LMFDB - Embedded newform 768.3.l.c.319.1

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			319.1
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	768.319
Dual form			768.3.l.c.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(-0.732051 - 0.732051i) q^{5} -11.9700 q^{7} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.22474 - 1.22474i) q^{3} +(-0.732051 - 0.732051i) q^{5} -11.9700 q^{7} +3.00000i q^{9} +(-14.4195 + 14.4195i) q^{11} +(7.39230 - 7.39230i) q^{13} +1.79315i q^{15} -5.60770 q^{17} +(11.9700 + 11.9700i) q^{19} +(14.6603 + 14.6603i) q^{21} +41.4655 q^{23} -23.9282i q^{25} +(3.67423 - 3.67423i) q^{27} +(4.19615 - 4.19615i) q^{29} -20.2523i q^{31} +35.3205 q^{33} +(8.76268 + 8.76268i) q^{35} +(17.9282 + 17.9282i) q^{37} -18.1074 q^{39} -39.1769i q^{41} +(-29.2923 + 29.2923i) q^{43} +(2.19615 - 2.19615i) q^{45} -56.5141i q^{47} +94.2820 q^{49} +(6.86800 + 6.86800i) q^{51} +(7.51666 + 7.51666i) q^{53} +21.1117 q^{55} -29.3205i q^{57} +(-33.5350 + 33.5350i) q^{59} +(39.4974 - 39.4974i) q^{61} -35.9101i q^{63} -10.8231 q^{65} +(-38.2853 - 38.2853i) q^{67} +(-50.7846 - 50.7846i) q^{69} +26.9716 q^{71} +85.5692i q^{73} +(-29.3059 + 29.3059i) q^{75} +(172.603 - 172.603i) q^{77} -66.8198i q^{79} -9.00000 q^{81} +(102.656 + 102.656i) q^{83} +(4.10512 + 4.10512i) q^{85} -10.2784 q^{87} +85.2154i q^{89} +(-88.4862 + 88.4862i) q^{91} +(-24.8038 + 24.8038i) q^{93} -17.5254i q^{95} +33.0718 q^{97} +(-43.2586 - 43.2586i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.732051	−	0.732051i	−0.146410	−	0.146410i	0.630102	−	0.776512i	$-0.283013\pi$
$5$	−0.732051	−	0.732051i	−0.146410	−	0.146410i	−0.776512	+	0.630102i	$0.783013\pi$
$6$		0			0
$7$	−11.9700			−1.71001			−0.855003	−	0.518622i	$-0.826445\pi$
$7$	−11.9700			−1.71001			−0.855003	+	0.518622i	$0.826445\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	−14.4195	+	14.4195i	−1.31087	+	1.31087i	−0.390090	+	0.920777i	$0.627556\pi$
$11$	−14.4195	+	14.4195i	−1.31087	+	1.31087i	−0.920777	+	0.390090i	$0.872444\pi$
$12$		0			0
$13$	7.39230	−	7.39230i	0.568639	−	0.568639i	−0.363108	−	0.931747i	$-0.618284\pi$
$13$	7.39230	−	7.39230i	0.568639	−	0.568639i	0.931747	+	0.363108i	$0.118284\pi$
$14$		0			0
$15$			1.79315i			0.119543i
$16$		0			0
$17$	−5.60770			−0.329864			−0.164932	−	0.986305i	$-0.552741\pi$
$17$	−5.60770			−0.329864			−0.164932	+	0.986305i	$0.552741\pi$
$18$		0			0
$19$	11.9700	+	11.9700i	0.630002	+	0.630002i	0.948069	−	0.318066i	$-0.103033\pi$
$19$	11.9700	+	11.9700i	0.630002	+	0.630002i	−0.318066	+	0.948069i	$0.603033\pi$
$20$		0			0
$21$	14.6603	+	14.6603i	0.698107	+	0.698107i
$22$		0			0
$23$	41.4655			1.80285			0.901423	−	0.432939i	$-0.142524\pi$
$23$	41.4655			1.80285			0.901423	+	0.432939i	$0.142524\pi$
$24$		0			0
$25$		−	23.9282i		−	0.957128i
$26$		0			0
$27$	3.67423	−	3.67423i	0.136083	−	0.136083i
$28$		0			0
$29$	4.19615	−	4.19615i	0.144695	−	0.144695i	−0.631048	−	0.775743i	$-0.717375\pi$
$29$	4.19615	−	4.19615i	0.144695	−	0.144695i	0.775743	+	0.631048i	$0.217375\pi$
$30$		0			0
$31$		−	20.2523i		−	0.653299i	−0.945146	−	0.326649i	$-0.894080\pi$
$31$		−	20.2523i		−	0.653299i	0.945146	−	0.326649i	$-0.105920\pi$
$32$		0			0
$33$	35.3205			1.07032
$34$		0			0
$35$	8.76268	+	8.76268i	0.250362	+	0.250362i
$36$		0			0
$37$	17.9282	+	17.9282i	0.484546	+	0.484546i	0.906580	−	0.422034i	$-0.138684\pi$
$37$	17.9282	+	17.9282i	0.484546	+	0.484546i	−0.422034	+	0.906580i	$0.638684\pi$
$38$		0			0
$39$	−18.1074			−0.464292
$40$		0			0
$41$		−	39.1769i		−	0.955535i	−0.878486	−	0.477767i	$-0.841446\pi$
$41$		−	39.1769i		−	0.955535i	0.878486	−	0.477767i	$-0.158554\pi$
$42$		0			0
$43$	−29.2923	+	29.2923i	−0.681217	+	0.681217i	−0.960274	−	0.279057i	$-0.909978\pi$
$43$	−29.2923	+	29.2923i	−0.681217	+	0.681217i	0.279057	+	0.960274i	$0.409978\pi$
$44$		0			0
$45$	2.19615	−	2.19615i	0.0488034	−	0.0488034i
$46$		0			0
$47$		−	56.5141i		−	1.20243i	−0.799088	−	0.601214i	$-0.794684\pi$
$47$		−	56.5141i		−	1.20243i	0.799088	−	0.601214i	$-0.205316\pi$
$48$		0			0
$49$	94.2820			1.92412
$50$		0			0
$51$	6.86800	+	6.86800i	0.134667	+	0.134667i
$52$		0			0
$53$	7.51666	+	7.51666i	0.141824	+	0.141824i	0.774454	−	0.632630i	$-0.218025\pi$
$53$	7.51666	+	7.51666i	0.141824	+	0.141824i	−0.632630	+	0.774454i	$0.718025\pi$
$54$		0			0
$55$	21.1117			0.383848
$56$		0			0
$57$		−	29.3205i		−	0.514395i
$58$		0			0
$59$	−33.5350	+	33.5350i	−0.568390	+	0.568390i	−0.931677	−	0.363288i	$-0.881654\pi$
$59$	−33.5350	+	33.5350i	−0.568390	+	0.568390i	0.363288	+	0.931677i	$0.381654\pi$
$60$		0			0
$61$	39.4974	−	39.4974i	0.647499	−	0.647499i	−0.304889	−	0.952388i	$-0.598619\pi$
$61$	39.4974	−	39.4974i	0.647499	−	0.647499i	0.952388	+	0.304889i	$0.0986194\pi$
$62$		0			0
$63$		−	35.9101i		−	0.570002i
$64$		0			0
$65$	−10.8231			−0.166509
$66$		0			0
$67$	−38.2853	−	38.2853i	−0.571422	−	0.571422i	0.361103	−	0.932526i	$-0.382400\pi$
$67$	−38.2853	−	38.2853i	−0.571422	−	0.571422i	−0.932526	+	0.361103i	$0.882400\pi$
$68$		0			0
$69$	−50.7846	−	50.7846i	−0.736009	−	0.736009i
$70$		0			0
$71$	26.9716			0.379882			0.189941	−	0.981796i	$-0.439170\pi$
$71$	26.9716			0.379882			0.189941	+	0.981796i	$0.439170\pi$
$72$		0			0
$73$			85.5692i			1.17218i	0.810246	+	0.586091i	$0.199334\pi$
$73$			85.5692i			1.17218i	−0.810246	+	0.586091i	$0.800666\pi$
$74$		0			0
$75$	−29.3059	+	29.3059i	−0.390746	+	0.390746i
$76$		0			0
$77$	172.603	−	172.603i	2.24159	−	2.24159i
$78$		0			0
$79$		−	66.8198i		−	0.845820i	−0.906172	−	0.422910i	$-0.861009\pi$
$79$		−	66.8198i		−	0.845820i	0.906172	−	0.422910i	$-0.138991\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	102.656	+	102.656i	1.23681	+	1.23681i	0.961296	+	0.275518i	$0.0888495\pi$
$83$	102.656	+	102.656i	1.23681	+	1.23681i	0.275518	+	0.961296i	$0.411150\pi$
$84$		0			0
$85$	4.10512	+	4.10512i	0.0482955	+	0.0482955i
$86$		0			0
$87$	−10.2784			−0.118143
$88$		0			0
$89$			85.2154i			0.957476i	0.877958	+	0.478738i	$0.158906\pi$
$89$			85.2154i			0.957476i	−0.877958	+	0.478738i	$0.841094\pi$
$90$		0			0
$91$	−88.4862	+	88.4862i	−0.972376	+	0.972376i
$92$		0			0
$93$	−24.8038	+	24.8038i	−0.266708	+	0.266708i
$94$		0			0
$95$		−	17.5254i		−	0.184478i
$96$		0			0
$97$	33.0718			0.340946			0.170473	−	0.985362i	$-0.445470\pi$
$97$	33.0718			0.340946			0.170473	+	0.985362i	$0.445470\pi$
$98$		0			0
$99$	−43.2586	−	43.2586i	−0.436956	−	0.436956i
$100$		0			0
$101$	91.1244	+	91.1244i	0.902221	+	0.902221i	0.995628	−	0.0934067i	$-0.0297757\pi$
$101$	91.1244	+	91.1244i	0.902221	+	0.902221i	−0.0934067	+	0.995628i	$0.529776\pi$
$102$		0			0
$103$	39.2390			0.380961			0.190480	−	0.981691i	$-0.438995\pi$
$103$	39.2390			0.380961			0.190480	+	0.981691i	$0.438995\pi$
$104$		0			0
$105$		−	21.4641i		−	0.204420i
$106$		0			0
$107$	119.552	−	119.552i	1.11731	−	1.11731i	0.125172	−	0.992135i	$-0.460052\pi$
$107$	119.552	−	119.552i	1.11731	−	1.11731i	0.992135	−	0.125172i	$-0.0399481\pi$
$108$		0			0
$109$	−63.6795	+	63.6795i	−0.584216	+	0.584216i	−0.936059	−	0.351843i	$-0.885555\pi$
$109$	−63.6795	+	63.6795i	−0.584216	+	0.584216i	0.351843	+	0.936059i	$0.385555\pi$
$110$		0			0
$111$		−	43.9149i		−	0.395630i
$112$		0			0
$113$	175.990			1.55743			0.778715	−	0.627377i	$-0.215872\pi$
$113$	175.990			1.55743			0.778715	+	0.627377i	$0.215872\pi$
$114$		0			0
$115$	−30.3548	−	30.3548i	−0.263955	−	0.263955i
$116$		0			0
$117$	22.1769	+	22.1769i	0.189546	+	0.189546i
$118$		0			0
$119$	67.1244			0.564070
$120$		0			0
$121$		−	294.846i		−	2.43674i
$122$		0			0
$123$	−47.9817	+	47.9817i	−0.390095	+	0.390095i
$124$		0			0
$125$	−35.8179	+	35.8179i	−0.286543	+	0.286543i
$126$		0			0
$127$		−	193.287i		−	1.52194i	−0.648786	−	0.760971i	$-0.724723\pi$
$127$		−	193.287i		−	1.52194i	0.648786	−	0.760971i	$-0.275277\pi$
$128$		0			0
$129$	71.7513			0.556212
$130$		0			0
$131$	88.7908	+	88.7908i	0.677793	+	0.677793i	0.959500	−	0.281708i	$-0.0909008\pi$
$131$	88.7908	+	88.7908i	0.677793	+	0.677793i	−0.281708	+	0.959500i	$0.590901\pi$
$132$		0			0
$133$	−143.282	−	143.282i	−1.07731	−	1.07731i
$134$		0			0
$135$	−5.37945			−0.0398478
$136$		0			0
$137$			89.5307i			0.653509i	0.945109	+	0.326755i	$0.105955\pi$
$137$			89.5307i			0.653509i	−0.945109	+	0.326755i	$0.894045\pi$
$138$		0			0
$139$	44.2395	−	44.2395i	0.318270	−	0.318270i	−0.529833	−	0.848102i	$-0.677745\pi$
$139$	44.2395	−	44.2395i	0.318270	−	0.318270i	0.848102	+	0.529833i	$0.177745\pi$
$140$		0			0
$141$	−69.2154	+	69.2154i	−0.490889	+	0.490889i
$142$		0			0
$143$			213.187i			1.49082i
$144$		0			0
$145$	−6.14359			−0.0423696
$146$		0			0
$147$	−115.471	−	115.471i	−0.785520	−	0.785520i
$148$		0			0
$149$	100.053	+	100.053i	0.671494	+	0.671494i	0.958060	−	0.286567i	$-0.0925140\pi$
$149$	100.053	+	100.053i	0.671494	+	0.671494i	−0.286567	+	0.958060i	$0.592514\pi$
$150$		0			0
$151$	−158.033			−1.04658			−0.523288	−	0.852156i	$-0.675295\pi$
$151$	−158.033			−1.04658			−0.523288	+	0.852156i	$0.675295\pi$
$152$		0			0
$153$		−	16.8231i		−	0.109955i
$154$		0			0
$155$	−14.8257	+	14.8257i	−0.0956496	+	0.0956496i
$156$		0			0
$157$	−53.8513	+	53.8513i	−0.343002	+	0.343002i	−0.857495	−	0.514493i	$-0.827980\pi$
$157$	−53.8513	+	53.8513i	−0.343002	+	0.343002i	0.514493	+	0.857495i	$0.327980\pi$
$158$		0			0
$159$		−	18.4120i		−	0.115799i
$160$		0			0
$161$	−496.344			−3.08288
$162$		0			0
$163$	98.2842	+	98.2842i	0.602971	+	0.602971i	0.941100	−	0.338129i	$-0.109794\pi$
$163$	98.2842	+	98.2842i	0.602971	+	0.602971i	−0.338129	+	0.941100i	$0.609794\pi$
$164$		0			0
$165$	−25.8564	−	25.8564i	−0.156705	−	0.156705i
$166$		0			0
$167$	229.860			1.37641			0.688205	−	0.725516i	$-0.258399\pi$
$167$	229.860			1.37641			0.688205	+	0.725516i	$0.258399\pi$
$168$		0			0
$169$			59.7077i			0.353300i
$170$		0			0
$171$	−35.9101	+	35.9101i	−0.210001	+	0.210001i
$172$		0			0
$173$	162.694	−	162.694i	0.940425	−	0.940425i	−0.0578973	−	0.998323i	$-0.518440\pi$
$173$	162.694	−	162.694i	0.940425	−	0.940425i	0.998323	+	0.0578973i	$0.0184396\pi$
$174$		0			0
$175$			286.422i			1.63670i
$176$		0			0
$177$	82.1436			0.464088
$178$		0			0
$179$	36.8240	+	36.8240i	0.205720	+	0.205720i	0.802446	−	0.596725i	$-0.203532\pi$
$179$	36.8240	+	36.8240i	0.205720	+	0.205720i	−0.596725	+	0.802446i	$0.703532\pi$
$180$		0			0
$181$	146.818	+	146.818i	0.811149	+	0.811149i	0.984806	−	0.173657i	$-0.0555585\pi$
$181$	146.818	+	146.818i	0.811149	+	0.811149i	−0.173657	+	0.984806i	$0.555559\pi$
$182$		0			0
$183$	−96.7485			−0.528681
$184$		0			0
$185$		−	26.2487i		−	0.141885i
$186$		0			0
$187$	80.8604	−	80.8604i	0.432408	−	0.432408i
$188$		0			0
$189$	−43.9808	+	43.9808i	−0.232702	+	0.232702i
$190$		0			0
$191$			169.151i			0.885606i	0.896619	+	0.442803i	$0.146016\pi$
$191$			169.151i			0.885606i	−0.896619	+	0.442803i	$0.853984\pi$
$192$		0			0
$193$	23.1487			0.119942			0.0599709	−	0.998200i	$-0.480899\pi$
$193$	23.1487			0.119942			0.0599709	+	0.998200i	$0.480899\pi$
$194$		0			0
$195$	13.2555	+	13.2555i	0.0679770	+	0.0679770i
$196$		0			0
$197$	−133.583	−	133.583i	−0.678088	−	0.678088i	0.281479	−	0.959567i	$-0.409175\pi$
$197$	−133.583	−	133.583i	−0.678088	−	0.678088i	−0.959567	+	0.281479i	$0.909175\pi$
$198$		0			0
$199$	−343.599			−1.72663			−0.863315	−	0.504665i	$-0.831616\pi$
$199$	−343.599			−1.72663			−0.863315	+	0.504665i	$0.831616\pi$
$200$		0			0
$201$			93.7795i			0.466564i
$202$		0			0
$203$	−50.2281	+	50.2281i	−0.247429	+	0.247429i
$204$		0			0
$205$	−28.6795	+	28.6795i	−0.139900	+	0.139900i
$206$		0			0
$207$			124.396i			0.600949i
$208$		0			0
$209$	−345.205			−1.65170
$210$		0			0
$211$	−276.170	−	276.170i	−1.30886	−	1.30886i	−0.922235	−	0.386630i	$-0.873639\pi$
$211$	−276.170	−	276.170i	−1.30886	−	1.30886i	−0.386630	−	0.922235i	$-0.626361\pi$
$212$		0			0
$213$	−33.0333	−	33.0333i	−0.155086	−	0.155086i
$214$		0			0
$215$	42.8870			0.199474
$216$		0			0
$217$			242.420i			1.11715i
$218$		0			0
$219$	104.800	−	104.800i	0.478541	−	0.478541i
$220$		0			0
$221$	−41.4538	+	41.4538i	−0.187574	+	0.187574i
$222$		0			0
$223$			233.885i			1.04881i	0.851468	+	0.524407i	$0.175713\pi$
$223$			233.885i			1.04881i	−0.851468	+	0.524407i	$0.824287\pi$
$224$		0			0
$225$	71.7846			0.319043
$226$		0			0
$227$	−51.6351	−	51.6351i	−0.227467	−	0.227467i	0.584167	−	0.811634i	$-0.301421\pi$
$227$	−51.6351	−	51.6351i	−0.227467	−	0.227467i	−0.811634	+	0.584167i	$0.801421\pi$
$228$		0			0
$229$	−120.244	−	120.244i	−0.525081	−	0.525081i	0.394021	−	0.919102i	$-0.371084\pi$
$229$	−120.244	−	120.244i	−0.525081	−	0.525081i	−0.919102	+	0.394021i	$0.871084\pi$
$230$		0			0
$231$	−422.788			−1.83025
$232$		0			0
$233$			167.779i			0.720083i	0.932936	+	0.360042i	$0.117238\pi$
$233$			167.779i			0.720083i	−0.932936	+	0.360042i	$0.882762\pi$
$234$		0			0
$235$	−41.3712	+	41.3712i	−0.176048	+	0.176048i
$236$		0			0
$237$	−81.8372	+	81.8372i	−0.345305	+	0.345305i
$238$		0			0
$239$		−	182.386i		−	0.763123i	−0.924343	−	0.381562i	$-0.875386\pi$
$239$		−	182.386i		−	0.763123i	0.924343	−	0.381562i	$-0.124614\pi$
$240$		0			0
$241$	165.503			0.686733			0.343366	−	0.939202i	$-0.388433\pi$
$241$	165.503			0.686733			0.343366	+	0.939202i	$0.388433\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.0453609	+	0.0453609i
$244$		0			0
$245$	−69.0192	−	69.0192i	−0.281711	−	0.281711i
$246$		0			0
$247$	176.972			0.716488
$248$		0			0
$249$		−	251.454i		−	1.00985i
$250$		0			0
$251$	253.157	−	253.157i	1.00859	−	1.00859i	0.00863029	−	0.999963i	$-0.497253\pi$
$251$	253.157	−	253.157i	1.00859	−	1.00859i	0.999963	−	0.00863029i	$-0.00274714\pi$
$252$		0			0
$253$	−597.913	+	597.913i	−2.36329	+	2.36329i
$254$		0			0
$255$		−	10.0554i		−	0.0394331i
$256$		0			0
$257$	−474.918			−1.84793			−0.923965	−	0.382478i	$-0.875071\pi$
$257$	−474.918			−1.84793			−0.923965	+	0.382478i	$0.875071\pi$
$258$		0			0
$259$	−214.601	−	214.601i	−0.828577	−	0.828577i
$260$		0			0
$261$	12.5885	+	12.5885i	0.0482316	+	0.0482316i
$262$		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
$265$		−	11.0052i		−	0.0415289i
$266$		0			0
$267$	104.367	−	104.367i	0.390888	−	0.390888i
$268$		0			0
$269$	−100.119	+	100.119i	−0.372190	+	0.372190i	−0.868274	−	0.496084i	$-0.834771\pi$
$269$	−100.119	+	100.119i	−0.372190	+	0.372190i	0.496084	+	0.868274i	$0.334771\pi$
$270$		0			0
$271$		−	167.033i		−	0.616358i	−0.951328	−	0.308179i	$-0.900280\pi$
$271$		−	167.033i		−	0.616358i	0.951328	−	0.308179i	$-0.0997197\pi$
$272$		0			0
$273$	216.746			0.793942
$274$		0			0
$275$	345.034	+	345.034i	1.25467	+	1.25467i
$276$		0			0
$277$	95.1718	+	95.1718i	0.343580	+	0.343580i	0.857712	−	0.514131i	$-0.171886\pi$
$277$	95.1718	+	95.1718i	0.343580	+	0.343580i	−0.514131	+	0.857712i	$0.671886\pi$
$278$		0			0
$279$	60.7568			0.217766
$280$		0			0
$281$		−	225.482i		−	0.802427i	−0.915985	−	0.401213i	$-0.868589\pi$
$281$		−	225.482i		−	0.802427i	0.915985	−	0.401213i	$-0.131411\pi$
$282$		0			0
$283$	278.241	−	278.241i	0.983184	−	0.983184i	−0.0166770	−	0.999861i	$-0.505309\pi$
$283$	278.241	−	278.241i	0.983184	−	0.983184i	0.999861	+	0.0166770i	$0.00530870\pi$
$284$		0			0
$285$	−21.4641	+	21.4641i	−0.0753126	+	0.0753126i
$286$		0			0
$287$			468.950i			1.63397i
$288$		0			0
$289$	−257.554			−0.891189
$290$		0			0
$291$	−40.5045	−	40.5045i	−0.139191	−	0.139191i
$292$		0			0
$293$	−146.291	−	146.291i	−0.499287	−	0.499287i	0.411929	−	0.911216i	$-0.364855\pi$
$293$	−146.291	−	146.291i	−0.499287	−	0.499287i	−0.911216	+	0.411929i	$0.864855\pi$
$294$		0			0
$295$	49.0986			0.166436
$296$		0			0
$297$			105.962i			0.356773i
$298$		0			0
$299$	306.525	−	306.525i	1.02517	−	1.02517i
$300$		0			0
$301$	350.631	−	350.631i	1.16489	−	1.16489i
$302$		0			0
$303$		−	223.208i		−	0.736661i
$304$		0			0
$305$	−57.8282			−0.189601
$306$		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
$309$	−48.0577	−	48.0577i	−0.155527	−	0.155527i
$310$		0			0
$311$	−38.1911			−0.122801			−0.0614004	−	0.998113i	$-0.519557\pi$
$311$	−38.1911			−0.122801			−0.0614004	+	0.998113i	$0.519557\pi$
$312$		0			0
$313$			136.718i			0.436799i	0.975859	+	0.218399i	$0.0700835\pi$
$313$			136.718i			0.436799i	−0.975859	+	0.218399i	$0.929916\pi$
$314$		0			0
$315$	−26.2880	+	26.2880i	−0.0834541	+	0.0834541i
$316$		0			0
$317$	69.7372	−	69.7372i	0.219991	−	0.219991i	−0.588503	−	0.808495i	$-0.700283\pi$
$317$	69.7372	−	69.7372i	0.219991	−	0.219991i	0.808495	+	0.588503i	$0.200283\pi$
$318$		0			0
$319$			121.013i			0.379352i
$320$		0			0
$321$	−292.841			−0.912277
$322$		0			0
$323$	−67.1244	−	67.1244i	−0.207815	−	0.207815i
$324$		0			0
$325$	−176.885	−	176.885i	−0.544260	−	0.544260i
$326$		0			0
$327$	155.982			0.477010
$328$		0			0
$329$			676.477i			2.05616i
$330$		0			0
$331$	46.9737	−	46.9737i	0.141914	−	0.141914i	−0.632580	−	0.774495i	$-0.718004\pi$
$331$	46.9737	−	46.9737i	0.141914	−	0.141914i	0.774495	+	0.632580i	$0.218004\pi$
$332$		0			0
$333$	−53.7846	+	53.7846i	−0.161515	+	0.161515i
$334$		0			0
$335$			56.0536i			0.167324i
$336$		0			0
$337$	82.8513			0.245849			0.122925	−	0.992416i	$-0.460773\pi$
$337$	82.8513			0.245849			0.122925	+	0.992416i	$0.460773\pi$
$338$		0			0
$339$	−215.542	−	215.542i	−0.635818	−	0.635818i
$340$		0			0
$341$	292.028	+	292.028i	0.856388	+	0.856388i
$342$		0			0
$343$	−542.028			−1.58026
$344$		0			0
$345$			74.3538i			0.215518i
$346$		0			0
$347$	−68.8631	+	68.8631i	−0.198453	+	0.198453i	−0.799336	−	0.600884i	$-0.794815\pi$
$347$	−68.8631	+	68.8631i	−0.198453	+	0.198453i	0.600884	+	0.799336i	$0.294815\pi$
$348$		0			0
$349$	9.86156	−	9.86156i	0.0282566	−	0.0282566i	−0.692837	−	0.721094i	$-0.743640\pi$
$349$	9.86156	−	9.86156i	0.0282566	−	0.0282566i	0.721094	+	0.692837i	$0.243640\pi$
$350$		0			0
$351$		−	54.3221i		−	0.154764i
$352$		0			0
$353$	337.215			0.955284			0.477642	−	0.878554i	$-0.341492\pi$
$353$	337.215			0.955284			0.477642	+	0.878554i	$0.341492\pi$
$354$		0			0
$355$	−19.7446	−	19.7446i	−0.0556185	−	0.0556185i
$356$		0			0
$357$	−82.2102	−	82.2102i	−0.230281	−	0.230281i
$358$		0			0
$359$	−8.07914			−0.0225046			−0.0112523	−	0.999937i	$-0.503582\pi$
$359$	−8.07914			−0.0225046			−0.0112523	+	0.999937i	$0.503582\pi$
$360$		0			0
$361$		−	74.4359i		−	0.206194i
$362$		0			0
$363$	−361.111	+	361.111i	−0.994797	+	0.994797i
$364$		0			0
$365$	62.6410	−	62.6410i	0.171619	−	0.171619i
$366$		0			0
$367$		−	239.448i		−	0.652447i	−0.945293	−	0.326224i	$-0.894224\pi$
$367$		−	239.448i		−	0.652447i	0.945293	−	0.326224i	$-0.105776\pi$
$368$		0			0
$369$	117.531			0.318512
$370$		0			0
$371$	−89.9748	−	89.9748i	−0.242520	−	0.242520i
$372$		0			0
$373$	−269.354	−	269.354i	−0.722128	−	0.722128i	0.246910	−	0.969038i	$-0.420585\pi$
$373$	−269.354	−	269.354i	−0.722128	−	0.722128i	−0.969038	+	0.246910i	$0.920585\pi$
$374$		0			0
$375$	87.7357			0.233962
$376$		0			0
$377$		−	62.0385i		−	0.164558i
$378$		0			0
$379$	−286.422	+	286.422i	−0.755730	+	0.755730i	−0.975542	−	0.219812i	$-0.929456\pi$
$379$	−286.422	+	286.422i	−0.755730	+	0.755730i	0.219812	+	0.975542i	$0.429456\pi$
$380$		0			0
$381$	−236.727	+	236.727i	−0.621330	+	0.621330i
$382$		0			0
$383$			295.117i			0.770541i	0.922804	+	0.385271i	$0.125892\pi$
$383$			295.117i			0.770541i	−0.922804	+	0.385271i	$0.874108\pi$
$384$		0			0
$385$	−252.708			−0.656384
$386$		0			0
$387$	−87.8770	−	87.8770i	−0.227072	−	0.227072i
$388$		0			0
$389$	111.401	+	111.401i	0.286378	+	0.286378i	0.835646	−	0.549268i	$-0.185093\pi$
$389$	111.401	+	111.401i	0.286378	+	0.286378i	−0.549268	+	0.835646i	$0.685093\pi$
$390$		0			0
$391$	−232.526			−0.594695
$392$		0			0
$393$		−	217.492i		−	0.553415i
$394$		0			0
$395$	−48.9155	+	48.9155i	−0.123837	+	0.123837i
$396$		0			0
$397$	6.13844	−	6.13844i	0.0154621	−	0.0154621i	−0.699333	−	0.714796i	$-0.746520\pi$
$397$	6.13844	−	6.13844i	0.0154621	−	0.0154621i	0.714796	+	0.699333i	$0.246520\pi$
$398$		0			0
$399$			350.968i			0.879619i
$400$		0			0
$401$	328.018			0.818000			0.409000	−	0.912534i	$-0.365878\pi$
$401$	328.018			0.818000			0.409000	+	0.912534i	$0.365878\pi$
$402$		0			0
$403$	−149.711	−	149.711i	−0.371491	−	0.371491i
$404$		0			0
$405$	6.58846	+	6.58846i	0.0162678	+	0.0162678i
$406$		0			0
$407$	−517.033			−1.27035
$408$		0			0
$409$		−	655.349i		−	1.60232i	−0.598451	−	0.801160i	$-0.704217\pi$
$409$		−	655.349i		−	1.60232i	0.598451	−	0.801160i	$-0.295783\pi$
$410$		0			0
$411$	109.652	−	109.652i	0.266794	−	0.266794i
$412$		0			0
$413$	401.415	−	401.415i	0.971950	−	0.971950i
$414$		0			0
$415$		−	150.298i		−	0.362164i
$416$		0			0
$417$	−108.364			−0.259866
$418$		0			0
$419$	−10.1842	−	10.1842i	−0.0243059	−	0.0243059i	0.694849	−	0.719155i	$-0.255471\pi$
$419$	−10.1842	−	10.1842i	−0.0243059	−	0.0243059i	−0.719155	+	0.694849i	$0.755471\pi$
$420$		0			0
$421$	−212.244	−	212.244i	−0.504141	−	0.504141i	0.408581	−	0.912722i	$-0.366024\pi$
$421$	−212.244	−	212.244i	−0.504141	−	0.504141i	−0.912722	+	0.408581i	$0.866024\pi$
$422$		0			0
$423$	169.542			0.400809
$424$		0			0
$425$			134.182i			0.315723i
$426$		0			0
$427$	−472.786	+	472.786i	−1.10723	+	1.10723i
$428$		0			0
$429$	261.100	−	261.100i	0.608625	−	0.608625i
$430$		0			0
$431$		−	721.666i		−	1.67440i	−0.546898	−	0.837199i	$-0.684191\pi$
$431$		−	721.666i		−	1.67440i	0.546898	−	0.837199i	$-0.315809\pi$
$432$		0			0
$433$	−50.2975			−0.116161			−0.0580803	−	0.998312i	$-0.518498\pi$
$433$	−50.2975			−0.116161			−0.0580803	+	0.998312i	$0.518498\pi$
$434$		0			0
$435$	7.52433	+	7.52433i	0.0172973	+	0.0172973i
$436$		0			0
$437$	496.344	+	496.344i	1.13580	+	1.13580i
$438$		0			0
$439$	865.408			1.97132			0.985658	−	0.168754i	$-0.0539745\pi$
$439$	865.408			1.97132			0.985658	+	0.168754i	$0.0539745\pi$
$440$		0			0
$441$			282.846i			0.641374i
$442$		0			0
$443$	−157.451	+	157.451i	−0.355420	+	0.355420i	−0.862121	−	0.506702i	$-0.830865\pi$
$443$	−157.451	+	157.451i	−0.355420	+	0.355420i	0.506702	+	0.862121i	$0.330865\pi$
$444$		0			0
$445$	62.3820	−	62.3820i	0.140184	−	0.140184i
$446$		0			0
$447$		−	245.078i		−	0.548272i
$448$		0			0
$449$	578.526			1.28848			0.644238	−	0.764825i	$-0.277175\pi$
$449$	578.526			1.28848			0.644238	+	0.764825i	$0.277175\pi$
$450$		0			0
$451$	564.913	+	564.913i	1.25258	+	1.25258i
$452$		0			0
$453$	193.550	+	193.550i	0.427263	+	0.427263i
$454$		0			0
$455$	129.553			0.284732
$456$		0			0
$457$			63.6565i			0.139292i	0.997572	+	0.0696460i	$0.0221870\pi$
$457$			63.6565i			0.139292i	−0.997572	+	0.0696460i	$0.977813\pi$
$458$		0			0
$459$	−20.6040	+	20.6040i	−0.0448889	+	0.0448889i
$460$		0			0
$461$	32.5781	−	32.5781i	0.0706684	−	0.0706684i	−0.670889	−	0.741558i	$-0.734087\pi$
$461$	32.5781	−	32.5781i	0.0706684	−	0.0706684i	0.741558	+	0.670889i	$0.234087\pi$
$462$		0			0
$463$			391.277i			0.845090i	0.906342	+	0.422545i	$0.138863\pi$
$463$			391.277i			0.845090i	−0.906342	+	0.422545i	$0.861137\pi$
$464$		0			0
$465$	36.3154			0.0780975
$466$		0			0
$467$	356.917	+	356.917i	0.764276	+	0.764276i	0.977092	−	0.212817i	$-0.0682637\pi$
$467$	356.917	+	356.917i	0.764276	+	0.764276i	−0.212817	+	0.977092i	$0.568264\pi$
$468$		0			0
$469$	458.277	+	458.277i	0.977136	+	0.977136i
$470$		0			0
$471$	131.908			0.280060
$472$		0			0
$473$		−	844.764i		−	1.78597i
$474$		0			0
$475$	286.422	−	286.422i	0.602993	−	0.602993i
$476$		0			0
$477$	−22.5500	+	22.5500i	−0.0472746	+	0.0472746i
$478$		0			0
$479$		−	207.367i		−	0.432917i	−0.976292	−	0.216458i	$-0.930549\pi$
$479$		−	207.367i		−	0.432917i	0.976292	−	0.216458i	$-0.0694506\pi$
$480$		0			0
$481$	265.061			0.551063
$482$		0			0
$483$	607.894	+	607.894i	1.25858	+	1.25858i
$484$		0			0
$485$	−24.2102	−	24.2102i	−0.0499180	−	0.0499180i
$486$		0			0
$487$	600.551			1.23316			0.616582	−	0.787291i	$-0.288517\pi$
$487$	600.551			1.23316			0.616582	+	0.787291i	$0.288517\pi$
$488$		0			0
$489$		−	240.746i		−	0.492323i
$490$		0			0
$491$	359.304	−	359.304i	0.731781	−	0.731781i	−0.239191	−	0.970972i	$-0.576882\pi$
$491$	359.304	−	359.304i	0.731781	−	0.731781i	0.970972	+	0.239191i	$0.0768823\pi$
$492$		0			0
$493$	−23.5307	+	23.5307i	−0.0477297	+	0.0477297i
$494$		0			0
$495$			63.3350i			0.127949i
$496$		0			0
$497$	−322.851			−0.649600
$498$		0			0
$499$	−114.301	−	114.301i	−0.229060	−	0.229060i	0.583240	−	0.812300i	$-0.301785\pi$
$499$	−114.301	−	114.301i	−0.229060	−	0.229060i	−0.812300	+	0.583240i	$0.801785\pi$
$500$		0			0
$501$	−281.520	−	281.520i	−0.561917	−	0.561917i
$502$		0			0
$503$	144.627			0.287528			0.143764	−	0.989612i	$-0.454079\pi$
$503$	144.627			0.287528			0.143764	+	0.989612i	$0.454079\pi$
$504$		0			0
$505$		−	133.415i		−	0.264189i
$506$		0			0
$507$	73.1266	−	73.1266i	0.144234	−	0.144234i
$508$		0			0
$509$	−399.986	+	399.986i	−0.785827	+	0.785827i	−0.980807	−	0.194980i	$-0.937536\pi$
$509$	−399.986	+	399.986i	−0.785827	+	0.785827i	0.194980	+	0.980807i	$0.437536\pi$
$510$		0			0
$511$		−	1024.27i		−	2.00444i
$512$		0			0
$513$	87.9615			0.171465
$514$		0			0
$515$	−28.7249	−	28.7249i	−0.0557765	−	0.0557765i
$516$		0			0
$517$	814.908	+	814.908i	1.57622	+	1.57622i
$518$		0			0
$519$	−398.516			−0.767854
$520$		0			0
$521$		−	769.300i		−	1.47658i	−0.674482	−	0.738292i	$-0.735633\pi$
$521$		−	769.300i		−	1.47658i	0.674482	−	0.738292i	$-0.264367\pi$
$522$		0			0
$523$	−220.664	+	220.664i	−0.421921	+	0.421921i	−0.885865	−	0.463944i	$-0.846434\pi$
$523$	−220.664	+	220.664i	−0.421921	+	0.421921i	0.463944	+	0.885865i	$0.346434\pi$
$524$		0			0
$525$	350.794	−	350.794i	0.668178	−	0.668178i
$526$		0			0
$527$			113.568i			0.215500i
$528$		0			0
$529$	1190.38			2.25025
$530$		0			0
$531$	−100.605	−	100.605i	−0.189463	−	0.189463i
$532$		0			0
$533$	−289.608	−	289.608i	−0.543354	−	0.543354i
$534$		0			0
$535$	−175.036			−0.327170
$536$		0			0
$537$		−	90.1999i		−	0.167970i
$538$		0			0
$539$	−1359.50	+	1359.50i	−2.52227	+	2.52227i
$540$		0			0
$541$	500.664	−	500.664i	0.925442	−	0.925442i	−0.0719653	−	0.997407i	$-0.522927\pi$
$541$	500.664	−	500.664i	0.925442	−	0.925442i	0.997407	+	0.0719653i	$0.0229271\pi$
$542$		0			0
$543$		−	359.629i		−	0.662300i
$544$		0			0
$545$	93.2332			0.171070
$546$		0			0
$547$	−0.899244	−	0.899244i	−0.00164396	−	0.00164396i	0.706284	−	0.707928i	$-0.250370\pi$
$547$	−0.899244	−	0.899244i	−0.00164396	−	0.00164396i	−0.707928	+	0.706284i	$0.750370\pi$
$548$		0			0
$549$	118.492	+	118.492i	0.215833	+	0.215833i
$550$		0			0
$551$	100.456			0.182316
$552$		0			0
$553$			799.836i			1.44636i
$554$		0			0
$555$	−32.1480	+	32.1480i	−0.0579243	+	0.0579243i
$556$		0			0
$557$	47.7166	−	47.7166i	0.0856671	−	0.0856671i	−0.662975	−	0.748642i	$-0.730706\pi$
$557$	47.7166	−	47.7166i	0.0856671	−	0.0856671i	0.748642	+	0.662975i	$0.230706\pi$
$558$		0			0
$559$			433.076i			0.774733i
$560$		0			0
$561$	−198.067			−0.353060
$562$		0			0
$563$	−629.947	−	629.947i	−1.11891	−	1.11891i	−0.991902	−	0.127009i	$-0.959462\pi$
$563$	−629.947	−	629.947i	−1.11891	−	1.11891i	−0.127009	−	0.991902i	$-0.540538\pi$
$564$		0			0
$565$	−128.833	−	128.833i	−0.228024	−	0.228024i
$566$		0			0
$567$	107.730			0.190001
$568$		0			0
$569$			368.018i			0.646780i	0.946266	+	0.323390i	$0.104823\pi$
$569$			368.018i			0.646780i	−0.946266	+	0.323390i	$0.895177\pi$
$570$		0			0
$571$	−212.875	+	212.875i	−0.372811	+	0.372811i	−0.868500	−	0.495689i	$-0.834916\pi$
$571$	−212.875	+	212.875i	−0.372811	+	0.372811i	0.495689	+	0.868500i	$0.334916\pi$
$572$		0			0
$573$	207.167	−	207.167i	0.361547	−	0.361547i
$574$		0			0
$575$		−	992.194i		−	1.72555i
$576$		0			0
$577$	34.2563			0.0593696			0.0296848	−	0.999559i	$-0.490550\pi$
$577$	34.2563			0.0593696			0.0296848	+	0.999559i	$0.490550\pi$
$578$		0			0
$579$	−28.3513	−	28.3513i	−0.0489660	−	0.0489660i
$580$		0			0
$581$	−1228.79	−	1228.79i	−2.11496	−	2.11496i
$582$		0			0
$583$	−216.774			−0.371824
$584$		0			0
$585$		−	32.4693i		−	0.0555030i
$586$		0			0
$587$	−419.242	+	419.242i	−0.714211	+	0.714211i	−0.967413	−	0.253203i	$-0.918516\pi$
$587$	−419.242	+	419.242i	−0.714211	+	0.714211i	0.253203	+	0.967413i	$0.418516\pi$
$588$		0			0
$589$	242.420	−	242.420i	0.411580	−	0.411580i
$590$		0			0
$591$			327.211i			0.553656i
$592$		0			0
$593$	755.703			1.27437			0.637186	−	0.770710i	$-0.280098\pi$
$593$	755.703			1.27437			0.637186	+	0.770710i	$0.280098\pi$
$594$		0			0
$595$	−49.1385	−	49.1385i	−0.0825856	−	0.0825856i
$596$		0			0
$597$	420.822	+	420.822i	0.704894	+	0.704894i
$598$		0			0
$599$	−868.367			−1.44969			−0.724847	−	0.688910i	$-0.758090\pi$
$599$	−868.367			−1.44969			−0.724847	+	0.688910i	$0.758090\pi$
$600$		0			0
$601$			464.728i			0.773258i	0.922235	+	0.386629i	$0.126361\pi$
$601$			464.728i			0.773258i	−0.922235	+	0.386629i	$0.873639\pi$
$602$		0			0
$603$	114.856	−	114.856i	0.190474	−	0.190474i
$604$		0			0
$605$	−215.842	+	215.842i	−0.356764	+	0.356764i
$606$		0			0
$607$		−	76.2262i		−	0.125579i	−0.998027	−	0.0627893i	$-0.980000\pi$
$607$		−	76.2262i		−	0.125579i	0.998027	−	0.0627893i	$-0.0199996\pi$
$608$		0			0
$609$	123.033			0.202025
$610$		0			0
$611$	−417.770	−	417.770i	−0.683747	−	0.683747i
$612$		0			0
$613$	570.272	+	570.272i	0.930296	+	0.930296i	0.997724	−	0.0674277i	$-0.0214792\pi$
$613$	570.272	+	570.272i	0.930296	+	0.930296i	−0.0674277	+	0.997724i	$0.521479\pi$
$614$		0			0
$615$	70.2501			0.114228
$616$		0			0
$617$			189.559i			0.307227i	0.988131	+	0.153613i	$0.0490910\pi$
$617$			189.559i			0.307227i	−0.988131	+	0.153613i	$0.950909\pi$
$618$		0			0
$619$	350.373	−	350.373i	0.566031	−	0.566031i	−0.364983	−	0.931014i	$-0.618925\pi$
$619$	350.373	−	350.373i	0.566031	−	0.566031i	0.931014	+	0.364983i	$0.118925\pi$
$620$		0			0
$621$	152.354	−	152.354i	0.245336	−	0.245336i
$622$		0			0
$623$		−	1020.03i		−	1.63729i
$624$		0			0
$625$	−545.764			−0.873222
$626$		0			0
$627$	422.788	+	422.788i	0.674303	+	0.674303i
$628$		0			0
$629$	−100.536	−	100.536i	−0.159834	−	0.159834i
$630$		0			0
$631$	622.598			0.986685			0.493343	−	0.869835i	$-0.335775\pi$
$631$	622.598			0.986685			0.493343	+	0.869835i	$0.335775\pi$
$632$		0			0
$633$			676.477i			1.06868i
$634$		0			0
$635$	−141.496	+	141.496i	−0.222828	+	0.222828i
$636$		0			0
$637$	696.962	−	696.962i	1.09413	−	1.09413i
$638$		0			0
$639$			80.9148i			0.126627i
$640$		0			0
$641$	491.177			0.766267			0.383133	−	0.923693i	$-0.374845\pi$
$641$	491.177			0.766267			0.383133	+	0.923693i	$0.374845\pi$
$642$		0			0
$643$	361.031	+	361.031i	0.561478	+	0.561478i	0.929727	−	0.368249i	$-0.120043\pi$
$643$	361.031	+	361.031i	0.561478	+	0.561478i	−0.368249	+	0.929727i	$0.620043\pi$
$644$		0			0
$645$	−52.5256	−	52.5256i	−0.0814350	−	0.0814350i
$646$		0			0
$647$	626.935			0.968988			0.484494	−	0.874795i	$-0.339004\pi$
$647$	626.935			0.968988			0.484494	+	0.874795i	$0.339004\pi$
$648$		0			0
$649$		−	967.118i		−	1.49017i
$650$		0			0
$651$	296.903	−	296.903i	0.456073	−	0.456073i
$652$		0			0
$653$	53.3064	−	53.3064i	0.0816331	−	0.0816331i	−0.665111	−	0.746744i	$-0.731616\pi$
$653$	53.3064	−	53.3064i	0.0816331	−	0.0816331i	0.746744	+	0.665111i	$0.231616\pi$
$654$		0			0
$655$		−	129.999i		−	0.198471i
$656$		0			0
$657$	−256.708			−0.390727
$658$		0			0
$659$	395.871	+	395.871i	0.600715	+	0.600715i	0.940502	−	0.339788i	$-0.110355\pi$
$659$	395.871	+	395.871i	0.600715	+	0.600715i	−0.339788	+	0.940502i	$0.610355\pi$
$660$		0			0
$661$	−692.251	−	692.251i	−1.04728	−	1.04728i	−0.998825	−	0.0484531i	$-0.984571\pi$
$661$	−692.251	−	692.251i	−1.04728	−	1.04728i	−0.0484531	−	0.998825i	$-0.515429\pi$
$662$		0			0
$663$	101.541			0.153153
$664$		0			0
$665$			209.779i			0.315458i
$666$		0			0
$667$	173.995	−	173.995i	0.260863	−	0.260863i
$668$		0			0
$669$	286.450	−	286.450i	0.428176	−	0.428176i
$670$		0			0
$671$			1139.07i			1.69757i
$672$		0			0
$673$	−673.261			−1.00039			−0.500194	−	0.865913i	$-0.666738\pi$
$673$	−673.261			−1.00039			−0.500194	+	0.865913i	$0.666738\pi$
$674$		0			0
$675$	−87.9178	−	87.9178i	−0.130249	−	0.130249i
$676$		0			0
$677$	660.922	+	660.922i	0.976251	+	0.976251i	0.999724	−	0.0234738i	$-0.00747264\pi$
$677$	660.922	+	660.922i	0.976251	+	0.976251i	−0.0234738	+	0.999724i	$0.507473\pi$
$678$		0			0
$679$	−395.871			−0.583021
$680$		0			0
$681$			126.480i			0.185726i
$682$		0			0
$683$	329.336	−	329.336i	0.482190	−	0.482190i	−0.423640	−	0.905830i	$-0.639248\pi$
$683$	329.336	−	329.336i	0.482190	−	0.482190i	0.905830	+	0.423640i	$0.139248\pi$
$684$		0			0
$685$	65.5411	−	65.5411i	0.0956804	−	0.0956804i
$686$		0			0
$687$			294.535i			0.428727i
$688$		0			0
$689$	111.131			0.161293
$690$		0			0
$691$	819.395	+	819.395i	1.18581	+	1.18581i	0.978215	+	0.207596i	$0.0665639\pi$
$691$	819.395	+	819.395i	1.18581	+	1.18581i	0.207596	+	0.978215i	$0.433436\pi$
$692$		0			0
$693$	517.808	+	517.808i	0.747197	+	0.747197i
$694$		0			0
$695$	−64.7711			−0.0931958
$696$		0			0
$697$			219.692i			0.315197i
$698$		0			0
$699$	205.487	−	205.487i	0.293973	−	0.293973i
$700$		0			0
$701$	74.2628	−	74.2628i	0.105938	−	0.105938i	−0.652151	−	0.758089i	$-0.726133\pi$
$701$	74.2628	−	74.2628i	0.105938	−	0.105938i	0.758089	+	0.652151i	$0.226133\pi$
$702$		0			0
$703$			429.203i			0.610530i
$704$		0			0
$705$	101.338			0.143742
$706$		0			0
$707$	−1090.76	−	1090.76i	−1.54280	−	1.54280i
$708$		0			0
$709$	−741.249	−	741.249i	−1.04548	−	1.04548i	−0.998915	−	0.0465697i	$-0.985171\pi$
$709$	−741.249	−	741.249i	−1.04548	−	1.04548i	−0.0465697	−	0.998915i	$-0.514829\pi$
$710$		0			0
$711$	200.459			0.281940
$712$		0			0
$713$		−	839.769i		−	1.17780i
$714$		0			0
$715$	156.064	−	156.064i	0.218271	−	0.218271i
$716$		0			0
$717$	−223.377	+	223.377i	−0.311544	+	0.311544i
$718$		0			0
$719$		−	722.873i		−	1.00539i	−0.864465	−	0.502694i	$-0.832342\pi$
$719$		−	722.873i		−	1.00539i	0.864465	−	0.502694i	$-0.167658\pi$
$720$		0			0
$721$	−469.692			−0.651445
$722$		0			0
$723$	−202.698	−	202.698i	−0.280357	−	0.280357i
$724$		0			0
$725$	−100.406	−	100.406i	−0.138492	−	0.138492i
$726$		0			0
$727$	565.138			0.777356			0.388678	−	0.921374i	$-0.372932\pi$
$727$	565.138			0.777356			0.388678	+	0.921374i	$0.372932\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	164.263	−	164.263i	0.224709	−	0.224709i
$732$		0			0
$733$	−469.315	+	469.315i	−0.640267	+	0.640267i	−0.950621	−	0.310354i	$-0.899552\pi$
$733$	−469.315	+	469.315i	−0.640267	+	0.640267i	0.310354	+	0.950621i	$0.399552\pi$
$734$		0			0
$735$			169.062i			0.230016i
$736$		0			0
$737$	1104.11			1.49812
$738$		0			0
$739$	−323.090	−	323.090i	−0.437199	−	0.437199i	0.453870	−	0.891068i	$-0.350043\pi$
$739$	−323.090	−	323.090i	−0.437199	−	0.437199i	−0.891068	+	0.453870i	$0.850043\pi$
$740$		0			0
$741$	−216.746	−	216.746i	−0.292505	−	0.292505i
$742$		0			0
$743$	−224.610			−0.302301			−0.151151	−	0.988511i	$-0.548298\pi$
$743$	−224.610			−0.302301			−0.151151	+	0.988511i	$0.548298\pi$
$744$		0			0
$745$		−	146.487i		−	0.196627i
$746$		0			0
$747$	−307.967	+	307.967i	−0.412271	+	0.412271i
$748$		0			0
$749$	−1431.04	+	1431.04i	−1.91060	+	1.91060i
$750$		0			0
$751$		−	1120.87i		−	1.49251i	−0.665662	−	0.746254i	$-0.731851\pi$
$751$		−	1120.87i		−	1.49251i	0.665662	−	0.746254i	$-0.268149\pi$
$752$		0			0
$753$	−620.105			−0.823513
$754$		0			0
$755$	115.688	+	115.688i	0.153229	+	0.153229i
$756$		0			0
$757$	63.3360	+	63.3360i	0.0836671	+	0.0836671i	0.747702	−	0.664035i	$-0.231157\pi$
$757$	63.3360	+	63.3360i	0.0836671	+	0.0836671i	−0.664035	+	0.747702i	$0.731157\pi$
$758$		0			0
$759$	1464.58			1.92962
$760$		0			0
$761$		−	168.879i		−	0.221918i	−0.993825	−	0.110959i	$-0.964608\pi$
$761$		−	168.879i		−	0.221918i	0.993825	−	0.110959i	$-0.0353922\pi$
$762$		0			0
$763$	762.247	−	762.247i	0.999012	−	0.999012i
$764$		0			0
$765$	−12.3154	+	12.3154i	−0.0160985	+	0.0160985i
$766$		0			0
$767$			495.802i			0.646417i
$768$		0			0
$769$	141.979			0.184629			0.0923143	−	0.995730i	$-0.470574\pi$
$769$	141.979			0.184629			0.0923143	+	0.995730i	$0.470574\pi$
$770$		0			0
$771$	581.653	+	581.653i	0.754414	+	0.754414i
$772$		0			0
$773$	548.729	+	548.729i	0.709870	+	0.709870i	0.966508	−	0.256638i	$-0.0826148\pi$
$773$	548.729	+	548.729i	0.709870	+	0.709870i	−0.256638	+	0.966508i	$0.582615\pi$
$774$		0			0
$775$	−484.600			−0.625290
$776$		0			0
$777$			525.664i			0.676530i
$778$		0			0
$779$	468.950	−	468.950i	0.601989	−	0.601989i
$780$		0			0
$781$	−388.918	+	388.918i	−0.497974	+	0.497974i
$782$		0			0
$783$		−	30.8353i		−	0.0393810i
$784$		0			0
$785$	78.8437			0.100438
$786$		0			0
$787$	303.407	+	303.407i	0.385523	+	0.385523i	0.873087	−	0.487564i	$-0.162114\pi$
$787$	303.407	+	303.407i	0.385523	+	0.385523i	−0.487564	+	0.873087i	$0.662114\pi$
$788$		0			0
$789$	−443.405	−	443.405i	−0.561984	−	0.561984i
$790$		0			0
$791$	−2106.60			−2.66322
$792$		0			0
$793$		−	583.954i		−	0.736386i
$794$		0			0
$795$	−13.4785	+	13.4785i	−0.0169541	+	0.0169541i
$796$		0			0
$797$	−925.678	+	925.678i	−1.16145	+	1.16145i	−0.177295	+	0.984158i	$0.556735\pi$
$797$	−925.678	+	925.678i	−1.16145	+	1.16145i	−0.984158	+	0.177295i	$0.943265\pi$
$798$		0			0
$799$			316.914i			0.396638i
$800$		0			0
$801$	−255.646			−0.319159
$802$		0			0
$803$	−1233.87	−	1233.87i	−1.53657	−	1.53657i
$804$		0			0
$805$	363.349	+	363.349i	0.451365	+	0.451365i
$806$		0			0
$807$	245.241			0.303892
$808$		0			0
$809$		−	212.382i		−	0.262524i	−0.991348	−	0.131262i	$-0.958097\pi$
$809$		−	212.382i		−	0.262524i	0.991348	−	0.131262i	$-0.0419029\pi$
$810$		0			0
$811$	−528.775	+	528.775i	−0.652003	+	0.652003i	−0.953475	−	0.301472i	$-0.902522\pi$
$811$	−528.775	+	528.775i	−0.652003	+	0.652003i	0.301472	+	0.953475i	$0.402522\pi$
$812$		0			0
$813$	−204.573	+	204.573i	−0.251627	+	0.251627i
$814$		0			0
$815$		−	143.898i		−	0.176562i
$816$		0			0
$817$	−701.261			−0.858337
$818$		0			0
$819$	−265.459	−	265.459i	−0.324125	−	0.324125i
$820$		0			0
$821$	−773.542	−	773.542i	−0.942195	−	0.942195i	0.0562233	−	0.998418i	$-0.482094\pi$
$821$	−773.542	−	773.542i	−0.942195	−	0.942195i	−0.998418	+	0.0562233i	$0.982094\pi$
$822$		0			0
$823$	709.750			0.862394			0.431197	−	0.902258i	$-0.358091\pi$
$823$	709.750			0.862394			0.431197	+	0.902258i	$0.358091\pi$
$824$		0			0
$825$		−	845.156i		−	1.02443i
$826$		0			0
$827$	155.643	−	155.643i	0.188202	−	0.188202i	−0.606716	−	0.794918i	$-0.707514\pi$
$827$	155.643	−	155.643i	0.188202	−	0.188202i	0.794918	+	0.606716i	$0.207514\pi$
$828$		0			0
$829$	403.238	−	403.238i	0.486415	−	0.486415i	−0.420758	−	0.907173i	$-0.638236\pi$
$829$	403.238	−	403.238i	0.486415	−	0.486415i	0.907173	+	0.420758i	$0.138236\pi$
$830$		0			0
$831$		−	233.122i		−	0.280532i
$832$		0			0
$833$	−528.705			−0.634700
$834$		0			0
$835$	−168.270	−	168.270i	−0.201520	−	0.201520i
$836$		0			0
$837$	−74.4115	−	74.4115i	−0.0889027	−	0.0889027i
$838$		0			0
$839$	−1590.26			−1.89543			−0.947714	−	0.319119i	$-0.896613\pi$
$839$	−1590.26			−1.89543			−0.947714	+	0.319119i	$0.896613\pi$
$840$		0			0
$841$			805.785i			0.958127i
$842$		0			0
$843$	−276.158	+	276.158i	−0.327589	+	0.327589i
$844$		0			0
$845$	43.7090	−	43.7090i	0.0517267	−	0.0517267i
$846$		0			0
$847$			3529.32i			4.16685i
$848$		0			0
$849$	−681.549			−0.802766
$850$		0			0
$851$	743.401	+	743.401i	0.873562	+	0.873562i
$852$		0			0
$853$	161.154	+	161.154i	0.188926	+	0.188926i	0.795232	−	0.606306i	$-0.207349\pi$
$853$	161.154	+	161.154i	0.188926	+	0.188926i	−0.606306	+	0.795232i	$0.707349\pi$
$854$		0			0
$855$	52.5761			0.0614925
$856$		0			0
$857$			1449.60i			1.69148i	0.533592	+	0.845742i	$0.320842\pi$
$857$			1449.60i			1.69148i	−0.533592	+	0.845742i	$0.679158\pi$
$858$		0			0
$859$	−330.215	+	330.215i	−0.384418	+	0.384418i	−0.872691	−	0.488273i	$-0.837627\pi$
$859$	−330.215	+	330.215i	−0.384418	+	0.384418i	0.488273	+	0.872691i	$0.337627\pi$
$860$		0			0
$861$	574.344	−	574.344i	0.667066	−	0.667066i
$862$		0			0
$863$			1065.50i			1.23465i	0.786709	+	0.617324i	$0.211783\pi$
$863$			1065.50i			1.23465i	−0.786709	+	0.617324i	$0.788217\pi$
$864$		0			0
$865$	−238.200			−0.275376
$866$		0			0
$867$	315.438	+	315.438i	0.363827	+	0.363827i
$868$		0			0
$869$	963.510	+	963.510i	1.10876	+	1.10876i
$870$		0			0
$871$	−566.033			−0.649866
$872$		0			0
$873$			99.2154i			0.113649i
$874$		0			0
$875$	428.742	−	428.742i	0.489991	−	0.489991i
$876$		0			0
$877$	−538.959	+	538.959i	−0.614548	+	0.614548i	−0.944128	−	0.329580i	$-0.893093\pi$
$877$	−538.959	+	538.959i	−0.614548	+	0.614548i	0.329580	+	0.944128i	$0.393093\pi$
$878$		0			0
$879$			358.338i			0.407666i
$880$		0			0
$881$	38.2515			0.0434182			0.0217091	−	0.999764i	$-0.493089\pi$
$881$	38.2515			0.0434182			0.0217091	+	0.999764i	$0.493089\pi$
$882$		0			0
$883$	−1018.18	−	1018.18i	−1.15309	−	1.15309i	−0.985929	−	0.167166i	$-0.946538\pi$
$883$	−1018.18	−	1018.18i	−1.15309	−	1.15309i	−0.167166	−	0.985929i	$-0.553462\pi$
$884$		0			0
$885$	−60.1333	−	60.1333i	−0.0679472	−	0.0679472i
$886$		0			0
$887$	−194.284			−0.219035			−0.109518	−	0.993985i	$-0.534931\pi$
$887$	−194.284			−0.219035			−0.109518	+	0.993985i	$0.534931\pi$
$888$		0			0
$889$			2313.65i			2.60253i
$890$		0			0
$891$	129.776	−	129.776i	0.145652	−	0.145652i
$892$		0			0
$893$	676.477	−	676.477i	0.757533	−	0.757533i
$894$		0			0
$895$		−	53.9140i		−	0.0602391i
$896$		0			0
$897$	−750.831			−0.837046
$898$		0			0
$899$	−84.9816	−	84.9816i	−0.0945290	−	0.0945290i
$900$		0			0
$901$	−42.1511	−	42.1511i	−0.0467826	−	0.0467826i
$902$		0			0
$903$	−858.866			−0.951125
$904$		0			0
$905$		−	214.956i		−	0.237521i
$906$		0			0
$907$	61.6381	−	61.6381i	0.0679582	−	0.0679582i	−0.672311	−	0.740269i	$-0.734698\pi$
$907$	61.6381	−	61.6381i	0.0679582	−	0.0679582i	0.740269	+	0.672311i	$0.234698\pi$
$908$		0			0
$909$	−273.373	+	273.373i	−0.300740	+	0.300740i
$910$		0			0
$911$			1378.63i			1.51331i	0.653812	+	0.756657i	$0.273169\pi$
$911$			1378.63i			1.51331i	−0.653812	+	0.756657i	$0.726831\pi$
$912$		0			0
$913$	−2960.49			−3.24260
$914$		0			0
$915$	70.8248	+	70.8248i	0.0774042	+	0.0774042i
$916$		0			0
$917$	−1062.83	−	1062.83i	−1.15903	−	1.15903i
$918$		0			0
$919$	486.051			0.528891			0.264445	−	0.964401i	$-0.414811\pi$
$919$	486.051			0.528891			0.264445	+	0.964401i	$0.414811\pi$
$920$		0			0
$921$		−	339.482i		−	0.368601i
$922$		0			0
$923$	199.382	−	199.382i	0.216015	−	0.216015i
$924$		0			0
$925$	428.990	−	428.990i	0.463773	−	0.463773i
$926$		0			0
$927$			117.717i			0.126987i
$928$		0			0
$929$	182.008			0.195918			0.0979589	−	0.995190i	$-0.468769\pi$
$929$	182.008			0.195918			0.0979589	+	0.995190i	$0.468769\pi$
$930$		0			0
$931$	1128.56	+	1128.56i	1.21220	+	1.21220i
$932$		0			0
$933$	46.7743	+	46.7743i	0.0501332	+	0.0501332i
$934$		0			0
$935$	−118.388			−0.126618
$936$		0			0
$937$		−	883.723i		−	0.943141i	−0.881828	−	0.471571i	$-0.843687\pi$
$937$		−	883.723i		−	0.943141i	0.881828	−	0.471571i	$-0.156313\pi$
$938$		0			0
$939$	167.445	−	167.445i	0.178322	−	0.178322i
$940$		0			0
$941$	−477.258	+	477.258i	−0.507181	+	0.507181i	−0.913660	−	0.406479i	$-0.866756\pi$
$941$	−477.258	+	477.258i	−0.507181	+	0.507181i	0.406479	+	0.913660i	$0.366756\pi$
$942$		0			0
$943$		−	1624.49i		−	1.72268i
$944$		0			0
$945$	64.3923			0.0681400
$946$		0			0
$947$	−41.7774	−	41.7774i	−0.0441155	−	0.0441155i	0.684705	−	0.728820i	$-0.259931\pi$
$947$	−41.7774	−	41.7774i	−0.0441155	−	0.0441155i	−0.728820	+	0.684705i	$0.759931\pi$
$948$		0			0
$949$	632.554	+	632.554i	0.666548	+	0.666548i
$950$		0			0
$951$	−170.821			−0.179622
$952$		0			0
$953$			84.1718i			0.0883229i	0.999024	+	0.0441615i	$0.0140616\pi$
$953$			84.1718i			0.0883229i	−0.999024	+	0.0441615i	$0.985938\pi$
$954$		0			0
$955$	123.827	−	123.827i	0.129662	−	0.129662i
$956$		0			0
$957$	148.210	−	148.210i	0.154870	−	0.154870i
$958$		0			0
$959$		−	1071.69i		−	1.11750i
$960$		0			0
$961$	550.846			0.573201
$962$		0			0
$963$	358.655	+	358.655i	0.372436	+	0.372436i
$964$		0			0
$965$	−16.9461	−	16.9461i	−0.0175607	−	0.0175607i
$966$		0			0
$967$	1276.14			1.31969			0.659844	−	0.751402i	$-0.270622\pi$
$967$	1276.14			1.31969			0.659844	+	0.751402i	$0.270622\pi$
$968$		0			0
$969$			164.420i			0.169681i
$970$		0			0
$971$	352.409	−	352.409i	0.362934	−	0.362934i	−0.501958	−	0.864892i	$-0.667387\pi$
$971$	352.409	−	352.409i	0.362934	−	0.362934i	0.864892	+	0.501958i	$0.167387\pi$
$972$		0			0
$973$	−529.549	+	529.549i	−0.544243	+	0.544243i
$974$		0			0
$975$			433.277i			0.444387i
$976$		0			0
$977$	711.808			0.728565			0.364282	−	0.931289i	$-0.381314\pi$
$977$	711.808			0.728565			0.364282	+	0.931289i	$0.381314\pi$
$978$		0			0
$979$	−1228.77	−	1228.77i	−1.25512	−	1.25512i
$980$		0			0
$981$	−191.038	−	191.038i	−0.194739	−	0.194739i
$982$		0			0
$983$	−895.864			−0.911357			−0.455679	−	0.890144i	$-0.650603\pi$
$983$	−895.864			−0.911357			−0.455679	+	0.890144i	$0.650603\pi$
$984$		0			0
$985$			195.580i			0.198558i
$986$		0			0
$987$	828.511	−	828.511i	0.839424	−	0.839424i
$988$		0			0
$989$	−1214.62	+	1214.62i	−1.22813	+	1.22813i
$990$		0			0
$991$			1277.48i			1.28908i	0.764570	+	0.644541i	$0.222952\pi$
$991$			1277.48i			1.28908i	−0.764570	+	0.644541i	$0.777048\pi$
$992$		0			0
$993$	−115.061			−0.115873
$994$		0			0
$995$	251.532	+	251.532i	0.252796	+	0.252796i
$996$		0			0
$997$	−1177.08	−	1177.08i	−1.18062	−	1.18062i	−0.979584	−	0.201035i	$-0.935570\pi$
$997$	−1177.08	−	1177.08i	−1.18062	−	1.18062i	−0.201035	−	0.979584i	$-0.564430\pi$
$998$		0			0
$999$	131.745			0.131877

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(-0.732051 - 0.732051i) q^{5} -11.9700 q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 - 1.22474i) q^{3} +(-0.732051 - 0.732051i) q^{5} -11.9700 q^{7} +3.00000i q^{9} +(-14.4195 + 14.4195i) q^{11} +(7.39230 - 7.39230i) q^{13} +1.79315i q^{15} -5.60770 q^{17} +(11.9700 + 11.9700i) q^{19} +(14.6603 + 14.6603i) q^{21} +41.4655 q^{23} -23.9282i q^{25} +(3.67423 - 3.67423i) q^{27} +(4.19615 - 4.19615i) q^{29} -20.2523i q^{31} +35.3205 q^{33} +(8.76268 + 8.76268i) q^{35} +(17.9282 + 17.9282i) q^{37} -18.1074 q^{39} -39.1769i q^{41} +(-29.2923 + 29.2923i) q^{43} +(2.19615 - 2.19615i) q^{45} -56.5141i q^{47} +94.2820 q^{49} +(6.86800 + 6.86800i) q^{51} +(7.51666 + 7.51666i) q^{53} +21.1117 q^{55} -29.3205i q^{57} +(-33.5350 + 33.5350i) q^{59} +(39.4974 - 39.4974i) q^{61} -35.9101i q^{63} -10.8231 q^{65} +(-38.2853 - 38.2853i) q^{67} +(-50.7846 - 50.7846i) q^{69} +26.9716 q^{71} +85.5692i q^{73} +(-29.3059 + 29.3059i) q^{75} +(172.603 - 172.603i) q^{77} -66.8198i q^{79} -9.00000 q^{81} +(102.656 + 102.656i) q^{83} +(4.10512 + 4.10512i) q^{85} -10.2784 q^{87} +85.2154i q^{89} +(-88.4862 + 88.4862i) q^{91} +(-24.8038 + 24.8038i) q^{93} -17.5254i q^{95} +33.0718 q^{97} +(-43.2586 - 43.2586i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more