LMFDB - Embedded newform 1080.2.q.d.721.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,2,Mod(361,1080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1080.361"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,3,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$8.62384341830$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.954288.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.3
Root			$-1.62241 - 0.606458i$ of defining polynomial
Character	$\chi$	$=$	1080.721
Dual form			1080.2.q.d.361.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(2.62241 - 4.54214i) q^{7} +(1.33641 - 2.31473i) q^{11} +(-1.90841 - 3.30545i) q^{13} -3.52884 q^{17} -4.67282 q^{19} +(-2.47842 - 4.29275i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.928007 + 1.60735i) q^{29} +(4.33641 + 7.51089i) q^{31} +5.24482 q^{35} -2.67282 q^{37} +(-1.83641 - 3.18076i) q^{41} +(-1.76442 + 3.05606i) q^{43} +(4.63164 - 8.02224i) q^{47} +(-10.2541 - 17.7605i) q^{49} +2.85601 q^{53} +2.67282 q^{55} +(2.10083 + 3.63875i) q^{59} +(3.98040 - 6.89425i) q^{61} +(1.90841 - 3.30545i) q^{65} +(0.429983 + 0.744753i) q^{67} +15.1625 q^{71} +6.28797 q^{73} +(-7.00924 - 12.1404i) q^{77} +(-2.81681 + 4.87886i) q^{79} +(-1.94958 + 3.37678i) q^{83} +(-1.76442 - 3.05606i) q^{85} +11.0000 q^{89} -20.0185 q^{91} +(-2.33641 - 4.04678i) q^{95} +(1.91764 - 3.32145i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5} + 5 q^{7} - 2 q^{11} - 4 q^{17} - 8 q^{19} - 7 q^{23} - 3 q^{25} - 7 q^{29} + 16 q^{31} + 10 q^{35} + 4 q^{37} - q^{41} - 2 q^{43} - 13 q^{47} - 10 q^{49} + 20 q^{53} - 4 q^{55} - 6 q^{59}+ \cdots - 30 q^{97}+O(q^{100}) $ 6 * q + 3 * q^5 + 5 * q^7 - 2 * q^11 - 4 * q^17 - 8 * q^19 - 7 * q^23 - 3 * q^25 - 7 * q^29 + 16 * q^31 + 10 * q^35 + 4 * q^37 - q^41 - 2 * q^43 - 13 * q^47 - 10 * q^49 + 20 * q^53 - 4 * q^55 - 6 * q^59 + 11 * q^61 - q^67 + 28 * q^71 + 32 * q^73 - 12 * q^77 + 6 * q^79 - 21 * q^83 - 2 * q^85 + 66 * q^89 - 60 * q^91 - 4 * q^95 - 30 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	2.62241	−	4.54214i	0.991177	−	1.71677i	0.380803	−	0.924656i	$-0.375648\pi$
$7$	2.62241	−	4.54214i	0.991177	−	1.71677i	0.610374	−	0.792113i	$-0.291019\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.33641	−	2.31473i	0.402943	−	0.697918i	−0.591136	−	0.806572i	$-0.701321\pi$
$11$	1.33641	−	2.31473i	0.402943	−	0.697918i	0.994080	+	0.108653i	$0.0346538\pi$
$12$		0			0
$13$	−1.90841	−	3.30545i	−0.529296	−	0.916768i	−0.999416	−	0.0341656i	$-0.989123\pi$
$13$	−1.90841	−	3.30545i	−0.529296	−	0.916768i	0.470120	−	0.882603i	$-0.344211\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.52884			−0.855869			−0.427934	−	0.903810i	$-0.640759\pi$
$17$	−3.52884			−0.855869			−0.427934	+	0.903810i	$0.640759\pi$
$18$		0			0
$19$	−4.67282			−1.07202			−0.536010	−	0.844212i	$-0.680069\pi$
$19$	−4.67282			−1.07202			−0.536010	+	0.844212i	$0.680069\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.47842	−	4.29275i	−0.516787	−	0.895101i	−0.999810	−	0.0194933i	$-0.993795\pi$
$23$	−2.47842	−	4.29275i	−0.516787	−	0.895101i	0.483023	−	0.875608i	$-0.339539\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.928007	+	1.60735i	−0.172327	+	0.298478i	−0.939233	−	0.343281i	$-0.888462\pi$
$29$	−0.928007	+	1.60735i	−0.172327	+	0.298478i	0.766906	+	0.641759i	$0.221795\pi$
$30$		0			0
$31$	4.33641	+	7.51089i	0.778843	+	1.34899i	0.932609	+	0.360888i	$0.117526\pi$
$31$	4.33641	+	7.51089i	0.778843	+	1.34899i	−0.153767	+	0.988107i	$0.549140\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.24482			0.886536
$36$		0			0
$37$	−2.67282			−0.439410			−0.219705	−	0.975566i	$-0.570509\pi$
$37$	−2.67282			−0.439410			−0.219705	+	0.975566i	$0.570509\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.83641	−	3.18076i	−0.286799	−	0.496751i	0.686245	−	0.727371i	$-0.259258\pi$
$41$	−1.83641	−	3.18076i	−0.286799	−	0.496751i	−0.973044	+	0.230620i	$0.925925\pi$
$42$		0			0
$43$	−1.76442	+	3.05606i	−0.269071	+	0.466045i	−0.968622	−	0.248538i	$-0.920050\pi$
$43$	−1.76442	+	3.05606i	−0.269071	+	0.466045i	0.699551	+	0.714583i	$0.253383\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.63164	−	8.02224i	0.675595	−	1.17016i	−0.300700	−	0.953719i	$-0.597220\pi$
$47$	4.63164	−	8.02224i	0.675595	−	1.17016i	0.976295	−	0.216446i	$-0.0694464\pi$
$48$		0			0
$49$	−10.2541	−	17.7605i	−1.46486	−	2.53722i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.85601			0.392304			0.196152	−	0.980574i	$-0.437155\pi$
$53$	2.85601			0.392304			0.196152	+	0.980574i	$0.437155\pi$
$54$		0			0
$55$	2.67282			0.360403
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.10083	+	3.63875i	0.273505	+	0.473724i	0.969757	−	0.244073i	$-0.0784837\pi$
$59$	2.10083	+	3.63875i	0.273505	+	0.473724i	−0.696252	+	0.717797i	$0.745150\pi$
$60$		0			0
$61$	3.98040	−	6.89425i	0.509638	−	0.882719i	−0.490300	−	0.871554i	$-0.663113\pi$
$61$	3.98040	−	6.89425i	0.509638	−	0.882719i	0.999938	−	0.0111647i	$-0.00355392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.90841	−	3.30545i	0.236709	−	0.409991i
$66$		0			0
$67$	0.429983	+	0.744753i	0.0525308	+	0.0909860i	0.891095	−	0.453817i	$-0.149938\pi$
$67$	0.429983	+	0.744753i	0.0525308	+	0.0909860i	−0.838564	+	0.544803i	$0.816605\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	15.1625			1.79945			0.899726	−	0.436454i	$-0.143766\pi$
$71$	15.1625			1.79945			0.899726	+	0.436454i	$0.143766\pi$
$72$		0			0
$73$	6.28797			0.735952			0.367976	−	0.929835i	$-0.380051\pi$
$73$	6.28797			0.735952			0.367976	+	0.929835i	$0.380051\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.00924	−	12.1404i	−0.798777	−	1.38352i
$78$		0			0
$79$	−2.81681	+	4.87886i	−0.316916	+	0.548914i	−0.979843	−	0.199770i	$-0.935981\pi$
$79$	−2.81681	+	4.87886i	−0.316916	+	0.548914i	0.662927	+	0.748684i	$0.269314\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.94958	+	3.37678i	−0.213995	+	0.370650i	−0.952961	−	0.303093i	$-0.901981\pi$
$83$	−1.94958	+	3.37678i	−0.213995	+	0.370650i	0.738966	+	0.673742i	$0.235314\pi$
$84$		0			0
$85$	−1.76442	−	3.05606i	−0.191378	−	0.331477i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.0000			1.16600			0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000			1.16600			0.582999	+	0.812473i	$0.301879\pi$
$90$		0			0
$91$	−20.0185			−2.09851
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.33641	−	4.04678i	−0.239711	−	0.415191i
$96$		0			0
$97$	1.91764	−	3.32145i	0.194707	−	0.337242i	−0.752097	−	0.659052i	$-0.770958\pi$
$97$	1.91764	−	3.32145i	0.194707	−	0.337242i	0.946804	+	0.321810i	$0.104291\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.24482	+	9.08429i	−0.521879	+	0.903921i	0.477797	+	0.878470i	$0.341435\pi$
$101$	−5.24482	+	9.08429i	−0.521879	+	0.903921i	−0.999676	+	0.0254505i	$0.991898\pi$
$102$		0			0
$103$	−2.05239	−	3.55485i	−0.202228	−	0.350269i	0.747018	−	0.664804i	$-0.231485\pi$
$103$	−2.05239	−	3.55485i	−0.202228	−	0.350269i	−0.949246	+	0.314534i	$0.898152\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	19.7345			1.90780			0.953901	−	0.300122i	$-0.0970275\pi$
$107$	19.7345			1.90780			0.953901	+	0.300122i	$0.0970275\pi$
$108$		0			0
$109$	−9.50811			−0.910711			−0.455356	−	0.890310i	$-0.650488\pi$
$109$	−9.50811			−0.910711			−0.455356	+	0.890310i	$0.650488\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.71598	−	8.16832i	−0.443642	−	0.768411i	0.554314	−	0.832307i	$-0.312981\pi$
$113$	−4.71598	−	8.16832i	−0.443642	−	0.768411i	−0.997957	+	0.0638967i	$0.979647\pi$
$114$		0			0
$115$	2.47842	−	4.29275i	0.231114	−	0.400301i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.25405	+	16.0285i	−0.848318	+	1.46933i
$120$		0			0
$121$	1.92801	+	3.33941i	0.175273	+	0.303582i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	18.6089			1.65128			0.825638	−	0.564200i	$-0.190815\pi$
$127$	18.6089			1.65128			0.825638	+	0.564200i	$0.190815\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.67282	−	13.2897i	−0.670378	−	1.16113i	−0.977797	−	0.209554i	$-0.932799\pi$
$131$	−7.67282	−	13.2897i	−0.670378	−	1.16113i	0.307419	−	0.951574i	$-0.400535\pi$
$132$		0			0
$133$	−12.2541	+	21.2246i	−1.06256	+	1.84041i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.244817	−	0.424035i	0.0209161	−	0.0362278i	−0.855378	−	0.518005i	$-0.826675\pi$
$137$	0.244817	−	0.424035i	0.0209161	−	0.0362278i	0.876294	+	0.481777i	$0.160008\pi$
$138$		0			0
$139$	5.34565	+	9.25893i	0.453412	+	0.785332i	0.998595	−	0.0529843i	$-0.0168733\pi$
$139$	5.34565	+	9.25893i	0.453412	+	0.785332i	−0.545183	+	0.838317i	$0.683540\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.2017			−0.853106
$144$		0			0
$145$	−1.85601			−0.154134
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1.93724	+	3.35540i	0.158705	+	0.274885i	0.934402	−	0.356220i	$-0.115935\pi$
$149$	1.93724	+	3.35540i	0.158705	+	0.274885i	−0.775697	+	0.631106i	$0.782601\pi$
$150$		0			0
$151$	7.33641	−	12.7070i	0.597029	−	1.03408i	−0.396228	−	0.918152i	$-0.629681\pi$
$151$	7.33641	−	12.7070i	0.597029	−	1.03408i	0.993257	−	0.115932i	$-0.0369855\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.33641	+	7.51089i	−0.348309	+	0.603289i
$156$		0			0
$157$	6.59046	+	11.4150i	0.525976	+	0.911018i	0.999542	+	0.0302592i	$0.00963329\pi$
$157$	6.59046	+	11.4150i	0.525976	+	0.911018i	−0.473566	+	0.880758i	$0.657033\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−25.9977			−2.04891
$162$		0			0
$163$	20.8745			1.63502			0.817508	−	0.575917i	$-0.195355\pi$
$163$	20.8745			1.63502			0.817508	+	0.575917i	$0.195355\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.00197644	+	0.00342329i	0.000152941	+	0.000264902i	0.866102	−	0.499868i	$-0.166618\pi$
$167$	0.00197644	+	0.00342329i	0.000152941	+	0.000264902i	−0.865949	+	0.500132i	$0.833285\pi$
$168$		0			0
$169$	−0.784020	+	1.35796i	−0.0603092	+	0.104459i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.3064	+	17.8513i	−0.783584	+	1.35721i	0.146257	+	0.989247i	$0.453277\pi$
$173$	−10.3064	+	17.8513i	−0.783584	+	1.35721i	−0.929841	+	0.367961i	$0.880056\pi$
$174$		0			0
$175$	2.62241	+	4.54214i	0.198235	+	0.343354i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.10478			−0.306806			−0.153403	−	0.988164i	$-0.549023\pi$
$179$	−4.10478			−0.306806			−0.153403	+	0.988164i	$0.549023\pi$
$180$		0			0
$181$	−4.43196			−0.329425			−0.164712	−	0.986342i	$-0.552670\pi$
$181$	−4.43196			−0.329425			−0.164712	+	0.986342i	$0.552670\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.33641	−	2.31473i	−0.0982550	−	0.170183i
$186$		0			0
$187$	−4.71598	+	8.16832i	−0.344867	+	0.597326i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.58123	+	14.8631i	−0.620916	+	1.07546i	0.368400	+	0.929668i	$0.379906\pi$
$191$	−8.58123	+	14.8631i	−0.620916	+	1.07546i	−0.989316	+	0.145790i	$0.953427\pi$
$192$		0			0
$193$	−2.94761	−	5.10541i	−0.212173	−	0.367495i	0.740221	−	0.672364i	$-0.234721\pi$
$193$	−2.94761	−	5.10541i	−0.212173	−	0.367495i	−0.952394	+	0.304868i	$0.901388\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	21.8353			1.55570			0.777850	−	0.628450i	$-0.216310\pi$
$197$	21.8353			1.55570			0.777850	+	0.628450i	$0.216310\pi$
$198$		0			0
$199$	−13.2488			−0.939180			−0.469590	−	0.882885i	$-0.655598\pi$
$199$	−13.2488			−0.939180			−0.469590	+	0.882885i	$0.655598\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	4.86723	+	8.43028i	0.341612	+	0.591690i
$204$		0			0
$205$	1.83641	−	3.18076i	0.128261	−	0.222154i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.24482	+	10.8163i	−0.431963	+	0.748182i
$210$		0			0
$211$	−0.192425	−	0.333290i	−0.0132471	−	0.0229447i	0.859326	−	0.511428i	$-0.170883\pi$
$211$	−0.192425	−	0.333290i	−0.0132471	−	0.0229447i	−0.872573	+	0.488484i	$0.837550\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.52884			−0.240665
$216$		0			0
$217$	45.4874			3.08788
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.73445	+	11.6644i	0.453008	+	0.784633i
$222$		0			0
$223$	5.71400	−	9.89694i	0.382638	−	0.662748i	−0.608800	−	0.793323i	$-0.708349\pi$
$223$	5.71400	−	9.89694i	0.382638	−	0.662748i	0.991438	+	0.130575i	$0.0416823\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	8.97003	−	15.5366i	0.595362	−	1.03120i	−0.398134	−	0.917327i	$-0.630342\pi$
$227$	8.97003	−	15.5366i	0.595362	−	1.03120i	0.993496	−	0.113870i	$-0.0363247\pi$
$228$		0			0
$229$	2.45684	+	4.25538i	0.162353	+	0.281203i	0.935712	−	0.352765i	$-0.114758\pi$
$229$	2.45684	+	4.25538i	0.162353	+	0.281203i	−0.773359	+	0.633968i	$0.781425\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.59442			−0.300990			−0.150495	−	0.988611i	$-0.548087\pi$
$233$	−4.59442			−0.300990			−0.150495	+	0.988611i	$0.548087\pi$
$234$		0			0
$235$	9.26329			0.604270
$236$		0			0
$237$		0			0
$238$		0			0
$239$	13.1193	+	22.7233i	0.848617	+	1.46985i	0.882443	+	0.470420i	$0.155898\pi$
$239$	13.1193	+	22.7233i	0.848617	+	1.46985i	−0.0338255	+	0.999428i	$0.510769\pi$
$240$		0			0
$241$	0.447608	−	0.775280i	0.0288330	−	0.0499402i	−0.851249	−	0.524762i	$-0.824154\pi$
$241$	0.447608	−	0.775280i	0.0288330	−	0.0499402i	0.880082	+	0.474822i	$0.157488\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.2541	−	17.7605i	0.655107	−	1.13468i
$246$		0			0
$247$	8.91764	+	15.4458i	0.567416	+	0.982793i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.10478			−0.132853			−0.0664264	−	0.997791i	$-0.521160\pi$
$251$	−2.10478			−0.132853			−0.0664264	+	0.997791i	$0.521160\pi$
$252$		0			0
$253$	−13.2488			−0.832943
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.34565	−	4.06278i	−0.146317	−	0.253429i	0.783546	−	0.621333i	$-0.213409\pi$
$257$	−2.34565	−	4.06278i	−0.146317	−	0.253429i	−0.929864	+	0.367904i	$0.880075\pi$
$258$		0			0
$259$	−7.00924	+	12.1404i	−0.435533	+	0.754365i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.90841	−	8.50161i	0.302665	−	0.524232i	−0.674074	−	0.738664i	$-0.735457\pi$
$263$	4.90841	−	8.50161i	0.302665	−	0.524232i	0.976739	+	0.214433i	$0.0687903\pi$
$264$		0			0
$265$	1.42801	+	2.47338i	0.0877218	+	0.151939i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−20.4504			−1.24688			−0.623442	−	0.781869i	$-0.714266\pi$
$269$	−20.4504			−1.24688			−0.623442	+	0.781869i	$0.714266\pi$
$270$		0			0
$271$	−5.03920			−0.306110			−0.153055	−	0.988218i	$-0.548911\pi$
$271$	−5.03920			−0.306110			−0.153055	+	0.988218i	$0.548911\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.33641	+	2.31473i	0.0805887	+	0.139584i
$276$		0			0
$277$	−0.523554	+	0.906823i	−0.0314573	+	0.0544857i	−0.881325	−	0.472510i	$-0.843348\pi$
$277$	−0.523554	+	0.906823i	−0.0314573	+	0.0544857i	0.849868	+	0.526995i	$0.176682\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.879568	−	1.52346i	0.0524706	−	0.0908818i	−0.838597	−	0.544752i	$-0.816624\pi$
$281$	0.879568	−	1.52346i	0.0524706	−	0.0908818i	0.891068	+	0.453870i	$0.149957\pi$
$282$		0			0
$283$	−1.43922	−	2.49280i	−0.0855527	−	0.148182i	0.820074	−	0.572258i	$-0.193932\pi$
$283$	−1.43922	−	2.49280i	−0.0855527	−	0.148182i	−0.905627	+	0.424076i	$0.860599\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−19.2633			−1.13708
$288$		0			0
$289$	−4.54731			−0.267489
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.4557	+	21.5739i	0.727671	+	1.26036i	0.957865	+	0.287218i	$0.0927303\pi$
$293$	12.4557	+	21.5739i	0.727671	+	1.26036i	−0.230195	+	0.973145i	$0.573936\pi$
$294$		0			0
$295$	−2.10083	+	3.63875i	−0.122315	+	0.211856i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.45967	+	16.3846i	−0.547067	+	0.947547i
$300$		0			0
$301$	9.25405	+	16.0285i	0.533395	+	0.923867i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.96080			0.455834
$306$		0			0
$307$	−1.32322			−0.0755203			−0.0377602	−	0.999287i	$-0.512022\pi$
$307$	−1.32322			−0.0755203			−0.0377602	+	0.999287i	$0.512022\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.47645	+	4.28933i	0.140426	+	0.243226i	0.927657	−	0.373433i	$-0.121819\pi$
$311$	2.47645	+	4.28933i	0.140426	+	0.243226i	−0.787231	+	0.616658i	$0.788486\pi$
$312$		0			0
$313$	−4.80757	+	8.32696i	−0.271740	+	0.470668i	−0.969308	−	0.245851i	$-0.920932\pi$
$313$	−4.80757	+	8.32696i	−0.271740	+	0.470668i	0.697567	+	0.716519i	$0.254266\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.15322	+	10.6577i	−0.345599	+	0.598596i	−0.985462	−	0.169893i	$-0.945658\pi$
$317$	−6.15322	+	10.6577i	−0.345599	+	0.598596i	0.639863	+	0.768489i	$0.278991\pi$
$318$		0			0
$319$	2.48040	+	4.29618i	0.138876	+	0.240540i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.4896			0.917508
$324$		0			0
$325$	3.81681			0.211719
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−24.2921	−	42.0752i	−1.33927	−	2.31968i
$330$		0			0
$331$	−0.773654	+	1.34001i	−0.0425239	+	0.0736535i	−0.886504	−	0.462721i	$-0.846873\pi$
$331$	−0.773654	+	1.34001i	−0.0425239	+	0.0736535i	0.843980	+	0.536374i	$0.180207\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.429983	+	0.744753i	−0.0234925	+	0.0406902i
$336$		0			0
$337$	−16.1585	−	27.9874i	−0.880210	−	1.52457i	−0.851107	−	0.524992i	$-0.824068\pi$
$337$	−16.1585	−	27.9874i	−0.880210	−	1.52457i	−0.0291025	−	0.999576i	$-0.509265\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	23.1809			1.25532
$342$		0			0
$343$	−70.8475			−3.82541
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.5420	−	21.7234i	−0.673291	−	1.16617i	−0.976965	−	0.213399i	$-0.931547\pi$
$347$	−12.5420	−	21.7234i	−0.673291	−	1.16617i	0.303674	−	0.952776i	$-0.401787\pi$
$348$		0			0
$349$	−10.3745	+	17.9691i	−0.555333	+	0.961866i	0.442544	+	0.896747i	$0.354076\pi$
$349$	−10.3745	+	17.9691i	−0.555333	+	0.961866i	−0.997878	+	0.0651190i	$0.979257\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.05767	+	15.6884i	−0.482091	+	0.835007i	−0.999789	−	0.0205571i	$-0.993456\pi$
$353$	−9.05767	+	15.6884i	−0.482091	+	0.835007i	0.517697	+	0.855564i	$0.326789\pi$
$354$		0			0
$355$	7.58123	+	13.1311i	0.402370	+	0.696925i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−25.3720			−1.33908			−0.669542	−	0.742774i	$-0.733510\pi$
$359$	−25.3720			−1.33908			−0.669542	+	0.742774i	$0.733510\pi$
$360$		0			0
$361$	2.83528			0.149225
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.14399	+	5.44554i	0.164564	+	0.285033i
$366$		0			0
$367$	−3.09159	+	5.35480i	−0.161380	+	0.279518i	−0.935364	−	0.353687i	$-0.884928\pi$
$367$	−3.09159	+	5.35480i	−0.161380	+	0.279518i	0.773984	+	0.633205i	$0.218261\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.48963	−	12.9724i	0.388842	−	0.673495i
$372$		0			0
$373$	8.40727	+	14.5618i	0.435312	+	0.753983i	0.997321	−	0.0731484i	$-0.0233047\pi$
$373$	8.40727	+	14.5618i	0.435312	+	0.753983i	−0.562009	+	0.827131i	$0.689971\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.08405			0.364847
$378$		0			0
$379$	22.0000			1.13006			0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000			1.13006			0.565032	+	0.825069i	$0.308864\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.39804	−	5.88558i	−0.173632	−	0.300739i	0.766055	−	0.642775i	$-0.222217\pi$
$383$	−3.39804	−	5.88558i	−0.173632	−	0.300739i	−0.939687	+	0.342036i	$0.888884\pi$
$384$		0			0
$385$	7.00924	−	12.1404i	0.357224	−	0.618730i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4.30757	+	7.46094i	−0.218403	+	0.378285i	−0.954320	−	0.298787i	$-0.903418\pi$
$389$	−4.30757	+	7.46094i	−0.218403	+	0.378285i	0.735917	+	0.677072i	$0.236751\pi$
$390$		0			0
$391$	8.74595	+	15.1484i	0.442302	+	0.766089i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.63362			−0.283458
$396$		0			0
$397$	22.4033			1.12439			0.562195	−	0.827005i	$-0.309957\pi$
$397$	22.4033			1.12439			0.562195	+	0.827005i	$0.309957\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.95684	+	12.0496i	0.347408	+	0.601729i	0.985788	−	0.167993i	$-0.0537285\pi$
$401$	6.95684	+	12.0496i	0.347408	+	0.601729i	−0.638380	+	0.769721i	$0.720395\pi$
$402$		0			0
$403$	16.5513	−	28.6676i	0.824477	−	1.42804i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.57199	+	6.18687i	−0.177057	+	0.306672i
$408$		0			0
$409$	−15.3064	−	26.5115i	−0.756855	−	1.31091i	−0.944447	−	0.328664i	$-0.893402\pi$
$409$	−15.3064	−	26.5115i	−0.756855	−	1.31091i	0.187592	−	0.982247i	$-0.439932\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	22.0369			1.08437
$414$		0			0
$415$	−3.89917			−0.191403
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.86525	−	11.8910i	−0.335389	−	0.580911i	0.648170	−	0.761496i	$-0.275535\pi$
$419$	−6.86525	−	11.8910i	−0.335389	−	0.580911i	−0.983560	+	0.180584i	$0.942201\pi$
$420$		0			0
$421$	−7.96080	+	13.7885i	−0.387985	+	0.672011i	−0.992178	−	0.124827i	$-0.960162\pi$
$421$	−7.96080	+	13.7885i	−0.387985	+	0.672011i	0.604193	+	0.796838i	$0.293496\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.76442	−	3.05606i	0.0855869	−	0.148241i
$426$		0			0
$427$	−20.8765	−	36.1591i	−1.01028	−	1.74986i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	25.2672			1.21708			0.608540	−	0.793523i	$-0.291755\pi$
$431$	25.2672			1.21708			0.608540	+	0.793523i	$0.291755\pi$
$432$		0			0
$433$	9.24086			0.444088			0.222044	−	0.975037i	$-0.428727\pi$
$433$	9.24086			0.444088			0.222044	+	0.975037i	$0.428727\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	11.5812	+	20.0593i	0.554005	+	0.959565i
$438$		0			0
$439$	8.71598	−	15.0965i	0.415991	−	0.720518i	−0.579541	−	0.814943i	$-0.696768\pi$
$439$	8.71598	−	15.0965i	0.415991	−	0.720518i	0.995532	+	0.0944256i	$0.0301014\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.81483	+	3.14338i	−0.0862254	+	0.149347i	−0.905913	−	0.423464i	$-0.860814\pi$
$443$	−1.81483	+	3.14338i	−0.0862254	+	0.149347i	0.819687	+	0.572811i	$0.194147\pi$
$444$		0			0
$445$	5.50000	+	9.52628i	0.260725	+	0.451589i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	39.6785			1.87254			0.936271	−	0.351278i	$-0.114253\pi$
$449$	39.6785			1.87254			0.936271	+	0.351278i	$0.114253\pi$
$450$		0			0
$451$	−9.81681			−0.462256
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−10.0092	−	17.3365i	−0.469240	−	0.812748i
$456$		0			0
$457$	−2.84678	+	4.93076i	−0.133167	+	0.230651i	−0.924896	−	0.380221i	$-0.875848\pi$
$457$	−2.84678	+	4.93076i	−0.133167	+	0.230651i	0.791729	+	0.610873i	$0.209181\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8.65850	−	14.9970i	0.403267	−	0.698479i	−0.590851	−	0.806780i	$-0.701208\pi$
$461$	8.65850	−	14.9970i	0.403267	−	0.698479i	0.994118	+	0.108302i	$0.0345413\pi$
$462$		0			0
$463$	−12.8837	−	22.3153i	−0.598757	−	1.03708i	−0.993005	−	0.118074i	$-0.962328\pi$
$463$	−12.8837	−	22.3153i	−0.598757	−	1.03708i	0.394248	−	0.919004i	$-0.371005\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.04711			0.0484544			0.0242272	−	0.999706i	$-0.492287\pi$
$467$	1.04711			0.0484544			0.0242272	+	0.999706i	$0.492287\pi$
$468$		0			0
$469$	4.51037			0.208269
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.71598	+	8.16832i	0.216841	+	0.375580i
$474$		0			0
$475$	2.33641	−	4.04678i	0.107202	−	0.185679i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.2880	−	19.5513i	0.515761	−	0.893324i	−0.484072	−	0.875028i	$-0.660843\pi$
$479$	11.2880	−	19.5513i	0.515761	−	0.893324i	0.999833	−	0.0182955i	$-0.00582395\pi$
$480$		0			0
$481$	5.10083	+	8.83490i	0.232578	+	0.402837i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.83528			0.174151
$486$		0			0
$487$	0.923855			0.0418638			0.0209319	−	0.999781i	$-0.493337\pi$
$487$	0.923855			0.0418638			0.0209319	+	0.999781i	$0.493337\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	4.04316	+	7.00295i	0.182465	+	0.316039i	0.942719	−	0.333587i	$-0.108259\pi$
$491$	4.04316	+	7.00295i	0.182465	+	0.316039i	−0.760254	+	0.649626i	$0.774926\pi$
$492$		0			0
$493$	3.27478	−	5.67209i	0.147489	−	0.255458i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	39.7622	−	68.8701i	1.78358	−	3.08925i
$498$		0			0
$499$	6.34169	+	10.9841i	0.283893	+	0.491718i	0.972340	−	0.233569i	$-0.0750405\pi$
$499$	6.34169	+	10.9841i	0.283893	+	0.491718i	−0.688447	+	0.725287i	$0.741707\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.93837			−0.220191			−0.110096	−	0.993921i	$-0.535116\pi$
$503$	−4.93837			−0.220191			−0.110096	+	0.993921i	$0.535116\pi$
$504$		0			0
$505$	−10.4896			−0.466783
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.60478	+	13.1719i	0.337076	+	0.583833i	0.983881	−	0.178822i	$-0.0572287\pi$
$509$	7.60478	+	13.1719i	0.337076	+	0.583833i	−0.646805	+	0.762655i	$0.723895\pi$
$510$		0			0
$511$	16.4896	−	28.5609i	0.729458	−	1.26346i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.05239	−	3.55485i	0.0904392	−	0.156645i
$516$		0			0
$517$	−12.3796	−	21.4420i	−0.544453	−	0.943020i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−22.1809			−0.971764			−0.485882	−	0.874024i	$-0.661502\pi$
$521$	−22.1809			−0.971764			−0.485882	+	0.874024i	$0.661502\pi$
$522$		0			0
$523$	−9.46495			−0.413873			−0.206937	−	0.978354i	$-0.566349\pi$
$523$	−9.46495			−0.413873			−0.206937	+	0.978354i	$0.566349\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−15.3025	−	26.5047i	−0.666587	−	1.15456i
$528$		0			0
$529$	−0.785151	+	1.35992i	−0.0341370	+	0.0591270i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.00924	+	12.1404i	−0.303604	+	0.525857i
$534$		0			0
$535$	9.86723	+	17.0905i	0.426597	+	0.738888i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−54.8145			−2.36103
$540$		0			0
$541$	−2.42405			−0.104218			−0.0521091	−	0.998641i	$-0.516594\pi$
$541$	−2.42405			−0.104218			−0.0521091	+	0.998641i	$0.516594\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.75405	−	8.23426i	−0.203641	−	0.352717i
$546$		0			0
$547$	7.64088	−	13.2344i	0.326700	−	0.565862i	−0.655155	−	0.755495i	$-0.727397\pi$
$547$	7.64088	−	13.2344i	0.326700	−	0.565862i	0.981855	+	0.189633i	$0.0607299\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4.33641	−	7.51089i	0.184737	−	0.319974i
$552$		0			0
$553$	14.7737	+	25.5887i	0.628240	+	1.08814i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−38.9793			−1.65160			−0.825802	−	0.563960i	$-0.809277\pi$
$557$	−38.9793			−1.65160			−0.825802	+	0.563960i	$0.809277\pi$
$558$		0			0
$559$	13.4689			0.569674
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.30447	−	3.99146i	−0.0971217	−	0.168220i	0.813370	−	0.581746i	$-0.197630\pi$
$563$	−2.30447	−	3.99146i	−0.0971217	−	0.168220i	−0.910492	+	0.413526i	$0.864297\pi$
$564$		0			0
$565$	4.71598	−	8.16832i	0.198403	−	0.343644i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.69129	−	9.85761i	0.238591	−	0.413253i	−0.721719	−	0.692186i	$-0.756648\pi$
$569$	5.69129	−	9.85761i	0.238591	−	0.413253i	0.960310	+	0.278934i	$0.0899810\pi$
$570$		0			0
$571$	−22.6129	−	39.1667i	−0.946320	−	1.63907i	−0.753087	−	0.657921i	$-0.771436\pi$
$571$	−22.6129	−	39.1667i	−0.946320	−	1.63907i	−0.193233	−	0.981153i	$-0.561897\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.95684			0.206715
$576$		0			0
$577$	5.14399			0.214147			0.107073	−	0.994251i	$-0.465852\pi$
$577$	5.14399			0.214147			0.107073	+	0.994251i	$0.465852\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10.2252	+	17.7106i	0.424213	+	0.734759i
$582$		0			0
$583$	3.81681	−	6.61091i	0.158076	−	0.273796i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−21.6409	+	37.4831i	−0.893215	+	1.54709i	−0.0572160	+	0.998362i	$0.518222\pi$
$587$	−21.6409	+	37.4831i	−0.893215	+	1.54709i	−0.835999	+	0.548731i	$0.815111\pi$
$588$		0			0
$589$	−20.2633	−	35.0970i	−0.834934	−	1.44615i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−33.3641			−1.37010			−0.685050	−	0.728496i	$-0.740220\pi$
$593$	−33.3641			−1.37010			−0.685050	+	0.728496i	$0.740220\pi$
$594$		0			0
$595$	−18.5081			−0.758758
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.0577	+	34.7409i	0.819534	+	1.41948i	0.906026	+	0.423223i	$0.139101\pi$
$599$	20.0577	+	34.7409i	0.819534	+	1.41948i	−0.0864914	+	0.996253i	$0.527566\pi$
$600$		0			0
$601$	2.98153	−	5.16416i	0.121619	−	0.210650i	−0.798787	−	0.601614i	$-0.794525\pi$
$601$	2.98153	−	5.16416i	0.121619	−	0.210650i	0.920406	+	0.390963i	$0.127858\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.92801	+	3.33941i	−0.0783846	+	0.135766i
$606$		0			0
$607$	7.57002	+	13.1117i	0.307258	+	0.532186i	0.977761	−	0.209720i	$-0.0672554\pi$
$607$	7.57002	+	13.1117i	0.307258	+	0.532186i	−0.670504	+	0.741906i	$0.733922\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−35.3562			−1.43036
$612$		0			0
$613$	28.5081			1.15143			0.575716	−	0.817650i	$-0.304723\pi$
$613$	28.5081			1.15143			0.575716	+	0.817650i	$0.304723\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.71598	+	15.0965i	0.350892	+	0.607763i	0.986406	−	0.164326i	$-0.0525450\pi$
$617$	8.71598	+	15.0965i	0.350892	+	0.607763i	−0.635514	+	0.772089i	$0.719212\pi$
$618$		0			0
$619$	−14.8260	+	25.6795i	−0.595909	+	1.03214i	0.397509	+	0.917598i	$0.369875\pi$
$619$	−14.8260	+	25.6795i	−0.595909	+	1.03214i	−0.993418	+	0.114546i	$0.963459\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	28.8465	−	49.9636i	1.15571	−	2.00175i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	9.43196			0.376077
$630$		0			0
$631$	−40.1233			−1.59728			−0.798641	−	0.601808i	$-0.794447\pi$
$631$	−40.1233			−1.59728			−0.798641	+	0.601808i	$0.794447\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.30447	+	16.1158i	0.369237	+	0.639536i
$636$		0			0
$637$	−39.1378	+	67.7886i	−1.55070	+	2.68588i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2.20674	−	3.82219i	0.0871612	−	0.150968i	−0.819149	−	0.573581i	$-0.805554\pi$
$641$	2.20674	−	3.82219i	0.0871612	−	0.150968i	0.906310	+	0.422613i	$0.138887\pi$
$642$		0			0
$643$	−9.91566	−	17.1744i	−0.391036	−	0.677294i	0.601551	−	0.798835i	$-0.294550\pi$
$643$	−9.91566	−	17.1744i	−0.391036	−	0.677294i	−0.992586	+	0.121541i	$0.961216\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	42.6873			1.67821			0.839106	−	0.543967i	$-0.183079\pi$
$647$	42.6873			1.67821			0.839106	+	0.543967i	$0.183079\pi$
$648$		0			0
$649$	11.2303			0.440828
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18.6821	−	32.3583i	−0.731085	−	1.26628i	−0.956420	−	0.291996i	$-0.905681\pi$
$653$	−18.6821	−	32.3583i	−0.731085	−	1.26628i	0.225334	−	0.974282i	$-0.427653\pi$
$654$		0			0
$655$	7.67282	−	13.2897i	0.299802	−	0.519272i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.71598	+	4.70422i	−0.105800	+	0.183250i	−0.914065	−	0.405569i	$-0.867074\pi$
$659$	−2.71598	+	4.70422i	−0.105800	+	0.183250i	0.808265	+	0.588819i	$0.200407\pi$
$660$		0			0
$661$	−8.54731	−	14.8044i	−0.332452	−	0.575823i	0.650540	−	0.759472i	$-0.274543\pi$
$661$	−8.54731	−	14.8044i	−0.332452	−	0.575823i	−0.982992	+	0.183648i	$0.941209\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−24.5081			−0.950384
$666$		0			0
$667$	9.19997			0.356224
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.6389	−	18.4271i	−0.410710	−	0.711371i
$672$		0			0
$673$	19.7005	−	34.1223i	0.759400	−	1.31532i	−0.183757	−	0.982972i	$-0.558826\pi$
$673$	19.7005	−	34.1223i	0.759400	−	1.31532i	0.943157	−	0.332347i	$-0.107841\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.13475	−	10.6257i	0.235778	−	0.408379i	−0.723721	−	0.690093i	$-0.757570\pi$
$677$	6.13475	−	10.6257i	0.235778	−	0.408379i	0.959498	+	0.281714i	$0.0909030\pi$
$678$		0			0
$679$	−10.0577	−	17.4204i	−0.385978	−	0.668534i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−11.8952			−0.455158			−0.227579	−	0.973760i	$-0.573081\pi$
$683$	−11.8952			−0.455158			−0.227579	+	0.973760i	$0.573081\pi$
$684$		0			0
$685$	0.489634			0.0187080
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−5.45043	−	9.44042i	−0.207645	−	0.359651i
$690$		0			0
$691$	2.63362	−	4.56156i	0.100188	−	0.173530i	−0.811574	−	0.584249i	$-0.801389\pi$
$691$	2.63362	−	4.56156i	0.100188	−	0.173530i	0.911762	+	0.410719i	$0.134722\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−5.34565	+	9.25893i	−0.202772	+	0.351211i
$696$		0			0
$697$	6.48040	+	11.2244i	0.245463	+	0.425154i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.1131			0.835200			0.417600	−	0.908631i	$-0.362871\pi$
$701$	22.1131			0.835200			0.417600	+	0.908631i	$0.362871\pi$
$702$		0			0
$703$	12.4896			0.471055
$704$		0			0
$705$		0			0
$706$		0			0
$707$	27.5081	+	47.6454i	1.03455	+	1.79189i
$708$		0			0
$709$	−8.70561	+	15.0786i	−0.326946	+	0.566287i	−0.981904	−	0.189378i	$-0.939353\pi$
$709$	−8.70561	+	15.0786i	−0.326946	+	0.566287i	0.654958	+	0.755665i	$0.272686\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21.4949	−	37.2303i	0.804991	−	1.39429i
$714$		0			0
$715$	−5.10083	−	8.83490i	−0.190760	−	0.330406i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	19.4689			0.726068			0.363034	−	0.931776i	$-0.381741\pi$
$719$	19.4689			0.726068			0.363034	+	0.931776i	$0.381741\pi$
$720$		0			0
$721$	−21.5288			−0.801776
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.928007	−	1.60735i	−0.0344653	−	0.0596957i
$726$		0			0
$727$	11.9628	−	20.7201i	0.443675	−	0.768467i	−0.554284	−	0.832328i	$-0.687008\pi$
$727$	11.9628	−	20.7201i	0.443675	−	0.768467i	0.997959	+	0.0638604i	$0.0203412\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.22635	−	10.7843i	0.230290	−	0.398873i
$732$		0			0
$733$	2.05767	+	3.56400i	0.0760019	+	0.131639i	0.901522	−	0.432734i	$-0.142451\pi$
$733$	2.05767	+	3.56400i	0.0760019	+	0.131639i	−0.825520	+	0.564373i	$0.809118\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.29854			0.0846678
$738$		0			0
$739$	16.8145			0.618533			0.309267	−	0.950975i	$-0.399916\pi$
$739$	16.8145			0.618533			0.309267	+	0.950975i	$0.399916\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	13.6801	+	23.6946i	0.501874	+	0.869271i	0.999998	+	0.00216476i	$0.000689064\pi$
$743$	13.6801	+	23.6946i	0.501874	+	0.869271i	−0.498124	+	0.867106i	$0.665978\pi$
$744$		0			0
$745$	−1.93724	+	3.35540i	−0.0709751	+	0.122932i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	51.7518	−	89.6367i	1.89097	−	3.27525i
$750$		0			0
$751$	−2.23558	−	3.87214i	−0.0815775	−	0.141296i	0.822350	−	0.568982i	$-0.192662\pi$
$751$	−2.23558	−	3.87214i	−0.0815775	−	0.141296i	−0.903928	+	0.427685i	$0.859329\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14.6728			0.533999
$756$		0			0
$757$	−17.7753			−0.646056			−0.323028	−	0.946389i	$-0.604701\pi$
$757$	−17.7753			−0.646056			−0.323028	+	0.946389i	$0.604701\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−20.1952	−	34.9792i	−0.732077	−	1.26799i	−0.955994	−	0.293387i	$-0.905218\pi$
$761$	−20.1952	−	34.9792i	−0.732077	−	1.26799i	0.223917	−	0.974608i	$-0.428116\pi$
$762$		0			0
$763$	−24.9341	+	43.1872i	−0.902676	+	1.56348i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.01847	−	13.8884i	0.289530	−	0.501481i
$768$		0			0
$769$	19.8496	+	34.3805i	0.715795	+	1.23979i	0.962652	+	0.270741i	$0.0872688\pi$
$769$	19.8496	+	34.3805i	0.715795	+	1.23979i	−0.246857	+	0.969052i	$0.579398\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8.97927			−0.322962			−0.161481	−	0.986876i	$-0.551627\pi$
$773$	−8.97927			−0.322962			−0.161481	+	0.986876i	$0.551627\pi$
$774$		0			0
$775$	−8.67282			−0.311537
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.58123	+	14.8631i	0.307454	+	0.532527i
$780$		0			0
$781$	20.2633	−	35.0970i	0.725077	−	1.25587i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.59046	+	11.4150i	−0.235224	+	0.407420i
$786$		0			0
$787$	−17.9476	−	31.0862i	−0.639763	−	1.10810i	−0.985485	−	0.169765i	$-0.945699\pi$
$787$	−17.9476	−	31.0862i	−0.639763	−	1.10810i	0.345721	−	0.938337i	$-0.387634\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−49.4689			−1.75891
$792$		0			0
$793$	−30.3849			−1.07900
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.9229	+	32.7755i	0.670284	+	1.16097i	0.977823	+	0.209431i	$0.0671612\pi$
$797$	18.9229	+	32.7755i	0.670284	+	1.16097i	−0.307539	+	0.951535i	$0.599505\pi$
$798$		0			0
$799$	−16.3443	+	28.3092i	−0.578220	+	1.00151i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.40332	−	14.5550i	0.296547	−	0.513634i
$804$		0			0
$805$	−12.9989	−	22.5147i	−0.458150	−	0.793539i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	27.8722			0.979935			0.489968	−	0.871741i	$-0.337009\pi$
$809$	27.8722			0.979935			0.489968	+	0.871741i	$0.337009\pi$
$810$		0			0
$811$	48.3249			1.69692			0.848459	−	0.529262i	$-0.177531\pi$
$811$	48.3249			1.69692			0.848459	+	0.529262i	$0.177531\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.4372	+	18.0778i	0.365601	+	0.633239i
$816$		0			0
$817$	8.24482	−	14.2804i	0.288450	−	0.499609i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−14.8549	+	25.7294i	−0.518439	+	0.897963i	0.481331	+	0.876539i	$0.340153\pi$
$821$	−14.8549	+	25.7294i	−0.518439	+	0.897963i	−0.999770	+	0.0214240i	$0.993180\pi$
$822$		0			0
$823$	15.1028	+	26.1588i	0.526451	+	0.911839i	0.999525	+	0.0308169i	$0.00981088\pi$
$823$	15.1028	+	26.1588i	0.526451	+	0.911839i	−0.473074	+	0.881023i	$0.656856\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.14003			0.0396429			0.0198214	−	0.999804i	$-0.493690\pi$
$827$	1.14003			0.0396429			0.0198214	+	0.999804i	$0.493690\pi$
$828$		0			0
$829$	−17.9608			−0.623804			−0.311902	−	0.950114i	$-0.600966\pi$
$829$	−17.9608			−0.623804			−0.311902	+	0.950114i	$0.600966\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	36.1849	+	62.6741i	1.25373	+	2.17153i
$834$		0			0
$835$	−0.00197644	+	0.00342329i	−6.83975e−5	+	0.000118468i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	0.605914	−	1.04947i	0.0209185	−	0.0362318i	−0.855377	−	0.518007i	$-0.826674\pi$
$839$	0.605914	−	1.04947i	0.0209185	−	0.0362318i	0.876295	+	0.481775i	$0.160008\pi$
$840$		0			0
$841$	12.7776	+	22.1315i	0.440607	+	0.763154i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.56804			−0.0539422
$846$		0			0
$847$	20.2241			0.694908
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.62438	+	11.4738i	0.227081	+	0.393316i
$852$		0			0
$853$	−17.6821	+	30.6262i	−0.605422	+	1.04862i	0.386562	+	0.922263i	$0.373662\pi$
$853$	−17.6821	+	30.6262i	−0.605422	+	1.04862i	−0.991985	+	0.126359i	$0.959671\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−20.9061	+	36.2105i	−0.714140	+	1.23693i	0.249150	+	0.968465i	$0.419849\pi$
$857$	−20.9061	+	36.2105i	−0.714140	+	1.23693i	−0.963290	+	0.268462i	$0.913485\pi$
$858$		0			0
$859$	5.88767	+	10.1977i	0.200885	+	0.347943i	0.948814	−	0.315836i	$-0.102285\pi$
$859$	5.88767	+	10.1977i	0.200885	+	0.347943i	−0.747929	+	0.663779i	$0.768952\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.1193			0.889111			0.444556	−	0.895751i	$-0.353362\pi$
$863$	26.1193			0.889111			0.444556	+	0.895751i	$0.353362\pi$
$864$		0			0
$865$	−20.6129			−0.700859
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.52884	+	13.0403i	0.255398	+	0.442363i
$870$		0			0
$871$	1.64116	−	2.84258i	0.0556087	−	0.0963171i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.62241	+	4.54214i	−0.0886536	+	0.153553i
$876$		0			0
$877$	26.3117	+	45.5732i	0.888484	+	1.53890i	0.841668	+	0.539996i	$0.181574\pi$
$877$	26.3117	+	45.5732i	0.888484	+	1.53890i	0.0468161	+	0.998904i	$0.485093\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	12.8089			0.431543			0.215771	−	0.976444i	$-0.430773\pi$
$881$	12.8089			0.431543			0.215771	+	0.976444i	$0.430773\pi$
$882$		0			0
$883$	21.7529			0.732044			0.366022	−	0.930606i	$-0.380719\pi$
$883$	21.7529			0.732044			0.366022	+	0.930606i	$0.380719\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16.5236	−	28.6196i	−0.554807	−	0.960953i	−0.997919	−	0.0644870i	$-0.979459\pi$
$887$	−16.5236	−	28.6196i	−0.554807	−	0.960953i	0.443112	−	0.896466i	$-0.353874\pi$
$888$		0			0
$889$	48.8002	−	84.5245i	1.63671	−	2.83486i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−21.6429	+	37.4865i	−0.724251	+	1.25444i
$894$		0			0
$895$	−2.05239	−	3.55485i	−0.0686039	−	0.118825i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−16.0969			−0.536861
$900$		0			0
$901$	−10.0784			−0.335760
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.21598	−	3.83819i	−0.0736617	−	0.127586i
$906$		0			0
$907$	−24.5924	+	42.5954i	−0.816579	+	1.41436i	0.0916102	+	0.995795i	$0.470799\pi$
$907$	−24.5924	+	42.5954i	−0.816579	+	1.41436i	−0.908189	+	0.418561i	$0.862535\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.22635	+	5.58819i	−0.106894	+	0.185145i	−0.914510	−	0.404563i	$-0.867424\pi$
$911$	−3.22635	+	5.58819i	−0.106894	+	0.185145i	0.807617	+	0.589708i	$0.200757\pi$
$912$		0			0
$913$	5.21090	+	9.02554i	0.172456	+	0.298702i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−80.4851			−2.65785
$918$		0			0
$919$	−19.5552			−0.645067			−0.322533	−	0.946558i	$-0.604534\pi$
$919$	−19.5552			−0.645067			−0.322533	+	0.946558i	$0.604534\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−28.9361	−	50.1188i	−0.952444	−	1.64968i
$924$		0			0
$925$	1.33641	−	2.31473i	0.0439410	−	0.0761080i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25.5865	+	44.3171i	−0.839466	+	1.45400i	0.0508754	+	0.998705i	$0.483799\pi$
$929$	−25.5865	+	44.3171i	−0.839466	+	1.45400i	−0.890342	+	0.455293i	$0.849534\pi$
$930$		0			0
$931$	47.9154	+	82.9919i	1.57036	+	2.71995i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9.43196			−0.308458
$936$		0			0
$937$	−37.6627			−1.23039			−0.615193	−	0.788377i	$-0.710922\pi$
$937$	−37.6627			−1.23039			−0.615193	+	0.788377i	$0.710922\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.9073	−	22.3561i	−0.420765	−	0.728787i	0.575249	−	0.817978i	$-0.304905\pi$
$941$	−12.9073	−	22.3561i	−0.420765	−	0.728787i	−0.996014	+	0.0891915i	$0.971572\pi$
$942$		0			0
$943$	−9.10281	+	15.7665i	−0.296428	+	0.513429i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.919617	+	1.59282i	−0.0298835	+	0.0517598i	−0.880580	−	0.473897i	$-0.842847\pi$
$947$	−0.919617	+	1.59282i	−0.0298835	+	0.0517598i	0.850697	+	0.525657i	$0.176180\pi$
$948$		0			0
$949$	−12.0000	−	20.7846i	−0.389536	−	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−39.7674			−1.28819			−0.644097	−	0.764944i	$-0.722767\pi$
$953$	−39.7674			−1.28819			−0.644097	+	0.764944i	$0.722767\pi$
$954$		0			0
$955$	−17.1625			−0.555364
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.28402	−	2.22399i	−0.0414632	−	0.0718163i
$960$		0			0
$961$	−22.1089	+	38.2938i	−0.713191	+	1.23528i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.94761	−	5.10541i	0.0948869	−	0.164349i
$966$		0			0
$967$	−4.85271	−	8.40514i	−0.156053	−	0.270291i	0.777389	−	0.629020i	$-0.216544\pi$
$967$	−4.85271	−	8.40514i	−0.156053	−	0.270291i	−0.933442	+	0.358729i	$0.883210\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−13.8247			−0.443656			−0.221828	−	0.975086i	$-0.571202\pi$
$971$	−13.8247			−0.443656			−0.221828	+	0.975086i	$0.571202\pi$
$972$		0			0
$973$	56.0739			1.79765
$974$		0			0
$975$		0			0
$976$		0			0
$977$	17.9885	+	31.1570i	0.575503	+	0.996801i	0.995987	+	0.0895006i	$0.0285271\pi$
$977$	17.9885	+	31.1570i	0.575503	+	0.996801i	−0.420484	+	0.907300i	$0.638140\pi$
$978$		0			0
$979$	14.7005	−	25.4621i	0.469831	−	0.813771i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−11.2745	+	19.5280i	−0.359601	+	0.622847i	−0.987894	−	0.155130i	$-0.950420\pi$
$983$	−11.2745	+	19.5280i	−0.359601	+	0.622847i	0.628293	+	0.777976i	$0.283754\pi$
$984$		0			0
$985$	10.9176	+	18.9099i	0.347865	+	0.602520i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17.4919			0.556210
$990$		0			0
$991$	−13.8767			−0.440809			−0.220405	−	0.975409i	$-0.570738\pi$
$991$	−13.8767			−0.440809			−0.220405	+	0.975409i	$0.570738\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−6.62438	−	11.4738i	−0.210007	−	0.363743i
$996$		0			0
$997$	−17.7529	+	30.7490i	−0.562241	+	0.973829i	0.435060	+	0.900402i	$0.356727\pi$
$997$	−17.7529	+	30.7490i	−0.562241	+	0.973829i	−0.997301	+	0.0734279i	$0.976606\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.2.q.d.721.3		6
3.2	odd	2		360.2.q.d.241.3	yes	6
4.3	odd	2		2160.2.q.j.721.1		6
9.2	odd	6		3240.2.a.r.1.1		3
9.4	even	3	inner	1080.2.q.d.361.3		6
9.5	odd	6		360.2.q.d.121.3	✓	6
9.7	even	3		3240.2.a.q.1.1		3
12.11	even	2		720.2.q.j.241.1		6
36.7	odd	6		6480.2.a.bu.1.3		3
36.11	even	6		6480.2.a.bx.1.3		3
36.23	even	6		720.2.q.j.481.1		6
36.31	odd	6		2160.2.q.j.1441.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.d.121.3	✓	6	9.5	odd	6
360.2.q.d.241.3	yes	6	3.2	odd	2
720.2.q.j.241.1		6	12.11	even	2
720.2.q.j.481.1		6	36.23	even	6
1080.2.q.d.361.3		6	9.4	even	3	inner
1080.2.q.d.721.3		6	1.1	even	1	trivial
2160.2.q.j.721.1		6	4.3	odd	2
2160.2.q.j.1441.1		6	36.31	odd	6
3240.2.a.q.1.1		3	9.7	even	3
3240.2.a.r.1.1		3	9.2	odd	6
6480.2.a.bu.1.3		3	36.7	odd	6
6480.2.a.bx.1.3		3	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(2.62241 - 4.54214i) q^{7} +(1.33641 - 2.31473i) q^{11} +(-1.90841 - 3.30545i) q^{13} -3.52884 q^{17} -4.67282 q^{19} +(-2.47842 - 4.29275i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.928007 + 1.60735i) q^{29} +(4.33641 + 7.51089i) q^{31} +5.24482 q^{35} -2.67282 q^{37} +(-1.83641 - 3.18076i) q^{41} +(-1.76442 + 3.05606i) q^{43} +(4.63164 - 8.02224i) q^{47} +(-10.2541 - 17.7605i) q^{49} +2.85601 q^{53} +2.67282 q^{55} +(2.10083 + 3.63875i) q^{59} +(3.98040 - 6.89425i) q^{61} +(1.90841 - 3.30545i) q^{65} +(0.429983 + 0.744753i) q^{67} +15.1625 q^{71} +6.28797 q^{73} +(-7.00924 - 12.1404i) q^{77} +(-2.81681 + 4.87886i) q^{79} +(-1.94958 + 3.37678i) q^{83} +(-1.76442 - 3.05606i) q^{85} +11.0000 q^{89} -20.0185 q^{91} +(-2.33641 - 4.04678i) q^{95} +(1.91764 - 3.32145i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} + 5 q^{7} - 2 q^{11} - 4 q^{17} - 8 q^{19} - 7 q^{23} - 3 q^{25} - 7 q^{29} + 16 q^{31} + 10 q^{35} + 4 q^{37} - q^{41} - 2 q^{43} - 13 q^{47} - 10 q^{49} + 20 q^{53} - 4 q^{55} - 6 q^{59}+ \cdots - 30 q^{97}+O(q^{100}) \) 6 * q + 3 * q^5 + 5 * q^7 - 2 * q^11 - 4 * q^17 - 8 * q^19 - 7 * q^23 - 3 * q^25 - 7 * q^29 + 16 * q^31 + 10 * q^35 + 4 * q^37 - q^41 - 2 * q^43 - 13 * q^47 - 10 * q^49 + 20 * q^53 - 4 * q^55 - 6 * q^59 + 11 * q^61 - q^67 + 28 * q^71 + 32 * q^73 - 12 * q^77 + 6 * q^79 - 21 * q^83 - 2 * q^85 + 66 * q^89 - 60 * q^91 - 4 * q^95 - 30 * q^97

Introduction

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