LMFDB - Embedded newform 768.2.k.c.191.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(191,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 2]))

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

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

chi := DirichletCharacter("768.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.k (of order $4$, degree $2$, 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:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			191.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	768.191
Dual form			768.2.k.c.575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.292893 + 1.70711i) q^{3} +(-1.41421 - 1.41421i) q^{5} -1.41421 q^{7} +(-2.82843 + 1.00000i) q^{9} +O(q^{10})$	\(q+(0.292893 + 1.70711i) q^{3} +(-1.41421 - 1.41421i) q^{5} -1.41421 q^{7} +(-2.82843 + 1.00000i) q^{9} +(4.00000 - 4.00000i) q^{11} +(-3.00000 - 3.00000i) q^{13} +(2.00000 - 2.82843i) q^{15} +2.82843i q^{17} +(2.82843 - 2.82843i) q^{19} +(-0.414214 - 2.41421i) q^{21} -1.00000i q^{25} +(-2.53553 - 4.53553i) q^{27} +(7.07107 - 7.07107i) q^{29} -4.24264i q^{31} +(8.00000 + 5.65685i) q^{33} +(2.00000 + 2.00000i) q^{35} +(-5.00000 + 5.00000i) q^{37} +(4.24264 - 6.00000i) q^{39} -2.82843 q^{41} +(-2.82843 - 2.82843i) q^{43} +(5.41421 + 2.58579i) q^{45} +12.0000 q^{47} -5.00000 q^{49} +(-4.82843 + 0.828427i) q^{51} +(4.24264 + 4.24264i) q^{53} -11.3137 q^{55} +(5.65685 + 4.00000i) q^{57} +(-2.00000 + 2.00000i) q^{59} +(-9.00000 - 9.00000i) q^{61} +(4.00000 - 1.41421i) q^{63} +8.48528i q^{65} +(7.07107 - 7.07107i) q^{67} -4.00000i q^{71} +4.00000i q^{73} +(1.70711 - 0.292893i) q^{75} +(-5.65685 + 5.65685i) q^{77} +9.89949i q^{79} +(7.00000 - 5.65685i) q^{81} +(4.00000 - 4.00000i) q^{85} +(14.1421 + 10.0000i) q^{87} +5.65685 q^{89} +(4.24264 + 4.24264i) q^{91} +(7.24264 - 1.24264i) q^{93} -8.00000 q^{95} -8.00000 q^{97} +(-7.31371 + 15.3137i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3}+O(q^{10}) $ 4 * q + 4 * q^3	$ 4 q + 4 q^{3} + 16 q^{11} - 12 q^{13} + 8 q^{15} + 4 q^{21} + 4 q^{27} + 32 q^{33} + 8 q^{35} - 20 q^{37} + 16 q^{45} + 48 q^{47} - 20 q^{49} - 8 q^{51} - 8 q^{59} - 36 q^{61} + 16 q^{63} + 4 q^{75} + 28 q^{81} + 16 q^{85} + 12 q^{93} - 32 q^{95} - 32 q^{97} + 16 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 16 * q^11 - 12 * q^13 + 8 * q^15 + 4 * q^21 + 4 * q^27 + 32 * q^33 + 8 * q^35 - 20 * q^37 + 16 * q^45 + 48 * q^47 - 20 * q^49 - 8 * q^51 - 8 * q^59 - 36 * q^61 + 16 * q^63 + 4 * q^75 + 28 * q^81 + 16 * q^85 + 12 * q^93 - 32 * q^95 - 32 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.292893	+	1.70711i	0.169102	+	0.985599i
$3$	0.292893	+	1.70711i	0.169102	+	0.985599i
$4$		0			0
$5$	−1.41421	−	1.41421i	−0.632456	−	0.632456i	0.316228	−	0.948683i	$-0.397584\pi$
$5$	−1.41421	−	1.41421i	−0.632456	−	0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$	−1.41421			−0.534522			−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421			−0.534522			−0.267261	+	0.963624i	$0.586119\pi$
$8$		0			0
$9$	−2.82843	+	1.00000i	−0.942809	+	0.333333i
$10$		0			0
$11$	4.00000	−	4.00000i	1.20605	−	1.20605i	0.233748	−	0.972297i	$-0.424901\pi$
$11$	4.00000	−	4.00000i	1.20605	−	1.20605i	0.972297	−	0.233748i	$-0.0750991\pi$
$12$		0			0
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	0.155747	−	0.987797i	$-0.450222\pi$
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	−0.987797	+	0.155747i	$0.950222\pi$
$14$		0			0
$15$	2.00000	−	2.82843i	0.516398	−	0.730297i
$16$		0			0
$17$			2.82843i			0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$			2.82843i			0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$		0			0
$19$	2.82843	−	2.82843i	0.648886	−	0.648886i	−0.303838	−	0.952724i	$-0.598268\pi$
$19$	2.82843	−	2.82843i	0.648886	−	0.648886i	0.952724	+	0.303838i	$0.0982682\pi$
$20$		0			0
$21$	−0.414214	−	2.41421i	−0.0903888	−	0.526825i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$		−	1.00000i		−	0.200000i
$26$		0			0
$27$	−2.53553	−	4.53553i	−0.487964	−	0.872864i
$28$		0			0
$29$	7.07107	−	7.07107i	1.31306	−	1.31306i	0.393919	−	0.919145i	$-0.371119\pi$
$29$	7.07107	−	7.07107i	1.31306	−	1.31306i	0.919145	−	0.393919i	$-0.128881\pi$
$30$		0			0
$31$		−	4.24264i		−	0.762001i	−0.924575	−	0.381000i	$-0.875580\pi$
$31$		−	4.24264i		−	0.762001i	0.924575	−	0.381000i	$-0.124420\pi$
$32$		0			0
$33$	8.00000	+	5.65685i	1.39262	+	0.984732i
$34$		0			0
$35$	2.00000	+	2.00000i	0.338062	+	0.338062i
$36$		0			0
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	−5.00000	+	5.00000i	−0.821995	+	0.821995i	0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$	4.24264	−	6.00000i	0.679366	−	0.960769i
$40$		0			0
$41$	−2.82843			−0.441726			−0.220863	−	0.975305i	$-0.570887\pi$
$41$	−2.82843			−0.441726			−0.220863	+	0.975305i	$0.570887\pi$
$42$		0			0
$43$	−2.82843	−	2.82843i	−0.431331	−	0.431331i	0.457750	−	0.889081i	$-0.348656\pi$
$43$	−2.82843	−	2.82843i	−0.431331	−	0.431331i	−0.889081	+	0.457750i	$0.848656\pi$
$44$		0			0
$45$	5.41421	+	2.58579i	0.807103	+	0.385466i
$46$		0			0
$47$	12.0000			1.75038			0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000			1.75038			0.875190	+	0.483779i	$0.160736\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$	−4.82843	+	0.828427i	−0.676115	+	0.116003i
$52$		0			0
$53$	4.24264	+	4.24264i	0.582772	+	0.582772i	0.935664	−	0.352892i	$-0.114802\pi$
$53$	4.24264	+	4.24264i	0.582772	+	0.582772i	−0.352892	+	0.935664i	$0.614802\pi$
$54$		0			0
$55$	−11.3137			−1.52554
$56$		0			0
$57$	5.65685	+	4.00000i	0.749269	+	0.529813i
$58$		0			0
$59$	−2.00000	+	2.00000i	−0.260378	+	0.260378i	−0.825208	−	0.564830i	$-0.808942\pi$
$59$	−2.00000	+	2.00000i	−0.260378	+	0.260378i	0.564830	+	0.825208i	$0.308942\pi$
$60$		0			0
$61$	−9.00000	−	9.00000i	−1.15233	−	1.15233i	−0.986084	−	0.166248i	$-0.946835\pi$
$61$	−9.00000	−	9.00000i	−1.15233	−	1.15233i	−0.166248	−	0.986084i	$-0.553165\pi$
$62$		0			0
$63$	4.00000	−	1.41421i	0.503953	−	0.178174i
$64$		0			0
$65$			8.48528i			1.05247i
$66$		0			0
$67$	7.07107	−	7.07107i	0.863868	−	0.863868i	−0.127917	−	0.991785i	$-0.540829\pi$
$67$	7.07107	−	7.07107i	0.863868	−	0.863868i	0.991785	+	0.127917i	$0.0408290\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.00000i		−	0.474713i	−0.971423	−	0.237356i	$-0.923719\pi$
$71$		−	4.00000i		−	0.474713i	0.971423	−	0.237356i	$-0.0762809\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$	1.70711	−	0.292893i	0.197120	−	0.0338204i
$76$		0			0
$77$	−5.65685	+	5.65685i	−0.644658	+	0.644658i
$78$		0			0
$79$			9.89949i			1.11378i	0.830586	+	0.556890i	$0.188006\pi$
$79$			9.89949i			1.11378i	−0.830586	+	0.556890i	$0.811994\pi$
$80$		0			0
$81$	7.00000	−	5.65685i	0.777778	−	0.628539i
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$	4.00000	−	4.00000i	0.433861	−	0.433861i
$86$		0			0
$87$	14.1421	+	10.0000i	1.51620	+	1.07211i
$88$		0			0
$89$	5.65685			0.599625			0.299813	−	0.953998i	$-0.403076\pi$
$89$	5.65685			0.599625			0.299813	+	0.953998i	$0.403076\pi$
$90$		0			0
$91$	4.24264	+	4.24264i	0.444750	+	0.444750i
$92$		0			0
$93$	7.24264	−	1.24264i	0.751027	−	0.128856i
$94$		0			0
$95$	−8.00000			−0.820783
$96$		0			0
$97$	−8.00000			−0.812277			−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000			−0.812277			−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$	−7.31371	+	15.3137i	−0.735055	+	1.53909i
$100$		0			0
$101$	−9.89949	−	9.89949i	−0.985037	−	0.985037i	0.0148531	−	0.999890i	$-0.495272\pi$
$101$	−9.89949	−	9.89949i	−0.985037	−	0.985037i	−0.999890	+	0.0148531i	$0.995272\pi$
$102$		0			0
$103$	−4.24264			−0.418040			−0.209020	−	0.977911i	$-0.567027\pi$
$103$	−4.24264			−0.418040			−0.209020	+	0.977911i	$0.567027\pi$
$104$		0			0
$105$	−2.82843	+	4.00000i	−0.276026	+	0.390360i
$106$		0			0
$107$	−2.00000	+	2.00000i	−0.193347	+	0.193347i	−0.797141	−	0.603793i	$-0.793655\pi$
$107$	−2.00000	+	2.00000i	−0.193347	+	0.193347i	0.603793	+	0.797141i	$0.293655\pi$
$108$		0			0
$109$	3.00000	+	3.00000i	0.287348	+	0.287348i	0.836031	−	0.548683i	$-0.184871\pi$
$109$	3.00000	+	3.00000i	0.287348	+	0.287348i	−0.548683	+	0.836031i	$0.684871\pi$
$110$		0			0
$111$	−10.0000	−	7.07107i	−0.949158	−	0.671156i
$112$		0			0
$113$		−	16.9706i		−	1.59646i	−0.602355	−	0.798228i	$-0.705771\pi$
$113$		−	16.9706i		−	1.59646i	0.602355	−	0.798228i	$-0.294229\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	11.4853	+	5.48528i	1.06181	+	0.507114i
$118$		0			0
$119$		−	4.00000i		−	0.366679i
$120$		0			0
$121$		−	21.0000i		−	1.90909i
$122$		0			0
$123$	−0.828427	−	4.82843i	−0.0746968	−	0.435365i
$124$		0			0
$125$	−8.48528	+	8.48528i	−0.758947	+	0.758947i
$126$		0			0
$127$			4.24264i			0.376473i	0.982124	+	0.188237i	$0.0602772\pi$
$127$			4.24264i			0.376473i	−0.982124	+	0.188237i	$0.939723\pi$
$128$		0			0
$129$	4.00000	−	5.65685i	0.352180	−	0.498058i
$130$		0			0
$131$	−6.00000	−	6.00000i	−0.524222	−	0.524222i	0.394621	−	0.918844i	$-0.370876\pi$
$131$	−6.00000	−	6.00000i	−0.524222	−	0.524222i	−0.918844	+	0.394621i	$0.870876\pi$
$132$		0			0
$133$	−4.00000	+	4.00000i	−0.346844	+	0.346844i
$134$		0			0
$135$	−2.82843	+	10.0000i	−0.243432	+	0.860663i
$136$		0			0
$137$	−14.1421			−1.20824			−0.604122	−	0.796892i	$-0.706476\pi$
$137$	−14.1421			−1.20824			−0.604122	+	0.796892i	$0.706476\pi$
$138$		0			0
$139$	9.89949	+	9.89949i	0.839664	+	0.839664i	0.988815	−	0.149150i	$-0.0476538\pi$
$139$	9.89949	+	9.89949i	0.839664	+	0.839664i	−0.149150	+	0.988815i	$0.547654\pi$
$140$		0			0
$141$	3.51472	+	20.4853i	0.295993	+	1.72517i
$142$		0			0
$143$	−24.0000			−2.00698
$144$		0			0
$145$	−20.0000			−1.66091
$146$		0			0
$147$	−1.46447	−	8.53553i	−0.120787	−	0.703999i
$148$		0			0
$149$	4.24264	+	4.24264i	0.347571	+	0.347571i	0.859204	−	0.511633i	$-0.170959\pi$
$149$	4.24264	+	4.24264i	0.347571	+	0.347571i	−0.511633	+	0.859204i	$0.670959\pi$
$150$		0			0
$151$	−4.24264			−0.345261			−0.172631	−	0.984987i	$-0.555227\pi$
$151$	−4.24264			−0.345261			−0.172631	+	0.984987i	$0.555227\pi$
$152$		0			0
$153$	−2.82843	−	8.00000i	−0.228665	−	0.646762i
$154$		0			0
$155$	−6.00000	+	6.00000i	−0.481932	+	0.481932i
$156$		0			0
$157$	−3.00000	−	3.00000i	−0.239426	−	0.239426i	0.577186	−	0.816612i	$-0.304151\pi$
$157$	−3.00000	−	3.00000i	−0.239426	−	0.239426i	−0.816612	+	0.577186i	$0.804151\pi$
$158$		0			0
$159$	−6.00000	+	8.48528i	−0.475831	+	0.672927i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.48528	−	8.48528i	0.664619	−	0.664619i	−0.291847	−	0.956465i	$-0.594270\pi$
$163$	8.48528	−	8.48528i	0.664619	−	0.664619i	0.956465	+	0.291847i	$0.0942697\pi$
$164$		0			0
$165$	−3.31371	−	19.3137i	−0.257972	−	1.50357i
$166$		0			0
$167$			16.0000i			1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$			16.0000i			1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	−5.17157	+	10.8284i	−0.395480	+	0.828071i
$172$		0			0
$173$	−7.07107	+	7.07107i	−0.537603	+	0.537603i	−0.922824	−	0.385221i	$-0.874125\pi$
$173$	−7.07107	+	7.07107i	−0.537603	+	0.537603i	0.385221	+	0.922824i	$0.374125\pi$
$174$		0			0
$175$			1.41421i			0.106904i
$176$		0			0
$177$	−4.00000	−	2.82843i	−0.300658	−	0.212598i
$178$		0			0
$179$	6.00000	+	6.00000i	0.448461	+	0.448461i	0.894843	−	0.446382i	$-0.147288\pi$
$179$	6.00000	+	6.00000i	0.448461	+	0.448461i	−0.446382	+	0.894843i	$0.647288\pi$
$180$		0			0
$181$	−3.00000	+	3.00000i	−0.222988	+	0.222988i	−0.809756	−	0.586767i	$-0.800400\pi$
$181$	−3.00000	+	3.00000i	−0.222988	+	0.222988i	0.586767	+	0.809756i	$0.300400\pi$
$182$		0			0
$183$	12.7279	−	18.0000i	0.940875	−	1.33060i
$184$		0			0
$185$	14.1421			1.03975
$186$		0			0
$187$	11.3137	+	11.3137i	0.827340	+	0.827340i
$188$		0			0
$189$	3.58579	+	6.41421i	0.260828	+	0.466565i
$190$		0			0
$191$	4.00000			0.289430			0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000			0.289430			0.144715	+	0.989473i	$0.453773\pi$
$192$		0			0
$193$	2.00000			0.143963			0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000			0.143963			0.0719816	+	0.997406i	$0.477068\pi$
$194$		0			0
$195$	−14.4853	+	2.48528i	−1.03731	+	0.177975i
$196$		0			0
$197$	12.7279	+	12.7279i	0.906827	+	0.906827i	0.996015	−	0.0891879i	$-0.0284272\pi$
$197$	12.7279	+	12.7279i	0.906827	+	0.906827i	−0.0891879	+	0.996015i	$0.528427\pi$
$198$		0			0
$199$	15.5563			1.10276			0.551380	−	0.834254i	$-0.314101\pi$
$199$	15.5563			1.10276			0.551380	+	0.834254i	$0.314101\pi$
$200$		0			0
$201$	14.1421	+	10.0000i	0.997509	+	0.705346i
$202$		0			0
$203$	−10.0000	+	10.0000i	−0.701862	+	0.701862i
$204$		0			0
$205$	4.00000	+	4.00000i	0.279372	+	0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	22.6274i		−	1.56517i
$210$		0			0
$211$	−12.7279	+	12.7279i	−0.876226	+	0.876226i	−0.993142	−	0.116916i	$-0.962699\pi$
$211$	−12.7279	+	12.7279i	−0.876226	+	0.876226i	0.116916	+	0.993142i	$0.462699\pi$
$212$		0			0
$213$	6.82843	−	1.17157i	0.467876	−	0.0802749i
$214$		0			0
$215$			8.00000i			0.545595i
$216$		0			0
$217$			6.00000i			0.407307i
$218$		0			0
$219$	−6.82843	+	1.17157i	−0.461422	+	0.0791676i
$220$		0			0
$221$	8.48528	−	8.48528i	0.570782	−	0.570782i
$222$		0			0
$223$			12.7279i			0.852325i	0.904647	+	0.426162i	$0.140135\pi$
$223$			12.7279i			0.852325i	−0.904647	+	0.426162i	$0.859865\pi$
$224$		0			0
$225$	1.00000	+	2.82843i	0.0666667	+	0.188562i
$226$		0			0
$227$	−8.00000	−	8.00000i	−0.530979	−	0.530979i	0.389885	−	0.920864i	$-0.372515\pi$
$227$	−8.00000	−	8.00000i	−0.530979	−	0.530979i	−0.920864	+	0.389885i	$0.872515\pi$
$228$		0			0
$229$	5.00000	−	5.00000i	0.330409	−	0.330409i	−0.522333	−	0.852742i	$-0.674938\pi$
$229$	5.00000	−	5.00000i	0.330409	−	0.330409i	0.852742	+	0.522333i	$0.174938\pi$
$230$		0			0
$231$	−11.3137	−	8.00000i	−0.744387	−	0.526361i
$232$		0			0
$233$	11.3137			0.741186			0.370593	−	0.928795i	$-0.379155\pi$
$233$	11.3137			0.741186			0.370593	+	0.928795i	$0.379155\pi$
$234$		0			0
$235$	−16.9706	−	16.9706i	−1.10704	−	1.10704i
$236$		0			0
$237$	−16.8995	+	2.89949i	−1.09774	+	0.188342i
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$		0			0
$243$	11.7071	+	10.2929i	0.751011	+	0.660289i
$244$		0			0
$245$	7.07107	+	7.07107i	0.451754	+	0.451754i
$246$		0			0
$247$	−16.9706			−1.07981
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.00000	−	8.00000i	0.504956	−	0.504956i	−0.408018	−	0.912974i	$-0.633780\pi$
$251$	8.00000	−	8.00000i	0.504956	−	0.504956i	0.912974	+	0.408018i	$0.133780\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	8.00000	+	5.65685i	0.500979	+	0.354246i
$256$		0			0
$257$			11.3137i			0.705730i	0.935674	+	0.352865i	$0.114792\pi$
$257$			11.3137i			0.705730i	−0.935674	+	0.352865i	$0.885208\pi$
$258$		0			0
$259$	7.07107	−	7.07107i	0.439375	−	0.439375i
$260$		0			0
$261$	−12.9289	+	27.0711i	−0.800281	+	1.67566i
$262$		0			0
$263$			4.00000i			0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$			4.00000i			0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$		0			0
$265$		−	12.0000i		−	0.737154i
$266$		0			0
$267$	1.65685	+	9.65685i	0.101398	+	0.590990i
$268$		0			0
$269$	4.24264	−	4.24264i	0.258678	−	0.258678i	−0.565838	−	0.824516i	$-0.691447\pi$
$269$	4.24264	−	4.24264i	0.258678	−	0.258678i	0.824516	+	0.565838i	$0.191447\pi$
$270$		0			0
$271$			21.2132i			1.28861i	0.764768	+	0.644305i	$0.222853\pi$
$271$			21.2132i			1.28861i	−0.764768	+	0.644305i	$0.777147\pi$
$272$		0			0
$273$	−6.00000	+	8.48528i	−0.363137	+	0.513553i
$274$		0			0
$275$	−4.00000	−	4.00000i	−0.241209	−	0.241209i
$276$		0			0
$277$	3.00000	−	3.00000i	0.180253	−	0.180253i	−0.611213	−	0.791466i	$-0.709318\pi$
$277$	3.00000	−	3.00000i	0.180253	−	0.180253i	0.791466	+	0.611213i	$0.209318\pi$
$278$		0			0
$279$	4.24264	+	12.0000i	0.254000	+	0.718421i
$280$		0			0
$281$	16.9706			1.01238			0.506189	−	0.862422i	$-0.331054\pi$
$281$	16.9706			1.01238			0.506189	+	0.862422i	$0.331054\pi$
$282$		0			0
$283$	9.89949	+	9.89949i	0.588464	+	0.588464i	0.937215	−	0.348751i	$-0.113394\pi$
$283$	9.89949	+	9.89949i	0.588464	+	0.588464i	−0.348751	+	0.937215i	$0.613394\pi$
$284$		0			0
$285$	−2.34315	−	13.6569i	−0.138796	−	0.808962i
$286$		0			0
$287$	4.00000			0.236113
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$	−2.34315	−	13.6569i	−0.137358	−	0.800579i
$292$		0			0
$293$	−9.89949	−	9.89949i	−0.578335	−	0.578335i	0.356110	−	0.934444i	$-0.384103\pi$
$293$	−9.89949	−	9.89949i	−0.578335	−	0.578335i	−0.934444	+	0.356110i	$0.884103\pi$
$294$		0			0
$295$	5.65685			0.329355
$296$		0			0
$297$	−28.2843	−	8.00000i	−1.64122	−	0.464207i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	4.00000	+	4.00000i	0.230556	+	0.230556i
$302$		0			0
$303$	14.0000	−	19.7990i	0.804279	−	1.13742i
$304$		0			0
$305$			25.4558i			1.45760i
$306$		0			0
$307$	−1.41421	+	1.41421i	−0.0807134	+	0.0807134i	−0.746311	−	0.665597i	$-0.768177\pi$
$307$	−1.41421	+	1.41421i	−0.0807134	+	0.0807134i	0.665597	+	0.746311i	$0.268177\pi$
$308$		0			0
$309$	−1.24264	−	7.24264i	−0.0706914	−	0.412019i
$310$		0			0
$311$		−	28.0000i		−	1.58773i	−0.608091	−	0.793867i	$-0.708065\pi$
$311$		−	28.0000i		−	1.58773i	0.608091	−	0.793867i	$-0.291935\pi$
$312$		0			0
$313$			22.0000i			1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$			22.0000i			1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$		0			0
$315$	−7.65685	−	3.65685i	−0.431415	−	0.206040i
$316$		0			0
$317$	4.24264	−	4.24264i	0.238290	−	0.238290i	−0.577851	−	0.816142i	$-0.696109\pi$
$317$	4.24264	−	4.24264i	0.238290	−	0.238290i	0.816142	+	0.577851i	$0.196109\pi$
$318$		0			0
$319$		−	56.5685i		−	3.16723i
$320$		0			0
$321$	−4.00000	−	2.82843i	−0.223258	−	0.157867i
$322$		0			0
$323$	8.00000	+	8.00000i	0.445132	+	0.445132i
$324$		0			0
$325$	−3.00000	+	3.00000i	−0.166410	+	0.166410i
$326$		0			0
$327$	−4.24264	+	6.00000i	−0.234619	+	0.331801i
$328$		0			0
$329$	−16.9706			−0.935617
$330$		0			0
$331$	−21.2132	−	21.2132i	−1.16598	−	1.16598i	−0.983143	−	0.182841i	$-0.941471\pi$
$331$	−21.2132	−	21.2132i	−1.16598	−	1.16598i	−0.182841	−	0.983143i	$-0.558529\pi$
$332$		0			0
$333$	9.14214	−	19.1421i	0.500986	−	1.04898i
$334$		0			0
$335$	−20.0000			−1.09272
$336$		0			0
$337$	20.0000			1.08947			0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000			1.08947			0.544735	+	0.838608i	$0.316630\pi$
$338$		0			0
$339$	28.9706	−	4.97056i	1.57346	−	0.269964i
$340$		0			0
$341$	−16.9706	−	16.9706i	−0.919007	−	0.919007i
$342$		0			0
$343$	16.9706			0.916324
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.0000	−	20.0000i	1.07366	−	1.07366i	0.0765939	−	0.997062i	$-0.475596\pi$
$347$	20.0000	−	20.0000i	1.07366	−	1.07366i	0.997062	−	0.0765939i	$-0.0244045\pi$
$348$		0			0
$349$	9.00000	+	9.00000i	0.481759	+	0.481759i	0.905693	−	0.423934i	$-0.139351\pi$
$349$	9.00000	+	9.00000i	0.481759	+	0.481759i	−0.423934	+	0.905693i	$0.639351\pi$
$350$		0			0
$351$	−6.00000	+	21.2132i	−0.320256	+	1.13228i
$352$		0			0
$353$		−	11.3137i		−	0.602168i	−0.953598	−	0.301084i	$-0.902652\pi$
$353$		−	11.3137i		−	0.602168i	0.953598	−	0.301084i	$-0.0973484\pi$
$354$		0			0
$355$	−5.65685	+	5.65685i	−0.300235	+	0.300235i
$356$		0			0
$357$	6.82843	−	1.17157i	0.361399	−	0.0620062i
$358$		0			0
$359$		−	20.0000i		−	1.05556i	−0.849381	−	0.527780i	$-0.823025\pi$
$359$		−	20.0000i		−	1.05556i	0.849381	−	0.527780i	$-0.176975\pi$
$360$		0			0
$361$			3.00000i			0.157895i
$362$		0			0
$363$	35.8492	−	6.15076i	1.88160	−	0.322831i
$364$		0			0
$365$	5.65685	−	5.65685i	0.296093	−	0.296093i
$366$		0			0
$367$		−	18.3848i		−	0.959678i	−0.877357	−	0.479839i	$-0.840695\pi$
$367$		−	18.3848i		−	0.959678i	0.877357	−	0.479839i	$-0.159305\pi$
$368$		0			0
$369$	8.00000	−	2.82843i	0.416463	−	0.147242i
$370$		0			0
$371$	−6.00000	−	6.00000i	−0.311504	−	0.311504i
$372$		0			0
$373$	9.00000	−	9.00000i	0.466002	−	0.466002i	−0.434614	−	0.900617i	$-0.643115\pi$
$373$	9.00000	−	9.00000i	0.466002	−	0.466002i	0.900617	+	0.434614i	$0.143115\pi$
$374$		0			0
$375$	−16.9706	−	12.0000i	−0.876356	−	0.619677i
$376$		0			0
$377$	−42.4264			−2.18507
$378$		0			0
$379$	−14.1421	−	14.1421i	−0.726433	−	0.726433i	0.243475	−	0.969907i	$-0.421713\pi$
$379$	−14.1421	−	14.1421i	−0.726433	−	0.726433i	−0.969907	+	0.243475i	$0.921713\pi$
$380$		0			0
$381$	−7.24264	+	1.24264i	−0.371052	+	0.0636624i
$382$		0			0
$383$	28.0000			1.43073			0.715367	−	0.698749i	$-0.246260\pi$
$383$	28.0000			1.43073			0.715367	+	0.698749i	$0.246260\pi$
$384$		0			0
$385$	16.0000			0.815436
$386$		0			0
$387$	10.8284	+	5.17157i	0.550440	+	0.262886i
$388$		0			0
$389$	−7.07107	−	7.07107i	−0.358517	−	0.358517i	0.504749	−	0.863266i	$-0.331585\pi$
$389$	−7.07107	−	7.07107i	−0.358517	−	0.358517i	−0.863266	+	0.504749i	$0.831585\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	8.48528	−	12.0000i	0.428026	−	0.605320i
$394$		0			0
$395$	14.0000	−	14.0000i	0.704416	−	0.704416i
$396$		0			0
$397$	−13.0000	−	13.0000i	−0.652451	−	0.652451i	0.301131	−	0.953583i	$-0.402636\pi$
$397$	−13.0000	−	13.0000i	−0.652451	−	0.652451i	−0.953583	+	0.301131i	$0.902636\pi$
$398$		0			0
$399$	−8.00000	−	5.65685i	−0.400501	−	0.283197i
$400$		0			0
$401$			31.1127i			1.55369i	0.629689	+	0.776847i	$0.283182\pi$
$401$			31.1127i			1.55369i	−0.629689	+	0.776847i	$0.716818\pi$
$402$		0			0
$403$	−12.7279	+	12.7279i	−0.634023	+	0.634023i
$404$		0			0
$405$	−17.8995	−	1.89949i	−0.889433	−	0.0943867i
$406$		0			0
$407$			40.0000i			1.98273i
$408$		0			0
$409$		−	8.00000i		−	0.395575i	−0.980245	−	0.197787i	$-0.936624\pi$
$409$		−	8.00000i		−	0.395575i	0.980245	−	0.197787i	$-0.0633755\pi$
$410$		0			0
$411$	−4.14214	−	24.1421i	−0.204316	−	1.19084i
$412$		0			0
$413$	2.82843	−	2.82843i	0.139178	−	0.139178i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−14.0000	+	19.7990i	−0.685583	+	0.969561i
$418$		0			0
$419$	24.0000	+	24.0000i	1.17248	+	1.17248i	0.981617	+	0.190859i	$0.0611274\pi$
$419$	24.0000	+	24.0000i	1.17248	+	1.17248i	0.190859	+	0.981617i	$0.438873\pi$
$420$		0			0
$421$	−11.0000	+	11.0000i	−0.536107	+	0.536107i	−0.922383	−	0.386276i	$-0.873761\pi$
$421$	−11.0000	+	11.0000i	−0.536107	+	0.536107i	0.386276	+	0.922383i	$0.373761\pi$
$422$		0			0
$423$	−33.9411	+	12.0000i	−1.65027	+	0.583460i
$424$		0			0
$425$	2.82843			0.137199
$426$		0			0
$427$	12.7279	+	12.7279i	0.615947	+	0.615947i
$428$		0			0
$429$	−7.02944	−	40.9706i	−0.339384	−	1.97808i
$430$		0			0
$431$	28.0000			1.34871			0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000			1.34871			0.674356	+	0.738406i	$0.264421\pi$
$432$		0			0
$433$	16.0000			0.768911			0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000			0.768911			0.384455	+	0.923144i	$0.374389\pi$
$434$		0			0
$435$	−5.85786	−	34.1421i	−0.280863	−	1.63699i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−4.24264			−0.202490			−0.101245	−	0.994862i	$-0.532283\pi$
$439$	−4.24264			−0.202490			−0.101245	+	0.994862i	$0.532283\pi$
$440$		0			0
$441$	14.1421	−	5.00000i	0.673435	−	0.238095i
$442$		0			0
$443$	−8.00000	+	8.00000i	−0.380091	+	0.380091i	−0.871135	−	0.491044i	$-0.836616\pi$
$443$	−8.00000	+	8.00000i	−0.380091	+	0.380091i	0.491044	+	0.871135i	$0.336616\pi$
$444$		0			0
$445$	−8.00000	−	8.00000i	−0.379236	−	0.379236i
$446$		0			0
$447$	−6.00000	+	8.48528i	−0.283790	+	0.401340i
$448$		0			0
$449$		−	2.82843i		−	0.133482i	−0.997770	−	0.0667409i	$-0.978740\pi$
$449$		−	2.82843i		−	0.133482i	0.997770	−	0.0667409i	$-0.0212601\pi$
$450$		0			0
$451$	−11.3137	+	11.3137i	−0.532742	+	0.532742i
$452$		0			0
$453$	−1.24264	−	7.24264i	−0.0583844	−	0.340289i
$454$		0			0
$455$		−	12.0000i		−	0.562569i
$456$		0			0
$457$			16.0000i			0.748448i	0.927338	+	0.374224i	$0.122091\pi$
$457$			16.0000i			0.748448i	−0.927338	+	0.374224i	$0.877909\pi$
$458$		0			0
$459$	12.8284	−	7.17157i	0.598780	−	0.334740i
$460$		0			0
$461$	−4.24264	+	4.24264i	−0.197599	+	0.197599i	−0.798970	−	0.601371i	$-0.794622\pi$
$461$	−4.24264	+	4.24264i	−0.197599	+	0.197599i	0.601371	+	0.798970i	$0.294622\pi$
$462$		0			0
$463$			7.07107i			0.328620i	0.986409	+	0.164310i	$0.0525398\pi$
$463$			7.07107i			0.328620i	−0.986409	+	0.164310i	$0.947460\pi$
$464$		0			0
$465$	−12.0000	−	8.48528i	−0.556487	−	0.393496i
$466$		0			0
$467$	8.00000	+	8.00000i	0.370196	+	0.370196i	0.867548	−	0.497353i	$-0.165694\pi$
$467$	8.00000	+	8.00000i	0.370196	+	0.370196i	−0.497353	+	0.867548i	$0.665694\pi$
$468$		0			0
$469$	−10.0000	+	10.0000i	−0.461757	+	0.461757i
$470$		0			0
$471$	4.24264	−	6.00000i	0.195491	−	0.276465i
$472$		0			0
$473$	−22.6274			−1.04041
$474$		0			0
$475$	−2.82843	−	2.82843i	−0.129777	−	0.129777i
$476$		0			0
$477$	−16.2426	−	7.75736i	−0.743699	−	0.355185i
$478$		0			0
$479$	−8.00000			−0.365529			−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000			−0.365529			−0.182765	+	0.983157i	$0.558505\pi$
$480$		0			0
$481$	30.0000			1.36788
$482$		0			0
$483$		0			0
$484$		0			0
$485$	11.3137	+	11.3137i	0.513729	+	0.513729i
$486$		0			0
$487$	35.3553			1.60210			0.801052	−	0.598595i	$-0.204274\pi$
$487$	35.3553			1.60210			0.801052	+	0.598595i	$0.204274\pi$
$488$		0			0
$489$	16.9706	+	12.0000i	0.767435	+	0.542659i
$490$		0			0
$491$	2.00000	−	2.00000i	0.0902587	−	0.0902587i	−0.660536	−	0.750795i	$-0.729671\pi$
$491$	2.00000	−	2.00000i	0.0902587	−	0.0902587i	0.750795	+	0.660536i	$0.229671\pi$
$492$		0			0
$493$	20.0000	+	20.0000i	0.900755	+	0.900755i
$494$		0			0
$495$	32.0000	−	11.3137i	1.43829	−	0.508513i
$496$		0			0
$497$			5.65685i			0.253745i
$498$		0			0
$499$	−9.89949	+	9.89949i	−0.443162	+	0.443162i	−0.893073	−	0.449911i	$-0.851456\pi$
$499$	−9.89949	+	9.89949i	−0.443162	+	0.443162i	0.449911	+	0.893073i	$0.351456\pi$
$500$		0			0
$501$	−27.3137	+	4.68629i	−1.22029	+	0.209368i
$502$		0			0
$503$		−	12.0000i		−	0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$		−	12.0000i		−	0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$		0			0
$505$			28.0000i			1.24598i
$506$		0			0
$507$	−8.53553	+	1.46447i	−0.379076	+	0.0650392i
$508$		0			0
$509$	4.24264	−	4.24264i	0.188052	−	0.188052i	−0.606802	−	0.794853i	$-0.707548\pi$
$509$	4.24264	−	4.24264i	0.188052	−	0.188052i	0.794853	+	0.606802i	$0.207548\pi$
$510$		0			0
$511$		−	5.65685i		−	0.250244i
$512$		0			0
$513$	−20.0000	−	5.65685i	−0.883022	−	0.249756i
$514$		0			0
$515$	6.00000	+	6.00000i	0.264392	+	0.264392i
$516$		0			0
$517$	48.0000	−	48.0000i	2.11104	−	2.11104i
$518$		0			0
$519$	−14.1421	−	10.0000i	−0.620771	−	0.438951i
$520$		0			0
$521$	36.7696			1.61090			0.805452	−	0.592661i	$-0.201923\pi$
$521$	36.7696			1.61090			0.805452	+	0.592661i	$0.201923\pi$
$522$		0			0
$523$	−25.4558	−	25.4558i	−1.11311	−	1.11311i	−0.992728	−	0.120378i	$-0.961589\pi$
$523$	−25.4558	−	25.4558i	−1.11311	−	1.11311i	−0.120378	−	0.992728i	$-0.538411\pi$
$524$		0			0
$525$	−2.41421	+	0.414214i	−0.105365	+	0.0180778i
$526$		0			0
$527$	12.0000			0.522728
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$	3.65685	−	7.65685i	0.158694	−	0.332279i
$532$		0			0
$533$	8.48528	+	8.48528i	0.367538	+	0.367538i
$534$		0			0
$535$	5.65685			0.244567
$536$		0			0
$537$	−8.48528	+	12.0000i	−0.366167	+	0.517838i
$538$		0			0
$539$	−20.0000	+	20.0000i	−0.861461	+	0.861461i
$540$		0			0
$541$	31.0000	+	31.0000i	1.33279	+	1.33279i	0.902861	+	0.429934i	$0.141463\pi$
$541$	31.0000	+	31.0000i	1.33279	+	1.33279i	0.429934	+	0.902861i	$0.358537\pi$
$542$		0			0
$543$	−6.00000	−	4.24264i	−0.257485	−	0.182069i
$544$		0			0
$545$		−	8.48528i		−	0.363470i
$546$		0			0
$547$	19.7990	−	19.7990i	0.846544	−	0.846544i	−0.143156	−	0.989700i	$-0.545725\pi$
$547$	19.7990	−	19.7990i	0.846544	−	0.846544i	0.989700	+	0.143156i	$0.0457252\pi$
$548$		0			0
$549$	34.4558	+	16.4558i	1.47054	+	0.702318i
$550$		0			0
$551$		−	40.0000i		−	1.70406i
$552$		0			0
$553$		−	14.0000i		−	0.595341i
$554$		0			0
$555$	4.14214	+	24.1421i	0.175824	+	1.02478i
$556$		0			0
$557$	18.3848	−	18.3848i	0.778988	−	0.778988i	−0.200671	−	0.979659i	$-0.564312\pi$
$557$	18.3848	−	18.3848i	0.778988	−	0.778988i	0.979659	+	0.200671i	$0.0643121\pi$
$558$		0			0
$559$			16.9706i			0.717778i
$560$		0			0
$561$	−16.0000	+	22.6274i	−0.675521	+	0.955330i
$562$		0			0
$563$	−20.0000	−	20.0000i	−0.842900	−	0.842900i	0.146336	−	0.989235i	$-0.453252\pi$
$563$	−20.0000	−	20.0000i	−0.842900	−	0.842900i	−0.989235	+	0.146336i	$0.953252\pi$
$564$		0			0
$565$	−24.0000	+	24.0000i	−1.00969	+	1.00969i
$566$		0			0
$567$	−9.89949	+	8.00000i	−0.415740	+	0.335968i
$568$		0			0
$569$	2.82843			0.118574			0.0592869	−	0.998241i	$-0.481117\pi$
$569$	2.82843			0.118574			0.0592869	+	0.998241i	$0.481117\pi$
$570$		0			0
$571$	18.3848	+	18.3848i	0.769379	+	0.769379i	0.977997	−	0.208618i	$-0.0668966\pi$
$571$	18.3848	+	18.3848i	0.769379	+	0.769379i	−0.208618	+	0.977997i	$0.566897\pi$
$572$		0			0
$573$	1.17157	+	6.82843i	0.0489432	+	0.285262i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−28.0000			−1.16566			−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000			−1.16566			−0.582828	+	0.812596i	$0.698054\pi$
$578$		0			0
$579$	0.585786	+	3.41421i	0.0243445	+	0.141890i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	33.9411			1.40570
$584$		0			0
$585$	−8.48528	−	24.0000i	−0.350823	−	0.992278i
$586$		0			0
$587$	−14.0000	+	14.0000i	−0.577842	+	0.577842i	−0.934308	−	0.356466i	$-0.883981\pi$
$587$	−14.0000	+	14.0000i	−0.577842	+	0.577842i	0.356466	+	0.934308i	$0.383981\pi$
$588$		0			0
$589$	−12.0000	−	12.0000i	−0.494451	−	0.494451i
$590$		0			0
$591$	−18.0000	+	25.4558i	−0.740421	+	1.04711i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$	−5.65685	+	5.65685i	−0.231908	+	0.231908i
$596$		0			0
$597$	4.55635	+	26.5563i	0.186479	+	1.08688i
$598$		0			0
$599$			32.0000i			1.30748i	0.756717	+	0.653742i	$0.226802\pi$
$599$			32.0000i			1.30748i	−0.756717	+	0.653742i	$0.773198\pi$
$600$		0			0
$601$		−	20.0000i		−	0.815817i	−0.913023	−	0.407909i	$-0.866258\pi$
$601$		−	20.0000i		−	0.815817i	0.913023	−	0.407909i	$-0.133742\pi$
$602$		0			0
$603$	−12.9289	+	27.0711i	−0.526507	+	1.10242i
$604$		0			0
$605$	−29.6985	+	29.6985i	−1.20742	+	1.20742i
$606$		0			0
$607$		−	29.6985i		−	1.20542i	−0.797959	−	0.602712i	$-0.794087\pi$
$607$		−	29.6985i		−	1.20542i	0.797959	−	0.602712i	$-0.205913\pi$
$608$		0			0
$609$	−20.0000	−	14.1421i	−0.810441	−	0.573068i
$610$		0			0
$611$	−36.0000	−	36.0000i	−1.45640	−	1.45640i
$612$		0			0
$613$	−25.0000	+	25.0000i	−1.00974	+	1.00974i	−0.00978840	+	0.999952i	$0.503116\pi$
$613$	−25.0000	+	25.0000i	−1.00974	+	1.00974i	−0.999952	+	0.00978840i	$0.996884\pi$
$614$		0			0
$615$	−5.65685	+	8.00000i	−0.228106	+	0.322591i
$616$		0			0
$617$	28.2843			1.13868			0.569341	−	0.822102i	$-0.307198\pi$
$617$	28.2843			1.13868			0.569341	+	0.822102i	$0.307198\pi$
$618$		0			0
$619$	−9.89949	−	9.89949i	−0.397894	−	0.397894i	0.479595	−	0.877490i	$-0.340783\pi$
$619$	−9.89949	−	9.89949i	−0.397894	−	0.397894i	−0.877490	+	0.479595i	$0.840783\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8.00000			−0.320513
$624$		0			0
$625$	19.0000			0.760000
$626$		0			0
$627$	38.6274	−	6.62742i	1.54263	−	0.264674i
$628$		0			0
$629$	−14.1421	−	14.1421i	−0.563884	−	0.563884i
$630$		0			0
$631$	−35.3553			−1.40747			−0.703737	−	0.710461i	$-0.748487\pi$
$631$	−35.3553			−1.40747			−0.703737	+	0.710461i	$0.748487\pi$
$632$		0			0
$633$	−25.4558	−	18.0000i	−1.01178	−	0.715436i
$634$		0			0
$635$	6.00000	−	6.00000i	0.238103	−	0.238103i
$636$		0			0
$637$	15.0000	+	15.0000i	0.594322	+	0.594322i
$638$		0			0
$639$	4.00000	+	11.3137i	0.158238	+	0.447563i
$640$		0			0
$641$		−	2.82843i		−	0.111716i	−0.998439	−	0.0558581i	$-0.982211\pi$
$641$		−	2.82843i		−	0.111716i	0.998439	−	0.0558581i	$-0.0177894\pi$
$642$		0			0
$643$	−19.7990	+	19.7990i	−0.780796	+	0.780796i	−0.979965	−	0.199169i	$-0.936176\pi$
$643$	−19.7990	+	19.7990i	−0.780796	+	0.780796i	0.199169	+	0.979965i	$0.436176\pi$
$644$		0			0
$645$	−13.6569	+	2.34315i	−0.537738	+	0.0922613i
$646$		0			0
$647$			40.0000i			1.57256i	0.617869	+	0.786281i	$0.287996\pi$
$647$			40.0000i			1.57256i	−0.617869	+	0.786281i	$0.712004\pi$
$648$		0			0
$649$			16.0000i			0.628055i
$650$		0			0
$651$	−10.2426	+	1.75736i	−0.401441	+	0.0688763i
$652$		0			0
$653$	7.07107	−	7.07107i	0.276712	−	0.276712i	−0.555083	−	0.831795i	$-0.687313\pi$
$653$	7.07107	−	7.07107i	0.276712	−	0.276712i	0.831795	+	0.555083i	$0.187313\pi$
$654$		0			0
$655$			16.9706i			0.663095i
$656$		0			0
$657$	−4.00000	−	11.3137i	−0.156055	−	0.441390i
$658$		0			0
$659$	−2.00000	−	2.00000i	−0.0779089	−	0.0779089i	0.667078	−	0.744987i	$-0.267545\pi$
$659$	−2.00000	−	2.00000i	−0.0779089	−	0.0779089i	−0.744987	+	0.667078i	$0.767545\pi$
$660$		0			0
$661$	23.0000	−	23.0000i	0.894596	−	0.894596i	−0.100355	−	0.994952i	$-0.531998\pi$
$661$	23.0000	−	23.0000i	0.894596	−	0.894596i	0.994952	+	0.100355i	$0.0319980\pi$
$662$		0			0
$663$	16.9706	+	12.0000i	0.659082	+	0.466041i
$664$		0			0
$665$	11.3137			0.438727
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−21.7279	+	3.72792i	−0.840050	+	0.144130i
$670$		0			0
$671$	−72.0000			−2.77953
$672$		0			0
$673$	−48.0000			−1.85026			−0.925132	−	0.379646i	$-0.876046\pi$
$673$	−48.0000			−1.85026			−0.925132	+	0.379646i	$0.876046\pi$
$674$		0			0
$675$	−4.53553	+	2.53553i	−0.174573	+	0.0975927i
$676$		0			0
$677$	29.6985	+	29.6985i	1.14141	+	1.14141i	0.988193	+	0.153212i	$0.0489618\pi$
$677$	29.6985	+	29.6985i	1.14141	+	1.14141i	0.153212	+	0.988193i	$0.451038\pi$
$678$		0			0
$679$	11.3137			0.434180
$680$		0			0
$681$	11.3137	−	16.0000i	0.433542	−	0.613121i
$682$		0			0
$683$	8.00000	−	8.00000i	0.306111	−	0.306111i	−0.537288	−	0.843399i	$-0.680551\pi$
$683$	8.00000	−	8.00000i	0.306111	−	0.306111i	0.843399	+	0.537288i	$0.180551\pi$
$684$		0			0
$685$	20.0000	+	20.0000i	0.764161	+	0.764161i
$686$		0			0
$687$	10.0000	+	7.07107i	0.381524	+	0.269778i
$688$		0			0
$689$		−	25.4558i		−	0.969790i
$690$		0			0
$691$	−25.4558	+	25.4558i	−0.968386	+	0.968386i	−0.999515	−	0.0311294i	$-0.990090\pi$
$691$	−25.4558	+	25.4558i	−0.968386	+	0.968386i	0.0311294	+	0.999515i	$0.490090\pi$
$692$		0			0
$693$	10.3431	−	21.6569i	0.392904	−	0.822676i
$694$		0			0
$695$		−	28.0000i		−	1.06210i
$696$		0			0
$697$		−	8.00000i		−	0.303022i
$698$		0			0
$699$	3.31371	+	19.3137i	0.125336	+	0.730512i
$700$		0			0
$701$	−21.2132	+	21.2132i	−0.801212	+	0.801212i	−0.983285	−	0.182073i	$-0.941719\pi$
$701$	−21.2132	+	21.2132i	−0.801212	+	0.801212i	0.182073	+	0.983285i	$0.441719\pi$
$702$		0			0
$703$			28.2843i			1.06676i
$704$		0			0
$705$	24.0000	−	33.9411i	0.903892	−	1.27830i
$706$		0			0
$707$	14.0000	+	14.0000i	0.526524	+	0.526524i
$708$		0			0
$709$	27.0000	−	27.0000i	1.01401	−	1.01401i	0.0141058	−	0.999901i	$-0.495510\pi$
$709$	27.0000	−	27.0000i	1.01401	−	1.01401i	0.999901	−	0.0141058i	$-0.00449016\pi$
$710$		0			0
$711$	−9.89949	−	28.0000i	−0.371260	−	1.05008i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	33.9411	+	33.9411i	1.26933	+	1.26933i
$716$		0			0
$717$	3.51472	+	20.4853i	0.131260	+	0.765037i
$718$		0			0
$719$	−16.0000			−0.596699			−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000			−0.596699			−0.298350	+	0.954457i	$0.596436\pi$
$720$		0			0
$721$	6.00000			0.223452
$722$		0			0
$723$	−0.585786	−	3.41421i	−0.0217856	−	0.126976i
$724$		0			0
$725$	−7.07107	−	7.07107i	−0.262613	−	0.262613i
$726$		0			0
$727$	38.1838			1.41616			0.708079	−	0.706133i	$-0.249562\pi$
$727$	38.1838			1.41616			0.708079	+	0.706133i	$0.249562\pi$
$728$		0			0
$729$	−14.1421	+	23.0000i	−0.523783	+	0.851852i
$730$		0			0
$731$	8.00000	−	8.00000i	0.295891	−	0.295891i
$732$		0			0
$733$	1.00000	+	1.00000i	0.0369358	+	0.0369358i	0.725333	−	0.688398i	$-0.241686\pi$
$733$	1.00000	+	1.00000i	0.0369358	+	0.0369358i	−0.688398	+	0.725333i	$0.741686\pi$
$734$		0			0
$735$	−10.0000	+	14.1421i	−0.368856	+	0.521641i
$736$		0			0
$737$		−	56.5685i		−	2.08373i
$738$		0			0
$739$	1.41421	−	1.41421i	0.0520227	−	0.0520227i	−0.680617	−	0.732640i	$-0.738288\pi$
$739$	1.41421	−	1.41421i	0.0520227	−	0.0520227i	0.732640	+	0.680617i	$0.238288\pi$
$740$		0			0
$741$	−4.97056	−	28.9706i	−0.182598	−	1.06426i
$742$		0			0
$743$			32.0000i			1.17397i	0.809599	+	0.586983i	$0.199684\pi$
$743$			32.0000i			1.17397i	−0.809599	+	0.586983i	$0.800316\pi$
$744$		0			0
$745$		−	12.0000i		−	0.439646i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.82843	−	2.82843i	0.103348	−	0.103348i
$750$		0			0
$751$		−	32.5269i		−	1.18692i	−0.804862	−	0.593462i	$-0.797761\pi$
$751$		−	32.5269i		−	1.18692i	0.804862	−	0.593462i	$-0.202239\pi$
$752$		0			0
$753$	16.0000	+	11.3137i	0.583072	+	0.412294i
$754$		0			0
$755$	6.00000	+	6.00000i	0.218362	+	0.218362i
$756$		0			0
$757$	5.00000	−	5.00000i	0.181728	−	0.181728i	−0.610380	−	0.792108i	$-0.708983\pi$
$757$	5.00000	−	5.00000i	0.181728	−	0.181728i	0.792108	+	0.610380i	$0.208983\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	8.48528			0.307591			0.153796	−	0.988103i	$-0.450850\pi$
$761$	8.48528			0.307591			0.153796	+	0.988103i	$0.450850\pi$
$762$		0			0
$763$	−4.24264	−	4.24264i	−0.153594	−	0.153594i
$764$		0			0
$765$	−7.31371	+	15.3137i	−0.264428	+	0.553668i
$766$		0			0
$767$	12.0000			0.433295
$768$		0			0
$769$	−18.0000			−0.649097			−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000			−0.649097			−0.324548	+	0.945869i	$0.605212\pi$
$770$		0			0
$771$	−19.3137	+	3.31371i	−0.695566	+	0.119340i
$772$		0			0
$773$	−4.24264	−	4.24264i	−0.152597	−	0.152597i	0.626680	−	0.779277i	$-0.284413\pi$
$773$	−4.24264	−	4.24264i	−0.152597	−	0.152597i	−0.779277	+	0.626680i	$0.784413\pi$
$774$		0			0
$775$	−4.24264			−0.152400
$776$		0			0
$777$	14.1421	+	10.0000i	0.507346	+	0.358748i
$778$		0			0
$779$	−8.00000	+	8.00000i	−0.286630	+	0.286630i
$780$		0			0
$781$	−16.0000	−	16.0000i	−0.572525	−	0.572525i
$782$		0			0
$783$	−50.0000	−	14.1421i	−1.78685	−	0.505399i
$784$		0			0
$785$			8.48528i			0.302853i
$786$		0			0
$787$	−8.48528	+	8.48528i	−0.302468	+	0.302468i	−0.841979	−	0.539511i	$-0.818609\pi$
$787$	−8.48528	+	8.48528i	−0.302468	+	0.302468i	0.539511	+	0.841979i	$0.318609\pi$
$788$		0			0
$789$	−6.82843	+	1.17157i	−0.243098	+	0.0417091i
$790$		0			0
$791$			24.0000i			0.853342i
$792$		0			0
$793$			54.0000i			1.91760i
$794$		0			0
$795$	20.4853	−	3.51472i	0.726538	−	0.124654i
$796$		0			0
$797$	−32.5269	+	32.5269i	−1.15216	+	1.15216i	−0.166044	+	0.986118i	$0.553099\pi$
$797$	−32.5269	+	32.5269i	−1.15216	+	1.15216i	−0.986118	+	0.166044i	$0.946901\pi$
$798$		0			0
$799$			33.9411i			1.20075i
$800$		0			0
$801$	−16.0000	+	5.65685i	−0.565332	+	0.199875i
$802$		0			0
$803$	16.0000	+	16.0000i	0.564628	+	0.564628i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	8.48528	+	6.00000i	0.298696	+	0.211210i
$808$		0			0
$809$	−36.7696			−1.29275			−0.646374	−	0.763020i	$-0.723716\pi$
$809$	−36.7696			−1.29275			−0.646374	+	0.763020i	$0.723716\pi$
$810$		0			0
$811$	14.1421	+	14.1421i	0.496598	+	0.496598i	0.910377	−	0.413780i	$-0.135792\pi$
$811$	14.1421	+	14.1421i	0.496598	+	0.496598i	−0.413780	+	0.910377i	$0.635792\pi$
$812$		0			0
$813$	−36.2132	+	6.21320i	−1.27005	+	0.217907i
$814$		0			0
$815$	−24.0000			−0.840683
$816$		0			0
$817$	−16.0000			−0.559769
$818$		0			0
$819$	−16.2426	−	7.75736i	−0.567564	−	0.271064i
$820$		0			0
$821$	−26.8701	−	26.8701i	−0.937771	−	0.937771i	0.0604026	−	0.998174i	$-0.480762\pi$
$821$	−26.8701	−	26.8701i	−0.937771	−	0.937771i	−0.998174	+	0.0604026i	$0.980762\pi$
$822$		0			0
$823$	−35.3553			−1.23241			−0.616205	−	0.787586i	$-0.711331\pi$
$823$	−35.3553			−1.23241			−0.616205	+	0.787586i	$0.711331\pi$
$824$		0			0
$825$	5.65685	−	8.00000i	0.196946	−	0.278524i
$826$		0			0
$827$	−30.0000	+	30.0000i	−1.04320	+	1.04320i	−0.0441786	+	0.999024i	$0.514067\pi$
$827$	−30.0000	+	30.0000i	−1.04320	+	1.04320i	−0.999024	+	0.0441786i	$0.985933\pi$
$828$		0			0
$829$	13.0000	+	13.0000i	0.451509	+	0.451509i	0.895855	−	0.444346i	$-0.146564\pi$
$829$	13.0000	+	13.0000i	0.451509	+	0.451509i	−0.444346	+	0.895855i	$0.646564\pi$
$830$		0			0
$831$	6.00000	+	4.24264i	0.208138	+	0.147176i
$832$		0			0
$833$		−	14.1421i		−	0.489996i
$834$		0			0
$835$	22.6274	−	22.6274i	0.783054	−	0.783054i
$836$		0			0
$837$	−19.2426	+	10.7574i	−0.665123	+	0.371829i
$838$		0			0
$839$			36.0000i			1.24286i	0.783470	+	0.621429i	$0.213448\pi$
$839$			36.0000i			1.24286i	−0.783470	+	0.621429i	$0.786552\pi$
$840$		0			0
$841$		−	71.0000i		−	2.44828i
$842$		0			0
$843$	4.97056	+	28.9706i	0.171195	+	0.997799i
$844$		0			0
$845$	7.07107	−	7.07107i	0.243252	−	0.243252i
$846$		0			0
$847$			29.6985i			1.02045i
$848$		0			0
$849$	−14.0000	+	19.7990i	−0.480479	+	0.679500i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−15.0000	+	15.0000i	−0.513590	+	0.513590i	−0.915625	−	0.402034i	$-0.868303\pi$
$853$	−15.0000	+	15.0000i	−0.513590	+	0.513590i	0.402034	+	0.915625i	$0.368303\pi$
$854$		0			0
$855$	22.6274	−	8.00000i	0.773841	−	0.273594i
$856$		0			0
$857$	−25.4558			−0.869555			−0.434778	−	0.900538i	$-0.643173\pi$
$857$	−25.4558			−0.869555			−0.434778	+	0.900538i	$0.643173\pi$
$858$		0			0
$859$	14.1421	+	14.1421i	0.482523	+	0.482523i	0.905937	−	0.423413i	$-0.139168\pi$
$859$	14.1421	+	14.1421i	0.482523	+	0.482523i	−0.423413	+	0.905937i	$0.639168\pi$
$860$		0			0
$861$	1.17157	+	6.82843i	0.0399271	+	0.232712i
$862$		0			0
$863$	28.0000			0.953131			0.476566	−	0.879139i	$-0.341881\pi$
$863$	28.0000			0.953131			0.476566	+	0.879139i	$0.341881\pi$
$864$		0			0
$865$	20.0000			0.680020
$866$		0			0
$867$	2.63604	+	15.3640i	0.0895246	+	0.521787i
$868$		0			0
$869$	39.5980	+	39.5980i	1.34327	+	1.34327i
$870$		0			0
$871$	−42.4264			−1.43756
$872$		0			0
$873$	22.6274	−	8.00000i	0.765822	−	0.270759i
$874$		0			0
$875$	12.0000	−	12.0000i	0.405674	−	0.405674i
$876$		0			0
$877$	−15.0000	−	15.0000i	−0.506514	−	0.506514i	0.406941	−	0.913455i	$-0.366596\pi$
$877$	−15.0000	−	15.0000i	−0.506514	−	0.506514i	−0.913455	+	0.406941i	$0.866596\pi$
$878$		0			0
$879$	14.0000	−	19.7990i	0.472208	−	0.667803i
$880$		0			0
$881$		−	16.9706i		−	0.571753i	−0.958267	−	0.285876i	$-0.907715\pi$
$881$		−	16.9706i		−	0.571753i	0.958267	−	0.285876i	$-0.0922847\pi$
$882$		0			0
$883$	31.1127	−	31.1127i	1.04703	−	1.04703i	0.0481873	−	0.998838i	$-0.484656\pi$
$883$	31.1127	−	31.1127i	1.04703	−	1.04703i	0.998838	−	0.0481873i	$-0.0153445\pi$
$884$		0			0
$885$	1.65685	+	9.65685i	0.0556945	+	0.324612i
$886$		0			0
$887$		−	44.0000i		−	1.47738i	−0.674048	−	0.738688i	$-0.735446\pi$
$887$		−	44.0000i		−	1.47738i	0.674048	−	0.738688i	$-0.264554\pi$
$888$		0			0
$889$		−	6.00000i		−	0.201234i
$890$		0			0
$891$	5.37258	−	50.6274i	0.179988	−	1.69608i
$892$		0			0
$893$	33.9411	−	33.9411i	1.13580	−	1.13580i
$894$		0			0
$895$		−	16.9706i		−	0.567263i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−30.0000	−	30.0000i	−1.00056	−	1.00056i
$900$		0			0
$901$	−12.0000	+	12.0000i	−0.399778	+	0.399778i
$902$		0			0
$903$	−5.65685	+	8.00000i	−0.188248	+	0.266223i
$904$		0			0
$905$	8.48528			0.282060
$906$		0			0
$907$	−31.1127	−	31.1127i	−1.03308	−	1.03308i	−0.999434	−	0.0336464i	$-0.989288\pi$
$907$	−31.1127	−	31.1127i	−1.03308	−	1.03308i	−0.0336464	−	0.999434i	$-0.510712\pi$
$908$		0			0
$909$	37.8995	+	18.1005i	1.25705	+	0.600356i
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−43.4558	+	7.45584i	−1.43661	+	0.246483i
$916$		0			0
$917$	8.48528	+	8.48528i	0.280209	+	0.280209i
$918$		0			0
$919$	21.2132			0.699759			0.349880	−	0.936795i	$-0.386223\pi$
$919$	21.2132			0.699759			0.349880	+	0.936795i	$0.386223\pi$
$920$		0			0
$921$	−2.82843	−	2.00000i	−0.0931998	−	0.0659022i
$922$		0			0
$923$	−12.0000	+	12.0000i	−0.394985	+	0.394985i
$924$		0			0
$925$	5.00000	+	5.00000i	0.164399	+	0.164399i
$926$		0			0
$927$	12.0000	−	4.24264i	0.394132	−	0.139347i
$928$		0			0
$929$			59.3970i			1.94875i	0.224927	+	0.974376i	$0.427786\pi$
$929$			59.3970i			1.94875i	−0.224927	+	0.974376i	$0.572214\pi$
$930$		0			0
$931$	−14.1421	+	14.1421i	−0.463490	+	0.463490i
$932$		0			0
$933$	47.7990	−	8.20101i	1.56487	−	0.268489i
$934$		0			0
$935$		−	32.0000i		−	1.04651i
$936$		0			0
$937$			18.0000i			0.588034i	0.955800	+	0.294017i	$0.0949923\pi$
$937$			18.0000i			0.588034i	−0.955800	+	0.294017i	$0.905008\pi$
$938$		0			0
$939$	−37.5563	+	6.44365i	−1.22561	+	0.210281i
$940$		0			0
$941$	−21.2132	+	21.2132i	−0.691531	+	0.691531i	−0.962569	−	0.271038i	$-0.912633\pi$
$941$	−21.2132	+	21.2132i	−0.691531	+	0.691531i	0.271038	+	0.962569i	$0.412633\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	4.00000	−	14.1421i	0.130120	−	0.460044i
$946$		0			0
$947$	34.0000	+	34.0000i	1.10485	+	1.10485i	0.993816	+	0.111035i	$0.0354166\pi$
$947$	34.0000	+	34.0000i	1.10485	+	1.10485i	0.111035	+	0.993816i	$0.464583\pi$
$948$		0			0
$949$	12.0000	−	12.0000i	0.389536	−	0.389536i
$950$		0			0
$951$	8.48528	+	6.00000i	0.275154	+	0.194563i
$952$		0			0
$953$	14.1421			0.458109			0.229054	−	0.973414i	$-0.426437\pi$
$953$	14.1421			0.458109			0.229054	+	0.973414i	$0.426437\pi$
$954$		0			0
$955$	−5.65685	−	5.65685i	−0.183052	−	0.183052i
$956$		0			0
$957$	96.5685	−	16.5685i	3.12162	−	0.535585i
$958$		0			0
$959$	20.0000			0.645834
$960$		0			0
$961$	13.0000			0.419355
$962$		0			0
$963$	3.65685	−	7.65685i	0.117840	−	0.246739i
$964$		0			0
$965$	−2.82843	−	2.82843i	−0.0910503	−	0.0910503i
$966$		0			0
$967$	1.41421			0.0454780			0.0227390	−	0.999741i	$-0.492761\pi$
$967$	1.41421			0.0454780			0.0227390	+	0.999741i	$0.492761\pi$
$968$		0			0
$969$	−11.3137	+	16.0000i	−0.363449	+	0.513994i
$970$		0			0
$971$	−12.0000	+	12.0000i	−0.385098	+	0.385098i	−0.872935	−	0.487837i	$-0.837786\pi$
$971$	−12.0000	+	12.0000i	−0.385098	+	0.385098i	0.487837	+	0.872935i	$0.337786\pi$
$972$		0			0
$973$	−14.0000	−	14.0000i	−0.448819	−	0.448819i
$974$		0			0
$975$	−6.00000	−	4.24264i	−0.192154	−	0.135873i
$976$		0			0
$977$			19.7990i			0.633426i	0.948521	+	0.316713i	$0.102579\pi$
$977$			19.7990i			0.633426i	−0.948521	+	0.316713i	$0.897421\pi$
$978$		0			0
$979$	22.6274	−	22.6274i	0.723175	−	0.723175i
$980$		0			0
$981$	−11.4853	−	5.48528i	−0.366697	−	0.175132i
$982$		0			0
$983$		−	28.0000i		−	0.893061i	−0.894768	−	0.446531i	$-0.852659\pi$
$983$		−	28.0000i		−	0.893061i	0.894768	−	0.446531i	$-0.147341\pi$
$984$		0			0
$985$		−	36.0000i		−	1.14706i
$986$		0			0
$987$	−4.97056	−	28.9706i	−0.158215	−	0.922143i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	12.7279i		−	0.404316i	−0.979353	−	0.202158i	$-0.935205\pi$
$991$		−	12.7279i		−	0.404316i	0.979353	−	0.202158i	$-0.0647954\pi$
$992$		0			0
$993$	30.0000	−	42.4264i	0.952021	−	1.34636i
$994$		0			0
$995$	−22.0000	−	22.0000i	−0.697447	−	0.697447i
$996$		0			0
$997$	−21.0000	+	21.0000i	−0.665077	+	0.665077i	−0.956572	−	0.291496i	$-0.905847\pi$
$997$	−21.0000	+	21.0000i	−0.665077	+	0.665077i	0.291496	+	0.956572i	$0.405847\pi$
$998$		0			0
$999$	35.3553	+	10.0000i	1.11859	+	0.316386i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.k.c.191.1	yes	4
3.2	odd	2		768.2.k.a.191.1	✓	4
4.3	odd	2		768.2.k.a.191.2	yes	4
8.3	odd	2		768.2.k.d.191.1	yes	4
8.5	even	2		768.2.k.b.191.2	yes	4
12.11	even	2	inner	768.2.k.c.191.2	yes	4
16.3	odd	4		768.2.k.d.575.2	yes	4
16.5	even	4	inner	768.2.k.c.575.2	yes	4
16.11	odd	4		768.2.k.a.575.1	yes	4
16.13	even	4		768.2.k.b.575.1	yes	4
24.5	odd	2		768.2.k.d.191.2	yes	4
24.11	even	2		768.2.k.b.191.1	yes	4
48.5	odd	4		768.2.k.a.575.2	yes	4
48.11	even	4	inner	768.2.k.c.575.1	yes	4
48.29	odd	4		768.2.k.d.575.1	yes	4
48.35	even	4		768.2.k.b.575.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.k.a.191.1	✓	4	3.2	odd	2
768.2.k.a.191.2	yes	4	4.3	odd	2
768.2.k.a.575.1	yes	4	16.11	odd	4
768.2.k.a.575.2	yes	4	48.5	odd	4
768.2.k.b.191.1	yes	4	24.11	even	2
768.2.k.b.191.2	yes	4	8.5	even	2
768.2.k.b.575.1	yes	4	16.13	even	4
768.2.k.b.575.2	yes	4	48.35	even	4
768.2.k.c.191.1	yes	4	1.1	even	1	trivial
768.2.k.c.191.2	yes	4	12.11	even	2	inner
768.2.k.c.575.1	yes	4	48.11	even	4	inner
768.2.k.c.575.2	yes	4	16.5	even	4	inner
768.2.k.d.191.1	yes	4	8.3	odd	2
768.2.k.d.191.2	yes	4	24.5	odd	2
768.2.k.d.575.1	yes	4	48.29	odd	4
768.2.k.d.575.2	yes	4	16.3	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.292893 + 1.70711i) q^{3} +(-1.41421 - 1.41421i) q^{5} -1.41421 q^{7} +(-2.82843 + 1.00000i) q^{9} +O(q^{10})\)	\(q+(0.292893 + 1.70711i) q^{3} +(-1.41421 - 1.41421i) q^{5} -1.41421 q^{7} +(-2.82843 + 1.00000i) q^{9} +(4.00000 - 4.00000i) q^{11} +(-3.00000 - 3.00000i) q^{13} +(2.00000 - 2.82843i) q^{15} +2.82843i q^{17} +(2.82843 - 2.82843i) q^{19} +(-0.414214 - 2.41421i) q^{21} -1.00000i q^{25} +(-2.53553 - 4.53553i) q^{27} +(7.07107 - 7.07107i) q^{29} -4.24264i q^{31} +(8.00000 + 5.65685i) q^{33} +(2.00000 + 2.00000i) q^{35} +(-5.00000 + 5.00000i) q^{37} +(4.24264 - 6.00000i) q^{39} -2.82843 q^{41} +(-2.82843 - 2.82843i) q^{43} +(5.41421 + 2.58579i) q^{45} +12.0000 q^{47} -5.00000 q^{49} +(-4.82843 + 0.828427i) q^{51} +(4.24264 + 4.24264i) q^{53} -11.3137 q^{55} +(5.65685 + 4.00000i) q^{57} +(-2.00000 + 2.00000i) q^{59} +(-9.00000 - 9.00000i) q^{61} +(4.00000 - 1.41421i) q^{63} +8.48528i q^{65} +(7.07107 - 7.07107i) q^{67} -4.00000i q^{71} +4.00000i q^{73} +(1.70711 - 0.292893i) q^{75} +(-5.65685 + 5.65685i) q^{77} +9.89949i q^{79} +(7.00000 - 5.65685i) q^{81} +(4.00000 - 4.00000i) q^{85} +(14.1421 + 10.0000i) q^{87} +5.65685 q^{89} +(4.24264 + 4.24264i) q^{91} +(7.24264 - 1.24264i) q^{93} -8.00000 q^{95} -8.00000 q^{97} +(-7.31371 + 15.3137i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3}+O(q^{10}) \) 4 * q + 4 * q^3	\( 4 q + 4 q^{3} + 16 q^{11} - 12 q^{13} + 8 q^{15} + 4 q^{21} + 4 q^{27} + 32 q^{33} + 8 q^{35} - 20 q^{37} + 16 q^{45} + 48 q^{47} - 20 q^{49} - 8 q^{51} - 8 q^{59} - 36 q^{61} + 16 q^{63} + 4 q^{75} + 28 q^{81} + 16 q^{85} + 12 q^{93} - 32 q^{95} - 32 q^{97} + 16 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 16 * q^11 - 12 * q^13 + 8 * q^15 + 4 * q^21 + 4 * q^27 + 32 * q^33 + 8 * q^35 - 20 * q^37 + 16 * q^45 + 48 * q^47 - 20 * q^49 - 8 * q^51 - 8 * q^59 - 36 * q^61 + 16 * q^63 + 4 * q^75 + 28 * q^81 + 16 * q^85 + 12 * q^93 - 32 * q^95 - 32 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more