LMFDB - Embedded newform 1008.3.cg.i.145.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(145,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))

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

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

chi := DirichletCharacter("1008.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.cg (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.2
Root			$0.707107 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1008.145
Dual form			1008.3.cg.i.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.24264 + 2.44949i) q^{5} +(-3.50000 + 6.06218i) q^{7} +O(q^{10})$	$q+(4.24264 + 2.44949i) q^{5} +(-3.50000 + 6.06218i) q^{7} +(-8.48528 - 14.6969i) q^{11} +1.73205i q^{13} +(-4.24264 + 2.44949i) q^{17} +(-25.5000 - 14.7224i) q^{19} +(-4.24264 + 7.34847i) q^{23} +(-0.500000 - 0.866025i) q^{25} -33.9411 q^{29} +(10.5000 - 6.06218i) q^{31} +(-29.6985 + 17.1464i) q^{35} +(23.5000 - 40.7032i) q^{37} -68.5857i q^{41} -31.0000 q^{43} +(72.1249 + 41.6413i) q^{47} +(-24.5000 - 42.4352i) q^{49} +(38.1838 + 66.1362i) q^{53} -83.1384i q^{55} +(72.1249 - 41.6413i) q^{59} +(-72.0000 - 41.5692i) q^{61} +(-4.24264 + 7.34847i) q^{65} +(-15.5000 - 26.8468i) q^{67} -59.3970 q^{71} +(-70.5000 + 40.7032i) q^{73} +118.794 q^{77} +(20.5000 - 35.5070i) q^{79} -4.89898i q^{83} -24.0000 q^{85} +(50.9117 + 29.3939i) q^{89} +(-10.5000 - 6.06218i) q^{91} +(-72.1249 - 124.924i) q^{95} -41.5692i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{7}+O(q^{10}) $ 4 * q - 14 * q^7	$ 4 q - 14 q^{7} - 102 q^{19} - 2 q^{25} + 42 q^{31} + 94 q^{37} - 124 q^{43} - 98 q^{49} - 288 q^{61} - 62 q^{67} - 282 q^{73} + 82 q^{79} - 96 q^{85} - 42 q^{91}+O(q^{100}) $ 4 * q - 14 * q^7 - 102 * q^19 - 2 * q^25 + 42 * q^31 + 94 * q^37 - 124 * q^43 - 98 * q^49 - 288 * q^61 - 62 * q^67 - 282 * q^73 + 82 * q^79 - 96 * q^85 - 42 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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
$5$	4.24264	+	2.44949i	0.848528	+	0.489898i	0.860154	−	0.510034i	$-0.170367\pi$
$5$	4.24264	+	2.44949i	0.848528	+	0.489898i	−0.0116258	+	0.999932i	$0.503701\pi$
$6$		0			0
$7$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$7$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−8.48528	−	14.6969i	−0.771389	−	1.33609i	−0.936802	−	0.349861i	$-0.886229\pi$
$11$	−8.48528	−	14.6969i	−0.771389	−	1.33609i	0.165412	−	0.986224i	$-0.447104\pi$
$12$		0			0
$13$			1.73205i			0.133235i	0.997779	+	0.0666173i	$0.0212207\pi$
$13$			1.73205i			0.133235i	−0.997779	+	0.0666173i	$0.978779\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.24264	+	2.44949i	−0.249567	+	0.144088i	−0.619566	−	0.784945i	$-0.712691\pi$
$17$	−4.24264	+	2.44949i	−0.249567	+	0.144088i	0.369999	+	0.929032i	$0.379358\pi$
$18$		0			0
$19$	−25.5000	−	14.7224i	−1.34211	−	0.774865i	−0.354989	−	0.934870i	$-0.615515\pi$
$19$	−25.5000	−	14.7224i	−1.34211	−	0.774865i	−0.987116	+	0.160006i	$0.948849\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.24264	+	7.34847i	−0.184463	+	0.319499i	−0.943395	−	0.331670i	$-0.892388\pi$
$23$	−4.24264	+	7.34847i	−0.184463	+	0.319499i	0.758933	+	0.651169i	$0.225721\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−33.9411			−1.17038			−0.585192	−	0.810895i	$-0.698981\pi$
$29$	−33.9411			−1.17038			−0.585192	+	0.810895i	$0.698981\pi$
$30$		0			0
$31$	10.5000	−	6.06218i	0.338710	−	0.195554i	−0.320992	−	0.947082i	$-0.604016\pi$
$31$	10.5000	−	6.06218i	0.338710	−	0.195554i	0.659701	+	0.751528i	$0.270683\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−29.6985	+	17.1464i	−0.848528	+	0.489898i
$36$		0			0
$37$	23.5000	−	40.7032i	0.635135	−	1.10009i	−0.351351	−	0.936244i	$-0.614278\pi$
$37$	23.5000	−	40.7032i	0.635135	−	1.10009i	0.986486	−	0.163843i	$-0.0523889\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	68.5857i		−	1.67282i	−0.548103	−	0.836411i	$-0.684650\pi$
$41$		−	68.5857i		−	1.67282i	0.548103	−	0.836411i	$-0.315350\pi$
$42$		0			0
$43$	−31.0000			−0.720930			−0.360465	−	0.932773i	$-0.617382\pi$
$43$	−31.0000			−0.720930			−0.360465	+	0.932773i	$0.617382\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	72.1249	+	41.6413i	1.53457	+	0.885986i	0.999142	+	0.0414059i	$0.0131837\pi$
$47$	72.1249	+	41.6413i	1.53457	+	0.885986i	0.535430	+	0.844580i	$0.320150\pi$
$48$		0			0
$49$	−24.5000	−	42.4352i	−0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	38.1838	+	66.1362i	0.720448	+	1.24785i	0.960820	+	0.277172i	$0.0893973\pi$
$53$	38.1838	+	66.1362i	0.720448	+	1.24785i	−0.240372	+	0.970681i	$0.577269\pi$
$54$		0			0
$55$		−	83.1384i		−	1.51161i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	72.1249	−	41.6413i	1.22246	−	0.705785i	0.257015	−	0.966407i	$-0.417261\pi$
$59$	72.1249	−	41.6413i	1.22246	−	0.705785i	0.965441	+	0.260622i	$0.0839277\pi$
$60$		0			0
$61$	−72.0000	−	41.5692i	−1.18033	−	0.681463i	−0.224237	−	0.974535i	$-0.571989\pi$
$61$	−72.0000	−	41.5692i	−1.18033	−	0.681463i	−0.956090	+	0.293072i	$0.905322\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.24264	+	7.34847i	−0.0652714	+	0.113053i
$66$		0			0
$67$	−15.5000	−	26.8468i	−0.231343	−	0.400698i	0.726860	−	0.686785i	$-0.240979\pi$
$67$	−15.5000	−	26.8468i	−0.231343	−	0.400698i	−0.958204	+	0.286087i	$0.907645\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−59.3970			−0.836577			−0.418289	−	0.908314i	$-0.637370\pi$
$71$	−59.3970			−0.836577			−0.418289	+	0.908314i	$0.637370\pi$
$72$		0			0
$73$	−70.5000	+	40.7032i	−0.965753	+	0.557578i	−0.897939	−	0.440120i	$-0.854936\pi$
$73$	−70.5000	+	40.7032i	−0.965753	+	0.557578i	−0.0678144	+	0.997698i	$0.521603\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	118.794			1.54278
$78$		0			0
$79$	20.5000	−	35.5070i	0.259494	−	0.449456i	−0.706613	−	0.707601i	$-0.749778\pi$
$79$	20.5000	−	35.5070i	0.259494	−	0.449456i	0.966106	+	0.258144i	$0.0831110\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	4.89898i		−	0.0590238i	−0.999564	−	0.0295119i	$-0.990605\pi$
$83$		−	4.89898i		−	0.0590238i	0.999564	−	0.0295119i	$-0.00939530\pi$
$84$		0			0
$85$	−24.0000			−0.282353
$86$		0			0
$87$		0			0
$88$		0			0
$89$	50.9117	+	29.3939i	0.572041	+	0.330268i	0.757964	−	0.652296i	$-0.226194\pi$
$89$	50.9117	+	29.3939i	0.572041	+	0.330268i	−0.185923	+	0.982564i	$0.559527\pi$
$90$		0			0
$91$	−10.5000	−	6.06218i	−0.115385	−	0.0666173i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−72.1249	−	124.924i	−0.759209	−	1.31499i
$96$		0			0
$97$		−	41.5692i		−	0.428549i	−0.976774	−	0.214274i	$-0.931261\pi$
$97$		−	41.5692i		−	0.428549i	0.976774	−	0.214274i	$-0.0687387\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−152.735	+	88.1816i	−1.51223	+	0.873085i	−0.512331	+	0.858788i	$0.671218\pi$
$101$	−152.735	+	88.1816i	−1.51223	+	0.873085i	−0.999898	+	0.0142971i	$0.995449\pi$
$102$		0			0
$103$	25.5000	+	14.7224i	0.247573	+	0.142936i	0.618652	−	0.785665i	$-0.287679\pi$
$103$	25.5000	+	14.7224i	0.247573	+	0.142936i	−0.371080	+	0.928601i	$0.621012\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−72.1249	+	124.924i	−0.674064	+	1.16751i	0.302677	+	0.953093i	$0.402120\pi$
$107$	−72.1249	+	124.924i	−0.674064	+	1.16751i	−0.976741	+	0.214421i	$0.931214\pi$
$108$		0			0
$109$	−84.5000	−	146.358i	−0.775229	−	1.34274i	−0.934665	−	0.355528i	$-0.884301\pi$
$109$	−84.5000	−	146.358i	−0.775229	−	1.34274i	0.159436	−	0.987208i	$-0.449032\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−59.3970			−0.525637			−0.262818	−	0.964845i	$-0.584652\pi$
$113$	−59.3970			−0.525637			−0.262818	+	0.964845i	$0.584652\pi$
$114$		0			0
$115$	−36.0000	+	20.7846i	−0.313043	+	0.180736i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	34.2929i		−	0.288175i
$120$		0			0
$121$	−83.5000	+	144.626i	−0.690083	+	1.19526i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	127.373i		−	1.01899i
$126$		0			0
$127$	−209.000			−1.64567			−0.822835	−	0.568281i	$-0.807609\pi$
$127$	−209.000			−1.64567			−0.822835	+	0.568281i	$0.807609\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	50.9117	+	29.3939i	0.388639	+	0.224381i	0.681570	−	0.731753i	$-0.261297\pi$
$131$	50.9117	+	29.3939i	0.388639	+	0.224381i	−0.292931	+	0.956133i	$0.594631\pi$
$132$		0			0
$133$	178.500	−	103.057i	1.34211	−	0.774865i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−76.3675	−	132.272i	−0.557427	−	0.965492i	−0.997710	−	0.0676333i	$-0.978455\pi$
$137$	−76.3675	−	132.272i	−0.557427	−	0.965492i	0.440283	−	0.897859i	$-0.354878\pi$
$138$		0			0
$139$		−	195.722i		−	1.40807i	−0.710165	−	0.704035i	$-0.751380\pi$
$139$		−	195.722i		−	1.40807i	0.710165	−	0.704035i	$-0.248620\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	25.4558	−	14.6969i	0.178013	−	0.102776i
$144$		0			0
$145$	−144.000	−	83.1384i	−0.993103	−	0.573369i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−25.4558	+	44.0908i	−0.170845	+	0.295912i	−0.938715	−	0.344693i	$-0.887983\pi$
$149$	−25.4558	+	44.0908i	−0.170845	+	0.295912i	0.767871	+	0.640605i	$0.221316\pi$
$150$		0			0
$151$	5.00000	+	8.66025i	0.0331126	+	0.0573527i	0.882107	−	0.471049i	$-0.156125\pi$
$151$	5.00000	+	8.66025i	0.0331126	+	0.0573527i	−0.848994	+	0.528402i	$0.822791\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	59.3970			0.383206
$156$		0			0
$157$	36.0000	−	20.7846i	0.229299	−	0.132386i	−0.380949	−	0.924596i	$-0.624403\pi$
$157$	36.0000	−	20.7846i	0.229299	−	0.132386i	0.610249	+	0.792210i	$0.291069\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−29.6985	−	51.4393i	−0.184463	−	0.319499i
$162$		0			0
$163$	43.0000	−	74.4782i	0.263804	−	0.456921i	−0.703446	−	0.710749i	$-0.748356\pi$
$163$	43.0000	−	74.4782i	0.263804	−	0.456921i	0.967250	+	0.253828i	$0.0816896\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			181.262i			1.08540i	0.839926	+	0.542701i	$0.182598\pi$
$167$			181.262i			1.08540i	−0.839926	+	0.542701i	$0.817402\pi$
$168$		0			0
$169$	166.000			0.982249
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−38.1838	−	22.0454i	−0.220715	−	0.127430i	0.385566	−	0.922680i	$-0.374006\pi$
$173$	−38.1838	−	22.0454i	−0.220715	−	0.127430i	−0.606281	+	0.795250i	$0.707340\pi$
$174$		0			0
$175$	7.00000			0.0400000
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.24264	−	7.34847i	−0.0237019	−	0.0410529i	0.853931	−	0.520386i	$-0.174212\pi$
$179$	−4.24264	−	7.34847i	−0.0237019	−	0.0410529i	−0.877633	+	0.479333i	$0.840879\pi$
$180$		0			0
$181$		−	43.3013i		−	0.239234i	−0.992820	−	0.119617i	$-0.961833\pi$
$181$		−	43.3013i		−	0.239234i	0.992820	−	0.119617i	$-0.0381666\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	199.404	−	115.126i	1.07786	−	0.622303i
$186$		0			0
$187$	72.0000	+	41.5692i	0.385027	+	0.222295i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	38.1838	−	66.1362i	0.199915	−	0.346263i	−0.748586	−	0.663038i	$-0.769267\pi$
$191$	38.1838	−	66.1362i	0.199915	−	0.346263i	0.948501	+	0.316775i	$0.102600\pi$
$192$		0			0
$193$	143.500	+	248.549i	0.743523	+	1.28782i	0.950882	+	0.309555i	$0.100180\pi$
$193$	143.500	+	248.549i	0.743523	+	1.28782i	−0.207358	+	0.978265i	$0.566487\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	127.279			0.646087			0.323044	−	0.946384i	$-0.395294\pi$
$197$	127.279			0.646087			0.323044	+	0.946384i	$0.395294\pi$
$198$		0			0
$199$	−180.000	+	103.923i	−0.904523	+	0.522226i	−0.878665	−	0.477439i	$-0.841565\pi$
$199$	−180.000	+	103.923i	−0.904523	+	0.522226i	−0.0258579	+	0.999666i	$0.508232\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	118.794	−	205.757i	0.585192	−	1.01358i
$204$		0			0
$205$	168.000	−	290.985i	0.819512	−	1.41944i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			499.696i			2.39089i
$210$		0			0
$211$	−82.0000			−0.388626			−0.194313	−	0.980940i	$-0.562248\pi$
$211$	−82.0000			−0.388626			−0.194313	+	0.980940i	$0.562248\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−131.522	−	75.9342i	−0.611730	−	0.353182i
$216$		0			0
$217$			84.8705i			0.391108i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.24264	−	7.34847i	−0.0191975	−	0.0332510i
$222$		0			0
$223$		−	41.5692i		−	0.186409i	−0.995647	−	0.0932045i	$-0.970289\pi$
$223$		−	41.5692i		−	0.186409i	0.995647	−	0.0932045i	$-0.0297110\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−330.926	+	191.060i	−1.45782	+	0.841675i	−0.998904	−	0.0468029i	$-0.985097\pi$
$227$	−330.926	+	191.060i	−1.45782	+	0.841675i	−0.458920	+	0.888478i	$0.651763\pi$
$228$		0			0
$229$	−70.5000	−	40.7032i	−0.307860	−	0.177743i	0.338108	−	0.941107i	$-0.390213\pi$
$229$	−70.5000	−	40.7032i	−0.307860	−	0.177743i	−0.645969	+	0.763364i	$0.723546\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−114.551	+	198.409i	−0.491636	+	0.851539i	−0.999954	−	0.00963059i	$-0.996934\pi$
$233$	−114.551	+	198.409i	−0.491636	+	0.851539i	0.508317	+	0.861170i	$0.330268\pi$
$234$		0			0
$235$	204.000	+	353.338i	0.868085	+	1.50357i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	67.8823			0.284026			0.142013	−	0.989865i	$-0.454642\pi$
$239$	67.8823			0.284026			0.142013	+	0.989865i	$0.454642\pi$
$240$		0			0
$241$	−396.000	+	228.631i	−1.64315	+	0.948675i	−0.663451	+	0.748220i	$0.730909\pi$
$241$	−396.000	+	228.631i	−1.64315	+	0.948675i	−0.979703	+	0.200455i	$0.935758\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	240.050i		−	0.979796i
$246$		0			0
$247$	25.5000	−	44.1673i	0.103239	−	0.178815i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	347.828i		−	1.38577i	−0.721050	−	0.692884i	$-0.756340\pi$
$251$		−	347.828i		−	1.38577i	0.721050	−	0.692884i	$-0.243660\pi$
$252$		0			0
$253$	144.000			0.569170
$254$		0			0
$255$		0			0
$256$		0			0
$257$	140.007	+	80.8332i	0.544775	+	0.314526i	0.747012	−	0.664811i	$-0.231488\pi$
$257$	140.007	+	80.8332i	0.544775	+	0.314526i	−0.202237	+	0.979337i	$0.564821\pi$
$258$		0			0
$259$	164.500	+	284.922i	0.635135	+	1.10009i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	127.279	+	220.454i	0.483951	+	0.838228i	0.999830	−	0.0184332i	$-0.00586781\pi$
$263$	127.279	+	220.454i	0.483951	+	0.838228i	−0.515879	+	0.856662i	$0.672534\pi$
$264$		0			0
$265$			374.123i			1.41178i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.9706	−	9.79796i	0.0630876	−	0.0364236i	−0.468124	−	0.883663i	$-0.655070\pi$
$269$	16.9706	−	9.79796i	0.0630876	−	0.0364236i	0.531212	+	0.847239i	$0.321737\pi$
$270$		0			0
$271$	−36.0000	−	20.7846i	−0.132841	−	0.0766960i	0.432107	−	0.901823i	$-0.357770\pi$
$271$	−36.0000	−	20.7846i	−0.132841	−	0.0766960i	−0.564948	+	0.825127i	$0.691104\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.48528	+	14.6969i	−0.0308556	+	0.0534434i
$276$		0			0
$277$	168.500	+	291.851i	0.608303	+	1.05361i	0.991520	+	0.129954i	$0.0414829\pi$
$277$	168.500	+	291.851i	0.608303	+	1.05361i	−0.383217	+	0.923658i	$0.625184\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	246.073			0.875705			0.437853	−	0.899047i	$-0.355739\pi$
$281$	246.073			0.875705			0.437853	+	0.899047i	$0.355739\pi$
$282$		0			0
$283$	169.500	−	97.8609i	0.598940	−	0.345798i	−0.169685	−	0.985498i	$-0.554275\pi$
$283$	169.500	−	97.8609i	0.598940	−	0.345798i	0.768624	+	0.639700i	$0.220942\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	415.779	+	240.050i	1.44871	+	0.836411i
$288$		0			0
$289$	−132.500	+	229.497i	−0.458478	+	0.794106i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	97.9796i		−	0.334401i	−0.985923	−	0.167201i	$-0.946527\pi$
$293$		−	97.9796i		−	0.334401i	0.985923	−	0.167201i	$-0.0534728\pi$
$294$		0			0
$295$	408.000			1.38305
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.7279	−	7.34847i	−0.0425683	−	0.0245768i
$300$		0			0
$301$	108.500	−	187.928i	0.360465	−	0.624344i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−203.647	−	352.727i	−0.667694	−	1.15648i
$306$		0			0
$307$		−	71.0141i		−	0.231316i	−0.993289	−	0.115658i	$-0.963102\pi$
$307$		−	71.0141i		−	0.231316i	0.993289	−	0.115658i	$-0.0368977\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−186.676	+	107.778i	−0.600245	+	0.346552i	−0.769138	−	0.639083i	$-0.779314\pi$
$311$	−186.676	+	107.778i	−0.600245	+	0.346552i	0.168893	+	0.985634i	$0.445981\pi$
$312$		0			0
$313$	253.500	+	146.358i	0.809904	+	0.467598i	0.846923	−	0.531716i	$-0.178453\pi$
$313$	253.500	+	146.358i	0.809904	+	0.467598i	−0.0370184	+	0.999315i	$0.511786\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	118.794	−	205.757i	0.374744	−	0.649076i	−0.615544	−	0.788102i	$-0.711064\pi$
$317$	118.794	−	205.757i	0.374744	−	0.649076i	0.990289	+	0.139026i	$0.0443972\pi$
$318$		0			0
$319$	288.000	+	498.831i	0.902821	+	1.56373i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	144.250			0.446594
$324$		0			0
$325$	1.50000	−	0.866025i	0.00461538	−	0.00266469i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−504.874	+	291.489i	−1.53457	+	0.885986i
$330$		0			0
$331$	−92.5000	+	160.215i	−0.279456	+	0.484032i	−0.971250	−	0.238063i	$-0.923488\pi$
$331$	−92.5000	+	160.215i	−0.279456	+	0.484032i	0.691794	+	0.722095i	$0.256821\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	151.868i		−	0.453338i
$336$		0			0
$337$	−359.000			−1.06528			−0.532641	−	0.846341i	$-0.678800\pi$
$337$	−359.000			−1.06528			−0.532641	+	0.846341i	$0.678800\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−178.191	−	102.879i	−0.522554	−	0.301697i
$342$		0			0
$343$	343.000			1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$	233.345	+	404.166i	0.672465	+	1.16474i	0.977203	+	0.212307i	$0.0680976\pi$
$347$	233.345	+	404.166i	0.672465	+	1.16474i	−0.304738	+	0.952436i	$0.598569\pi$
$348$		0			0
$349$		−	581.969i		−	1.66753i	−0.552117	−	0.833767i	$-0.686180\pi$
$349$		−	581.969i		−	1.66753i	0.552117	−	0.833767i	$-0.313820\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	250.316	−	144.520i	0.709110	−	0.409405i	−0.101621	−	0.994823i	$-0.532403\pi$
$353$	250.316	−	144.520i	0.709110	−	0.409405i	0.810731	+	0.585418i	$0.199070\pi$
$354$		0			0
$355$	−252.000	−	145.492i	−0.709859	−	0.409837i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	169.706	−	293.939i	0.472718	−	0.818771i	−0.526795	−	0.849992i	$-0.676606\pi$
$359$	169.706	−	293.939i	0.472718	−	0.818771i	0.999512	+	0.0312215i	$0.00993973\pi$
$360$		0			0
$361$	253.000	+	438.209i	0.700831	+	1.21387i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−398.808			−1.09263
$366$		0			0
$367$	−133.500	+	77.0763i	−0.363760	+	0.210017i	−0.670729	−	0.741703i	$-0.734019\pi$
$367$	−133.500	+	77.0763i	−0.363760	+	0.210017i	0.306969	+	0.951720i	$0.400685\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−534.573			−1.44090
$372$		0			0
$373$	144.500	−	250.281i	0.387399	−	0.670996i	−0.604699	−	0.796454i	$-0.706707\pi$
$373$	144.500	−	250.281i	0.387399	−	0.670996i	0.992099	+	0.125458i	$0.0400401\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	58.7878i		−	0.155936i
$378$		0			0
$379$	−7.00000			−0.0184697			−0.00923483	−	0.999957i	$-0.502940\pi$
$379$	−7.00000			−0.0184697			−0.00923483	+	0.999957i	$0.502940\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−428.507	−	247.398i	−1.11882	−	0.645949i	−0.177718	−	0.984081i	$-0.556871\pi$
$383$	−428.507	−	247.398i	−1.11882	−	0.645949i	−0.941099	+	0.338132i	$0.890205\pi$
$384$		0			0
$385$	504.000	+	290.985i	1.30909	+	0.755804i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	114.551	+	198.409i	0.294476	+	0.510048i	0.974863	−	0.222805i	$-0.0715214\pi$
$389$	114.551	+	198.409i	0.294476	+	0.510048i	−0.680387	+	0.732853i	$0.738188\pi$
$390$		0			0
$391$		−	41.5692i		−	0.106315i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	173.948	−	100.429i	0.440375	−	0.254251i
$396$		0			0
$397$	70.5000	+	40.7032i	0.177582	+	0.102527i	0.586156	−	0.810198i	$-0.300641\pi$
$397$	70.5000	+	40.7032i	0.177582	+	0.102527i	−0.408574	+	0.912725i	$0.633974\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	46.6690	−	80.8332i	0.116382	−	0.201579i	−0.801950	−	0.597392i	$-0.796204\pi$
$401$	46.6690	−	80.8332i	0.116382	−	0.201579i	0.918331	+	0.395813i	$0.129537\pi$
$402$		0			0
$403$	10.5000	+	18.1865i	0.0260546	+	0.0451279i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−797.616			−1.95975
$408$		0			0
$409$	−361.500	+	208.712i	−0.883863	+	0.510299i	−0.871930	−	0.489630i	$-0.837132\pi$
$409$	−361.500	+	208.712i	−0.883863	+	0.510299i	−0.0119329	+	0.999929i	$0.503798\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			582.979i			1.41157i
$414$		0			0
$415$	12.0000	−	20.7846i	0.0289157	−	0.0500834i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			19.5959i			0.0467683i	0.999727	+	0.0233842i	$0.00744408\pi$
$419$			19.5959i			0.0467683i	−0.999727	+	0.0233842i	$0.992556\pi$
$420$		0			0
$421$	407.000			0.966746			0.483373	−	0.875415i	$-0.339412\pi$
$421$	407.000			0.966746			0.483373	+	0.875415i	$0.339412\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.24264	+	2.44949i	0.00998268	+	0.00576351i
$426$		0			0
$427$	504.000	−	290.985i	1.18033	−	0.681463i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−80.6102	−	139.621i	−0.187031	−	0.323946i	0.757228	−	0.653150i	$-0.226553\pi$
$431$	−80.6102	−	139.621i	−0.187031	−	0.323946i	−0.944259	+	0.329204i	$0.893220\pi$
$432$		0			0
$433$			168.009i			0.388011i	0.981000	+	0.194006i	$0.0621480\pi$
$433$			168.009i			0.388011i	−0.981000	+	0.194006i	$0.937852\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	216.375	−	124.924i	0.495137	−	0.285867i
$438$		0			0
$439$	468.000	+	270.200i	1.06606	+	0.615490i	0.927102	−	0.374809i	$-0.122292\pi$
$439$	468.000	+	270.200i	1.06606	+	0.615490i	0.138957	+	0.990298i	$0.455625\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	63.6396	−	110.227i	0.143656	−	0.248819i	−0.785215	−	0.619224i	$-0.787447\pi$
$443$	63.6396	−	110.227i	0.143656	−	0.248819i	0.928871	+	0.370404i	$0.120781\pi$
$444$		0			0
$445$	144.000	+	249.415i	0.323596	+	0.560484i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	110.309			0.245676			0.122838	−	0.992427i	$-0.460800\pi$
$449$	110.309			0.245676			0.122838	+	0.992427i	$0.460800\pi$
$450$		0			0
$451$	−1008.00	+	581.969i	−2.23503	+	1.29040i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−29.6985	−	51.4393i	−0.0652714	−	0.113053i
$456$		0			0
$457$	−12.5000	+	21.6506i	−0.0273523	+	0.0473756i	−0.879378	−	0.476125i	$-0.842041\pi$
$457$	−12.5000	+	21.6506i	−0.0273523	+	0.0473756i	0.852025	+	0.523501i	$0.175374\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			78.3837i			0.170030i	0.996380	+	0.0850148i	$0.0270938\pi$
$461$			78.3837i			0.170030i	−0.996380	+	0.0850148i	$0.972906\pi$
$462$		0			0
$463$	−521.000			−1.12527			−0.562635	−	0.826705i	$-0.690212\pi$
$463$	−521.000			−1.12527			−0.562635	+	0.826705i	$0.690212\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	190.919	+	110.227i	0.408820	+	0.236032i	0.690283	−	0.723540i	$-0.257486\pi$
$467$	190.919	+	110.227i	0.408820	+	0.236032i	−0.281463	+	0.959572i	$0.590820\pi$
$468$		0			0
$469$	217.000			0.462687
$470$		0			0
$471$		0			0
$472$		0			0
$473$	263.044	+	455.605i	0.556118	+	0.963224i
$474$		0			0
$475$			29.4449i			0.0619892i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	759.433	−	438.459i	1.58545	−	0.915363i	0.591412	−	0.806370i	$-0.298571\pi$
$479$	759.433	−	438.459i	1.58545	−	0.915363i	0.994043	−	0.108993i	$-0.0347626\pi$
$480$		0			0
$481$	70.5000	+	40.7032i	0.146570	+	0.0846220i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	101.823	−	176.363i	0.209945	−	0.363636i
$486$		0			0
$487$	63.5000	+	109.985i	0.130390	+	0.225842i	0.923827	−	0.382810i	$-0.125044\pi$
$487$	63.5000	+	109.985i	0.130390	+	0.225842i	−0.793437	+	0.608653i	$0.791710\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	627.911			1.27884			0.639420	−	0.768857i	$-0.279174\pi$
$491$	627.911			1.27884			0.639420	+	0.768857i	$0.279174\pi$
$492$		0			0
$493$	144.000	−	83.1384i	0.292089	−	0.168638i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	207.889	−	360.075i	0.418289	−	0.724497i
$498$		0			0
$499$	−116.500	+	201.784i	−0.233467	+	0.404377i	−0.958826	−	0.283994i	$-0.908340\pi$
$499$	−116.500	+	201.784i	−0.233467	+	0.404377i	0.725359	+	0.688371i	$0.241674\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			538.888i			1.07135i	0.844425	+	0.535674i	$0.179942\pi$
$503$			538.888i			1.07135i	−0.844425	+	0.535674i	$0.820058\pi$
$504$		0			0
$505$	−864.000			−1.71089
$506$		0			0
$507$		0			0
$508$		0			0
$509$	275.772	+	159.217i	0.541791	+	0.312803i	0.745805	−	0.666165i	$-0.232065\pi$
$509$	275.772	+	159.217i	0.541791	+	0.312803i	−0.204013	+	0.978968i	$0.565399\pi$
$510$		0			0
$511$		−	569.845i		−	1.11516i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	72.1249	+	124.924i	0.140048	+	0.242571i
$516$		0			0
$517$		−	1413.35i		−	2.73376i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−492.146	+	284.141i	−0.944619	+	0.545376i	−0.891405	−	0.453207i	$-0.850280\pi$
$521$	−492.146	+	284.141i	−0.944619	+	0.545376i	−0.0532135	+	0.998583i	$0.516946\pi$
$522$		0			0
$523$	−457.500	−	264.138i	−0.874761	−	0.505043i	−0.00583355	−	0.999983i	$-0.501857\pi$
$523$	−457.500	−	264.138i	−0.874761	−	0.505043i	−0.868927	+	0.494939i	$0.835190\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−29.6985	+	51.4393i	−0.0563539	+	0.0976078i
$528$		0			0
$529$	228.500	+	395.774i	0.431947	+	0.748154i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	118.794			0.222878
$534$		0			0
$535$	−612.000	+	353.338i	−1.14393	+	0.660446i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−415.779	+	720.150i	−0.771389	+	1.33609i
$540$		0			0
$541$	−167.500	+	290.119i	−0.309612	+	0.536263i	−0.978277	−	0.207300i	$-0.933532\pi$
$541$	−167.500	+	290.119i	−0.309612	+	0.536263i	0.668666	+	0.743563i	$0.266866\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	827.928i		−	1.51913i
$546$		0			0
$547$	658.000			1.20293			0.601463	−	0.798901i	$-0.294585\pi$
$547$	658.000			1.20293			0.601463	+	0.798901i	$0.294585\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	865.499	+	499.696i	1.57078	+	0.906889i
$552$		0			0
$553$	143.500	+	248.549i	0.259494	+	0.449456i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	135.765	+	235.151i	0.243742	+	0.422174i	0.961777	−	0.273833i	$-0.0882915\pi$
$557$	135.765	+	235.151i	0.243742	+	0.422174i	−0.718035	+	0.696007i	$0.754958\pi$
$558$		0			0
$559$		−	53.6936i		−	0.0960529i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.7279	−	7.34847i	0.0226073	−	0.0130523i	−0.488654	−	0.872478i	$-0.662512\pi$
$563$	12.7279	−	7.34847i	0.0226073	−	0.0130523i	0.511261	+	0.859425i	$0.329179\pi$
$564$		0			0
$565$	−252.000	−	145.492i	−0.446018	−	0.257508i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	424.264	−	734.847i	0.745631	−	1.29147i	−0.204268	−	0.978915i	$-0.565481\pi$
$569$	424.264	−	734.847i	0.745631	−	1.29147i	0.949899	−	0.312556i	$-0.101185\pi$
$570$		0			0
$571$	224.500	+	388.845i	0.393170	+	0.680990i	0.992866	−	0.119238i	$-0.0380451\pi$
$571$	224.500	+	388.845i	0.393170	+	0.680990i	−0.599696	+	0.800228i	$0.704712\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.48528			0.0147570
$576$		0			0
$577$	−253.500	+	146.358i	−0.439341	+	0.253654i	−0.703318	−	0.710875i	$-0.748299\pi$
$577$	−253.500	+	146.358i	−0.439341	+	0.253654i	0.263977	+	0.964529i	$0.414966\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	29.6985	+	17.1464i	0.0511162	+	0.0295119i
$582$		0			0
$583$	648.000	−	1122.37i	1.11149	−	1.92516i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	529.090i		−	0.901345i	−0.892689	−	0.450673i	$-0.851184\pi$
$587$		−	529.090i		−	0.901345i	0.892689	−	0.450673i	$-0.148816\pi$
$588$		0			0
$589$	−357.000			−0.606112
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−907.925	−	524.191i	−1.53107	−	0.883964i	−0.999313	−	0.0370681i	$-0.988198\pi$
$593$	−907.925	−	524.191i	−1.53107	−	0.883964i	−0.531758	−	0.846896i	$-0.678469\pi$
$594$		0			0
$595$	84.0000	−	145.492i	0.141176	−	0.244525i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−322.441	−	558.484i	−0.538298	−	0.932360i	−0.998996	−	0.0448028i	$-0.985734\pi$
$599$	−322.441	−	558.484i	−0.538298	−	0.932360i	0.460698	−	0.887557i	$-0.347599\pi$
$600$		0			0
$601$		−	458.993i		−	0.763716i	−0.924221	−	0.381858i	$-0.875284\pi$
$601$		−	458.993i		−	0.763716i	0.924221	−	0.381858i	$-0.124716\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−708.521	+	409.065i	−1.17111	+	0.676140i
$606$		0			0
$607$	−910.500	−	525.677i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$607$	−910.500	−	525.677i	−1.50000	−	0.866025i	−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−72.1249	+	124.924i	−0.118044	+	0.204458i
$612$		0			0
$613$	−145.000	−	251.147i	−0.236542	−	0.409702i	0.723178	−	0.690662i	$-0.242681\pi$
$613$	−145.000	−	251.147i	−0.236542	−	0.409702i	−0.959720	+	0.280960i	$0.909347\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−729.734			−1.18271			−0.591357	−	0.806410i	$-0.701407\pi$
$617$	−729.734			−1.18271			−0.591357	+	0.806410i	$0.701407\pi$
$618$		0			0
$619$	709.500	−	409.630i	1.14620	−	0.661761i	0.198244	−	0.980153i	$-0.436476\pi$
$619$	709.500	−	409.630i	1.14620	−	0.661761i	0.947959	+	0.318392i	$0.103143\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−356.382	+	205.757i	−0.572041	+	0.330268i
$624$		0			0
$625$	299.500	−	518.749i	0.479200	−	0.829999i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			230.252i			0.366060i
$630$		0			0
$631$	58.0000			0.0919176			0.0459588	−	0.998943i	$-0.485366\pi$
$631$	58.0000			0.0919176			0.0459588	+	0.998943i	$0.485366\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−886.712	−	511.943i	−1.39640	−	0.806210i
$636$		0			0
$637$	73.5000	−	42.4352i	0.115385	−	0.0666173i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	479.418	+	830.377i	0.747923	+	1.29544i	0.948817	+	0.315827i	$0.102282\pi$
$641$	479.418	+	830.377i	0.747923	+	1.29544i	−0.200894	+	0.979613i	$0.564385\pi$
$642$		0			0
$643$		−	760.370i		−	1.18254i	−0.806475	−	0.591268i	$-0.798628\pi$
$643$		−	760.370i		−	1.18254i	0.806475	−	0.591268i	$-0.201372\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−305.470	+	176.363i	−0.472133	+	0.272586i	−0.717132	−	0.696937i	$-0.754546\pi$
$647$	−305.470	+	176.363i	−0.472133	+	0.272586i	0.244999	+	0.969523i	$0.421212\pi$
$648$		0			0
$649$	−1224.00	−	706.677i	−1.88598	−	1.08887i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−220.617	+	382.120i	−0.337852	+	0.585177i	−0.984028	−	0.178011i	$-0.943034\pi$
$653$	−220.617	+	382.120i	−0.337852	+	0.585177i	0.646177	+	0.763188i	$0.276367\pi$
$654$		0			0
$655$	144.000	+	249.415i	0.219847	+	0.380787i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	161.220			0.244644			0.122322	−	0.992490i	$-0.460966\pi$
$659$	161.220			0.244644			0.122322	+	0.992490i	$0.460966\pi$
$660$		0			0
$661$	−721.500	+	416.558i	−1.09153	+	0.630194i	−0.933983	−	0.357318i	$-0.883691\pi$
$661$	−721.500	+	416.558i	−1.09153	+	0.630194i	−0.157545	+	0.987512i	$0.550358\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1009.75			1.51842
$666$		0			0
$667$	144.000	−	249.415i	0.215892	−	0.373936i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1410.91i			2.10269i
$672$		0			0
$673$	−263.000			−0.390788			−0.195394	−	0.980725i	$-0.562598\pi$
$673$	−263.000			−0.390788			−0.195394	+	0.980725i	$0.562598\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−432.749	−	249.848i	−0.639216	−	0.369052i	0.145096	−	0.989418i	$-0.453651\pi$
$677$	−432.749	−	249.848i	−0.639216	−	0.369052i	−0.784313	+	0.620366i	$0.786984\pi$
$678$		0			0
$679$	252.000	+	145.492i	0.371134	+	0.214274i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−479.418	−	830.377i	−0.701930	−	1.21578i	−0.967788	−	0.251767i	$-0.918988\pi$
$683$	−479.418	−	830.377i	−0.701930	−	1.21578i	0.265858	−	0.964012i	$-0.414345\pi$
$684$		0			0
$685$		−	748.246i		−	1.09233i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−114.551	+	66.1362i	−0.166257	+	0.0959887i
$690$		0			0
$691$	−1069.50	−	617.476i	−1.54776	−	0.893598i	−0.998313	−	0.0580674i	$-0.981506\pi$
$691$	−1069.50	−	617.476i	−1.54776	−	0.893598i	−0.549444	−	0.835530i	$-0.685161\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	479.418	−	830.377i	0.689811	−	1.19479i
$696$		0			0
$697$	168.000	+	290.985i	0.241033	+	0.417481i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	975.807			1.39202			0.696011	−	0.718031i	$-0.254956\pi$
$701$	975.807			1.39202			0.696011	+	0.718031i	$0.254956\pi$
$702$		0			0
$703$	−1198.50	+	691.954i	−1.70484	+	0.984288i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	1234.54i		−	1.74617i
$708$		0			0
$709$	553.000	−	957.824i	0.779972	−	1.35095i	−0.151986	−	0.988383i	$-0.548567\pi$
$709$	553.000	−	957.824i	0.779972	−	1.35095i	0.931957	−	0.362568i	$-0.118100\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			102.879i			0.144290i
$714$		0			0
$715$	144.000			0.201399
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−593.970	−	342.929i	−0.826105	−	0.476952i	0.0264120	−	0.999651i	$-0.491592\pi$
$719$	−593.970	−	342.929i	−0.826105	−	0.476952i	−0.852517	+	0.522699i	$0.824925\pi$
$720$		0			0
$721$	−178.500	+	103.057i	−0.247573	+	0.142936i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	16.9706	+	29.3939i	0.0234077	+	0.0405433i
$726$		0			0
$727$		−	427.817i		−	0.588468i	−0.955733	−	0.294234i	$-0.904935\pi$
$727$		−	427.817i		−	0.588468i	0.955733	−	0.294234i	$-0.0950646\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	131.522	−	75.9342i	0.179920	−	0.103877i
$732$		0			0
$733$	34.5000	+	19.9186i	0.0470668	+	0.0271741i	0.523349	−	0.852119i	$-0.324682\pi$
$733$	34.5000	+	19.9186i	0.0470668	+	0.0271741i	−0.476282	+	0.879293i	$0.658016\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−263.044	+	455.605i	−0.356911	+	0.618189i
$738$		0			0
$739$	−243.500	−	421.754i	−0.329499	−	0.570710i	0.652913	−	0.757433i	$-0.273547\pi$
$739$	−243.500	−	421.754i	−0.329499	−	0.570710i	−0.982413	+	0.186723i	$0.940213\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	509.117			0.685218			0.342609	−	0.939478i	$-0.388689\pi$
$743$	509.117			0.685218			0.342609	+	0.939478i	$0.388689\pi$
$744$		0			0
$745$	−216.000	+	124.708i	−0.289933	+	0.167393i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−504.874	−	874.468i	−0.674064	−	1.16751i
$750$		0			0
$751$	−272.500	+	471.984i	−0.362850	+	0.628474i	−0.988429	−	0.151687i	$-0.951529\pi$
$751$	−272.500	+	471.984i	−0.362850	+	0.628474i	0.625579	+	0.780161i	$0.284863\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			48.9898i			0.0648871i
$756$		0			0
$757$	−770.000			−1.01717			−0.508587	−	0.861011i	$-0.669832\pi$
$757$	−770.000			−1.01717			−0.508587	+	0.861011i	$0.669832\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	148.492	+	85.7321i	0.195128	+	0.112657i	0.594381	−	0.804184i	$-0.297397\pi$
$761$	148.492	+	85.7321i	0.195128	+	0.112657i	−0.399253	+	0.916841i	$0.630730\pi$
$762$		0			0
$763$	1183.00			1.55046
$764$		0			0
$765$		0			0
$766$		0			0
$767$	72.1249	+	124.924i	0.0940351	+	0.162874i
$768$		0			0
$769$		−	704.945i		−	0.916703i	−0.888771	−	0.458352i	$-0.848440\pi$
$769$		−	704.945i		−	0.916703i	0.888771	−	0.458352i	$-0.151560\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	797.616	−	460.504i	1.03185	−	0.595736i	0.114333	−	0.993443i	$-0.463527\pi$
$773$	797.616	−	460.504i	1.03185	−	0.595736i	0.917513	+	0.397706i	$0.130194\pi$
$774$		0			0
$775$	−10.5000	−	6.06218i	−0.0135484	−	0.00782216i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1009.75	+	1748.94i	−1.29621	+	2.24510i
$780$		0			0
$781$	504.000	+	872.954i	0.645327	+	1.11774i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	203.647			0.259423
$786$		0			0
$787$	−396.000	+	228.631i	−0.503177	+	0.290509i	−0.730024	−	0.683421i	$-0.760491\pi$
$787$	−396.000	+	228.631i	−0.503177	+	0.290509i	0.226848	+	0.973930i	$0.427158\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	207.889	−	360.075i	0.262818	−	0.455215i
$792$		0			0
$793$	72.0000	−	124.708i	0.0907945	−	0.157261i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			14.6969i			0.0184403i	0.999957	+	0.00922016i	$0.00293491\pi$
$797$			14.6969i			0.0184403i	−0.999957	+	0.00922016i	$0.997065\pi$
$798$		0			0
$799$	−408.000			−0.510638
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1196.42	+	690.756i	1.48994	+	0.860219i
$804$		0			0
$805$		−	290.985i		−	0.361471i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−470.933	−	815.680i	−0.582118	−	1.00826i	−0.995228	−	0.0975763i	$-0.968891\pi$
$809$	−470.933	−	815.680i	−0.582118	−	1.00826i	0.413110	−	0.910681i	$-0.364442\pi$
$810$		0			0
$811$		−	498.831i		−	0.615081i	−0.951535	−	0.307540i	$-0.900494\pi$
$811$		−	498.831i		−	0.615081i	0.951535	−	0.307540i	$-0.0995059\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	364.867	−	210.656i	0.447690	−	0.258474i
$816$		0			0
$817$	790.500	+	456.395i	0.967564	+	0.558623i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−301.227	+	521.741i	−0.366903	+	0.635495i	−0.989080	−	0.147382i	$-0.952915\pi$
$821$	−301.227	+	521.741i	−0.366903	+	0.635495i	0.622176	+	0.782877i	$0.286249\pi$
$822$		0			0
$823$	19.0000	+	32.9090i	0.0230863	+	0.0399866i	0.877338	−	0.479873i	$-0.159317\pi$
$823$	19.0000	+	32.9090i	0.0230863	+	0.0399866i	−0.854251	+	0.519860i	$0.825984\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	687.308			0.831086			0.415543	−	0.909574i	$-0.363592\pi$
$827$	687.308			0.831086			0.415543	+	0.909574i	$0.363592\pi$
$828$		0			0
$829$	−721.500	+	416.558i	−0.870326	+	0.502483i	−0.867456	−	0.497513i	$-0.834247\pi$
$829$	−721.500	+	416.558i	−0.870326	+	0.502483i	−0.00286924	+	0.999996i	$0.500913\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	207.889	+	120.025i	0.249567	+	0.144088i
$834$		0			0
$835$	−444.000	+	769.031i	−0.531737	+	0.920995i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	244.949i		−	0.291953i	−0.989288	−	0.145977i	$-0.953368\pi$
$839$		−	244.949i		−	0.291953i	0.989288	−	0.145977i	$-0.0466325\pi$
$840$		0			0
$841$	311.000			0.369798
$842$		0			0
$843$		0			0
$844$		0			0
$845$	704.278	+	406.615i	0.833466	+	0.481202i
$846$		0			0
$847$	−584.500	−	1012.38i	−0.690083	−	1.19526i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	199.404	+	345.378i	0.234317	+	0.405850i
$852$		0			0
$853$			1245.34i			1.45996i	0.683469	+	0.729979i	$0.260470\pi$
$853$			1245.34i			1.45996i	−0.683469	+	0.729979i	$0.739530\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1060.66	+	612.372i	−1.23764	+	0.714554i	−0.968612	−	0.248578i	$-0.920037\pi$
$857$	−1060.66	+	612.372i	−1.23764	+	0.714554i	−0.269031	+	0.963131i	$0.586703\pi$
$858$		0			0
$859$	−216.000	−	124.708i	−0.251455	−	0.145178i	0.368975	−	0.929439i	$-0.379709\pi$
$859$	−216.000	−	124.708i	−0.251455	−	0.145178i	−0.620430	+	0.784262i	$0.713042\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−330.926	+	573.181i	−0.383460	+	0.664172i	−0.991554	−	0.129693i	$-0.958601\pi$
$863$	−330.926	+	573.181i	−0.383460	+	0.664172i	0.608094	+	0.793865i	$0.291934\pi$
$864$		0			0
$865$	−108.000	−	187.061i	−0.124855	−	0.216256i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−695.793			−0.800682
$870$		0			0
$871$	46.5000	−	26.8468i	0.0533869	−	0.0308229i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	772.161	+	445.807i	0.882469	+	0.509494i
$876$		0			0
$877$	287.000	−	497.099i	0.327252	−	0.566817i	−0.654714	−	0.755877i	$-0.727211\pi$
$877$	287.000	−	497.099i	0.327252	−	0.566817i	0.981966	+	0.189060i	$0.0605441\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			161.666i			0.183503i	0.995782	+	0.0917516i	$0.0292466\pi$
$881$			161.666i			0.183503i	−0.995782	+	0.0917516i	$0.970753\pi$
$882$		0			0
$883$	−1735.00			−1.96489			−0.982446	−	0.186546i	$-0.940271\pi$
$883$	−1735.00			−1.96489			−0.982446	+	0.186546i	$0.940271\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−169.706	−	97.9796i	−0.191325	−	0.110462i	0.401277	−	0.915957i	$-0.368566\pi$
$887$	−169.706	−	97.9796i	−0.191325	−	0.110462i	−0.592603	+	0.805495i	$0.701900\pi$
$888$		0			0
$889$	731.500	−	1267.00i	0.822835	−	1.42519i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1226.12	−	2123.71i	−1.37304	−	2.37817i
$894$		0			0
$895$		−	41.5692i		−	0.0464461i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−356.382	+	205.757i	−0.396420	+	0.228873i
$900$		0			0
$901$	−324.000	−	187.061i	−0.359600	−	0.207615i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	106.066	−	183.712i	0.117200	−	0.202996i
$906$		0			0
$907$	375.500	+	650.385i	0.414002	+	0.717073i	0.995323	−	0.0966015i	$-0.0307972\pi$
$907$	375.500	+	650.385i	0.414002	+	0.717073i	−0.581321	+	0.813674i	$0.697464\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1247.34			−1.36919			−0.684597	−	0.728921i	$-0.740022\pi$
$911$	−1247.34			−1.36919			−0.684597	+	0.728921i	$0.740022\pi$
$912$		0			0
$913$	−72.0000	+	41.5692i	−0.0788609	+	0.0455304i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−356.382	+	205.757i	−0.388639	+	0.224381i
$918$		0			0
$919$	507.500	−	879.016i	0.552231	−	0.956492i	−0.445883	−	0.895091i	$-0.647110\pi$
$919$	507.500	−	879.016i	0.552231	−	0.956492i	0.998113	−	0.0614001i	$-0.0195566\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	102.879i		−	0.111461i
$924$		0			0
$925$	−47.0000			−0.0508108
$926$		0			0
$927$		0			0
$928$		0			0
$929$	946.109	+	546.236i	1.01842	+	0.587983i	0.913645	−	0.406513i	$-0.133256\pi$
$929$	946.109	+	546.236i	1.01842	+	0.587983i	0.104772	+	0.994496i	$0.466589\pi$
$930$		0			0
$931$			1442.80i			1.54973i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	203.647	+	352.727i	0.217804	+	0.377248i
$936$		0			0
$937$		−	1747.64i		−	1.86514i	−0.360985	−	0.932572i	$-0.617559\pi$
$937$		−	1747.64i		−	1.86514i	0.360985	−	0.932572i	$-0.382441\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1073.39	−	619.721i	1.14069	−	0.658577i	0.194088	−	0.980984i	$-0.437825\pi$
$941$	1073.39	−	619.721i	1.14069	−	0.658577i	0.946601	+	0.322407i	$0.104492\pi$
$942$		0			0
$943$	504.000	+	290.985i	0.534464	+	0.308573i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	602.455	−	1043.48i	0.636172	−	1.10188i	−0.350093	−	0.936715i	$-0.613850\pi$
$947$	602.455	−	1043.48i	0.636172	−	1.10188i	0.986266	−	0.165168i	$-0.0528165\pi$
$948$		0			0
$949$	−70.5000	−	122.110i	−0.0742887	−	0.128672i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1026.72			1.07735			0.538677	−	0.842512i	$-0.318924\pi$
$953$	1026.72			1.07735			0.538677	+	0.842512i	$0.318924\pi$
$954$		0			0
$955$	324.000	−	187.061i	0.339267	−	0.195876i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1069.15			1.11485
$960$		0			0
$961$	−407.000	+	704.945i	−0.423517	+	0.733553i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			1406.01i			1.45700i
$966$		0			0
$967$	895.000			0.925543			0.462771	−	0.886478i	$-0.346855\pi$
$967$	895.000			0.925543			0.462771	+	0.886478i	$0.346855\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	182.434	+	105.328i	0.187882	+	0.108474i	0.590991	−	0.806678i	$-0.298737\pi$
$971$	182.434	+	105.328i	0.187882	+	0.108474i	−0.403109	+	0.915152i	$0.632070\pi$
$972$		0			0
$973$	1186.50	+	685.026i	1.21942	+	0.704035i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	627.911	+	1087.57i	0.642693	+	1.11318i	0.984829	+	0.173526i	$0.0555161\pi$
$977$	627.911	+	1087.57i	0.642693	+	1.11318i	−0.342136	+	0.939650i	$0.611151\pi$
$978$		0			0
$979$		−	997.661i		−	1.01906i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1022.48	+	590.327i	−1.04016	+	0.600536i	−0.919878	−	0.392204i	$-0.871713\pi$
$983$	−1022.48	+	590.327i	−1.04016	+	0.600536i	−0.120281	+	0.992740i	$0.538379\pi$
$984$		0			0
$985$	540.000	+	311.769i	0.548223	+	0.316517i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	131.522	−	227.803i	0.132985	−	0.230336i
$990$		0			0
$991$	327.500	+	567.247i	0.330474	+	0.572398i	0.982605	−	0.185708i	$-0.0594580\pi$
$991$	327.500	+	567.247i	0.330474	+	0.572398i	−0.652131	+	0.758107i	$0.726125\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1018.23			−1.02335
$996$		0			0
$997$	397.500	−	229.497i	0.398696	−	0.230187i	−0.287225	−	0.957863i	$-0.592733\pi$
$997$	397.500	−	229.497i	0.398696	−	0.230187i	0.685921	+	0.727676i	$0.259399\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.cg.i.145.2		4
3.2	odd	2	inner	1008.3.cg.i.145.1		4
4.3	odd	2		126.3.n.b.19.1	✓	4
7.3	odd	6	inner	1008.3.cg.i.577.2		4
12.11	even	2		126.3.n.b.19.2	yes	4
21.17	even	6	inner	1008.3.cg.i.577.1		4
28.3	even	6		126.3.n.b.73.1	yes	4
28.11	odd	6		882.3.n.c.325.1		4
28.19	even	6		882.3.c.c.685.4		4
28.23	odd	6		882.3.c.c.685.3		4
28.27	even	2		882.3.n.c.19.1		4
84.11	even	6		882.3.n.c.325.2		4
84.23	even	6		882.3.c.c.685.2		4
84.47	odd	6		882.3.c.c.685.1		4
84.59	odd	6		126.3.n.b.73.2	yes	4
84.83	odd	2		882.3.n.c.19.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.n.b.19.1	✓	4	4.3	odd	2
126.3.n.b.19.2	yes	4	12.11	even	2
126.3.n.b.73.1	yes	4	28.3	even	6
126.3.n.b.73.2	yes	4	84.59	odd	6
882.3.c.c.685.1		4	84.47	odd	6
882.3.c.c.685.2		4	84.23	even	6
882.3.c.c.685.3		4	28.23	odd	6
882.3.c.c.685.4		4	28.19	even	6
882.3.n.c.19.1		4	28.27	even	2
882.3.n.c.19.2		4	84.83	odd	2
882.3.n.c.325.1		4	28.11	odd	6
882.3.n.c.325.2		4	84.11	even	6
1008.3.cg.i.145.1		4	3.2	odd	2	inner
1008.3.cg.i.145.2		4	1.1	even	1	trivial
1008.3.cg.i.577.1		4	21.17	even	6	inner
1008.3.cg.i.577.2		4	7.3	odd	6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.cg (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.24264 + 2.44949i) q^{5} +(-3.50000 + 6.06218i) q^{7} +O(q^{10})\)	\(q+(4.24264 + 2.44949i) q^{5} +(-3.50000 + 6.06218i) q^{7} +(-8.48528 - 14.6969i) q^{11} +1.73205i q^{13} +(-4.24264 + 2.44949i) q^{17} +(-25.5000 - 14.7224i) q^{19} +(-4.24264 + 7.34847i) q^{23} +(-0.500000 - 0.866025i) q^{25} -33.9411 q^{29} +(10.5000 - 6.06218i) q^{31} +(-29.6985 + 17.1464i) q^{35} +(23.5000 - 40.7032i) q^{37} -68.5857i q^{41} -31.0000 q^{43} +(72.1249 + 41.6413i) q^{47} +(-24.5000 - 42.4352i) q^{49} +(38.1838 + 66.1362i) q^{53} -83.1384i q^{55} +(72.1249 - 41.6413i) q^{59} +(-72.0000 - 41.5692i) q^{61} +(-4.24264 + 7.34847i) q^{65} +(-15.5000 - 26.8468i) q^{67} -59.3970 q^{71} +(-70.5000 + 40.7032i) q^{73} +118.794 q^{77} +(20.5000 - 35.5070i) q^{79} -4.89898i q^{83} -24.0000 q^{85} +(50.9117 + 29.3939i) q^{89} +(-10.5000 - 6.06218i) q^{91} +(-72.1249 - 124.924i) q^{95} -41.5692i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{7}+O(q^{10}) \) 4 * q - 14 * q^7	\( 4 q - 14 q^{7} - 102 q^{19} - 2 q^{25} + 42 q^{31} + 94 q^{37} - 124 q^{43} - 98 q^{49} - 288 q^{61} - 62 q^{67} - 282 q^{73} + 82 q^{79} - 96 q^{85} - 42 q^{91}+O(q^{100}) \) 4 * q - 14 * q^7 - 102 * q^19 - 2 * q^25 + 42 * q^31 + 94 * q^37 - 124 * q^43 - 98 * q^49 - 288 * q^61 - 62 * q^67 - 282 * q^73 + 82 * q^79 - 96 * q^85 - 42 * q^91

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