LMFDB - Embedded newform 768.3.l.b.319.3

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			319.3
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	768.319
Dual form			768.3.l.b.703.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +(-2.73205 - 2.73205i) q^{5} +2.17209 q^{7} +3.00000i q^{9} +O(q^{10})$	\(q+(1.22474 + 1.22474i) q^{3} +(-2.73205 - 2.73205i) q^{5} +2.17209 q^{7} +3.00000i q^{9} +(0.277401 - 0.277401i) q^{11} +(13.3923 - 13.3923i) q^{13} -6.69213i q^{15} -26.3923 q^{17} +(2.17209 + 2.17209i) q^{19} +(2.66025 + 2.66025i) q^{21} +7.52433 q^{23} -10.0718i q^{25} +(-3.67423 + 3.67423i) q^{27} +(6.19615 - 6.19615i) q^{29} -28.7375i q^{31} +0.679492 q^{33} +(-5.93426 - 5.93426i) q^{35} +(-4.07180 - 4.07180i) q^{37} +32.8043 q^{39} +23.1769i q^{41} +(49.0913 - 49.0913i) q^{43} +(8.19615 - 8.19615i) q^{45} -90.4553i q^{47} -44.2820 q^{49} +(-32.3238 - 32.3238i) q^{51} +(37.5167 + 37.5167i) q^{53} -1.51575 q^{55} +5.32051i q^{57} +(44.8487 - 44.8487i) q^{59} +(57.4974 - 57.4974i) q^{61} +6.51626i q^{63} -73.1769 q^{65} +(-57.8812 - 57.8812i) q^{67} +(9.21539 + 9.21539i) q^{69} -46.5675 q^{71} +2.43078i q^{73} +(12.3354 - 12.3354i) q^{75} +(0.602540 - 0.602540i) q^{77} -1.76594i q^{79} -9.00000 q^{81} +(58.5648 + 58.5648i) q^{83} +(72.1051 + 72.1051i) q^{85} +15.1774 q^{87} +126.785i q^{89} +(29.0893 - 29.0893i) q^{91} +(35.1962 - 35.1962i) q^{93} -11.8685i q^{95} +46.9282 q^{97} +(0.832204 + 0.832204i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

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

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	0.378990	−	0.925401i	$-0.376271\pi$
$5$	−2.73205	−	2.73205i	−0.546410	−	0.546410i	−0.925401	+	0.378990i	$0.876271\pi$
$6$		0			0
$7$	2.17209			0.310298			0.155149	−	0.987891i	$-0.450414\pi$
$7$	2.17209			0.310298			0.155149	+	0.987891i	$0.450414\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	0.277401	−	0.277401i	0.0252183	−	0.0252183i	−0.694385	−	0.719604i	$-0.744324\pi$
$11$	0.277401	−	0.277401i	0.0252183	−	0.0252183i	0.719604	+	0.694385i	$0.244324\pi$
$12$		0			0
$13$	13.3923	−	13.3923i	1.03018	−	1.03018i	0.0306470	−	0.999530i	$-0.490243\pi$
$13$	13.3923	−	13.3923i	1.03018	−	1.03018i	0.999530	−	0.0306470i	$-0.00975678\pi$
$14$		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	$-0.224241\pi$
$19$	2.17209	+	2.17209i	0.114320	+	0.114320i	−0.647632	+	0.761953i	$0.724241\pi$
$20$		0			0
$21$	2.66025	+	2.66025i	0.126679	+	0.126679i
$22$		0			0
$23$	7.52433			0.327145			0.163572	−	0.986531i	$-0.447698\pi$
$23$	7.52433			0.327145			0.163572	+	0.986531i	$0.447698\pi$
$24$		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	−0.592160	−	0.805820i	$-0.701725\pi$
$29$	6.19615	−	6.19615i	0.213660	−	0.213660i	0.805820	+	0.592160i	$0.201725\pi$
$30$		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	0.649938	−	0.759987i	$-0.274795\pi$
$37$	−4.07180	−	4.07180i	−0.110049	−	0.110049i	−0.759987	+	0.649938i	$0.774795\pi$
$38$		0			0
$39$	32.8043			0.841136
$40$		0			0
$41$			23.1769i			0.565291i	0.959224	+	0.282645i	$0.0912119\pi$
$41$			23.1769i			0.565291i	−0.959224	+	0.282645i	$0.908788\pi$
$42$		0			0
$43$	49.0913	−	49.0913i	1.14166	−	1.14166i	0.153512	−	0.988147i	$-0.450942\pi$
$43$	49.0913	−	49.0913i	1.14166	−	1.14166i	0.988147	−	0.153512i	$-0.0490584\pi$
$44$		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	0.966085	−	0.258224i	$-0.0831371\pi$
$53$	37.5167	+	37.5167i	0.707862	+	0.707862i	−0.258224	+	0.966085i	$0.583137\pi$
$54$		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	$-0.569367\pi$
$59$	44.8487	−	44.8487i	0.760147	−	0.760147i	0.976349	+	0.216201i	$0.0693668\pi$
$60$		0			0
$61$	57.4974	−	57.4974i	0.942581	−	0.942581i	−0.0558580	−	0.998439i	$-0.517789\pi$
$61$	57.4974	−	57.4974i	0.942581	−	0.942581i	0.998439	+	0.0558580i	$0.0177894\pi$
$62$		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	$-0.459180\pi$
$67$	−57.8812	−	57.8812i	−0.863899	−	0.863899i	−0.991788	+	0.127890i	$0.959180\pi$
$68$		0			0
$69$	9.21539	+	9.21539i	0.133556	+	0.133556i
$70$		0			0
$71$	−46.5675			−0.655880			−0.327940	−	0.944698i	$-0.606354\pi$
$71$	−46.5675			−0.655880			−0.327940	+	0.944698i	$0.606354\pi$
$72$		0			0
$73$			2.43078i			0.0332984i	0.999861	+	0.0166492i	$0.00529984\pi$
$73$			2.43078i			0.0332984i	−0.999861	+	0.0166492i	$0.994700\pi$
$74$		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	$-0.0837249\pi$
$83$	58.5648	+	58.5648i	0.705600	+	0.705600i	−0.260007	+	0.965607i	$0.583725\pi$
$84$		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	$0.252334\pi$
$89$			126.785i			1.42455i	−0.701902	+	0.712273i	$0.747666\pi$
$90$		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	0.293748	−	0.955883i	$-0.405097\pi$
$101$	−66.8756	−	66.8756i	−0.662135	−	0.662135i	−0.955883	+	0.293748i	$0.905097\pi$
$102$		0			0
$103$	166.518			1.61668			0.808341	−	0.588715i	$-0.200366\pi$
$103$	166.518			1.61668			0.808341	+	0.588715i	$0.200366\pi$
$104$		0			0
$105$		−	14.5359i		−	0.138437i
$106$		0			0
$107$	129.350	−	129.350i	1.20888	−	1.20888i	0.237485	−	0.971391i	$-0.423677\pi$
$107$	129.350	−	129.350i	1.20888	−	1.20888i	0.971391	−	0.237485i	$-0.0763231\pi$
$108$		0			0
$109$	98.3205	−	98.3205i	0.902023	−	0.902023i	−0.0935880	−	0.995611i	$-0.529834\pi$
$109$	98.3205	−	98.3205i	0.902023	−	0.902023i	0.995611	+	0.0935880i	$0.0298336\pi$
$110$		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	$-0.198222\pi$
$131$	30.0031	+	30.0031i	0.229031	+	0.229031i	−0.583257	+	0.812288i	$0.698222\pi$
$132$		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	$-0.884157\pi$
$137$		−	97.5307i		−	0.711903i	0.934504	−	0.355952i	$-0.115843\pi$
$138$		0			0
$139$	−151.720	+	151.720i	−1.09151	+	1.09151i	−0.0961409	+	0.995368i	$0.530650\pi$
$139$	−151.720	+	151.720i	−1.09151	+	1.09151i	−0.995368	+	0.0961409i	$0.969350\pi$
$140$		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	0.467983	−	0.883738i	$-0.344981\pi$
$149$	−61.9474	−	61.9474i	−0.415755	−	0.415755i	−0.883738	+	0.467983i	$0.844981\pi$
$150$		0			0
$151$	−47.7242			−0.316055			−0.158027	−	0.987435i	$-0.550513\pi$
$151$	−47.7242			−0.316055			−0.158027	+	0.987435i	$0.550513\pi$
$152$		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	−0.0716892	+	0.997427i	$0.522839\pi$
$157$	−167.851	+	167.851i	−1.06912	+	1.06912i	−0.997427	+	0.0716892i	$0.977161\pi$
$158$		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	$-0.303956\pi$
$163$	−38.8872	−	38.8872i	−0.238572	−	0.238572i	−0.816259	+	0.577687i	$0.803956\pi$
$164$		0			0
$165$	−1.85641	−	1.85641i	−0.0112509	−	0.0112509i
$166$		0			0
$167$	−239.658			−1.43508			−0.717540	−	0.696517i	$-0.754732\pi$
$167$	−239.658			−1.43508			−0.717540	+	0.696517i	$0.754732\pi$
$168$		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	−0.848648	−	0.528958i	$-0.822583\pi$
$173$	−55.3064	+	55.3064i	−0.319690	+	0.319690i	0.528958	+	0.848648i	$0.322583\pi$
$174$		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	0.999631	+	0.0271470i	$0.00864223\pi$
$179$	183.793	+	183.793i	1.02678	+	1.02678i	0.0271470	+	0.999631i	$0.491358\pi$
$180$		0			0
$181$	−15.1821	−	15.1821i	−0.0838788	−	0.0838788i	0.663923	−	0.747801i	$-0.268890\pi$
$181$	−15.1821	−	15.1821i	−0.0838788	−	0.0838788i	−0.747801	+	0.663923i	$0.768890\pi$
$182$		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	0.435364	−	0.900254i	$-0.356620\pi$
$197$	−91.5833	−	91.5833i	−0.464890	−	0.464890i	−0.900254	+	0.435364i	$0.856620\pi$
$198$		0			0
$199$	196.630			0.988091			0.494045	−	0.869436i	$-0.335518\pi$
$199$	196.630			0.988091			0.494045	+	0.869436i	$0.335518\pi$
$200$		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	$-0.336629\pi$
$211$	−80.2113	−	80.2113i	−0.380148	−	0.380148i	−0.871156	+	0.491007i	$0.836629\pi$
$212$		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	0.950293	+	0.311357i	$0.100783\pi$
$227$	286.395	+	286.395i	1.26165	+	1.26165i	0.311357	+	0.950293i	$0.399217\pi$
$228$		0			0
$229$	−122.244	−	122.244i	−0.533815	−	0.533815i	0.387891	−	0.921705i	$-0.373204\pi$
$229$	−122.244	−	122.244i	−0.533815	−	0.533815i	−0.921705	+	0.387891i	$0.873204\pi$
$230$		0			0
$231$	1.47592			0.00638925
$232$		0			0
$233$		−	67.7795i		−	0.290899i	−0.989366	−	0.145449i	$-0.953537\pi$
$233$		−	67.7795i		−	0.290899i	0.989366	−	0.145449i	$-0.0464628\pi$
$234$		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	$-0.965119\pi$
$251$	−222.044	+	222.044i	−0.884638	+	0.884638i	0.109364	+	0.994002i	$0.465119\pi$
$252$		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	$-0.741637\pi$
$263$	−362.039			−1.37657			−0.688286	+	0.725439i	$0.741637\pi$
$264$		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	−0.891711	−	0.452606i	$-0.850494\pi$
$269$	−118.119	+	118.119i	−0.439105	+	0.439105i	0.452606	+	0.891711i	$0.350494\pi$
$270$		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	0.935411	−	0.353563i	$-0.115030\pi$
$277$	161.172	+	161.172i	0.581848	+	0.581848i	−0.353563	+	0.935411i	$0.615030\pi$
$278$		0			0
$279$	86.2126			0.309006
$280$		0			0
$281$			453.482i			1.61381i	0.590678	+	0.806907i	$0.298860\pi$
$281$			453.482i			1.61381i	−0.590678	+	0.806907i	$0.701140\pi$
$282$		0			0
$283$	72.4839	−	72.4839i	0.256127	−	0.256127i	−0.567350	−	0.823477i	$-0.692031\pi$
$283$	72.4839	−	72.4839i	0.256127	−	0.256127i	0.823477	+	0.567350i	$0.192031\pi$
$284$		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	−0.991674	−	0.128773i	$-0.958896\pi$
$293$	−328.291	−	328.291i	−1.12045	−	1.12045i	−0.128773	−	0.991674i	$-0.541104\pi$
$294$		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	$-0.146580\pi$
$307$	138.593	+	138.593i	0.451443	+	0.451443i	−0.444390	+	0.895833i	$0.646580\pi$
$308$		0			0
$309$	203.942	+	203.942i	0.660007	+	0.660007i
$310$		0			0
$311$	312.534			1.00493			0.502466	−	0.864597i	$-0.332426\pi$
$311$	312.534			1.00493			0.502466	+	0.864597i	$0.332426\pi$
$312$		0			0
$313$			275.282i			0.879495i	0.898121	+	0.439748i	$0.144932\pi$
$313$			275.282i			0.879495i	−0.898121	+	0.439748i	$0.855068\pi$
$314$		0			0
$315$	17.8028	−	17.8028i	0.0565167	−	0.0565167i
$316$		0			0
$317$	−260.263	+	260.263i	−0.821018	+	0.821018i	−0.986254	−	0.165236i	$-0.947161\pi$
$317$	−260.263	+	260.263i	−0.821018	+	0.821018i	0.165236	+	0.986254i	$0.447161\pi$
$318$		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	$-0.677456\pi$
$331$	105.761	−	105.761i	0.319521	−	0.319521i	0.848583	+	0.529062i	$0.177456\pi$
$332$		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	$-0.695899\pi$
$347$	83.0052	−	83.0052i	0.239208	−	0.239208i	0.816522	+	0.577314i	$0.195899\pi$
$348$		0			0
$349$	−176.138	+	176.138i	−0.504695	+	0.504695i	−0.912893	−	0.408199i	$-0.866157\pi$
$349$	−176.138	+	176.138i	−0.504695	+	0.504695i	0.408199	+	0.912893i	$0.366157\pi$
$350$		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	$-0.531218\pi$
$359$	−70.3045			−0.195834			−0.0979172	+	0.995195i	$0.531218\pi$
$360$		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	0.873899	−	0.486108i	$-0.161584\pi$
$373$	144.646	+	144.646i	0.387791	+	0.387791i	−0.486108	+	0.873899i	$0.661584\pi$
$374$		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	$-0.762996\pi$
$379$	−21.8768	+	21.8768i	−0.0577225	+	0.0577225i	0.677656	+	0.735379i	$0.262996\pi$
$380$		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	0.948282	−	0.317430i	$-0.102820\pi$
$389$	245.401	+	245.401i	0.630851	+	0.630851i	−0.317430	+	0.948282i	$0.602820\pi$
$390$		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	−0.476048	−	0.879419i	$-0.657931\pi$
$397$	160.138	−	160.138i	0.403371	−	0.403371i	0.879419	+	0.476048i	$0.157931\pi$
$398$		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	$-0.864986\pi$
$409$		−	336.651i		−	0.823108i	0.911385	−	0.411554i	$-0.135014\pi$
$410$		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	$-0.400492\pi$
$419$	−269.830	−	269.830i	−0.643986	−	0.643986i	−0.951533	+	0.307547i	$0.900492\pi$
$420$		0			0
$421$	−30.2436	−	30.2436i	−0.0718374	−	0.0718374i	0.670275	−	0.742113i	$-0.266176\pi$
$421$	−30.2436	−	30.2436i	−0.0718374	−	0.0718374i	−0.742113	+	0.670275i	$0.766176\pi$
$422$		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	$-0.232902\pi$
$439$	653.276			1.48810			0.744050	+	0.668124i	$0.232902\pi$
$440$		0			0
$441$		−	132.846i		−	0.301238i
$442$		0			0
$443$	92.3971	−	92.3971i	0.208571	−	0.208571i	−0.595089	−	0.803660i	$-0.702883\pi$
$443$	92.3971	−	92.3971i	0.208571	−	0.208571i	0.803660	+	0.595089i	$0.202883\pi$
$444$		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	$0.217180\pi$
$457$			576.344i			1.26115i	−0.776130	+	0.630573i	$0.782820\pi$
$458$		0			0
$459$	96.9715	−	96.9715i	0.211267	−	0.211267i
$460$		0			0
$461$	386.578	−	386.578i	0.838564	−	0.838564i	−0.150106	−	0.988670i	$-0.547961\pi$
$461$	386.578	−	386.578i	0.838564	−	0.838564i	0.988670	+	0.150106i	$0.0479614\pi$
$462$		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	0.989938	+	0.141498i	$0.0451920\pi$
$467$	528.381	+	528.381i	1.13144	+	1.13144i	0.141498	+	0.989938i	$0.454808\pi$
$468$		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	$-0.234138\pi$
$487$	722.173			1.48290			0.741451	+	0.671007i	$0.234138\pi$
$488$		0			0
$489$		−	95.2539i		−	0.194793i
$490$		0			0
$491$	104.558	−	104.558i	0.212948	−	0.212948i	−0.592570	−	0.805519i	$-0.701887\pi$
$491$	104.558	−	104.558i	0.212948	−	0.212948i	0.805519	+	0.592570i	$0.201887\pi$
$492$		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	$-0.342634\pi$
$499$	−202.483	−	202.483i	−0.405777	−	0.405777i	−0.880263	+	0.474486i	$0.842634\pi$
$500$		0			0
$501$	−293.520	−	293.520i	−0.585869	−	0.585869i
$502$		0			0
$503$	874.361			1.73829			0.869146	−	0.494555i	$-0.164669\pi$
$503$	874.361			1.73829			0.869146	+	0.494555i	$0.164669\pi$
$504$		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	0.0671155	−	0.997745i	$-0.478620\pi$
$509$	542.014	−	542.014i	1.06486	−	1.06486i	0.997745	−	0.0671155i	$-0.0213796\pi$
$510$		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	$0.0126196\pi$
$521$			41.2999i			0.0792704i	−0.999214	+	0.0396352i	$0.987380\pi$
$522$		0			0
$523$	−93.2910	+	93.2910i	−0.178377	+	0.178377i	−0.790648	−	0.612271i	$-0.790256\pi$
$523$	−93.2910	+	93.2910i	−0.178377	+	0.178377i	0.612271	+	0.790648i	$0.290256\pi$
$524$		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	−0.662663	−	0.748918i	$-0.730574\pi$
$541$	46.6640	−	46.6640i	0.0862551	−	0.0862551i	0.748918	+	0.662663i	$0.230574\pi$
$542$		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	$-0.0905863\pi$
$547$	371.423	+	371.423i	0.679019	+	0.679019i	−0.280760	+	0.959778i	$0.590586\pi$
$548$		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	−0.0340406	−	0.999420i	$-0.510838\pi$
$557$	537.717	−	537.717i	0.965380	−	0.965380i	0.999420	+	0.0340406i	$0.0108375\pi$
$558$		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	$-0.361113\pi$
$563$	−272.321	−	272.321i	−0.483697	−	0.483697i	−0.906310	+	0.422613i	$0.861113\pi$
$564$		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	$-0.913918\pi$
$569$		−	304.018i		−	0.534302i	0.963655	−	0.267151i	$-0.0860823\pi$
$570$		0			0
$571$	326.012	−	326.012i	0.570950	−	0.570950i	−0.361444	−	0.932394i	$-0.617716\pi$
$571$	326.012	−	326.012i	0.570950	−	0.570950i	0.932394	+	0.361444i	$0.117716\pi$
$572$		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	$-0.583991\pi$
$587$	413.585	−	413.585i	0.704574	−	0.704574i	0.965389	+	0.260815i	$0.0839913\pi$
$588$		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	$-0.372639\pi$
$599$	466.651			0.779049			0.389525	+	0.921016i	$0.372639\pi$
$600$		0			0
$601$			991.272i			1.64937i	0.565592	+	0.824685i	$0.308648\pi$
$601$			991.272i			1.64937i	−0.565592	+	0.824685i	$0.691352\pi$
$602$		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	0.670539	−	0.741874i	$-0.266063\pi$
$613$	−43.7283	−	43.7283i	−0.0713349	−	0.0713349i	−0.741874	+	0.670539i	$0.766063\pi$
$614$		0			0
$615$	155.103			0.252200
$616$		0			0
$617$		−	281.559i		−	0.456335i	−0.973622	−	0.228168i	$-0.926727\pi$
$617$		−	281.559i		−	0.456335i	0.973622	−	0.228168i	$-0.0732734\pi$
$618$		0			0
$619$	418.959	−	418.959i	0.676832	−	0.676832i	−0.282450	−	0.959282i	$-0.591147\pi$
$619$	418.959	−	418.959i	0.676832	−	0.676832i	0.959282	+	0.282450i	$0.0911472\pi$
$620$		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	$-0.571131\pi$
$631$	−279.670			−0.443217			−0.221608	+	0.975136i	$0.571131\pi$
$632$		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	$-0.0894658\pi$
$643$	439.414	+	439.414i	0.683381	+	0.683381i	−0.277379	+	0.960761i	$0.589466\pi$
$644$		0			0
$645$	−328.526	−	328.526i	−0.509342	−	0.509342i
$646$		0			0
$647$	−29.2598			−0.0452238			−0.0226119	−	0.999744i	$-0.507198\pi$
$647$	−29.2598			−0.0452238			−0.0226119	+	0.999744i	$0.507198\pi$
$648$		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	−0.819362	−	0.573277i	$-0.805672\pi$
$653$	−160.694	+	160.694i	−0.246085	+	0.246085i	0.573277	+	0.819362i	$0.305672\pi$
$654$		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	$-0.215116\pi$
$659$	101.932	+	101.932i	0.154677	+	0.154677i	−0.625526	+	0.780203i	$0.715116\pi$
$660$		0			0
$661$	−610.251	−	610.251i	−0.923224	−	0.923224i	0.0740318	−	0.997256i	$-0.476413\pi$
$661$	−610.251	−	610.251i	−0.923224	−	0.923224i	−0.997256	+	0.0740318i	$0.976413\pi$
$662$		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	0.878296	−	0.478117i	$-0.158680\pi$
$677$	270.922	+	270.922i	0.400180	+	0.400180i	−0.478117	+	0.878296i	$0.658680\pi$
$678$		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	$-0.627140\pi$
$683$	363.629	−	363.629i	0.532399	−	0.532399i	0.921286	+	0.388886i	$0.127140\pi$
$684$		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	−0.999648	−	0.0265262i	$-0.991555\pi$
$691$	−709.086	−	709.086i	−1.02617	−	1.02617i	−0.0265262	−	0.999648i	$-0.508445\pi$
$692$		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	−0.619301	−	0.785154i	$-0.712584\pi$
$701$	116.263	−	116.263i	0.165853	−	0.165853i	0.785154	+	0.619301i	$0.212584\pi$
$702$		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	0.999743	−	0.0226610i	$-0.00721384\pi$
$709$	692.751	+	692.751i	0.977082	+	0.977082i	−0.0226610	+	0.999743i	$0.507214\pi$
$710$		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	$-0.703869\pi$
$727$	−868.875			−1.19515			−0.597575	+	0.801813i	$0.703869\pi$
$728$		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	−0.523605	−	0.851961i	$-0.675413\pi$
$733$	240.685	−	240.685i	0.328356	−	0.328356i	0.851961	+	0.523605i	$0.175413\pi$
$734$		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	$-0.261866\pi$
$739$	−38.9489	−	38.9489i	−0.0527049	−	0.0527049i	−0.732968	+	0.680263i	$0.761866\pi$
$740$		0			0
$741$	71.2539	+	71.2539i	0.0961591	+	0.0961591i
$742$		0			0
$743$	312.791			0.420984			0.210492	−	0.977596i	$-0.432493\pi$
$743$	312.791			0.420984			0.210492	+	0.977596i	$0.432493\pi$
$744$		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	0.177442	−	0.984131i	$-0.443218\pi$
$757$	−610.664	−	610.664i	−0.806690	−	0.806690i	−0.984131	+	0.177442i	$0.943218\pi$
$758$		0			0
$759$	5.11272			0.00673613
$760$		0			0
$761$			336.879i			0.442680i	0.975197	+	0.221340i	$0.0710430\pi$
$761$			336.879i			0.442680i	−0.975197	+	0.221340i	$0.928957\pi$
$762$		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	0.982718	+	0.185107i	$0.0592633\pi$
$773$	902.729	+	902.729i	1.16783	+	1.16783i	0.185107	+	0.982718i	$0.440737\pi$
$774$		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	$-0.0729788\pi$
$787$	587.548	+	587.548i	0.746566	+	0.746566i	−0.227266	+	0.973833i	$0.572979\pi$
$788$		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	−0.543133	−	0.839647i	$-0.682762\pi$
$797$	236.322	−	236.322i	0.296514	−	0.296514i	0.839647	+	0.543133i	$0.182762\pi$
$798$		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	$0.0387298\pi$
$809$			196.382i			0.242747i	−0.992607	+	0.121373i	$0.961270\pi$
$810$		0			0
$811$	−910.895	+	910.895i	−1.12317	+	1.12317i	−0.131914	+	0.991261i	$0.542112\pi$
$811$	−910.895	+	910.895i	−1.12317	+	1.12317i	−0.991261	+	0.131914i	$0.957888\pi$
$812$		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	−0.966798	−	0.255544i	$-0.917746\pi$
$821$	−1003.54	−	1003.54i	−1.22234	−	1.22234i	−0.255544	−	0.966798i	$-0.582254\pi$
$822$		0			0
$823$	554.187			0.673374			0.336687	−	0.941617i	$-0.390694\pi$
$823$	554.187			0.673374			0.336687	+	0.941617i	$0.390694\pi$
$824$		0			0
$825$		−	6.84370i		−	0.00829540i
$826$		0			0
$827$	−647.789	+	647.789i	−0.783300	+	0.783300i	−0.980386	−	0.197086i	$-0.936852\pi$
$827$	−647.789	+	647.789i	−0.783300	+	0.783300i	0.197086	+	0.980386i	$0.436852\pi$
$828$		0			0
$829$	33.2384	−	33.2384i	0.0400946	−	0.0400946i	−0.686775	−	0.726870i	$-0.740974\pi$
$829$	33.2384	−	33.2384i	0.0400946	−	0.0400946i	0.726870	+	0.686775i	$0.240974\pi$
$830$		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	$-0.318810\pi$
$839$	904.408			1.07796			0.538980	+	0.842319i	$0.318810\pi$
$840$		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	0.282895	−	0.959151i	$-0.408705\pi$
$853$	−576.846	−	576.846i	−0.676256	−	0.676256i	−0.959151	+	0.282895i	$0.908705\pi$
$854$		0			0
$855$	35.6055			0.0416439
$856$		0			0
$857$		−	1425.60i		−	1.66348i	−0.555166	−	0.831740i	$-0.687345\pi$
$857$		−	1425.60i		−	1.66348i	0.555166	−	0.831740i	$-0.312655\pi$
$858$		0			0
$859$	−193.044	+	193.044i	−0.224731	+	0.224731i	−0.810487	−	0.585756i	$-0.800798\pi$
$859$	−193.044	+	193.044i	−0.224731	+	0.224731i	0.585756	+	0.810487i	$0.300798\pi$
$860$		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	−0.169496	+	0.985531i	$0.554214\pi$
$877$	−1012.96	+	1012.96i	−1.15503	+	1.15503i	−0.985531	+	0.169496i	$0.945786\pi$
$878$		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	0.999758	+	0.0220057i	$0.00700520\pi$
$883$	902.217	+	902.217i	1.02176	+	1.02176i	0.0220057	+	0.999758i	$0.492995\pi$
$884$		0			0
$885$	−300.133	−	300.133i	−0.339134	−	0.339134i
$886$		0			0
$887$	1389.64			1.56667			0.783334	−	0.621601i	$-0.213517\pi$
$887$	1389.64			1.56667			0.783334	+	0.621601i	$0.213517\pi$
$888$		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	$-0.545949\pi$
$907$	767.091	−	767.091i	0.845745	−	0.845745i	0.989599	+	0.143854i	$0.0459494\pi$
$908$		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	$-0.700723\pi$
$919$	−1083.73			−1.17925			−0.589623	+	0.807679i	$0.700723\pi$
$920$		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	$-0.775176\pi$
$937$		−	1216.28i		−	1.29805i	0.760765	−	0.649027i	$-0.224824\pi$
$938$		0			0
$939$	−337.150	+	337.150i	−0.359052	+	0.359052i
$940$		0			0
$941$	92.7424	−	92.7424i	0.0985572	−	0.0985572i	−0.656109	−	0.754666i	$-0.727799\pi$
$941$	92.7424	−	92.7424i	0.0985572	−	0.0985572i	0.754666	+	0.656109i	$0.227799\pi$
$942$		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	$-0.328249\pi$
$947$	−325.918	−	325.918i	−0.344159	−	0.344159i	−0.857928	+	0.513770i	$0.828249\pi$
$948$		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	$-0.971207\pi$
$953$		−	172.172i		−	0.180663i	0.995912	−	0.0903315i	$-0.0287926\pi$
$954$		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	$-0.479417\pi$
$967$	124.969			0.129234			0.0646169	+	0.997910i	$0.479417\pi$
$968$		0			0
$969$		−	140.420i		−	0.144913i
$970$		0			0
$971$	63.3695	−	63.3695i	0.0652621	−	0.0652621i	−0.673722	−	0.738985i	$-0.735306\pi$
$971$	63.3695	−	63.3695i	0.0652621	−	0.0652621i	0.738985	+	0.673722i	$0.235306\pi$
$972$		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	$-0.600936\pi$
$983$	−613.021			−0.623623			−0.311812	+	0.950144i	$0.600936\pi$
$984$		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	0.827237	+	0.561854i	$0.189912\pi$
$997$	1384.92	+	1384.92i	1.38909	+	1.38909i	0.561854	+	0.827237i	$0.310088\pi$
$998$		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.319.3	yes	8
4.3	odd	2	inner	768.3.l.b.319.1	✓	8
8.3	odd	2		768.3.l.c.319.4	yes	8
8.5	even	2		768.3.l.c.319.2	yes	8
16.3	odd	4	inner	768.3.l.b.703.3	yes	8
16.5	even	4		768.3.l.c.703.4	yes	8
16.11	odd	4		768.3.l.c.703.2	yes	8
16.13	even	4	inner	768.3.l.b.703.1	yes	8

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
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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