LMFDB - Embedded newform 768.3.l.b.703.1

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:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			703.1
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	768.703
Dual form			768.3.l.b.319.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 + 1.22474i) q^{3} +(-2.73205 + 2.73205i) q^{5} -2.17209 q^{7} -3.00000i q^{9} +O(q^{10})$	\(q+(-1.22474 + 1.22474i) q^{3} +(-2.73205 + 2.73205i) q^{5} -2.17209 q^{7} -3.00000i q^{9} +(-0.277401 - 0.277401i) q^{11} +(13.3923 + 13.3923i) q^{13} -6.69213i q^{15} -26.3923 q^{17} +(-2.17209 + 2.17209i) q^{19} +(2.66025 - 2.66025i) q^{21} -7.52433 q^{23} +10.0718i q^{25} +(3.67423 + 3.67423i) q^{27} +(6.19615 + 6.19615i) q^{29} -28.7375i q^{31} +0.679492 q^{33} +(5.93426 - 5.93426i) q^{35} +(-4.07180 + 4.07180i) q^{37} -32.8043 q^{39} -23.1769i q^{41} +(-49.0913 - 49.0913i) q^{43} +(8.19615 + 8.19615i) q^{45} -90.4553i q^{47} -44.2820 q^{49} +(32.3238 - 32.3238i) q^{51} +(37.5167 - 37.5167i) q^{53} +1.51575 q^{55} -5.32051i q^{57} +(-44.8487 - 44.8487i) q^{59} +(57.4974 + 57.4974i) q^{61} +6.51626i q^{63} -73.1769 q^{65} +(57.8812 - 57.8812i) q^{67} +(9.21539 - 9.21539i) q^{69} +46.5675 q^{71} -2.43078i q^{73} +(-12.3354 - 12.3354i) q^{75} +(0.602540 + 0.602540i) q^{77} -1.76594i q^{79} -9.00000 q^{81} +(-58.5648 + 58.5648i) q^{83} +(72.1051 - 72.1051i) q^{85} -15.1774 q^{87} -126.785i q^{89} +(-29.0893 - 29.0893i) q^{91} +(35.1962 + 35.1962i) q^{93} -11.8685i q^{95} +46.9282 q^{97} +(-0.832204 + 0.832204i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} + 24 q^{13} - 128 q^{17} - 48 q^{21} + 8 q^{29} + 144 q^{33} - 88 q^{37} + 24 q^{45} + 200 q^{49} + 120 q^{53} + 72 q^{61} - 336 q^{65} + 240 q^{69} - 688 q^{77} - 72 q^{81} + 272 q^{85} + 240 q^{93} + 320 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 + 24 * q^13 - 128 * q^17 - 48 * q^21 + 8 * q^29 + 144 * q^33 - 88 * q^37 + 24 * q^45 + 200 * q^49 + 120 * q^53 + 72 * q^61 - 336 * q^65 + 240 * q^69 - 688 * q^77 - 72 * q^81 + 272 * q^85 + 240 * q^93 + 320 * 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
$5$	−2.73205	+	2.73205i	−0.546410	+	0.546410i	−0.925401	−	0.378990i	$-0.876271\pi$
$5$	−2.73205	+	2.73205i	−0.546410	+	0.546410i	0.378990	+	0.925401i	$0.376271\pi$
$6$		0			0
$7$	−2.17209			−0.310298			−0.155149	−	0.987891i	$-0.549586\pi$
$7$	−2.17209			−0.310298			−0.155149	+	0.987891i	$0.549586\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	−0.277401	−	0.277401i	−0.0252183	−	0.0252183i	0.694385	−	0.719604i	$-0.255676\pi$
$11$	−0.277401	−	0.277401i	−0.0252183	−	0.0252183i	−0.719604	+	0.694385i	$0.755676\pi$
$12$		0			0
$13$	13.3923	+	13.3923i	1.03018	+	1.03018i	0.999530	+	0.0306470i	$0.00975678\pi$
$13$	13.3923	+	13.3923i	1.03018	+	1.03018i	0.0306470	+	0.999530i	$0.490243\pi$
$14$		0			0
$15$		−	6.69213i		−	0.446142i
$16$		0			0
$17$	−26.3923			−1.55249			−0.776244	−	0.630432i	$-0.782878\pi$
$17$	−26.3923			−1.55249			−0.776244	+	0.630432i	$0.782878\pi$
$18$		0			0
$19$	−2.17209	+	2.17209i	−0.114320	+	0.114320i	−0.761953	−	0.647632i	$-0.775759\pi$
$19$	−2.17209	+	2.17209i	−0.114320	+	0.114320i	0.647632	+	0.761953i	$0.275759\pi$
$20$		0			0
$21$	2.66025	−	2.66025i	0.126679	−	0.126679i
$22$		0			0
$23$	−7.52433			−0.327145			−0.163572	−	0.986531i	$-0.552302\pi$
$23$	−7.52433			−0.327145			−0.163572	+	0.986531i	$0.552302\pi$
$24$		0			0
$25$			10.0718i			0.402872i
$26$		0			0
$27$	3.67423	+	3.67423i	0.136083	+	0.136083i
$28$		0			0
$29$	6.19615	+	6.19615i	0.213660	+	0.213660i	0.805820	−	0.592160i	$-0.201725\pi$
$29$	6.19615	+	6.19615i	0.213660	+	0.213660i	−0.592160	+	0.805820i	$0.701725\pi$
$30$		0			0
$31$		−	28.7375i		−	0.927017i	−0.886092	−	0.463509i	$-0.846590\pi$
$31$		−	28.7375i		−	0.927017i	0.886092	−	0.463509i	$-0.153410\pi$
$32$		0			0
$33$	0.679492			0.0205907
$34$		0			0
$35$	5.93426	−	5.93426i	0.169550	−	0.169550i
$36$		0			0
$37$	−4.07180	+	4.07180i	−0.110049	+	0.110049i	−0.759987	−	0.649938i	$-0.774795\pi$
$37$	−4.07180	+	4.07180i	−0.110049	+	0.110049i	0.649938	+	0.759987i	$0.274795\pi$
$38$		0			0
$39$	−32.8043			−0.841136
$40$		0			0
$41$		−	23.1769i		−	0.565291i	−0.959224	−	0.282645i	$-0.908788\pi$
$41$		−	23.1769i		−	0.565291i	0.959224	−	0.282645i	$-0.0912119\pi$
$42$		0			0
$43$	−49.0913	−	49.0913i	−1.14166	−	1.14166i	−0.988147	−	0.153512i	$-0.950942\pi$
$43$	−49.0913	−	49.0913i	−1.14166	−	1.14166i	−0.153512	−	0.988147i	$-0.549058\pi$
$44$		0			0
$45$	8.19615	+	8.19615i	0.182137	+	0.182137i
$46$		0			0
$47$		−	90.4553i		−	1.92458i	−0.272026	−	0.962290i	$-0.587694\pi$
$47$		−	90.4553i		−	1.92458i	0.272026	−	0.962290i	$-0.412306\pi$
$48$		0			0
$49$	−44.2820			−0.903715
$50$		0			0
$51$	32.3238	−	32.3238i	0.633801	−	0.633801i
$52$		0			0
$53$	37.5167	−	37.5167i	0.707862	−	0.707862i	−0.258224	−	0.966085i	$-0.583137\pi$
$53$	37.5167	−	37.5167i	0.707862	−	0.707862i	0.966085	+	0.258224i	$0.0831371\pi$
$54$		0			0
$55$	1.51575			0.0275591
$56$		0			0
$57$		−	5.32051i		−	0.0933422i
$58$		0			0
$59$	−44.8487	−	44.8487i	−0.760147	−	0.760147i	0.216201	−	0.976349i	$-0.430633\pi$
$59$	−44.8487	−	44.8487i	−0.760147	−	0.760147i	−0.976349	+	0.216201i	$0.930633\pi$
$60$		0			0
$61$	57.4974	+	57.4974i	0.942581	+	0.942581i	0.998439	−	0.0558580i	$-0.0177894\pi$
$61$	57.4974	+	57.4974i	0.942581	+	0.942581i	−0.0558580	+	0.998439i	$0.517789\pi$
$62$		0			0
$63$			6.51626i			0.103433i
$64$		0			0
$65$	−73.1769			−1.12580
$66$		0			0
$67$	57.8812	−	57.8812i	0.863899	−	0.863899i	−0.127890	−	0.991788i	$-0.540820\pi$
$67$	57.8812	−	57.8812i	0.863899	−	0.863899i	0.991788	+	0.127890i	$0.0408203\pi$
$68$		0			0
$69$	9.21539	−	9.21539i	0.133556	−	0.133556i
$70$		0			0
$71$	46.5675			0.655880			0.327940	−	0.944698i	$-0.393646\pi$
$71$	46.5675			0.655880			0.327940	+	0.944698i	$0.393646\pi$
$72$		0			0
$73$		−	2.43078i		−	0.0332984i	−0.999861	−	0.0166492i	$-0.994700\pi$
$73$		−	2.43078i		−	0.0332984i	0.999861	−	0.0166492i	$-0.00529984\pi$
$74$		0			0
$75$	−12.3354	−	12.3354i	−0.164472	−	0.164472i
$76$		0			0
$77$	0.602540	+	0.602540i	0.00782520	+	0.00782520i
$78$		0			0
$79$		−	1.76594i		−	0.0223537i	−0.999938	−	0.0111769i	$-0.996442\pi$
$79$		−	1.76594i		−	0.0223537i	0.999938	−	0.0111769i	$-0.00355778\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−58.5648	+	58.5648i	−0.705600	+	0.705600i	−0.965607	−	0.260007i	$-0.916275\pi$
$83$	−58.5648	+	58.5648i	−0.705600	+	0.705600i	0.260007	+	0.965607i	$0.416275\pi$
$84$		0			0
$85$	72.1051	−	72.1051i	0.848296	−	0.848296i
$86$		0			0
$87$	−15.1774			−0.174453
$88$		0			0
$89$		−	126.785i		−	1.42455i	−0.701902	−	0.712273i	$-0.747666\pi$
$89$		−	126.785i		−	1.42455i	0.701902	−	0.712273i	$-0.252334\pi$
$90$		0			0
$91$	−29.0893	−	29.0893i	−0.319662	−	0.319662i
$92$		0			0
$93$	35.1962	+	35.1962i	0.378453	+	0.378453i
$94$		0			0
$95$		−	11.8685i		−	0.124932i
$96$		0			0
$97$	46.9282			0.483796			0.241898	−	0.970302i	$-0.422230\pi$
$97$	46.9282			0.483796			0.241898	+	0.970302i	$0.422230\pi$
$98$		0			0
$99$	−0.832204	+	0.832204i	−0.00840610	+	0.00840610i
$100$		0			0
$101$	−66.8756	+	66.8756i	−0.662135	+	0.662135i	−0.955883	−	0.293748i	$-0.905097\pi$
$101$	−66.8756	+	66.8756i	−0.662135	+	0.662135i	0.293748	+	0.955883i	$0.405097\pi$
$102$		0			0
$103$	−166.518			−1.61668			−0.808341	−	0.588715i	$-0.799634\pi$
$103$	−166.518			−1.61668			−0.808341	+	0.588715i	$0.799634\pi$
$104$		0			0
$105$			14.5359i			0.138437i
$106$		0			0
$107$	−129.350	−	129.350i	−1.20888	−	1.20888i	−0.971391	−	0.237485i	$-0.923677\pi$
$107$	−129.350	−	129.350i	−1.20888	−	1.20888i	−0.237485	−	0.971391i	$-0.576323\pi$
$108$		0			0
$109$	98.3205	+	98.3205i	0.902023	+	0.902023i	0.995611	−	0.0935880i	$-0.0298336\pi$
$109$	98.3205	+	98.3205i	0.902023	+	0.902023i	−0.0935880	+	0.995611i	$0.529834\pi$
$110$		0			0
$111$		−	9.97382i		−	0.0898543i
$112$		0			0
$113$	−211.990			−1.87601			−0.938007	−	0.346615i	$-0.887331\pi$
$113$	−211.990			−1.87601			−0.938007	+	0.346615i	$0.887331\pi$
$114$		0			0
$115$	20.5569	−	20.5569i	0.178755	−	0.178755i
$116$		0			0
$117$	40.1769	−	40.1769i	0.343392	−	0.343392i
$118$		0			0
$119$	57.3264			0.481735
$120$		0			0
$121$		−	120.846i		−	0.998728i
$122$		0			0
$123$	28.3858	+	28.3858i	0.230779	+	0.230779i
$124$		0			0
$125$	−95.8179	−	95.8179i	−0.766543	−	0.766543i
$126$		0			0
$127$		−	32.0664i		−	0.252491i	−0.991999	−	0.126245i	$-0.959707\pi$
$127$		−	32.0664i		−	0.252491i	0.991999	−	0.126245i	$-0.0402927\pi$
$128$		0			0
$129$	120.249			0.932161
$130$		0			0
$131$	−30.0031	+	30.0031i	−0.229031	+	0.229031i	−0.812288	−	0.583257i	$-0.801778\pi$
$131$	−30.0031	+	30.0031i	−0.229031	+	0.229031i	0.583257	+	0.812288i	$0.301778\pi$
$132$		0			0
$133$	4.71797	−	4.71797i	0.0354734	−	0.0354734i
$134$		0			0
$135$	−20.0764			−0.148714
$136$		0			0
$137$			97.5307i			0.711903i	0.934504	+	0.355952i	$0.115843\pi$
$137$			97.5307i			0.711903i	−0.934504	+	0.355952i	$0.884157\pi$
$138$		0			0
$139$	151.720	+	151.720i	1.09151	+	1.09151i	0.995368	+	0.0961409i	$0.0306500\pi$
$139$	151.720	+	151.720i	1.09151	+	1.09151i	0.0961409	+	0.995368i	$0.469350\pi$
$140$		0			0
$141$	110.785	+	110.785i	0.785706	+	0.785706i
$142$		0			0
$143$		−	7.43009i		−	0.0519587i
$144$		0			0
$145$	−33.8564			−0.233492
$146$		0			0
$147$	54.2342	−	54.2342i	0.368940	−	0.368940i
$148$		0			0
$149$	−61.9474	+	61.9474i	−0.415755	+	0.415755i	−0.883738	−	0.467983i	$-0.844981\pi$
$149$	−61.9474	+	61.9474i	−0.415755	+	0.415755i	0.467983	+	0.883738i	$0.344981\pi$
$150$		0			0
$151$	47.7242			0.316055			0.158027	−	0.987435i	$-0.449487\pi$
$151$	47.7242			0.316055			0.158027	+	0.987435i	$0.449487\pi$
$152$		0			0
$153$			79.1769i			0.517496i
$154$		0			0
$155$	78.5124	+	78.5124i	0.506532	+	0.506532i
$156$		0			0
$157$	−167.851	−	167.851i	−1.06912	−	1.06912i	−0.997427	−	0.0716892i	$-0.977161\pi$
$157$	−167.851	−	167.851i	−1.06912	−	1.06912i	−0.0716892	−	0.997427i	$-0.522839\pi$
$158$		0			0
$159$			91.8967i			0.577967i
$160$		0			0
$161$	16.3435			0.101513
$162$		0			0
$163$	38.8872	−	38.8872i	0.238572	−	0.238572i	−0.577687	−	0.816259i	$-0.696044\pi$
$163$	38.8872	−	38.8872i	0.238572	−	0.238572i	0.816259	+	0.577687i	$0.196044\pi$
$164$		0			0
$165$	−1.85641	+	1.85641i	−0.0112509	+	0.0112509i
$166$		0			0
$167$	239.658			1.43508			0.717540	−	0.696517i	$-0.245268\pi$
$167$	239.658			1.43508			0.717540	+	0.696517i	$0.245268\pi$
$168$		0			0
$169$			189.708i			1.12253i
$170$		0			0
$171$	6.51626	+	6.51626i	0.0381068	+	0.0381068i
$172$		0			0
$173$	−55.3064	−	55.3064i	−0.319690	−	0.319690i	0.528958	−	0.848648i	$-0.322583\pi$
$173$	−55.3064	−	55.3064i	−0.319690	−	0.319690i	−0.848648	+	0.528958i	$0.822583\pi$
$174$		0			0
$175$		−	21.8768i		−	0.125010i
$176$		0			0
$177$	109.856			0.620658
$178$		0			0
$179$	−183.793	+	183.793i	−1.02678	+	1.02678i	−0.0271470	+	0.999631i	$0.508642\pi$
$179$	−183.793	+	183.793i	−1.02678	+	1.02678i	−0.999631	+	0.0271470i	$0.991358\pi$
$180$		0			0
$181$	−15.1821	+	15.1821i	−0.0838788	+	0.0838788i	−0.747801	−	0.663923i	$-0.768890\pi$
$181$	−15.1821	+	15.1821i	−0.0838788	+	0.0838788i	0.663923	+	0.747801i	$0.268890\pi$
$182$		0			0
$183$	−140.839			−0.769614
$184$		0			0
$185$		−	22.2487i		−	0.120263i
$186$		0			0
$187$	7.32126	+	7.32126i	0.0391511	+	0.0391511i
$188$		0			0
$189$	−7.98076	−	7.98076i	−0.0422263	−	0.0422263i
$190$		0			0
$191$		−	198.545i		−	1.03950i	−0.854318	−	0.519751i	$-0.826025\pi$
$191$		−	198.545i		−	1.03950i	0.854318	−	0.519751i	$-0.173975\pi$
$192$		0			0
$193$	244.851			1.26866			0.634330	−	0.773063i	$-0.281276\pi$
$193$	244.851			1.26866			0.634330	+	0.773063i	$0.281276\pi$
$194$		0			0
$195$	89.6231	−	89.6231i	0.459605	−	0.459605i
$196$		0			0
$197$	−91.5833	+	91.5833i	−0.464890	+	0.464890i	−0.900254	−	0.435364i	$-0.856620\pi$
$197$	−91.5833	+	91.5833i	−0.464890	+	0.464890i	0.435364	+	0.900254i	$0.356620\pi$
$198$		0			0
$199$	−196.630			−0.988091			−0.494045	−	0.869436i	$-0.664482\pi$
$199$	−196.630			−0.988091			−0.494045	+	0.869436i	$0.664482\pi$
$200$		0			0
$201$			141.779i			0.705370i
$202$		0			0
$203$	−13.4586	−	13.4586i	−0.0662985	−	0.0662985i
$204$		0			0
$205$	63.3205	+	63.3205i	0.308881	+	0.308881i
$206$		0			0
$207$			22.5730i			0.109048i
$208$		0			0
$209$	1.20508			0.00576594
$210$		0			0
$211$	80.2113	−	80.2113i	0.380148	−	0.380148i	−0.491007	−	0.871156i	$-0.663371\pi$
$211$	80.2113	−	80.2113i	0.380148	−	0.380148i	0.871156	+	0.491007i	$0.163371\pi$
$212$		0			0
$213$	−57.0333	+	57.0333i	−0.267762	+	0.267762i
$214$		0			0
$215$	268.240			1.24763
$216$		0			0
$217$			62.4205i			0.287652i
$218$		0			0
$219$	2.97709	+	2.97709i	0.0135940	+	0.0135940i
$220$		0			0
$221$	−353.454	−	353.454i	−1.59934	−	1.59934i
$222$		0			0
$223$			344.194i			1.54347i	0.635943	+	0.771736i	$0.280611\pi$
$223$			344.194i			1.54347i	−0.635943	+	0.771736i	$0.719389\pi$
$224$		0			0
$225$	30.2154			0.134291
$226$		0			0
$227$	−286.395	+	286.395i	−1.26165	+	1.26165i	−0.311357	+	0.950293i	$0.600783\pi$
$227$	−286.395	+	286.395i	−1.26165	+	1.26165i	−0.950293	+	0.311357i	$0.899217\pi$
$228$		0			0
$229$	−122.244	+	122.244i	−0.533815	+	0.533815i	−0.921705	−	0.387891i	$-0.873204\pi$
$229$	−122.244	+	122.244i	−0.533815	+	0.533815i	0.387891	+	0.921705i	$0.373204\pi$
$230$		0			0
$231$	−1.47592			−0.00638925
$232$		0			0
$233$			67.7795i			0.290899i	0.989366	+	0.145449i	$0.0464628\pi$
$233$			67.7795i			0.290899i	−0.989366	+	0.145449i	$0.953537\pi$
$234$		0			0
$235$	247.128	+	247.128i	1.05161	+	1.05161i
$236$		0			0
$237$	2.16283	+	2.16283i	0.00912587	+	0.00912587i
$238$		0			0
$239$			309.760i			1.29607i	0.761612	+	0.648033i	$0.224408\pi$
$239$			309.760i			1.29607i	−0.761612	+	0.648033i	$0.775592\pi$
$240$		0			0
$241$	262.497			1.08920			0.544600	−	0.838696i	$-0.316681\pi$
$241$	262.497			1.08920			0.544600	+	0.838696i	$0.316681\pi$
$242$		0			0
$243$	11.0227	−	11.0227i	0.0453609	−	0.0453609i
$244$		0			0
$245$	120.981	−	120.981i	0.493799	−	0.493799i
$246$		0			0
$247$	−58.1785			−0.235541
$248$		0			0
$249$		−	143.454i		−	0.576120i
$250$		0			0
$251$	222.044	+	222.044i	0.884638	+	0.884638i	0.994002	−	0.109364i	$-0.0348813\pi$
$251$	222.044	+	222.044i	0.884638	+	0.884638i	−0.109364	+	0.994002i	$0.534881\pi$
$252$		0			0
$253$	2.08726	+	2.08726i	0.00825004	+	0.00825004i
$254$		0			0
$255$			176.621i			0.692630i
$256$		0			0
$257$	−73.0821			−0.284366			−0.142183	−	0.989840i	$-0.545412\pi$
$257$	−73.0821			−0.284366			−0.142183	+	0.989840i	$0.545412\pi$
$258$		0			0
$259$	8.84430	−	8.84430i	0.0341479	−	0.0341479i
$260$		0			0
$261$	18.5885	−	18.5885i	0.0712201	−	0.0712201i
$262$		0			0
$263$	362.039			1.37657			0.688286	−	0.725439i	$-0.258363\pi$
$263$	362.039			1.37657			0.688286	+	0.725439i	$0.258363\pi$
$264$		0			0
$265$			204.995i			0.773565i
$266$		0			0
$267$	155.279	+	155.279i	0.581569	+	0.581569i
$268$		0			0
$269$	−118.119	−	118.119i	−0.439105	−	0.439105i	0.452606	−	0.891711i	$-0.350494\pi$
$269$	−118.119	−	118.119i	−0.439105	−	0.439105i	−0.891711	+	0.452606i	$0.850494\pi$
$270$		0			0
$271$			333.598i			1.23099i	0.788141	+	0.615495i	$0.211044\pi$
$271$			333.598i			1.23099i	−0.788141	+	0.615495i	$0.788956\pi$
$272$		0			0
$273$	71.2539			0.261003
$274$		0			0
$275$	2.79393	−	2.79393i	0.0101597	−	0.0101597i
$276$		0			0
$277$	161.172	−	161.172i	0.581848	−	0.581848i	−0.353563	−	0.935411i	$-0.615030\pi$
$277$	161.172	−	161.172i	0.581848	−	0.581848i	0.935411	+	0.353563i	$0.115030\pi$
$278$		0			0
$279$	−86.2126			−0.309006
$280$		0			0
$281$		−	453.482i		−	1.61381i	−0.590678	−	0.806907i	$-0.701140\pi$
$281$		−	453.482i		−	1.61381i	0.590678	−	0.806907i	$-0.298860\pi$
$282$		0			0
$283$	−72.4839	−	72.4839i	−0.256127	−	0.256127i	0.567350	−	0.823477i	$-0.307969\pi$
$283$	−72.4839	−	72.4839i	−0.256127	−	0.256127i	−0.823477	+	0.567350i	$0.807969\pi$
$284$		0			0
$285$	14.5359	+	14.5359i	0.0510032	+	0.0510032i
$286$		0			0
$287$			50.3423i			0.175409i
$288$		0			0
$289$	407.554			1.41022
$290$		0			0
$291$	−57.4751	+	57.4751i	−0.197509	+	0.197509i
$292$		0			0
$293$	−328.291	+	328.291i	−1.12045	+	1.12045i	−0.128773	+	0.991674i	$0.541104\pi$
$293$	−328.291	+	328.291i	−1.12045	+	1.12045i	−0.991674	+	0.128773i	$0.958896\pi$
$294$		0			0
$295$	245.058			0.830704
$296$		0			0
$297$		−	2.03848i		−	0.00686355i
$298$		0			0
$299$	−100.768	−	100.768i	−0.337017	−	0.337017i
$300$		0			0
$301$	106.631	+	106.631i	0.354255	+	0.354255i
$302$		0			0
$303$		−	163.811i		−	0.540631i
$304$		0			0
$305$	−314.172			−1.03007
$306$		0			0
$307$	−138.593	+	138.593i	−0.451443	+	0.451443i	−0.895833	−	0.444390i	$-0.853420\pi$
$307$	−138.593	+	138.593i	−0.451443	+	0.451443i	0.444390	+	0.895833i	$0.353420\pi$
$308$		0			0
$309$	203.942	−	203.942i	0.660007	−	0.660007i
$310$		0			0
$311$	−312.534			−1.00493			−0.502466	−	0.864597i	$-0.667574\pi$
$311$	−312.534			−1.00493			−0.502466	+	0.864597i	$0.667574\pi$
$312$		0			0
$313$		−	275.282i		−	0.879495i	−0.898121	−	0.439748i	$-0.855068\pi$
$313$		−	275.282i		−	0.879495i	0.898121	−	0.439748i	$-0.144932\pi$
$314$		0			0
$315$	−17.8028	−	17.8028i	−0.0565167	−	0.0565167i
$316$		0			0
$317$	−260.263	−	260.263i	−0.821018	−	0.821018i	0.165236	−	0.986254i	$-0.447161\pi$
$317$	−260.263	−	260.263i	−0.821018	−	0.821018i	−0.986254	+	0.165236i	$0.947161\pi$
$318$		0			0
$319$		−	3.43764i		−	0.0107763i
$320$		0			0
$321$	316.841			0.987043
$322$		0			0
$323$	57.3264	−	57.3264i	0.177481	−	0.177481i
$324$		0			0
$325$	−134.885	+	134.885i	−0.415029	+	0.415029i
$326$		0			0
$327$	−240.835			−0.736499
$328$		0			0
$329$			196.477i			0.597194i
$330$		0			0
$331$	−105.761	−	105.761i	−0.319521	−	0.319521i	0.529062	−	0.848583i	$-0.322544\pi$
$331$	−105.761	−	105.761i	−0.319521	−	0.319521i	−0.848583	+	0.529062i	$0.822544\pi$
$332$		0			0
$333$	12.2154	+	12.2154i	0.0366829	+	0.0366829i
$334$		0			0
$335$			316.269i			0.944086i
$336$		0			0
$337$	−138.851			−0.412022			−0.206011	−	0.978550i	$-0.566048\pi$
$337$	−138.851			−0.412022			−0.206011	+	0.978550i	$0.566048\pi$
$338$		0			0
$339$	259.633	−	259.633i	0.765880	−	0.765880i
$340$		0			0
$341$	−7.97183	+	7.97183i	−0.0233778	+	0.0233778i
$342$		0			0
$343$	202.617			0.590720
$344$		0			0
$345$			50.3538i			0.145953i
$346$		0			0
$347$	−83.0052	−	83.0052i	−0.239208	−	0.239208i	0.577314	−	0.816522i	$-0.304101\pi$
$347$	−83.0052	−	83.0052i	−0.239208	−	0.239208i	−0.816522	+	0.577314i	$0.804101\pi$
$348$		0			0
$349$	−176.138	−	176.138i	−0.504695	−	0.504695i	0.408199	−	0.912893i	$-0.366157\pi$
$349$	−176.138	−	176.138i	−0.504695	−	0.504695i	−0.912893	+	0.408199i	$0.866157\pi$
$350$		0			0
$351$			98.4129i			0.280379i
$352$		0			0
$353$	378.785			1.07304			0.536522	−	0.843886i	$-0.319738\pi$
$353$	378.785			1.07304			0.536522	+	0.843886i	$0.319738\pi$
$354$		0			0
$355$	−127.225	+	127.225i	−0.358380	+	0.358380i
$356$		0			0
$357$	−70.2102	+	70.2102i	−0.196667	+	0.196667i
$358$		0			0
$359$	70.3045			0.195834			0.0979172	−	0.995195i	$-0.468782\pi$
$359$	70.3045			0.195834			0.0979172	+	0.995195i	$0.468782\pi$
$360$		0			0
$361$			351.564i			0.973862i
$362$		0			0
$363$	148.006	+	148.006i	0.407729	+	0.407729i
$364$		0			0
$365$	6.64102	+	6.64102i	0.0181946	+	0.0181946i
$366$		0			0
$367$		−	83.8846i		−	0.228568i	−0.993448	−	0.114284i	$-0.963543\pi$
$367$		−	83.8846i		−	0.228568i	0.993448	−	0.114284i	$-0.0364574\pi$
$368$		0			0
$369$	−69.5307			−0.188430
$370$		0			0
$371$	−81.4895	+	81.4895i	−0.219648	+	0.219648i
$372$		0			0
$373$	144.646	−	144.646i	0.387791	−	0.387791i	−0.486108	−	0.873899i	$-0.661584\pi$
$373$	144.646	−	144.646i	0.387791	−	0.387791i	0.873899	+	0.486108i	$0.161584\pi$
$374$		0			0
$375$	234.705			0.625880
$376$		0			0
$377$			165.962i			0.440216i
$378$		0			0
$379$	21.8768	+	21.8768i	0.0577225	+	0.0577225i	0.735379	−	0.677656i	$-0.237004\pi$
$379$	21.8768	+	21.8768i	0.0577225	+	0.0577225i	−0.677656	+	0.735379i	$0.737004\pi$
$380$		0			0
$381$	39.2731	+	39.2731i	0.103079	+	0.103079i
$382$		0			0
$383$		−	344.107i		−	0.898452i	−0.893418	−	0.449226i	$-0.851700\pi$
$383$		−	344.107i		−	0.898452i	0.893418	−	0.449226i	$-0.148300\pi$
$384$		0			0
$385$	−3.29234			−0.00855154
$386$		0			0
$387$	−147.274	+	147.274i	−0.380553	+	0.380553i
$388$		0			0
$389$	245.401	−	245.401i	0.630851	−	0.630851i	−0.317430	−	0.948282i	$-0.602820\pi$
$389$	245.401	−	245.401i	0.630851	−	0.630851i	0.948282	+	0.317430i	$0.102820\pi$
$390$		0			0
$391$	198.585			0.507889
$392$		0			0
$393$		−	73.4923i		−	0.187003i
$394$		0			0
$395$	4.82465	+	4.82465i	0.0122143	+	0.0122143i
$396$		0			0
$397$	160.138	+	160.138i	0.403371	+	0.403371i	0.879419	−	0.476048i	$-0.157931\pi$
$397$	160.138	+	160.138i	0.403371	+	0.403371i	−0.476048	+	0.879419i	$0.657931\pi$
$398$		0			0
$399$			11.5566i			0.0289639i
$400$		0			0
$401$	−344.018			−0.857900			−0.428950	−	0.903328i	$-0.641116\pi$
$401$	−344.018			−0.857900			−0.428950	+	0.903328i	$0.641116\pi$
$402$		0			0
$403$	384.862	−	384.862i	0.954992	−	0.954992i
$404$		0			0
$405$	24.5885	−	24.5885i	0.0607122	−	0.0607122i
$406$		0			0
$407$	2.25904			0.00555048
$408$		0			0
$409$			336.651i			0.823108i	0.911385	+	0.411554i	$0.135014\pi$
$409$			336.651i			0.823108i	−0.911385	+	0.411554i	$0.864986\pi$
$410$		0			0
$411$	−119.450	−	119.450i	−0.290633	−	0.290633i
$412$		0			0
$413$	97.4153	+	97.4153i	0.235872	+	0.235872i
$414$		0			0
$415$		−	320.004i		−	0.771094i
$416$		0			0
$417$	−371.636			−0.891213
$418$		0			0
$419$	269.830	−	269.830i	0.643986	−	0.643986i	−0.307547	−	0.951533i	$-0.599508\pi$
$419$	269.830	−	269.830i	0.643986	−	0.643986i	0.951533	+	0.307547i	$0.0995081\pi$
$420$		0			0
$421$	−30.2436	+	30.2436i	−0.0718374	+	0.0718374i	−0.742113	−	0.670275i	$-0.766176\pi$
$421$	−30.2436	+	30.2436i	−0.0718374	+	0.0718374i	0.670275	+	0.742113i	$0.266176\pi$
$422$		0			0
$423$	−271.366			−0.641527
$424$		0			0
$425$		−	265.818i		−	0.625454i
$426$		0			0
$427$	−124.889	−	124.889i	−0.292481	−	0.292481i
$428$		0			0
$429$	9.09996	+	9.09996i	0.0212120	+	0.0212120i
$430$		0			0
$431$		−	699.038i		−	1.62190i	−0.585116	−	0.810949i	$-0.698951\pi$
$431$		−	699.038i		−	1.62190i	0.585116	−	0.810949i	$-0.301049\pi$
$432$		0			0
$433$	−493.703			−1.14019			−0.570095	−	0.821579i	$-0.693094\pi$
$433$	−493.703			−1.14019			−0.570095	+	0.821579i	$0.693094\pi$
$434$		0			0
$435$	41.4655	−	41.4655i	0.0953229	−	0.0953229i
$436$		0			0
$437$	16.3435	−	16.3435i	0.0373994	−	0.0373994i
$438$		0			0
$439$	−653.276			−1.48810			−0.744050	−	0.668124i	$-0.767098\pi$
$439$	−653.276			−1.48810			−0.744050	+	0.668124i	$0.767098\pi$
$440$		0			0
$441$			132.846i			0.301238i
$442$		0			0
$443$	−92.3971	−	92.3971i	−0.208571	−	0.208571i	0.595089	−	0.803660i	$-0.297117\pi$
$443$	−92.3971	−	92.3971i	−0.208571	−	0.208571i	−0.803660	+	0.595089i	$0.797117\pi$
$444$		0			0
$445$	346.382	+	346.382i	0.778387	+	0.778387i
$446$		0			0
$447$		−	151.740i		−	0.339462i
$448$		0			0
$449$	197.474			0.439809			0.219905	−	0.975521i	$-0.429425\pi$
$449$	197.474			0.439809			0.219905	+	0.975521i	$0.429425\pi$
$450$		0			0
$451$	−6.42931	+	6.42931i	−0.0142557	+	0.0142557i
$452$		0			0
$453$	−58.4500	+	58.4500i	−0.129029	+	0.129029i
$454$		0			0
$455$	158.947			0.349333
$456$		0			0
$457$		−	576.344i		−	1.26115i	−0.776130	−	0.630573i	$-0.782820\pi$
$457$		−	576.344i		−	1.26115i	0.776130	−	0.630573i	$-0.217180\pi$
$458$		0			0
$459$	−96.9715	−	96.9715i	−0.211267	−	0.211267i
$460$		0			0
$461$	386.578	+	386.578i	0.838564	+	0.838564i	0.988670	−	0.150106i	$-0.0479614\pi$
$461$	386.578	+	386.578i	0.838564	+	0.838564i	−0.150106	+	0.988670i	$0.547961\pi$
$462$		0			0
$463$		−	165.924i		−	0.358366i	−0.983816	−	0.179183i	$-0.942655\pi$
$463$		−	165.924i		−	0.358366i	0.983816	−	0.179183i	$-0.0573454\pi$
$464$		0			0
$465$	−192.315			−0.413581
$466$		0			0
$467$	−528.381	+	528.381i	−1.13144	+	1.13144i	−0.141498	+	0.989938i	$0.545192\pi$
$467$	−528.381	+	528.381i	−1.13144	+	1.13144i	−0.989938	+	0.141498i	$0.954808\pi$
$468$		0			0
$469$	−125.723	+	125.723i	−0.268066	+	0.268066i
$470$		0			0
$471$	411.150			0.872930
$472$		0			0
$473$			27.2360i			0.0575814i
$474$		0			0
$475$	−21.8768	−	21.8768i	−0.0460565	−	0.0460565i
$476$		0			0
$477$	−112.550	−	112.550i	−0.235954	−	0.235954i
$478$		0			0
$479$		−	439.298i		−	0.917115i	−0.888665	−	0.458558i	$-0.848366\pi$
$479$		−	439.298i		−	0.917115i	0.888665	−	0.458558i	$-0.151634\pi$
$480$		0			0
$481$	−109.061			−0.226739
$482$		0			0
$483$	−20.0166	+	20.0166i	−0.0414423	+	0.0414423i
$484$		0			0
$485$	−128.210	+	128.210i	−0.264351	+	0.264351i
$486$		0			0
$487$	−722.173			−1.48290			−0.741451	−	0.671007i	$-0.765862\pi$
$487$	−722.173			−1.48290			−0.741451	+	0.671007i	$0.765862\pi$
$488$		0			0
$489$			95.2539i			0.194793i
$490$		0			0
$491$	−104.558	−	104.558i	−0.212948	−	0.212948i	0.592570	−	0.805519i	$-0.298113\pi$
$491$	−104.558	−	104.558i	−0.212948	−	0.212948i	−0.805519	+	0.592570i	$0.798113\pi$
$492$		0			0
$493$	−163.531	−	163.531i	−0.331705	−	0.331705i
$494$		0			0
$495$		−	4.54725i		−	0.00918636i
$496$		0			0
$497$	−101.149			−0.203519
$498$		0			0
$499$	202.483	−	202.483i	0.405777	−	0.405777i	−0.474486	−	0.880263i	$-0.657366\pi$
$499$	202.483	−	202.483i	0.405777	−	0.405777i	0.880263	+	0.474486i	$0.157366\pi$
$500$		0			0
$501$	−293.520	+	293.520i	−0.585869	+	0.585869i
$502$		0			0
$503$	−874.361			−1.73829			−0.869146	−	0.494555i	$-0.835331\pi$
$503$	−874.361			−1.73829			−0.869146	+	0.494555i	$0.835331\pi$
$504$		0			0
$505$		−	365.415i		−	0.723595i
$506$		0			0
$507$	−232.343	−	232.343i	−0.458271	−	0.458271i
$508$		0			0
$509$	542.014	+	542.014i	1.06486	+	1.06486i	0.997745	+	0.0671155i	$0.0213796\pi$
$509$	542.014	+	542.014i	1.06486	+	1.06486i	0.0671155	+	0.997745i	$0.478620\pi$
$510$		0			0
$511$			5.27987i			0.0103324i
$512$		0			0
$513$	−15.9615			−0.0311141
$514$		0			0
$515$	454.936	−	454.936i	0.883371	−	0.883371i
$516$		0			0
$517$	−25.0924	+	25.0924i	−0.0485347	+	0.0485347i
$518$		0			0
$519$	135.473			0.261026
$520$		0			0
$521$		−	41.2999i		−	0.0792704i	−0.999214	−	0.0396352i	$-0.987380\pi$
$521$		−	41.2999i		−	0.0792704i	0.999214	−	0.0396352i	$-0.0126196\pi$
$522$		0			0
$523$	93.2910	+	93.2910i	0.178377	+	0.178377i	0.790648	−	0.612271i	$-0.209744\pi$
$523$	93.2910	+	93.2910i	0.178377	+	0.178377i	−0.612271	+	0.790648i	$0.709744\pi$
$524$		0			0
$525$	26.7935	+	26.7935i	0.0510353	+	0.0510353i
$526$		0			0
$527$			758.450i			1.43918i
$528$		0			0
$529$	−472.384			−0.892976
$530$		0			0
$531$	−134.546	+	134.546i	−0.253382	+	0.253382i
$532$		0			0
$533$	310.392	−	310.392i	0.582350	−	0.582350i
$534$		0			0
$535$	706.780			1.32108
$536$		0			0
$537$		−	450.200i		−	0.838361i
$538$		0			0
$539$	12.2839	+	12.2839i	0.0227902	+	0.0227902i
$540$		0			0
$541$	46.6640	+	46.6640i	0.0862551	+	0.0862551i	0.748918	−	0.662663i	$-0.230574\pi$
$541$	46.6640	+	46.6640i	0.0862551	+	0.0862551i	−0.662663	+	0.748918i	$0.730574\pi$
$542$		0			0
$543$		−	37.1883i		−	0.0684868i
$544$		0			0
$545$	−537.233			−0.985749
$546$		0			0
$547$	−371.423	+	371.423i	−0.679019	+	0.679019i	−0.959778	−	0.280760i	$-0.909414\pi$
$547$	−371.423	+	371.423i	−0.679019	+	0.679019i	0.280760	+	0.959778i	$0.409414\pi$
$548$		0			0
$549$	172.492	−	172.492i	0.314194	−	0.314194i
$550$		0			0
$551$	−26.9172			−0.0488515
$552$		0			0
$553$			3.83579i			0.00693632i
$554$		0			0
$555$	27.2490	+	27.2490i	0.0490973	+	0.0490973i
$556$		0			0
$557$	537.717	+	537.717i	0.965380	+	0.965380i	0.999420	−	0.0340406i	$-0.0108375\pi$
$557$	537.717	+	537.717i	0.965380	+	0.965380i	−0.0340406	+	0.999420i	$0.510838\pi$
$558$		0			0
$559$		−	1314.89i		−	2.35222i
$560$		0			0
$561$	−17.9334			−0.0319668
$562$		0			0
$563$	272.321	−	272.321i	0.483697	−	0.483697i	−0.422613	−	0.906310i	$-0.638887\pi$
$563$	272.321	−	272.321i	0.483697	−	0.483697i	0.906310	+	0.422613i	$0.138887\pi$
$564$		0			0
$565$	579.167	−	579.167i	1.02507	−	1.02507i
$566$		0			0
$567$	19.5488			0.0344776
$568$		0			0
$569$			304.018i			0.534302i	0.963655	+	0.267151i	$0.0860823\pi$
$569$			304.018i			0.534302i	−0.963655	+	0.267151i	$0.913918\pi$
$570$		0			0
$571$	−326.012	−	326.012i	−0.570950	−	0.570950i	0.361444	−	0.932394i	$-0.382284\pi$
$571$	−326.012	−	326.012i	−0.570950	−	0.570950i	−0.932394	+	0.361444i	$0.882284\pi$
$572$		0			0
$573$	243.167	+	243.167i	0.424375	+	0.424375i
$574$		0			0
$575$		−	75.7836i		−	0.131798i
$576$		0			0
$577$	−1074.26			−1.86180			−0.930898	−	0.365279i	$-0.880973\pi$
$577$	−1074.26			−1.86180			−0.930898	+	0.365279i	$0.880973\pi$
$578$		0			0
$579$	−299.880	+	299.880i	−0.517928	+	0.517928i
$580$		0			0
$581$	127.208	−	127.208i	0.218946	−	0.218946i
$582$		0			0
$583$	−20.8143			−0.0357021
$584$		0			0
$585$			219.531i			0.375266i
$586$		0			0
$587$	−413.585	−	413.585i	−0.704574	−	0.704574i	0.260815	−	0.965389i	$-0.416009\pi$
$587$	−413.585	−	413.585i	−0.704574	−	0.704574i	−0.965389	+	0.260815i	$0.916009\pi$
$588$		0			0
$589$	62.4205	+	62.4205i	0.105977	+	0.105977i
$590$		0			0
$591$		−	224.332i		−	0.379581i
$592$		0			0
$593$	312.297			0.526640			0.263320	−	0.964709i	$-0.415183\pi$
$593$	312.297			0.526640			0.263320	+	0.964709i	$0.415183\pi$
$594$		0			0
$595$	−156.619	+	156.619i	−0.263225	+	0.263225i
$596$		0			0
$597$	240.822	−	240.822i	0.403386	−	0.403386i
$598$		0			0
$599$	−466.651			−0.779049			−0.389525	−	0.921016i	$-0.627361\pi$
$599$	−466.651			−0.779049			−0.389525	+	0.921016i	$0.627361\pi$
$600$		0			0
$601$		−	991.272i		−	1.64937i	−0.565592	−	0.824685i	$-0.691352\pi$
$601$		−	991.272i		−	1.64937i	0.565592	−	0.824685i	$-0.308648\pi$
$602$		0			0
$603$	−173.644	−	173.644i	−0.287966	−	0.287966i
$604$		0			0
$605$	330.158	+	330.158i	0.545715	+	0.545715i
$606$		0			0
$607$			458.347i			0.755101i	0.925989	+	0.377551i	$0.123234\pi$
$607$			458.347i			0.755101i	−0.925989	+	0.377551i	$0.876766\pi$
$608$		0			0
$609$	32.9667			0.0541325
$610$		0			0
$611$	1211.40	−	1211.40i	1.98266	−	1.98266i
$612$		0			0
$613$	−43.7283	+	43.7283i	−0.0713349	+	0.0713349i	−0.741874	−	0.670539i	$-0.766063\pi$
$613$	−43.7283	+	43.7283i	−0.0713349	+	0.0713349i	0.670539	+	0.741874i	$0.266063\pi$
$614$		0			0
$615$	−155.103			−0.252200
$616$		0			0
$617$			281.559i			0.456335i	0.973622	+	0.228168i	$0.0732734\pi$
$617$			281.559i			0.456335i	−0.973622	+	0.228168i	$0.926727\pi$
$618$		0			0
$619$	−418.959	−	418.959i	−0.676832	−	0.676832i	0.282450	−	0.959282i	$-0.408853\pi$
$619$	−418.959	−	418.959i	−0.676832	−	0.676832i	−0.959282	+	0.282450i	$0.908853\pi$
$620$		0			0
$621$	−27.6462	−	27.6462i	−0.0445188	−	0.0445188i
$622$		0			0
$623$			275.387i			0.442034i
$624$		0			0
$625$	271.764			0.434822
$626$		0			0
$627$	−1.47592	+	1.47592i	−0.00235393	+	0.00235393i
$628$		0			0
$629$	107.464	−	107.464i	0.170849	−	0.170849i
$630$		0			0
$631$	279.670			0.443217			0.221608	−	0.975136i	$-0.428869\pi$
$631$	279.670			0.443217			0.221608	+	0.975136i	$0.428869\pi$
$632$		0			0
$633$			196.477i			0.310390i
$634$		0			0
$635$	87.6069	+	87.6069i	0.137964	+	0.137964i
$636$		0			0
$637$	−593.038	−	593.038i	−0.930987	−	0.930987i
$638$		0			0
$639$		−	139.703i		−	0.218627i
$640$		0			0
$641$	428.823			0.668991			0.334495	−	0.942397i	$-0.391434\pi$
$641$	428.823			0.668991			0.334495	+	0.942397i	$0.391434\pi$
$642$		0			0
$643$	−439.414	+	439.414i	−0.683381	+	0.683381i	−0.960761	−	0.277379i	$-0.910534\pi$
$643$	−439.414	+	439.414i	−0.683381	+	0.683381i	0.277379	+	0.960761i	$0.410534\pi$
$644$		0			0
$645$	−328.526	+	328.526i	−0.509342	+	0.509342i
$646$		0			0
$647$	29.2598			0.0452238			0.0226119	−	0.999744i	$-0.492802\pi$
$647$	29.2598			0.0452238			0.0226119	+	0.999744i	$0.492802\pi$
$648$		0			0
$649$			24.8822i			0.0383393i
$650$		0			0
$651$	−76.4492	−	76.4492i	−0.117433	−	0.117433i
$652$		0			0
$653$	−160.694	−	160.694i	−0.246085	−	0.246085i	0.573277	−	0.819362i	$-0.305672\pi$
$653$	−160.694	−	160.694i	−0.246085	−	0.246085i	−0.819362	+	0.573277i	$0.805672\pi$
$654$		0			0
$655$		−	163.940i		−	0.250290i
$656$		0			0
$657$	−7.29234			−0.0110995
$658$		0			0
$659$	−101.932	+	101.932i	−0.154677	+	0.154677i	−0.780203	−	0.625526i	$-0.784884\pi$
$659$	−101.932	+	101.932i	−0.154677	+	0.154677i	0.625526	+	0.780203i	$0.284884\pi$
$660$		0			0
$661$	−610.251	+	610.251i	−0.923224	+	0.923224i	−0.997256	−	0.0740318i	$-0.976413\pi$
$661$	−610.251	+	610.251i	−0.923224	+	0.923224i	0.0740318	+	0.997256i	$0.476413\pi$
$662$		0			0
$663$	865.781			1.30585
$664$		0			0
$665$			25.7795i			0.0387661i
$666$		0			0
$667$	−46.6219	−	46.6219i	−0.0698979	−	0.0698979i
$668$		0			0
$669$	−421.550	−	421.550i	−0.630120	−	0.630120i
$670$		0			0
$671$		−	31.8997i		−	0.0475406i
$672$		0			0
$673$	241.261			0.358486			0.179243	−	0.983805i	$-0.442635\pi$
$673$	241.261			0.358486			0.179243	+	0.983805i	$0.442635\pi$
$674$		0			0
$675$	−37.0061	+	37.0061i	−0.0548239	+	0.0548239i
$676$		0			0
$677$	270.922	−	270.922i	0.400180	−	0.400180i	−0.478117	−	0.878296i	$-0.658680\pi$
$677$	270.922	−	270.922i	0.400180	−	0.400180i	0.878296	+	0.478117i	$0.158680\pi$
$678$		0			0
$679$	−101.932			−0.150121
$680$		0			0
$681$		−	701.520i		−	1.03013i
$682$		0			0
$683$	−363.629	−	363.629i	−0.532399	−	0.532399i	0.388886	−	0.921286i	$-0.372860\pi$
$683$	−363.629	−	363.629i	−0.532399	−	0.532399i	−0.921286	+	0.388886i	$0.872860\pi$
$684$		0			0
$685$	−266.459	−	266.459i	−0.388991	−	0.388991i
$686$		0			0
$687$		−	299.434i		−	0.435858i
$688$		0			0
$689$	1004.87			1.45845
$690$		0			0
$691$	709.086	−	709.086i	1.02617	−	1.02617i	0.0265262	−	0.999648i	$-0.491555\pi$
$691$	709.086	−	709.086i	1.02617	−	1.02617i	0.999648	−	0.0265262i	$-0.00844455\pi$
$692$		0			0
$693$	1.80762	−	1.80762i	0.00260840	−	0.00260840i
$694$		0			0
$695$	−829.012			−1.19282
$696$		0			0
$697$			611.692i			0.877607i
$698$		0			0
$699$	−83.0125	−	83.0125i	−0.118759	−	0.118759i
$700$		0			0
$701$	116.263	+	116.263i	0.165853	+	0.165853i	0.785154	−	0.619301i	$-0.212584\pi$
$701$	116.263	+	116.263i	0.165853	+	0.165853i	−0.619301	+	0.785154i	$0.712584\pi$
$702$		0			0
$703$		−	17.6886i		−	0.0251616i
$704$		0			0
$705$	−605.338			−0.858636
$706$		0			0
$707$	145.260	−	145.260i	0.205459	−	0.205459i
$708$		0			0
$709$	692.751	−	692.751i	0.977082	−	0.977082i	−0.0226610	−	0.999743i	$-0.507214\pi$
$709$	692.751	−	692.751i	0.977082	−	0.977082i	0.999743	+	0.0226610i	$0.00721384\pi$
$710$		0			0
$711$	−5.29783			−0.00745124
$712$		0			0
$713$			216.231i			0.303269i
$714$		0			0
$715$	20.2994	+	20.2994i	0.0283907	+	0.0283907i
$716$		0			0
$717$	−379.377	−	379.377i	−0.529117	−	0.529117i
$718$		0			0
$719$			1036.41i			1.44146i	0.693217	+	0.720729i	$0.256193\pi$
$719$			1036.41i			1.44146i	−0.693217	+	0.720729i	$0.743807\pi$
$720$		0			0
$721$	361.692			0.501654
$722$		0			0
$723$	−321.492	+	321.492i	−0.444664	+	0.444664i
$724$		0			0
$725$	−62.4064	+	62.4064i	−0.0860778	+	0.0860778i
$726$		0			0
$727$	868.875			1.19515			0.597575	−	0.801813i	$-0.296131\pi$
$727$	868.875			1.19515			0.597575	+	0.801813i	$0.296131\pi$
$728$		0			0
$729$			27.0000i			0.0370370i
$730$		0			0
$731$	1295.63	+	1295.63i	1.77241	+	1.77241i
$732$		0			0
$733$	240.685	+	240.685i	0.328356	+	0.328356i	0.851961	−	0.523605i	$-0.175413\pi$
$733$	240.685	+	240.685i	0.328356	+	0.328356i	−0.523605	+	0.851961i	$0.675413\pi$
$734$		0			0
$735$			296.341i			0.403185i
$736$		0			0
$737$	−32.1127			−0.0435721
$738$		0			0
$739$	38.9489	−	38.9489i	0.0527049	−	0.0527049i	−0.680263	−	0.732968i	$-0.738134\pi$
$739$	38.9489	−	38.9489i	0.0527049	−	0.0527049i	0.732968	+	0.680263i	$0.238134\pi$
$740$		0			0
$741$	71.2539	−	71.2539i	0.0961591	−	0.0961591i
$742$		0			0
$743$	−312.791			−0.420984			−0.210492	−	0.977596i	$-0.567507\pi$
$743$	−312.791			−0.420984			−0.210492	+	0.977596i	$0.567507\pi$
$744$		0			0
$745$		−	338.487i		−	0.454345i
$746$		0			0
$747$	175.694	+	175.694i	0.235200	+	0.235200i
$748$		0			0
$749$	280.959	+	280.959i	0.375112	+	0.375112i
$750$		0			0
$751$		−	495.790i		−	0.660174i	−0.943951	−	0.330087i	$-0.892922\pi$
$751$		−	495.790i		−	0.660174i	0.943951	−	0.330087i	$-0.107078\pi$
$752$		0			0
$753$	−543.895			−0.722304
$754$		0			0
$755$	−130.385	+	130.385i	−0.172695	+	0.172695i
$756$		0			0
$757$	−610.664	+	610.664i	−0.806690	+	0.806690i	−0.984131	−	0.177442i	$-0.943218\pi$
$757$	−610.664	+	610.664i	−0.806690	+	0.806690i	0.177442	+	0.984131i	$0.443218\pi$
$758$		0			0
$759$	−5.11272			−0.00673613
$760$		0			0
$761$		−	336.879i		−	0.442680i	−0.975197	−	0.221340i	$-0.928957\pi$
$761$		−	336.879i		−	0.442680i	0.975197	−	0.221340i	$-0.0710430\pi$
$762$		0			0
$763$	−213.561	−	213.561i	−0.279896	−	0.279896i
$764$		0			0
$765$	−216.315	−	216.315i	−0.282765	−	0.282765i
$766$		0			0
$767$		−	1201.25i		−	1.56617i
$768$		0			0
$769$	−633.979			−0.824421			−0.412210	−	0.911089i	$-0.635243\pi$
$769$	−633.979			−0.824421			−0.412210	+	0.911089i	$0.635243\pi$
$770$		0			0
$771$	89.5069	−	89.5069i	0.116092	−	0.116092i
$772$		0			0
$773$	902.729	−	902.729i	1.16783	−	1.16783i	0.185107	−	0.982718i	$-0.440737\pi$
$773$	902.729	−	902.729i	1.16783	−	1.16783i	0.982718	−	0.185107i	$-0.0592633\pi$
$774$		0			0
$775$	289.439			0.373469
$776$		0			0
$777$			21.6640i			0.0278816i
$778$		0			0
$779$	50.3423	+	50.3423i	0.0646243	+	0.0646243i
$780$		0			0
$781$	−12.9179	−	12.9179i	−0.0165402	−	0.0165402i
$782$		0			0
$783$			45.5322i			0.0581510i
$784$		0			0
$785$	917.156			1.16835
$786$		0			0
$787$	−587.548	+	587.548i	−0.746566	+	0.746566i	−0.973833	−	0.227266i	$-0.927021\pi$
$787$	−587.548	+	587.548i	−0.746566	+	0.746566i	0.227266	+	0.973833i	$0.427021\pi$
$788$		0			0
$789$	−443.405	+	443.405i	−0.561984	+	0.561984i
$790$		0			0
$791$	460.460			0.582124
$792$		0			0
$793$			1540.05i			1.94205i
$794$		0			0
$795$	−251.066	−	251.066i	−0.315807	−	0.315807i
$796$		0			0
$797$	236.322	+	236.322i	0.296514	+	0.296514i	0.839647	−	0.543133i	$-0.182762\pi$
$797$	236.322	+	236.322i	0.296514	+	0.296514i	−0.543133	+	0.839647i	$0.682762\pi$
$798$		0			0
$799$			2387.32i			2.98789i
$800$		0			0
$801$	−380.354			−0.474849
$802$		0			0
$803$	−0.674302	+	0.674302i	−0.000839729	+	0.000839729i
$804$		0			0
$805$	−44.6513	+	44.6513i	−0.0554675	+	0.0554675i
$806$		0			0
$807$	289.332			0.358528
$808$		0			0
$809$		−	196.382i		−	0.242747i	−0.992607	−	0.121373i	$-0.961270\pi$
$809$		−	196.382i		−	0.242747i	0.992607	−	0.121373i	$-0.0387298\pi$
$810$		0			0
$811$	910.895	+	910.895i	1.12317	+	1.12317i	0.991261	+	0.131914i	$0.0421122\pi$
$811$	910.895	+	910.895i	1.12317	+	1.12317i	0.131914	+	0.991261i	$0.457888\pi$
$812$		0			0
$813$	−408.573	−	408.573i	−0.502550	−	0.502550i
$814$		0			0
$815$			212.484i			0.260716i
$816$		0			0
$817$	213.261			0.261030
$818$		0			0
$819$	−87.2678	+	87.2678i	−0.106554	+	0.106554i
$820$		0			0
$821$	−1003.54	+	1003.54i	−1.22234	+	1.22234i	−0.255544	+	0.966798i	$0.582254\pi$
$821$	−1003.54	+	1003.54i	−1.22234	+	1.22234i	−0.966798	+	0.255544i	$0.917746\pi$
$822$		0			0
$823$	−554.187			−0.673374			−0.336687	−	0.941617i	$-0.609306\pi$
$823$	−554.187			−0.673374			−0.336687	+	0.941617i	$0.609306\pi$
$824$		0			0
$825$			6.84370i			0.00829540i
$826$		0			0
$827$	647.789	+	647.789i	0.783300	+	0.783300i	0.980386	−	0.197086i	$-0.0631478\pi$
$827$	647.789	+	647.789i	0.783300	+	0.783300i	−0.197086	+	0.980386i	$0.563148\pi$
$828$		0			0
$829$	33.2384	+	33.2384i	0.0400946	+	0.0400946i	0.726870	−	0.686775i	$-0.240974\pi$
$829$	33.2384	+	33.2384i	0.0400946	+	0.0400946i	−0.686775	+	0.726870i	$0.740974\pi$
$830$		0			0
$831$			394.789i			0.475077i
$832$		0			0
$833$	1168.70			1.40301
$834$		0			0
$835$	−654.759	+	654.759i	−0.784143	+	0.784143i
$836$		0			0
$837$	105.588	−	105.588i	0.126151	−	0.126151i
$838$		0			0
$839$	−904.408			−1.07796			−0.538980	−	0.842319i	$-0.681190\pi$
$839$	−904.408			−1.07796			−0.538980	+	0.842319i	$0.681190\pi$
$840$		0			0
$841$		−	764.215i		−	0.908698i
$842$		0			0
$843$	555.400	+	555.400i	0.658837	+	0.658837i
$844$		0			0
$845$	−518.291	−	518.291i	−0.613362	−	0.613362i
$846$		0			0
$847$			262.488i			0.309904i
$848$		0			0
$849$	177.549			0.209127
$850$		0			0
$851$	30.6376	−	30.6376i	0.0360018	−	0.0360018i
$852$		0			0
$853$	−576.846	+	576.846i	−0.676256	+	0.676256i	−0.959151	−	0.282895i	$-0.908705\pi$
$853$	−576.846	+	576.846i	−0.676256	+	0.676256i	0.282895	+	0.959151i	$0.408705\pi$
$854$		0			0
$855$	−35.6055			−0.0416439
$856$		0			0
$857$			1425.60i			1.66348i	0.555166	+	0.831740i	$0.312655\pi$
$857$			1425.60i			1.66348i	−0.555166	+	0.831740i	$0.687345\pi$
$858$		0			0
$859$	193.044	+	193.044i	0.224731	+	0.224731i	0.810487	−	0.585756i	$-0.199202\pi$
$859$	193.044	+	193.044i	0.224731	+	0.224731i	−0.585756	+	0.810487i	$0.699202\pi$
$860$		0			0
$861$	−61.6565	−	61.6565i	−0.0716103	−	0.0716103i
$862$		0			0
$863$			1139.04i			1.31986i	0.751327	+	0.659930i	$0.229414\pi$
$863$			1139.04i			1.31986i	−0.751327	+	0.659930i	$0.770586\pi$
$864$		0			0
$865$	302.200			0.349364
$866$		0			0
$867$	−499.149	+	499.149i	−0.575720	+	0.575720i
$868$		0			0
$869$	−0.489875	+	0.489875i	−0.000563723	+	0.000563723i
$870$		0			0
$871$	1550.33			1.77994
$872$		0			0
$873$		−	140.785i		−	0.161265i
$874$		0			0
$875$	208.125	+	208.125i	0.237857	+	0.237857i
$876$		0			0
$877$	−1012.96	−	1012.96i	−1.15503	−	1.15503i	−0.985531	−	0.169496i	$-0.945786\pi$
$877$	−1012.96	−	1012.96i	−1.15503	−	1.15503i	−0.169496	−	0.985531i	$-0.554214\pi$
$878$		0			0
$879$		−	804.145i		−	0.914841i
$880$		0			0
$881$	1437.75			1.63195			0.815975	−	0.578087i	$-0.196200\pi$
$881$	1437.75			1.63195			0.815975	+	0.578087i	$0.196200\pi$
$882$		0			0
$883$	−902.217	+	902.217i	−1.02176	+	1.02176i	−0.0220057	+	0.999758i	$0.507005\pi$
$883$	−902.217	+	902.217i	−1.02176	+	1.02176i	−0.999758	+	0.0220057i	$0.992995\pi$
$884$		0			0
$885$	−300.133	+	300.133i	−0.339134	+	0.339134i
$886$		0			0
$887$	−1389.64			−1.56667			−0.783334	−	0.621601i	$-0.786483\pi$
$887$	−1389.64			−1.56667			−0.783334	+	0.621601i	$0.786483\pi$
$888$		0			0
$889$			69.6510i			0.0783475i
$890$		0			0
$891$	2.49661	+	2.49661i	0.00280203	+	0.00280203i
$892$		0			0
$893$	196.477	+	196.477i	0.220019	+	0.220019i
$894$		0			0
$895$		−	1004.27i		−	1.12208i
$896$		0			0
$897$	246.831			0.275174
$898$		0			0
$899$	178.062	−	178.062i	0.198067	−	0.198067i
$900$		0			0
$901$	−990.151	+	990.151i	−1.09895	+	1.09895i
$902$		0			0
$903$	−261.191			−0.289248
$904$		0			0
$905$		−	82.9564i		−	0.0916645i
$906$		0			0
$907$	−767.091	−	767.091i	−0.845745	−	0.845745i	0.143854	−	0.989599i	$-0.454051\pi$
$907$	−767.091	−	767.091i	−0.845745	−	0.845745i	−0.989599	+	0.143854i	$0.954051\pi$
$908$		0			0
$909$	200.627	+	200.627i	0.220712	+	0.220712i
$910$		0			0
$911$			1531.36i			1.68097i	0.541834	+	0.840485i	$0.317730\pi$
$911$			1531.36i			1.68097i	−0.541834	+	0.840485i	$0.682270\pi$
$912$		0			0
$913$	32.4919			0.0355881
$914$		0			0
$915$	384.780	−	384.780i	0.420525	−	0.420525i
$916$		0			0
$917$	65.1694	−	65.1694i	0.0710680	−	0.0710680i
$918$		0			0
$919$	1083.73			1.17925			0.589623	−	0.807679i	$-0.299277\pi$
$919$	1083.73			1.17925			0.589623	+	0.807679i	$0.299277\pi$
$920$		0			0
$921$		−	339.482i		−	0.368601i
$922$		0			0
$923$	623.646	+	623.646i	0.675673	+	0.675673i
$924$		0			0
$925$	−41.0103	−	41.0103i	−0.0443355	−	0.0443355i
$926$		0			0
$927$			499.555i			0.538894i
$928$		0			0
$929$	−878.008			−0.945110			−0.472555	−	0.881301i	$-0.656668\pi$
$929$	−878.008			−0.945110			−0.472555	+	0.881301i	$0.656668\pi$
$930$		0			0
$931$	96.1845	−	96.1845i	0.103313	−	0.103313i
$932$		0			0
$933$	382.774	−	382.774i	0.410262	−	0.410262i
$934$		0			0
$935$	−40.0041			−0.0427852
$936$		0			0
$937$			1216.28i			1.29805i	0.760765	+	0.649027i	$0.224824\pi$
$937$			1216.28i			1.29805i	−0.760765	+	0.649027i	$0.775176\pi$
$938$		0			0
$939$	337.150	+	337.150i	0.359052	+	0.359052i
$940$		0			0
$941$	92.7424	+	92.7424i	0.0985572	+	0.0985572i	0.754666	−	0.656109i	$-0.227799\pi$
$941$	92.7424	+	92.7424i	0.0985572	+	0.0985572i	−0.656109	+	0.754666i	$0.727799\pi$
$942$		0			0
$943$			174.391i			0.184932i
$944$		0			0
$945$	43.6077			0.0461457
$946$		0			0
$947$	325.918	−	325.918i	0.344159	−	0.344159i	−0.513770	−	0.857928i	$-0.671751\pi$
$947$	325.918	−	325.918i	0.344159	−	0.344159i	0.857928	+	0.513770i	$0.171751\pi$
$948$		0			0
$949$	32.5538	−	32.5538i	0.0343032	−	0.0343032i
$950$		0			0
$951$	637.511			0.670359
$952$		0			0
$953$			172.172i			0.180663i	0.995912	+	0.0903315i	$0.0287926\pi$
$953$			172.172i			0.180663i	−0.995912	+	0.0903315i	$0.971207\pi$
$954$		0			0
$955$	542.434	+	542.434i	0.567994	+	0.567994i
$956$		0			0
$957$	4.21024	+	4.21024i	0.00439941	+	0.00439941i
$958$		0			0
$959$		−	211.845i		−	0.220902i
$960$		0			0
$961$	135.154			0.140639
$962$		0			0
$963$	−388.049	+	388.049i	−0.402959	+	0.402959i
$964$		0			0
$965$	−668.946	+	668.946i	−0.693208	+	0.693208i
$966$		0			0
$967$	−124.969			−0.129234			−0.0646169	−	0.997910i	$-0.520583\pi$
$967$	−124.969			−0.129234			−0.0646169	+	0.997910i	$0.520583\pi$
$968$		0			0
$969$			140.420i			0.144913i
$970$		0			0
$971$	−63.3695	−	63.3695i	−0.0652621	−	0.0652621i	0.673722	−	0.738985i	$-0.264694\pi$
$971$	−63.3695	−	63.3695i	−0.0652621	−	0.0652621i	−0.738985	+	0.673722i	$0.764694\pi$
$972$		0			0
$973$	−329.549	−	329.549i	−0.338693	−	0.338693i
$974$		0			0
$975$		−	330.398i		−	0.338870i
$976$		0			0
$977$	192.192			0.196717			0.0983584	−	0.995151i	$-0.468641\pi$
$977$	192.192			0.196717			0.0983584	+	0.995151i	$0.468641\pi$
$978$		0			0
$979$	−35.1702	+	35.1702i	−0.0359246	+	0.0359246i
$980$		0			0
$981$	294.962	−	294.962i	0.300674	−	0.300674i
$982$		0			0
$983$	613.021			0.623623			0.311812	−	0.950144i	$-0.399064\pi$
$983$	613.021			0.623623			0.311812	+	0.950144i	$0.399064\pi$
$984$		0			0
$985$		−	500.420i		−	0.508041i
$986$		0			0
$987$	−240.634	−	240.634i	−0.243803	−	0.243803i
$988$		0			0
$989$	369.380	+	369.380i	0.373488	+	0.373488i
$990$		0			0
$991$		−	954.148i		−	0.962813i	−0.876497	−	0.481407i	$-0.840126\pi$
$991$		−	954.148i		−	0.962813i	0.876497	−	0.481407i	$-0.159874\pi$
$992$		0			0
$993$	259.061			0.260888
$994$		0			0
$995$	537.203	−	537.203i	0.539903	−	0.539903i
$996$		0			0
$997$	1384.92	−	1384.92i	1.38909	−	1.38909i	0.561854	−	0.827237i	$-0.310088\pi$
$997$	1384.92	−	1384.92i	1.38909	−	1.38909i	0.827237	−	0.561854i	$-0.189912\pi$
$998$		0			0
$999$	−29.9215			−0.0299514

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.l.b.703.1	yes	8
4.3	odd	2	inner	768.3.l.b.703.3	yes	8
8.3	odd	2		768.3.l.c.703.2	yes	8
8.5	even	2		768.3.l.c.703.4	yes	8
16.3	odd	4		768.3.l.c.319.4	yes	8
16.5	even	4	inner	768.3.l.b.319.3	yes	8
16.11	odd	4	inner	768.3.l.b.319.1	✓	8
16.13	even	4		768.3.l.c.319.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.3.l.b.319.1	✓	8	16.11	odd	4	inner
768.3.l.b.319.3	yes	8	16.5	even	4	inner
768.3.l.b.703.1	yes	8	1.1	even	1	trivial
768.3.l.b.703.3	yes	8	4.3	odd	2	inner
768.3.l.c.319.2	yes	8	16.13	even	4
768.3.l.c.319.4	yes	8	16.3	odd	4
768.3.l.c.703.2	yes	8	8.3	odd	2
768.3.l.c.703.4	yes	8	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 + 1.22474i) q^{3} +(-2.73205 + 2.73205i) q^{5} -2.17209 q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 + 1.22474i) q^{3} +(-2.73205 + 2.73205i) q^{5} -2.17209 q^{7} -3.00000i q^{9} +(-0.277401 - 0.277401i) q^{11} +(13.3923 + 13.3923i) q^{13} -6.69213i q^{15} -26.3923 q^{17} +(-2.17209 + 2.17209i) q^{19} +(2.66025 - 2.66025i) q^{21} -7.52433 q^{23} +10.0718i q^{25} +(3.67423 + 3.67423i) q^{27} +(6.19615 + 6.19615i) q^{29} -28.7375i q^{31} +0.679492 q^{33} +(5.93426 - 5.93426i) q^{35} +(-4.07180 + 4.07180i) q^{37} -32.8043 q^{39} -23.1769i q^{41} +(-49.0913 - 49.0913i) q^{43} +(8.19615 + 8.19615i) q^{45} -90.4553i q^{47} -44.2820 q^{49} +(32.3238 - 32.3238i) q^{51} +(37.5167 - 37.5167i) q^{53} +1.51575 q^{55} -5.32051i q^{57} +(-44.8487 - 44.8487i) q^{59} +(57.4974 + 57.4974i) q^{61} +6.51626i q^{63} -73.1769 q^{65} +(57.8812 - 57.8812i) q^{67} +(9.21539 - 9.21539i) q^{69} +46.5675 q^{71} -2.43078i q^{73} +(-12.3354 - 12.3354i) q^{75} +(0.602540 + 0.602540i) q^{77} -1.76594i q^{79} -9.00000 q^{81} +(-58.5648 + 58.5648i) q^{83} +(72.1051 - 72.1051i) q^{85} -15.1774 q^{87} -126.785i q^{89} +(-29.0893 - 29.0893i) q^{91} +(35.1962 + 35.1962i) q^{93} -11.8685i q^{95} +46.9282 q^{97} +(-0.832204 + 0.832204i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} + 24 q^{13} - 128 q^{17} - 48 q^{21} + 8 q^{29} + 144 q^{33} - 88 q^{37} + 24 q^{45} + 200 q^{49} + 120 q^{53} + 72 q^{61} - 336 q^{65} + 240 q^{69} - 688 q^{77} - 72 q^{81} + 272 q^{85} + 240 q^{93} + 320 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 + 24 * q^13 - 128 * q^17 - 48 * q^21 + 8 * q^29 + 144 * q^33 - 88 * q^37 + 24 * q^45 + 200 * q^49 + 120 * q^53 + 72 * q^61 - 336 * q^65 + 240 * q^69 - 688 * q^77 - 72 * q^81 + 272 * q^85 + 240 * q^93 + 320 * q^97

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