LMFDB - Embedded newform 1520.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.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:	$12.1372611072$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -1.00000 q^{5} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} -1.00000 q^{5} +4.46410 q^{9} -2.00000 q^{11} -0.732051 q^{13} +2.73205 q^{15} +7.46410 q^{17} -1.00000 q^{19} +1.46410 q^{23} +1.00000 q^{25} -4.00000 q^{27} -3.46410 q^{29} -4.00000 q^{31} +5.46410 q^{33} +7.66025 q^{37} +2.00000 q^{39} +0.535898 q^{41} +2.92820 q^{43} -4.46410 q^{45} +2.92820 q^{47} -7.00000 q^{49} -20.3923 q^{51} -11.6603 q^{53} +2.00000 q^{55} +2.73205 q^{57} -1.46410 q^{59} -6.53590 q^{61} +0.732051 q^{65} +4.19615 q^{67} -4.00000 q^{69} -10.9282 q^{71} -10.3923 q^{73} -2.73205 q^{75} +8.39230 q^{79} -2.46410 q^{81} +5.46410 q^{83} -7.46410 q^{85} +9.46410 q^{87} -3.46410 q^{89} +10.9282 q^{93} +1.00000 q^{95} -15.6603 q^{97} -8.92820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 8 q^{17} - 2 q^{19} - 4 q^{23} + 2 q^{25} - 8 q^{27} - 8 q^{31} + 4 q^{33} - 2 q^{37} + 4 q^{39} + 8 q^{41} - 8 q^{43} - 2 q^{45} - 8 q^{47} - 14 q^{49} - 20 q^{51} - 6 q^{53} + 4 q^{55} + 2 q^{57} + 4 q^{59} - 20 q^{61} - 2 q^{65} - 2 q^{67} - 8 q^{69} - 8 q^{71} - 2 q^{75} - 4 q^{79} + 2 q^{81} + 4 q^{83} - 8 q^{85} + 12 q^{87} + 8 q^{93} + 2 q^{95} - 14 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^11 + 2 * q^13 + 2 * q^15 + 8 * q^17 - 2 * q^19 - 4 * q^23 + 2 * q^25 - 8 * q^27 - 8 * q^31 + 4 * q^33 - 2 * q^37 + 4 * q^39 + 8 * q^41 - 8 * q^43 - 2 * q^45 - 8 * q^47 - 14 * q^49 - 20 * q^51 - 6 * q^53 + 4 * q^55 + 2 * q^57 + 4 * q^59 - 20 * q^61 - 2 * q^65 - 2 * q^67 - 8 * q^69 - 8 * q^71 - 2 * q^75 - 4 * q^79 + 2 * q^81 + 4 * q^83 - 8 * q^85 + 12 * q^87 + 8 * q^93 + 2 * q^95 - 14 * q^97 - 4 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−0.732051	−0.203034	−0.101517	−	0.994834i	$-0.532370\pi$
$13$	−0.732051	−0.203034	−0.101517	+	0.994834i	$0.532370\pi$
$14$	0	0
$15$	2.73205	0.705412
$16$	0	0
$17$	7.46410	1.81031	0.905155	−	0.425081i	$-0.139754\pi$
$17$	7.46410	1.81031	0.905155	+	0.425081i	$0.139754\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.46410	0.305286	0.152643	−	0.988281i	$-0.451221\pi$
$23$	1.46410	0.305286	0.152643	+	0.988281i	$0.451221\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	5.46410	0.951178
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.66025	1.25934	0.629669	−	0.776864i	$-0.283191\pi$
$37$	7.66025	1.25934	0.629669	+	0.776864i	$0.283191\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	0.535898	0.0836933	0.0418466	−	0.999124i	$-0.486676\pi$
$41$	0.535898	0.0836933	0.0418466	+	0.999124i	$0.486676\pi$
$42$	0	0
$43$	2.92820	0.446547	0.223273	−	0.974756i	$-0.428326\pi$
$43$	2.92820	0.446547	0.223273	+	0.974756i	$0.428326\pi$
$44$	0	0
$45$	−4.46410	−0.665469
$46$	0	0
$47$	2.92820	0.427122	0.213561	−	0.976930i	$-0.431494\pi$
$47$	2.92820	0.427122	0.213561	+	0.976930i	$0.431494\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−20.3923	−2.85549
$52$	0	0
$53$	−11.6603	−1.60166	−0.800830	−	0.598892i	$-0.795608\pi$
$53$	−11.6603	−1.60166	−0.800830	+	0.598892i	$0.795608\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	2.73205	0.361869
$58$	0	0
$59$	−1.46410	−0.190610	−0.0953049	−	0.995448i	$-0.530383\pi$
$59$	−1.46410	−0.190610	−0.0953049	+	0.995448i	$0.530383\pi$
$60$	0	0
$61$	−6.53590	−0.836836	−0.418418	−	0.908255i	$-0.637415\pi$
$61$	−6.53590	−0.836836	−0.418418	+	0.908255i	$0.637415\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.732051	0.0907997
$66$	0	0
$67$	4.19615	0.512642	0.256321	−	0.966592i	$-0.417490\pi$
$67$	4.19615	0.512642	0.256321	+	0.966592i	$0.417490\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−10.9282	−1.29694	−0.648470	−	0.761241i	$-0.724591\pi$
$71$	−10.9282	−1.29694	−0.648470	+	0.761241i	$0.724591\pi$
$72$	0	0
$73$	−10.3923	−1.21633	−0.608164	−	0.793812i	$-0.708094\pi$
$73$	−10.3923	−1.21633	−0.608164	+	0.793812i	$0.708094\pi$
$74$	0	0
$75$	−2.73205	−0.315470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.39230	0.944208	0.472104	−	0.881543i	$-0.343495\pi$
$79$	8.39230	0.944208	0.472104	+	0.881543i	$0.343495\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	5.46410	0.599763	0.299882	−	0.953976i	$-0.403053\pi$
$83$	5.46410	0.599763	0.299882	+	0.953976i	$0.403053\pi$
$84$	0	0
$85$	−7.46410	−0.809595
$86$	0	0
$87$	9.46410	1.01466
$88$	0	0
$89$	−3.46410	−0.367194	−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410	−0.367194	−0.183597	+	0.983002i	$0.558774\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	10.9282	1.13320
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−15.6603	−1.59006	−0.795029	−	0.606571i	$-0.792544\pi$
$97$	−15.6603	−1.59006	−0.795029	+	0.606571i	$0.792544\pi$
$98$	0	0
$99$	−8.92820	−0.897318
$100$	0	0
$101$	1.46410	0.145684	0.0728418	−	0.997344i	$-0.476793\pi$
$101$	1.46410	0.145684	0.0728418	+	0.997344i	$0.476793\pi$
$102$	0	0
$103$	−5.66025	−0.557721	−0.278861	−	0.960332i	$-0.589957\pi$
$103$	−5.66025	−0.557721	−0.278861	+	0.960332i	$0.589957\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.1244	−1.46213	−0.731063	−	0.682310i	$-0.760976\pi$
$107$	−15.1244	−1.46213	−0.731063	+	0.682310i	$0.760976\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−20.9282	−1.98642
$112$	0	0
$113$	16.7321	1.57402	0.787009	−	0.616941i	$-0.211628\pi$
$113$	16.7321	1.57402	0.787009	+	0.616941i	$0.211628\pi$
$114$	0	0
$115$	−1.46410	−0.136528
$116$	0	0
$117$	−3.26795	−0.302122
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−1.46410	−0.132014
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.6603	−1.56709	−0.783547	−	0.621332i	$-0.786592\pi$
$127$	−17.6603	−1.56709	−0.783547	+	0.621332i	$0.786592\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	0.535898	0.0457849	0.0228924	−	0.999738i	$-0.492712\pi$
$137$	0.535898	0.0457849	0.0228924	+	0.999738i	$0.492712\pi$
$138$	0	0
$139$	−7.85641	−0.666372	−0.333186	−	0.942861i	$-0.608124\pi$
$139$	−7.85641	−0.666372	−0.333186	+	0.942861i	$0.608124\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	1.46410	0.122434
$144$	0	0
$145$	3.46410	0.287678
$146$	0	0
$147$	19.1244	1.57735
$148$	0	0
$149$	15.3205	1.25510	0.627552	−	0.778574i	$-0.284057\pi$
$149$	15.3205	1.25510	0.627552	+	0.778574i	$0.284057\pi$
$150$	0	0
$151$	5.46410	0.444662	0.222331	−	0.974971i	$-0.428633\pi$
$151$	5.46410	0.444662	0.222331	+	0.974971i	$0.428633\pi$
$152$	0	0
$153$	33.3205	2.69380
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−7.46410	−0.595700	−0.297850	−	0.954613i	$-0.596270\pi$
$157$	−7.46410	−0.595700	−0.297850	+	0.954613i	$0.596270\pi$
$158$	0	0
$159$	31.8564	2.52638
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	−5.46410	−0.425380
$166$	0	0
$167$	6.73205	0.520942	0.260471	−	0.965482i	$-0.416122\pi$
$167$	6.73205	0.520942	0.260471	+	0.965482i	$0.416122\pi$
$168$	0	0
$169$	−12.4641	−0.958777
$170$	0	0
$171$	−4.46410	−0.341378
$172$	0	0
$173$	−17.1244	−1.30194	−0.650970	−	0.759103i	$-0.725638\pi$
$173$	−17.1244	−1.30194	−0.650970	+	0.759103i	$0.725638\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	9.46410	0.707380	0.353690	−	0.935363i	$-0.384927\pi$
$179$	9.46410	0.707380	0.353690	+	0.935363i	$0.384927\pi$
$180$	0	0
$181$	−15.8564	−1.17860	−0.589299	−	0.807915i	$-0.700596\pi$
$181$	−15.8564	−1.17860	−0.589299	+	0.807915i	$0.700596\pi$
$182$	0	0
$183$	17.8564	1.31998
$184$	0	0
$185$	−7.66025	−0.563193
$186$	0	0
$187$	−14.9282	−1.09166
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.9282	0.790737	0.395369	−	0.918523i	$-0.370617\pi$
$191$	10.9282	0.790737	0.395369	+	0.918523i	$0.370617\pi$
$192$	0	0
$193$	14.1962	1.02186	0.510931	−	0.859622i	$-0.329301\pi$
$193$	14.1962	1.02186	0.510931	+	0.859622i	$0.329301\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	11.8564	0.844734	0.422367	−	0.906425i	$-0.361199\pi$
$197$	11.8564	0.844734	0.422367	+	0.906425i	$0.361199\pi$
$198$	0	0
$199$	18.9282	1.34178	0.670892	−	0.741555i	$-0.265911\pi$
$199$	18.9282	1.34178	0.670892	+	0.741555i	$0.265911\pi$
$200$	0	0
$201$	−11.4641	−0.808615
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.535898	−0.0374288
$206$	0	0
$207$	6.53590	0.454276
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	16.3923	1.12849	0.564246	−	0.825606i	$-0.309167\pi$
$211$	16.3923	1.12849	0.564246	+	0.825606i	$0.309167\pi$
$212$	0	0
$213$	29.8564	2.04573
$214$	0	0
$215$	−2.92820	−0.199702
$216$	0	0
$217$	0	0
$218$	0	0
$219$	28.3923	1.91857
$220$	0	0
$221$	−5.46410	−0.367555
$222$	0	0
$223$	−26.0526	−1.74461	−0.872304	−	0.488964i	$-0.837375\pi$
$223$	−26.0526	−1.74461	−0.872304	+	0.488964i	$0.837375\pi$
$224$	0	0
$225$	4.46410	0.297607
$226$	0	0
$227$	−1.26795	−0.0841567	−0.0420784	−	0.999114i	$-0.513398\pi$
$227$	−1.26795	−0.0841567	−0.0420784	+	0.999114i	$0.513398\pi$
$228$	0	0
$229$	5.46410	0.361078	0.180539	−	0.983568i	$-0.442216\pi$
$229$	5.46410	0.361078	0.180539	+	0.983568i	$0.442216\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	−2.92820	−0.191015
$236$	0	0
$237$	−22.9282	−1.48935
$238$	0	0
$239$	18.9282	1.22436	0.612182	−	0.790717i	$-0.290292\pi$
$239$	18.9282	1.22436	0.612182	+	0.790717i	$0.290292\pi$
$240$	0	0
$241$	−10.3923	−0.669427	−0.334714	−	0.942320i	$-0.608640\pi$
$241$	−10.3923	−0.669427	−0.334714	+	0.942320i	$0.608640\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	7.00000	0.447214
$246$	0	0
$247$	0.732051	0.0465793
$248$	0	0
$249$	−14.9282	−0.946036
$250$	0	0
$251$	1.85641	0.117175	0.0585877	−	0.998282i	$-0.481340\pi$
$251$	1.85641	0.117175	0.0585877	+	0.998282i	$0.481340\pi$
$252$	0	0
$253$	−2.92820	−0.184095
$254$	0	0
$255$	20.3923	1.27702
$256$	0	0
$257$	24.7321	1.54274	0.771371	−	0.636385i	$-0.219571\pi$
$257$	24.7321	1.54274	0.771371	+	0.636385i	$0.219571\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−15.4641	−0.957204
$262$	0	0
$263$	−19.3205	−1.19135	−0.595677	−	0.803224i	$-0.703116\pi$
$263$	−19.3205	−1.19135	−0.595677	+	0.803224i	$0.703116\pi$
$264$	0	0
$265$	11.6603	0.716284
$266$	0	0
$267$	9.46410	0.579194
$268$	0	0
$269$	−8.92820	−0.544362	−0.272181	−	0.962246i	$-0.587745\pi$
$269$	−8.92820	−0.544362	−0.272181	+	0.962246i	$0.587745\pi$
$270$	0	0
$271$	−14.0000	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−20.5359	−1.23388	−0.616941	−	0.787009i	$-0.711628\pi$
$277$	−20.5359	−1.23388	−0.616941	+	0.787009i	$0.711628\pi$
$278$	0	0
$279$	−17.8564	−1.06904
$280$	0	0
$281$	−22.3923	−1.33581	−0.667906	−	0.744245i	$-0.732809\pi$
$281$	−22.3923	−1.33581	−0.667906	+	0.744245i	$0.732809\pi$
$282$	0	0
$283$	9.46410	0.562582	0.281291	−	0.959622i	$-0.409237\pi$
$283$	9.46410	0.562582	0.281291	+	0.959622i	$0.409237\pi$
$284$	0	0
$285$	−2.73205	−0.161833
$286$	0	0
$287$	0	0
$288$	0	0
$289$	38.7128	2.27722
$290$	0	0
$291$	42.7846	2.50808
$292$	0	0
$293$	18.1962	1.06303	0.531515	−	0.847049i	$-0.321623\pi$
$293$	18.1962	1.06303	0.531515	+	0.847049i	$0.321623\pi$
$294$	0	0
$295$	1.46410	0.0852433
$296$	0	0
$297$	8.00000	0.464207
$298$	0	0
$299$	−1.07180	−0.0619836
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	6.53590	0.374244
$306$	0	0
$307$	−25.2679	−1.44212	−0.721059	−	0.692874i	$-0.756344\pi$
$307$	−25.2679	−1.44212	−0.721059	+	0.692874i	$0.756344\pi$
$308$	0	0
$309$	15.4641	0.879722
$310$	0	0
$311$	−19.8564	−1.12595	−0.562977	−	0.826473i	$-0.690344\pi$
$311$	−19.8564	−1.12595	−0.562977	+	0.826473i	$0.690344\pi$
$312$	0	0
$313$	−15.8564	−0.896257	−0.448129	−	0.893969i	$-0.647909\pi$
$313$	−15.8564	−0.896257	−0.448129	+	0.893969i	$0.647909\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.58846	−0.370045	−0.185022	−	0.982734i	$-0.559236\pi$
$317$	−6.58846	−0.370045	−0.185022	+	0.982734i	$0.559236\pi$
$318$	0	0
$319$	6.92820	0.387905
$320$	0	0
$321$	41.3205	2.30629
$322$	0	0
$323$	−7.46410	−0.415314
$324$	0	0
$325$	−0.732051	−0.0406069
$326$	0	0
$327$	38.2487	2.11516
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.2487	−0.783180	−0.391590	−	0.920140i	$-0.628075\pi$
$331$	−14.2487	−0.783180	−0.391590	+	0.920140i	$0.628075\pi$
$332$	0	0
$333$	34.1962	1.87394
$334$	0	0
$335$	−4.19615	−0.229260
$336$	0	0
$337$	11.6603	0.635175	0.317587	−	0.948229i	$-0.397127\pi$
$337$	11.6603	0.635175	0.317587	+	0.948229i	$0.397127\pi$
$338$	0	0
$339$	−45.7128	−2.48278
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	8.39230	0.450523	0.225261	−	0.974298i	$-0.427676\pi$
$347$	8.39230	0.450523	0.225261	+	0.974298i	$0.427676\pi$
$348$	0	0
$349$	−22.7846	−1.21963	−0.609816	−	0.792543i	$-0.708757\pi$
$349$	−22.7846	−1.21963	−0.609816	+	0.792543i	$0.708757\pi$
$350$	0	0
$351$	2.92820	0.156296
$352$	0	0
$353$	32.2487	1.71643	0.858213	−	0.513294i	$-0.171575\pi$
$353$	32.2487	1.71643	0.858213	+	0.513294i	$0.171575\pi$
$354$	0	0
$355$	10.9282	0.580009
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8.92820	−0.471213	−0.235606	−	0.971849i	$-0.575708\pi$
$359$	−8.92820	−0.471213	−0.235606	+	0.971849i	$0.575708\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	19.1244	1.00377
$364$	0	0
$365$	10.3923	0.543958
$366$	0	0
$367$	−10.9282	−0.570448	−0.285224	−	0.958461i	$-0.592068\pi$
$367$	−10.9282	−0.570448	−0.285224	+	0.958461i	$0.592068\pi$
$368$	0	0
$369$	2.39230	0.124538
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.1962	0.527937	0.263968	−	0.964531i	$-0.414969\pi$
$373$	10.1962	0.527937	0.263968	+	0.964531i	$0.414969\pi$
$374$	0	0
$375$	2.73205	0.141082
$376$	0	0
$377$	2.53590	0.130605
$378$	0	0
$379$	34.6410	1.77939	0.889695	−	0.456556i	$-0.150917\pi$
$379$	34.6410	1.77939	0.889695	+	0.456556i	$0.150917\pi$
$380$	0	0
$381$	48.2487	2.47186
$382$	0	0
$383$	−32.1962	−1.64515	−0.822573	−	0.568659i	$-0.807462\pi$
$383$	−32.1962	−1.64515	−0.822573	+	0.568659i	$0.807462\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.0718	0.664477
$388$	0	0
$389$	−15.8564	−0.803952	−0.401976	−	0.915650i	$-0.631676\pi$
$389$	−15.8564	−0.803952	−0.401976	+	0.915650i	$0.631676\pi$
$390$	0	0
$391$	10.9282	0.552663
$392$	0	0
$393$	32.7846	1.65376
$394$	0	0
$395$	−8.39230	−0.422263
$396$	0	0
$397$	−19.4641	−0.976875	−0.488438	−	0.872599i	$-0.662433\pi$
$397$	−19.4641	−0.976875	−0.488438	+	0.872599i	$0.662433\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.0000	0.699127	0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000	0.699127	0.349563	+	0.936913i	$0.386330\pi$
$402$	0	0
$403$	2.92820	0.145864
$404$	0	0
$405$	2.46410	0.122442
$406$	0	0
$407$	−15.3205	−0.759409
$408$	0	0
$409$	15.0718	0.745252	0.372626	−	0.927982i	$-0.378457\pi$
$409$	15.0718	0.745252	0.372626	+	0.927982i	$0.378457\pi$
$410$	0	0
$411$	−1.46410	−0.0722188
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.46410	−0.268222
$416$	0	0
$417$	21.4641	1.05110
$418$	0	0
$419$	−25.8564	−1.26317	−0.631584	−	0.775307i	$-0.717595\pi$
$419$	−25.8564	−1.26317	−0.631584	+	0.775307i	$0.717595\pi$
$420$	0	0
$421$	16.2487	0.791914	0.395957	−	0.918269i	$-0.370413\pi$
$421$	16.2487	0.791914	0.395957	+	0.918269i	$0.370413\pi$
$422$	0	0
$423$	13.0718	0.635573
$424$	0	0
$425$	7.46410	0.362062
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−28.3923	−1.36761	−0.683805	−	0.729665i	$-0.739676\pi$
$431$	−28.3923	−1.36761	−0.683805	+	0.729665i	$0.739676\pi$
$432$	0	0
$433$	−26.9808	−1.29661	−0.648306	−	0.761380i	$-0.724522\pi$
$433$	−26.9808	−1.29661	−0.648306	+	0.761380i	$0.724522\pi$
$434$	0	0
$435$	−9.46410	−0.453769
$436$	0	0
$437$	−1.46410	−0.0700375
$438$	0	0
$439$	−26.2487	−1.25278	−0.626391	−	0.779509i	$-0.715469\pi$
$439$	−26.2487	−1.25278	−0.626391	+	0.779509i	$0.715469\pi$
$440$	0	0
$441$	−31.2487	−1.48803
$442$	0	0
$443$	33.4641	1.58993	0.794964	−	0.606657i	$-0.207490\pi$
$443$	33.4641	1.58993	0.794964	+	0.606657i	$0.207490\pi$
$444$	0	0
$445$	3.46410	0.164214
$446$	0	0
$447$	−41.8564	−1.97974
$448$	0	0
$449$	−36.6410	−1.72920	−0.864598	−	0.502464i	$-0.832427\pi$
$449$	−36.6410	−1.72920	−0.864598	+	0.502464i	$0.832427\pi$
$450$	0	0
$451$	−1.07180	−0.0504689
$452$	0	0
$453$	−14.9282	−0.701388
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.9282	1.35320	0.676602	−	0.736349i	$-0.263452\pi$
$457$	28.9282	1.35320	0.676602	+	0.736349i	$0.263452\pi$
$458$	0	0
$459$	−29.8564	−1.39358
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	−10.2487	−0.476298	−0.238149	−	0.971229i	$-0.576541\pi$
$463$	−10.2487	−0.476298	−0.238149	+	0.971229i	$0.576541\pi$
$464$	0	0
$465$	−10.9282	−0.506783
$466$	0	0
$467$	35.3205	1.63444	0.817219	−	0.576327i	$-0.195515\pi$
$467$	35.3205	1.63444	0.817219	+	0.576327i	$0.195515\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	20.3923	0.939628
$472$	0	0
$473$	−5.85641	−0.269278
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−52.0526	−2.38332
$478$	0	0
$479$	−19.8564	−0.907262	−0.453631	−	0.891190i	$-0.649872\pi$
$479$	−19.8564	−0.907262	−0.453631	+	0.891190i	$0.649872\pi$
$480$	0	0
$481$	−5.60770	−0.255689
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.6603	0.711096
$486$	0	0
$487$	−42.7321	−1.93637	−0.968187	−	0.250228i	$-0.919495\pi$
$487$	−42.7321	−1.93637	−0.968187	+	0.250228i	$0.919495\pi$
$488$	0	0
$489$	−10.9282	−0.494190
$490$	0	0
$491$	1.85641	0.0837785	0.0418892	−	0.999122i	$-0.486662\pi$
$491$	1.85641	0.0837785	0.0418892	+	0.999122i	$0.486662\pi$
$492$	0	0
$493$	−25.8564	−1.16451
$494$	0	0
$495$	8.92820	0.401293
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−30.7846	−1.37811	−0.689054	−	0.724710i	$-0.741974\pi$
$499$	−30.7846	−1.37811	−0.689054	+	0.724710i	$0.741974\pi$
$500$	0	0
$501$	−18.3923	−0.821708
$502$	0	0
$503$	−23.3205	−1.03981	−0.519905	−	0.854224i	$-0.674033\pi$
$503$	−23.3205	−1.03981	−0.519905	+	0.854224i	$0.674033\pi$
$504$	0	0
$505$	−1.46410	−0.0651517
$506$	0	0
$507$	34.0526	1.51233
$508$	0	0
$509$	2.39230	0.106037	0.0530185	−	0.998594i	$-0.483116\pi$
$509$	2.39230	0.106037	0.0530185	+	0.998594i	$0.483116\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	5.66025	0.249421
$516$	0	0
$517$	−5.85641	−0.257564
$518$	0	0
$519$	46.7846	2.05362
$520$	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
$523$	18.7321	0.819095	0.409548	−	0.912289i	$-0.365687\pi$
$523$	18.7321	0.819095	0.409548	+	0.912289i	$0.365687\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−29.8564	−1.30057
$528$	0	0
$529$	−20.8564	−0.906800
$530$	0	0
$531$	−6.53590	−0.283634
$532$	0	0
$533$	−0.392305	−0.0169926
$534$	0	0
$535$	15.1244	0.653883
$536$	0	0
$537$	−25.8564	−1.11579
$538$	0	0
$539$	14.0000	0.603023
$540$	0	0
$541$	−23.6077	−1.01497	−0.507487	−	0.861659i	$-0.669425\pi$
$541$	−23.6077	−1.01497	−0.507487	+	0.861659i	$0.669425\pi$
$542$	0	0
$543$	43.3205	1.85906
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	−16.5885	−0.709271	−0.354636	−	0.935005i	$-0.615395\pi$
$547$	−16.5885	−0.709271	−0.354636	+	0.935005i	$0.615395\pi$
$548$	0	0
$549$	−29.1769	−1.24524
$550$	0	0
$551$	3.46410	0.147576
$552$	0	0
$553$	0	0
$554$	0	0
$555$	20.9282	0.888353
$556$	0	0
$557$	12.2487	0.518995	0.259497	−	0.965744i	$-0.416443\pi$
$557$	12.2487	0.518995	0.259497	+	0.965744i	$0.416443\pi$
$558$	0	0
$559$	−2.14359	−0.0906643
$560$	0	0
$561$	40.7846	1.72193
$562$	0	0
$563$	8.58846	0.361960	0.180980	−	0.983487i	$-0.442073\pi$
$563$	8.58846	0.361960	0.180980	+	0.983487i	$0.442073\pi$
$564$	0	0
$565$	−16.7321	−0.703923
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.7846	0.619803	0.309902	−	0.950769i	$-0.399704\pi$
$569$	14.7846	0.619803	0.309902	+	0.950769i	$0.399704\pi$
$570$	0	0
$571$	30.7846	1.28830	0.644148	−	0.764901i	$-0.277212\pi$
$571$	30.7846	1.28830	0.644148	+	0.764901i	$0.277212\pi$
$572$	0	0
$573$	−29.8564	−1.24727
$574$	0	0
$575$	1.46410	0.0610573
$576$	0	0
$577$	35.8564	1.49272	0.746361	−	0.665541i	$-0.231799\pi$
$577$	35.8564	1.49272	0.746361	+	0.665541i	$0.231799\pi$
$578$	0	0
$579$	−38.7846	−1.61183
$580$	0	0
$581$	0	0
$582$	0	0
$583$	23.3205	0.965837
$584$	0	0
$585$	3.26795	0.135113
$586$	0	0
$587$	−36.1051	−1.49022	−0.745109	−	0.666943i	$-0.767602\pi$
$587$	−36.1051	−1.49022	−0.745109	+	0.666943i	$0.767602\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−32.3923	−1.33244
$592$	0	0
$593$	−35.8564	−1.47245	−0.736223	−	0.676739i	$-0.763393\pi$
$593$	−35.8564	−1.47245	−0.736223	+	0.676739i	$0.763393\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−51.7128	−2.11646
$598$	0	0
$599$	43.0333	1.75829	0.879147	−	0.476551i	$-0.158113\pi$
$599$	43.0333	1.75829	0.879147	+	0.476551i	$0.158113\pi$
$600$	0	0
$601$	−36.5359	−1.49033	−0.745165	−	0.666880i	$-0.767629\pi$
$601$	−36.5359	−1.49033	−0.745165	+	0.666880i	$0.767629\pi$
$602$	0	0
$603$	18.7321	0.762828
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−38.4449	−1.56043	−0.780214	−	0.625512i	$-0.784890\pi$
$607$	−38.4449	−1.56043	−0.780214	+	0.625512i	$0.784890\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.14359	−0.0867205
$612$	0	0
$613$	44.2487	1.78719	0.893594	−	0.448875i	$-0.148175\pi$
$613$	44.2487	1.78719	0.893594	+	0.448875i	$0.148175\pi$
$614$	0	0
$615$	1.46410	0.0590383
$616$	0	0
$617$	3.46410	0.139459	0.0697297	−	0.997566i	$-0.477786\pi$
$617$	3.46410	0.139459	0.0697297	+	0.997566i	$0.477786\pi$
$618$	0	0
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	0	0
$621$	−5.85641	−0.235009
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.46410	−0.218215
$628$	0	0
$629$	57.1769	2.27979
$630$	0	0
$631$	−20.6410	−0.821706	−0.410853	−	0.911702i	$-0.634769\pi$
$631$	−20.6410	−0.821706	−0.410853	+	0.911702i	$0.634769\pi$
$632$	0	0
$633$	−44.7846	−1.78003
$634$	0	0
$635$	17.6603	0.700826
$636$	0	0
$637$	5.12436	0.203034
$638$	0	0
$639$	−48.7846	−1.92989
$640$	0	0
$641$	45.0333	1.77871	0.889355	−	0.457218i	$-0.151154\pi$
$641$	45.0333	1.77871	0.889355	+	0.457218i	$0.151154\pi$
$642$	0	0
$643$	−1.75129	−0.0690641	−0.0345320	−	0.999404i	$-0.510994\pi$
$643$	−1.75129	−0.0690641	−0.0345320	+	0.999404i	$0.510994\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	47.3205	1.86036	0.930181	−	0.367102i	$-0.119650\pi$
$647$	47.3205	1.86036	0.930181	+	0.367102i	$0.119650\pi$
$648$	0	0
$649$	2.92820	0.114942
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−34.3923	−1.34587	−0.672937	−	0.739699i	$-0.734968\pi$
$653$	−34.3923	−1.34587	−0.672937	+	0.739699i	$0.734968\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	−46.3923	−1.80994
$658$	0	0
$659$	−32.3923	−1.26183	−0.630913	−	0.775854i	$-0.717319\pi$
$659$	−32.3923	−1.26183	−0.630913	+	0.775854i	$0.717319\pi$
$660$	0	0
$661$	−16.5359	−0.643172	−0.321586	−	0.946880i	$-0.604216\pi$
$661$	−16.5359	−0.643172	−0.321586	+	0.946880i	$0.604216\pi$
$662$	0	0
$663$	14.9282	0.579763
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.07180	−0.196381
$668$	0	0
$669$	71.1769	2.75186
$670$	0	0
$671$	13.0718	0.504631
$672$	0	0
$673$	−33.8038	−1.30304	−0.651521	−	0.758630i	$-0.725869\pi$
$673$	−33.8038	−1.30304	−0.651521	+	0.758630i	$0.725869\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	29.9090	1.14949	0.574747	−	0.818331i	$-0.305100\pi$
$677$	29.9090	1.14949	0.574747	+	0.818331i	$0.305100\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.46410	0.132745
$682$	0	0
$683$	15.8038	0.604717	0.302359	−	0.953194i	$-0.402226\pi$
$683$	15.8038	0.604717	0.302359	+	0.953194i	$0.402226\pi$
$684$	0	0
$685$	−0.535898	−0.0204756
$686$	0	0
$687$	−14.9282	−0.569546
$688$	0	0
$689$	8.53590	0.325192
$690$	0	0
$691$	38.7846	1.47544	0.737718	−	0.675109i	$-0.235903\pi$
$691$	38.7846	1.47544	0.737718	+	0.675109i	$0.235903\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.85641	0.298010
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	−38.2487	−1.44670
$700$	0	0
$701$	−22.2487	−0.840322	−0.420161	−	0.907450i	$-0.638026\pi$
$701$	−22.2487	−0.840322	−0.420161	+	0.907450i	$0.638026\pi$
$702$	0	0
$703$	−7.66025	−0.288912
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	0	0
$708$	0	0
$709$	18.7846	0.705471	0.352735	−	0.935723i	$-0.385252\pi$
$709$	18.7846	0.705471	0.352735	+	0.935723i	$0.385252\pi$
$710$	0	0
$711$	37.4641	1.40501
$712$	0	0
$713$	−5.85641	−0.219324
$714$	0	0
$715$	−1.46410	−0.0547543
$716$	0	0
$717$	−51.7128	−1.93125
$718$	0	0
$719$	47.5692	1.77403	0.887016	−	0.461738i	$-0.152774\pi$
$719$	47.5692	1.77403	0.887016	+	0.461738i	$0.152774\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	28.3923	1.05592
$724$	0	0
$725$	−3.46410	−0.128654
$726$	0	0
$727$	−21.0718	−0.781510	−0.390755	−	0.920495i	$-0.627786\pi$
$727$	−21.0718	−0.781510	−0.390755	+	0.920495i	$0.627786\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	21.8564	0.808388
$732$	0	0
$733$	−16.1436	−0.596277	−0.298139	−	0.954523i	$-0.596366\pi$
$733$	−16.1436	−0.596277	−0.298139	+	0.954523i	$0.596366\pi$
$734$	0	0
$735$	−19.1244	−0.705412
$736$	0	0
$737$	−8.39230	−0.309135
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	−29.3731	−1.07759	−0.538797	−	0.842436i	$-0.681121\pi$
$743$	−29.3731	−1.07759	−0.538797	+	0.842436i	$0.681121\pi$
$744$	0	0
$745$	−15.3205	−0.561300
$746$	0	0
$747$	24.3923	0.892468
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25.4641	−0.929198	−0.464599	−	0.885521i	$-0.653802\pi$
$751$	−25.4641	−0.929198	−0.464599	+	0.885521i	$0.653802\pi$
$752$	0	0
$753$	−5.07180	−0.184827
$754$	0	0
$755$	−5.46410	−0.198859
$756$	0	0
$757$	−1.21539	−0.0441741	−0.0220871	−	0.999756i	$-0.507031\pi$
$757$	−1.21539	−0.0441741	−0.0220871	+	0.999756i	$0.507031\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	24.3923	0.884220	0.442110	−	0.896961i	$-0.354230\pi$
$761$	24.3923	0.884220	0.442110	+	0.896961i	$0.354230\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−33.3205	−1.20471
$766$	0	0
$767$	1.07180	0.0387003
$768$	0	0
$769$	24.3923	0.879609	0.439805	−	0.898094i	$-0.355048\pi$
$769$	24.3923	0.879609	0.439805	+	0.898094i	$0.355048\pi$
$770$	0	0
$771$	−67.5692	−2.43345
$772$	0	0
$773$	10.1962	0.366730	0.183365	−	0.983045i	$-0.441301\pi$
$773$	10.1962	0.366730	0.183365	+	0.983045i	$0.441301\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.535898	−0.0192006
$780$	0	0
$781$	21.8564	0.782084
$782$	0	0
$783$	13.8564	0.495188
$784$	0	0
$785$	7.46410	0.266405
$786$	0	0
$787$	0.196152	0.00699208	0.00349604	−	0.999994i	$-0.498887\pi$
$787$	0.196152	0.00699208	0.00349604	+	0.999994i	$0.498887\pi$
$788$	0	0
$789$	52.7846	1.87918
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.78461	0.169906
$794$	0	0
$795$	−31.8564	−1.12983
$796$	0	0
$797$	16.0526	0.568611	0.284305	−	0.958734i	$-0.408237\pi$
$797$	16.0526	0.568611	0.284305	+	0.958734i	$0.408237\pi$
$798$	0	0
$799$	21.8564	0.773224
$800$	0	0
$801$	−15.4641	−0.546397
$802$	0	0
$803$	20.7846	0.733473
$804$	0	0
$805$	0	0
$806$	0	0
$807$	24.3923	0.858650
$808$	0	0
$809$	9.71281	0.341484	0.170742	−	0.985316i	$-0.445383\pi$
$809$	9.71281	0.341484	0.170742	+	0.985316i	$0.445383\pi$
$810$	0	0
$811$	−13.8564	−0.486564	−0.243282	−	0.969956i	$-0.578224\pi$
$811$	−13.8564	−0.486564	−0.243282	+	0.969956i	$0.578224\pi$
$812$	0	0
$813$	38.2487	1.34144
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−2.92820	−0.102445
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.7846	0.795188	0.397594	−	0.917561i	$-0.369845\pi$
$821$	22.7846	0.795188	0.397594	+	0.917561i	$0.369845\pi$
$822$	0	0
$823$	26.9282	0.938658	0.469329	−	0.883023i	$-0.344496\pi$
$823$	26.9282	0.938658	0.469329	+	0.883023i	$0.344496\pi$
$824$	0	0
$825$	5.46410	0.190236
$826$	0	0
$827$	−35.9090	−1.24868	−0.624339	−	0.781154i	$-0.714631\pi$
$827$	−35.9090	−1.24868	−0.624339	+	0.781154i	$0.714631\pi$
$828$	0	0
$829$	−33.7128	−1.17089	−0.585447	−	0.810711i	$-0.699081\pi$
$829$	−33.7128	−1.17089	−0.585447	+	0.810711i	$0.699081\pi$
$830$	0	0
$831$	56.1051	1.94626
$832$	0	0
$833$	−52.2487	−1.81031
$834$	0	0
$835$	−6.73205	−0.232972
$836$	0	0
$837$	16.0000	0.553041
$838$	0	0
$839$	−23.6077	−0.815028	−0.407514	−	0.913199i	$-0.633604\pi$
$839$	−23.6077	−0.815028	−0.407514	+	0.913199i	$0.633604\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	61.1769	2.10704
$844$	0	0
$845$	12.4641	0.428778
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−25.8564	−0.887390
$850$	0	0
$851$	11.2154	0.384459
$852$	0	0
$853$	14.7846	0.506215	0.253108	−	0.967438i	$-0.418547\pi$
$853$	14.7846	0.506215	0.253108	+	0.967438i	$0.418547\pi$
$854$	0	0
$855$	4.46410	0.152669
$856$	0	0
$857$	−29.1244	−0.994869	−0.497435	−	0.867502i	$-0.665725\pi$
$857$	−29.1244	−0.994869	−0.497435	+	0.867502i	$0.665725\pi$
$858$	0	0
$859$	−30.1436	−1.02849	−0.514243	−	0.857644i	$-0.671927\pi$
$859$	−30.1436	−1.02849	−0.514243	+	0.857644i	$0.671927\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.2679	0.860131	0.430065	−	0.902798i	$-0.358490\pi$
$863$	25.2679	0.860131	0.430065	+	0.902798i	$0.358490\pi$
$864$	0	0
$865$	17.1244	0.582246
$866$	0	0
$867$	−105.765	−3.59198
$868$	0	0
$869$	−16.7846	−0.569379
$870$	0	0
$871$	−3.07180	−0.104084
$872$	0	0
$873$	−69.9090	−2.36606
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.4449	−0.420233	−0.210117	−	0.977676i	$-0.567384\pi$
$877$	−12.4449	−0.420233	−0.210117	+	0.977676i	$0.567384\pi$
$878$	0	0
$879$	−49.7128	−1.67677
$880$	0	0
$881$	12.3923	0.417507	0.208754	−	0.977968i	$-0.433059\pi$
$881$	12.3923	0.417507	0.208754	+	0.977968i	$0.433059\pi$
$882$	0	0
$883$	−38.6410	−1.30037	−0.650187	−	0.759774i	$-0.725309\pi$
$883$	−38.6410	−1.30037	−0.650187	+	0.759774i	$0.725309\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	20.8756	0.700936	0.350468	−	0.936575i	$-0.386023\pi$
$887$	20.8756	0.700936	0.350468	+	0.936575i	$0.386023\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4.92820	0.165101
$892$	0	0
$893$	−2.92820	−0.0979886
$894$	0	0
$895$	−9.46410	−0.316350
$896$	0	0
$897$	2.92820	0.0977699
$898$	0	0
$899$	13.8564	0.462137
$900$	0	0
$901$	−87.0333	−2.89950
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15.8564	0.527085
$906$	0	0
$907$	34.4449	1.14372	0.571861	−	0.820350i	$-0.306221\pi$
$907$	34.4449	1.14372	0.571861	+	0.820350i	$0.306221\pi$
$908$	0	0
$909$	6.53590	0.216782
$910$	0	0
$911$	−49.4641	−1.63882	−0.819409	−	0.573209i	$-0.805698\pi$
$911$	−49.4641	−1.63882	−0.819409	+	0.573209i	$0.805698\pi$
$912$	0	0
$913$	−10.9282	−0.361671
$914$	0	0
$915$	−17.8564	−0.590315
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−2.92820	−0.0965925	−0.0482963	−	0.998833i	$-0.515379\pi$
$919$	−2.92820	−0.0965925	−0.0482963	+	0.998833i	$0.515379\pi$
$920$	0	0
$921$	69.0333	2.27473
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	7.66025	0.251868
$926$	0	0
$927$	−25.2679	−0.829908
$928$	0	0
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	0	0
$933$	54.2487	1.77602
$934$	0	0
$935$	14.9282	0.488204
$936$	0	0
$937$	29.6077	0.967241	0.483621	−	0.875278i	$-0.339321\pi$
$937$	29.6077	0.967241	0.483621	+	0.875278i	$0.339321\pi$
$938$	0	0
$939$	43.3205	1.41371
$940$	0	0
$941$	35.0718	1.14331	0.571654	−	0.820495i	$-0.306302\pi$
$941$	35.0718	1.14331	0.571654	+	0.820495i	$0.306302\pi$
$942$	0	0
$943$	0.784610	0.0255504
$944$	0	0
$945$	0	0
$946$	0	0
$947$	50.2487	1.63286	0.816432	−	0.577442i	$-0.195949\pi$
$947$	50.2487	1.63286	0.816432	+	0.577442i	$0.195949\pi$
$948$	0	0
$949$	7.60770	0.246956
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−17.4115	−0.564015	−0.282008	−	0.959412i	$-0.591000\pi$
$953$	−17.4115	−0.564015	−0.282008	+	0.959412i	$0.591000\pi$
$954$	0	0
$955$	−10.9282	−0.353628
$956$	0	0
$957$	−18.9282	−0.611862
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−67.5167	−2.17569
$964$	0	0
$965$	−14.1962	−0.456990
$966$	0	0
$967$	57.1769	1.83869	0.919343	−	0.393457i	$-0.128721\pi$
$967$	57.1769	1.83869	0.919343	+	0.393457i	$0.128721\pi$
$968$	0	0
$969$	20.3923	0.655095
$970$	0	0
$971$	−6.92820	−0.222337	−0.111168	−	0.993802i	$-0.535459\pi$
$971$	−6.92820	−0.222337	−0.111168	+	0.993802i	$0.535459\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	−8.44486	−0.270175	−0.135088	−	0.990834i	$-0.543132\pi$
$977$	−8.44486	−0.270175	−0.135088	+	0.990834i	$0.543132\pi$
$978$	0	0
$979$	6.92820	0.221426
$980$	0	0
$981$	−62.4974	−1.99539
$982$	0	0
$983$	3.51666	0.112164	0.0560820	−	0.998426i	$-0.482139\pi$
$983$	3.51666	0.112164	0.0560820	+	0.998426i	$0.482139\pi$
$984$	0	0
$985$	−11.8564	−0.377777
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.28719	0.136325
$990$	0	0
$991$	3.32051	0.105479	0.0527397	−	0.998608i	$-0.483205\pi$
$991$	3.32051	0.105479	0.0527397	+	0.998608i	$0.483205\pi$
$992$	0	0
$993$	38.9282	1.23535
$994$	0	0
$995$	−18.9282	−0.600064
$996$	0	0
$997$	−38.3923	−1.21590	−0.607948	−	0.793977i	$-0.708007\pi$
$997$	−38.3923	−1.21590	−0.607948	+	0.793977i	$0.708007\pi$
$998$	0	0
$999$	−30.6410	−0.969439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.a.k.1.1		2
4.3	odd	2		760.2.a.g.1.2	✓	2
5.4	even	2		7600.2.a.bd.1.2		2
8.3	odd	2		6080.2.a.ba.1.1		2
8.5	even	2		6080.2.a.bk.1.2		2
12.11	even	2		6840.2.a.y.1.1		2
20.3	even	4		3800.2.d.h.3649.4		4
20.7	even	4		3800.2.d.h.3649.1		4
20.19	odd	2		3800.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.g.1.2	✓	2	4.3	odd	2
1520.2.a.k.1.1		2	1.1	even	1	trivial
3800.2.a.j.1.1		2	20.19	odd	2
3800.2.d.h.3649.1		4	20.7	even	4
3800.2.d.h.3649.4		4	20.3	even	4
6080.2.a.ba.1.1		2	8.3	odd	2
6080.2.a.bk.1.2		2	8.5	even	2
6840.2.a.y.1.1		2	12.11	even	2
7600.2.a.bd.1.2		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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -1.00000 q^{5} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} -1.00000 q^{5} +4.46410 q^{9} -2.00000 q^{11} -0.732051 q^{13} +2.73205 q^{15} +7.46410 q^{17} -1.00000 q^{19} +1.46410 q^{23} +1.00000 q^{25} -4.00000 q^{27} -3.46410 q^{29} -4.00000 q^{31} +5.46410 q^{33} +7.66025 q^{37} +2.00000 q^{39} +0.535898 q^{41} +2.92820 q^{43} -4.46410 q^{45} +2.92820 q^{47} -7.00000 q^{49} -20.3923 q^{51} -11.6603 q^{53} +2.00000 q^{55} +2.73205 q^{57} -1.46410 q^{59} -6.53590 q^{61} +0.732051 q^{65} +4.19615 q^{67} -4.00000 q^{69} -10.9282 q^{71} -10.3923 q^{73} -2.73205 q^{75} +8.39230 q^{79} -2.46410 q^{81} +5.46410 q^{83} -7.46410 q^{85} +9.46410 q^{87} -3.46410 q^{89} +10.9282 q^{93} +1.00000 q^{95} -15.6603 q^{97} -8.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 8 q^{17} - 2 q^{19} - 4 q^{23} + 2 q^{25} - 8 q^{27} - 8 q^{31} + 4 q^{33} - 2 q^{37} + 4 q^{39} + 8 q^{41} - 8 q^{43} - 2 q^{45} - 8 q^{47} - 14 q^{49} - 20 q^{51} - 6 q^{53} + 4 q^{55} + 2 q^{57} + 4 q^{59} - 20 q^{61} - 2 q^{65} - 2 q^{67} - 8 q^{69} - 8 q^{71} - 2 q^{75} - 4 q^{79} + 2 q^{81} + 4 q^{83} - 8 q^{85} + 12 q^{87} + 8 q^{93} + 2 q^{95} - 14 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 - 4 * q^11 + 2 * q^13 + 2 * q^15 + 8 * q^17 - 2 * q^19 - 4 * q^23 + 2 * q^25 - 8 * q^27 - 8 * q^31 + 4 * q^33 - 2 * q^37 + 4 * q^39 + 8 * q^41 - 8 * q^43 - 2 * q^45 - 8 * q^47 - 14 * q^49 - 20 * q^51 - 6 * q^53 + 4 * q^55 + 2 * q^57 + 4 * q^59 - 20 * q^61 - 2 * q^65 - 2 * q^67 - 8 * q^69 - 8 * q^71 - 2 * q^75 - 4 * q^79 + 2 * q^81 + 4 * q^83 - 8 * q^85 + 12 * q^87 + 8 * q^93 + 2 * q^95 - 14 * q^97 - 4 * q^99

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