LMFDB - Embedded newform 1344.3.d.e.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,3,Mod(449,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1344.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1344.d (of order $2$, degree $1$, not 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:	$36.6213475300$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-2.57794i$ of defining polynomial
Character	$\chi$	$=$	1344.449
Dual form			1344.3.d.e.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.64575 + 1.41421i) q^{3} -6.57008i q^{5} -2.64575 q^{7} +(5.00000 - 7.48331i) q^{9} +O(q^{10})$	\(q+(-2.64575 + 1.41421i) q^{3} -6.57008i q^{5} -2.64575 q^{7} +(5.00000 - 7.48331i) q^{9} +0.412247i q^{11} -20.5830 q^{13} +(9.29150 + 17.3828i) q^{15} +15.8799i q^{17} -16.0000 q^{19} +(7.00000 - 3.74166i) q^{21} +36.0024i q^{23} -18.1660 q^{25} +(-2.64575 + 26.8701i) q^{27} -20.8010i q^{29} +5.54249 q^{31} +(-0.583005 - 1.09070i) q^{33} +17.3828i q^{35} -20.0000 q^{37} +(54.4575 - 29.1088i) q^{39} -76.1013i q^{41} +51.7490 q^{43} +(-49.1660 - 32.8504i) q^{45} +8.48528i q^{47} +7.00000 q^{49} +(-22.4575 - 42.0142i) q^{51} +50.9117i q^{53} +2.70850 q^{55} +(42.3320 - 22.6274i) q^{57} -1.64899i q^{59} +66.9150 q^{61} +(-13.2288 + 19.7990i) q^{63} +135.232i q^{65} +49.4170 q^{67} +(-50.9150 - 95.2533i) q^{69} -87.7385i q^{71} -12.3320 q^{73} +(48.0627 - 25.6906i) q^{75} -1.09070i q^{77} +84.9150 q^{79} +(-31.0000 - 74.8331i) q^{81} +4.12247i q^{83} +104.332 q^{85} +(29.4170 + 55.0342i) q^{87} +31.2014i q^{89} +54.4575 q^{91} +(-14.6640 + 7.83826i) q^{93} +105.121i q^{95} -68.8340 q^{97} +(3.08497 + 2.06123i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 20 q^{9}+O(q^{10}) $ 4 * q + 20 * q^9	$ 4 q + 20 q^{9} - 40 q^{13} + 16 q^{15} - 64 q^{19} + 28 q^{21} + 12 q^{25} + 128 q^{31} + 40 q^{33} - 80 q^{37} + 112 q^{39} + 80 q^{43} - 112 q^{45} + 28 q^{49} + 16 q^{51} + 32 q^{55} + 56 q^{61} + 240 q^{67} + 8 q^{69} + 120 q^{73} + 224 q^{75} + 128 q^{79} - 124 q^{81} + 248 q^{85} + 160 q^{87} + 112 q^{91} + 280 q^{93} - 360 q^{97} + 224 q^{99}+O(q^{100}) $ 4 * q + 20 * q^9 - 40 * q^13 + 16 * q^15 - 64 * q^19 + 28 * q^21 + 12 * q^25 + 128 * q^31 + 40 * q^33 - 80 * q^37 + 112 * q^39 + 80 * q^43 - 112 * q^45 + 28 * q^49 + 16 * q^51 + 32 * q^55 + 56 * q^61 + 240 * q^67 + 8 * q^69 + 120 * q^73 + 224 * q^75 + 128 * q^79 - 124 * q^81 + 248 * q^85 + 160 * q^87 + 112 * q^91 + 280 * q^93 - 360 * q^97 + 224 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	−2.64575	+	1.41421i	−0.881917	+	0.471405i
$3$	−2.64575	+	1.41421i	−0.881917	+	0.471405i
$4$		0			0
$5$		−	6.57008i		−	1.31402i	−0.753883	−	0.657008i	$-0.771822\pi$
$5$		−	6.57008i		−	1.31402i	0.753883	−	0.657008i	$-0.228178\pi$
$6$		0			0
$7$	−2.64575			−0.377964
$7$	−2.64575			−0.377964
$8$		0			0
$9$	5.00000	−	7.48331i	0.555556	−	0.831479i
$10$		0			0
$11$			0.412247i			0.0374770i	0.999824	+	0.0187385i	$0.00596500\pi$
$11$			0.412247i			0.0374770i	−0.999824	+	0.0187385i	$0.994035\pi$
$12$		0			0
$13$	−20.5830			−1.58331			−0.791654	−	0.610970i	$-0.790780\pi$
$13$	−20.5830			−1.58331			−0.791654	+	0.610970i	$0.790780\pi$
$14$		0			0
$15$	9.29150	+	17.3828i	0.619434	+	1.15885i
$16$		0			0
$17$			15.8799i			0.934109i	0.884228	+	0.467055i	$0.154685\pi$
$17$			15.8799i			0.934109i	−0.884228	+	0.467055i	$0.845315\pi$
$18$		0			0
$19$	−16.0000			−0.842105			−0.421053	−	0.907036i	$-0.638339\pi$
$19$	−16.0000			−0.842105			−0.421053	+	0.907036i	$0.638339\pi$
$20$		0			0
$21$	7.00000	−	3.74166i	0.333333	−	0.178174i
$22$		0			0
$23$			36.0024i			1.56532i	0.622449	+	0.782660i	$0.286138\pi$
$23$			36.0024i			1.56532i	−0.622449	+	0.782660i	$0.713862\pi$
$24$		0			0
$25$	−18.1660			−0.726640
$26$		0			0
$27$	−2.64575	+	26.8701i	−0.0979908	+	0.995187i
$28$		0			0
$29$		−	20.8010i		−	0.717274i	−0.933477	−	0.358637i	$-0.883242\pi$
$29$		−	20.8010i		−	0.717274i	0.933477	−	0.358637i	$-0.116758\pi$
$30$		0			0
$31$	5.54249			0.178790			0.0893949	−	0.995996i	$-0.471507\pi$
$31$	5.54249			0.178790			0.0893949	+	0.995996i	$0.471507\pi$
$32$		0			0
$33$	−0.583005	−	1.09070i	−0.0176668	−	0.0330516i
$34$		0			0
$35$			17.3828i			0.496652i
$36$		0			0
$37$	−20.0000			−0.540541			−0.270270	−	0.962784i	$-0.587113\pi$
$37$	−20.0000			−0.540541			−0.270270	+	0.962784i	$0.587113\pi$
$38$		0			0
$39$	54.4575	−	29.1088i	1.39635	−	0.746379i
$40$		0			0
$41$		−	76.1013i		−	1.85613i	−0.372418	−	0.928065i	$-0.621471\pi$
$41$		−	76.1013i		−	1.85613i	0.372418	−	0.928065i	$-0.378529\pi$
$42$		0			0
$43$	51.7490			1.20347			0.601733	−	0.798698i	$-0.294477\pi$
$43$	51.7490			1.20347			0.601733	+	0.798698i	$0.294477\pi$
$44$		0			0
$45$	−49.1660	−	32.8504i	−1.09258	−	0.730009i
$46$		0			0
$47$			8.48528i			0.180538i	0.995917	+	0.0902690i	$0.0287727\pi$
$47$			8.48528i			0.180538i	−0.995917	+	0.0902690i	$0.971227\pi$
$48$		0			0
$49$	7.00000			0.142857
$50$		0			0
$51$	−22.4575	−	42.0142i	−0.440343	−	0.823807i
$52$		0			0
$53$			50.9117i			0.960598i	0.877105	+	0.480299i	$0.159472\pi$
$53$			50.9117i			0.960598i	−0.877105	+	0.480299i	$0.840528\pi$
$54$		0			0
$55$	2.70850			0.0492454
$56$		0			0
$57$	42.3320	−	22.6274i	0.742667	−	0.396972i
$58$		0			0
$59$		−	1.64899i		−	0.0279489i	−0.999902	−	0.0139745i	$-0.995552\pi$
$59$		−	1.64899i		−	0.0279489i	0.999902	−	0.0139745i	$-0.00444836\pi$
$60$		0			0
$61$	66.9150			1.09697			0.548484	−	0.836161i	$-0.315205\pi$
$61$	66.9150			1.09697			0.548484	+	0.836161i	$0.315205\pi$
$62$		0			0
$63$	−13.2288	+	19.7990i	−0.209980	+	0.314270i
$64$		0			0
$65$			135.232i			2.08049i
$66$		0			0
$67$	49.4170			0.737567			0.368784	−	0.929515i	$-0.379774\pi$
$67$	49.4170			0.737567			0.368784	+	0.929515i	$0.379774\pi$
$68$		0			0
$69$	−50.9150	−	95.2533i	−0.737899	−	1.38048i
$70$		0			0
$71$		−	87.7385i		−	1.23575i	−0.786275	−	0.617877i	$-0.787993\pi$
$71$		−	87.7385i		−	1.23575i	0.786275	−	0.617877i	$-0.212007\pi$
$72$		0			0
$73$	−12.3320			−0.168932			−0.0844659	−	0.996426i	$-0.526918\pi$
$73$	−12.3320			−0.168932			−0.0844659	+	0.996426i	$0.526918\pi$
$74$		0			0
$75$	48.0627	−	25.6906i	0.640837	−	0.342542i
$76$		0			0
$77$		−	1.09070i		−	0.0141650i
$78$		0			0
$79$	84.9150			1.07487			0.537437	−	0.843304i	$-0.319393\pi$
$79$	84.9150			1.07487			0.537437	+	0.843304i	$0.319393\pi$
$80$		0			0
$81$	−31.0000	−	74.8331i	−0.382716	−	0.923866i
$82$		0			0
$83$			4.12247i			0.0496683i	0.999692	+	0.0248342i	$0.00790577\pi$
$83$			4.12247i			0.0496683i	−0.999692	+	0.0248342i	$0.992094\pi$
$84$		0			0
$85$	104.332			1.22744
$86$		0			0
$87$	29.4170	+	55.0342i	0.338126	+	0.632577i
$88$		0			0
$89$			31.2014i			0.350578i	0.984517	+	0.175289i	$0.0560860\pi$
$89$			31.2014i			0.350578i	−0.984517	+	0.175289i	$0.943914\pi$
$90$		0			0
$91$	54.4575			0.598434
$92$		0			0
$93$	−14.6640	+	7.83826i	−0.157678	+	0.0842824i
$94$		0			0
$95$			105.121i			1.10654i
$96$		0			0
$97$	−68.8340			−0.709629			−0.354814	−	0.934937i	$-0.615456\pi$
$97$	−68.8340			−0.709629			−0.354814	+	0.934937i	$0.615456\pi$
$98$		0			0
$99$	3.08497	+	2.06123i	0.0311614	+	0.0208206i
$100$		0			0
$101$			140.153i			1.38766i	0.720141	+	0.693828i	$0.244077\pi$
$101$			140.153i			1.38766i	−0.720141	+	0.693828i	$0.755923\pi$
$102$		0			0
$103$	−39.3725			−0.382258			−0.191129	−	0.981565i	$-0.561215\pi$
$103$	−39.3725			−0.382258			−0.191129	+	0.981565i	$0.561215\pi$
$104$		0			0
$105$	−24.5830	−	45.9906i	−0.234124	−	0.438006i
$106$		0			0
$107$			108.007i			1.00941i	0.863291	+	0.504706i	$0.168399\pi$
$107$			108.007i			1.00941i	−0.863291	+	0.504706i	$0.831601\pi$
$108$		0			0
$109$	123.830			1.13606			0.568028	−	0.823009i	$-0.307707\pi$
$109$	123.830			1.13606			0.568028	+	0.823009i	$0.307707\pi$
$110$		0			0
$111$	52.9150	−	28.2843i	0.476712	−	0.254813i
$112$		0			0
$113$		−	80.4900i		−	0.712301i	−0.934429	−	0.356150i	$-0.884089\pi$
$113$		−	80.4900i		−	0.712301i	0.934429	−	0.356150i	$-0.115911\pi$
$114$		0			0
$115$	236.539			2.05686
$116$		0			0
$117$	−102.915	+	154.029i	−0.879616	+	1.31649i
$118$		0			0
$119$		−	42.0142i		−	0.353060i
$120$		0			0
$121$	120.830			0.998595
$122$		0			0
$123$	107.624	+	201.345i	0.874988	+	1.63695i
$124$		0			0
$125$		−	44.8999i		−	0.359199i
$126$		0			0
$127$	132.915			1.04658			0.523288	−	0.852156i	$-0.324705\pi$
$127$	132.915			1.04658			0.523288	+	0.852156i	$0.324705\pi$
$128$		0			0
$129$	−136.915	+	73.1842i	−1.06136	+	0.567319i
$130$		0			0
$131$		−	4.12247i		−	0.0314692i	−0.999876	−	0.0157346i	$-0.994991\pi$
$131$		−	4.12247i		−	0.0314692i	0.999876	−	0.0157346i	$-0.00500869\pi$
$132$		0			0
$133$	42.3320			0.318286
$134$		0			0
$135$	176.539	+	17.3828i	1.30769	+	0.128762i
$136$		0			0
$137$			99.6420i			0.727314i	0.931533	+	0.363657i	$0.118472\pi$
$137$			99.6420i			0.727314i	−0.931533	+	0.363657i	$0.881528\pi$
$138$		0			0
$139$	93.5425			0.672968			0.336484	−	0.941689i	$-0.390762\pi$
$139$	93.5425			0.672968			0.336484	+	0.941689i	$0.390762\pi$
$140$		0			0
$141$	−12.0000	−	22.4499i	−0.0851064	−	0.159219i
$142$		0			0
$143$		−	8.48528i		−	0.0593376i
$144$		0			0
$145$	−136.664			−0.942511
$146$		0			0
$147$	−18.5203	+	9.89949i	−0.125988	+	0.0673435i
$148$		0			0
$149$			49.8469i			0.334543i	0.985911	+	0.167271i	$0.0534956\pi$
$149$			49.8469i			0.334543i	−0.985911	+	0.167271i	$0.946504\pi$
$150$		0			0
$151$	−211.660			−1.40172			−0.700861	−	0.713298i	$-0.747201\pi$
$151$	−211.660			−1.40172			−0.700861	+	0.713298i	$0.747201\pi$
$152$		0			0
$153$	118.834	+	79.3993i	0.776693	+	0.518950i
$154$		0			0
$155$		−	36.4146i		−	0.234933i
$156$		0			0
$157$	108.745			0.692644			0.346322	−	0.938116i	$-0.387430\pi$
$157$	108.745			0.692644			0.346322	+	0.938116i	$0.387430\pi$
$158$		0			0
$159$	−72.0000	−	134.700i	−0.452830	−	0.847168i
$160$		0			0
$161$		−	95.2533i		−	0.591635i
$162$		0			0
$163$	13.0039			0.0797788			0.0398894	−	0.999204i	$-0.487299\pi$
$163$	13.0039			0.0797788			0.0398894	+	0.999204i	$0.487299\pi$
$164$		0			0
$165$	−7.16601	+	3.83039i	−0.0434304	+	0.0232145i
$166$		0			0
$167$			156.858i			0.939267i	0.882862	+	0.469633i	$0.155614\pi$
$167$			156.858i			0.939267i	−0.882862	+	0.469633i	$0.844386\pi$
$168$		0			0
$169$	254.660			1.50686
$170$		0			0
$171$	−80.0000	+	119.733i	−0.467836	+	0.700193i
$172$		0			0
$173$			5.45351i			0.0315232i	0.999876	+	0.0157616i	$0.00501728\pi$
$173$			5.45351i			0.0315232i	−0.999876	+	0.0157616i	$0.994983\pi$
$174$		0			0
$175$	48.0627			0.274644
$176$		0			0
$177$	2.33202	+	4.36281i	0.0131753	+	0.0246487i
$178$		0			0
$179$			1.23674i			0.00690917i	0.999994	+	0.00345458i	$0.00109963\pi$
$179$			1.23674i			0.00690917i	−0.999994	+	0.00345458i	$0.998900\pi$
$180$		0			0
$181$	186.915			1.03268			0.516340	−	0.856384i	$-0.327294\pi$
$181$	186.915			1.03268			0.516340	+	0.856384i	$0.327294\pi$
$182$		0			0
$183$	−177.041	+	94.6321i	−0.967435	+	0.517116i
$184$		0			0
$185$			131.402i			0.710279i
$186$		0			0
$187$	−6.54642			−0.0350076
$188$		0			0
$189$	7.00000	−	71.0915i	0.0370370	−	0.376145i
$190$		0			0
$191$			350.542i			1.83530i	0.397392	+	0.917649i	$0.369915\pi$
$191$			350.542i			1.83530i	−0.397392	+	0.917649i	$0.630085\pi$
$192$		0			0
$193$	270.494			1.40152			0.700762	−	0.713395i	$-0.252844\pi$
$193$	270.494			1.40152			0.700762	+	0.713395i	$0.252844\pi$
$194$		0			0
$195$	−191.247	−	357.790i	−0.980754	−	1.83482i
$196$		0			0
$197$		−	63.5194i		−	0.322434i	−0.986919	−	0.161217i	$-0.948458\pi$
$197$		−	63.5194i		−	0.322434i	0.986919	−	0.161217i	$-0.0515419\pi$
$198$		0			0
$199$	88.0000			0.442211			0.221106	−	0.975250i	$-0.429033\pi$
$199$	88.0000			0.442211			0.221106	+	0.975250i	$0.429033\pi$
$200$		0			0
$201$	−130.745	+	69.8862i	−0.650473	+	0.347692i
$202$		0			0
$203$			55.0342i			0.271104i
$204$		0			0
$205$	−499.992			−2.43899
$206$		0			0
$207$	269.417	+	180.012i	1.30153	+	0.869622i
$208$		0			0
$209$		−	6.59595i		−	0.0315596i
$210$		0			0
$211$	−378.745			−1.79500			−0.897500	−	0.441014i	$-0.854619\pi$
$211$	−378.745			−1.79500			−0.897500	+	0.441014i	$0.854619\pi$
$212$		0			0
$213$	124.081	+	232.134i	0.582540	+	1.08983i
$214$		0			0
$215$		−	339.995i		−	1.58137i
$216$		0			0
$217$	−14.6640			−0.0675762
$218$		0			0
$219$	32.6275	−	17.4401i	0.148984	−	0.0796352i
$220$		0			0
$221$		−	326.855i		−	1.47898i
$222$		0			0
$223$	230.494			1.03361			0.516803	−	0.856104i	$-0.327122\pi$
$223$	230.494			1.03361			0.516803	+	0.856104i	$0.327122\pi$
$224$		0			0
$225$	−90.8301	+	135.942i	−0.403689	+	0.604187i
$226$		0			0
$227$			258.441i			1.13850i	0.822163	+	0.569252i	$0.192767\pi$
$227$			258.441i			1.13850i	−0.822163	+	0.569252i	$0.807233\pi$
$228$		0			0
$229$	37.0850			0.161943			0.0809716	−	0.996716i	$-0.474198\pi$
$229$	37.0850			0.161943			0.0809716	+	0.996716i	$0.474198\pi$
$230$		0			0
$231$	1.54249	+	2.88573i	0.00667743	+	0.0124923i
$232$		0			0
$233$		−	82.6714i		−	0.354813i	−0.984138	−	0.177406i	$-0.943229\pi$
$233$		−	82.6714i		−	0.354813i	0.984138	−	0.177406i	$-0.0567707\pi$
$234$		0			0
$235$	55.7490			0.237230
$236$		0			0
$237$	−224.664	+	120.088i	−0.947950	+	0.506700i
$238$		0			0
$239$			168.469i			0.704891i	0.935832	+	0.352445i	$0.114650\pi$
$239$			168.469i			0.704891i	−0.935832	+	0.352445i	$0.885350\pi$
$240$		0			0
$241$	−130.000			−0.539419			−0.269710	−	0.962942i	$-0.586928\pi$
$241$	−130.000			−0.539419			−0.269710	+	0.962942i	$0.586928\pi$
$242$		0			0
$243$	187.848	+	154.149i	0.773038	+	0.634359i
$244$		0			0
$245$		−	45.9906i		−	0.187717i
$246$		0			0
$247$	329.328			1.33331
$248$		0			0
$249$	−5.83005	−	10.9070i	−0.0234139	−	0.0438033i
$250$		0			0
$251$			119.859i			0.477525i	0.971078	+	0.238763i	$0.0767417\pi$
$251$			119.859i			0.477525i	−0.971078	+	0.238763i	$0.923258\pi$
$252$		0			0
$253$	−14.8419			−0.0586635
$254$		0			0
$255$	−276.037	+	147.548i	−1.08250	+	0.578619i
$256$		0			0
$257$		−	223.889i		−	0.871165i	−0.900149	−	0.435583i	$-0.856542\pi$
$257$		−	223.889i		−	0.871165i	0.900149	−	0.435583i	$-0.143458\pi$
$258$		0			0
$259$	52.9150			0.204305
$260$		0			0
$261$	−155.660	−	104.005i	−0.596399	−	0.398486i
$262$		0			0
$263$		−	214.434i		−	0.815337i	−0.913130	−	0.407668i	$-0.866342\pi$
$263$		−	214.434i		−	0.815337i	0.913130	−	0.407668i	$-0.133658\pi$
$264$		0			0
$265$	334.494			1.26224
$266$		0			0
$267$	−44.1255	−	82.5512i	−0.165264	−	0.309181i
$268$		0			0
$269$			382.156i			1.42065i	0.703872	+	0.710326i	$0.251453\pi$
$269$			382.156i			1.42065i	−0.703872	+	0.710326i	$0.748547\pi$
$270$		0			0
$271$	−114.458			−0.422352			−0.211176	−	0.977448i	$-0.567729\pi$
$271$	−114.458			−0.422352			−0.211176	+	0.977448i	$0.567729\pi$
$272$		0			0
$273$	−144.081	+	77.0146i	−0.527769	+	0.282105i
$274$		0			0
$275$		−	7.48888i		−	0.0272323i
$276$		0			0
$277$	230.494			0.832109			0.416054	−	0.909340i	$-0.363413\pi$
$277$	230.494			0.832109			0.416054	+	0.909340i	$0.363413\pi$
$278$		0			0
$279$	27.7124	−	41.4762i	0.0993277	−	0.148660i
$280$		0			0
$281$			73.9458i			0.263152i	0.991306	+	0.131576i	$0.0420038\pi$
$281$			73.9458i			0.263152i	−0.991306	+	0.131576i	$0.957996\pi$
$282$		0			0
$283$	141.166			0.498820			0.249410	−	0.968398i	$-0.419763\pi$
$283$	141.166			0.498820			0.249410	+	0.968398i	$0.419763\pi$
$284$		0			0
$285$	−148.664	−	278.125i	−0.521628	−	0.975877i
$286$		0			0
$287$			201.345i			0.701551i
$288$		0			0
$289$	36.8301			0.127440
$290$		0			0
$291$	182.118	−	97.3460i	0.625834	−	0.334522i
$292$		0			0
$293$		−	329.595i		−	1.12490i	−0.826832	−	0.562449i	$-0.809859\pi$
$293$		−	329.595i		−	1.12490i	0.826832	−	0.562449i	$-0.190141\pi$
$294$		0			0
$295$	−10.8340			−0.0367254
$296$		0			0
$297$	−11.0771	−	1.09070i	−0.0372966	−	0.00367240i
$298$		0			0
$299$		−	741.037i		−	2.47838i
$300$		0			0
$301$	−136.915			−0.454867
$302$		0			0
$303$	−198.207	−	370.810i	−0.654147	−	1.22380i
$304$		0			0
$305$		−	439.637i		−	1.44143i
$306$		0			0
$307$	−105.830			−0.344723			−0.172362	−	0.985034i	$-0.555140\pi$
$307$	−105.830			−0.344723			−0.172362	+	0.985034i	$0.555140\pi$
$308$		0			0
$309$	104.170	−	55.6812i	0.337120	−	0.180198i
$310$		0			0
$311$		−	385.136i		−	1.23838i	−0.785242	−	0.619189i	$-0.787461\pi$
$311$		−	385.136i		−	1.23838i	0.785242	−	0.619189i	$-0.212539\pi$
$312$		0			0
$313$	−79.3281			−0.253444			−0.126722	−	0.991938i	$-0.540446\pi$
$313$	−79.3281			−0.253444			−0.126722	+	0.991938i	$0.540446\pi$
$314$		0			0
$315$	130.081	+	86.9140i	0.412956	+	0.275918i
$316$		0			0
$317$		−	411.176i		−	1.29708i	−0.761179	−	0.648542i	$-0.775379\pi$
$317$		−	411.176i		−	1.29708i	0.761179	−	0.648542i	$-0.224621\pi$
$318$		0			0
$319$	8.57513			0.0268813
$320$		0			0
$321$	−152.745	−	285.760i	−0.475841	−	0.890218i
$322$		0			0
$323$		−	254.078i		−	0.786618i
$324$		0			0
$325$	373.911			1.15050
$326$		0			0
$327$	−327.624	+	175.122i	−1.00191	+	0.535542i
$328$		0			0
$329$		−	22.4499i		−	0.0682369i
$330$		0			0
$331$	−489.490			−1.47882			−0.739411	−	0.673254i	$-0.764896\pi$
$331$	−489.490			−1.47882			−0.739411	+	0.673254i	$0.764896\pi$
$332$		0			0
$333$	−100.000	+	149.666i	−0.300300	+	0.449448i
$334$		0			0
$335$		−	324.674i		−	0.969176i
$336$		0			0
$337$	500.316			1.48462			0.742309	−	0.670058i	$-0.233731\pi$
$337$	500.316			1.48462			0.742309	+	0.670058i	$0.233731\pi$
$338$		0			0
$339$	113.830	+	212.957i	0.335782	+	0.628190i
$340$		0			0
$341$			2.28487i			0.00670051i
$342$		0			0
$343$	−18.5203			−0.0539949
$344$		0			0
$345$	−625.822	+	334.516i	−1.81398	+	0.969612i
$346$		0			0
$347$			192.860i			0.555792i	0.960611	+	0.277896i	$0.0896371\pi$
$347$			192.860i			0.555792i	−0.960611	+	0.277896i	$0.910363\pi$
$348$		0			0
$349$	−148.405			−0.425230			−0.212615	−	0.977136i	$-0.568198\pi$
$349$	−148.405			−0.425230			−0.212615	+	0.977136i	$0.568198\pi$
$350$		0			0
$351$	54.4575	−	553.067i	0.155150	−	1.57569i
$352$		0			0
$353$		−	163.771i		−	0.463942i	−0.972723	−	0.231971i	$-0.925483\pi$
$353$		−	163.771i		−	0.463942i	0.972723	−	0.231971i	$-0.0745174\pi$
$354$		0			0
$355$	−576.450			−1.62380
$356$		0			0
$357$	59.4170	+	111.159i	0.166434	+	0.311370i
$358$		0			0
$359$			334.877i			0.932804i	0.884573	+	0.466402i	$0.154450\pi$
$359$			334.877i			0.932804i	−0.884573	+	0.466402i	$0.845550\pi$
$360$		0			0
$361$	−105.000			−0.290859
$362$		0			0
$363$	−319.686	+	170.879i	−0.880678	+	0.470742i
$364$		0			0
$365$			81.0224i			0.221979i
$366$		0			0
$367$	−326.996			−0.890997			−0.445499	−	0.895283i	$-0.646974\pi$
$367$	−326.996			−0.890997			−0.445499	+	0.895283i	$0.646974\pi$
$368$		0			0
$369$	−569.490	−	380.507i	−1.54333	−	1.03118i
$370$		0			0
$371$		−	134.700i		−	0.363072i
$372$		0			0
$373$	305.336			0.818595			0.409298	−	0.912401i	$-0.365774\pi$
$373$	305.336			0.818595			0.409298	+	0.912401i	$0.365774\pi$
$374$		0			0
$375$	63.4980	+	118.794i	0.169328	+	0.316784i
$376$		0			0
$377$			428.146i			1.13567i
$378$		0			0
$379$	199.660			0.526808			0.263404	−	0.964686i	$-0.415155\pi$
$379$	199.660			0.526808			0.263404	+	0.964686i	$0.415155\pi$
$380$		0			0
$381$	−351.660	+	187.970i	−0.922992	+	0.493360i
$382$		0			0
$383$		−	529.489i		−	1.38248i	−0.722626	−	0.691239i	$-0.757065\pi$
$383$		−	529.489i		−	1.38248i	0.722626	−	0.691239i	$-0.242935\pi$
$384$		0			0
$385$	−7.16601			−0.0186130
$386$		0			0
$387$	258.745	−	387.254i	0.668592	−	1.00066i
$388$		0			0
$389$			238.704i			0.613636i	0.951768	+	0.306818i	$0.0992643\pi$
$389$			238.704i			0.613636i	−0.951768	+	0.306818i	$0.900736\pi$
$390$		0			0
$391$	−571.712			−1.46218
$392$		0			0
$393$	5.83005	+	10.9070i	0.0148347	+	0.0277533i
$394$		0			0
$395$		−	557.899i		−	1.41240i
$396$		0			0
$397$	−320.583			−0.807514			−0.403757	−	0.914866i	$-0.632296\pi$
$397$	−320.583			−0.807514			−0.403757	+	0.914866i	$0.632296\pi$
$398$		0			0
$399$	−112.000	+	59.8665i	−0.280702	+	0.150041i
$400$		0			0
$401$		−	232.108i		−	0.578824i	−0.957205	−	0.289412i	$-0.906540\pi$
$401$		−	232.108i		−	0.578824i	0.957205	−	0.289412i	$-0.0934598\pi$
$402$		0			0
$403$	−114.081			−0.283079
$404$		0			0
$405$	−491.660	+	203.673i	−1.21398	+	0.502895i
$406$		0			0
$407$		−	8.24494i		−	0.0202578i
$408$		0			0
$409$	528.980			1.29335			0.646675	−	0.762765i	$-0.276159\pi$
$409$	528.980			1.29335			0.646675	+	0.762765i	$0.276159\pi$
$410$		0			0
$411$	−140.915	−	263.628i	−0.342859	−	0.641430i
$412$		0			0
$413$			4.36281i			0.0105637i
$414$		0			0
$415$	27.0850			0.0652650
$416$		0			0
$417$	−247.490	+	132.289i	−0.593502	+	0.317240i
$418$		0			0
$419$			89.7998i			0.214319i	0.994242	+	0.107160i	$0.0341756\pi$
$419$			89.7998i			0.214319i	−0.994242	+	0.107160i	$0.965824\pi$
$420$		0			0
$421$	777.150			1.84596			0.922981	−	0.384845i	$-0.125745\pi$
$421$	777.150			1.84596			0.922981	+	0.384845i	$0.125745\pi$
$422$		0			0
$423$	63.4980	+	42.4264i	0.150114	+	0.100299i
$424$		0			0
$425$		−	288.474i		−	0.678762i
$426$		0			0
$427$	−177.041			−0.414615
$428$		0			0
$429$	12.0000	+	22.4499i	0.0279720	+	0.0523309i
$430$		0			0
$431$			245.901i			0.570537i	0.958448	+	0.285268i	$0.0920827\pi$
$431$			245.901i			0.570537i	−0.958448	+	0.285268i	$0.907917\pi$
$432$		0			0
$433$	−796.996			−1.84064			−0.920319	−	0.391169i	$-0.872071\pi$
$433$	−796.996			−1.84064			−0.920319	+	0.391169i	$0.872071\pi$
$434$		0			0
$435$	361.579	−	193.272i	0.831216	−	0.444304i
$436$		0			0
$437$		−	576.038i		−	1.31816i
$438$		0			0
$439$	−553.830			−1.26157			−0.630786	−	0.775957i	$-0.717267\pi$
$439$	−553.830			−1.26157			−0.630786	+	0.775957i	$0.717267\pi$
$440$		0			0
$441$	35.0000	−	52.3832i	0.0793651	−	0.118783i
$442$		0			0
$443$			773.981i			1.74714i	0.486702	+	0.873568i	$0.338200\pi$
$443$			773.981i			1.74714i	−0.486702	+	0.873568i	$0.661800\pi$
$444$		0			0
$445$	204.996			0.460665
$446$		0			0
$447$	−70.4941	−	131.882i	−0.157705	−	0.295039i
$448$		0			0
$449$			677.174i			1.50818i	0.656770	+	0.754091i	$0.271922\pi$
$449$			677.174i			1.50818i	−0.656770	+	0.754091i	$0.728078\pi$
$450$		0			0
$451$	31.3725			0.0695622
$452$		0			0
$453$	560.000	−	299.333i	1.23620	−	0.660778i
$454$		0			0
$455$		−	357.790i		−	0.786353i
$456$		0			0
$457$	834.664			1.82640			0.913199	−	0.407514i	$-0.133604\pi$
$457$	834.664			1.82640			0.913199	+	0.407514i	$0.133604\pi$
$458$		0			0
$459$	−426.693	−	42.0142i	−0.929614	−	0.0915341i
$460$		0			0
$461$			347.150i			0.753036i	0.926409	+	0.376518i	$0.122879\pi$
$461$			347.150i			0.753036i	−0.926409	+	0.376518i	$0.877121\pi$
$462$		0			0
$463$	−317.668			−0.686108			−0.343054	−	0.939316i	$-0.611461\pi$
$463$	−317.668			−0.686108			−0.343054	+	0.939316i	$0.611461\pi$
$464$		0			0
$465$	51.4980	+	96.3440i	0.110748	+	0.207191i
$466$		0			0
$467$			547.421i			1.17221i	0.810236	+	0.586104i	$0.199339\pi$
$467$			547.421i			1.17221i	−0.810236	+	0.586104i	$0.800661\pi$
$468$		0			0
$469$	−130.745			−0.278774
$470$		0			0
$471$	−287.712	+	153.789i	−0.610854	+	0.326515i
$472$		0			0
$473$			21.3334i			0.0451023i
$474$		0			0
$475$	290.656			0.611908
$476$		0			0
$477$	380.988	+	254.558i	0.798717	+	0.533665i
$478$		0			0
$479$			135.524i			0.282931i	0.989943	+	0.141466i	$0.0451815\pi$
$479$			135.524i			0.282931i	−0.989943	+	0.141466i	$0.954818\pi$
$480$		0			0
$481$	411.660			0.855842
$482$		0			0
$483$	134.708	+	252.017i	0.278900	+	0.521773i
$484$		0			0
$485$			452.245i			0.932464i
$486$		0			0
$487$	−23.4980			−0.0482506			−0.0241253	−	0.999709i	$-0.507680\pi$
$487$	−23.4980			−0.0482506			−0.0241253	+	0.999709i	$0.507680\pi$
$488$		0			0
$489$	−34.4052	+	18.3903i	−0.0703582	+	0.0376081i
$490$		0			0
$491$			103.404i			0.210599i	0.994441	+	0.105299i	$0.0335801\pi$
$491$			103.404i			0.210599i	−0.994441	+	0.105299i	$0.966420\pi$
$492$		0			0
$493$	330.316			0.670013
$494$		0			0
$495$	13.5425	−	20.2685i	0.0273586	−	0.0409465i
$496$		0			0
$497$			232.134i			0.467071i
$498$		0			0
$499$	−64.3399			−0.128938			−0.0644688	−	0.997920i	$-0.520535\pi$
$499$	−64.3399			−0.128938			−0.0644688	+	0.997920i	$0.520535\pi$
$500$		0			0
$501$	−221.830	−	415.006i	−0.442775	−	0.828355i
$502$		0			0
$503$			546.940i			1.08736i	0.839294	+	0.543678i	$0.182969\pi$
$503$			546.940i			1.08736i	−0.839294	+	0.543678i	$0.817031\pi$
$504$		0			0
$505$	920.818			1.82340
$506$		0			0
$507$	−673.767	+	360.144i	−1.32893	+	0.710343i
$508$		0			0
$509$		−	63.5453i		−	0.124843i	−0.998050	−	0.0624217i	$-0.980118\pi$
$509$		−	63.5453i		−	0.124843i	0.998050	−	0.0624217i	$-0.0198824\pi$
$510$		0			0
$511$	32.6275			0.0638502
$512$		0			0
$513$	42.3320	−	429.921i	0.0825186	−	0.838052i
$514$		0			0
$515$			258.681i			0.502293i
$516$		0			0
$517$	−3.49803			−0.00676602
$518$		0			0
$519$	−7.71243	−	14.4286i	−0.0148602	−	0.0278009i
$520$		0			0
$521$			642.297i			1.23282i	0.787427	+	0.616408i	$0.211413\pi$
$521$			642.297i			1.23282i	−0.787427	+	0.616408i	$0.788587\pi$
$522$		0			0
$523$	−112.376			−0.214869			−0.107434	−	0.994212i	$-0.534264\pi$
$523$	−112.376			−0.214869			−0.107434	+	0.994212i	$0.534264\pi$
$524$		0			0
$525$	−127.162	+	67.9710i	−0.242213	+	0.129469i
$526$		0			0
$527$			88.0139i			0.167009i
$528$		0			0
$529$	−767.170			−1.45023
$530$		0			0
$531$	−12.3399	−	8.24494i	−0.0232390	−	0.0155272i
$532$		0			0
$533$			1566.39i			2.93883i
$534$		0			0
$535$	709.616			1.32638
$536$		0			0
$537$	−1.74902	−	3.27211i	−0.00325701	−	0.00609331i
$538$		0			0
$539$			2.88573i			0.00535386i
$540$		0			0
$541$	33.1503			0.0612759			0.0306380	−	0.999531i	$-0.490246\pi$
$541$	33.1503			0.0612759			0.0306380	+	0.999531i	$0.490246\pi$
$542$		0			0
$543$	−494.531	+	264.338i	−0.910738	+	0.486810i
$544$		0			0
$545$		−	813.574i		−	1.49280i
$546$		0			0
$547$	−919.911			−1.68174			−0.840869	−	0.541238i	$-0.817956\pi$
$547$	−919.911			−1.68174			−0.840869	+	0.541238i	$0.817956\pi$
$548$		0			0
$549$	334.575	−	500.746i	0.609426	−	0.912106i
$550$		0			0
$551$			332.815i			0.604021i
$552$		0			0
$553$	−224.664			−0.406264
$554$		0			0
$555$	−185.830	−	347.656i	−0.334829	−	0.626408i
$556$		0			0
$557$			725.371i			1.30228i	0.758957	+	0.651141i	$0.225709\pi$
$557$			725.371i			1.30228i	−0.758957	+	0.651141i	$0.774291\pi$
$558$		0			0
$559$	−1065.15			−1.90546
$560$		0			0
$561$	17.3202	−	9.25804i	0.0308738	−	0.0165027i
$562$		0			0
$563$		−	728.773i		−	1.29445i	−0.762301	−	0.647223i	$-0.775930\pi$
$563$		−	728.773i		−	1.29445i	0.762301	−	0.647223i	$-0.224070\pi$
$564$		0			0
$565$	−528.826			−0.935975
$566$		0			0
$567$	82.0183	+	197.990i	0.144653	+	0.349189i
$568$		0			0
$569$		−	167.044i		−	0.293574i	−0.989168	−	0.146787i	$-0.953107\pi$
$569$		−	167.044i		−	0.293574i	0.989168	−	0.146787i	$-0.0468932\pi$
$570$		0			0
$571$	−72.9150			−0.127697			−0.0638485	−	0.997960i	$-0.520337\pi$
$571$	−72.9150			−0.127697			−0.0638485	+	0.997960i	$0.520337\pi$
$572$		0			0
$573$	−495.741	−	927.447i	−0.865168	−	1.61858i
$574$		0			0
$575$		−	654.019i		−	1.13742i
$576$		0			0
$577$	552.154			0.956940			0.478470	−	0.878104i	$-0.341192\pi$
$577$	552.154			0.956940			0.478470	+	0.878104i	$0.341192\pi$
$578$		0			0
$579$	−715.660	+	382.536i	−1.23603	+	0.660685i
$580$		0			0
$581$		−	10.9070i		−	0.0187729i
$582$		0			0
$583$	−20.9882			−0.0360003
$584$		0			0
$585$	1011.98	+	676.160i	1.72989	+	1.15583i
$586$		0			0
$587$		−	1115.21i		−	1.89985i	−0.312474	−	0.949926i	$-0.601158\pi$
$587$		−	1115.21i		−	1.89985i	0.312474	−	0.949926i	$-0.398842\pi$
$588$		0			0
$589$	−88.6798			−0.150560
$590$		0			0
$591$	89.8301	+	168.057i	0.151997	+	0.284360i
$592$		0			0
$593$			957.506i			1.61468i	0.590086	+	0.807340i	$0.299094\pi$
$593$			957.506i			1.61468i	−0.590086	+	0.807340i	$0.700906\pi$
$594$		0			0
$595$	−276.037			−0.463927
$596$		0			0
$597$	−232.826	+	124.451i	−0.389993	+	0.208460i
$598$		0			0
$599$		−	610.047i		−	1.01844i	−0.860636	−	0.509221i	$-0.829933\pi$
$599$		−	610.047i		−	1.01844i	0.860636	−	0.509221i	$-0.170067\pi$
$600$		0			0
$601$	−974.470			−1.62142			−0.810708	−	0.585451i	$-0.800917\pi$
$601$	−974.470			−1.62142			−0.810708	+	0.585451i	$0.800917\pi$
$602$		0			0
$603$	247.085	−	369.803i	0.409759	−	0.613272i
$604$		0			0
$605$		−	793.864i		−	1.31217i
$606$		0			0
$607$	1096.15			1.80584			0.902921	−	0.429806i	$-0.141418\pi$
$607$	1096.15			1.80584			0.902921	+	0.429806i	$0.141418\pi$
$608$		0			0
$609$	−77.8301	−	145.607i	−0.127800	−	0.239091i
$610$		0			0
$611$		−	174.653i		−	0.285847i
$612$		0			0
$613$	311.166			0.507612			0.253806	−	0.967255i	$-0.418318\pi$
$613$	311.166			0.507612			0.253806	+	0.967255i	$0.418318\pi$
$614$		0			0
$615$	1322.85	−	707.096i	2.15098	−	1.14975i
$616$		0			0
$617$			905.503i			1.46759i	0.679371	+	0.733795i	$0.262253\pi$
$617$			905.503i			1.46759i	−0.679371	+	0.733795i	$0.737747\pi$
$618$		0			0
$619$	54.7974			0.0885257			0.0442628	−	0.999020i	$-0.485906\pi$
$619$	54.7974			0.0885257			0.0442628	+	0.999020i	$0.485906\pi$
$620$		0			0
$621$	−967.385	−	95.2533i	−1.55779	−	0.153387i
$622$		0			0
$623$		−	82.5512i		−	0.132506i
$624$		0			0
$625$	−749.146			−1.19863
$626$		0			0
$627$	9.32808	+	17.4512i	0.0148773	+	0.0278329i
$628$		0			0
$629$		−	317.597i		−	0.504924i
$630$		0			0
$631$	−181.490			−0.287623			−0.143812	−	0.989605i	$-0.545936\pi$
$631$	−181.490			−0.287623			−0.143812	+	0.989605i	$0.545936\pi$
$632$		0			0
$633$	1002.07	−	535.626i	1.58304	−	0.846171i
$634$		0			0
$635$		−	873.263i		−	1.37522i
$636$		0			0
$637$	−144.081			−0.226187
$638$		0			0
$639$	−656.575	−	438.693i	−1.02750	−	0.686530i
$640$		0			0
$641$			837.621i			1.30674i	0.757038	+	0.653371i	$0.226646\pi$
$641$			837.621i			1.30674i	−0.757038	+	0.653371i	$0.773354\pi$
$642$		0			0
$643$	59.0118			0.0917758			0.0458879	−	0.998947i	$-0.485388\pi$
$643$	59.0118			0.0917758			0.0458879	+	0.998947i	$0.485388\pi$
$644$		0			0
$645$	480.826	+	899.543i	0.745467	+	1.39464i
$646$		0			0
$647$			547.661i			0.846462i	0.906022	+	0.423231i	$0.139104\pi$
$647$			547.661i			0.846462i	−0.906022	+	0.423231i	$0.860896\pi$
$648$		0			0
$649$	0.679790			0.00104744
$650$		0			0
$651$	38.7974	−	20.7381i	0.0595966	−	0.0318557i
$652$		0			0
$653$		−	764.895i		−	1.17136i	−0.810544	−	0.585678i	$-0.800828\pi$
$653$		−	764.895i		−	1.17136i	0.810544	−	0.585678i	$-0.199172\pi$
$654$		0			0
$655$	−27.0850			−0.0413511
$656$		0			0
$657$	−61.6601	+	92.2844i	−0.0938510	+	0.140463i
$658$		0			0
$659$			1050.80i			1.59454i	0.603623	+	0.797270i	$0.293723\pi$
$659$			1050.80i			1.59454i	−0.603623	+	0.797270i	$0.706277\pi$
$660$		0			0
$661$	145.085			0.219493			0.109747	−	0.993960i	$-0.464996\pi$
$661$	145.085			0.219493			0.109747	+	0.993960i	$0.464996\pi$
$662$		0			0
$663$	462.243	+	864.778i	0.697199	+	1.30434i
$664$		0			0
$665$		−	278.125i		−	0.418233i
$666$		0			0
$667$	748.884			1.12276
$668$		0			0
$669$	−609.830	+	325.968i	−0.911555	+	0.487246i
$670$		0			0
$671$			27.5855i			0.0411111i
$672$		0			0
$673$	323.498			0.480681			0.240340	−	0.970689i	$-0.422741\pi$
$673$	323.498			0.480681			0.240340	+	0.970689i	$0.422741\pi$
$674$		0			0
$675$	48.0627	−	488.122i	0.0712041	−	0.723143i
$676$		0			0
$677$		−	166.434i		−	0.245840i	−0.992417	−	0.122920i	$-0.960774\pi$
$677$		−	166.434i		−	0.245840i	0.992417	−	0.122920i	$-0.0392258\pi$
$678$		0			0
$679$	182.118			0.268214
$680$		0			0
$681$	−365.490	−	683.769i	−0.536696	−	1.00407i
$682$		0			0
$683$		−	973.506i		−	1.42534i	−0.701501	−	0.712669i	$-0.747486\pi$
$683$		−	973.506i		−	1.42534i	0.701501	−	0.712669i	$-0.252514\pi$
$684$		0			0
$685$	654.656			0.955702
$686$		0			0
$687$	−98.1176	+	52.4461i	−0.142820	+	0.0763407i
$688$		0			0
$689$		−	1047.92i		−	1.52092i
$690$		0			0
$691$	1268.86			1.83627			0.918135	−	0.396268i	$-0.129695\pi$
$691$	1268.86			1.83627			0.918135	+	0.396268i	$0.129695\pi$
$692$		0			0
$693$	−8.16207	−	5.45351i	−0.0117779	−	0.00786943i
$694$		0			0
$695$		−	614.582i		−	0.884291i
$696$		0			0
$697$	1208.48			1.73383
$698$		0			0
$699$	116.915	+	218.728i	0.167260	+	0.312916i
$700$		0			0
$701$		−	798.940i		−	1.13971i	−0.821744	−	0.569857i	$-0.806998\pi$
$701$		−	798.940i		−	1.13971i	0.821744	−	0.569857i	$-0.193002\pi$
$702$		0			0
$703$	320.000			0.455192
$704$		0			0
$705$	−147.498	+	78.8410i	−0.209217	+	0.111831i
$706$		0			0
$707$		−	370.810i		−	0.524484i
$708$		0			0
$709$	651.490			0.918886			0.459443	−	0.888207i	$-0.348049\pi$
$709$	651.490			0.918886			0.459443	+	0.888207i	$0.348049\pi$
$710$		0			0
$711$	424.575	−	635.446i	0.597152	−	0.893735i
$712$		0			0
$713$			199.543i			0.279863i
$714$		0			0
$715$	−55.7490			−0.0779707
$716$		0			0
$717$	−238.251	−	445.727i	−0.332289	−	0.621655i
$718$		0			0
$719$		−	878.587i		−	1.22196i	−0.791647	−	0.610979i	$-0.790776\pi$
$719$		−	878.587i		−	1.22196i	0.791647	−	0.610979i	$-0.209224\pi$
$720$		0			0
$721$	104.170			0.144480
$722$		0			0
$723$	343.948	−	183.848i	0.475723	−	0.254285i
$724$		0			0
$725$			377.870i			0.521201i
$726$		0			0
$727$	−442.782			−0.609053			−0.304527	−	0.952504i	$-0.598498\pi$
$727$	−442.782			−0.609053			−0.304527	+	0.952504i	$0.598498\pi$
$728$		0			0
$729$	−715.000	−	142.183i	−0.980796	−	0.195038i
$730$		0			0
$731$			821.767i			1.12417i
$732$		0			0
$733$	−962.559			−1.31318			−0.656589	−	0.754249i	$-0.728001\pi$
$733$	−962.559			−1.31318			−0.656589	+	0.754249i	$0.728001\pi$
$734$		0			0
$735$	65.0405	+	121.680i	0.0884905	+	0.165551i
$736$		0			0
$737$			20.3720i			0.0276418i
$738$		0			0
$739$	1224.81			1.65739			0.828694	−	0.559701i	$-0.189084\pi$
$739$	1224.81			1.65739			0.828694	+	0.559701i	$0.189084\pi$
$740$		0			0
$741$	−871.320	+	465.740i	−1.17587	+	0.628529i
$742$		0			0
$743$			1447.24i			1.94783i	0.226908	+	0.973916i	$0.427138\pi$
$743$			1447.24i			1.94783i	−0.226908	+	0.973916i	$0.572862\pi$
$744$		0			0
$745$	327.498			0.439595
$746$		0			0
$747$	30.8497	+	20.6123i	0.0412982	+	0.0275935i
$748$		0			0
$749$		−	285.760i		−	0.381522i
$750$		0			0
$751$	684.915			0.912004			0.456002	−	0.889979i	$-0.349281\pi$
$751$	684.915			0.912004			0.456002	+	0.889979i	$0.349281\pi$
$752$		0			0
$753$	−169.506	−	317.117i	−0.225107	−	0.421137i
$754$		0			0
$755$			1390.62i			1.84189i
$756$		0			0
$757$	907.135			1.19833			0.599164	−	0.800626i	$-0.295500\pi$
$757$	907.135			1.19833			0.599164	+	0.800626i	$0.295500\pi$
$758$		0			0
$759$	39.2679	−	20.9896i	0.0517363	−	0.0276542i
$760$		0			0
$761$		−	1451.51i		−	1.90737i	−0.300808	−	0.953685i	$-0.597256\pi$
$761$		−	1451.51i		−	1.90737i	0.300808	−	0.953685i	$-0.402744\pi$
$762$		0			0
$763$	−327.624			−0.429389
$764$		0			0
$765$	521.660	−	780.749i	0.681909	−	1.02059i
$766$		0			0
$767$			33.9411i			0.0442518i
$768$		0			0
$769$	−1089.32			−1.41654			−0.708271	−	0.705941i	$-0.750524\pi$
$769$	−1089.32			−1.41654			−0.708271	+	0.705941i	$0.750524\pi$
$770$		0			0
$771$	316.627	+	592.356i	0.410671	+	0.768295i
$772$		0			0
$773$			1019.58i			1.31900i	0.751707	+	0.659498i	$0.229231\pi$
$773$			1019.58i			1.31900i	−0.751707	+	0.659498i	$0.770769\pi$
$774$		0			0
$775$	−100.685			−0.129916
$776$		0			0
$777$	−140.000	+	74.8331i	−0.180180	+	0.0963104i
$778$		0			0
$779$			1217.62i			1.56306i
$780$		0			0
$781$	36.1699			0.0463124
$782$		0			0
$783$	558.923	+	55.0342i	0.713822	+	0.0702863i
$784$		0			0
$785$		−	714.464i		−	0.910146i
$786$		0			0
$787$	−1267.00			−1.60991			−0.804956	−	0.593334i	$-0.797811\pi$
$787$	−1267.00			−1.60991			−0.804956	+	0.593334i	$0.797811\pi$
$788$		0			0
$789$	303.255	+	567.338i	0.384354	+	0.719060i
$790$		0			0
$791$			212.957i			0.269224i
$792$		0			0
$793$	−1377.31			−1.73684
$794$		0			0
$795$	−884.988	+	473.046i	−1.11319	+	0.595027i
$796$		0			0
$797$			922.123i			1.15699i	0.815685	+	0.578496i	$0.196360\pi$
$797$			922.123i			1.15699i	−0.815685	+	0.578496i	$0.803640\pi$
$798$		0			0
$799$	−134.745			−0.168642
$800$		0			0
$801$	233.490	+	156.007i	0.291498	+	0.194766i
$802$		0			0
$803$		−	5.08384i		−	0.00633106i
$804$		0			0
$805$	−625.822			−0.777419
$806$		0			0
$807$	−540.450	−	1011.09i	−0.669702	−	1.25290i
$808$		0			0
$809$			706.855i			0.873740i	0.899525	+	0.436870i	$0.143913\pi$
$809$			706.855i			0.873740i	−0.899525	+	0.436870i	$0.856087\pi$
$810$		0			0
$811$	833.778			1.02809			0.514043	−	0.857764i	$-0.328147\pi$
$811$	833.778			1.02809			0.514043	+	0.857764i	$0.328147\pi$
$812$		0			0
$813$	302.826	−	161.867i	0.372480	−	0.199099i
$814$		0			0
$815$		−	85.4370i		−	0.104831i
$816$		0			0
$817$	−827.984			−1.01344
$818$		0			0
$819$	272.288	−	407.523i	0.332463	−	0.497586i
$820$		0			0
$821$		−	251.260i		−	0.306042i	−0.988223	−	0.153021i	$-0.951100\pi$
$821$		−	251.260i		−	0.306042i	0.988223	−	0.153021i	$-0.0489002\pi$
$822$		0			0
$823$	−38.9229			−0.0472939			−0.0236470	−	0.999720i	$-0.507528\pi$
$823$	−38.9229			−0.0472939			−0.0236470	+	0.999720i	$0.507528\pi$
$824$		0			0
$825$	10.5909	+	19.8137i	0.0128374	+	0.0240166i
$826$		0			0
$827$		−	108.007i		−	0.130601i	−0.997866	−	0.0653005i	$-0.979199\pi$
$827$		−	108.007i		−	0.130601i	0.997866	−	0.0653005i	$-0.0208006\pi$
$828$		0			0
$829$	1410.58			1.70154			0.850769	−	0.525540i	$-0.176137\pi$
$829$	1410.58			1.70154			0.850769	+	0.525540i	$0.176137\pi$
$830$		0			0
$831$	−609.830	+	325.968i	−0.733851	+	0.392260i
$832$		0			0
$833$			111.159i			0.133444i
$834$		0			0
$835$	1030.57			1.23421
$836$		0			0
$837$	−14.6640	+	148.927i	−0.0175198	+	0.177929i
$838$		0			0
$839$			299.906i			0.357456i	0.983899	+	0.178728i	$0.0571982\pi$
$839$			299.906i			0.357456i	−0.983899	+	0.178728i	$0.942802\pi$
$840$		0			0
$841$	408.320			0.485517
$842$		0			0
$843$	−104.575	−	195.642i	−0.124051	−	0.232078i
$844$		0			0
$845$		−	1673.14i		−	1.98005i
$846$		0			0
$847$	−319.686			−0.377434
$848$		0			0
$849$	−373.490	+	199.639i	−0.439918	+	0.235146i
$850$		0			0
$851$		−	720.047i		−	0.846119i
$852$		0			0
$853$	13.7648			0.0161369			0.00806844	−	0.999967i	$-0.497432\pi$
$853$	13.7648			0.0161369			0.00806844	+	0.999967i	$0.497432\pi$
$854$		0			0
$855$	786.656	+	525.607i	0.920066	+	0.614745i
$856$		0			0
$857$		−	252.506i		−	0.294640i	−0.989089	−	0.147320i	$-0.952935\pi$
$857$		−	252.506i		−	0.294640i	0.989089	−	0.147320i	$-0.0470647\pi$
$858$		0			0
$859$	674.510			0.785227			0.392613	−	0.919704i	$-0.371571\pi$
$859$	674.510			0.785227			0.392613	+	0.919704i	$0.371571\pi$
$860$		0			0
$861$	−284.745	−	532.709i	−0.330714	−	0.618710i
$862$		0			0
$863$			477.718i			0.553555i	0.960934	+	0.276777i	$0.0892664\pi$
$863$			477.718i			0.553555i	−0.960934	+	0.276777i	$0.910734\pi$
$864$		0			0
$865$	35.8301			0.0414220
$866$		0			0
$867$	−97.4432	+	52.0856i	−0.112391	+	0.0600756i
$868$		0			0
$869$			35.0060i			0.0402830i
$870$		0			0
$871$	−1017.15			−1.16780
$872$		0			0
$873$	−344.170	+	515.106i	−0.394238	+	0.590042i
$874$		0			0
$875$			118.794i			0.135765i
$876$		0			0
$877$	−1533.14			−1.74817			−0.874083	−	0.485776i	$-0.838537\pi$
$877$	−1533.14			−1.74817			−0.874083	+	0.485776i	$0.838537\pi$
$878$		0			0
$879$	466.118	+	872.026i	0.530282	+	0.992066i
$880$		0			0
$881$		−	1368.30i		−	1.55313i	−0.630039	−	0.776563i	$-0.716961\pi$
$881$		−	1368.30i		−	1.55313i	0.630039	−	0.776563i	$-0.283039\pi$
$882$		0			0
$883$	944.486			1.06963			0.534817	−	0.844968i	$-0.320381\pi$
$883$	944.486			1.06963			0.534817	+	0.844968i	$0.320381\pi$
$884$		0			0
$885$	28.6640	−	15.3216i	0.0323887	−	0.0173125i
$886$		0			0
$887$			1326.86i			1.49590i	0.663754	+	0.747951i	$0.268962\pi$
$887$			1326.86i			1.49590i	−0.663754	+	0.747951i	$0.731038\pi$
$888$		0			0
$889$	−351.660			−0.395568
$890$		0			0
$891$	30.8497	−	12.7797i	0.0346237	−	0.0143430i
$892$		0			0
$893$		−	135.765i		−	0.152032i
$894$		0			0
$895$	8.12549			0.00907876
$896$		0			0
$897$	1047.98	+	1960.60i	1.16832	+	2.18573i
$898$		0			0
$899$		−	115.289i		−	0.128241i
$900$		0			0
$901$	−808.470			−0.897304
$902$		0			0
$903$	362.243	−	193.627i	0.401155	−	0.214426i
$904$		0			0
$905$		−	1228.05i		−	1.35696i
$906$		0			0
$907$	−1219.64			−1.34470			−0.672351	−	0.740233i	$-0.734715\pi$
$907$	−1219.64			−1.34470			−0.672351	+	0.740233i	$0.734715\pi$
$908$		0			0
$909$	1048.81	+	700.766i	1.15381	+	0.770920i
$910$		0			0
$911$			63.8282i			0.0700639i	0.999386	+	0.0350320i	$0.0111533\pi$
$911$			63.8282i			0.0700639i	−0.999386	+	0.0350320i	$0.988847\pi$
$912$		0			0
$913$	−1.69948			−0.00186142
$914$		0			0
$915$	621.741	+	1163.17i	0.679499	+	1.27123i
$916$		0			0
$917$			10.9070i			0.0118943i
$918$		0			0
$919$	−981.385			−1.06788			−0.533942	−	0.845521i	$-0.679290\pi$
$919$	−981.385			−1.06788			−0.533942	+	0.845521i	$0.679290\pi$
$920$		0			0
$921$	280.000	−	149.666i	0.304017	−	0.162504i
$922$		0			0
$923$			1805.92i			1.95658i
$924$		0			0
$925$	363.320			0.392779
$926$		0			0
$927$	−196.863	+	294.637i	−0.212365	+	0.317839i
$928$		0			0
$929$		−	340.931i		−	0.366987i	−0.983021	−	0.183493i	$-0.941259\pi$
$929$		−	340.931i		−	0.366987i	0.983021	−	0.183493i	$-0.0587406\pi$
$930$		0			0
$931$	−112.000			−0.120301
$932$		0			0
$933$	544.664	+	1018.97i	0.583777	+	1.09215i
$934$		0			0
$935$			43.0106i			0.0460006i
$936$		0			0
$937$	1010.00			1.07791			0.538954	−	0.842335i	$-0.318820\pi$
$937$	1010.00			1.07791			0.538954	+	0.842335i	$0.318820\pi$
$938$		0			0
$939$	209.882	−	112.187i	0.223517	−	0.119475i
$940$		0			0
$941$		−	289.058i		−	0.307182i	−0.988135	−	0.153591i	$-0.950916\pi$
$941$		−	289.058i		−	0.307182i	0.988135	−	0.153591i	$-0.0490838\pi$
$942$		0			0
$943$	2739.83			2.90544
$944$		0			0
$945$	−467.077	−	45.9906i	−0.494261	−	0.0486673i
$946$		0			0
$947$		−	828.054i		−	0.874397i	−0.899365	−	0.437199i	$-0.855971\pi$
$947$		−	828.054i		−	0.874397i	0.899365	−	0.437199i	$-0.144029\pi$
$948$		0			0
$949$	253.830			0.267471
$950$		0			0
$951$	581.490	+	1087.87i	0.611451	+	1.14392i
$952$		0			0
$953$			102.785i			0.107854i	0.998545	+	0.0539269i	$0.0171738\pi$
$953$			102.785i			0.107854i	−0.998545	+	0.0539269i	$0.982826\pi$
$954$		0			0
$955$	2303.09			2.41161
$956$		0			0
$957$	−22.6877	+	12.1271i	−0.0237071	+	0.0126720i
$958$		0			0
$959$		−	263.628i		−	0.274899i
$960$		0			0
$961$	−930.281			−0.968034
$962$		0			0
$963$	808.251	+	540.035i	0.839305	+	0.560784i
$964$		0			0
$965$		−	1777.17i		−	1.84163i
$966$		0			0
$967$	184.753			0.191058			0.0955289	−	0.995427i	$-0.469546\pi$
$967$	184.753			0.191058			0.0955289	+	0.995427i	$0.469546\pi$
$968$		0			0
$969$	359.320	+	672.227i	0.370815	+	0.693732i
$970$		0			0
$971$			1469.33i			1.51321i	0.653871	+	0.756606i	$0.273144\pi$
$971$			1469.33i			1.51321i	−0.653871	+	0.756606i	$0.726856\pi$
$972$		0			0
$973$	−247.490			−0.254358
$974$		0			0
$975$	−989.276	+	528.790i	−1.01464	+	0.542349i
$976$		0			0
$977$			663.072i			0.678682i	0.940663	+	0.339341i	$0.110204\pi$
$977$			663.072i			0.678682i	−0.940663	+	0.339341i	$0.889796\pi$
$978$		0			0
$979$	−12.8627			−0.0131386
$980$		0			0
$981$	619.150	−	926.659i	0.631142	−	0.944607i
$982$		0			0
$983$		−	1899.26i		−	1.93211i	−0.258342	−	0.966053i	$-0.583176\pi$
$983$		−	1899.26i		−	1.93211i	0.258342	−	0.966053i	$-0.416824\pi$
$984$		0			0
$985$	−417.328			−0.423683
$986$		0			0
$987$	31.7490	+	59.3970i	0.0321672	+	0.0601793i
$988$		0			0
$989$			1863.09i			1.88381i
$990$		0			0
$991$	257.725			0.260066			0.130033	−	0.991510i	$-0.458492\pi$
$991$	257.725			0.260066			0.130033	+	0.991510i	$0.458492\pi$
$992$		0			0
$993$	1295.07	−	692.244i	1.30420	−	0.697123i
$994$		0			0
$995$		−	578.167i		−	0.581073i
$996$		0			0
$997$	−1235.74			−1.23946			−0.619730	−	0.784815i	$-0.712758\pi$
$997$	−1235.74			−1.23946			−0.619730	+	0.784815i	$0.712758\pi$
$998$		0			0
$999$	52.9150	−	537.401i	0.0529680	−	0.537939i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.d.e.449.2		4
3.2	odd	2	inner	1344.3.d.e.449.1		4
4.3	odd	2		1344.3.d.c.449.3		4
8.3	odd	2		42.3.b.a.29.3	yes	4
8.5	even	2		336.3.d.b.113.3		4
12.11	even	2		1344.3.d.c.449.4		4
24.5	odd	2		336.3.d.b.113.4		4
24.11	even	2		42.3.b.a.29.1	✓	4
40.3	even	4		1050.3.c.a.449.5		8
40.19	odd	2		1050.3.e.a.701.2		4
40.27	even	4		1050.3.c.a.449.3		8
56.3	even	6		294.3.h.g.275.1		8
56.11	odd	6		294.3.h.d.275.2		8
56.19	even	6		294.3.h.g.263.3		8
56.27	even	2		294.3.b.h.197.4		4
56.51	odd	6		294.3.h.d.263.4		8
72.11	even	6		1134.3.q.a.701.2		8
72.43	odd	6		1134.3.q.a.701.3		8
72.59	even	6		1134.3.q.a.1079.3		8
72.67	odd	6		1134.3.q.a.1079.2		8
120.59	even	2		1050.3.e.a.701.4		4
120.83	odd	4		1050.3.c.a.449.2		8
120.107	odd	4		1050.3.c.a.449.8		8
168.11	even	6		294.3.h.d.275.4		8
168.59	odd	6		294.3.h.g.275.3		8
168.83	odd	2		294.3.b.h.197.2		4
168.107	even	6		294.3.h.d.263.2		8
168.131	odd	6		294.3.h.g.263.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.b.a.29.1	✓	4	24.11	even	2
42.3.b.a.29.3	yes	4	8.3	odd	2
294.3.b.h.197.2		4	168.83	odd	2
294.3.b.h.197.4		4	56.27	even	2
294.3.h.d.263.2		8	168.107	even	6
294.3.h.d.263.4		8	56.51	odd	6
294.3.h.d.275.2		8	56.11	odd	6
294.3.h.d.275.4		8	168.11	even	6
294.3.h.g.263.1		8	168.131	odd	6
294.3.h.g.263.3		8	56.19	even	6
294.3.h.g.275.1		8	56.3	even	6
294.3.h.g.275.3		8	168.59	odd	6
336.3.d.b.113.3		4	8.5	even	2
336.3.d.b.113.4		4	24.5	odd	2
1050.3.c.a.449.2		8	120.83	odd	4
1050.3.c.a.449.3		8	40.27	even	4
1050.3.c.a.449.5		8	40.3	even	4
1050.3.c.a.449.8		8	120.107	odd	4
1050.3.e.a.701.2		4	40.19	odd	2
1050.3.e.a.701.4		4	120.59	even	2
1134.3.q.a.701.2		8	72.11	even	6
1134.3.q.a.701.3		8	72.43	odd	6
1134.3.q.a.1079.2		8	72.67	odd	6
1134.3.q.a.1079.3		8	72.59	even	6
1344.3.d.c.449.3		4	4.3	odd	2
1344.3.d.c.449.4		4	12.11	even	2
1344.3.d.e.449.1		4	3.2	odd	2	inner
1344.3.d.e.449.2		4	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1344.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.64575 + 1.41421i) q^{3} -6.57008i q^{5} -2.64575 q^{7} +(5.00000 - 7.48331i) q^{9} +O(q^{10})\)	\(q+(-2.64575 + 1.41421i) q^{3} -6.57008i q^{5} -2.64575 q^{7} +(5.00000 - 7.48331i) q^{9} +0.412247i q^{11} -20.5830 q^{13} +(9.29150 + 17.3828i) q^{15} +15.8799i q^{17} -16.0000 q^{19} +(7.00000 - 3.74166i) q^{21} +36.0024i q^{23} -18.1660 q^{25} +(-2.64575 + 26.8701i) q^{27} -20.8010i q^{29} +5.54249 q^{31} +(-0.583005 - 1.09070i) q^{33} +17.3828i q^{35} -20.0000 q^{37} +(54.4575 - 29.1088i) q^{39} -76.1013i q^{41} +51.7490 q^{43} +(-49.1660 - 32.8504i) q^{45} +8.48528i q^{47} +7.00000 q^{49} +(-22.4575 - 42.0142i) q^{51} +50.9117i q^{53} +2.70850 q^{55} +(42.3320 - 22.6274i) q^{57} -1.64899i q^{59} +66.9150 q^{61} +(-13.2288 + 19.7990i) q^{63} +135.232i q^{65} +49.4170 q^{67} +(-50.9150 - 95.2533i) q^{69} -87.7385i q^{71} -12.3320 q^{73} +(48.0627 - 25.6906i) q^{75} -1.09070i q^{77} +84.9150 q^{79} +(-31.0000 - 74.8331i) q^{81} +4.12247i q^{83} +104.332 q^{85} +(29.4170 + 55.0342i) q^{87} +31.2014i q^{89} +54.4575 q^{91} +(-14.6640 + 7.83826i) q^{93} +105.121i q^{95} -68.8340 q^{97} +(3.08497 + 2.06123i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 20 q^{9}+O(q^{10}) \) 4 * q + 20 * q^9	\( 4 q + 20 q^{9} - 40 q^{13} + 16 q^{15} - 64 q^{19} + 28 q^{21} + 12 q^{25} + 128 q^{31} + 40 q^{33} - 80 q^{37} + 112 q^{39} + 80 q^{43} - 112 q^{45} + 28 q^{49} + 16 q^{51} + 32 q^{55} + 56 q^{61} + 240 q^{67} + 8 q^{69} + 120 q^{73} + 224 q^{75} + 128 q^{79} - 124 q^{81} + 248 q^{85} + 160 q^{87} + 112 q^{91} + 280 q^{93} - 360 q^{97} + 224 q^{99}+O(q^{100}) \) 4 * q + 20 * q^9 - 40 * q^13 + 16 * q^15 - 64 * q^19 + 28 * q^21 + 12 * q^25 + 128 * q^31 + 40 * q^33 - 80 * q^37 + 112 * q^39 + 80 * q^43 - 112 * q^45 + 28 * q^49 + 16 * q^51 + 32 * q^55 + 56 * q^61 + 240 * q^67 + 8 * q^69 + 120 * q^73 + 224 * q^75 + 128 * q^79 - 124 * q^81 + 248 * q^85 + 160 * q^87 + 112 * q^91 + 280 * q^93 - 360 * q^97 + 224 * q^99

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