LMFDB - Embedded newform 3800.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, 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("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
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	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{3} -2.46410 q^{9} +O(q^{10})$	$q+0.732051 q^{3} -2.46410 q^{9} +2.00000 q^{11} -2.73205 q^{13} -0.535898 q^{17} +1.00000 q^{19} -5.46410 q^{23} -4.00000 q^{27} +3.46410 q^{29} +4.00000 q^{31} +1.46410 q^{33} +9.66025 q^{37} -2.00000 q^{39} +7.46410 q^{41} -10.9282 q^{43} -10.9282 q^{47} -7.00000 q^{49} -0.392305 q^{51} -5.66025 q^{53} +0.732051 q^{57} -5.46410 q^{59} -13.4641 q^{61} -6.19615 q^{67} -4.00000 q^{69} -2.92820 q^{71} -10.3923 q^{73} +12.3923 q^{79} +4.46410 q^{81} -1.46410 q^{83} +2.53590 q^{87} +3.46410 q^{89} +2.92820 q^{93} -1.66025 q^{97} -4.92820 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} + 4 q^{11} - 2 q^{13} - 8 q^{17} + 2 q^{19} - 4 q^{23} - 8 q^{27} + 8 q^{31} - 4 q^{33} + 2 q^{37} - 4 q^{39} + 8 q^{41} - 8 q^{43} - 8 q^{47} - 14 q^{49} + 20 q^{51} + 6 q^{53} - 2 q^{57} - 4 q^{59} - 20 q^{61} - 2 q^{67} - 8 q^{69} + 8 q^{71} + 4 q^{79} + 2 q^{81} + 4 q^{83} + 12 q^{87} - 8 q^{93} + 14 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^11 - 2 * q^13 - 8 * q^17 + 2 * q^19 - 4 * q^23 - 8 * q^27 + 8 * q^31 - 4 * q^33 + 2 * q^37 - 4 * q^39 + 8 * q^41 - 8 * q^43 - 8 * q^47 - 14 * q^49 + 20 * q^51 + 6 * q^53 - 2 * q^57 - 4 * q^59 - 20 * q^61 - 2 * q^67 - 8 * q^69 + 8 * q^71 + 4 * q^79 + 2 * q^81 + 4 * q^83 + 12 * q^87 - 8 * q^93 + 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
$5$	0	0
$5$	0	0
$6$	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	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−2.73205	−0.757735	−0.378867	−	0.925451i	$-0.623686\pi$
$13$	−2.73205	−0.757735	−0.378867	+	0.925451i	$0.623686\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.535898	−0.129974	−0.0649872	−	0.997886i	$-0.520701\pi$
$17$	−0.535898	−0.129974	−0.0649872	+	0.997886i	$0.520701\pi$
$18$	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
$25$	0	0
$26$	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	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	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	$-0.207959\pi$
$37$	9.66025	1.58814	0.794068	+	0.607829i	$0.207959\pi$
$38$	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
$45$	0	0
$46$	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	$-0.627092\pi$
$53$	−5.66025	−0.777496	−0.388748	+	0.921344i	$0.627092\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.732051	0.0969625
$58$	0	0
$59$	−5.46410	−0.711365	−0.355683	−	0.934607i	$-0.615752\pi$
$59$	−5.46410	−0.711365	−0.355683	+	0.934607i	$0.615752\pi$
$60$	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
$65$	0	0
$66$	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	$-0.555591\pi$
$71$	−2.92820	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$72$	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
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.3923	1.39424	0.697122	−	0.716953i	$-0.254464\pi$
$79$	12.3923	1.39424	0.697122	+	0.716953i	$0.254464\pi$
$80$	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	0
$86$	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
$95$	0	0
$96$	0	0
$97$	−1.66025	−0.168573	−0.0842866	−	0.996442i	$-0.526861\pi$
$97$	−1.66025	−0.168573	−0.0842866	+	0.996442i	$0.526861\pi$
$98$	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	$-0.714523\pi$
$113$	−13.2679	−1.24814	−0.624072	+	0.781367i	$0.714523\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	0	0
$126$	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	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.46410	−0.637701	−0.318851	−	0.947805i	$-0.603297\pi$
$137$	−7.46410	−0.637701	−0.318851	+	0.947805i	$0.603297\pi$
$138$	0	0
$139$	−19.8564	−1.68420	−0.842099	−	0.539323i	$-0.818680\pi$
$139$	−19.8564	−1.68420	−0.842099	+	0.539323i	$0.818680\pi$
$140$	0	0
$141$	−8.00000	−0.673722
$142$	0	0
$143$	−5.46410	−0.456931
$144$	0	0
$145$	0	0
$146$	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	$-0.481026\pi$
$151$	1.46410	0.119147	0.0595734	+	0.998224i	$0.481026\pi$
$152$	0	0
$153$	1.32051	0.106757
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0.535898	0.0427693	0.0213847	−	0.999771i	$-0.493193\pi$
$157$	0.535898	0.0427693	0.0213847	+	0.999771i	$0.493193\pi$
$158$	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
$165$	0	0
$166$	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	$-0.587297\pi$
$173$	−7.12436	−0.541655	−0.270827	+	0.962628i	$0.587297\pi$
$174$	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	$-0.530212\pi$
$179$	−2.53590	−0.189542	−0.0947710	+	0.995499i	$0.530212\pi$
$180$	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
$185$	0	0
$186$	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	$-0.466215\pi$
$191$	2.92820	0.211877	0.105939	+	0.994373i	$0.466215\pi$
$192$	0	0
$193$	−3.80385	−0.273807	−0.136903	−	0.990584i	$-0.543715\pi$
$193$	−3.80385	−0.273807	−0.136903	+	0.990584i	$0.543715\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.8564	1.12972	0.564861	−	0.825186i	$-0.308930\pi$
$197$	15.8564	1.12972	0.564861	+	0.825186i	$0.308930\pi$
$198$	0	0
$199$	−5.07180	−0.359530	−0.179765	−	0.983710i	$-0.557534\pi$
$199$	−5.07180	−0.359530	−0.179765	+	0.983710i	$0.557534\pi$
$200$	0	0
$201$	−4.53590	−0.319938
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	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	$-0.451690\pi$
$211$	4.39230	0.302379	0.151189	+	0.988505i	$0.451690\pi$
$212$	0	0
$213$	−2.14359	−0.146877
$214$	0	0
$215$	0	0
$216$	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
$225$	0	0
$226$	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	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	9.07180	0.589277
$238$	0	0
$239$	−5.07180	−0.328067	−0.164034	−	0.986455i	$-0.552451\pi$
$239$	−5.07180	−0.328067	−0.164034	+	0.986455i	$0.552451\pi$
$240$	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
$245$	0	0
$246$	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	$-0.196175\pi$
$251$	25.8564	1.63204	0.816021	+	0.578022i	$0.196175\pi$
$252$	0	0
$253$	−10.9282	−0.687050
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.2679	−1.32666	−0.663329	−	0.748328i	$-0.730857\pi$
$257$	−21.2679	−1.32666	−0.663329	+	0.748328i	$0.730857\pi$
$258$	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
$265$	0	0
$266$	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	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	27.4641	1.65016	0.825079	−	0.565017i	$-0.191131\pi$
$277$	27.4641	1.65016	0.825079	+	0.565017i	$0.191131\pi$
$278$	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	0
$286$	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	$-0.573203\pi$
$293$	−7.80385	−0.455906	−0.227953	+	0.973672i	$0.573203\pi$
$294$	0	0
$295$	0	0
$296$	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
$305$	0	0
$306$	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	$-0.571503\pi$
$311$	−7.85641	−0.445496	−0.222748	+	0.974876i	$0.571503\pi$
$312$	0	0
$313$	−11.8564	−0.670164	−0.335082	−	0.942189i	$-0.608764\pi$
$313$	−11.8564	−0.670164	−0.335082	+	0.942189i	$0.608764\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.5885	−1.38103	−0.690513	−	0.723320i	$-0.742615\pi$
$317$	−24.5885	−1.38103	−0.690513	+	0.723320i	$0.742615\pi$
$318$	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
$325$	0	0
$326$	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	$-0.890338\pi$
$331$	−34.2487	−1.88248	−0.941240	+	0.337739i	$0.890338\pi$
$332$	0	0
$333$	−23.8038	−1.30444
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.66025	0.308334	0.154167	−	0.988045i	$-0.450731\pi$
$337$	5.66025	0.308334	0.154167	+	0.988045i	$0.450731\pi$
$338$	0	0
$339$	−9.71281	−0.527528
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	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	$-0.357661\pi$
$353$	16.2487	0.864832	0.432416	+	0.901674i	$0.357661\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.92820	−0.260101	−0.130050	−	0.991507i	$-0.541514\pi$
$359$	−4.92820	−0.260101	−0.130050	+	0.991507i	$0.541514\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−5.12436	−0.268959
$364$	0	0
$365$	0	0
$366$	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	$-0.498384\pi$
$373$	0.196152	0.0101564	0.00507819	+	0.999987i	$0.498384\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.46410	−0.487426
$378$	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
$381$	−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
$395$	0	0
$396$	0	0
$397$	12.5359	0.629159	0.314579	−	0.949231i	$-0.398137\pi$
$397$	12.5359	0.629159	0.314579	+	0.949231i	$0.398137\pi$
$398$	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
$405$	0	0
$406$	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
$415$	0	0
$416$	0	0
$417$	−14.5359	−0.711826
$418$	0	0
$419$	−1.85641	−0.0906914	−0.0453457	−	0.998971i	$-0.514439\pi$
$419$	−1.85641	−0.0906914	−0.0453457	+	0.998971i	$0.514439\pi$
$420$	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	0
$426$	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	$-0.441346\pi$
$431$	7.60770	0.366450	0.183225	+	0.983071i	$0.441346\pi$
$432$	0	0
$433$	−24.9808	−1.20050	−0.600249	−	0.799813i	$-0.704932\pi$
$433$	−24.9808	−1.20050	−0.600249	+	0.799813i	$0.704932\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.46410	−0.261383
$438$	0	0
$439$	−22.2487	−1.06187	−0.530937	−	0.847412i	$-0.678160\pi$
$439$	−22.2487	−1.06187	−0.530937	+	0.847412i	$0.678160\pi$
$440$	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
$445$	0	0
$446$	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	$-0.614673\pi$
$457$	−15.0718	−0.705029	−0.352514	+	0.935806i	$0.614673\pi$
$458$	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
$465$	0	0
$466$	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
$475$	0	0
$476$	0	0
$477$	13.9474	0.638609
$478$	0	0
$479$	−7.85641	−0.358968	−0.179484	−	0.983761i	$-0.557443\pi$
$479$	−7.85641	−0.358968	−0.179484	+	0.983761i	$0.557443\pi$
$480$	0	0
$481$	−26.3923	−1.20339
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	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	$-0.301706\pi$
$491$	25.8564	1.16688	0.583442	+	0.812155i	$0.301706\pi$
$492$	0	0
$493$	−1.85641	−0.0836083
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−10.7846	−0.482785	−0.241393	−	0.970428i	$-0.577604\pi$
$499$	−10.7846	−0.482785	−0.241393	+	0.970428i	$0.577604\pi$
$500$	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
$505$	0	0
$506$	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
$515$	0	0
$516$	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
$535$	0	0
$536$	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
$545$	0	0
$546$	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
$555$	0	0
$556$	0	0
$557$	36.2487	1.53591	0.767954	−	0.640505i	$-0.221275\pi$
$557$	36.2487	1.53591	0.767954	+	0.640505i	$0.221275\pi$
$558$	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
$565$	0	0
$566$	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	$-0.427546\pi$
$571$	10.7846	0.451322	0.225661	+	0.974206i	$0.427546\pi$
$572$	0	0
$573$	2.14359	0.0895499
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8.14359	−0.339022	−0.169511	−	0.985528i	$-0.554219\pi$
$577$	−8.14359	−0.339022	−0.169511	+	0.985528i	$0.554219\pi$
$578$	0	0
$579$	−2.78461	−0.115724
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−11.3205	−0.468848
$584$	0	0
$585$	0	0
$586$	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	$-0.446525\pi$
$593$	8.14359	0.334417	0.167209	+	0.985922i	$0.446525\pi$
$594$	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	$-0.0893462\pi$
$599$	47.0333	1.92173	0.960865	+	0.277018i	$0.0893462\pi$
$600$	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
$605$	0	0
$606$	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	$-0.472655\pi$
$613$	4.24871	0.171604	0.0858019	+	0.996312i	$0.472655\pi$
$614$	0	0
$615$	0	0
$616$	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	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	21.8564	0.877067
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	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	$-0.919491\pi$
$631$	−48.6410	−1.93637	−0.968184	+	0.250239i	$0.919491\pi$
$632$	0	0
$633$	3.21539	0.127800
$634$	0	0
$635$	0	0
$636$	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
$645$	0	0
$646$	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	$-0.414214\pi$
$653$	13.6077	0.532510	0.266255	+	0.963903i	$0.414214\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	25.6077	0.999051
$658$	0	0
$659$	11.6077	0.452172	0.226086	−	0.974107i	$-0.427407\pi$
$659$	11.6077	0.452172	0.226086	+	0.974107i	$0.427407\pi$
$660$	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	$-0.175500\pi$
$673$	44.1962	1.70364	0.851818	+	0.523837i	$0.175500\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	35.9090	1.38009	0.690047	−	0.723765i	$-0.257590\pi$
$677$	35.9090	1.38009	0.690047	+	0.723765i	$0.257590\pi$
$678$	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
$685$	0	0
$686$	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	$-0.483133\pi$
$691$	2.78461	0.105932	0.0529658	+	0.998596i	$0.483133\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	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
$705$	0	0
$706$	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
$715$	0	0
$716$	0	0
$717$	−3.71281	−0.138658
$718$	0	0
$719$	35.5692	1.32651	0.663254	−	0.748394i	$-0.269175\pi$
$719$	35.5692	1.32651	0.663254	+	0.748394i	$0.269175\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.60770	0.282933
$724$	0	0
$725$	0	0
$726$	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	$-0.199501\pi$
$733$	43.8564	1.61987	0.809937	+	0.586517i	$0.199501\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.3923	−0.456476
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	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
$745$	0	0
$746$	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	$-0.390185\pi$
$751$	18.5359	0.676385	0.338192	+	0.941077i	$0.390185\pi$
$752$	0	0
$753$	18.9282	0.689782
$754$	0	0
$755$	0	0
$756$	0	0
$757$	42.7846	1.55503	0.777517	−	0.628862i	$-0.216479\pi$
$757$	42.7846	1.55503	0.777517	+	0.628862i	$0.216479\pi$
$758$	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
$765$	0	0
$766$	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	$-0.498877\pi$
$773$	0.196152	0.00705511	0.00352756	+	0.999994i	$0.498877\pi$
$774$	0	0
$775$	0	0
$776$	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	0
$786$	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
$795$	0	0
$796$	0	0
$797$	22.0526	0.781142	0.390571	−	0.920573i	$-0.372278\pi$
$797$	22.0526	0.781142	0.390571	+	0.920573i	$0.372278\pi$
$798$	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	$-0.578224\pi$
$811$	−13.8564	−0.486564	−0.243282	+	0.969956i	$0.578224\pi$
$812$	0	0
$813$	10.2487	0.359438
$814$	0	0
$815$	0	0
$816$	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
$825$	0	0
$826$	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
$835$	0	0
$836$	0	0
$837$	−16.0000	−0.553041
$838$	0	0
$839$	44.3923	1.53259	0.766296	−	0.642487i	$-0.222097\pi$
$839$	44.3923	1.53259	0.766296	+	0.642487i	$0.222097\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	−1.17691	−0.0405351
$844$	0	0
$845$	0	0
$846$	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	$-0.348371\pi$
$853$	26.7846	0.917088	0.458544	+	0.888672i	$0.348371\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.87564	0.166549	0.0832744	−	0.996527i	$-0.473462\pi$
$857$	4.87564	0.166549	0.0832744	+	0.996527i	$0.473462\pi$
$858$	0	0
$859$	57.8564	1.97404	0.987018	−	0.160612i	$-0.0513469\pi$
$859$	57.8564	1.97404	0.987018	+	0.160612i	$0.0513469\pi$
$860$	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
$865$	0	0
$866$	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	$-0.786909\pi$
$877$	−46.4449	−1.56833	−0.784166	+	0.620551i	$0.786909\pi$
$878$	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
$885$	0	0
$886$	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
$895$	0	0
$896$	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
$905$	0	0
$906$	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	$-0.251109\pi$
$911$	42.5359	1.40928	0.704639	+	0.709566i	$0.251109\pi$
$912$	0	0
$913$	−2.92820	−0.0969094
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−10.9282	−0.360488	−0.180244	−	0.983622i	$-0.557689\pi$
$919$	−10.9282	−0.360488	−0.180244	+	0.983622i	$0.557689\pi$
$920$	0	0
$921$	−21.0333	−0.693071
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	0	0
$926$	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
$935$	0	0
$936$	0	0
$937$	−50.3923	−1.64624	−0.823122	−	0.567864i	$-0.807770\pi$
$937$	−50.3923	−1.64624	−0.823122	+	0.567864i	$0.807770\pi$
$938$	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	$-0.211650\pi$
$953$	48.5885	1.57393	0.786967	+	0.616995i	$0.211650\pi$
$954$	0	0
$955$	0	0
$956$	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
$965$	0	0
$966$	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	$-0.535459\pi$
$971$	−6.92820	−0.222337	−0.111168	+	0.993802i	$0.535459\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.4449	−1.61387	−0.806937	−	0.590637i	$-0.798876\pi$
$977$	−50.4449	−1.61387	−0.806937	+	0.590637i	$0.798876\pi$
$978$	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
$985$	0	0
$986$	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	$-0.334265\pi$
$991$	31.3205	0.994929	0.497464	+	0.867484i	$0.334265\pi$
$992$	0	0
$993$	−25.0718	−0.795629
$994$	0	0
$995$	0	0
$996$	0	0
$997$	17.6077	0.557641	0.278821	−	0.960343i	$-0.410056\pi$
$997$	17.6077	0.557641	0.278821	+	0.960343i	$0.410056\pi$
$998$	0	0
$999$	−38.6410	−1.22255

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.g.1.1	✓	2	5.4	even	2
1520.2.a.k.1.2		2	20.19	odd	2
3800.2.a.j.1.2		2	1.1	even	1	trivial
3800.2.d.h.3649.2		4	5.2	odd	4
3800.2.d.h.3649.3		4	5.3	odd	4
6080.2.a.ba.1.2		2	40.29	even	2
6080.2.a.bk.1.1		2	40.19	odd	2
6840.2.a.y.1.2		2	15.14	odd	2
7600.2.a.bd.1.1		2	4.3	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{3} -2.46410 q^{9} +O(q^{10})\)	\(q+0.732051 q^{3} -2.46410 q^{9} +2.00000 q^{11} -2.73205 q^{13} -0.535898 q^{17} +1.00000 q^{19} -5.46410 q^{23} -4.00000 q^{27} +3.46410 q^{29} +4.00000 q^{31} +1.46410 q^{33} +9.66025 q^{37} -2.00000 q^{39} +7.46410 q^{41} -10.9282 q^{43} -10.9282 q^{47} -7.00000 q^{49} -0.392305 q^{51} -5.66025 q^{53} +0.732051 q^{57} -5.46410 q^{59} -13.4641 q^{61} -6.19615 q^{67} -4.00000 q^{69} -2.92820 q^{71} -10.3923 q^{73} +12.3923 q^{79} +4.46410 q^{81} -1.46410 q^{83} +2.53590 q^{87} +3.46410 q^{89} +2.92820 q^{93} -1.66025 q^{97} -4.92820 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} + 4 q^{11} - 2 q^{13} - 8 q^{17} + 2 q^{19} - 4 q^{23} - 8 q^{27} + 8 q^{31} - 4 q^{33} + 2 q^{37} - 4 q^{39} + 8 q^{41} - 8 q^{43} - 8 q^{47} - 14 q^{49} + 20 q^{51} + 6 q^{53} - 2 q^{57} - 4 q^{59} - 20 q^{61} - 2 q^{67} - 8 q^{69} + 8 q^{71} + 4 q^{79} + 2 q^{81} + 4 q^{83} + 12 q^{87} - 8 q^{93} + 14 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^11 - 2 * q^13 - 8 * q^17 + 2 * q^19 - 4 * q^23 - 8 * q^27 + 8 * q^31 - 4 * q^33 + 2 * q^37 - 4 * q^39 + 8 * q^41 - 8 * q^43 - 8 * q^47 - 14 * q^49 + 20 * q^51 + 6 * q^53 - 2 * q^57 - 4 * q^59 - 20 * q^61 - 2 * q^67 - 8 * q^69 + 8 * q^71 + 4 * q^79 + 2 * q^81 + 4 * q^83 + 12 * q^87 - 8 * q^93 + 14 * q^97 + 4 * q^99

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