LMFDB - Embedded newform 768.3.l.f.703.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(319,768)]

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, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("768.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.l (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:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 52 x^{14} + 1743 x^{12} - 34996 x^{10} + 513409 x^{8} - 5039424 x^{6} + 36142848 x^{4} + \cdots + 429981696 $ x^16 - 52x^14 + 1743x^12 - 34996x^10 + 513409x^8 - 5039424x^6 + 36142848x^4 - 155271168*x^2 + 429981696
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			703.2
Root			$-3.95096 - 2.28109i$ of defining polynomial
Character	$\chi$	$=$	768.703
Dual form			768.3.l.f.319.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 + 1.22474i) q^{3} +(-3.71984 + 3.71984i) q^{5} +5.63959 q^{7} -3.00000i q^{9} +O(q^{10})$	\(q+(-1.22474 + 1.22474i) q^{3} +(-3.71984 + 3.71984i) q^{5} +5.63959 q^{7} -3.00000i q^{9} +(-5.26065 - 5.26065i) q^{11} +(-1.74099 - 1.74099i) q^{13} -9.11171i q^{15} -1.43968 q^{17} +(-22.1170 + 22.1170i) q^{19} +(-6.90705 + 6.90705i) q^{21} +2.27362 q^{23} -2.67442i q^{25} +(3.67423 + 3.67423i) q^{27} +(-35.9507 - 35.9507i) q^{29} +13.3072i q^{31} +12.8859 q^{33} +(-20.9784 + 20.9784i) q^{35} +(36.1679 - 36.1679i) q^{37} +4.26454 q^{39} -63.2115i q^{41} +(46.5429 + 46.5429i) q^{43} +(11.1595 + 11.1595i) q^{45} -61.0614i q^{47} -17.1951 q^{49} +(1.76324 - 1.76324i) q^{51} +(49.0582 - 49.0582i) q^{53} +39.1375 q^{55} -54.1754i q^{57} +(-41.6432 - 41.6432i) q^{59} +(-40.4833 - 40.4833i) q^{61} -16.9188i q^{63} +12.9524 q^{65} +(-36.0209 + 36.0209i) q^{67} +(-2.78461 + 2.78461i) q^{69} -8.24767 q^{71} -122.728i q^{73} +(3.27548 + 3.27548i) q^{75} +(-29.6679 - 29.6679i) q^{77} -93.0832i q^{79} -9.00000 q^{81} +(-21.6857 + 21.6857i) q^{83} +(5.35538 - 5.35538i) q^{85} +88.0608 q^{87} +119.561i q^{89} +(-9.81846 - 9.81846i) q^{91} +(-16.2979 - 16.2979i) q^{93} -164.544i q^{95} -115.028 q^{97} +(-15.7819 + 15.7819i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{5}+O(q^{10}) $ 16 * q + 8 * q^5	$ 16 q + 8 q^{5} - 32 q^{13} + 112 q^{17} + 72 q^{21} - 56 q^{29} - 48 q^{33} + 272 q^{37} - 24 q^{45} + 304 q^{49} + 504 q^{53} - 176 q^{61} + 624 q^{65} + 288 q^{69} - 848 q^{77} - 144 q^{81} + 880 q^{85} + 216 q^{93} - 160 q^{97}+O(q^{100}) $ 16 * q + 8 * q^5 - 32 * q^13 + 112 * q^17 + 72 * q^21 - 56 * q^29 - 48 * q^33 + 272 * q^37 - 24 * q^45 + 304 * q^49 + 504 * q^53 - 176 * q^61 + 624 * q^65 + 288 * q^69 - 848 * q^77 - 144 * q^81 + 880 * q^85 + 216 * q^93 - 160 * q^97

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
$3$	−1.22474	+	1.22474i	−0.408248	+	0.408248i
$3$	−1.22474	+	1.22474i	−0.408248	+	0.408248i
$4$		0			0
$5$	−3.71984	+	3.71984i	−0.743968	+	0.743968i	−0.973339	−	0.229371i	$-0.926333\pi$
$5$	−3.71984	+	3.71984i	−0.743968	+	0.743968i	0.229371	+	0.973339i	$0.426333\pi$
$6$		0			0
$7$	5.63959			0.805655			0.402828	−	0.915276i	$-0.368027\pi$
$7$	5.63959			0.805655			0.402828	+	0.915276i	$0.368027\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	−5.26065	−	5.26065i	−0.478241	−	0.478241i	0.426328	−	0.904569i	$-0.359807\pi$
$11$	−5.26065	−	5.26065i	−0.478241	−	0.478241i	−0.904569	+	0.426328i	$0.859807\pi$
$12$		0			0
$13$	−1.74099	−	1.74099i	−0.133922	−	0.133922i	0.636968	−	0.770890i	$-0.280188\pi$
$13$	−1.74099	−	1.74099i	−0.133922	−	0.133922i	−0.770890	+	0.636968i	$0.780188\pi$
$14$		0			0
$15$		−	9.11171i		−	0.607447i
$16$		0			0
$17$	−1.43968			−0.0846870			−0.0423435	−	0.999103i	$-0.513482\pi$
$17$	−1.43968			−0.0846870			−0.0423435	+	0.999103i	$0.513482\pi$
$18$		0			0
$19$	−22.1170	+	22.1170i	−1.16405	+	1.16405i	−0.180475	+	0.983580i	$0.557763\pi$
$19$	−22.1170	+	22.1170i	−1.16405	+	1.16405i	−0.983580	+	0.180475i	$0.942237\pi$
$20$		0			0
$21$	−6.90705	+	6.90705i	−0.328907	+	0.328907i
$22$		0			0
$23$	2.27362			0.0988532			0.0494266	−	0.998778i	$-0.484261\pi$
$23$	2.27362			0.0988532			0.0494266	+	0.998778i	$0.484261\pi$
$24$		0			0
$25$		−	2.67442i		−	0.106977i
$26$		0			0
$27$	3.67423	+	3.67423i	0.136083	+	0.136083i
$28$		0			0
$29$	−35.9507	−	35.9507i	−1.23968	−	1.23968i	−0.960132	−	0.279546i	$-0.909816\pi$
$29$	−35.9507	−	35.9507i	−1.23968	−	1.23968i	−0.279546	−	0.960132i	$-0.590184\pi$
$30$		0			0
$31$			13.3072i			0.429264i	0.976695	+	0.214632i	$0.0688551\pi$
$31$			13.3072i			0.429264i	−0.976695	+	0.214632i	$0.931145\pi$
$32$		0			0
$33$	12.8859			0.390482
$34$		0			0
$35$	−20.9784	+	20.9784i	−0.599382	+	0.599382i
$36$		0			0
$37$	36.1679	−	36.1679i	0.977510	−	0.977510i	−0.0222427	−	0.999753i	$-0.507081\pi$
$37$	36.1679	−	36.1679i	0.977510	−	0.977510i	0.999753	+	0.0222427i	$0.00708066\pi$
$38$		0			0
$39$	4.26454			0.109347
$40$		0			0
$41$		−	63.2115i		−	1.54174i	−0.636990	−	0.770872i	$-0.719821\pi$
$41$		−	63.2115i		−	1.54174i	0.636990	−	0.770872i	$-0.280179\pi$
$42$		0			0
$43$	46.5429	+	46.5429i	1.08239	+	1.08239i	0.996286	+	0.0861060i	$0.0274424\pi$
$43$	46.5429	+	46.5429i	1.08239	+	1.08239i	0.0861060	+	0.996286i	$0.472558\pi$
$44$		0			0
$45$	11.1595	+	11.1595i	0.247989	+	0.247989i
$46$		0			0
$47$		−	61.0614i		−	1.29918i	−0.760285	−	0.649589i	$-0.774941\pi$
$47$		−	61.0614i		−	1.29918i	0.760285	−	0.649589i	$-0.225059\pi$
$48$		0			0
$49$	−17.1951			−0.350920
$50$		0			0
$51$	1.76324	−	1.76324i	0.0345733	−	0.0345733i
$52$		0			0
$53$	49.0582	−	49.0582i	0.925627	−	0.925627i	−0.0717927	−	0.997420i	$-0.522872\pi$
$53$	49.0582	−	49.0582i	0.925627	−	0.925627i	0.997420	+	0.0717927i	$0.0228720\pi$
$54$		0			0
$55$	39.1375			0.711592
$56$		0			0
$57$		−	54.1754i		−	0.950446i
$58$		0			0
$59$	−41.6432	−	41.6432i	−0.705818	−	0.705818i	0.259835	−	0.965653i	$-0.416332\pi$
$59$	−41.6432	−	41.6432i	−0.705818	−	0.705818i	−0.965653	+	0.259835i	$0.916332\pi$
$60$		0			0
$61$	−40.4833	−	40.4833i	−0.663661	−	0.663661i	0.292580	−	0.956241i	$-0.405486\pi$
$61$	−40.4833	−	40.4833i	−0.663661	−	0.663661i	−0.956241	+	0.292580i	$0.905486\pi$
$62$		0			0
$63$		−	16.9188i		−	0.268552i
$64$		0			0
$65$	12.9524			0.199268
$66$		0			0
$67$	−36.0209	+	36.0209i	−0.537626	+	0.537626i	−0.922831	−	0.385205i	$-0.874131\pi$
$67$	−36.0209	+	36.0209i	−0.537626	+	0.537626i	0.385205	+	0.922831i	$0.374131\pi$
$68$		0			0
$69$	−2.78461	+	2.78461i	−0.0403567	+	0.0403567i
$70$		0			0
$71$	−8.24767			−0.116164			−0.0580822	−	0.998312i	$-0.518499\pi$
$71$	−8.24767			−0.116164			−0.0580822	+	0.998312i	$0.518499\pi$
$72$		0			0
$73$		−	122.728i		−	1.68121i	−0.541649	−	0.840604i	$-0.682200\pi$
$73$		−	122.728i		−	1.68121i	0.541649	−	0.840604i	$-0.317800\pi$
$74$		0			0
$75$	3.27548	+	3.27548i	0.0436730	+	0.0436730i
$76$		0			0
$77$	−29.6679	−	29.6679i	−0.385297	−	0.385297i
$78$		0			0
$79$		−	93.0832i		−	1.17827i	−0.808035	−	0.589134i	$-0.799469\pi$
$79$		−	93.0832i		−	1.17827i	0.808035	−	0.589134i	$-0.200531\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−21.6857	+	21.6857i	−0.261273	+	0.261273i	−0.825571	−	0.564298i	$-0.809147\pi$
$83$	−21.6857	+	21.6857i	−0.261273	+	0.261273i	0.564298	+	0.825571i	$0.309147\pi$
$84$		0			0
$85$	5.35538	−	5.35538i	0.0630044	−	0.0630044i
$86$		0			0
$87$	88.0608			1.01219
$88$		0			0
$89$			119.561i			1.34339i	0.740829	+	0.671693i	$0.234433\pi$
$89$			119.561i			1.34339i	−0.740829	+	0.671693i	$0.765567\pi$
$90$		0			0
$91$	−9.81846	−	9.81846i	−0.107895	−	0.107895i
$92$		0			0
$93$	−16.2979	−	16.2979i	−0.175246	−	0.175246i
$94$		0			0
$95$		−	164.544i		−	1.73204i
$96$		0			0
$97$	−115.028			−1.18585			−0.592926	−	0.805257i	$-0.702027\pi$
$97$	−115.028			−1.18585			−0.592926	+	0.805257i	$0.702027\pi$
$98$		0			0
$99$	−15.7819	+	15.7819i	−0.159414	+	0.159414i
$100$		0			0
$101$	37.6316	−	37.6316i	0.372590	−	0.372590i	−0.495829	−	0.868420i	$-0.665136\pi$
$101$	37.6316	−	37.6316i	0.372590	−	0.372590i	0.868420	+	0.495829i	$0.165136\pi$
$102$		0			0
$103$	130.007			1.26220			0.631100	−	0.775702i	$-0.282604\pi$
$103$	130.007			1.26220			0.631100	+	0.775702i	$0.282604\pi$
$104$		0			0
$105$		−	51.3863i		−	0.489393i
$106$		0			0
$107$	−28.6866	−	28.6866i	−0.268099	−	0.268099i	0.560235	−	0.828334i	$-0.310711\pi$
$107$	−28.6866	−	28.6866i	−0.268099	−	0.268099i	−0.828334	+	0.560235i	$0.810711\pi$
$108$		0			0
$109$	41.4538	+	41.4538i	0.380310	+	0.380310i	0.871214	−	0.490904i	$-0.163333\pi$
$109$	41.4538	+	41.4538i	0.380310	+	0.380310i	−0.490904	+	0.871214i	$0.663333\pi$
$110$		0			0
$111$			88.5928i			0.798133i
$112$		0			0
$113$	−16.4099			−0.145220			−0.0726101	−	0.997360i	$-0.523133\pi$
$113$	−16.4099			−0.145220			−0.0726101	+	0.997360i	$0.523133\pi$
$114$		0			0
$115$	−8.45752	+	8.45752i	−0.0735436	+	0.0735436i
$116$		0			0
$117$	−5.22297	+	5.22297i	−0.0446408	+	0.0446408i
$118$		0			0
$119$	−8.11920			−0.0682285
$120$		0			0
$121$		−	65.6512i		−	0.542572i
$122$		0			0
$123$	77.4179	+	77.4179i	0.629414	+	0.629414i
$124$		0			0
$125$	−83.0476	−	83.0476i	−0.664381	−	0.664381i
$126$		0			0
$127$			96.2701i			0.758033i	0.925390	+	0.379016i	$0.123738\pi$
$127$			96.2701i			0.758033i	−0.925390	+	0.379016i	$0.876262\pi$
$128$		0			0
$129$	−114.006			−0.883769
$130$		0			0
$131$	−157.643	+	157.643i	−1.20338	+	1.20338i	−0.230249	+	0.973132i	$0.573954\pi$
$131$	−157.643	+	157.643i	−1.20338	+	1.20338i	−0.973132	+	0.230249i	$0.926046\pi$
$132$		0			0
$133$	−124.731	+	124.731i	−0.937826	+	0.937826i
$134$		0			0
$135$	−27.3351			−0.202482
$136$		0			0
$137$			129.255i			0.943471i	0.881740	+	0.471735i	$0.156372\pi$
$137$			129.255i			0.943471i	−0.881740	+	0.471735i	$0.843628\pi$
$138$		0			0
$139$	−3.93636	−	3.93636i	−0.0283191	−	0.0283191i	0.692805	−	0.721125i	$-0.256374\pi$
$139$	−3.93636	−	3.93636i	−0.0283191	−	0.0283191i	−0.721125	+	0.692805i	$0.756374\pi$
$140$		0			0
$141$	74.7846	+	74.7846i	0.530387	+	0.530387i
$142$		0			0
$143$			18.3175i			0.128094i
$144$		0			0
$145$	267.461			1.84456
$146$		0			0
$147$	21.0596	−	21.0596i	0.143262	−	0.143262i
$148$		0			0
$149$	170.277	−	170.277i	1.14280	−	1.14280i	0.154863	−	0.987936i	$-0.450506\pi$
$149$	170.277	−	170.277i	1.14280	−	1.14280i	0.987936	−	0.154863i	$-0.0494937\pi$
$150$		0			0
$151$	−22.9645			−0.152083			−0.0760414	−	0.997105i	$-0.524228\pi$
$151$	−22.9645			−0.152083			−0.0760414	+	0.997105i	$0.524228\pi$
$152$		0			0
$153$			4.31904i			0.0282290i
$154$		0			0
$155$	−49.5005	−	49.5005i	−0.319358	−	0.319358i
$156$		0			0
$157$	28.6344	+	28.6344i	0.182385	+	0.182385i	0.792394	−	0.610009i	$-0.208834\pi$
$157$	28.6344	+	28.6344i	0.182385	+	0.182385i	−0.610009	+	0.792394i	$0.708834\pi$
$158$		0			0
$159$			120.168i			0.755771i
$160$		0			0
$161$	12.8223			0.0796416
$162$		0			0
$163$	156.712	−	156.712i	0.961421	−	0.961421i	−0.0378616	−	0.999283i	$-0.512055\pi$
$163$	156.712	−	156.712i	0.961421	−	0.961421i	0.999283	+	0.0378616i	$0.0120546\pi$
$164$		0			0
$165$	−47.9335	+	47.9335i	−0.290506	+	0.290506i
$166$		0			0
$167$	−315.505			−1.88925			−0.944626	−	0.328148i	$-0.893575\pi$
$167$	−315.505			−1.88925			−0.944626	+	0.328148i	$0.893575\pi$
$168$		0			0
$169$		−	162.938i		−	0.964130i
$170$		0			0
$171$	66.3511	+	66.3511i	0.388018	+	0.388018i
$172$		0			0
$173$	−120.104	−	120.104i	−0.694243	−	0.694243i	0.268919	−	0.963163i	$-0.413334\pi$
$173$	−120.104	−	120.104i	−0.694243	−	0.694243i	−0.963163	+	0.268919i	$0.913334\pi$
$174$		0			0
$175$		−	15.0826i		−	0.0861863i
$176$		0			0
$177$	102.005			0.576298
$178$		0			0
$179$	181.133	−	181.133i	1.01192	−	1.01192i	0.0119883	−	0.999928i	$-0.496184\pi$
$179$	181.133	−	181.133i	1.01192	−	1.01192i	0.999928	−	0.0119883i	$-0.00381607\pi$
$180$		0			0
$181$	−20.9668	+	20.9668i	−0.115839	+	0.115839i	−0.762650	−	0.646811i	$-0.776102\pi$
$181$	−20.9668	+	20.9668i	−0.115839	+	0.115839i	0.646811	+	0.762650i	$0.276102\pi$
$182$		0			0
$183$	99.1634			0.541877
$184$		0			0
$185$			269.077i			1.45447i
$186$		0			0
$187$	7.57365	+	7.57365i	0.0405008	+	0.0405008i
$188$		0			0
$189$	20.7212	+	20.7212i	0.109636	+	0.109636i
$190$		0			0
$191$			40.2387i			0.210674i	0.994437	+	0.105337i	$0.0335921\pi$
$191$			40.2387i			0.210674i	−0.994437	+	0.105337i	$0.966408\pi$
$192$		0			0
$193$	−29.7137			−0.153957			−0.0769784	−	0.997033i	$-0.524527\pi$
$193$	−29.7137			−0.153957			−0.0769784	+	0.997033i	$0.524527\pi$
$194$		0			0
$195$	−15.8634	+	15.8634i	−0.0813507	+	0.0813507i
$196$		0			0
$197$	−135.692	+	135.692i	−0.688790	+	0.688790i	−0.961964	−	0.273175i	$-0.911926\pi$
$197$	−135.692	+	135.692i	−0.688790	+	0.688790i	0.273175	+	0.961964i	$0.411926\pi$
$198$		0			0
$199$	−158.309			−0.795524			−0.397762	−	0.917489i	$-0.630213\pi$
$199$	−158.309			−0.795524			−0.397762	+	0.917489i	$0.630213\pi$
$200$		0			0
$201$		−	88.2329i		−	0.438970i
$202$		0			0
$203$	−202.747	−	202.747i	−0.998753	−	0.998753i
$204$		0			0
$205$	235.137	+	235.137i	1.14701	+	1.14701i
$206$		0			0
$207$		−	6.82087i		−	0.0329511i
$208$		0			0
$209$	232.700			1.11340
$210$		0			0
$211$	−36.9270	+	36.9270i	−0.175010	+	0.175010i	−0.789176	−	0.614167i	$-0.789492\pi$
$211$	−36.9270	+	36.9270i	−0.175010	+	0.175010i	0.614167	+	0.789176i	$0.289492\pi$
$212$		0			0
$213$	10.1013	−	10.1013i	0.0474239	−	0.0474239i
$214$		0			0
$215$	−346.264			−1.61053
$216$		0			0
$217$			75.0469i			0.345838i
$218$		0			0
$219$	150.311	+	150.311i	0.686351	+	0.686351i
$220$		0			0
$221$	2.50647	+	2.50647i	0.0113415	+	0.0113415i
$222$		0			0
$223$			185.284i			0.830869i	0.909623	+	0.415435i	$0.136370\pi$
$223$			185.284i			0.830869i	−0.909623	+	0.415435i	$0.863630\pi$
$224$		0			0
$225$	−8.02325			−0.0356589
$226$		0			0
$227$	165.315	−	165.315i	0.728261	−	0.728261i	−0.242012	−	0.970273i	$-0.577807\pi$
$227$	165.315	−	165.315i	0.728261	−	0.728261i	0.970273	+	0.242012i	$0.0778075\pi$
$228$		0			0
$229$	−291.789	+	291.789i	−1.27419	+	1.27419i	−0.330321	+	0.943869i	$0.607157\pi$
$229$	−291.789	+	291.789i	−1.27419	+	1.27419i	−0.943869	+	0.330321i	$0.892843\pi$
$230$		0			0
$231$	72.6712			0.314594
$232$		0			0
$233$			265.531i			1.13962i	0.821778	+	0.569808i	$0.192982\pi$
$233$			265.531i			1.13962i	−0.821778	+	0.569808i	$0.807018\pi$
$234$		0			0
$235$	227.139	+	227.139i	0.966547	+	0.966547i
$236$		0			0
$237$	114.003	+	114.003i	0.481026	+	0.481026i
$238$		0			0
$239$			284.025i			1.18839i	0.804321	+	0.594195i	$0.202529\pi$
$239$			284.025i			1.18839i	−0.804321	+	0.594195i	$0.797471\pi$
$240$		0			0
$241$	−39.1098			−0.162281			−0.0811406	−	0.996703i	$-0.525856\pi$
$241$	−39.1098			−0.162281			−0.0811406	+	0.996703i	$0.525856\pi$
$242$		0			0
$243$	11.0227	−	11.0227i	0.0453609	−	0.0453609i
$244$		0			0
$245$	63.9629	−	63.9629i	0.261073	−	0.261073i
$246$		0			0
$247$	77.0111			0.311786
$248$		0			0
$249$		−	53.1188i		−	0.213329i
$250$		0			0
$251$	−304.283	−	304.283i	−1.21228	−	1.21228i	−0.970275	−	0.242007i	$-0.922194\pi$
$251$	−304.283	−	304.283i	−1.21228	−	1.21228i	−0.242007	−	0.970275i	$-0.577806\pi$
$252$		0			0
$253$	−11.9607	−	11.9607i	−0.0472756	−	0.0472756i
$254$		0			0
$255$			13.1179i			0.0514429i
$256$		0			0
$257$	−425.226			−1.65458			−0.827288	−	0.561778i	$-0.810117\pi$
$257$	−425.226			−1.65458			−0.827288	+	0.561778i	$0.810117\pi$
$258$		0			0
$259$	203.972	−	203.972i	0.787536	−	0.787536i
$260$		0			0
$261$	−107.852	+	107.852i	−0.413226	+	0.413226i
$262$		0			0
$263$	192.884			0.733399			0.366700	−	0.930339i	$-0.380488\pi$
$263$	192.884			0.733399			0.366700	+	0.930339i	$0.380488\pi$
$264$		0			0
$265$			364.977i			1.37727i
$266$		0			0
$267$	−146.432	−	146.432i	−0.548435	−	0.548435i
$268$		0			0
$269$	111.994	+	111.994i	0.416335	+	0.416335i	0.883938	−	0.467603i	$-0.154882\pi$
$269$	111.994	+	111.994i	0.416335	+	0.416335i	−0.467603	+	0.883938i	$0.654882\pi$
$270$		0			0
$271$			201.397i			0.743161i	0.928401	+	0.371581i	$0.121184\pi$
$271$			201.397i			0.743161i	−0.928401	+	0.371581i	$0.878816\pi$
$272$		0			0
$273$	24.0502			0.0880960
$274$		0			0
$275$	−14.0692	+	14.0692i	−0.0511606	+	0.0511606i
$276$		0			0
$277$	199.341	−	199.341i	0.719642	−	0.719642i	−0.248890	−	0.968532i	$-0.580066\pi$
$277$	199.341	−	199.341i	0.719642	−	0.719642i	0.968532	+	0.248890i	$0.0800657\pi$
$278$		0			0
$279$	39.9215			0.143088
$280$		0			0
$281$			526.542i			1.87381i	0.349578	+	0.936907i	$0.386325\pi$
$281$			526.542i			1.87381i	−0.349578	+	0.936907i	$0.613675\pi$
$282$		0			0
$283$	113.365	+	113.365i	0.400585	+	0.400585i	0.878439	−	0.477854i	$-0.158585\pi$
$283$	113.365	+	113.365i	0.400585	+	0.400585i	−0.477854	+	0.878439i	$0.658585\pi$
$284$		0			0
$285$	201.524	+	201.524i	0.707102	+	0.707102i
$286$		0			0
$287$		−	356.487i		−	1.24211i
$288$		0			0
$289$	−286.927			−0.992828
$290$		0			0
$291$	140.880	−	140.880i	0.484122	−	0.484122i
$292$		0			0
$293$	−29.7487	+	29.7487i	−0.101531	+	0.101531i	−0.756048	−	0.654516i	$-0.772872\pi$
$293$	−29.7487	+	29.7487i	−0.101531	+	0.101531i	0.654516	+	0.756048i	$0.272872\pi$
$294$		0			0
$295$	309.812			1.05021
$296$		0			0
$297$		−	38.6577i		−	0.130161i
$298$		0			0
$299$	−3.95836	−	3.95836i	−0.0132387	−	0.0132387i
$300$		0			0
$301$	262.482	+	262.482i	0.872035	+	0.872035i
$302$		0			0
$303$			92.1783i			0.304219i
$304$		0			0
$305$	301.183			0.987485
$306$		0			0
$307$	−52.8719	+	52.8719i	−0.172221	+	0.172221i	−0.787954	−	0.615733i	$-0.788860\pi$
$307$	−52.8719	+	52.8719i	−0.172221	+	0.172221i	0.615733	+	0.787954i	$0.288860\pi$
$308$		0			0
$309$	−159.225	+	159.225i	−0.515291	+	0.515291i
$310$		0			0
$311$	−350.713			−1.12770			−0.563848	−	0.825879i	$-0.690680\pi$
$311$	−350.713			−1.12770			−0.563848	+	0.825879i	$0.690680\pi$
$312$		0			0
$313$			393.542i			1.25732i	0.777679	+	0.628662i	$0.216397\pi$
$313$			393.542i			1.25732i	−0.777679	+	0.628662i	$0.783603\pi$
$314$		0			0
$315$	62.9351	+	62.9351i	0.199794	+	0.199794i
$316$		0			0
$317$	−330.939	−	330.939i	−1.04397	−	1.04397i	−0.998988	−	0.0449847i	$-0.985676\pi$
$317$	−330.939	−	330.939i	−1.04397	−	1.04397i	−0.0449847	−	0.998988i	$-0.514324\pi$
$318$		0			0
$319$			378.248i			1.18573i
$320$		0			0
$321$	70.2675			0.218902
$322$		0			0
$323$	31.8414	−	31.8414i	0.0985803	−	0.0985803i
$324$		0			0
$325$	−4.65613	+	4.65613i	−0.0143266	+	0.0143266i
$326$		0			0
$327$	−101.541			−0.310522
$328$		0			0
$329$		−	344.361i		−	1.04669i
$330$		0			0
$331$	−181.290	−	181.290i	−0.547703	−	0.547703i	0.378073	−	0.925776i	$-0.376587\pi$
$331$	−181.290	−	181.290i	−0.547703	−	0.547703i	−0.925776	+	0.378073i	$0.876587\pi$
$332$		0			0
$333$	−108.504	−	108.504i	−0.325837	−	0.325837i
$334$		0			0
$335$		−	267.984i		−	0.799953i
$336$		0			0
$337$	−136.580			−0.405282			−0.202641	−	0.979253i	$-0.564952\pi$
$337$	−136.580			−0.405282			−0.202641	+	0.979253i	$0.564952\pi$
$338$		0			0
$339$	20.0979	−	20.0979i	0.0592859	−	0.0592859i
$340$		0			0
$341$	70.0043	−	70.0043i	0.205291	−	0.205291i
$342$		0			0
$343$	−373.313			−1.08838
$344$		0			0
$345$		−	20.7166i		−	0.0600481i
$346$		0			0
$347$	−135.037	−	135.037i	−0.389157	−	0.389157i	0.485230	−	0.874387i	$-0.338736\pi$
$347$	−135.037	−	135.037i	−0.389157	−	0.389157i	−0.874387	+	0.485230i	$0.838736\pi$
$348$		0			0
$349$	53.9888	+	53.9888i	0.154696	+	0.154696i	0.780212	−	0.625516i	$-0.215111\pi$
$349$	53.9888	+	53.9888i	0.154696	+	0.154696i	−0.625516	+	0.780212i	$0.715111\pi$
$350$		0			0
$351$		−	12.7936i		−	0.0364490i
$352$		0			0
$353$	137.350			0.389092			0.194546	−	0.980893i	$-0.437677\pi$
$353$	137.350			0.389092			0.194546	+	0.980893i	$0.437677\pi$
$354$		0			0
$355$	30.6800	−	30.6800i	0.0864226	−	0.0864226i
$356$		0			0
$357$	9.94394	−	9.94394i	0.0278542	−	0.0278542i
$358$		0			0
$359$	122.293			0.340649			0.170324	−	0.985388i	$-0.445518\pi$
$359$	122.293			0.340649			0.170324	+	0.985388i	$0.445518\pi$
$360$		0			0
$361$		−	617.326i		−	1.71005i
$362$		0			0
$363$	80.4059	+	80.4059i	0.221504	+	0.221504i
$364$		0			0
$365$	456.529	+	456.529i	1.25077	+	1.25077i
$366$		0			0
$367$			59.0682i			0.160949i	0.996757	+	0.0804744i	$0.0256435\pi$
$367$			59.0682i			0.160949i	−0.996757	+	0.0804744i	$0.974356\pi$
$368$		0			0
$369$	−189.634			−0.513915
$370$		0			0
$371$	276.668	−	276.668i	0.745736	−	0.745736i
$372$		0			0
$373$	101.421	−	101.421i	0.271906	−	0.271906i	−0.557961	−	0.829867i	$-0.688416\pi$
$373$	101.421	−	101.421i	0.271906	−	0.271906i	0.829867	+	0.557961i	$0.188416\pi$
$374$		0			0
$375$	203.424			0.542465
$376$		0			0
$377$			125.180i			0.332041i
$378$		0			0
$379$	272.095	+	272.095i	0.717929	+	0.717929i	0.968181	−	0.250252i	$-0.0805135\pi$
$379$	272.095	+	272.095i	0.717929	+	0.717929i	−0.250252	+	0.968181i	$0.580514\pi$
$380$		0			0
$381$	−117.906	−	117.906i	−0.309466	−	0.309466i
$382$		0			0
$383$		−	315.688i		−	0.824251i	−0.911127	−	0.412125i	$-0.864787\pi$
$383$		−	315.688i		−	0.824251i	0.911127	−	0.412125i	$-0.135213\pi$
$384$		0			0
$385$	220.719			0.573297
$386$		0			0
$387$	139.629	−	139.629i	0.360797	−	0.360797i
$388$		0			0
$389$	−71.8537	+	71.8537i	−0.184714	+	0.184714i	−0.793406	−	0.608692i	$-0.791694\pi$
$389$	−71.8537	+	71.8537i	−0.184714	+	0.184714i	0.608692	+	0.793406i	$0.291694\pi$
$390$		0			0
$391$	−3.27329			−0.00837159
$392$		0			0
$393$		−	386.145i		−	0.982556i
$394$		0			0
$395$	346.255	+	346.255i	0.876594	+	0.876594i
$396$		0			0
$397$	−269.316	−	269.316i	−0.678378	−	0.678378i	0.281255	−	0.959633i	$-0.409249\pi$
$397$	−269.316	−	269.316i	−0.678378	−	0.678378i	−0.959633	+	0.281255i	$0.909249\pi$
$398$		0			0
$399$		−	305.527i		−	0.765732i
$400$		0			0
$401$	159.680			0.398204			0.199102	−	0.979979i	$-0.436197\pi$
$401$	159.680			0.398204			0.199102	+	0.979979i	$0.436197\pi$
$402$		0			0
$403$	23.1676	−	23.1676i	0.0574880	−	0.0574880i
$404$		0			0
$405$	33.4786	−	33.4786i	0.0826631	−	0.0826631i
$406$		0			0
$407$	−380.533			−0.934970
$408$		0			0
$409$		−	190.873i		−	0.466683i	−0.972395	−	0.233342i	$-0.925034\pi$
$409$		−	190.873i		−	0.466683i	0.972395	−	0.233342i	$-0.0749661\pi$
$410$		0			0
$411$	−158.305	−	158.305i	−0.385170	−	0.385170i
$412$		0			0
$413$	−234.851	−	234.851i	−0.568646	−	0.568646i
$414$		0			0
$415$		−	161.334i		−	0.388758i
$416$		0			0
$417$	9.64207			0.0231225
$418$		0			0
$419$	−370.260	+	370.260i	−0.883676	+	0.883676i	−0.993906	−	0.110230i	$-0.964841\pi$
$419$	−370.260	+	370.260i	−0.883676	+	0.883676i	0.110230	+	0.993906i	$0.464841\pi$
$420$		0			0
$421$	273.190	−	273.190i	0.648907	−	0.648907i	−0.303822	−	0.952729i	$-0.598263\pi$
$421$	273.190	−	273.190i	0.648907	−	0.648907i	0.952729	+	0.303822i	$0.0982628\pi$
$422$		0			0
$423$	−183.184			−0.433059
$424$		0			0
$425$			3.85030i			0.00905954i
$426$		0			0
$427$	−228.309	−	228.309i	−0.534682	−	0.534682i
$428$		0			0
$429$	−22.4342	−	22.4342i	−0.0522942	−	0.0522942i
$430$		0			0
$431$			507.863i			1.17834i	0.808011	+	0.589168i	$0.200544\pi$
$431$			507.863i			1.17834i	−0.808011	+	0.589168i	$0.799456\pi$
$432$		0			0
$433$	352.417			0.813895			0.406948	−	0.913452i	$-0.366593\pi$
$433$	352.417			0.813895			0.406948	+	0.913452i	$0.366593\pi$
$434$		0			0
$435$	−327.572	+	327.572i	−0.753039	+	0.753039i
$436$		0			0
$437$	−50.2858	+	50.2858i	−0.115071	+	0.115071i
$438$		0			0
$439$	776.156			1.76801			0.884005	−	0.467477i	$-0.154837\pi$
$439$	776.156			1.76801			0.884005	+	0.467477i	$0.154837\pi$
$440$		0			0
$441$			51.5852i			0.116973i
$442$		0			0
$443$	77.8171	+	77.8171i	0.175659	+	0.175659i	0.789461	−	0.613801i	$-0.210360\pi$
$443$	77.8171	+	77.8171i	0.175659	+	0.175659i	−0.613801	+	0.789461i	$0.710360\pi$
$444$		0			0
$445$	−444.749	−	444.749i	−0.999437	−	0.999437i
$446$		0			0
$447$			417.092i			0.933092i
$448$		0			0
$449$	−135.576			−0.301952			−0.150976	−	0.988537i	$-0.548242\pi$
$449$	−135.576			−0.301952			−0.150976	+	0.988537i	$0.548242\pi$
$450$		0			0
$451$	−332.533	+	332.533i	−0.737325	+	0.737325i
$452$		0			0
$453$	28.1257	−	28.1257i	0.0620875	−	0.0620875i
$454$		0			0
$455$	73.0462			0.160541
$456$		0			0
$457$		−	265.075i		−	0.580032i	−0.957022	−	0.290016i	$-0.906339\pi$
$457$		−	265.075i		−	0.580032i	0.957022	−	0.290016i	$-0.0936607\pi$
$458$		0			0
$459$	−5.28972	−	5.28972i	−0.0115244	−	0.0115244i
$460$		0			0
$461$	−57.6181	−	57.6181i	−0.124985	−	0.124985i	0.641847	−	0.766832i	$-0.278168\pi$
$461$	−57.6181	−	57.6181i	−0.124985	−	0.124985i	−0.766832	+	0.641847i	$0.778168\pi$
$462$		0			0
$463$		−	343.191i		−	0.741234i	−0.928786	−	0.370617i	$-0.879146\pi$
$463$		−	343.191i		−	0.741234i	0.928786	−	0.370617i	$-0.120854\pi$
$464$		0			0
$465$	121.251			0.260755
$466$		0			0
$467$	−402.281	+	402.281i	−0.861416	+	0.861416i	−0.991503	−	0.130086i	$-0.958475\pi$
$467$	−402.281	+	402.281i	−0.861416	+	0.861416i	0.130086	+	0.991503i	$0.458475\pi$
$468$		0			0
$469$	−203.143	+	203.143i	−0.433141	+	0.433141i
$470$		0			0
$471$	−70.1397			−0.148917
$472$		0			0
$473$		−	489.691i		−	1.03529i
$474$		0			0
$475$	59.1502	+	59.1502i	0.124527	+	0.124527i
$476$		0			0
$477$	−147.175	−	147.175i	−0.308542	−	0.308542i
$478$		0			0
$479$		−	148.717i		−	0.310474i	−0.987877	−	0.155237i	$-0.950386\pi$
$479$		−	148.717i		−	0.310474i	0.987877	−	0.155237i	$-0.0496141\pi$
$480$		0			0
$481$	−125.936			−0.261821
$482$		0			0
$483$	−15.7040	+	15.7040i	−0.0325135	+	0.0325135i
$484$		0			0
$485$	427.884	−	427.884i	0.882236	−	0.882236i
$486$		0			0
$487$	−631.177			−1.29605			−0.648025	−	0.761619i	$-0.724405\pi$
$487$	−631.177			−1.29605			−0.648025	+	0.761619i	$0.724405\pi$
$488$		0			0
$489$			383.864i			0.784997i
$490$		0			0
$491$	2.03842	+	2.03842i	0.00415157	+	0.00415157i	0.709180	−	0.705028i	$-0.249066\pi$
$491$	2.03842	+	2.03842i	0.00415157	+	0.00415157i	−0.705028	+	0.709180i	$0.749066\pi$
$492$		0			0
$493$	51.7575	+	51.7575i	0.104985	+	0.104985i
$494$		0			0
$495$		−	117.413i		−	0.237197i
$496$		0			0
$497$	−46.5134			−0.0935884
$498$		0			0
$499$	−366.034	+	366.034i	−0.733535	+	0.733535i	−0.971318	−	0.237783i	$-0.923579\pi$
$499$	−366.034	+	366.034i	−0.733535	+	0.733535i	0.237783	+	0.971318i	$0.423579\pi$
$500$		0			0
$501$	386.413	−	386.413i	0.771284	−	0.771284i
$502$		0			0
$503$	835.830			1.66169			0.830845	−	0.556504i	$-0.187858\pi$
$503$	835.830			1.66169			0.830845	+	0.556504i	$0.187858\pi$
$504$		0			0
$505$			279.967i			0.554391i
$506$		0			0
$507$	199.557	+	199.557i	0.393604	+	0.393604i
$508$		0			0
$509$	−47.1170	−	47.1170i	−0.0925679	−	0.0925679i	0.659306	−	0.751874i	$-0.270850\pi$
$509$	−47.1170	−	47.1170i	−0.0925679	−	0.0925679i	−0.751874	+	0.659306i	$0.770850\pi$
$510$		0			0
$511$		−	692.136i		−	1.35447i
$512$		0			0
$513$	−162.526			−0.316815
$514$		0			0
$515$	−483.603	+	483.603i	−0.939036	+	0.939036i
$516$		0			0
$517$	−321.222	+	321.222i	−0.621320	+	0.621320i
$518$		0			0
$519$	294.194			0.566847
$520$		0			0
$521$		−	546.253i		−	1.04847i	−0.851574	−	0.524235i	$-0.824351\pi$
$521$		−	546.253i		−	1.04847i	0.851574	−	0.524235i	$-0.175649\pi$
$522$		0			0
$523$	−331.446	−	331.446i	−0.633740	−	0.633740i	0.315264	−	0.949004i	$-0.397907\pi$
$523$	−331.446	−	331.446i	−0.633740	−	0.633740i	−0.949004	+	0.315264i	$0.897907\pi$
$524$		0			0
$525$	18.4723	+	18.4723i	0.0351854	+	0.0351854i
$526$		0			0
$527$		−	19.1581i		−	0.0363531i
$528$		0			0
$529$	−523.831			−0.990228
$530$		0			0
$531$	−124.930	+	124.930i	−0.235273	+	0.235273i
$532$		0			0
$533$	−110.051	+	110.051i	−0.206474	+	0.206474i
$534$		0			0
$535$	213.419			0.398914
$536$		0			0
$537$			443.684i			0.826226i
$538$		0			0
$539$	90.4572	+	90.4572i	0.167824	+	0.167824i
$540$		0			0
$541$	−34.7238	−	34.7238i	−0.0641844	−	0.0641844i	0.674286	−	0.738470i	$-0.264452\pi$
$541$	−34.7238	−	34.7238i	−0.0641844	−	0.0641844i	−0.738470	+	0.674286i	$0.764452\pi$
$542$		0			0
$543$		−	51.3580i		−	0.0945820i
$544$		0			0
$545$	−308.403			−0.565877
$546$		0			0
$547$	11.2062	−	11.2062i	0.0204867	−	0.0204867i	−0.696789	−	0.717276i	$-0.745389\pi$
$547$	11.2062	−	11.2062i	0.0204867	−	0.0204867i	0.717276	+	0.696789i	$0.245389\pi$
$548$		0			0
$549$	−121.450	+	121.450i	−0.221220	+	0.221220i
$550$		0			0
$551$	1590.24			2.88611
$552$		0			0
$553$		−	524.951i		−	0.949278i
$554$		0			0
$555$	−329.551	−	329.551i	−0.593786	−	0.593786i
$556$		0			0
$557$	−485.377	−	485.377i	−0.871414	−	0.871414i	0.121213	−	0.992627i	$-0.461322\pi$
$557$	−485.377	−	485.377i	−0.871414	−	0.871414i	−0.992627	+	0.121213i	$0.961322\pi$
$558$		0			0
$559$		−	162.061i		−	0.289913i
$560$		0			0
$561$	−18.5516			−0.0330688
$562$		0			0
$563$	140.768	−	140.768i	0.250032	−	0.250032i	−0.570951	−	0.820984i	$-0.693426\pi$
$563$	140.768	−	140.768i	0.250032	−	0.250032i	0.820984	+	0.570951i	$0.193426\pi$
$564$		0			0
$565$	61.0421	−	61.0421i	0.108039	−	0.108039i
$566$		0			0
$567$	−50.7563			−0.0895172
$568$		0			0
$569$		−	953.637i		−	1.67599i	−0.545680	−	0.837993i	$-0.683729\pi$
$569$		−	953.637i		−	1.67599i	0.545680	−	0.837993i	$-0.316271\pi$
$570$		0			0
$571$	−51.7112	−	51.7112i	−0.0905624	−	0.0905624i	0.660374	−	0.750937i	$-0.270398\pi$
$571$	−51.7112	−	51.7112i	−0.0905624	−	0.0905624i	−0.750937	+	0.660374i	$0.770398\pi$
$572$		0			0
$573$	−49.2821	−	49.2821i	−0.0860072	−	0.0860072i
$574$		0			0
$575$		−	6.08062i		−	0.0105750i
$576$		0			0
$577$	511.951			0.887263			0.443632	−	0.896209i	$-0.353690\pi$
$577$	511.951			0.887263			0.443632	+	0.896209i	$0.353690\pi$
$578$		0			0
$579$	36.3917	−	36.3917i	0.0628526	−	0.0628526i
$580$		0			0
$581$	−122.298	+	122.298i	−0.210496	+	0.210496i
$582$		0			0
$583$	−516.156			−0.885345
$584$		0			0
$585$		−	38.8572i		−	0.0664226i
$586$		0			0
$587$	738.225	+	738.225i	1.25762	+	1.25762i	0.952224	+	0.305399i	$0.0987900\pi$
$587$	738.225	+	738.225i	1.25762	+	1.25762i	0.305399	+	0.952224i	$0.401210\pi$
$588$		0			0
$589$	−294.315	−	294.315i	−0.499686	−	0.499686i
$590$		0			0
$591$		−	332.375i		−	0.562395i
$592$		0			0
$593$	−712.871			−1.20214			−0.601071	−	0.799195i	$-0.705259\pi$
$593$	−712.871			−1.20214			−0.601071	+	0.799195i	$0.705259\pi$
$594$		0			0
$595$	30.2021	−	30.2021i	0.0507599	−	0.0507599i
$596$		0			0
$597$	193.889	−	193.889i	0.324771	−	0.324771i
$598$		0			0
$599$	−706.968			−1.18025			−0.590123	−	0.807313i	$-0.700921\pi$
$599$	−706.968			−1.18025			−0.590123	+	0.807313i	$0.700921\pi$
$600$		0			0
$601$		−	405.158i		−	0.674139i	−0.941480	−	0.337069i	$-0.890564\pi$
$601$		−	405.158i		−	0.674139i	0.941480	−	0.337069i	$-0.109436\pi$
$602$		0			0
$603$	108.063	+	108.063i	0.179209	+	0.179209i
$604$		0			0
$605$	244.212	+	244.212i	0.403656	+	0.403656i
$606$		0			0
$607$		−	1065.53i		−	1.75541i	−0.479202	−	0.877705i	$-0.659074\pi$
$607$		−	1065.53i		−	1.75541i	0.479202	−	0.877705i	$-0.340926\pi$
$608$		0			0
$609$	496.626			0.815479
$610$		0			0
$611$	−106.307	+	106.307i	−0.173989	+	0.173989i
$612$		0			0
$613$	71.1894	−	71.1894i	0.116133	−	0.116133i	−0.646652	−	0.762785i	$-0.723831\pi$
$613$	71.1894	−	71.1894i	0.116133	−	0.116133i	0.762785	+	0.646652i	$0.223831\pi$
$614$		0			0
$615$	−575.965			−0.936528
$616$		0			0
$617$			36.7627i			0.0595830i	0.999556	+	0.0297915i	$0.00948433\pi$
$617$			36.7627i			0.0595830i	−0.999556	+	0.0297915i	$0.990516\pi$
$618$		0			0
$619$	−227.779	−	227.779i	−0.367980	−	0.367980i	0.498760	−	0.866740i	$-0.333789\pi$
$619$	−227.779	−	227.779i	−0.367980	−	0.367980i	−0.866740	+	0.498760i	$0.833789\pi$
$620$		0			0
$621$	8.35383	+	8.35383i	0.0134522	+	0.0134522i
$622$		0			0
$623$			674.277i			1.08231i
$624$		0			0
$625$	684.708			1.09553
$626$		0			0
$627$	−284.998	+	284.998i	−0.454542	+	0.454542i
$628$		0			0
$629$	−52.0701	+	52.0701i	−0.0827824	+	0.0827824i
$630$		0			0
$631$	994.938			1.57676			0.788382	−	0.615186i	$-0.210919\pi$
$631$	994.938			1.57676			0.788382	+	0.615186i	$0.210919\pi$
$632$		0			0
$633$		−	90.4524i		−	0.142895i
$634$		0			0
$635$	−358.110	−	358.110i	−0.563952	−	0.563952i
$636$		0			0
$637$	29.9365	+	29.9365i	0.0469960	+	0.0469960i
$638$		0			0
$639$			24.7430i			0.0387215i
$640$		0			0
$641$	312.367			0.487311			0.243656	−	0.969862i	$-0.421653\pi$
$641$	312.367			0.487311			0.243656	+	0.969862i	$0.421653\pi$
$642$		0			0
$643$	98.2755	−	98.2755i	0.152839	−	0.152839i	−0.626546	−	0.779385i	$-0.715532\pi$
$643$	98.2755	−	98.2755i	0.152839	−	0.152839i	0.779385	+	0.626546i	$0.215532\pi$
$644$		0			0
$645$	424.085	−	424.085i	0.657496	−	0.657496i
$646$		0			0
$647$	123.488			0.190863			0.0954313	−	0.995436i	$-0.469577\pi$
$647$	123.488			0.190863			0.0954313	+	0.995436i	$0.469577\pi$
$648$		0			0
$649$			438.141i			0.675102i
$650$		0			0
$651$	−91.9133	−	91.9133i	−0.141188	−	0.141188i
$652$		0			0
$653$	474.759	+	474.759i	0.727043	+	0.727043i	0.970030	−	0.242987i	$-0.0781272\pi$
$653$	474.759	+	474.759i	0.727043	+	0.727043i	−0.242987	+	0.970030i	$0.578127\pi$
$654$		0			0
$655$		−	1172.81i		−	1.79055i
$656$		0			0
$657$	−368.185			−0.560403
$658$		0			0
$659$	848.321	−	848.321i	1.28729	−	1.28729i	0.350856	−	0.936429i	$-0.385891\pi$
$659$	848.321	−	848.321i	1.28729	−	1.28729i	0.936429	−	0.350856i	$-0.114109\pi$
$660$		0			0
$661$	734.819	−	734.819i	1.11168	−	1.11168i	0.118754	−	0.992924i	$-0.462110\pi$
$661$	734.819	−	734.819i	1.11168	−	1.11168i	0.992924	−	0.118754i	$-0.0378901\pi$
$662$		0			0
$663$	−6.13957			−0.00926028
$664$		0			0
$665$		−	927.958i		−	1.39543i
$666$		0			0
$667$	−81.7383	−	81.7383i	−0.122546	−	0.122546i
$668$		0			0
$669$	−226.925	−	226.925i	−0.339201	−	0.339201i
$670$		0			0
$671$			425.937i			0.634779i
$672$		0			0
$673$	−1060.24			−1.57540			−0.787700	−	0.616059i	$-0.788728\pi$
$673$	−1060.24			−1.57540			−0.787700	+	0.616059i	$0.788728\pi$
$674$		0			0
$675$	9.82644	−	9.82644i	0.0145577	−	0.0145577i
$676$		0			0
$677$	−5.38518	+	5.38518i	−0.00795448	+	0.00795448i	−0.711073	−	0.703118i	$-0.751790\pi$
$677$	−5.38518	+	5.38518i	−0.00795448	+	0.00795448i	0.703118	+	0.711073i	$0.251790\pi$
$678$		0			0
$679$	−648.708			−0.955388
$680$		0			0
$681$			404.938i			0.594622i
$682$		0			0
$683$	669.250	+	669.250i	0.979868	+	0.979868i	0.999801	−	0.0199331i	$-0.00634533\pi$
$683$	669.250	+	669.250i	0.979868	+	0.979868i	−0.0199331	+	0.999801i	$0.506345\pi$
$684$		0			0
$685$	−480.810	−	480.810i	−0.701912	−	0.701912i
$686$		0			0
$687$		−	714.735i		−	1.04037i
$688$		0			0
$689$	−170.820			−0.247924
$690$		0			0
$691$	394.907	−	394.907i	0.571501	−	0.571501i	−0.361047	−	0.932548i	$-0.617581\pi$
$691$	394.907	−	394.907i	0.571501	−	0.571501i	0.932548	+	0.361047i	$0.117581\pi$
$692$		0			0
$693$	−89.0036	+	89.0036i	−0.128432	+	0.128432i
$694$		0			0
$695$	29.2852			0.0421370
$696$		0			0
$697$			91.0043i			0.130566i
$698$		0			0
$699$	−325.207	−	325.207i	−0.465246	−	0.465246i
$700$		0			0
$701$	769.452	+	769.452i	1.09765	+	1.09765i	0.994685	+	0.102963i	$0.0328324\pi$
$701$	769.452	+	769.452i	1.09765	+	1.09765i	0.102963	+	0.994685i	$0.467168\pi$
$702$		0			0
$703$			1599.85i			2.27575i
$704$		0			0
$705$	−556.374			−0.789182
$706$		0			0
$707$	212.227	−	212.227i	0.300179	−	0.300179i
$708$		0			0
$709$	818.749	−	818.749i	1.15479	−	1.15479i	0.169215	−	0.985579i	$-0.445877\pi$
$709$	818.749	−	818.749i	1.15479	−	1.15479i	0.985579	−	0.169215i	$-0.0541232\pi$
$710$		0			0
$711$	−279.250			−0.392756
$712$		0			0
$713$			30.2555i			0.0424341i
$714$		0			0
$715$	−68.1381	−	68.1381i	−0.0952980	−	0.0952980i
$716$		0			0
$717$	−347.858	−	347.858i	−0.485158	−	0.485158i
$718$		0			0
$719$		−	294.229i		−	0.409219i	−0.978844	−	0.204610i	$-0.934408\pi$
$719$		−	294.229i		−	0.409219i	0.978844	−	0.204610i	$-0.0655925\pi$
$720$		0			0
$721$	733.183			1.01690
$722$		0			0
$723$	47.8995	−	47.8995i	0.0662510	−	0.0662510i
$724$		0			0
$725$	−96.1471	+	96.1471i	−0.132617	+	0.132617i
$726$		0			0
$727$	−720.844			−0.991532			−0.495766	−	0.868456i	$-0.665113\pi$
$727$	−720.844			−0.991532			−0.495766	+	0.868456i	$0.665113\pi$
$728$		0			0
$729$			27.0000i			0.0370370i
$730$		0			0
$731$	−67.0068	−	67.0068i	−0.0916646	−	0.0916646i
$732$		0			0
$733$	459.987	+	459.987i	0.627541	+	0.627541i	0.947449	−	0.319908i	$-0.103652\pi$
$733$	459.987	+	459.987i	0.627541	+	0.627541i	−0.319908	+	0.947449i	$0.603652\pi$
$734$		0			0
$735$			156.677i			0.213165i
$736$		0			0
$737$	378.987			0.514229
$738$		0			0
$739$	−817.319	+	817.319i	−1.10598	+	1.10598i	−0.112306	+	0.993674i	$0.535824\pi$
$739$	−817.319	+	817.319i	−1.10598	+	1.10598i	−0.993674	+	0.112306i	$0.964176\pi$
$740$		0			0
$741$	−94.3189	+	94.3189i	−0.127286	+	0.127286i
$742$		0			0
$743$	874.150			1.17651			0.588257	−	0.808674i	$-0.299814\pi$
$743$	874.150			1.17651			0.588257	+	0.808674i	$0.299814\pi$
$744$		0			0
$745$			1266.81i			1.70041i
$746$		0			0
$747$	65.0570	+	65.0570i	0.0870910	+	0.0870910i
$748$		0			0
$749$	−161.780	−	161.780i	−0.215995	−	0.215995i
$750$		0			0
$751$			1278.46i			1.70234i	0.524892	+	0.851169i	$0.324106\pi$
$751$			1278.46i			1.70234i	−0.524892	+	0.851169i	$0.675894\pi$
$752$		0			0
$753$	745.337			0.989824
$754$		0			0
$755$	85.4243	−	85.4243i	0.113145	−	0.113145i
$756$		0			0
$757$	−543.081	+	543.081i	−0.717412	+	0.717412i	−0.968075	−	0.250662i	$-0.919352\pi$
$757$	−543.081	+	543.081i	−0.717412	+	0.717412i	0.250662	+	0.968075i	$0.419352\pi$
$758$		0			0
$759$	29.2977			0.0386004
$760$		0			0
$761$		−	121.613i		−	0.159806i	−0.996803	−	0.0799031i	$-0.974539\pi$
$761$		−	121.613i		−	0.159806i	0.996803	−	0.0799031i	$-0.0254611\pi$
$762$		0			0
$763$	233.782	+	233.782i	0.306399	+	0.306399i
$764$		0			0
$765$	−16.0661	−	16.0661i	−0.0210015	−	0.0210015i
$766$		0			0
$767$			145.001i			0.189049i
$768$		0			0
$769$	368.174			0.478769			0.239385	−	0.970925i	$-0.423054\pi$
$769$	368.174			0.478769			0.239385	+	0.970925i	$0.423054\pi$
$770$		0			0
$771$	520.793	−	520.793i	0.675478	−	0.675478i
$772$		0			0
$773$	−871.868	+	871.868i	−1.12790	+	1.12790i	−0.137384	+	0.990518i	$0.543869\pi$
$773$	−871.868	+	871.868i	−1.12790	+	1.12790i	−0.990518	+	0.137384i	$0.956131\pi$
$774$		0			0
$775$	35.5889			0.0459212
$776$		0			0
$777$			499.627i			0.643020i
$778$		0			0
$779$	1398.05	+	1398.05i	1.79467	+	1.79467i
$780$		0			0
$781$	43.3881	+	43.3881i	0.0555545	+	0.0555545i
$782$		0			0
$783$		−	264.182i		−	0.337398i
$784$		0			0
$785$	−213.031			−0.271377
$786$		0			0
$787$	138.396	−	138.396i	0.175853	−	0.175853i	−0.613692	−	0.789545i	$-0.710316\pi$
$787$	138.396	−	138.396i	0.175853	−	0.175853i	0.789545	+	0.613692i	$0.210316\pi$
$788$		0			0
$789$	−236.234	+	236.234i	−0.299409	+	0.299409i
$790$		0			0
$791$	−92.5449			−0.116997
$792$		0			0
$793$			140.962i			0.177758i
$794$		0			0
$795$	−447.004	−	447.004i	−0.562270	−	0.562270i
$796$		0			0
$797$	419.719	+	419.719i	0.526623	+	0.526623i	0.919564	−	0.392940i	$-0.128542\pi$
$797$	419.719	+	419.719i	0.526623	+	0.526623i	−0.392940	+	0.919564i	$0.628542\pi$
$798$		0			0
$799$			87.9088i			0.110024i
$800$		0			0
$801$	358.684			0.447796
$802$		0			0
$803$	−645.630	+	645.630i	−0.804023	+	0.804023i
$804$		0			0
$805$	−47.6969	+	47.6969i	−0.0592508	+	0.0592508i
$806$		0			0
$807$	−274.328			−0.339936
$808$		0			0
$809$		−	340.167i		−	0.420478i	−0.977650	−	0.210239i	$-0.932576\pi$
$809$		−	340.167i		−	0.420478i	0.977650	−	0.210239i	$-0.0674242\pi$
$810$		0			0
$811$	−808.987	−	808.987i	−0.997518	−	0.997518i	0.00247919	−	0.999997i	$-0.499211\pi$
$811$	−808.987	−	808.987i	−0.997518	−	0.997518i	−0.999997	+	0.00247919i	$0.999211\pi$
$812$		0			0
$813$	−246.660	−	246.660i	−0.303394	−	0.303394i
$814$		0			0
$815$			1165.88i			1.43053i
$816$		0			0
$817$	−2058.78			−2.51993
$818$		0			0
$819$	−29.4554	+	29.4554i	−0.0359651	+	0.0359651i
$820$		0			0
$821$	−504.072	+	504.072i	−0.613973	+	0.613973i	−0.943979	−	0.330006i	$-0.892949\pi$
$821$	−504.072	+	504.072i	−0.613973	+	0.613973i	0.330006	+	0.943979i	$0.392949\pi$
$822$		0			0
$823$	−312.505			−0.379715			−0.189858	−	0.981812i	$-0.560803\pi$
$823$	−312.505			−0.379715			−0.189858	+	0.981812i	$0.560803\pi$
$824$		0			0
$825$		−	34.4623i		−	0.0417725i
$826$		0			0
$827$	−336.658	−	336.658i	−0.407083	−	0.407083i	0.473637	−	0.880720i	$-0.342941\pi$
$827$	−336.658	−	336.658i	−0.407083	−	0.407083i	−0.880720	+	0.473637i	$0.842941\pi$
$828$		0			0
$829$	65.0235	+	65.0235i	0.0784360	+	0.0784360i	0.745236	−	0.666800i	$-0.232337\pi$
$829$	65.0235	+	65.0235i	0.0784360	+	0.0784360i	−0.666800	+	0.745236i	$0.732337\pi$
$830$		0			0
$831$			488.283i			0.587585i
$832$		0			0
$833$	24.7554			0.0297184
$834$		0			0
$835$	1173.63	−	1173.63i	1.40554	−	1.40554i
$836$		0			0
$837$	−48.8937	+	48.8937i	−0.0584154	+	0.0584154i
$838$		0			0
$839$	−153.170			−0.182563			−0.0912815	−	0.995825i	$-0.529096\pi$
$839$	−153.170			−0.182563			−0.0912815	+	0.995825i	$0.529096\pi$
$840$		0			0
$841$			1743.90i			2.07360i
$842$		0			0
$843$	−644.879	−	644.879i	−0.764981	−	0.764981i
$844$		0			0
$845$	606.103	+	606.103i	0.717282	+	0.717282i
$846$		0			0
$847$		−	370.245i		−	0.437126i
$848$		0			0
$849$	−277.687			−0.327076
$850$		0			0
$851$	82.2321	−	82.2321i	0.0966300	−	0.0966300i
$852$		0			0
$853$	−505.256	+	505.256i	−0.592328	+	0.592328i	−0.938260	−	0.345932i	$-0.887563\pi$
$853$	−505.256	+	505.256i	−0.592328	+	0.592328i	0.345932	+	0.938260i	$0.387563\pi$
$854$		0			0
$855$	−493.631			−0.577346
$856$		0			0
$857$			390.376i			0.455514i	0.973718	+	0.227757i	$0.0731392\pi$
$857$			390.376i			0.455514i	−0.973718	+	0.227757i	$0.926861\pi$
$858$		0			0
$859$	155.878	+	155.878i	0.181464	+	0.181464i	0.791994	−	0.610529i	$-0.209043\pi$
$859$	155.878	+	155.878i	0.181464	+	0.181464i	−0.610529	+	0.791994i	$0.709043\pi$
$860$		0			0
$861$	436.605	+	436.605i	0.507091	+	0.507091i
$862$		0			0
$863$		−	1035.85i		−	1.20029i	−0.799893	−	0.600143i	$-0.795110\pi$
$863$		−	1035.85i		−	1.20029i	0.799893	−	0.600143i	$-0.204890\pi$
$864$		0			0
$865$	893.536			1.03299
$866$		0			0
$867$	351.413	−	351.413i	0.405320	−	0.405320i
$868$		0			0
$869$	−489.678	+	489.678i	−0.563496	+	0.563496i
$870$		0			0
$871$	125.424			0.144000
$872$		0			0
$873$			345.083i			0.395284i
$874$		0			0
$875$	−468.354	−	468.354i	−0.535262	−	0.535262i
$876$		0			0
$877$	−501.050	−	501.050i	−0.571323	−	0.571323i	0.361175	−	0.932498i	$-0.382376\pi$
$877$	−501.050	−	501.050i	−0.571323	−	0.571323i	−0.932498	+	0.361175i	$0.882376\pi$
$878$		0			0
$879$		−	72.8690i		−	0.0828999i
$880$		0			0
$881$	990.013			1.12374			0.561869	−	0.827226i	$-0.310083\pi$
$881$	990.013			1.12374			0.561869	+	0.827226i	$0.310083\pi$
$882$		0			0
$883$	−933.600	+	933.600i	−1.05731	+	1.05731i	−0.0590501	+	0.998255i	$0.518807\pi$
$883$	−933.600	+	933.600i	−1.05731	+	1.05731i	−0.998255	+	0.0590501i	$0.981193\pi$
$884$		0			0
$885$	−379.441	+	379.441i	−0.428747	+	0.428747i
$886$		0			0
$887$	572.252			0.645155			0.322578	−	0.946543i	$-0.395451\pi$
$887$	572.252			0.645155			0.322578	+	0.946543i	$0.395451\pi$
$888$		0			0
$889$			542.924i			0.610713i
$890$		0			0
$891$	47.3458	+	47.3458i	0.0531379	+	0.0531379i
$892$		0			0
$893$	1350.50	+	1350.50i	1.51231	+	1.51231i
$894$		0			0
$895$			1347.57i			1.50567i
$896$		0			0
$897$	9.69595			0.0108093
$898$		0			0
$899$	478.402	−	478.402i	0.532149	−	0.532149i
$900$		0			0
$901$	−70.6281	+	70.6281i	−0.0783886	+	0.0783886i
$902$		0			0
$903$	−642.948			−0.712013
$904$		0			0
$905$		−	155.986i		−	0.172361i
$906$		0			0
$907$	356.673	+	356.673i	0.393245	+	0.393245i	0.875842	−	0.482598i	$-0.160307\pi$
$907$	356.673	+	356.673i	0.393245	+	0.393245i	−0.482598	+	0.875842i	$0.660307\pi$
$908$		0			0
$909$	−112.895	−	112.895i	−0.124197	−	0.124197i
$910$		0			0
$911$			656.235i			0.720345i	0.932886	+	0.360173i	$0.117282\pi$
$911$			656.235i			0.720345i	−0.932886	+	0.360173i	$0.882718\pi$
$912$		0			0
$913$	228.161			0.249903
$914$		0			0
$915$	−368.872	+	368.872i	−0.403139	+	0.403139i
$916$		0			0
$917$	−889.040	+	889.040i	−0.969510	+	0.969510i
$918$		0			0
$919$	−947.482			−1.03099			−0.515496	−	0.856892i	$-0.672392\pi$
$919$	−947.482			−1.03099			−0.515496	+	0.856892i	$0.672392\pi$
$920$		0			0
$921$		−	129.509i		−	0.140618i
$922$		0			0
$923$	14.3591	+	14.3591i	0.0155570	+	0.0155570i
$924$		0			0
$925$	−96.7280	−	96.7280i	−0.104571	−	0.104571i
$926$		0			0
$927$		−	390.020i		−	0.420733i
$928$		0			0
$929$	628.551			0.676589			0.338295	−	0.941040i	$-0.390150\pi$
$929$	628.551			0.676589			0.338295	+	0.941040i	$0.390150\pi$
$930$		0			0
$931$	380.304	−	380.304i	0.408490	−	0.408490i
$932$		0			0
$933$	429.534	−	429.534i	0.460380	−	0.460380i
$934$		0			0
$935$	−56.3455			−0.0602626
$936$		0			0
$937$			910.799i			0.972037i	0.873949	+	0.486018i	$0.161551\pi$
$937$			910.799i			0.972037i	−0.873949	+	0.486018i	$0.838449\pi$
$938$		0			0
$939$	−481.989	−	481.989i	−0.513300	−	0.513300i
$940$		0			0
$941$	−304.500	−	304.500i	−0.323592	−	0.323592i	0.526551	−	0.850143i	$-0.323485\pi$
$941$	−304.500	−	304.500i	−0.323592	−	0.323592i	−0.850143	+	0.526551i	$0.823485\pi$
$942$		0			0
$943$		−	143.719i		−	0.152406i
$944$		0			0
$945$	−154.159			−0.163131
$946$		0			0
$947$	720.759	−	720.759i	0.761097	−	0.761097i	−0.215423	−	0.976521i	$-0.569113\pi$
$947$	720.759	−	720.759i	0.761097	−	0.761097i	0.976521	+	0.215423i	$0.0691131\pi$
$948$		0			0
$949$	−213.669	+	213.669i	−0.225151	+	0.225151i
$950$		0			0
$951$	810.632			0.852400
$952$		0			0
$953$			372.120i			0.390473i	0.980756	+	0.195236i	$0.0625474\pi$
$953$			372.120i			0.390473i	−0.980756	+	0.195236i	$0.937453\pi$
$954$		0			0
$955$	−149.681	−	149.681i	−0.156735	−	0.156735i
$956$		0			0
$957$	−463.257	−	463.257i	−0.484072	−	0.484072i
$958$		0			0
$959$			728.947i			0.760112i
$960$		0			0
$961$	783.919			0.815733
$962$		0			0
$963$	−86.0597	+	86.0597i	−0.0893663	+	0.0893663i
$964$		0			0
$965$	110.530	−	110.530i	0.114539	−	0.114539i
$966$		0			0
$967$	−1575.78			−1.62956			−0.814780	−	0.579771i	$-0.803142\pi$
$967$	−1575.78			−1.62956			−0.814780	+	0.579771i	$0.803142\pi$
$968$		0			0
$969$			77.9953i			0.0804905i
$970$		0			0
$971$	48.8117	+	48.8117i	0.0502695	+	0.0502695i	0.731795	−	0.681525i	$-0.238683\pi$
$971$	48.8117	+	48.8117i	0.0502695	+	0.0502695i	−0.681525	+	0.731795i	$0.738683\pi$
$972$		0			0
$973$	−22.1994	−	22.1994i	−0.0228154	−	0.0228154i
$974$		0			0
$975$		−	11.4051i		−	0.0116976i
$976$		0			0
$977$	−83.5759			−0.0855434			−0.0427717	−	0.999085i	$-0.513619\pi$
$977$	−83.5759			−0.0855434			−0.0427717	+	0.999085i	$0.513619\pi$
$978$		0			0
$979$	628.970	−	628.970i	0.642462	−	0.642462i
$980$		0			0
$981$	124.361	−	124.361i	0.126770	−	0.126770i
$982$		0			0
$983$	−1864.55			−1.89679			−0.948397	−	0.317086i	$-0.897296\pi$
$983$	−1864.55			−1.89679			−0.948397	+	0.317086i	$0.897296\pi$
$984$		0			0
$985$		−	1009.50i		−	1.02488i
$986$		0			0
$987$	421.754	+	421.754i	0.427309	+	0.427309i
$988$		0			0
$989$	105.821	+	105.821i	0.106998	+	0.106998i
$990$		0			0
$991$			491.954i			0.496422i	0.968706	+	0.248211i	$0.0798427\pi$
$991$			491.954i			0.496422i	−0.968706	+	0.248211i	$0.920157\pi$
$992$		0			0
$993$	444.068			0.447198
$994$		0			0
$995$	588.885	−	588.885i	0.591844	−	0.591844i
$996$		0			0
$997$	708.553	−	708.553i	0.710685	−	0.710685i	−0.255993	−	0.966679i	$-0.582403\pi$
$997$	708.553	−	708.553i	0.710685	−	0.710685i	0.966679	+	0.255993i	$0.0824025\pi$
$998$		0			0
$999$	265.778			0.266044

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.l.f.703.2	yes	16
4.3	odd	2	inner	768.3.l.f.703.6	yes	16
8.3	odd	2		768.3.l.e.703.3	yes	16
8.5	even	2		768.3.l.e.703.7	yes	16
16.3	odd	4		768.3.l.e.319.7	yes	16
16.5	even	4	inner	768.3.l.f.319.6	yes	16
16.11	odd	4	inner	768.3.l.f.319.2	yes	16
16.13	even	4		768.3.l.e.319.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.3.l.e.319.3	✓	16	16.13	even	4
768.3.l.e.319.7	yes	16	16.3	odd	4
768.3.l.e.703.3	yes	16	8.3	odd	2
768.3.l.e.703.7	yes	16	8.5	even	2
768.3.l.f.319.2	yes	16	16.11	odd	4	inner
768.3.l.f.319.6	yes	16	16.5	even	4	inner
768.3.l.f.703.2	yes	16	1.1	even	1	trivial
768.3.l.f.703.6	yes	16	4.3	odd	2	inner

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 1.22474i) q^{3} +(-3.71984 + 3.71984i) q^{5} +5.63959 q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 + 1.22474i) q^{3} +(-3.71984 + 3.71984i) q^{5} +5.63959 q^{7} -3.00000i q^{9} +(-5.26065 - 5.26065i) q^{11} +(-1.74099 - 1.74099i) q^{13} -9.11171i q^{15} -1.43968 q^{17} +(-22.1170 + 22.1170i) q^{19} +(-6.90705 + 6.90705i) q^{21} +2.27362 q^{23} -2.67442i q^{25} +(3.67423 + 3.67423i) q^{27} +(-35.9507 - 35.9507i) q^{29} +13.3072i q^{31} +12.8859 q^{33} +(-20.9784 + 20.9784i) q^{35} +(36.1679 - 36.1679i) q^{37} +4.26454 q^{39} -63.2115i q^{41} +(46.5429 + 46.5429i) q^{43} +(11.1595 + 11.1595i) q^{45} -61.0614i q^{47} -17.1951 q^{49} +(1.76324 - 1.76324i) q^{51} +(49.0582 - 49.0582i) q^{53} +39.1375 q^{55} -54.1754i q^{57} +(-41.6432 - 41.6432i) q^{59} +(-40.4833 - 40.4833i) q^{61} -16.9188i q^{63} +12.9524 q^{65} +(-36.0209 + 36.0209i) q^{67} +(-2.78461 + 2.78461i) q^{69} -8.24767 q^{71} -122.728i q^{73} +(3.27548 + 3.27548i) q^{75} +(-29.6679 - 29.6679i) q^{77} -93.0832i q^{79} -9.00000 q^{81} +(-21.6857 + 21.6857i) q^{83} +(5.35538 - 5.35538i) q^{85} +88.0608 q^{87} +119.561i q^{89} +(-9.81846 - 9.81846i) q^{91} +(-16.2979 - 16.2979i) q^{93} -164.544i q^{95} -115.028 q^{97} +(-15.7819 + 15.7819i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{5}+O(q^{10}) \) 16 * q + 8 * q^5	\( 16 q + 8 q^{5} - 32 q^{13} + 112 q^{17} + 72 q^{21} - 56 q^{29} - 48 q^{33} + 272 q^{37} - 24 q^{45} + 304 q^{49} + 504 q^{53} - 176 q^{61} + 624 q^{65} + 288 q^{69} - 848 q^{77} - 144 q^{81} + 880 q^{85} + 216 q^{93} - 160 q^{97}+O(q^{100}) \) 16 * q + 8 * q^5 - 32 * q^13 + 112 * q^17 + 72 * q^21 - 56 * q^29 - 48 * q^33 + 272 * q^37 - 24 * q^45 + 304 * q^49 + 504 * q^53 - 176 * q^61 + 624 * q^65 + 288 * q^69 - 848 * q^77 - 144 * q^81 + 880 * q^85 + 216 * q^93 - 160 * q^97

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