LMFDB - Embedded newform 756.3.bk.g.53.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,3,Mod(53,756)] 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("756.53"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	756.bk (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,24,0,0,0,0,0,88] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$20.5995079856$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 46 x^{14} + 1437 x^{12} - 24668 x^{10} + 309582 x^{8} - 2188585 x^{6} + 10478650 x^{4} + \cdots + 194481 $ x^16 - 46x^14 + 1437x^12 - 24668x^10 + 309582x^8 - 2188585x^6 + 10478650x^4 - 1447803*x^2 + 194481
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.4
Root			$0.321999 + 0.185906i$ of defining polynomial
Character	$\chi$	$=$	756.53
Dual form			756.3.bk.g.485.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.181791 + 0.104957i) q^{5} +(6.74497 + 1.87226i) q^{7} +(16.4658 + 9.50651i) q^{11} -7.60538 q^{13} +(4.04837 + 2.33733i) q^{17} +(-9.11147 - 15.7815i) q^{19} +(3.20049 - 1.84780i) q^{23} +(-12.4780 + 21.6125i) q^{25} +25.8897i q^{29} +(4.91813 - 8.51844i) q^{31} +(-1.42268 + 0.367572i) q^{35} +(-3.12741 - 5.41684i) q^{37} +17.8687i q^{41} +41.0391 q^{43} +(25.7037 - 14.8400i) q^{47} +(41.9893 + 25.2567i) q^{49} +(69.4573 + 40.1012i) q^{53} -3.99109 q^{55} +(69.3661 + 40.0485i) q^{59} +(-29.2889 - 50.7298i) q^{61} +(1.38259 - 0.798236i) q^{65} +(-45.4046 + 78.6431i) q^{67} +88.2331i q^{71} +(10.0019 - 17.3238i) q^{73} +(93.2625 + 94.9493i) q^{77} +(-32.1803 - 55.7379i) q^{79} +37.3016i q^{83} -0.981273 q^{85} +(123.592 - 71.3556i) q^{89} +(-51.2981 - 14.2392i) q^{91} +(3.31276 + 1.91262i) q^{95} -44.6187 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 24 q^{7} + 88 q^{13} + 14 q^{19} + 36 q^{25} - 68 q^{31} - 76 q^{37} - 292 q^{43} - 20 q^{49} - 272 q^{55} - 110 q^{61} - 72 q^{67} + 60 q^{73} + 154 q^{79} + 700 q^{85} - 74 q^{91} + 264 q^{97}+O(q^{100}) $ 16 * q + 24 * q^7 + 88 * q^13 + 14 * q^19 + 36 * q^25 - 68 * q^31 - 76 * q^37 - 292 * q^43 - 20 * q^49 - 272 * q^55 - 110 * q^61 - 72 * q^67 + 60 * q^73 + 154 * q^79 + 700 * q^85 - 74 * q^91 + 264 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.181791	+	0.104957i	−0.0363581	+	0.0209914i	−0.518069	−	0.855339i	$-0.673349\pi$
$5$	−0.181791	+	0.104957i	−0.0363581	+	0.0209914i	0.481711	+	0.876330i	$0.340016\pi$
$6$		0			0
$7$	6.74497	+	1.87226i	0.963567	+	0.267466i
$7$	6.74497	+	1.87226i	0.963567	+	0.267466i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	16.4658	+	9.50651i	1.49689	+	0.864228i	0.999994	−	0.00358274i	$-0.00114042\pi$
$11$	16.4658	+	9.50651i	1.49689	+	0.864228i	0.496894	+	0.867811i	$0.334474\pi$
$12$		0			0
$13$	−7.60538			−0.585029			−0.292515	−	0.956261i	$-0.594492\pi$
$13$	−7.60538			−0.585029			−0.292515	+	0.956261i	$0.594492\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.04837	+	2.33733i	0.238139	+	0.137490i	0.614321	−	0.789056i	$-0.289430\pi$
$17$	4.04837	+	2.33733i	0.238139	+	0.137490i	−0.376182	+	0.926546i	$0.622763\pi$
$18$		0			0
$19$	−9.11147	−	15.7815i	−0.479551	−	0.830607i	0.520174	−	0.854060i	$-0.325867\pi$
$19$	−9.11147	−	15.7815i	−0.479551	−	0.830607i	−0.999725	+	0.0234536i	$0.992534\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.20049	−	1.84780i	0.139152	−	0.0803393i	−0.428808	−	0.903396i	$-0.641066\pi$
$23$	3.20049	−	1.84780i	0.139152	−	0.0803393i	0.567960	+	0.823056i	$0.307733\pi$
$24$		0			0
$25$	−12.4780	+	21.6125i	−0.499119	+	0.864499i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			25.8897i			0.892747i	0.894847	+	0.446373i	$0.147285\pi$
$29$			25.8897i			0.892747i	−0.894847	+	0.446373i	$0.852715\pi$
$30$		0			0
$31$	4.91813	−	8.51844i	0.158649	−	0.274789i	−0.775733	−	0.631062i	$-0.782619\pi$
$31$	4.91813	−	8.51844i	0.158649	−	0.274789i	0.934382	+	0.356273i	$0.115953\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.42268	+	0.367572i	−0.0406479	+	0.0105021i
$36$		0			0
$37$	−3.12741	−	5.41684i	−0.0845246	−	0.146401i	0.820664	−	0.571411i	$-0.193604\pi$
$37$	−3.12741	−	5.41684i	−0.0845246	−	0.146401i	−0.905189	+	0.425010i	$0.860271\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			17.8687i			0.435822i	0.975969	+	0.217911i	$0.0699242\pi$
$41$			17.8687i			0.435822i	−0.975969	+	0.217911i	$0.930076\pi$
$42$		0			0
$43$	41.0391			0.954397			0.477199	−	0.878795i	$-0.341652\pi$
$43$	41.0391			0.954397			0.477199	+	0.878795i	$0.341652\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	25.7037	−	14.8400i	0.546886	−	0.315745i	−0.200979	−	0.979596i	$-0.564412\pi$
$47$	25.7037	−	14.8400i	0.546886	−	0.315745i	0.747865	+	0.663851i	$0.231079\pi$
$48$		0			0
$49$	41.9893	+	25.2567i	0.856924	+	0.515442i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	69.4573	+	40.1012i	1.31052	+	0.756627i	0.982181	−	0.187935i	$-0.0601794\pi$
$53$	69.4573	+	40.1012i	1.31052	+	0.756627i	0.328334	+	0.944562i	$0.393513\pi$
$54$		0			0
$55$	−3.99109			−0.0725653
$56$		0			0
$57$		0			0
$58$		0			0
$59$	69.3661	+	40.0485i	1.17570	+	0.678788i	0.955015	−	0.296558i	$-0.0958387\pi$
$59$	69.3661	+	40.0485i	1.17570	+	0.678788i	0.220681	+	0.975346i	$0.429172\pi$
$60$		0			0
$61$	−29.2889	−	50.7298i	−0.480145	−	0.831636i	0.519595	−	0.854413i	$-0.326083\pi$
$61$	−29.2889	−	50.7298i	−0.480145	−	0.831636i	−0.999741	+	0.0227763i	$0.992749\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.38259	−	0.798236i	0.0212706	−	0.0122806i
$66$		0			0
$67$	−45.4046	+	78.6431i	−0.677680	+	1.17378i	0.297997	+	0.954567i	$0.403681\pi$
$67$	−45.4046	+	78.6431i	−0.677680	+	1.17378i	−0.975678	+	0.219210i	$0.929652\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			88.2331i			1.24272i	0.783525	+	0.621360i	$0.213420\pi$
$71$			88.2331i			1.24272i	−0.783525	+	0.621360i	$0.786580\pi$
$72$		0			0
$73$	10.0019	−	17.3238i	0.137013	−	0.237313i	−0.789352	−	0.613941i	$-0.789583\pi$
$73$	10.0019	−	17.3238i	0.137013	−	0.237313i	0.926364	+	0.376628i	$0.122917\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	93.2625	+	94.9493i	1.21120	+	1.23311i
$78$		0			0
$79$	−32.1803	−	55.7379i	−0.407346	−	0.705543i	0.587246	−	0.809409i	$-0.300212\pi$
$79$	−32.1803	−	55.7379i	−0.407346	−	0.705543i	−0.994591	+	0.103865i	$0.966879\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			37.3016i			0.449416i	0.974426	+	0.224708i	$0.0721429\pi$
$83$			37.3016i			0.449416i	−0.974426	+	0.224708i	$0.927857\pi$
$84$		0			0
$85$	−0.981273			−0.0115444
$86$		0			0
$87$		0			0
$88$		0			0
$89$	123.592	−	71.3556i	1.38867	−	0.801748i	0.395503	−	0.918465i	$-0.370570\pi$
$89$	123.592	−	71.3556i	1.38867	−	0.801748i	0.993165	+	0.116716i	$0.0372369\pi$
$90$		0			0
$91$	−51.2981	−	14.2392i	−0.563715	−	0.156475i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.31276	+	1.91262i	0.0348711	+	0.0201329i
$96$		0			0
$97$	−44.6187			−0.459987			−0.229993	−	0.973192i	$-0.573870\pi$
$97$	−44.6187			−0.459987			−0.229993	+	0.973192i	$0.573870\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.3713	+	10.6067i	0.181894	+	0.105017i	0.588182	−	0.808728i	$-0.299844\pi$
$101$	18.3713	+	10.6067i	0.181894	+	0.105017i	−0.406288	+	0.913745i	$0.633177\pi$
$102$		0			0
$103$	62.7498	+	108.686i	0.609221	+	1.05520i	0.991369	+	0.131101i	$0.0418512\pi$
$103$	62.7498	+	108.686i	0.609221	+	1.05520i	−0.382148	+	0.924101i	$0.624815\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−94.0831	+	54.3189i	−0.879281	+	0.507653i	−0.870421	−	0.492307i	$-0.836154\pi$
$107$	−94.0831	+	54.3189i	−0.879281	+	0.507653i	−0.00886005	+	0.999961i	$0.502820\pi$
$108$		0			0
$109$	88.7759	−	153.764i	0.814458	−	1.41068i	−0.0952585	−	0.995453i	$-0.530368\pi$
$109$	88.7759	−	153.764i	0.814458	−	1.41068i	0.909716	−	0.415230i	$-0.136299\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	150.170i		−	1.32894i	−0.747314	−	0.664471i	$-0.768657\pi$
$113$		−	150.170i		−	1.32894i	0.747314	−	0.664471i	$-0.231343\pi$
$114$		0			0
$115$	−0.387879	+	0.671827i	−0.00337286	+	0.00584197i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	22.9300	+	23.3448i	0.192689	+	0.196175i
$120$		0			0
$121$	120.248	+	208.275i	0.993782	+	1.72128i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	10.4864i		−	0.0838914i
$126$		0			0
$127$	−101.926			−0.802564			−0.401282	−	0.915955i	$-0.631435\pi$
$127$	−101.926			−0.802564			−0.401282	+	0.915955i	$0.631435\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−211.426	+	122.067i	−1.61394	+	0.931806i	−0.625490	+	0.780232i	$0.715101\pi$
$131$	−211.426	+	122.067i	−1.61394	+	0.931806i	−0.988446	+	0.151574i	$0.951566\pi$
$132$		0			0
$133$	−31.9095	−	123.505i	−0.239921	−	0.928609i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	109.199	+	63.0463i	0.797076	+	0.460192i	0.842448	−	0.538778i	$-0.181114\pi$
$137$	109.199	+	63.0463i	0.797076	+	0.460192i	−0.0453718	+	0.998970i	$0.514447\pi$
$138$		0			0
$139$	176.319			1.26848			0.634241	−	0.773136i	$-0.281313\pi$
$139$	176.319			1.26848			0.634241	+	0.773136i	$0.281313\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−125.228	−	72.3006i	−0.875723	−	0.505599i
$144$		0			0
$145$	−2.71730	−	4.70649i	−0.0187400	−	0.0324586i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	21.4047	−	12.3580i	0.143656	−	0.0829396i	−0.426450	−	0.904511i	$-0.640236\pi$
$149$	21.4047	−	12.3580i	0.143656	−	0.0829396i	0.570105	+	0.821572i	$0.306902\pi$
$150$		0			0
$151$	51.0267	−	88.3808i	0.337925	−	0.585303i	−0.646117	−	0.763238i	$-0.723608\pi$
$151$	51.0267	−	88.3808i	0.337925	−	0.585303i	0.984042	+	0.177935i	$0.0569416\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			2.06476i			0.0133211i
$156$		0			0
$157$	−104.088	+	180.286i	−0.662984	+	1.14832i	0.316844	+	0.948478i	$0.397377\pi$
$157$	−104.088	+	180.286i	−0.662984	+	1.14832i	−0.979828	+	0.199844i	$0.935957\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	25.0468	−	6.47124i	0.155570	−	0.0401940i
$162$		0			0
$163$	−101.170	−	175.231i	−0.620674	−	1.07504i	−0.989360	−	0.145485i	$-0.953526\pi$
$163$	−101.170	−	175.231i	−0.620674	−	1.07504i	0.368687	−	0.929554i	$-0.379808\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	274.788i		−	1.64544i	−0.568449	−	0.822719i	$-0.692456\pi$
$167$		−	274.788i		−	1.64544i	0.568449	−	0.822719i	$-0.307544\pi$
$168$		0			0
$169$	−111.158			−0.657741
$170$		0			0
$171$		0			0
$172$		0			0
$173$	100.783	−	58.1871i	0.582560	−	0.336341i	−0.179590	−	0.983742i	$-0.557477\pi$
$173$	100.783	−	58.1871i	0.582560	−	0.336341i	0.762150	+	0.647400i	$0.224144\pi$
$174$		0			0
$175$	−124.628	+	122.414i	−0.712158	+	0.699506i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	23.1847	+	13.3857i	0.129523	+	0.0747803i	0.563362	−	0.826210i	$-0.309508\pi$
$179$	23.1847	+	13.3857i	0.129523	+	0.0747803i	−0.433838	+	0.900991i	$0.642841\pi$
$180$		0			0
$181$	−105.045			−0.580359			−0.290180	−	0.956972i	$-0.593715\pi$
$181$	−105.045			−0.580359			−0.290180	+	0.956972i	$0.593715\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.13707	+	0.656486i	0.00614631	+	0.00354857i
$186$		0			0
$187$	44.4396	+	76.9717i	0.237645	+	0.411613i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−271.187	+	156.570i	−1.41983	+	0.819737i	−0.996283	−	0.0861385i	$-0.972547\pi$
$191$	−271.187	+	156.570i	−1.41983	+	0.819737i	−0.423543	+	0.905876i	$0.639214\pi$
$192$		0			0
$193$	10.8132	−	18.7290i	0.0560270	−	0.0970416i	−0.836652	−	0.547735i	$-0.815490\pi$
$193$	10.8132	−	18.7290i	0.0560270	−	0.0970416i	0.892679	+	0.450694i	$0.148823\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	102.841i		−	0.522038i	−0.965334	−	0.261019i	$-0.915942\pi$
$197$		−	102.841i		−	0.522038i	0.965334	−	0.261019i	$-0.0840585\pi$
$198$		0			0
$199$	97.1497	−	168.268i	0.488189	−	0.845569i	−0.511718	−	0.859153i	$-0.670991\pi$
$199$	97.1497	−	168.268i	0.488189	−	0.845569i	0.999908	+	0.0135845i	$0.00432421\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−48.4721	+	174.625i	−0.238779	+	0.860222i
$204$		0			0
$205$	−1.87544	−	3.24836i	−0.00914850	−	0.0158457i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	346.473i		−	1.65777i
$210$		0			0
$211$	62.0446			0.294050			0.147025	−	0.989133i	$-0.453030\pi$
$211$	62.0446			0.294050			0.147025	+	0.989133i	$0.453030\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−7.46052	+	4.30733i	−0.0347001	+	0.0200341i
$216$		0			0
$217$	49.1214	−	48.2487i	0.226366	−	0.222344i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−30.7894	−	17.7762i	−0.139318	−	0.0804355i
$222$		0			0
$223$	180.161			0.807899			0.403949	−	0.914781i	$-0.367637\pi$
$223$	180.161			0.807899			0.403949	+	0.914781i	$0.367637\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−390.063	−	225.203i	−1.71834	−	0.992083i	−0.921960	−	0.387285i	$-0.873413\pi$
$227$	−390.063	−	225.203i	−1.71834	−	0.992083i	−0.796378	−	0.604799i	$-0.793254\pi$
$228$		0			0
$229$	100.935	+	174.825i	0.440765	+	0.763428i	0.997746	−	0.0670971i	$-0.0213737\pi$
$229$	100.935	+	174.825i	0.440765	+	0.763428i	−0.556981	+	0.830525i	$0.688040\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−198.611	+	114.668i	−0.852408	+	0.492138i	−0.861463	−	0.507821i	$-0.830451\pi$
$233$	−198.611	+	114.668i	−0.852408	+	0.492138i	0.00905441	+	0.999959i	$0.497118\pi$
$234$		0			0
$235$	−3.11512	+	5.39555i	−0.0132558	+	0.0229598i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	80.7464i		−	0.337851i	−0.985629	−	0.168925i	$-0.945970\pi$
$239$		−	80.7464i		−	0.337851i	0.985629	−	0.168925i	$-0.0540297\pi$
$240$		0			0
$241$	−5.63130	+	9.75370i	−0.0233664	+	0.0404718i	−0.877472	−	0.479628i	$-0.840772\pi$
$241$	−5.63130	+	9.75370i	−0.0233664	+	0.0404718i	0.854106	+	0.520099i	$0.174105\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−10.2841	−	0.184361i	−0.0419760	−	0.000752492i
$246$		0			0
$247$	69.2962	+	120.025i	0.280551	+	0.485929i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	323.254i		−	1.28786i	−0.765083	−	0.643932i	$-0.777302\pi$
$251$		−	323.254i		−	1.28786i	0.765083	−	0.643932i	$-0.222698\pi$
$252$		0			0
$253$	70.2647			0.277726
$254$		0			0
$255$		0			0
$256$		0			0
$257$	235.895	−	136.194i	0.917878	−	0.529937i	0.0349203	−	0.999390i	$-0.488882\pi$
$257$	235.895	−	136.194i	0.917878	−	0.529937i	0.882957	+	0.469453i	$0.155549\pi$
$258$		0			0
$259$	−10.9526	−	42.3917i	−0.0422880	−	0.163675i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−147.756	−	85.3071i	−0.561811	−	0.324362i	0.192061	−	0.981383i	$-0.438483\pi$
$263$	−147.756	−	85.3071i	−0.561811	−	0.324362i	−0.753872	+	0.657021i	$0.771816\pi$
$264$		0			0
$265$	−16.8356			−0.0635305
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−196.531	−	113.467i	−0.730599	−	0.421812i	0.0880422	−	0.996117i	$-0.471939\pi$
$269$	−196.531	−	113.467i	−0.730599	−	0.421812i	−0.818641	+	0.574305i	$0.805272\pi$
$270$		0			0
$271$	−150.201	−	260.156i	−0.554249	−	0.959987i	−0.997962	−	0.0638179i	$-0.979672\pi$
$271$	−150.201	−	260.156i	−0.554249	−	0.959987i	0.443713	−	0.896169i	$-0.353661\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−410.919	+	237.244i	−1.49425	+	0.862705i
$276$		0			0
$277$	−60.2976	+	104.439i	−0.217681	+	0.377034i	−0.954099	−	0.299493i	$-0.903183\pi$
$277$	−60.2976	+	104.439i	−0.217681	+	0.377034i	0.736418	+	0.676527i	$0.236516\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	174.327i		−	0.620381i	−0.950674	−	0.310191i	$-0.899607\pi$
$281$		−	174.327i		−	0.620381i	0.950674	−	0.310191i	$-0.100393\pi$
$282$		0			0
$283$	−80.3886	+	139.237i	−0.284059	+	0.492004i	−0.972380	−	0.233401i	$-0.925014\pi$
$283$	−80.3886	+	139.237i	−0.284059	+	0.492004i	0.688322	+	0.725405i	$0.258348\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−33.4548	+	120.524i	−0.116567	+	0.419944i
$288$		0			0
$289$	−133.574	−	231.357i	−0.462193	−	0.800542i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	169.668i		−	0.579071i	−0.957167	−	0.289536i	$-0.906499\pi$
$293$		−	169.668i		−	0.579071i	0.957167	−	0.289536i	$-0.0935009\pi$
$294$		0			0
$295$	−16.8135			−0.0569948
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−24.3409	+	14.0533i	−0.0814079	+	0.0470008i
$300$		0			0
$301$	276.807	+	76.8358i	0.919626	+	0.255268i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	10.6489	+	6.14813i	0.0349144	+	0.0201578i
$306$		0			0
$307$	−466.100			−1.51824			−0.759120	−	0.650950i	$-0.774371\pi$
$307$	−466.100			−1.51824			−0.759120	+	0.650950i	$0.774371\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−499.817	−	288.569i	−1.60713	−	0.927876i	−0.990009	−	0.141006i	$-0.954966\pi$
$311$	−499.817	−	288.569i	−1.60713	−	0.927876i	−0.617119	−	0.786870i	$-0.711700\pi$
$312$		0			0
$313$	−237.831	−	411.936i	−0.759845	−	1.31609i	−0.942929	−	0.332993i	$-0.891942\pi$
$313$	−237.831	−	411.936i	−0.759845	−	1.31609i	0.183085	−	0.983097i	$-0.441392\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−154.787	+	89.3665i	−0.488288	+	0.281913i	−0.723864	−	0.689943i	$-0.757636\pi$
$317$	−154.787	+	89.3665i	−0.488288	+	0.281913i	0.235576	+	0.971856i	$0.424302\pi$
$318$		0			0
$319$	−246.120	+	426.293i	−0.771537	+	1.33634i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	85.1859i		−	0.263733i
$324$		0			0
$325$	94.8997	−	164.371i	0.291999	−	0.505757i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	201.155	−	51.9716i	0.611413	−	0.157968i
$330$		0			0
$331$	29.0733	+	50.3565i	0.0878348	+	0.152134i	0.906596	−	0.422000i	$-0.138672\pi$
$331$	29.0733	+	50.3565i	0.0878348	+	0.152134i	−0.818761	+	0.574135i	$0.805339\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	19.0621i		−	0.0569017i
$336$		0			0
$337$	321.818			0.954948			0.477474	−	0.878646i	$-0.341552\pi$
$337$	321.818			0.954948			0.477474	+	0.878646i	$0.341552\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	161.961	−	93.5085i	0.474960	−	0.274218i
$342$		0			0
$343$	235.930	+	248.970i	0.687841	+	0.725861i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	232.474	+	134.219i	0.669953	+	0.386798i	0.796059	−	0.605219i	$-0.206915\pi$
$347$	232.474	+	134.219i	0.669953	+	0.386798i	−0.126106	+	0.992017i	$0.540248\pi$
$348$		0			0
$349$	−9.87359			−0.0282911			−0.0141455	−	0.999900i	$-0.504503\pi$
$349$	−9.87359			−0.0282911			−0.0141455	+	0.999900i	$0.504503\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	237.031	+	136.850i	0.671476	+	0.387677i	0.796636	−	0.604460i	$-0.206611\pi$
$353$	237.031	+	136.850i	0.671476	+	0.387677i	−0.125160	+	0.992137i	$0.539944\pi$
$354$		0			0
$355$	−9.26067	−	16.0399i	−0.0260864	−	0.0451829i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−247.509	+	142.899i	−0.689440	+	0.398048i	−0.803402	−	0.595437i	$-0.796979\pi$
$359$	−247.509	+	142.899i	−0.689440	+	0.398048i	0.113962	+	0.993485i	$0.463646\pi$
$360$		0			0
$361$	14.4622	−	25.0493i	0.0400616	−	0.0693887i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			4.19908i			0.0115043i
$366$		0			0
$367$	139.261	−	241.207i	0.379457	−	0.657239i	−0.611526	−	0.791224i	$-0.709444\pi$
$367$	139.261	−	241.207i	0.379457	−	0.657239i	0.990983	+	0.133985i	$0.0427775\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	393.408	+	400.524i	1.06040	+	1.07958i
$372$		0			0
$373$	118.591	+	205.405i	0.317937	+	0.550684i	0.980058	−	0.198714i	$-0.0636765\pi$
$373$	118.591	+	205.405i	0.317937	+	0.550684i	−0.662120	+	0.749398i	$0.730343\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	196.901i		−	0.522283i
$378$		0			0
$379$	472.380			1.24639			0.623193	−	0.782068i	$-0.285835\pi$
$379$	472.380			1.24639			0.623193	+	0.782068i	$0.285835\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	358.666	−	207.076i	0.936465	−	0.540668i	0.0476147	−	0.998866i	$-0.484838\pi$
$383$	358.666	−	207.076i	0.936465	−	0.540668i	0.888850	+	0.458197i	$0.151505\pi$
$384$		0			0
$385$	−26.9198	−	7.47236i	−0.0699216	−	0.0194087i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	614.253	+	354.639i	1.57906	+	0.911669i	0.994991	+	0.0999627i	$0.0318724\pi$
$389$	614.253	+	354.639i	1.57906	+	0.911669i	0.584066	+	0.811706i	$0.301461\pi$
$390$		0			0
$391$	17.2757			0.0441833
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11.7001	+	6.75508i	0.0296206	+	0.0171015i
$396$		0			0
$397$	125.857	+	217.990i	0.317019	+	0.549093i	0.979865	−	0.199663i	$-0.0639847\pi$
$397$	125.857	+	217.990i	0.317019	+	0.549093i	−0.662845	+	0.748756i	$0.730651\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−496.267	+	286.520i	−1.23757	+	0.714514i	−0.968598	−	0.248633i	$-0.920019\pi$
$401$	−496.267	+	286.520i	−1.23757	+	0.714514i	−0.268977	+	0.963147i	$0.586685\pi$
$402$		0			0
$403$	−37.4042	+	64.7860i	−0.0928144	+	0.160759i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	118.923i		−	0.292194i
$408$		0			0
$409$	58.5577	−	101.425i	0.143173	−	0.247983i	−0.785517	−	0.618840i	$-0.787603\pi$
$409$	58.5577	−	101.425i	0.143173	−	0.247983i	0.928690	+	0.370858i	$0.120936\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	392.891	+	399.997i	0.951310	+	0.968516i
$414$		0			0
$415$	−3.91505	−	6.78107i	−0.00943386	−	0.0163399i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	30.5477i		−	0.0729062i	−0.999335	−	0.0364531i	$-0.988394\pi$
$419$		−	30.5477i		−	0.0729062i	0.999335	−	0.0364531i	$-0.0116059\pi$
$420$		0			0
$421$	−402.732			−0.956608			−0.478304	−	0.878194i	$-0.658748\pi$
$421$	−402.732			−0.956608			−0.478304	+	0.878194i	$0.658748\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−101.031	+	58.3301i	−0.237719	+	0.137247i
$426$		0			0
$427$	−102.573	−	397.008i	−0.240218	−	0.929760i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	330.346	+	190.725i	0.766464	+	0.442518i	0.831612	−	0.555357i	$-0.187418\pi$
$431$	330.346	+	190.725i	0.766464	+	0.442518i	−0.0651475	+	0.997876i	$0.520752\pi$
$432$		0			0
$433$	−177.293			−0.409453			−0.204727	−	0.978819i	$-0.565631\pi$
$433$	−177.293			−0.409453			−0.204727	+	0.978819i	$0.565631\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−58.3224	−	33.6724i	−0.133461	−	0.0770536i
$438$		0			0
$439$	−31.5890	−	54.7138i	−0.0719567	−	0.124633i	0.827802	−	0.561020i	$-0.189591\pi$
$439$	−31.5890	−	54.7138i	−0.0719567	−	0.124633i	−0.899759	+	0.436388i	$0.856258\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−285.537	+	164.855i	−0.644553	+	0.372133i	−0.786366	−	0.617761i	$-0.788040\pi$
$443$	−285.537	+	164.855i	−0.644553	+	0.372133i	0.141813	+	0.989893i	$0.454707\pi$
$444$		0			0
$445$	−14.9785	+	25.9435i	−0.0336596	+	0.0583001i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	335.346i		−	0.746872i	−0.927656	−	0.373436i	$-0.878180\pi$
$449$		−	335.346i		−	0.746872i	0.927656	−	0.373436i	$-0.121820\pi$
$450$		0			0
$451$	−169.869	+	294.222i	−0.376650	+	0.652377i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	10.8200	−	2.79552i	0.0237802	−	0.00614401i
$456$		0			0
$457$	−62.0641	−	107.498i	−0.135808	−	0.235226i	0.790098	−	0.612981i	$-0.210030\pi$
$457$	−62.0641	−	107.498i	−0.135808	−	0.235226i	−0.925906	+	0.377755i	$0.876696\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	776.337i		−	1.68403i	−0.539456	−	0.842014i	$-0.681370\pi$
$461$		−	776.337i		−	1.68403i	0.539456	−	0.842014i	$-0.318630\pi$
$462$		0			0
$463$	−595.677			−1.28656			−0.643280	−	0.765631i	$-0.722427\pi$
$463$	−595.677			−1.28656			−0.643280	+	0.765631i	$0.722427\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−595.122	+	343.594i	−1.27435	+	0.735747i	−0.975804	−	0.218647i	$-0.929836\pi$
$467$	−595.122	+	343.594i	−1.27435	+	0.735747i	−0.298548	+	0.954395i	$0.596502\pi$
$468$		0			0
$469$	−453.493	+	445.436i	−0.966936	+	0.949757i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	675.740	+	390.139i	1.42863	+	0.824817i
$474$		0			0
$475$	454.771			0.957412
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−38.0835	−	21.9875i	−0.0795062	−	0.0459029i	0.459720	−	0.888064i	$-0.347950\pi$
$479$	−38.0835	−	21.9875i	−0.0795062	−	0.0459029i	−0.539226	+	0.842161i	$0.681283\pi$
$480$		0			0
$481$	23.7852	+	41.1971i	0.0494494	+	0.0856488i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.11126	−	4.68304i	0.0167242	−	0.00965575i
$486$		0			0
$487$	400.853	−	694.298i	0.823107	−	1.42566i	−0.0802496	−	0.996775i	$-0.525572\pi$
$487$	400.853	−	694.298i	0.823107	−	1.42566i	0.903357	−	0.428889i	$-0.141095\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			949.361i			1.93353i	0.255676	+	0.966763i	$0.417702\pi$
$491$			949.361i			1.93353i	−0.255676	+	0.966763i	$0.582298\pi$
$492$		0			0
$493$	−60.5126	+	104.811i	−0.122744	+	0.212598i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−165.195	+	595.130i	−0.332385	+	1.19744i
$498$		0			0
$499$	−289.433	−	501.313i	−0.580026	−	1.00463i	−0.995476	−	0.0950181i	$-0.969709\pi$
$499$	−289.433	−	501.313i	−0.580026	−	1.00463i	0.415450	−	0.909616i	$-0.363624\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	439.255i		−	0.873271i	−0.899639	−	0.436635i	$-0.856170\pi$
$503$		−	439.255i		−	0.873271i	0.899639	−	0.436635i	$-0.143830\pi$
$504$		0			0
$505$	−4.45297			−0.00881776
$506$		0			0
$507$		0			0
$508$		0			0
$509$	762.229	−	440.073i	1.49750	−	0.864584i	0.497507	−	0.867460i	$-0.334249\pi$
$509$	762.229	−	440.073i	1.49750	−	0.864584i	0.999996	+	0.00287592i	$0.000915436\pi$
$510$		0			0
$511$	99.8974	−	98.1226i	0.195494	−	0.192021i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−22.8146	−	13.1720i	−0.0443003	−	0.0255768i
$516$		0			0
$517$	564.307			1.09150
$518$		0			0
$519$		0			0
$520$		0			0
$521$	860.620	+	496.879i	1.65186	+	0.953703i	0.976306	+	0.216395i	$0.0694298\pi$
$521$	860.620	+	496.879i	1.65186	+	0.953703i	0.675556	+	0.737308i	$0.263903\pi$
$522$		0			0
$523$	389.830	+	675.205i	0.745372	+	1.29102i	0.950021	+	0.312187i	$0.101062\pi$
$523$	389.830	+	675.205i	0.745372	+	1.29102i	−0.204649	+	0.978836i	$0.565605\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	39.8207	−	22.9905i	0.0755612	−	0.0436253i
$528$		0			0
$529$	−257.671	+	446.300i	−0.487091	+	0.843667i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	135.898i		−	0.254969i
$534$		0			0
$535$	11.4023	−	19.7493i	0.0213127	−	0.0369146i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	451.283	+	815.042i	0.837260	+	1.51214i
$540$		0			0
$541$	13.3902	+	23.1924i	0.0247507	+	0.0428695i	0.878135	−	0.478412i	$-0.158787\pi$
$541$	13.3902	+	23.1924i	0.0247507	+	0.0428695i	−0.853385	+	0.521282i	$0.825454\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			37.2705i			0.0683863i
$546$		0			0
$547$	359.940			0.658025			0.329013	−	0.944326i	$-0.393284\pi$
$547$	359.940			0.658025			0.329013	+	0.944326i	$0.393284\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	408.578	−	235.893i	0.741522	−	0.428118i
$552$		0			0
$553$	−112.699	−	436.201i	−0.203796	−	0.788790i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	315.324	+	182.052i	0.566111	+	0.326845i	0.755595	−	0.655039i	$-0.227348\pi$
$557$	315.324	+	182.052i	0.566111	+	0.326845i	−0.189483	+	0.981884i	$0.560681\pi$
$558$		0			0
$559$	−312.118			−0.558350
$560$		0			0
$561$		0			0
$562$		0			0
$563$	61.4640	+	35.4863i	0.109172	+	0.0630307i	0.553592	−	0.832788i	$-0.313257\pi$
$563$	61.4640	+	35.4863i	0.109172	+	0.0630307i	−0.444420	+	0.895819i	$0.646590\pi$
$564$		0			0
$565$	15.7614	+	27.2995i	0.0278963	+	0.0483178i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−239.749	+	138.419i	−0.421351	+	0.243267i	−0.695655	−	0.718376i	$-0.744886\pi$
$569$	−239.749	+	138.419i	−0.421351	+	0.243267i	0.274304	+	0.961643i	$0.411552\pi$
$570$		0			0
$571$	347.610	−	602.078i	0.608773	−	1.05443i	−0.382669	−	0.923885i	$-0.624995\pi$
$571$	347.610	−	602.078i	0.608773	−	1.05443i	0.991443	−	0.130541i	$-0.0416715\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			92.2274i			0.160395i
$576$		0			0
$577$	−81.1614	+	140.576i	−0.140661	+	0.243632i	−0.927746	−	0.373213i	$-0.878256\pi$
$577$	−81.1614	+	140.576i	−0.140661	+	0.243632i	0.787085	+	0.616845i	$0.211589\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−69.8382	+	251.598i	−0.120203	+	0.433043i
$582$		0			0
$583$	762.445	+	1320.59i	1.30780	+	2.26517i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			544.350i			0.927342i	0.886007	+	0.463671i	$0.153468\pi$
$587$			544.350i			0.927342i	−0.886007	+	0.463671i	$0.846532\pi$
$588$		0			0
$589$	−179.245			−0.304322
$590$		0			0
$591$		0			0
$592$		0			0
$593$	269.171	−	155.406i	0.453914	−	0.262067i	−0.255568	−	0.966791i	$-0.582262\pi$
$593$	269.171	−	155.406i	0.453914	−	0.262067i	0.709482	+	0.704724i	$0.248929\pi$
$594$		0			0
$595$	−6.61866	−	1.83720i	−0.0111238	−	0.00308773i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	646.818	+	373.440i	1.07983	+	0.623439i	0.930850	−	0.365401i	$-0.119068\pi$
$599$	646.818	+	373.440i	1.07983	+	0.623439i	0.148979	+	0.988840i	$0.452401\pi$
$600$		0			0
$601$	452.880			0.753544			0.376772	−	0.926306i	$-0.377034\pi$
$601$	452.880			0.753544			0.376772	+	0.926306i	$0.377034\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−43.7197	−	25.2416i	−0.0722640	−	0.0417217i
$606$		0			0
$607$	−450.859	−	780.910i	−0.742766	−	1.28651i	−0.951231	−	0.308478i	$-0.900180\pi$
$607$	−450.859	−	780.910i	−0.742766	−	1.28651i	0.208466	−	0.978030i	$-0.433153\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−195.486	+	112.864i	−0.319944	+	0.184720i
$612$		0			0
$613$	423.312	−	733.198i	0.690558	−	1.19608i	−0.281097	−	0.959679i	$-0.590698\pi$
$613$	423.312	−	733.198i	0.690558	−	1.19608i	0.971655	−	0.236403i	$-0.0759685\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	5.96933i		−	0.00967477i	−0.999988	−	0.00483739i	$-0.998460\pi$
$617$		−	5.96933i		−	0.00967477i	0.999988	−	0.00483739i	$-0.00153979\pi$
$618$		0			0
$619$	−387.612	+	671.364i	−0.626191	+	1.08459i	0.362118	+	0.932132i	$0.382054\pi$
$619$	−387.612	+	671.364i	−0.626191	+	1.08459i	−0.988309	+	0.152463i	$0.951280\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	967.217	−	249.896i	1.55252	−	0.401117i
$624$		0			0
$625$	−310.849	−	538.406i	−0.497358	−	0.861449i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	29.2391i		−	0.0464851i
$630$		0			0
$631$	−619.610			−0.981950			−0.490975	−	0.871174i	$-0.663359\pi$
$631$	−619.610			−0.981950			−0.490975	+	0.871174i	$0.663359\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.5291	−	10.6978i	0.0291797	−	0.0168469i
$636$		0			0
$637$	−319.345	−	192.087i	−0.501326	−	0.301549i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	581.796	+	335.900i	0.907638	+	0.524025i	0.879670	−	0.475584i	$-0.157763\pi$
$641$	581.796	+	335.900i	0.907638	+	0.524025i	0.0279676	+	0.999609i	$0.491096\pi$
$642$		0			0
$643$	−1042.23			−1.62088			−0.810441	−	0.585820i	$-0.800772\pi$
$643$	−1042.23			−1.62088			−0.810441	+	0.585820i	$0.800772\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−791.505	−	456.976i	−1.22335	−	0.706300i	−0.257717	−	0.966220i	$-0.582970\pi$
$647$	−791.505	−	456.976i	−1.22335	−	0.706300i	−0.965630	+	0.259921i	$0.916303\pi$
$648$		0			0
$649$	761.443	+	1318.86i	1.17326	+	2.03214i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−792.106	+	457.322i	−1.21303	+	0.700341i	−0.963417	−	0.268006i	$-0.913635\pi$
$653$	−792.106	+	457.322i	−1.21303	+	0.700341i	−0.249608	+	0.968347i	$0.580302\pi$
$654$		0			0
$655$	25.6234	−	44.3811i	0.0391198	−	0.0677574i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			201.561i			0.305858i	0.988237	+	0.152929i	$0.0488706\pi$
$659$			201.561i			0.305858i	−0.988237	+	0.152929i	$0.951129\pi$
$660$		0			0
$661$	333.315	−	577.319i	0.504259	−	0.873402i	−0.495729	−	0.868477i	$-0.665099\pi$
$661$	333.315	−	577.319i	0.504259	−	0.873402i	0.999988	−	0.00492495i	$-0.00156767\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	18.7635	+	19.1029i	0.0282158	+	0.0287262i
$666$		0			0
$667$	47.8390	+	82.8596i	0.0717227	+	0.124227i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1113.74i		−	1.65982i
$672$		0			0
$673$	−369.069			−0.548394			−0.274197	−	0.961674i	$-0.588412\pi$
$673$	−369.069			−0.548394			−0.274197	+	0.961674i	$0.588412\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−266.790	+	154.031i	−0.394077	+	0.227521i	−0.683925	−	0.729552i	$-0.739729\pi$
$677$	−266.790	+	154.031i	−0.394077	+	0.227521i	0.289848	+	0.957073i	$0.406395\pi$
$678$		0			0
$679$	−300.952	−	83.5378i	−0.443228	−	0.123031i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	105.264	+	60.7745i	0.154121	+	0.0889817i	0.575077	−	0.818099i	$-0.304972\pi$
$683$	105.264	+	60.7745i	0.154121	+	0.0889817i	−0.420956	+	0.907081i	$0.638305\pi$
$684$		0			0
$685$	−26.4685			−0.0386402
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−528.249	−	304.985i	−0.766690	−	0.442649i
$690$		0			0
$691$	86.0268	+	149.003i	0.124496	+	0.215634i	0.921536	−	0.388293i	$-0.126935\pi$
$691$	86.0268	+	149.003i	0.124496	+	0.215634i	−0.797040	+	0.603927i	$0.793602\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−32.0531	+	18.5059i	−0.0461196	+	0.0266272i
$696$		0			0
$697$	−41.7650	+	72.3390i	−0.0599210	+	0.103786i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	602.280i		−	0.859173i	−0.903026	−	0.429587i	$-0.858659\pi$
$701$		−	602.280i		−	0.859173i	0.903026	−	0.429587i	$-0.141341\pi$
$702$		0			0
$703$	−56.9906	+	98.7107i	−0.0810678	+	0.140413i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	104.055	+	105.938i	0.147179	+	0.149841i
$708$		0			0
$709$	−417.436	−	723.020i	−0.588767	−	1.01977i	−0.994394	−	0.105736i	$-0.966280\pi$
$709$	−417.436	−	723.020i	−0.588767	−	1.01977i	0.405627	−	0.914039i	$-0.367053\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	36.3509i		−	0.0509831i
$714$		0			0
$715$	30.3538			0.0424528
$716$		0			0
$717$		0			0
$718$		0			0
$719$	774.168	−	446.966i	1.07673	−	0.621650i	0.146717	−	0.989179i	$-0.453129\pi$
$719$	774.168	−	446.966i	1.07673	−	0.621650i	0.930012	+	0.367529i	$0.119796\pi$
$720$		0			0
$721$	219.758	+	850.567i	0.304796	+	1.17970i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−559.540	−	323.050i	−0.771779	−	0.445587i
$726$		0			0
$727$	862.131			1.18587			0.592937	−	0.805249i	$-0.297968\pi$
$727$	862.131			1.18587			0.592937	+	0.805249i	$0.297968\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	166.141	+	95.9217i	0.227279	+	0.131220i
$732$		0			0
$733$	−658.287	−	1140.19i	−0.898072	−	1.55551i	−0.829956	−	0.557829i	$-0.811635\pi$
$733$	−658.287	−	1140.19i	−0.898072	−	1.55551i	−0.0681159	−	0.997677i	$-0.521699\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1495.24	+	863.279i	−2.02882	+	1.17134i
$738$		0			0
$739$	−318.373	+	551.438i	−0.430816	+	0.746195i	−0.996944	−	0.0781223i	$-0.975108\pi$
$739$	−318.373	+	551.438i	−0.430816	+	0.746195i	0.566128	+	0.824317i	$0.308441\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	720.253i		−	0.969385i	−0.874685	−	0.484692i	$-0.838932\pi$
$743$		−	720.253i		−	0.969385i	0.874685	−	0.484692i	$-0.161068\pi$
$744$		0			0
$745$	−2.59411	+	4.49313i	−0.00348203	+	0.00603105i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−736.287	+	190.232i	−0.983027	+	0.253981i
$750$		0			0
$751$	114.613	+	198.515i	0.152614	+	0.264335i	0.932188	−	0.361976i	$-0.117898\pi$
$751$	114.613	+	198.515i	0.152614	+	0.264335i	−0.779574	+	0.626310i	$0.784564\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			21.4224i			0.0283740i
$756$		0			0
$757$	1207.52			1.59514			0.797570	−	0.603226i	$-0.206118\pi$
$757$	1207.52			1.59514			0.797570	+	0.603226i	$0.206118\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−417.256	+	240.903i	−0.548300	+	0.316561i	−0.748436	−	0.663207i	$-0.769195\pi$
$761$	−417.256	+	240.903i	−0.548300	+	0.316561i	0.200136	+	0.979768i	$0.435862\pi$
$762$		0			0
$763$	886.678	−	870.925i	1.16209	−	1.14145i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−527.555	−	304.584i	−0.687816	−	0.397111i
$768$		0			0
$769$	1187.92			1.54475			0.772377	−	0.635165i	$-0.219068\pi$
$769$	1187.92			1.54475			0.772377	+	0.635165i	$0.219068\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−912.742	−	526.972i	−1.18078	−	0.681723i	−0.224584	−	0.974455i	$-0.572102\pi$
$773$	−912.742	−	526.972i	−1.18078	−	0.681723i	−0.956195	+	0.292732i	$0.905436\pi$
$774$		0			0
$775$	122.736	+	212.586i	0.158370	+	0.274304i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	281.995	−	162.810i	0.361997	−	0.208999i
$780$		0			0
$781$	−838.789	+	1452.83i	−1.07399	+	1.86021i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	43.6992i		−	0.0556677i
$786$		0			0
$787$	198.710	−	344.175i	0.252490	−	0.437325i	−0.711721	−	0.702462i	$-0.752084\pi$
$787$	198.710	−	344.175i	0.252490	−	0.437325i	0.964211	+	0.265137i	$0.0854172\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	281.158	−	1012.89i	0.355446	−	1.28052i
$792$		0			0
$793$	222.753	+	385.820i	0.280899	+	0.486532i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			538.927i			0.676194i	0.941111	+	0.338097i	$0.109783\pi$
$797$			538.927i			0.676194i	−0.941111	+	0.338097i	$0.890217\pi$
$798$		0			0
$799$	138.744			0.173647
$800$		0			0
$801$		0			0
$802$		0			0
$803$	329.378	−	190.167i	0.410185	−	0.236820i
$804$		0			0
$805$	−3.87407	+	3.80524i	−0.00481251	+	0.00472701i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−105.177	−	60.7238i	−0.130008	−	0.0750603i	0.433585	−	0.901113i	$-0.357248\pi$
$809$	−105.177	−	60.7238i	−0.130008	−	0.0750603i	−0.563594	+	0.826052i	$0.690582\pi$
$810$		0			0
$811$	−1552.10			−1.91381			−0.956904	−	0.290405i	$-0.906210\pi$
$811$	−1552.10			−1.91381			−0.956904	+	0.290405i	$0.906210\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	36.7834	+	21.2369i	0.0451330	+	0.0260576i
$816$		0			0
$817$	−373.926	−	647.659i	−0.457682	−	0.792729i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−275.818	+	159.243i	−0.335953	+	0.193963i	−0.658481	−	0.752597i	$-0.728801\pi$
$821$	−275.818	+	159.243i	−0.335953	+	0.193963i	0.322528	+	0.946560i	$0.395467\pi$
$822$		0			0
$823$	35.2055	−	60.9778i	0.0427771	−	0.0740921i	−0.843844	−	0.536588i	$-0.819713\pi$
$823$	35.2055	−	60.9778i	0.0427771	−	0.0740921i	0.886621	+	0.462496i	$0.153046\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			848.986i			1.02659i	0.858214	+	0.513293i	$0.171575\pi$
$827$			848.986i			1.02659i	−0.858214	+	0.513293i	$0.828425\pi$
$828$		0			0
$829$	362.558	−	627.968i	0.437343	−	0.757501i	−0.560140	−	0.828398i	$-0.689253\pi$
$829$	362.558	−	627.968i	0.437343	−	0.757501i	0.997484	+	0.0708969i	$0.0225861\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	110.955	+	200.391i	0.133199	+	0.240565i
$834$		0			0
$835$	28.8409	+	49.9539i	0.0345400	+	0.0598250i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1406.16i		−	1.67600i	−0.545674	−	0.837998i	$-0.683726\pi$
$839$		−	1406.16i		−	1.67600i	0.545674	−	0.837998i	$-0.316274\pi$
$840$		0			0
$841$	170.726			0.203003
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.2075	−	11.6668i	0.0239142	−	0.0138069i
$846$		0			0
$847$	421.122	+	1629.94i	0.497192	+	1.92437i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−20.0185	−	11.5577i	−0.0235235	−	0.0135813i
$852$		0			0
$853$	−1076.57			−1.26210			−0.631051	−	0.775742i	$-0.717376\pi$
$853$	−1076.57			−1.26210			−0.631051	+	0.775742i	$0.717376\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1068.50	−	616.900i	−1.24679	−	0.719837i	−0.276325	−	0.961064i	$-0.589117\pi$
$857$	−1068.50	−	616.900i	−1.24679	−	0.719837i	−0.970469	+	0.241227i	$0.922450\pi$
$858$		0			0
$859$	−538.904	−	933.409i	−0.627362	−	1.08662i	−0.988079	−	0.153947i	$-0.950801\pi$
$859$	−538.904	−	933.409i	−0.627362	−	1.08662i	0.360717	−	0.932675i	$-0.382532\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	986.156	−	569.358i	1.14271	−	0.659742i	0.195608	−	0.980682i	$-0.437332\pi$
$863$	986.156	−	569.358i	1.14271	−	0.659742i	0.947100	+	0.320940i	$0.103999\pi$
$864$		0			0
$865$	−12.2143	+	21.1557i	−0.0141205	+	0.0244575i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1223.69i		−	1.40816i
$870$		0			0
$871$	345.319	−	598.110i	0.396463	−	0.686694i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	19.6333	−	70.7307i	0.0224381	−	0.0808351i
$876$		0			0
$877$	−232.103	−	402.015i	−0.264656	−	0.458398i	0.702817	−	0.711370i	$-0.251925\pi$
$877$	−232.103	−	402.015i	−0.264656	−	0.458398i	−0.967473	+	0.252972i	$0.918592\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	343.027i		−	0.389361i	−0.980867	−	0.194680i	$-0.937633\pi$
$881$		−	343.027i		−	0.389361i	0.980867	−	0.194680i	$-0.0623669\pi$
$882$		0			0
$883$	821.673			0.930547			0.465274	−	0.885167i	$-0.345956\pi$
$883$	821.673			0.930547			0.465274	+	0.885167i	$0.345956\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1002.29	+	578.670i	−1.12997	+	0.652390i	−0.943928	−	0.330151i	$-0.892900\pi$
$887$	−1002.29	+	578.670i	−1.12997	+	0.652390i	−0.186045	+	0.982541i	$0.559567\pi$
$888$		0			0
$889$	−687.485	−	190.831i	−0.773324	−	0.214658i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−468.396	−	270.429i	−0.524520	−	0.302832i
$894$		0			0
$895$	−5.61967			−0.00627896
$896$		0			0
$897$		0			0
$898$		0			0
$899$	220.540	+	127.329i	0.245317	+	0.141634i
$900$		0			0
$901$	187.459	+	324.689i	0.208057	+	0.360365i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	19.0962	−	11.0252i	0.0211008	−	0.0121825i
$906$		0			0
$907$	−404.722	+	700.998i	−0.446220	+	0.772876i	−0.998136	−	0.0610237i	$-0.980563\pi$
$907$	−404.722	+	700.998i	−0.446220	+	0.772876i	0.551916	+	0.833900i	$0.313897\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	278.914i		−	0.306162i	−0.988214	−	0.153081i	$-0.951080\pi$
$911$		−	278.914i		−	0.306162i	0.988214	−	0.153081i	$-0.0489196\pi$
$912$		0			0
$913$	−354.608	+	614.199i	−0.388398	+	0.672726i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1654.60	+	427.492i	−1.80436	+	0.466186i
$918$		0			0
$919$	−523.336	−	906.445i	−0.569463	−	0.986338i	−0.996619	−	0.0821605i	$-0.973818\pi$
$919$	−523.336	−	906.445i	−0.569463	−	0.986338i	0.427156	−	0.904178i	$-0.359515\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	671.046i		−	0.727028i
$924$		0			0
$925$	156.095			0.168751
$926$		0			0
$927$		0			0
$928$		0			0
$929$	154.140	−	88.9926i	0.165920	−	0.0957939i	−0.414741	−	0.909940i	$-0.636128\pi$
$929$	154.140	−	88.9926i	0.165920	−	0.0957939i	0.580661	+	0.814146i	$0.302794\pi$
$930$		0			0
$931$	16.0046	−	892.781i	0.0171908	−	0.958948i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−16.1574	−	9.32848i	−0.0172806	−	0.00997699i
$936$		0			0
$937$	−1333.75			−1.42343			−0.711713	−	0.702470i	$-0.752080\pi$
$937$	−1333.75			−1.42343			−0.711713	+	0.702470i	$0.752080\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	842.430	+	486.377i	0.895250	+	0.516873i	0.875656	−	0.482935i	$-0.160429\pi$
$941$	842.430	+	486.377i	0.895250	+	0.516873i	0.0195937	+	0.999808i	$0.493763\pi$
$942$		0			0
$943$	33.0179	+	57.1886i	0.0350136	+	0.0606454i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−849.798	+	490.631i	−0.897358	+	0.518090i	−0.876342	−	0.481690i	$-0.840023\pi$
$947$	−849.798	+	490.631i	−0.897358	+	0.518090i	−0.0210156	+	0.999779i	$0.506690\pi$
$948$		0			0
$949$	−76.0684	+	131.754i	−0.0801564	+	0.138835i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			294.776i			0.309314i	0.987968	+	0.154657i	$0.0494272\pi$
$953$			294.776i			0.309314i	−0.987968	+	0.154657i	$0.950573\pi$
$954$		0			0
$955$	32.8661	−	56.9258i	0.0344148	−	0.0596082i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	618.508	+	629.695i	0.644951	+	0.656616i
$960$		0			0
$961$	432.124	+	748.461i	0.449661	+	0.778835i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			4.53968i			0.00470433i
$966$		0			0
$967$	−1314.45			−1.35930			−0.679651	−	0.733535i	$-0.737869\pi$
$967$	−1314.45			−1.35930			−0.679651	+	0.733535i	$0.737869\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	44.1535	−	25.4920i	0.0454722	−	0.0262534i	−0.477092	−	0.878854i	$-0.658309\pi$
$971$	44.1535	−	25.4920i	0.0454722	−	0.0262534i	0.522564	+	0.852600i	$0.324976\pi$
$972$		0			0
$973$	1189.27	+	330.115i	1.22227	+	0.339275i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1392.84	−	804.159i	−1.42563	−	0.823090i	−0.428862	−	0.903370i	$-0.641085\pi$
$977$	−1392.84	−	804.159i	−1.42563	−	0.823090i	−0.996772	+	0.0802800i	$0.974419\pi$
$978$		0			0
$979$	2713.37			2.77157
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−863.213	−	498.376i	−0.878142	−	0.506995i	−0.00809650	−	0.999967i	$-0.502577\pi$
$983$	−863.213	−	498.376i	−0.878142	−	0.506995i	−0.870045	+	0.492972i	$0.835911\pi$
$984$		0			0
$985$	10.7939	+	18.6956i	0.0109583	+	0.0189803i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	131.345	−	75.8322i	0.132806	−	0.0766756i
$990$		0			0
$991$	438.637	−	759.741i	0.442620	−	0.766641i	−0.555263	−	0.831675i	$-0.687382\pi$
$991$	438.637	−	759.741i	0.442620	−	0.766641i	0.997883	+	0.0650341i	$0.0207156\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			40.7861i			0.0409910i
$996$		0			0
$997$	−860.736	+	1490.84i	−0.863326	+	1.49533i	0.00537302	+	0.999986i	$0.498290\pi$
$997$	−860.736	+	1490.84i	−0.863326	+	1.49533i	−0.868699	+	0.495340i	$0.835044\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.3.bk.g.53.4	✓	16
3.2	odd	2	inner	756.3.bk.g.53.5	yes	16
7.2	even	3	inner	756.3.bk.g.485.5	yes	16
21.2	odd	6	inner	756.3.bk.g.485.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.3.bk.g.53.4	✓	16	1.1	even	1	trivial
756.3.bk.g.53.5	yes	16	3.2	odd	2	inner
756.3.bk.g.485.4	yes	16	21.2	odd	6	inner
756.3.bk.g.485.5	yes	16	7.2	even	3	inner

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:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	756.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.181791 + 0.104957i) q^{5} +(6.74497 + 1.87226i) q^{7} +(16.4658 + 9.50651i) q^{11} -7.60538 q^{13} +(4.04837 + 2.33733i) q^{17} +(-9.11147 - 15.7815i) q^{19} +(3.20049 - 1.84780i) q^{23} +(-12.4780 + 21.6125i) q^{25} +25.8897i q^{29} +(4.91813 - 8.51844i) q^{31} +(-1.42268 + 0.367572i) q^{35} +(-3.12741 - 5.41684i) q^{37} +17.8687i q^{41} +41.0391 q^{43} +(25.7037 - 14.8400i) q^{47} +(41.9893 + 25.2567i) q^{49} +(69.4573 + 40.1012i) q^{53} -3.99109 q^{55} +(69.3661 + 40.0485i) q^{59} +(-29.2889 - 50.7298i) q^{61} +(1.38259 - 0.798236i) q^{65} +(-45.4046 + 78.6431i) q^{67} +88.2331i q^{71} +(10.0019 - 17.3238i) q^{73} +(93.2625 + 94.9493i) q^{77} +(-32.1803 - 55.7379i) q^{79} +37.3016i q^{83} -0.981273 q^{85} +(123.592 - 71.3556i) q^{89} +(-51.2981 - 14.2392i) q^{91} +(3.31276 + 1.91262i) q^{95} -44.6187 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 24 q^{7} + 88 q^{13} + 14 q^{19} + 36 q^{25} - 68 q^{31} - 76 q^{37} - 292 q^{43} - 20 q^{49} - 272 q^{55} - 110 q^{61} - 72 q^{67} + 60 q^{73} + 154 q^{79} + 700 q^{85} - 74 q^{91} + 264 q^{97}+O(q^{100}) \) 16 * q + 24 * q^7 + 88 * q^13 + 14 * q^19 + 36 * q^25 - 68 * q^31 - 76 * q^37 - 292 * q^43 - 20 * q^49 - 272 * q^55 - 110 * q^61 - 72 * q^67 + 60 * q^73 + 154 * q^79 + 700 * q^85 - 74 * q^91 + 264 * q^97

Introduction

L-functions

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