LMFDB - Embedded newform 1080.2.q.e.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 = [8,0,0,0,4,0,1] 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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.856615824.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 $ x^8 + 11x^6 + 36x^4 + 32*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.3
Root			$0.385731i$ of defining polynomial
Character	$\chi$	$=$	1080.721
Dual form			1080.2.q.e.361.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(0.165947 - 0.287429i) q^{7} +(2.20380 - 3.81710i) q^{11} +(-3.49031 - 6.04539i) q^{13} +3.07139 q^{17} -7.55364 q^{19} +(0.834053 + 1.44462i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(1.78651 - 3.09432i) q^{29} +(-2.82220 - 4.88820i) q^{31} +0.331895 q^{35} +7.64441 q^{37} +(2.74919 + 4.76174i) q^{41} +(3.53570 - 6.12401i) q^{43} +(3.98815 - 6.90768i) q^{47} +(3.44492 + 5.96678i) q^{49} +5.47900 q^{53} +4.40761 q^{55} +(-3.69411 - 6.39839i) q^{59} +(-0.296197 + 0.513029i) q^{61} +(3.49031 - 6.04539i) q^{65} +(-2.61087 - 4.52216i) q^{67} -8.98062 q^{71} -8.05202 q^{73} +(-0.731430 - 1.26687i) q^{77} +(5.49838 - 9.52347i) q^{79} +(-3.14657 + 5.45002i) q^{83} +(1.53570 + 2.65991i) q^{85} -11.9060 q^{89} -2.31683 q^{91} +(-3.77682 - 6.54164i) q^{95} +(0.622718 - 1.07858i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} + q^{7} + q^{11} - 4 q^{13} - 10 q^{17} + 2 q^{19} + 7 q^{23} - 4 q^{25} + 7 q^{29} + 2 q^{31} + 2 q^{35} + 12 q^{37} + 12 q^{41} + 11 q^{43} + 7 q^{47} - 3 q^{49} - 24 q^{53} + 2 q^{55}+ \cdots - q^{97}+O(q^{100}) $ 8 * q + 4 * q^5 + q^7 + q^11 - 4 * q^13 - 10 * q^17 + 2 * q^19 + 7 * q^23 - 4 * q^25 + 7 * q^29 + 2 * q^31 + 2 * q^35 + 12 * q^37 + 12 * q^41 + 11 * q^43 + 7 * q^47 - 3 * q^49 - 24 * q^53 + 2 * q^55 + 11 * q^59 - 19 * q^61 + 4 * q^65 + 10 * q^67 - 24 * q^71 + 18 * q^73 + 32 * q^77 + 24 * q^79 + 23 * q^83 - 5 * q^85 - 42 * q^89 + 28 * q^91 + q^95 - 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
$7$	0.165947	−	0.287429i	0.0627222	−	0.108638i	−0.832959	−	0.553335i	$-0.813355\pi$
$7$	0.165947	−	0.287429i	0.0627222	−	0.108638i	0.895681	+	0.444696i	$0.146688\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.20380	−	3.81710i	0.664472	−	1.15090i	−0.314957	−	0.949106i	$-0.601990\pi$
$11$	2.20380	−	3.81710i	0.664472	−	1.15090i	0.979428	−	0.201792i	$-0.0646766\pi$
$12$		0			0
$13$	−3.49031	−	6.04539i	−0.968038	−	1.67669i	−0.701222	−	0.712943i	$-0.747362\pi$
$13$	−3.49031	−	6.04539i	−0.968038	−	1.67669i	−0.266816	−	0.963747i	$-0.585972\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.07139			0.744923			0.372461	−	0.928048i	$-0.378514\pi$
$17$	3.07139			0.744923			0.372461	+	0.928048i	$0.378514\pi$
$18$		0			0
$19$	−7.55364			−1.73292			−0.866461	−	0.499244i	$-0.833611\pi$
$19$	−7.55364			−1.73292			−0.866461	+	0.499244i	$0.833611\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.834053	+	1.44462i	0.173912	+	0.301224i	0.939784	−	0.341768i	$-0.111026\pi$
$23$	0.834053	+	1.44462i	0.173912	+	0.301224i	−0.765872	+	0.642993i	$0.777693\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.78651	−	3.09432i	0.331746	−	0.574601i	−0.651108	−	0.758985i	$-0.725696\pi$
$29$	1.78651	−	3.09432i	0.331746	−	0.574601i	0.982854	+	0.184384i	$0.0590289\pi$
$30$		0			0
$31$	−2.82220	−	4.88820i	−0.506883	−	0.877947i	−0.999968	−	0.00796608i	$-0.997464\pi$
$31$	−2.82220	−	4.88820i	−0.506883	−	0.877947i	0.493085	−	0.869981i	$-0.335869\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.331895			0.0561004
$36$		0			0
$37$	7.64441			1.25673			0.628367	−	0.777917i	$-0.283724\pi$
$37$	7.64441			1.25673			0.628367	+	0.777917i	$0.283724\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.74919	+	4.76174i	0.429351	+	0.743658i	0.996816	−	0.0797396i	$-0.0254089\pi$
$41$	2.74919	+	4.76174i	0.429351	+	0.743658i	−0.567464	+	0.823398i	$0.692076\pi$
$42$		0			0
$43$	3.53570	−	6.12401i	0.539189	−	0.933902i	−0.459759	−	0.888044i	$-0.652064\pi$
$43$	3.53570	−	6.12401i	0.539189	−	0.933902i	0.998948	−	0.0458587i	$-0.0146024\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.98815	−	6.90768i	0.581732	−	1.00759i	−0.413542	−	0.910485i	$-0.635709\pi$
$47$	3.98815	−	6.90768i	0.581732	−	1.00759i	0.995274	−	0.0971042i	$-0.0309580\pi$
$48$		0			0
$49$	3.44492	+	5.96678i	0.492132	+	0.852397i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.47900			0.752599			0.376299	−	0.926498i	$-0.377196\pi$
$53$	5.47900			0.752599			0.376299	+	0.926498i	$0.377196\pi$
$54$		0			0
$55$	4.40761			0.594321
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−3.69411	−	6.39839i	−0.480933	−	0.833000i	0.518828	−	0.854879i	$-0.326368\pi$
$59$	−3.69411	−	6.39839i	−0.480933	−	0.833000i	−0.999761	+	0.0218790i	$0.993035\pi$
$60$		0			0
$61$	−0.296197	+	0.513029i	−0.0379242	+	0.0656866i	−0.884364	−	0.466797i	$-0.845408\pi$
$61$	−0.296197	+	0.513029i	−0.0379242	+	0.0656866i	0.846440	+	0.532484i	$0.178741\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	3.49031	−	6.04539i	0.432920	−	0.749839i
$66$		0			0
$67$	−2.61087	−	4.52216i	−0.318969	−	0.552470i	0.661305	−	0.750118i	$-0.270003\pi$
$67$	−2.61087	−	4.52216i	−0.318969	−	0.552470i	−0.980273	+	0.197648i	$0.936670\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.98062			−1.06580			−0.532902	−	0.846177i	$-0.678898\pi$
$71$	−8.98062			−1.06580			−0.532902	+	0.846177i	$0.678898\pi$
$72$		0			0
$73$	−8.05202			−0.942417			−0.471209	−	0.882022i	$-0.656182\pi$
$73$	−8.05202			−0.942417			−0.471209	+	0.882022i	$0.656182\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.731430	−	1.26687i	−0.0833542	−	0.144374i
$78$		0			0
$79$	5.49838	−	9.52347i	0.618616	−	1.07147i	−0.371122	−	0.928584i	$-0.621027\pi$
$79$	5.49838	−	9.52347i	0.618616	−	1.07147i	0.989739	−	0.142891i	$-0.0456397\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.14657	+	5.45002i	−0.345381	+	0.598217i	−0.985423	−	0.170123i	$-0.945584\pi$
$83$	−3.14657	+	5.45002i	−0.345381	+	0.598217i	0.640042	+	0.768340i	$0.278917\pi$
$84$		0			0
$85$	1.53570	+	2.65991i	0.166570	+	0.288507i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.9060			−1.26203			−0.631016	−	0.775770i	$-0.717362\pi$
$89$	−11.9060			−1.26203			−0.631016	+	0.775770i	$0.717362\pi$
$90$		0			0
$91$	−2.31683			−0.242870
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.77682	−	6.54164i	−0.387493	−	0.671158i
$96$		0			0
$97$	0.622718	−	1.07858i	0.0632274	−	0.109513i	−0.832679	−	0.553756i	$-0.813194\pi$
$97$	0.622718	−	1.07858i	0.0632274	−	0.109513i	0.895906	+	0.444243i	$0.146527\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.33189	−	4.03896i	0.232032	−	0.401892i	−0.726374	−	0.687300i	$-0.758796\pi$
$101$	2.33189	−	4.03896i	0.232032	−	0.401892i	0.958406	+	0.285408i	$0.0921292\pi$
$102$		0			0
$103$	2.08270	+	3.60735i	0.205215	+	0.355443i	0.950201	−	0.311637i	$-0.100877\pi$
$103$	2.08270	+	3.60735i	0.205215	+	0.355443i	−0.744986	+	0.667080i	$0.767544\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.57409			0.635541			0.317771	−	0.948168i	$-0.397066\pi$
$107$	6.57409			0.635541			0.317771	+	0.948168i	$0.397066\pi$
$108$		0			0
$109$	0.743816			0.0712446			0.0356223	−	0.999365i	$-0.488659\pi$
$109$	0.743816			0.0712446			0.0356223	+	0.999365i	$0.488659\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	8.31252	+	14.3977i	0.781976	+	1.35442i	0.930789	+	0.365556i	$0.119121\pi$
$113$	8.31252	+	14.3977i	0.781976	+	1.35442i	−0.148814	+	0.988865i	$0.547545\pi$
$114$		0			0
$115$	−0.834053	+	1.44462i	−0.0777758	+	0.134712i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.509690	−	0.882809i	0.0467232	−	0.0809269i
$120$		0			0
$121$	−4.21349	−	7.29798i	−0.383045	−	0.663453i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	4.79152			0.425178			0.212589	−	0.977142i	$-0.431810\pi$
$127$	4.79152			0.425178			0.212589	+	0.977142i	$0.431810\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.88985	+	17.1297i	0.864080	+	1.49663i	0.867958	+	0.496638i	$0.165432\pi$
$131$	9.88985	+	17.1297i	0.864080	+	1.49663i	−0.00387809	+	0.999992i	$0.501234\pi$
$132$		0			0
$133$	−1.25351	+	2.17114i	−0.108693	+	0.188261i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.26713	−	16.0511i	0.791744	−	1.37134i	−0.133142	−	0.991097i	$-0.542507\pi$
$137$	9.26713	−	16.0511i	0.791744	−	1.37134i	0.924886	−	0.380244i	$-0.124160\pi$
$138$		0			0
$139$	−2.68980	−	4.65887i	−0.228146	−	0.395160i	0.729113	−	0.684393i	$-0.239933\pi$
$139$	−2.68980	−	4.65887i	−0.228146	−	0.395160i	−0.957259	+	0.289234i	$0.906600\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−30.7678			−2.57293
$144$		0			0
$145$	3.57301			0.296723
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.51794	+	9.55735i	0.452047	+	0.782969i	0.998513	−	0.0545128i	$-0.0173606\pi$
$149$	5.51794	+	9.55735i	0.452047	+	0.782969i	−0.546466	+	0.837481i	$0.684027\pi$
$150$		0			0
$151$	0.750810	−	1.30044i	0.0611001	−	0.105828i	−0.833857	−	0.551980i	$-0.813872\pi$
$151$	0.750810	−	1.30044i	0.0611001	−	0.105828i	0.894957	+	0.446152i	$0.147206\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.82220	−	4.88820i	0.226685	−	0.392630i
$156$		0			0
$157$	11.8109	+	20.4571i	0.942612	+	1.63265i	0.760463	+	0.649381i	$0.224972\pi$
$157$	11.8109	+	20.4571i	0.942612	+	1.63265i	0.182149	+	0.983271i	$0.441695\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.553635			0.0436326
$162$		0			0
$163$	−9.16217			−0.717636			−0.358818	−	0.933407i	$-0.616820\pi$
$163$	−9.16217			−0.717636			−0.358818	+	0.933407i	$0.616820\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.324363	−	0.561813i	−0.0250999	−	0.0434744i	0.853203	−	0.521580i	$-0.174657\pi$
$167$	−0.324363	−	0.561813i	−0.0250999	−	0.0434744i	−0.878303	+	0.478105i	$0.841324\pi$
$168$		0			0
$169$	−17.8645	+	30.9423i	−1.37419	+	2.38017i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.40761	−	7.63420i	0.335104	−	0.580417i	−0.648401	−	0.761299i	$-0.724562\pi$
$173$	4.40761	−	7.63420i	0.335104	−	0.580417i	0.983505	+	0.180882i	$0.0578952\pi$
$174$		0			0
$175$	0.165947	+	0.287429i	0.0125444	+	0.0217276i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.16217			−0.684813			−0.342406	−	0.939552i	$-0.611242\pi$
$179$	−9.16217			−0.684813			−0.342406	+	0.939552i	$0.611242\pi$
$180$		0			0
$181$	13.5730			1.00887			0.504437	−	0.863448i	$-0.331700\pi$
$181$	13.5730			1.00887			0.504437	+	0.863448i	$0.331700\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.82220	+	6.62025i	0.281014	+	0.486731i
$186$		0			0
$187$	6.76875	−	11.7238i	0.494980	−	0.857330i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−5.13472	+	8.89360i	−0.371535	+	0.643518i	−0.989802	−	0.142450i	$-0.954502\pi$
$191$	−5.13472	+	8.89360i	−0.371535	+	0.643518i	0.618267	+	0.785968i	$0.287835\pi$
$192$		0			0
$193$	−4.53570	−	7.85606i	−0.326487	−	0.565491i	0.655325	−	0.755347i	$-0.272531\pi$
$193$	−4.53570	−	7.85606i	−0.326487	−	0.565491i	−0.981812	+	0.189855i	$0.939198\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.80658			0.342455			0.171227	−	0.985232i	$-0.445227\pi$
$197$	4.80658			0.342455			0.171227	+	0.985232i	$0.445227\pi$
$198$		0			0
$199$	−18.4596			−1.30857			−0.654284	−	0.756249i	$-0.727030\pi$
$199$	−18.4596			−1.30857			−0.654284	+	0.756249i	$0.727030\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.592932	−	1.02699i	−0.0416157	−	0.0720805i
$204$		0			0
$205$	−2.74919	+	4.76174i	−0.192012	+	0.332574i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.6467	+	28.8330i	−1.15148	+	1.99442i
$210$		0			0
$211$	−0.177795	−	0.307950i	−0.0122399	−	0.0212002i	0.859841	−	0.510563i	$-0.170563\pi$
$211$	−0.177795	−	0.307950i	−0.0122399	−	0.0212002i	−0.872080	+	0.489362i	$0.837230\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.07139			0.482265
$216$		0			0
$217$	−1.87335			−0.127171
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−10.7201	−	18.5678i	−0.721113	−	1.24900i
$222$		0			0
$223$	−1.65194	+	2.86125i	−0.110622	+	0.191603i	−0.916021	−	0.401130i	$-0.868618\pi$
$223$	−1.65194	+	2.86125i	−0.110622	+	0.191603i	0.805399	+	0.592733i	$0.201951\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.36921	+	4.10360i	−0.157250	+	0.272365i	−0.933876	−	0.357597i	$-0.883596\pi$
$227$	−2.36921	+	4.10360i	−0.157250	+	0.272365i	0.776626	+	0.629962i	$0.216930\pi$
$228$		0			0
$229$	−9.53032	−	16.5070i	−0.629781	−	1.09081i	−0.987595	−	0.157021i	$-0.949811\pi$
$229$	−9.53032	−	16.5070i	−0.629781	−	1.09081i	0.357814	−	0.933793i	$-0.383522\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.3721			0.876035			0.438017	−	0.898967i	$-0.355681\pi$
$233$	13.3721			0.876035			0.438017	+	0.898967i	$0.355681\pi$
$234$		0			0
$235$	7.97630			0.520317
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.64873	+	16.7121i	0.624124	+	1.08101i	0.988710	+	0.149845i	$0.0478774\pi$
$239$	9.64873	+	16.7121i	0.624124	+	1.08101i	−0.364585	+	0.931170i	$0.618789\pi$
$240$		0			0
$241$	−9.99031	+	17.3037i	−0.643532	+	1.11463i	0.341106	+	0.940025i	$0.389199\pi$
$241$	−9.99031	+	17.3037i	−0.643532	+	1.11463i	−0.984638	+	0.174606i	$0.944135\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.44492	+	5.96678i	−0.220088	+	0.381204i
$246$		0			0
$247$	26.3645	+	45.6647i	1.67753	+	2.90558i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.889846			0.0561666			0.0280833	−	0.999606i	$-0.491060\pi$
$251$	0.889846			0.0561666			0.0280833	+	0.999606i	$0.491060\pi$
$252$		0			0
$253$	7.35235			0.462238
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.36222	+	14.4838i	0.521621	+	0.903474i	0.999684	+	0.0251482i	$0.00800576\pi$
$257$	8.36222	+	14.4838i	0.521621	+	0.903474i	−0.478063	+	0.878326i	$0.658661\pi$
$258$		0			0
$259$	1.26857	−	2.19723i	0.0788251	−	0.136529i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.56170	+	6.16905i	−0.219624	+	0.380400i	−0.954693	−	0.297592i	$-0.903816\pi$
$263$	−3.56170	+	6.16905i	−0.219624	+	0.380400i	0.735069	+	0.677992i	$0.237150\pi$
$264$		0			0
$265$	2.73950	+	4.74495i	0.168286	+	0.291480i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−23.5030			−1.43300			−0.716502	−	0.697585i	$-0.754258\pi$
$269$	−23.5030			−1.43300			−0.716502	+	0.697585i	$0.754258\pi$
$270$		0			0
$271$	−8.49838			−0.516240			−0.258120	−	0.966113i	$-0.583103\pi$
$271$	−8.49838			−0.516240			−0.258120	+	0.966113i	$0.583103\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.20380	+	3.81710i	0.132894	+	0.230180i
$276$		0			0
$277$	11.1153	−	19.2523i	0.667856	−	1.15676i	−0.310646	−	0.950526i	$-0.600545\pi$
$277$	11.1153	−	19.2523i	0.667856	−	1.15676i	0.978502	−	0.206235i	$-0.0661212\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−5.87029	+	10.1676i	−0.350192	+	0.606550i	−0.986283	−	0.165064i	$-0.947217\pi$
$281$	−5.87029	+	10.1676i	−0.350192	+	0.606550i	0.636091	+	0.771614i	$0.280550\pi$
$282$		0			0
$283$	11.8667	+	20.5537i	0.705401	+	1.22179i	0.966546	+	0.256491i	$0.0825666\pi$
$283$	11.8667	+	20.5537i	0.705401	+	1.22179i	−0.261145	+	0.965300i	$0.584100\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.82488			0.107719
$288$		0			0
$289$	−7.56653			−0.445090
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−10.1347	−	17.5538i	−0.592077	−	1.02551i	−0.993952	−	0.109813i	$-0.964975\pi$
$293$	−10.1347	−	17.5538i	−0.592077	−	1.02551i	0.401876	−	0.915694i	$-0.368358\pi$
$294$		0			0
$295$	3.69411	−	6.39839i	0.215080	−	0.372529i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.82220	−	10.0844i	0.336707	−	0.583193i
$300$		0			0
$301$	−1.17348	−	2.03253i	−0.0676382	−	0.117153i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.592395			−0.0339204
$306$		0			0
$307$	2.05094			0.117053			0.0585267	−	0.998286i	$-0.481360\pi$
$307$	2.05094			0.117053			0.0585267	+	0.998286i	$0.481360\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.89792	−	17.1437i	−0.561259	−	0.972130i	−0.997387	−	0.0722448i	$-0.976984\pi$
$311$	−9.89792	−	17.1437i	−0.561259	−	0.972130i	0.436128	−	0.899885i	$-0.356350\pi$
$312$		0			0
$313$	0.0577727	−	0.100065i	0.00326551	−	0.00565603i	−0.864388	−	0.502825i	$-0.832294\pi$
$313$	0.0577727	−	0.100065i	0.00326551	−	0.00565603i	0.867654	+	0.497169i	$0.165627\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.8936	−	20.6003i	0.668011	−	1.15703i	−0.310448	−	0.950590i	$-0.600479\pi$
$317$	11.8936	−	20.6003i	0.668011	−	1.15703i	0.978459	−	0.206439i	$-0.0661875\pi$
$318$		0			0
$319$	−7.87422	−	13.6385i	−0.440872	−	0.763612i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−23.2002			−1.29089
$324$		0			0
$325$	6.98062			0.387215
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.32365	−	2.29262i	−0.0729750	−	0.126396i
$330$		0			0
$331$	−17.3125	+	29.9862i	−0.951582	+	1.64819i	−0.209579	+	0.977792i	$0.567209\pi$
$331$	−17.3125	+	29.9862i	−0.951582	+	1.64819i	−0.742003	+	0.670396i	$0.766124\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.61087	−	4.52216i	0.142647	−	0.247072i
$336$		0			0
$337$	5.26713	+	9.12293i	0.286919	+	0.496958i	0.973073	−	0.230499i	$-0.0740357\pi$
$337$	5.26713	+	9.12293i	0.286919	+	0.496958i	−0.686154	+	0.727456i	$0.740702\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−24.8783			−1.34724
$342$		0			0
$343$	4.60997			0.248915
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.94654	−	5.10356i	−0.158179	−	0.273974i	0.776033	−	0.630692i	$-0.217229\pi$
$347$	−2.94654	−	5.10356i	−0.158179	−	0.273974i	−0.934212	+	0.356718i	$0.883896\pi$
$348$		0			0
$349$	11.5066	−	19.9301i	0.615936	−	1.06683i	−0.374284	−	0.927314i	$-0.622111\pi$
$349$	11.5066	−	19.9301i	0.615936	−	1.06683i	0.990220	−	0.139518i	$-0.0445552\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.36222	+	12.7517i	−0.391851	+	0.678706i	−0.992694	−	0.120661i	$-0.961499\pi$
$353$	−7.36222	+	12.7517i	−0.391851	+	0.678706i	0.600842	+	0.799367i	$0.294832\pi$
$354$		0			0
$355$	−4.49031	−	7.77745i	−0.238321	−	0.412784i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.02262			−0.212306			−0.106153	−	0.994350i	$-0.533853\pi$
$359$	−4.02262			−0.212306			−0.106153	+	0.994350i	$0.533853\pi$
$360$		0			0
$361$	38.0574			2.00302
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.02601	−	6.97325i	−0.210731	−	0.364997i
$366$		0			0
$367$	4.43830	−	7.68735i	0.231677	−	0.401277i	−0.726625	−	0.687035i	$-0.758912\pi$
$367$	4.43830	−	7.68735i	0.231677	−	0.401277i	0.958302	+	0.285758i	$0.0922454\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.909226	−	1.57482i	0.0472046	−	0.0817608i
$372$		0			0
$373$	−15.3645	−	26.6121i	−0.795545	−	1.37792i	−0.922492	−	0.386015i	$-0.873851\pi$
$373$	−15.3645	−	26.6121i	−0.795545	−	1.37792i	0.126947	−	0.991909i	$-0.459482\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−24.9419			−1.28457
$378$		0			0
$379$	−4.24004			−0.217796			−0.108898	−	0.994053i	$-0.534732\pi$
$379$	−4.24004			−0.217796			−0.108898	+	0.994053i	$0.534732\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	16.0439	+	27.7889i	0.819807	+	1.41995i	0.905824	+	0.423654i	$0.139253\pi$
$383$	16.0439	+	27.7889i	0.819807	+	1.41995i	−0.0860167	+	0.996294i	$0.527414\pi$
$384$		0			0
$385$	0.731430	−	1.26687i	0.0372771	−	0.0645659i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.1667	+	24.5374i	−0.718278	+	1.24409i	0.243403	+	0.969925i	$0.421736\pi$
$389$	−14.1667	+	24.5374i	−0.718278	+	1.24409i	−0.961681	+	0.274169i	$0.911597\pi$
$390$		0			0
$391$	2.56170	+	4.43700i	0.129551	+	0.224389i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.9968			0.553307
$396$		0			0
$397$	13.1395			0.659455			0.329728	−	0.944076i	$-0.393043\pi$
$397$	13.1395			0.659455			0.329728	+	0.944076i	$0.393043\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.59471	+	13.1544i	0.379262	+	0.656900i	0.990955	−	0.134195i	$-0.0428447\pi$
$401$	7.59471	+	13.1544i	0.379262	+	0.656900i	−0.611693	+	0.791095i	$0.709511\pi$
$402$		0			0
$403$	−19.7007	+	34.1227i	−0.981364	+	1.69977i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	16.8468	−	29.1795i	0.835063	−	1.44637i
$408$		0			0
$409$	5.10064	+	8.83457i	0.252211	+	0.436841i	0.964134	−	0.265415i	$-0.0855091\pi$
$409$	5.10064	+	8.83457i	0.252211	+	0.436841i	−0.711924	+	0.702257i	$0.752176\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.45211			−0.120661
$414$		0			0
$415$	−6.29314			−0.308918
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.32058	+	10.9476i	0.308781	+	0.534824i	0.978096	−	0.208155i	$-0.0667457\pi$
$419$	6.32058	+	10.9476i	0.308781	+	0.534824i	−0.669315	+	0.742979i	$0.733412\pi$
$420$		0			0
$421$	−0.501620	+	0.868832i	−0.0244475	+	0.0423443i	−0.877990	−	0.478678i	$-0.841116\pi$
$421$	−0.501620	+	0.868832i	−0.0244475	+	0.0423443i	0.853543	+	0.521023i	$0.174449\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.53570	+	2.65991i	−0.0744923	+	0.129024i
$426$		0			0
$427$	0.0983063	+	0.170272i	0.00475738	+	0.00824002i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	39.0685			1.88186			0.940932	−	0.338596i	$-0.109952\pi$
$431$	39.0685			1.88186			0.940932	+	0.338596i	$0.109952\pi$
$432$		0			0
$433$	−29.6663			−1.42567			−0.712836	−	0.701331i	$-0.752589\pi$
$433$	−29.6663			−1.42567			−0.712836	+	0.701331i	$0.752589\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.30013	−	10.9121i	−0.301376	−	0.521999i
$438$		0			0
$439$	16.1223	−	27.9247i	0.769477	−	1.33277i	−0.168370	−	0.985724i	$-0.553850\pi$
$439$	16.1223	−	27.9247i	0.769477	−	1.33277i	0.937847	−	0.347049i	$-0.112816\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.22372	+	3.85160i	−0.105652	+	0.182995i	−0.914004	−	0.405704i	$-0.867026\pi$
$443$	−2.22372	+	3.85160i	−0.105652	+	0.182995i	0.808352	+	0.588699i	$0.200360\pi$
$444$		0			0
$445$	−5.95299	−	10.3109i	−0.282199	−	0.488783i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	13.0639			0.616523			0.308261	−	0.951302i	$-0.400253\pi$
$449$	13.0639			0.616523			0.308261	+	0.951302i	$0.400253\pi$
$450$		0			0
$451$	24.2347			1.14117
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.15842	−	2.00643i	−0.0543074	−	0.0940631i
$456$		0			0
$457$	−8.16504	+	14.1423i	−0.381945	+	0.661547i	−0.991340	−	0.131319i	$-0.958079\pi$
$457$	−8.16504	+	14.1423i	−0.381945	+	0.661547i	0.609396	+	0.792866i	$0.291412\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	10.7639	−	18.6436i	0.501324	−	0.868319i	−0.498675	−	0.866789i	$-0.666180\pi$
$461$	10.7639	−	18.6436i	0.501324	−	0.868319i	0.999999	−	0.00152975i	$-0.000486935\pi$
$462$		0			0
$463$	11.1584	+	19.3269i	0.518576	+	0.898199i	0.999767	+	0.0215837i	$0.00687083\pi$
$463$	11.1584	+	19.3269i	0.518576	+	0.898199i	−0.481192	+	0.876616i	$0.659796\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	36.5203			1.68996			0.844978	−	0.534801i	$-0.179613\pi$
$467$	36.5203			1.68996			0.844978	+	0.534801i	$0.179613\pi$
$468$		0			0
$469$	−1.73307			−0.0800256
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−15.5840	−	26.9922i	−0.716551	−	1.24110i
$474$		0			0
$475$	3.77682	−	6.54164i	0.173292	−	0.300151i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.47900	−	7.75786i	0.204651	−	0.354465i	−0.745371	−	0.666650i	$-0.767727\pi$
$479$	4.47900	−	7.75786i	0.204651	−	0.354465i	0.950021	+	0.312185i	$0.101061\pi$
$480$		0			0
$481$	−26.6814	−	46.2135i	−1.21657	−	2.10715i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.24544			0.0565523
$486$		0			0
$487$	−29.5016			−1.33685			−0.668423	−	0.743781i	$-0.733030\pi$
$487$	−29.5016			−1.33685			−0.668423	+	0.743781i	$0.733030\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.17635	−	8.96571i	−0.233606	−	0.404617i	0.725261	−	0.688474i	$-0.241719\pi$
$491$	−5.17635	−	8.96571i	−0.233606	−	0.404617i	−0.958867	+	0.283857i	$0.908386\pi$
$492$		0			0
$493$	5.48707	−	9.50388i	0.247125	−	0.428033i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.49031	+	2.58129i	−0.0668495	+	0.115787i
$498$		0			0
$499$	−11.1925	−	19.3860i	−0.501045	−	0.867835i	−0.999999	−	0.00120683i	$-0.999616\pi$
$499$	−11.1925	−	19.3860i	−0.501045	−	0.867835i	0.498954	−	0.866628i	$-0.333717\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	42.8654			1.91127			0.955637	−	0.294546i	$-0.0951685\pi$
$503$	42.8654			1.91127			0.955637	+	0.294546i	$0.0951685\pi$
$504$		0			0
$505$	4.66379			0.207536
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.54539	−	6.14079i	−0.157147	−	0.272186i	0.776692	−	0.629881i	$-0.216896\pi$
$509$	−3.54539	−	6.14079i	−0.157147	−	0.272186i	−0.933839	+	0.357695i	$0.883563\pi$
$510$		0			0
$511$	−1.33621	+	2.31438i	−0.0591105	+	0.102382i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.08270	+	3.60735i	−0.0917749	+	0.158959i
$516$		0			0
$517$	−17.5782	−	30.4463i	−0.773088	−	1.33903i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	39.5863			1.73431			0.867153	−	0.498042i	$-0.165947\pi$
$521$	39.5863			1.73431			0.867153	+	0.498042i	$0.165947\pi$
$522$		0			0
$523$	25.9375			1.13417			0.567085	−	0.823659i	$-0.308071\pi$
$523$	25.9375			1.13417			0.567085	+	0.823659i	$0.308071\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.66811	−	15.0136i	−0.377589	−	0.654003i
$528$		0			0
$529$	10.1087	−	17.5088i	0.439509	−	0.761252i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	19.1911	−	33.2399i	0.831257	−	1.43978i
$534$		0			0
$535$	3.28705	+	5.69333i	0.142111	+	0.246144i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	30.3677			1.30803
$540$		0			0
$541$	−38.7929			−1.66784			−0.833920	−	0.551886i	$-0.813908\pi$
$541$	−38.7929			−1.66784			−0.833920	+	0.551886i	$0.813908\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.371908	+	0.644164i	0.0159308	+	0.0275929i
$546$		0			0
$547$	5.02655	−	8.70623i	0.214920	−	0.372252i	−0.738328	−	0.674442i	$-0.764384\pi$
$547$	5.02655	−	8.70623i	0.214920	−	0.372252i	0.953248	+	0.302190i	$0.0977177\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−13.4946	+	23.3734i	−0.574890	+	0.995739i
$552$		0			0
$553$	−1.82488	−	3.16079i	−0.0776019	−	0.134410i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	23.4316			0.992829			0.496415	−	0.868086i	$-0.334650\pi$
$557$	23.4316			0.992829			0.496415	+	0.868086i	$0.334650\pi$
$558$		0			0
$559$	−49.3627			−2.08782
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.7580	−	35.9539i	−0.874844	−	1.51527i	−0.856928	−	0.515435i	$-0.827630\pi$
$563$	−20.7580	−	35.9539i	−0.874844	−	1.51527i	−0.0179159	−	0.999839i	$-0.505703\pi$
$564$		0			0
$565$	−8.31252	+	14.3977i	−0.349710	+	0.605716i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17.5990	+	30.4824i	−0.737789	+	1.27789i	0.215699	+	0.976460i	$0.430797\pi$
$569$	−17.5990	+	30.4824i	−0.737789	+	1.27789i	−0.953489	+	0.301429i	$0.902536\pi$
$570$		0			0
$571$	0.689797	+	1.19476i	0.0288671	+	0.0499993i	0.880098	−	0.474792i	$-0.157477\pi$
$571$	0.689797	+	1.19476i	0.0288671	+	0.0499993i	−0.851231	+	0.524791i	$0.824143\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.66811			−0.0695648
$576$		0			0
$577$	−17.3322			−0.721549			−0.360775	−	0.932653i	$-0.617488\pi$
$577$	−17.3322			−0.721549			−0.360775	+	0.932653i	$0.617488\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.04433	+	1.80883i	0.0433261	+	0.0750430i
$582$		0			0
$583$	12.0746	−	20.9139i	0.500080	−	0.866164i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.8450	+	39.5687i	−0.942914	+	1.63317i	−0.183039	+	0.983106i	$0.558593\pi$
$587$	−22.8450	+	39.5687i	−0.942914	+	1.63317i	−0.759875	+	0.650069i	$0.774740\pi$
$588$		0			0
$589$	21.3179	+	36.9237i	0.878389	+	1.52141i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.4661			−0.429791			−0.214896	−	0.976637i	$-0.568941\pi$
$593$	−10.4661			−0.429791			−0.214896	+	0.976637i	$0.568941\pi$
$594$		0			0
$595$	1.01938			0.0417905
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.5698	+	18.3074i	0.431869	+	0.748020i	0.997034	−	0.0769583i	$-0.0245208\pi$
$599$	10.5698	+	18.3074i	0.431869	+	0.748020i	−0.565165	+	0.824978i	$0.691188\pi$
$600$		0			0
$601$	5.33421	−	9.23912i	0.217587	−	0.376871i	−0.736483	−	0.676456i	$-0.763515\pi$
$601$	5.33421	−	9.23912i	0.217587	−	0.376871i	0.954070	+	0.299585i	$0.0968481\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.21349	−	7.29798i	0.171303	−	0.296705i
$606$		0			0
$607$	−0.651942	−	1.12920i	−0.0264615	−	0.0458327i	0.852491	−	0.522741i	$-0.175091\pi$
$607$	−0.651942	−	1.12920i	−0.0264615	−	0.0458327i	−0.878953	+	0.476909i	$0.841757\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−55.6796			−2.25255
$612$		0			0
$613$	30.9741			1.25103			0.625517	−	0.780211i	$-0.284888\pi$
$613$	30.9741			1.25103			0.625517	+	0.780211i	$0.284888\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.29190	−	2.23763i	−0.0520099	−	0.0900838i	0.838848	−	0.544365i	$-0.183229\pi$
$617$	−1.29190	−	2.23763i	−0.0520099	−	0.0900838i	−0.890858	+	0.454281i	$0.849896\pi$
$618$		0			0
$619$	0.465378	−	0.806058i	0.0187051	−	0.0323982i	−0.856521	−	0.516112i	$-0.827379\pi$
$619$	0.465378	−	0.806058i	0.0187051	−	0.0323982i	0.875226	+	0.483713i	$0.160712\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.97577	+	3.42213i	−0.0791574	+	0.137105i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	23.4790			0.936169
$630$		0			0
$631$	20.6250			0.821069			0.410535	−	0.911845i	$-0.365342\pi$
$631$	20.6250			0.821069			0.410535	+	0.911845i	$0.365342\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2.39576	+	4.14957i	0.0950727	+	0.164671i
$636$		0			0
$637$	24.0477	−	41.6518i	0.952805	−	1.65031i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	11.9516	−	20.7007i	0.472058	−	0.817628i	−0.527431	−	0.849598i	$-0.676845\pi$
$641$	11.9516	−	20.7007i	0.472058	−	0.817628i	0.999489	+	0.0319697i	$0.0101780\pi$
$642$		0			0
$643$	−1.12863	−	1.95484i	−0.0445088	−	0.0770915i	0.842913	−	0.538050i	$-0.180839\pi$
$643$	−1.12863	−	1.95484i	−0.0445088	−	0.0770915i	−0.887422	+	0.460959i	$0.847506\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	12.7262			0.500320			0.250160	−	0.968204i	$-0.419517\pi$
$647$	12.7262			0.500320			0.250160	+	0.968204i	$0.419517\pi$
$648$		0			0
$649$	−32.5644			−1.27826
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.585400	+	1.01394i	0.0229085	+	0.0396787i	0.877252	−	0.480029i	$-0.159374\pi$
$653$	0.585400	+	1.01394i	0.0229085	+	0.0396787i	−0.854344	+	0.519708i	$0.826041\pi$
$654$		0			0
$655$	−9.88985	+	17.1297i	−0.386428	+	0.669313i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	17.3678	−	30.0819i	0.676552	−	1.17182i	−0.299460	−	0.954109i	$-0.596807\pi$
$659$	17.3678	−	30.0819i	0.676552	−	1.17182i	0.976013	−	0.217714i	$-0.0698601\pi$
$660$		0			0
$661$	−17.5730	−	30.4374i	−0.683511	−	1.18388i	−0.973902	−	0.226968i	$-0.927119\pi$
$661$	−17.5730	−	30.4374i	−0.683511	−	1.18388i	0.290391	−	0.956908i	$-0.406215\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.50701			−0.0972177
$666$		0			0
$667$	5.96017			0.230779
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.30552	+	2.26123i	0.0503991	+	0.0872938i
$672$		0			0
$673$	13.6374	−	23.6207i	0.525684	−	0.910511i	−0.473869	−	0.880596i	$-0.657143\pi$
$673$	13.6374	−	23.6207i	0.525684	−	0.910511i	0.999552	−	0.0299154i	$-0.00952380\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4.56602	−	7.90858i	0.175486	−	0.303951i	−0.764843	−	0.644217i	$-0.777184\pi$
$677$	4.56602	−	7.90858i	0.175486	−	0.303951i	0.940330	+	0.340265i	$0.110517\pi$
$678$		0			0
$679$	−0.206677	−	0.357975i	−0.00793153	−	0.0137378i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	29.0326			1.11090			0.555451	−	0.831549i	$-0.312545\pi$
$683$	29.0326			1.11090			0.555451	+	0.831549i	$0.312545\pi$
$684$		0			0
$685$	18.5343			0.708158
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−19.1234	−	33.1227i	−0.728544	−	1.26188i
$690$		0			0
$691$	8.38823	−	14.5288i	0.319103	−	0.552703i	−0.661198	−	0.750211i	$-0.729952\pi$
$691$	8.38823	−	14.5288i	0.319103	−	0.552703i	0.980301	+	0.197509i	$0.0632850\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.68980	−	4.65887i	0.102030	−	0.176721i
$696$		0			0
$697$	8.44385	+	14.6252i	0.319834	+	0.553968i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.93400			−0.110816			−0.0554078	−	0.998464i	$-0.517646\pi$
$701$	−2.93400			−0.110816			−0.0554078	+	0.998464i	$0.517646\pi$
$702$		0			0
$703$	−57.7431			−2.17782
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.773944	−	1.34051i	−0.0291071	−	0.0504150i
$708$		0			0
$709$	−13.9293	+	24.1263i	−0.523126	+	0.906080i	0.476512	+	0.879168i	$0.341901\pi$
$709$	−13.9293	+	24.1263i	−0.523126	+	0.906080i	−0.999638	+	0.0269124i	$0.991432\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.70773	−	8.15404i	0.176306	−	0.305371i
$714$		0			0
$715$	−15.3839	−	26.6457i	−0.575326	−	0.996493i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.95585			−0.259409			−0.129705	−	0.991553i	$-0.541403\pi$
$719$	−6.95585			−0.259409			−0.129705	+	0.991553i	$0.541403\pi$
$720$		0			0
$721$	1.38248			0.0514861
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.78651	+	3.09432i	0.0663492	+	0.114920i
$726$		0			0
$727$	2.51023	−	4.34784i	0.0930992	−	0.161253i	−0.815715	−	0.578455i	$-0.803656\pi$
$727$	2.51023	−	4.34784i	0.0930992	−	0.161253i	0.908814	+	0.417202i	$0.136989\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.8595	−	18.8092i	0.401654	−	0.695685i
$732$		0			0
$733$	10.2368	+	17.7307i	0.378105	+	0.654897i	0.990787	−	0.135433i	$-0.0432425\pi$
$733$	10.2368	+	17.7307i	0.378105	+	0.654897i	−0.612682	+	0.790330i	$0.709909\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−23.0154			−0.847782
$738$		0			0
$739$	51.1614			1.88200			0.941002	−	0.338401i	$-0.109886\pi$
$739$	51.1614			1.88200			0.941002	+	0.338401i	$0.109886\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.07086	−	5.31888i	−0.112659	−	0.195131i	0.804183	−	0.594382i	$-0.202603\pi$
$743$	−3.07086	−	5.31888i	−0.112659	−	0.195131i	−0.916841	+	0.399251i	$0.869270\pi$
$744$		0			0
$745$	−5.51794	+	9.55735i	−0.202162	+	0.350154i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.09095	−	1.88959i	0.0398626	−	0.0690440i
$750$		0			0
$751$	7.54557	+	13.0693i	0.275342	+	0.476906i	0.970221	−	0.242220i	$-0.0778757\pi$
$751$	7.54557	+	13.0693i	0.275342	+	0.476906i	−0.694880	+	0.719126i	$0.744542\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.50162			0.0546496
$756$		0			0
$757$	−2.87659			−0.104551			−0.0522757	−	0.998633i	$-0.516647\pi$
$757$	−2.87659			−0.104551			−0.0522757	+	0.998633i	$0.516647\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.119478	+	0.206941i	0.00433106	+	0.00750162i	0.868183	−	0.496245i	$-0.165288\pi$
$761$	0.119478	+	0.206941i	0.00433106	+	0.00750162i	−0.863852	+	0.503746i	$0.831955\pi$
$762$		0			0
$763$	0.123434	−	0.213794i	0.00446862	−	0.00773988i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−25.7872	+	44.6647i	−0.931122	+	1.61275i
$768$		0			0
$769$	1.92930	+	3.34164i	0.0695722	+	0.120503i	0.898713	−	0.438537i	$-0.144503\pi$
$769$	1.92930	+	3.34164i	0.0695722	+	0.120503i	−0.829141	+	0.559040i	$0.811170\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.22355			0.187878			0.0939390	−	0.995578i	$-0.470054\pi$
$773$	5.22355			0.187878			0.0939390	+	0.995578i	$0.470054\pi$
$774$		0			0
$775$	5.64441			0.202753
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−20.7664	−	35.9684i	−0.744033	−	1.28870i
$780$		0			0
$781$	−19.7915	+	34.2799i	−0.708196	+	1.22663i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.8109	+	20.4571i	−0.421549	+	0.730144i
$786$		0			0
$787$	10.8818	+	18.8478i	0.387893	+	0.671851i	0.992166	−	0.124927i	$-0.0398695\pi$
$787$	10.8818	+	18.8478i	0.387893	+	0.671851i	−0.604273	+	0.796778i	$0.706536\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.51776			0.196189
$792$		0			0
$793$	4.13528			0.146848
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.7228	+	22.0365i	0.450665	+	0.780574i	0.998427	−	0.0560596i	$-0.0178537\pi$
$797$	12.7228	+	22.0365i	0.450665	+	0.780574i	−0.547763	+	0.836634i	$0.684520\pi$
$798$		0			0
$799$	12.2492	−	21.2162i	0.433345	−	0.750576i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−17.7451	+	30.7353i	−0.626209	+	1.08463i
$804$		0			0
$805$	0.276818	+	0.479462i	0.00975654	+	0.0168988i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	11.0854			0.389741			0.194871	−	0.980829i	$-0.437571\pi$
$809$	11.0854			0.389741			0.194871	+	0.980829i	$0.437571\pi$
$810$		0			0
$811$	42.4869			1.49192			0.745958	−	0.665993i	$-0.231992\pi$
$811$	42.4869			1.49192			0.745958	+	0.665993i	$0.231992\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.58108	−	7.93467i	−0.160468	−	0.277939i
$816$		0			0
$817$	−26.7074	+	46.2585i	−0.934373	+	1.61838i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−15.0673	+	26.0973i	−0.525851	+	0.910800i	0.473696	+	0.880689i	$0.342920\pi$
$821$	−15.0673	+	26.0973i	−0.525851	+	0.910800i	−0.999547	+	0.0301119i	$0.990414\pi$
$822$		0			0
$823$	−10.8109	−	18.7251i	−0.376845	−	0.652715i	0.613756	−	0.789495i	$-0.289658\pi$
$823$	−10.8109	−	18.7251i	−0.376845	−	0.652715i	−0.990601	+	0.136781i	$0.956324\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1.82488			0.0634574			0.0317287	−	0.999497i	$-0.489899\pi$
$827$	1.82488			0.0634574			0.0317287	+	0.999497i	$0.489899\pi$
$828$		0			0
$829$	30.4374			1.05713			0.528567	−	0.848892i	$-0.322730\pi$
$829$	30.4374			1.05713			0.528567	+	0.848892i	$0.322730\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	10.5807	+	18.3263i	0.366600	+	0.634970i
$834$		0			0
$835$	0.324363	−	0.561813i	0.0112250	−	0.0194423i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.7834	−	29.0698i	0.579429	−	1.00360i	−0.416116	−	0.909312i	$-0.636609\pi$
$839$	16.7834	−	29.0698i	0.579429	−	1.00360i	0.995545	−	0.0942887i	$-0.0300577\pi$
$840$		0			0
$841$	8.11678	+	14.0587i	0.279889	+	0.484782i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−35.7291			−1.22912
$846$		0			0
$847$	−2.79687			−0.0961016
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.37584	+	11.0433i	0.218561	+	0.378559i
$852$		0			0
$853$	11.8775	−	20.5724i	0.406676	−	0.704384i	−0.587839	−	0.808978i	$-0.700021\pi$
$853$	11.8775	−	20.5724i	0.406676	−	0.704384i	0.994515	+	0.104594i	$0.0333543\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−13.0052	+	22.5256i	−0.444249	+	0.769461i	−0.998000	−	0.0632211i	$-0.979863\pi$
$857$	−13.0052	+	22.5256i	−0.444249	+	0.769461i	0.553751	+	0.832682i	$0.313196\pi$
$858$		0			0
$859$	25.3835	+	43.9656i	0.866075	+	1.50009i	0.865975	+	0.500086i	$0.166698\pi$
$859$	25.3835	+	43.9656i	0.866075	+	1.50009i	9.98605e−5		1.00000i	$0.499968\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−32.9343			−1.12110			−0.560548	−	0.828122i	$-0.689410\pi$
$863$	−32.9343			−1.12110			−0.560548	+	0.828122i	$0.689410\pi$
$864$		0			0
$865$	8.81521			0.299726
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24.2347	−	41.9757i	−0.822105	−	1.42393i
$870$		0			0
$871$	−18.2255	+	31.5675i	−0.617547	+	1.06962i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.165947	+	0.287429i	−0.00561004	+	0.00971688i
$876$		0			0
$877$	−8.56170	−	14.8293i	−0.289108	−	0.500750i	0.684489	−	0.729023i	$-0.260025\pi$
$877$	−8.56170	−	14.8293i	−0.289108	−	0.500750i	−0.973597	+	0.228273i	$0.926692\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	46.7617			1.57544			0.787721	−	0.616032i	$-0.211261\pi$
$881$	46.7617			1.57544			0.787721	+	0.616032i	$0.211261\pi$
$882$		0			0
$883$	−21.1356			−0.711269			−0.355635	−	0.934625i	$-0.615735\pi$
$883$	−21.1356			−0.711269			−0.355635	+	0.934625i	$0.615735\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−22.9887	−	39.8176i	−0.771885	−	1.33694i	−0.936529	−	0.350590i	$-0.885981\pi$
$887$	−22.9887	−	39.8176i	−0.771885	−	1.33694i	0.164644	−	0.986353i	$-0.447352\pi$
$888$		0			0
$889$	0.795139	−	1.37722i	0.0266681	−	0.0461905i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−30.1250	+	52.1781i	−1.00810	+	1.74607i
$894$		0			0
$895$	−4.58108	−	7.93467i	−0.153129	−	0.265227i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−20.1676			−0.672626
$900$		0			0
$901$	16.8282			0.560628
$902$		0			0
$903$		0			0
$904$		0			0
$905$	6.78651	+	11.7546i	0.225591	+	0.390735i
$906$		0			0
$907$	3.29404	−	5.70544i	0.109377	−	0.189446i	−0.806141	−	0.591723i	$-0.798448\pi$
$907$	3.29404	−	5.70544i	0.109377	−	0.189446i	0.915518	+	0.402277i	$0.131781\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−11.8855	+	20.5863i	−0.393785	+	0.682056i	−0.992945	−	0.118573i	$-0.962168\pi$
$911$	−11.8855	+	20.5863i	−0.393785	+	0.682056i	0.599160	+	0.800629i	$0.295501\pi$
$912$		0			0
$913$	13.8688	+	24.0215i	0.458991	+	0.794996i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6.56477			0.216788
$918$		0			0
$919$	46.1305			1.52171			0.760853	−	0.648924i	$-0.224781\pi$
$919$	46.1305			1.52171			0.760853	+	0.648924i	$0.224781\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	31.3452	+	54.2914i	1.03174	+	1.78702i
$924$		0			0
$925$	−3.82220	+	6.62025i	−0.125673	+	0.217673i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11.3082	−	19.5864i	0.371010	−	0.642608i	−0.618711	−	0.785619i	$-0.712345\pi$
$929$	11.3082	−	19.5864i	0.371010	−	0.642608i	0.989721	+	0.143010i	$0.0456782\pi$
$930$		0			0
$931$	−26.0217	−	45.0709i	−0.852826	−	1.47714i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	13.5375			0.442723
$936$		0			0
$937$	−47.5830			−1.55447			−0.777235	−	0.629211i	$-0.783378\pi$
$937$	−47.5830			−1.55447			−0.777235	+	0.629211i	$0.783378\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21.4991	−	37.2375i	−0.700850	−	1.21391i	−0.968169	−	0.250299i	$-0.919471\pi$
$941$	−21.4991	−	37.2375i	−0.700850	−	1.21391i	0.267319	−	0.963608i	$-0.413862\pi$
$942$		0			0
$943$	−4.58594	+	7.94308i	−0.149339	+	0.258662i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24.0274	+	41.6167i	−0.780786	+	1.35236i	0.150698	+	0.988580i	$0.451848\pi$
$947$	−24.0274	+	41.6167i	−0.780786	+	1.35236i	−0.931484	+	0.363781i	$0.881486\pi$
$948$		0			0
$949$	28.1040	+	48.6776i	0.912295	+	1.58014i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	39.0413			1.26467			0.632335	−	0.774695i	$-0.282096\pi$
$953$	39.0413			1.26467			0.632335	+	0.774695i	$0.282096\pi$
$954$		0			0
$955$	−10.2694			−0.332311
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.07571	−	5.32729i	−0.0993199	−	0.172027i
$960$		0			0
$961$	−0.429681	+	0.744229i	−0.0138607	+	0.0240074i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	4.53570	−	7.85606i	0.146009	−	0.252895i
$966$		0			0
$967$	1.02423	+	1.77402i	0.0329371	+	0.0570488i	0.882024	−	0.471204i	$-0.156181\pi$
$967$	1.02423	+	1.77402i	0.0329371	+	0.0570488i	−0.849087	+	0.528253i	$0.822847\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−27.4068			−0.879527			−0.439764	−	0.898114i	$-0.644938\pi$
$971$	−27.4068			−0.879527			−0.439764	+	0.898114i	$0.644938\pi$
$972$		0			0
$973$	−1.78546			−0.0572392
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2.04914	−	3.54922i	−0.0655578	−	0.113549i	0.831383	−	0.555699i	$-0.187549\pi$
$977$	−2.04914	−	3.54922i	−0.0655578	−	0.113549i	−0.896941	+	0.442150i	$0.854216\pi$
$978$		0			0
$979$	−26.2384	+	45.4463i	−0.838584	+	1.45247i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.05903	+	5.29840i	−0.0975680	+	0.168993i	−0.910677	−	0.413118i	$-0.864440\pi$
$983$	−3.05903	+	5.29840i	−0.0975680	+	0.168993i	0.813109	+	0.582111i	$0.197773\pi$
$984$		0			0
$985$	2.40329	+	4.16262i	0.0765752	+	0.132632i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	11.7958			0.375086
$990$		0			0
$991$	14.9881			0.476114			0.238057	−	0.971251i	$-0.423490\pi$
$991$	14.9881			0.476114			0.238057	+	0.971251i	$0.423490\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9.22981	−	15.9865i	−0.292605	−	0.506806i
$996$		0			0
$997$	6.74813	−	11.6881i	0.213715	−	0.370166i	−0.739159	−	0.673531i	$-0.764777\pi$
$997$	6.74813	−	11.6881i	0.213715	−	0.370166i	0.952874	+	0.303365i	$0.0981101\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.2.q.e.721.3		8
3.2	odd	2		360.2.q.e.241.1	yes	8
4.3	odd	2		2160.2.q.l.721.2		8
9.2	odd	6		3240.2.a.u.1.2		4
9.4	even	3	inner	1080.2.q.e.361.3		8
9.5	odd	6		360.2.q.e.121.1	✓	8
9.7	even	3		3240.2.a.s.1.2		4
12.11	even	2		720.2.q.l.241.4		8
36.7	odd	6		6480.2.a.bz.1.3		4
36.11	even	6		6480.2.a.cb.1.3		4
36.23	even	6		720.2.q.l.481.4		8
36.31	odd	6		2160.2.q.l.1441.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.e.121.1	✓	8	9.5	odd	6
360.2.q.e.241.1	yes	8	3.2	odd	2
720.2.q.l.241.4		8	12.11	even	2
720.2.q.l.481.4		8	36.23	even	6
1080.2.q.e.361.3		8	9.4	even	3	inner
1080.2.q.e.721.3		8	1.1	even	1	trivial
2160.2.q.l.721.2		8	4.3	odd	2
2160.2.q.l.1441.2		8	36.31	odd	6
3240.2.a.s.1.2		4	9.7	even	3
3240.2.a.u.1.2		4	9.2	odd	6
6480.2.a.bz.1.3		4	36.7	odd	6
6480.2.a.cb.1.3		4	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)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(0.165947 - 0.287429i) q^{7} +(2.20380 - 3.81710i) q^{11} +(-3.49031 - 6.04539i) q^{13} +3.07139 q^{17} -7.55364 q^{19} +(0.834053 + 1.44462i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(1.78651 - 3.09432i) q^{29} +(-2.82220 - 4.88820i) q^{31} +0.331895 q^{35} +7.64441 q^{37} +(2.74919 + 4.76174i) q^{41} +(3.53570 - 6.12401i) q^{43} +(3.98815 - 6.90768i) q^{47} +(3.44492 + 5.96678i) q^{49} +5.47900 q^{53} +4.40761 q^{55} +(-3.69411 - 6.39839i) q^{59} +(-0.296197 + 0.513029i) q^{61} +(3.49031 - 6.04539i) q^{65} +(-2.61087 - 4.52216i) q^{67} -8.98062 q^{71} -8.05202 q^{73} +(-0.731430 - 1.26687i) q^{77} +(5.49838 - 9.52347i) q^{79} +(-3.14657 + 5.45002i) q^{83} +(1.53570 + 2.65991i) q^{85} -11.9060 q^{89} -2.31683 q^{91} +(-3.77682 - 6.54164i) q^{95} +(0.622718 - 1.07858i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} + q^{7} + q^{11} - 4 q^{13} - 10 q^{17} + 2 q^{19} + 7 q^{23} - 4 q^{25} + 7 q^{29} + 2 q^{31} + 2 q^{35} + 12 q^{37} + 12 q^{41} + 11 q^{43} + 7 q^{47} - 3 q^{49} - 24 q^{53} + 2 q^{55}+ \cdots - q^{97}+O(q^{100}) \) 8 * q + 4 * q^5 + q^7 + q^11 - 4 * q^13 - 10 * q^17 + 2 * q^19 + 7 * q^23 - 4 * q^25 + 7 * q^29 + 2 * q^31 + 2 * q^35 + 12 * q^37 + 12 * q^41 + 11 * q^43 + 7 * q^47 - 3 * q^49 - 24 * q^53 + 2 * q^55 + 11 * q^59 - 19 * q^61 + 4 * q^65 + 10 * q^67 - 24 * q^71 + 18 * q^73 + 32 * q^77 + 24 * q^79 + 23 * q^83 - 5 * q^85 - 42 * q^89 + 28 * q^91 + q^95 - q^97

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