LMFDB - Embedded newform 1792.2.f.f.1791.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(1791,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1792.1791");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.f (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:	$14.3091920422$
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			1791.4
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1792.1791
Dual form			1792.2.f.f.1791.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{3} +2.44949i q^{5} +(2.44949 + 1.00000i) q^{7} +3.00000 q^{9} +O(q^{10})$	$q+2.44949 q^{3} +2.44949i q^{5} +(2.44949 + 1.00000i) q^{7} +3.00000 q^{9} +2.00000i q^{11} +2.44949i q^{13} +6.00000i q^{15} -4.89898i q^{17} -2.44949 q^{19} +(6.00000 + 2.44949i) q^{21} +4.00000i q^{23} -1.00000 q^{25} +4.00000 q^{29} +4.89898 q^{31} +4.89898i q^{33} +(-2.44949 + 6.00000i) q^{35} -8.00000 q^{37} +6.00000i q^{39} -6.00000i q^{43} +7.34847i q^{45} -4.89898 q^{47} +(5.00000 + 4.89898i) q^{49} -12.0000i q^{51} +4.00000 q^{53} -4.89898 q^{55} -6.00000 q^{57} +2.44949 q^{59} +7.34847i q^{61} +(7.34847 + 3.00000i) q^{63} -6.00000 q^{65} +2.00000i q^{67} +9.79796i q^{69} -10.0000i q^{71} -14.6969i q^{73} -2.44949 q^{75} +(-2.00000 + 4.89898i) q^{77} -6.00000i q^{79} -9.00000 q^{81} +2.44949 q^{83} +12.0000 q^{85} +9.79796 q^{87} +14.6969i q^{89} +(-2.44949 + 6.00000i) q^{91} +12.0000 q^{93} -6.00000i q^{95} -4.89898i q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} + 24 q^{21} - 4 q^{25} + 16 q^{29} - 32 q^{37} + 20 q^{49} + 16 q^{53} - 24 q^{57} - 24 q^{65} - 8 q^{77} - 36 q^{81} + 48 q^{85} + 48 q^{93}+O(q^{100}) $ 4 * q + 12 * q^9 + 24 * q^21 - 4 * q^25 + 16 * q^29 - 32 * q^37 + 20 * q^49 + 16 * q^53 - 24 * q^57 - 24 * q^65 - 8 * q^77 - 36 * q^81 + 48 * q^85 + 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	2.44949			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.44949			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$4$		0			0
$5$			2.44949i			1.09545i	0.836660	+	0.547723i	$0.184505\pi$
$5$			2.44949i			1.09545i	−0.836660	+	0.547723i	$0.815495\pi$
$6$		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
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$			2.44949i			0.679366i	0.940540	+	0.339683i	$0.110320\pi$
$13$			2.44949i			0.679366i	−0.940540	+	0.339683i	$0.889680\pi$
$14$		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
$19$	−2.44949			−0.561951			−0.280976	−	0.959715i	$-0.590658\pi$
$19$	−2.44949			−0.561951			−0.280976	+	0.959715i	$0.590658\pi$
$20$		0			0
$21$	6.00000	+	2.44949i	1.30931	+	0.534522i
$22$		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
$29$	4.00000			0.742781			0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000			0.742781			0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$	4.89898			0.879883			0.439941	−	0.898027i	$-0.354999\pi$
$31$	4.89898			0.879883			0.439941	+	0.898027i	$0.354999\pi$
$32$		0			0
$33$			4.89898i			0.852803i
$34$		0			0
$35$	−2.44949	+	6.00000i	−0.414039	+	1.01419i
$36$		0			0
$37$	−8.00000			−1.31519			−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000			−1.31519			−0.657596	+	0.753371i	$0.728427\pi$
$38$		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
$43$		−	6.00000i		−	0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$		−	6.00000i		−	0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$		0			0
$45$			7.34847i			1.09545i
$46$		0			0
$47$	−4.89898			−0.714590			−0.357295	−	0.933992i	$-0.616301\pi$
$47$	−4.89898			−0.714590			−0.357295	+	0.933992i	$0.616301\pi$
$48$		0			0
$49$	5.00000	+	4.89898i	0.714286	+	0.699854i
$50$		0			0
$51$		−	12.0000i		−	1.68034i
$52$		0			0
$53$	4.00000			0.549442			0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000			0.549442			0.274721	+	0.961524i	$0.411414\pi$
$54$		0			0
$55$	−4.89898			−0.660578
$56$		0			0
$57$	−6.00000			−0.794719
$58$		0			0
$59$	2.44949			0.318896			0.159448	−	0.987206i	$-0.449029\pi$
$59$	2.44949			0.318896			0.159448	+	0.987206i	$0.449029\pi$
$60$		0			0
$61$			7.34847i			0.940875i	0.882433	+	0.470438i	$0.155904\pi$
$61$			7.34847i			0.940875i	−0.882433	+	0.470438i	$0.844096\pi$
$62$		0			0
$63$	7.34847	+	3.00000i	0.925820	+	0.377964i
$64$		0			0
$65$	−6.00000			−0.744208
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$			9.79796i			1.17954i
$70$		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	$-0.670418\pi$
$73$		−	14.6969i		−	1.72015i	0.510171	−	0.860073i	$-0.329582\pi$
$74$		0			0
$75$	−2.44949			−0.282843
$76$		0			0
$77$	−2.00000	+	4.89898i	−0.227921	+	0.558291i
$78$		0			0
$79$		−	6.00000i		−	0.675053i	−0.941316	−	0.337526i	$-0.890410\pi$
$79$		−	6.00000i		−	0.675053i	0.941316	−	0.337526i	$-0.109590\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$	2.44949			0.268866			0.134433	−	0.990923i	$-0.457079\pi$
$83$	2.44949			0.268866			0.134433	+	0.990923i	$0.457079\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$		0			0
$87$	9.79796			1.05045
$88$		0			0
$89$			14.6969i			1.55787i	0.627103	+	0.778936i	$0.284240\pi$
$89$			14.6969i			1.55787i	−0.627103	+	0.778936i	$0.715760\pi$
$90$		0			0
$91$	−2.44949	+	6.00000i	−0.256776	+	0.628971i
$92$		0			0
$93$	12.0000			1.24434
$94$		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
$99$			6.00000i			0.603023i
$100$		0			0
$101$			17.1464i			1.70613i	0.521802	+	0.853067i	$0.325260\pi$
$101$			17.1464i			1.70613i	−0.521802	+	0.853067i	$0.674740\pi$
$102$		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
$107$		−	2.00000i		−	0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$		−	2.00000i		−	0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$		0			0
$109$	4.00000			0.383131			0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000			0.383131			0.191565	+	0.981480i	$0.438644\pi$
$110$		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
$115$	−9.79796			−0.913664
$116$		0			0
$117$			7.34847i			0.679366i
$118$		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
$125$			9.79796i			0.876356i
$126$		0			0
$127$		−	12.0000i		−	1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$		−	12.0000i		−	1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$		0			0
$129$		−	14.6969i		−	1.29399i
$130$		0			0
$131$	12.2474			1.07006			0.535032	−	0.844832i	$-0.320299\pi$
$131$	12.2474			1.07006			0.535032	+	0.844832i	$0.320299\pi$
$132$		0			0
$133$	−6.00000	−	2.44949i	−0.520266	−	0.212398i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.00000			0.170872			0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000			0.170872			0.0854358	+	0.996344i	$0.472772\pi$
$138$		0			0
$139$	2.44949			0.207763			0.103882	−	0.994590i	$-0.466874\pi$
$139$	2.44949			0.207763			0.103882	+	0.994590i	$0.466874\pi$
$140$		0			0
$141$	−12.0000			−1.01058
$142$		0			0
$143$	−4.89898			−0.409673
$144$		0			0
$145$			9.79796i			0.813676i
$146$		0			0
$147$	12.2474	+	12.0000i	1.01015	+	0.989743i
$148$		0			0
$149$	16.0000			1.31077			0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000			1.31077			0.655386	+	0.755295i	$0.272506\pi$
$150$		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
$155$			12.0000i			0.963863i
$156$		0			0
$157$		−	7.34847i		−	0.586472i	−0.956040	−	0.293236i	$-0.905268\pi$
$157$		−	7.34847i		−	0.586472i	0.956040	−	0.293236i	$-0.0947321\pi$
$158$		0			0
$159$	9.79796			0.777029
$160$		0			0
$161$	−4.00000	+	9.79796i	−0.315244	+	0.772187i
$162$		0			0
$163$		−	14.0000i		−	1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$		−	14.0000i		−	1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$		0			0
$165$	−12.0000			−0.934199
$166$		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
$171$	−7.34847			−0.561951
$172$		0			0
$173$			2.44949i			0.186231i	0.995655	+	0.0931156i	$0.0296826\pi$
$173$			2.44949i			0.186231i	−0.995655	+	0.0931156i	$0.970317\pi$
$174$		0			0
$175$	−2.44949	−	1.00000i	−0.185164	−	0.0755929i
$176$		0			0
$177$	6.00000			0.450988
$178$		0			0
$179$		−	10.0000i		−	0.747435i	−0.927543	−	0.373718i	$-0.878083\pi$
$179$		−	10.0000i		−	0.747435i	0.927543	−	0.373718i	$-0.121917\pi$
$180$		0			0
$181$		−	7.34847i		−	0.546207i	−0.961985	−	0.273104i	$-0.911950\pi$
$181$		−	7.34847i		−	0.546207i	0.961985	−	0.273104i	$-0.0880502\pi$
$182$		0			0
$183$			18.0000i			1.33060i
$184$		0			0
$185$		−	19.5959i		−	1.44072i
$186$		0			0
$187$	9.79796			0.716498
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	10.0000i		−	0.723575i	−0.932261	−	0.361787i	$-0.882167\pi$
$191$		−	10.0000i		−	0.723575i	0.932261	−	0.361787i	$-0.117833\pi$
$192$		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
$195$	−14.6969			−1.05247
$196$		0			0
$197$	−8.00000			−0.569976			−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000			−0.569976			−0.284988	+	0.958531i	$0.591990\pi$
$198$		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
$203$	9.79796	+	4.00000i	0.687682	+	0.280745i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			12.0000i			0.834058i
$208$		0			0
$209$		−	4.89898i		−	0.338869i
$210$		0			0
$211$			18.0000i			1.23917i	0.784929	+	0.619586i	$0.212699\pi$
$211$			18.0000i			1.23917i	−0.784929	+	0.619586i	$0.787301\pi$
$212$		0			0
$213$		−	24.4949i		−	1.67836i
$214$		0			0
$215$	14.6969			1.00232
$216$		0			0
$217$	12.0000	+	4.89898i	0.814613	+	0.332564i
$218$		0			0
$219$		−	36.0000i		−	2.43265i
$220$		0			0
$221$	12.0000			0.807207
$222$		0			0
$223$	9.79796			0.656120			0.328060	−	0.944657i	$-0.393605\pi$
$223$	9.79796			0.656120			0.328060	+	0.944657i	$0.393605\pi$
$224$		0			0
$225$	−3.00000			−0.200000
$226$		0			0
$227$	7.34847			0.487735			0.243868	−	0.969809i	$-0.421584\pi$
$227$	7.34847			0.487735			0.243868	+	0.969809i	$0.421584\pi$
$228$		0			0
$229$		−	12.2474i		−	0.809334i	−0.914464	−	0.404667i	$-0.867387\pi$
$229$		−	12.2474i		−	0.809334i	0.914464	−	0.404667i	$-0.132613\pi$
$230$		0			0
$231$	−4.89898	+	12.0000i	−0.322329	+	0.789542i
$232$		0			0
$233$	−14.0000			−0.917170			−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000			−0.917170			−0.458585	+	0.888650i	$0.651644\pi$
$234$		0			0
$235$		−	12.0000i		−	0.782794i
$236$		0			0
$237$		−	14.6969i		−	0.954669i
$238$		0			0
$239$			4.00000i			0.258738i	0.991596	+	0.129369i	$0.0412952\pi$
$239$			4.00000i			0.258738i	−0.991596	+	0.129369i	$0.958705\pi$
$240$		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
$243$	−22.0454			−1.41421
$244$		0			0
$245$	−12.0000	+	12.2474i	−0.766652	+	0.782461i
$246$		0			0
$247$		−	6.00000i		−	0.381771i
$248$		0			0
$249$	6.00000			0.380235
$250$		0			0
$251$	−12.2474			−0.773052			−0.386526	−	0.922278i	$-0.626325\pi$
$251$	−12.2474			−0.773052			−0.386526	+	0.922278i	$0.626325\pi$
$252$		0			0
$253$	−8.00000			−0.502956
$254$		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
$259$	−19.5959	−	8.00000i	−1.21763	−	0.497096i
$260$		0			0
$261$	12.0000			0.742781
$262$		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
$267$			36.0000i			2.20316i
$268$		0			0
$269$			12.2474i			0.746740i	0.927682	+	0.373370i	$0.121798\pi$
$269$			12.2474i			0.746740i	−0.927682	+	0.373370i	$0.878202\pi$
$270$		0			0
$271$	−19.5959			−1.19037			−0.595184	−	0.803590i	$-0.702921\pi$
$271$	−19.5959			−1.19037			−0.595184	+	0.803590i	$0.702921\pi$
$272$		0			0
$273$	−6.00000	+	14.6969i	−0.363137	+	0.889499i
$274$		0			0
$275$		−	2.00000i		−	0.120605i
$276$		0			0
$277$	12.0000			0.721010			0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000			0.721010			0.360505	+	0.932757i	$0.382604\pi$
$278$		0			0
$279$	14.6969			0.879883
$280$		0			0
$281$	−2.00000			−0.119310			−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000			−0.119310			−0.0596550	+	0.998219i	$0.519000\pi$
$282$		0			0
$283$	−2.44949			−0.145607			−0.0728035	−	0.997346i	$-0.523195\pi$
$283$	−2.44949			−0.145607			−0.0728035	+	0.997346i	$0.523195\pi$
$284$		0			0
$285$		−	14.6969i		−	0.870572i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−7.00000			−0.411765
$290$		0			0
$291$		−	12.0000i		−	0.703452i
$292$		0			0
$293$		−	2.44949i		−	0.143101i	−0.997437	−	0.0715504i	$-0.977205\pi$
$293$		−	2.44949i		−	0.143101i	0.997437	−	0.0715504i	$-0.0227947\pi$
$294$		0			0
$295$			6.00000i			0.349334i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−9.79796			−0.566631
$300$		0			0
$301$	6.00000	−	14.6969i	0.345834	−	0.847117i
$302$		0			0
$303$			42.0000i			2.41284i
$304$		0			0
$305$	−18.0000			−1.03068
$306$		0			0
$307$	31.8434			1.81740			0.908698	−	0.417453i	$-0.137077\pi$
$307$	31.8434			1.81740			0.908698	+	0.417453i	$0.137077\pi$
$308$		0			0
$309$	−24.0000			−1.36531
$310$		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	$0.0893093\pi$
$313$			9.79796i			0.553813i	−0.960897	+	0.276907i	$0.910691\pi$
$314$		0			0
$315$	−7.34847	+	18.0000i	−0.414039	+	1.01419i
$316$		0			0
$317$	−32.0000			−1.79730			−0.898650	−	0.438667i	$-0.855451\pi$
$317$	−32.0000			−1.79730			−0.898650	+	0.438667i	$0.855451\pi$
$318$		0			0
$319$			8.00000i			0.447914i
$320$		0			0
$321$		−	4.89898i		−	0.273434i
$322$		0			0
$323$			12.0000i			0.667698i
$324$		0			0
$325$		−	2.44949i		−	0.135873i
$326$		0			0
$327$	9.79796			0.541828
$328$		0			0
$329$	−12.0000	−	4.89898i	−0.661581	−	0.270089i
$330$		0			0
$331$		−	18.0000i		−	0.989369i	−0.869072	−	0.494685i	$-0.835284\pi$
$331$		−	18.0000i		−	0.989369i	0.869072	−	0.494685i	$-0.164716\pi$
$332$		0			0
$333$	−24.0000			−1.31519
$334$		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
$339$	9.79796			0.532152
$340$		0			0
$341$			9.79796i			0.530589i
$342$		0			0
$343$	7.34847	+	17.0000i	0.396780	+	0.917914i
$344$		0			0
$345$	−24.0000			−1.29212
$346$		0			0
$347$		−	22.0000i		−	1.18102i	−0.807030	−	0.590511i	$-0.798926\pi$
$347$		−	22.0000i		−	1.18102i	0.807030	−	0.590511i	$-0.201074\pi$
$348$		0			0
$349$		−	12.2474i		−	0.655591i	−0.944749	−	0.327795i	$-0.893694\pi$
$349$		−	12.2474i		−	0.655591i	0.944749	−	0.327795i	$-0.106306\pi$
$350$		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
$355$	24.4949			1.30005
$356$		0			0
$357$	12.0000	−	29.3939i	0.635107	−	1.55569i
$358$		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
$363$	17.1464			0.899954
$364$		0			0
$365$	36.0000			1.88433
$366$		0			0
$367$	−29.3939			−1.53435			−0.767174	−	0.641439i	$-0.778338\pi$
$367$	−29.3939			−1.53435			−0.767174	+	0.641439i	$0.778338\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.79796	+	4.00000i	0.508685	+	0.207670i
$372$		0			0
$373$	−36.0000			−1.86401			−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−36.0000			−1.86401			−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$			24.0000i			1.23935i
$376$		0			0
$377$			9.79796i			0.504621i
$378$		0			0
$379$			30.0000i			1.54100i	0.637442	+	0.770498i	$0.279993\pi$
$379$			30.0000i			1.54100i	−0.637442	+	0.770498i	$0.720007\pi$
$380$		0			0
$381$		−	29.3939i		−	1.50589i
$382$		0			0
$383$	34.2929			1.75228			0.876142	−	0.482054i	$-0.160109\pi$
$383$	34.2929			1.75228			0.876142	+	0.482054i	$0.160109\pi$
$384$		0			0
$385$	−12.0000	−	4.89898i	−0.611577	−	0.249675i
$386$		0			0
$387$		−	18.0000i		−	0.914991i
$388$		0			0
$389$	16.0000			0.811232			0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000			0.811232			0.405616	+	0.914044i	$0.367057\pi$
$390$		0			0
$391$	19.5959			0.991008
$392$		0			0
$393$	30.0000			1.51330
$394$		0			0
$395$	14.6969			0.739483
$396$		0			0
$397$		−	31.8434i		−	1.59817i	−0.601216	−	0.799086i	$-0.705317\pi$
$397$		−	31.8434i		−	1.59817i	0.601216	−	0.799086i	$-0.294683\pi$
$398$		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
$403$			12.0000i			0.597763i
$404$		0			0
$405$		−	22.0454i		−	1.09545i
$406$		0			0
$407$		−	16.0000i		−	0.793091i
$408$		0			0
$409$		−	9.79796i		−	0.484478i	−0.970217	−	0.242239i	$-0.922118\pi$
$409$		−	9.79796i		−	0.484478i	0.970217	−	0.242239i	$-0.0778818\pi$
$410$		0			0
$411$	4.89898			0.241649
$412$		0			0
$413$	6.00000	+	2.44949i	0.295241	+	0.120532i
$414$		0			0
$415$			6.00000i			0.294528i
$416$		0			0
$417$	6.00000			0.293821
$418$		0			0
$419$	−26.9444			−1.31632			−0.658160	−	0.752878i	$-0.728665\pi$
$419$	−26.9444			−1.31632			−0.658160	+	0.752878i	$0.728665\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$	−14.6969			−0.714590
$424$		0			0
$425$			4.89898i			0.237635i
$426$		0			0
$427$	−7.34847	+	18.0000i	−0.355617	+	0.871081i
$428$		0			0
$429$	−12.0000			−0.579365
$430$		0			0
$431$		−	20.0000i		−	0.963366i	−0.876346	−	0.481683i	$-0.840026\pi$
$431$		−	20.0000i		−	0.963366i	0.876346	−	0.481683i	$-0.159974\pi$
$432$		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
$435$			24.0000i			1.15071i
$436$		0			0
$437$		−	9.79796i		−	0.468700i
$438$		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
$443$		−	26.0000i		−	1.23530i	−0.786454	−	0.617649i	$-0.788085\pi$
$443$		−	26.0000i		−	1.23530i	0.786454	−	0.617649i	$-0.211915\pi$
$444$		0			0
$445$	−36.0000			−1.70656
$446$		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
$453$		−	48.9898i		−	2.30174i
$454$		0			0
$455$	−14.6969	−	6.00000i	−0.689003	−	0.281284i
$456$		0			0
$457$	12.0000			0.561336			0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000			0.561336			0.280668	+	0.959805i	$0.409444\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			7.34847i			0.342252i	0.985249	+	0.171126i	$0.0547406\pi$
$461$			7.34847i			0.342252i	−0.985249	+	0.171126i	$0.945259\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$		0			0
$465$			29.3939i			1.36311i
$466$		0			0
$467$	−41.6413			−1.92693			−0.963465	−	0.267833i	$-0.913692\pi$
$467$	−41.6413			−1.92693			−0.963465	+	0.267833i	$0.913692\pi$
$468$		0			0
$469$	−2.00000	+	4.89898i	−0.0923514	+	0.226214i
$470$		0			0
$471$		−	18.0000i		−	0.829396i
$472$		0			0
$473$	12.0000			0.551761
$474$		0			0
$475$	2.44949			0.112390
$476$		0			0
$477$	12.0000			0.549442
$478$		0			0
$479$	−24.4949			−1.11920			−0.559600	−	0.828763i	$-0.689045\pi$
$479$	−24.4949			−1.11920			−0.559600	+	0.828763i	$0.689045\pi$
$480$		0			0
$481$		−	19.5959i		−	0.893497i
$482$		0			0
$483$	−9.79796	+	24.0000i	−0.445823	+	1.09204i
$484$		0			0
$485$	12.0000			0.544892
$486$		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
$491$			2.00000i			0.0902587i	0.998981	+	0.0451294i	$0.0143700\pi$
$491$			2.00000i			0.0902587i	−0.998981	+	0.0451294i	$0.985630\pi$
$492$		0			0
$493$		−	19.5959i		−	0.882556i
$494$		0			0
$495$	−14.6969			−0.660578
$496$		0			0
$497$	10.0000	−	24.4949i	0.448561	−	1.09875i
$498$		0			0
$499$		−	30.0000i		−	1.34298i	−0.741012	−	0.671492i	$-0.765654\pi$
$499$		−	30.0000i		−	1.34298i	0.741012	−	0.671492i	$-0.234346\pi$
$500$		0			0
$501$	−48.0000			−2.14448
$502$		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
$507$	17.1464			0.761500
$508$		0			0
$509$			36.7423i			1.62858i	0.580461	+	0.814288i	$0.302872\pi$
$509$			36.7423i			1.62858i	−0.580461	+	0.814288i	$0.697128\pi$
$510$		0			0
$511$	14.6969	−	36.0000i	0.650154	−	1.59255i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	24.0000i		−	1.05757i
$516$		0			0
$517$		−	9.79796i		−	0.430914i
$518$		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
$523$	−26.9444			−1.17820			−0.589098	−	0.808062i	$-0.700517\pi$
$523$	−26.9444			−1.17820			−0.589098	+	0.808062i	$0.700517\pi$
$524$		0			0
$525$	−6.00000	−	2.44949i	−0.261861	−	0.106904i
$526$		0			0
$527$		−	24.0000i		−	1.04546i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	7.34847			0.318896
$532$		0			0
$533$		0			0
$534$		0			0
$535$	4.89898			0.211801
$536$		0			0
$537$		−	24.4949i		−	1.05703i
$538$		0			0
$539$	−9.79796	+	10.0000i	−0.422028	+	0.430730i
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		−	18.0000i		−	0.772454i
$544$		0			0
$545$			9.79796i			0.419698i
$546$		0			0
$547$			2.00000i			0.0855138i	0.999086	+	0.0427569i	$0.0136141\pi$
$547$			2.00000i			0.0855138i	−0.999086	+	0.0427569i	$0.986386\pi$
$548$		0			0
$549$			22.0454i			0.940875i
$550$		0			0
$551$	−9.79796			−0.417407
$552$		0			0
$553$	6.00000	−	14.6969i	0.255146	−	0.624977i
$554$		0			0
$555$		−	48.0000i		−	2.03749i
$556$		0			0
$557$	28.0000			1.18640			0.593199	−	0.805056i	$-0.297865\pi$
$557$	28.0000			1.18640			0.593199	+	0.805056i	$0.297865\pi$
$558$		0			0
$559$	14.6969			0.621614
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$	−22.0454			−0.929103			−0.464552	−	0.885546i	$-0.653784\pi$
$563$	−22.0454			−0.929103			−0.464552	+	0.885546i	$0.653784\pi$
$564$		0			0
$565$			9.79796i			0.412203i
$566$		0			0
$567$	−22.0454	−	9.00000i	−0.925820	−	0.377964i
$568$		0			0
$569$	20.0000			0.838444			0.419222	−	0.907884i	$-0.362303\pi$
$569$	20.0000			0.838444			0.419222	+	0.907884i	$0.362303\pi$
$570$		0			0
$571$			2.00000i			0.0836974i	0.999124	+	0.0418487i	$0.0133247\pi$
$571$			2.00000i			0.0836974i	−0.999124	+	0.0418487i	$0.986675\pi$
$572$		0			0
$573$		−	24.4949i		−	1.02329i
$574$		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
$579$	58.7878			2.44314
$580$		0			0
$581$	6.00000	+	2.44949i	0.248922	+	0.101622i
$582$		0			0
$583$			8.00000i			0.331326i
$584$		0			0
$585$	−18.0000			−0.744208
$586$		0			0
$587$	−7.34847			−0.303304			−0.151652	−	0.988434i	$-0.548459\pi$
$587$	−7.34847			−0.303304			−0.151652	+	0.988434i	$0.548459\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		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
$595$	29.3939	+	12.0000i	1.20503	+	0.491952i
$596$		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	$0.166514\pi$
$601$			24.4949i			0.999168i	−0.866266	+	0.499584i	$0.833486\pi$
$602$		0			0
$603$			6.00000i			0.244339i
$604$		0			0
$605$			17.1464i			0.697101i
$606$		0			0
$607$	−29.3939			−1.19306			−0.596530	−	0.802591i	$-0.703454\pi$
$607$	−29.3939			−1.19306			−0.596530	+	0.802591i	$0.703454\pi$
$608$		0			0
$609$	24.0000	+	9.79796i	0.972529	+	0.397033i
$610$		0			0
$611$		−	12.0000i		−	0.485468i
$612$		0			0
$613$	24.0000			0.969351			0.484675	−	0.874694i	$-0.338938\pi$
$613$	24.0000			0.969351			0.484675	+	0.874694i	$0.338938\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	32.0000			1.28827			0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000			1.28827			0.644136	+	0.764911i	$0.277217\pi$
$618$		0			0
$619$	−22.0454			−0.886080			−0.443040	−	0.896502i	$-0.646100\pi$
$619$	−22.0454			−0.886080			−0.443040	+	0.896502i	$0.646100\pi$
$620$		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
$627$		−	12.0000i		−	0.479234i
$628$		0			0
$629$			39.1918i			1.56268i
$630$		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
$635$	29.3939			1.16646
$636$		0			0
$637$	−12.0000	+	12.2474i	−0.475457	+	0.485262i
$638$		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
$643$	−22.0454			−0.869386			−0.434693	−	0.900579i	$-0.643143\pi$
$643$	−22.0454			−0.869386			−0.434693	+	0.900579i	$0.643143\pi$
$644$		0			0
$645$	36.0000			1.41750
$646$		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
$651$	29.3939	+	12.0000i	1.15204	+	0.470317i
$652$		0			0
$653$	16.0000			0.626128			0.313064	−	0.949732i	$-0.398644\pi$
$653$	16.0000			0.626128			0.313064	+	0.949732i	$0.398644\pi$
$654$		0			0
$655$			30.0000i			1.17220i
$656$		0			0
$657$		−	44.0908i		−	1.72015i
$658$		0			0
$659$		−	10.0000i		−	0.389545i	−0.980848	−	0.194772i	$-0.937603\pi$
$659$		−	10.0000i		−	0.389545i	0.980848	−	0.194772i	$-0.0623968\pi$
$660$		0			0
$661$		−	31.8434i		−	1.23856i	−0.785169	−	0.619282i	$-0.787424\pi$
$661$		−	31.8434i		−	1.23856i	0.785169	−	0.619282i	$-0.212576\pi$
$662$		0			0
$663$	29.3939			1.14156
$664$		0			0
$665$	6.00000	−	14.6969i	0.232670	−	0.569923i
$666$		0			0
$667$			16.0000i			0.619522i
$668$		0			0
$669$	24.0000			0.927894
$670$		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
$677$			31.8434i			1.22384i	0.790920	+	0.611920i	$0.209603\pi$
$677$			31.8434i			1.22384i	−0.790920	+	0.611920i	$0.790397\pi$
$678$		0			0
$679$	4.89898	−	12.0000i	0.188006	−	0.460518i
$680$		0			0
$681$	18.0000			0.689761
$682$		0			0
$683$			14.0000i			0.535695i	0.963461	+	0.267848i	$0.0863124\pi$
$683$			14.0000i			0.535695i	−0.963461	+	0.267848i	$0.913688\pi$
$684$		0			0
$685$			4.89898i			0.187180i
$686$		0			0
$687$		−	30.0000i		−	1.14457i
$688$		0			0
$689$			9.79796i			0.373273i
$690$		0			0
$691$	−36.7423			−1.39774			−0.698872	−	0.715246i	$-0.746314\pi$
$691$	−36.7423			−1.39774			−0.698872	+	0.715246i	$0.746314\pi$
$692$		0			0
$693$	−6.00000	+	14.6969i	−0.227921	+	0.558291i
$694$		0			0
$695$			6.00000i			0.227593i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−34.2929			−1.29707
$700$		0			0
$701$	−40.0000			−1.51078			−0.755390	−	0.655276i	$-0.772552\pi$
$701$	−40.0000			−1.51078			−0.755390	+	0.655276i	$0.772552\pi$
$702$		0			0
$703$	19.5959			0.739074
$704$		0			0
$705$		−	29.3939i		−	1.10704i
$706$		0			0
$707$	−17.1464	+	42.0000i	−0.644858	+	1.57957i
$708$		0			0
$709$	−4.00000			−0.150223			−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000			−0.150223			−0.0751116	+	0.997175i	$0.523931\pi$
$710$		0			0
$711$		−	18.0000i		−	0.675053i
$712$		0			0
$713$			19.5959i			0.733873i
$714$		0			0
$715$		−	12.0000i		−	0.448775i
$716$		0			0
$717$			9.79796i			0.365911i
$718$		0			0
$719$	24.4949			0.913506			0.456753	−	0.889594i	$-0.349012\pi$
$719$	24.4949			0.913506			0.456753	+	0.889594i	$0.349012\pi$
$720$		0			0
$721$	−24.0000	−	9.79796i	−0.893807	−	0.364895i
$722$		0			0
$723$		−	60.0000i		−	2.23142i
$724$		0			0
$725$	−4.00000			−0.148556
$726$		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
$731$	−29.3939			−1.08717
$732$		0			0
$733$		−	22.0454i		−	0.814266i	−0.913369	−	0.407133i	$-0.866529\pi$
$733$		−	22.0454i		−	0.814266i	0.913369	−	0.407133i	$-0.133471\pi$
$734$		0			0
$735$	−29.3939	+	30.0000i	−1.08421	+	1.10657i
$736$		0			0
$737$	−4.00000			−0.147342
$738$		0			0
$739$			50.0000i			1.83928i	0.392763	+	0.919640i	$0.371519\pi$
$739$			50.0000i			1.83928i	−0.392763	+	0.919640i	$0.628481\pi$
$740$		0			0
$741$		−	14.6969i		−	0.539906i
$742$		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
$747$	7.34847			0.268866
$748$		0			0
$749$	2.00000	−	4.89898i	0.0730784	−	0.179005i
$750$		0			0
$751$		−	20.0000i		−	0.729810i	−0.931045	−	0.364905i	$-0.881101\pi$
$751$		−	20.0000i		−	0.729810i	0.931045	−	0.364905i	$-0.118899\pi$
$752$		0			0
$753$	−30.0000			−1.09326
$754$		0			0
$755$	48.9898			1.78292
$756$		0			0
$757$	−8.00000			−0.290765			−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000			−0.290765			−0.145382	+	0.989376i	$0.546441\pi$
$758$		0			0
$759$	−19.5959			−0.711287
$760$		0			0
$761$		−	48.9898i		−	1.77588i	−0.459961	−	0.887939i	$-0.652136\pi$
$761$		−	48.9898i		−	1.77588i	0.459961	−	0.887939i	$-0.347864\pi$
$762$		0			0
$763$	9.79796	+	4.00000i	0.354710	+	0.144810i
$764$		0			0
$765$	36.0000			1.30158
$766$		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
$771$			48.0000i			1.72868i
$772$		0			0
$773$			22.0454i			0.792918i	0.918052	+	0.396459i	$0.129761\pi$
$773$			22.0454i			0.792918i	−0.918052	+	0.396459i	$0.870239\pi$
$774$		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
$781$	20.0000			0.715656
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.0000			0.642448
$786$		0			0
$787$	7.34847			0.261945			0.130972	−	0.991386i	$-0.458190\pi$
$787$	7.34847			0.261945			0.130972	+	0.991386i	$0.458190\pi$
$788$		0			0
$789$			34.2929i			1.22086i
$790$		0			0
$791$	9.79796	+	4.00000i	0.348375	+	0.142224i
$792$		0			0
$793$	−18.0000			−0.639199
$794$		0			0
$795$			24.0000i			0.851192i
$796$		0			0
$797$			41.6413i			1.47501i	0.675341	+	0.737506i	$0.263997\pi$
$797$			41.6413i			1.47501i	−0.675341	+	0.737506i	$0.736003\pi$
$798$		0			0
$799$			24.0000i			0.849059i
$800$		0			0
$801$			44.0908i			1.55787i
$802$		0			0
$803$	29.3939			1.03729
$804$		0			0
$805$	−24.0000	−	9.79796i	−0.845889	−	0.345333i
$806$		0			0
$807$			30.0000i			1.05605i
$808$		0			0
$809$	−20.0000			−0.703163			−0.351581	−	0.936157i	$-0.614356\pi$
$809$	−20.0000			−0.703163			−0.351581	+	0.936157i	$0.614356\pi$
$810$		0			0
$811$	36.7423			1.29020			0.645099	−	0.764099i	$-0.276816\pi$
$811$	36.7423			1.29020			0.645099	+	0.764099i	$0.276816\pi$
$812$		0			0
$813$	−48.0000			−1.68343
$814$		0			0
$815$	34.2929			1.20123
$816$		0			0
$817$			14.6969i			0.514181i
$818$		0			0
$819$	−7.34847	+	18.0000i	−0.256776	+	0.628971i
$820$		0			0
$821$	20.0000			0.698005			0.349002	−	0.937122i	$-0.386521\pi$
$821$	20.0000			0.698005			0.349002	+	0.937122i	$0.386521\pi$
$822$		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
$827$		−	22.0000i		−	0.765015i	−0.923952	−	0.382507i	$-0.875061\pi$
$827$		−	22.0000i		−	0.765015i	0.923952	−	0.382507i	$-0.124939\pi$
$828$		0			0
$829$			36.7423i			1.27611i	0.769989	+	0.638057i	$0.220262\pi$
$829$			36.7423i			1.27611i	−0.769989	+	0.638057i	$0.779738\pi$
$830$		0			0
$831$	29.3939			1.01966
$832$		0			0
$833$	24.0000	−	24.4949i	0.831551	−	0.848698i
$834$		0			0
$835$		−	48.0000i		−	1.66111i
$836$		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
$843$	−4.89898			−0.168730
$844$		0			0
$845$			17.1464i			0.589855i
$846$		0			0
$847$	17.1464	+	7.00000i	0.589158	+	0.240523i
$848$		0			0
$849$	−6.00000			−0.205919
$850$		0			0
$851$		−	32.0000i		−	1.09695i
$852$		0			0
$853$		−	2.44949i		−	0.0838689i	−0.999120	−	0.0419345i	$-0.986648\pi$
$853$		−	2.44949i		−	0.0838689i	0.999120	−	0.0419345i	$-0.0133521\pi$
$854$		0			0
$855$		−	18.0000i		−	0.615587i
$856$		0			0
$857$			29.3939i			1.00408i	0.864846	+	0.502038i	$0.167416\pi$
$857$			29.3939i			1.00408i	−0.864846	+	0.502038i	$0.832584\pi$
$858$		0			0
$859$	26.9444			0.919331			0.459665	−	0.888092i	$-0.347969\pi$
$859$	26.9444			0.919331			0.459665	+	0.888092i	$0.347969\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			46.0000i			1.56586i	0.622111	+	0.782929i	$0.286275\pi$
$863$			46.0000i			1.56586i	−0.622111	+	0.782929i	$0.713725\pi$
$864$		0			0
$865$	−6.00000			−0.204006
$866$		0			0
$867$	−17.1464			−0.582323
$868$		0			0
$869$	12.0000			0.407072
$870$		0			0
$871$	−4.89898			−0.165996
$872$		0			0
$873$		−	14.6969i		−	0.497416i
$874$		0			0
$875$	−9.79796	+	24.0000i	−0.331231	+	0.811348i
$876$		0			0
$877$	48.0000			1.62084			0.810422	−	0.585846i	$-0.199238\pi$
$877$	48.0000			1.62084			0.810422	+	0.585846i	$0.199238\pi$
$878$		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
$883$			6.00000i			0.201916i	0.994891	+	0.100958i	$0.0321908\pi$
$883$			6.00000i			0.201916i	−0.994891	+	0.100958i	$0.967809\pi$
$884$		0			0
$885$			14.6969i			0.494032i
$886$		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
$891$		−	18.0000i		−	0.603023i
$892$		0			0
$893$	12.0000			0.401565
$894$		0			0
$895$	24.4949			0.818774
$896$		0			0
$897$	−24.0000			−0.801337
$898$		0			0
$899$	19.5959			0.653560
$900$		0			0
$901$		−	19.5959i		−	0.652835i
$902$		0			0
$903$	14.6969	−	36.0000i	0.489083	−	1.19800i
$904$		0			0
$905$	18.0000			0.598340
$906$		0			0
$907$		−	42.0000i		−	1.39459i	−0.716786	−	0.697294i	$-0.754387\pi$
$907$		−	42.0000i		−	1.39459i	0.716786	−	0.697294i	$-0.245613\pi$
$908$		0			0
$909$			51.4393i			1.70613i
$910$		0			0
$911$			20.0000i			0.662630i	0.943520	+	0.331315i	$0.107492\pi$
$911$			20.0000i			0.662630i	−0.943520	+	0.331315i	$0.892508\pi$
$912$		0			0
$913$			4.89898i			0.162133i
$914$		0			0
$915$	−44.0908			−1.45760
$916$		0			0
$917$	30.0000	+	12.2474i	0.990687	+	0.404446i
$918$		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
$923$	24.4949			0.806259
$924$		0			0
$925$	8.00000			0.263038
$926$		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
$931$	−12.2474	−	12.0000i	−0.401394	−	0.393284i
$932$		0			0
$933$	−12.0000			−0.392862
$934$		0			0
$935$			24.0000i			0.784884i
$936$		0			0
$937$			4.89898i			0.160043i	0.996793	+	0.0800213i	$0.0254988\pi$
$937$			4.89898i			0.160043i	−0.996793	+	0.0800213i	$0.974501\pi$
$938$		0			0
$939$			24.0000i			0.783210i
$940$		0			0
$941$			31.8434i			1.03806i	0.854755	+	0.519032i	$0.173707\pi$
$941$			31.8434i			1.03806i	−0.854755	+	0.519032i	$0.826293\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			22.0000i			0.714904i	0.933932	+	0.357452i	$0.116354\pi$
$947$			22.0000i			0.714904i	−0.933932	+	0.357452i	$0.883646\pi$
$948$		0			0
$949$	36.0000			1.16861
$950$		0			0
$951$	−78.3837			−2.54176
$952$		0			0
$953$	−34.0000			−1.10137			−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000			−1.10137			−0.550684	+	0.834714i	$0.685633\pi$
$954$		0			0
$955$	24.4949			0.792636
$956$		0			0
$957$			19.5959i			0.633446i
$958$		0			0
$959$	4.89898	+	2.00000i	0.158196	+	0.0645834i
$960$		0			0
$961$	−7.00000			−0.225806
$962$		0			0
$963$		−	6.00000i		−	0.193347i
$964$		0			0
$965$			58.7878i			1.89244i
$966$		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
$971$	−36.7423			−1.17912			−0.589559	−	0.807725i	$-0.700698\pi$
$971$	−36.7423			−1.17912			−0.589559	+	0.807725i	$0.700698\pi$
$972$		0			0
$973$	6.00000	+	2.44949i	0.192351	+	0.0785270i
$974$		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
$979$	−29.3939			−0.939432
$980$		0			0
$981$	12.0000			0.383131
$982$		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
$987$	−29.3939	−	12.0000i	−0.935617	−	0.381964i
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$			10.0000i			0.317660i	0.987306	+	0.158830i	$0.0507723\pi$
$991$			10.0000i			0.317660i	−0.987306	+	0.158830i	$0.949228\pi$
$992$		0			0
$993$		−	44.0908i		−	1.39918i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			31.8434i			1.00849i	0.863561	+	0.504245i	$0.168229\pi$
$997$			31.8434i			1.00849i	−0.863561	+	0.504245i	$0.831771\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.f.f.1791.4		4
4.3	odd	2	inner	1792.2.f.f.1791.2		4
7.6	odd	2	inner	1792.2.f.f.1791.1		4
8.3	odd	2		1792.2.f.e.1791.3		4
8.5	even	2		1792.2.f.e.1791.1		4
16.3	odd	4		224.2.e.b.111.2		4
16.5	even	4		224.2.e.b.111.1		4
16.11	odd	4		56.2.e.b.27.4	yes	4
16.13	even	4		56.2.e.b.27.2	yes	4
28.27	even	2	inner	1792.2.f.f.1791.3		4
48.5	odd	4		2016.2.p.e.559.3		4
48.11	even	4		504.2.p.f.307.2		4
48.29	odd	4		504.2.p.f.307.3		4
48.35	even	4		2016.2.p.e.559.2		4
56.13	odd	2		1792.2.f.e.1791.4		4
56.27	even	2		1792.2.f.e.1791.2		4
112.3	even	12		1568.2.q.e.1391.4		8
112.5	odd	12		1568.2.q.e.815.1		8
112.11	odd	12		392.2.m.f.19.2		8
112.13	odd	4		56.2.e.b.27.1	✓	4
112.19	even	12		1568.2.q.e.815.2		8
112.27	even	4		56.2.e.b.27.3	yes	4
112.37	even	12		1568.2.q.e.815.4		8
112.45	odd	12		392.2.m.f.19.3		8
112.51	odd	12		1568.2.q.e.815.3		8
112.53	even	12		1568.2.q.e.1391.2		8
112.59	even	12		392.2.m.f.19.1		8
112.61	odd	12		392.2.m.f.227.2		8
112.67	odd	12		1568.2.q.e.1391.1		8
112.69	odd	4		224.2.e.b.111.4		4
112.75	even	12		392.2.m.f.227.4		8
112.83	even	4		224.2.e.b.111.3		4
112.93	even	12		392.2.m.f.227.1		8
112.101	odd	12		1568.2.q.e.1391.3		8
112.107	odd	12		392.2.m.f.227.3		8
112.109	even	12		392.2.m.f.19.4		8
336.83	odd	4		2016.2.p.e.559.4		4
336.125	even	4		504.2.p.f.307.4		4
336.251	odd	4		504.2.p.f.307.1		4
336.293	even	4		2016.2.p.e.559.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.e.b.27.1	✓	4	112.13	odd	4
56.2.e.b.27.2	yes	4	16.13	even	4
56.2.e.b.27.3	yes	4	112.27	even	4
56.2.e.b.27.4	yes	4	16.11	odd	4
224.2.e.b.111.1		4	16.5	even	4
224.2.e.b.111.2		4	16.3	odd	4
224.2.e.b.111.3		4	112.83	even	4
224.2.e.b.111.4		4	112.69	odd	4
392.2.m.f.19.1		8	112.59	even	12
392.2.m.f.19.2		8	112.11	odd	12
392.2.m.f.19.3		8	112.45	odd	12
392.2.m.f.19.4		8	112.109	even	12
392.2.m.f.227.1		8	112.93	even	12
392.2.m.f.227.2		8	112.61	odd	12
392.2.m.f.227.3		8	112.107	odd	12
392.2.m.f.227.4		8	112.75	even	12
504.2.p.f.307.1		4	336.251	odd	4
504.2.p.f.307.2		4	48.11	even	4
504.2.p.f.307.3		4	48.29	odd	4
504.2.p.f.307.4		4	336.125	even	4
1568.2.q.e.815.1		8	112.5	odd	12
1568.2.q.e.815.2		8	112.19	even	12
1568.2.q.e.815.3		8	112.51	odd	12
1568.2.q.e.815.4		8	112.37	even	12
1568.2.q.e.1391.1		8	112.67	odd	12
1568.2.q.e.1391.2		8	112.53	even	12
1568.2.q.e.1391.3		8	112.101	odd	12
1568.2.q.e.1391.4		8	112.3	even	12
1792.2.f.e.1791.1		4	8.5	even	2
1792.2.f.e.1791.2		4	56.27	even	2
1792.2.f.e.1791.3		4	8.3	odd	2
1792.2.f.e.1791.4		4	56.13	odd	2
1792.2.f.f.1791.1		4	7.6	odd	2	inner
1792.2.f.f.1791.2		4	4.3	odd	2	inner
1792.2.f.f.1791.3		4	28.27	even	2	inner
1792.2.f.f.1791.4		4	1.1	even	1	trivial
2016.2.p.e.559.1		4	336.293	even	4
2016.2.p.e.559.2		4	48.35	even	4
2016.2.p.e.559.3		4	48.5	odd	4
2016.2.p.e.559.4		4	336.83	odd	4

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.44949 q^{3} +2.44949i q^{5} +(2.44949 + 1.00000i) q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+2.44949 q^{3} +2.44949i q^{5} +(2.44949 + 1.00000i) q^{7} +3.00000 q^{9} +2.00000i q^{11} +2.44949i q^{13} +6.00000i q^{15} -4.89898i q^{17} -2.44949 q^{19} +(6.00000 + 2.44949i) q^{21} +4.00000i q^{23} -1.00000 q^{25} +4.00000 q^{29} +4.89898 q^{31} +4.89898i q^{33} +(-2.44949 + 6.00000i) q^{35} -8.00000 q^{37} +6.00000i q^{39} -6.00000i q^{43} +7.34847i q^{45} -4.89898 q^{47} +(5.00000 + 4.89898i) q^{49} -12.0000i q^{51} +4.00000 q^{53} -4.89898 q^{55} -6.00000 q^{57} +2.44949 q^{59} +7.34847i q^{61} +(7.34847 + 3.00000i) q^{63} -6.00000 q^{65} +2.00000i q^{67} +9.79796i q^{69} -10.0000i q^{71} -14.6969i q^{73} -2.44949 q^{75} +(-2.00000 + 4.89898i) q^{77} -6.00000i q^{79} -9.00000 q^{81} +2.44949 q^{83} +12.0000 q^{85} +9.79796 q^{87} +14.6969i q^{89} +(-2.44949 + 6.00000i) q^{91} +12.0000 q^{93} -6.00000i q^{95} -4.89898i q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} + 24 q^{21} - 4 q^{25} + 16 q^{29} - 32 q^{37} + 20 q^{49} + 16 q^{53} - 24 q^{57} - 24 q^{65} - 8 q^{77} - 36 q^{81} + 48 q^{85} + 48 q^{93}+O(q^{100}) \) 4 * q + 12 * q^9 + 24 * q^21 - 4 * q^25 + 16 * q^29 - 32 * q^37 + 20 * q^49 + 16 * q^53 - 24 * q^57 - 24 * q^65 - 8 * q^77 - 36 * q^81 + 48 * q^85 + 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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