LMFDB - Embedded newform 224.2.e.b.111.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,2,Mod(111,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("224.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	224.e (of order $2$, degree $1$, 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:	$1.78864900528$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.2
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	224.111
Dual form			224.2.e.b.111.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949i q^{3} +2.44949 q^{5} +(2.44949 + 1.00000i) q^{7} -3.00000 q^{9} +O(q^{10})$	$q-2.44949i q^{3} +2.44949 q^{5} +(2.44949 + 1.00000i) q^{7} -3.00000 q^{9} -2.00000 q^{11} -2.44949 q^{13} -6.00000i q^{15} -4.89898i q^{17} +2.44949i q^{19} +(2.44949 - 6.00000i) q^{21} +4.00000i q^{23} +1.00000 q^{25} +4.00000i q^{29} -4.89898 q^{31} +4.89898i q^{33} +(6.00000 + 2.44949i) q^{35} +8.00000i q^{37} +6.00000i q^{39} +6.00000 q^{43} -7.34847 q^{45} +4.89898 q^{47} +(5.00000 + 4.89898i) q^{49} -12.0000 q^{51} -4.00000i q^{53} -4.89898 q^{55} +6.00000 q^{57} +2.44949i q^{59} -7.34847 q^{61} +(-7.34847 - 3.00000i) q^{63} -6.00000 q^{65} +2.00000 q^{67} +9.79796 q^{69} -10.0000i q^{71} +14.6969i q^{73} -2.44949i q^{75} +(-4.89898 - 2.00000i) q^{77} +6.00000i q^{79} -9.00000 q^{81} -2.44949i q^{83} -12.0000i q^{85} +9.79796 q^{87} -14.6969i q^{89} +(-6.00000 - 2.44949i) q^{91} +12.0000i q^{93} +6.00000i q^{95} -4.89898i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} - 8 q^{11} + 4 q^{25} + 24 q^{35} + 24 q^{43} + 20 q^{49} - 48 q^{51} + 24 q^{57} - 24 q^{65} + 8 q^{67} - 36 q^{81} - 24 q^{91} + 24 q^{99}+O(q^{100}) $ 4 * q - 12 * q^9 - 8 * q^11 + 4 * q^25 + 24 * q^35 + 24 * q^43 + 20 * q^49 - 48 * q^51 + 24 * q^57 - 24 * q^65 + 8 * q^67 - 36 * q^81 - 24 * q^91 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$		−	2.44949i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$3$		−	2.44949i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$4$		0			0
$5$	2.44949			1.09545			0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949			1.09545			0.547723	+	0.836660i	$0.315495\pi$
$6$		0			0
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$7$	2.44949	+	1.00000i	0.925820	+	0.377964i
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$	−2.44949			−0.679366			−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949			−0.679366			−0.339683	+	0.940540i	$0.610320\pi$
$14$		0			0
$15$		−	6.00000i		−	1.54919i
$16$		0			0
$17$		−	4.89898i		−	1.18818i	−0.804400	−	0.594089i	$-0.797513\pi$
$17$		−	4.89898i		−	1.18818i	0.804400	−	0.594089i	$-0.202487\pi$
$18$		0			0
$19$			2.44949i			0.561951i	0.959715	+	0.280976i	$0.0906580\pi$
$19$			2.44949i			0.561951i	−0.959715	+	0.280976i	$0.909342\pi$
$20$		0			0
$21$	2.44949	−	6.00000i	0.534522	−	1.30931i
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$	−4.89898			−0.879883			−0.439941	−	0.898027i	$-0.645001\pi$
$31$	−4.89898			−0.879883			−0.439941	+	0.898027i	$0.645001\pi$
$32$		0			0
$33$			4.89898i			0.852803i
$34$		0			0
$35$	6.00000	+	2.44949i	1.01419	+	0.414039i
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$			6.00000i			0.960769i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	6.00000			0.914991			0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000			0.914991			0.457496	+	0.889212i	$0.348747\pi$
$44$		0			0
$45$	−7.34847			−1.09545
$46$		0			0
$47$	4.89898			0.714590			0.357295	−	0.933992i	$-0.383699\pi$
$47$	4.89898			0.714590			0.357295	+	0.933992i	$0.383699\pi$
$48$		0			0
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$		0			0
$51$	−12.0000			−1.68034
$52$		0			0
$53$		−	4.00000i		−	0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$		−	4.00000i		−	0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$		0			0
$55$	−4.89898			−0.660578
$56$		0			0
$57$	6.00000			0.794719
$58$		0			0
$59$			2.44949i			0.318896i	0.987206	+	0.159448i	$0.0509715\pi$
$59$			2.44949i			0.318896i	−0.987206	+	0.159448i	$0.949029\pi$
$60$		0			0
$61$	−7.34847			−0.940875			−0.470438	−	0.882433i	$-0.655904\pi$
$61$	−7.34847			−0.940875			−0.470438	+	0.882433i	$0.655904\pi$
$62$		0			0
$63$	−7.34847	−	3.00000i	−0.925820	−	0.377964i
$64$		0			0
$65$	−6.00000			−0.744208
$66$		0			0
$67$	2.00000			0.244339			0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000			0.244339			0.122169	+	0.992509i	$0.461015\pi$
$68$		0			0
$69$	9.79796			1.17954
$70$		0			0
$71$		−	10.0000i		−	1.18678i	−0.804914	−	0.593391i	$-0.797789\pi$
$71$		−	10.0000i		−	1.18678i	0.804914	−	0.593391i	$-0.202211\pi$
$72$		0			0
$73$			14.6969i			1.72015i	0.510171	+	0.860073i	$0.329582\pi$
$73$			14.6969i			1.72015i	−0.510171	+	0.860073i	$0.670418\pi$
$74$		0			0
$75$		−	2.44949i		−	0.282843i
$76$		0			0
$77$	−4.89898	−	2.00000i	−0.558291	−	0.227921i
$78$		0			0
$79$			6.00000i			0.675053i	0.941316	+	0.337526i	$0.109590\pi$
$79$			6.00000i			0.675053i	−0.941316	+	0.337526i	$0.890410\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$		−	2.44949i		−	0.268866i	−0.990923	−	0.134433i	$-0.957079\pi$
$83$		−	2.44949i		−	0.268866i	0.990923	−	0.134433i	$-0.0429214\pi$
$84$		0			0
$85$		−	12.0000i		−	1.30158i
$86$		0			0
$87$	9.79796			1.05045
$88$		0			0
$89$		−	14.6969i		−	1.55787i	−0.627103	−	0.778936i	$-0.715760\pi$
$89$		−	14.6969i		−	1.55787i	0.627103	−	0.778936i	$-0.284240\pi$
$90$		0			0
$91$	−6.00000	−	2.44949i	−0.628971	−	0.256776i
$92$		0			0
$93$			12.0000i			1.24434i
$94$		0			0
$95$			6.00000i			0.615587i
$96$		0			0
$97$		−	4.89898i		−	0.497416i	−0.968579	−	0.248708i	$-0.919994\pi$
$97$		−	4.89898i		−	0.497416i	0.968579	−	0.248708i	$-0.0800060\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	17.1464			1.70613			0.853067	−	0.521802i	$-0.174740\pi$
$101$	17.1464			1.70613			0.853067	+	0.521802i	$0.174740\pi$
$102$		0			0
$103$	−9.79796			−0.965422			−0.482711	−	0.875780i	$-0.660348\pi$
$103$	−9.79796			−0.965422			−0.482711	+	0.875780i	$0.660348\pi$
$104$		0			0
$105$	6.00000	−	14.6969i	0.585540	−	1.43427i
$106$		0			0
$107$	2.00000			0.193347			0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000			0.193347			0.0966736	+	0.995316i	$0.469180\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$	19.5959			1.85996
$112$		0			0
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$			9.79796i			0.913664i
$116$		0			0
$117$	7.34847			0.679366
$118$		0			0
$119$	4.89898	−	12.0000i	0.449089	−	1.10004i
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−9.79796			−0.876356
$126$		0			0
$127$			12.0000i			1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$			12.0000i			1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$		0			0
$129$		−	14.6969i		−	1.29399i
$130$		0			0
$131$		−	12.2474i		−	1.07006i	−0.844832	−	0.535032i	$-0.820299\pi$
$131$		−	12.2474i		−	1.07006i	0.844832	−	0.535032i	$-0.179701\pi$
$132$		0			0
$133$	−2.44949	+	6.00000i	−0.212398	+	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.00000			−0.170872			−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000			−0.170872			−0.0854358	+	0.996344i	$0.527228\pi$
$138$		0			0
$139$			2.44949i			0.207763i	0.994590	+	0.103882i	$0.0331263\pi$
$139$			2.44949i			0.207763i	−0.994590	+	0.103882i	$0.966874\pi$
$140$		0			0
$141$		−	12.0000i		−	1.01058i
$142$		0			0
$143$	4.89898			0.409673
$144$		0			0
$145$			9.79796i			0.813676i
$146$		0			0
$147$	12.0000	−	12.2474i	0.989743	−	1.01015i
$148$		0			0
$149$		−	16.0000i		−	1.31077i	−0.755295	−	0.655386i	$-0.772506\pi$
$149$		−	16.0000i		−	1.31077i	0.755295	−	0.655386i	$-0.227494\pi$
$150$		0			0
$151$		−	20.0000i		−	1.62758i	−0.581161	−	0.813788i	$-0.697401\pi$
$151$		−	20.0000i		−	1.62758i	0.581161	−	0.813788i	$-0.302599\pi$
$152$		0			0
$153$			14.6969i			1.18818i
$154$		0			0
$155$	−12.0000			−0.963863
$156$		0			0
$157$	7.34847			0.586472			0.293236	−	0.956040i	$-0.405268\pi$
$157$	7.34847			0.586472			0.293236	+	0.956040i	$0.405268\pi$
$158$		0			0
$159$	−9.79796			−0.777029
$160$		0			0
$161$	−4.00000	+	9.79796i	−0.315244	+	0.772187i
$162$		0			0
$163$	−14.0000			−1.09656			−0.548282	−	0.836293i	$-0.684718\pi$
$163$	−14.0000			−1.09656			−0.548282	+	0.836293i	$0.684718\pi$
$164$		0			0
$165$			12.0000i			0.934199i
$166$		0			0
$167$	−19.5959			−1.51638			−0.758189	−	0.652035i	$-0.773915\pi$
$167$	−19.5959			−1.51638			−0.758189	+	0.652035i	$0.773915\pi$
$168$		0			0
$169$	−7.00000			−0.538462
$170$		0			0
$171$		−	7.34847i		−	0.561951i
$172$		0			0
$173$	−2.44949			−0.186231			−0.0931156	−	0.995655i	$-0.529683\pi$
$173$	−2.44949			−0.186231			−0.0931156	+	0.995655i	$0.529683\pi$
$174$		0			0
$175$	2.44949	+	1.00000i	0.185164	+	0.0755929i
$176$		0			0
$177$	6.00000			0.450988
$178$		0			0
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$	−7.34847			−0.546207			−0.273104	−	0.961985i	$-0.588050\pi$
$181$	−7.34847			−0.546207			−0.273104	+	0.961985i	$0.588050\pi$
$182$		0			0
$183$			18.0000i			1.33060i
$184$		0			0
$185$			19.5959i			1.44072i
$186$		0			0
$187$			9.79796i			0.716498i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			10.0000i			0.723575i	0.932261	+	0.361787i	$0.117833\pi$
$191$			10.0000i			0.723575i	−0.932261	+	0.361787i	$0.882167\pi$
$192$		0			0
$193$	24.0000			1.72756			0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000			1.72756			0.863779	+	0.503871i	$0.168091\pi$
$194$		0			0
$195$			14.6969i			1.05247i
$196$		0			0
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		−	4.89898i		−	0.345547i
$202$		0			0
$203$	−4.00000	+	9.79796i	−0.280745	+	0.687682i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	12.0000i		−	0.834058i
$208$		0			0
$209$		−	4.89898i		−	0.338869i
$210$		0			0
$211$	18.0000			1.23917			0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000			1.23917			0.619586	+	0.784929i	$0.287301\pi$
$212$		0			0
$213$	−24.4949			−1.67836
$214$		0			0
$215$	14.6969			1.00232
$216$		0			0
$217$	−12.0000	−	4.89898i	−0.814613	−	0.332564i
$218$		0			0
$219$	36.0000			2.43265
$220$		0			0
$221$			12.0000i			0.807207i
$222$		0			0
$223$	−9.79796			−0.656120			−0.328060	−	0.944657i	$-0.606395\pi$
$223$	−9.79796			−0.656120			−0.328060	+	0.944657i	$0.606395\pi$
$224$		0			0
$225$	−3.00000			−0.200000
$226$		0			0
$227$		−	7.34847i		−	0.487735i	−0.969809	−	0.243868i	$-0.921584\pi$
$227$		−	7.34847i		−	0.487735i	0.969809	−	0.243868i	$-0.0784162\pi$
$228$		0			0
$229$	−12.2474			−0.809334			−0.404667	−	0.914464i	$-0.632613\pi$
$229$	−12.2474			−0.809334			−0.404667	+	0.914464i	$0.632613\pi$
$230$		0			0
$231$	−4.89898	+	12.0000i	−0.322329	+	0.789542i
$232$		0			0
$233$	14.0000			0.917170			0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000			0.917170			0.458585	+	0.888650i	$0.348356\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$		0			0
$237$	14.6969			0.954669
$238$		0			0
$239$		−	4.00000i		−	0.258738i	−0.991596	−	0.129369i	$-0.958705\pi$
$239$		−	4.00000i		−	0.258738i	0.991596	−	0.129369i	$-0.0412952\pi$
$240$		0			0
$241$		−	24.4949i		−	1.57786i	−0.614486	−	0.788928i	$-0.710637\pi$
$241$		−	24.4949i		−	1.57786i	0.614486	−	0.788928i	$-0.289363\pi$
$242$		0			0
$243$			22.0454i			1.41421i
$244$		0			0
$245$	12.2474	+	12.0000i	0.782461	+	0.766652i
$246$		0			0
$247$		−	6.00000i		−	0.381771i
$248$		0			0
$249$	−6.00000			−0.380235
$250$		0			0
$251$		−	12.2474i		−	0.773052i	−0.922278	−	0.386526i	$-0.873675\pi$
$251$		−	12.2474i		−	0.773052i	0.922278	−	0.386526i	$-0.126325\pi$
$252$		0			0
$253$		−	8.00000i		−	0.502956i
$254$		0			0
$255$	−29.3939			−1.84072
$256$		0			0
$257$			19.5959i			1.22236i	0.791492	+	0.611180i	$0.209305\pi$
$257$			19.5959i			1.22236i	−0.791492	+	0.611180i	$0.790695\pi$
$258$		0			0
$259$	−8.00000	+	19.5959i	−0.497096	+	1.21763i
$260$		0			0
$261$		−	12.0000i		−	0.742781i
$262$		0			0
$263$			14.0000i			0.863277i	0.902047	+	0.431638i	$0.142064\pi$
$263$			14.0000i			0.863277i	−0.902047	+	0.431638i	$0.857936\pi$
$264$		0			0
$265$		−	9.79796i		−	0.601884i
$266$		0			0
$267$	−36.0000			−2.20316
$268$		0			0
$269$	−12.2474			−0.746740			−0.373370	−	0.927682i	$-0.621798\pi$
$269$	−12.2474			−0.746740			−0.373370	+	0.927682i	$0.621798\pi$
$270$		0			0
$271$	19.5959			1.19037			0.595184	−	0.803590i	$-0.297079\pi$
$271$	19.5959			1.19037			0.595184	+	0.803590i	$0.297079\pi$
$272$		0			0
$273$	−6.00000	+	14.6969i	−0.363137	+	0.889499i
$274$		0			0
$275$	−2.00000			−0.120605
$276$		0			0
$277$		−	12.0000i		−	0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$		−	12.0000i		−	0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$		0			0
$279$	14.6969			0.879883
$280$		0			0
$281$	2.00000			0.119310			0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000			0.119310			0.0596550	+	0.998219i	$0.481000\pi$
$282$		0			0
$283$		−	2.44949i		−	0.145607i	−0.997346	−	0.0728035i	$-0.976805\pi$
$283$		−	2.44949i		−	0.145607i	0.997346	−	0.0728035i	$-0.0231946\pi$
$284$		0			0
$285$	14.6969			0.870572
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$	−12.0000			−0.703452
$292$		0			0
$293$	−2.44949			−0.143101			−0.0715504	−	0.997437i	$-0.522795\pi$
$293$	−2.44949			−0.143101			−0.0715504	+	0.997437i	$0.522795\pi$
$294$		0			0
$295$			6.00000i			0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	9.79796i		−	0.566631i
$300$		0			0
$301$	14.6969	+	6.00000i	0.847117	+	0.345834i
$302$		0			0
$303$		−	42.0000i		−	2.41284i
$304$		0			0
$305$	−18.0000			−1.03068
$306$		0			0
$307$		−	31.8434i		−	1.81740i	−0.417453	−	0.908698i	$-0.637077\pi$
$307$		−	31.8434i		−	1.81740i	0.417453	−	0.908698i	$-0.362923\pi$
$308$		0			0
$309$			24.0000i			1.36531i
$310$		0			0
$311$	−4.89898			−0.277796			−0.138898	−	0.990307i	$-0.544356\pi$
$311$	−4.89898			−0.277796			−0.138898	+	0.990307i	$0.544356\pi$
$312$		0			0
$313$		−	9.79796i		−	0.553813i	−0.960897	−	0.276907i	$-0.910691\pi$
$313$		−	9.79796i		−	0.553813i	0.960897	−	0.276907i	$-0.0893093\pi$
$314$		0			0
$315$	−18.0000	−	7.34847i	−1.01419	−	0.414039i
$316$		0			0
$317$		−	32.0000i		−	1.79730i	−0.438667	−	0.898650i	$-0.644549\pi$
$317$		−	32.0000i		−	1.79730i	0.438667	−	0.898650i	$-0.355451\pi$
$318$		0			0
$319$		−	8.00000i		−	0.447914i
$320$		0			0
$321$		−	4.89898i		−	0.273434i
$322$		0			0
$323$	12.0000			0.667698
$324$		0			0
$325$	−2.44949			−0.135873
$326$		0			0
$327$	9.79796			0.541828
$328$		0			0
$329$	12.0000	+	4.89898i	0.661581	+	0.270089i
$330$		0			0
$331$	18.0000			0.989369			0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000			0.989369			0.494685	+	0.869072i	$0.335284\pi$
$332$		0			0
$333$		−	24.0000i		−	1.31519i
$334$		0			0
$335$	4.89898			0.267660
$336$		0			0
$337$	−12.0000			−0.653682			−0.326841	−	0.945079i	$-0.605984\pi$
$337$	−12.0000			−0.653682			−0.326841	+	0.945079i	$0.605984\pi$
$338$		0			0
$339$		−	9.79796i		−	0.532152i
$340$		0			0
$341$	9.79796			0.530589
$342$		0			0
$343$	7.34847	+	17.0000i	0.396780	+	0.917914i
$344$		0			0
$345$	24.0000			1.29212
$346$		0			0
$347$	22.0000			1.18102			0.590511	−	0.807030i	$-0.298926\pi$
$347$	22.0000			1.18102			0.590511	+	0.807030i	$0.298926\pi$
$348$		0			0
$349$	12.2474			0.655591			0.327795	−	0.944749i	$-0.393694\pi$
$349$	12.2474			0.655591			0.327795	+	0.944749i	$0.393694\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	9.79796i		−	0.521493i	−0.965407	−	0.260746i	$-0.916031\pi$
$353$		−	9.79796i		−	0.521493i	0.965407	−	0.260746i	$-0.0839686\pi$
$354$		0			0
$355$		−	24.4949i		−	1.30005i
$356$		0			0
$357$	−29.3939	−	12.0000i	−1.55569	−	0.635107i
$358$		0			0
$359$		−	4.00000i		−	0.211112i	−0.994413	−	0.105556i	$-0.966338\pi$
$359$		−	4.00000i		−	0.211112i	0.994413	−	0.105556i	$-0.0336622\pi$
$360$		0			0
$361$	13.0000			0.684211
$362$		0			0
$363$			17.1464i			0.899954i
$364$		0			0
$365$			36.0000i			1.88433i
$366$		0			0
$367$	29.3939			1.53435			0.767174	−	0.641439i	$-0.221662\pi$
$367$	29.3939			1.53435			0.767174	+	0.641439i	$0.221662\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.00000	−	9.79796i	0.207670	−	0.508685i
$372$		0			0
$373$			36.0000i			1.86401i	0.362446	+	0.932005i	$0.381942\pi$
$373$			36.0000i			1.86401i	−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$			24.0000i			1.23935i
$376$		0			0
$377$		−	9.79796i		−	0.504621i
$378$		0			0
$379$	−30.0000			−1.54100			−0.770498	−	0.637442i	$-0.779993\pi$
$379$	−30.0000			−1.54100			−0.770498	+	0.637442i	$0.779993\pi$
$380$		0			0
$381$	29.3939			1.50589
$382$		0			0
$383$	−34.2929			−1.75228			−0.876142	−	0.482054i	$-0.839891\pi$
$383$	−34.2929			−1.75228			−0.876142	+	0.482054i	$0.839891\pi$
$384$		0			0
$385$	−12.0000	−	4.89898i	−0.611577	−	0.249675i
$386$		0			0
$387$	−18.0000			−0.914991
$388$		0			0
$389$		−	16.0000i		−	0.811232i	−0.914044	−	0.405616i	$-0.867057\pi$
$389$		−	16.0000i		−	0.811232i	0.914044	−	0.405616i	$-0.132943\pi$
$390$		0			0
$391$	19.5959			0.991008
$392$		0			0
$393$	−30.0000			−1.51330
$394$		0			0
$395$			14.6969i			0.739483i
$396$		0			0
$397$	31.8434			1.59817			0.799086	−	0.601216i	$-0.205317\pi$
$397$	31.8434			1.59817			0.799086	+	0.601216i	$0.205317\pi$
$398$		0			0
$399$	14.6969	+	6.00000i	0.735767	+	0.300376i
$400$		0			0
$401$	32.0000			1.59800			0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000			1.59800			0.799002	+	0.601329i	$0.205362\pi$
$402$		0			0
$403$	12.0000			0.597763
$404$		0			0
$405$	−22.0454			−1.09545
$406$		0			0
$407$		−	16.0000i		−	0.793091i
$408$		0			0
$409$			9.79796i			0.484478i	0.970217	+	0.242239i	$0.0778818\pi$
$409$			9.79796i			0.484478i	−0.970217	+	0.242239i	$0.922118\pi$
$410$		0			0
$411$			4.89898i			0.241649i
$412$		0			0
$413$	−2.44949	+	6.00000i	−0.120532	+	0.295241i
$414$		0			0
$415$		−	6.00000i		−	0.294528i
$416$		0			0
$417$	6.00000			0.293821
$418$		0			0
$419$			26.9444i			1.31632i	0.752878	+	0.658160i	$0.228665\pi$
$419$			26.9444i			1.31632i	−0.752878	+	0.658160i	$0.771335\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	−14.6969			−0.714590
$424$		0			0
$425$		−	4.89898i		−	0.237635i
$426$		0			0
$427$	−18.0000	−	7.34847i	−0.871081	−	0.355617i
$428$		0			0
$429$		−	12.0000i		−	0.579365i
$430$		0			0
$431$			20.0000i			0.963366i	0.876346	+	0.481683i	$0.159974\pi$
$431$			20.0000i			0.963366i	−0.876346	+	0.481683i	$0.840026\pi$
$432$		0			0
$433$			14.6969i			0.706290i	0.935569	+	0.353145i	$0.114888\pi$
$433$			14.6969i			0.706290i	−0.935569	+	0.353145i	$0.885112\pi$
$434$		0			0
$435$	24.0000			1.15071
$436$		0			0
$437$	−9.79796			−0.468700
$438$		0			0
$439$	−24.4949			−1.16908			−0.584539	−	0.811366i	$-0.698725\pi$
$439$	−24.4949			−1.16908			−0.584539	+	0.811366i	$0.698725\pi$
$440$		0			0
$441$	−15.0000	−	14.6969i	−0.714286	−	0.699854i
$442$		0			0
$443$	26.0000			1.23530			0.617649	−	0.786454i	$-0.288085\pi$
$443$	26.0000			1.23530			0.617649	+	0.786454i	$0.288085\pi$
$444$		0			0
$445$		−	36.0000i		−	1.70656i
$446$		0			0
$447$	−39.1918			−1.85371
$448$		0			0
$449$	−10.0000			−0.471929			−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000			−0.471929			−0.235965	+	0.971762i	$0.575825\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−48.9898			−2.30174
$454$		0			0
$455$	−14.6969	−	6.00000i	−0.689003	−	0.281284i
$456$		0			0
$457$	−12.0000			−0.561336			−0.280668	−	0.959805i	$-0.590556\pi$
$457$	−12.0000			−0.561336			−0.280668	+	0.959805i	$0.590556\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.34847			−0.342252			−0.171126	−	0.985249i	$-0.554741\pi$
$461$	−7.34847			−0.342252			−0.171126	+	0.985249i	$0.554741\pi$
$462$		0			0
$463$		−	6.00000i		−	0.278844i	−0.990233	−	0.139422i	$-0.955476\pi$
$463$		−	6.00000i		−	0.278844i	0.990233	−	0.139422i	$-0.0445244\pi$
$464$		0			0
$465$			29.3939i			1.36311i
$466$		0			0
$467$			41.6413i			1.92693i	0.267833	+	0.963465i	$0.413692\pi$
$467$			41.6413i			1.92693i	−0.267833	+	0.963465i	$0.586308\pi$
$468$		0			0
$469$	4.89898	+	2.00000i	0.226214	+	0.0923514i
$470$		0			0
$471$		−	18.0000i		−	0.829396i
$472$		0			0
$473$	−12.0000			−0.551761
$474$		0			0
$475$			2.44949i			0.112390i
$476$		0			0
$477$			12.0000i			0.549442i
$478$		0			0
$479$	24.4949			1.11920			0.559600	−	0.828763i	$-0.310955\pi$
$479$	24.4949			1.11920			0.559600	+	0.828763i	$0.310955\pi$
$480$		0			0
$481$		−	19.5959i		−	0.893497i
$482$		0			0
$483$	24.0000	+	9.79796i	1.09204	+	0.445823i
$484$		0			0
$485$		−	12.0000i		−	0.544892i
$486$		0			0
$487$		−	28.0000i		−	1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$		−	28.0000i		−	1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$		0			0
$489$			34.2929i			1.55078i
$490$		0			0
$491$	−2.00000			−0.0902587			−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000			−0.0902587			−0.0451294	+	0.998981i	$0.514370\pi$
$492$		0			0
$493$	19.5959			0.882556
$494$		0			0
$495$	14.6969			0.660578
$496$		0			0
$497$	10.0000	−	24.4949i	0.448561	−	1.09875i
$498$		0			0
$499$	−30.0000			−1.34298			−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000			−1.34298			−0.671492	+	0.741012i	$0.734346\pi$
$500$		0			0
$501$			48.0000i			2.14448i
$502$		0			0
$503$	14.6969			0.655304			0.327652	−	0.944798i	$-0.393743\pi$
$503$	14.6969			0.655304			0.327652	+	0.944798i	$0.393743\pi$
$504$		0			0
$505$	42.0000			1.86898
$506$		0			0
$507$			17.1464i			0.761500i
$508$		0			0
$509$	−36.7423			−1.62858			−0.814288	−	0.580461i	$-0.802872\pi$
$509$	−36.7423			−1.62858			−0.814288	+	0.580461i	$0.802872\pi$
$510$		0			0
$511$	−14.6969	+	36.0000i	−0.650154	+	1.59255i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−24.0000			−1.05757
$516$		0			0
$517$	−9.79796			−0.430914
$518$		0			0
$519$			6.00000i			0.263371i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		−	26.9444i		−	1.17820i	−0.808062	−	0.589098i	$-0.799483\pi$
$523$		−	26.9444i		−	1.17820i	0.808062	−	0.589098i	$-0.200517\pi$
$524$		0			0
$525$	2.44949	−	6.00000i	0.106904	−	0.261861i
$526$		0			0
$527$			24.0000i			1.04546i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$		−	7.34847i		−	0.318896i
$532$		0			0
$533$		0			0
$534$		0			0
$535$	4.89898			0.211801
$536$		0			0
$537$			24.4949i			1.05703i
$538$		0			0
$539$	−10.0000	−	9.79796i	−0.430730	−	0.422028i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$			18.0000i			0.772454i
$544$		0			0
$545$			9.79796i			0.419698i
$546$		0			0
$547$	2.00000			0.0855138			0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000			0.0855138			0.0427569	+	0.999086i	$0.486386\pi$
$548$		0			0
$549$	22.0454			0.940875
$550$		0			0
$551$	−9.79796			−0.417407
$552$		0			0
$553$	−6.00000	+	14.6969i	−0.255146	+	0.624977i
$554$		0			0
$555$	48.0000			2.03749
$556$		0			0
$557$			28.0000i			1.18640i	0.805056	+	0.593199i	$0.202135\pi$
$557$			28.0000i			1.18640i	−0.805056	+	0.593199i	$0.797865\pi$
$558$		0			0
$559$	−14.6969			−0.621614
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$			22.0454i			0.929103i	0.885546	+	0.464552i	$0.153784\pi$
$563$			22.0454i			0.929103i	−0.885546	+	0.464552i	$0.846216\pi$
$564$		0			0
$565$	9.79796			0.412203
$566$		0			0
$567$	−22.0454	−	9.00000i	−0.925820	−	0.377964i
$568$		0			0
$569$	−20.0000			−0.838444			−0.419222	−	0.907884i	$-0.637697\pi$
$569$	−20.0000			−0.838444			−0.419222	+	0.907884i	$0.637697\pi$
$570$		0			0
$571$	−2.00000			−0.0836974			−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000			−0.0836974			−0.0418487	+	0.999124i	$0.513325\pi$
$572$		0			0
$573$	24.4949			1.02329
$574$		0			0
$575$			4.00000i			0.166812i
$576$		0			0
$577$			19.5959i			0.815789i	0.913029	+	0.407894i	$0.133737\pi$
$577$			19.5959i			0.815789i	−0.913029	+	0.407894i	$0.866263\pi$
$578$		0			0
$579$		−	58.7878i		−	2.44314i
$580$		0			0
$581$	2.44949	−	6.00000i	0.101622	−	0.248922i
$582$		0			0
$583$			8.00000i			0.331326i
$584$		0			0
$585$	18.0000			0.744208
$586$		0			0
$587$		−	7.34847i		−	0.303304i	−0.988434	−	0.151652i	$-0.951541\pi$
$587$		−	7.34847i		−	0.303304i	0.988434	−	0.151652i	$-0.0484593\pi$
$588$		0			0
$589$		−	12.0000i		−	0.494451i
$590$		0			0
$591$	19.5959			0.806068
$592$		0			0
$593$		−	9.79796i		−	0.402354i	−0.979555	−	0.201177i	$-0.935523\pi$
$593$		−	9.79796i		−	0.402354i	0.979555	−	0.201177i	$-0.0644766\pi$
$594$		0			0
$595$	12.0000	−	29.3939i	0.491952	−	1.20503i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	14.0000i		−	0.572024i	−0.958226	−	0.286012i	$-0.907670\pi$
$599$		−	14.0000i		−	0.572024i	0.958226	−	0.286012i	$-0.0923298\pi$
$600$		0			0
$601$		−	24.4949i		−	0.999168i	−0.866266	−	0.499584i	$-0.833486\pi$
$601$		−	24.4949i		−	0.999168i	0.866266	−	0.499584i	$-0.166514\pi$
$602$		0			0
$603$	−6.00000			−0.244339
$604$		0			0
$605$	−17.1464			−0.697101
$606$		0			0
$607$	29.3939			1.19306			0.596530	−	0.802591i	$-0.296546\pi$
$607$	29.3939			1.19306			0.596530	+	0.802591i	$0.296546\pi$
$608$		0			0
$609$	24.0000	+	9.79796i	0.972529	+	0.397033i
$610$		0			0
$611$	−12.0000			−0.485468
$612$		0			0
$613$		−	24.0000i		−	0.969351i	−0.874694	−	0.484675i	$-0.838938\pi$
$613$		−	24.0000i		−	0.969351i	0.874694	−	0.484675i	$-0.161062\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−32.0000			−1.28827			−0.644136	−	0.764911i	$-0.722783\pi$
$617$	−32.0000			−1.28827			−0.644136	+	0.764911i	$0.722783\pi$
$618$		0			0
$619$		−	22.0454i		−	0.886080i	−0.896502	−	0.443040i	$-0.853900\pi$
$619$		−	22.0454i		−	0.886080i	0.896502	−	0.443040i	$-0.146100\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	14.6969	−	36.0000i	0.588820	−	1.44231i
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$	−12.0000			−0.479234
$628$		0			0
$629$	39.1918			1.56268
$630$		0			0
$631$		−	30.0000i		−	1.19428i	−0.802137	−	0.597141i	$-0.796303\pi$
$631$		−	30.0000i		−	1.19428i	0.802137	−	0.597141i	$-0.203697\pi$
$632$		0			0
$633$		−	44.0908i		−	1.75245i
$634$		0			0
$635$			29.3939i			1.16646i
$636$		0			0
$637$	−12.2474	−	12.0000i	−0.485262	−	0.475457i
$638$		0			0
$639$			30.0000i			1.18678i
$640$		0			0
$641$	−8.00000			−0.315981			−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000			−0.315981			−0.157991	+	0.987441i	$0.550502\pi$
$642$		0			0
$643$			22.0454i			0.869386i	0.900579	+	0.434693i	$0.143143\pi$
$643$			22.0454i			0.869386i	−0.900579	+	0.434693i	$0.856857\pi$
$644$		0			0
$645$		−	36.0000i		−	1.41750i
$646$		0			0
$647$	−19.5959			−0.770395			−0.385198	−	0.922834i	$-0.625867\pi$
$647$	−19.5959			−0.770395			−0.385198	+	0.922834i	$0.625867\pi$
$648$		0			0
$649$		−	4.89898i		−	0.192302i
$650$		0			0
$651$	−12.0000	+	29.3939i	−0.470317	+	1.15204i
$652$		0			0
$653$			16.0000i			0.626128i	0.949732	+	0.313064i	$0.101356\pi$
$653$			16.0000i			0.626128i	−0.949732	+	0.313064i	$0.898644\pi$
$654$		0			0
$655$		−	30.0000i		−	1.17220i
$656$		0			0
$657$		−	44.0908i		−	1.72015i
$658$		0			0
$659$	−10.0000			−0.389545			−0.194772	−	0.980848i	$-0.562397\pi$
$659$	−10.0000			−0.389545			−0.194772	+	0.980848i	$0.562397\pi$
$660$		0			0
$661$	−31.8434			−1.23856			−0.619282	−	0.785169i	$-0.712576\pi$
$661$	−31.8434			−1.23856			−0.619282	+	0.785169i	$0.712576\pi$
$662$		0			0
$663$	29.3939			1.14156
$664$		0			0
$665$	−6.00000	+	14.6969i	−0.232670	+	0.569923i
$666$		0			0
$667$	−16.0000			−0.619522
$668$		0			0
$669$			24.0000i			0.927894i
$670$		0			0
$671$	14.6969			0.567369
$672$		0			0
$673$	−6.00000			−0.231283			−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000			−0.231283			−0.115642	+	0.993291i	$0.536892\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.8434			1.22384			0.611920	−	0.790920i	$-0.290397\pi$
$677$	31.8434			1.22384			0.611920	+	0.790920i	$0.290397\pi$
$678$		0			0
$679$	4.89898	−	12.0000i	0.188006	−	0.460518i
$680$		0			0
$681$	−18.0000			−0.689761
$682$		0			0
$683$	−14.0000			−0.535695			−0.267848	−	0.963461i	$-0.586312\pi$
$683$	−14.0000			−0.535695			−0.267848	+	0.963461i	$0.586312\pi$
$684$		0			0
$685$	−4.89898			−0.187180
$686$		0			0
$687$			30.0000i			1.14457i
$688$		0			0
$689$			9.79796i			0.373273i
$690$		0			0
$691$			36.7423i			1.39774i	0.715246	+	0.698872i	$0.246314\pi$
$691$			36.7423i			1.39774i	−0.715246	+	0.698872i	$0.753686\pi$
$692$		0			0
$693$	14.6969	+	6.00000i	0.558291	+	0.227921i
$694$		0			0
$695$			6.00000i			0.227593i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		−	34.2929i		−	1.29707i
$700$		0			0
$701$		−	40.0000i		−	1.51078i	−0.655276	−	0.755390i	$-0.727448\pi$
$701$		−	40.0000i		−	1.51078i	0.655276	−	0.755390i	$-0.272552\pi$
$702$		0			0
$703$	−19.5959			−0.739074
$704$		0			0
$705$		−	29.3939i		−	1.10704i
$706$		0			0
$707$	42.0000	+	17.1464i	1.57957	+	0.644858i
$708$		0			0
$709$			4.00000i			0.150223i	0.997175	+	0.0751116i	$0.0239313\pi$
$709$			4.00000i			0.150223i	−0.997175	+	0.0751116i	$0.976069\pi$
$710$		0			0
$711$		−	18.0000i		−	0.675053i
$712$		0			0
$713$		−	19.5959i		−	0.733873i
$714$		0			0
$715$	12.0000			0.448775
$716$		0			0
$717$	−9.79796			−0.365911
$718$		0			0
$719$	−24.4949			−0.913506			−0.456753	−	0.889594i	$-0.650988\pi$
$719$	−24.4949			−0.913506			−0.456753	+	0.889594i	$0.650988\pi$
$720$		0			0
$721$	−24.0000	−	9.79796i	−0.893807	−	0.364895i
$722$		0			0
$723$	−60.0000			−2.23142
$724$		0			0
$725$			4.00000i			0.148556i
$726$		0			0
$727$	29.3939			1.09016			0.545079	−	0.838385i	$-0.316500\pi$
$727$	29.3939			1.09016			0.545079	+	0.838385i	$0.316500\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		−	29.3939i		−	1.08717i
$732$		0			0
$733$	22.0454			0.814266			0.407133	−	0.913369i	$-0.366529\pi$
$733$	22.0454			0.814266			0.407133	+	0.913369i	$0.366529\pi$
$734$		0			0
$735$	29.3939	−	30.0000i	1.08421	−	1.10657i
$736$		0			0
$737$	−4.00000			−0.147342
$738$		0			0
$739$	50.0000			1.83928			0.919640	−	0.392763i	$-0.128481\pi$
$739$	50.0000			1.83928			0.919640	+	0.392763i	$0.128481\pi$
$740$		0			0
$741$	−14.6969			−0.539906
$742$		0			0
$743$			44.0000i			1.61420i	0.590412	+	0.807102i	$0.298965\pi$
$743$			44.0000i			1.61420i	−0.590412	+	0.807102i	$0.701035\pi$
$744$		0			0
$745$		−	39.1918i		−	1.43588i
$746$		0			0
$747$			7.34847i			0.268866i
$748$		0			0
$749$	4.89898	+	2.00000i	0.179005	+	0.0730784i
$750$		0			0
$751$			20.0000i			0.729810i	0.931045	+	0.364905i	$0.118899\pi$
$751$			20.0000i			0.729810i	−0.931045	+	0.364905i	$0.881101\pi$
$752$		0			0
$753$	−30.0000			−1.09326
$754$		0			0
$755$		−	48.9898i		−	1.78292i
$756$		0			0
$757$			8.00000i			0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$			8.00000i			0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$		0			0
$759$	−19.5959			−0.711287
$760$		0			0
$761$			48.9898i			1.77588i	0.459961	+	0.887939i	$0.347864\pi$
$761$			48.9898i			1.77588i	−0.459961	+	0.887939i	$0.652136\pi$
$762$		0			0
$763$	−4.00000	+	9.79796i	−0.144810	+	0.354710i
$764$		0			0
$765$			36.0000i			1.30158i
$766$		0			0
$767$		−	6.00000i		−	0.216647i
$768$		0			0
$769$			34.2929i			1.23663i	0.785930	+	0.618316i	$0.212185\pi$
$769$			34.2929i			1.23663i	−0.785930	+	0.618316i	$0.787815\pi$
$770$		0			0
$771$	48.0000			1.72868
$772$		0			0
$773$	22.0454			0.792918			0.396459	−	0.918052i	$-0.370239\pi$
$773$	22.0454			0.792918			0.396459	+	0.918052i	$0.370239\pi$
$774$		0			0
$775$	−4.89898			−0.175977
$776$		0			0
$777$	48.0000	+	19.5959i	1.72199	+	0.703000i
$778$		0			0
$779$		0			0
$780$		0			0
$781$			20.0000i			0.715656i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.0000			0.642448
$786$		0			0
$787$		−	7.34847i		−	0.261945i	−0.991386	−	0.130972i	$-0.958190\pi$
$787$		−	7.34847i		−	0.261945i	0.991386	−	0.130972i	$-0.0418099\pi$
$788$		0			0
$789$	34.2929			1.22086
$790$		0			0
$791$	9.79796	+	4.00000i	0.348375	+	0.142224i
$792$		0			0
$793$	18.0000			0.639199
$794$		0			0
$795$	−24.0000			−0.851192
$796$		0			0
$797$	−41.6413			−1.47501			−0.737506	−	0.675341i	$-0.763997\pi$
$797$	−41.6413			−1.47501			−0.737506	+	0.675341i	$0.763997\pi$
$798$		0			0
$799$		−	24.0000i		−	0.849059i
$800$		0			0
$801$			44.0908i			1.55787i
$802$		0			0
$803$		−	29.3939i		−	1.03729i
$804$		0			0
$805$	−9.79796	+	24.0000i	−0.345333	+	0.845889i
$806$		0			0
$807$			30.0000i			1.05605i
$808$		0			0
$809$	20.0000			0.703163			0.351581	−	0.936157i	$-0.385644\pi$
$809$	20.0000			0.703163			0.351581	+	0.936157i	$0.385644\pi$
$810$		0			0
$811$			36.7423i			1.29020i	0.764099	+	0.645099i	$0.223184\pi$
$811$			36.7423i			1.29020i	−0.764099	+	0.645099i	$0.776816\pi$
$812$		0			0
$813$		−	48.0000i		−	1.68343i
$814$		0			0
$815$	−34.2929			−1.20123
$816$		0			0
$817$			14.6969i			0.514181i
$818$		0			0
$819$	18.0000	+	7.34847i	0.628971	+	0.256776i
$820$		0			0
$821$		−	20.0000i		−	0.698005i	−0.937122	−	0.349002i	$-0.886521\pi$
$821$		−	20.0000i		−	0.698005i	0.937122	−	0.349002i	$-0.113479\pi$
$822$		0			0
$823$			54.0000i			1.88232i	0.337959	+	0.941161i	$0.390263\pi$
$823$			54.0000i			1.88232i	−0.337959	+	0.941161i	$0.609737\pi$
$824$		0			0
$825$			4.89898i			0.170561i
$826$		0			0
$827$	22.0000			0.765015			0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000			0.765015			0.382507	+	0.923952i	$0.375061\pi$
$828$		0			0
$829$	−36.7423			−1.27611			−0.638057	−	0.769989i	$-0.720262\pi$
$829$	−36.7423			−1.27611			−0.638057	+	0.769989i	$0.720262\pi$
$830$		0			0
$831$	−29.3939			−1.01966
$832$		0			0
$833$	24.0000	−	24.4949i	0.831551	−	0.848698i
$834$		0			0
$835$	−48.0000			−1.66111
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	13.0000			0.448276
$842$		0			0
$843$		−	4.89898i		−	0.168730i
$844$		0			0
$845$	−17.1464			−0.589855
$846$		0			0
$847$	−17.1464	−	7.00000i	−0.589158	−	0.240523i
$848$		0			0
$849$	−6.00000			−0.205919
$850$		0			0
$851$	−32.0000			−1.09695
$852$		0			0
$853$	−2.44949			−0.0838689			−0.0419345	−	0.999120i	$-0.513352\pi$
$853$	−2.44949			−0.0838689			−0.0419345	+	0.999120i	$0.513352\pi$
$854$		0			0
$855$		−	18.0000i		−	0.615587i
$856$		0			0
$857$		−	29.3939i		−	1.00408i	−0.864846	−	0.502038i	$-0.832584\pi$
$857$		−	29.3939i		−	1.00408i	0.864846	−	0.502038i	$-0.167416\pi$
$858$		0			0
$859$			26.9444i			0.919331i	0.888092	+	0.459665i	$0.152031\pi$
$859$			26.9444i			0.919331i	−0.888092	+	0.459665i	$0.847969\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	46.0000i		−	1.56586i	−0.622111	−	0.782929i	$-0.713725\pi$
$863$		−	46.0000i		−	1.56586i	0.622111	−	0.782929i	$-0.286275\pi$
$864$		0			0
$865$	−6.00000			−0.204006
$866$		0			0
$867$			17.1464i			0.582323i
$868$		0			0
$869$		−	12.0000i		−	0.407072i
$870$		0			0
$871$	−4.89898			−0.165996
$872$		0			0
$873$			14.6969i			0.497416i
$874$		0			0
$875$	−24.0000	−	9.79796i	−0.811348	−	0.331231i
$876$		0			0
$877$			48.0000i			1.62084i	0.585846	+	0.810422i	$0.300762\pi$
$877$			48.0000i			1.62084i	−0.585846	+	0.810422i	$0.699238\pi$
$878$		0			0
$879$			6.00000i			0.202375i
$880$		0			0
$881$			48.9898i			1.65051i	0.564762	+	0.825254i	$0.308968\pi$
$881$			48.9898i			1.65051i	−0.564762	+	0.825254i	$0.691032\pi$
$882$		0			0
$883$	6.00000			0.201916			0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000			0.201916			0.100958	+	0.994891i	$0.467809\pi$
$884$		0			0
$885$	14.6969			0.494032
$886$		0			0
$887$	29.3939			0.986950			0.493475	−	0.869760i	$-0.335726\pi$
$887$	29.3939			0.986950			0.493475	+	0.869760i	$0.335726\pi$
$888$		0			0
$889$	−12.0000	+	29.3939i	−0.402467	+	0.985839i
$890$		0			0
$891$	18.0000			0.603023
$892$		0			0
$893$			12.0000i			0.401565i
$894$		0			0
$895$	−24.4949			−0.818774
$896$		0			0
$897$	−24.0000			−0.801337
$898$		0			0
$899$		−	19.5959i		−	0.653560i
$900$		0			0
$901$	−19.5959			−0.652835
$902$		0			0
$903$	14.6969	−	36.0000i	0.489083	−	1.19800i
$904$		0			0
$905$	−18.0000			−0.598340
$906$		0			0
$907$	42.0000			1.39459			0.697294	−	0.716786i	$-0.254387\pi$
$907$	42.0000			1.39459			0.697294	+	0.716786i	$0.254387\pi$
$908$		0			0
$909$	−51.4393			−1.70613
$910$		0			0
$911$		−	20.0000i		−	0.662630i	−0.943520	−	0.331315i	$-0.892508\pi$
$911$		−	20.0000i		−	0.662630i	0.943520	−	0.331315i	$-0.107492\pi$
$912$		0			0
$913$			4.89898i			0.162133i
$914$		0			0
$915$			44.0908i			1.45760i
$916$		0			0
$917$	12.2474	−	30.0000i	0.404446	−	0.990687i
$918$		0			0
$919$			46.0000i			1.51740i	0.651440	+	0.758700i	$0.274165\pi$
$919$			46.0000i			1.51740i	−0.651440	+	0.758700i	$0.725835\pi$
$920$		0			0
$921$	−78.0000			−2.57019
$922$		0			0
$923$			24.4949i			0.806259i
$924$		0			0
$925$			8.00000i			0.263038i
$926$		0			0
$927$	29.3939			0.965422
$928$		0			0
$929$		−	14.6969i		−	0.482191i	−0.970501	−	0.241095i	$-0.922493\pi$
$929$		−	14.6969i		−	0.482191i	0.970501	−	0.241095i	$-0.0775067\pi$
$930$		0			0
$931$	−12.0000	+	12.2474i	−0.393284	+	0.401394i
$932$		0			0
$933$			12.0000i			0.392862i
$934$		0			0
$935$			24.0000i			0.784884i
$936$		0			0
$937$		−	4.89898i		−	0.160043i	−0.996793	−	0.0800213i	$-0.974501\pi$
$937$		−	4.89898i		−	0.160043i	0.996793	−	0.0800213i	$-0.0254988\pi$
$938$		0			0
$939$	−24.0000			−0.783210
$940$		0			0
$941$	−31.8434			−1.03806			−0.519032	−	0.854755i	$-0.673707\pi$
$941$	−31.8434			−1.03806			−0.519032	+	0.854755i	$0.673707\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	22.0000			0.714904			0.357452	−	0.933932i	$-0.383646\pi$
$947$	22.0000			0.714904			0.357452	+	0.933932i	$0.383646\pi$
$948$		0			0
$949$		−	36.0000i		−	1.16861i
$950$		0			0
$951$	−78.3837			−2.54176
$952$		0			0
$953$	34.0000			1.10137			0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000			1.10137			0.550684	+	0.834714i	$0.314367\pi$
$954$		0			0
$955$			24.4949i			0.792636i
$956$		0			0
$957$	−19.5959			−0.633446
$958$		0			0
$959$	−4.89898	−	2.00000i	−0.158196	−	0.0645834i
$960$		0			0
$961$	−7.00000			−0.225806
$962$		0			0
$963$	−6.00000			−0.193347
$964$		0			0
$965$	58.7878			1.89244
$966$		0			0
$967$			12.0000i			0.385894i	0.981209	+	0.192947i	$0.0618045\pi$
$967$			12.0000i			0.385894i	−0.981209	+	0.192947i	$0.938195\pi$
$968$		0			0
$969$		−	29.3939i		−	0.944267i
$970$		0			0
$971$		−	36.7423i		−	1.17912i	−0.807725	−	0.589559i	$-0.799302\pi$
$971$		−	36.7423i		−	1.17912i	0.807725	−	0.589559i	$-0.200698\pi$
$972$		0			0
$973$	−2.44949	+	6.00000i	−0.0785270	+	0.192351i
$974$		0			0
$975$			6.00000i			0.192154i
$976$		0			0
$977$	38.0000			1.21573			0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000			1.21573			0.607864	+	0.794041i	$0.292027\pi$
$978$		0			0
$979$			29.3939i			0.939432i
$980$		0			0
$981$		−	12.0000i		−	0.383131i
$982$		0			0
$983$	39.1918			1.25003			0.625013	−	0.780615i	$-0.285094\pi$
$983$	39.1918			1.25003			0.625013	+	0.780615i	$0.285094\pi$
$984$		0			0
$985$			19.5959i			0.624378i
$986$		0			0
$987$	12.0000	−	29.3939i	0.381964	−	0.935617i
$988$		0			0
$989$			24.0000i			0.763156i
$990$		0			0
$991$		−	10.0000i		−	0.317660i	−0.987306	−	0.158830i	$-0.949228\pi$
$991$		−	10.0000i		−	0.317660i	0.987306	−	0.158830i	$-0.0507723\pi$
$992$		0			0
$993$		−	44.0908i		−	1.39918i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	31.8434			1.00849			0.504245	−	0.863561i	$-0.331771\pi$
$997$	31.8434			1.00849			0.504245	+	0.863561i	$0.331771\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.2.e.b.111.2		4
3.2	odd	2		2016.2.p.e.559.2		4
4.3	odd	2		56.2.e.b.27.2	yes	4
7.2	even	3		1568.2.q.e.815.3		8
7.3	odd	6		1568.2.q.e.1391.4		8
7.4	even	3		1568.2.q.e.1391.1		8
7.5	odd	6		1568.2.q.e.815.2		8
7.6	odd	2	inner	224.2.e.b.111.3		4
8.3	odd	2	inner	224.2.e.b.111.1		4
8.5	even	2		56.2.e.b.27.4	yes	4
12.11	even	2		504.2.p.f.307.3		4
16.3	odd	4		1792.2.f.e.1791.1		4
16.5	even	4		1792.2.f.f.1791.2		4
16.11	odd	4		1792.2.f.f.1791.4		4
16.13	even	4		1792.2.f.e.1791.3		4
21.20	even	2		2016.2.p.e.559.4		4
24.5	odd	2		504.2.p.f.307.2		4
24.11	even	2		2016.2.p.e.559.3		4
28.3	even	6		392.2.m.f.19.3		8
28.11	odd	6		392.2.m.f.19.4		8
28.19	even	6		392.2.m.f.227.2		8
28.23	odd	6		392.2.m.f.227.1		8
28.27	even	2		56.2.e.b.27.1	✓	4
56.3	even	6		1568.2.q.e.1391.3		8
56.5	odd	6		392.2.m.f.227.4		8
56.11	odd	6		1568.2.q.e.1391.2		8
56.13	odd	2		56.2.e.b.27.3	yes	4
56.19	even	6		1568.2.q.e.815.1		8
56.27	even	2	inner	224.2.e.b.111.4		4
56.37	even	6		392.2.m.f.227.3		8
56.45	odd	6		392.2.m.f.19.1		8
56.51	odd	6		1568.2.q.e.815.4		8
56.53	even	6		392.2.m.f.19.2		8
84.83	odd	2		504.2.p.f.307.4		4
112.13	odd	4		1792.2.f.e.1791.2		4
112.27	even	4		1792.2.f.f.1791.1		4
112.69	odd	4		1792.2.f.f.1791.3		4
112.83	even	4		1792.2.f.e.1791.4		4
168.83	odd	2		2016.2.p.e.559.1		4
168.125	even	2		504.2.p.f.307.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.e.b.27.1	✓	4	28.27	even	2
56.2.e.b.27.2	yes	4	4.3	odd	2
56.2.e.b.27.3	yes	4	56.13	odd	2
56.2.e.b.27.4	yes	4	8.5	even	2
224.2.e.b.111.1		4	8.3	odd	2	inner
224.2.e.b.111.2		4	1.1	even	1	trivial
224.2.e.b.111.3		4	7.6	odd	2	inner
224.2.e.b.111.4		4	56.27	even	2	inner
392.2.m.f.19.1		8	56.45	odd	6
392.2.m.f.19.2		8	56.53	even	6
392.2.m.f.19.3		8	28.3	even	6
392.2.m.f.19.4		8	28.11	odd	6
392.2.m.f.227.1		8	28.23	odd	6
392.2.m.f.227.2		8	28.19	even	6
392.2.m.f.227.3		8	56.37	even	6
392.2.m.f.227.4		8	56.5	odd	6
504.2.p.f.307.1		4	168.125	even	2
504.2.p.f.307.2		4	24.5	odd	2
504.2.p.f.307.3		4	12.11	even	2
504.2.p.f.307.4		4	84.83	odd	2
1568.2.q.e.815.1		8	56.19	even	6
1568.2.q.e.815.2		8	7.5	odd	6
1568.2.q.e.815.3		8	7.2	even	3
1568.2.q.e.815.4		8	56.51	odd	6
1568.2.q.e.1391.1		8	7.4	even	3
1568.2.q.e.1391.2		8	56.11	odd	6
1568.2.q.e.1391.3		8	56.3	even	6
1568.2.q.e.1391.4		8	7.3	odd	6
1792.2.f.e.1791.1		4	16.3	odd	4
1792.2.f.e.1791.2		4	112.13	odd	4
1792.2.f.e.1791.3		4	16.13	even	4
1792.2.f.e.1791.4		4	112.83	even	4
1792.2.f.f.1791.1		4	112.27	even	4
1792.2.f.f.1791.2		4	16.5	even	4
1792.2.f.f.1791.3		4	112.69	odd	4
1792.2.f.f.1791.4		4	16.11	odd	4
2016.2.p.e.559.1		4	168.83	odd	2
2016.2.p.e.559.2		4	3.2	odd	2
2016.2.p.e.559.3		4	24.11	even	2
2016.2.p.e.559.4		4	21.20	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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.44949i q^{3} +2.44949 q^{5} +(2.44949 + 1.00000i) q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-2.44949i q^{3} +2.44949 q^{5} +(2.44949 + 1.00000i) q^{7} -3.00000 q^{9} -2.00000 q^{11} -2.44949 q^{13} -6.00000i q^{15} -4.89898i q^{17} +2.44949i q^{19} +(2.44949 - 6.00000i) q^{21} +4.00000i q^{23} +1.00000 q^{25} +4.00000i q^{29} -4.89898 q^{31} +4.89898i q^{33} +(6.00000 + 2.44949i) q^{35} +8.00000i q^{37} +6.00000i q^{39} +6.00000 q^{43} -7.34847 q^{45} +4.89898 q^{47} +(5.00000 + 4.89898i) q^{49} -12.0000 q^{51} -4.00000i q^{53} -4.89898 q^{55} +6.00000 q^{57} +2.44949i q^{59} -7.34847 q^{61} +(-7.34847 - 3.00000i) q^{63} -6.00000 q^{65} +2.00000 q^{67} +9.79796 q^{69} -10.0000i q^{71} +14.6969i q^{73} -2.44949i q^{75} +(-4.89898 - 2.00000i) q^{77} +6.00000i q^{79} -9.00000 q^{81} -2.44949i q^{83} -12.0000i q^{85} +9.79796 q^{87} -14.6969i q^{89} +(-6.00000 - 2.44949i) q^{91} +12.0000i q^{93} +6.00000i q^{95} -4.89898i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} - 8 q^{11} + 4 q^{25} + 24 q^{35} + 24 q^{43} + 20 q^{49} - 48 q^{51} + 24 q^{57} - 24 q^{65} + 8 q^{67} - 36 q^{81} - 24 q^{91} + 24 q^{99}+O(q^{100}) \) 4 * q - 12 * q^9 - 8 * q^11 + 4 * q^25 + 24 * q^35 + 24 * q^43 + 20 * q^49 - 48 * q^51 + 24 * q^57 - 24 * q^65 + 8 * q^67 - 36 * q^81 - 24 * q^91 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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