LMFDB - Embedded newform 132.3.e.b.89.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [132,3,Mod(89,132)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(132, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("132.89");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 132 = 2^{2} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	132.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$3.59673948956$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 11x^{2} - 10x + 3 $ x^4 - 2x^3 + 11x^2 - 10*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.3
Root			$0.500000 + 3.07253i$ of defining polynomial
Character	$\chi$	$=$	132.89
Dual form			132.3.e.b.89.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 2.82843i) q^{3} -9.46168i q^{5} +9.38083 q^{7} +(-7.00000 - 5.65685i) q^{9} +3.31662i q^{11} +21.3808 q^{13} +(26.7617 + 9.46168i) q^{15} -1.95279i q^{17} +4.76166 q^{19} +(-9.38083 + 26.5330i) q^{21} -11.4145i q^{23} -64.5233 q^{25} +(23.0000 - 14.1421i) q^{27} -18.9234i q^{29} -12.7617 q^{31} +(-9.38083 - 3.31662i) q^{33} -88.7584i q^{35} +11.2383 q^{37} +(-21.3808 + 60.4741i) q^{39} +52.8645i q^{41} -48.7617 q^{43} +(-53.5233 + 66.2317i) q^{45} +26.4322i q^{47} +39.0000 q^{49} +(5.52333 + 1.95279i) q^{51} +32.2906i q^{53} +31.3808 q^{55} +(-4.76166 + 13.4680i) q^{57} +28.0828i q^{59} -2.61917 q^{61} +(-65.6658 - 53.0660i) q^{63} -202.299i q^{65} +109.047 q^{67} +(32.2850 + 11.4145i) q^{69} +53.1668i q^{71} -119.808 q^{73} +(64.5233 - 182.500i) q^{75} +31.1127i q^{77} -38.6192 q^{79} +(17.0000 + 79.1960i) q^{81} +116.841i q^{83} -18.4767 q^{85} +(53.5233 + 18.9234i) q^{87} +15.0178i q^{89} +200.570 q^{91} +(12.7617 - 36.0954i) q^{93} -45.0533i q^{95} -52.4767 q^{97} +(18.7617 - 23.2164i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 28 q^{9} + 48 q^{13} + 32 q^{15} - 56 q^{19} - 108 q^{25} + 92 q^{27} + 24 q^{31} + 120 q^{37} - 48 q^{39} - 120 q^{43} - 64 q^{45} + 156 q^{49} - 128 q^{51} + 88 q^{55} + 56 q^{57} - 48 q^{61}+ \cdots - 360 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 28 * q^9 + 48 * q^13 + 32 * q^15 - 56 * q^19 - 108 * q^25 + 92 * q^27 + 24 * q^31 + 120 * q^37 - 48 * q^39 - 120 * q^43 - 64 * q^45 + 156 * q^49 - 128 * q^51 + 88 * q^55 + 56 * q^57 - 48 * q^61 + 136 * q^67 - 96 * q^69 - 104 * q^73 + 108 * q^75 - 192 * q^79 + 68 * q^81 - 224 * q^85 + 64 * q^87 + 352 * q^91 - 24 * q^93 - 360 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/132\mathbb{Z}\right)^\times$.

$n$	$13$	$67$	$89$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$3$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$4$		0			0
$5$		−	9.46168i		−	1.89234i	−0.323677	−	0.946168i	$-0.604919\pi$
$5$		−	9.46168i		−	1.89234i	0.323677	−	0.946168i	$-0.395081\pi$
$6$		0			0
$7$	9.38083			1.34012			0.670059	−	0.742307i	$-0.266269\pi$
$7$	9.38083			1.34012			0.670059	+	0.742307i	$0.266269\pi$
$8$		0			0
$9$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$10$		0			0
$11$			3.31662i			0.301511i
$11$			3.31662i			0.301511i
$12$		0			0
$13$	21.3808			1.64468			0.822340	−	0.568997i	$-0.192668\pi$
$13$	21.3808			1.64468			0.822340	+	0.568997i	$0.192668\pi$
$14$		0			0
$15$	26.7617	+	9.46168i	1.78411	+	0.630778i
$16$		0			0
$17$		−	1.95279i		−	0.114870i	−0.998349	−	0.0574350i	$-0.981708\pi$
$17$		−	1.95279i		−	0.114870i	0.998349	−	0.0574350i	$-0.0182922\pi$
$18$		0			0
$19$	4.76166			0.250614			0.125307	−	0.992118i	$-0.460008\pi$
$19$	4.76166			0.250614			0.125307	+	0.992118i	$0.460008\pi$
$20$		0			0
$21$	−9.38083	+	26.5330i	−0.446706	+	1.26348i
$22$		0			0
$23$		−	11.4145i		−	0.496281i	−0.968724	−	0.248141i	$-0.920181\pi$
$23$		−	11.4145i		−	0.496281i	0.968724	−	0.248141i	$-0.0798195\pi$
$24$		0			0
$25$	−64.5233			−2.58093
$26$		0			0
$27$	23.0000	−	14.1421i	0.851852	−	0.523783i
$28$		0			0
$29$		−	18.9234i		−	0.652529i	−0.945278	−	0.326265i	$-0.894210\pi$
$29$		−	18.9234i		−	0.652529i	0.945278	−	0.326265i	$-0.105790\pi$
$30$		0			0
$31$	−12.7617			−0.411667			−0.205833	−	0.978587i	$-0.565990\pi$
$31$	−12.7617			−0.411667			−0.205833	+	0.978587i	$0.565990\pi$
$32$		0			0
$33$	−9.38083	−	3.31662i	−0.284268	−	0.100504i
$34$		0			0
$35$		−	88.7584i		−	2.53595i
$36$		0			0
$37$	11.2383			0.303739			0.151869	−	0.988401i	$-0.451471\pi$
$37$	11.2383			0.303739			0.151869	+	0.988401i	$0.451471\pi$
$38$		0			0
$39$	−21.3808	+	60.4741i	−0.548226	+	1.55062i
$40$		0			0
$41$			52.8645i			1.28938i	0.764445	+	0.644689i	$0.223013\pi$
$41$			52.8645i			1.28938i	−0.764445	+	0.644689i	$0.776987\pi$
$42$		0			0
$43$	−48.7617			−1.13399			−0.566996	−	0.823720i	$-0.691895\pi$
$43$	−48.7617			−1.13399			−0.566996	+	0.823720i	$0.691895\pi$
$44$		0			0
$45$	−53.5233	+	66.2317i	−1.18941	+	1.47182i
$46$		0			0
$47$			26.4322i			0.562388i	0.959651	+	0.281194i	$0.0907305\pi$
$47$			26.4322i			0.562388i	−0.959651	+	0.281194i	$0.909270\pi$
$48$		0			0
$49$	39.0000			0.795918
$50$		0			0
$51$	5.52333	+	1.95279i	0.108301	+	0.0382900i
$52$		0			0
$53$			32.2906i			0.609257i	0.952471	+	0.304628i	$0.0985323\pi$
$53$			32.2906i			0.609257i	−0.952471	+	0.304628i	$0.901468\pi$
$54$		0			0
$55$	31.3808			0.570561
$56$		0			0
$57$	−4.76166	+	13.4680i	−0.0835379	+	0.236281i
$58$		0			0
$59$			28.0828i			0.475979i	0.971268	+	0.237989i	$0.0764884\pi$
$59$			28.0828i			0.475979i	−0.971268	+	0.237989i	$0.923512\pi$
$60$		0			0
$61$	−2.61917			−0.0429372			−0.0214686	−	0.999770i	$-0.506834\pi$
$61$	−2.61917			−0.0429372			−0.0214686	+	0.999770i	$0.506834\pi$
$62$		0			0
$63$	−65.6658	−	53.0660i	−1.04231	−	0.842317i
$64$		0			0
$65$		−	202.299i		−	3.11228i
$66$		0			0
$67$	109.047			1.62756			0.813781	−	0.581172i	$-0.197406\pi$
$67$	109.047			1.62756			0.813781	+	0.581172i	$0.197406\pi$
$68$		0			0
$69$	32.2850	+	11.4145i	0.467898	+	0.165427i
$70$		0			0
$71$			53.1668i			0.748828i	0.927262	+	0.374414i	$0.122156\pi$
$71$			53.1668i			0.748828i	−0.927262	+	0.374414i	$0.877844\pi$
$72$		0			0
$73$	−119.808			−1.64121			−0.820605	−	0.571496i	$-0.806363\pi$
$73$	−119.808			−1.64121			−0.820605	+	0.571496i	$0.806363\pi$
$74$		0			0
$75$	64.5233	−	182.500i	0.860311	−	2.43333i
$76$		0			0
$77$			31.1127i			0.404061i
$78$		0			0
$79$	−38.6192			−0.488850			−0.244425	−	0.969668i	$-0.578599\pi$
$79$	−38.6192			−0.488850			−0.244425	+	0.969668i	$0.578599\pi$
$80$		0			0
$81$	17.0000	+	79.1960i	0.209877	+	0.977728i
$82$		0			0
$83$			116.841i			1.40772i	0.710336	+	0.703862i	$0.248543\pi$
$83$			116.841i			1.40772i	−0.710336	+	0.703862i	$0.751457\pi$
$84$		0			0
$85$	−18.4767			−0.217373
$86$		0			0
$87$	53.5233	+	18.9234i	0.615211	+	0.217510i
$88$		0			0
$89$			15.0178i			0.168739i	0.996435	+	0.0843695i	$0.0268876\pi$
$89$			15.0178i			0.168739i	−0.996435	+	0.0843695i	$0.973112\pi$
$90$		0			0
$91$	200.570			2.20407
$92$		0			0
$93$	12.7617	−	36.0954i	0.137222	−	0.388123i
$94$		0			0
$95$		−	45.0533i		−	0.474245i
$96$		0			0
$97$	−52.4767			−0.540997			−0.270498	−	0.962720i	$-0.587188\pi$
$97$	−52.4767			−0.540997			−0.270498	+	0.962720i	$0.587188\pi$
$98$		0			0
$99$	18.7617	−	23.2164i	0.189512	−	0.234509i
$100$		0			0
$101$		−	139.670i		−	1.38287i	−0.722438	−	0.691436i	$-0.756978\pi$
$101$		−	139.670i		−	1.38287i	0.722438	−	0.691436i	$-0.243022\pi$
$102$		0			0
$103$	−33.0467			−0.320841			−0.160421	−	0.987049i	$-0.551285\pi$
$103$	−33.0467			−0.320841			−0.160421	+	0.987049i	$0.551285\pi$
$104$		0			0
$105$	251.047	+	88.7584i	2.39092	+	0.845318i
$106$		0			0
$107$			119.398i			1.11587i	0.829883	+	0.557937i	$0.188407\pi$
$107$			119.398i			1.11587i	−0.829883	+	0.557937i	$0.811593\pi$
$108$		0			0
$109$	172.712			1.58452			0.792259	−	0.610185i	$-0.208905\pi$
$109$	172.712			1.58452			0.792259	+	0.610185i	$0.208905\pi$
$110$		0			0
$111$	−11.2383	+	31.7868i	−0.101246	+	0.286368i
$112$		0			0
$113$		−	124.652i		−	1.10312i	−0.834136	−	0.551559i	$-0.814033\pi$
$113$		−	124.652i		−	1.10312i	0.834136	−	0.551559i	$-0.185967\pi$
$114$		0			0
$115$	−108.000			−0.939130
$116$		0			0
$117$	−149.666	−	120.948i	−1.27920	−	1.03375i
$118$		0			0
$119$		−	18.3188i		−	0.153939i
$120$		0			0
$121$	−11.0000			−0.0909091
$122$		0			0
$123$	−149.523	−	52.8645i	−1.21564	−	0.429793i
$124$		0			0
$125$			373.957i			2.99166i
$126$		0			0
$127$	−112.427			−0.885256			−0.442628	−	0.896705i	$-0.645954\pi$
$127$	−112.427			−0.885256			−0.442628	+	0.896705i	$0.645954\pi$
$128$		0			0
$129$	48.7617	−	137.919i	0.377997	−	1.06914i
$130$		0			0
$131$			68.4868i			0.522800i	0.965231	+	0.261400i	$0.0841842\pi$
$131$			68.4868i			0.522800i	−0.965231	+	0.261400i	$0.915816\pi$
$132$		0			0
$133$	44.6684			0.335852
$134$		0			0
$135$	−133.808	−	217.619i	−0.991173	−	1.61199i
$136$		0			0
$137$			161.894i			1.18171i	0.806777	+	0.590856i	$0.201210\pi$
$137$			161.894i			1.18171i	−0.806777	+	0.590856i	$0.798790\pi$
$138$		0			0
$139$	169.047			1.21616			0.608081	−	0.793875i	$-0.291939\pi$
$139$	169.047			1.21616			0.608081	+	0.793875i	$0.291939\pi$
$140$		0			0
$141$	−74.7617	−	26.4322i	−0.530225	−	0.187463i
$142$		0			0
$143$			70.9122i			0.495889i
$144$		0			0
$145$	−179.047			−1.23480
$146$		0			0
$147$	−39.0000	+	110.309i	−0.265306	+	0.750399i
$148$		0			0
$149$		−	60.6756i		−	0.407219i	−0.979052	−	0.203610i	$-0.934733\pi$
$149$		−	60.6756i		−	0.407219i	0.979052	−	0.203610i	$-0.0652673\pi$
$150$		0			0
$151$	11.2875			0.0747518			0.0373759	−	0.999301i	$-0.488100\pi$
$151$	11.2875			0.0747518			0.0373759	+	0.999301i	$0.488100\pi$
$152$		0			0
$153$	−11.0467	+	13.6695i	−0.0722003	+	0.0893434i
$154$		0			0
$155$			120.747i			0.779011i
$156$		0			0
$157$	−23.8083			−0.151645			−0.0758227	−	0.997121i	$-0.524158\pi$
$157$	−23.8083			−0.151645			−0.0758227	+	0.997121i	$0.524158\pi$
$158$		0			0
$159$	−91.3316	−	32.2906i	−0.574413	−	0.203086i
$160$		0			0
$161$		−	107.077i		−	0.665076i
$162$		0			0
$163$	−69.0467			−0.423599			−0.211800	−	0.977313i	$-0.567932\pi$
$163$	−69.0467			−0.423599			−0.211800	+	0.977313i	$0.567932\pi$
$164$		0			0
$165$	−31.3808	+	88.7584i	−0.190187	+	0.537930i
$166$		0			0
$167$		−	56.1655i		−	0.336320i	−0.985760	−	0.168160i	$-0.946217\pi$
$167$		−	56.1655i		−	0.336320i	0.985760	−	0.168160i	$-0.0537826\pi$
$168$		0			0
$169$	288.140			1.70497
$170$		0			0
$171$	−33.3316	−	26.9360i	−0.194922	−	0.157521i
$172$		0			0
$173$		−	158.593i		−	0.916725i	−0.888765	−	0.458363i	$-0.848436\pi$
$173$		−	158.593i		−	0.916725i	0.888765	−	0.458363i	$-0.151564\pi$
$174$		0			0
$175$	−605.282			−3.45876
$176$		0			0
$177$	−79.4300	−	28.0828i	−0.448757	−	0.158660i
$178$		0			0
$179$			198.997i			1.11172i	0.831277	+	0.555859i	$0.187611\pi$
$179$			198.997i			1.11172i	−0.831277	+	0.555859i	$0.812389\pi$
$180$		0			0
$181$	−214.000			−1.18232			−0.591160	−	0.806554i	$-0.701330\pi$
$181$	−214.000			−1.18232			−0.591160	+	0.806554i	$0.701330\pi$
$182$		0			0
$183$	2.61917	−	7.40813i	0.0143124	−	0.0404816i
$184$		0			0
$185$		−	106.334i		−	0.574776i
$186$		0			0
$187$	6.47667			0.0346346
$188$		0			0
$189$	215.759	−	132.665i	1.14158	−	0.701931i
$190$		0			0
$191$			71.4856i			0.374270i	0.982334	+	0.187135i	$0.0599202\pi$
$191$			71.4856i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$192$		0			0
$193$	138.570			0.717979			0.358990	−	0.933342i	$-0.383121\pi$
$193$	138.570			0.717979			0.358990	+	0.933342i	$0.383121\pi$
$194$		0			0
$195$	572.187	+	202.299i	2.93429	+	1.03743i
$196$		0			0
$197$		−	101.823i		−	0.516870i	−0.966029	−	0.258435i	$-0.916793\pi$
$197$		−	101.823i		−	0.516870i	0.966029	−	0.258435i	$-0.0832068\pi$
$198$		0			0
$199$	77.7150			0.390528			0.195264	−	0.980751i	$-0.437444\pi$
$199$	77.7150			0.390528			0.195264	+	0.980751i	$0.437444\pi$
$200$		0			0
$201$	−109.047	+	308.431i	−0.542521	+	1.53448i
$202$		0			0
$203$		−	177.517i		−	0.874467i
$204$		0			0
$205$	500.187			2.43993
$206$		0			0
$207$	−64.5700	+	79.9013i	−0.311932	+	0.385996i
$208$		0			0
$209$			15.7926i			0.0755629i
$210$		0			0
$211$	−70.9533			−0.336272			−0.168136	−	0.985764i	$-0.553775\pi$
$211$	−70.9533			−0.336272			−0.168136	+	0.985764i	$0.553775\pi$
$212$		0			0
$213$	−150.378	−	53.1668i	−0.706001	−	0.249609i
$214$		0			0
$215$			461.367i			2.14589i
$216$		0			0
$217$	−119.715			−0.551682
$218$		0			0
$219$	119.808	−	338.869i	0.547070	−	1.54735i
$220$		0			0
$221$		−	41.7523i		−	0.188924i
$222$		0			0
$223$	−158.570			−0.711076			−0.355538	−	0.934662i	$-0.615702\pi$
$223$	−158.570			−0.711076			−0.355538	+	0.934662i	$0.615702\pi$
$224$		0			0
$225$	451.663	+	364.999i	2.00739	+	1.62222i
$226$		0			0
$227$		−	324.254i		−	1.42843i	−0.699925	−	0.714217i	$-0.746783\pi$
$227$		−	324.254i		−	1.42843i	0.699925	−	0.714217i	$-0.253217\pi$
$228$		0			0
$229$	51.8083			0.226237			0.113119	−	0.993581i	$-0.463916\pi$
$229$	51.8083			0.226237			0.113119	+	0.993581i	$0.463916\pi$
$230$		0			0
$231$	−88.0000	−	31.1127i	−0.380952	−	0.134687i
$232$		0			0
$233$			124.652i			0.534988i	0.963559	+	0.267494i	$0.0861956\pi$
$233$			124.652i			0.534988i	−0.963559	+	0.267494i	$0.913804\pi$
$234$		0			0
$235$	250.093			1.06423
$236$		0			0
$237$	38.6192	−	109.232i	0.162950	−	0.460892i
$238$		0			0
$239$		−	360.148i		−	1.50690i	−0.657507	−	0.753448i	$-0.728389\pi$
$239$		−	360.148i		−	1.50690i	0.657507	−	0.753448i	$-0.271611\pi$
$240$		0			0
$241$	−3.52333			−0.0146196			−0.00730981	−	0.999973i	$-0.502327\pi$
$241$	−3.52333			−0.0146196			−0.00730981	+	0.999973i	$0.502327\pi$
$242$		0			0
$243$	−241.000	−	31.1127i	−0.991770	−	0.128036i
$244$		0			0
$245$		−	369.005i		−	1.50614i
$246$		0			0
$247$	101.808			0.412179
$248$		0			0
$249$	−330.477	−	116.841i	−1.32722	−	0.469242i
$250$		0			0
$251$		−	341.969i		−	1.36242i	−0.732086	−	0.681212i	$-0.761453\pi$
$251$		−	341.969i		−	1.36242i	0.732086	−	0.681212i	$-0.238547\pi$
$252$		0			0
$253$	37.8575			0.149634
$254$		0			0
$255$	18.4767	−	52.2599i	0.0724575	−	0.204941i
$256$		0			0
$257$		−	375.166i		−	1.45979i	−0.683559	−	0.729895i	$-0.739569\pi$
$257$		−	375.166i		−	1.45979i	0.683559	−	0.729895i	$-0.260431\pi$
$258$		0			0
$259$	105.425			0.407046
$260$		0			0
$261$	−107.047	+	132.463i	−0.410140	+	0.507523i
$262$		0			0
$263$			256.511i			0.975328i	0.873031	+	0.487664i	$0.162151\pi$
$263$			256.511i			0.975328i	−0.873031	+	0.487664i	$0.837849\pi$
$264$		0			0
$265$	305.523			1.15292
$266$		0			0
$267$	−42.4767	−	15.0178i	−0.159089	−	0.0562463i
$268$		0			0
$269$			179.772i			0.668297i	0.942520	+	0.334148i	$0.108449\pi$
$269$			179.772i			0.668297i	−0.942520	+	0.334148i	$0.891551\pi$
$270$		0			0
$271$	0.240875			0.000888838			0.000444419	−	1.00000i	$-0.499859\pi$
$271$	0.240875			0.000888838			0.000444419		1.00000i	$0.499859\pi$
$272$		0			0
$273$	−200.570	+	567.298i	−0.734689	+	2.07801i
$274$		0			0
$275$		−	214.000i		−	0.778181i
$276$		0			0
$277$	375.858			1.35689			0.678443	−	0.734653i	$-0.262655\pi$
$277$	375.858			1.35689			0.678443	+	0.734653i	$0.262655\pi$
$278$		0			0
$279$	89.3316	+	72.1909i	0.320185	+	0.258749i
$280$		0			0
$281$		−	103.776i		−	0.369310i	−0.982803	−	0.184655i	$-0.940883\pi$
$281$		−	103.776i		−	0.369310i	0.982803	−	0.184655i	$-0.0591168\pi$
$282$		0			0
$283$	−469.617			−1.65942			−0.829711	−	0.558193i	$-0.811495\pi$
$283$	−469.617			−1.65942			−0.829711	+	0.558193i	$0.811495\pi$
$284$		0			0
$285$	127.430	+	45.0533i	0.447123	+	0.158082i
$286$		0			0
$287$			495.913i			1.72792i
$288$		0			0
$289$	285.187			0.986805
$290$		0			0
$291$	52.4767	−	148.426i	0.180332	−	0.510057i
$292$		0			0
$293$			416.314i			1.42087i	0.703765	+	0.710433i	$0.251501\pi$
$293$			416.314i			1.42087i	−0.703765	+	0.710433i	$0.748499\pi$
$294$		0			0
$295$	265.710			0.900712
$296$		0			0
$297$	46.9042	+	76.2824i	0.157926	+	0.256843i
$298$		0			0
$299$		−	244.051i		−	0.816223i
$300$		0			0
$301$	−457.425			−1.51968
$302$		0			0
$303$	395.047	+	139.670i	1.30378	+	0.460957i
$304$		0			0
$305$			24.7817i			0.0812516i
$306$		0			0
$307$	287.140			0.935309			0.467655	−	0.883911i	$-0.345099\pi$
$307$	287.140			0.935309			0.467655	+	0.883911i	$0.345099\pi$
$308$		0			0
$309$	33.0467	−	93.4700i	0.106947	−	0.302492i
$310$		0			0
$311$			380.861i			1.22463i	0.790612	+	0.612317i	$0.209762\pi$
$311$			380.861i			1.22463i	−0.790612	+	0.612317i	$0.790238\pi$
$312$		0			0
$313$	−341.233			−1.09020			−0.545101	−	0.838370i	$-0.683509\pi$
$313$	−341.233			−1.09020			−0.545101	+	0.838370i	$0.683509\pi$
$314$		0			0
$315$	−502.093	+	621.309i	−1.59395	+	1.97241i
$316$		0			0
$317$		−	513.186i		−	1.61888i	−0.587201	−	0.809441i	$-0.699770\pi$
$317$		−	513.186i		−	1.61888i	0.587201	−	0.809441i	$-0.300230\pi$
$318$		0			0
$319$	62.7617			0.196745
$320$		0			0
$321$	−337.710	−	119.398i	−1.05206	−	0.371958i
$322$		0			0
$323$		−	9.29853i		−	0.0287880i
$324$		0			0
$325$	−1379.56			−4.24481
$326$		0			0
$327$	−172.712	+	488.505i	−0.528173	+	1.49390i
$328$		0			0
$329$			247.956i			0.753667i
$330$		0			0
$331$	254.000			0.767372			0.383686	−	0.923464i	$-0.374655\pi$
$331$	254.000			0.767372			0.383686	+	0.923464i	$0.374655\pi$
$332$		0			0
$333$	−78.6684	−	63.5736i	−0.236241	−	0.190912i
$334$		0			0
$335$		−	1031.76i		−	3.07989i
$336$		0			0
$337$	−195.425			−0.579896			−0.289948	−	0.957042i	$-0.593638\pi$
$337$	−195.425			−0.579896			−0.289948	+	0.957042i	$0.593638\pi$
$338$		0			0
$339$	352.570	+	124.652i	1.04003	+	0.367706i
$340$		0			0
$341$		−	42.3256i		−	0.124122i
$342$		0			0
$343$	−93.8083			−0.273494
$344$		0			0
$345$	108.000	−	305.470i	0.313043	−	0.885421i
$346$		0			0
$347$		−	488.706i		−	1.40838i	−0.710014	−	0.704188i	$-0.751311\pi$
$347$		−	488.706i		−	1.40838i	0.710014	−	0.704188i	$-0.248689\pi$
$348$		0			0
$349$	−619.189			−1.77418			−0.887090	−	0.461596i	$-0.847277\pi$
$349$	−619.189			−1.77418			−0.887090	+	0.461596i	$0.847277\pi$
$350$		0			0
$351$	491.759	−	302.371i	1.40102	−	0.861455i
$352$		0			0
$353$		−	419.010i		−	1.18700i	−0.804835	−	0.593499i	$-0.797746\pi$
$353$		−	419.010i		−	1.18700i	0.804835	−	0.593499i	$-0.202254\pi$
$354$		0			0
$355$	503.047			1.41703
$356$		0			0
$357$	51.8134	+	18.3188i	0.145136	+	0.0513132i
$358$		0			0
$359$			366.146i			1.01990i	0.860203	+	0.509952i	$0.170337\pi$
$359$			366.146i			1.01990i	−0.860203	+	0.509952i	$0.829663\pi$
$360$		0			0
$361$	−338.327			−0.937193
$362$		0			0
$363$	11.0000	−	31.1127i	0.0303030	−	0.0857099i
$364$		0			0
$365$			1133.59i			3.10572i
$366$		0			0
$367$	−376.280			−1.02529			−0.512643	−	0.858602i	$-0.671334\pi$
$367$	−376.280			−1.02529			−0.512643	+	0.858602i	$0.671334\pi$
$368$		0			0
$369$	299.047	−	370.051i	0.810425	−	1.00285i
$370$		0			0
$371$			302.913i			0.816477i
$372$		0			0
$373$	−65.4792			−0.175547			−0.0877737	−	0.996140i	$-0.527975\pi$
$373$	−65.4792			−0.175547			−0.0877737	+	0.996140i	$0.527975\pi$
$374$		0			0
$375$	−1057.71	−	373.957i	−2.82056	−	0.997218i
$376$		0			0
$377$		−	404.597i		−	1.07320i
$378$		0			0
$379$	285.233			0.752594			0.376297	−	0.926499i	$-0.377197\pi$
$379$	285.233			0.752594			0.376297	+	0.926499i	$0.377197\pi$
$380$		0			0
$381$	112.427	−	317.993i	0.295085	−	0.834627i
$382$		0			0
$383$			657.505i			1.71672i	0.513045	+	0.858362i	$0.328517\pi$
$383$			657.505i			1.71672i	−0.513045	+	0.858362i	$0.671483\pi$
$384$		0			0
$385$	294.378			0.764619
$386$		0			0
$387$	341.332	+	275.838i	0.881994	+	0.712759i
$388$		0			0
$389$			317.350i			0.815810i	0.913024	+	0.407905i	$0.133741\pi$
$389$			317.350i			0.815810i	−0.913024	+	0.407905i	$0.866259\pi$
$390$		0			0
$391$	−22.2901			−0.0570078
$392$		0			0
$393$	−193.710	−	68.4868i	−0.492901	−	0.174267i
$394$		0			0
$395$			365.402i			0.925069i
$396$		0			0
$397$	245.617			0.618682			0.309341	−	0.950951i	$-0.399892\pi$
$397$	245.617			0.618682			0.309341	+	0.950951i	$0.399892\pi$
$398$		0			0
$399$	−44.6684	+	126.341i	−0.111951	+	0.316645i
$400$		0			0
$401$			248.700i			0.620200i	0.950704	+	0.310100i	$0.100362\pi$
$401$			248.700i			0.620200i	−0.950704	+	0.310100i	$0.899638\pi$
$402$		0			0
$403$	−272.855			−0.677059
$404$		0			0
$405$	749.327	−	160.849i	1.85019	−	0.397157i
$406$		0			0
$407$			37.2733i			0.0915807i
$408$		0			0
$409$	358.285			0.876002			0.438001	−	0.898974i	$-0.355687\pi$
$409$	358.285			0.876002			0.438001	+	0.898974i	$0.355687\pi$
$410$		0			0
$411$	−457.907	−	161.894i	−1.11413	−	0.393904i
$412$		0			0
$413$			263.440i			0.637868i
$414$		0			0
$415$	1105.51			2.66389
$416$		0			0
$417$	−169.047	+	478.136i	−0.405388	+	1.14661i
$418$		0			0
$419$		−	630.329i		−	1.50437i	−0.658955	−	0.752183i	$-0.729001\pi$
$419$		−	630.329i		−	1.50437i	0.658955	−	0.752183i	$-0.270999\pi$
$420$		0			0
$421$	7.62171			0.0181038			0.00905191	−	0.999959i	$-0.497119\pi$
$421$	7.62171			0.0181038			0.00905191	+	0.999959i	$0.497119\pi$
$422$		0			0
$423$	149.523	−	185.026i	0.353483	−	0.437413i
$424$		0			0
$425$			126.001i			0.296472i
$426$		0			0
$427$	−24.5700			−0.0575409
$428$		0			0
$429$	−200.570	−	70.9122i	−0.467529	−	0.165296i
$430$		0			0
$431$		−	411.804i		−	0.955461i	−0.878506	−	0.477730i	$-0.841460\pi$
$431$		−	411.804i		−	0.955461i	0.878506	−	0.477730i	$-0.158540\pi$
$432$		0			0
$433$	655.327			1.51346			0.756728	−	0.653730i	$-0.226797\pi$
$433$	655.327			1.51346			0.756728	+	0.653730i	$0.226797\pi$
$434$		0			0
$435$	179.047	−	506.420i	0.411601	−	1.16418i
$436$		0			0
$437$		−	54.3518i		−	0.124375i
$438$		0			0
$439$	169.951			0.387132			0.193566	−	0.981087i	$-0.437995\pi$
$439$	169.951			0.387132			0.193566	+	0.981087i	$0.437995\pi$
$440$		0			0
$441$	−273.000	−	220.617i	−0.619048	−	0.500266i
$442$		0			0
$443$			391.532i			0.883820i	0.897060	+	0.441910i	$0.145699\pi$
$443$			391.532i			0.883820i	−0.897060	+	0.441910i	$0.854301\pi$
$444$		0			0
$445$	142.093			0.319311
$446$		0			0
$447$	171.617	+	60.6756i	0.383930	+	0.135740i
$448$		0			0
$449$			680.032i			1.51455i	0.653098	+	0.757274i	$0.273469\pi$
$449$			680.032i			1.51455i	−0.653098	+	0.757274i	$0.726531\pi$
$450$		0			0
$451$	−175.332			−0.388762
$452$		0			0
$453$	−11.2875	+	31.9259i	−0.0249173	+	0.0704767i
$454$		0			0
$455$		−	1897.73i		−	4.17083i
$456$		0			0
$457$	−251.052			−0.549347			−0.274674	−	0.961538i	$-0.588570\pi$
$457$	−251.052			−0.549347			−0.274674	+	0.961538i	$0.588570\pi$
$458$		0			0
$459$	−27.6166	−	44.9142i	−0.0601670	−	0.0978523i
$460$		0			0
$461$			21.0153i			0.0455863i	0.999740	+	0.0227931i	$0.00725591\pi$
$461$			21.0153i			0.0455863i	−0.999740	+	0.0227931i	$0.992744\pi$
$462$		0			0
$463$	672.093			1.45161			0.725803	−	0.687903i	$-0.241469\pi$
$463$	672.093			1.45161			0.725803	+	0.687903i	$0.241469\pi$
$464$		0			0
$465$	−341.523	−	120.747i	−0.734459	−	0.259670i
$466$		0			0
$467$			339.877i			0.727787i	0.931441	+	0.363894i	$0.118553\pi$
$467$			339.877i			0.727787i	−0.931441	+	0.363894i	$0.881447\pi$
$468$		0			0
$469$	1022.95			2.18113
$470$		0			0
$471$	23.8083	−	67.3401i	0.0505484	−	0.142973i
$472$		0			0
$473$		−	161.724i		−	0.341912i
$474$		0			0
$475$	−307.238			−0.646818
$476$		0			0
$477$	182.663	−	226.034i	0.382942	−	0.473866i
$478$		0			0
$479$		−	414.500i		−	0.865345i	−0.901551	−	0.432672i	$-0.857571\pi$
$479$		−	414.500i		−	0.865345i	0.901551	−	0.432672i	$-0.142429\pi$
$480$		0			0
$481$	240.285			0.499553
$482$		0			0
$483$	302.860	+	107.077i	0.627039	+	0.221692i
$484$		0			0
$485$			496.517i			1.02375i
$486$		0			0
$487$	−453.902			−0.932036			−0.466018	−	0.884775i	$-0.654312\pi$
$487$	−453.902			−0.932036			−0.466018	+	0.884775i	$0.654312\pi$
$488$		0			0
$489$	69.0467	−	195.293i	0.141200	−	0.399373i
$490$		0			0
$491$			447.093i			0.910576i	0.890344	+	0.455288i	$0.150464\pi$
$491$			447.093i			0.910576i	−0.890344	+	0.455288i	$0.849536\pi$
$492$		0			0
$493$	−36.9533			−0.0749561
$494$		0			0
$495$	−219.666	−	177.517i	−0.443769	−	0.358620i
$496$		0			0
$497$			498.748i			1.00352i
$498$		0			0
$499$	−333.047			−0.667428			−0.333714	−	0.942674i	$-0.608302\pi$
$499$	−333.047			−0.667428			−0.333714	+	0.942674i	$0.608302\pi$
$500$		0			0
$501$	158.860	+	56.1655i	0.317086	+	0.112107i
$502$		0			0
$503$			207.552i			0.412629i	0.978486	+	0.206314i	$0.0661470\pi$
$503$			207.552i			0.412629i	−0.978486	+	0.206314i	$0.933853\pi$
$504$		0			0
$505$	−1321.51			−2.61686
$506$		0			0
$507$	−288.140	+	814.983i	−0.568323	+	1.60746i
$508$		0			0
$509$			204.414i			0.401600i	0.979632	+	0.200800i	$0.0643542\pi$
$509$			204.414i			0.401600i	−0.979632	+	0.200800i	$0.935646\pi$
$510$		0			0
$511$	−1123.90			−2.19942
$512$		0			0
$513$	109.518	−	67.3401i	0.213486	−	0.131267i
$514$		0			0
$515$			312.677i			0.607139i
$516$		0			0
$517$	−87.6658			−0.169566
$518$		0			0
$519$	448.570	+	158.593i	0.864297	+	0.305575i
$520$		0			0
$521$			522.647i			1.00316i	0.865111	+	0.501581i	$0.167248\pi$
$521$			522.647i			1.00316i	−0.865111	+	0.501581i	$0.832752\pi$
$522$		0			0
$523$	−209.135			−0.399875			−0.199938	−	0.979809i	$-0.564074\pi$
$523$	−209.135			−0.399875			−0.199938	+	0.979809i	$0.564074\pi$
$524$		0			0
$525$	605.282	−	1712.00i	1.15292	−	3.26095i
$526$		0			0
$527$			24.9209i			0.0472882i
$528$		0			0
$529$	398.710			0.753705
$530$		0			0
$531$	158.860	−	196.579i	0.299171	−	0.370206i
$532$		0			0
$533$			1130.29i			2.12061i
$534$		0			0
$535$	1129.71			2.11161
$536$		0			0
$537$	−562.850	−	198.997i	−1.04814	−	0.370573i
$538$		0			0
$539$			129.348i			0.239978i
$540$		0			0
$541$	1025.47			1.89551			0.947753	−	0.319004i	$-0.103348\pi$
$541$	1025.47			1.89551			0.947753	+	0.319004i	$0.103348\pi$
$542$		0			0
$543$	214.000	−	605.283i	0.394107	−	1.11470i
$544$		0			0
$545$		−	1634.15i		−	2.99844i
$546$		0			0
$547$	−641.332			−1.17245			−0.586226	−	0.810147i	$-0.699387\pi$
$547$	−641.332			−1.17245			−0.586226	+	0.810147i	$0.699387\pi$
$548$		0			0
$549$	18.3342	+	14.8163i	0.0333956	+	0.0269877i
$550$		0			0
$551$		−	90.1066i		−	0.163533i
$552$		0			0
$553$	−362.280			−0.655117
$554$		0			0
$555$	300.757	+	106.334i	0.541904	+	0.191592i
$556$		0			0
$557$		−	695.654i		−	1.24893i	−0.781053	−	0.624465i	$-0.785317\pi$
$557$		−	695.654i		−	1.24893i	0.781053	−	0.624465i	$-0.214683\pi$
$558$		0			0
$559$	−1042.56			−1.86505
$560$		0			0
$561$	−6.47667	+	18.3188i	−0.0115449	+	0.0326538i
$562$		0			0
$563$			674.173i			1.19747i	0.800949	+	0.598733i	$0.204329\pi$
$563$			674.173i			1.19747i	−0.800949	+	0.598733i	$0.795671\pi$
$564$		0			0
$565$	−1179.42			−2.08747
$566$		0			0
$567$	159.474	+	742.924i	0.281259	+	1.31027i
$568$		0			0
$569$		−	507.164i		−	0.891325i	−0.895201	−	0.445663i	$-0.852968\pi$
$569$		−	507.164i		−	0.891325i	0.895201	−	0.445663i	$-0.147032\pi$
$570$		0			0
$571$	−43.1400			−0.0755516			−0.0377758	−	0.999286i	$-0.512027\pi$
$571$	−43.1400			−0.0755516			−0.0377758	+	0.999286i	$0.512027\pi$
$572$		0			0
$573$	−202.192	−	71.4856i	−0.352865	−	0.124757i
$574$		0			0
$575$			736.499i			1.28087i
$576$		0			0
$577$	−83.9067			−0.145419			−0.0727094	−	0.997353i	$-0.523165\pi$
$577$	−83.9067			−0.145419			−0.0727094	+	0.997353i	$0.523165\pi$
$578$		0			0
$579$	−138.570	+	391.935i	−0.239326	+	0.676917i
$580$		0			0
$581$			1096.07i			1.88652i
$582$		0			0
$583$	−107.096			−0.183698
$584$		0			0
$585$	−1144.37	+	1416.09i	−1.95619	+	2.42067i
$586$		0			0
$587$		−	350.384i		−	0.596907i	−0.954424	−	0.298453i	$-0.903529\pi$
$587$		−	350.384i		−	0.596907i	0.954424	−	0.298453i	$-0.0964708\pi$
$588$		0			0
$589$	−60.7667			−0.103169
$590$		0			0
$591$	288.000	+	101.823i	0.487310	+	0.172290i
$592$		0			0
$593$		−	555.240i		−	0.936324i	−0.883643	−	0.468162i	$-0.844916\pi$
$593$		−	555.240i		−	0.936324i	0.883643	−	0.468162i	$-0.155084\pi$
$594$		0			0
$595$	−173.327			−0.291305
$596$		0			0
$597$	−77.7150	+	219.811i	−0.130176	+	0.368193i
$598$		0			0
$599$		−	282.943i		−	0.472360i	−0.971709	−	0.236180i	$-0.924105\pi$
$599$		−	282.943i		−	0.472360i	0.971709	−	0.236180i	$-0.0758955\pi$
$600$		0			0
$601$	255.052			0.424379			0.212189	−	0.977229i	$-0.431941\pi$
$601$	255.052			0.424379			0.212189	+	0.977229i	$0.431941\pi$
$602$		0			0
$603$	−763.327	−	616.861i	−1.26588	−	1.02299i
$604$		0			0
$605$			104.078i			0.172030i
$606$		0			0
$607$	−328.427			−0.541067			−0.270533	−	0.962711i	$-0.587200\pi$
$607$	−328.427			−0.541067			−0.270533	+	0.962711i	$0.587200\pi$
$608$		0			0
$609$	502.093	+	177.517i	0.824455	+	0.291489i
$610$		0			0
$611$			565.143i			0.924948i
$612$		0			0
$613$	119.288			0.194596			0.0972981	−	0.995255i	$-0.468980\pi$
$613$	119.288			0.194596			0.0972981	+	0.995255i	$0.468980\pi$
$614$		0			0
$615$	−500.187	+	1414.74i	−0.813312	+	2.30039i
$616$		0			0
$617$		−	1047.39i		−	1.69755i	−0.528757	−	0.848773i	$-0.677342\pi$
$617$		−	1047.39i		−	1.69755i	0.528757	−	0.848773i	$-0.322658\pi$
$618$		0			0
$619$	125.813			0.203253			0.101626	−	0.994823i	$-0.467595\pi$
$619$	125.813			0.203253			0.101626	+	0.994823i	$0.467595\pi$
$620$		0			0
$621$	−161.425	−	262.533i	−0.259944	−	0.422758i
$622$		0			0
$623$			140.879i			0.226130i
$624$		0			0
$625$	1925.18			3.08028
$626$		0			0
$627$	−44.6684	−	15.7926i	−0.0712414	−	0.0251876i
$628$		0			0
$629$		−	21.9461i		−	0.0348905i
$630$		0			0
$631$	524.182			0.830716			0.415358	−	0.909658i	$-0.363656\pi$
$631$	524.182			0.830716			0.415358	+	0.909658i	$0.363656\pi$
$632$		0			0
$633$	70.9533	−	200.686i	0.112091	−	0.317040i
$634$		0			0
$635$			1063.75i			1.67520i
$636$		0			0
$637$	833.852			1.30903
$638$		0			0
$639$	300.757	−	372.167i	0.470668	−	0.582421i
$640$		0			0
$641$		−	658.412i		−	1.02716i	−0.858041	−	0.513582i	$-0.828318\pi$
$641$		−	658.412i		−	1.02716i	0.858041	−	0.513582i	$-0.171682\pi$
$642$		0			0
$643$	−1081.42			−1.68183			−0.840917	−	0.541163i	$-0.817984\pi$
$643$	−1081.42			−1.68183			−0.840917	+	0.541163i	$0.817984\pi$
$644$		0			0
$645$	−1304.94	−	461.367i	−2.02317	−	0.715298i
$646$		0			0
$647$		−	141.786i		−	0.219144i	−0.993979	−	0.109572i	$-0.965052\pi$
$647$		−	141.786i		−	0.219144i	0.993979	−	0.109572i	$-0.0349480\pi$
$648$		0			0
$649$	−93.1400			−0.143513
$650$		0			0
$651$	119.715	−	338.605i	0.183894	−	0.520131i
$652$		0			0
$653$		−	697.304i		−	1.06785i	−0.845533	−	0.533924i	$-0.820717\pi$
$653$		−	697.304i		−	1.06785i	0.845533	−	0.533924i	$-0.179283\pi$
$654$		0			0
$655$	648.000			0.989313
$656$		0			0
$657$	838.658	+	677.738i	1.27650	+	1.03156i
$658$		0			0
$659$		−	986.245i		−	1.49658i	−0.663372	−	0.748289i	$-0.730875\pi$
$659$		−	986.245i		−	1.49658i	0.663372	−	0.748289i	$-0.269125\pi$
$660$		0			0
$661$	−1114.66			−1.68632			−0.843161	−	0.537662i	$-0.819308\pi$
$661$	−1114.66			−1.68632			−0.843161	+	0.537662i	$0.819308\pi$
$662$		0			0
$663$	118.093	+	41.7523i	0.178120	+	0.0629748i
$664$		0			0
$665$		−	422.638i		−	0.635545i
$666$		0			0
$667$	−216.000			−0.323838
$668$		0			0
$669$	158.570	−	448.504i	0.237025	−	0.670409i
$670$		0			0
$671$		−	8.68680i		−	0.0129460i
$672$		0			0
$673$	−418.855			−0.622370			−0.311185	−	0.950349i	$-0.600726\pi$
$673$	−418.855			−0.622370			−0.311185	+	0.950349i	$0.600726\pi$
$674$		0			0
$675$	−1484.04	+	912.498i	−2.19857	+	1.35185i
$676$		0			0
$677$			51.0508i			0.0754074i	0.999289	+	0.0377037i	$0.0120043\pi$
$677$			51.0508i			0.0754074i	−0.999289	+	0.0377037i	$0.987996\pi$
$678$		0			0
$679$	−492.275			−0.725000
$680$		0			0
$681$	917.130	+	324.254i	1.34674	+	0.476144i
$682$		0			0
$683$		−	1177.29i		−	1.72371i	−0.507156	−	0.861854i	$-0.669303\pi$
$683$		−	1177.29i		−	1.72371i	0.507156	−	0.861854i	$-0.330697\pi$
$684$		0			0
$685$	1531.79			2.23619
$686$		0			0
$687$	−51.8083	+	146.536i	−0.0754124	+	0.213298i
$688$		0			0
$689$			690.400i			1.00203i
$690$		0			0
$691$	177.233			0.256488			0.128244	−	0.991743i	$-0.459066\pi$
$691$	177.233			0.256488			0.128244	+	0.991743i	$0.459066\pi$
$692$		0			0
$693$	176.000	−	217.789i	0.253968	−	0.314270i
$694$		0			0
$695$		−	1599.46i		−	2.30139i
$696$		0			0
$697$	103.233			0.148111
$698$		0			0
$699$	−352.570	−	124.652i	−0.504392	−	0.178329i
$700$		0			0
$701$		−	373.352i		−	0.532600i	−0.963890	−	0.266300i	$-0.914199\pi$
$701$		−	373.352i		−	0.532600i	0.963890	−	0.266300i	$-0.0858011\pi$
$702$		0			0
$703$	53.5132			0.0761212
$704$		0			0
$705$	−250.093	+	707.371i	−0.354742	+	1.00336i
$706$		0			0
$707$		−	1310.22i		−	1.85321i
$708$		0			0
$709$	701.518			0.989447			0.494724	−	0.869050i	$-0.335269\pi$
$709$	701.518			0.989447			0.494724	+	0.869050i	$0.335269\pi$
$710$		0			0
$711$	270.334	+	218.463i	0.380217	+	0.307262i
$712$		0			0
$713$			145.668i			0.204302i
$714$		0			0
$715$	670.948			0.938389
$716$		0			0
$717$	1018.65	+	360.148i	1.42072	+	0.502299i
$718$		0			0
$719$			1411.74i			1.96348i	0.190226	+	0.981740i	$0.439078\pi$
$719$			1411.74i			1.96348i	−0.190226	+	0.981740i	$0.560922\pi$
$720$		0			0
$721$	−310.005			−0.429965
$722$		0			0
$723$	3.52333	−	9.96547i	0.00487320	−	0.0137835i
$724$		0			0
$725$			1221.00i			1.68413i
$726$		0			0
$727$	1061.52			1.46014			0.730068	−	0.683375i	$-0.239489\pi$
$727$	1061.52			1.46014			0.730068	+	0.683375i	$0.239489\pi$
$728$		0			0
$729$	329.000	−	650.538i	0.451303	−	0.892371i
$730$		0			0
$731$			95.2213i			0.130262i
$732$		0			0
$733$	−853.764			−1.16475			−0.582377	−	0.812919i	$-0.697877\pi$
$733$	−853.764			−1.16475			−0.582377	+	0.812919i	$0.697877\pi$
$734$		0			0
$735$	1043.70	+	369.005i	1.42001	+	0.502048i
$736$		0			0
$737$			361.667i			0.490728i
$738$		0			0
$739$	−1198.56			−1.62187			−0.810934	−	0.585138i	$-0.801040\pi$
$739$	−1198.56			−1.62187			−0.810934	+	0.585138i	$0.801040\pi$
$740$		0			0
$741$	−101.808	+	287.957i	−0.137393	+	0.388606i
$742$		0			0
$743$		−	1368.20i		−	1.84145i	−0.390207	−	0.920727i	$-0.627596\pi$
$743$		−	1368.20i		−	1.84145i	0.390207	−	0.920727i	$-0.372404\pi$
$744$		0			0
$745$	−574.093			−0.770595
$746$		0			0
$747$	660.953	−	817.888i	0.884810	−	1.09490i
$748$		0			0
$749$			1120.06i			1.49540i
$750$		0			0
$751$	544.368			0.724858			0.362429	−	0.932011i	$-0.381948\pi$
$751$	544.368			0.724858			0.362429	+	0.932011i	$0.381948\pi$
$752$		0			0
$753$	967.233	+	341.969i	1.28451	+	0.454142i
$754$		0			0
$755$		−	106.799i		−	0.141456i
$756$		0			0
$757$	190.482			0.251627			0.125814	−	0.992054i	$-0.459846\pi$
$757$	190.482			0.251627			0.125814	+	0.992054i	$0.459846\pi$
$758$		0			0
$759$	−37.8575	+	107.077i	−0.0498781	+	0.141077i
$760$		0			0
$761$			318.861i			0.419003i	0.977808	+	0.209502i	$0.0671841\pi$
$761$			318.861i			0.419003i	−0.977808	+	0.209502i	$0.932816\pi$
$762$		0			0
$763$	1620.19			2.12344
$764$		0			0
$765$	129.337	+	104.520i	0.169068	+	0.136627i
$766$		0			0
$767$			600.433i			0.782833i
$768$		0			0
$769$	369.233			0.480147			0.240074	−	0.970755i	$-0.422828\pi$
$769$	369.233			0.480147			0.240074	+	0.970755i	$0.422828\pi$
$770$		0			0
$771$	1061.13	+	375.166i	1.37630	+	0.486597i
$772$		0			0
$773$		−	217.014i		−	0.280743i	−0.990099	−	0.140371i	$-0.955170\pi$
$773$		−	217.014i		−	0.280743i	0.990099	−	0.140371i	$-0.0448296\pi$
$774$		0			0
$775$	823.425			1.06248
$776$		0			0
$777$	−105.425	+	298.187i	−0.135682	+	0.383767i
$778$		0			0
$779$			251.723i			0.323136i
$780$		0			0
$781$	−176.334			−0.225780
$782$		0			0
$783$	−267.617	−	435.237i	−0.341784	−	0.555858i
$784$		0			0
$785$			225.267i			0.286964i
$786$		0			0
$787$	−840.565			−1.06806			−0.534031	−	0.845465i	$-0.679323\pi$
$787$	−840.565			−1.06806			−0.534031	+	0.845465i	$0.679323\pi$
$788$		0			0
$789$	−725.523	−	256.511i	−0.919548	−	0.325109i
$790$		0			0
$791$		−	1169.34i		−	1.47831i
$792$		0			0
$793$	−56.0000			−0.0706179
$794$		0			0
$795$	−305.523	+	864.150i	−0.384306	+	1.08698i
$796$		0			0
$797$		−	625.517i		−	0.784839i	−0.919786	−	0.392419i	$-0.871638\pi$
$797$		−	625.517i		−	0.784839i	0.919786	−	0.392419i	$-0.128362\pi$
$798$		0			0
$799$	51.6166			0.0646015
$800$		0			0
$801$	84.9533	−	105.124i	0.106059	−	0.131241i
$802$		0			0
$803$		−	397.359i		−	0.494843i
$804$		0			0
$805$	−1013.13			−1.25855
$806$		0			0
$807$	−508.472	−	179.772i	−0.630076	−	0.222766i
$808$		0			0
$809$			435.237i			0.537994i	0.963141	+	0.268997i	$0.0866922\pi$
$809$			435.237i			0.537994i	−0.963141	+	0.268997i	$0.913308\pi$
$810$		0			0
$811$	−1052.95			−1.29833			−0.649167	−	0.760646i	$-0.724882\pi$
$811$	−1052.95			−1.29833			−0.649167	+	0.760646i	$0.724882\pi$
$812$		0			0
$813$	−0.240875	+	0.681298i	−0.000296279	+	0.000838004i
$814$		0			0
$815$			653.297i			0.801592i
$816$		0			0
$817$	−232.187			−0.284194
$818$		0			0
$819$	−1403.99	−	1134.60i	−1.71427	−	1.38534i
$820$		0			0
$821$		−	968.670i		−	1.17987i	−0.807452	−	0.589933i	$-0.799154\pi$
$821$		−	968.670i		−	1.17987i	0.807452	−	0.589933i	$-0.200846\pi$
$822$		0			0
$823$	−1164.66			−1.41514			−0.707572	−	0.706641i	$-0.750209\pi$
$823$	−1164.66			−1.41514			−0.707572	+	0.706641i	$0.750209\pi$
$824$		0			0
$825$	605.282	+	214.000i	0.733676	+	0.259394i
$826$		0			0
$827$		−	1199.05i		−	1.44988i	−0.688811	−	0.724940i	$-0.741867\pi$
$827$		−	1199.05i		−	1.44988i	0.688811	−	0.724940i	$-0.258133\pi$
$828$		0			0
$829$	−894.943			−1.07955			−0.539773	−	0.841811i	$-0.681490\pi$
$829$	−894.943			−1.07955			−0.539773	+	0.841811i	$0.681490\pi$
$830$		0			0
$831$	−375.858	+	1063.09i	−0.452295	+	1.27928i
$832$		0			0
$833$		−	76.1588i		−	0.0914272i
$834$		0			0
$835$	−531.420			−0.636431
$836$		0			0
$837$	−293.518	+	180.477i	−0.350679	+	0.215624i
$838$		0			0
$839$			1032.07i			1.23011i	0.788482	+	0.615057i	$0.210867\pi$
$839$			1032.07i			1.23011i	−0.788482	+	0.615057i	$0.789133\pi$
$840$		0			0
$841$	482.907			0.574205
$842$		0			0
$843$	293.523	+	103.776i	0.348189	+	0.123103i
$844$		0			0
$845$		−	2726.29i		−	3.22638i
$846$		0			0
$847$	−103.189			−0.121829
$848$		0			0
$849$	469.617	−	1328.28i	0.553141	−	1.56452i
$850$		0			0
$851$		−	128.280i		−	0.150740i
$852$		0			0
$853$	1193.47			1.39914			0.699572	−	0.714563i	$-0.253374\pi$
$853$	1193.47			1.39914			0.699572	+	0.714563i	$0.253374\pi$
$854$		0			0
$855$	−254.860	+	315.373i	−0.298082	+	0.368858i
$856$		0			0
$857$		−	727.781i		−	0.849220i	−0.905376	−	0.424610i	$-0.860411\pi$
$857$		−	727.781i		−	0.849220i	0.905376	−	0.424610i	$-0.139589\pi$
$858$		0			0
$859$	−582.187			−0.677749			−0.338875	−	0.940832i	$-0.610046\pi$
$859$	−582.187			−0.677749			−0.338875	+	0.940832i	$0.610046\pi$
$860$		0			0
$861$	−1402.65	−	495.913i	−1.62910	−	0.575973i
$862$		0			0
$863$		−	112.029i		−	0.129813i	−0.997891	−	0.0649066i	$-0.979325\pi$
$863$		−	112.029i		−	0.129813i	0.997891	−	0.0649066i	$-0.0206749\pi$
$864$		0			0
$865$	−1500.56			−1.73475
$866$		0			0
$867$	−285.187	+	806.630i	−0.328935	+	0.930369i
$868$		0			0
$869$		−	128.085i		−	0.147394i
$870$		0			0
$871$	2331.51			2.67682
$872$		0			0
$873$	367.337	+	296.853i	0.420775	+	0.340038i
$874$		0			0
$875$			3508.03i			4.00917i
$876$		0			0
$877$	353.666			0.403268			0.201634	−	0.979461i	$-0.435375\pi$
$877$	353.666			0.403268			0.201634	+	0.979461i	$0.435375\pi$
$878$		0			0
$879$	−1177.51	−	416.314i	−1.33961	−	0.473622i
$880$		0			0
$881$			638.279i			0.724494i	0.932082	+	0.362247i	$0.117990\pi$
$881$			638.279i			0.724494i	−0.932082	+	0.362247i	$0.882010\pi$
$882$		0			0
$883$	330.570			0.374371			0.187186	−	0.982325i	$-0.440063\pi$
$883$	330.570			0.374371			0.187186	+	0.982325i	$0.440063\pi$
$884$		0			0
$885$	−265.710	+	751.541i	−0.300237	+	0.849199i
$886$		0			0
$887$			215.968i			0.243481i	0.992562	+	0.121741i	$0.0388476\pi$
$887$			215.968i			0.243481i	−0.992562	+	0.121741i	$0.961152\pi$
$888$		0			0
$889$	−1054.66			−1.18635
$890$		0			0
$891$	−262.663	+	56.3826i	−0.294796	+	0.0632802i
$892$		0			0
$893$			125.861i			0.140942i
$894$		0			0
$895$	1882.85			2.10374
$896$		0			0
$897$	690.280	+	244.051i	0.769543	+	0.272074i
$898$		0			0
$899$			241.493i			0.268625i
$900$		0			0
$901$	63.0568			0.0699854
$902$		0			0
$903$	457.425	−	1293.79i	0.506561	−	1.43277i
$904$		0			0
$905$			2024.80i			2.23735i
$906$		0			0
$907$	243.907			0.268916			0.134458	−	0.990919i	$-0.457071\pi$
$907$	243.907			0.268916			0.134458	+	0.990919i	$0.457071\pi$
$908$		0			0
$909$	−790.093	+	977.691i	−0.869190	+	1.07557i
$910$		0			0
$911$		−	932.940i		−	1.02408i	−0.858961	−	0.512041i	$-0.828889\pi$
$911$		−	932.940i		−	1.02408i	0.858961	−	0.512041i	$-0.171111\pi$
$912$		0			0
$913$	−387.518			−0.424445
$914$		0			0
$915$	−70.0933	−	24.7817i	−0.0766047	−	0.0270839i
$916$		0			0
$917$			642.463i			0.700614i
$918$		0			0
$919$	−379.847			−0.413327			−0.206663	−	0.978412i	$-0.566261\pi$
$919$	−379.847			−0.413327			−0.206663	+	0.978412i	$0.566261\pi$
$920$		0			0
$921$	−287.140	+	812.154i	−0.311770	+	0.881818i
$922$		0			0
$923$			1136.75i			1.23158i
$924$		0			0
$925$	−725.135			−0.783930
$926$		0			0
$927$	231.327	+	186.940i	0.249543	+	0.201661i
$928$		0			0
$929$			842.857i			0.907273i	0.891187	+	0.453637i	$0.149874\pi$
$929$			842.857i			0.907273i	−0.891187	+	0.453637i	$0.850126\pi$
$930$		0			0
$931$	185.705			0.199468
$932$		0			0
$933$	−1077.24	−	380.861i	−1.15460	−	0.408211i
$934$		0			0
$935$		−	61.2802i		−	0.0655403i
$936$		0			0
$937$	537.233			0.573355			0.286677	−	0.958027i	$-0.407449\pi$
$937$	537.233			0.573355			0.286677	+	0.958027i	$0.407449\pi$
$938$		0			0
$939$	341.233	−	965.153i	0.363401	−	1.02785i
$940$		0			0
$941$			1676.04i			1.78113i	0.454858	+	0.890564i	$0.349690\pi$
$941$			1676.04i			1.78113i	−0.454858	+	0.890564i	$0.650310\pi$
$942$		0			0
$943$	603.420			0.639894
$944$		0			0
$945$	−1255.23	−	2041.44i	−1.32829	−	2.16026i
$946$		0			0
$947$			939.705i			0.992296i	0.868238	+	0.496148i	$0.165253\pi$
$947$			939.705i			0.992296i	−0.868238	+	0.496148i	$0.834747\pi$
$948$		0			0
$949$	−2561.60			−2.69926
$950$		0			0
$951$	1451.51	+	513.186i	1.52630	+	0.539627i
$952$		0			0
$953$		−	1531.58i		−	1.60712i	−0.595226	−	0.803559i	$-0.702937\pi$
$953$		−	1531.58i		−	1.60712i	0.595226	−	0.803559i	$-0.297063\pi$
$954$		0			0
$955$	676.373			0.708244
$956$		0			0
$957$	−62.7617	+	177.517i	−0.0655817	+	0.185493i
$958$		0			0
$959$			1518.70i			1.58363i
$960$		0			0
$961$	−798.140			−0.830531
$962$		0			0
$963$	675.420	−	835.789i	0.701371	−	0.867902i
$964$		0			0
$965$		−	1311.10i		−	1.35866i
$966$		0			0
$967$	−448.427			−0.463731			−0.231865	−	0.972748i	$-0.574483\pi$
$967$	−448.427			−0.463731			−0.231865	+	0.972748i	$0.574483\pi$
$968$		0			0
$969$	26.3002	+	9.29853i	0.0271416	+	0.00959601i
$970$		0			0
$971$		−	1630.24i		−	1.67893i	−0.543411	−	0.839467i	$-0.682868\pi$
$971$		−	1630.24i		−	1.67893i	0.543411	−	0.839467i	$-0.317132\pi$
$972$		0			0
$973$	1585.80			1.62980
$974$		0			0
$975$	1379.56	−	3901.99i	1.41494	−	4.00204i
$976$		0			0
$977$			482.382i			0.493738i	0.969049	+	0.246869i	$0.0794018\pi$
$977$			482.382i			0.493738i	−0.969049	+	0.246869i	$0.920598\pi$
$978$		0			0
$979$	−49.8083			−0.0508767
$980$		0			0
$981$	−1208.99	−	977.009i	−1.23240	−	0.995932i
$982$		0			0
$983$			1632.22i			1.66045i	0.557430	+	0.830224i	$0.311788\pi$
$983$			1632.22i			1.66045i	−0.557430	+	0.830224i	$0.688212\pi$
$984$		0			0
$985$	−963.420			−0.978091
$986$		0			0
$987$	−701.327	−	247.956i	−0.710564	−	0.251222i
$988$		0			0
$989$			556.588i			0.562779i
$990$		0			0
$991$	−138.482			−0.139739			−0.0698697	−	0.997556i	$-0.522258\pi$
$991$	−138.482			−0.139739			−0.0698697	+	0.997556i	$0.522258\pi$
$992$		0			0
$993$	−254.000	+	718.420i	−0.255791	+	0.723485i
$994$		0			0
$995$		−	735.314i		−	0.739009i
$996$		0			0
$997$	1420.42			1.42469			0.712346	−	0.701829i	$-0.247633\pi$
$997$	1420.42			1.42469			0.712346	+	0.701829i	$0.247633\pi$
$998$		0			0
$999$	258.482	−	158.934i	0.258740	−	0.159093i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	132.3.e.b.89.3	yes	4
3.2	odd	2	inner	132.3.e.b.89.2	✓	4
4.3	odd	2		528.3.i.c.353.1		4
11.10	odd	2		1452.3.e.h.485.3		4
12.11	even	2		528.3.i.c.353.4		4
33.32	even	2		1452.3.e.h.485.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.3.e.b.89.2	✓	4	3.2	odd	2	inner
132.3.e.b.89.3	yes	4	1.1	even	1	trivial
528.3.i.c.353.1		4	4.3	odd	2
528.3.i.c.353.4		4	12.11	even	2
1452.3.e.h.485.2		4	33.32	even	2
1452.3.e.h.485.3		4	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 132 = 2^{2} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	132.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 2.82843i) q^{3} -9.46168i q^{5} +9.38083 q^{7} +(-7.00000 - 5.65685i) q^{9} +3.31662i q^{11} +21.3808 q^{13} +(26.7617 + 9.46168i) q^{15} -1.95279i q^{17} +4.76166 q^{19} +(-9.38083 + 26.5330i) q^{21} -11.4145i q^{23} -64.5233 q^{25} +(23.0000 - 14.1421i) q^{27} -18.9234i q^{29} -12.7617 q^{31} +(-9.38083 - 3.31662i) q^{33} -88.7584i q^{35} +11.2383 q^{37} +(-21.3808 + 60.4741i) q^{39} +52.8645i q^{41} -48.7617 q^{43} +(-53.5233 + 66.2317i) q^{45} +26.4322i q^{47} +39.0000 q^{49} +(5.52333 + 1.95279i) q^{51} +32.2906i q^{53} +31.3808 q^{55} +(-4.76166 + 13.4680i) q^{57} +28.0828i q^{59} -2.61917 q^{61} +(-65.6658 - 53.0660i) q^{63} -202.299i q^{65} +109.047 q^{67} +(32.2850 + 11.4145i) q^{69} +53.1668i q^{71} -119.808 q^{73} +(64.5233 - 182.500i) q^{75} +31.1127i q^{77} -38.6192 q^{79} +(17.0000 + 79.1960i) q^{81} +116.841i q^{83} -18.4767 q^{85} +(53.5233 + 18.9234i) q^{87} +15.0178i q^{89} +200.570 q^{91} +(12.7617 - 36.0954i) q^{93} -45.0533i q^{95} -52.4767 q^{97} +(18.7617 - 23.2164i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 28 q^{9} + 48 q^{13} + 32 q^{15} - 56 q^{19} - 108 q^{25} + 92 q^{27} + 24 q^{31} + 120 q^{37} - 48 q^{39} - 120 q^{43} - 64 q^{45} + 156 q^{49} - 128 q^{51} + 88 q^{55} + 56 q^{57} - 48 q^{61}+ \cdots - 360 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 28 * q^9 + 48 * q^13 + 32 * q^15 - 56 * q^19 - 108 * q^25 + 92 * q^27 + 24 * q^31 + 120 * q^37 - 48 * q^39 - 120 * q^43 - 64 * q^45 + 156 * q^49 - 128 * q^51 + 88 * q^55 + 56 * q^57 - 48 * q^61 + 136 * q^67 - 96 * q^69 - 104 * q^73 + 108 * q^75 - 192 * q^79 + 68 * q^81 - 224 * q^85 + 64 * q^87 + 352 * q^91 - 24 * q^93 - 360 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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