−1.70294
$786$	0	0
$787$	−15.0526	−0.536566	−0.268283	−	0.963340i	$-0.586456\pi$
$787$	−15.0526	−0.536566	−0.268283	+	0.963340i	$0.586456\pi$
$788$	−11.2154	−0.399532
$789$	0	0
$790$	4.53590	0.161380
$791$	1.51666	0.0539262
$792$	0	0
$793$	28.8564	1.02472
$794$	−0.287187	−0.0101919
$795$	0	0
$796$	21.9615	0.778406
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	0	0
$799$	−1.21539	−0.0429974
$800$	14.4308	0.510205
$801$	0	0
$802$	9.85641	0.348042
$803$	7.19615	0.253947
$804$	0	0
$805$	−5.85641	−0.206411
$806$	−20.3923	−0.718288
$807$	0	0
$808$	25.8564	0.909625
$809$	−54.5885	−1.91923	−0.959614	−	0.281320i	$-0.909228\pi$
$809$	−54.5885	−1.91923	−0.959614	+	0.281320i	$0.909228\pi$
$810$	0	0
$811$	−0.392305	−0.0137757	−0.00688784	−	0.999976i	$-0.502192\pi$
$811$	−0.392305	−0.0137757	−0.00688784	+	0.999976i	$0.502192\pi$
$812$	1.85641	0.0651471
$813$	0	0
$814$	−0.339746	−0.0119081
$815$	−51.9090	−1.81829
$816$	0	0
$817$	−18.0000	−0.629740
$818$	−24.0910	−0.842323
$819$	0	0
$820$	21.8564	0.763259
$821$	−8.39230	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$821$	−8.39230	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$822$	0	0
$823$	−6.71281	−0.233994	−0.116997	−	0.993132i	$-0.537327\pi$
$823$	−6.71281	−0.233994	−0.116997	+	0.993132i	$0.537327\pi$
$824$	37.6743	1.31245
$825$	0	0
$826$	−0.0384758	−0.00133874
$827$	−6.98076	−0.242745	−0.121372	−	0.992607i	$-0.538730\pi$
$827$	−6.98076	−0.242745	−0.121372	+	0.992607i	$0.538730\pi$
$828$	0	0
$829$	−21.3923	−0.742985	−0.371493	−	0.928436i	$-0.621154\pi$
$829$	−21.3923	−0.742985	−0.371493	+	0.928436i	$0.621154\pi$
$830$	0	0
$831$	0	0
$832$	−8.00000	−0.277350
$833$	42.9282	1.48737
$834$	0	0
$835$	52.6410	1.82172
$836$	7.60770	0.263118
$837$	0	0
$838$	23.2154	0.801962
$839$	−55.0333	−1.89996	−0.949981	−	0.312309i	$-0.898898\pi$
$839$	−55.0333	−1.89996	−0.949981	+	0.312309i	$0.898898\pi$
$840$	0	0
$841$	−6.60770	−0.227852
$842$	−5.01924	−0.172974
$843$	0	0
$844$	39.0333	1.34358
$845$	−2.53590	−0.0872376
$846$	0	0
$847$	−0.267949	−0.00920684
$848$	−1.35898	−0.0466677
$849$	0	0
$850$	−11.1769	−0.383365
$851$	−3.71281	−0.127274
$852$	0	0
$853$	15.7321	0.538655	0.269328	−	0.963049i	$-0.413199\pi$
$853$	15.7321	0.538655	0.269328	+	0.963049i	$0.413199\pi$
$854$	1.51666	0.0518991
$855$	0	0
$856$	−29.5692	−1.01066
$857$	−34.1962	−1.16812	−0.584059	−	0.811711i	$-0.698536\pi$
$857$	−34.1962	−1.16812	−0.584059	+	0.811711i	$0.698536\pi$
$858$	0	0
$859$	1.14359	0.0390189	0.0195095	−	0.999810i	$-0.493790\pi$
$859$	1.14359	0.0390189	0.0195095	+	0.999810i	$0.493790\pi$
$860$	−13.8564	−0.472500
$861$	0	0
$862$	−15.7128	−0.535181
$863$	−47.9090	−1.63084	−0.815420	−	0.578870i	$-0.803494\pi$
$863$	−47.9090	−1.63084	−0.815420	+	0.578870i	$0.803494\pi$
$864$	0	0
$865$	44.7846	1.52272
$866$	−17.8564	−0.606785
$867$	0	0
$868$	2.92820	0.0993897
$869$	2.26795	0.0769349
$870$	0	0
$871$	29.5885	1.00257
$872$	40.2102	1.36169
$873$	0	0
$874$	−30.4308	−1.02934
$875$	−1.85641	−0.0627580
$876$	0	0
$877$	−0.803848	−0.0271440	−0.0135720	−	0.999908i	$-0.504320\pi$
$877$	−0.803848	−0.0271440	−0.0135720	+	0.999908i	$0.504320\pi$
$878$	27.4256	0.925570
$879$	0	0
$880$	2.92820	0.0987097
$881$	16.6795	0.561946	0.280973	−	0.959716i	$-0.409343\pi$
$881$	16.6795	0.561946	0.280973	+	0.959716i	$0.409343\pi$
$882$	0	0
$883$	−40.3205	−1.35689	−0.678447	−	0.734650i	$-0.737347\pi$
$883$	−40.3205	−1.35689	−0.678447	+	0.734650i	$0.737347\pi$
$884$	−33.8564	−1.13871
$885$	0	0
$886$	4.92820	0.165566
$887$	−3.26795	−0.109727	−0.0548635	−	0.998494i	$-0.517472\pi$
$887$	−3.26795	−0.109727	−0.0548635	+	0.998494i	$0.517472\pi$
$888$	0	0
$889$	0.392305	0.0131575
$890$	−35.3205	−1.18395
$891$	0	0
$892$	14.4308	0.483178
$893$	1.01924	0.0341075
$894$	0	0
$895$	−10.3923	−0.347376
$896$	2.71797	0.0908009
$897$	0	0
$898$	−28.0000	−0.934372
$899$	35.3205	1.17800
$900$	0	0
$901$	7.85641	0.261735
$902$	−4.00000	−0.133185
$903$	0	0
$904$	14.3538	0.477401
$905$	−50.0526	−1.66380
$906$	0	0
$907$	5.53590	0.183816	0.0919082	−	0.995767i	$-0.470703\pi$
$907$	5.53590	0.183816	0.0919082	+	0.995767i	$0.470703\pi$
$908$	−18.0666	−0.599563
$909$	0	0
$910$	−2.00000	−0.0662994
$911$	2.53590	0.0840181	0.0420090	−	0.999117i	$-0.486624\pi$
$911$	2.53590	0.0840181	0.0420090	+	0.999117i	$0.486624\pi$
$912$	0	0
$913$	0	0
$914$	−18.2487	−0.603614
$915$	0	0
$916$	−6.43078	−0.212479
$917$	2.44486	0.0807365
$918$	0	0
$919$	−8.67949	−0.286310	−0.143155	−	0.989700i	$-0.545725\pi$
$919$	−8.67949	−0.286310	−0.143155	+	0.989700i	$0.545725\pi$
$920$	−55.4256	−1.82733
$921$	0	0
$922$	9.03332	0.297497
$923$	51.7128	1.70215
$924$	0	0
$925$	1.14359	0.0376011
$926$	11.5551	0.379725
$927$	0	0
$928$	27.7128	0.909718
$929$	−46.7321	−1.53323	−0.766614	−	0.642108i	$-0.778060\pi$
$929$	−46.7321	−1.53323	−0.766614	+	0.642108i	$0.778060\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	−34.6410	−1.13470
$933$	0	0
$934$	3.71281	0.121487
$935$	−16.9282	−0.553611
$936$	0	0
$937$	−34.8038	−1.13699	−0.568496	−	0.822686i	$-0.692475\pi$
$937$	−34.8038	−1.13699	−0.568496	+	0.822686i	$0.692475\pi$
$938$	1.55514	0.0507770
$939$	0	0
$940$	0.784610	0.0255911
$941$	−33.4641	−1.09090	−0.545449	−	0.838144i	$-0.683641\pi$
$941$	−33.4641	−1.09090	−0.545449	+	0.838144i	$0.683641\pi$
$942$	0	0
$943$	−43.7128	−1.42349
$944$	0.210236	0.00684258
$945$	0	0
$946$	2.53590	0.0824492
$947$	6.33975	0.206014	0.103007	−	0.994681i	$-0.467154\pi$
$947$	6.33975	0.206014	0.103007	+	0.994681i	$0.467154\pi$
$948$	0	0
$949$	26.8564	0.871796
$950$	9.37307	0.304102
$951$	0	0
$952$	−4.21024	−0.136455
$953$	33.8564	1.09672	0.548358	−	0.836243i	$-0.315253\pi$
$953$	33.8564	1.09672	0.548358	+	0.836243i	$0.315253\pi$
$954$	0	0
$955$	−10.9282	−0.353628
$956$	−6.43078	−0.207986
$957$	0	0
$958$	−24.5359	−0.792719
$959$	3.32051	0.107225
$960$	0	0
$961$	24.7128	0.797188
$962$	−1.26795	−0.0408803
$963$	0	0
$964$	−7.03332	−0.226528
$965$	36.0526	1.16057
$966$	0	0
$967$	61.5885	1.98055	0.990276	−	0.139119i	$-0.0444271\pi$
$967$	61.5885	1.98055	0.990276	+	0.139119i	$0.0444271\pi$
$968$	−2.53590	−0.0815069
$969$	0	0
$970$	−30.7846	−0.988434
$971$	28.5885	0.917447	0.458724	−	0.888579i	$-0.348307\pi$
$971$	28.5885	0.917447	0.458724	+	0.888579i	$0.348307\pi$
$972$	0	0
$973$	−0.856406	−0.0274551
$974$	−6.48334	−0.207740
$975$	0	0
$976$	−8.28719	−0.265266
$977$	18.3397	0.586740	0.293370	−	0.955999i	$-0.405223\pi$
$977$	18.3397	0.586740	0.293370	+	0.955999i	$0.405223\pi$
$978$	0	0
$979$	−17.6603	−0.564424
$980$	−27.7128	−0.885253
$981$	0	0
$982$	−5.89488	−0.188113
$983$	−22.0526	−0.703367	−0.351684	−	0.936119i	$-0.614391\pi$
$983$	−22.0526	−0.703367	−0.351684	+	0.936119i	$0.614391\pi$
$984$	0	0
$985$	−20.9282	−0.666828
$986$	−21.4641	−0.683556
$987$	0	0
$988$	28.3923	0.903280
$989$	27.7128	0.881216
$990$	0	0
$991$	−19.5359	−0.620578	−0.310289	−	0.950642i	$-0.600426\pi$
$991$	−19.5359	−0.620578	−0.310289	+	0.950642i	$0.600426\pi$
$992$	43.7128	1.38788
$993$	0	0
$994$	2.71797	0.0862087
$995$	40.9808	1.29918
$996$	0	0
$997$	27.8564	0.882221	0.441111	−	0.897453i	$-0.354585\pi$
$997$	27.8564	0.882221	0.441111	+	0.897453i	$0.354585\pi$
$998$	−9.17691	−0.290490
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	297.2.a.e.1.2	✓	2
3.2	odd	2		297.2.a.f.1.1	yes	2
4.3	odd	2		4752.2.a.w.1.1		2
5.4	even	2		7425.2.a.bl.1.1		2
9.2	odd	6		891.2.e.m.595.2		4
9.4	even	3		891.2.e.p.298.1		4
9.5	odd	6		891.2.e.m.298.2		4
9.7	even	3		891.2.e.p.595.1		4
11.10	odd	2		3267.2.a.q.1.1		2
12.11	even	2		4752.2.a.bf.1.2		2
15.14	odd	2		7425.2.a.z.1.2		2
33.32	even	2		3267.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.a.e.1.2	✓	2	1.1	even	1	trivial
297.2.a.f.1.1	yes	2	3.2	odd	2
891.2.e.m.298.2		4	9.5	odd	6
891.2.e.m.595.2		4	9.2	odd	6
891.2.e.p.298.1		4	9.4	even	3
891.2.e.p.595.1		4	9.7	even	3
3267.2.a.l.1.2		2	33.32	even	2
3267.2.a.q.1.1		2	11.10	odd	2
4752.2.a.w.1.1		2	4.3	odd	2
4752.2.a.bf.1.2		2	12.11	even	2
7425.2.a.z.1.2		2	15.14	odd	2
7425.2.a.bl.1.1		2	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{2} -1.46410 q^{4} -2.73205 q^{5} -0.267949 q^{7} -2.53590 q^{8} +O(q^{10})\)	\(q+0.732051 q^{2} -1.46410 q^{4} -2.73205 q^{5} -0.267949 q^{7} -2.53590 q^{8} -2.00000 q^{10} -1.00000 q^{11} -3.73205 q^{13} -0.196152 q^{14} +1.07180 q^{16} -6.19615 q^{17} +5.19615 q^{19} +4.00000 q^{20} -0.732051 q^{22} -8.00000 q^{23} +2.46410 q^{25} -2.73205 q^{26} +0.392305 q^{28} +4.73205 q^{29} +7.46410 q^{31} +5.85641 q^{32} -4.53590 q^{34} +0.732051 q^{35} +0.464102 q^{37} +3.80385 q^{38} +6.92820 q^{40} +5.46410 q^{41} -3.46410 q^{43} +1.46410 q^{44} -5.85641 q^{46} +0.196152 q^{47} -6.92820 q^{49} +1.80385 q^{50} +5.46410 q^{52} -1.26795 q^{53} +2.73205 q^{55} +0.679492 q^{56} +3.46410 q^{58} +0.196152 q^{59} -7.73205 q^{61} +5.46410 q^{62} +2.14359 q^{64} +10.1962 q^{65} -7.92820 q^{67} +9.07180 q^{68} +0.535898 q^{70} -13.8564 q^{71} -7.19615 q^{73} +0.339746 q^{74} -7.60770 q^{76} +0.267949 q^{77} -2.26795 q^{79} -2.92820 q^{80} +4.00000 q^{82} +16.9282 q^{85} -2.53590 q^{86} +2.53590 q^{88} +17.6603 q^{89} +1.00000 q^{91} +11.7128 q^{92} +0.143594 q^{94} -14.1962 q^{95} +15.3923 q^{97} -5.07180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8	\( 2 q - 2 q^{2} + 4 q^{4} - 2 q^{5} - 4 q^{7} - 12 q^{8} - 4 q^{10} - 2 q^{11} - 4 q^{13} + 10 q^{14} + 16 q^{16} - 2 q^{17} + 8 q^{20} + 2 q^{22} - 16 q^{23} - 2 q^{25} - 2 q^{26} - 20 q^{28} + 6 q^{29} + 8 q^{31} - 16 q^{32} - 16 q^{34} - 2 q^{35} - 6 q^{37} + 18 q^{38} + 4 q^{41} - 4 q^{44} + 16 q^{46} - 10 q^{47} + 14 q^{50} + 4 q^{52} - 6 q^{53} + 2 q^{55} + 36 q^{56} - 10 q^{59} - 12 q^{61} + 4 q^{62} + 32 q^{64} + 10 q^{65} - 2 q^{67} + 32 q^{68} + 8 q^{70} - 4 q^{73} + 18 q^{74} - 36 q^{76} + 4 q^{77} - 8 q^{79} + 8 q^{80} + 8 q^{82} + 20 q^{85} - 12 q^{86} + 12 q^{88} + 18 q^{89} + 2 q^{91} - 32 q^{92} + 28 q^{94} - 18 q^{95} + 10 q^{97} - 24 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^4 - 2 * q^5 - 4 * q^7 - 12 * q^8 - 4 * q^10 - 2 * q^11 - 4 * q^13 + 10 * q^14 + 16 * q^16 - 2 * q^17 + 8 * q^20 + 2 * q^22 - 16 * q^23 - 2 * q^25 - 2 * q^26 - 20 * q^28 + 6 * q^29 + 8 * q^31 - 16 * q^32 - 16 * q^34 - 2 * q^35 - 6 * q^37 + 18 * q^38 + 4 * q^41 - 4 * q^44 + 16 * q^46 - 10 * q^47 + 14 * q^50 + 4 * q^52 - 6 * q^53 + 2 * q^55 + 36 * q^56 - 10 * q^59 - 12 * q^61 + 4 * q^62 + 32 * q^64 + 10 * q^65 - 2 * q^67 + 32 * q^68 + 8 * q^70 - 4 * q^73 + 18 * q^74 - 36 * q^76 + 4 * q^77 - 8 * q^79 + 8 * q^80 + 8 * q^82 + 20 * q^85 - 12 * q^86 + 12 * q^88 + 18 * q^89 + 2 * q^91 - 32 * q^92 + 28 * q^94 - 18 * q^95 + 10 * q^97 - 24 * q^98

Introduction

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