LMFDB - Embedded newform 1344.3.d.g.449.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,3,Mod(449,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,6,0,0,0,0,0,-4,0,0,0,-28,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$36.6213475300$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.116032.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 8x^{2} + 14x + 21 $ x^4 - 2x^3 + 8x^2 + 14*x + 21
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.82288 - 2.99477i$ of defining polynomial
Character	$\chi$	$=$	1344.449
Dual form			1344.3.d.g.449.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.177124 + 2.99477i) q^{3} -3.86775i q^{5} +2.64575 q^{7} +(-8.93725 + 1.06089i) q^{9} +14.1009i q^{11} -20.2288 q^{13} +(11.5830 - 0.685072i) q^{15} -21.8363i q^{17} +6.93725 q^{19} +(0.468627 + 7.92341i) q^{21} -7.73550i q^{23} +10.0405 q^{25} +(-4.76013 - 26.5771i) q^{27} +37.3073i q^{29} -15.2915 q^{31} +(-42.2288 + 2.49760i) q^{33} -10.2331i q^{35} +12.1255 q^{37} +(-3.58301 - 60.5804i) q^{39} -42.3026i q^{41} +16.1255 q^{43} +(4.10326 + 34.5671i) q^{45} -71.8744i q^{47} +7.00000 q^{49} +(65.3948 - 3.86775i) q^{51} -101.446i q^{53} +54.5385 q^{55} +(1.22876 + 20.7755i) q^{57} -25.7041i q^{59} -78.1033 q^{61} +(-23.6458 + 2.80686i) q^{63} +78.2398i q^{65} -123.166 q^{67} +(23.1660 - 1.37014i) q^{69} +34.5671i q^{71} -100.539 q^{73} +(1.77842 + 30.0690i) q^{75} +37.3073i q^{77} +42.5830 q^{79} +(78.7490 - 18.9629i) q^{81} -82.1075i q^{83} -84.4575 q^{85} +(-111.727 + 6.60804i) q^{87} -42.3026i q^{89} -53.5203 q^{91} +(-2.70850 - 45.7945i) q^{93} -26.8316i q^{95} +48.3320 q^{97} +(-14.9595 - 126.023i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{3} - 4 q^{9} - 28 q^{13} + 4 q^{15} - 4 q^{19} - 14 q^{21} - 108 q^{25} + 18 q^{27} - 40 q^{31} - 116 q^{33} + 112 q^{37} + 28 q^{39} + 128 q^{43} - 100 q^{45} + 28 q^{49} + 124 q^{51} - 184 q^{55}+ \cdots - 208 q^{99}+O(q^{100}) $ 4 * q + 6 * q^3 - 4 * q^9 - 28 * q^13 + 4 * q^15 - 4 * q^19 - 14 * q^21 - 108 * q^25 + 18 * q^27 - 40 * q^31 - 116 * q^33 + 112 * q^37 + 28 * q^39 + 128 * q^43 - 100 * q^45 + 28 * q^49 + 124 * q^51 - 184 * q^55 - 48 * q^57 - 196 * q^61 - 84 * q^63 - 408 * q^67 + 8 * q^69 - 358 * q^75 + 128 * q^79 + 188 * q^81 - 232 * q^85 - 140 * q^87 - 140 * q^91 - 32 * q^93 + 24 * q^97 - 208 * q^99

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
$3$	0.177124	+	2.99477i	0.0590414	+	0.998256i
$3$	0.177124	+	2.99477i	0.0590414	+	0.998256i
$4$		0			0
$5$		−	3.86775i		−	0.773550i	−0.922174	−	0.386775i	$-0.873589\pi$
$5$		−	3.86775i		−	0.773550i	0.922174	−	0.386775i	$-0.126411\pi$
$6$		0			0
$7$	2.64575			0.377964
$7$	2.64575			0.377964
$8$		0			0
$9$	−8.93725	+	1.06089i	−0.993028	+	0.117877i
$10$		0			0
$11$			14.1009i			1.28190i	0.767585	+	0.640948i	$0.221458\pi$
$11$			14.1009i			1.28190i	−0.767585	+	0.640948i	$0.778542\pi$
$12$		0			0
$13$	−20.2288			−1.55606			−0.778029	−	0.628228i	$-0.783780\pi$
$13$	−20.2288			−1.55606			−0.778029	+	0.628228i	$0.783780\pi$
$14$		0			0
$15$	11.5830	−	0.685072i	0.772200	−	0.0456715i
$16$		0			0
$17$		−	21.8363i		−	1.28449i	−0.766499	−	0.642246i	$-0.778003\pi$
$17$		−	21.8363i		−	1.28449i	0.766499	−	0.642246i	$-0.221997\pi$
$18$		0			0
$19$	6.93725			0.365119			0.182559	−	0.983195i	$-0.441562\pi$
$19$	6.93725			0.365119			0.182559	+	0.983195i	$0.441562\pi$
$20$		0			0
$21$	0.468627	+	7.92341i	0.0223156	+	0.377305i
$22$		0			0
$23$		−	7.73550i		−	0.336326i	−0.985759	−	0.168163i	$-0.946217\pi$
$23$		−	7.73550i		−	0.336326i	0.985759	−	0.168163i	$-0.0537835\pi$
$24$		0			0
$25$	10.0405			0.401621
$26$		0			0
$27$	−4.76013	−	26.5771i	−0.176301	−	0.984336i
$28$		0			0
$29$			37.3073i			1.28646i	0.765673	+	0.643230i	$0.222406\pi$
$29$			37.3073i			1.28646i	−0.765673	+	0.643230i	$0.777594\pi$
$30$		0			0
$31$	−15.2915			−0.493274			−0.246637	−	0.969108i	$-0.579326\pi$
$31$	−15.2915			−0.493274			−0.246637	+	0.969108i	$0.579326\pi$
$32$		0			0
$33$	−42.2288	+	2.49760i	−1.27966	+	0.0756850i
$34$		0			0
$35$		−	10.2331i		−	0.292374i
$36$		0			0
$37$	12.1255			0.327716			0.163858	−	0.986484i	$-0.447606\pi$
$37$	12.1255			0.327716			0.163858	+	0.986484i	$0.447606\pi$
$38$		0			0
$39$	−3.58301	−	60.5804i	−0.0918719	−	1.55334i
$40$		0			0
$41$		−	42.3026i		−	1.03177i	−0.856658	−	0.515885i	$-0.827463\pi$
$41$		−	42.3026i		−	1.03177i	0.856658	−	0.515885i	$-0.172537\pi$
$42$		0			0
$43$	16.1255			0.375011			0.187506	−	0.982264i	$-0.439960\pi$
$43$	16.1255			0.375011			0.187506	+	0.982264i	$0.439960\pi$
$44$		0			0
$45$	4.10326	+	34.5671i	0.0911837	+	0.768157i
$46$		0			0
$47$		−	71.8744i		−	1.52924i	−0.644480	−	0.764621i	$-0.722926\pi$
$47$		−	71.8744i		−	1.52924i	0.644480	−	0.764621i	$-0.277074\pi$
$48$		0			0
$49$	7.00000			0.142857
$50$		0			0
$51$	65.3948	−	3.86775i	1.28225	−	0.0758382i
$52$		0			0
$53$		−	101.446i		−	1.91408i	−0.289956	−	0.957040i	$-0.593641\pi$
$53$		−	101.446i		−	1.91408i	0.289956	−	0.957040i	$-0.406359\pi$
$54$		0			0
$55$	54.5385			0.991610
$56$		0			0
$57$	1.22876	+	20.7755i	0.0215571	+	0.364482i
$58$		0			0
$59$		−	25.7041i		−	0.435663i	−0.975986	−	0.217831i	$-0.930102\pi$
$59$		−	25.7041i		−	0.435663i	0.975986	−	0.217831i	$-0.0698983\pi$
$60$		0			0
$61$	−78.1033			−1.28038			−0.640191	−	0.768216i	$-0.721145\pi$
$61$	−78.1033			−1.28038			−0.640191	+	0.768216i	$0.721145\pi$
$62$		0			0
$63$	−23.6458	+	2.80686i	−0.375329	+	0.0445533i
$64$		0			0
$65$			78.2398i			1.20369i
$66$		0			0
$67$	−123.166			−1.83830			−0.919149	−	0.393909i	$-0.871122\pi$
$67$	−123.166			−1.83830			−0.919149	+	0.393909i	$0.871122\pi$
$68$		0			0
$69$	23.1660	−	1.37014i	0.335739	−	0.0198572i
$70$		0			0
$71$			34.5671i			0.486860i	0.969918	+	0.243430i	$0.0782726\pi$
$71$			34.5671i			0.486860i	−0.969918	+	0.243430i	$0.921727\pi$
$72$		0			0
$73$	−100.539			−1.37724			−0.688620	−	0.725122i	$-0.741783\pi$
$73$	−100.539			−1.37724			−0.688620	+	0.725122i	$0.741783\pi$
$74$		0			0
$75$	1.77842	+	30.0690i	0.0237123	+	0.400920i
$76$		0			0
$77$			37.3073i			0.484511i
$78$		0			0
$79$	42.5830			0.539025			0.269513	−	0.962997i	$-0.413137\pi$
$79$	42.5830			0.539025			0.269513	+	0.962997i	$0.413137\pi$
$80$		0			0
$81$	78.7490	−	18.9629i	0.972210	−	0.234110i
$82$		0			0
$83$		−	82.1075i		−	0.989247i	−0.869107	−	0.494623i	$-0.835306\pi$
$83$		−	82.1075i		−	0.989247i	0.869107	−	0.494623i	$-0.164694\pi$
$84$		0			0
$85$	−84.4575			−0.993618
$86$		0			0
$87$	−111.727	+	6.60804i	−1.28422	+	0.0759545i
$88$		0			0
$89$		−	42.3026i		−	0.475310i	−0.971350	−	0.237655i	$-0.923621\pi$
$89$		−	42.3026i		−	0.475310i	0.971350	−	0.237655i	$-0.0763787\pi$
$90$		0			0
$91$	−53.5203			−0.588135
$92$		0			0
$93$	−2.70850	−	45.7945i	−0.0291236	−	0.492414i
$94$		0			0
$95$		−	26.8316i		−	0.282437i
$96$		0			0
$97$	48.3320			0.498268			0.249134	−	0.968469i	$-0.419854\pi$
$97$	48.3320			0.498268			0.249134	+	0.968469i	$0.419854\pi$
$98$		0			0
$99$	−14.9595	−	126.023i	−0.151106	−	1.27296i
$100$		0			0
$101$			80.7374i			0.799380i	0.916650	+	0.399690i	$0.130882\pi$
$101$			80.7374i			0.799380i	−0.916650	+	0.399690i	$0.869118\pi$
$102$		0			0
$103$	−48.2510			−0.468456			−0.234228	−	0.972182i	$-0.575256\pi$
$103$	−48.2510			−0.468456			−0.234228	+	0.972182i	$0.575256\pi$
$104$		0			0
$105$	30.6458	−	1.81253i	0.291864	−	0.0172622i
$106$		0			0
$107$		−	196.527i		−	1.83670i	−0.395767	−	0.918351i	$-0.629521\pi$
$107$		−	196.527i		−	1.83670i	0.395767	−	0.918351i	$-0.370479\pi$
$108$		0			0
$109$	53.2915			0.488913			0.244456	−	0.969660i	$-0.421390\pi$
$109$	53.2915			0.488913			0.244456	+	0.969660i	$0.421390\pi$
$110$		0			0
$111$	2.14772	+	36.3130i	0.0193488	+	0.327144i
$112$		0			0
$113$			140.124i			1.24003i	0.784589	+	0.620017i	$0.212874\pi$
$113$			140.124i			1.24003i	−0.784589	+	0.620017i	$0.787126\pi$
$114$		0			0
$115$	−29.9190			−0.260165
$116$		0			0
$117$	180.790	−	21.4605i	1.54521	−	0.183423i
$118$		0			0
$119$		−	57.7735i		−	0.485492i
$120$		0			0
$121$	−77.8340			−0.643256
$122$		0			0
$123$	126.686	−	7.49281i	1.02997	−	0.0609172i
$124$		0			0
$125$		−	135.528i		−	1.08422i
$126$		0			0
$127$	115.579			0.910071			0.455036	−	0.890473i	$-0.349626\pi$
$127$	115.579			0.910071			0.455036	+	0.890473i	$0.349626\pi$
$128$		0			0
$129$	2.85622	+	48.2921i	0.0221412	+	0.374357i
$130$		0			0
$131$		−	82.5929i		−	0.630480i	−0.949012	−	0.315240i	$-0.897915\pi$
$131$		−	82.5929i		−	0.630480i	0.949012	−	0.315240i	$-0.102085\pi$
$132$		0			0
$133$	18.3542			0.138002
$134$		0			0
$135$	−102.793	+	18.4110i	−0.761433	+	0.136378i
$136$		0			0
$137$		−	153.739i		−	1.12218i	−0.827753	−	0.561092i	$-0.810381\pi$
$137$		−	153.739i		−	1.12218i	0.827753	−	0.561092i	$-0.189619\pi$
$138$		0			0
$139$	220.516			1.58645			0.793224	−	0.608930i	$-0.208401\pi$
$139$	220.516			1.58645			0.793224	+	0.608930i	$0.208401\pi$
$140$		0			0
$141$	215.247	−	12.7307i	1.52657	−	0.0902887i
$142$		0			0
$143$		−	285.243i		−	1.99470i
$144$		0			0
$145$	144.295			0.995141
$146$		0			0
$147$	1.23987	+	20.9634i	0.00843449	+	0.142608i
$148$		0			0
$149$		−	194.272i		−	1.30384i	−0.758288	−	0.651920i	$-0.773964\pi$
$149$		−	194.272i		−	1.30384i	0.758288	−	0.651920i	$-0.226036\pi$
$150$		0			0
$151$	−128.081			−0.848219			−0.424109	−	0.905611i	$-0.639413\pi$
$151$	−128.081			−0.848219			−0.424109	+	0.905611i	$0.639413\pi$
$152$		0			0
$153$	23.1660	+	195.157i	0.151412	+	1.27554i
$154$		0			0
$155$			59.1437i			0.381572i
$156$		0			0
$157$	−147.682			−0.940652			−0.470326	−	0.882493i	$-0.655864\pi$
$157$	−147.682			−0.940652			−0.470326	+	0.882493i	$0.655864\pi$
$158$		0			0
$159$	303.808	−	17.9686i	1.91074	−	0.113010i
$160$		0			0
$161$		−	20.4662i		−	0.127119i
$162$		0			0
$163$	−210.369			−1.29060			−0.645302	−	0.763927i	$-0.723269\pi$
$163$	−210.369			−1.29060			−0.645302	+	0.763927i	$0.723269\pi$
$164$		0			0
$165$	9.66010	+	163.330i	0.0585461	+	0.989880i
$166$		0			0
$167$		−	85.0905i		−	0.509524i	−0.967004	−	0.254762i	$-0.918003\pi$
$167$		−	85.0905i		−	0.509524i	0.967004	−	0.254762i	$-0.0819971\pi$
$168$		0			0
$169$	240.203			1.42132
$170$		0			0
$171$	−62.0000	+	7.35968i	−0.362573	+	0.0430391i
$172$		0			0
$173$			93.4681i			0.540278i	0.962821	+	0.270139i	$0.0870696\pi$
$173$			93.4681i			0.540278i	−0.962821	+	0.270139i	$0.912930\pi$
$174$		0			0
$175$	26.5647			0.151798
$176$		0			0
$177$	76.9778	−	4.55282i	0.434903	−	0.0257222i
$178$		0			0
$179$		−	196.527i		−	1.09792i	−0.835850	−	0.548958i	$-0.815025\pi$
$179$		−	196.527i		−	1.09792i	0.835850	−	0.548958i	$-0.184975\pi$
$180$		0			0
$181$	164.022			0.906200			0.453100	−	0.891460i	$-0.350318\pi$
$181$	164.022			0.906200			0.453100	+	0.891460i	$0.350318\pi$
$182$		0			0
$183$	−13.8340	−	233.901i	−0.0755956	−	1.27815i
$184$		0			0
$185$		−	46.8984i		−	0.253505i
$186$		0			0
$187$	307.911			1.64658
$188$		0			0
$189$	−12.5941	−	70.3163i	−0.0666356	−	0.372044i
$190$		0			0
$191$			230.209i			1.20528i	0.798011	+	0.602642i	$0.205885\pi$
$191$			230.209i			1.20528i	−0.798011	+	0.602642i	$0.794115\pi$
$192$		0			0
$193$	−184.620			−0.956578			−0.478289	−	0.878202i	$-0.658743\pi$
$193$	−184.620			−0.956578			−0.478289	+	0.878202i	$0.658743\pi$
$194$		0			0
$195$	−234.310	+	13.8582i	−1.20159	+	0.0710675i
$196$		0			0
$197$			91.4558i			0.464243i	0.972687	+	0.232121i	$0.0745667\pi$
$197$			91.4558i			0.464243i	−0.972687	+	0.232121i	$0.925433\pi$
$198$		0			0
$199$	203.749			1.02386			0.511932	−	0.859026i	$-0.328930\pi$
$199$	203.749			1.02386			0.511932	+	0.859026i	$0.328930\pi$
$200$		0			0
$201$	−21.8157	−	368.853i	−0.108536	−	1.83509i
$202$		0			0
$203$			98.7060i			0.486236i
$204$		0			0
$205$	−163.616			−0.798125
$206$		0			0
$207$	8.20653	+	69.1341i	0.0396451	+	0.333981i
$208$		0			0
$209$			97.8212i			0.468044i
$210$		0			0
$211$	188.243			0.892147			0.446074	−	0.894996i	$-0.352822\pi$
$211$	188.243			0.892147			0.446074	+	0.894996i	$0.352822\pi$
$212$		0			0
$213$	−103.520	+	6.12267i	−0.486011	+	0.0287449i
$214$		0			0
$215$		−	62.3694i		−	0.290090i
$216$		0			0
$217$	−40.4575			−0.186440
$218$		0			0
$219$	−17.8078	−	301.089i	−0.0813143	−	1.37484i
$220$		0			0
$221$			441.722i			1.99874i
$222$		0			0
$223$	−154.125			−0.691146			−0.345573	−	0.938392i	$-0.612315\pi$
$223$	−154.125			−0.691146			−0.345573	+	0.938392i	$0.612315\pi$
$224$		0			0
$225$	−89.7347	+	10.6519i	−0.398821	+	0.0473418i
$226$		0			0
$227$			233.592i			1.02904i	0.857479	+	0.514519i	$0.172030\pi$
$227$			233.592i			1.02904i	−0.857479	+	0.514519i	$0.827970\pi$
$228$		0			0
$229$	−275.269			−1.20205			−0.601025	−	0.799231i	$-0.705241\pi$
$229$	−275.269			−1.20205			−0.601025	+	0.799231i	$0.705241\pi$
$230$		0			0
$231$	−111.727	+	6.60804i	−0.483666	+	0.0286062i
$232$		0			0
$233$			35.4518i			0.152154i	0.997102	+	0.0760769i	$0.0242394\pi$
$233$			35.4518i			0.152154i	−0.997102	+	0.0760769i	$0.975761\pi$
$234$		0			0
$235$	−277.992			−1.18295
$236$		0			0
$237$	7.54249	+	127.526i	0.0318248	+	0.538085i
$238$		0			0
$239$			56.8888i			0.238028i	0.992893	+	0.119014i	$0.0379734\pi$
$239$			56.8888i			0.238028i	−0.992893	+	0.119014i	$0.962027\pi$
$240$		0			0
$241$	−209.749			−0.870328			−0.435164	−	0.900351i	$-0.643310\pi$
$241$	−209.749			−0.870328			−0.435164	+	0.900351i	$0.643310\pi$
$242$		0			0
$243$	70.7379	+	232.476i	0.291102	+	0.956692i
$244$		0			0
$245$		−	27.0742i		−	0.110507i
$246$		0			0
$247$	−140.332			−0.568146
$248$		0			0
$249$	245.893	−	14.5432i	0.987521	−	0.0584066i
$250$		0			0
$251$			77.1123i			0.307220i	0.988132	+	0.153610i	$0.0490900\pi$
$251$			77.1123i			0.307220i	−0.988132	+	0.153610i	$0.950910\pi$
$252$		0			0
$253$	109.077			0.431135
$254$		0			0
$255$	−14.9595	−	252.931i	−0.0586646	−	0.991884i
$256$		0			0
$257$		−	158.335i		−	0.616090i	−0.951372	−	0.308045i	$-0.900325\pi$
$257$		−	158.335i		−	0.616090i	0.951372	−	0.308045i	$-0.0996747\pi$
$258$		0			0
$259$	32.0810			0.123865
$260$		0			0
$261$	−39.5791	−	333.425i	−0.151644	−	1.27749i
$262$		0			0
$263$		−	150.114i		−	0.570776i	−0.958412	−	0.285388i	$-0.907878\pi$
$263$		−	150.114i		−	0.570776i	0.958412	−	0.285388i	$-0.0921225\pi$
$264$		0			0
$265$	−392.369			−1.48064
$266$		0			0
$267$	126.686	−	7.49281i	0.474480	−	0.0280630i
$268$		0			0
$269$			190.804i			0.709308i	0.934998	+	0.354654i	$0.115401\pi$
$269$			190.804i			0.709308i	−0.934998	+	0.354654i	$0.884599\pi$
$270$		0			0
$271$	342.450			1.26365			0.631826	−	0.775110i	$-0.282306\pi$
$271$	342.450			1.26365			0.631826	+	0.775110i	$0.282306\pi$
$272$		0			0
$273$	−9.47974	−	160.281i	−0.0347243	−	0.587109i
$274$		0			0
$275$			141.580i			0.514836i
$276$		0			0
$277$	93.2470			0.336632			0.168316	−	0.985733i	$-0.446167\pi$
$277$	93.2470			0.336632			0.168316	+	0.985733i	$0.446167\pi$
$278$		0			0
$279$	136.664	−	16.2226i	0.489835	−	0.0581456i
$280$		0			0
$281$			251.075i			0.893505i	0.894657	+	0.446753i	$0.147420\pi$
$281$			251.075i			0.893505i	−0.894657	+	0.446753i	$0.852580\pi$
$282$		0			0
$283$	16.6497			0.0588328			0.0294164	−	0.999567i	$-0.490635\pi$
$283$	16.6497			0.0588328			0.0294164	+	0.999567i	$0.490635\pi$
$284$		0			0
$285$	80.3542	−	4.75252i	0.281945	−	0.0166755i
$286$		0			0
$287$		−	111.922i		−	0.389972i
$288$		0			0
$289$	−187.826			−0.649917
$290$		0			0
$291$	8.56078	+	144.743i	0.0294185	+	0.497399i
$292$		0			0
$293$		−	307.322i		−	1.04888i	−0.851448	−	0.524440i	$-0.824275\pi$
$293$		−	307.322i		−	1.04888i	0.851448	−	0.524440i	$-0.175725\pi$
$294$		0			0
$295$	−99.4170			−0.337007
$296$		0			0
$297$	374.759	−	67.1219i	1.26182	−	0.226000i
$298$		0			0
$299$			156.480i			0.523343i
$300$		0			0
$301$	42.6640			0.141741
$302$		0			0
$303$	−241.790	+	14.3006i	−0.797985	+	0.0471965i
$304$		0			0
$305$			302.084i			0.990439i
$306$		0			0
$307$	126.775			0.412948			0.206474	−	0.978452i	$-0.433801\pi$
$307$	126.775			0.412948			0.206474	+	0.978452i	$0.433801\pi$
$308$		0			0
$309$	−8.54642	−	144.500i	−0.0276583	−	0.467639i
$310$		0			0
$311$			328.915i			1.05761i	0.848745	+	0.528803i	$0.177359\pi$
$311$			328.915i			1.05761i	−0.848745	+	0.528803i	$0.822641\pi$
$312$		0			0
$313$	−259.284			−0.828382			−0.414191	−	0.910190i	$-0.635935\pi$
$313$	−259.284			−0.828382			−0.414191	+	0.910190i	$0.635935\pi$
$314$		0			0
$315$	10.8562	+	91.4558i	0.0344642	+	0.290336i
$316$		0			0
$317$		−	8.62027i		−	0.0271933i	−0.999908	−	0.0135966i	$-0.995672\pi$
$317$		−	8.62027i		−	0.0271933i	0.999908	−	0.0135966i	$-0.00432808\pi$
$318$		0			0
$319$	−526.065			−1.64911
$320$		0			0
$321$	588.553	−	34.8097i	1.83350	−	0.108442i
$322$		0			0
$323$		−	151.484i		−	0.468992i
$324$		0			0
$325$	−203.107			−0.624945
$326$		0			0
$327$	9.43922	+	159.596i	0.0288661	+	0.488060i
$328$		0			0
$329$		−	190.162i		−	0.577999i
$330$		0			0
$331$	−474.369			−1.43314			−0.716569	−	0.697516i	$-0.754288\pi$
$331$	−474.369			−1.43314			−0.716569	+	0.697516i	$0.754288\pi$
$332$		0			0
$333$	−108.369	+	12.8638i	−0.325431	+	0.0386301i
$334$		0			0
$335$			476.375i			1.42202i
$336$		0			0
$337$	−609.195			−1.80770			−0.903850	−	0.427850i	$-0.859271\pi$
$337$	−609.195			−1.80770			−0.903850	+	0.427850i	$0.859271\pi$
$338$		0			0
$339$	−419.638	+	24.8193i	−1.23787	+	0.0732133i
$340$		0			0
$341$		−	215.623i		−	0.632326i
$342$		0			0
$343$	18.5203			0.0539949
$344$		0			0
$345$	−5.29938	−	89.6003i	−0.0153605	−	0.259711i
$346$		0			0
$347$			427.136i			1.23094i	0.788161	+	0.615470i	$0.211034\pi$
$347$			427.136i			1.23094i	−0.788161	+	0.615470i	$0.788966\pi$
$348$		0			0
$349$	−316.759			−0.907620			−0.453810	−	0.891098i	$-0.649936\pi$
$349$	−316.759			−0.907620			−0.453810	+	0.891098i	$0.649936\pi$
$350$		0			0
$351$	96.2915	+	537.621i	0.274335	+	1.53168i
$352$		0			0
$353$			86.4606i			0.244931i	0.992473	+	0.122465i	$0.0390801\pi$
$353$			86.4606i			0.244931i	−0.992473	+	0.122465i	$0.960920\pi$
$354$		0			0
$355$	133.697			0.376610
$356$		0			0
$357$	173.018	−	10.2331i	0.484645	−	0.0286642i
$358$		0			0
$359$			286.213i			0.797252i	0.917114	+	0.398626i	$0.130513\pi$
$359$			286.213i			0.797252i	−0.917114	+	0.398626i	$0.869487\pi$
$360$		0			0
$361$	−312.875			−0.866688
$362$		0			0
$363$	−13.7863	−	233.095i	−0.0379788	−	0.642134i
$364$		0			0
$365$			388.858i			1.06536i
$366$		0			0
$367$	−111.616			−0.304130			−0.152065	−	0.988371i	$-0.548592\pi$
$367$	−111.616			−0.304130			−0.152065	+	0.988371i	$0.548592\pi$
$368$		0			0
$369$	44.8784	+	378.069i	0.121622	+	1.02458i
$370$		0			0
$371$		−	268.402i		−	0.723454i
$372$		0			0
$373$	−85.0850			−0.228110			−0.114055	−	0.993474i	$-0.536384\pi$
$373$	−85.0850			−0.228110			−0.114055	+	0.993474i	$0.536384\pi$
$374$		0			0
$375$	405.875	−	24.0053i	1.08233	−	0.0640141i
$376$		0			0
$377$		−	754.681i		−	2.00181i
$378$		0			0
$379$	−26.5464			−0.0700433			−0.0350217	−	0.999387i	$-0.511150\pi$
$379$	−26.5464			−0.0700433			−0.0350217	+	0.999387i	$0.511150\pi$
$380$		0			0
$381$	20.4719	+	346.132i	0.0537319	+	0.908484i
$382$		0			0
$383$		−	115.547i		−	0.301690i	−0.988557	−	0.150845i	$-0.951801\pi$
$383$		−	115.547i		−	0.301690i	0.988557	−	0.150845i	$-0.0481994\pi$
$384$		0			0
$385$	144.295			0.374793
$386$		0			0
$387$	−144.118	+	17.1074i	−0.372397	+	0.0442052i
$388$		0			0
$389$		−	385.718i		−	0.991563i	−0.868447	−	0.495782i	$-0.834882\pi$
$389$		−	385.718i		−	0.991563i	0.868447	−	0.495782i	$-0.165118\pi$
$390$		0			0
$391$	−168.915			−0.432008
$392$		0			0
$393$	247.346	−	14.6292i	0.629380	−	0.0372244i
$394$		0			0
$395$		−	164.700i		−	0.416963i
$396$		0			0
$397$	−47.5934			−0.119883			−0.0599413	−	0.998202i	$-0.519091\pi$
$397$	−47.5934			−0.119883			−0.0599413	+	0.998202i	$0.519091\pi$
$398$		0			0
$399$	3.25098	+	54.9667i	0.00814783	+	0.137761i
$400$		0			0
$401$			458.563i			1.14355i	0.820411	+	0.571775i	$0.193745\pi$
$401$			458.563i			1.14355i	−0.820411	+	0.571775i	$0.806255\pi$
$402$		0			0
$403$	309.328			0.767563
$404$		0			0
$405$	−73.3438	−	304.581i	−0.181096	−	0.752053i
$406$		0			0
$407$			170.980i			0.420098i
$408$		0			0
$409$	−57.8666			−0.141483			−0.0707416	−	0.997495i	$-0.522537\pi$
$409$	−57.8666			−0.141483			−0.0707416	+	0.997495i	$0.522537\pi$
$410$		0			0
$411$	460.413	−	27.2310i	1.12023	−	0.0662554i
$412$		0			0
$413$		−	68.0066i		−	0.164665i
$414$		0			0
$415$	−317.571			−0.765232
$416$		0			0
$417$	39.0588	+	660.395i	0.0936662	+	1.58368i
$418$		0			0
$419$		−	115.304i		−	0.275190i	−0.990489	−	0.137595i	$-0.956063\pi$
$419$		−	115.304i		−	0.275190i	0.990489	−	0.137595i	$-0.0439372\pi$
$420$		0			0
$421$	−622.664			−1.47901			−0.739506	−	0.673150i	$-0.764941\pi$
$421$	−622.664			−1.47901			−0.739506	+	0.673150i	$0.764941\pi$
$422$		0			0
$423$	76.2510	+	642.360i	0.180262	+	1.51858i
$424$		0			0
$425$		−	219.248i		−	0.515878i
$426$		0			0
$427$	−206.642			−0.483939
$428$		0			0
$429$	854.235	−	50.5234i	1.99122	−	0.117770i
$430$		0			0
$431$		−	170.666i		−	0.395978i	−0.980204	−	0.197989i	$-0.936559\pi$
$431$		−	170.666i		−	0.395978i	0.980204	−	0.197989i	$-0.0634410\pi$
$432$		0			0
$433$	−251.992			−0.581968			−0.290984	−	0.956728i	$-0.593983\pi$
$433$	−251.992			−0.581968			−0.290984	+	0.956728i	$0.593983\pi$
$434$		0			0
$435$	25.5582	+	432.131i	0.0587546	+	0.993405i
$436$		0			0
$437$		−	53.6631i		−	0.122799i
$438$		0			0
$439$	452.089			1.02982			0.514908	−	0.857246i	$-0.327826\pi$
$439$	452.089			1.02982			0.514908	+	0.857246i	$0.327826\pi$
$440$		0			0
$441$	−62.5608	+	7.42624i	−0.141861	+	0.0168396i
$442$		0			0
$443$			109.182i			0.246460i	0.992378	+	0.123230i	$0.0393253\pi$
$443$			109.182i			0.246460i	−0.992378	+	0.123230i	$0.960675\pi$
$444$		0			0
$445$	−163.616			−0.367676
$446$		0			0
$447$	581.800	−	34.4103i	1.30157	−	0.0769806i
$448$		0			0
$449$			565.005i			1.25836i	0.777259	+	0.629181i	$0.216610\pi$
$449$			565.005i			1.25836i	−0.777259	+	0.629181i	$0.783390\pi$
$450$		0			0
$451$	596.502			1.32262
$452$		0			0
$453$	−22.6863	−	383.573i	−0.0500801	−	0.846739i
$454$		0			0
$455$			207.003i			0.454951i
$456$		0			0
$457$	595.705			1.30351			0.651756	−	0.758429i	$-0.274033\pi$
$457$	595.705			1.30351			0.651756	+	0.758429i	$0.274033\pi$
$458$		0			0
$459$	−580.346	+	103.944i	−1.26437	+	0.226457i
$460$		0			0
$461$		−	445.590i		−	0.966573i	−0.875462	−	0.483286i	$-0.839443\pi$
$461$		−	445.590i		−	0.966573i	0.875462	−	0.483286i	$-0.160557\pi$
$462$		0			0
$463$	333.919			0.721207			0.360604	−	0.932719i	$-0.382571\pi$
$463$	333.919			0.721207			0.360604	+	0.932719i	$0.382571\pi$
$464$		0			0
$465$	−177.122	+	10.4758i	−0.380907	+	0.0225286i
$466$		0			0
$467$			448.244i			0.959838i	0.877313	+	0.479919i	$0.159334\pi$
$467$			448.244i			0.959838i	−0.877313	+	0.479919i	$0.840666\pi$
$468$		0			0
$469$	−325.867			−0.694812
$470$		0			0
$471$	−26.1581	−	442.274i	−0.0555374	−	0.939011i
$472$		0			0
$473$			227.383i			0.480725i
$474$		0			0
$475$	69.6536			0.146639
$476$		0			0
$477$	107.624	+	906.651i	0.225626	+	1.90074i
$478$		0			0
$479$		−	13.7014i		−	0.0286043i	−0.999898	−	0.0143021i	$-0.995447\pi$
$479$		−	13.7014i		−	0.0286043i	0.999898	−	0.0143021i	$-0.00455267\pi$
$480$		0			0
$481$	−245.284			−0.509945
$482$		0			0
$483$	61.2915	−	3.62506i	0.126898	−	0.00750531i
$484$		0			0
$485$		−	186.936i		−	0.385435i
$486$		0			0
$487$	−11.2549			−0.0231107			−0.0115554	−	0.999933i	$-0.503678\pi$
$487$	−11.2549			−0.0231107			−0.0115554	+	0.999933i	$0.503678\pi$
$488$		0			0
$489$	−37.2614	−	630.005i	−0.0761992	−	1.28835i
$490$		0			0
$491$			285.642i			0.581756i	0.956760	+	0.290878i	$0.0939473\pi$
$491$			285.642i			0.581756i	−0.956760	+	0.290878i	$0.906053\pi$
$492$		0			0
$493$	814.656			1.65245
$494$		0			0
$495$	−487.425	+	57.8595i	−0.984697	+	0.116888i
$496$		0			0
$497$			91.4558i			0.184016i
$498$		0			0
$499$	780.818			1.56477			0.782383	−	0.622798i	$-0.214004\pi$
$499$	780.818			1.56477			0.782383	+	0.622798i	$0.214004\pi$
$500$		0			0
$501$	254.826	−	15.0716i	0.508635	−	0.0300830i
$502$		0			0
$503$			420.771i			0.836522i	0.908327	+	0.418261i	$0.137360\pi$
$503$			420.771i			0.836522i	−0.908327	+	0.418261i	$0.862640\pi$
$504$		0			0
$505$	312.272			0.618360
$506$		0			0
$507$	42.5457	+	719.351i	0.0839166	+	1.41884i
$508$		0			0
$509$		−	102.973i		−	0.202305i	−0.994871	−	0.101152i	$-0.967747\pi$
$509$		−	102.973i		−	0.202305i	0.994871	−	0.101152i	$-0.0322530\pi$
$510$		0			0
$511$	−266.000			−0.520548
$512$		0			0
$513$	−33.0222	−	184.372i	−0.0643708	−	0.359400i
$514$		0			0
$515$			186.623i			0.362374i
$516$		0			0
$517$	1013.49			1.96033
$518$		0			0
$519$	−279.915	+	16.5555i	−0.539335	+	0.0318988i
$520$		0			0
$521$		−	76.7836i		−	0.147377i	−0.997281	−	0.0736887i	$-0.976523\pi$
$521$		−	76.7836i		−	0.147377i	0.997281	−	0.0736887i	$-0.0234771\pi$
$522$		0			0
$523$	−404.561			−0.773539			−0.386769	−	0.922176i	$-0.626409\pi$
$523$	−404.561			−0.773539			−0.386769	+	0.922176i	$0.626409\pi$
$524$		0			0
$525$	4.70526	+	79.5551i	0.00896240	+	0.151534i
$526$		0			0
$527$			333.911i			0.633606i
$528$		0			0
$529$	469.162			0.886885
$530$		0			0
$531$	27.2693	+	229.724i	0.0513546	+	0.432625i
$532$		0			0
$533$			855.728i			1.60549i
$534$		0			0
$535$	−760.118			−1.42078
$536$		0			0
$537$	588.553	−	34.8097i	1.09600	−	0.0648226i
$538$		0			0
$539$			98.7060i			0.183128i
$540$		0			0
$541$	−592.907			−1.09595			−0.547973	−	0.836496i	$-0.684600\pi$
$541$	−592.907			−1.09595			−0.547973	+	0.836496i	$0.684600\pi$
$542$		0			0
$543$	29.0523	+	491.208i	0.0535034	+	0.904619i
$544$		0			0
$545$		−	206.118i		−	0.378198i
$546$		0			0
$547$	−793.689			−1.45099			−0.725493	−	0.688230i	$-0.758388\pi$
$547$	−793.689			−1.45099			−0.725493	+	0.688230i	$0.758388\pi$
$548$		0			0
$549$	698.029	−	82.8591i	1.27145	−	0.150927i
$550$		0			0
$551$			258.811i			0.469711i
$552$		0			0
$553$	112.664			0.203732
$554$		0			0
$555$	140.450	−	8.30684i	0.253062	−	0.0149673i
$556$		0			0
$557$			368.963i			0.662411i	0.943559	+	0.331206i	$0.107455\pi$
$557$			368.963i			0.662411i	−0.943559	+	0.331206i	$0.892545\pi$
$558$		0			0
$559$	−326.199			−0.583540
$560$		0			0
$561$	54.5385	+	922.122i	0.0972167	+	1.64371i
$562$		0			0
$563$		−	529.796i		−	0.941022i	−0.882394	−	0.470511i	$-0.844070\pi$
$563$		−	529.796i		−	0.941022i	0.882394	−	0.470511i	$-0.155930\pi$
$564$		0			0
$565$	541.963			0.959227
$566$		0			0
$567$	208.350	−	50.1712i	0.367461	−	0.0884853i
$568$		0			0
$569$		−	1068.04i		−	1.87705i	−0.345216	−	0.938523i	$-0.612194\pi$
$569$		−	1068.04i		−	1.87705i	0.345216	−	0.938523i	$-0.387806\pi$
$570$		0			0
$571$	−118.191			−0.206989			−0.103495	−	0.994630i	$-0.533002\pi$
$571$	−118.191			−0.206989			−0.103495	+	0.994630i	$0.533002\pi$
$572$		0			0
$573$	−689.423	+	40.7757i	−1.20318	+	0.0711618i
$574$		0			0
$575$		−	77.6684i		−	0.135075i
$576$		0			0
$577$	586.559			1.01657			0.508284	−	0.861190i	$-0.330280\pi$
$577$	586.559			1.01657			0.508284	+	0.861190i	$0.330280\pi$
$578$		0			0
$579$	−32.7006	−	552.893i	−0.0564778	−	0.954909i
$580$		0			0
$581$		−	217.236i		−	0.373900i
$582$		0			0
$583$	1430.48			2.45365
$584$		0			0
$585$	−83.0039	−	699.249i	−0.141887	−	1.19530i
$586$		0			0
$587$			809.558i			1.37914i	0.724217	+	0.689572i	$0.242201\pi$
$587$			809.558i			1.37914i	−0.724217	+	0.689572i	$0.757799\pi$
$588$		0			0
$589$	−106.081			−0.180104
$590$		0			0
$591$	−273.889	+	16.1991i	−0.463433	+	0.0274096i
$592$		0			0
$593$		−	480.314i		−	0.809973i	−0.914323	−	0.404986i	$-0.867276\pi$
$593$		−	480.314i		−	0.809973i	0.914323	−	0.404986i	$-0.132724\pi$
$594$		0			0
$595$	−223.454			−0.375552
$596$		0			0
$597$	36.0889	+	610.181i	0.0604504	+	1.02208i
$598$		0			0
$599$			386.689i			0.645557i	0.946474	+	0.322779i	$0.104617\pi$
$599$			386.689i			0.645557i	−0.946474	+	0.322779i	$0.895383\pi$
$600$		0			0
$601$	−474.369			−0.789299			−0.394649	−	0.918832i	$-0.629134\pi$
$601$	−474.369			−0.789299			−0.394649	+	0.918832i	$0.629134\pi$
$602$		0			0
$603$	1100.77	−	130.666i	1.82548	−	0.216693i
$604$		0			0
$605$			301.042i			0.497591i
$606$		0			0
$607$	434.199			0.715319			0.357660	−	0.933852i	$-0.383575\pi$
$607$	434.199			0.715319			0.357660	+	0.933852i	$0.383575\pi$
$608$		0			0
$609$	−295.601	+	17.4832i	−0.485388	+	0.0287081i
$610$		0			0
$611$			1453.93i			2.37959i
$612$		0			0
$613$	−614.782			−1.00291			−0.501453	−	0.865185i	$-0.667201\pi$
$613$	−614.782			−1.00291			−0.501453	+	0.865185i	$0.667201\pi$
$614$		0			0
$615$	−28.9803	−	489.991i	−0.0471225	−	0.796733i
$616$		0			0
$617$			527.697i			0.855263i	0.903953	+	0.427632i	$0.140652\pi$
$617$			527.697i			0.855263i	−0.903953	+	0.427632i	$0.859348\pi$
$618$		0			0
$619$	−679.970			−1.09850			−0.549249	−	0.835659i	$-0.685086\pi$
$619$	−679.970			−1.09850			−0.549249	+	0.835659i	$0.685086\pi$
$620$		0			0
$621$	−205.587	+	36.8220i	−0.331058	+	0.0592946i
$622$		0			0
$623$		−	111.922i		−	0.179650i
$624$		0			0
$625$	−273.175			−0.437080
$626$		0			0
$627$	−292.952	+	17.3265i	−0.467227	+	0.0276340i
$628$		0			0
$629$		−	264.776i		−	0.420948i
$630$		0			0
$631$	929.239			1.47265			0.736323	−	0.676631i	$-0.236561\pi$
$631$	929.239			1.47265			0.736323	+	0.676631i	$0.236561\pi$
$632$		0			0
$633$	33.3424	+	563.744i	0.0526737	+	0.890591i
$634$		0			0
$635$		−	447.031i		−	0.703986i
$636$		0			0
$637$	−141.601			−0.222294
$638$		0			0
$639$	−36.6719	−	308.935i	−0.0573895	−	0.483466i
$640$		0			0
$641$		−	798.925i		−	1.24637i	−0.782073	−	0.623187i	$-0.785838\pi$
$641$		−	798.925i		−	1.24637i	0.782073	−	0.623187i	$-0.214162\pi$
$642$		0			0
$643$	306.051			0.475973			0.237987	−	0.971268i	$-0.423513\pi$
$643$	306.051			0.475973			0.237987	+	0.971268i	$0.423513\pi$
$644$		0			0
$645$	186.782	−	11.0471i	0.289584	−	0.0171273i
$646$		0			0
$647$		−	856.527i		−	1.32384i	−0.749573	−	0.661922i	$-0.769741\pi$
$647$		−	856.527i		−	1.32384i	0.749573	−	0.661922i	$-0.230259\pi$
$648$		0			0
$649$	362.450			0.558474
$650$		0			0
$651$	−7.16601	−	121.161i	−0.0110077	−	0.186115i
$652$		0			0
$653$			610.533i			0.934966i	0.884002	+	0.467483i	$0.154839\pi$
$653$			610.533i			0.934966i	−0.884002	+	0.467483i	$0.845161\pi$
$654$		0			0
$655$	−319.448			−0.487708
$656$		0			0
$657$	898.539	−	106.661i	1.36764	−	0.162345i
$658$		0			0
$659$		−	41.3318i		−	0.0627190i	−0.999508	−	0.0313595i	$-0.990016\pi$
$659$		−	41.3318i		−	0.0627190i	0.999508	−	0.0313595i	$-0.00998367\pi$
$660$		0			0
$661$	−259.285			−0.392262			−0.196131	−	0.980578i	$-0.562838\pi$
$661$	−259.285			−0.392262			−0.196131	+	0.980578i	$0.562838\pi$
$662$		0			0
$663$	−1322.85	+	78.2398i	−1.99526	+	0.118009i
$664$		0			0
$665$		−	70.9896i		−	0.106751i
$666$		0			0
$667$	288.591			0.432670
$668$		0			0
$669$	−27.2994	−	461.570i	−0.0408062	−	0.689940i
$670$		0			0
$671$		−	1101.32i		−	1.64132i
$672$		0			0
$673$	113.498			0.168645			0.0843225	−	0.996439i	$-0.473127\pi$
$673$	113.498			0.168645			0.0843225	+	0.996439i	$0.473127\pi$
$674$		0			0
$675$	−47.7942	−	266.848i	−0.0708062	−	0.395330i
$676$		0			0
$677$		−	1239.43i		−	1.83077i	−0.402575	−	0.915387i	$-0.631885\pi$
$677$		−	1239.43i		−	1.83077i	0.402575	−	0.915387i	$-0.368115\pi$
$678$		0			0
$679$	127.875			0.188328
$680$		0			0
$681$	−699.553	+	41.3748i	−1.02724	+	0.0607559i
$682$		0			0
$683$		−	1286.58i		−	1.88371i	−0.336017	−	0.941856i	$-0.609080\pi$
$683$		−	1286.58i		−	1.88371i	0.336017	−	0.941856i	$-0.390920\pi$
$684$		0			0
$685$	−594.625			−0.868065
$686$		0			0
$687$	−48.7569	−	824.367i	−0.0709707	−	1.19995i
$688$		0			0
$689$			2052.13i			2.97842i
$690$		0			0
$691$	−888.278			−1.28550			−0.642748	−	0.766077i	$-0.722206\pi$
$691$	−888.278			−1.28550			−0.642748	+	0.766077i	$0.722206\pi$
$692$		0			0
$693$	−39.5791	−	333.425i	−0.0571127	−	0.481133i
$694$		0			0
$695$		−	852.902i		−	1.22720i
$696$		0			0
$697$	−923.733			−1.32530
$698$		0			0
$699$	−106.170	+	6.27938i	−0.151888	+	0.00898338i
$700$		0			0
$701$			270.171i			0.385408i	0.981257	+	0.192704i	$0.0617257\pi$
$701$			270.171i			0.385408i	−0.981257	+	0.192704i	$0.938274\pi$
$702$		0			0
$703$	84.1176			0.119655
$704$		0			0
$705$	−49.2392	−	832.522i	−0.0698428	−	1.18088i
$706$		0			0
$707$			213.611i			0.302137i
$708$		0			0
$709$	−81.8824			−0.115490			−0.0577450	−	0.998331i	$-0.518391\pi$
$709$	−81.8824			−0.115490			−0.0577450	+	0.998331i	$0.518391\pi$
$710$		0			0
$711$	−380.575	+	45.1760i	−0.535267	+	0.0635386i
$712$		0			0
$713$			118.287i			0.165901i
$714$		0			0
$715$	−1103.25			−1.54300
$716$		0			0
$717$	−170.369	+	10.0764i	−0.237613	+	0.0140535i
$718$		0			0
$719$			680.552i			0.946526i	0.880921	+	0.473263i	$0.156924\pi$
$719$			680.552i			0.946526i	−0.880921	+	0.473263i	$0.843076\pi$
$720$		0			0
$721$	−127.660			−0.177060
$722$		0			0
$723$	−37.1517	−	628.149i	−0.0513854	−	0.868810i
$724$		0			0
$725$			374.585i			0.516669i
$726$		0			0
$727$	−333.344			−0.458520			−0.229260	−	0.973365i	$-0.573631\pi$
$727$	−333.344			−0.458520			−0.229260	+	0.973365i	$0.573631\pi$
$728$		0			0
$729$	−683.682	+	253.021i	−0.937836	+	0.347079i
$730$		0			0
$731$		−	352.122i		−	0.481699i
$732$		0			0
$733$	825.261			1.12587			0.562934	−	0.826502i	$-0.309672\pi$
$733$	825.261			1.12587			0.562934	+	0.826502i	$0.309672\pi$
$734$		0			0
$735$	81.0810	−	4.79551i	0.110314	−	0.00652450i
$736$		0			0
$737$		−	1736.75i		−	2.35651i
$738$		0			0
$739$	1074.77			1.45435			0.727176	−	0.686451i	$-0.240832\pi$
$739$	1074.77			1.45435			0.727176	+	0.686451i	$0.240832\pi$
$740$		0			0
$741$	−24.8562	−	420.262i	−0.0335442	−	0.567155i
$742$		0			0
$743$		−	171.465i		−	0.230774i	−0.993321	−	0.115387i	$-0.963189\pi$
$743$		−	171.465i		−	0.230774i	0.993321	−	0.115387i	$-0.0368108\pi$
$744$		0			0
$745$	−751.396			−1.00859
$746$		0			0
$747$	87.1072	+	733.816i	0.116609	+	0.982350i
$748$		0			0
$749$		−	519.962i		−	0.694208i
$750$		0			0
$751$	−352.640			−0.469561			−0.234781	−	0.972048i	$-0.575437\pi$
$751$	−352.640			−0.469561			−0.234781	+	0.972048i	$0.575437\pi$
$752$		0			0
$753$	−230.933	+	13.6585i	−0.306684	+	0.0181387i
$754$		0			0
$755$			495.385i			0.656139i
$756$		0			0
$757$	−624.790			−0.825349			−0.412675	−	0.910878i	$-0.635405\pi$
$757$	−624.790			−0.825349			−0.412675	+	0.910878i	$0.635405\pi$
$758$		0			0
$759$	19.3202	+	326.660i	0.0254548	+	0.430383i
$760$		0			0
$761$		−	539.144i		−	0.708468i	−0.935157	−	0.354234i	$-0.884742\pi$
$761$		−	539.144i		−	0.708468i	0.935157	−	0.354234i	$-0.115258\pi$
$762$		0			0
$763$	140.996			0.184792
$764$		0			0
$765$	754.818	−	89.6003i	0.986691	−	0.117125i
$766$		0			0
$767$			519.962i			0.677916i
$768$		0			0
$769$	−357.409			−0.464771			−0.232386	−	0.972624i	$-0.574653\pi$
$769$	−357.409			−0.464771			−0.232386	+	0.972624i	$0.574653\pi$
$770$		0			0
$771$	474.176	−	28.0450i	0.615015	−	0.0363748i
$772$		0			0
$773$			232.707i			0.301044i	0.988607	+	0.150522i	$0.0480954\pi$
$773$			232.707i			0.301044i	−0.988607	+	0.150522i	$0.951905\pi$
$774$		0			0
$775$	−153.535			−0.198109
$776$		0			0
$777$	5.68233	+	96.0752i	0.00731317	+	0.123649i
$778$		0			0
$779$		−	293.464i		−	0.376718i
$780$		0			0
$781$	−487.425			−0.624104
$782$		0			0
$783$	991.520	−	177.588i	1.26631	−	0.226804i
$784$		0			0
$785$			571.198i			0.727641i
$786$		0			0
$787$	1202.71			1.52823			0.764114	−	0.645082i	$-0.223177\pi$
$787$	1202.71			1.52823			0.764114	+	0.645082i	$0.223177\pi$
$788$		0			0
$789$	449.557	−	26.5889i	0.569781	−	0.0336995i
$790$		0			0
$791$			370.733i			0.468688i
$792$		0			0
$793$	1579.93			1.99235
$794$		0			0
$795$	−69.4980	−	1175.05i	−0.0874189	−	1.47805i
$796$		0			0
$797$			1108.42i			1.39074i	0.718654	+	0.695368i	$0.244758\pi$
$797$			1108.42i			1.39074i	−0.718654	+	0.695368i	$0.755242\pi$
$798$		0			0
$799$	−1569.47			−1.96430
$800$		0			0
$801$	44.8784	+	378.069i	0.0560280	+	0.471996i
$802$		0			0
$803$		−	1417.68i		−	1.76548i
$804$		0			0
$805$	−79.1581			−0.0983331
$806$		0			0
$807$	−571.413	+	33.7960i	−0.708071	+	0.0418786i
$808$		0			0
$809$			867.089i			1.07180i	0.844281	+	0.535901i	$0.180028\pi$
$809$			867.089i			1.07180i	−0.844281	+	0.535901i	$0.819972\pi$
$810$		0			0
$811$	−1008.35			−1.24335			−0.621673	−	0.783277i	$-0.713547\pi$
$811$	−1008.35			−1.24335			−0.621673	+	0.783277i	$0.713547\pi$
$812$		0			0
$813$	60.6562	+	1025.56i	0.0746078	+	1.26145i
$814$		0			0
$815$			813.653i			0.998347i
$816$		0			0
$817$	111.867			0.136924
$818$		0			0
$819$	478.324	−	56.7792i	0.584034	−	0.0693275i
$820$		0			0
$821$		−	1316.06i		−	1.60300i	−0.597996	−	0.801499i	$-0.704036\pi$
$821$		−	1316.06i		−	1.60300i	0.597996	−	0.801499i	$-0.295964\pi$
$822$		0			0
$823$	992.146			1.20552			0.602762	−	0.797921i	$-0.294067\pi$
$823$	992.146			1.20552			0.602762	+	0.797921i	$0.294067\pi$
$824$		0			0
$825$	−423.999	+	25.0772i	−0.513938	+	0.0303967i
$826$		0			0
$827$			39.2488i			0.0474593i	0.999718	+	0.0237296i	$0.00755409\pi$
$827$			39.2488i			0.0474593i	−0.999718	+	0.0237296i	$0.992446\pi$
$828$		0			0
$829$	−1083.74			−1.30729			−0.653643	−	0.756803i	$-0.726760\pi$
$829$	−1083.74			−1.30729			−0.653643	+	0.756803i	$0.726760\pi$
$830$		0			0
$831$	16.5163	+	279.253i	0.0198752	+	0.336045i
$832$		0			0
$833$		−	152.854i		−	0.183499i
$834$		0			0
$835$	−329.109			−0.394142
$836$		0			0
$837$	72.7895	+	406.403i	0.0869648	+	0.485548i
$838$		0			0
$839$			570.971i			0.680537i	0.940328	+	0.340269i	$0.110518\pi$
$839$			570.971i			0.680537i	−0.940328	+	0.340269i	$0.889482\pi$
$840$		0			0
$841$	−550.838			−0.654980
$842$		0			0
$843$	−751.911	+	44.4715i	−0.891947	+	0.0527539i
$844$		0			0
$845$		−	929.043i		−	1.09946i
$846$		0			0
$847$	−205.929			−0.243128
$848$		0			0
$849$	2.94907	+	49.8619i	0.00347357	+	0.0587302i
$850$		0			0
$851$		−	93.7967i		−	0.110219i
$852$		0			0
$853$	381.290			0.446999			0.223499	−	0.974704i	$-0.428252\pi$
$853$	381.290			0.446999			0.223499	+	0.974704i	$0.428252\pi$
$854$		0			0
$855$	28.4654	+	239.800i	0.0332929	+	0.280468i
$856$		0			0
$857$			342.531i			0.399686i	0.979828	+	0.199843i	$0.0640432\pi$
$857$			342.531i			0.399686i	−0.979828	+	0.199843i	$0.935957\pi$
$858$		0			0
$859$	522.051			0.607743			0.303871	−	0.952713i	$-0.401721\pi$
$859$	522.051			0.607743			0.303871	+	0.952713i	$0.401721\pi$
$860$		0			0
$861$	335.180	−	19.8241i	0.389292	−	0.0230245i
$862$		0			0
$863$		−	118.373i		−	0.137165i	−0.997645	−	0.0685825i	$-0.978152\pi$
$863$		−	118.373i		−	0.137165i	0.997645	−	0.0685825i	$-0.0218476\pi$
$864$		0			0
$865$	361.511			0.417932
$866$		0			0
$867$	−33.2686	−	562.495i	−0.0383721	−	0.648784i
$868$		0			0
$869$			600.457i			0.690974i
$870$		0			0
$871$	2491.50			2.86050
$872$		0			0
$873$	−431.956	+	51.2751i	−0.494794	+	0.0587343i
$874$		0			0
$875$		−	358.573i		−	0.409798i
$876$		0			0
$877$	−60.7321			−0.0692499			−0.0346249	−	0.999400i	$-0.511024\pi$
$877$	−60.7321			−0.0692499			−0.0346249	+	0.999400i	$0.511024\pi$
$878$		0			0
$879$	920.357	−	54.4342i	1.04705	−	0.0619274i
$880$		0			0
$881$			817.935i			0.928417i	0.885726	+	0.464208i	$0.153661\pi$
$881$			817.935i			0.928417i	−0.885726	+	0.464208i	$0.846339\pi$
$882$		0			0
$883$	19.9764			0.0226233			0.0113117	−	0.999936i	$-0.496399\pi$
$883$	19.9764			0.0226233			0.0113117	+	0.999936i	$0.496399\pi$
$884$		0			0
$885$	−17.6092	−	297.731i	−0.0198974	−	0.336419i
$886$		0			0
$887$		−	682.008i		−	0.768893i	−0.923147	−	0.384446i	$-0.874392\pi$
$887$		−	682.008i		−	0.768893i	0.923147	−	0.384446i	$-0.125608\pi$
$888$		0			0
$889$	305.793			0.343975
$890$		0			0
$891$	267.393	+	1110.43i	0.300105	+	1.24627i
$892$		0			0
$893$		−	498.611i		−	0.558355i
$894$		0			0
$895$	−760.118			−0.849293
$896$		0			0
$897$	−468.620	+	27.7163i	−0.522430	+	0.0308989i
$898$		0			0
$899$		−	570.485i		−	0.634578i
$900$		0			0
$901$	−2215.22			−2.45862
$902$		0			0
$903$	7.55684	+	127.769i	0.00836859	+	0.141494i
$904$		0			0
$905$		−	634.397i		−	0.700991i
$906$		0			0
$907$	649.895			0.716533			0.358266	−	0.933619i	$-0.383368\pi$
$907$	649.895			0.716533			0.358266	+	0.933619i	$0.383368\pi$
$908$		0			0
$909$	−85.6536	−	721.570i	−0.0942284	−	0.793807i
$910$		0			0
$911$		−	1522.75i		−	1.67152i	−0.549099	−	0.835758i	$-0.685029\pi$
$911$		−	1522.75i		−	1.67152i	0.549099	−	0.835758i	$-0.314971\pi$
$912$		0			0
$913$	1157.79			1.26811
$914$		0			0
$915$	−904.671	+	53.5064i	−0.988711	+	0.0584769i
$916$		0			0
$917$		−	218.520i		−	0.238299i
$918$		0			0
$919$	753.652			0.820079			0.410039	−	0.912068i	$-0.365515\pi$
$919$	753.652			0.820079			0.410039	+	0.912068i	$0.365515\pi$
$920$		0			0
$921$	22.4550	+	379.662i	0.0243811	+	0.412228i
$922$		0			0
$923$		−	699.249i		−	0.757582i
$924$		0			0
$925$	121.746			0.131618
$926$		0			0
$927$	431.231	−	51.1891i	0.465190	−	0.0552202i
$928$		0			0
$929$			555.100i			0.597525i	0.954328	+	0.298762i	$0.0965738\pi$
$929$			555.100i			0.597525i	−0.954328	+	0.298762i	$0.903426\pi$
$930$		0			0
$931$	48.5608			0.0521598
$932$		0			0
$933$	−985.025	+	58.2589i	−1.05576	+	0.0624426i
$934$		0			0
$935$		−	1190.92i		−	1.27371i
$936$		0			0
$937$	−231.357			−0.246912			−0.123456	−	0.992350i	$-0.539398\pi$
$937$	−231.357			−0.246912			−0.123456	+	0.992350i	$0.539398\pi$
$938$		0			0
$939$	−45.9254	−	776.494i	−0.0489089	−	0.826937i
$940$		0			0
$941$			525.513i			0.558463i	0.960224	+	0.279231i	$0.0900796\pi$
$941$			525.513i			0.558463i	−0.960224	+	0.279231i	$0.909920\pi$
$942$		0			0
$943$	−327.231			−0.347011
$944$		0			0
$945$	−271.966	+	48.7109i	−0.287795	+	0.0515459i
$946$		0			0
$947$		−	571.855i		−	0.603860i	−0.953330	−	0.301930i	$-0.902369\pi$
$947$		−	571.855i		−	0.603860i	0.953330	−	0.301930i	$-0.0976309\pi$
$948$		0			0
$949$	2033.77			2.14307
$950$		0			0
$951$	25.8157	−	1.52686i	0.0271458	−	0.00160553i
$952$		0			0
$953$			596.119i			0.625518i	0.949833	+	0.312759i	$0.101253\pi$
$953$			596.119i			0.625518i	−0.949833	+	0.312759i	$0.898747\pi$
$954$		0			0
$955$	890.392			0.932348
$956$		0			0
$957$	−93.1790	−	1575.44i	−0.0973657	−	1.64623i
$958$		0			0
$959$		−	406.756i		−	0.424146i
$960$		0			0
$961$	−727.170			−0.756680
$962$		0			0
$963$	208.494	+	1756.41i	0.216505	+	1.82390i
$964$		0			0
$965$			714.062i			0.739961i
$966$		0			0
$967$	−1519.98			−1.57185			−0.785924	−	0.618324i	$-0.787812\pi$
$967$	−1519.98			−1.57185			−0.785924	+	0.618324i	$0.787812\pi$
$968$		0			0
$969$	453.660	−	26.8316i	0.468173	−	0.0276899i
$970$		0			0
$971$		−	1077.39i		−	1.10957i	−0.831995	−	0.554783i	$-0.812801\pi$
$971$		−	1077.39i		−	1.10957i	0.831995	−	0.554783i	$-0.187199\pi$
$972$		0			0
$973$	583.431			0.599621
$974$		0			0
$975$	−35.9752	−	608.259i	−0.0368977	−	0.623855i
$976$		0			0
$977$			1800.57i			1.84296i	0.388427	+	0.921480i	$0.373019\pi$
$977$			1800.57i			1.84296i	−0.388427	+	0.921480i	$0.626981\pi$
$978$		0			0
$979$	596.502			0.609297
$980$		0			0
$981$	−476.280	+	56.5365i	−0.485504	+	0.0576315i
$982$		0			0
$983$		−	512.798i		−	0.521666i	−0.965384	−	0.260833i	$-0.916003\pi$
$983$		−	512.798i		−	0.521666i	0.965384	−	0.260833i	$-0.0839972\pi$
$984$		0			0
$985$	353.728			0.359115
$986$		0			0
$987$	569.490	−	33.6823i	0.576991	−	0.0341259i
$988$		0			0
$989$		−	124.739i		−	0.126126i
$990$		0			0
$991$	191.749			0.193490			0.0967452	−	0.995309i	$-0.469157\pi$
$991$	191.749			0.193490			0.0967452	+	0.995309i	$0.469157\pi$
$992$		0			0
$993$	−84.0222	−	1420.62i	−0.0846145	−	1.43064i
$994$		0			0
$995$		−	788.050i		−	0.792010i
$996$		0			0
$997$	1677.56			1.68260			0.841302	−	0.540565i	$-0.181789\pi$
$997$	1677.56			1.68260			0.841302	+	0.540565i	$0.181789\pi$
$998$		0			0
$999$	−57.7189	−	322.260i	−0.0577767	−	0.322583i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.d.g.449.2		4
3.2	odd	2	inner	1344.3.d.g.449.1		4
4.3	odd	2		1344.3.d.a.449.3		4
8.3	odd	2		84.3.c.a.29.2	yes	4
8.5	even	2		336.3.d.a.113.3		4
12.11	even	2		1344.3.d.a.449.4		4
24.5	odd	2		336.3.d.a.113.4		4
24.11	even	2		84.3.c.a.29.1	✓	4
40.3	even	4		2100.3.e.a.449.2		8
40.19	odd	2		2100.3.g.a.701.3		4
40.27	even	4		2100.3.e.a.449.7		8
56.3	even	6		588.3.p.h.569.1		8
56.11	odd	6		588.3.p.f.569.4		8
56.19	even	6		588.3.p.h.557.4		8
56.27	even	2		588.3.c.h.197.3		4
56.51	odd	6		588.3.p.f.557.1		8
72.11	even	6		2268.3.bg.a.701.3		8
72.43	odd	6		2268.3.bg.a.701.2		8
72.59	even	6		2268.3.bg.a.2213.2		8
72.67	odd	6		2268.3.bg.a.2213.3		8
120.59	even	2		2100.3.g.a.701.4		4
120.83	odd	4		2100.3.e.a.449.8		8
120.107	odd	4		2100.3.e.a.449.1		8
168.11	even	6		588.3.p.f.569.1		8
168.59	odd	6		588.3.p.h.569.4		8
168.83	odd	2		588.3.c.h.197.4		4
168.107	even	6		588.3.p.f.557.4		8
168.131	odd	6		588.3.p.h.557.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.3.c.a.29.1	✓	4	24.11	even	2
84.3.c.a.29.2	yes	4	8.3	odd	2
336.3.d.a.113.3		4	8.5	even	2
336.3.d.a.113.4		4	24.5	odd	2
588.3.c.h.197.3		4	56.27	even	2
588.3.c.h.197.4		4	168.83	odd	2
588.3.p.f.557.1		8	56.51	odd	6
588.3.p.f.557.4		8	168.107	even	6
588.3.p.f.569.1		8	168.11	even	6
588.3.p.f.569.4		8	56.11	odd	6
588.3.p.h.557.1		8	168.131	odd	6
588.3.p.h.557.4		8	56.19	even	6
588.3.p.h.569.1		8	56.3	even	6
588.3.p.h.569.4		8	168.59	odd	6
1344.3.d.a.449.3		4	4.3	odd	2
1344.3.d.a.449.4		4	12.11	even	2
1344.3.d.g.449.1		4	3.2	odd	2	inner
1344.3.d.g.449.2		4	1.1	even	1	trivial
2100.3.e.a.449.1		8	120.107	odd	4
2100.3.e.a.449.2		8	40.3	even	4
2100.3.e.a.449.7		8	40.27	even	4
2100.3.e.a.449.8		8	120.83	odd	4
2100.3.g.a.701.3		4	40.19	odd	2
2100.3.g.a.701.4		4	120.59	even	2
2268.3.bg.a.701.2		8	72.43	odd	6
2268.3.bg.a.701.3		8	72.11	even	6
2268.3.bg.a.2213.2		8	72.59	even	6
2268.3.bg.a.2213.3		8	72.67	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1344.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.177124 + 2.99477i) q^{3} -3.86775i q^{5} +2.64575 q^{7} +(-8.93725 + 1.06089i) q^{9} +14.1009i q^{11} -20.2288 q^{13} +(11.5830 - 0.685072i) q^{15} -21.8363i q^{17} +6.93725 q^{19} +(0.468627 + 7.92341i) q^{21} -7.73550i q^{23} +10.0405 q^{25} +(-4.76013 - 26.5771i) q^{27} +37.3073i q^{29} -15.2915 q^{31} +(-42.2288 + 2.49760i) q^{33} -10.2331i q^{35} +12.1255 q^{37} +(-3.58301 - 60.5804i) q^{39} -42.3026i q^{41} +16.1255 q^{43} +(4.10326 + 34.5671i) q^{45} -71.8744i q^{47} +7.00000 q^{49} +(65.3948 - 3.86775i) q^{51} -101.446i q^{53} +54.5385 q^{55} +(1.22876 + 20.7755i) q^{57} -25.7041i q^{59} -78.1033 q^{61} +(-23.6458 + 2.80686i) q^{63} +78.2398i q^{65} -123.166 q^{67} +(23.1660 - 1.37014i) q^{69} +34.5671i q^{71} -100.539 q^{73} +(1.77842 + 30.0690i) q^{75} +37.3073i q^{77} +42.5830 q^{79} +(78.7490 - 18.9629i) q^{81} -82.1075i q^{83} -84.4575 q^{85} +(-111.727 + 6.60804i) q^{87} -42.3026i q^{89} -53.5203 q^{91} +(-2.70850 - 45.7945i) q^{93} -26.8316i q^{95} +48.3320 q^{97} +(-14.9595 - 126.023i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{3} - 4 q^{9} - 28 q^{13} + 4 q^{15} - 4 q^{19} - 14 q^{21} - 108 q^{25} + 18 q^{27} - 40 q^{31} - 116 q^{33} + 112 q^{37} + 28 q^{39} + 128 q^{43} - 100 q^{45} + 28 q^{49} + 124 q^{51} - 184 q^{55}+ \cdots - 208 q^{99}+O(q^{100}) \) 4 * q + 6 * q^3 - 4 * q^9 - 28 * q^13 + 4 * q^15 - 4 * q^19 - 14 * q^21 - 108 * q^25 + 18 * q^27 - 40 * q^31 - 116 * q^33 + 112 * q^37 + 28 * q^39 + 128 * q^43 - 100 * q^45 + 28 * q^49 + 124 * q^51 - 184 * q^55 - 48 * q^57 - 196 * q^61 - 84 * q^63 - 408 * q^67 + 8 * q^69 - 358 * q^75 + 128 * q^79 + 188 * q^81 - 232 * q^85 - 140 * q^87 - 140 * q^91 - 32 * q^93 + 24 * q^97 - 208 * q^99

Introduction

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