LMFDB - Embedded newform 768.2.f.g.383.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,2,Mod(383,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.383"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.3
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	768.383
Dual form			768.2.f.g.383.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.41421i) q^{3} -2.82843 q^{5} -2.00000i q^{7} +(-1.00000 + 2.82843i) q^{9} +2.82843i q^{11} +2.00000i q^{13} +(-2.82843 - 4.00000i) q^{15} -6.00000 q^{19} +(2.82843 - 2.00000i) q^{21} -5.65685 q^{23} +3.00000 q^{25} +(-5.00000 + 1.41421i) q^{27} -2.82843 q^{29} +2.00000i q^{31} +(-4.00000 + 2.82843i) q^{33} +5.65685i q^{35} +6.00000i q^{37} +(-2.82843 + 2.00000i) q^{39} -5.65685i q^{41} -2.00000 q^{43} +(2.82843 - 8.00000i) q^{45} -11.3137 q^{47} +3.00000 q^{49} +8.48528 q^{53} -8.00000i q^{55} +(-6.00000 - 8.48528i) q^{57} +2.82843i q^{59} +2.00000i q^{61} +(5.65685 + 2.00000i) q^{63} -5.65685i q^{65} +2.00000 q^{67} +(-5.65685 - 8.00000i) q^{69} +5.65685 q^{71} +6.00000 q^{73} +(3.00000 + 4.24264i) q^{75} +5.65685 q^{77} -14.0000i q^{79} +(-7.00000 - 5.65685i) q^{81} +2.82843i q^{83} +(-2.82843 - 4.00000i) q^{87} +16.9706i q^{89} +4.00000 q^{91} +(-2.82843 + 2.00000i) q^{93} +16.9706 q^{95} +10.0000 q^{97} +(-8.00000 - 2.82843i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{9} - 24 q^{19} + 12 q^{25} - 20 q^{27} - 16 q^{33} - 8 q^{43} + 12 q^{49} - 24 q^{57} + 8 q^{67} + 24 q^{73} + 12 q^{75} - 28 q^{81} + 16 q^{91} + 40 q^{97} - 32 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^9 - 24 * q^19 + 12 * q^25 - 20 * q^27 - 16 * q^33 - 8 * q^43 + 12 * q^49 - 24 * q^57 + 8 * q^67 + 24 * q^73 + 12 * q^75 - 28 * q^81 + 16 * q^91 + 40 * q^97 - 32 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	1.00000	+	1.41421i	0.577350	+	0.816497i
$3$	1.00000	+	1.41421i	0.577350	+	0.816497i
$4$		0			0
$5$	−2.82843			−1.26491			−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843			−1.26491			−0.632456	+	0.774597i	$0.717953\pi$
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$		0			0
$11$			2.82843i			0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$			2.82843i			0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$	−2.82843	−	4.00000i	−0.730297	−	1.03280i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$		0			0
$21$	2.82843	−	2.00000i	0.617213	−	0.436436i
$22$		0			0
$23$	−5.65685			−1.17954			−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685			−1.17954			−0.589768	+	0.807573i	$0.700781\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	−5.00000	+	1.41421i	−0.962250	+	0.272166i
$28$		0			0
$29$	−2.82843			−0.525226			−0.262613	−	0.964901i	$-0.584584\pi$
$29$	−2.82843			−0.525226			−0.262613	+	0.964901i	$0.584584\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$		0			0
$33$	−4.00000	+	2.82843i	−0.696311	+	0.492366i
$34$		0			0
$35$			5.65685i			0.956183i
$36$		0			0
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$		0			0
$39$	−2.82843	+	2.00000i	−0.452911	+	0.320256i
$40$		0			0
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$	−2.00000			−0.304997			−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		0			0
$45$	2.82843	−	8.00000i	0.421637	−	1.19257i
$46$		0			0
$47$	−11.3137			−1.65027			−0.825137	−	0.564933i	$-0.808902\pi$
$47$	−11.3137			−1.65027			−0.825137	+	0.564933i	$0.808902\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.48528			1.16554			0.582772	−	0.812636i	$-0.301968\pi$
$53$	8.48528			1.16554			0.582772	+	0.812636i	$0.301968\pi$
$54$		0			0
$55$		−	8.00000i		−	1.07872i
$56$		0			0
$57$	−6.00000	−	8.48528i	−0.794719	−	1.12390i
$58$		0			0
$59$			2.82843i			0.368230i	0.982905	+	0.184115i	$0.0589419\pi$
$59$			2.82843i			0.368230i	−0.982905	+	0.184115i	$0.941058\pi$
$60$		0			0
$61$			2.00000i			0.256074i	0.991769	+	0.128037i	$0.0408676\pi$
$61$			2.00000i			0.256074i	−0.991769	+	0.128037i	$0.959132\pi$
$62$		0			0
$63$	5.65685	+	2.00000i	0.712697	+	0.251976i
$64$		0			0
$65$		−	5.65685i		−	0.701646i
$66$		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
$69$	−5.65685	−	8.00000i	−0.681005	−	0.963087i
$70$		0			0
$71$	5.65685			0.671345			0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685			0.671345			0.335673	+	0.941979i	$0.391036\pi$
$72$		0			0
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$	3.00000	+	4.24264i	0.346410	+	0.489898i
$76$		0			0
$77$	5.65685			0.644658
$78$		0			0
$79$		−	14.0000i		−	1.57512i	−0.616236	−	0.787562i	$-0.711343\pi$
$79$		−	14.0000i		−	1.57512i	0.616236	−	0.787562i	$-0.288657\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$			2.82843i			0.310460i	0.987878	+	0.155230i	$0.0496119\pi$
$83$			2.82843i			0.310460i	−0.987878	+	0.155230i	$0.950388\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.82843	−	4.00000i	−0.303239	−	0.428845i
$88$		0			0
$89$			16.9706i			1.79888i	0.437048	+	0.899438i	$0.356024\pi$
$89$			16.9706i			1.79888i	−0.437048	+	0.899438i	$0.643976\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$	−2.82843	+	2.00000i	−0.293294	+	0.207390i
$94$		0			0
$95$	16.9706			1.74114
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$	−8.00000	−	2.82843i	−0.804030	−	0.284268i
$100$		0			0
$101$	−2.82843			−0.281439			−0.140720	−	0.990050i	$-0.544942\pi$
$101$	−2.82843			−0.281439			−0.140720	+	0.990050i	$0.544942\pi$
$102$		0			0
$103$			14.0000i			1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$			14.0000i			1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$		0			0
$105$	−8.00000	+	5.65685i	−0.780720	+	0.552052i
$106$		0			0
$107$		−	8.48528i		−	0.820303i	−0.912017	−	0.410152i	$-0.865476\pi$
$107$		−	8.48528i		−	0.820303i	0.912017	−	0.410152i	$-0.134524\pi$
$108$		0			0
$109$			18.0000i			1.72409i	0.506834	+	0.862044i	$0.330816\pi$
$109$			18.0000i			1.72409i	−0.506834	+	0.862044i	$0.669184\pi$
$110$		0			0
$111$	−8.48528	+	6.00000i	−0.805387	+	0.569495i
$112$		0			0
$113$			11.3137i			1.06430i	0.846649	+	0.532152i	$0.178617\pi$
$113$			11.3137i			1.06430i	−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$	16.0000			1.49201
$116$		0			0
$117$	−5.65685	−	2.00000i	−0.522976	−	0.184900i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.00000			0.272727
$122$		0			0
$123$	8.00000	−	5.65685i	0.721336	−	0.510061i
$124$		0			0
$125$	5.65685			0.505964
$126$		0			0
$127$			18.0000i			1.59724i	0.601834	+	0.798621i	$0.294437\pi$
$127$			18.0000i			1.59724i	−0.601834	+	0.798621i	$0.705563\pi$
$128$		0			0
$129$	−2.00000	−	2.82843i	−0.176090	−	0.249029i
$130$		0			0
$131$		−	19.7990i		−	1.72985i	−0.501905	−	0.864923i	$-0.667367\pi$
$131$		−	19.7990i		−	1.72985i	0.501905	−	0.864923i	$-0.332633\pi$
$132$		0			0
$133$			12.0000i			1.04053i
$134$		0			0
$135$	14.1421	−	4.00000i	1.21716	−	0.344265i
$136$		0			0
$137$		−	5.65685i		−	0.483298i	−0.970364	−	0.241649i	$-0.922312\pi$
$137$		−	5.65685i		−	0.483298i	0.970364	−	0.241649i	$-0.0776882\pi$
$138$		0			0
$139$	−10.0000			−0.848189			−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000			−0.848189			−0.424094	+	0.905618i	$0.639408\pi$
$140$		0			0
$141$	−11.3137	−	16.0000i	−0.952786	−	1.34744i
$142$		0			0
$143$	−5.65685			−0.473050
$144$		0			0
$145$	8.00000			0.664364
$146$		0			0
$147$	3.00000	+	4.24264i	0.247436	+	0.349927i
$148$		0			0
$149$	−14.1421			−1.15857			−0.579284	−	0.815125i	$-0.696668\pi$
$149$	−14.1421			−1.15857			−0.579284	+	0.815125i	$0.696668\pi$
$150$		0			0
$151$			14.0000i			1.13930i	0.821886	+	0.569652i	$0.192922\pi$
$151$			14.0000i			1.13930i	−0.821886	+	0.569652i	$0.807078\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	5.65685i		−	0.454369i
$156$		0			0
$157$		−	14.0000i		−	1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$		−	14.0000i		−	1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$		0			0
$159$	8.48528	+	12.0000i	0.672927	+	0.951662i
$160$		0			0
$161$			11.3137i			0.891645i
$162$		0			0
$163$	−6.00000			−0.469956			−0.234978	−	0.972001i	$-0.575502\pi$
$163$	−6.00000			−0.469956			−0.234978	+	0.972001i	$0.575502\pi$
$164$		0			0
$165$	11.3137	−	8.00000i	0.880771	−	0.622799i
$166$		0			0
$167$	5.65685			0.437741			0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685			0.437741			0.218870	+	0.975754i	$0.429763\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	6.00000	−	16.9706i	0.458831	−	1.29777i
$172$		0			0
$173$	−14.1421			−1.07521			−0.537603	−	0.843198i	$-0.680670\pi$
$173$	−14.1421			−1.07521			−0.537603	+	0.843198i	$0.680670\pi$
$174$		0			0
$175$		−	6.00000i		−	0.453557i
$176$		0			0
$177$	−4.00000	+	2.82843i	−0.300658	+	0.212598i
$178$		0			0
$179$		−	8.48528i		−	0.634220i	−0.948389	−	0.317110i	$-0.897288\pi$
$179$		−	8.48528i		−	0.634220i	0.948389	−	0.317110i	$-0.102712\pi$
$180$		0			0
$181$			6.00000i			0.445976i	0.974821	+	0.222988i	$0.0715812\pi$
$181$			6.00000i			0.445976i	−0.974821	+	0.222988i	$0.928419\pi$
$182$		0			0
$183$	−2.82843	+	2.00000i	−0.209083	+	0.147844i
$184$		0			0
$185$		−	16.9706i		−	1.24770i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	2.82843	+	10.0000i	0.205738	+	0.727393i
$190$		0			0
$191$	22.6274			1.63726			0.818631	−	0.574320i	$-0.194733\pi$
$191$	22.6274			1.63726			0.818631	+	0.574320i	$0.194733\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$	8.00000	−	5.65685i	0.572892	−	0.405096i
$196$		0			0
$197$	19.7990			1.41062			0.705310	−	0.708899i	$-0.250808\pi$
$197$	19.7990			1.41062			0.705310	+	0.708899i	$0.250808\pi$
$198$		0			0
$199$		−	2.00000i		−	0.141776i	−0.997484	−	0.0708881i	$-0.977417\pi$
$199$		−	2.00000i		−	0.141776i	0.997484	−	0.0708881i	$-0.0225833\pi$
$200$		0			0
$201$	2.00000	+	2.82843i	0.141069	+	0.199502i
$202$		0			0
$203$			5.65685i			0.397033i
$204$		0			0
$205$			16.0000i			1.11749i
$206$		0			0
$207$	5.65685	−	16.0000i	0.393179	−	1.11208i
$208$		0			0
$209$		−	16.9706i		−	1.17388i
$210$		0			0
$211$	2.00000			0.137686			0.0688428	−	0.997628i	$-0.478069\pi$
$211$	2.00000			0.137686			0.0688428	+	0.997628i	$0.478069\pi$
$212$		0			0
$213$	5.65685	+	8.00000i	0.387601	+	0.548151i
$214$		0			0
$215$	5.65685			0.385794
$216$		0			0
$217$	4.00000			0.271538
$218$		0			0
$219$	6.00000	+	8.48528i	0.405442	+	0.573382i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			2.00000i			0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$			2.00000i			0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$		0			0
$225$	−3.00000	+	8.48528i	−0.200000	+	0.565685i
$226$		0			0
$227$			14.1421i			0.938647i	0.883026	+	0.469323i	$0.155502\pi$
$227$			14.1421i			0.938647i	−0.883026	+	0.469323i	$0.844498\pi$
$228$		0			0
$229$		−	10.0000i		−	0.660819i	−0.943838	−	0.330409i	$-0.892813\pi$
$229$		−	10.0000i		−	0.660819i	0.943838	−	0.330409i	$-0.107187\pi$
$230$		0			0
$231$	5.65685	+	8.00000i	0.372194	+	0.526361i
$232$		0			0
$233$			5.65685i			0.370593i	0.982683	+	0.185296i	$0.0593245\pi$
$233$			5.65685i			0.370593i	−0.982683	+	0.185296i	$0.940675\pi$
$234$		0			0
$235$	32.0000			2.08745
$236$		0			0
$237$	19.7990	−	14.0000i	1.28608	−	0.909398i
$238$		0			0
$239$	−11.3137			−0.731823			−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137			−0.731823			−0.365911	+	0.930650i	$0.619243\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$	1.00000	−	15.5563i	0.0641500	−	0.997940i
$244$		0			0
$245$	−8.48528			−0.542105
$246$		0			0
$247$		−	12.0000i		−	0.763542i
$248$		0			0
$249$	−4.00000	+	2.82843i	−0.253490	+	0.179244i
$250$		0			0
$251$		−	8.48528i		−	0.535586i	−0.963476	−	0.267793i	$-0.913706\pi$
$251$		−	8.48528i		−	0.535586i	0.963476	−	0.267793i	$-0.0862944\pi$
$252$		0			0
$253$		−	16.0000i		−	1.00591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	22.6274i		−	1.41146i	−0.708481	−	0.705730i	$-0.750619\pi$
$257$		−	22.6274i		−	1.41146i	0.708481	−	0.705730i	$-0.249381\pi$
$258$		0			0
$259$	12.0000			0.745644
$260$		0			0
$261$	2.82843	−	8.00000i	0.175075	−	0.495188i
$262$		0			0
$263$	5.65685			0.348817			0.174408	−	0.984673i	$-0.444199\pi$
$263$	5.65685			0.348817			0.174408	+	0.984673i	$0.444199\pi$
$264$		0			0
$265$	−24.0000			−1.47431
$266$		0			0
$267$	−24.0000	+	16.9706i	−1.46878	+	1.03858i
$268$		0			0
$269$	8.48528			0.517357			0.258678	−	0.965964i	$-0.416713\pi$
$269$	8.48528			0.517357			0.258678	+	0.965964i	$0.416713\pi$
$270$		0			0
$271$			18.0000i			1.09342i	0.837321	+	0.546711i	$0.184120\pi$
$271$			18.0000i			1.09342i	−0.837321	+	0.546711i	$0.815880\pi$
$272$		0			0
$273$	4.00000	+	5.65685i	0.242091	+	0.342368i
$274$		0			0
$275$			8.48528i			0.511682i
$276$		0			0
$277$		−	10.0000i		−	0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$		−	10.0000i		−	0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$		0			0
$279$	−5.65685	−	2.00000i	−0.338667	−	0.119737i
$280$		0			0
$281$		−	5.65685i		−	0.337460i	−0.985662	−	0.168730i	$-0.946033\pi$
$281$		−	5.65685i		−	0.337460i	0.985662	−	0.168730i	$-0.0539665\pi$
$282$		0			0
$283$	−10.0000			−0.594438			−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000			−0.594438			−0.297219	+	0.954809i	$0.596059\pi$
$284$		0			0
$285$	16.9706	+	24.0000i	1.00525	+	1.42164i
$286$		0			0
$287$	−11.3137			−0.667827
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$	10.0000	+	14.1421i	0.586210	+	0.829027i
$292$		0			0
$293$	−2.82843			−0.165238			−0.0826192	−	0.996581i	$-0.526329\pi$
$293$	−2.82843			−0.165238			−0.0826192	+	0.996581i	$0.526329\pi$
$294$		0			0
$295$		−	8.00000i		−	0.465778i
$296$		0			0
$297$	−4.00000	−	14.1421i	−0.232104	−	0.820610i
$298$		0			0
$299$		−	11.3137i		−	0.654289i
$300$		0			0
$301$			4.00000i			0.230556i
$302$		0			0
$303$	−2.82843	−	4.00000i	−0.162489	−	0.229794i
$304$		0			0
$305$		−	5.65685i		−	0.323911i
$306$		0			0
$307$	−30.0000			−1.71219			−0.856095	−	0.516818i	$-0.827116\pi$
$307$	−30.0000			−1.71219			−0.856095	+	0.516818i	$0.827116\pi$
$308$		0			0
$309$	−19.7990	+	14.0000i	−1.12633	+	0.796432i
$310$		0			0
$311$	−5.65685			−0.320771			−0.160385	−	0.987054i	$-0.551274\pi$
$311$	−5.65685			−0.320771			−0.160385	+	0.987054i	$0.551274\pi$
$312$		0			0
$313$	−2.00000			−0.113047			−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000			−0.113047			−0.0565233	+	0.998401i	$0.518002\pi$
$314$		0			0
$315$	−16.0000	−	5.65685i	−0.901498	−	0.318728i
$316$		0			0
$317$	−2.82843			−0.158860			−0.0794301	−	0.996840i	$-0.525310\pi$
$317$	−2.82843			−0.158860			−0.0794301	+	0.996840i	$0.525310\pi$
$318$		0			0
$319$		−	8.00000i		−	0.447914i
$320$		0			0
$321$	12.0000	−	8.48528i	0.669775	−	0.473602i
$322$		0			0
$323$		0			0
$324$		0			0
$325$			6.00000i			0.332820i
$326$		0			0
$327$	−25.4558	+	18.0000i	−1.40771	+	0.995402i
$328$		0			0
$329$			22.6274i			1.24749i
$330$		0			0
$331$	−10.0000			−0.549650			−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000			−0.549650			−0.274825	+	0.961494i	$0.588620\pi$
$332$		0			0
$333$	−16.9706	−	6.00000i	−0.929981	−	0.328798i
$334$		0			0
$335$	−5.65685			−0.309067
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$	−16.0000	+	11.3137i	−0.869001	+	0.614476i
$340$		0			0
$341$	−5.65685			−0.306336
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$	16.0000	+	22.6274i	0.861411	+	1.21822i
$346$		0			0
$347$			14.1421i			0.759190i	0.925153	+	0.379595i	$0.123937\pi$
$347$			14.1421i			0.759190i	−0.925153	+	0.379595i	$0.876063\pi$
$348$		0			0
$349$		−	14.0000i		−	0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$		−	14.0000i		−	0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$		0			0
$351$	−2.82843	−	10.0000i	−0.150970	−	0.533761i
$352$		0			0
$353$			22.6274i			1.20434i	0.798369	+	0.602168i	$0.205696\pi$
$353$			22.6274i			1.20434i	−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$	−16.0000			−0.849192
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.9706			−0.895672			−0.447836	−	0.894116i	$-0.647805\pi$
$359$	−16.9706			−0.895672			−0.447836	+	0.894116i	$0.647805\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$		0			0
$363$	3.00000	+	4.24264i	0.157459	+	0.222681i
$364$		0			0
$365$	−16.9706			−0.888280
$366$		0			0
$367$		−	14.0000i		−	0.730794i	−0.930852	−	0.365397i	$-0.880933\pi$
$367$		−	14.0000i		−	0.730794i	0.930852	−	0.365397i	$-0.119067\pi$
$368$		0			0
$369$	16.0000	+	5.65685i	0.832927	+	0.294484i
$370$		0			0
$371$		−	16.9706i		−	0.881068i
$372$		0			0
$373$		−	26.0000i		−	1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$		−	26.0000i		−	1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$		0			0
$375$	5.65685	+	8.00000i	0.292119	+	0.413118i
$376$		0			0
$377$		−	5.65685i		−	0.291343i
$378$		0			0
$379$	−2.00000			−0.102733			−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000			−0.102733			−0.0513665	+	0.998680i	$0.516358\pi$
$380$		0			0
$381$	−25.4558	+	18.0000i	−1.30414	+	0.922168i
$382$		0			0
$383$	−22.6274			−1.15621			−0.578103	−	0.815963i	$-0.696207\pi$
$383$	−22.6274			−1.15621			−0.578103	+	0.815963i	$0.696207\pi$
$384$		0			0
$385$	−16.0000			−0.815436
$386$		0			0
$387$	2.00000	−	5.65685i	0.101666	−	0.287554i
$388$		0			0
$389$	−2.82843			−0.143407			−0.0717035	−	0.997426i	$-0.522844\pi$
$389$	−2.82843			−0.143407			−0.0717035	+	0.997426i	$0.522844\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	28.0000	−	19.7990i	1.41241	−	0.998727i
$394$		0			0
$395$			39.5980i			1.99239i
$396$		0			0
$397$			18.0000i			0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$			18.0000i			0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$		0			0
$399$	−16.9706	+	12.0000i	−0.849591	+	0.600751i
$400$		0			0
$401$			22.6274i			1.12996i	0.825105	+	0.564980i	$0.191116\pi$
$401$			22.6274i			1.12996i	−0.825105	+	0.564980i	$0.808884\pi$
$402$		0			0
$403$	−4.00000			−0.199254
$404$		0			0
$405$	19.7990	+	16.0000i	0.983820	+	0.795046i
$406$		0			0
$407$	−16.9706			−0.841200
$408$		0			0
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$		0			0
$411$	8.00000	−	5.65685i	0.394611	−	0.279032i
$412$		0			0
$413$	5.65685			0.278356
$414$		0			0
$415$		−	8.00000i		−	0.392705i
$416$		0			0
$417$	−10.0000	−	14.1421i	−0.489702	−	0.692543i
$418$		0			0
$419$			36.7696i			1.79631i	0.439679	+	0.898155i	$0.355092\pi$
$419$			36.7696i			1.79631i	−0.439679	+	0.898155i	$0.644908\pi$
$420$		0			0
$421$			22.0000i			1.07221i	0.844150	+	0.536107i	$0.180106\pi$
$421$			22.0000i			1.07221i	−0.844150	+	0.536107i	$0.819894\pi$
$422$		0			0
$423$	11.3137	−	32.0000i	0.550091	−	1.55589i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.00000			0.193574
$428$		0			0
$429$	−5.65685	−	8.00000i	−0.273115	−	0.386244i
$430$		0			0
$431$	33.9411			1.63489			0.817443	−	0.576009i	$-0.195391\pi$
$431$	33.9411			1.63489			0.817443	+	0.576009i	$0.195391\pi$
$432$		0			0
$433$	−6.00000			−0.288342			−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000			−0.288342			−0.144171	+	0.989553i	$0.546051\pi$
$434$		0			0
$435$	8.00000	+	11.3137i	0.383571	+	0.542451i
$436$		0			0
$437$	33.9411			1.62362
$438$		0			0
$439$			14.0000i			0.668184i	0.942541	+	0.334092i	$0.108430\pi$
$439$			14.0000i			0.668184i	−0.942541	+	0.334092i	$0.891570\pi$
$440$		0			0
$441$	−3.00000	+	8.48528i	−0.142857	+	0.404061i
$442$		0			0
$443$			14.1421i			0.671913i	0.941877	+	0.335957i	$0.109060\pi$
$443$			14.1421i			0.671913i	−0.941877	+	0.335957i	$0.890940\pi$
$444$		0			0
$445$		−	48.0000i		−	2.27542i
$446$		0			0
$447$	−14.1421	−	20.0000i	−0.668900	−	0.945968i
$448$		0			0
$449$			33.9411i			1.60178i	0.598811	+	0.800890i	$0.295640\pi$
$449$			33.9411i			1.60178i	−0.598811	+	0.800890i	$0.704360\pi$
$450$		0			0
$451$	16.0000			0.753411
$452$		0			0
$453$	−19.7990	+	14.0000i	−0.930238	+	0.657777i
$454$		0			0
$455$	−11.3137			−0.530395
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	31.1127			1.44906			0.724531	−	0.689242i	$-0.242056\pi$
$461$	31.1127			1.44906			0.724531	+	0.689242i	$0.242056\pi$
$462$		0			0
$463$			2.00000i			0.0929479i	0.998920	+	0.0464739i	$0.0147984\pi$
$463$			2.00000i			0.0929479i	−0.998920	+	0.0464739i	$0.985202\pi$
$464$		0			0
$465$	8.00000	−	5.65685i	0.370991	−	0.262330i
$466$		0			0
$467$			25.4558i			1.17796i	0.808149	+	0.588978i	$0.200470\pi$
$467$			25.4558i			1.17796i	−0.808149	+	0.588978i	$0.799530\pi$
$468$		0			0
$469$		−	4.00000i		−	0.184703i
$470$		0			0
$471$	19.7990	−	14.0000i	0.912289	−	0.645086i
$472$		0			0
$473$		−	5.65685i		−	0.260102i
$474$		0			0
$475$	−18.0000			−0.825897
$476$		0			0
$477$	−8.48528	+	24.0000i	−0.388514	+	1.09888i
$478$		0			0
$479$	−22.6274			−1.03387			−0.516937	−	0.856024i	$-0.672928\pi$
$479$	−22.6274			−1.03387			−0.516937	+	0.856024i	$0.672928\pi$
$480$		0			0
$481$	−12.0000			−0.547153
$482$		0			0
$483$	−16.0000	+	11.3137i	−0.728025	+	0.514792i
$484$		0			0
$485$	−28.2843			−1.28432
$486$		0			0
$487$		−	18.0000i		−	0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$		−	18.0000i		−	0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$		0			0
$489$	−6.00000	−	8.48528i	−0.271329	−	0.383718i
$490$		0			0
$491$		−	31.1127i		−	1.40410i	−0.712129	−	0.702048i	$-0.752269\pi$
$491$		−	31.1127i		−	1.40410i	0.712129	−	0.702048i	$-0.247731\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	22.6274	+	8.00000i	1.01703	+	0.359573i
$496$		0			0
$497$		−	11.3137i		−	0.507489i
$498$		0			0
$499$	34.0000			1.52205			0.761025	−	0.648723i	$-0.224697\pi$
$499$	34.0000			1.52205			0.761025	+	0.648723i	$0.224697\pi$
$500$		0			0
$501$	5.65685	+	8.00000i	0.252730	+	0.357414i
$502$		0			0
$503$	−5.65685			−0.252227			−0.126113	−	0.992016i	$-0.540250\pi$
$503$	−5.65685			−0.252227			−0.126113	+	0.992016i	$0.540250\pi$
$504$		0			0
$505$	8.00000			0.355995
$506$		0			0
$507$	9.00000	+	12.7279i	0.399704	+	0.565267i
$508$		0			0
$509$	−2.82843			−0.125368			−0.0626839	−	0.998033i	$-0.519966\pi$
$509$	−2.82843			−0.125368			−0.0626839	+	0.998033i	$0.519966\pi$
$510$		0			0
$511$		−	12.0000i		−	0.530849i
$512$		0			0
$513$	30.0000	−	8.48528i	1.32453	−	0.374634i
$514$		0			0
$515$		−	39.5980i		−	1.74490i
$516$		0			0
$517$		−	32.0000i		−	1.40736i
$518$		0			0
$519$	−14.1421	−	20.0000i	−0.620771	−	0.877903i
$520$		0			0
$521$		−	5.65685i		−	0.247831i	−0.992293	−	0.123916i	$-0.960455\pi$
$521$		−	5.65685i		−	0.247831i	0.992293	−	0.123916i	$-0.0395452\pi$
$522$		0			0
$523$	−34.0000			−1.48672			−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000			−1.48672			−0.743358	+	0.668894i	$0.766768\pi$
$524$		0			0
$525$	8.48528	−	6.00000i	0.370328	−	0.261861i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	9.00000			0.391304
$530$		0			0
$531$	−8.00000	−	2.82843i	−0.347170	−	0.122743i
$532$		0			0
$533$	11.3137			0.490051
$534$		0			0
$535$			24.0000i			1.03761i
$536$		0			0
$537$	12.0000	−	8.48528i	0.517838	−	0.366167i
$538$		0			0
$539$			8.48528i			0.365487i
$540$		0			0
$541$			2.00000i			0.0859867i	0.999075	+	0.0429934i	$0.0136894\pi$
$541$			2.00000i			0.0859867i	−0.999075	+	0.0429934i	$0.986311\pi$
$542$		0			0
$543$	−8.48528	+	6.00000i	−0.364138	+	0.257485i
$544$		0			0
$545$		−	50.9117i		−	2.18082i
$546$		0			0
$547$	−38.0000			−1.62476			−0.812381	−	0.583127i	$-0.801829\pi$
$547$	−38.0000			−1.62476			−0.812381	+	0.583127i	$0.801829\pi$
$548$		0			0
$549$	−5.65685	−	2.00000i	−0.241429	−	0.0853579i
$550$		0			0
$551$	16.9706			0.722970
$552$		0			0
$553$	−28.0000			−1.19068
$554$		0			0
$555$	24.0000	−	16.9706i	1.01874	−	0.720360i
$556$		0			0
$557$	8.48528			0.359533			0.179766	−	0.983709i	$-0.442466\pi$
$557$	8.48528			0.359533			0.179766	+	0.983709i	$0.442466\pi$
$558$		0			0
$559$		−	4.00000i		−	0.169182i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			2.82843i			0.119204i	0.998222	+	0.0596020i	$0.0189832\pi$
$563$			2.82843i			0.119204i	−0.998222	+	0.0596020i	$0.981017\pi$
$564$		0			0
$565$		−	32.0000i		−	1.34625i
$566$		0			0
$567$	−11.3137	+	14.0000i	−0.475131	+	0.587945i
$568$		0			0
$569$			5.65685i			0.237148i	0.992945	+	0.118574i	$0.0378322\pi$
$569$			5.65685i			0.237148i	−0.992945	+	0.118574i	$0.962168\pi$
$570$		0			0
$571$	22.0000			0.920671			0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000			0.920671			0.460336	+	0.887745i	$0.347729\pi$
$572$		0			0
$573$	22.6274	+	32.0000i	0.945274	+	1.33682i
$574$		0			0
$575$	−16.9706			−0.707721
$576$		0			0
$577$	−6.00000			−0.249783			−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000			−0.249783			−0.124892	+	0.992170i	$0.539858\pi$
$578$		0			0
$579$	−14.0000	−	19.7990i	−0.581820	−	0.822818i
$580$		0			0
$581$	5.65685			0.234686
$582$		0			0
$583$			24.0000i			0.993978i
$584$		0			0
$585$	16.0000	+	5.65685i	0.661519	+	0.233882i
$586$		0			0
$587$			14.1421i			0.583708i	0.956463	+	0.291854i	$0.0942722\pi$
$587$			14.1421i			0.583708i	−0.956463	+	0.291854i	$0.905728\pi$
$588$		0			0
$589$		−	12.0000i		−	0.494451i
$590$		0			0
$591$	19.7990	+	28.0000i	0.814422	+	1.15177i
$592$		0			0
$593$		−	33.9411i		−	1.39379i	−0.717171	−	0.696897i	$-0.754563\pi$
$593$		−	33.9411i		−	1.39379i	0.717171	−	0.696897i	$-0.245437\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.82843	−	2.00000i	0.115760	−	0.0818546i
$598$		0			0
$599$	39.5980			1.61793			0.808965	−	0.587857i	$-0.200028\pi$
$599$	39.5980			1.61793			0.808965	+	0.587857i	$0.200028\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$	−2.00000	+	5.65685i	−0.0814463	+	0.230365i
$604$		0			0
$605$	−8.48528			−0.344976
$606$		0			0
$607$		−	14.0000i		−	0.568242i	−0.958788	−	0.284121i	$-0.908298\pi$
$607$		−	14.0000i		−	0.568242i	0.958788	−	0.284121i	$-0.0917018\pi$
$608$		0			0
$609$	−8.00000	+	5.65685i	−0.324176	+	0.229227i
$610$		0			0
$611$		−	22.6274i		−	0.915407i
$612$		0			0
$613$			6.00000i			0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$			6.00000i			0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$		0			0
$615$	−22.6274	+	16.0000i	−0.912426	+	0.645182i
$616$		0			0
$617$			28.2843i			1.13868i	0.822102	+	0.569341i	$0.192802\pi$
$617$			28.2843i			1.13868i	−0.822102	+	0.569341i	$0.807198\pi$
$618$		0			0
$619$	−10.0000			−0.401934			−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000			−0.401934			−0.200967	+	0.979598i	$0.564408\pi$
$620$		0			0
$621$	28.2843	−	8.00000i	1.13501	−	0.321029i
$622$		0			0
$623$	33.9411			1.35982
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	24.0000	−	16.9706i	0.958468	−	0.677739i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			30.0000i			1.19428i	0.802137	+	0.597141i	$0.203697\pi$
$631$			30.0000i			1.19428i	−0.802137	+	0.597141i	$0.796303\pi$
$632$		0			0
$633$	2.00000	+	2.82843i	0.0794929	+	0.112420i
$634$		0			0
$635$		−	50.9117i		−	2.02037i
$636$		0			0
$637$			6.00000i			0.237729i
$638$		0			0
$639$	−5.65685	+	16.0000i	−0.223782	+	0.632950i
$640$		0			0
$641$			11.3137i			0.446865i	0.974719	+	0.223432i	$0.0717262\pi$
$641$			11.3137i			0.446865i	−0.974719	+	0.223432i	$0.928274\pi$
$642$		0			0
$643$	26.0000			1.02534			0.512670	−	0.858586i	$-0.328656\pi$
$643$	26.0000			1.02534			0.512670	+	0.858586i	$0.328656\pi$
$644$		0			0
$645$	5.65685	+	8.00000i	0.222738	+	0.315000i
$646$		0			0
$647$	−16.9706			−0.667182			−0.333591	−	0.942718i	$-0.608260\pi$
$647$	−16.9706			−0.667182			−0.333591	+	0.942718i	$0.608260\pi$
$648$		0			0
$649$	−8.00000			−0.314027
$650$		0			0
$651$	4.00000	+	5.65685i	0.156772	+	0.221710i
$652$		0			0
$653$	−14.1421			−0.553425			−0.276712	−	0.960953i	$-0.589245\pi$
$653$	−14.1421			−0.553425			−0.276712	+	0.960953i	$0.589245\pi$
$654$		0			0
$655$			56.0000i			2.18810i
$656$		0			0
$657$	−6.00000	+	16.9706i	−0.234082	+	0.662085i
$658$		0			0
$659$			14.1421i			0.550899i	0.961315	+	0.275450i	$0.0888267\pi$
$659$			14.1421i			0.550899i	−0.961315	+	0.275450i	$0.911173\pi$
$660$		0			0
$661$		−	10.0000i		−	0.388955i	−0.980907	−	0.194477i	$-0.937699\pi$
$661$		−	10.0000i		−	0.388955i	0.980907	−	0.194477i	$-0.0623011\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	33.9411i		−	1.31618i
$666$		0			0
$667$	16.0000			0.619522
$668$		0			0
$669$	−2.82843	+	2.00000i	−0.109353	+	0.0773245i
$670$		0			0
$671$	−5.65685			−0.218380
$672$		0			0
$673$	10.0000			0.385472			0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000			0.385472			0.192736	+	0.981251i	$0.438264\pi$
$674$		0			0
$675$	−15.0000	+	4.24264i	−0.577350	+	0.163299i
$676$		0			0
$677$	−2.82843			−0.108705			−0.0543526	−	0.998522i	$-0.517310\pi$
$677$	−2.82843			−0.108705			−0.0543526	+	0.998522i	$0.517310\pi$
$678$		0			0
$679$		−	20.0000i		−	0.767530i
$680$		0			0
$681$	−20.0000	+	14.1421i	−0.766402	+	0.541928i
$682$		0			0
$683$		−	42.4264i		−	1.62340i	−0.584074	−	0.811701i	$-0.698542\pi$
$683$		−	42.4264i		−	1.62340i	0.584074	−	0.811701i	$-0.301458\pi$
$684$		0			0
$685$			16.0000i			0.611329i
$686$		0			0
$687$	14.1421	−	10.0000i	0.539556	−	0.381524i
$688$		0			0
$689$			16.9706i			0.646527i
$690$		0			0
$691$	26.0000			0.989087			0.494543	−	0.869153i	$-0.335335\pi$
$691$	26.0000			0.989087			0.494543	+	0.869153i	$0.335335\pi$
$692$		0			0
$693$	−5.65685	+	16.0000i	−0.214886	+	0.607790i
$694$		0			0
$695$	28.2843			1.07288
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−8.00000	+	5.65685i	−0.302588	+	0.213962i
$700$		0			0
$701$	−25.4558			−0.961454			−0.480727	−	0.876870i	$-0.659627\pi$
$701$	−25.4558			−0.961454			−0.480727	+	0.876870i	$0.659627\pi$
$702$		0			0
$703$		−	36.0000i		−	1.35777i
$704$		0			0
$705$	32.0000	+	45.2548i	1.20519	+	1.70440i
$706$		0			0
$707$			5.65685i			0.212748i
$708$		0			0
$709$			38.0000i			1.42712i	0.700594	+	0.713560i	$0.252918\pi$
$709$			38.0000i			1.42712i	−0.700594	+	0.713560i	$0.747082\pi$
$710$		0			0
$711$	39.5980	+	14.0000i	1.48504	+	0.525041i
$712$		0			0
$713$		−	11.3137i		−	0.423702i
$714$		0			0
$715$	16.0000			0.598366
$716$		0			0
$717$	−11.3137	−	16.0000i	−0.422518	−	0.597531i
$718$		0			0
$719$	33.9411			1.26579			0.632895	−	0.774237i	$-0.281866\pi$
$719$	33.9411			1.26579			0.632895	+	0.774237i	$0.281866\pi$
$720$		0			0
$721$	28.0000			1.04277
$722$		0			0
$723$	2.00000	+	2.82843i	0.0743808	+	0.105190i
$724$		0			0
$725$	−8.48528			−0.315135
$726$		0			0
$727$		−	34.0000i		−	1.26099i	−0.776193	−	0.630495i	$-0.782852\pi$
$727$		−	34.0000i		−	1.26099i	0.776193	−	0.630495i	$-0.217148\pi$
$728$		0			0
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	46.0000i		−	1.69905i	−0.527549	−	0.849524i	$-0.676889\pi$
$733$		−	46.0000i		−	1.69905i	0.527549	−	0.849524i	$-0.323111\pi$
$734$		0			0
$735$	−8.48528	−	12.0000i	−0.312984	−	0.442627i
$736$		0			0
$737$			5.65685i			0.208373i
$738$		0			0
$739$	34.0000			1.25071			0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000			1.25071			0.625355	+	0.780340i	$0.284954\pi$
$740$		0			0
$741$	16.9706	−	12.0000i	0.623429	−	0.440831i
$742$		0			0
$743$	5.65685			0.207530			0.103765	−	0.994602i	$-0.466911\pi$
$743$	5.65685			0.207530			0.103765	+	0.994602i	$0.466911\pi$
$744$		0			0
$745$	40.0000			1.46549
$746$		0			0
$747$	−8.00000	−	2.82843i	−0.292705	−	0.103487i
$748$		0			0
$749$	−16.9706			−0.620091
$750$		0			0
$751$			34.0000i			1.24068i	0.784334	+	0.620339i	$0.213005\pi$
$751$			34.0000i			1.24068i	−0.784334	+	0.620339i	$0.786995\pi$
$752$		0			0
$753$	12.0000	−	8.48528i	0.437304	−	0.309221i
$754$		0			0
$755$		−	39.5980i		−	1.44112i
$756$		0			0
$757$			38.0000i			1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$			38.0000i			1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$		0			0
$759$	22.6274	−	16.0000i	0.821323	−	0.580763i
$760$		0			0
$761$			28.2843i			1.02530i	0.858596	+	0.512652i	$0.171337\pi$
$761$			28.2843i			1.02530i	−0.858596	+	0.512652i	$0.828663\pi$
$762$		0			0
$763$	36.0000			1.30329
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.65685			−0.204257
$768$		0			0
$769$	−46.0000			−1.65880			−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000			−1.65880			−0.829401	+	0.558653i	$0.811318\pi$
$770$		0			0
$771$	32.0000	−	22.6274i	1.15245	−	0.814907i
$772$		0			0
$773$	−25.4558			−0.915583			−0.457792	−	0.889060i	$-0.651359\pi$
$773$	−25.4558			−0.915583			−0.457792	+	0.889060i	$0.651359\pi$
$774$		0			0
$775$			6.00000i			0.215526i
$776$		0			0
$777$	12.0000	+	16.9706i	0.430498	+	0.608816i
$778$		0			0
$779$			33.9411i			1.21607i
$780$		0			0
$781$			16.0000i			0.572525i
$782$		0			0
$783$	14.1421	−	4.00000i	0.505399	−	0.142948i
$784$		0			0
$785$			39.5980i			1.41331i
$786$		0			0
$787$	−38.0000			−1.35455			−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000			−1.35455			−0.677277	+	0.735728i	$0.736840\pi$
$788$		0			0
$789$	5.65685	+	8.00000i	0.201389	+	0.284808i
$790$		0			0
$791$	22.6274			0.804538
$792$		0			0
$793$	−4.00000			−0.142044
$794$		0			0
$795$	−24.0000	−	33.9411i	−0.851192	−	1.20377i
$796$		0			0
$797$	−2.82843			−0.100188			−0.0500940	−	0.998745i	$-0.515952\pi$
$797$	−2.82843			−0.100188			−0.0500940	+	0.998745i	$0.515952\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−48.0000	−	16.9706i	−1.69600	−	0.599625i
$802$		0			0
$803$			16.9706i			0.598878i
$804$		0			0
$805$		−	32.0000i		−	1.12785i
$806$		0			0
$807$	8.48528	+	12.0000i	0.298696	+	0.422420i
$808$		0			0
$809$		−	50.9117i		−	1.78996i	−0.446107	−	0.894980i	$-0.647190\pi$
$809$		−	50.9117i		−	1.78996i	0.446107	−	0.894980i	$-0.352810\pi$
$810$		0			0
$811$	30.0000			1.05344			0.526721	−	0.850038i	$-0.323421\pi$
$811$	30.0000			1.05344			0.526721	+	0.850038i	$0.323421\pi$
$812$		0			0
$813$	−25.4558	+	18.0000i	−0.892775	+	0.631288i
$814$		0			0
$815$	16.9706			0.594453
$816$		0			0
$817$	12.0000			0.419827
$818$		0			0
$819$	−4.00000	+	11.3137i	−0.139771	+	0.395333i
$820$		0			0
$821$	−14.1421			−0.493564			−0.246782	−	0.969071i	$-0.579373\pi$
$821$	−14.1421			−0.493564			−0.246782	+	0.969071i	$0.579373\pi$
$822$		0			0
$823$		−	2.00000i		−	0.0697156i	−0.999392	−	0.0348578i	$-0.988902\pi$
$823$		−	2.00000i		−	0.0697156i	0.999392	−	0.0348578i	$-0.0110978\pi$
$824$		0			0
$825$	−12.0000	+	8.48528i	−0.417786	+	0.295420i
$826$		0			0
$827$			25.4558i			0.885186i	0.896723	+	0.442593i	$0.145941\pi$
$827$			25.4558i			0.885186i	−0.896723	+	0.442593i	$0.854059\pi$
$828$		0			0
$829$			18.0000i			0.625166i	0.949890	+	0.312583i	$0.101194\pi$
$829$			18.0000i			0.625166i	−0.949890	+	0.312583i	$0.898806\pi$
$830$		0			0
$831$	14.1421	−	10.0000i	0.490585	−	0.346896i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−16.0000			−0.553703
$836$		0			0
$837$	−2.82843	−	10.0000i	−0.0977647	−	0.345651i
$838$		0			0
$839$	−39.5980			−1.36707			−0.683537	−	0.729916i	$-0.739559\pi$
$839$	−39.5980			−1.36707			−0.683537	+	0.729916i	$0.739559\pi$
$840$		0			0
$841$	−21.0000			−0.724138
$842$		0			0
$843$	8.00000	−	5.65685i	0.275535	−	0.194832i
$844$		0			0
$845$	−25.4558			−0.875708
$846$		0			0
$847$		−	6.00000i		−	0.206162i
$848$		0			0
$849$	−10.0000	−	14.1421i	−0.343199	−	0.485357i
$850$		0			0
$851$		−	33.9411i		−	1.16349i
$852$		0			0
$853$		−	26.0000i		−	0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$		−	26.0000i		−	0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$		0			0
$855$	−16.9706	+	48.0000i	−0.580381	+	1.64157i
$856$		0			0
$857$			28.2843i			0.966172i	0.875573	+	0.483086i	$0.160484\pi$
$857$			28.2843i			0.966172i	−0.875573	+	0.483086i	$0.839516\pi$
$858$		0			0
$859$	−2.00000			−0.0682391			−0.0341196	−	0.999418i	$-0.510863\pi$
$859$	−2.00000			−0.0682391			−0.0341196	+	0.999418i	$0.510863\pi$
$860$		0			0
$861$	−11.3137	−	16.0000i	−0.385570	−	0.545279i
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	40.0000			1.36004
$866$		0			0
$867$	17.0000	+	24.0416i	0.577350	+	0.816497i
$868$		0			0
$869$	39.5980			1.34327
$870$		0			0
$871$			4.00000i			0.135535i
$872$		0			0
$873$	−10.0000	+	28.2843i	−0.338449	+	0.957278i
$874$		0			0
$875$		−	11.3137i		−	0.382473i
$876$		0			0
$877$			2.00000i			0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$			2.00000i			0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$		0			0
$879$	−2.82843	−	4.00000i	−0.0954005	−	0.134917i
$880$		0			0
$881$		−	33.9411i		−	1.14351i	−0.820426	−	0.571753i	$-0.806264\pi$
$881$		−	33.9411i		−	1.14351i	0.820426	−	0.571753i	$-0.193736\pi$
$882$		0			0
$883$	−6.00000			−0.201916			−0.100958	−	0.994891i	$-0.532191\pi$
$883$	−6.00000			−0.201916			−0.100958	+	0.994891i	$0.532191\pi$
$884$		0			0
$885$	11.3137	−	8.00000i	0.380306	−	0.268917i
$886$		0			0
$887$	−5.65685			−0.189939			−0.0949693	−	0.995480i	$-0.530275\pi$
$887$	−5.65685			−0.189939			−0.0949693	+	0.995480i	$0.530275\pi$
$888$		0			0
$889$	36.0000			1.20740
$890$		0			0
$891$	16.0000	−	19.7990i	0.536020	−	0.663291i
$892$		0			0
$893$	67.8823			2.27159
$894$		0			0
$895$			24.0000i			0.802232i
$896$		0			0
$897$	16.0000	−	11.3137i	0.534224	−	0.377754i
$898$		0			0
$899$		−	5.65685i		−	0.188667i
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−5.65685	+	4.00000i	−0.188248	+	0.133112i
$904$		0			0
$905$		−	16.9706i		−	0.564121i
$906$		0			0
$907$	−2.00000			−0.0664089			−0.0332045	−	0.999449i	$-0.510571\pi$
$907$	−2.00000			−0.0664089			−0.0332045	+	0.999449i	$0.510571\pi$
$908$		0			0
$909$	2.82843	−	8.00000i	0.0938130	−	0.265343i
$910$		0			0
$911$	11.3137			0.374840			0.187420	−	0.982280i	$-0.439987\pi$
$911$	11.3137			0.374840			0.187420	+	0.982280i	$0.439987\pi$
$912$		0			0
$913$	−8.00000			−0.264761
$914$		0			0
$915$	8.00000	−	5.65685i	0.264472	−	0.187010i
$916$		0			0
$917$	−39.5980			−1.30764
$918$		0			0
$919$			30.0000i			0.989609i	0.869004	+	0.494804i	$0.164760\pi$
$919$			30.0000i			0.989609i	−0.869004	+	0.494804i	$0.835240\pi$
$920$		0			0
$921$	−30.0000	−	42.4264i	−0.988534	−	1.39800i
$922$		0			0
$923$			11.3137i			0.372395i
$924$		0			0
$925$			18.0000i			0.591836i
$926$		0			0
$927$	−39.5980	−	14.0000i	−1.30057	−	0.459820i
$928$		0			0
$929$		−	11.3137i		−	0.371191i	−0.982626	−	0.185595i	$-0.940579\pi$
$929$		−	11.3137i		−	0.371191i	0.982626	−	0.185595i	$-0.0594214\pi$
$930$		0			0
$931$	−18.0000			−0.589926
$932$		0			0
$933$	−5.65685	−	8.00000i	−0.185197	−	0.261908i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	30.0000			0.980057			0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000			0.980057			0.490029	+	0.871706i	$0.336986\pi$
$938$		0			0
$939$	−2.00000	−	2.82843i	−0.0652675	−	0.0923022i
$940$		0			0
$941$	53.7401			1.75188			0.875939	−	0.482422i	$-0.160243\pi$
$941$	53.7401			1.75188			0.875939	+	0.482422i	$0.160243\pi$
$942$		0			0
$943$			32.0000i			1.04206i
$944$		0			0
$945$	−8.00000	−	28.2843i	−0.260240	−	0.920087i
$946$		0			0
$947$			14.1421i			0.459558i	0.973243	+	0.229779i	$0.0738003\pi$
$947$			14.1421i			0.459558i	−0.973243	+	0.229779i	$0.926200\pi$
$948$		0			0
$949$			12.0000i			0.389536i
$950$		0			0
$951$	−2.82843	−	4.00000i	−0.0917180	−	0.129709i
$952$		0			0
$953$			50.9117i			1.64919i	0.565723	+	0.824596i	$0.308597\pi$
$953$			50.9117i			1.64919i	−0.565723	+	0.824596i	$0.691403\pi$
$954$		0			0
$955$	−64.0000			−2.07099
$956$		0			0
$957$	11.3137	−	8.00000i	0.365720	−	0.258603i
$958$		0			0
$959$	−11.3137			−0.365339
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	24.0000	+	8.48528i	0.773389	+	0.273434i
$964$		0			0
$965$	39.5980			1.27470
$966$		0			0
$967$		−	50.0000i		−	1.60789i	−0.594703	−	0.803946i	$-0.702730\pi$
$967$		−	50.0000i		−	1.60789i	0.594703	−	0.803946i	$-0.297270\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	19.7990i		−	0.635380i	−0.948195	−	0.317690i	$-0.897093\pi$
$971$		−	19.7990i		−	0.635380i	0.948195	−	0.317690i	$-0.102907\pi$
$972$		0			0
$973$			20.0000i			0.641171i
$974$		0			0
$975$	−8.48528	+	6.00000i	−0.271746	+	0.192154i
$976$		0			0
$977$			45.2548i			1.44783i	0.689889	+	0.723915i	$0.257659\pi$
$977$			45.2548i			1.44783i	−0.689889	+	0.723915i	$0.742341\pi$
$978$		0			0
$979$	−48.0000			−1.53409
$980$		0			0
$981$	−50.9117	−	18.0000i	−1.62549	−	0.574696i
$982$		0			0
$983$	−28.2843			−0.902128			−0.451064	−	0.892492i	$-0.648955\pi$
$983$	−28.2843			−0.902128			−0.451064	+	0.892492i	$0.648955\pi$
$984$		0			0
$985$	−56.0000			−1.78431
$986$		0			0
$987$	−32.0000	+	22.6274i	−1.01857	+	0.720239i
$988$		0			0
$989$	11.3137			0.359755
$990$		0			0
$991$			18.0000i			0.571789i	0.958261	+	0.285894i	$0.0922907\pi$
$991$			18.0000i			0.571789i	−0.958261	+	0.285894i	$0.907709\pi$
$992$		0			0
$993$	−10.0000	−	14.1421i	−0.317340	−	0.448787i
$994$		0			0
$995$			5.65685i			0.179334i
$996$		0			0
$997$			38.0000i			1.20347i	0.798695	+	0.601736i	$0.205524\pi$
$997$			38.0000i			1.20347i	−0.798695	+	0.601736i	$0.794476\pi$
$998$		0			0
$999$	−8.48528	−	30.0000i	−0.268462	−	0.949158i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.f.g.383.3		4
3.2	odd	2	inner	768.2.f.g.383.2		4
4.3	odd	2		768.2.f.a.383.1		4
8.3	odd	2	inner	768.2.f.g.383.4		4
8.5	even	2		768.2.f.a.383.2		4
12.11	even	2		768.2.f.a.383.4		4
16.3	odd	4		96.2.c.a.95.3	yes	4
16.5	even	4		192.2.c.b.191.3		4
16.11	odd	4		192.2.c.b.191.2		4
16.13	even	4		96.2.c.a.95.2	yes	4
24.5	odd	2		768.2.f.a.383.3		4
24.11	even	2	inner	768.2.f.g.383.1		4
48.5	odd	4		192.2.c.b.191.1		4
48.11	even	4		192.2.c.b.191.4		4
48.29	odd	4		96.2.c.a.95.4	yes	4
48.35	even	4		96.2.c.a.95.1	✓	4
80.3	even	4		2400.2.o.h.2399.3		4
80.13	odd	4		2400.2.o.a.2399.2		4
80.19	odd	4		2400.2.h.c.1151.2		4
80.29	even	4		2400.2.h.c.1151.3		4
80.67	even	4		2400.2.o.a.2399.1		4
80.77	odd	4		2400.2.o.h.2399.4		4
144.13	even	12		2592.2.s.e.1727.3		8
144.29	odd	12		2592.2.s.e.863.4		8
144.61	even	12		2592.2.s.e.863.2		8
144.67	odd	12		2592.2.s.e.1727.4		8
144.77	odd	12		2592.2.s.e.1727.1		8
144.83	even	12		2592.2.s.e.863.3		8
144.115	odd	12		2592.2.s.e.863.1		8
144.131	even	12		2592.2.s.e.1727.2		8
240.29	odd	4		2400.2.h.c.1151.1		4
240.77	even	4		2400.2.o.h.2399.1		4
240.83	odd	4		2400.2.o.h.2399.2		4
240.173	even	4		2400.2.o.a.2399.3		4
240.179	even	4		2400.2.h.c.1151.4		4
240.227	odd	4		2400.2.o.a.2399.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.2.c.a.95.1	✓	4	48.35	even	4
96.2.c.a.95.2	yes	4	16.13	even	4
96.2.c.a.95.3	yes	4	16.3	odd	4
96.2.c.a.95.4	yes	4	48.29	odd	4
192.2.c.b.191.1		4	48.5	odd	4
192.2.c.b.191.2		4	16.11	odd	4
192.2.c.b.191.3		4	16.5	even	4
192.2.c.b.191.4		4	48.11	even	4
768.2.f.a.383.1		4	4.3	odd	2
768.2.f.a.383.2		4	8.5	even	2
768.2.f.a.383.3		4	24.5	odd	2
768.2.f.a.383.4		4	12.11	even	2
768.2.f.g.383.1		4	24.11	even	2	inner
768.2.f.g.383.2		4	3.2	odd	2	inner
768.2.f.g.383.3		4	1.1	even	1	trivial
768.2.f.g.383.4		4	8.3	odd	2	inner
2400.2.h.c.1151.1		4	240.29	odd	4
2400.2.h.c.1151.2		4	80.19	odd	4
2400.2.h.c.1151.3		4	80.29	even	4
2400.2.h.c.1151.4		4	240.179	even	4
2400.2.o.a.2399.1		4	80.67	even	4
2400.2.o.a.2399.2		4	80.13	odd	4
2400.2.o.a.2399.3		4	240.173	even	4
2400.2.o.a.2399.4		4	240.227	odd	4
2400.2.o.h.2399.1		4	240.77	even	4
2400.2.o.h.2399.2		4	240.83	odd	4
2400.2.o.h.2399.3		4	80.3	even	4
2400.2.o.h.2399.4		4	80.77	odd	4
2592.2.s.e.863.1		8	144.115	odd	12
2592.2.s.e.863.2		8	144.61	even	12
2592.2.s.e.863.3		8	144.83	even	12
2592.2.s.e.863.4		8	144.29	odd	12
2592.2.s.e.1727.1		8	144.77	odd	12
2592.2.s.e.1727.2		8	144.131	even	12
2592.2.s.e.1727.3		8	144.13	even	12
2592.2.s.e.1727.4		8	144.67	odd	12

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.41421i) q^{3} -2.82843 q^{5} -2.00000i q^{7} +(-1.00000 + 2.82843i) q^{9} +2.82843i q^{11} +2.00000i q^{13} +(-2.82843 - 4.00000i) q^{15} -6.00000 q^{19} +(2.82843 - 2.00000i) q^{21} -5.65685 q^{23} +3.00000 q^{25} +(-5.00000 + 1.41421i) q^{27} -2.82843 q^{29} +2.00000i q^{31} +(-4.00000 + 2.82843i) q^{33} +5.65685i q^{35} +6.00000i q^{37} +(-2.82843 + 2.00000i) q^{39} -5.65685i q^{41} -2.00000 q^{43} +(2.82843 - 8.00000i) q^{45} -11.3137 q^{47} +3.00000 q^{49} +8.48528 q^{53} -8.00000i q^{55} +(-6.00000 - 8.48528i) q^{57} +2.82843i q^{59} +2.00000i q^{61} +(5.65685 + 2.00000i) q^{63} -5.65685i q^{65} +2.00000 q^{67} +(-5.65685 - 8.00000i) q^{69} +5.65685 q^{71} +6.00000 q^{73} +(3.00000 + 4.24264i) q^{75} +5.65685 q^{77} -14.0000i q^{79} +(-7.00000 - 5.65685i) q^{81} +2.82843i q^{83} +(-2.82843 - 4.00000i) q^{87} +16.9706i q^{89} +4.00000 q^{91} +(-2.82843 + 2.00000i) q^{93} +16.9706 q^{95} +10.0000 q^{97} +(-8.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{9} - 24 q^{19} + 12 q^{25} - 20 q^{27} - 16 q^{33} - 8 q^{43} + 12 q^{49} - 24 q^{57} + 8 q^{67} + 24 q^{73} + 12 q^{75} - 28 q^{81} + 16 q^{91} + 40 q^{97} - 32 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^9 - 24 * q^19 + 12 * q^25 - 20 * q^27 - 16 * q^33 - 8 * q^43 + 12 * q^49 - 24 * q^57 + 8 * q^67 + 24 * q^73 + 12 * q^75 - 28 * q^81 + 16 * q^91 + 40 * q^97 - 32 * q^99

Introduction

L-functions

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