LMFDB - Embedded newform 272.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,2,Mod(1,272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("272.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	272.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:	$2.17193093498$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 68)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	272.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -3.46410 q^{5} +2.73205 q^{7} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} -3.46410 q^{5} +2.73205 q^{7} +4.46410 q^{9} +1.26795 q^{11} +5.46410 q^{13} +9.46410 q^{15} -1.00000 q^{17} +1.46410 q^{19} -7.46410 q^{21} +1.26795 q^{23} +7.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} -4.19615 q^{31} -3.46410 q^{33} -9.46410 q^{35} +4.53590 q^{37} -14.9282 q^{39} -6.00000 q^{41} +8.39230 q^{43} -15.4641 q^{45} +6.92820 q^{47} +0.464102 q^{49} +2.73205 q^{51} +12.9282 q^{53} -4.39230 q^{55} -4.00000 q^{57} -2.53590 q^{59} -0.535898 q^{61} +12.1962 q^{63} -18.9282 q^{65} -14.9282 q^{67} -3.46410 q^{69} +8.19615 q^{71} +2.00000 q^{73} -19.1244 q^{75} +3.46410 q^{77} +12.1962 q^{79} -2.46410 q^{81} +2.53590 q^{83} +3.46410 q^{85} -9.46410 q^{87} +2.53590 q^{89} +14.9282 q^{91} +11.4641 q^{93} -5.07180 q^{95} -4.92820 q^{97} +5.66025 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} + 6 q^{11} + 4 q^{13} + 12 q^{15} - 2 q^{17} - 4 q^{19} - 8 q^{21} + 6 q^{23} + 14 q^{25} - 8 q^{27} + 2 q^{31} - 12 q^{35} + 16 q^{37} - 16 q^{39} - 12 q^{41} - 4 q^{43} - 24 q^{45} - 6 q^{49} + 2 q^{51} + 12 q^{53} + 12 q^{55} - 8 q^{57} - 12 q^{59} - 8 q^{61} + 14 q^{63} - 24 q^{65} - 16 q^{67} + 6 q^{71} + 4 q^{73} - 14 q^{75} + 14 q^{79} + 2 q^{81} + 12 q^{83} - 12 q^{87} + 12 q^{89} + 16 q^{91} + 16 q^{93} - 24 q^{95} + 4 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 + 6 * q^11 + 4 * q^13 + 12 * q^15 - 2 * q^17 - 4 * q^19 - 8 * q^21 + 6 * q^23 + 14 * q^25 - 8 * q^27 + 2 * q^31 - 12 * q^35 + 16 * q^37 - 16 * q^39 - 12 * q^41 - 4 * q^43 - 24 * q^45 - 6 * q^49 + 2 * q^51 + 12 * q^53 + 12 * q^55 - 8 * q^57 - 12 * q^59 - 8 * q^61 + 14 * q^63 - 24 * q^65 - 16 * q^67 + 6 * q^71 + 4 * q^73 - 14 * q^75 + 14 * q^79 + 2 * q^81 + 12 * q^83 - 12 * q^87 + 12 * q^89 + 16 * q^91 + 16 * q^93 - 24 * q^95 + 4 * q^97 - 6 * q^99

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	0
$2$	0	0
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	0	0
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	2.73205	1.03262	0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205	1.03262	0.516309	+	0.856402i	$0.327306\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	0	0
$15$	9.46410	2.44362
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	1.46410	0.335888	0.167944	−	0.985797i	$-0.446287\pi$
$19$	1.46410	0.335888	0.167944	+	0.985797i	$0.446287\pi$
$20$	0	0
$21$	−7.46410	−1.62880
$22$	0	0
$23$	1.26795	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$23$	1.26795	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	−4.19615	−0.753651	−0.376826	−	0.926284i	$-0.622984\pi$
$31$	−4.19615	−0.753651	−0.376826	+	0.926284i	$0.622984\pi$
$32$	0	0
$33$	−3.46410	−0.603023
$34$	0	0
$35$	−9.46410	−1.59973
$36$	0	0
$37$	4.53590	0.745697	0.372849	−	0.927892i	$-0.378381\pi$
$37$	4.53590	0.745697	0.372849	+	0.927892i	$0.378381\pi$
$38$	0	0
$39$	−14.9282	−2.39043
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	8.39230	1.27981	0.639907	−	0.768452i	$-0.278973\pi$
$43$	8.39230	1.27981	0.639907	+	0.768452i	$0.278973\pi$
$44$	0	0
$45$	−15.4641	−2.30525
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	2.73205	0.382564
$52$	0	0
$53$	12.9282	1.77583	0.887913	−	0.460012i	$-0.152155\pi$
$53$	12.9282	1.77583	0.887913	+	0.460012i	$0.152155\pi$
$54$	0	0
$55$	−4.39230	−0.592258
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	0	0
$61$	−0.535898	−0.0686148	−0.0343074	−	0.999411i	$-0.510923\pi$
$61$	−0.535898	−0.0686148	−0.0343074	+	0.999411i	$0.510923\pi$
$62$	0	0
$63$	12.1962	1.53657
$64$	0	0
$65$	−18.9282	−2.34775
$66$	0	0
$67$	−14.9282	−1.82377	−0.911885	−	0.410445i	$-0.865373\pi$
$67$	−14.9282	−1.82377	−0.911885	+	0.410445i	$0.865373\pi$
$68$	0	0
$69$	−3.46410	−0.417029
$70$	0	0
$71$	8.19615	0.972704	0.486352	−	0.873763i	$-0.338327\pi$
$71$	8.19615	0.972704	0.486352	+	0.873763i	$0.338327\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−19.1244	−2.20829
$76$	0	0
$77$	3.46410	0.394771
$78$	0	0
$79$	12.1962	1.37217	0.686087	−	0.727519i	$-0.259327\pi$
$79$	12.1962	1.37217	0.686087	+	0.727519i	$0.259327\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	2.53590	0.278351	0.139176	−	0.990268i	$-0.455555\pi$
$83$	2.53590	0.278351	0.139176	+	0.990268i	$0.455555\pi$
$84$	0	0
$85$	3.46410	0.375735
$86$	0	0
$87$	−9.46410	−1.01466
$88$	0	0
$89$	2.53590	0.268805	0.134402	−	0.990927i	$-0.457089\pi$
$89$	2.53590	0.268805	0.134402	+	0.990927i	$0.457089\pi$
$90$	0	0
$91$	14.9282	1.56490
$92$	0	0
$93$	11.4641	1.18877
$94$	0	0
$95$	−5.07180	−0.520355
$96$	0	0
$97$	−4.92820	−0.500383	−0.250192	−	0.968196i	$-0.580494\pi$
$97$	−4.92820	−0.500383	−0.250192	+	0.968196i	$0.580494\pi$
$98$	0	0
$99$	5.66025	0.568877
$100$	0	0
$101$	−2.53590	−0.252331	−0.126166	−	0.992009i	$-0.540267\pi$
$101$	−2.53590	−0.252331	−0.126166	+	0.992009i	$0.540267\pi$
$102$	0	0
$103$	10.9282	1.07679	0.538394	−	0.842693i	$-0.319031\pi$
$103$	10.9282	1.07679	0.538394	+	0.842693i	$0.319031\pi$
$104$	0	0
$105$	25.8564	2.52333
$106$	0	0
$107$	−10.7321	−1.03751	−0.518753	−	0.854924i	$-0.673604\pi$
$107$	−10.7321	−1.03751	−0.518753	+	0.854924i	$0.673604\pi$
$108$	0	0
$109$	−14.3923	−1.37853	−0.689266	−	0.724508i	$-0.742067\pi$
$109$	−14.3923	−1.37853	−0.689266	+	0.724508i	$0.742067\pi$
$110$	0	0
$111$	−12.3923	−1.17623
$112$	0	0
$113$	19.8564	1.86793	0.933967	−	0.357360i	$-0.116323\pi$
$113$	19.8564	1.86793	0.933967	+	0.357360i	$0.116323\pi$
$114$	0	0
$115$	−4.39230	−0.409585
$116$	0	0
$117$	24.3923	2.25507
$118$	0	0
$119$	−2.73205	−0.250447
$120$	0	0
$121$	−9.39230	−0.853846
$122$	0	0
$123$	16.3923	1.47804
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−0.392305	−0.0348114	−0.0174057	−	0.999849i	$-0.505541\pi$
$127$	−0.392305	−0.0348114	−0.0174057	+	0.999849i	$0.505541\pi$
$128$	0	0
$129$	−22.9282	−2.01872
$130$	0	0
$131$	−5.66025	−0.494539	−0.247269	−	0.968947i	$-0.579533\pi$
$131$	−5.66025	−0.494539	−0.247269	+	0.968947i	$0.579533\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	13.8564	1.19257
$136$	0	0
$137$	2.53590	0.216656	0.108328	−	0.994115i	$-0.465450\pi$
$137$	2.53590	0.216656	0.108328	+	0.994115i	$0.465450\pi$
$138$	0	0
$139$	14.7321	1.24956	0.624778	−	0.780802i	$-0.285189\pi$
$139$	14.7321	1.24956	0.624778	+	0.780802i	$0.285189\pi$
$140$	0	0
$141$	−18.9282	−1.59404
$142$	0	0
$143$	6.92820	0.579365
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	−1.26795	−0.104579
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	13.4641	1.09569	0.547847	−	0.836579i	$-0.315448\pi$
$151$	13.4641	1.09569	0.547847	+	0.836579i	$0.315448\pi$
$152$	0	0
$153$	−4.46410	−0.360901
$154$	0	0
$155$	14.5359	1.16755
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	−35.3205	−2.80110
$160$	0	0
$161$	3.46410	0.273009
$162$	0	0
$163$	−20.5885	−1.61261	−0.806306	−	0.591498i	$-0.798537\pi$
$163$	−20.5885	−1.61261	−0.806306	+	0.591498i	$0.798537\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	−5.66025	−0.438004	−0.219002	−	0.975724i	$-0.570280\pi$
$167$	−5.66025	−0.438004	−0.219002	+	0.975724i	$0.570280\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	6.53590	0.499813
$172$	0	0
$173$	−15.4641	−1.17571	−0.587857	−	0.808965i	$-0.700028\pi$
$173$	−15.4641	−1.17571	−0.587857	+	0.808965i	$0.700028\pi$
$174$	0	0
$175$	19.1244	1.44567
$176$	0	0
$177$	6.92820	0.520756
$178$	0	0
$179$	−14.5359	−1.08646	−0.543232	−	0.839583i	$-0.682800\pi$
$179$	−14.5359	−1.08646	−0.543232	+	0.839583i	$0.682800\pi$
$180$	0	0
$181$	4.53590	0.337151	0.168575	−	0.985689i	$-0.446083\pi$
$181$	4.53590	0.337151	0.168575	+	0.985689i	$0.446083\pi$
$182$	0	0
$183$	1.46410	0.108230
$184$	0	0
$185$	−15.7128	−1.15523
$186$	0	0
$187$	−1.26795	−0.0927216
$188$	0	0
$189$	−10.9282	−0.794910
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	20.9282	1.50645	0.753223	−	0.657766i	$-0.228498\pi$
$193$	20.9282	1.50645	0.753223	+	0.657766i	$0.228498\pi$
$194$	0	0
$195$	51.7128	3.70323
$196$	0	0
$197$	3.46410	0.246807	0.123404	−	0.992357i	$-0.460619\pi$
$197$	3.46410	0.246807	0.123404	+	0.992357i	$0.460619\pi$
$198$	0	0
$199$	−4.19615	−0.297457	−0.148729	−	0.988878i	$-0.547518\pi$
$199$	−4.19615	−0.297457	−0.148729	+	0.988878i	$0.547518\pi$
$200$	0	0
$201$	40.7846	2.87672
$202$	0	0
$203$	9.46410	0.664250
$204$	0	0
$205$	20.7846	1.45166
$206$	0	0
$207$	5.66025	0.393415
$208$	0	0
$209$	1.85641	0.128410
$210$	0	0
$211$	−2.33975	−0.161075	−0.0805374	−	0.996752i	$-0.525664\pi$
$211$	−2.33975	−0.161075	−0.0805374	+	0.996752i	$0.525664\pi$
$212$	0	0
$213$	−22.3923	−1.53430
$214$	0	0
$215$	−29.0718	−1.98268
$216$	0	0
$217$	−11.4641	−0.778234
$218$	0	0
$219$	−5.46410	−0.369230
$220$	0	0
$221$	−5.46410	−0.367555
$222$	0	0
$223$	−12.3923	−0.829850	−0.414925	−	0.909856i	$-0.636192\pi$
$223$	−12.3923	−0.829850	−0.414925	+	0.909856i	$0.636192\pi$
$224$	0	0
$225$	31.2487	2.08325
$226$	0	0
$227$	20.1962	1.34047	0.670233	−	0.742151i	$-0.266194\pi$
$227$	20.1962	1.34047	0.670233	+	0.742151i	$0.266194\pi$
$228$	0	0
$229$	−20.3923	−1.34756	−0.673781	−	0.738931i	$-0.735331\pi$
$229$	−20.3923	−1.34756	−0.673781	+	0.738931i	$0.735331\pi$
$230$	0	0
$231$	−9.46410	−0.622692
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	0	0
$237$	−33.3205	−2.16440
$238$	0	0
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	−1.60770	−0.102712
$246$	0	0
$247$	8.00000	0.509028
$248$	0	0
$249$	−6.92820	−0.439057
$250$	0	0
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	0	0
$253$	1.60770	0.101075
$254$	0	0
$255$	−9.46410	−0.592665
$256$	0	0
$257$	−28.3923	−1.77106	−0.885532	−	0.464579i	$-0.846206\pi$
$257$	−28.3923	−1.77106	−0.885532	+	0.464579i	$0.846206\pi$
$258$	0	0
$259$	12.3923	0.770020
$260$	0	0
$261$	15.4641	0.957204
$262$	0	0
$263$	−11.3205	−0.698052	−0.349026	−	0.937113i	$-0.613488\pi$
$263$	−11.3205	−0.698052	−0.349026	+	0.937113i	$0.613488\pi$
$264$	0	0
$265$	−44.7846	−2.75110
$266$	0	0
$267$	−6.92820	−0.423999
$268$	0	0
$269$	−27.4641	−1.67452	−0.837258	−	0.546808i	$-0.815843\pi$
$269$	−27.4641	−1.67452	−0.837258	+	0.546808i	$0.815843\pi$
$270$	0	0
$271$	−1.07180	−0.0651070	−0.0325535	−	0.999470i	$-0.510364\pi$
$271$	−1.07180	−0.0651070	−0.0325535	+	0.999470i	$0.510364\pi$
$272$	0	0
$273$	−40.7846	−2.46840
$274$	0	0
$275$	8.87564	0.535221
$276$	0	0
$277$	6.39230	0.384076	0.192038	−	0.981387i	$-0.438490\pi$
$277$	6.39230	0.384076	0.192038	+	0.981387i	$0.438490\pi$
$278$	0	0
$279$	−18.7321	−1.12146
$280$	0	0
$281$	−19.8564	−1.18453	−0.592267	−	0.805742i	$-0.701767\pi$
$281$	−19.8564	−1.18453	−0.592267	+	0.805742i	$0.701767\pi$
$282$	0	0
$283$	−20.5885	−1.22386	−0.611928	−	0.790913i	$-0.709606\pi$
$283$	−20.5885	−1.22386	−0.611928	+	0.790913i	$0.709606\pi$
$284$	0	0
$285$	13.8564	0.820783
$286$	0	0
$287$	−16.3923	−0.967607
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	13.4641	0.789280
$292$	0	0
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	8.78461	0.511460
$296$	0	0
$297$	−5.07180	−0.294295
$298$	0	0
$299$	6.92820	0.400668
$300$	0	0
$301$	22.9282	1.32156
$302$	0	0
$303$	6.92820	0.398015
$304$	0	0
$305$	1.85641	0.106298
$306$	0	0
$307$	24.7846	1.41453	0.707266	−	0.706947i	$-0.249928\pi$
$307$	24.7846	1.41453	0.707266	+	0.706947i	$0.249928\pi$
$308$	0	0
$309$	−29.8564	−1.69847
$310$	0	0
$311$	3.12436	0.177166	0.0885830	−	0.996069i	$-0.471766\pi$
$311$	3.12436	0.177166	0.0885830	+	0.996069i	$0.471766\pi$
$312$	0	0
$313$	20.9282	1.18293	0.591466	−	0.806330i	$-0.298549\pi$
$313$	20.9282	1.18293	0.591466	+	0.806330i	$0.298549\pi$
$314$	0	0
$315$	−42.2487	−2.38045
$316$	0	0
$317$	8.53590	0.479424	0.239712	−	0.970844i	$-0.422947\pi$
$317$	8.53590	0.479424	0.239712	+	0.970844i	$0.422947\pi$
$318$	0	0
$319$	4.39230	0.245922
$320$	0	0
$321$	29.3205	1.63651
$322$	0	0
$323$	−1.46410	−0.0814648
$324$	0	0
$325$	38.2487	2.12166
$326$	0	0
$327$	39.3205	2.17443
$328$	0	0
$329$	18.9282	1.04355
$330$	0	0
$331$	−19.3205	−1.06195	−0.530976	−	0.847387i	$-0.678174\pi$
$331$	−19.3205	−1.06195	−0.530976	+	0.847387i	$0.678174\pi$
$332$	0	0
$333$	20.2487	1.10962
$334$	0	0
$335$	51.7128	2.82537
$336$	0	0
$337$	−6.78461	−0.369581	−0.184791	−	0.982778i	$-0.559161\pi$
$337$	−6.78461	−0.369581	−0.184791	+	0.982778i	$0.559161\pi$
$338$	0	0
$339$	−54.2487	−2.94639
$340$	0	0
$341$	−5.32051	−0.288122
$342$	0	0
$343$	−17.8564	−0.964155
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	34.0526	1.82804	0.914019	−	0.405672i	$-0.132963\pi$
$347$	34.0526	1.82804	0.914019	+	0.405672i	$0.132963\pi$
$348$	0	0
$349$	10.7846	0.577287	0.288643	−	0.957437i	$-0.406796\pi$
$349$	10.7846	0.577287	0.288643	+	0.957437i	$0.406796\pi$
$350$	0	0
$351$	−21.8564	−1.16661
$352$	0	0
$353$	−19.8564	−1.05685	−0.528425	−	0.848980i	$-0.677217\pi$
$353$	−19.8564	−1.05685	−0.528425	+	0.848980i	$0.677217\pi$
$354$	0	0
$355$	−28.3923	−1.50691
$356$	0	0
$357$	7.46410	0.395042
$358$	0	0
$359$	23.3205	1.23081	0.615405	−	0.788211i	$-0.288993\pi$
$359$	23.3205	1.23081	0.615405	+	0.788211i	$0.288993\pi$
$360$	0	0
$361$	−16.8564	−0.887179
$362$	0	0
$363$	25.6603	1.34681
$364$	0	0
$365$	−6.92820	−0.362639
$366$	0	0
$367$	−2.33975	−0.122134	−0.0610669	−	0.998134i	$-0.519450\pi$
$367$	−2.33975	−0.122134	−0.0610669	+	0.998134i	$0.519450\pi$
$368$	0	0
$369$	−26.7846	−1.39435
$370$	0	0
$371$	35.3205	1.83375
$372$	0	0
$373$	−8.39230	−0.434537	−0.217269	−	0.976112i	$-0.569715\pi$
$373$	−8.39230	−0.434537	−0.217269	+	0.976112i	$0.569715\pi$
$374$	0	0
$375$	18.9282	0.977448
$376$	0	0
$377$	18.9282	0.974852
$378$	0	0
$379$	5.26795	0.270596	0.135298	−	0.990805i	$-0.456801\pi$
$379$	5.26795	0.270596	0.135298	+	0.990805i	$0.456801\pi$
$380$	0	0
$381$	1.07180	0.0549098
$382$	0	0
$383$	2.53590	0.129578	0.0647892	−	0.997899i	$-0.479363\pi$
$383$	2.53590	0.129578	0.0647892	+	0.997899i	$0.479363\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	37.4641	1.90441
$388$	0	0
$389$	37.1769	1.88494	0.942472	−	0.334285i	$-0.108495\pi$
$389$	37.1769	1.88494	0.942472	+	0.334285i	$0.108495\pi$
$390$	0	0
$391$	−1.26795	−0.0641229
$392$	0	0
$393$	15.4641	0.780061
$394$	0	0
$395$	−42.2487	−2.12576
$396$	0	0
$397$	−5.60770	−0.281442	−0.140721	−	0.990049i	$-0.544942\pi$
$397$	−5.60770	−0.281442	−0.140721	+	0.990049i	$0.544942\pi$
$398$	0	0
$399$	−10.9282	−0.547094
$400$	0	0
$401$	31.8564	1.59083	0.795417	−	0.606063i	$-0.207252\pi$
$401$	31.8564	1.59083	0.795417	+	0.606063i	$0.207252\pi$
$402$	0	0
$403$	−22.9282	−1.14214
$404$	0	0
$405$	8.53590	0.424152
$406$	0	0
$407$	5.75129	0.285081
$408$	0	0
$409$	17.7128	0.875842	0.437921	−	0.899013i	$-0.355715\pi$
$409$	17.7128	0.875842	0.437921	+	0.899013i	$0.355715\pi$
$410$	0	0
$411$	−6.92820	−0.341743
$412$	0	0
$413$	−6.92820	−0.340915
$414$	0	0
$415$	−8.78461	−0.431220
$416$	0	0
$417$	−40.2487	−1.97099
$418$	0	0
$419$	24.5885	1.20122	0.600612	−	0.799540i	$-0.294924\pi$
$419$	24.5885	1.20122	0.600612	+	0.799540i	$0.294924\pi$
$420$	0	0
$421$	−22.2487	−1.08434	−0.542168	−	0.840270i	$-0.682396\pi$
$421$	−22.2487	−1.08434	−0.542168	+	0.840270i	$0.682396\pi$
$422$	0	0
$423$	30.9282	1.50378
$424$	0	0
$425$	−7.00000	−0.339550
$426$	0	0
$427$	−1.46410	−0.0708528
$428$	0	0
$429$	−18.9282	−0.913862
$430$	0	0
$431$	−17.6603	−0.850665	−0.425332	−	0.905037i	$-0.639843\pi$
$431$	−17.6603	−0.850665	−0.425332	+	0.905037i	$0.639843\pi$
$432$	0	0
$433$	3.60770	0.173375	0.0866874	−	0.996236i	$-0.472372\pi$
$433$	3.60770	0.173375	0.0866874	+	0.996236i	$0.472372\pi$
$434$	0	0
$435$	32.7846	1.57190
$436$	0	0
$437$	1.85641	0.0888040
$438$	0	0
$439$	24.1962	1.15482	0.577410	−	0.816455i	$-0.304064\pi$
$439$	24.1962	1.15482	0.577410	+	0.816455i	$0.304064\pi$
$440$	0	0
$441$	2.07180	0.0986570
$442$	0	0
$443$	20.7846	0.987507	0.493753	−	0.869602i	$-0.335625\pi$
$443$	20.7846	0.987507	0.493753	+	0.869602i	$0.335625\pi$
$444$	0	0
$445$	−8.78461	−0.416430
$446$	0	0
$447$	−16.3923	−0.775329
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−7.60770	−0.358232
$452$	0	0
$453$	−36.7846	−1.72829
$454$	0	0
$455$	−51.7128	−2.42433
$456$	0	0
$457$	−20.3923	−0.953912	−0.476956	−	0.878927i	$-0.658260\pi$
$457$	−20.3923	−0.953912	−0.476956	+	0.878927i	$0.658260\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	26.7846	1.24748	0.623742	−	0.781630i	$-0.285612\pi$
$461$	26.7846	1.24748	0.623742	+	0.781630i	$0.285612\pi$
$462$	0	0
$463$	2.14359	0.0996212	0.0498106	−	0.998759i	$-0.484138\pi$
$463$	2.14359	0.0996212	0.0498106	+	0.998759i	$0.484138\pi$
$464$	0	0
$465$	−39.7128	−1.84164
$466$	0	0
$467$	−14.5359	−0.672641	−0.336321	−	0.941748i	$-0.609183\pi$
$467$	−14.5359	−0.672641	−0.336321	+	0.941748i	$0.609183\pi$
$468$	0	0
$469$	−40.7846	−1.88326
$470$	0	0
$471$	−5.46410	−0.251773
$472$	0	0
$473$	10.6410	0.489274
$474$	0	0
$475$	10.2487	0.470243
$476$	0	0
$477$	57.7128	2.64249
$478$	0	0
$479$	17.6603	0.806918	0.403459	−	0.914998i	$-0.367808\pi$
$479$	17.6603	0.806918	0.403459	+	0.914998i	$0.367808\pi$
$480$	0	0
$481$	24.7846	1.13008
$482$	0	0
$483$	−9.46410	−0.430632
$484$	0	0
$485$	17.0718	0.775190
$486$	0	0
$487$	−6.05256	−0.274268	−0.137134	−	0.990553i	$-0.543789\pi$
$487$	−6.05256	−0.274268	−0.137134	+	0.990553i	$0.543789\pi$
$488$	0	0
$489$	56.2487	2.54365
$490$	0	0
$491$	37.1769	1.67777	0.838885	−	0.544308i	$-0.183208\pi$
$491$	37.1769	1.67777	0.838885	+	0.544308i	$0.183208\pi$
$492$	0	0
$493$	−3.46410	−0.156015
$494$	0	0
$495$	−19.6077	−0.881300
$496$	0	0
$497$	22.3923	1.00443
$498$	0	0
$499$	−24.9808	−1.11829	−0.559146	−	0.829069i	$-0.688871\pi$
$499$	−24.9808	−1.11829	−0.559146	+	0.829069i	$0.688871\pi$
$500$	0	0
$501$	15.4641	0.690885
$502$	0	0
$503$	−27.1244	−1.20942	−0.604708	−	0.796448i	$-0.706710\pi$
$503$	−27.1244	−1.20942	−0.604708	+	0.796448i	$0.706710\pi$
$504$	0	0
$505$	8.78461	0.390910
$506$	0	0
$507$	−46.0526	−2.04527
$508$	0	0
$509$	−21.7128	−0.962404	−0.481202	−	0.876610i	$-0.659800\pi$
$509$	−21.7128	−0.962404	−0.481202	+	0.876610i	$0.659800\pi$
$510$	0	0
$511$	5.46410	0.241718
$512$	0	0
$513$	−5.85641	−0.258567
$514$	0	0
$515$	−37.8564	−1.66815
$516$	0	0
$517$	8.78461	0.386347
$518$	0	0
$519$	42.2487	1.85451
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	12.7846	0.559032	0.279516	−	0.960141i	$-0.409826\pi$
$523$	12.7846	0.559032	0.279516	+	0.960141i	$0.409826\pi$
$524$	0	0
$525$	−52.2487	−2.28032
$526$	0	0
$527$	4.19615	0.182787
$528$	0	0
$529$	−21.3923	−0.930100
$530$	0	0
$531$	−11.3205	−0.491268
$532$	0	0
$533$	−32.7846	−1.42006
$534$	0	0
$535$	37.1769	1.60730
$536$	0	0
$537$	39.7128	1.71373
$538$	0	0
$539$	0.588457	0.0253466
$540$	0	0
$541$	18.3923	0.790747	0.395373	−	0.918520i	$-0.370615\pi$
$541$	18.3923	0.790747	0.395373	+	0.918520i	$0.370615\pi$
$542$	0	0
$543$	−12.3923	−0.531805
$544$	0	0
$545$	49.8564	2.13561
$546$	0	0
$547$	−42.7321	−1.82709	−0.913545	−	0.406737i	$-0.866667\pi$
$547$	−42.7321	−1.82709	−0.913545	+	0.406737i	$0.866667\pi$
$548$	0	0
$549$	−2.39230	−0.102101
$550$	0	0
$551$	5.07180	0.216066
$552$	0	0
$553$	33.3205	1.41693
$554$	0	0
$555$	42.9282	1.82220
$556$	0	0
$557$	−26.5359	−1.12436	−0.562181	−	0.827014i	$-0.690038\pi$
$557$	−26.5359	−1.12436	−0.562181	+	0.827014i	$0.690038\pi$
$558$	0	0
$559$	45.8564	1.93952
$560$	0	0
$561$	3.46410	0.146254
$562$	0	0
$563$	35.3205	1.48858	0.744291	−	0.667855i	$-0.232788\pi$
$563$	35.3205	1.48858	0.744291	+	0.667855i	$0.232788\pi$
$564$	0	0
$565$	−68.7846	−2.89379
$566$	0	0
$567$	−6.73205	−0.282720
$568$	0	0
$569$	−7.85641	−0.329358	−0.164679	−	0.986347i	$-0.552659\pi$
$569$	−7.85641	−0.329358	−0.164679	+	0.986347i	$0.552659\pi$
$570$	0	0
$571$	−11.1244	−0.465540	−0.232770	−	0.972532i	$-0.574779\pi$
$571$	−11.1244	−0.465540	−0.232770	+	0.972532i	$0.574779\pi$
$572$	0	0
$573$	32.7846	1.36960
$574$	0	0
$575$	8.87564	0.370140
$576$	0	0
$577$	−13.4641	−0.560518	−0.280259	−	0.959924i	$-0.590420\pi$
$577$	−13.4641	−0.560518	−0.280259	+	0.959924i	$0.590420\pi$
$578$	0	0
$579$	−57.1769	−2.37619
$580$	0	0
$581$	6.92820	0.287430
$582$	0	0
$583$	16.3923	0.678900
$584$	0	0
$585$	−84.4974	−3.49354
$586$	0	0
$587$	−28.3923	−1.17188	−0.585938	−	0.810356i	$-0.699274\pi$
$587$	−28.3923	−1.17188	−0.585938	+	0.810356i	$0.699274\pi$
$588$	0	0
$589$	−6.14359	−0.253142
$590$	0	0
$591$	−9.46410	−0.389301
$592$	0	0
$593$	−4.14359	−0.170157	−0.0850785	−	0.996374i	$-0.527114\pi$
$593$	−4.14359	−0.170157	−0.0850785	+	0.996374i	$0.527114\pi$
$594$	0	0
$595$	9.46410	0.387990
$596$	0	0
$597$	11.4641	0.469194
$598$	0	0
$599$	27.7128	1.13231	0.566157	−	0.824297i	$-0.308429\pi$
$599$	27.7128	1.13231	0.566157	+	0.824297i	$0.308429\pi$
$600$	0	0
$601$	10.7846	0.439913	0.219957	−	0.975510i	$-0.429408\pi$
$601$	10.7846	0.439913	0.219957	+	0.975510i	$0.429408\pi$
$602$	0	0
$603$	−66.6410	−2.71383
$604$	0	0
$605$	32.5359	1.32277
$606$	0	0
$607$	38.7321	1.57209	0.786043	−	0.618172i	$-0.212127\pi$
$607$	38.7321	1.57209	0.786043	+	0.618172i	$0.212127\pi$
$608$	0	0
$609$	−25.8564	−1.04775
$610$	0	0
$611$	37.8564	1.53151
$612$	0	0
$613$	−47.8564	−1.93290	−0.966451	−	0.256851i	$-0.917315\pi$
$613$	−47.8564	−1.93290	−0.966451	+	0.256851i	$0.917315\pi$
$614$	0	0
$615$	−56.7846	−2.28978
$616$	0	0
$617$	−35.5692	−1.43196	−0.715981	−	0.698119i	$-0.754020\pi$
$617$	−35.5692	−1.43196	−0.715981	+	0.698119i	$0.754020\pi$
$618$	0	0
$619$	−11.8038	−0.474437	−0.237218	−	0.971456i	$-0.576236\pi$
$619$	−11.8038	−0.474437	−0.237218	+	0.971456i	$0.576236\pi$
$620$	0	0
$621$	−5.07180	−0.203524
$622$	0	0
$623$	6.92820	0.277573
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	−5.07180	−0.202548
$628$	0	0
$629$	−4.53590	−0.180858
$630$	0	0
$631$	−41.4641	−1.65066	−0.825330	−	0.564651i	$-0.809011\pi$
$631$	−41.4641	−1.65066	−0.825330	+	0.564651i	$0.809011\pi$
$632$	0	0
$633$	6.39230	0.254071
$634$	0	0
$635$	1.35898	0.0539296
$636$	0	0
$637$	2.53590	0.100476
$638$	0	0
$639$	36.5885	1.44742
$640$	0	0
$641$	−12.9282	−0.510633	−0.255317	−	0.966857i	$-0.582180\pi$
$641$	−12.9282	−0.510633	−0.255317	+	0.966857i	$0.582180\pi$
$642$	0	0
$643$	−32.5885	−1.28516	−0.642582	−	0.766217i	$-0.722137\pi$
$643$	−32.5885	−1.28516	−0.642582	+	0.766217i	$0.722137\pi$
$644$	0	0
$645$	79.4256	3.12738
$646$	0	0
$647$	−17.0718	−0.671162	−0.335581	−	0.942011i	$-0.608933\pi$
$647$	−17.0718	−0.671162	−0.335581	+	0.942011i	$0.608933\pi$
$648$	0	0
$649$	−3.21539	−0.126215
$650$	0	0
$651$	31.3205	1.22755
$652$	0	0
$653$	−10.3923	−0.406682	−0.203341	−	0.979108i	$-0.565180\pi$
$653$	−10.3923	−0.406682	−0.203341	+	0.979108i	$0.565180\pi$
$654$	0	0
$655$	19.6077	0.766136
$656$	0	0
$657$	8.92820	0.348322
$658$	0	0
$659$	−42.9282	−1.67225	−0.836123	−	0.548542i	$-0.815183\pi$
$659$	−42.9282	−1.67225	−0.836123	+	0.548542i	$0.815183\pi$
$660$	0	0
$661$	−15.0718	−0.586225	−0.293112	−	0.956078i	$-0.594691\pi$
$661$	−15.0718	−0.586225	−0.293112	+	0.956078i	$0.594691\pi$
$662$	0	0
$663$	14.9282	0.579763
$664$	0	0
$665$	−13.8564	−0.537328
$666$	0	0
$667$	4.39230	0.170071
$668$	0	0
$669$	33.8564	1.30896
$670$	0	0
$671$	−0.679492	−0.0262315
$672$	0	0
$673$	0.143594	0.00553512	0.00276756	−	0.999996i	$-0.499119\pi$
$673$	0.143594	0.00553512	0.00276756	+	0.999996i	$0.499119\pi$
$674$	0	0
$675$	−28.0000	−1.07772
$676$	0	0
$677$	−24.2487	−0.931954	−0.465977	−	0.884797i	$-0.654297\pi$
$677$	−24.2487	−0.931954	−0.465977	+	0.884797i	$0.654297\pi$
$678$	0	0
$679$	−13.4641	−0.516705
$680$	0	0
$681$	−55.1769	−2.11438
$682$	0	0
$683$	46.0526	1.76215	0.881076	−	0.472975i	$-0.156820\pi$
$683$	46.0526	1.76215	0.881076	+	0.472975i	$0.156820\pi$
$684$	0	0
$685$	−8.78461	−0.335643
$686$	0	0
$687$	55.7128	2.12558
$688$	0	0
$689$	70.6410	2.69121
$690$	0	0
$691$	36.1962	1.37697	0.688483	−	0.725252i	$-0.258277\pi$
$691$	36.1962	1.37697	0.688483	+	0.725252i	$0.258277\pi$
$692$	0	0
$693$	15.4641	0.587433
$694$	0	0
$695$	−51.0333	−1.93580
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	−16.3923	−0.620014
$700$	0	0
$701$	−16.3923	−0.619129	−0.309564	−	0.950878i	$-0.600183\pi$
$701$	−16.3923	−0.619129	−0.309564	+	0.950878i	$0.600183\pi$
$702$	0	0
$703$	6.64102	0.250471
$704$	0	0
$705$	65.5692	2.46948
$706$	0	0
$707$	−6.92820	−0.260562
$708$	0	0
$709$	−16.2487	−0.610233	−0.305117	−	0.952315i	$-0.598695\pi$
$709$	−16.2487	−0.610233	−0.305117	+	0.952315i	$0.598695\pi$
$710$	0	0
$711$	54.4449	2.04184
$712$	0	0
$713$	−5.32051	−0.199255
$714$	0	0
$715$	−24.0000	−0.897549
$716$	0	0
$717$	56.7846	2.12066
$718$	0	0
$719$	−43.5167	−1.62290	−0.811449	−	0.584424i	$-0.801321\pi$
$719$	−43.5167	−1.62290	−0.811449	+	0.584424i	$0.801321\pi$
$720$	0	0
$721$	29.8564	1.11191
$722$	0	0
$723$	−5.46410	−0.203212
$724$	0	0
$725$	24.2487	0.900575
$726$	0	0
$727$	−4.78461	−0.177451	−0.0887257	−	0.996056i	$-0.528279\pi$
$727$	−4.78461	−0.177451	−0.0887257	+	0.996056i	$0.528279\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	−8.39230	−0.310401
$732$	0	0
$733$	−18.7846	−0.693825	−0.346913	−	0.937897i	$-0.612770\pi$
$733$	−18.7846	−0.693825	−0.346913	+	0.937897i	$0.612770\pi$
$734$	0	0
$735$	4.39230	0.162013
$736$	0	0
$737$	−18.9282	−0.697229
$738$	0	0
$739$	−12.3923	−0.455858	−0.227929	−	0.973678i	$-0.573195\pi$
$739$	−12.3923	−0.455858	−0.227929	+	0.973678i	$0.573195\pi$
$740$	0	0
$741$	−21.8564	−0.802915
$742$	0	0
$743$	−23.9090	−0.877135	−0.438567	−	0.898698i	$-0.644514\pi$
$743$	−23.9090	−0.877135	−0.438567	+	0.898698i	$0.644514\pi$
$744$	0	0
$745$	−20.7846	−0.761489
$746$	0	0
$747$	11.3205	0.414196
$748$	0	0
$749$	−29.3205	−1.07135
$750$	0	0
$751$	−18.7321	−0.683542	−0.341771	−	0.939783i	$-0.611027\pi$
$751$	−18.7321	−0.683542	−0.341771	+	0.939783i	$0.611027\pi$
$752$	0	0
$753$	18.9282	0.689782
$754$	0	0
$755$	−46.6410	−1.69744
$756$	0	0
$757$	17.4641	0.634744	0.317372	−	0.948301i	$-0.397200\pi$
$757$	17.4641	0.634744	0.317372	+	0.948301i	$0.397200\pi$
$758$	0	0
$759$	−4.39230	−0.159431
$760$	0	0
$761$	16.3923	0.594221	0.297110	−	0.954843i	$-0.403977\pi$
$761$	16.3923	0.594221	0.297110	+	0.954843i	$0.403977\pi$
$762$	0	0
$763$	−39.3205	−1.42350
$764$	0	0
$765$	15.4641	0.559106
$766$	0	0
$767$	−13.8564	−0.500326
$768$	0	0
$769$	0.392305	0.0141469	0.00707344	−	0.999975i	$-0.497748\pi$
$769$	0.392305	0.0141469	0.00707344	+	0.999975i	$0.497748\pi$
$770$	0	0
$771$	77.5692	2.79359
$772$	0	0
$773$	−0.679492	−0.0244396	−0.0122198	−	0.999925i	$-0.503890\pi$
$773$	−0.679492	−0.0244396	−0.0122198	+	0.999925i	$0.503890\pi$
$774$	0	0
$775$	−29.3731	−1.05511
$776$	0	0
$777$	−33.8564	−1.21459
$778$	0	0
$779$	−8.78461	−0.314741
$780$	0	0
$781$	10.3923	0.371866
$782$	0	0
$783$	−13.8564	−0.495188
$784$	0	0
$785$	−6.92820	−0.247278
$786$	0	0
$787$	−23.1244	−0.824294	−0.412147	−	0.911117i	$-0.635221\pi$
$787$	−23.1244	−0.824294	−0.412147	+	0.911117i	$0.635221\pi$
$788$	0	0
$789$	30.9282	1.10107
$790$	0	0
$791$	54.2487	1.92886
$792$	0	0
$793$	−2.92820	−0.103984
$794$	0	0
$795$	122.354	4.33944
$796$	0	0
$797$	−2.78461	−0.0986359	−0.0493180	−	0.998783i	$-0.515705\pi$
$797$	−2.78461	−0.0986359	−0.0493180	+	0.998783i	$0.515705\pi$
$798$	0	0
$799$	−6.92820	−0.245102
$800$	0	0
$801$	11.3205	0.399990
$802$	0	0
$803$	2.53590	0.0894899
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	75.0333	2.64130
$808$	0	0
$809$	−7.85641	−0.276217	−0.138108	−	0.990417i	$-0.544102\pi$
$809$	−7.85641	−0.276217	−0.138108	+	0.990417i	$0.544102\pi$
$810$	0	0
$811$	37.3731	1.31235	0.656173	−	0.754611i	$-0.272174\pi$
$811$	37.3731	1.31235	0.656173	+	0.754611i	$0.272174\pi$
$812$	0	0
$813$	2.92820	0.102697
$814$	0	0
$815$	71.3205	2.49825
$816$	0	0
$817$	12.2872	0.429874
$818$	0	0
$819$	66.6410	2.32863
$820$	0	0
$821$	45.0333	1.57167	0.785837	−	0.618434i	$-0.212233\pi$
$821$	45.0333	1.57167	0.785837	+	0.618434i	$0.212233\pi$
$822$	0	0
$823$	−1.66025	−0.0578728	−0.0289364	−	0.999581i	$-0.509212\pi$
$823$	−1.66025	−0.0578728	−0.0289364	+	0.999581i	$0.509212\pi$
$824$	0	0
$825$	−24.2487	−0.844232
$826$	0	0
$827$	19.5167	0.678661	0.339330	−	0.940667i	$-0.389800\pi$
$827$	19.5167	0.678661	0.339330	+	0.940667i	$0.389800\pi$
$828$	0	0
$829$	39.8564	1.38427	0.692135	−	0.721768i	$-0.256670\pi$
$829$	39.8564	1.38427	0.692135	+	0.721768i	$0.256670\pi$
$830$	0	0
$831$	−17.4641	−0.605823
$832$	0	0
$833$	−0.464102	−0.0160802
$834$	0	0
$835$	19.6077	0.678552
$836$	0	0
$837$	16.7846	0.580161
$838$	0	0
$839$	−4.48334	−0.154782	−0.0773910	−	0.997001i	$-0.524659\pi$
$839$	−4.48334	−0.154782	−0.0773910	+	0.997001i	$0.524659\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	54.2487	1.86842
$844$	0	0
$845$	−58.3923	−2.00876
$846$	0	0
$847$	−25.6603	−0.881697
$848$	0	0
$849$	56.2487	1.93045
$850$	0	0
$851$	5.75129	0.197152
$852$	0	0
$853$	8.24871	0.282430	0.141215	−	0.989979i	$-0.454899\pi$
$853$	8.24871	0.282430	0.141215	+	0.989979i	$0.454899\pi$
$854$	0	0
$855$	−22.6410	−0.774306
$856$	0	0
$857$	16.6410	0.568446	0.284223	−	0.958758i	$-0.408264\pi$
$857$	16.6410	0.568446	0.284223	+	0.958758i	$0.408264\pi$
$858$	0	0
$859$	27.3205	0.932164	0.466082	−	0.884742i	$-0.345665\pi$
$859$	27.3205	0.932164	0.466082	+	0.884742i	$0.345665\pi$
$860$	0	0
$861$	44.7846	1.52626
$862$	0	0
$863$	−10.1436	−0.345292	−0.172646	−	0.984984i	$-0.555232\pi$
$863$	−10.1436	−0.345292	−0.172646	+	0.984984i	$0.555232\pi$
$864$	0	0
$865$	53.5692	1.82141
$866$	0	0
$867$	−2.73205	−0.0927853
$868$	0	0
$869$	15.4641	0.524584
$870$	0	0
$871$	−81.5692	−2.76387
$872$	0	0
$873$	−22.0000	−0.744587
$874$	0	0
$875$	−18.9282	−0.639890
$876$	0	0
$877$	9.60770	0.324429	0.162214	−	0.986756i	$-0.448136\pi$
$877$	9.60770	0.324429	0.162214	+	0.986756i	$0.448136\pi$
$878$	0	0
$879$	−81.9615	−2.76449
$880$	0	0
$881$	−38.7846	−1.30669	−0.653343	−	0.757062i	$-0.726634\pi$
$881$	−38.7846	−1.30669	−0.653343	+	0.757062i	$0.726634\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	−11.4115	−0.383162	−0.191581	−	0.981477i	$-0.561361\pi$
$887$	−11.4115	−0.383162	−0.191581	+	0.981477i	$0.561361\pi$
$888$	0	0
$889$	−1.07180	−0.0359469
$890$	0	0
$891$	−3.12436	−0.104670
$892$	0	0
$893$	10.1436	0.339442
$894$	0	0
$895$	50.3538	1.68314
$896$	0	0
$897$	−18.9282	−0.631994
$898$	0	0
$899$	−14.5359	−0.484799
$900$	0	0
$901$	−12.9282	−0.430701
$902$	0	0
$903$	−62.6410	−2.08456
$904$	0	0
$905$	−15.7128	−0.522312
$906$	0	0
$907$	−34.4449	−1.14372	−0.571861	−	0.820350i	$-0.693779\pi$
$907$	−34.4449	−1.14372	−0.571861	+	0.820350i	$0.693779\pi$
$908$	0	0
$909$	−11.3205	−0.375478
$910$	0	0
$911$	15.8038	0.523605	0.261802	−	0.965121i	$-0.415683\pi$
$911$	15.8038	0.523605	0.261802	+	0.965121i	$0.415683\pi$
$912$	0	0
$913$	3.21539	0.106414
$914$	0	0
$915$	−5.07180	−0.167668
$916$	0	0
$917$	−15.4641	−0.510670
$918$	0	0
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	0	0
$921$	−67.7128	−2.23121
$922$	0	0
$923$	44.7846	1.47410
$924$	0	0
$925$	31.7513	1.04398
$926$	0	0
$927$	48.7846	1.60230
$928$	0	0
$929$	−9.71281	−0.318667	−0.159334	−	0.987225i	$-0.550935\pi$
$929$	−9.71281	−0.318667	−0.159334	+	0.987225i	$0.550935\pi$
$930$	0	0
$931$	0.679492	0.0222694
$932$	0	0
$933$	−8.53590	−0.279453
$934$	0	0
$935$	4.39230	0.143644
$936$	0	0
$937$	0.143594	0.00469100	0.00234550	−	0.999997i	$-0.499253\pi$
$937$	0.143594	0.00469100	0.00234550	+	0.999997i	$0.499253\pi$
$938$	0	0
$939$	−57.1769	−1.86590
$940$	0	0
$941$	41.3205	1.34701	0.673505	−	0.739183i	$-0.264788\pi$
$941$	41.3205	1.34701	0.673505	+	0.739183i	$0.264788\pi$
$942$	0	0
$943$	−7.60770	−0.247741
$944$	0	0
$945$	37.8564	1.23147
$946$	0	0
$947$	−49.2679	−1.60099	−0.800497	−	0.599337i	$-0.795431\pi$
$947$	−49.2679	−1.60099	−0.800497	+	0.599337i	$0.795431\pi$
$948$	0	0
$949$	10.9282	0.354744
$950$	0	0
$951$	−23.3205	−0.756219
$952$	0	0
$953$	6.24871	0.202416	0.101208	−	0.994865i	$-0.467729\pi$
$953$	6.24871	0.202416	0.101208	+	0.994865i	$0.467729\pi$
$954$	0	0
$955$	41.5692	1.34515
$956$	0	0
$957$	−12.0000	−0.387905
$958$	0	0
$959$	6.92820	0.223723
$960$	0	0
$961$	−13.3923	−0.432010
$962$	0	0
$963$	−47.9090	−1.54384
$964$	0	0
$965$	−72.4974	−2.33377
$966$	0	0
$967$	−27.6077	−0.887804	−0.443902	−	0.896075i	$-0.646406\pi$
$967$	−27.6077	−0.887804	−0.443902	+	0.896075i	$0.646406\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−20.1051	−0.645204	−0.322602	−	0.946535i	$-0.604558\pi$
$971$	−20.1051	−0.645204	−0.322602	+	0.946535i	$0.604558\pi$
$972$	0	0
$973$	40.2487	1.29031
$974$	0	0
$975$	−104.497	−3.34660
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	3.21539	0.102764
$980$	0	0
$981$	−64.2487	−2.05130
$982$	0	0
$983$	35.9090	1.14532	0.572659	−	0.819794i	$-0.305912\pi$
$983$	35.9090	1.14532	0.572659	+	0.819794i	$0.305912\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	−51.7128	−1.64604
$988$	0	0
$989$	10.6410	0.338365
$990$	0	0
$991$	0.196152	0.00623099	0.00311549	−	0.999995i	$-0.499008\pi$
$991$	0.196152	0.00623099	0.00311549	+	0.999995i	$0.499008\pi$
$992$	0	0
$993$	52.7846	1.67507
$994$	0	0
$995$	14.5359	0.460819
$996$	0	0
$997$	21.6077	0.684323	0.342161	−	0.939641i	$-0.388841\pi$
$997$	21.6077	0.684323	0.342161	+	0.939641i	$0.388841\pi$
$998$	0	0
$999$	−18.1436	−0.574038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.2.a.e.1.1		2
3.2	odd	2		2448.2.a.y.1.2		2
4.3	odd	2		68.2.a.a.1.2	✓	2
5.4	even	2		6800.2.a.bh.1.2		2
8.3	odd	2		1088.2.a.p.1.1		2
8.5	even	2		1088.2.a.t.1.2		2
12.11	even	2		612.2.a.e.1.2		2
17.16	even	2		4624.2.a.x.1.2		2
20.3	even	4		1700.2.e.c.749.4		4
20.7	even	4		1700.2.e.c.749.1		4
20.19	odd	2		1700.2.a.d.1.1		2
24.5	odd	2		9792.2.a.cs.1.1		2
24.11	even	2		9792.2.a.cr.1.1		2
28.27	even	2		3332.2.a.h.1.1		2
44.43	even	2		8228.2.a.k.1.2		2
68.3	even	16		1156.2.h.f.757.4		16
68.7	even	16		1156.2.h.f.1001.1		16
68.11	even	16		1156.2.h.f.733.1		16
68.15	odd	8		1156.2.e.d.905.1		8
68.19	odd	8		1156.2.e.d.905.4		8
68.23	even	16		1156.2.h.f.733.4		16
68.27	even	16		1156.2.h.f.1001.4		16
68.31	even	16		1156.2.h.f.757.1		16
68.39	even	16		1156.2.h.f.977.1		16
68.43	odd	8		1156.2.e.d.829.4		8
68.47	odd	4		1156.2.b.c.577.4		4
68.55	odd	4		1156.2.b.c.577.1		4
68.59	odd	8		1156.2.e.d.829.1		8
68.63	even	16		1156.2.h.f.977.4		16
68.67	odd	2		1156.2.a.a.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.a.a.1.2	✓	2	4.3	odd	2
272.2.a.e.1.1		2	1.1	even	1	trivial
612.2.a.e.1.2		2	12.11	even	2
1088.2.a.p.1.1		2	8.3	odd	2
1088.2.a.t.1.2		2	8.5	even	2
1156.2.a.a.1.1		2	68.67	odd	2
1156.2.b.c.577.1		4	68.55	odd	4
1156.2.b.c.577.4		4	68.47	odd	4
1156.2.e.d.829.1		8	68.59	odd	8
1156.2.e.d.829.4		8	68.43	odd	8
1156.2.e.d.905.1		8	68.15	odd	8
1156.2.e.d.905.4		8	68.19	odd	8
1156.2.h.f.733.1		16	68.11	even	16
1156.2.h.f.733.4		16	68.23	even	16
1156.2.h.f.757.1		16	68.31	even	16
1156.2.h.f.757.4		16	68.3	even	16
1156.2.h.f.977.1		16	68.39	even	16
1156.2.h.f.977.4		16	68.63	even	16
1156.2.h.f.1001.1		16	68.7	even	16
1156.2.h.f.1001.4		16	68.27	even	16
1700.2.a.d.1.1		2	20.19	odd	2
1700.2.e.c.749.1		4	20.7	even	4
1700.2.e.c.749.4		4	20.3	even	4
2448.2.a.y.1.2		2	3.2	odd	2
3332.2.a.h.1.1		2	28.27	even	2
4624.2.a.x.1.2		2	17.16	even	2
6800.2.a.bh.1.2		2	5.4	even	2
8228.2.a.k.1.2		2	44.43	even	2
9792.2.a.cr.1.1		2	24.11	even	2
9792.2.a.cs.1.1		2	24.5	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:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	272.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -3.46410 q^{5} +2.73205 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} -3.46410 q^{5} +2.73205 q^{7} +4.46410 q^{9} +1.26795 q^{11} +5.46410 q^{13} +9.46410 q^{15} -1.00000 q^{17} +1.46410 q^{19} -7.46410 q^{21} +1.26795 q^{23} +7.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} -4.19615 q^{31} -3.46410 q^{33} -9.46410 q^{35} +4.53590 q^{37} -14.9282 q^{39} -6.00000 q^{41} +8.39230 q^{43} -15.4641 q^{45} +6.92820 q^{47} +0.464102 q^{49} +2.73205 q^{51} +12.9282 q^{53} -4.39230 q^{55} -4.00000 q^{57} -2.53590 q^{59} -0.535898 q^{61} +12.1962 q^{63} -18.9282 q^{65} -14.9282 q^{67} -3.46410 q^{69} +8.19615 q^{71} +2.00000 q^{73} -19.1244 q^{75} +3.46410 q^{77} +12.1962 q^{79} -2.46410 q^{81} +2.53590 q^{83} +3.46410 q^{85} -9.46410 q^{87} +2.53590 q^{89} +14.9282 q^{91} +11.4641 q^{93} -5.07180 q^{95} -4.92820 q^{97} +5.66025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} + 6 q^{11} + 4 q^{13} + 12 q^{15} - 2 q^{17} - 4 q^{19} - 8 q^{21} + 6 q^{23} + 14 q^{25} - 8 q^{27} + 2 q^{31} - 12 q^{35} + 16 q^{37} - 16 q^{39} - 12 q^{41} - 4 q^{43} - 24 q^{45} - 6 q^{49} + 2 q^{51} + 12 q^{53} + 12 q^{55} - 8 q^{57} - 12 q^{59} - 8 q^{61} + 14 q^{63} - 24 q^{65} - 16 q^{67} + 6 q^{71} + 4 q^{73} - 14 q^{75} + 14 q^{79} + 2 q^{81} + 12 q^{83} - 12 q^{87} + 12 q^{89} + 16 q^{91} + 16 q^{93} - 24 q^{95} + 4 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 + 6 * q^11 + 4 * q^13 + 12 * q^15 - 2 * q^17 - 4 * q^19 - 8 * q^21 + 6 * q^23 + 14 * q^25 - 8 * q^27 + 2 * q^31 - 12 * q^35 + 16 * q^37 - 16 * q^39 - 12 * q^41 - 4 * q^43 - 24 * q^45 - 6 * q^49 + 2 * q^51 + 12 * q^53 + 12 * q^55 - 8 * q^57 - 12 * q^59 - 8 * q^61 + 14 * q^63 - 24 * q^65 - 16 * q^67 + 6 * q^71 + 4 * q^73 - 14 * q^75 + 14 * q^79 + 2 * q^81 + 12 * q^83 - 12 * q^87 + 12 * q^89 + 16 * q^91 + 16 * q^93 - 24 * q^95 + 4 * q^97 - 6 * q^99

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