LMFDB - Embedded newform 672.2.b.a.223.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(223,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("672.223");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.b (of order $2$, degree $1$, 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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.836829184.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 $ x^8 + 14x^6 + 61x^4 + 84*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.7
Root			$2.63640i$ of defining polynomial
Character	$\chi$	$=$	672.223
Dual form			672.2.b.a.223.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.31423i q^{5} +(0.222191 + 2.63640i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.31423i q^{5} +(0.222191 + 2.63640i) q^{7} +1.00000 q^{9} -3.58704i q^{11} +2.82843i q^{13} -2.31423i q^{15} +2.31423i q^{17} -7.90126 q^{19} +(-0.222191 - 2.63640i) q^{21} +0.130157i q^{23} -0.355642 q^{25} -1.00000 q^{27} +0.199975 q^{29} -3.45688 q^{31} +3.58704i q^{33} +(-6.10124 + 0.514201i) q^{35} -11.5581 q^{37} -2.82843i q^{39} +7.97108i q^{41} +4.38404i q^{43} +2.31423i q^{45} +6.54562 q^{47} +(-6.90126 + 1.17157i) q^{49} -2.31423i q^{51} +10.3456 q^{53} +8.30121 q^{55} +7.90126 q^{57} -9.65685 q^{59} +6.19998i q^{61} +(0.222191 + 2.63640i) q^{63} -6.54562 q^{65} +9.01250i q^{67} -0.130157i q^{69} +4.75861i q^{71} -10.2853i q^{73} +0.355642 q^{75} +(9.45688 - 0.797008i) q^{77} +4.24441i q^{79} +1.00000 q^{81} +0.768089 q^{83} -5.35564 q^{85} -0.199975 q^{87} -17.7486i q^{89} +(-7.45688 + 0.628452i) q^{91} +3.45688 q^{93} -18.2853i q^{95} +9.02840i q^{97} -3.58704i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 - 4 * q^7 + 8 * q^9	$ 8 q - 8 q^{3} - 4 q^{7} + 8 q^{9} - 8 q^{19} + 4 q^{21} - 16 q^{25} - 8 q^{27} + 16 q^{31} + 8 q^{35} + 8 q^{37} - 16 q^{47} + 16 q^{53} + 8 q^{55} + 8 q^{57} - 32 q^{59} - 4 q^{63} + 16 q^{65} + 16 q^{75} + 32 q^{77} + 8 q^{81} - 16 q^{83} - 56 q^{85} - 16 q^{91} - 16 q^{93}+O(q^{100}) $ 8 * q - 8 * q^3 - 4 * q^7 + 8 * q^9 - 8 * q^19 + 4 * q^21 - 16 * q^25 - 8 * q^27 + 16 * q^31 + 8 * q^35 + 8 * q^37 - 16 * q^47 + 16 * q^53 + 8 * q^55 + 8 * q^57 - 32 * q^59 - 4 * q^63 + 16 * q^65 + 16 * q^75 + 32 * q^77 + 8 * q^81 - 16 * q^83 - 56 * q^85 - 16 * q^91 - 16 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$			2.31423i			1.03495i	0.855697	+	0.517477i	$0.173129\pi$
$5$			2.31423i			1.03495i	−0.855697	+	0.517477i	$0.826871\pi$
$6$		0			0
$7$	0.222191	+	2.63640i	0.0839804	+	0.996467i
$7$	0.222191	+	2.63640i	0.0839804	+	0.996467i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	3.58704i		−	1.08153i	−0.841173	−	0.540766i	$-0.818134\pi$
$11$		−	3.58704i		−	1.08153i	0.841173	−	0.540766i	$-0.181866\pi$
$12$		0			0
$13$			2.82843i			0.784465i	0.919866	+	0.392232i	$0.128297\pi$
$13$			2.82843i			0.784465i	−0.919866	+	0.392232i	$0.871703\pi$
$14$		0			0
$15$		−	2.31423i		−	0.597531i
$16$		0			0
$17$			2.31423i			0.561282i	0.959813	+	0.280641i	$0.0905471\pi$
$17$			2.31423i			0.561282i	−0.959813	+	0.280641i	$0.909453\pi$
$18$		0			0
$19$	−7.90126			−1.81267			−0.906337	−	0.422556i	$-0.861133\pi$
$19$	−7.90126			−1.81267			−0.906337	+	0.422556i	$0.861133\pi$
$20$		0			0
$21$	−0.222191	−	2.63640i	−0.0484861	−	0.575311i
$22$		0			0
$23$			0.130157i			0.0271395i	0.999908	+	0.0135698i	$0.00431953\pi$
$23$			0.130157i			0.0271395i	−0.999908	+	0.0135698i	$0.995680\pi$
$24$		0			0
$25$	−0.355642			−0.0711284
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	0.199975			0.0371344			0.0185672	−	0.999828i	$-0.494090\pi$
$29$	0.199975			0.0371344			0.0185672	+	0.999828i	$0.494090\pi$
$30$		0			0
$31$	−3.45688			−0.620874			−0.310437	−	0.950594i	$-0.600475\pi$
$31$	−3.45688			−0.620874			−0.310437	+	0.950594i	$0.600475\pi$
$32$		0			0
$33$			3.58704i			0.624423i
$34$		0			0
$35$	−6.10124	+	0.514201i	−1.03130	+	0.0869158i
$36$		0			0
$37$	−11.5581			−1.90014			−0.950071	−	0.312033i	$-0.898990\pi$
$37$	−11.5581			−1.90014			−0.950071	+	0.312033i	$0.898990\pi$
$38$		0			0
$39$		−	2.82843i		−	0.452911i
$40$		0			0
$41$			7.97108i			1.24487i	0.782670	+	0.622437i	$0.213857\pi$
$41$			7.97108i			1.24487i	−0.782670	+	0.622437i	$0.786143\pi$
$42$		0			0
$43$			4.38404i			0.668560i	0.942474	+	0.334280i	$0.108493\pi$
$43$			4.38404i			0.668560i	−0.942474	+	0.334280i	$0.891507\pi$
$44$		0			0
$45$			2.31423i			0.344984i
$46$		0			0
$47$	6.54562			0.954777			0.477388	−	0.878692i	$-0.341583\pi$
$47$	6.54562			0.954777			0.477388	+	0.878692i	$0.341583\pi$
$48$		0			0
$49$	−6.90126	+	1.17157i	−0.985895	+	0.167368i
$50$		0			0
$51$		−	2.31423i		−	0.324056i
$52$		0			0
$53$	10.3456			1.42108			0.710542	−	0.703655i	$-0.248450\pi$
$53$	10.3456			1.42108			0.710542	+	0.703655i	$0.248450\pi$
$54$		0			0
$55$	8.30121			1.11934
$56$		0			0
$57$	7.90126			1.04655
$58$		0			0
$59$	−9.65685			−1.25722			−0.628608	−	0.777723i	$-0.716375\pi$
$59$	−9.65685			−1.25722			−0.628608	+	0.777723i	$0.716375\pi$
$60$		0			0
$61$			6.19998i			0.793825i	0.917856	+	0.396913i	$0.129918\pi$
$61$			6.19998i			0.793825i	−0.917856	+	0.396913i	$0.870082\pi$
$62$		0			0
$63$	0.222191	+	2.63640i	0.0279935	+	0.332156i
$64$		0			0
$65$	−6.54562			−0.811884
$66$		0			0
$67$			9.01250i			1.10105i	0.834818	+	0.550526i	$0.185573\pi$
$67$			9.01250i			1.10105i	−0.834818	+	0.550526i	$0.814427\pi$
$68$		0			0
$69$		−	0.130157i		−	0.0156690i
$70$		0			0
$71$			4.75861i			0.564743i	0.959305	+	0.282371i	$0.0911211\pi$
$71$			4.75861i			0.564743i	−0.959305	+	0.282371i	$0.908879\pi$
$72$		0			0
$73$		−	10.2853i		−	1.20380i	−0.798570	−	0.601902i	$-0.794410\pi$
$73$		−	10.2853i		−	1.20380i	0.798570	−	0.601902i	$-0.205590\pi$
$74$		0			0
$75$	0.355642			0.0410660
$76$		0			0
$77$	9.45688	−	0.797008i	1.07771	−	0.0908275i
$78$		0			0
$79$			4.24441i			0.477533i	0.971077	+	0.238767i	$0.0767431\pi$
$79$			4.24441i			0.477533i	−0.971077	+	0.238767i	$0.923257\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	0.768089			0.0843087			0.0421543	−	0.999111i	$-0.486578\pi$
$83$	0.768089			0.0843087			0.0421543	+	0.999111i	$0.486578\pi$
$84$		0			0
$85$	−5.35564			−0.580901
$86$		0			0
$87$	−0.199975			−0.0214396
$88$		0			0
$89$		−	17.7486i		−	1.88135i	−0.339310	−	0.940675i	$-0.610194\pi$
$89$		−	17.7486i		−	1.88135i	0.339310	−	0.940675i	$-0.389806\pi$
$90$		0			0
$91$	−7.45688	+	0.628452i	−0.781693	+	0.0658797i
$92$		0			0
$93$	3.45688			0.358462
$94$		0			0
$95$		−	18.2853i		−	1.87603i
$96$		0			0
$97$			9.02840i			0.916695i	0.888773	+	0.458348i	$0.151559\pi$
$97$			9.02840i			0.916695i	−0.888773	+	0.458348i	$0.848441\pi$
$98$		0			0
$99$		−	3.58704i		−	0.360511i
$100$		0			0
$101$		−	13.6279i		−	1.35603i	−0.735048	−	0.678015i	$-0.762840\pi$
$101$		−	13.6279i		−	1.35603i	0.735048	−	0.678015i	$-0.237160\pi$
$102$		0			0
$103$	11.4569			1.12888			0.564440	−	0.825474i	$-0.309092\pi$
$103$	11.4569			1.12888			0.564440	+	0.825474i	$0.309092\pi$
$104$		0			0
$105$	6.10124	−	0.514201i	0.595420	−	0.0501809i
$106$		0			0
$107$		−	6.95858i		−	0.672712i	−0.941735	−	0.336356i	$-0.890806\pi$
$107$		−	6.95858i		−	0.672712i	0.941735	−	0.336356i	$-0.109194\pi$
$108$		0			0
$109$	−2.88877			−0.276694			−0.138347	−	0.990384i	$-0.544179\pi$
$109$	−2.88877			−0.276694			−0.138347	+	0.990384i	$0.544179\pi$
$110$		0			0
$111$	11.5581			1.09705
$112$		0			0
$113$	2.36814			0.222776			0.111388	−	0.993777i	$-0.464470\pi$
$113$	2.36814			0.222776			0.111388	+	0.993777i	$0.464470\pi$
$114$		0			0
$115$	−0.301212			−0.0280882
$116$		0			0
$117$			2.82843i			0.261488i
$118$		0			0
$119$	−6.10124	+	0.514201i	−0.559299	+	0.0471367i
$120$		0			0
$121$	−1.86683			−0.169712
$122$		0			0
$123$		−	7.97108i		−	0.718728i
$124$		0			0
$125$			10.7481i			0.961339i
$126$		0			0
$127$			7.55812i			0.670674i	0.942098	+	0.335337i	$0.108850\pi$
$127$			7.55812i			0.670674i	−0.942098	+	0.335337i	$0.891150\pi$
$128$		0			0
$129$		−	4.38404i		−	0.385994i
$130$		0			0
$131$	17.4594			1.52543			0.762716	−	0.646733i	$-0.223865\pi$
$131$	17.4594			1.52543			0.762716	+	0.646733i	$0.223865\pi$
$132$		0			0
$133$	−1.75559	−	20.8309i	−0.152229	−	1.80627i
$134$		0			0
$135$		−	2.31423i		−	0.199177i
$136$		0			0
$137$	−18.2025			−1.55514			−0.777571	−	0.628795i	$-0.783548\pi$
$137$	−18.2025			−1.55514			−0.777571	+	0.628795i	$0.783548\pi$
$138$		0			0
$139$	15.3137			1.29889			0.649446	−	0.760408i	$-0.275001\pi$
$139$	15.3137			1.29889			0.649446	+	0.760408i	$0.275001\pi$
$140$		0			0
$141$	−6.54562			−0.551241
$142$		0			0
$143$	10.1457			0.848424
$144$		0			0
$145$			0.462787i			0.0384324i
$146$		0			0
$147$	6.90126	−	1.17157i	0.569206	−	0.0966297i
$148$		0			0
$149$	22.5481			1.84721			0.923607	−	0.383341i	$-0.125227\pi$
$149$	22.5481			1.84721			0.923607	+	0.383341i	$0.125227\pi$
$150$		0			0
$151$			17.8434i			1.45208i	0.687654	+	0.726039i	$0.258641\pi$
$151$			17.8434i			1.45208i	−0.687654	+	0.726039i	$0.741359\pi$
$152$		0			0
$153$			2.31423i			0.187094i
$154$		0			0
$155$		−	8.00000i		−	0.642575i
$156$		0			0
$157$		−	5.11373i		−	0.408120i	−0.978958	−	0.204060i	$-0.934586\pi$
$157$		−	5.11373i		−	0.408120i	0.978958	−	0.204060i	$-0.0654138\pi$
$158$		0			0
$159$	−10.3456			−0.820463
$160$		0			0
$161$	−0.343146	+	0.0289197i	−0.0270437	+	0.00227919i
$162$		0			0
$163$		−	10.3012i		−	0.806853i	−0.915012	−	0.403427i	$-0.867819\pi$
$163$		−	10.3012i		−	0.806853i	0.915012	−	0.403427i	$-0.132181\pi$
$164$		0			0
$165$	−8.30121			−0.646248
$166$		0			0
$167$	14.0250			1.08529			0.542643	−	0.839963i	$-0.317424\pi$
$167$	14.0250			1.08529			0.542643	+	0.839963i	$0.317424\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$	−7.90126			−0.604225
$172$		0			0
$173$			8.02892i			0.610427i	0.952284	+	0.305214i	$0.0987279\pi$
$173$			8.02892i			0.610427i	−0.952284	+	0.305214i	$0.901272\pi$
$174$		0			0
$175$	−0.0790206	−	0.937617i	−0.00597340	−	0.0708772i
$176$		0			0
$177$	9.65685			0.725854
$178$		0			0
$179$			21.1043i			1.57741i	0.614774	+	0.788703i	$0.289247\pi$
$179$			21.1043i			1.57741i	−0.614774	+	0.788703i	$0.710753\pi$
$180$		0			0
$181$		−	16.2060i		−	1.20458i	−0.798276	−	0.602292i	$-0.794254\pi$
$181$		−	16.2060i		−	1.20458i	0.798276	−	0.602292i	$-0.205746\pi$
$182$		0			0
$183$		−	6.19998i		−	0.458315i
$184$		0			0
$185$		−	26.7481i		−	1.96656i
$186$		0			0
$187$	8.30121			0.607045
$188$		0			0
$189$	−0.222191	−	2.63640i	−0.0161620	−	0.191770i
$190$		0			0
$191$			7.60953i			0.550606i	0.961357	+	0.275303i	$0.0887782\pi$
$191$			7.60953i			0.550606i	−0.961357	+	0.275303i	$0.911222\pi$
$192$		0			0
$193$	10.6694			0.767997			0.383998	−	0.923334i	$-0.374547\pi$
$193$	10.6694			0.767997			0.383998	+	0.923334i	$0.374547\pi$
$194$		0			0
$195$	6.54562			0.468742
$196$		0			0
$197$	4.68879			0.334062			0.167031	−	0.985952i	$-0.446582\pi$
$197$	4.68879			0.334062			0.167031	+	0.985952i	$0.446582\pi$
$198$		0			0
$199$	−22.4694			−1.59281			−0.796406	−	0.604762i	$-0.793268\pi$
$199$	−22.4694			−1.59281			−0.796406	+	0.604762i	$0.793268\pi$
$200$		0			0
$201$		−	9.01250i		−	0.635692i
$202$		0			0
$203$	0.0444327	+	0.527215i	0.00311857	+	0.0370032i
$204$		0			0
$205$	−18.4469			−1.28839
$206$		0			0
$207$			0.130157i			0.00904652i
$208$		0			0
$209$			28.3421i			1.96046i
$210$		0			0
$211$			15.4185i			1.06145i	0.847543	+	0.530726i	$0.178081\pi$
$211$			15.4185i			1.06145i	−0.847543	+	0.530726i	$0.821919\pi$
$212$		0			0
$213$		−	4.75861i		−	0.326055i
$214$		0			0
$215$	−10.1457			−0.691929
$216$		0			0
$217$	−0.768089	−	9.11373i	−0.0521413	−	0.618681i
$218$		0			0
$219$			10.2853i			0.695017i
$220$		0			0
$221$	−6.54562			−0.440306
$222$		0			0
$223$	2.86933			0.192144			0.0960721	−	0.995374i	$-0.469372\pi$
$223$	2.86933			0.192144			0.0960721	+	0.995374i	$0.469372\pi$
$224$		0			0
$225$	−0.355642			−0.0237095
$226$		0			0
$227$	−2.91376			−0.193393			−0.0966965	−	0.995314i	$-0.530828\pi$
$227$	−2.91376			−0.193393			−0.0966965	+	0.995314i	$0.530828\pi$
$228$		0			0
$229$			20.0853i			1.32728i	0.748054	+	0.663638i	$0.230988\pi$
$229$			20.0853i			1.32728i	−0.748054	+	0.663638i	$0.769012\pi$
$230$		0			0
$231$	−9.45688	+	0.797008i	−0.622217	+	0.0524393i
$232$		0			0
$233$	−0.343146			−0.0224802			−0.0112401	−	0.999937i	$-0.503578\pi$
$233$	−0.343146			−0.0224802			−0.0112401	+	0.999937i	$0.503578\pi$
$234$		0			0
$235$			15.1480i			0.988149i
$236$		0			0
$237$		−	4.24441i		−	0.275704i
$238$		0			0
$239$		−	8.56113i		−	0.553774i	−0.960903	−	0.276887i	$-0.910697\pi$
$239$		−	8.56113i		−	0.553774i	0.960903	−	0.276887i	$-0.0893027\pi$
$240$		0			0
$241$			6.40598i			0.412646i	0.978484	+	0.206323i	$0.0661497\pi$
$241$			6.40598i			0.412646i	−0.978484	+	0.206323i	$0.933850\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	−2.71128	−	15.9711i	−0.173218	−	1.02035i
$246$		0			0
$247$		−	22.3481i		−	1.42198i
$248$		0			0
$249$	−0.768089			−0.0486756
$250$		0			0
$251$	24.2025			1.52765			0.763823	−	0.645425i	$-0.223320\pi$
$251$	24.2025			1.52765			0.763823	+	0.645425i	$0.223320\pi$
$252$		0			0
$253$	0.466877			0.0293523
$254$		0			0
$255$	5.35564			0.335383
$256$		0			0
$257$			1.22798i			0.0765996i	0.999266	+	0.0382998i	$0.0121942\pi$
$257$			1.22798i			0.0765996i	−0.999266	+	0.0382998i	$0.987806\pi$
$258$		0			0
$259$	−2.56811	−	30.4719i	−0.159575	−	1.89343i
$260$		0			0
$261$	0.199975			0.0123781
$262$		0			0
$263$			6.01551i			0.370932i	0.982651	+	0.185466i	$0.0593795\pi$
$263$			6.01551i			0.370932i	−0.982651	+	0.185466i	$0.940620\pi$
$264$		0			0
$265$			23.9422i			1.47076i
$266$		0			0
$267$			17.7486i			1.08620i
$268$		0			0
$269$			22.3771i			1.36435i	0.731187	+	0.682177i	$0.238967\pi$
$269$			22.3771i			1.36435i	−0.731187	+	0.682177i	$0.761033\pi$
$270$		0			0
$271$	−9.63436			−0.585246			−0.292623	−	0.956228i	$-0.594528\pi$
$271$	−9.63436			−0.585246			−0.292623	+	0.956228i	$0.594528\pi$
$272$		0			0
$273$	7.45688	−	0.628452i	0.451311	−	0.0380357i
$274$		0			0
$275$			1.27570i			0.0769277i
$276$		0			0
$277$	11.9581			0.718491			0.359245	−	0.933243i	$-0.383034\pi$
$277$	11.9581			0.718491			0.359245	+	0.933243i	$0.383034\pi$
$278$		0			0
$279$	−3.45688			−0.206958
$280$		0			0
$281$	−17.6818			−1.05481			−0.527405	−	0.849614i	$-0.676835\pi$
$281$	−17.6818			−1.05481			−0.527405	+	0.849614i	$0.676835\pi$
$282$		0			0
$283$	3.50131			0.208131			0.104066	−	0.994570i	$-0.466815\pi$
$283$	3.50131			0.208131			0.104066	+	0.994570i	$0.466815\pi$
$284$		0			0
$285$			18.2853i			1.08313i
$286$		0			0
$287$	−21.0150	+	1.77111i	−1.24048	+	0.104545i
$288$		0			0
$289$	11.6444			0.684962
$290$		0			0
$291$		−	9.02840i		−	0.529254i
$292$		0			0
$293$		−	3.40047i		−	0.198657i	−0.995055	−	0.0993287i	$-0.968330\pi$
$293$		−	3.40047i		−	0.198657i	0.995055	−	0.0993287i	$-0.0316695\pi$
$294$		0			0
$295$		−	22.3481i		−	1.30116i
$296$		0			0
$297$			3.58704i			0.208141i
$298$		0			0
$299$	−0.368139			−0.0212900
$300$		0			0
$301$	−11.5581	+	0.974097i	−0.666199	+	0.0561460i
$302$		0			0
$303$			13.6279i			0.782904i
$304$		0			0
$305$	−14.3481			−0.821572
$306$		0			0
$307$	12.1037			0.690797			0.345398	−	0.938456i	$-0.387744\pi$
$307$	12.1037			0.690797			0.345398	+	0.938456i	$0.387744\pi$
$308$		0			0
$309$	−11.4569			−0.651759
$310$		0			0
$311$	−26.7481			−1.51675			−0.758373	−	0.651821i	$-0.774005\pi$
$311$	−26.7481			−1.51675			−0.758373	+	0.651821i	$0.774005\pi$
$312$		0			0
$313$		−	27.8343i		−	1.57329i	−0.617406	−	0.786645i	$-0.711816\pi$
$313$		−	27.8343i		−	1.57329i	0.617406	−	0.786645i	$-0.288184\pi$
$314$		0			0
$315$	−6.10124	+	0.514201i	−0.343766	+	0.0289719i
$316$		0			0
$317$	5.85683			0.328952			0.164476	−	0.986381i	$-0.447407\pi$
$317$	5.85683			0.328952			0.164476	+	0.986381i	$0.447407\pi$
$318$		0			0
$319$		−	0.717318i		−	0.0401621i
$320$		0			0
$321$			6.95858i			0.388390i
$322$		0			0
$323$		−	18.2853i		−	1.01742i
$324$		0			0
$325$		−	1.00591i		−	0.0557977i
$326$		0			0
$327$	2.88877			0.159749
$328$		0			0
$329$	1.45438	+	17.2569i	0.0801826	+	0.951404i
$330$		0			0
$331$			2.84787i			0.156533i	0.996932	+	0.0782665i	$0.0249385\pi$
$331$			2.84787i			0.156533i	−0.996932	+	0.0782665i	$0.975062\pi$
$332$		0			0
$333$	−11.5581			−0.633381
$334$		0			0
$335$	−20.8570			−1.13954
$336$		0			0
$337$	−10.4888			−0.571362			−0.285681	−	0.958325i	$-0.592220\pi$
$337$	−10.4888			−0.571362			−0.285681	+	0.958325i	$0.592220\pi$
$338$		0			0
$339$	−2.36814			−0.128620
$340$		0			0
$341$			12.4000i			0.671495i
$342$		0			0
$343$	−4.62214	−	17.9342i	−0.249572	−	0.968356i
$344$		0			0
$345$	0.301212			0.0162167
$346$		0			0
$347$		−	21.8145i		−	1.17106i	−0.810649	−	0.585532i	$-0.800886\pi$
$347$		−	21.8145i		−	1.17106i	0.810649	−	0.585532i	$-0.199114\pi$
$348$		0			0
$349$			28.7188i			1.53728i	0.639681	+	0.768641i	$0.279066\pi$
$349$			28.7188i			1.53728i	−0.639681	+	0.768641i	$0.720934\pi$
$350$		0			0
$351$		−	2.82843i		−	0.150970i
$352$		0			0
$353$		−	5.91428i		−	0.314785i	−0.987536	−	0.157393i	$-0.949691\pi$
$353$		−	5.91428i		−	0.314785i	0.987536	−	0.157393i	$-0.0503088\pi$
$354$		0			0
$355$	−11.0125			−0.584483
$356$		0			0
$357$	6.10124	−	0.514201i	0.322912	−	0.0272144i
$358$		0			0
$359$			20.4723i			1.08048i	0.841509	+	0.540242i	$0.181667\pi$
$359$			20.4723i			1.08048i	−0.841509	+	0.540242i	$0.818333\pi$
$360$		0			0
$361$	43.4299			2.28579
$362$		0			0
$363$	1.86683			0.0979830
$364$		0			0
$365$	23.8025			1.24588
$366$		0			0
$367$	−28.4494			−1.48505			−0.742523	−	0.669821i	$-0.766371\pi$
$367$	−28.4494			−1.48505			−0.742523	+	0.669821i	$0.766371\pi$
$368$		0			0
$369$			7.97108i			0.414958i
$370$		0			0
$371$	2.29871	+	27.2753i	0.119343	+	1.41606i
$372$		0			0
$373$	26.2913			1.36131			0.680657	−	0.732602i	$-0.261694\pi$
$373$	26.2913			1.36131			0.680657	+	0.732602i	$0.261694\pi$
$374$		0			0
$375$		−	10.7481i		−	0.555029i
$376$		0			0
$377$			0.565615i			0.0291306i
$378$		0			0
$379$		−	9.44347i		−	0.485079i	−0.970142	−	0.242539i	$-0.922020\pi$
$379$		−	9.44347i		−	0.485079i	0.970142	−	0.242539i	$-0.0779803\pi$
$380$		0			0
$381$		−	7.55812i		−	0.387214i
$382$		0			0
$383$	2.14567			0.109639			0.0548193	−	0.998496i	$-0.482542\pi$
$383$	2.14567			0.109639			0.0548193	+	0.998496i	$0.482542\pi$
$384$		0			0
$385$	1.84446	+	21.8854i	0.0940023	+	1.11538i
$386$		0			0
$387$			4.38404i			0.222853i
$388$		0			0
$389$	−20.7388			−1.05150			−0.525749	−	0.850640i	$-0.676215\pi$
$389$	−20.7388			−1.05150			−0.525749	+	0.850640i	$0.676215\pi$
$390$		0			0
$391$	−0.301212			−0.0152329
$392$		0			0
$393$	−17.4594			−0.880709
$394$		0			0
$395$	−9.82252			−0.494225
$396$		0			0
$397$			19.8568i			0.996586i	0.867009	+	0.498293i	$0.166040\pi$
$397$			19.8568i			0.996586i	−0.867009	+	0.498293i	$0.833960\pi$
$398$		0			0
$399$	1.75559	+	20.8309i	0.0878895	+	1.04285i
$400$		0			0
$401$	12.4249			0.620472			0.310236	−	0.950660i	$-0.399592\pi$
$401$	12.4249			0.620472			0.310236	+	0.950660i	$0.399592\pi$
$402$		0			0
$403$		−	9.77753i		−	0.487054i
$404$		0			0
$405$			2.31423i			0.114995i
$406$		0			0
$407$			41.4594i			2.05507i
$408$		0			0
$409$		−	9.54903i		−	0.472169i	−0.971733	−	0.236085i	$-0.924136\pi$
$409$		−	9.54903i		−	0.472169i	0.971733	−	0.236085i	$-0.0758642\pi$
$410$		0			0
$411$	18.2025			0.897862
$412$		0			0
$413$	−2.14567	−	25.4594i	−0.105582	−	1.25277i
$414$		0			0
$415$			1.77753i			0.0872556i
$416$		0			0
$417$	−15.3137			−0.749916
$418$		0			0
$419$	−21.2569			−1.03847			−0.519234	−	0.854632i	$-0.673783\pi$
$419$	−21.2569			−1.03847			−0.519234	+	0.854632i	$0.673783\pi$
$420$		0			0
$421$	19.1582			0.933712			0.466856	−	0.884333i	$-0.345387\pi$
$421$	19.1582			0.933712			0.466856	+	0.884333i	$0.345387\pi$
$422$		0			0
$423$	6.54562			0.318259
$424$		0			0
$425$		−	0.823037i		−	0.0399231i
$426$		0			0
$427$	−16.3456	+	1.37758i	−0.791021	+	0.0666658i
$428$		0			0
$429$	−10.1457			−0.489838
$430$		0			0
$431$		−	23.3042i		−	1.12253i	−0.827638	−	0.561263i	$-0.810316\pi$
$431$		−	23.3042i		−	1.12253i	0.827638	−	0.561263i	$-0.189684\pi$
$432$		0			0
$433$			33.9482i			1.63145i	0.578443	+	0.815723i	$0.303661\pi$
$433$			33.9482i			1.63145i	−0.578443	+	0.815723i	$0.696339\pi$
$434$		0			0
$435$		−	0.462787i		−	0.0221890i
$436$		0			0
$437$		−	1.02840i		−	0.0491951i
$438$		0			0
$439$	9.13567			0.436022			0.218011	−	0.975946i	$-0.430043\pi$
$439$	9.13567			0.436022			0.218011	+	0.975946i	$0.430043\pi$
$440$		0			0
$441$	−6.90126	+	1.17157i	−0.328632	+	0.0557892i
$442$		0			0
$443$			28.5258i			1.35530i	0.735384	+	0.677651i	$0.237002\pi$
$443$			28.5258i			1.35530i	−0.735384	+	0.677651i	$0.762998\pi$
$444$		0			0
$445$	41.0743			1.94711
$446$		0			0
$447$	−22.5481			−1.06649
$448$		0			0
$449$	−34.3731			−1.62217			−0.811084	−	0.584929i	$-0.801122\pi$
$449$	−34.3731			−1.62217			−0.811084	+	0.584929i	$0.801122\pi$
$450$		0			0
$451$	28.5926			1.34637
$452$		0			0
$453$		−	17.8434i		−	0.838357i
$454$		0			0
$455$	−1.45438	−	17.2569i	−0.0681824	−	0.809016i
$456$		0			0
$457$	−34.8938			−1.63226			−0.816131	−	0.577867i	$-0.803885\pi$
$457$	−34.8938			−1.63226			−0.816131	+	0.577867i	$0.803885\pi$
$458$		0			0
$459$		−	2.31423i		−	0.108019i
$460$		0			0
$461$		−	21.2908i		−	0.991612i	−0.868433	−	0.495806i	$-0.834873\pi$
$461$		−	21.2908i		−	0.991612i	0.868433	−	0.495806i	$-0.165127\pi$
$462$		0			0
$463$			1.33565i			0.0620728i	0.999518	+	0.0310364i	$0.00988078\pi$
$463$			1.33565i			0.0620728i	−0.999518	+	0.0310364i	$0.990119\pi$
$464$		0			0
$465$			8.00000i			0.370991i
$466$		0			0
$467$	−9.98000			−0.461820			−0.230910	−	0.972975i	$-0.574170\pi$
$467$	−9.98000			−0.461820			−0.230910	+	0.972975i	$0.574170\pi$
$468$		0			0
$469$	−23.7606	+	2.00250i	−1.09716	+	0.0924668i
$470$		0			0
$471$			5.11373i			0.235628i
$472$		0			0
$473$	15.7257			0.723070
$474$		0			0
$475$	2.81002			0.128933
$476$		0			0
$477$	10.3456			0.473695
$478$		0			0
$479$	3.42939			0.156693			0.0783464	−	0.996926i	$-0.475036\pi$
$479$	3.42939			0.156693			0.0783464	+	0.996926i	$0.475036\pi$
$480$		0			0
$481$		−	32.6913i		−	1.49059i
$482$		0			0
$483$	0.343146	−	0.0289197i	0.0156137	−	0.00131589i
$484$		0			0
$485$	−20.8938			−0.948737
$486$		0			0
$487$			20.7522i			0.940371i	0.882568	+	0.470186i	$0.155813\pi$
$487$			20.7522i			0.940371i	−0.882568	+	0.470186i	$0.844187\pi$
$488$		0			0
$489$			10.3012i			0.465837i
$490$		0			0
$491$		−	36.6154i		−	1.65243i	−0.563354	−	0.826216i	$-0.690489\pi$
$491$		−	36.6154i		−	1.65243i	0.563354	−	0.826216i	$-0.309511\pi$
$492$		0			0
$493$			0.462787i			0.0208429i
$494$		0			0
$495$	8.30121			0.373112
$496$		0			0
$497$	−12.5456	+	1.05732i	−0.562748	+	0.0474274i
$498$		0			0
$499$		−	34.7004i		−	1.55340i	−0.629869	−	0.776701i	$-0.716892\pi$
$499$		−	34.7004i		−	1.55340i	0.629869	−	0.776701i	$-0.283108\pi$
$500$		0			0
$501$	−14.0250			−0.626590
$502$		0			0
$503$	41.8593			1.86642			0.933208	−	0.359338i	$-0.116997\pi$
$503$	41.8593			1.86642			0.933208	+	0.359338i	$0.116997\pi$
$504$		0			0
$505$	31.5381			1.40343
$506$		0			0
$507$	−5.00000			−0.222058
$508$		0			0
$509$			37.2908i			1.65289i	0.563020	+	0.826443i	$0.309639\pi$
$509$			37.2908i			1.65289i	−0.563020	+	0.826443i	$0.690361\pi$
$510$		0			0
$511$	27.1162	−	2.28531i	1.19955	−	0.101096i
$512$		0			0
$513$	7.90126			0.348849
$514$		0			0
$515$			26.5138i			1.16834i
$516$		0			0
$517$		−	23.4794i		−	1.03262i
$518$		0			0
$519$		−	8.02892i		−	0.352430i
$520$		0			0
$521$			7.46330i			0.326973i	0.986546	+	0.163487i	$0.0522741\pi$
$521$			7.46330i			0.326973i	−0.986546	+	0.163487i	$0.947726\pi$
$522$		0			0
$523$	−2.17748			−0.0952146			−0.0476073	−	0.998866i	$-0.515160\pi$
$523$	−2.17748			−0.0952146			−0.0476073	+	0.998866i	$0.515160\pi$
$524$		0			0
$525$	0.0790206	+	0.937617i	0.00344874	+	0.0409210i
$526$		0			0
$527$		−	8.00000i		−	0.348485i
$528$		0			0
$529$	22.9831			0.999263
$530$		0			0
$531$	−9.65685			−0.419072
$532$		0			0
$533$	−22.5456			−0.976559
$534$		0			0
$535$	16.1037			0.696225
$536$		0			0
$537$		−	21.1043i		−	0.910716i
$538$		0			0
$539$	4.20247	+	24.7551i	0.181013	+	1.06628i
$540$		0			0
$541$	−16.1555			−0.694581			−0.347291	−	0.937758i	$-0.612898\pi$
$541$	−16.1555			−0.694581			−0.347291	+	0.937758i	$0.612898\pi$
$542$		0			0
$543$			16.2060i			0.695466i
$544$		0			0
$545$		−	6.68526i		−	0.286365i
$546$		0			0
$547$			3.06930i			0.131234i	0.997845	+	0.0656169i	$0.0209015\pi$
$547$			3.06930i			0.131234i	−0.997845	+	0.0656169i	$0.979098\pi$
$548$		0			0
$549$			6.19998i			0.264608i
$550$		0			0
$551$	−1.58005			−0.0673126
$552$		0			0
$553$	−11.1900	+	0.943071i	−0.475846	+	0.0401034i
$554$		0			0
$555$			26.7481i			1.13539i
$556$		0			0
$557$	−12.7656			−0.540895			−0.270448	−	0.962735i	$-0.587172\pi$
$557$	−12.7656			−0.540895			−0.270448	+	0.962735i	$0.587172\pi$
$558$		0			0
$559$	−12.4000			−0.524462
$560$		0			0
$561$	−8.30121			−0.350477
$562$		0			0
$563$	17.5044			0.737721			0.368861	−	0.929485i	$-0.379748\pi$
$563$	17.5044			0.737721			0.368861	+	0.929485i	$0.379748\pi$
$564$		0			0
$565$			5.48041i			0.230563i
$566$		0			0
$567$	0.222191	+	2.63640i	0.00933116	+	0.110719i
$568$		0			0
$569$	−5.47937			−0.229707			−0.114854	−	0.993382i	$-0.536640\pi$
$569$	−5.47937			−0.229707			−0.114854	+	0.993382i	$0.536640\pi$
$570$		0			0
$571$		−	3.67380i		−	0.153744i	−0.997041	−	0.0768718i	$-0.975507\pi$
$571$		−	3.67380i		−	0.153744i	0.997041	−	0.0768718i	$-0.0244932\pi$
$572$		0			0
$573$		−	7.60953i		−	0.317893i
$574$		0			0
$575$		−	0.0462892i		−	0.00193039i
$576$		0			0
$577$		−	36.0050i		−	1.49891i	−0.662057	−	0.749454i	$-0.730316\pi$
$577$		−	36.0050i		−	1.49891i	0.662057	−	0.749454i	$-0.269684\pi$
$578$		0			0
$579$	−10.6694			−0.443403
$580$		0			0
$581$	0.170663	+	2.02499i	0.00708028	+	0.0840109i
$582$		0			0
$583$		−	37.1102i		−	1.53695i
$584$		0			0
$585$	−6.54562			−0.270628
$586$		0			0
$587$	18.2663			0.753933			0.376966	−	0.926227i	$-0.376967\pi$
$587$	18.2663			0.753933			0.376966	+	0.926227i	$0.376967\pi$
$588$		0			0
$589$	27.3137			1.12544
$590$		0			0
$591$	−4.68879			−0.192871
$592$		0			0
$593$		−	24.4789i		−	1.00523i	−0.864511	−	0.502613i	$-0.832372\pi$
$593$		−	24.4789i		−	1.00523i	0.864511	−	0.502613i	$-0.167628\pi$
$594$		0			0
$595$	−1.18998	−	14.1196i	−0.0487843	−	0.578849i
$596$		0			0
$597$	22.4694			0.919610
$598$		0			0
$599$		−	27.1257i		−	1.10833i	−0.832408	−	0.554163i	$-0.813038\pi$
$599$		−	27.1257i		−	1.10833i	0.832408	−	0.554163i	$-0.186962\pi$
$600$		0			0
$601$		−	46.5188i		−	1.89754i	−0.315965	−	0.948771i	$-0.602328\pi$
$601$		−	46.5188i		−	1.89754i	0.315965	−	0.948771i	$-0.397672\pi$
$602$		0			0
$603$			9.01250i			0.367017i
$604$		0			0
$605$		−	4.32026i		−	0.175644i
$606$		0			0
$607$	15.9606			0.647819			0.323910	−	0.946088i	$-0.395003\pi$
$607$	15.9606			0.647819			0.323910	+	0.946088i	$0.395003\pi$
$608$		0			0
$609$	−0.0444327	−	0.527215i	−0.00180050	−	0.0213638i
$610$		0			0
$611$			18.5138i			0.748989i
$612$		0			0
$613$	7.33870			0.296407			0.148204	−	0.988957i	$-0.452651\pi$
$613$	7.33870			0.296407			0.148204	+	0.988957i	$0.452651\pi$
$614$		0			0
$615$	18.4469			0.743850
$616$		0			0
$617$	38.6524			1.55609			0.778044	−	0.628210i	$-0.216212\pi$
$617$	38.6524			1.55609			0.778044	+	0.628210i	$0.216212\pi$
$618$		0			0
$619$	−26.8688			−1.07995			−0.539974	−	0.841682i	$-0.681566\pi$
$619$	−26.8688			−1.07995			−0.539974	+	0.841682i	$0.681566\pi$
$620$		0			0
$621$		−	0.130157i		−	0.00522301i
$622$		0			0
$623$	46.7925	−	3.94359i	1.87470	−	0.157997i
$624$		0			0
$625$	−26.6517			−1.06607
$626$		0			0
$627$		−	28.3421i		−	1.13187i
$628$		0			0
$629$		−	26.7481i		−	1.06652i
$630$		0			0
$631$		−	22.7900i		−	0.907257i	−0.891191	−	0.453628i	$-0.850129\pi$
$631$		−	22.7900i		−	0.907257i	0.891191	−	0.453628i	$-0.149871\pi$
$632$		0			0
$633$		−	15.4185i		−	0.612830i
$634$		0			0
$635$	−17.4912			−0.694117
$636$		0			0
$637$	−3.31371	−	19.5197i	−0.131294	−	0.773399i
$638$		0			0
$639$			4.75861i			0.188248i
$640$		0			0
$641$	15.3776			0.607378			0.303689	−	0.952771i	$-0.401782\pi$
$641$	15.3776			0.607378			0.303689	+	0.952771i	$0.401782\pi$
$642$		0			0
$643$	−29.1900			−1.15114			−0.575570	−	0.817752i	$-0.695220\pi$
$643$	−29.1900			−1.15114			−0.575570	+	0.817752i	$0.695220\pi$
$644$		0			0
$645$	10.1457			0.399485
$646$		0			0
$647$	10.2275			0.402083			0.201042	−	0.979583i	$-0.435567\pi$
$647$	10.2275			0.402083			0.201042	+	0.979583i	$0.435567\pi$
$648$		0			0
$649$			34.6395i			1.35972i
$650$		0			0
$651$	0.768089	+	9.11373i	0.0301038	+	0.357195i
$652$		0			0
$653$	7.48187			0.292788			0.146394	−	0.989226i	$-0.453233\pi$
$653$	7.48187			0.292788			0.146394	+	0.989226i	$0.453233\pi$
$654$		0			0
$655$			40.4049i			1.57875i
$656$		0			0
$657$		−	10.2853i		−	0.401268i
$658$		0			0
$659$			27.9602i			1.08917i	0.838704	+	0.544587i	$0.183314\pi$
$659$			27.9602i			1.08917i	−0.838704	+	0.544587i	$0.816686\pi$
$660$		0			0
$661$			9.23441i			0.359177i	0.983742	+	0.179588i	$0.0574766\pi$
$661$			9.23441i			0.359177i	−0.983742	+	0.179588i	$0.942523\pi$
$662$		0			0
$663$	6.54562			0.254211
$664$		0			0
$665$	48.2075	−	4.06284i	1.86941	−	0.157550i
$666$		0			0
$667$			0.0260281i			0.00100781i
$668$		0			0
$669$	−2.86933			−0.110935
$670$		0			0
$671$	22.2395			0.858548
$672$		0			0
$673$	14.9607			0.576692			0.288346	−	0.957526i	$-0.406895\pi$
$673$	14.9607			0.576692			0.288346	+	0.957526i	$0.406895\pi$
$674$		0			0
$675$	0.355642			0.0136887
$676$		0			0
$677$			5.45727i			0.209740i	0.994486	+	0.104870i	$0.0334426\pi$
$677$			5.45727i			0.209740i	−0.994486	+	0.104870i	$0.966557\pi$
$678$		0			0
$679$	−23.8025	+	2.00603i	−0.913457	+	0.0769845i
$680$		0			0
$681$	2.91376			0.111655
$682$		0			0
$683$		−	25.4274i		−	0.972953i	−0.873694	−	0.486476i	$-0.838282\pi$
$683$		−	25.4274i		−	0.972953i	0.873694	−	0.486476i	$-0.161718\pi$
$684$		0			0
$685$		−	42.1246i		−	1.60950i
$686$		0			0
$687$		−	20.0853i		−	0.766303i
$688$		0			0
$689$			29.2619i			1.11479i
$690$		0			0
$691$	29.0962			1.10687			0.553437	−	0.832891i	$-0.313316\pi$
$691$	29.0962			1.10687			0.553437	+	0.832891i	$0.313316\pi$
$692$		0			0
$693$	9.45688	−	0.797008i	0.359237	−	0.0302758i
$694$		0			0
$695$			35.4394i			1.34429i
$696$		0			0
$697$	−18.4469			−0.698725
$698$		0			0
$699$	0.343146			0.0129790
$700$		0			0
$701$	−45.8618			−1.73218			−0.866089	−	0.499890i	$-0.833374\pi$
$701$	−45.8618			−1.73218			−0.866089	+	0.499890i	$0.833374\pi$
$702$		0			0
$703$	91.3237			3.44434
$704$		0			0
$705$		−	15.1480i		−	0.570508i
$706$		0			0
$707$	35.9288	−	3.02801i	1.35124	−	0.113880i
$708$		0			0
$709$	12.1775			0.457335			0.228667	−	0.973505i	$-0.426563\pi$
$709$	12.1775			0.457335			0.228667	+	0.973505i	$0.426563\pi$
$710$		0			0
$711$			4.24441i			0.159178i
$712$		0			0
$713$		−	0.449936i		−	0.0168502i
$714$		0			0
$715$			23.4794i			0.878079i
$716$		0			0
$717$			8.56113i			0.319721i
$718$		0			0
$719$	−12.5706			−0.468805			−0.234402	−	0.972140i	$-0.575313\pi$
$719$	−12.5706			−0.468805			−0.234402	+	0.972140i	$0.575313\pi$
$720$		0			0
$721$	2.54562	+	30.2050i	0.0948038	+	1.12489i
$722$		0			0
$723$		−	6.40598i		−	0.238241i
$724$		0			0
$725$	−0.0711196			−0.00264131
$726$		0			0
$727$	9.22941			0.342300			0.171150	−	0.985245i	$-0.445252\pi$
$727$	9.22941			0.342300			0.171150	+	0.985245i	$0.445252\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−10.1457			−0.375251
$732$		0			0
$733$		−	25.2216i		−	0.931580i	−0.884895	−	0.465790i	$-0.845770\pi$
$733$		−	25.2216i		−	0.931580i	0.884895	−	0.465790i	$-0.154230\pi$
$734$		0			0
$735$	2.71128	+	15.9711i	0.100007	+	0.589102i
$736$		0			0
$737$	32.3281			1.19082
$738$		0			0
$739$		−	4.93070i		−	0.181379i	−0.995879	−	0.0906894i	$-0.971093\pi$
$739$		−	4.93070i		−	0.181379i	0.995879	−	0.0906894i	$-0.0289070\pi$
$740$		0			0
$741$			22.3481i			0.820980i
$742$		0			0
$743$			50.3126i			1.84579i	0.385050	+	0.922896i	$0.374184\pi$
$743$			50.3126i			1.84579i	−0.385050	+	0.922896i	$0.625816\pi$
$744$		0			0
$745$			52.1814i			1.91178i
$746$		0			0
$747$	0.768089			0.0281029
$748$		0			0
$749$	18.3456	−	1.54614i	0.670335	−	0.0564946i
$750$		0			0
$751$		−	27.1321i		−	0.990066i	−0.868874	−	0.495033i	$-0.835156\pi$
$751$		−	27.1321i		−	0.990066i	0.868874	−	0.495033i	$-0.164844\pi$
$752$		0			0
$753$	−24.2025			−0.881987
$754$		0			0
$755$	−41.2937			−1.50283
$756$		0			0
$757$	−34.6913			−1.26088			−0.630438	−	0.776239i	$-0.717125\pi$
$757$	−34.6913			−1.26088			−0.630438	+	0.776239i	$0.717125\pi$
$758$		0			0
$759$	−0.466877			−0.0169466
$760$		0			0
$761$		−	19.9082i		−	0.721673i	−0.932629	−	0.360837i	$-0.882491\pi$
$761$		−	19.9082i		−	0.721673i	0.932629	−	0.360837i	$-0.117509\pi$
$762$		0			0
$763$	−0.641859	−	7.61596i	−0.0232368	−	0.275716i
$764$		0			0
$765$	−5.35564			−0.193634
$766$		0			0
$767$		−	27.3137i		−	0.986241i
$768$		0			0
$769$		−	3.83434i		−	0.138270i	−0.997607	−	0.0691348i	$-0.977976\pi$
$769$		−	3.83434i		−	0.138270i	0.997607	−	0.0691348i	$-0.0220239\pi$
$770$		0			0
$771$		−	1.22798i		−	0.0442248i
$772$		0			0
$773$			1.33660i			0.0480740i	0.999711	+	0.0240370i	$0.00765195\pi$
$773$			1.33660i			0.0480740i	−0.999711	+	0.0240370i	$0.992348\pi$
$774$		0			0
$775$	1.22941			0.0441618
$776$		0			0
$777$	2.56811	+	30.4719i	0.0921306	+	1.09317i
$778$		0			0
$779$		−	62.9816i		−	2.25655i
$780$		0			0
$781$	17.0693			0.610788
$782$		0			0
$783$	−0.199975			−0.00714652
$784$		0			0
$785$	11.8343			0.422386
$786$		0			0
$787$	−2.11386			−0.0753509			−0.0376755	−	0.999290i	$-0.511995\pi$
$787$	−2.11386			−0.0753509			−0.0376755	+	0.999290i	$0.511995\pi$
$788$		0			0
$789$		−	6.01551i		−	0.214158i
$790$		0			0
$791$	0.526180	+	6.24337i	0.0187088	+	0.221989i
$792$		0			0
$793$	−17.5362			−0.622728
$794$		0			0
$795$		−	23.9422i		−	0.849141i
$796$		0			0
$797$		−	31.1770i		−	1.10434i	−0.833730	−	0.552172i	$-0.813799\pi$
$797$		−	31.1770i		−	1.10434i	0.833730	−	0.552172i	$-0.186201\pi$
$798$		0			0
$799$			15.1480i			0.535899i
$800$		0			0
$801$		−	17.7486i		−	0.627116i
$802$		0			0
$803$	−36.8938			−1.30195
$804$		0			0
$805$	−0.0669267	−	0.794117i	−0.00235886	−	0.0279889i
$806$		0			0
$807$		−	22.3771i		−	0.787710i
$808$		0			0
$809$	−40.6395			−1.42881			−0.714404	−	0.699733i	$-0.753302\pi$
$809$	−40.6395			−1.42881			−0.714404	+	0.699733i	$0.753302\pi$
$810$		0			0
$811$	−11.3187			−0.397454			−0.198727	−	0.980055i	$-0.563681\pi$
$811$	−11.3187			−0.397454			−0.198727	+	0.980055i	$0.563681\pi$
$812$		0			0
$813$	9.63436			0.337892
$814$		0			0
$815$	23.8393			0.835055
$816$		0			0
$817$		−	34.6395i		−	1.21188i
$818$		0			0
$819$	−7.45688	+	0.628452i	−0.260564	+	0.0219599i
$820$		0			0
$821$	−37.2594			−1.30036			−0.650181	−	0.759779i	$-0.725307\pi$
$821$	−37.2594			−1.30036			−0.650181	+	0.759779i	$0.725307\pi$
$822$		0			0
$823$			50.9975i			1.77766i	0.458236	+	0.888831i	$0.348481\pi$
$823$			50.9975i			1.77766i	−0.458236	+	0.888831i	$0.651519\pi$
$824$		0			0
$825$		−	1.27570i		−	0.0444142i
$826$		0			0
$827$		−	41.2439i		−	1.43419i	−0.696975	−	0.717095i	$-0.745471\pi$
$827$		−	41.2439i		−	1.43419i	0.696975	−	0.717095i	$-0.254529\pi$
$828$		0			0
$829$			10.8734i			0.377649i	0.982011	+	0.188825i	$0.0604678\pi$
$829$			10.8734i			0.377649i	−0.982011	+	0.188825i	$0.939532\pi$
$830$		0			0
$831$	−11.9581			−0.414821
$832$		0			0
$833$	−2.71128	−	15.9711i	−0.0939404	−	0.553365i
$834$		0			0
$835$			32.4570i			1.12322i
$836$		0			0
$837$	3.45688			0.119487
$838$		0			0
$839$	−36.5438			−1.26163			−0.630816	−	0.775932i	$-0.717280\pi$
$839$	−36.5438			−1.26163			−0.630816	+	0.775932i	$0.717280\pi$
$840$		0			0
$841$	−28.9600			−0.998621
$842$		0			0
$843$	17.6818			0.608995
$844$		0			0
$845$			11.5711i			0.398059i
$846$		0			0
$847$	−0.414793	−	4.92171i	−0.0142524	−	0.169112i
$848$		0			0
$849$	−3.50131			−0.120165
$850$		0			0
$851$		−	1.50437i		−	0.0515690i
$852$		0			0
$853$			4.09818i			0.140319i	0.997536	+	0.0701596i	$0.0223508\pi$
$853$			4.09818i			0.140319i	−0.997536	+	0.0701596i	$0.977649\pi$
$854$		0			0
$855$		−	18.2853i		−	0.625344i
$856$		0			0
$857$			7.40547i			0.252966i	0.991969	+	0.126483i	$0.0403689\pi$
$857$			7.40547i			0.252966i	−0.991969	+	0.126483i	$0.959631\pi$
$858$		0			0
$859$	−22.5287			−0.768669			−0.384334	−	0.923194i	$-0.625569\pi$
$859$	−22.5287			−0.768669			−0.384334	+	0.923194i	$0.625569\pi$
$860$		0			0
$861$	21.0150	−	1.77111i	0.716189	−	0.0603591i
$862$		0			0
$863$		−	24.2748i		−	0.826324i	−0.910658	−	0.413162i	$-0.864424\pi$
$863$		−	24.2748i		−	0.826324i	0.910658	−	0.413162i	$-0.135576\pi$
$864$		0			0
$865$	−18.5807			−0.631764
$866$		0			0
$867$	−11.6444			−0.395463
$868$		0			0
$869$	15.2248			0.516467
$870$		0			0
$871$	−25.4912			−0.863736
$872$		0			0
$873$			9.02840i			0.305565i
$874$		0			0
$875$	−28.3363	+	2.38813i	−0.957943	+	0.0807337i
$876$		0			0
$877$	−4.46993			−0.150939			−0.0754694	−	0.997148i	$-0.524046\pi$
$877$	−4.46993			−0.150939			−0.0754694	+	0.997148i	$0.524046\pi$
$878$		0			0
$879$			3.40047i			0.114695i
$880$		0			0
$881$		−	19.8504i		−	0.668777i	−0.942435	−	0.334389i	$-0.891470\pi$
$881$		−	19.8504i		−	0.668777i	0.942435	−	0.334389i	$-0.108530\pi$
$882$		0			0
$883$			42.2116i			1.42053i	0.703933	+	0.710266i	$0.251425\pi$
$883$			42.2116i			1.42053i	−0.703933	+	0.710266i	$0.748575\pi$
$884$		0			0
$885$			22.3481i			0.751225i
$886$		0			0
$887$	5.22509			0.175441			0.0877207	−	0.996145i	$-0.472042\pi$
$887$	5.22509			0.175441			0.0877207	+	0.996145i	$0.472042\pi$
$888$		0			0
$889$	−19.9263	+	1.67935i	−0.668305	+	0.0563235i
$890$		0			0
$891$		−	3.58704i		−	0.120170i
$892$		0			0
$893$	−51.7187			−1.73070
$894$		0			0
$895$	−48.8400			−1.63254
$896$		0			0
$897$	0.368139			0.0122918
$898$		0			0
$899$	−0.691289			−0.0230558
$900$		0			0
$901$			23.9422i			0.797629i
$902$		0			0
$903$	11.5581	−	0.974097i	0.384630	−	0.0324159i
$904$		0			0
$905$	37.5044			1.24669
$906$		0			0
$907$			51.1371i			1.69798i	0.528408	+	0.848990i	$0.322789\pi$
$907$			51.1371i			1.69798i	−0.528408	+	0.848990i	$0.677211\pi$
$908$		0			0
$909$		−	13.6279i		−	0.452010i
$910$		0			0
$911$			18.9811i			0.628871i	0.949279	+	0.314436i	$0.101815\pi$
$911$			18.9811i			0.628871i	−0.949279	+	0.314436i	$0.898185\pi$
$912$		0			0
$913$		−	2.75516i		−	0.0911826i
$914$		0			0
$915$	14.3481			0.474335
$916$		0			0
$917$	3.87932	+	46.0300i	0.128107	+	1.52004i
$918$		0			0
$919$		−	12.3890i		−	0.408677i	−0.978900	−	0.204338i	$-0.934496\pi$
$919$		−	12.3890i		−	0.408677i	0.978900	−	0.204338i	$-0.0655043\pi$
$920$		0			0
$921$	−12.1037			−0.398832
$922$		0			0
$923$	−13.4594			−0.443021
$924$		0			0
$925$	4.11055			0.135154
$926$		0			0
$927$	11.4569			0.376293
$928$		0			0
$929$			2.87984i			0.0944845i	0.998883	+	0.0472423i	$0.0150433\pi$
$929$			2.87984i			0.0944845i	−0.998883	+	0.0472423i	$0.984957\pi$
$930$		0			0
$931$	54.5287	−	9.25690i	1.78711	−	0.303383i
$932$		0			0
$933$	26.7481			0.875693
$934$		0			0
$935$			19.2109i			0.628263i
$936$		0			0
$937$			27.5551i			0.900185i	0.892982	+	0.450092i	$0.148609\pi$
$937$			27.5551i			0.900185i	−0.892982	+	0.450092i	$0.851391\pi$
$938$		0			0
$939$			27.8343i			0.908339i
$940$		0			0
$941$			0.707358i			0.0230592i	0.999934	+	0.0115296i	$0.00367007\pi$
$941$			0.707358i			0.0230592i	−0.999934	+	0.0115296i	$0.996330\pi$
$942$		0			0
$943$	−1.03749			−0.0337853
$944$		0			0
$945$	6.10124	−	0.514201i	0.198473	−	0.0167270i
$946$		0			0
$947$			4.64725i			0.151015i	0.997145	+	0.0755077i	$0.0240577\pi$
$947$			4.64725i			0.151015i	−0.997145	+	0.0755077i	$0.975942\pi$
$948$		0			0
$949$	29.0912			0.944342
$950$		0			0
$951$	−5.85683			−0.189921
$952$		0			0
$953$	6.77516			0.219469			0.109734	−	0.993961i	$-0.465000\pi$
$953$	6.77516			0.219469			0.109734	+	0.993961i	$0.465000\pi$
$954$		0			0
$955$	−17.6102			−0.569852
$956$		0			0
$957$			0.717318i			0.0231876i
$958$		0			0
$959$	−4.04443	−	47.9891i	−0.130602	−	1.54965i
$960$		0			0
$961$	−19.0500			−0.614516
$962$		0			0
$963$		−	6.95858i		−	0.224237i
$964$		0			0
$965$			24.6913i			0.794841i
$966$		0			0
$967$			15.0375i			0.483573i	0.970329	+	0.241787i	$0.0777334\pi$
$967$			15.0375i			0.483573i	−0.970329	+	0.241787i	$0.922267\pi$
$968$		0			0
$969$			18.2853i			0.587409i
$970$		0			0
$971$	35.5601			1.14118			0.570588	−	0.821236i	$-0.306715\pi$
$971$	35.5601			1.14118			0.570588	+	0.821236i	$0.306715\pi$
$972$		0			0
$973$	3.40257	+	40.3731i	0.109082	+	1.29430i
$974$		0			0
$975$			1.00591i			0.0322148i
$976$		0			0
$977$	44.8370			1.43446			0.717231	−	0.696836i	$-0.245409\pi$
$977$	44.8370			1.43446			0.717231	+	0.696836i	$0.245409\pi$
$978$		0			0
$979$	−63.6649			−2.03474
$980$		0			0
$981$	−2.88877			−0.0922312
$982$		0			0
$983$	18.1386			0.578532			0.289266	−	0.957249i	$-0.406589\pi$
$983$	18.1386			0.578532			0.289266	+	0.957249i	$0.406589\pi$
$984$		0			0
$985$			10.8509i			0.345739i
$986$		0			0
$987$	−1.45438	−	17.2569i	−0.0462934	−	0.549293i
$988$		0			0
$989$	−0.570613			−0.0181444
$990$		0			0
$991$			40.0659i			1.27273i	0.771386	+	0.636367i	$0.219564\pi$
$991$			40.0659i			1.27273i	−0.771386	+	0.636367i	$0.780436\pi$
$992$		0			0
$993$		−	2.84787i		−	0.0903743i
$994$		0			0
$995$		−	51.9992i		−	1.64849i
$996$		0			0
$997$			44.7824i			1.41827i	0.705071	+	0.709136i	$0.250915\pi$
$997$			44.7824i			1.41827i	−0.705071	+	0.709136i	$0.749085\pi$
$998$		0			0
$999$	11.5581			0.365683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.b.a.223.7	yes	8
3.2	odd	2		2016.2.b.a.1567.2		8
4.3	odd	2		672.2.b.b.223.7	yes	8
7.6	odd	2		672.2.b.b.223.2	yes	8
8.3	odd	2		1344.2.b.g.895.2		8
8.5	even	2		1344.2.b.h.895.2		8
12.11	even	2		2016.2.b.c.1567.2		8
21.20	even	2		2016.2.b.c.1567.7		8
24.5	odd	2		4032.2.b.o.3583.7		8
24.11	even	2		4032.2.b.q.3583.7		8
28.27	even	2	inner	672.2.b.a.223.2	✓	8
56.13	odd	2		1344.2.b.g.895.7		8
56.27	even	2		1344.2.b.h.895.7		8
84.83	odd	2		2016.2.b.a.1567.7		8
168.83	odd	2		4032.2.b.o.3583.2		8
168.125	even	2		4032.2.b.q.3583.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.b.a.223.2	✓	8	28.27	even	2	inner
672.2.b.a.223.7	yes	8	1.1	even	1	trivial
672.2.b.b.223.2	yes	8	7.6	odd	2
672.2.b.b.223.7	yes	8	4.3	odd	2
1344.2.b.g.895.2		8	8.3	odd	2
1344.2.b.g.895.7		8	56.13	odd	2
1344.2.b.h.895.2		8	8.5	even	2
1344.2.b.h.895.7		8	56.27	even	2
2016.2.b.a.1567.2		8	3.2	odd	2
2016.2.b.a.1567.7		8	84.83	odd	2
2016.2.b.c.1567.2		8	12.11	even	2
2016.2.b.c.1567.7		8	21.20	even	2
4032.2.b.o.3583.2		8	168.83	odd	2
4032.2.b.o.3583.7		8	24.5	odd	2
4032.2.b.q.3583.2		8	168.125	even	2
4032.2.b.q.3583.7		8	24.11	even	2

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.31423i q^{5} +(0.222191 + 2.63640i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.31423i q^{5} +(0.222191 + 2.63640i) q^{7} +1.00000 q^{9} -3.58704i q^{11} +2.82843i q^{13} -2.31423i q^{15} +2.31423i q^{17} -7.90126 q^{19} +(-0.222191 - 2.63640i) q^{21} +0.130157i q^{23} -0.355642 q^{25} -1.00000 q^{27} +0.199975 q^{29} -3.45688 q^{31} +3.58704i q^{33} +(-6.10124 + 0.514201i) q^{35} -11.5581 q^{37} -2.82843i q^{39} +7.97108i q^{41} +4.38404i q^{43} +2.31423i q^{45} +6.54562 q^{47} +(-6.90126 + 1.17157i) q^{49} -2.31423i q^{51} +10.3456 q^{53} +8.30121 q^{55} +7.90126 q^{57} -9.65685 q^{59} +6.19998i q^{61} +(0.222191 + 2.63640i) q^{63} -6.54562 q^{65} +9.01250i q^{67} -0.130157i q^{69} +4.75861i q^{71} -10.2853i q^{73} +0.355642 q^{75} +(9.45688 - 0.797008i) q^{77} +4.24441i q^{79} +1.00000 q^{81} +0.768089 q^{83} -5.35564 q^{85} -0.199975 q^{87} -17.7486i q^{89} +(-7.45688 + 0.628452i) q^{91} +3.45688 q^{93} -18.2853i q^{95} +9.02840i q^{97} -3.58704i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 - 4 * q^7 + 8 * q^9	\( 8 q - 8 q^{3} - 4 q^{7} + 8 q^{9} - 8 q^{19} + 4 q^{21} - 16 q^{25} - 8 q^{27} + 16 q^{31} + 8 q^{35} + 8 q^{37} - 16 q^{47} + 16 q^{53} + 8 q^{55} + 8 q^{57} - 32 q^{59} - 4 q^{63} + 16 q^{65} + 16 q^{75} + 32 q^{77} + 8 q^{81} - 16 q^{83} - 56 q^{85} - 16 q^{91} - 16 q^{93}+O(q^{100}) \) 8 * q - 8 * q^3 - 4 * q^7 + 8 * q^9 - 8 * q^19 + 4 * q^21 - 16 * q^25 - 8 * q^27 + 16 * q^31 + 8 * q^35 + 8 * q^37 - 16 * q^47 + 16 * q^53 + 8 * q^55 + 8 * q^57 - 32 * q^59 - 4 * q^63 + 16 * q^65 + 16 * q^75 + 32 * q^77 + 8 * q^81 - 16 * q^83 - 56 * q^85 - 16 * q^91 - 16 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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