LMFDB - Embedded newform 2736.3.m.e.1711.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(1711,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2736.1711");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.m (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} + \cdots + 2052928 $ x^12 - 2x^11 - 2x^10 - 28x^9 - 400x^8 - 520x^7 + 17067x^6 - 3250x^5 - 195494x^4 + 302996x^3 + 602332x^2 - 2263536*x + 2052928
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1711.4
Root			$-3.01700 - 0.588235i$ of defining polynomial
Character	$\chi$	$=$	2736.1711
Dual form			2736.3.m.e.1711.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.69015 q^{5} +3.50408i q^{7} +O(q^{10})$	$q-4.69015 q^{5} +3.50408i q^{7} +15.3257i q^{11} +1.31231 q^{13} +29.5521 q^{17} +4.35890i q^{19} +39.1830i q^{23} -3.00246 q^{25} -13.2264 q^{29} -18.9119i q^{31} -16.4347i q^{35} +14.7234 q^{37} -33.8469 q^{41} -36.4663i q^{43} +39.0754i q^{47} +36.7214 q^{49} +69.6801 q^{53} -71.8799i q^{55} -32.4721i q^{59} +69.1928 q^{61} -6.15491 q^{65} -88.5136i q^{67} +93.7996i q^{71} -67.5164 q^{73} -53.7025 q^{77} +79.8522i q^{79} -41.1508i q^{83} -138.604 q^{85} +94.9959 q^{89} +4.59843i q^{91} -20.4439i q^{95} -134.948 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 10 q^{5}+O(q^{10}) $ 12 * q + 10 * q^5	$ 12 q + 10 q^{5} + 36 q^{13} - 14 q^{17} + 10 q^{25} + 108 q^{29} - 16 q^{37} - 16 q^{41} - 118 q^{49} - 220 q^{53} + 366 q^{61} - 140 q^{65} - 158 q^{73} + 286 q^{77} + 98 q^{85} + 396 q^{89} + 224 q^{97}+O(q^{100}) $ 12 * q + 10 * q^5 + 36 * q^13 - 14 * q^17 + 10 * q^25 + 108 * q^29 - 16 * q^37 - 16 * q^41 - 118 * q^49 - 220 * q^53 + 366 * q^61 - 140 * q^65 - 158 * q^73 + 286 * q^77 + 98 * q^85 + 396 * q^89 + 224 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$1$	$-1$	$1$

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	0
$3$	0	0
$4$	0	0
$5$	−4.69015	−0.938031	−0.469015	−	0.883190i	$-0.655391\pi$
$5$	−4.69015	−0.938031	−0.469015	+	0.883190i	$0.655391\pi$
$6$	0	0
$7$	3.50408i	0.500583i	0.968170	+	0.250292i	$0.0805265\pi$
$7$	3.50408i	0.500583i	−0.968170	+	0.250292i	$0.919474\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.3257i	1.39325i	0.717438	+	0.696623i	$0.245315\pi$
$11$	15.3257i	1.39325i	−0.717438	+	0.696623i	$0.754685\pi$
$12$	0	0
$13$	1.31231	0.100947	0.0504733	−	0.998725i	$-0.483927\pi$
$13$	1.31231	0.100947	0.0504733	+	0.998725i	$0.483927\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	29.5521	1.73836	0.869179	−	0.494498i	$-0.164648\pi$
$17$	29.5521	1.73836	0.869179	+	0.494498i	$0.164648\pi$
$18$	0	0
$19$	4.35890i	0.229416i
$19$	4.35890i	0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	39.1830i	1.70361i	0.523860	+	0.851804i	$0.324492\pi$
$23$	39.1830i	1.70361i	−0.523860	+	0.851804i	$0.675508\pi$
$24$	0	0
$25$	−3.00246	−0.120098
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−13.2264	−0.456083	−0.228042	−	0.973651i	$-0.573232\pi$
$29$	−13.2264	−0.456083	−0.228042	+	0.973651i	$0.573232\pi$
$30$	0	0
$31$	− 18.9119i	− 0.610062i	−0.952342	−	0.305031i	$-0.901333\pi$
$31$	− 18.9119i	− 0.610062i	0.952342	−	0.305031i	$-0.0986668\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 16.4347i	− 0.469562i
$36$	0	0
$37$	14.7234	0.397931	0.198965	−	0.980007i	$-0.436242\pi$
$37$	14.7234	0.397931	0.198965	+	0.980007i	$0.436242\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−33.8469	−0.825535	−0.412768	−	0.910836i	$-0.635438\pi$
$41$	−33.8469	−0.825535	−0.412768	+	0.910836i	$0.635438\pi$
$42$	0	0
$43$	− 36.4663i	− 0.848053i	−0.905650	−	0.424026i	$-0.860616\pi$
$43$	− 36.4663i	− 0.848053i	0.905650	−	0.424026i	$-0.139384\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	39.0754i	0.831393i	0.909503	+	0.415696i	$0.136462\pi$
$47$	39.0754i	0.831393i	−0.909503	+	0.415696i	$0.863538\pi$
$48$	0	0
$49$	36.7214	0.749416
$50$	0	0
$51$	0	0
$52$	0	0
$53$	69.6801	1.31472	0.657359	−	0.753577i	$-0.271673\pi$
$53$	69.6801	1.31472	0.657359	+	0.753577i	$0.271673\pi$
$54$	0	0
$55$	− 71.8799i	− 1.30691i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 32.4721i	− 0.550374i	−0.961391	−	0.275187i	$-0.911260\pi$
$59$	− 32.4721i	− 0.550374i	0.961391	−	0.275187i	$-0.0887398\pi$
$60$	0	0
$61$	69.1928	1.13431	0.567155	−	0.823611i	$-0.308044\pi$
$61$	69.1928	1.13431	0.567155	+	0.823611i	$0.308044\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.15491	−0.0946910
$66$	0	0
$67$	− 88.5136i	− 1.32110i	−0.750782	−	0.660550i	$-0.770323\pi$
$67$	− 88.5136i	− 1.32110i	0.750782	−	0.660550i	$-0.229677\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	93.7996i	1.32112i	0.750773	+	0.660561i	$0.229681\pi$
$71$	93.7996i	1.32112i	−0.750773	+	0.660561i	$0.770319\pi$
$72$	0	0
$73$	−67.5164	−0.924883	−0.462441	−	0.886650i	$-0.653026\pi$
$73$	−67.5164	−0.924883	−0.462441	+	0.886650i	$0.653026\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−53.7025	−0.697435
$78$	0	0
$79$	79.8522i	1.01079i	0.862889	+	0.505394i	$0.168653\pi$
$79$	79.8522i	1.01079i	−0.862889	+	0.505394i	$0.831347\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 41.1508i	− 0.495793i	−0.968787	−	0.247896i	$-0.920261\pi$
$83$	− 41.1508i	− 0.495793i	0.968787	−	0.247896i	$-0.0797392\pi$
$84$	0	0
$85$	−138.604	−1.63063
$86$	0	0
$87$	0	0
$88$	0	0
$89$	94.9959	1.06737	0.533685	−	0.845683i	$-0.320807\pi$
$89$	94.9959	1.06737	0.533685	+	0.845683i	$0.320807\pi$
$90$	0	0
$91$	4.59843i	0.0505321i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 20.4439i	− 0.215199i
$96$	0	0
$97$	−134.948	−1.39122	−0.695608	−	0.718421i	$-0.744865\pi$
$97$	−134.948	−1.39122	−0.695608	+	0.718421i	$0.744865\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−20.2349	−0.200346	−0.100173	−	0.994970i	$-0.531940\pi$
$101$	−20.2349	−0.200346	−0.100173	+	0.994970i	$0.531940\pi$
$102$	0	0
$103$	140.433i	1.36342i	0.731621	+	0.681712i	$0.238764\pi$
$103$	140.433i	1.36342i	−0.731621	+	0.681712i	$0.761236\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	148.928i	1.39185i	0.718116	+	0.695924i	$0.245005\pi$
$107$	148.928i	1.39185i	−0.718116	+	0.695924i	$0.754995\pi$
$108$	0	0
$109$	−103.930	−0.953487	−0.476743	−	0.879043i	$-0.658183\pi$
$109$	−103.930	−0.953487	−0.476743	+	0.879043i	$0.658183\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−117.740	−1.04195	−0.520975	−	0.853572i	$-0.674432\pi$
$113$	−117.740	−1.04195	−0.520975	+	0.853572i	$0.674432\pi$
$114$	0	0
$115$	− 183.774i	− 1.59804i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	103.553i	0.870193i
$120$	0	0
$121$	−113.877	−0.941132
$122$	0	0
$123$	0	0
$124$	0	0
$125$	131.336	1.05069
$126$	0	0
$127$	− 147.584i	− 1.16208i	−0.813876	−	0.581039i	$-0.802647\pi$
$127$	− 147.584i	− 1.16208i	0.813876	−	0.581039i	$-0.197353\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	93.7476i	0.715631i	0.933792	+	0.357815i	$0.116478\pi$
$131$	93.7476i	0.715631i	−0.933792	+	0.357815i	$0.883522\pi$
$132$	0	0
$133$	−15.2739	−0.114842
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−189.966	−1.38662	−0.693308	−	0.720641i	$-0.743847\pi$
$137$	−189.966	−1.38662	−0.693308	+	0.720641i	$0.743847\pi$
$138$	0	0
$139$	24.4698i	0.176042i	0.996119	+	0.0880210i	$0.0280543\pi$
$139$	24.4698i	0.176042i	−0.996119	+	0.0880210i	$0.971946\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	20.1120i	0.140643i
$144$	0	0
$145$	62.0339	0.427820
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−210.466	−1.41252	−0.706260	−	0.707952i	$-0.749619\pi$
$149$	−210.466	−1.41252	−0.706260	+	0.707952i	$0.749619\pi$
$150$	0	0
$151$	− 67.1756i	− 0.444872i	−0.974947	−	0.222436i	$-0.928599\pi$
$151$	− 67.1756i	− 0.444872i	0.974947	−	0.222436i	$-0.0714008\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	88.6998i	0.572257i
$156$	0	0
$157$	−121.092	−0.771285	−0.385643	−	0.922648i	$-0.626020\pi$
$157$	−121.092	−0.771285	−0.385643	+	0.922648i	$0.626020\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−137.300	−0.852798
$162$	0	0
$163$	129.972i	0.797374i	0.917087	+	0.398687i	$0.130534\pi$
$163$	129.972i	0.797374i	−0.917087	+	0.398687i	$0.869466\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	69.9751i	0.419013i	0.977807	+	0.209506i	$0.0671857\pi$
$167$	69.9751i	0.419013i	−0.977807	+	0.209506i	$0.932814\pi$
$168$	0	0
$169$	−167.278	−0.989810
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−93.3582	−0.539643	−0.269821	−	0.962910i	$-0.586965\pi$
$173$	−93.3582	−0.539643	−0.269821	+	0.962910i	$0.586965\pi$
$174$	0	0
$175$	− 10.5209i	− 0.0601192i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 235.391i	− 1.31503i	−0.753441	−	0.657516i	$-0.771607\pi$
$179$	− 235.391i	− 1.31503i	0.753441	−	0.657516i	$-0.228393\pi$
$180$	0	0
$181$	87.0308	0.480833	0.240416	−	0.970670i	$-0.422716\pi$
$181$	87.0308	0.480833	0.240416	+	0.970670i	$0.422716\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−69.0552	−0.373271
$186$	0	0
$187$	452.906i	2.42196i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	91.1219i	0.477078i	0.971133	+	0.238539i	$0.0766684\pi$
$191$	91.1219i	0.477078i	−0.971133	+	0.238539i	$0.923332\pi$
$192$	0	0
$193$	366.918	1.90113	0.950564	−	0.310527i	$-0.100506\pi$
$193$	366.918	1.90113	0.950564	+	0.310527i	$0.100506\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	40.6287	0.206237	0.103118	−	0.994669i	$-0.467118\pi$
$197$	40.6287	0.206237	0.103118	+	0.994669i	$0.467118\pi$
$198$	0	0
$199$	343.074i	1.72399i	0.506918	+	0.861994i	$0.330785\pi$
$199$	343.074i	1.72399i	−0.506918	+	0.861994i	$0.669215\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 46.3464i	− 0.228308i
$204$	0	0
$205$	158.747	0.774377
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−66.8032	−0.319632
$210$	0	0
$211$	− 82.1408i	− 0.389293i	−0.980873	−	0.194647i	$-0.937644\pi$
$211$	− 82.1408i	− 0.389293i	0.980873	−	0.194647i	$-0.0623560\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	171.032i	0.795500i
$216$	0	0
$217$	66.2689	0.305387
$218$	0	0
$219$	0	0
$220$	0	0
$221$	38.7814	0.175481
$222$	0	0
$223$	312.579i	1.40170i	0.713310	+	0.700849i	$0.247195\pi$
$223$	312.579i	1.40170i	−0.713310	+	0.700849i	$0.752805\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	207.701i	0.914983i	0.889214	+	0.457492i	$0.151252\pi$
$227$	207.701i	0.914983i	−0.889214	+	0.457492i	$0.848748\pi$
$228$	0	0
$229$	−316.012	−1.37996	−0.689982	−	0.723827i	$-0.742382\pi$
$229$	−316.012	−1.37996	−0.689982	+	0.723827i	$0.742382\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−363.003	−1.55795	−0.778977	−	0.627052i	$-0.784261\pi$
$233$	−363.003	−1.55795	−0.778977	+	0.627052i	$0.784261\pi$
$234$	0	0
$235$	− 183.270i	− 0.779872i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 85.1763i	− 0.356386i	−0.983996	−	0.178193i	$-0.942975\pi$
$239$	− 85.1763i	− 0.356386i	0.983996	−	0.178193i	$-0.0570252\pi$
$240$	0	0
$241$	379.566	1.57496	0.787482	−	0.616337i	$-0.211384\pi$
$241$	379.566	1.57496	0.787482	+	0.616337i	$0.211384\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−172.229	−0.702976
$246$	0	0
$247$	5.72021i	0.0231587i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 377.646i	− 1.50457i	−0.658840	−	0.752283i	$-0.728952\pi$
$251$	− 377.646i	− 1.50457i	0.658840	−	0.752283i	$-0.271048\pi$
$252$	0	0
$253$	−600.507	−2.37354
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−323.569	−1.25902	−0.629511	−	0.776992i	$-0.716745\pi$
$257$	−323.569	−1.25902	−0.629511	+	0.776992i	$0.716745\pi$
$258$	0	0
$259$	51.5921i	0.199197i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	325.791i	1.23875i	0.785095	+	0.619375i	$0.212614\pi$
$263$	325.791i	1.23875i	−0.785095	+	0.619375i	$0.787386\pi$
$264$	0	0
$265$	−326.810	−1.23325
$266$	0	0
$267$	0	0
$268$	0	0
$269$	144.231	0.536176	0.268088	−	0.963394i	$-0.413608\pi$
$269$	144.231	0.536176	0.268088	+	0.963394i	$0.413608\pi$
$270$	0	0
$271$	− 39.0143i	− 0.143964i	−0.997406	−	0.0719822i	$-0.977068\pi$
$271$	− 39.0143i	− 0.143964i	0.997406	−	0.0719822i	$-0.0229325\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 46.0148i	− 0.167326i
$276$	0	0
$277$	−428.413	−1.54662	−0.773309	−	0.634030i	$-0.781400\pi$
$277$	−428.413	−1.54662	−0.773309	+	0.634030i	$0.781400\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−109.094	−0.388235	−0.194118	−	0.980978i	$-0.562184\pi$
$281$	−109.094	−0.388235	−0.194118	+	0.980978i	$0.562184\pi$
$282$	0	0
$283$	45.1854i	0.159666i	0.996808	+	0.0798329i	$0.0254387\pi$
$283$	45.1854i	0.159666i	−0.996808	+	0.0798329i	$0.974561\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 118.602i	− 0.413249i
$288$	0	0
$289$	584.326	2.02189
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−241.451	−0.824063	−0.412032	−	0.911170i	$-0.635181\pi$
$293$	−241.451	−0.824063	−0.412032	+	0.911170i	$0.635181\pi$
$294$	0	0
$295$	152.299i	0.516268i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	51.4201i	0.171973i
$300$	0	0
$301$	127.781	0.424521
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−324.525	−1.06402
$306$	0	0
$307$	294.343i	0.958772i	0.877604	+	0.479386i	$0.159141\pi$
$307$	294.343i	0.958772i	−0.877604	+	0.479386i	$0.840859\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 411.051i	− 1.32171i	−0.750515	−	0.660854i	$-0.770194\pi$
$311$	− 411.051i	− 1.32171i	0.750515	−	0.660854i	$-0.229806\pi$
$312$	0	0
$313$	18.5564	0.0592858	0.0296429	−	0.999561i	$-0.490563\pi$
$313$	18.5564	0.0592858	0.0296429	+	0.999561i	$0.490563\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−413.612	−1.30477	−0.652384	−	0.757888i	$-0.726231\pi$
$317$	−413.612	−1.30477	−0.652384	+	0.757888i	$0.726231\pi$
$318$	0	0
$319$	− 202.704i	− 0.635436i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	128.815i	0.398807i
$324$	0	0
$325$	−3.94014	−0.0121235
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−136.924	−0.416181
$330$	0	0
$331$	− 342.853i	− 1.03581i	−0.855439	−	0.517904i	$-0.826712\pi$
$331$	− 342.853i	− 1.03581i	0.855439	−	0.517904i	$-0.173288\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	415.143i	1.23923i
$336$	0	0
$337$	111.053	0.329534	0.164767	−	0.986333i	$-0.447313\pi$
$337$	111.053	0.329534	0.164767	+	0.986333i	$0.447313\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	289.838	0.849965
$342$	0	0
$343$	300.375i	0.875728i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 378.847i	− 1.09178i	−0.837858	−	0.545888i	$-0.816192\pi$
$347$	− 378.847i	− 1.09178i	0.837858	−	0.545888i	$-0.183808\pi$
$348$	0	0
$349$	−435.797	−1.24870	−0.624350	−	0.781144i	$-0.714636\pi$
$349$	−435.797	−1.24870	−0.624350	+	0.781144i	$0.714636\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	374.264	1.06024	0.530119	−	0.847923i	$-0.322147\pi$
$353$	374.264	1.06024	0.530119	+	0.847923i	$0.322147\pi$
$354$	0	0
$355$	− 439.935i	− 1.23925i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 286.053i	− 0.796804i	−0.917211	−	0.398402i	$-0.869565\pi$
$359$	− 286.053i	− 0.796804i	0.917211	−	0.398402i	$-0.130435\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	316.662	0.867568
$366$	0	0
$367$	− 232.679i	− 0.634003i	−0.948425	−	0.317002i	$-0.897324\pi$
$367$	− 232.679i	− 0.634003i	0.948425	−	0.317002i	$-0.102676\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	244.165i	0.658126i
$372$	0	0
$373$	262.525	0.703820	0.351910	−	0.936034i	$-0.385532\pi$
$373$	262.525	0.703820	0.351910	+	0.936034i	$0.385532\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−17.3571	−0.0460400
$378$	0	0
$379$	29.9226i	0.0789516i	0.999221	+	0.0394758i	$0.0125688\pi$
$379$	29.9226i	0.0789516i	−0.999221	+	0.0394758i	$0.987431\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	21.0261i	0.0548985i	0.999623	+	0.0274493i	$0.00873847\pi$
$383$	21.0261i	0.0548985i	−0.999623	+	0.0274493i	$0.991262\pi$
$384$	0	0
$385$	251.873	0.654216
$386$	0	0
$387$	0	0
$388$	0	0
$389$	77.9673	0.200430	0.100215	−	0.994966i	$-0.468047\pi$
$389$	77.9673	0.200430	0.100215	+	0.994966i	$0.468047\pi$
$390$	0	0
$391$	1157.94i	2.96148i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 374.519i	− 0.948150i
$396$	0	0
$397$	244.627	0.616189	0.308095	−	0.951356i	$-0.400309\pi$
$397$	244.627	0.616189	0.308095	+	0.951356i	$0.400309\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	190.828	0.475880	0.237940	−	0.971280i	$-0.423528\pi$
$401$	190.828	0.475880	0.237940	+	0.971280i	$0.423528\pi$
$402$	0	0
$403$	− 24.8182i	− 0.0615836i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	225.647i	0.554415i
$408$	0	0
$409$	−270.277	−0.660825	−0.330412	−	0.943837i	$-0.607188\pi$
$409$	−270.277	−0.660825	−0.330412	+	0.943837i	$0.607188\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	113.785	0.275508
$414$	0	0
$415$	193.004i	0.465069i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 247.653i	− 0.591058i	−0.955334	−	0.295529i	$-0.904504\pi$
$419$	− 247.653i	− 0.591058i	0.955334	−	0.295529i	$-0.0954959\pi$
$420$	0	0
$421$	551.992	1.31115	0.655573	−	0.755132i	$-0.272427\pi$
$421$	551.992	1.31115	0.655573	+	0.755132i	$0.272427\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−88.7289	−0.208774
$426$	0	0
$427$	242.457i	0.567816i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	231.366i	0.536812i	0.963306	+	0.268406i	$0.0864969\pi$
$431$	231.366i	0.536812i	−0.963306	+	0.268406i	$0.913503\pi$
$432$	0	0
$433$	390.273	0.901323	0.450662	−	0.892695i	$-0.351188\pi$
$433$	390.273	0.901323	0.450662	+	0.892695i	$0.351188\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−170.795	−0.390835
$438$	0	0
$439$	247.991i	0.564900i	0.959282	+	0.282450i	$0.0911471\pi$
$439$	247.991i	0.564900i	−0.959282	+	0.282450i	$0.908853\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 264.789i	− 0.597717i	−0.954297	−	0.298859i	$-0.903394\pi$
$443$	− 264.789i	− 0.597717i	0.954297	−	0.298859i	$-0.0966059\pi$
$444$	0	0
$445$	−445.545	−1.00123
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−824.108	−1.83543	−0.917715	−	0.397239i	$-0.869968\pi$
$449$	−824.108	−1.83543	−0.917715	+	0.397239i	$0.869968\pi$
$450$	0	0
$451$	− 518.728i	− 1.15017i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 21.5673i	− 0.0474007i
$456$	0	0
$457$	571.149	1.24978	0.624890	−	0.780713i	$-0.285144\pi$
$457$	571.149	1.24978	0.624890	+	0.780713i	$0.285144\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−605.221	−1.31284	−0.656421	−	0.754394i	$-0.727931\pi$
$461$	−605.221	−1.31284	−0.656421	+	0.754394i	$0.727931\pi$
$462$	0	0
$463$	− 258.444i	− 0.558195i	−0.960263	−	0.279098i	$-0.909965\pi$
$463$	− 258.444i	− 0.558195i	0.960263	−	0.279098i	$-0.0900353\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	118.201i	0.253106i	0.991960	+	0.126553i	$0.0403915\pi$
$467$	118.201i	0.253106i	−0.991960	+	0.126553i	$0.959609\pi$
$468$	0	0
$469$	310.159	0.661320
$470$	0	0
$471$	0	0
$472$	0	0
$473$	558.871	1.18155
$474$	0	0
$475$	− 13.0874i	− 0.0275525i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	398.490i	0.831921i	0.909383	+	0.415960i	$0.136555\pi$
$479$	398.490i	0.831921i	−0.909383	+	0.415960i	$0.863445\pi$
$480$	0	0
$481$	19.3216	0.0401697
$482$	0	0
$483$	0	0
$484$	0	0
$485$	632.927	1.30500
$486$	0	0
$487$	19.9057i	0.0408740i	0.999791	+	0.0204370i	$0.00650576\pi$
$487$	19.9057i	0.0408740i	−0.999791	+	0.0204370i	$0.993494\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	322.529i	0.656881i	0.944525	+	0.328440i	$0.106523\pi$
$491$	322.529i	0.656881i	−0.944525	+	0.328440i	$0.893477\pi$
$492$	0	0
$493$	−390.868	−0.792836
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−328.682	−0.661331
$498$	0	0
$499$	529.427i	1.06098i	0.847692	+	0.530488i	$0.177991\pi$
$499$	529.427i	1.06098i	−0.847692	+	0.530488i	$0.822009\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	355.262i	0.706287i	0.935569	+	0.353144i	$0.114887\pi$
$503$	355.262i	0.706287i	−0.935569	+	0.353144i	$0.885113\pi$
$504$	0	0
$505$	94.9048	0.187930
$506$	0	0
$507$	0	0
$508$	0	0
$509$	216.593	0.425526	0.212763	−	0.977104i	$-0.431754\pi$
$509$	216.593	0.425526	0.212763	+	0.977104i	$0.431754\pi$
$510$	0	0
$511$	− 236.583i	− 0.462981i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 658.651i	− 1.27893i
$516$	0	0
$517$	−598.858	−1.15833
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.78403	−0.0168599	−0.00842997	−	0.999964i	$-0.502683\pi$
$521$	−8.78403	−0.0168599	−0.00842997	+	0.999964i	$0.502683\pi$
$522$	0	0
$523$	− 1035.75i	− 1.98040i	−0.139643	−	0.990202i	$-0.544595\pi$
$523$	− 1035.75i	− 1.98040i	0.139643	−	0.990202i	$-0.455405\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 558.886i	− 1.06051i
$528$	0	0
$529$	−1006.31	−1.90228
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−44.4175	−0.0833349
$534$	0	0
$535$	− 698.494i	− 1.30560i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	562.781i	1.04412i
$540$	0	0
$541$	97.9390	0.181033	0.0905166	−	0.995895i	$-0.471148\pi$
$541$	97.9390	0.181033	0.0905166	+	0.995895i	$0.471148\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	487.448	0.894400
$546$	0	0
$547$	− 118.529i	− 0.216690i	−0.994113	−	0.108345i	$-0.965445\pi$
$547$	− 118.529i	− 0.216690i	0.994113	−	0.108345i	$-0.0345551\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 57.6526i	− 0.104633i
$552$	0	0
$553$	−279.809	−0.505983
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−198.759	−0.356839	−0.178419	−	0.983955i	$-0.557098\pi$
$557$	−198.759	−0.356839	−0.178419	+	0.983955i	$0.557098\pi$
$558$	0	0
$559$	− 47.8549i	− 0.0856080i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	636.666i	1.13085i	0.824801	+	0.565423i	$0.191287\pi$
$563$	636.666i	1.13085i	−0.824801	+	0.565423i	$0.808713\pi$
$564$	0	0
$565$	552.221	0.977382
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−372.294	−0.654295	−0.327148	−	0.944973i	$-0.606087\pi$
$569$	−372.294	−0.654295	−0.327148	+	0.944973i	$0.606087\pi$
$570$	0	0
$571$	− 135.453i	− 0.237221i	−0.992941	−	0.118610i	$-0.962156\pi$
$571$	− 135.453i	− 0.237221i	0.992941	−	0.118610i	$-0.0378439\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 117.645i	− 0.204601i
$576$	0	0
$577$	−202.495	−0.350945	−0.175473	−	0.984484i	$-0.556145\pi$
$577$	−202.495	−0.350945	−0.175473	+	0.984484i	$0.556145\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	144.196	0.248185
$582$	0	0
$583$	1067.90i	1.83172i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 460.765i	− 0.784948i	−0.919763	−	0.392474i	$-0.871619\pi$
$587$	− 460.765i	− 0.784948i	0.919763	−	0.392474i	$-0.128381\pi$
$588$	0	0
$589$	82.4351	0.139958
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−430.579	−0.726103	−0.363051	−	0.931769i	$-0.618265\pi$
$593$	−430.579	−0.726103	−0.363051	+	0.931769i	$0.618265\pi$
$594$	0	0
$595$	− 485.679i	− 0.816268i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	899.653i	1.50192i	0.660345	+	0.750962i	$0.270410\pi$
$599$	899.653i	1.50192i	−0.660345	+	0.750962i	$0.729590\pi$
$600$	0	0
$601$	−764.763	−1.27248	−0.636242	−	0.771490i	$-0.719512\pi$
$601$	−764.763	−1.27248	−0.636242	+	0.771490i	$0.719512\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	534.100	0.882810
$606$	0	0
$607$	− 1171.39i	− 1.92981i	−0.262607	−	0.964903i	$-0.584582\pi$
$607$	− 1171.39i	− 1.92981i	0.262607	−	0.964903i	$-0.415418\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	51.2789i	0.0839262i
$612$	0	0
$613$	−790.790	−1.29003	−0.645016	−	0.764169i	$-0.723149\pi$
$613$	−790.790	−1.29003	−0.645016	+	0.764169i	$0.723149\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	276.328	0.447857	0.223929	−	0.974606i	$-0.428112\pi$
$617$	276.328	0.447857	0.223929	+	0.974606i	$0.428112\pi$
$618$	0	0
$619$	784.071i	1.26667i	0.773876	+	0.633337i	$0.218315\pi$
$619$	784.071i	1.26667i	−0.773876	+	0.633337i	$0.781685\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	332.874i	0.534307i
$624$	0	0
$625$	−540.924	−0.865478
$626$	0	0
$627$	0	0
$628$	0	0
$629$	435.108	0.691746
$630$	0	0
$631$	− 653.266i	− 1.03529i	−0.855597	−	0.517643i	$-0.826810\pi$
$631$	− 653.266i	− 1.03529i	0.855597	−	0.517643i	$-0.173190\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	692.191i	1.09006i
$636$	0	0
$637$	48.1897	0.0756510
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−395.997	−0.617780	−0.308890	−	0.951098i	$-0.599957\pi$
$641$	−395.997	−0.617780	−0.308890	+	0.951098i	$0.599957\pi$
$642$	0	0
$643$	279.749i	0.435069i	0.976053	+	0.217534i	$0.0698015\pi$
$643$	279.749i	0.435069i	−0.976053	+	0.217534i	$0.930199\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	596.479i	0.921916i	0.887422	+	0.460958i	$0.152494\pi$
$647$	596.479i	0.921916i	−0.887422	+	0.460958i	$0.847506\pi$
$648$	0	0
$649$	497.657	0.766806
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1080.76	1.65506	0.827532	−	0.561418i	$-0.189744\pi$
$653$	1080.76	1.65506	0.827532	+	0.561418i	$0.189744\pi$
$654$	0	0
$655$	− 439.691i	− 0.671284i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 455.705i	− 0.691510i	−0.938325	−	0.345755i	$-0.887623\pi$
$659$	− 455.705i	− 0.691510i	0.938325	−	0.345755i	$-0.112377\pi$
$660$	0	0
$661$	1052.38	1.59210	0.796051	−	0.605229i	$-0.206919\pi$
$661$	1052.38	1.59210	0.796051	+	0.605229i	$0.206919\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	71.6371	0.107725
$666$	0	0
$667$	− 518.251i	− 0.776987i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1060.43i	1.58037i
$672$	0	0
$673$	−687.887	−1.02212	−0.511060	−	0.859545i	$-0.670747\pi$
$673$	−687.887	−1.02212	−0.511060	+	0.859545i	$0.670747\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.1309	0.0666631	0.0333315	−	0.999444i	$-0.489388\pi$
$677$	45.1309	0.0666631	0.0333315	+	0.999444i	$0.489388\pi$
$678$	0	0
$679$	− 472.869i	− 0.696419i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 504.393i	− 0.738497i	−0.929331	−	0.369249i	$-0.879615\pi$
$683$	− 504.393i	− 0.738497i	0.929331	−	0.369249i	$-0.120385\pi$
$684$	0	0
$685$	890.972	1.30069
$686$	0	0
$687$	0	0
$688$	0	0
$689$	91.4415	0.132716
$690$	0	0
$691$	− 340.792i	− 0.493187i	−0.969119	−	0.246594i	$-0.920689\pi$
$691$	− 340.792i	− 0.493187i	0.969119	−	0.246594i	$-0.0793112\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 114.767i	− 0.165133i
$696$	0	0
$697$	−1000.25	−1.43508
$698$	0	0
$699$	0	0
$700$	0	0
$701$	621.657	0.886815	0.443407	−	0.896320i	$-0.353770\pi$
$701$	621.657	0.886815	0.443407	+	0.896320i	$0.353770\pi$
$702$	0	0
$703$	64.1780i	0.0912916i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 70.9047i	− 0.100290i
$708$	0	0
$709$	1218.54	1.71867	0.859335	−	0.511414i	$-0.170878\pi$
$709$	1218.54	1.71867	0.859335	+	0.511414i	$0.170878\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	741.025	1.03931
$714$	0	0
$715$	− 94.3283i	− 0.131928i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 204.859i	− 0.284922i	−0.989800	−	0.142461i	$-0.954498\pi$
$719$	− 204.859i	− 0.284922i	0.989800	−	0.142461i	$-0.0455016\pi$
$720$	0	0
$721$	−492.087	−0.682507
$722$	0	0
$723$	0	0
$724$	0	0
$725$	39.7118	0.0547748
$726$	0	0
$727$	− 1211.72i	− 1.66674i	−0.552718	−	0.833368i	$-0.686409\pi$
$727$	− 1211.72i	− 1.66674i	0.552718	−	0.833368i	$-0.313591\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1077.65i	− 1.47422i
$732$	0	0
$733$	−1134.29	−1.54746	−0.773731	−	0.633514i	$-0.781612\pi$
$733$	−1134.29	−1.54746	−0.773731	+	0.633514i	$0.781612\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1356.53	1.84061
$738$	0	0
$739$	937.472i	1.26857i	0.773100	+	0.634284i	$0.218705\pi$
$739$	937.472i	1.26857i	−0.773100	+	0.634284i	$0.781295\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 445.904i	− 0.600141i	−0.953917	−	0.300070i	$-0.902990\pi$
$743$	− 445.904i	− 0.600141i	0.953917	−	0.300070i	$-0.0970101\pi$
$744$	0	0
$745$	987.116	1.32499
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−521.855	−0.696735
$750$	0	0
$751$	606.110i	0.807070i	0.914964	+	0.403535i	$0.132219\pi$
$751$	606.110i	0.807070i	−0.914964	+	0.403535i	$0.867781\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	315.064i	0.417303i
$756$	0	0
$757$	−405.982	−0.536304	−0.268152	−	0.963377i	$-0.586413\pi$
$757$	−405.982	−0.536304	−0.268152	+	0.963377i	$0.586413\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−141.386	−0.185790	−0.0928951	−	0.995676i	$-0.529612\pi$
$761$	−141.386	−0.185790	−0.0928951	+	0.995676i	$0.529612\pi$
$762$	0	0
$763$	− 364.179i	− 0.477299i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 42.6133i	− 0.0555584i
$768$	0	0
$769$	1452.65	1.88901	0.944507	−	0.328490i	$-0.106540\pi$
$769$	1452.65	1.88901	0.944507	+	0.328490i	$0.106540\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	786.603	1.01760	0.508799	−	0.860886i	$-0.330090\pi$
$773$	786.603	1.01760	0.508799	+	0.860886i	$0.330090\pi$
$774$	0	0
$775$	56.7822i	0.0732674i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 147.535i	− 0.189391i
$780$	0	0
$781$	−1437.54	−1.84065
$782$	0	0
$783$	0	0
$784$	0	0
$785$	567.939	0.723489
$786$	0	0
$787$	309.598i	0.393390i	0.980465	+	0.196695i	$0.0630209\pi$
$787$	309.598i	0.393390i	−0.980465	+	0.196695i	$0.936979\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 412.572i	− 0.521583i
$792$	0	0
$793$	90.8021	0.114505
$794$	0	0
$795$	0	0
$796$	0	0
$797$	479.138	0.601177	0.300588	−	0.953754i	$-0.402817\pi$
$797$	479.138	0.601177	0.300588	+	0.953754i	$0.402817\pi$
$798$	0	0
$799$	1154.76i	1.44526i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1034.74i	− 1.28859i
$804$	0	0
$805$	643.960	0.799951
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−451.634	−0.558262	−0.279131	−	0.960253i	$-0.590046\pi$
$809$	−451.634	−0.558262	−0.279131	+	0.960253i	$0.590046\pi$
$810$	0	0
$811$	409.759i	0.505252i	0.967564	+	0.252626i	$0.0812942\pi$
$811$	409.759i	0.505252i	−0.967564	+	0.252626i	$0.918706\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 609.589i	− 0.747962i
$816$	0	0
$817$	158.953	0.194557
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1158.36	−1.41092	−0.705458	−	0.708752i	$-0.749258\pi$
$821$	−1158.36	−1.41092	−0.705458	+	0.708752i	$0.749258\pi$
$822$	0	0
$823$	662.667i	0.805185i	0.915379	+	0.402592i	$0.131891\pi$
$823$	662.667i	0.805185i	−0.915379	+	0.402592i	$0.868109\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 759.815i	− 0.918761i	−0.888240	−	0.459380i	$-0.848071\pi$
$827$	− 759.815i	− 0.918761i	0.888240	−	0.459380i	$-0.151929\pi$
$828$	0	0
$829$	1096.44	1.32260	0.661300	−	0.750121i	$-0.270005\pi$
$829$	1096.44	1.32260	0.661300	+	0.750121i	$0.270005\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1085.19	1.30275
$834$	0	0
$835$	− 328.194i	− 0.393047i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	491.205i	0.585465i	0.956194	+	0.292732i	$0.0945645\pi$
$839$	491.205i	0.585465i	−0.956194	+	0.292732i	$0.905435\pi$
$840$	0	0
$841$	−666.062	−0.791988
$842$	0	0
$843$	0	0
$844$	0	0
$845$	784.559	0.928472
$846$	0	0
$847$	− 399.034i	− 0.471115i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	576.908i	0.677918i
$852$	0	0
$853$	−638.068	−0.748029	−0.374014	−	0.927423i	$-0.622019\pi$
$853$	−638.068	−0.748029	−0.374014	+	0.927423i	$0.622019\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1352.76	1.57849	0.789244	−	0.614080i	$-0.210473\pi$
$857$	1352.76	1.57849	0.789244	+	0.614080i	$0.210473\pi$
$858$	0	0
$859$	519.672i	0.604973i	0.953154	+	0.302486i	$0.0978167\pi$
$859$	519.672i	0.604973i	−0.953154	+	0.302486i	$0.902183\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	300.161i	0.347811i	0.984762	+	0.173905i	$0.0556387\pi$
$863$	300.161i	0.347811i	−0.984762	+	0.173905i	$0.944361\pi$
$864$	0	0
$865$	437.864	0.506202
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1223.79	−1.40827
$870$	0	0
$871$	− 116.157i	− 0.133360i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	460.212i	0.525956i
$876$	0	0
$877$	−811.129	−0.924890	−0.462445	−	0.886648i	$-0.653028\pi$
$877$	−811.129	−0.924890	−0.462445	+	0.886648i	$0.653028\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−153.562	−0.174304	−0.0871522	−	0.996195i	$-0.527777\pi$
$881$	−153.562	−0.174304	−0.0871522	+	0.996195i	$0.527777\pi$
$882$	0	0
$883$	851.900i	0.964779i	0.875957	+	0.482390i	$0.160231\pi$
$883$	851.900i	0.964779i	−0.875957	+	0.482390i	$0.839769\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1225.40i	1.38151i	0.723091	+	0.690753i	$0.242721\pi$
$887$	1225.40i	1.38151i	−0.723091	+	0.690753i	$0.757279\pi$
$888$	0	0
$889$	517.146	0.581716
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−170.326	−0.190735
$894$	0	0
$895$	1104.02i	1.23354i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	250.137i	0.278239i
$900$	0	0
$901$	2059.19	2.28545
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−408.188	−0.451036
$906$	0	0
$907$	− 1357.06i	− 1.49621i	−0.663580	−	0.748106i	$-0.730964\pi$
$907$	− 1357.06i	− 1.49621i	0.663580	−	0.748106i	$-0.269036\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 726.531i	− 0.797510i	−0.917058	−	0.398755i	$-0.869442\pi$
$911$	− 726.531i	− 0.797510i	0.917058	−	0.398755i	$-0.130558\pi$
$912$	0	0
$913$	630.664	0.690761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−328.499	−0.358233
$918$	0	0
$919$	− 884.110i	− 0.962035i	−0.876711	−	0.481017i	$-0.840267\pi$
$919$	− 884.110i	− 0.962035i	0.876711	−	0.481017i	$-0.159733\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	123.094i	0.133363i
$924$	0	0
$925$	−44.2065	−0.0477908
$926$	0	0
$927$	0	0
$928$	0	0
$929$	221.433	0.238356	0.119178	−	0.992873i	$-0.461974\pi$
$929$	221.433	0.238356	0.119178	+	0.992873i	$0.461974\pi$
$930$	0	0
$931$	160.065i	0.171928i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 2124.20i	− 2.27187i
$936$	0	0
$937$	1343.79	1.43414	0.717072	−	0.696999i	$-0.245482\pi$
$937$	1343.79	1.43414	0.717072	+	0.696999i	$0.245482\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1265.40	1.34474	0.672370	−	0.740215i	$-0.265277\pi$
$941$	1265.40	1.34474	0.672370	+	0.740215i	$0.265277\pi$
$942$	0	0
$943$	− 1326.22i	− 1.40639i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1391.94i	− 1.46984i	−0.678156	−	0.734918i	$-0.737220\pi$
$947$	− 1391.94i	− 1.46984i	0.678156	−	0.734918i	$-0.262780\pi$
$948$	0	0
$949$	−88.6022	−0.0933637
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1561.62	1.63863	0.819315	−	0.573343i	$-0.194354\pi$
$953$	1561.62	1.63863	0.819315	+	0.573343i	$0.194354\pi$
$954$	0	0
$955$	− 427.375i	− 0.447514i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 665.658i	− 0.694117i
$960$	0	0
$961$	603.340	0.627825
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1720.90	−1.78332
$966$	0	0
$967$	543.080i	0.561613i	0.959764	+	0.280806i	$0.0906019\pi$
$967$	543.080i	0.561613i	−0.959764	+	0.280806i	$0.909398\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 210.540i	− 0.216828i	−0.994106	−	0.108414i	$-0.965423\pi$
$971$	− 210.540i	− 0.216828i	0.994106	−	0.108414i	$-0.0345773\pi$
$972$	0	0
$973$	−85.7444	−0.0881237
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1723.36	−1.76393	−0.881963	−	0.471319i	$-0.843778\pi$
$977$	−1723.36	−1.76393	−0.881963	+	0.471319i	$0.843778\pi$
$978$	0	0
$979$	1455.88i	1.48711i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	222.883i	0.226737i	0.993553	+	0.113369i	$0.0361641\pi$
$983$	222.883i	0.226737i	−0.993553	+	0.113369i	$0.963836\pi$
$984$	0	0
$985$	−190.555	−0.193457
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1428.86	1.44475
$990$	0	0
$991$	− 1457.96i	− 1.47120i	−0.677415	−	0.735601i	$-0.736900\pi$
$991$	− 1457.96i	− 1.47120i	0.677415	−	0.735601i	$-0.263100\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1609.07i	− 1.61715i
$996$	0	0
$997$	823.554	0.826032	0.413016	−	0.910724i	$-0.364475\pi$
$997$	823.554	0.826032	0.413016	+	0.910724i	$0.364475\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.m.e.1711.4		12
3.2	odd	2		912.3.m.b.799.5	✓	12
4.3	odd	2	inner	2736.3.m.e.1711.3		12
12.11	even	2		912.3.m.b.799.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.3.m.b.799.5	✓	12	3.2	odd	2
912.3.m.b.799.11	yes	12	12.11	even	2
2736.3.m.e.1711.3		12	4.3	odd	2	inner
2736.3.m.e.1711.4		12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.69015 q^{5} +3.50408i q^{7} +O(q^{10})\)	\(q-4.69015 q^{5} +3.50408i q^{7} +15.3257i q^{11} +1.31231 q^{13} +29.5521 q^{17} +4.35890i q^{19} +39.1830i q^{23} -3.00246 q^{25} -13.2264 q^{29} -18.9119i q^{31} -16.4347i q^{35} +14.7234 q^{37} -33.8469 q^{41} -36.4663i q^{43} +39.0754i q^{47} +36.7214 q^{49} +69.6801 q^{53} -71.8799i q^{55} -32.4721i q^{59} +69.1928 q^{61} -6.15491 q^{65} -88.5136i q^{67} +93.7996i q^{71} -67.5164 q^{73} -53.7025 q^{77} +79.8522i q^{79} -41.1508i q^{83} -138.604 q^{85} +94.9959 q^{89} +4.59843i q^{91} -20.4439i q^{95} -134.948 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{5}+O(q^{10}) \) 12 * q + 10 * q^5	\( 12 q + 10 q^{5} + 36 q^{13} - 14 q^{17} + 10 q^{25} + 108 q^{29} - 16 q^{37} - 16 q^{41} - 118 q^{49} - 220 q^{53} + 366 q^{61} - 140 q^{65} - 158 q^{73} + 286 q^{77} + 98 q^{85} + 396 q^{89} + 224 q^{97}+O(q^{100}) \) 12 * q + 10 * q^5 + 36 * q^13 - 14 * q^17 + 10 * q^25 + 108 * q^29 - 16 * q^37 - 16 * q^41 - 118 * q^49 - 220 * q^53 + 366 * q^61 - 140 * q^65 - 158 * q^73 + 286 * q^77 + 98 * q^85 + 396 * q^89 + 224 * q^97

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