LMFDB - Embedded newform 1520.2.a.k.1.2

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			1.2
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)

$f(q)$	$=$	$q+0.732051 q^{3} -1.00000 q^{5} -2.46410 q^{9} +O(q^{10})$	$q+0.732051 q^{3} -1.00000 q^{5} -2.46410 q^{9} -2.00000 q^{11} +2.73205 q^{13} -0.732051 q^{15} +0.535898 q^{17} -1.00000 q^{19} -5.46410 q^{23} +1.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} -4.00000 q^{31} -1.46410 q^{33} -9.66025 q^{37} +2.00000 q^{39} +7.46410 q^{41} -10.9282 q^{43} +2.46410 q^{45} -10.9282 q^{47} -7.00000 q^{49} +0.392305 q^{51} +5.66025 q^{53} +2.00000 q^{55} -0.732051 q^{57} +5.46410 q^{59} -13.4641 q^{61} -2.73205 q^{65} -6.19615 q^{67} -4.00000 q^{69} +2.92820 q^{71} +10.3923 q^{73} +0.732051 q^{75} -12.3923 q^{79} +4.46410 q^{81} -1.46410 q^{83} -0.535898 q^{85} +2.53590 q^{87} +3.46410 q^{89} -2.92820 q^{93} +1.00000 q^{95} +1.66025 q^{97} +4.92820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	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
$9$	−2.46410	−0.821367
$10$	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
$13$	2.73205	0.757735	0.378867	−	0.925451i	$-0.376314\pi$
$13$	2.73205	0.757735	0.378867	+	0.925451i	$0.376314\pi$
$14$	0	0
$15$	−0.732051	−0.189015
$16$	0	0
$17$	0.535898	0.129974	0.0649872	−	0.997886i	$-0.479299\pi$
$17$	0.535898	0.129974	0.0649872	+	0.997886i	$0.479299\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.46410	−1.13934	−0.569672	−	0.821872i	$-0.692930\pi$
$23$	−5.46410	−1.13934	−0.569672	+	0.821872i	$0.692930\pi$
$24$	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	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	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
$33$	−1.46410	−0.254867
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.66025	−1.58814	−0.794068	−	0.607829i	$-0.792041\pi$
$37$	−9.66025	−1.58814	−0.794068	+	0.607829i	$0.792041\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	7.46410	1.16570	0.582848	−	0.812581i	$-0.301938\pi$
$41$	7.46410	1.16570	0.582848	+	0.812581i	$0.301938\pi$
$42$	0	0
$43$	−10.9282	−1.66654	−0.833268	−	0.552870i	$-0.813533\pi$
$43$	−10.9282	−1.66654	−0.833268	+	0.552870i	$0.813533\pi$
$44$	0	0
$45$	2.46410	0.367327
$46$	0	0
$47$	−10.9282	−1.59404	−0.797021	−	0.603951i	$-0.793592\pi$
$47$	−10.9282	−1.59404	−0.797021	+	0.603951i	$0.793592\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0.392305	0.0549337
$52$	0	0
$53$	5.66025	0.777496	0.388748	−	0.921344i	$-0.372908\pi$
$53$	5.66025	0.777496	0.388748	+	0.921344i	$0.372908\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	−0.732051	−0.0969625
$58$	0	0
$59$	5.46410	0.711365	0.355683	−	0.934607i	$-0.384248\pi$
$59$	5.46410	0.711365	0.355683	+	0.934607i	$0.384248\pi$
$60$	0	0
$61$	−13.4641	−1.72390	−0.861951	−	0.506992i	$-0.830757\pi$
$61$	−13.4641	−1.72390	−0.861951	+	0.506992i	$0.830757\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.73205	−0.338869
$66$	0	0
$67$	−6.19615	−0.756980	−0.378490	−	0.925605i	$-0.623557\pi$
$67$	−6.19615	−0.756980	−0.378490	+	0.925605i	$0.623557\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	2.92820	0.347514	0.173757	−	0.984789i	$-0.444409\pi$
$71$	2.92820	0.347514	0.173757	+	0.984789i	$0.444409\pi$
$72$	0	0
$73$	10.3923	1.21633	0.608164	−	0.793812i	$-0.291906\pi$
$73$	10.3923	1.21633	0.608164	+	0.793812i	$0.291906\pi$
$74$	0	0
$75$	0.732051	0.0845299
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.3923	−1.39424	−0.697122	−	0.716953i	$-0.745536\pi$
$79$	−12.3923	−1.39424	−0.697122	+	0.716953i	$0.745536\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−1.46410	−0.160706	−0.0803530	−	0.996766i	$-0.525605\pi$
$83$	−1.46410	−0.160706	−0.0803530	+	0.996766i	$0.525605\pi$
$84$	0	0
$85$	−0.535898	−0.0581263
$86$	0	0
$87$	2.53590	0.271877
$88$	0	0
$89$	3.46410	0.367194	0.183597	−	0.983002i	$-0.441226\pi$
$89$	3.46410	0.367194	0.183597	+	0.983002i	$0.441226\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.92820	−0.303641
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	1.66025	0.168573	0.0842866	−	0.996442i	$-0.473139\pi$
$97$	1.66025	0.168573	0.0842866	+	0.996442i	$0.473139\pi$
$98$	0	0
$99$	4.92820	0.495303
$100$	0	0
$101$	−5.46410	−0.543698	−0.271849	−	0.962340i	$-0.587635\pi$
$101$	−5.46410	−0.543698	−0.271849	+	0.962340i	$0.587635\pi$
$102$	0	0
$103$	11.6603	1.14892	0.574459	−	0.818533i	$-0.305212\pi$
$103$	11.6603	1.14892	0.574459	+	0.818533i	$0.305212\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.12436	0.882085	0.441042	−	0.897486i	$-0.354609\pi$
$107$	9.12436	0.882085	0.441042	+	0.897486i	$0.354609\pi$
$108$	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
$111$	−7.07180	−0.671225
$112$	0	0
$113$	13.2679	1.24814	0.624072	−	0.781367i	$-0.285477\pi$
$113$	13.2679	1.24814	0.624072	+	0.781367i	$0.285477\pi$
$114$	0	0
$115$	5.46410	0.509530
$116$	0	0
$117$	−6.73205	−0.622378
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	5.46410	0.492681
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.339746	−0.0301476	−0.0150738	−	0.999886i	$-0.504798\pi$
$127$	−0.339746	−0.0301476	−0.0150738	+	0.999886i	$0.504798\pi$
$128$	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
$137$	7.46410	0.637701	0.318851	−	0.947805i	$-0.396703\pi$
$137$	7.46410	0.637701	0.318851	+	0.947805i	$0.396703\pi$
$138$	0	0
$139$	19.8564	1.68420	0.842099	−	0.539323i	$-0.181320\pi$
$139$	19.8564	1.68420	0.842099	+	0.539323i	$0.181320\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−5.46410	−0.456931
$144$	0	0
$145$	−3.46410	−0.287678
$146$	0	0
$147$	−5.12436	−0.422650
$148$	0	0
$149$	−19.3205	−1.58280	−0.791399	−	0.611300i	$-0.790647\pi$
$149$	−19.3205	−1.58280	−0.791399	+	0.611300i	$0.790647\pi$
$150$	0	0
$151$	−1.46410	−0.119147	−0.0595734	−	0.998224i	$-0.518974\pi$
$151$	−1.46410	−0.119147	−0.0595734	+	0.998224i	$0.518974\pi$
$152$	0	0
$153$	−1.32051	−0.106757
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	−0.535898	−0.0427693	−0.0213847	−	0.999771i	$-0.506807\pi$
$157$	−0.535898	−0.0427693	−0.0213847	+	0.999771i	$0.506807\pi$
$158$	0	0
$159$	4.14359	0.328608
$160$	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
$165$	1.46410	0.113980
$166$	0	0
$167$	3.26795	0.252882	0.126441	−	0.991974i	$-0.459645\pi$
$167$	3.26795	0.252882	0.126441	+	0.991974i	$0.459645\pi$
$168$	0	0
$169$	−5.53590	−0.425838
$170$	0	0
$171$	2.46410	0.188435
$172$	0	0
$173$	7.12436	0.541655	0.270827	−	0.962628i	$-0.412703\pi$
$173$	7.12436	0.541655	0.270827	+	0.962628i	$0.412703\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	2.53590	0.189542	0.0947710	−	0.995499i	$-0.469788\pi$
$179$	2.53590	0.189542	0.0947710	+	0.995499i	$0.469788\pi$
$180$	0	0
$181$	11.8564	0.881280	0.440640	−	0.897684i	$-0.354752\pi$
$181$	11.8564	0.881280	0.440640	+	0.897684i	$0.354752\pi$
$182$	0	0
$183$	−9.85641	−0.728607
$184$	0	0
$185$	9.66025	0.710236
$186$	0	0
$187$	−1.07180	−0.0783775
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.92820	−0.211877	−0.105939	−	0.994373i	$-0.533785\pi$
$191$	−2.92820	−0.211877	−0.105939	+	0.994373i	$0.533785\pi$
$192$	0	0
$193$	3.80385	0.273807	0.136903	−	0.990584i	$-0.456285\pi$
$193$	3.80385	0.273807	0.136903	+	0.990584i	$0.456285\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−15.8564	−1.12972	−0.564861	−	0.825186i	$-0.691070\pi$
$197$	−15.8564	−1.12972	−0.564861	+	0.825186i	$0.691070\pi$
$198$	0	0
$199$	5.07180	0.359530	0.179765	−	0.983710i	$-0.442466\pi$
$199$	5.07180	0.359530	0.179765	+	0.983710i	$0.442466\pi$
$200$	0	0
$201$	−4.53590	−0.319938
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−7.46410	−0.521315
$206$	0	0
$207$	13.4641	0.935820
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	−4.39230	−0.302379	−0.151189	−	0.988505i	$-0.548310\pi$
$211$	−4.39230	−0.302379	−0.151189	+	0.988505i	$0.548310\pi$
$212$	0	0
$213$	2.14359	0.146877
$214$	0	0
$215$	10.9282	0.745297
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.60770	0.514080
$220$	0	0
$221$	1.46410	0.0984861
$222$	0	0
$223$	12.0526	0.807099	0.403550	−	0.914958i	$-0.367776\pi$
$223$	12.0526	0.807099	0.403550	+	0.914958i	$0.367776\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	−4.73205	−0.314077	−0.157039	−	0.987592i	$-0.550195\pi$
$227$	−4.73205	−0.314077	−0.157039	+	0.987592i	$0.550195\pi$
$228$	0	0
$229$	−1.46410	−0.0967506	−0.0483753	−	0.998829i	$-0.515404\pi$
$229$	−1.46410	−0.0967506	−0.0483753	+	0.998829i	$0.515404\pi$
$230$	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
$235$	10.9282	0.712877
$236$	0	0
$237$	−9.07180	−0.589277
$238$	0	0
$239$	5.07180	0.328067	0.164034	−	0.986455i	$-0.447549\pi$
$239$	5.07180	0.328067	0.164034	+	0.986455i	$0.447549\pi$
$240$	0	0
$241$	10.3923	0.669427	0.334714	−	0.942320i	$-0.391360\pi$
$241$	10.3923	0.669427	0.334714	+	0.942320i	$0.391360\pi$
$242$	0	0
$243$	15.2679	0.979439
$244$	0	0
$245$	7.00000	0.447214
$246$	0	0
$247$	−2.73205	−0.173836
$248$	0	0
$249$	−1.07180	−0.0679224
$250$	0	0
$251$	−25.8564	−1.63204	−0.816021	−	0.578022i	$-0.803825\pi$
$251$	−25.8564	−1.63204	−0.816021	+	0.578022i	$0.803825\pi$
$252$	0	0
$253$	10.9282	0.687050
$254$	0	0
$255$	−0.392305	−0.0245671
$256$	0	0
$257$	21.2679	1.32666	0.663329	−	0.748328i	$-0.269143\pi$
$257$	21.2679	1.32666	0.663329	+	0.748328i	$0.269143\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−8.53590	−0.528359
$262$	0	0
$263$	15.3205	0.944703	0.472351	−	0.881410i	$-0.343405\pi$
$263$	15.3205	0.944703	0.472351	+	0.881410i	$0.343405\pi$
$264$	0	0
$265$	−5.66025	−0.347707
$266$	0	0
$267$	2.53590	0.155194
$268$	0	0
$269$	4.92820	0.300478	0.150239	−	0.988650i	$-0.451996\pi$
$269$	4.92820	0.300478	0.150239	+	0.988650i	$0.451996\pi$
$270$	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
$277$	−27.4641	−1.65016	−0.825079	−	0.565017i	$-0.808869\pi$
$277$	−27.4641	−1.65016	−0.825079	+	0.565017i	$0.808869\pi$
$278$	0	0
$279$	9.85641	0.590088
$280$	0	0
$281$	−1.60770	−0.0959071	−0.0479535	−	0.998850i	$-0.515270\pi$
$281$	−1.60770	−0.0959071	−0.0479535	+	0.998850i	$0.515270\pi$
$282$	0	0
$283$	2.53590	0.150744	0.0753718	−	0.997156i	$-0.475986\pi$
$283$	2.53590	0.150744	0.0753718	+	0.997156i	$0.475986\pi$
$284$	0	0
$285$	0.732051	0.0433629
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.7128	−0.983107
$290$	0	0
$291$	1.21539	0.0712474
$292$	0	0
$293$	7.80385	0.455906	0.227953	−	0.973672i	$-0.426797\pi$
$293$	7.80385	0.455906	0.227953	+	0.973672i	$0.426797\pi$
$294$	0	0
$295$	−5.46410	−0.318132
$296$	0	0
$297$	8.00000	0.464207
$298$	0	0
$299$	−14.9282	−0.863320
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−4.00000	−0.229794
$304$	0	0
$305$	13.4641	0.770952
$306$	0	0
$307$	−28.7321	−1.63982	−0.819912	−	0.572489i	$-0.805978\pi$
$307$	−28.7321	−1.63982	−0.819912	+	0.572489i	$0.805978\pi$
$308$	0	0
$309$	8.53590	0.485590
$310$	0	0
$311$	7.85641	0.445496	0.222748	−	0.974876i	$-0.428497\pi$
$311$	7.85641	0.445496	0.222748	+	0.974876i	$0.428497\pi$
$312$	0	0
$313$	11.8564	0.670164	0.335082	−	0.942189i	$-0.391236\pi$
$313$	11.8564	0.670164	0.335082	+	0.942189i	$0.391236\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.5885	1.38103	0.690513	−	0.723320i	$-0.257385\pi$
$317$	24.5885	1.38103	0.690513	+	0.723320i	$0.257385\pi$
$318$	0	0
$319$	−6.92820	−0.387905
$320$	0	0
$321$	6.67949	0.372813
$322$	0	0
$323$	−0.535898	−0.0298182
$324$	0	0
$325$	2.73205	0.151547
$326$	0	0
$327$	−10.2487	−0.566755
$328$	0	0
$329$	0	0
$330$	0	0
$331$	34.2487	1.88248	0.941240	−	0.337739i	$-0.109662\pi$
$331$	34.2487	1.88248	0.941240	+	0.337739i	$0.109662\pi$
$332$	0	0
$333$	23.8038	1.30444
$334$	0	0
$335$	6.19615	0.338532
$336$	0	0
$337$	−5.66025	−0.308334	−0.154167	−	0.988045i	$-0.549269\pi$
$337$	−5.66025	−0.308334	−0.154167	+	0.988045i	$0.549269\pi$
$338$	0	0
$339$	9.71281	0.527528
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	−12.3923	−0.665254	−0.332627	−	0.943059i	$-0.607935\pi$
$347$	−12.3923	−0.665254	−0.332627	+	0.943059i	$0.607935\pi$
$348$	0	0
$349$	18.7846	1.00552	0.502759	−	0.864427i	$-0.332318\pi$
$349$	18.7846	1.00552	0.502759	+	0.864427i	$0.332318\pi$
$350$	0	0
$351$	−10.9282	−0.583304
$352$	0	0
$353$	−16.2487	−0.864832	−0.432416	−	0.901674i	$-0.642339\pi$
$353$	−16.2487	−0.864832	−0.432416	+	0.901674i	$0.642339\pi$
$354$	0	0
$355$	−2.92820	−0.155413
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4.92820	0.260101	0.130050	−	0.991507i	$-0.458486\pi$
$359$	4.92820	0.260101	0.130050	+	0.991507i	$0.458486\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−5.12436	−0.268959
$364$	0	0
$365$	−10.3923	−0.543958
$366$	0	0
$367$	2.92820	0.152851	0.0764255	−	0.997075i	$-0.475649\pi$
$367$	2.92820	0.152851	0.0764255	+	0.997075i	$0.475649\pi$
$368$	0	0
$369$	−18.3923	−0.957465
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−0.196152	−0.0101564	−0.00507819	−	0.999987i	$-0.501616\pi$
$373$	−0.196152	−0.0101564	−0.00507819	+	0.999987i	$0.501616\pi$
$374$	0	0
$375$	−0.732051	−0.0378029
$376$	0	0
$377$	9.46410	0.487426
$378$	0	0
$379$	−34.6410	−1.77939	−0.889695	−	0.456556i	$-0.849083\pi$
$379$	−34.6410	−1.77939	−0.889695	+	0.456556i	$0.849083\pi$
$380$	0	0
$381$	−0.248711	−0.0127419
$382$	0	0
$383$	−21.8038	−1.11412	−0.557062	−	0.830471i	$-0.688072\pi$
$383$	−21.8038	−1.11412	−0.557062	+	0.830471i	$0.688072\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	26.9282	1.36884
$388$	0	0
$389$	11.8564	0.601144	0.300572	−	0.953759i	$-0.402822\pi$
$389$	11.8564	0.601144	0.300572	+	0.953759i	$0.402822\pi$
$390$	0	0
$391$	−2.92820	−0.148086
$392$	0	0
$393$	−8.78461	−0.443125
$394$	0	0
$395$	12.3923	0.623525
$396$	0	0
$397$	−12.5359	−0.629159	−0.314579	−	0.949231i	$-0.601863\pi$
$397$	−12.5359	−0.629159	−0.314579	+	0.949231i	$0.601863\pi$
$398$	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
$403$	−10.9282	−0.544373
$404$	0	0
$405$	−4.46410	−0.221823
$406$	0	0
$407$	19.3205	0.957682
$408$	0	0
$409$	28.9282	1.43041	0.715204	−	0.698916i	$-0.246334\pi$
$409$	28.9282	1.43041	0.715204	+	0.698916i	$0.246334\pi$
$410$	0	0
$411$	5.46410	0.269524
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1.46410	0.0718699
$416$	0	0
$417$	14.5359	0.711826
$418$	0	0
$419$	1.85641	0.0906914	0.0453457	−	0.998971i	$-0.485561\pi$
$419$	1.85641	0.0906914	0.0453457	+	0.998971i	$0.485561\pi$
$420$	0	0
$421$	−32.2487	−1.57171	−0.785853	−	0.618413i	$-0.787776\pi$
$421$	−32.2487	−1.57171	−0.785853	+	0.618413i	$0.787776\pi$
$422$	0	0
$423$	26.9282	1.30929
$424$	0	0
$425$	0.535898	0.0259949
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−7.60770	−0.366450	−0.183225	−	0.983071i	$-0.558654\pi$
$431$	−7.60770	−0.366450	−0.183225	+	0.983071i	$0.558654\pi$
$432$	0	0
$433$	24.9808	1.20050	0.600249	−	0.799813i	$-0.295068\pi$
$433$	24.9808	1.20050	0.600249	+	0.799813i	$0.295068\pi$
$434$	0	0
$435$	−2.53590	−0.121587
$436$	0	0
$437$	5.46410	0.261383
$438$	0	0
$439$	22.2487	1.06187	0.530937	−	0.847412i	$-0.321840\pi$
$439$	22.2487	1.06187	0.530937	+	0.847412i	$0.321840\pi$
$440$	0	0
$441$	17.2487	0.821367
$442$	0	0
$443$	26.5359	1.26076	0.630379	−	0.776287i	$-0.282899\pi$
$443$	26.5359	1.26076	0.630379	+	0.776287i	$0.282899\pi$
$444$	0	0
$445$	−3.46410	−0.164214
$446$	0	0
$447$	−14.1436	−0.668969
$448$	0	0
$449$	32.6410	1.54042	0.770212	−	0.637787i	$-0.220150\pi$
$449$	32.6410	1.54042	0.770212	+	0.637787i	$0.220150\pi$
$450$	0	0
$451$	−14.9282	−0.702942
$452$	0	0
$453$	−1.07180	−0.0503574
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.0718	0.705029	0.352514	−	0.935806i	$-0.385327\pi$
$457$	15.0718	0.705029	0.352514	+	0.935806i	$0.385327\pi$
$458$	0	0
$459$	−2.14359	−0.100054
$460$	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
$463$	38.2487	1.77757	0.888784	−	0.458326i	$-0.151551\pi$
$463$	38.2487	1.77757	0.888784	+	0.458326i	$0.151551\pi$
$464$	0	0
$465$	2.92820	0.135792
$466$	0	0
$467$	0.679492	0.0314431	0.0157216	−	0.999876i	$-0.494995\pi$
$467$	0.679492	0.0314431	0.0157216	+	0.999876i	$0.494995\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.392305	−0.0180765
$472$	0	0
$473$	21.8564	1.00496
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−13.9474	−0.638609
$478$	0	0
$479$	7.85641	0.358968	0.179484	−	0.983761i	$-0.442557\pi$
$479$	7.85641	0.358968	0.179484	+	0.983761i	$0.442557\pi$
$480$	0	0
$481$	−26.3923	−1.20339
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.66025	−0.0753883
$486$	0	0
$487$	−39.2679	−1.77940	−0.889700	−	0.456545i	$-0.849087\pi$
$487$	−39.2679	−1.77940	−0.889700	+	0.456545i	$0.849087\pi$
$488$	0	0
$489$	2.92820	0.132418
$490$	0	0
$491$	−25.8564	−1.16688	−0.583442	−	0.812155i	$-0.698294\pi$
$491$	−25.8564	−1.16688	−0.583442	+	0.812155i	$0.698294\pi$
$492$	0	0
$493$	1.85641	0.0836083
$494$	0	0
$495$	−4.92820	−0.221506
$496$	0	0
$497$	0	0
$498$	0	0
$499$	10.7846	0.482785	0.241393	−	0.970428i	$-0.422396\pi$
$499$	10.7846	0.482785	0.241393	+	0.970428i	$0.422396\pi$
$500$	0	0
$501$	2.39230	0.106880
$502$	0	0
$503$	11.3205	0.504757	0.252378	−	0.967629i	$-0.418787\pi$
$503$	11.3205	0.504757	0.252378	+	0.967629i	$0.418787\pi$
$504$	0	0
$505$	5.46410	0.243149
$506$	0	0
$507$	−4.05256	−0.179980
$508$	0	0
$509$	−18.3923	−0.815225	−0.407612	−	0.913155i	$-0.633639\pi$
$509$	−18.3923	−0.815225	−0.407612	+	0.913155i	$0.633639\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	−11.6603	−0.513812
$516$	0	0
$517$	21.8564	0.961244
$518$	0	0
$519$	5.21539	0.228930
$520$	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
$523$	15.2679	0.667621	0.333810	−	0.942640i	$-0.391665\pi$
$523$	15.2679	0.667621	0.333810	+	0.942640i	$0.391665\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.14359	−0.0933764
$528$	0	0
$529$	6.85641	0.298105
$530$	0	0
$531$	−13.4641	−0.584292
$532$	0	0
$533$	20.3923	0.883289
$534$	0	0
$535$	−9.12436	−0.394480
$536$	0	0
$537$	1.85641	0.0801099
$538$	0	0
$539$	14.0000	0.603023
$540$	0	0
$541$	−44.3923	−1.90857	−0.954287	−	0.298891i	$-0.903383\pi$
$541$	−44.3923	−1.90857	−0.954287	+	0.298891i	$0.903383\pi$
$542$	0	0
$543$	8.67949	0.372473
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	14.5885	0.623757	0.311879	−	0.950122i	$-0.399042\pi$
$547$	14.5885	0.623757	0.311879	+	0.950122i	$0.399042\pi$
$548$	0	0
$549$	33.1769	1.41596
$550$	0	0
$551$	−3.46410	−0.147576
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.07180	0.300181
$556$	0	0
$557$	−36.2487	−1.53591	−0.767954	−	0.640505i	$-0.778725\pi$
$557$	−36.2487	−1.53591	−0.767954	+	0.640505i	$0.778725\pi$
$558$	0	0
$559$	−29.8564	−1.26279
$560$	0	0
$561$	−0.784610	−0.0331262
$562$	0	0
$563$	−22.5885	−0.951990	−0.475995	−	0.879448i	$-0.657912\pi$
$563$	−22.5885	−0.951990	−0.475995	+	0.879448i	$0.657912\pi$
$564$	0	0
$565$	−13.2679	−0.558187
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−26.7846	−1.12287	−0.561435	−	0.827521i	$-0.689750\pi$
$569$	−26.7846	−1.12287	−0.561435	+	0.827521i	$0.689750\pi$
$570$	0	0
$571$	−10.7846	−0.451322	−0.225661	−	0.974206i	$-0.572454\pi$
$571$	−10.7846	−0.451322	−0.225661	+	0.974206i	$0.572454\pi$
$572$	0	0
$573$	−2.14359	−0.0895499
$574$	0	0
$575$	−5.46410	−0.227869
$576$	0	0
$577$	8.14359	0.339022	0.169511	−	0.985528i	$-0.445781\pi$
$577$	8.14359	0.339022	0.169511	+	0.985528i	$0.445781\pi$
$578$	0	0
$579$	2.78461	0.115724
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−11.3205	−0.468848
$584$	0	0
$585$	6.73205	0.278336
$586$	0	0
$587$	40.1051	1.65532	0.827658	−	0.561233i	$-0.189673\pi$
$587$	40.1051	1.65532	0.827658	+	0.561233i	$0.189673\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	−11.6077	−0.477477
$592$	0	0
$593$	−8.14359	−0.334417	−0.167209	−	0.985922i	$-0.553475\pi$
$593$	−8.14359	−0.334417	−0.167209	+	0.985922i	$0.553475\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.71281	0.151955
$598$	0	0
$599$	−47.0333	−1.92173	−0.960865	−	0.277018i	$-0.910654\pi$
$599$	−47.0333	−1.92173	−0.960865	+	0.277018i	$0.910654\pi$
$600$	0	0
$601$	−43.4641	−1.77294	−0.886469	−	0.462788i	$-0.846849\pi$
$601$	−43.4641	−1.77294	−0.886469	+	0.462788i	$0.846849\pi$
$602$	0	0
$603$	15.2679	0.621759
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	20.4449	0.829831	0.414916	−	0.909860i	$-0.363811\pi$
$607$	20.4449	0.829831	0.414916	+	0.909860i	$0.363811\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−29.8564	−1.20786
$612$	0	0
$613$	−4.24871	−0.171604	−0.0858019	−	0.996312i	$-0.527345\pi$
$613$	−4.24871	−0.171604	−0.0858019	+	0.996312i	$0.527345\pi$
$614$	0	0
$615$	−5.46410	−0.220334
$616$	0	0
$617$	−3.46410	−0.139459	−0.0697297	−	0.997566i	$-0.522214\pi$
$617$	−3.46410	−0.139459	−0.0697297	+	0.997566i	$0.522214\pi$
$618$	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
$621$	21.8564	0.877067
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.46410	0.0584706
$628$	0	0
$629$	−5.17691	−0.206417
$630$	0	0
$631$	48.6410	1.93637	0.968184	−	0.250239i	$-0.0805091\pi$
$631$	48.6410	1.93637	0.968184	+	0.250239i	$0.0805091\pi$
$632$	0	0
$633$	−3.21539	−0.127800
$634$	0	0
$635$	0.339746	0.0134824
$636$	0	0
$637$	−19.1244	−0.757735
$638$	0	0
$639$	−7.21539	−0.285436
$640$	0	0
$641$	−45.0333	−1.77871	−0.889355	−	0.457218i	$-0.848846\pi$
$641$	−45.0333	−1.77871	−0.889355	+	0.457218i	$0.848846\pi$
$642$	0	0
$643$	−50.2487	−1.98162	−0.990808	−	0.135277i	$-0.956808\pi$
$643$	−50.2487	−1.98162	−0.990808	+	0.135277i	$0.956808\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	12.6795	0.498482	0.249241	−	0.968441i	$-0.419819\pi$
$647$	12.6795	0.498482	0.249241	+	0.968441i	$0.419819\pi$
$648$	0	0
$649$	−10.9282	−0.428969
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13.6077	−0.532510	−0.266255	−	0.963903i	$-0.585786\pi$
$653$	−13.6077	−0.532510	−0.266255	+	0.963903i	$0.585786\pi$
$654$	0	0
$655$	12.0000	0.468879
$656$	0	0
$657$	−25.6077	−0.999051
$658$	0	0
$659$	−11.6077	−0.452172	−0.226086	−	0.974107i	$-0.572593\pi$
$659$	−11.6077	−0.452172	−0.226086	+	0.974107i	$0.572593\pi$
$660$	0	0
$661$	−23.4641	−0.912648	−0.456324	−	0.889814i	$-0.650834\pi$
$661$	−23.4641	−0.912648	−0.456324	+	0.889814i	$0.650834\pi$
$662$	0	0
$663$	1.07180	0.0416251
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−18.9282	−0.732903
$668$	0	0
$669$	8.82309	0.341120
$670$	0	0
$671$	26.9282	1.03955
$672$	0	0
$673$	−44.1962	−1.70364	−0.851818	−	0.523837i	$-0.824500\pi$
$673$	−44.1962	−1.70364	−0.851818	+	0.523837i	$0.824500\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−35.9090	−1.38009	−0.690047	−	0.723765i	$-0.742410\pi$
$677$	−35.9090	−1.38009	−0.690047	+	0.723765i	$0.742410\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.46410	−0.132745
$682$	0	0
$683$	26.1962	1.00237	0.501184	−	0.865341i	$-0.332898\pi$
$683$	26.1962	1.00237	0.501184	+	0.865341i	$0.332898\pi$
$684$	0	0
$685$	−7.46410	−0.285189
$686$	0	0
$687$	−1.07180	−0.0408916
$688$	0	0
$689$	15.4641	0.589135
$690$	0	0
$691$	−2.78461	−0.105932	−0.0529658	−	0.998596i	$-0.516867\pi$
$691$	−2.78461	−0.105932	−0.0529658	+	0.998596i	$0.516867\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−19.8564	−0.753196
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	10.2487	0.387642
$700$	0	0
$701$	26.2487	0.991400	0.495700	−	0.868494i	$-0.334912\pi$
$701$	26.2487	0.991400	0.495700	+	0.868494i	$0.334912\pi$
$702$	0	0
$703$	9.66025	0.364343
$704$	0	0
$705$	8.00000	0.301297
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−22.7846	−0.855694	−0.427847	−	0.903851i	$-0.640728\pi$
$709$	−22.7846	−0.855694	−0.427847	+	0.903851i	$0.640728\pi$
$710$	0	0
$711$	30.5359	1.14519
$712$	0	0
$713$	21.8564	0.818529
$714$	0	0
$715$	5.46410	0.204346
$716$	0	0
$717$	3.71281	0.138658
$718$	0	0
$719$	−35.5692	−1.32651	−0.663254	−	0.748394i	$-0.730825\pi$
$719$	−35.5692	−1.32651	−0.663254	+	0.748394i	$0.730825\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.60770	0.282933
$724$	0	0
$725$	3.46410	0.128654
$726$	0	0
$727$	−34.9282	−1.29542	−0.647708	−	0.761889i	$-0.724272\pi$
$727$	−34.9282	−1.29542	−0.647708	+	0.761889i	$0.724272\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−5.85641	−0.216607
$732$	0	0
$733$	−43.8564	−1.61987	−0.809937	−	0.586517i	$-0.800499\pi$
$733$	−43.8564	−1.61987	−0.809937	+	0.586517i	$0.800499\pi$
$734$	0	0
$735$	5.12436	0.189015
$736$	0	0
$737$	12.3923	0.456476
$738$	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
$743$	43.3731	1.59120	0.795602	−	0.605820i	$-0.207155\pi$
$743$	43.3731	1.59120	0.795602	+	0.605820i	$0.207155\pi$
$744$	0	0
$745$	19.3205	0.707849
$746$	0	0
$747$	3.60770	0.131999
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−18.5359	−0.676385	−0.338192	−	0.941077i	$-0.609815\pi$
$751$	−18.5359	−0.676385	−0.338192	+	0.941077i	$0.609815\pi$
$752$	0	0
$753$	−18.9282	−0.689782
$754$	0	0
$755$	1.46410	0.0532841
$756$	0	0
$757$	−42.7846	−1.55503	−0.777517	−	0.628862i	$-0.783521\pi$
$757$	−42.7846	−1.55503	−0.777517	+	0.628862i	$0.783521\pi$
$758$	0	0
$759$	8.00000	0.290382
$760$	0	0
$761$	3.60770	0.130779	0.0653894	−	0.997860i	$-0.479171\pi$
$761$	3.60770	0.130779	0.0653894	+	0.997860i	$0.479171\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.32051	0.0477431
$766$	0	0
$767$	14.9282	0.539026
$768$	0	0
$769$	3.60770	0.130097	0.0650484	−	0.997882i	$-0.479280\pi$
$769$	3.60770	0.130097	0.0650484	+	0.997882i	$0.479280\pi$
$770$	0	0
$771$	15.5692	0.560712
$772$	0	0
$773$	−0.196152	−0.00705511	−0.00352756	−	0.999994i	$-0.501123\pi$
$773$	−0.196152	−0.00705511	−0.00352756	+	0.999994i	$0.501123\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.46410	−0.267429
$780$	0	0
$781$	−5.85641	−0.209559
$782$	0	0
$783$	−13.8564	−0.495188
$784$	0	0
$785$	0.535898	0.0191270
$786$	0	0
$787$	−10.1962	−0.363454	−0.181727	−	0.983349i	$-0.558169\pi$
$787$	−10.1962	−0.363454	−0.181727	+	0.983349i	$0.558169\pi$
$788$	0	0
$789$	11.2154	0.399278
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−36.7846	−1.30626
$794$	0	0
$795$	−4.14359	−0.146958
$796$	0	0
$797$	−22.0526	−0.781142	−0.390571	−	0.920573i	$-0.627722\pi$
$797$	−22.0526	−0.781142	−0.390571	+	0.920573i	$0.627722\pi$
$798$	0	0
$799$	−5.85641	−0.207185
$800$	0	0
$801$	−8.53590	−0.301601
$802$	0	0
$803$	−20.7846	−0.733473
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.60770	0.126997
$808$	0	0
$809$	−45.7128	−1.60718	−0.803588	−	0.595185i	$-0.797079\pi$
$809$	−45.7128	−1.60718	−0.803588	+	0.595185i	$0.797079\pi$
$810$	0	0
$811$	13.8564	0.486564	0.243282	−	0.969956i	$-0.421776\pi$
$811$	13.8564	0.486564	0.243282	+	0.969956i	$0.421776\pi$
$812$	0	0
$813$	−10.2487	−0.359438
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	10.9282	0.382329
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.7846	−0.655587	−0.327794	−	0.944749i	$-0.606305\pi$
$821$	−18.7846	−0.655587	−0.327794	+	0.944749i	$0.606305\pi$
$822$	0	0
$823$	13.0718	0.455654	0.227827	−	0.973702i	$-0.426838\pi$
$823$	13.0718	0.455654	0.227827	+	0.973702i	$0.426838\pi$
$824$	0	0
$825$	−1.46410	−0.0509735
$826$	0	0
$827$	29.9090	1.04004	0.520018	−	0.854155i	$-0.325925\pi$
$827$	29.9090	1.04004	0.520018	+	0.854155i	$0.325925\pi$
$828$	0	0
$829$	21.7128	0.754117	0.377059	−	0.926189i	$-0.376936\pi$
$829$	21.7128	0.754117	0.377059	+	0.926189i	$0.376936\pi$
$830$	0	0
$831$	−20.1051	−0.697439
$832$	0	0
$833$	−3.75129	−0.129974
$834$	0	0
$835$	−3.26795	−0.113092
$836$	0	0
$837$	16.0000	0.553041
$838$	0	0
$839$	−44.3923	−1.53259	−0.766296	−	0.642487i	$-0.777903\pi$
$839$	−44.3923	−1.53259	−0.766296	+	0.642487i	$0.777903\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	−1.17691	−0.0405351
$844$	0	0
$845$	5.53590	0.190441
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.85641	0.0637117
$850$	0	0
$851$	52.7846	1.80943
$852$	0	0
$853$	−26.7846	−0.917088	−0.458544	−	0.888672i	$-0.651629\pi$
$853$	−26.7846	−0.917088	−0.458544	+	0.888672i	$0.651629\pi$
$854$	0	0
$855$	−2.46410	−0.0842705
$856$	0	0
$857$	−4.87564	−0.166549	−0.0832744	−	0.996527i	$-0.526538\pi$
$857$	−4.87564	−0.166549	−0.0832744	+	0.996527i	$0.526538\pi$
$858$	0	0
$859$	−57.8564	−1.97404	−0.987018	−	0.160612i	$-0.948653\pi$
$859$	−57.8564	−1.97404	−0.987018	+	0.160612i	$0.948653\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.7321	0.978050	0.489025	−	0.872270i	$-0.337353\pi$
$863$	28.7321	0.978050	0.489025	+	0.872270i	$0.337353\pi$
$864$	0	0
$865$	−7.12436	−0.242235
$866$	0	0
$867$	−12.2346	−0.415510
$868$	0	0
$869$	24.7846	0.840760
$870$	0	0
$871$	−16.9282	−0.573590
$872$	0	0
$873$	−4.09103	−0.138461
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.4449	1.56833	0.784166	−	0.620551i	$-0.213091\pi$
$877$	46.4449	1.56833	0.784166	+	0.620551i	$0.213091\pi$
$878$	0	0
$879$	5.71281	0.192688
$880$	0	0
$881$	−8.39230	−0.282744	−0.141372	−	0.989957i	$-0.545151\pi$
$881$	−8.39230	−0.282744	−0.141372	+	0.989957i	$0.545151\pi$
$882$	0	0
$883$	30.6410	1.03115	0.515576	−	0.856844i	$-0.327578\pi$
$883$	30.6410	1.03115	0.515576	+	0.856844i	$0.327578\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	45.1244	1.51513	0.757564	−	0.652761i	$-0.226389\pi$
$887$	45.1244	1.51513	0.757564	+	0.652761i	$0.226389\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−8.92820	−0.299106
$892$	0	0
$893$	10.9282	0.365698
$894$	0	0
$895$	−2.53590	−0.0847657
$896$	0	0
$897$	−10.9282	−0.364882
$898$	0	0
$899$	−13.8564	−0.462137
$900$	0	0
$901$	3.03332	0.101055
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11.8564	−0.394120
$906$	0	0
$907$	−24.4449	−0.811678	−0.405839	−	0.913945i	$-0.633021\pi$
$907$	−24.4449	−0.811678	−0.405839	+	0.913945i	$0.633021\pi$
$908$	0	0
$909$	13.4641	0.446576
$910$	0	0
$911$	−42.5359	−1.40928	−0.704639	−	0.709566i	$-0.748891\pi$
$911$	−42.5359	−1.40928	−0.704639	+	0.709566i	$0.748891\pi$
$912$	0	0
$913$	2.92820	0.0969094
$914$	0	0
$915$	9.85641	0.325843
$916$	0	0
$917$	0	0
$918$	0	0
$919$	10.9282	0.360488	0.180244	−	0.983622i	$-0.442311\pi$
$919$	10.9282	0.360488	0.180244	+	0.983622i	$0.442311\pi$
$920$	0	0
$921$	−21.0333	−0.693071
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	−9.66025	−0.317627
$926$	0	0
$927$	−28.7321	−0.943684
$928$	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
$933$	5.75129	0.188289
$934$	0	0
$935$	1.07180	0.0350515
$936$	0	0
$937$	50.3923	1.64624	0.823122	−	0.567864i	$-0.192230\pi$
$937$	50.3923	1.64624	0.823122	+	0.567864i	$0.192230\pi$
$938$	0	0
$939$	8.67949	0.283245
$940$	0	0
$941$	48.9282	1.59501	0.797507	−	0.603310i	$-0.206152\pi$
$941$	48.9282	1.59501	0.797507	+	0.603310i	$0.206152\pi$
$942$	0	0
$943$	−40.7846	−1.32813
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.75129	0.0569092	0.0284546	−	0.999595i	$-0.490941\pi$
$947$	1.75129	0.0569092	0.0284546	+	0.999595i	$0.490941\pi$
$948$	0	0
$949$	28.3923	0.921653
$950$	0	0
$951$	18.0000	0.583690
$952$	0	0
$953$	−48.5885	−1.57393	−0.786967	−	0.616995i	$-0.788350\pi$
$953$	−48.5885	−1.57393	−0.786967	+	0.616995i	$0.788350\pi$
$954$	0	0
$955$	2.92820	0.0947544
$956$	0	0
$957$	−5.07180	−0.163948
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	−22.4833	−0.724515
$964$	0	0
$965$	−3.80385	−0.122450
$966$	0	0
$967$	−5.17691	−0.166478	−0.0832392	−	0.996530i	$-0.526527\pi$
$967$	−5.17691	−0.166478	−0.0832392	+	0.996530i	$0.526527\pi$
$968$	0	0
$969$	−0.392305	−0.0126026
$970$	0	0
$971$	6.92820	0.222337	0.111168	−	0.993802i	$-0.464541\pi$
$971$	6.92820	0.222337	0.111168	+	0.993802i	$0.464541\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.00000	0.0640513
$976$	0	0
$977$	50.4449	1.61387	0.806937	−	0.590637i	$-0.201124\pi$
$977$	50.4449	1.61387	0.806937	+	0.590637i	$0.201124\pi$
$978$	0	0
$979$	−6.92820	−0.221426
$980$	0	0
$981$	34.4974	1.10142
$982$	0	0
$983$	−41.5167	−1.32418	−0.662088	−	0.749426i	$-0.730329\pi$
$983$	−41.5167	−1.32418	−0.662088	+	0.749426i	$0.730329\pi$
$984$	0	0
$985$	15.8564	0.505227
$986$	0	0
$987$	0	0
$988$	0	0
$989$	59.7128	1.89876
$990$	0	0
$991$	−31.3205	−0.994929	−0.497464	−	0.867484i	$-0.665735\pi$
$991$	−31.3205	−0.994929	−0.497464	+	0.867484i	$0.665735\pi$
$992$	0	0
$993$	25.0718	0.795629
$994$	0	0
$995$	−5.07180	−0.160787
$996$	0	0
$997$	−17.6077	−0.557641	−0.278821	−	0.960343i	$-0.589944\pi$
$997$	−17.6077	−0.557641	−0.278821	+	0.960343i	$0.589944\pi$
$998$	0	0
$999$	38.6410	1.22255

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.a.k.1.2		2
4.3	odd	2		760.2.a.g.1.1	✓	2
5.4	even	2		7600.2.a.bd.1.1		2
8.3	odd	2		6080.2.a.ba.1.2		2
8.5	even	2		6080.2.a.bk.1.1		2
12.11	even	2		6840.2.a.y.1.2		2
20.3	even	4		3800.2.d.h.3649.2		4
20.7	even	4		3800.2.d.h.3649.3		4
20.19	odd	2		3800.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.g.1.1	✓	2	4.3	odd	2
1520.2.a.k.1.2		2	1.1	even	1	trivial
3800.2.a.j.1.2		2	20.19	odd	2
3800.2.d.h.3649.2		4	20.3	even	4
3800.2.d.h.3649.3		4	20.7	even	4
6080.2.a.ba.1.2		2	8.3	odd	2
6080.2.a.bk.1.1		2	8.5	even	2
6840.2.a.y.1.2		2	12.11	even	2
7600.2.a.bd.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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} -1.00000 q^{5} -2.46410 q^{9} +O(q^{10})\)	\(q+0.732051 q^{3} -1.00000 q^{5} -2.46410 q^{9} -2.00000 q^{11} +2.73205 q^{13} -0.732051 q^{15} +0.535898 q^{17} -1.00000 q^{19} -5.46410 q^{23} +1.00000 q^{25} -4.00000 q^{27} +3.46410 q^{29} -4.00000 q^{31} -1.46410 q^{33} -9.66025 q^{37} +2.00000 q^{39} +7.46410 q^{41} -10.9282 q^{43} +2.46410 q^{45} -10.9282 q^{47} -7.00000 q^{49} +0.392305 q^{51} +5.66025 q^{53} +2.00000 q^{55} -0.732051 q^{57} +5.46410 q^{59} -13.4641 q^{61} -2.73205 q^{65} -6.19615 q^{67} -4.00000 q^{69} +2.92820 q^{71} +10.3923 q^{73} +0.732051 q^{75} -12.3923 q^{79} +4.46410 q^{81} -1.46410 q^{83} -0.535898 q^{85} +2.53590 q^{87} +3.46410 q^{89} -2.92820 q^{93} +1.00000 q^{95} +1.66025 q^{97} +4.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 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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