LMFDB - Embedded newform 1296.3.o.be.271.2

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			271.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1296.271
Dual form			1296.3.o.be.703.2

$q$-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 $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

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

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	−0.162593	−	0.986693i	$-0.551986\pi$
$5$	3.86603	−	6.69615i	0.773205	−	1.33923i	0.935798	−	0.352537i	$-0.114681\pi$
$6$		0			0
$7$	−8.19615	+	4.73205i	−1.17088	+	0.676007i	−0.953887	−	0.300166i	$-0.902958\pi$
$7$	−8.19615	+	4.73205i	−1.17088	+	0.676007i	−0.216992	+	0.976173i	$0.569625\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	14.1962	−	8.19615i	1.29056	−	0.745105i	0.311806	−	0.950146i	$-0.399066\pi$
$11$	14.1962	−	8.19615i	1.29056	−	0.745105i	0.978753	+	0.205041i	$0.0657328\pi$
$12$		0			0
$13$	−8.69615	+	15.0622i	−0.668935	+	1.15863i	0.309268	+	0.950975i	$0.399916\pi$
$13$	−8.69615	+	15.0622i	−0.668935	+	1.15863i	−0.978202	+	0.207654i	$0.933417\pi$
$14$		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	$0.342883\pi$
$19$			33.4641i			1.76127i	−0.473797	+	0.880634i	$0.657117\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.58846	+	3.80385i	0.286455	+	0.165385i	0.636342	−	0.771407i	$-0.280447\pi$
$23$	6.58846	+	3.80385i	0.286455	+	0.165385i	−0.349887	+	0.936792i	$0.613780\pi$
$24$		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	0.791640	−	0.610988i	$-0.209228\pi$
$29$	−3.86603	−	6.69615i	−0.133311	−	0.230902i	−0.924951	+	0.380086i	$0.875894\pi$
$30$		0			0
$31$	28.3923	+	16.3923i	0.915881	+	0.528784i	0.882319	−	0.470653i	$-0.155981\pi$
$31$	28.3923	+	16.3923i	0.915881	+	0.528784i	0.0335622	+	0.999437i	$0.489315\pi$
$32$		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	0.493026	+	0.870015i	$0.335891\pi$
$41$	−20.7846	+	36.0000i	−0.506942	+	0.878049i	−0.999968	+	0.00803422i	$0.997443\pi$
$42$		0			0
$43$	−32.1962	+	18.5885i	−0.748748	+	0.432290i	−0.825241	−	0.564780i	$-0.808961\pi$
$43$	−32.1962	+	18.5885i	−0.748748	+	0.432290i	0.0764935	+	0.997070i	$0.475628\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	20.7846	−	12.0000i	0.442226	−	0.255319i	−0.262316	−	0.964982i	$-0.584486\pi$
$47$	20.7846	−	12.0000i	0.442226	−	0.255319i	0.704541	+	0.709663i	$0.251153\pi$
$48$		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	0.995304	−	0.0967985i	$-0.0308602\pi$
$59$	83.1384	+	48.0000i	1.40913	+	0.813559i	0.413822	+	0.910358i	$0.364194\pi$
$60$		0			0
$61$	17.4808	+	30.2776i	0.286570	+	0.496354i	0.972989	−	0.230853i	$-0.0741516\pi$
$61$	17.4808	+	30.2776i	0.286570	+	0.496354i	−0.686419	+	0.727206i	$0.740818\pi$
$62$		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	0.740418	−	0.672147i	$-0.234628\pi$
$67$	35.4115	+	20.4449i	0.528530	+	0.305147i	−0.211887	+	0.977294i	$0.567961\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	51.5307i		−	0.725785i	−0.931831	−	0.362893i	$-0.881789\pi$
$71$		−	51.5307i		−	0.725785i	0.931831	−	0.362893i	$-0.118211\pi$
$72$		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	−0.746920	−	0.664914i	$-0.768468\pi$
$79$	−43.0192	+	24.8372i	−0.544547	+	0.314395i	0.202373	+	0.979309i	$0.435135\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−79.6077	+	45.9615i	−0.959129	+	0.553753i	−0.895905	−	0.444246i	$-0.853472\pi$
$83$	−79.6077	+	45.9615i	−0.959129	+	0.553753i	−0.0632240	+	0.997999i	$0.520138\pi$
$84$		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	0.956194	−	0.292732i	$-0.0945645\pi$
$97$	21.7846	+	37.7321i	0.224584	+	0.388990i	−0.731611	+	0.681723i	$0.761231\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−85.1769	−	147.531i	−0.843336	−	1.46070i	−0.887059	−	0.461657i	$-0.847255\pi$
$101$	−85.1769	−	147.531i	−0.843336	−	1.46070i	0.0437229	−	0.999044i	$-0.486078\pi$
$102$		0			0
$103$	34.8231	+	20.1051i	0.338088	+	0.195195i	0.659426	−	0.751769i	$-0.270799\pi$
$103$	34.8231	+	20.1051i	0.338088	+	0.195195i	−0.321338	+	0.946965i	$0.604133\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			152.785i			1.42789i	0.700200	+	0.713947i	$0.253094\pi$
$107$			152.785i			1.42789i	−0.700200	+	0.713947i	$0.746906\pi$
$108$		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	−0.806094	−	0.591788i	$-0.798422\pi$
$113$	12.3686	−	21.4230i	0.109457	−	0.189584i	0.915550	+	0.402204i	$0.131756\pi$
$114$		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	$0.0326885\pi$
$127$			26.0385i			0.205027i	−0.994732	+	0.102514i	$0.967311\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	188.081	+	108.588i	1.43573	+	0.828920i	0.997549	−	0.0699670i	$-0.0222894\pi$
$131$	188.081	+	108.588i	1.43573	+	0.828920i	0.438181	+	0.898887i	$0.355623\pi$
$132$		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	0.848127	−	0.529793i	$-0.177731\pi$
$137$	−4.76091	−	8.24613i	−0.0347512	−	0.0601908i	−0.882878	+	0.469603i	$0.844397\pi$
$138$		0			0
$139$	19.6077	+	11.3205i	0.141063	+	0.0814425i	0.568870	−	0.822427i	$-0.307381\pi$
$139$	19.6077	+	11.3205i	0.141063	+	0.0814425i	−0.427808	+	0.903870i	$0.640714\pi$
$140$		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	−0.953735	−	0.300648i	$-0.902797\pi$
$149$	−32.2583	+	55.8731i	−0.216499	+	0.374987i	0.737236	+	0.675635i	$0.236130\pi$
$150$		0			0
$151$	−171.531	+	99.0333i	−1.13597	+	0.655850i	−0.945428	−	0.325830i	$-0.894356\pi$
$151$	−171.531	+	99.0333i	−1.13597	+	0.655850i	−0.190537	+	0.981680i	$0.561023\pi$
$152$		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	−0.0835606	+	0.996503i	$0.526629\pi$
$157$	−142.050	+	246.038i	−0.904777	+	1.56712i	−0.821216	+	0.570617i	$0.806704\pi$
$158$		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	$0.0224812\pi$
$163$			23.0052i			0.141136i	−0.997507	+	0.0705680i	$0.977519\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−188.627	−	108.904i	−1.12950	−	0.652119i	−0.185693	−	0.982608i	$-0.559453\pi$
$167$	−188.627	−	108.904i	−1.12950	−	0.652119i	−0.943810	+	0.330489i	$0.892786\pi$
$168$		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	0.980002	−	0.198989i	$-0.0637660\pi$
$173$	54.9571	+	95.1884i	0.317671	+	0.550222i	−0.662331	+	0.749212i	$0.730433\pi$
$174$		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	$-0.821629\pi$
$179$		−	190.277i		−	1.06300i	0.847059	−	0.531500i	$-0.178371\pi$
$180$		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	−0.825184	−	0.564864i	$-0.808929\pi$
$191$	−142.981	+	82.5500i	−0.748590	+	0.432199i	0.0765940	+	0.997062i	$0.475595\pi$
$192$		0			0
$193$	−21.4615	+	37.1725i	−0.111200	+	0.192603i	−0.916254	−	0.400597i	$-0.868803\pi$
$193$	−21.4615	+	37.1725i	−0.111200	+	0.192603i	0.805055	+	0.593201i	$0.202136\pi$
$194$		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	$0.247098\pi$
$199$			278.851i			1.40126i	−0.713524	+	0.700631i	$0.752902\pi$
$200$		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	0.745497	−	0.666509i	$-0.232212\pi$
$211$	114.158	+	65.9090i	0.541032	+	0.312365i	−0.204465	+	0.978874i	$0.565546\pi$
$212$		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	−0.566092	−	0.824342i	$-0.691545\pi$
$223$	−30.1577	+	17.4115i	−0.135236	+	0.0780787i	0.430855	+	0.902421i	$0.358212\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	37.0192	−	21.3731i	0.163080	−	0.0941545i	−0.416239	−	0.909255i	$-0.636652\pi$
$227$	37.0192	−	21.3731i	0.163080	−	0.0941545i	0.579319	+	0.815101i	$0.303319\pi$
$228$		0			0
$229$	96.4038	−	166.976i	0.420977	−	0.729154i	−0.575058	−	0.818113i	$-0.695021\pi$
$229$	96.4038	−	166.976i	0.420977	−	0.729154i	0.996035	+	0.0889585i	$0.0283539\pi$
$230$		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	0.867991	−	0.496581i	$-0.165411\pi$
$239$	208.392	+	120.315i	0.871934	+	0.503412i	0.00394377	+	0.999992i	$0.498745\pi$
$240$		0			0
$241$	147.031	+	254.665i	0.610086	+	1.05670i	0.991225	+	0.132182i	$0.0421985\pi$
$241$	147.031	+	254.665i	0.610086	+	1.05670i	−0.381139	+	0.924518i	$0.624468\pi$
$242$		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	$0.0327324\pi$
$251$			51.5307i			0.205302i	−0.994717	+	0.102651i	$0.967268\pi$
$252$		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	−0.0598552	+	0.998207i	$0.519064\pi$
$257$	−229.861	+	398.131i	−0.894400	+	1.54915i	−0.834545	+	0.550940i	$0.814269\pi$
$258$		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	−0.950964	−	0.309301i	$-0.899905\pi$
$263$	−304.708	+	175.923i	−1.15858	+	0.668909i	−0.207620	+	0.978210i	$0.566572\pi$
$264$		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	$-0.772487\pi$
$271$		−	355.244i		−	1.31086i	0.755255	−	0.655431i	$-0.227513\pi$
$272$		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	−0.947896	−	0.318581i	$-0.896794\pi$
$277$	−207.708	−	359.760i	−0.749847	−	1.29877i	0.198048	−	0.980192i	$-0.436540\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−120.493	−	208.700i	−0.428801	−	0.742704i	0.567966	−	0.823052i	$-0.307730\pi$
$281$	−120.493	−	208.700i	−0.428801	−	0.742704i	−0.996767	+	0.0803474i	$0.974397\pi$
$282$		0			0
$283$	92.7846	+	53.5692i	0.327861	+	0.189291i	0.654891	−	0.755723i	$-0.272715\pi$
$283$	92.7846	+	53.5692i	0.327861	+	0.189291i	−0.327030	+	0.945014i	$0.606048\pi$
$284$		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	−0.362747	−	0.931888i	$-0.618161\pi$
$293$	183.320	−	317.519i	0.625665	−	1.08368i	0.988412	−	0.151796i	$-0.0485056\pi$
$294$		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	$0.0911007\pi$
$307$			173.338i			0.564620i	−0.959323	+	0.282310i	$0.908899\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	93.3308	+	53.8846i	0.300099	+	0.173262i	0.642487	−	0.766296i	$-0.277902\pi$
$311$	93.3308	+	53.8846i	0.300099	+	0.173262i	−0.342388	+	0.939559i	$0.611236\pi$
$312$		0			0
$313$	32.9693	+	57.1044i	0.105333	+	0.182442i	0.913874	−	0.405997i	$-0.133076\pi$
$313$	32.9693	+	57.1044i	0.105333	+	0.182442i	−0.808541	+	0.588440i	$0.799742\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−273.942	−	474.481i	−0.864169	−	1.49678i	−0.867870	−	0.496792i	$-0.834511\pi$
$317$	−273.942	−	474.481i	−0.864169	−	1.49678i	0.00370080	−	0.999993i	$-0.498822\pi$
$318$		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	−0.277112	−	0.960838i	$-0.589377\pi$
$331$	137.842	−	79.5833i	0.416442	−	0.240433i	0.693554	+	0.720405i	$0.256044\pi$
$332$		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	−0.178148	+	0.984004i	$0.557010\pi$
$337$	−317.200	+	549.406i	−0.941246	+	1.63029i	−0.763098	+	0.646282i	$0.776323\pi$
$338$		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	0.460846	−	0.887480i	$-0.347546\pi$
$347$	−26.8269	−	15.4885i	−0.0773108	−	0.0446354i	−0.538157	+	0.842845i	$0.680879\pi$
$348$		0			0
$349$	−88.2154	−	152.794i	−0.252766	−	0.437804i	0.711520	−	0.702666i	$-0.248007\pi$
$349$	−88.2154	−	152.794i	−0.252766	−	0.437804i	−0.964286	+	0.264862i	$0.914674\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.2923	−	33.4153i	−0.0546525	−	0.0946610i	0.837405	−	0.546583i	$-0.184072\pi$
$353$	−19.2923	−	33.4153i	−0.0546525	−	0.0946610i	−0.892057	+	0.451922i	$0.850738\pi$
$354$		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	$-0.705791\pi$
$359$		−	573.100i		−	1.59638i	0.602407	−	0.798189i	$-0.294209\pi$
$360$		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	−0.141950	−	0.989874i	$-0.545337\pi$
$367$	236.469	−	136.526i	0.644330	−	0.372004i	0.786281	+	0.617869i	$0.212004\pi$
$368$		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	−0.689693	−	0.724102i	$-0.742255\pi$
$373$	105.277	−	182.345i	0.282244	−	0.488860i	0.971937	+	0.235241i	$0.0755880\pi$
$374$		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	$0.134407\pi$
$379$			310.641i			0.819633i	−0.912168	+	0.409817i	$0.865593\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−403.535	−	232.981i	−1.05361	−	0.608305i	−0.129956	−	0.991520i	$-0.541484\pi$
$383$	−403.535	−	232.981i	−1.05361	−	0.608305i	−0.923659	+	0.383215i	$0.874817\pi$
$384$		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	0.802216	−	0.597034i	$-0.203654\pi$
$389$	−45.1000	−	78.1154i	−0.115938	−	0.200811i	−0.918154	+	0.396223i	$0.870321\pi$
$390$		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	−0.221005	−	0.975273i	$-0.570934\pi$
$401$	294.378	−	509.877i	0.734109	−	1.27151i	0.955113	−	0.296241i	$-0.0957330\pi$
$402$		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	0.0325694	−	0.999469i	$-0.489631\pi$
$409$	360.677	−	624.711i	0.881851	−	1.52741i	0.849281	−	0.527941i	$-0.177036\pi$
$410$		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	−0.981426	−	0.191840i	$-0.938555\pi$
$419$	−686.438	−	396.315i	−1.63828	−	0.945860i	−0.656851	−	0.754020i	$-0.728112\pi$
$420$		0			0
$421$	−198.973	−	344.631i	−0.472620	−	0.818602i	0.526889	−	0.849934i	$-0.323358\pi$
$421$	−198.973	−	344.631i	−0.472620	−	0.818602i	−0.999509	+	0.0313322i	$0.990025\pi$
$422$		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	$0.0179674\pi$
$431$			48.6307i			0.112832i	−0.998407	+	0.0564161i	$0.982033\pi$
$432$		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	−0.704528	−	0.709677i	$-0.748841\pi$
$439$	−194.123	+	112.077i	−0.442194	+	0.255301i	0.262334	+	0.964977i	$0.415508\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−259.061	+	149.569i	−0.584789	+	0.337628i	−0.763034	−	0.646358i	$-0.776291\pi$
$443$	−259.061	+	149.569i	−0.584789	+	0.337628i	0.178245	+	0.983986i	$0.442958\pi$
$444$		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	0.841425	−	0.540374i	$-0.181717\pi$
$457$	−21.6000	−	37.4122i	−0.0472647	−	0.0818648i	−0.888690	+	0.458509i	$0.848384\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.7616	+	27.2999i	0.0341900	+	0.0592188i	0.882614	−	0.470098i	$-0.155782\pi$
$461$	15.7616	+	27.2999i	0.0341900	+	0.0592188i	−0.848424	+	0.529317i	$0.822448\pi$
$462$		0			0
$463$	5.33836	+	3.08211i	0.0115299	+	0.00665682i	0.505754	−	0.862678i	$-0.331214\pi$
$463$	5.33836	+	3.08211i	0.0115299	+	0.00665682i	−0.494224	+	0.869335i	$0.664548\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			330.200i			0.707066i	0.935422	+	0.353533i	$0.115020\pi$
$467$			330.200i			0.707066i	−0.935422	+	0.353533i	$0.884980\pi$
$468$		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	−0.438447	−	0.898757i	$-0.644471\pi$
$479$	57.8038	−	33.3731i	0.120676	−	0.0696724i	0.559123	+	0.829085i	$0.311138\pi$
$480$		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	$0.104262\pi$
$487$			313.359i			0.643448i	−0.946834	+	0.321724i	$0.895738\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	149.096	+	86.0807i	0.303658	+	0.175317i	0.644085	−	0.764954i	$-0.277238\pi$
$491$	149.096	+	86.0807i	0.303658	+	0.175317i	−0.340427	+	0.940271i	$0.610572\pi$
$492$		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	0.517459	−	0.855708i	$-0.326878\pi$
$499$	17.5270	+	10.1192i	0.0351242	+	0.0202790i	−0.482335	+	0.875987i	$0.660211\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	368.785i		−	0.733170i	−0.930385	−	0.366585i	$-0.880527\pi$
$503$		−	368.785i		−	0.733170i	0.930385	−	0.366585i	$-0.119473\pi$
$504$		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	−0.727766	−	0.685825i	$-0.759441\pi$
$509$	117.100	−	202.823i	0.230059	−	0.398474i	0.957825	+	0.287352i	$0.0927748\pi$
$510$		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	$0.167674\pi$
$523$			525.864i			1.00548i	−0.864439	+	0.502738i	$0.832326\pi$
$524$		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	0.487157	−	0.873314i	$-0.338034\pi$
$547$	813.415	−	469.626i	1.48705	−	0.858548i	0.999891	+	0.0147667i	$0.00470056\pi$
$548$		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	0.570754	−	0.821121i	$-0.306651\pi$
$563$	81.6462	+	47.1384i	0.145020	+	0.0837272i	−0.425734	+	0.904848i	$0.639984\pi$
$564$		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	0.884274	−	0.466969i	$-0.154654\pi$
$569$	21.4686	+	37.1846i	0.0377303	+	0.0653509i	−0.846544	+	0.532319i	$0.821321\pi$
$570$		0			0
$571$	−366.431	−	211.559i	−0.641735	−	0.370506i	0.143547	−	0.989643i	$-0.454149\pi$
$571$	−366.431	−	211.559i	−0.641735	−	0.370506i	−0.785283	+	0.619137i	$0.787482\pi$
$572$		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	−0.874382	−	0.485238i	$-0.838733\pi$
$587$	−523.219	+	302.081i	−0.891344	+	0.514618i	−0.0169624	+	0.999856i	$0.505400\pi$
$588$		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	0.991192	−	0.132436i	$-0.0422799\pi$
$599$	821.885	+	474.515i	1.37209	+	0.792179i	0.380903	+	0.924615i	$0.375613\pi$
$600$		0			0
$601$	181.362	+	314.127i	0.301766	+	0.522675i	0.976536	−	0.215354i	$-0.0690904\pi$
$601$	181.362	+	314.127i	0.301766	+	0.522675i	−0.674770	+	0.738028i	$0.735757\pi$
$602$		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	−0.499841	−	0.866117i	$-0.666608\pi$
$607$	−910.404	−	525.622i	−1.49984	−	0.865934i	−1.00000		0.000183305i	$0.999942\pi$
$608$		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	0.438546	+	0.898709i	$0.355494\pi$
$617$	−344.922	+	597.423i	−0.559031	+	0.968271i	−0.997578	+	0.0695621i	$0.977840\pi$
$618$		0			0
$619$	−766.823	+	442.726i	−1.23881	+	0.715227i	−0.968851	−	0.247645i	$-0.920343\pi$
$619$	−766.823	+	442.726i	−1.23881	+	0.715227i	−0.269959	+	0.962872i	$0.587010\pi$
$620$		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	$0.215363\pi$
$631$			790.179i			1.25227i	−0.779717	+	0.626133i	$0.784637\pi$
$632$		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	0.972464	−	0.233055i	$-0.0748722\pi$
$641$	182.301	+	315.754i	0.284400	+	0.492596i	−0.688063	+	0.725651i	$0.741539\pi$
$642$		0			0
$643$	−1030.13	−	594.746i	−1.60207	−	0.924955i	−0.991073	−	0.133322i	$-0.957436\pi$
$643$	−1030.13	−	594.746i	−1.60207	−	0.924955i	−0.610996	−	0.791633i	$-0.709231\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	68.0075i		−	0.105112i	−0.998618	−	0.0525561i	$-0.983263\pi$
$647$		−	68.0075i		−	0.105112i	0.998618	−	0.0525561i	$-0.0167368\pi$
$648$		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	−0.980307	−	0.197480i	$-0.936724\pi$
$653$	−208.392	+	360.946i	−0.319131	+	0.552750i	0.661176	+	0.750231i	$0.270058\pi$
$654$		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	−0.588204	+	0.808712i	$0.700165\pi$
$659$	−1042.98	+	602.165i	−1.58267	+	0.913756i	−0.994468	+	0.105044i	$0.966502\pi$
$660$		0			0
$661$	−602.758	+	1044.01i	−0.911888	+	1.57944i	−0.100492	+	0.994938i	$0.532042\pi$
$661$	−602.758	+	1044.01i	−0.911888	+	1.57944i	−0.811395	+	0.584498i	$0.801292\pi$
$662$		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	0.850134	−	0.526566i	$-0.176521\pi$
$673$	−20.8308	−	36.0800i	−0.0309522	−	0.0536107i	−0.881087	+	0.472955i	$0.843187\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.60770	+	13.1769i	0.0112374	+	0.0194637i	0.871589	−	0.490237i	$-0.163090\pi$
$677$	7.60770	+	13.1769i	0.0112374	+	0.0194637i	−0.860352	+	0.509700i	$0.829756\pi$
$678$		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	$-0.754751\pi$
$683$		−	951.384i		−	1.39295i	0.717581	−	0.696475i	$-0.245249\pi$
$684$		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	−0.415358	−	0.909658i	$-0.636344\pi$
$691$	113.842	−	65.7269i	0.164750	−	0.0951185i	0.580108	+	0.814540i	$0.303010\pi$
$692$		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	0.727361	−	0.686255i	$-0.240747\pi$
$709$	−163.519	−	283.224i	−0.230634	−	0.399469i	−0.957995	+	0.286786i	$0.907413\pi$
$710$		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	$0.154369\pi$
$719$			670.361i			0.932352i	−0.884692	+	0.466176i	$0.845631\pi$
$720$		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	−0.852522	−	0.522691i	$-0.824928\pi$
$727$	−600.588	+	346.750i	−0.826119	+	0.476960i	0.0264030	+	0.999651i	$0.491595\pi$
$728$		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	−0.738833	−	0.673889i	$-0.764623\pi$
$733$	157.000	−	271.932i	0.214188	−	0.370985i	0.953021	+	0.302904i	$0.0979561\pi$
$734$		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	$-0.949486\pi$
$739$		−	233.569i		−	0.316061i	0.987434	−	0.158031i	$-0.0505145\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	803.538	+	463.923i	1.08148	+	0.624392i	0.931295	−	0.364266i	$-0.118680\pi$
$743$	803.538	+	463.923i	1.08148	+	0.624392i	0.150183	+	0.988658i	$0.452014\pi$
$744$		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	0.726289	−	0.687390i	$-0.241244\pi$
$751$	371.096	+	214.252i	0.494136	+	0.285290i	−0.232153	+	0.972679i	$0.574577\pi$
$752$		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	−0.978068	−	0.208287i	$-0.933211\pi$
$761$	−234.884	+	406.831i	−0.308652	+	0.534600i	0.669416	+	0.742888i	$0.266544\pi$
$762$		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	−0.940856	−	0.338806i	$-0.889977\pi$
$769$	−136.123	+	235.772i	−0.177013	+	0.306596i	0.763843	+	0.645402i	$0.223310\pi$
$770$		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	0.798176	−	0.602424i	$-0.205798\pi$
$787$	531.657	+	306.953i	0.675550	+	0.390029i	−0.122627	+	0.992453i	$0.539132\pi$
$788$		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	−0.990234	−	0.139414i	$-0.955478\pi$
$797$	−298.381	+	516.812i	−0.374381	+	0.648446i	0.615854	+	0.787861i	$0.288811\pi$
$798$		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	$-0.890280\pi$
$811$		−	548.092i		−	0.675822i	0.941178	−	0.337911i	$-0.109720\pi$
$812$		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	0.993627	+	0.112718i	$0.0359558\pi$
$821$	488.027	+	845.288i	0.594431	+	1.02958i	−0.399196	+	0.916865i	$0.630711\pi$
$822$		0			0
$823$	1177.49	+	679.825i	1.43073	+	0.826033i	0.997176	−	0.0750936i	$-0.0239255\pi$
$823$	1177.49	+	679.825i	1.43073	+	0.826033i	0.433555	+	0.901127i	$0.357259\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			663.762i			0.802614i	0.915944	+	0.401307i	$0.131444\pi$
$827$			663.762i			0.802614i	−0.915944	+	0.401307i	$0.868556\pi$
$828$		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	0.136667	−	0.990617i	$-0.456361\pi$
$839$	891.773	−	514.865i	1.06290	−	0.613665i	0.926233	+	0.376952i	$0.123028\pi$
$840$		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	0.695895	−	0.718144i	$-0.255008\pi$
$853$	−233.708	−	404.794i	−0.273983	−	0.474553i	−0.969878	+	0.243591i	$0.921675\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	240.178	+	416.000i	0.280254	+	0.485414i	0.971447	−	0.237256i	$-0.0762480\pi$
$857$	240.178	+	416.000i	0.280254	+	0.485414i	−0.691193	+	0.722670i	$0.742915\pi$
$858$		0			0
$859$	297.015	+	171.482i	0.345769	+	0.199630i	0.662820	−	0.748779i	$-0.269359\pi$
$859$	297.015	+	171.482i	0.345769	+	0.199630i	−0.317051	+	0.948408i	$0.602693\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			888.084i			1.02907i	0.857470	+	0.514533i	$0.172035\pi$
$863$			888.084i			1.02907i	−0.857470	+	0.514533i	$0.827965\pi$
$864$		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	−0.589535	−	0.807743i	$-0.700689\pi$
$877$	354.973	−	614.831i	0.404758	−	0.701062i	0.994293	+	0.106681i	$0.0340223\pi$
$878$		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	$0.169696\pi$
$883$			897.513i			1.01644i	−0.861228	+	0.508218i	$0.830304\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1098.53	+	634.235i	1.23847	+	0.715033i	0.968782	−	0.247913i	$-0.0797448\pi$
$887$	1098.53	+	634.235i	1.23847	+	0.715033i	0.269692	+	0.962947i	$0.413078\pi$
$888$		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	−0.524998	−	0.851103i	$-0.675934\pi$
$907$	−45.7307	+	26.4026i	−0.0504197	+	0.0291098i	0.474578	+	0.880213i	$0.342601\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−545.096	+	314.711i	−0.598349	+	0.345457i	−0.768392	−	0.639980i	$-0.778943\pi$
$911$	−545.096	+	314.711i	−0.598349	+	0.345457i	0.170043	+	0.985437i	$0.445609\pi$
$912$		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	$-0.793133\pi$
$919$		−	1112.18i		−	1.21020i	0.796148	−	0.605101i	$-0.206867\pi$
$920$		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	0.747310	−	0.664475i	$-0.231345\pi$
$929$	−187.470	−	324.707i	−0.201798	−	0.349524i	−0.949108	+	0.314952i	$0.898012\pi$
$930$		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	−0.985082	−	0.172086i	$-0.944949\pi$
$941$	−323.243	+	559.873i	−0.343510	+	0.594977i	0.641572	+	0.767063i	$0.278283\pi$
$942$		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	−0.378391	−	0.925646i	$-0.623522\pi$
$947$	221.642	−	127.965i	0.234047	−	0.135127i	0.612438	+	0.790519i	$0.290189\pi$
$948$		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	0.884300	−	0.466919i	$-0.154636\pi$
$967$	891.657	+	514.799i	0.922086	+	0.532367i	0.0377863	+	0.999286i	$0.487969\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	405.731i		−	0.417848i	−0.977932	−	0.208924i	$-0.933004\pi$
$971$		−	405.731i		−	0.417848i	0.977932	−	0.208924i	$-0.0669962\pi$
$972$		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	−0.437561	−	0.899189i	$-0.644157\pi$
$977$	547.061	−	947.538i	0.559940	−	0.969845i	0.997501	−	0.0706558i	$-0.0225092\pi$
$978$		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	−0.667750	−	0.744385i	$-0.732743\pi$
$983$	−350.900	+	202.592i	−0.356969	+	0.206096i	0.310782	+	0.950481i	$0.399409\pi$
$984$		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	$0.0292552\pi$
$991$			181.905i			0.183557i	−0.995779	+	0.0917786i	$0.970745\pi$
$992$		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	0.557838	−	0.829950i	$-0.311631\pi$
$997$	−438.519	−	759.538i	−0.439839	−	0.761823i	−0.997677	+	0.0681268i	$0.978298\pi$
$998$		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.271.2		4
3.2	odd	2		1296.3.o.q.271.1		4
4.3	odd	2		1296.3.o.bf.271.2		4
9.2	odd	6		1296.3.o.r.703.1		4
9.4	even	3		1296.3.g.c.1135.1	✓	4
9.5	odd	6		1296.3.g.i.1135.3	yes	4
9.7	even	3		1296.3.o.bf.703.2		4
12.11	even	2		1296.3.o.r.271.1		4
36.7	odd	6	inner	1296.3.o.be.703.2		4
36.11	even	6		1296.3.o.q.703.1		4
36.23	even	6		1296.3.g.i.1135.4	yes	4
36.31	odd	6		1296.3.g.c.1135.2	yes	4

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

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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