LMFDB - Embedded newform 1296.3.o.be.703.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(271,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 2]))

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

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

chi := DirichletCharacter("1296.271");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.o (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			703.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1296.703
Dual form			1296.3.o.be.271.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.86603 + 6.69615i) q^{5} +(-8.19615 - 4.73205i) q^{7} +O(q^{10})$	$q+(3.86603 + 6.69615i) q^{5} +(-8.19615 - 4.73205i) q^{7} +(14.1962 + 8.19615i) q^{11} +(-8.69615 - 15.0622i) q^{13} -20.6603 q^{17} -33.4641i q^{19} +(6.58846 - 3.80385i) q^{23} +(-17.3923 + 30.1244i) q^{25} +(-3.86603 + 6.69615i) q^{29} +(28.3923 - 16.3923i) q^{31} -73.1769i q^{35} -26.1769 q^{37} +(-20.7846 - 36.0000i) q^{41} +(-32.1962 - 18.5885i) q^{43} +(20.7846 + 12.0000i) q^{47} +(20.2846 + 35.1340i) q^{49} +15.2154 q^{53} +126.746i q^{55} +(83.1384 - 48.0000i) q^{59} +(17.4808 - 30.2776i) q^{61} +(67.2391 - 116.462i) q^{65} +(35.4115 - 20.4449i) q^{67} +51.5307i q^{71} +132.138 q^{73} +(-77.5692 - 134.354i) q^{77} +(-43.0192 - 24.8372i) q^{79} +(-79.6077 - 45.9615i) q^{83} +(-79.8731 - 138.344i) q^{85} -77.6936 q^{89} +164.603i q^{91} +(224.081 - 129.373i) q^{95} +(21.7846 - 37.7321i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{5} - 12 q^{7}+O(q^{10}) $ 4 * q + 12 * q^5 - 12 * q^7	$ 4 q + 12 q^{5} - 12 q^{7} + 36 q^{11} - 14 q^{13} - 48 q^{17} - 36 q^{23} - 28 q^{25} - 12 q^{29} + 72 q^{31} + 20 q^{37} - 108 q^{43} - 2 q^{49} + 144 q^{53} - 34 q^{61} + 120 q^{65} + 204 q^{67} + 196 q^{73} - 144 q^{77} - 276 q^{79} - 360 q^{83} - 174 q^{85} - 96 q^{89} + 252 q^{95} + 4 q^{97}+O(q^{100}) $ 4 * q + 12 * q^5 - 12 * q^7 + 36 * q^11 - 14 * q^13 - 48 * q^17 - 36 * q^23 - 28 * q^25 - 12 * q^29 + 72 * q^31 + 20 * q^37 - 108 * q^43 - 2 * q^49 + 144 * q^53 - 34 * q^61 + 120 * q^65 + 204 * q^67 + 196 * q^73 - 144 * q^77 - 276 * q^79 - 360 * q^83 - 174 * q^85 - 96 * q^89 + 252 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	3.86603	+	6.69615i	0.773205	+	1.33923i	0.935798	+	0.352537i	$0.114681\pi$
$5$	3.86603	+	6.69615i	0.773205	+	1.33923i	−0.162593	+	0.986693i	$0.551986\pi$
$6$		0			0
$7$	−8.19615	−	4.73205i	−1.17088	−	0.676007i	−0.216992	−	0.976173i	$-0.569625\pi$
$7$	−8.19615	−	4.73205i	−1.17088	−	0.676007i	−0.953887	+	0.300166i	$0.902958\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	14.1962	+	8.19615i	1.29056	+	0.745105i	0.978753	−	0.205041i	$-0.0657328\pi$
$11$	14.1962	+	8.19615i	1.29056	+	0.745105i	0.311806	+	0.950146i	$0.399066\pi$
$12$		0			0
$13$	−8.69615	−	15.0622i	−0.668935	−	1.15863i	−0.978202	−	0.207654i	$-0.933417\pi$
$13$	−8.69615	−	15.0622i	−0.668935	−	1.15863i	0.309268	−	0.950975i	$-0.399916\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−20.6603			−1.21531			−0.607655	−	0.794201i	$-0.707889\pi$
$17$	−20.6603			−1.21531			−0.607655	+	0.794201i	$0.707889\pi$
$18$		0			0
$19$		−	33.4641i		−	1.76127i	−0.473797	−	0.880634i	$-0.657117\pi$
$19$		−	33.4641i		−	1.76127i	0.473797	−	0.880634i	$-0.342883\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.58846	−	3.80385i	0.286455	−	0.165385i	−0.349887	−	0.936792i	$-0.613780\pi$
$23$	6.58846	−	3.80385i	0.286455	−	0.165385i	0.636342	+	0.771407i	$0.280447\pi$
$24$		0			0
$25$	−17.3923	+	30.1244i	−0.695692	+	1.20497i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.86603	+	6.69615i	−0.133311	+	0.230902i	−0.924951	−	0.380086i	$-0.875894\pi$
$29$	−3.86603	+	6.69615i	−0.133311	+	0.230902i	0.791640	+	0.610988i	$0.209228\pi$
$30$		0			0
$31$	28.3923	−	16.3923i	0.915881	−	0.528784i	0.0335622	−	0.999437i	$-0.489315\pi$
$31$	28.3923	−	16.3923i	0.915881	−	0.528784i	0.882319	+	0.470653i	$0.155981\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	73.1769i		−	2.09077i
$36$		0			0
$37$	−26.1769			−0.707484			−0.353742	−	0.935343i	$-0.615091\pi$
$37$	−26.1769			−0.707484			−0.353742	+	0.935343i	$0.615091\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−20.7846	−	36.0000i	−0.506942	−	0.878049i	−0.999968	−	0.00803422i	$-0.997443\pi$
$41$	−20.7846	−	36.0000i	−0.506942	−	0.878049i	0.493026	−	0.870015i	$-0.335891\pi$
$42$		0			0
$43$	−32.1962	−	18.5885i	−0.748748	−	0.432290i	0.0764935	−	0.997070i	$-0.475628\pi$
$43$	−32.1962	−	18.5885i	−0.748748	−	0.432290i	−0.825241	+	0.564780i	$0.808961\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	20.7846	+	12.0000i	0.442226	+	0.255319i	0.704541	−	0.709663i	$-0.251153\pi$
$47$	20.7846	+	12.0000i	0.442226	+	0.255319i	−0.262316	+	0.964982i	$0.584486\pi$
$48$		0			0
$49$	20.2846	+	35.1340i	0.413972	+	0.717020i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	15.2154			0.287083			0.143541	−	0.989644i	$-0.454151\pi$
$53$	15.2154			0.287083			0.143541	+	0.989644i	$0.454151\pi$
$54$		0			0
$55$			126.746i			2.30448i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	83.1384	−	48.0000i	1.40913	−	0.813559i	0.413822	−	0.910358i	$-0.364194\pi$
$59$	83.1384	−	48.0000i	1.40913	−	0.813559i	0.995304	+	0.0967985i	$0.0308602\pi$
$60$		0			0
$61$	17.4808	−	30.2776i	0.286570	−	0.496354i	−0.686419	−	0.727206i	$-0.740818\pi$
$61$	17.4808	−	30.2776i	0.286570	−	0.496354i	0.972989	+	0.230853i	$0.0741516\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	67.2391	−	116.462i	1.03445	−	1.79172i
$66$		0			0
$67$	35.4115	−	20.4449i	0.528530	−	0.305147i	−0.211887	−	0.977294i	$-0.567961\pi$
$67$	35.4115	−	20.4449i	0.528530	−	0.305147i	0.740418	+	0.672147i	$0.234628\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			51.5307i			0.725785i	0.931831	+	0.362893i	$0.118211\pi$
$71$			51.5307i			0.725785i	−0.931831	+	0.362893i	$0.881789\pi$
$72$		0			0
$73$	132.138			1.81012			0.905058	−	0.425289i	$-0.139827\pi$
$73$	132.138			1.81012			0.905058	+	0.425289i	$0.139827\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−77.5692	−	134.354i	−1.00739	−	1.74485i
$78$		0			0
$79$	−43.0192	−	24.8372i	−0.544547	−	0.314395i	0.202373	−	0.979309i	$-0.435135\pi$
$79$	−43.0192	−	24.8372i	−0.544547	−	0.314395i	−0.746920	+	0.664914i	$0.768468\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−79.6077	−	45.9615i	−0.959129	−	0.553753i	−0.0632240	−	0.997999i	$-0.520138\pi$
$83$	−79.6077	−	45.9615i	−0.959129	−	0.553753i	−0.895905	+	0.444246i	$0.853472\pi$
$84$		0			0
$85$	−79.8731	−	138.344i	−0.939683	−	1.62758i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−77.6936			−0.872962			−0.436481	−	0.899714i	$-0.643775\pi$
$89$	−77.6936			−0.872962			−0.436481	+	0.899714i	$0.643775\pi$
$90$		0			0
$91$			164.603i			1.80882i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	224.081	−	129.373i	2.35874	−	1.36182i
$96$		0			0
$97$	21.7846	−	37.7321i	0.224584	−	0.388990i	−0.731611	−	0.681723i	$-0.761231\pi$
$97$	21.7846	−	37.7321i	0.224584	−	0.388990i	0.956194	+	0.292732i	$0.0945645\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−85.1769	+	147.531i	−0.843336	+	1.46070i	0.0437229	+	0.999044i	$0.486078\pi$
$101$	−85.1769	+	147.531i	−0.843336	+	1.46070i	−0.887059	+	0.461657i	$0.847255\pi$
$102$		0			0
$103$	34.8231	−	20.1051i	0.338088	−	0.195195i	−0.321338	−	0.946965i	$-0.604133\pi$
$103$	34.8231	−	20.1051i	0.338088	−	0.195195i	0.659426	+	0.751769i	$0.270799\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	152.785i		−	1.42789i	−0.700200	−	0.713947i	$-0.746906\pi$
$107$		−	152.785i		−	1.42789i	0.700200	−	0.713947i	$-0.253094\pi$
$108$		0			0
$109$	80.6077			0.739520			0.369760	−	0.929127i	$-0.379440\pi$
$109$	80.6077			0.739520			0.369760	+	0.929127i	$0.379440\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.3686	+	21.4230i	0.109457	+	0.189584i	0.915550	−	0.402204i	$-0.131756\pi$
$113$	12.3686	+	21.4230i	0.109457	+	0.189584i	−0.806094	+	0.591788i	$0.798422\pi$
$114$		0			0
$115$	50.9423	+	29.4115i	0.442976	+	0.255753i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	169.335	+	97.7654i	1.42298	+	0.821558i
$120$		0			0
$121$	73.8538	+	127.919i	0.610362	+	1.05718i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−75.6551			−0.605241
$126$		0			0
$127$		−	26.0385i		−	0.205027i	−0.994732	−	0.102514i	$-0.967311\pi$
$127$		−	26.0385i		−	0.205027i	0.994732	−	0.102514i	$-0.0326885\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	188.081	−	108.588i	1.43573	−	0.828920i	0.438181	−	0.898887i	$-0.355623\pi$
$131$	188.081	−	108.588i	1.43573	−	0.828920i	0.997549	+	0.0699670i	$0.0222894\pi$
$132$		0			0
$133$	−158.354	+	274.277i	−1.19063	+	2.06223i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.76091	+	8.24613i	−0.0347512	+	0.0601908i	−0.882878	−	0.469603i	$-0.844397\pi$
$137$	−4.76091	+	8.24613i	−0.0347512	+	0.0601908i	0.848127	+	0.529793i	$0.177731\pi$
$138$		0			0
$139$	19.6077	−	11.3205i	0.141063	−	0.0814425i	−0.427808	−	0.903870i	$-0.640714\pi$
$139$	19.6077	−	11.3205i	0.141063	−	0.0814425i	0.568870	+	0.822427i	$0.307381\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	285.100i		−	1.99371i
$144$		0			0
$145$	−59.7846			−0.412308
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−32.2583	−	55.8731i	−0.216499	−	0.374987i	0.737236	−	0.675635i	$-0.236130\pi$
$149$	−32.2583	−	55.8731i	−0.216499	−	0.374987i	−0.953735	+	0.300648i	$0.902797\pi$
$150$		0			0
$151$	−171.531	−	99.0333i	−1.13597	−	0.655850i	−0.190537	−	0.981680i	$-0.561023\pi$
$151$	−171.531	−	99.0333i	−1.13597	−	0.655850i	−0.945428	+	0.325830i	$0.894356\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	219.531	+	126.746i	1.41633	+	0.817717i
$156$		0			0
$157$	−142.050	−	246.038i	−0.904777	−	1.56712i	−0.821216	−	0.570617i	$-0.806704\pi$
$157$	−142.050	−	246.038i	−0.904777	−	1.56712i	−0.0835606	−	0.996503i	$-0.526629\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−72.0000			−0.447205
$162$		0			0
$163$		−	23.0052i		−	0.141136i	−0.997507	−	0.0705680i	$-0.977519\pi$
$163$		−	23.0052i		−	0.141136i	0.997507	−	0.0705680i	$-0.0224812\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−188.627	+	108.904i	−1.12950	+	0.652119i	−0.943810	−	0.330489i	$-0.892786\pi$
$167$	−188.627	+	108.904i	−1.12950	+	0.652119i	−0.185693	+	0.982608i	$0.559453\pi$
$168$		0			0
$169$	−66.7461	+	115.608i	−0.394948	+	0.684069i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	54.9571	−	95.1884i	0.317671	−	0.550222i	−0.662331	−	0.749212i	$-0.730433\pi$
$173$	54.9571	−	95.1884i	0.317671	−	0.550222i	0.980002	+	0.198989i	$0.0637660\pi$
$174$		0			0
$175$	285.100	−	164.603i	1.62914	−	0.940586i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			190.277i			1.06300i	0.847059	+	0.531500i	$0.178371\pi$
$179$			190.277i			1.06300i	−0.847059	+	0.531500i	$0.821629\pi$
$180$		0			0
$181$	34.8616			0.192605			0.0963027	−	0.995352i	$-0.469298\pi$
$181$	34.8616			0.192605			0.0963027	+	0.995352i	$0.469298\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−101.201	−	175.285i	−0.547030	−	0.947484i
$186$		0			0
$187$	−293.296	−	169.335i	−1.56843	−	0.905533i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−142.981	−	82.5500i	−0.748590	−	0.432199i	0.0765940	−	0.997062i	$-0.475595\pi$
$191$	−142.981	−	82.5500i	−0.748590	−	0.432199i	−0.825184	+	0.564864i	$0.808929\pi$
$192$		0			0
$193$	−21.4615	−	37.1725i	−0.111200	−	0.192603i	0.805055	−	0.593201i	$-0.202136\pi$
$193$	−21.4615	−	37.1725i	−0.111200	−	0.192603i	−0.916254	+	0.400597i	$0.868803\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−147.406			−0.748256			−0.374128	−	0.927377i	$-0.622058\pi$
$197$	−147.406			−0.748256			−0.374128	+	0.927377i	$0.622058\pi$
$198$		0			0
$199$		−	278.851i		−	1.40126i	−0.713524	−	0.700631i	$-0.752902\pi$
$199$		−	278.851i		−	1.40126i	0.713524	−	0.700631i	$-0.247098\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	63.3731	−	36.5885i	0.312183	−	0.180239i
$204$		0			0
$205$	160.708	−	278.354i	0.783940	−	1.35782i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	274.277	−	475.061i	1.31233	−	2.27302i
$210$		0			0
$211$	114.158	−	65.9090i	0.541032	−	0.312365i	−0.204465	−	0.978874i	$-0.565546\pi$
$211$	114.158	−	65.9090i	0.541032	−	0.312365i	0.745497	+	0.666509i	$0.232212\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	287.454i		−	1.33699i
$216$		0			0
$217$	−310.277			−1.42985
$218$		0			0
$219$		0			0
$220$		0			0
$221$	179.665	+	311.188i	0.812963	+	1.40809i
$222$		0			0
$223$	−30.1577	−	17.4115i	−0.135236	−	0.0780787i	0.430855	−	0.902421i	$-0.358212\pi$
$223$	−30.1577	−	17.4115i	−0.135236	−	0.0780787i	−0.566092	+	0.824342i	$0.691545\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	37.0192	+	21.3731i	0.163080	+	0.0941545i	0.579319	−	0.815101i	$-0.303319\pi$
$227$	37.0192	+	21.3731i	0.163080	+	0.0941545i	−0.416239	+	0.909255i	$0.636652\pi$
$228$		0			0
$229$	96.4038	+	166.976i	0.420977	+	0.729154i	0.996035	−	0.0889585i	$-0.0283539\pi$
$229$	96.4038	+	166.976i	0.420977	+	0.729154i	−0.575058	+	0.818113i	$0.695021\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	304.583			1.30722			0.653612	−	0.756830i	$-0.273253\pi$
$233$	304.583			1.30722			0.653612	+	0.756830i	$0.273253\pi$
$234$		0			0
$235$			185.569i			0.789656i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	208.392	−	120.315i	0.871934	−	0.503412i	0.00394377	−	0.999992i	$-0.498745\pi$
$239$	208.392	−	120.315i	0.871934	−	0.503412i	0.867991	+	0.496581i	$0.165411\pi$
$240$		0			0
$241$	147.031	−	254.665i	0.610086	−	1.05670i	−0.381139	−	0.924518i	$-0.624468\pi$
$241$	147.031	−	254.665i	0.610086	−	1.05670i	0.991225	−	0.132182i	$-0.0421985\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−156.842	+	271.658i	−0.640170	+	1.10881i
$246$		0			0
$247$	−504.042	+	291.009i	−2.04066	+	1.17817i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	51.5307i		−	0.205302i	−0.994717	−	0.102651i	$-0.967268\pi$
$251$		−	51.5307i		−	0.205302i	0.994717	−	0.102651i	$-0.0327324\pi$
$252$		0			0
$253$	124.708			0.492916
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−229.861	−	398.131i	−0.894400	−	1.54915i	−0.834545	−	0.550940i	$-0.814269\pi$
$257$	−229.861	−	398.131i	−0.894400	−	1.54915i	−0.0598552	−	0.998207i	$-0.519064\pi$
$258$		0			0
$259$	214.550	+	123.870i	0.828378	+	0.478264i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−304.708	−	175.923i	−1.15858	−	0.668909i	−0.207620	−	0.978210i	$-0.566572\pi$
$263$	−304.708	−	175.923i	−1.15858	−	0.668909i	−0.950964	+	0.309301i	$0.899905\pi$
$264$		0			0
$265$	58.8231	+	101.885i	0.221974	+	0.384470i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−219.655			−0.816562			−0.408281	−	0.912856i	$-0.633872\pi$
$269$	−219.655			−0.816562			−0.408281	+	0.912856i	$0.633872\pi$
$270$		0			0
$271$			355.244i			1.31086i	0.755255	+	0.655431i	$0.227513\pi$
$271$			355.244i			1.31086i	−0.755255	+	0.655431i	$0.772487\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−493.808	+	285.100i	−1.79566	+	1.03673i
$276$		0			0
$277$	−207.708	+	359.760i	−0.749847	+	1.29877i	0.198048	+	0.980192i	$0.436540\pi$
$277$	−207.708	+	359.760i	−0.749847	+	1.29877i	−0.947896	+	0.318581i	$0.896794\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−120.493	+	208.700i	−0.428801	+	0.742704i	−0.996767	−	0.0803474i	$-0.974397\pi$
$281$	−120.493	+	208.700i	−0.428801	+	0.742704i	0.567966	+	0.823052i	$0.307730\pi$
$282$		0			0
$283$	92.7846	−	53.5692i	0.327861	−	0.189291i	−0.327030	−	0.945014i	$-0.606048\pi$
$283$	92.7846	−	53.5692i	0.327861	−	0.189291i	0.654891	+	0.755723i	$0.272715\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			393.415i			1.37079i
$288$		0			0
$289$	137.846			0.476976
$290$		0			0
$291$		0			0
$292$		0			0
$293$	183.320	+	317.519i	0.625665	+	1.08368i	0.988412	+	0.151796i	$0.0485056\pi$
$293$	183.320	+	317.519i	0.625665	+	1.08368i	−0.362747	+	0.931888i	$0.618161\pi$
$294$		0			0
$295$	642.831	+	371.138i	2.17909	+	1.25810i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−114.588	−	66.1577i	−0.383239	−	0.221263i
$300$		0			0
$301$	175.923	+	304.708i	0.584462	+	1.01232i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	270.324			0.886309
$306$		0			0
$307$		−	173.338i		−	0.564620i	−0.959323	−	0.282310i	$-0.908899\pi$
$307$		−	173.338i		−	0.564620i	0.959323	−	0.282310i	$-0.0911007\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	93.3308	−	53.8846i	0.300099	−	0.173262i	−0.342388	−	0.939559i	$-0.611236\pi$
$311$	93.3308	−	53.8846i	0.300099	−	0.173262i	0.642487	+	0.766296i	$0.277902\pi$
$312$		0			0
$313$	32.9693	−	57.1044i	0.105333	−	0.182442i	−0.808541	−	0.588440i	$-0.799742\pi$
$313$	32.9693	−	57.1044i	0.105333	−	0.182442i	0.913874	+	0.405997i	$0.133076\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−273.942	+	474.481i	−0.864169	+	1.49678i	0.00370080	+	0.999993i	$0.498822\pi$
$317$	−273.942	+	474.481i	−0.864169	+	1.49678i	−0.867870	+	0.496792i	$0.834511\pi$
$318$		0			0
$319$	−109.765	+	63.3731i	−0.344092	+	0.198662i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			691.377i			2.14049i
$324$		0			0
$325$	604.985			1.86149
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−113.569	−	196.708i	−0.345195	−	0.597896i
$330$		0			0
$331$	137.842	+	79.5833i	0.416442	+	0.240433i	0.693554	−	0.720405i	$-0.256044\pi$
$331$	137.842	+	79.5833i	0.416442	+	0.240433i	−0.277112	+	0.960838i	$0.589377\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	273.804	+	158.081i	0.817325	+	0.471883i
$336$		0			0
$337$	−317.200	−	549.406i	−0.941246	−	1.63029i	−0.763098	−	0.646282i	$-0.776323\pi$
$337$	−317.200	−	549.406i	−0.941246	−	1.63029i	−0.178148	−	0.984004i	$-0.557010\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	537.415			1.57600
$342$		0			0
$343$			79.7898i			0.232623i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−26.8269	+	15.4885i	−0.0773108	+	0.0446354i	−0.538157	−	0.842845i	$-0.680879\pi$
$347$	−26.8269	+	15.4885i	−0.0773108	+	0.0446354i	0.460846	+	0.887480i	$0.347546\pi$
$348$		0			0
$349$	−88.2154	+	152.794i	−0.252766	+	0.437804i	−0.964286	−	0.264862i	$-0.914674\pi$
$349$	−88.2154	+	152.794i	−0.252766	+	0.437804i	0.711520	+	0.702666i	$0.248007\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.2923	+	33.4153i	−0.0546525	+	0.0946610i	−0.892057	−	0.451922i	$-0.850738\pi$
$353$	−19.2923	+	33.4153i	−0.0546525	+	0.0946610i	0.837405	+	0.546583i	$0.184072\pi$
$354$		0			0
$355$	−345.058	+	199.219i	−0.971994	+	0.561181i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			573.100i			1.59638i	0.602407	+	0.798189i	$0.294209\pi$
$359$			573.100i			1.59638i	−0.602407	+	0.798189i	$0.705791\pi$
$360$		0			0
$361$	−758.846			−2.10207
$362$		0			0
$363$		0			0
$364$		0			0
$365$	510.851	+	884.819i	1.39959	+	2.42416i
$366$		0			0
$367$	236.469	+	136.526i	0.644330	+	0.372004i	0.786281	−	0.617869i	$-0.212004\pi$
$367$	236.469	+	136.526i	0.644330	+	0.372004i	−0.141950	+	0.989874i	$0.545337\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−124.708	−	72.0000i	−0.336139	−	0.194070i
$372$		0			0
$373$	105.277	+	182.345i	0.282244	+	0.488860i	0.971937	−	0.235241i	$-0.0755880\pi$
$373$	105.277	+	182.345i	0.282244	+	0.488860i	−0.689693	+	0.724102i	$0.742255\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	134.478			0.356706
$378$		0			0
$379$		−	310.641i		−	0.819633i	−0.912168	−	0.409817i	$-0.865593\pi$
$379$		−	310.641i		−	0.819633i	0.912168	−	0.409817i	$-0.134407\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−403.535	+	232.981i	−1.05361	+	0.608305i	−0.923659	−	0.383215i	$-0.874817\pi$
$383$	−403.535	+	232.981i	−1.05361	+	0.608305i	−0.129956	+	0.991520i	$0.541484\pi$
$384$		0			0
$385$	599.769	−	1038.83i	1.55784	−	2.69826i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−45.1000	+	78.1154i	−0.115938	+	0.200811i	−0.918154	−	0.396223i	$-0.870321\pi$
$389$	−45.1000	+	78.1154i	−0.115938	+	0.200811i	0.802216	+	0.597034i	$0.203654\pi$
$390$		0			0
$391$	−136.119	+	78.5885i	−0.348131	+	0.200993i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	384.084i		−	0.972366i
$396$		0			0
$397$	89.7461			0.226061			0.113030	−	0.993592i	$-0.463944\pi$
$397$	89.7461			0.226061			0.113030	+	0.993592i	$0.463944\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	294.378	+	509.877i	0.734109	+	1.27151i	0.955113	+	0.296241i	$0.0957330\pi$
$401$	294.378	+	509.877i	0.734109	+	1.27151i	−0.221005	+	0.975273i	$0.570934\pi$
$402$		0			0
$403$	−493.808	−	285.100i	−1.22533	−	0.707444i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−371.611	−	214.550i	−0.913050	−	0.527150i
$408$		0			0
$409$	360.677	+	624.711i	0.881851	+	1.52741i	0.849281	+	0.527941i	$0.177036\pi$
$409$	360.677	+	624.711i	0.881851	+	1.52741i	0.0325694	+	0.999469i	$0.489631\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−908.554			−2.19989
$414$		0			0
$415$		−	710.754i		−	1.71266i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−686.438	+	396.315i	−1.63828	+	0.945860i	−0.656851	+	0.754020i	$0.728112\pi$
$419$	−686.438	+	396.315i	−1.63828	+	0.945860i	−0.981426	+	0.191840i	$0.938555\pi$
$420$		0			0
$421$	−198.973	+	344.631i	−0.472620	+	0.818602i	−0.999509	−	0.0313322i	$-0.990025\pi$
$421$	−198.973	+	344.631i	−0.472620	+	0.818602i	0.526889	+	0.849934i	$0.323358\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	359.329	−	622.377i	0.845481	−	1.46442i
$426$		0			0
$427$	−286.550	+	165.440i	−0.671077	+	0.387447i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	48.6307i		−	0.112832i	−0.998407	−	0.0564161i	$-0.982033\pi$
$431$		−	48.6307i		−	0.112832i	0.998407	−	0.0564161i	$-0.0179674\pi$
$432$		0			0
$433$	−379.200			−0.875750			−0.437875	−	0.899036i	$-0.644269\pi$
$433$	−379.200			−0.875750			−0.437875	+	0.899036i	$0.644269\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−127.292	−	220.477i	−0.291287	−	0.504524i
$438$		0			0
$439$	−194.123	−	112.077i	−0.442194	−	0.255301i	0.262334	−	0.964977i	$-0.415508\pi$
$439$	−194.123	−	112.077i	−0.442194	−	0.255301i	−0.704528	+	0.709677i	$0.748841\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−259.061	−	149.569i	−0.584789	−	0.337628i	0.178245	−	0.983986i	$-0.442958\pi$
$443$	−259.061	−	149.569i	−0.584789	−	0.337628i	−0.763034	+	0.646358i	$0.776291\pi$
$444$		0			0
$445$	−300.365	−	520.248i	−0.674978	−	1.16910i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	396.400			0.882850			0.441425	−	0.897298i	$-0.354473\pi$
$449$	396.400			0.882850			0.441425	+	0.897298i	$0.354473\pi$
$450$		0			0
$451$		−	681.415i		−	1.51090i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1102.20	+	636.358i	−2.42243	+	1.39859i
$456$		0			0
$457$	−21.6000	+	37.4122i	−0.0472647	+	0.0818648i	−0.888690	−	0.458509i	$-0.848384\pi$
$457$	−21.6000	+	37.4122i	−0.0472647	+	0.0818648i	0.841425	+	0.540374i	$0.181717\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.7616	−	27.2999i	0.0341900	−	0.0592188i	−0.848424	−	0.529317i	$-0.822448\pi$
$461$	15.7616	−	27.2999i	0.0341900	−	0.0592188i	0.882614	+	0.470098i	$0.155782\pi$
$462$		0			0
$463$	5.33836	−	3.08211i	0.0115299	−	0.00665682i	−0.494224	−	0.869335i	$-0.664548\pi$
$463$	5.33836	−	3.08211i	0.0115299	−	0.00665682i	0.505754	+	0.862678i	$0.331214\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	330.200i		−	0.707066i	−0.935422	−	0.353533i	$-0.884980\pi$
$467$		−	330.200i		−	0.707066i	0.935422	−	0.353533i	$-0.115020\pi$
$468$		0			0
$469$	−386.985			−0.825127
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−304.708	−	527.769i	−0.644202	−	1.11579i
$474$		0			0
$475$	1008.08	+	582.018i	2.12228	+	1.22530i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	57.8038	+	33.3731i	0.120676	+	0.0696724i	0.559123	−	0.829085i	$-0.311138\pi$
$479$	57.8038	+	33.3731i	0.120676	+	0.0696724i	−0.438447	+	0.898757i	$0.644471\pi$
$480$		0			0
$481$	227.638	+	394.281i	0.473261	+	0.819712i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	336.879			0.694597
$486$		0			0
$487$		−	313.359i		−	0.643448i	−0.946834	−	0.321724i	$-0.895738\pi$
$487$		−	313.359i		−	0.643448i	0.946834	−	0.321724i	$-0.104262\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	149.096	−	86.0807i	0.303658	−	0.175317i	−0.340427	−	0.940271i	$-0.610572\pi$
$491$	149.096	−	86.0807i	0.303658	−	0.175317i	0.644085	+	0.764954i	$0.277238\pi$
$492$		0			0
$493$	79.8731	−	138.344i	0.162014	−	0.280617i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	243.846	−	422.354i	0.490636	−	0.849806i
$498$		0			0
$499$	17.5270	−	10.1192i	0.0351242	−	0.0202790i	−0.482335	−	0.875987i	$-0.660211\pi$
$499$	17.5270	−	10.1192i	0.0351242	−	0.0202790i	0.517459	+	0.855708i	$0.326878\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			368.785i			0.733170i	0.930385	+	0.366585i	$0.119473\pi$
$503$			368.785i			0.733170i	−0.930385	+	0.366585i	$0.880527\pi$
$504$		0			0
$505$	−1317.18			−2.60829
$506$		0			0
$507$		0			0
$508$		0			0
$509$	117.100	+	202.823i	0.230059	+	0.398474i	0.957825	−	0.287352i	$-0.0927748\pi$
$509$	117.100	+	202.823i	0.230059	+	0.398474i	−0.727766	+	0.685825i	$0.759441\pi$
$510$		0			0
$511$	−1083.03	−	625.286i	−2.11943	−	1.22365i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	269.254	+	155.454i	0.522823	+	0.301852i
$516$		0			0
$517$	196.708	+	340.708i	0.380479	+	0.659009i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1010.98			1.94047			0.970235	−	0.242167i	$-0.0778581\pi$
$521$	1010.98			1.94047			0.970235	+	0.242167i	$0.0778581\pi$
$522$		0			0
$523$		−	525.864i		−	1.00548i	−0.864439	−	0.502738i	$-0.832326\pi$
$523$		−	525.864i		−	1.00548i	0.864439	−	0.502738i	$-0.167674\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−586.592	+	338.669i	−1.11308	+	0.642636i
$528$		0			0
$529$	−235.561	+	408.004i	−0.445296	+	0.771275i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−361.492	+	626.123i	−0.678222	+	1.17471i
$534$		0			0
$535$	1023.07	−	590.669i	1.91228	−	1.10405i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			665.023i			1.23381i
$540$		0			0
$541$	412.531			0.762534			0.381267	−	0.924465i	$-0.375488\pi$
$541$	412.531			0.762534			0.381267	+	0.924465i	$0.375488\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	311.631	+	539.761i	0.571801	+	0.990388i
$546$		0			0
$547$	813.415	+	469.626i	1.48705	+	0.858548i	0.999891	−	0.0147667i	$-0.00470056\pi$
$547$	813.415	+	469.626i	1.48705	+	0.858548i	0.487157	+	0.873314i	$0.338034\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	224.081	+	129.373i	0.406680	+	0.234797i
$552$		0			0
$553$	235.061	+	407.138i	0.425066	+	0.736236i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−631.022			−1.13289			−0.566447	−	0.824098i	$-0.691682\pi$
$557$	−631.022			−1.13289			−0.566447	+	0.824098i	$0.691682\pi$
$558$		0			0
$559$			646.592i			1.15669i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	81.6462	−	47.1384i	0.145020	−	0.0837272i	−0.425734	−	0.904848i	$-0.639984\pi$
$563$	81.6462	−	47.1384i	0.145020	−	0.0837272i	0.570754	+	0.821121i	$0.306651\pi$
$564$		0			0
$565$	−95.6347	+	165.644i	−0.169265	+	0.293175i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.4686	−	37.1846i	0.0377303	−	0.0653509i	−0.846544	−	0.532319i	$-0.821321\pi$
$569$	21.4686	−	37.1846i	0.0377303	−	0.0653509i	0.884274	+	0.466969i	$0.154654\pi$
$570$		0			0
$571$	−366.431	+	211.559i	−0.641735	+	0.370506i	−0.785283	−	0.619137i	$-0.787482\pi$
$571$	−366.431	+	211.559i	−0.641735	+	0.370506i	0.143547	+	0.989643i	$0.454149\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			264.631i			0.460227i
$576$		0			0
$577$	937.523			1.62482			0.812411	−	0.583085i	$-0.198154\pi$
$577$	937.523			1.62482			0.812411	+	0.583085i	$0.198154\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	434.985	+	753.415i	0.748683	+	1.29676i
$582$		0			0
$583$	216.000	+	124.708i	0.370497	+	0.213907i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−523.219	−	302.081i	−0.891344	−	0.514618i	−0.0169624	−	0.999856i	$-0.505400\pi$
$587$	−523.219	−	302.081i	−0.891344	−	0.514618i	−0.874382	+	0.485238i	$0.838733\pi$
$588$		0			0
$589$	−548.554	−	950.123i	−0.931331	−	1.61311i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	91.5678			0.154415			0.0772073	−	0.997015i	$-0.475400\pi$
$593$	91.5678			0.154415			0.0772073	+	0.997015i	$0.475400\pi$
$594$		0			0
$595$			1511.85i			2.54093i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	821.885	−	474.515i	1.37209	−	0.792179i	0.380903	−	0.924615i	$-0.375613\pi$
$599$	821.885	−	474.515i	1.37209	−	0.792179i	0.991192	+	0.132436i	$0.0422799\pi$
$600$		0			0
$601$	181.362	−	314.127i	0.301766	−	0.522675i	−0.674770	−	0.738028i	$-0.735757\pi$
$601$	181.362	−	314.127i	0.301766	−	0.522675i	0.976536	+	0.215354i	$0.0690904\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−571.042	+	989.073i	−0.943870	+	1.63483i
$606$		0			0
$607$	−910.404	+	525.622i	−1.49984	+	0.865934i	−1.00000		0.000183305i	$-0.999942\pi$
$607$	−910.404	+	525.622i	−1.49984	+	0.865934i	−0.499841	+	0.866117i	$0.666608\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	417.415i		−	0.683167i
$612$		0			0
$613$	208.585			0.340269			0.170134	−	0.985421i	$-0.445580\pi$
$613$	208.585			0.340269			0.170134	+	0.985421i	$0.445580\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−344.922	−	597.423i	−0.559031	−	0.968271i	−0.997578	−	0.0695621i	$-0.977840\pi$
$617$	−344.922	−	597.423i	−0.559031	−	0.968271i	0.438546	−	0.898709i	$-0.355494\pi$
$618$		0			0
$619$	−766.823	−	442.726i	−1.23881	−	0.715227i	−0.269959	−	0.962872i	$-0.587010\pi$
$619$	−766.823	−	442.726i	−1.23881	−	0.715227i	−0.968851	+	0.247645i	$0.920343\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	636.788	+	367.650i	1.02213	+	0.590128i
$624$		0			0
$625$	142.323	+	246.511i	0.227717	+	0.394417i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	540.822			0.859812
$630$		0			0
$631$		−	790.179i		−	1.25227i	−0.779717	−	0.626133i	$-0.784637\pi$
$631$		−	790.179i		−	1.25227i	0.779717	−	0.626133i	$-0.215363\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	174.358	−	100.665i	0.274579	−	0.158528i
$636$		0			0
$637$	352.796	−	611.061i	0.553840	−	0.959279i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	182.301	−	315.754i	0.284400	−	0.492596i	−0.688063	−	0.725651i	$-0.741539\pi$
$641$	182.301	−	315.754i	0.284400	−	0.492596i	0.972464	+	0.233055i	$0.0748722\pi$
$642$		0			0
$643$	−1030.13	+	594.746i	−1.60207	+	0.924955i	−0.610996	+	0.791633i	$0.709231\pi$
$643$	−1030.13	+	594.746i	−1.60207	+	0.924955i	−0.991073	+	0.133322i	$0.957436\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			68.0075i			0.105112i	0.998618	+	0.0525561i	$0.0167368\pi$
$647$			68.0075i			0.105112i	−0.998618	+	0.0525561i	$0.983263\pi$
$648$		0			0
$649$	1573.66			2.42475
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−208.392	−	360.946i	−0.319131	−	0.552750i	0.661176	−	0.750231i	$-0.270058\pi$
$653$	−208.392	−	360.946i	−0.319131	−	0.552750i	−0.980307	+	0.197480i	$0.936724\pi$
$654$		0			0
$655$	1454.25	+	839.611i	2.22023	+	1.28185i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1042.98	−	602.165i	−1.58267	−	0.913756i	−0.994468	−	0.105044i	$-0.966502\pi$
$659$	−1042.98	−	602.165i	−1.58267	−	0.913756i	−0.588204	−	0.808712i	$-0.700165\pi$
$660$		0			0
$661$	−602.758	−	1044.01i	−0.911888	−	1.57944i	−0.811395	−	0.584498i	$-0.801292\pi$
$661$	−602.758	−	1044.01i	−0.911888	−	1.57944i	−0.100492	−	0.994938i	$-0.532042\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2448.80			−3.68241
$666$		0			0
$667$			58.8231i			0.0881905i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	496.319	−	286.550i	0.739671	−	0.427049i
$672$		0			0
$673$	−20.8308	+	36.0800i	−0.0309522	+	0.0536107i	−0.881087	−	0.472955i	$-0.843187\pi$
$673$	−20.8308	+	36.0800i	−0.0309522	+	0.0536107i	0.850134	+	0.526566i	$0.176521\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.60770	−	13.1769i	0.0112374	−	0.0194637i	−0.860352	−	0.509700i	$-0.829756\pi$
$677$	7.60770	−	13.1769i	0.0112374	−	0.0194637i	0.871589	+	0.490237i	$0.163090\pi$
$678$		0			0
$679$	−357.100	+	206.172i	−0.525920	+	0.303640i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			951.384i			1.39295i	0.717581	+	0.696475i	$0.245249\pi$
$683$			951.384i			1.39295i	−0.717581	+	0.696475i	$0.754751\pi$
$684$		0			0
$685$	−73.6232			−0.107479
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−132.315	−	229.177i	−0.192040	−	0.332623i
$690$		0			0
$691$	113.842	+	65.7269i	0.164750	+	0.0951185i	0.580108	−	0.814540i	$-0.303010\pi$
$691$	113.842	+	65.7269i	0.164750	+	0.0951185i	−0.415358	+	0.909658i	$0.636344\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	151.608	+	87.5307i	0.218141	+	0.125944i
$696$		0			0
$697$	429.415	+	743.769i	0.616091	+	1.06710i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	466.930			0.666091			0.333045	−	0.942911i	$-0.391924\pi$
$701$	466.930			0.666091			0.333045	+	0.942911i	$0.391924\pi$
$702$		0			0
$703$			875.987i			1.24607i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1396.25	−	806.123i	1.97489	−	1.14020i
$708$		0			0
$709$	−163.519	+	283.224i	−0.230634	+	0.399469i	−0.957995	−	0.286786i	$-0.907413\pi$
$709$	−163.519	+	283.224i	−0.230634	+	0.399469i	0.727361	+	0.686255i	$0.240747\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	124.708	−	216.000i	0.174906	−	0.302945i
$714$		0			0
$715$	1909.07	−	1102.20i	2.67003	−	1.54154i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	670.361i		−	0.932352i	−0.884692	−	0.466176i	$-0.845631\pi$
$719$		−	670.361i		−	0.932352i	0.884692	−	0.466176i	$-0.154369\pi$
$720$		0			0
$721$	−380.554			−0.527814
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−134.478	−	232.923i	−0.185487	−	0.321273i
$726$		0			0
$727$	−600.588	−	346.750i	−0.826119	−	0.476960i	0.0264030	−	0.999651i	$-0.491595\pi$
$727$	−600.588	−	346.750i	−0.826119	−	0.476960i	−0.852522	+	0.522691i	$0.824928\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	665.181	+	384.042i	0.909960	+	0.525366i
$732$		0			0
$733$	157.000	+	271.932i	0.214188	+	0.370985i	0.953021	−	0.302904i	$-0.0979561\pi$
$733$	157.000	+	271.932i	0.214188	+	0.370985i	−0.738833	+	0.673889i	$0.764623\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	670.277			0.909467
$738$		0			0
$739$			233.569i			0.316061i	0.987434	+	0.158031i	$0.0505145\pi$
$739$			233.569i			0.316061i	−0.987434	+	0.158031i	$0.949486\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	803.538	−	463.923i	1.08148	−	0.624392i	0.150183	−	0.988658i	$-0.452014\pi$
$743$	803.538	−	463.923i	1.08148	−	0.624392i	0.931295	+	0.364266i	$0.118680\pi$
$744$		0			0
$745$	249.423	−	432.013i	0.334796	−	0.579884i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−722.985	+	1252.25i	−0.965266	+	1.67189i
$750$		0			0
$751$	371.096	−	214.252i	0.494136	−	0.285290i	−0.232153	−	0.972679i	$-0.574577\pi$
$751$	371.096	−	214.252i	0.494136	−	0.285290i	0.726289	+	0.687390i	$0.241244\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1531.46i		−	2.02843i
$756$		0			0
$757$	−342.246			−0.452108			−0.226054	−	0.974115i	$-0.572583\pi$
$757$	−342.246			−0.452108			−0.226054	+	0.974115i	$0.572583\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−234.884	−	406.831i	−0.308652	−	0.534600i	0.669416	−	0.742888i	$-0.266544\pi$
$761$	−234.884	−	406.831i	−0.308652	−	0.534600i	−0.978068	+	0.208287i	$0.933211\pi$
$762$		0			0
$763$	−660.673	−	381.440i	−0.865889	−	0.499921i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1445.97	−	834.831i	−1.88523	−	1.08844i
$768$		0			0
$769$	−136.123	−	235.772i	−0.177013	−	0.306596i	0.763843	−	0.645402i	$-0.223310\pi$
$769$	−136.123	−	235.772i	−0.177013	−	0.306596i	−0.940856	+	0.338806i	$0.889977\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1135.27			1.46866			0.734328	−	0.678795i	$-0.237498\pi$
$773$	1135.27			1.46866			0.734328	+	0.678795i	$0.237498\pi$
$774$		0			0
$775$			1140.40i			1.47148i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1204.71	+	695.538i	−1.54648	+	0.892860i
$780$		0			0
$781$	−422.354	+	731.538i	−0.540786	+	0.936669i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1098.34	−	1902.38i	1.39916	−	2.42341i
$786$		0			0
$787$	531.657	−	306.953i	0.675550	−	0.390029i	−0.122627	−	0.992453i	$-0.539132\pi$
$787$	531.657	−	306.953i	0.675550	−	0.390029i	0.798176	+	0.602424i	$0.205798\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	234.115i		−	0.295974i
$792$		0			0
$793$	−608.061			−0.766786
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−298.381	−	516.812i	−0.374381	−	0.648446i	0.615854	−	0.787861i	$-0.288811\pi$
$797$	−298.381	−	516.812i	−0.374381	−	0.648446i	−0.990234	+	0.139414i	$0.955478\pi$
$798$		0			0
$799$	−429.415	−	247.923i	−0.537441	−	0.310292i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1875.86	+	1083.03i	2.33606	+	1.34873i
$804$		0			0
$805$	−278.354	−	482.123i	−0.345781	−	0.598911i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−23.8935			−0.0295346			−0.0147673	−	0.999891i	$-0.504701\pi$
$809$	−23.8935			−0.0295346			−0.0147673	+	0.999891i	$0.504701\pi$
$810$		0			0
$811$			548.092i			0.675822i	0.941178	+	0.337911i	$0.109720\pi$
$811$			548.092i			0.675822i	−0.941178	+	0.337911i	$0.890280\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	154.046	−	88.9385i	0.189014	−	0.109127i
$816$		0			0
$817$	−622.046	+	1077.42i	−0.761378	+	1.31875i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	488.027	−	845.288i	0.594431	−	1.02958i	−0.399196	−	0.916865i	$-0.630711\pi$
$821$	488.027	−	845.288i	0.594431	−	1.02958i	0.993627	−	0.112718i	$-0.0359558\pi$
$822$		0			0
$823$	1177.49	−	679.825i	1.43073	−	0.826033i	0.433555	−	0.901127i	$-0.357259\pi$
$823$	1177.49	−	679.825i	1.43073	−	0.826033i	0.997176	+	0.0750936i	$0.0239255\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	663.762i		−	0.802614i	−0.915944	−	0.401307i	$-0.868556\pi$
$827$		−	663.762i		−	0.802614i	0.915944	−	0.401307i	$-0.131444\pi$
$828$		0			0
$829$	−403.108			−0.486258			−0.243129	−	0.969994i	$-0.578174\pi$
$829$	−403.108			−0.486258			−0.243129	+	0.969994i	$0.578174\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−419.085	−	725.877i	−0.503103	−	0.871401i
$834$		0			0
$835$	−1458.47	−	842.050i	−1.74667	−	1.00844i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	891.773	+	514.865i	1.06290	+	0.613665i	0.926233	−	0.376952i	$-0.123028\pi$
$839$	891.773	+	514.865i	1.06290	+	0.613665i	0.136667	+	0.990617i	$0.456361\pi$
$840$		0			0
$841$	390.608	+	676.552i	0.464456	+	0.804462i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1032.17			−1.22150
$846$		0			0
$847$		−	1397.92i		−	1.65044i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−172.465	+	99.5730i	−0.202662	+	0.117007i
$852$		0			0
$853$	−233.708	+	404.794i	−0.273983	+	0.474553i	−0.969878	−	0.243591i	$-0.921675\pi$
$853$	−233.708	+	404.794i	−0.273983	+	0.474553i	0.695895	+	0.718144i	$0.255008\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	240.178	−	416.000i	0.280254	−	0.485414i	−0.691193	−	0.722670i	$-0.742915\pi$
$857$	240.178	−	416.000i	0.280254	−	0.485414i	0.971447	+	0.237256i	$0.0762480\pi$
$858$		0			0
$859$	297.015	−	171.482i	0.345769	−	0.199630i	−0.317051	−	0.948408i	$-0.602693\pi$
$859$	297.015	−	171.482i	0.345769	−	0.199630i	0.662820	+	0.748779i	$0.269359\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	888.084i		−	1.02907i	−0.857470	−	0.514533i	$-0.827965\pi$
$863$		−	888.084i		−	1.02907i	0.857470	−	0.514533i	$-0.172035\pi$
$864$		0			0
$865$	849.862			0.982499
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−407.138	−	705.184i	−0.468514	−	0.811490i
$870$		0			0
$871$	−615.888	−	355.583i	−0.707105	−	0.408247i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	620.081	+	358.004i	0.708664	+	0.409147i
$876$		0			0
$877$	354.973	+	614.831i	0.404758	+	0.701062i	0.994293	−	0.106681i	$-0.0340223\pi$
$877$	354.973	+	614.831i	0.404758	+	0.701062i	−0.589535	+	0.807743i	$0.700689\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−996.862			−1.13151			−0.565756	−	0.824573i	$-0.691415\pi$
$881$	−996.862			−1.13151			−0.565756	+	0.824573i	$0.691415\pi$
$882$		0			0
$883$		−	897.513i		−	1.01644i	−0.861228	−	0.508218i	$-0.830304\pi$
$883$		−	897.513i		−	1.01644i	0.861228	−	0.508218i	$-0.169696\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1098.53	−	634.235i	1.23847	−	0.715033i	0.269692	−	0.962947i	$-0.413078\pi$
$887$	1098.53	−	634.235i	1.23847	−	0.715033i	0.968782	+	0.247913i	$0.0797448\pi$
$888$		0			0
$889$	−123.215	+	213.415i	−0.138600	+	0.240062i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	401.569	−	695.538i	0.449686	−	0.778878i
$894$		0			0
$895$	−1274.12	+	735.615i	−1.42360	+	0.821916i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			253.492i			0.281971i
$900$		0			0
$901$	−314.354			−0.348894
$902$		0			0
$903$		0			0
$904$		0			0
$905$	134.776	+	233.438i	0.148923	+	0.257943i
$906$		0			0
$907$	−45.7307	−	26.4026i	−0.0504197	−	0.0291098i	0.474578	−	0.880213i	$-0.342601\pi$
$907$	−45.7307	−	26.4026i	−0.0504197	−	0.0291098i	−0.524998	+	0.851103i	$0.675934\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−545.096	−	314.711i	−0.598349	−	0.345457i	0.170043	−	0.985437i	$-0.445609\pi$
$911$	−545.096	−	314.711i	−0.598349	−	0.345457i	−0.768392	+	0.639980i	$0.778943\pi$
$912$		0			0
$913$	−753.415	−	1304.95i	−0.825208	−	1.42930i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2055.38			−2.24142
$918$		0			0
$919$			1112.18i			1.21020i	0.796148	+	0.605101i	$0.206867\pi$
$919$			1112.18i			1.21020i	−0.796148	+	0.605101i	$0.793133\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	776.165	−	448.119i	0.840916	−	0.485503i
$924$		0			0
$925$	455.277	−	788.563i	0.492191	−	0.852500i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−187.470	+	324.707i	−0.201798	+	0.349524i	−0.949108	−	0.314952i	$-0.898012\pi$
$929$	−187.470	+	324.707i	−0.201798	+	0.349524i	0.747310	+	0.664475i	$0.231345\pi$
$930$		0			0
$931$	1175.73	−	678.806i	1.26286	−	0.729115i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	2618.61i		−	2.80065i
$936$		0			0
$937$	555.246			0.592578			0.296289	−	0.955098i	$-0.404251\pi$
$937$	555.246			0.592578			0.296289	+	0.955098i	$0.404251\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−323.243	−	559.873i	−0.343510	−	0.594977i	0.641572	−	0.767063i	$-0.278283\pi$
$941$	−323.243	−	559.873i	−0.343510	−	0.594977i	−0.985082	+	0.172086i	$0.944949\pi$
$942$		0			0
$943$	−273.877	−	158.123i	−0.290432	−	0.167681i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	221.642	+	127.965i	0.234047	+	0.135127i	0.612438	−	0.790519i	$-0.290189\pi$
$947$	221.642	+	127.965i	0.234047	+	0.135127i	−0.378391	+	0.925646i	$0.623522\pi$
$948$		0			0
$949$	−1149.10	−	1990.29i	−1.21085	−	2.09725i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	45.1220			0.0473473			0.0236736	−	0.999720i	$-0.492464\pi$
$953$	45.1220			0.0473473			0.0236736	+	0.999720i	$0.492464\pi$
$954$		0			0
$955$		−	1276.56i		−	1.33671i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	78.0422	−	45.0577i	0.0813788	−	0.0469841i
$960$		0			0
$961$	56.9153	−	98.5802i	0.0592251	−	0.102581i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	165.942	−	287.419i	0.171960	−	0.297844i
$966$		0			0
$967$	891.657	−	514.799i	0.922086	−	0.532367i	0.0377863	−	0.999286i	$-0.487969\pi$
$967$	891.657	−	514.799i	0.922086	−	0.532367i	0.884300	+	0.466919i	$0.154636\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			405.731i			0.417848i	0.977932	+	0.208924i	$0.0669962\pi$
$971$			405.731i			0.417848i	−0.977932	+	0.208924i	$0.933004\pi$
$972$		0			0
$973$	−214.277			−0.220223
$974$		0			0
$975$		0			0
$976$		0			0
$977$	547.061	+	947.538i	0.559940	+	0.969845i	0.997501	+	0.0706558i	$0.0225092\pi$
$977$	547.061	+	947.538i	0.559940	+	0.969845i	−0.437561	+	0.899189i	$0.644157\pi$
$978$		0			0
$979$	−1102.95	−	636.788i	−1.12661	−	0.650448i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−350.900	−	202.592i	−0.356969	−	0.206096i	0.310782	−	0.950481i	$-0.399409\pi$
$983$	−350.900	−	202.592i	−0.356969	−	0.206096i	−0.667750	+	0.744385i	$0.732743\pi$
$984$		0			0
$985$	−569.877	−	987.056i	−0.578555	−	1.00209i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−282.831			−0.285976
$990$		0			0
$991$		−	181.905i		−	0.183557i	−0.995779	−	0.0917786i	$-0.970745\pi$
$991$		−	181.905i		−	0.183557i	0.995779	−	0.0917786i	$-0.0292552\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1867.23	−	1078.05i	1.87661	−	1.08346i
$996$		0			0
$997$	−438.519	+	759.538i	−0.439839	+	0.761823i	−0.997677	−	0.0681268i	$-0.978298\pi$
$997$	−438.519	+	759.538i	−0.439839	+	0.761823i	0.557838	+	0.829950i	$0.311631\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.o.be.703.2		4
3.2	odd	2		1296.3.o.q.703.1		4
4.3	odd	2		1296.3.o.bf.703.2		4
9.2	odd	6		1296.3.g.i.1135.4	yes	4
9.4	even	3		1296.3.o.bf.271.2		4
9.5	odd	6		1296.3.o.r.271.1		4
9.7	even	3		1296.3.g.c.1135.2	yes	4
12.11	even	2		1296.3.o.r.703.1		4
36.7	odd	6		1296.3.g.c.1135.1	✓	4
36.11	even	6		1296.3.g.i.1135.3	yes	4
36.23	even	6		1296.3.o.q.271.1		4
36.31	odd	6	inner	1296.3.o.be.271.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1296.3.g.c.1135.1	✓	4	36.7	odd	6
1296.3.g.c.1135.2	yes	4	9.7	even	3
1296.3.g.i.1135.3	yes	4	36.11	even	6
1296.3.g.i.1135.4	yes	4	9.2	odd	6
1296.3.o.q.271.1		4	36.23	even	6
1296.3.o.q.703.1		4	3.2	odd	2
1296.3.o.r.271.1		4	9.5	odd	6
1296.3.o.r.703.1		4	12.11	even	2
1296.3.o.be.271.2		4	36.31	odd	6	inner
1296.3.o.be.703.2		4	1.1	even	1	trivial
1296.3.o.bf.271.2		4	9.4	even	3
1296.3.o.bf.703.2		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.86603 + 6.69615i) q^{5} +(-8.19615 - 4.73205i) q^{7} +O(q^{10})\)	\(q+(3.86603 + 6.69615i) q^{5} +(-8.19615 - 4.73205i) q^{7} +(14.1962 + 8.19615i) q^{11} +(-8.69615 - 15.0622i) q^{13} -20.6603 q^{17} -33.4641i q^{19} +(6.58846 - 3.80385i) q^{23} +(-17.3923 + 30.1244i) q^{25} +(-3.86603 + 6.69615i) q^{29} +(28.3923 - 16.3923i) q^{31} -73.1769i q^{35} -26.1769 q^{37} +(-20.7846 - 36.0000i) q^{41} +(-32.1962 - 18.5885i) q^{43} +(20.7846 + 12.0000i) q^{47} +(20.2846 + 35.1340i) q^{49} +15.2154 q^{53} +126.746i q^{55} +(83.1384 - 48.0000i) q^{59} +(17.4808 - 30.2776i) q^{61} +(67.2391 - 116.462i) q^{65} +(35.4115 - 20.4449i) q^{67} +51.5307i q^{71} +132.138 q^{73} +(-77.5692 - 134.354i) q^{77} +(-43.0192 - 24.8372i) q^{79} +(-79.6077 - 45.9615i) q^{83} +(-79.8731 - 138.344i) q^{85} -77.6936 q^{89} +164.603i q^{91} +(224.081 - 129.373i) q^{95} +(21.7846 - 37.7321i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{5} - 12 q^{7}+O(q^{10}) \) 4 * q + 12 * q^5 - 12 * q^7	\( 4 q + 12 q^{5} - 12 q^{7} + 36 q^{11} - 14 q^{13} - 48 q^{17} - 36 q^{23} - 28 q^{25} - 12 q^{29} + 72 q^{31} + 20 q^{37} - 108 q^{43} - 2 q^{49} + 144 q^{53} - 34 q^{61} + 120 q^{65} + 204 q^{67} + 196 q^{73} - 144 q^{77} - 276 q^{79} - 360 q^{83} - 174 q^{85} - 96 q^{89} + 252 q^{95} + 4 q^{97}+O(q^{100}) \) 4 * q + 12 * q^5 - 12 * q^7 + 36 * q^11 - 14 * q^13 - 48 * q^17 - 36 * q^23 - 28 * q^25 - 12 * q^29 + 72 * q^31 + 20 * q^37 - 108 * q^43 - 2 * q^49 + 144 * q^53 - 34 * q^61 + 120 * q^65 + 204 * q^67 + 196 * q^73 - 144 * q^77 - 276 * q^79 - 360 * q^83 - 174 * q^85 - 96 * q^89 + 252 * q^95 + 4 * q^97

Introduction

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