LMFDB - Embedded newform 1008.3.cg.i.145.1

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			145.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1008.145
Dual form			1008.3.cg.i.577.1

$q$-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	0.0116258	−	0.999932i	$-0.496299\pi$
$5$	−4.24264	−	2.44949i	−0.848528	−	0.489898i	−0.860154	+	0.510034i	$0.829633\pi$
$6$		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	$0.113771\pi$
$11$	8.48528	+	14.6969i	0.771389	+	1.33609i	−0.165412	+	0.986224i	$0.552896\pi$
$12$		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	−0.369999	−	0.929032i	$-0.620642\pi$
$17$	4.24264	−	2.44949i	0.249567	−	0.144088i	0.619566	+	0.784945i	$0.287309\pi$
$18$		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	−0.758933	−	0.651169i	$-0.774279\pi$
$23$	4.24264	−	7.34847i	0.184463	−	0.319499i	0.943395	+	0.331670i	$0.107612\pi$
$24$		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	$-0.301019\pi$
$29$	33.9411			1.17038			0.585192	+	0.810895i	$0.301019\pi$
$30$		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	$0.315350\pi$
$41$			68.5857i			1.67282i	−0.548103	+	0.836411i	$0.684650\pi$
$42$		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	$-0.986816\pi$
$47$	−72.1249	−	41.6413i	−1.53457	−	0.885986i	−0.535430	−	0.844580i	$-0.679850\pi$
$48$		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	$-0.910603\pi$
$53$	−38.1838	−	66.1362i	−0.720448	−	1.24785i	0.240372	−	0.970681i	$-0.422731\pi$
$54$		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	−0.965441	−	0.260622i	$-0.916072\pi$
$59$	−72.1249	+	41.6413i	−1.22246	+	0.705785i	−0.257015	+	0.966407i	$0.582739\pi$
$60$		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	$-0.362630\pi$
$71$	59.3970			0.836577			0.418289	+	0.908314i	$0.362630\pi$
$72$		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	$0.00939530\pi$
$83$			4.89898i			0.0590238i	−0.999564	+	0.0295119i	$0.990605\pi$
$84$		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	0.185923	−	0.982564i	$-0.440473\pi$
$89$	−50.9117	−	29.3939i	−0.572041	−	0.330268i	−0.757964	+	0.652296i	$0.773806\pi$
$90$		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	$-0.328782\pi$
$101$	152.735	−	88.1816i	1.51223	−	0.873085i	0.999898	−	0.0142971i	$-0.00455107\pi$
$102$		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	$-0.597880\pi$
$107$	72.1249	−	124.924i	0.674064	−	1.16751i	0.976741	−	0.214421i	$-0.0687863\pi$
$108$		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	$-0.415348\pi$
$113$	59.3970			0.525637			0.262818	+	0.964845i	$0.415348\pi$
$114$		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	0.292931	−	0.956133i	$-0.405369\pi$
$131$	−50.9117	−	29.3939i	−0.388639	−	0.224381i	−0.681570	+	0.731753i	$0.738703\pi$
$132$		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	$0.0215448\pi$
$137$	76.3675	+	132.272i	0.557427	+	0.965492i	−0.440283	+	0.897859i	$0.645122\pi$
$138$		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	−0.767871	−	0.640605i	$-0.778684\pi$
$149$	25.4558	−	44.0908i	0.170845	−	0.295912i	0.938715	+	0.344693i	$0.112017\pi$
$150$		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	$-0.817402\pi$
$167$		−	181.262i		−	1.08540i	0.839926	−	0.542701i	$-0.182598\pi$
$168$		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	0.606281	−	0.795250i	$-0.292660\pi$
$173$	38.1838	+	22.0454i	0.220715	+	0.127430i	−0.385566	+	0.922680i	$0.625994\pi$
$174$		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	0.877633	−	0.479333i	$-0.159121\pi$
$179$	4.24264	+	7.34847i	0.0237019	+	0.0410529i	−0.853931	+	0.520386i	$0.825788\pi$
$180$		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	−0.948501	−	0.316775i	$-0.897400\pi$
$191$	−38.1838	+	66.1362i	−0.199915	+	0.346263i	0.748586	+	0.663038i	$0.230733\pi$
$192$		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	$-0.604706\pi$
$197$	−127.279			−0.646087			−0.323044	+	0.946384i	$0.604706\pi$
$198$		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	0.458920	−	0.888478i	$-0.348237\pi$
$227$	330.926	−	191.060i	1.45782	−	0.841675i	0.998904	+	0.0468029i	$0.0149033\pi$
$228$		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	−0.508317	−	0.861170i	$-0.669732\pi$
$233$	114.551	−	198.409i	0.491636	−	0.851539i	0.999954	+	0.00963059i	$0.00306556\pi$
$234$		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	$-0.545358\pi$
$239$	−67.8823			−0.284026			−0.142013	+	0.989865i	$0.545358\pi$
$240$		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	$0.243660\pi$
$251$			347.828i			1.38577i	−0.721050	+	0.692884i	$0.756340\pi$
$252$		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	0.202237	−	0.979337i	$-0.435179\pi$
$257$	−140.007	−	80.8332i	−0.544775	−	0.314526i	−0.747012	+	0.664811i	$0.768512\pi$
$258$		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	0.515879	−	0.856662i	$-0.327466\pi$
$263$	−127.279	−	220.454i	−0.483951	−	0.838228i	−0.999830	+	0.0184332i	$0.994132\pi$
$264$		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	−0.531212	−	0.847239i	$-0.678263\pi$
$269$	−16.9706	+	9.79796i	−0.0630876	+	0.0364236i	0.468124	+	0.883663i	$0.344930\pi$
$270$		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	$-0.644261\pi$
$281$	−246.073			−0.875705			−0.437853	+	0.899047i	$0.644261\pi$
$282$		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	$0.0534728\pi$
$293$			97.9796i			0.334401i	−0.985923	+	0.167201i	$0.946527\pi$
$294$		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	−0.168893	−	0.985634i	$-0.554019\pi$
$311$	186.676	−	107.778i	0.600245	−	0.346552i	0.769138	+	0.639083i	$0.220686\pi$
$312$		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	−0.990289	−	0.139026i	$-0.955603\pi$
$317$	−118.794	+	205.757i	−0.374744	+	0.649076i	0.615544	+	0.788102i	$0.288936\pi$
$318$		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	$-0.931902\pi$
$347$	−233.345	−	404.166i	−0.672465	−	1.16474i	0.304738	−	0.952436i	$-0.401431\pi$
$348$		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	−0.810731	−	0.585418i	$-0.800930\pi$
$353$	−250.316	+	144.520i	−0.709110	+	0.409405i	0.101621	+	0.994823i	$0.467597\pi$
$354$		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	−0.999512	−	0.0312215i	$-0.990060\pi$
$359$	−169.706	+	293.939i	−0.472718	+	0.818771i	0.526795	+	0.849992i	$0.323394\pi$
$360$		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	0.941099	−	0.338132i	$-0.109795\pi$
$383$	428.507	+	247.398i	1.11882	+	0.645949i	0.177718	+	0.984081i	$0.443129\pi$
$384$		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	0.680387	−	0.732853i	$-0.261812\pi$
$389$	−114.551	−	198.409i	−0.294476	−	0.510048i	−0.974863	+	0.222805i	$0.928479\pi$
$390$		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	−0.918331	−	0.395813i	$-0.870463\pi$
$401$	−46.6690	+	80.8332i	−0.116382	+	0.201579i	0.801950	+	0.597392i	$0.203796\pi$
$402$		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	$-0.992556\pi$
$419$		−	19.5959i		−	0.0467683i	0.999727	−	0.0233842i	$-0.00744408\pi$
$420$		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	0.944259	−	0.329204i	$-0.106780\pi$
$431$	80.6102	+	139.621i	0.187031	+	0.323946i	−0.757228	+	0.653150i	$0.773447\pi$
$432$		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	−0.928871	−	0.370404i	$-0.879219\pi$
$443$	−63.6396	+	110.227i	−0.143656	+	0.248819i	0.785215	+	0.619224i	$0.212553\pi$
$444$		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	$-0.539200\pi$
$449$	−110.309			−0.245676			−0.122838	+	0.992427i	$0.539200\pi$
$450$		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	$-0.972906\pi$
$461$		−	78.3837i		−	0.170030i	0.996380	−	0.0850148i	$-0.0270938\pi$
$462$		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	0.281463	−	0.959572i	$-0.409180\pi$
$467$	−190.919	−	110.227i	−0.408820	−	0.236032i	−0.690283	+	0.723540i	$0.742514\pi$
$468$		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	$0.701429\pi$
$479$	−759.433	+	438.459i	−1.58545	+	0.915363i	−0.994043	+	0.108993i	$0.965237\pi$
$480$		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	$-0.720826\pi$
$491$	−627.911			−1.27884			−0.639420	+	0.768857i	$0.720826\pi$
$492$		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	$-0.820058\pi$
$503$		−	538.888i		−	1.07135i	0.844425	−	0.535674i	$-0.179942\pi$
$504$		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	0.204013	−	0.978968i	$-0.434601\pi$
$509$	−275.772	−	159.217i	−0.541791	−	0.312803i	−0.745805	+	0.666165i	$0.767935\pi$
$510$		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	0.0532135	−	0.998583i	$-0.483054\pi$
$521$	492.146	−	284.141i	0.944619	−	0.545376i	0.891405	+	0.453207i	$0.149720\pi$
$522$		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	0.718035	−	0.696007i	$-0.245042\pi$
$557$	−135.765	−	235.151i	−0.243742	−	0.422174i	−0.961777	+	0.273833i	$0.911708\pi$
$558$		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	−0.511261	−	0.859425i	$-0.670821\pi$
$563$	−12.7279	+	7.34847i	−0.0226073	+	0.0130523i	0.488654	+	0.872478i	$0.337488\pi$
$564$		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	$0.434519\pi$
$569$	−424.264	+	734.847i	−0.745631	+	1.29147i	−0.949899	+	0.312556i	$0.898815\pi$
$570$		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	$0.148816\pi$
$587$			529.090i			0.901345i	−0.892689	+	0.450673i	$0.851184\pi$
$588$		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	$0.0118018\pi$
$593$	907.925	+	524.191i	1.53107	+	0.883964i	0.531758	+	0.846896i	$0.321531\pi$
$594$		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	$0.0142660\pi$
$599$	322.441	+	558.484i	0.538298	+	0.932360i	−0.460698	+	0.887557i	$0.652401\pi$
$600$		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	$-0.298593\pi$
$617$	729.734			1.18271			0.591357	+	0.806410i	$0.298593\pi$
$618$		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	$-0.897718\pi$
$641$	−479.418	−	830.377i	−0.747923	−	1.29544i	0.200894	−	0.979613i	$-0.435615\pi$
$642$		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	−0.244999	−	0.969523i	$-0.578788\pi$
$647$	305.470	−	176.363i	0.472133	−	0.272586i	0.717132	+	0.696937i	$0.245454\pi$
$648$		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	−0.646177	−	0.763188i	$-0.723633\pi$
$653$	220.617	−	382.120i	0.337852	−	0.585177i	0.984028	+	0.178011i	$0.0569664\pi$
$654$		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	$-0.539034\pi$
$659$	−161.220			−0.244644			−0.122322	+	0.992490i	$0.539034\pi$
$660$		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	0.784313	−	0.620366i	$-0.213016\pi$
$677$	432.749	+	249.848i	0.639216	+	0.369052i	−0.145096	+	0.989418i	$0.546349\pi$
$678$		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	$0.0810115\pi$
$683$	479.418	+	830.377i	0.701930	+	1.21578i	−0.265858	+	0.964012i	$0.585655\pi$
$684$		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	$-0.745044\pi$
$701$	−975.807			−1.39202			−0.696011	+	0.718031i	$0.745044\pi$
$702$		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	0.852517	−	0.522699i	$-0.175075\pi$
$719$	593.970	+	342.929i	0.826105	+	0.476952i	−0.0264120	+	0.999651i	$0.508408\pi$
$720$		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	$-0.611311\pi$
$743$	−509.117			−0.685218			−0.342609	+	0.939478i	$0.611311\pi$
$744$		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	0.399253	−	0.916841i	$-0.369270\pi$
$761$	−148.492	−	85.7321i	−0.195128	−	0.112657i	−0.594381	+	0.804184i	$0.702603\pi$
$762$		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	−0.917513	−	0.397706i	$-0.869806\pi$
$773$	−797.616	+	460.504i	−1.03185	+	0.595736i	−0.114333	+	0.993443i	$0.536473\pi$
$774$		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	$-0.997065\pi$
$797$		−	14.6969i		−	0.0184403i	0.999957	−	0.00922016i	$-0.00293491\pi$
$798$		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	$0.0311090\pi$
$809$	470.933	+	815.680i	0.582118	+	1.00826i	−0.413110	+	0.910681i	$0.635558\pi$
$810$		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	−0.622176	−	0.782877i	$-0.713751\pi$
$821$	301.227	−	521.741i	0.366903	−	0.635495i	0.989080	+	0.147382i	$0.0470847\pi$
$822$		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	$-0.636408\pi$
$827$	−687.308			−0.831086			−0.415543	+	0.909574i	$0.636408\pi$
$828$		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	$0.0466325\pi$
$839$			244.949i			0.291953i	−0.989288	+	0.145977i	$0.953368\pi$
$840$		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	0.269031	−	0.963131i	$-0.413297\pi$
$857$	1060.66	−	612.372i	1.23764	−	0.714554i	0.968612	+	0.248578i	$0.0799632\pi$
$858$		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	−0.608094	−	0.793865i	$-0.708066\pi$
$863$	330.926	−	573.181i	0.383460	−	0.664172i	0.991554	+	0.129693i	$0.0413991\pi$
$864$		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	$-0.970753\pi$
$881$		−	161.666i		−	0.183503i	0.995782	−	0.0917516i	$-0.0292466\pi$
$882$		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	0.592603	−	0.805495i	$-0.298100\pi$
$887$	169.706	+	97.9796i	0.191325	+	0.110462i	−0.401277	+	0.915957i	$0.631434\pi$
$888$		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	$-0.259978\pi$
$911$	1247.34			1.36919			0.684597	+	0.728921i	$0.259978\pi$
$912$		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	−0.104772	−	0.994496i	$-0.533411\pi$
$929$	−946.109	−	546.236i	−1.01842	−	0.587983i	−0.913645	+	0.406513i	$0.866744\pi$
$930$		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	−0.946601	−	0.322407i	$-0.895508\pi$
$941$	−1073.39	+	619.721i	−1.14069	+	0.658577i	−0.194088	+	0.980984i	$0.562175\pi$
$942$		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	$0.386150\pi$
$947$	−602.455	+	1043.48i	−0.636172	+	1.10188i	−0.986266	+	0.165168i	$0.947183\pi$
$948$		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	$-0.681076\pi$
$953$	−1026.72			−1.07735			−0.538677	+	0.842512i	$0.681076\pi$
$954$		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	0.403109	−	0.915152i	$-0.367930\pi$
$971$	−182.434	−	105.328i	−0.187882	−	0.108474i	−0.590991	+	0.806678i	$0.701263\pi$
$972$		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	$-0.944484\pi$
$977$	−627.911	−	1087.57i	−0.642693	−	1.11318i	0.342136	−	0.939650i	$-0.388849\pi$
$978$		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	0.120281	−	0.992740i	$-0.461621\pi$
$983$	1022.48	−	590.327i	1.04016	−	0.600536i	0.919878	+	0.392204i	$0.128287\pi$
$984$		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.145.1		4
3.2	odd	2	inner	1008.3.cg.i.145.2		4
4.3	odd	2		126.3.n.b.19.2	yes	4
7.3	odd	6	inner	1008.3.cg.i.577.1		4
12.11	even	2		126.3.n.b.19.1	✓	4
21.17	even	6	inner	1008.3.cg.i.577.2		4
28.3	even	6		126.3.n.b.73.2	yes	4
28.11	odd	6		882.3.n.c.325.2		4
28.19	even	6		882.3.c.c.685.1		4
28.23	odd	6		882.3.c.c.685.2		4
28.27	even	2		882.3.n.c.19.2		4
84.11	even	6		882.3.n.c.325.1		4
84.23	even	6		882.3.c.c.685.3		4
84.47	odd	6		882.3.c.c.685.4		4
84.59	odd	6		126.3.n.b.73.1	yes	4
84.83	odd	2		882.3.n.c.19.1		4

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

Introduction

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