LMFDB - Embedded newform 768.3.l.f.319.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("768.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 52 x^{14} + 1743 x^{12} - 34996 x^{10} + 513409 x^{8} - 5039424 x^{6} + 36142848 x^{4} + \cdots + 429981696 $ x^16 - 52x^14 + 1743x^12 - 34996x^10 + 513409x^8 - 5039424x^6 + 36142848x^4 - 155271168*x^2 + 429981696
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			319.8
Root			$-2.27793 + 1.31516i$ of defining polynomial
Character	$\chi$	$=$	768.319
Dual form			768.3.l.f.703.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +(6.45189 + 6.45189i) q^{5} +8.74541 q^{7} +3.00000i q^{9} +O(q^{10})$	\(q+(1.22474 + 1.22474i) q^{3} +(6.45189 + 6.45189i) q^{5} +8.74541 q^{7} +3.00000i q^{9} +(-9.12435 + 9.12435i) q^{11} +(-9.18721 + 9.18721i) q^{13} +15.8038i q^{15} +18.9038 q^{17} +(-2.79852 - 2.79852i) q^{19} +(10.7109 + 10.7109i) q^{21} -2.27362 q^{23} +58.2538i q^{25} +(-3.67423 + 3.67423i) q^{27} +(-10.8865 + 10.8865i) q^{29} -50.9089i q^{31} -22.3500 q^{33} +(56.4245 + 56.4245i) q^{35} +(-47.2012 - 47.2012i) q^{37} -22.5040 q^{39} -27.6038i q^{41} +(35.9127 - 35.9127i) q^{43} +(-19.3557 + 19.3557i) q^{45} -61.0614i q^{47} +27.4823 q^{49} +(23.1523 + 23.1523i) q^{51} +(53.7789 + 53.7789i) q^{53} -117.739 q^{55} -6.85494i q^{57} +(20.5821 - 20.5821i) q^{59} +(-12.6936 + 12.6936i) q^{61} +26.2362i q^{63} -118.550 q^{65} +(43.7298 + 43.7298i) q^{67} +(-2.78461 - 2.78461i) q^{69} -20.5223 q^{71} +107.836i q^{73} +(-71.3460 + 71.3460i) q^{75} +(-79.7962 + 79.7962i) q^{77} -49.9282i q^{79} -9.00000 q^{81} +(15.0096 + 15.0096i) q^{83} +(121.965 + 121.965i) q^{85} -26.6664 q^{87} +62.0691i q^{89} +(-80.3460 + 80.3460i) q^{91} +(62.3504 - 62.3504i) q^{93} -36.1114i q^{95} -22.7518 q^{97} +(-27.3731 - 27.3731i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{5}+O(q^{10}) $ 16 * q + 8 * q^5	$ 16 q + 8 q^{5} - 32 q^{13} + 112 q^{17} + 72 q^{21} - 56 q^{29} - 48 q^{33} + 272 q^{37} - 24 q^{45} + 304 q^{49} + 504 q^{53} - 176 q^{61} + 624 q^{65} + 288 q^{69} - 848 q^{77} - 144 q^{81} + 880 q^{85} + 216 q^{93} - 160 q^{97}+O(q^{100}) $ 16 * q + 8 * q^5 - 32 * q^13 + 112 * q^17 + 72 * q^21 - 56 * q^29 - 48 * q^33 + 272 * q^37 - 24 * q^45 + 304 * q^49 + 504 * q^53 - 176 * q^61 + 624 * q^65 + 288 * q^69 - 848 * q^77 - 144 * q^81 + 880 * q^85 + 216 * q^93 - 160 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$5$	6.45189	+	6.45189i	1.29038	+	1.29038i	0.934552	+	0.355826i	$0.115800\pi$
$5$	6.45189	+	6.45189i	1.29038	+	1.29038i	0.355826	+	0.934552i	$0.384200\pi$
$6$		0			0
$7$	8.74541			1.24934			0.624672	−	0.780887i	$-0.285233\pi$
$7$	8.74541			1.24934			0.624672	+	0.780887i	$0.285233\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	−9.12435	+	9.12435i	−0.829486	+	0.829486i	−0.987446	−	0.157959i	$-0.949509\pi$
$11$	−9.12435	+	9.12435i	−0.829486	+	0.829486i	0.157959	+	0.987446i	$0.449509\pi$
$12$		0			0
$13$	−9.18721	+	9.18721i	−0.706709	+	0.706709i	−0.965842	−	0.259133i	$-0.916563\pi$
$13$	−9.18721	+	9.18721i	−0.706709	+	0.706709i	0.259133	+	0.965842i	$0.416563\pi$
$14$		0			0
$15$			15.8038i			1.05359i
$16$		0			0
$17$	18.9038			1.11199			0.555994	−	0.831187i	$-0.312338\pi$
$17$	18.9038			1.11199			0.555994	+	0.831187i	$0.312338\pi$
$18$		0			0
$19$	−2.79852	−	2.79852i	−0.147290	−	0.147290i	0.629616	−	0.776906i	$-0.283212\pi$
$19$	−2.79852	−	2.79852i	−0.147290	−	0.147290i	−0.776906	+	0.629616i	$0.783212\pi$
$20$		0			0
$21$	10.7109	+	10.7109i	0.510043	+	0.510043i
$22$		0			0
$23$	−2.27362			−0.0988532			−0.0494266	−	0.998778i	$-0.515739\pi$
$23$	−2.27362			−0.0988532			−0.0494266	+	0.998778i	$0.515739\pi$
$24$		0			0
$25$			58.2538i			2.33015i
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$		0			0
$29$	−10.8865	+	10.8865i	−0.375396	+	0.375396i	−0.869438	−	0.494042i	$-0.835519\pi$
$29$	−10.8865	+	10.8865i	−0.375396	+	0.375396i	0.494042	+	0.869438i	$0.335519\pi$
$30$		0			0
$31$		−	50.9089i		−	1.64222i	−0.570767	−	0.821112i	$-0.693354\pi$
$31$		−	50.9089i		−	1.64222i	0.570767	−	0.821112i	$-0.306646\pi$
$32$		0			0
$33$	−22.3500			−0.677273
$34$		0			0
$35$	56.4245	+	56.4245i	1.61213	+	1.61213i
$36$		0			0
$37$	−47.2012	−	47.2012i	−1.27571	−	1.27571i	−0.943046	−	0.332661i	$-0.892053\pi$
$37$	−47.2012	−	47.2012i	−1.27571	−	1.27571i	−0.332661	−	0.943046i	$-0.607947\pi$
$38$		0			0
$39$	−22.5040			−0.577025
$40$		0			0
$41$		−	27.6038i		−	0.673263i	−0.941636	−	0.336632i	$-0.890712\pi$
$41$		−	27.6038i		−	0.673263i	0.941636	−	0.336632i	$-0.109288\pi$
$42$		0			0
$43$	35.9127	−	35.9127i	0.835179	−	0.835179i	−0.153041	−	0.988220i	$-0.548907\pi$
$43$	35.9127	−	35.9127i	0.835179	−	0.835179i	0.988220	+	0.153041i	$0.0489066\pi$
$44$		0			0
$45$	−19.3557	+	19.3557i	−0.430126	+	0.430126i
$46$		0			0
$47$		−	61.0614i		−	1.29918i	−0.760285	−	0.649589i	$-0.774941\pi$
$47$		−	61.0614i		−	1.29918i	0.760285	−	0.649589i	$-0.225059\pi$
$48$		0			0
$49$	27.4823			0.560863
$50$		0			0
$51$	23.1523	+	23.1523i	0.453967	+	0.453967i
$52$		0			0
$53$	53.7789	+	53.7789i	1.01470	+	1.01470i	0.999890	+	0.0148067i	$0.00471328\pi$
$53$	53.7789	+	53.7789i	1.01470	+	1.01470i	0.0148067	+	0.999890i	$0.495287\pi$
$54$		0			0
$55$	−117.739			−2.14070
$56$		0			0
$57$		−	6.85494i		−	0.120262i
$58$		0			0
$59$	20.5821	−	20.5821i	0.348850	−	0.348850i	−0.510831	−	0.859681i	$-0.670662\pi$
$59$	20.5821	−	20.5821i	0.348850	−	0.348850i	0.859681	+	0.510831i	$0.170662\pi$
$60$		0			0
$61$	−12.6936	+	12.6936i	−0.208092	+	0.208092i	−0.803456	−	0.595364i	$-0.797008\pi$
$61$	−12.6936	+	12.6936i	−0.208092	+	0.208092i	0.595364	+	0.803456i	$0.297008\pi$
$62$		0			0
$63$			26.2362i			0.416448i
$64$		0			0
$65$	−118.550			−1.82384
$66$		0			0
$67$	43.7298	+	43.7298i	0.652684	+	0.652684i	0.953639	−	0.300954i	$-0.0973052\pi$
$67$	43.7298	+	43.7298i	0.652684	+	0.652684i	−0.300954	+	0.953639i	$0.597305\pi$
$68$		0			0
$69$	−2.78461	−	2.78461i	−0.0403567	−	0.0403567i
$70$		0			0
$71$	−20.5223			−0.289047			−0.144523	−	0.989501i	$-0.546165\pi$
$71$	−20.5223			−0.289047			−0.144523	+	0.989501i	$0.546165\pi$
$72$		0			0
$73$			107.836i			1.47720i	0.674142	+	0.738601i	$0.264513\pi$
$73$			107.836i			1.47720i	−0.674142	+	0.738601i	$0.735487\pi$
$74$		0			0
$75$	−71.3460	+	71.3460i	−0.951280	+	0.951280i
$76$		0			0
$77$	−79.7962	+	79.7962i	−1.03631	+	1.03631i
$78$		0			0
$79$		−	49.9282i		−	0.632003i	−0.948759	−	0.316001i	$-0.897660\pi$
$79$		−	49.9282i		−	0.632003i	0.948759	−	0.316001i	$-0.102340\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	15.0096	+	15.0096i	0.180838	+	0.180838i	0.791721	−	0.610883i	$-0.209185\pi$
$83$	15.0096	+	15.0096i	0.180838	+	0.180838i	−0.610883	+	0.791721i	$0.709185\pi$
$84$		0			0
$85$	121.965	+	121.965i	1.43488	+	1.43488i
$86$		0			0
$87$	−26.6664			−0.306510
$88$		0			0
$89$			62.0691i			0.697406i	0.937233	+	0.348703i	$0.113378\pi$
$89$			62.0691i			0.697406i	−0.937233	+	0.348703i	$0.886622\pi$
$90$		0			0
$91$	−80.3460	+	80.3460i	−0.882923	+	0.882923i
$92$		0			0
$93$	62.3504	−	62.3504i	0.670435	−	0.670435i
$94$		0			0
$95$		−	36.1114i		−	0.380120i
$96$		0			0
$97$	−22.7518			−0.234555			−0.117277	−	0.993099i	$-0.537417\pi$
$97$	−22.7518			−0.234555			−0.117277	+	0.993099i	$0.537417\pi$
$98$		0			0
$99$	−27.3731	−	27.3731i	−0.276495	−	0.276495i
$100$		0			0
$101$	73.5978	+	73.5978i	0.728691	+	0.728691i	0.970359	−	0.241668i	$-0.0776944\pi$
$101$	73.5978	+	73.5978i	0.728691	+	0.728691i	−0.241668	+	0.970359i	$0.577694\pi$
$102$		0			0
$103$	78.0596			0.757860			0.378930	−	0.925425i	$-0.376292\pi$
$103$	78.0596			0.757860			0.378930	+	0.925425i	$0.376292\pi$
$104$		0			0
$105$			138.211i			1.31630i
$106$		0			0
$107$	70.8088	−	70.8088i	0.661764	−	0.661764i	−0.294032	−	0.955796i	$-0.594997\pi$
$107$	70.8088	−	70.8088i	0.661764	−	0.661764i	0.955796	+	0.294032i	$0.0949971\pi$
$108$		0			0
$109$	48.9000	−	48.9000i	0.448624	−	0.448624i	−0.446273	−	0.894897i	$-0.647249\pi$
$109$	48.9000	−	48.9000i	0.448624	−	0.448624i	0.894897	+	0.446273i	$0.147249\pi$
$110$		0			0
$111$		−	115.619i		−	1.04161i
$112$		0			0
$113$	176.123			1.55861			0.779304	−	0.626646i	$-0.215573\pi$
$113$	176.123			1.55861			0.779304	+	0.626646i	$0.215573\pi$
$114$		0			0
$115$	−14.6692	−	14.6692i	−0.127558	−	0.127558i
$116$		0			0
$117$	−27.5616	−	27.5616i	−0.235570	−	0.235570i
$118$		0			0
$119$	165.321			1.38926
$120$		0			0
$121$		−	45.5076i		−	0.376096i
$122$		0			0
$123$	33.8076	−	33.8076i	0.274859	−	0.274859i
$124$		0			0
$125$	−214.550	+	214.550i	−1.71640	+	1.71640i
$126$		0			0
$127$			58.7585i			0.462665i	0.972875	+	0.231333i	$0.0743085\pi$
$127$			58.7585i			0.462665i	−0.972875	+	0.231333i	$0.925691\pi$
$128$		0			0
$129$	87.9678			0.681921
$130$		0			0
$131$	−127.992	−	127.992i	−0.977034	−	0.977034i	0.0227077	−	0.999742i	$-0.492771\pi$
$131$	−127.992	−	127.992i	−0.977034	−	0.977034i	−0.999742	+	0.0227077i	$0.992771\pi$
$132$		0			0
$133$	−24.4742	−	24.4742i	−0.184016	−	0.184016i
$134$		0			0
$135$	−47.4115			−0.351196
$136$		0			0
$137$		−	19.5573i		−	0.142754i	−0.997449	−	0.0713771i	$-0.977261\pi$
$137$		−	19.5573i		−	0.142754i	0.997449	−	0.0713771i	$-0.0227394\pi$
$138$		0			0
$139$	−166.618	+	166.618i	−1.19869	+	1.19869i	−0.224132	+	0.974559i	$0.571955\pi$
$139$	−166.618	+	166.618i	−1.19869	+	1.19869i	−0.974559	+	0.224132i	$0.928045\pi$
$140$		0			0
$141$	74.7846	−	74.7846i	0.530387	−	0.530387i
$142$		0			0
$143$		−	167.655i		−	1.17241i
$144$		0			0
$145$	−140.477			−0.968807
$146$		0			0
$147$	33.6588	+	33.6588i	0.228971	+	0.228971i
$148$		0			0
$149$	−52.7707	−	52.7707i	−0.354166	−	0.354166i	0.507491	−	0.861657i	$-0.330573\pi$
$149$	−52.7707	−	52.7707i	−0.354166	−	0.354166i	−0.861657	+	0.507491i	$0.830573\pi$
$150$		0			0
$151$	−158.397			−1.04899			−0.524494	−	0.851414i	$-0.675745\pi$
$151$	−158.397			−1.04899			−0.524494	+	0.851414i	$0.675745\pi$
$152$		0			0
$153$			56.7113i			0.370662i
$154$		0			0
$155$	328.459	−	328.459i	2.11909	−	2.11909i
$156$		0			0
$157$	−24.9498	+	24.9498i	−0.158916	+	0.158916i	−0.782086	−	0.623170i	$-0.785844\pi$
$157$	−24.9498	+	24.9498i	−0.158916	+	0.158916i	0.623170	+	0.782086i	$0.285844\pi$
$158$		0			0
$159$			131.731i			0.828497i
$160$		0			0
$161$	−19.8838			−0.123502
$162$		0			0
$163$	211.378	+	211.378i	1.29680	+	1.29680i	0.930495	+	0.366304i	$0.119377\pi$
$163$	211.378	+	211.378i	1.29680	+	1.29680i	0.366304	+	0.930495i	$0.380623\pi$
$164$		0			0
$165$	−144.200	−	144.200i	−0.873938	−	0.873938i
$166$		0			0
$167$	−27.6692			−0.165684			−0.0828420	−	0.996563i	$-0.526400\pi$
$167$	−27.6692			−0.165684			−0.0828420	+	0.996563i	$0.526400\pi$
$168$		0			0
$169$			0.190221i			0.00112557i
$170$		0			0
$171$	8.39555	−	8.39555i	0.0490968	−	0.0490968i
$172$		0			0
$173$	223.544	−	223.544i	1.29216	−	1.29216i	0.358713	−	0.933448i	$-0.383216\pi$
$173$	223.544	−	223.544i	1.29216	−	1.29216i	0.933448	−	0.358713i	$-0.116784\pi$
$174$		0			0
$175$			509.453i			2.91116i
$176$		0			0
$177$	50.4158			0.284835
$178$		0			0
$179$	140.980	+	140.980i	0.787599	+	0.787599i	0.981100	−	0.193501i	$-0.0619842\pi$
$179$	140.980	+	140.980i	0.787599	+	0.787599i	−0.193501	+	0.981100i	$0.561984\pi$
$180$		0			0
$181$	−124.679	−	124.679i	−0.688836	−	0.688836i	0.273139	−	0.961975i	$-0.411938\pi$
$181$	−124.679	−	124.679i	−0.688836	−	0.688836i	−0.961975	+	0.273139i	$0.911938\pi$
$182$		0			0
$183$	−31.0929			−0.169906
$184$		0			0
$185$		−	609.074i		−	3.29229i
$186$		0			0
$187$	−172.485	+	172.485i	−0.922378	+	0.922378i
$188$		0			0
$189$	−32.1327	+	32.1327i	−0.170014	+	0.170014i
$190$		0			0
$191$		−	57.3579i		−	0.300303i	−0.988663	−	0.150152i	$-0.952024\pi$
$191$		−	57.3579i		−	0.300303i	0.988663	−	0.150152i	$-0.0479762\pi$
$192$		0			0
$193$	−117.999			−0.611394			−0.305697	−	0.952129i	$-0.598890\pi$
$193$	−117.999			−0.611394			−0.305697	+	0.952129i	$0.598890\pi$
$194$		0			0
$195$	−145.193	−	145.193i	−0.744581	−	0.744581i
$196$		0			0
$197$	−69.9405	−	69.9405i	−0.355028	−	0.355028i	0.506948	−	0.861976i	$-0.330773\pi$
$197$	−69.9405	−	69.9405i	−0.355028	−	0.355028i	−0.861976	+	0.506948i	$0.830773\pi$
$198$		0			0
$199$	−34.3390			−0.172558			−0.0862789	−	0.996271i	$-0.527498\pi$
$199$	−34.3390			−0.172558			−0.0862789	+	0.996271i	$0.527498\pi$
$200$		0			0
$201$			107.116i			0.532914i
$202$		0			0
$203$	−95.2069	+	95.2069i	−0.469000	+	0.469000i
$204$		0			0
$205$	178.097	−	178.097i	0.868764	−	0.868764i
$206$		0			0
$207$		−	6.82087i		−	0.0329511i
$208$		0			0
$209$	51.0693			0.244351
$210$		0			0
$211$	−248.707	−	248.707i	−1.17871	−	1.17871i	−0.980074	−	0.198634i	$-0.936349\pi$
$211$	−248.707	−	248.707i	−1.17871	−	1.17871i	−0.198634	−	0.980074i	$-0.563651\pi$
$212$		0			0
$213$	−25.1346	−	25.1346i	−0.118003	−	0.118003i
$214$		0			0
$215$	463.410			2.15539
$216$		0			0
$217$		−	445.220i		−	2.05170i
$218$		0			0
$219$	−132.071	+	132.071i	−0.603066	+	0.603066i
$220$		0			0
$221$	−173.673	+	173.673i	−0.785851	+	0.785851i
$222$		0			0
$223$			40.4010i			0.181171i	0.995889	+	0.0905853i	$0.0288738\pi$
$223$			40.4010i			0.181171i	−0.995889	+	0.0905853i	$0.971126\pi$
$224$		0			0
$225$	−174.761			−0.776717
$226$		0			0
$227$	98.2253	+	98.2253i	0.432711	+	0.432711i	0.889549	−	0.456839i	$-0.151018\pi$
$227$	98.2253	+	98.2253i	0.432711	+	0.432711i	−0.456839	+	0.889549i	$0.651018\pi$
$228$		0			0
$229$	−21.3388	−	21.3388i	−0.0931823	−	0.0931823i	0.658979	−	0.752161i	$-0.270989\pi$
$229$	−21.3388	−	21.3388i	−0.0931823	−	0.0931823i	−0.752161	+	0.658979i	$0.770989\pi$
$230$		0			0
$231$	−195.460			−0.846147
$232$		0			0
$233$		−	113.685i		−	0.487918i	−0.969786	−	0.243959i	$-0.921554\pi$
$233$		−	113.685i		−	0.487918i	0.969786	−	0.243959i	$-0.0784462\pi$
$234$		0			0
$235$	393.961	−	393.961i	1.67643	−	1.67643i
$236$		0			0
$237$	61.1493	−	61.1493i	0.258014	−	0.258014i
$238$		0			0
$239$			51.7997i			0.216735i	0.994111	+	0.108368i	$0.0345623\pi$
$239$			51.7997i			0.216735i	−0.994111	+	0.108368i	$0.965438\pi$
$240$		0			0
$241$	320.161			1.32847			0.664234	−	0.747524i	$-0.268758\pi$
$241$	320.161			1.32847			0.664234	+	0.747524i	$0.268758\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$		0			0
$245$	177.313	+	177.313i	0.723725	+	0.723725i
$246$		0			0
$247$	51.4211			0.208183
$248$		0			0
$249$			36.7658i			0.147654i
$250$		0			0
$251$	59.7377	−	59.7377i	0.237999	−	0.237999i	−0.578022	−	0.816021i	$-0.696175\pi$
$251$	59.7377	−	59.7377i	0.237999	−	0.237999i	0.816021	+	0.578022i	$0.196175\pi$
$252$		0			0
$253$	20.7453	−	20.7453i	0.0819974	−	0.0819974i
$254$		0			0
$255$			298.752i			1.17158i
$256$		0			0
$257$	266.452			1.03678			0.518388	−	0.855145i	$-0.326532\pi$
$257$	266.452			1.03678			0.518388	+	0.855145i	$0.326532\pi$
$258$		0			0
$259$	−412.794	−	412.794i	−1.59380	−	1.59380i
$260$		0			0
$261$	−32.6595	−	32.6595i	−0.125132	−	0.125132i
$262$		0			0
$263$	520.169			1.97783			0.988915	−	0.148484i	$-0.0474392\pi$
$263$	520.169			1.97783			0.988915	+	0.148484i	$0.0474392\pi$
$264$		0			0
$265$			693.952i			2.61869i
$266$		0			0
$267$	−76.0189	+	76.0189i	−0.284715	+	0.284715i
$268$		0			0
$269$	187.187	−	187.187i	0.695861	−	0.695861i	−0.267654	−	0.963515i	$-0.586248\pi$
$269$	187.187	−	187.187i	0.695861	−	0.695861i	0.963515	+	0.267654i	$0.0862484\pi$
$270$		0			0
$271$		−	71.9182i		−	0.265381i	−0.991158	−	0.132690i	$-0.957638\pi$
$271$		−	71.9182i		−	0.265381i	0.991158	−	0.132690i	$-0.0423616\pi$
$272$		0			0
$273$	−196.807			−0.720904
$274$		0			0
$275$	−531.528	−	531.528i	−1.93283	−	1.93283i
$276$		0			0
$277$	−30.4229	−	30.4229i	−0.109830	−	0.109830i	0.650056	−	0.759886i	$-0.274745\pi$
$277$	−30.4229	−	30.4229i	−0.109830	−	0.109830i	−0.759886	+	0.650056i	$0.774745\pi$
$278$		0			0
$279$	152.727			0.547408
$280$		0			0
$281$		−	277.361i		−	0.987048i	−0.869732	−	0.493524i	$-0.835708\pi$
$281$		−	277.361i		−	0.987048i	0.869732	−	0.493524i	$-0.164292\pi$
$282$		0			0
$283$	−134.427	+	134.427i	−0.475005	+	0.475005i	−0.903530	−	0.428525i	$-0.859034\pi$
$283$	−134.427	+	134.427i	−0.475005	+	0.475005i	0.428525	+	0.903530i	$0.359034\pi$
$284$		0			0
$285$	44.2273	−	44.2273i	0.155184	−	0.155184i
$286$		0			0
$287$		−	241.407i		−	0.841138i
$288$		0			0
$289$	68.3529			0.236515
$290$		0			0
$291$	−27.8651	−	27.8651i	−0.0957565	−	0.0957565i
$292$		0			0
$293$	43.9833	+	43.9833i	0.150114	+	0.150114i	0.778169	−	0.628055i	$-0.216149\pi$
$293$	43.9833	+	43.9833i	0.150114	+	0.150114i	−0.628055	+	0.778169i	$0.716149\pi$
$294$		0			0
$295$	265.588			0.900297
$296$		0			0
$297$		−	67.0500i		−	0.225758i
$298$		0			0
$299$	20.8883	−	20.8883i	0.0698604	−	0.0698604i
$300$		0			0
$301$	314.071	−	314.071i	1.04343	−	1.04343i
$302$		0			0
$303$			180.277i			0.594974i
$304$		0			0
$305$	−163.796			−0.537035
$306$		0			0
$307$	−39.0814	−	39.0814i	−0.127301	−	0.127301i	0.640586	−	0.767887i	$-0.278692\pi$
$307$	−39.0814	−	39.0814i	−0.127301	−	0.127301i	−0.767887	+	0.640586i	$0.778692\pi$
$308$		0			0
$309$	95.6030	+	95.6030i	0.309395	+	0.309395i
$310$		0			0
$311$	−335.635			−1.07921			−0.539607	−	0.841917i	$-0.681427\pi$
$311$	−335.635			−1.07921			−0.539607	+	0.841917i	$0.681427\pi$
$312$		0			0
$313$			250.537i			0.800438i	0.916420	+	0.400219i	$0.131066\pi$
$313$			250.537i			0.800438i	−0.916420	+	0.400219i	$0.868934\pi$
$314$		0			0
$315$	−169.273	+	169.273i	−0.537376	+	0.537376i
$316$		0			0
$317$	26.1406	−	26.1406i	0.0824623	−	0.0824623i	−0.664673	−	0.747135i	$-0.731429\pi$
$317$	26.1406	−	26.1406i	0.0824623	−	0.0824623i	0.747135	+	0.664673i	$0.231429\pi$
$318$		0			0
$319$		−	198.664i		−	0.622772i
$320$		0			0
$321$	173.445			0.540328
$322$		0			0
$323$	−52.9025	−	52.9025i	−0.163785	−	0.163785i
$324$		0			0
$325$	−535.190	−	535.190i	−1.64674	−	1.64674i
$326$		0			0
$327$	119.780			0.366300
$328$		0			0
$329$		−	534.007i		−	1.62312i
$330$		0			0
$331$	68.2754	−	68.2754i	0.206270	−	0.206270i	−0.596410	−	0.802680i	$-0.703407\pi$
$331$	68.2754	−	68.2754i	0.206270	−	0.206270i	0.802680	+	0.596410i	$0.203407\pi$
$332$		0			0
$333$	141.604	−	141.604i	0.425236	−	0.425236i
$334$		0			0
$335$			564.280i			1.68442i
$336$		0			0
$337$	360.713			1.07037			0.535183	−	0.844736i	$-0.320243\pi$
$337$	360.713			1.07037			0.535183	+	0.844736i	$0.320243\pi$
$338$		0			0
$339$	215.705	+	215.705i	0.636299	+	0.636299i
$340$		0			0
$341$	464.511	+	464.511i	1.36220	+	1.36220i
$342$		0			0
$343$	−188.182			−0.548634
$344$		0			0
$345$		−	35.9320i		−	0.104151i
$346$		0			0
$347$	59.5347	−	59.5347i	0.171570	−	0.171570i	−0.616099	−	0.787669i	$-0.711288\pi$
$347$	59.5347	−	59.5347i	0.171570	−	0.171570i	0.787669	+	0.616099i	$0.211288\pi$
$348$		0			0
$349$	148.260	−	148.260i	0.424813	−	0.424813i	−0.462044	−	0.886857i	$-0.652884\pi$
$349$	148.260	−	148.260i	0.424813	−	0.424813i	0.886857	+	0.462044i	$0.152884\pi$
$350$		0			0
$351$		−	67.5120i		−	0.192342i
$352$		0			0
$353$	−643.683			−1.82346			−0.911732	−	0.410786i	$-0.865254\pi$
$353$	−643.683			−1.82346			−0.911732	+	0.410786i	$0.865254\pi$
$354$		0			0
$355$	−132.408	−	132.408i	−0.372980	−	0.372980i
$356$		0			0
$357$	202.477	+	202.477i	0.567161	+	0.567161i
$358$		0			0
$359$	−511.721			−1.42541			−0.712703	−	0.701466i	$-0.752529\pi$
$359$	−511.721			−1.42541			−0.712703	+	0.701466i	$0.752529\pi$
$360$		0			0
$361$		−	345.337i		−	0.956611i
$362$		0			0
$363$	55.7352	−	55.7352i	0.153540	−	0.153540i
$364$		0			0
$365$	−695.745	+	695.745i	−1.90615	+	1.90615i
$366$		0			0
$367$		−	417.149i		−	1.13665i	−0.822806	−	0.568323i	$-0.807593\pi$
$367$		−	417.149i		−	1.13665i	0.822806	−	0.568323i	$-0.192407\pi$
$368$		0			0
$369$	82.8114			0.224421
$370$		0			0
$371$	470.319	+	470.319i	1.26771	+	1.26771i
$372$		0			0
$373$	−125.813	−	125.813i	−0.337301	−	0.337301i	0.518050	−	0.855350i	$-0.326658\pi$
$373$	−125.813	−	125.813i	−0.337301	−	0.337301i	−0.855350	+	0.518050i	$0.826658\pi$
$374$		0			0
$375$	−525.538			−1.40143
$376$		0			0
$377$		−	200.033i		−	0.530592i
$378$		0			0
$379$	−95.6205	+	95.6205i	−0.252297	+	0.252297i	−0.821912	−	0.569615i	$-0.807092\pi$
$379$	−95.6205	+	95.6205i	−0.252297	+	0.252297i	0.569615	+	0.821912i	$0.307092\pi$
$380$		0			0
$381$	−71.9641	+	71.9641i	−0.188882	+	0.188882i
$382$		0			0
$383$		−	34.1848i		−	0.0892554i	−0.999004	−	0.0446277i	$-0.985790\pi$
$383$		−	34.1848i		−	0.0892554i	0.999004	−	0.0446277i	$-0.0142102\pi$
$384$		0			0
$385$	−1029.67			−2.67448
$386$		0			0
$387$	107.738	+	107.738i	0.278393	+	0.278393i
$388$		0			0
$389$	−476.532	−	476.532i	−1.22502	−	1.22502i	−0.965826	−	0.259192i	$-0.916544\pi$
$389$	−476.532	−	476.532i	−1.22502	−	1.22502i	−0.259192	−	0.965826i	$-0.583456\pi$
$390$		0			0
$391$	−42.9801			−0.109924
$392$		0			0
$393$		−	313.514i		−	0.797745i
$394$		0			0
$395$	322.131	−	322.131i	0.815523	−	0.815523i
$396$		0			0
$397$	−193.928	+	193.928i	−0.488483	+	0.488483i	−0.907827	−	0.419344i	$-0.862260\pi$
$397$	−193.928	+	193.928i	−0.488483	+	0.488483i	0.419344	+	0.907827i	$0.362260\pi$
$398$		0			0
$399$		−	59.9493i		−	0.150249i
$400$		0			0
$401$	458.989			1.14461			0.572306	−	0.820040i	$-0.306049\pi$
$401$	458.989			1.14461			0.572306	+	0.820040i	$0.306049\pi$
$402$		0			0
$403$	467.711	+	467.711i	1.16057	+	1.16057i
$404$		0			0
$405$	−58.0670	−	58.0670i	−0.143375	−	0.143375i
$406$		0			0
$407$	861.360			2.11636
$408$		0			0
$409$		−	93.9350i		−	0.229670i	−0.993385	−	0.114835i	$-0.963366\pi$
$409$		−	93.9350i		−	0.229670i	0.993385	−	0.114835i	$-0.0366339\pi$
$410$		0			0
$411$	23.9527	−	23.9527i	0.0582791	−	0.0582791i
$412$		0			0
$413$	179.999	−	179.999i	0.435834	−	0.435834i
$414$		0			0
$415$			193.680i			0.466699i
$416$		0			0
$417$	−408.129			−0.978727
$418$		0			0
$419$	263.922	+	263.922i	0.629885	+	0.629885i	0.948039	−	0.318154i	$-0.103063\pi$
$419$	263.922	+	263.922i	0.629885	+	0.629885i	−0.318154	+	0.948039i	$0.603063\pi$
$420$		0			0
$421$	180.379	+	180.379i	0.428455	+	0.428455i	0.888102	−	0.459647i	$-0.152024\pi$
$421$	180.379	+	180.379i	0.428455	+	0.428455i	−0.459647	+	0.888102i	$0.652024\pi$
$422$		0			0
$423$	183.184			0.433059
$424$		0			0
$425$			1101.22i			2.59110i
$426$		0			0
$427$	−111.011	+	111.011i	−0.259979	+	0.259979i
$428$		0			0
$429$	205.334	−	205.334i	0.478635	−	0.478635i
$430$		0			0
$431$			523.280i			1.21411i	0.794661	+	0.607054i	$0.207649\pi$
$431$			523.280i			1.21411i	−0.794661	+	0.607054i	$0.792351\pi$
$432$		0			0
$433$	477.399			1.10254			0.551269	−	0.834328i	$-0.314144\pi$
$433$	477.399			1.10254			0.551269	+	0.834328i	$0.314144\pi$
$434$		0			0
$435$	−172.048	−	172.048i	−0.395514	−	0.395514i
$436$		0			0
$437$	6.36278	+	6.36278i	0.0145601	+	0.0145601i
$438$		0			0
$439$	−295.808			−0.673823			−0.336911	−	0.941536i	$-0.609382\pi$
$439$	−295.808			−0.673823			−0.336911	+	0.941536i	$0.609382\pi$
$440$		0			0
$441$			82.4468i			0.186954i
$442$		0			0
$443$	156.953	−	156.953i	0.354297	−	0.354297i	−0.507409	−	0.861705i	$-0.669397\pi$
$443$	156.953	−	156.953i	0.354297	−	0.354297i	0.861705	+	0.507409i	$0.169397\pi$
$444$		0			0
$445$	−400.463	+	400.463i	−0.899918	+	0.899918i
$446$		0			0
$447$		−	129.261i		−	0.289175i
$448$		0			0
$449$	−109.390			−0.243631			−0.121816	−	0.992553i	$-0.538872\pi$
$449$	−109.390			−0.243631			−0.121816	+	0.992553i	$0.538872\pi$
$450$		0			0
$451$	251.867	+	251.867i	0.558463	+	0.558463i
$452$		0			0
$453$	−193.996	−	193.996i	−0.428248	−	0.428248i
$454$		0			0
$455$	−1036.77			−2.27861
$456$		0			0
$457$		−	146.854i		−	0.321344i	−0.987008	−	0.160672i	$-0.948634\pi$
$457$		−	146.854i		−	0.321344i	0.987008	−	0.160672i	$-0.0513661\pi$
$458$		0			0
$459$	−69.4569	+	69.4569i	−0.151322	+	0.151322i
$460$		0			0
$461$	235.902	−	235.902i	0.511717	−	0.511717i	−0.403335	−	0.915052i	$-0.632149\pi$
$461$	235.902	−	235.902i	0.511717	−	0.511717i	0.915052	+	0.403335i	$0.132149\pi$
$462$		0			0
$463$		−	685.886i		−	1.48140i	−0.671838	−	0.740698i	$-0.734495\pi$
$463$		−	685.886i		−	1.48140i	0.671838	−	0.740698i	$-0.265505\pi$
$464$		0			0
$465$	804.557			1.73023
$466$		0			0
$467$	−379.119	−	379.119i	−0.811818	−	0.811818i	0.173088	−	0.984906i	$-0.444625\pi$
$467$	−379.119	−	379.119i	−0.811818	−	0.811818i	−0.984906	+	0.173088i	$0.944625\pi$
$468$		0			0
$469$	382.435	+	382.435i	0.815427	+	0.815427i
$470$		0			0
$471$	−61.1142			−0.129754
$472$		0			0
$473$			655.360i			1.38554i
$474$		0			0
$475$	163.024	−	163.024i	0.343209	−	0.343209i
$476$		0			0
$477$	−161.337	+	161.337i	−0.338232	+	0.338232i
$478$		0			0
$479$		−	837.131i		−	1.74766i	−0.486228	−	0.873832i	$-0.661627\pi$
$479$		−	837.131i		−	1.74766i	0.486228	−	0.873832i	$-0.338373\pi$
$480$		0			0
$481$	867.295			1.80311
$482$		0			0
$483$	−24.3526	−	24.3526i	−0.0504194	−	0.0504194i
$484$		0			0
$485$	−146.792	−	146.792i	−0.302664	−	0.302664i
$486$		0			0
$487$	−73.6884			−0.151311			−0.0756554	−	0.997134i	$-0.524105\pi$
$487$	−73.6884			−0.151311			−0.0756554	+	0.997134i	$0.524105\pi$
$488$		0			0
$489$			517.769i			1.05883i
$490$		0			0
$491$	−651.908	+	651.908i	−1.32772	+	1.32772i	−0.420356	+	0.907359i	$0.638095\pi$
$491$	−651.908	+	651.908i	−1.32772	+	1.32772i	−0.907359	+	0.420356i	$0.861905\pi$
$492$		0			0
$493$	−205.796	+	205.796i	−0.417436	+	0.417436i
$494$		0			0
$495$		−	353.216i		−	0.713567i
$496$		0			0
$497$	−179.476			−0.361119
$498$		0			0
$499$	413.800	+	413.800i	0.829258	+	0.829258i	0.987414	−	0.158156i	$-0.0505550\pi$
$499$	413.800	+	413.800i	0.829258	+	0.829258i	−0.158156	+	0.987414i	$0.550555\pi$
$500$		0			0
$501$	−33.8877	−	33.8877i	−0.0676402	−	0.0676402i
$502$		0			0
$503$	293.356			0.583212			0.291606	−	0.956539i	$-0.405810\pi$
$503$	293.356			0.583212			0.291606	+	0.956539i	$0.405810\pi$
$504$		0			0
$505$			949.690i			1.88058i
$506$		0			0
$507$	−0.232972	+	0.232972i	−0.000459510	+	0.000459510i
$508$		0			0
$509$	−188.400	+	188.400i	−0.370137	+	0.370137i	−0.867527	−	0.497390i	$-0.834292\pi$
$509$	−188.400	+	188.400i	−0.370137	+	0.370137i	0.497390	+	0.867527i	$0.334292\pi$
$510$		0			0
$511$			943.069i			1.84554i
$512$		0			0
$513$	20.5648			0.0400874
$514$		0			0
$515$	503.632	+	503.632i	0.977926	+	0.977926i
$516$		0			0
$517$	557.145	+	557.145i	1.07765	+	1.07765i
$518$		0			0
$519$	547.568			1.05504
$520$		0			0
$521$			566.596i			1.08752i	0.839242	+	0.543758i	$0.182999\pi$
$521$			566.596i			1.08752i	−0.839242	+	0.543758i	$0.817001\pi$
$522$		0			0
$523$	−514.447	+	514.447i	−0.983647	+	0.983647i	−0.999868	−	0.0162219i	$-0.994836\pi$
$523$	−514.447	+	514.447i	−0.983647	+	0.983647i	0.0162219	+	0.999868i	$0.494836\pi$
$524$		0			0
$525$	−623.951	+	623.951i	−1.18848	+	1.18848i
$526$		0			0
$527$		−	962.371i		−	1.82613i
$528$		0			0
$529$	−523.831			−0.990228
$530$		0			0
$531$	61.7464	+	61.7464i	0.116283	+	0.116283i
$532$		0			0
$533$	253.602	+	253.602i	0.475801	+	0.475801i
$534$		0			0
$535$	913.701			1.70785
$536$		0			0
$537$			345.330i			0.643072i
$538$		0			0
$539$	−250.758	+	250.758i	−0.465228	+	0.465228i
$540$		0			0
$541$	−215.820	+	215.820i	−0.398927	+	0.398927i	−0.877855	−	0.478927i	$-0.841026\pi$
$541$	−215.820	+	215.820i	−0.398927	+	0.398927i	0.478927	+	0.877855i	$0.341026\pi$
$542$		0			0
$543$		−	305.401i		−	0.562432i
$544$		0			0
$545$	630.995			1.15779
$546$		0			0
$547$	−457.897	−	457.897i	−0.837107	−	0.837107i	0.151371	−	0.988477i	$-0.451631\pi$
$547$	−457.897	−	457.897i	−0.837107	−	0.837107i	−0.988477	+	0.151371i	$0.951631\pi$
$548$		0			0
$549$	−38.0808	−	38.0808i	−0.0693640	−	0.0693640i
$550$		0			0
$551$	60.9321			0.110585
$552$		0			0
$553$		−	436.643i		−	0.789589i
$554$		0			0
$555$	745.960	−	745.960i	1.34407	−	1.34407i
$556$		0			0
$557$	8.65563	−	8.65563i	0.0155397	−	0.0155397i	−0.699294	−	0.714834i	$-0.746502\pi$
$557$	8.65563	−	8.65563i	0.0155397	−	0.0155397i	0.714834	+	0.699294i	$0.246502\pi$
$558$		0			0
$559$			659.875i			1.18046i
$560$		0			0
$561$	−422.500			−0.753119
$562$		0			0
$563$	−498.328	−	498.328i	−0.885129	−	0.885129i	0.108921	−	0.994050i	$-0.465260\pi$
$563$	−498.328	−	498.328i	−0.885129	−	0.885129i	−0.994050	+	0.108921i	$0.965260\pi$
$564$		0			0
$565$	1136.32	+	1136.32i	2.01119	+	2.01119i
$566$		0			0
$567$	−78.7087			−0.138816
$568$		0			0
$569$			349.853i			0.614856i	0.951571	+	0.307428i	$0.0994683\pi$
$569$			349.853i			0.614856i	−0.951571	+	0.307428i	$0.900532\pi$
$570$		0			0
$571$	548.510	−	548.510i	0.960613	−	0.960613i	−0.0386402	−	0.999253i	$-0.512303\pi$
$571$	548.510	−	548.510i	0.960613	−	0.960613i	0.999253	+	0.0386402i	$0.0123026\pi$
$572$		0			0
$573$	70.2488	−	70.2488i	0.122598	−	0.122598i
$574$		0			0
$575$		−	132.447i		−	0.230343i
$576$		0			0
$577$	−620.371			−1.07517			−0.537583	−	0.843211i	$-0.680663\pi$
$577$	−620.371			−1.07517			−0.537583	+	0.843211i	$0.680663\pi$
$578$		0			0
$579$	−144.519	−	144.519i	−0.249601	−	0.249601i
$580$		0			0
$581$	131.265	+	131.265i	0.225929	+	0.225929i
$582$		0			0
$583$	−981.396			−1.68335
$584$		0			0
$585$		−	355.649i		−	0.607948i
$586$		0			0
$587$	−247.623	+	247.623i	−0.421845	+	0.421845i	−0.885839	−	0.463994i	$-0.846416\pi$
$587$	−247.623	+	247.623i	−0.421845	+	0.421845i	0.463994	+	0.885839i	$0.346416\pi$
$588$		0			0
$589$	−142.469	+	142.469i	−0.241884	+	0.241884i
$590$		0			0
$591$		−	171.319i		−	0.289879i
$592$		0			0
$593$	146.614			0.247242			0.123621	−	0.992330i	$-0.460549\pi$
$593$	146.614			0.247242			0.123621	+	0.992330i	$0.460549\pi$
$594$		0			0
$595$	1066.64	+	1066.64i	1.79266	+	1.79266i
$596$		0			0
$597$	−42.0565	−	42.0565i	−0.0704464	−	0.0704464i
$598$		0			0
$599$	−849.637			−1.41842			−0.709212	−	0.704995i	$-0.750949\pi$
$599$	−849.637			−1.41842			−0.709212	+	0.704995i	$0.750949\pi$
$600$		0			0
$601$		−	661.753i		−	1.10109i	−0.834807	−	0.550543i	$-0.814421\pi$
$601$		−	661.753i		−	1.10109i	0.834807	−	0.550543i	$-0.185579\pi$
$602$		0			0
$603$	−131.189	+	131.189i	−0.217561	+	0.217561i
$604$		0			0
$605$	293.610	−	293.610i	0.485306	−	0.485306i
$606$		0			0
$607$			12.7879i			0.0210674i	0.999945	+	0.0105337i	$0.00335304\pi$
$607$			12.7879i			0.0210674i	−0.999945	+	0.0105337i	$0.996647\pi$
$608$		0			0
$609$	−233.208			−0.382937
$610$		0			0
$611$	560.984	+	560.984i	0.918141	+	0.918141i
$612$		0			0
$613$	−22.0125	−	22.0125i	−0.0359094	−	0.0359094i	0.688924	−	0.724833i	$-0.258083\pi$
$613$	−22.0125	−	22.0125i	−0.0359094	−	0.0359094i	−0.724833	+	0.688924i	$0.758083\pi$
$614$		0			0
$615$	436.246			0.709343
$616$		0			0
$617$			340.322i			0.551575i	0.961219	+	0.275787i	$0.0889386\pi$
$617$			340.322i			0.551575i	−0.961219	+	0.275787i	$0.911061\pi$
$618$		0			0
$619$	−412.316	+	412.316i	−0.666100	+	0.666100i	−0.956811	−	0.290711i	$-0.906108\pi$
$619$	−412.316	+	412.316i	−0.666100	+	0.666100i	0.290711	+	0.956811i	$0.406108\pi$
$620$		0			0
$621$	8.35383	−	8.35383i	0.0134522	−	0.0134522i
$622$		0			0
$623$			542.820i			0.871301i
$624$		0			0
$625$	−1312.16			−2.09945
$626$		0			0
$627$	62.5469	+	62.5469i	0.0997558	+	0.0997558i
$628$		0			0
$629$	−892.281	−	892.281i	−1.41857	−	1.41857i
$630$		0			0
$631$	−462.693			−0.733270			−0.366635	−	0.930365i	$-0.619490\pi$
$631$	−462.693			−0.733270			−0.366635	+	0.930365i	$0.619490\pi$
$632$		0			0
$633$		−	609.206i		−	0.962411i
$634$		0			0
$635$	−379.103	+	379.103i	−0.597013	+	0.597013i
$636$		0			0
$637$	−252.485	+	252.485i	−0.396366	+	0.396366i
$638$		0			0
$639$		−	61.5670i		−	0.0963490i
$640$		0			0
$641$	419.144			0.653890			0.326945	−	0.945043i	$-0.393981\pi$
$641$	419.144			0.653890			0.326945	+	0.945043i	$0.393981\pi$
$642$		0			0
$643$	100.772	+	100.772i	0.156722	+	0.156722i	0.781112	−	0.624391i	$-0.214653\pi$
$643$	100.772	+	100.772i	0.156722	+	0.156722i	−0.624391	+	0.781112i	$0.714653\pi$
$644$		0			0
$645$	567.558	+	567.558i	0.879936	+	0.879936i
$646$		0			0
$647$	−200.577			−0.310011			−0.155006	−	0.987914i	$-0.549540\pi$
$647$	−200.577			−0.310011			−0.155006	+	0.987914i	$0.549540\pi$
$648$		0			0
$649$			375.597i			0.578733i
$650$		0			0
$651$	545.280	−	545.280i	0.837604	−	0.837604i
$652$		0			0
$653$	206.642	−	206.642i	0.316451	−	0.316451i	−0.530951	−	0.847402i	$-0.678165\pi$
$653$	206.642	−	206.642i	0.316451	−	0.316451i	0.847402	+	0.530951i	$0.178165\pi$
$654$		0			0
$655$		−	1651.57i		−	2.52149i
$656$		0			0
$657$	−323.507			−0.492401
$658$		0			0
$659$	−56.6668	−	56.6668i	−0.0859891	−	0.0859891i	0.662804	−	0.748793i	$-0.269366\pi$
$659$	−56.6668	−	56.6668i	−0.0859891	−	0.0859891i	−0.748793	+	0.662804i	$0.769366\pi$
$660$		0			0
$661$	−781.642	−	781.642i	−1.18251	−	1.18251i	−0.979091	−	0.203424i	$-0.934793\pi$
$661$	−781.642	−	781.642i	−1.18251	−	1.18251i	−0.203424	−	0.979091i	$-0.565207\pi$
$662$		0			0
$663$	−425.410			−0.641645
$664$		0			0
$665$		−	315.810i		−	0.474902i
$666$		0			0
$667$	24.7518	−	24.7518i	0.0371091	−	0.0371091i
$668$		0			0
$669$	−49.4810	+	49.4810i	−0.0739626	+	0.0739626i
$670$		0			0
$671$		−	231.642i		−	0.345219i
$672$		0			0
$673$	376.839			0.559939			0.279969	−	0.960009i	$-0.409676\pi$
$673$	376.839			0.559939			0.279969	+	0.960009i	$0.409676\pi$
$674$		0			0
$675$	−214.038	−	214.038i	−0.317093	−	0.317093i
$676$		0			0
$677$	307.017	+	307.017i	0.453497	+	0.453497i	0.896513	−	0.443017i	$-0.146092\pi$
$677$	307.017	+	307.017i	0.453497	+	0.453497i	−0.443017	+	0.896513i	$0.646092\pi$
$678$		0			0
$679$	−198.974			−0.293040
$680$		0			0
$681$			240.602i			0.353307i
$682$		0			0
$683$	−598.837	+	598.837i	−0.876775	+	0.876775i	−0.993200	−	0.116425i	$-0.962857\pi$
$683$	−598.837	+	598.837i	−0.876775	+	0.876775i	0.116425	+	0.993200i	$0.462857\pi$
$684$		0			0
$685$	126.182	−	126.182i	0.184207	−	0.184207i
$686$		0			0
$687$		−	52.2691i		−	0.0760831i
$688$		0			0
$689$	−988.157			−1.43419
$690$		0			0
$691$	−590.377	−	590.377i	−0.854381	−	0.854381i	0.136288	−	0.990669i	$-0.456483\pi$
$691$	−590.377	−	590.377i	−0.854381	−	0.854381i	−0.990669	+	0.136288i	$0.956483\pi$
$692$		0			0
$693$	−239.389	−	239.389i	−0.345438	−	0.345438i
$694$		0			0
$695$	−2150.00			−3.09353
$696$		0			0
$697$		−	521.816i		−	0.748660i
$698$		0			0
$699$	139.235	−	139.235i	0.199192	−	0.199192i
$700$		0			0
$701$	783.614	−	783.614i	1.11785	−	1.11785i	0.125795	−	0.992056i	$-0.459852\pi$
$701$	783.614	−	783.614i	1.11785	−	1.11785i	0.992056	−	0.125795i	$-0.0401481\pi$
$702$		0			0
$703$			264.187i			0.375799i
$704$		0			0
$705$	965.004			1.36880
$706$		0			0
$707$	643.644	+	643.644i	0.910387	+	0.910387i
$708$		0			0
$709$	49.1534	+	49.1534i	0.0693277	+	0.0693277i	0.740920	−	0.671593i	$-0.234390\pi$
$709$	49.1534	+	49.1534i	0.0693277	+	0.0693277i	−0.671593	+	0.740920i	$0.734390\pi$
$710$		0			0
$711$	149.785			0.210668
$712$		0			0
$713$			115.748i			0.162339i
$714$		0			0
$715$	1081.69	−	1081.69i	1.51285	−	1.51285i
$716$		0			0
$717$	−63.4414	+	63.4414i	−0.0884817	+	0.0884817i
$718$		0			0
$719$			727.032i			1.01117i	0.862776	+	0.505586i	$0.168724\pi$
$719$			727.032i			1.01117i	−0.862776	+	0.505586i	$0.831276\pi$
$720$		0			0
$721$	682.663			0.946828
$722$		0			0
$723$	392.115	+	392.115i	0.542345	+	0.542345i
$724$		0			0
$725$	−634.180	−	634.180i	−0.874730	−	0.874730i
$726$		0			0
$727$	948.459			1.30462			0.652310	−	0.757952i	$-0.273800\pi$
$727$	948.459			1.30462			0.652310	+	0.757952i	$0.273800\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	678.886	−	678.886i	0.928708	−	0.928708i
$732$		0			0
$733$	−267.069	+	267.069i	−0.364351	+	0.364351i	−0.865412	−	0.501061i	$-0.832943\pi$
$733$	−267.069	+	267.069i	−0.364351	+	0.364351i	0.501061	+	0.865412i	$0.332943\pi$
$734$		0			0
$735$			434.325i			0.590919i
$736$		0			0
$737$	−798.013			−1.08279
$738$		0			0
$739$	−591.857	−	591.857i	−0.800889	−	0.800889i	0.182345	−	0.983235i	$-0.441631\pi$
$739$	−591.857	−	591.857i	−0.800889	−	0.800889i	−0.983235	+	0.182345i	$0.941631\pi$
$740$		0			0
$741$	62.9778	+	62.9778i	0.0849902	+	0.0849902i
$742$		0			0
$743$	−78.9176			−0.106215			−0.0531074	−	0.998589i	$-0.516913\pi$
$743$	−78.9176			−0.106215			−0.0531074	+	0.998589i	$0.516913\pi$
$744$		0			0
$745$		−	680.942i		−	0.914016i
$746$		0			0
$747$	−45.0287	+	45.0287i	−0.0602794	+	0.0602794i
$748$		0			0
$749$	619.252	−	619.252i	0.826772	−	0.826772i
$750$		0			0
$751$		−	540.177i		−	0.719276i	−0.933092	−	0.359638i	$-0.882900\pi$
$751$		−	540.177i		−	0.719276i	0.933092	−	0.359638i	$-0.117100\pi$
$752$		0			0
$753$	146.327			0.194325
$754$		0			0
$755$	−1021.96	−	1021.96i	−1.35359	−	1.35359i
$756$		0			0
$757$	−925.760	−	925.760i	−1.22293	−	1.22293i	−0.966585	−	0.256348i	$-0.917481\pi$
$757$	−925.760	−	925.760i	−1.22293	−	1.22293i	−0.256348	−	0.966585i	$-0.582519\pi$
$758$		0			0
$759$	50.8155			0.0669506
$760$		0			0
$761$			1394.49i			1.83244i	0.400675	+	0.916220i	$0.368776\pi$
$761$			1394.49i			1.83244i	−0.400675	+	0.916220i	$0.631224\pi$
$762$		0			0
$763$	427.651	−	427.651i	0.560486	−	0.560486i
$764$		0			0
$765$	−365.895	+	365.895i	−0.478295	+	0.478295i
$766$		0			0
$767$			378.185i			0.493071i
$768$		0			0
$769$	754.308			0.980895			0.490447	−	0.871471i	$-0.336833\pi$
$769$	754.308			0.980895			0.490447	+	0.871471i	$0.336833\pi$
$770$		0			0
$771$	326.335	+	326.335i	0.423262	+	0.423262i
$772$		0			0
$773$	161.605	+	161.605i	0.209063	+	0.209063i	0.803869	−	0.594806i	$-0.202771\pi$
$773$	161.605	+	161.605i	0.209063	+	0.209063i	−0.594806	+	0.803869i	$0.702771\pi$
$774$		0			0
$775$	2965.64			3.82663
$776$		0			0
$777$		−	1011.13i		−	1.30133i
$778$		0			0
$779$	−77.2497	+	77.2497i	−0.0991652	+	0.0991652i
$780$		0			0
$781$	187.253	−	187.253i	0.239760	−	0.239760i
$782$		0			0
$783$		−	79.9991i		−	0.102170i
$784$		0			0
$785$	−321.946			−0.410123
$786$		0			0
$787$	−172.533	−	172.533i	−0.219229	−	0.219229i	0.588945	−	0.808173i	$-0.299544\pi$
$787$	−172.533	−	172.533i	−0.219229	−	0.219229i	−0.808173	+	0.588945i	$0.799544\pi$
$788$		0			0
$789$	637.075	+	637.075i	0.807446	+	0.807446i
$790$		0			0
$791$	1540.27			1.94724
$792$		0			0
$793$		−	233.238i		−	0.294121i
$794$		0			0
$795$	−849.914	+	849.914i	−1.06907	+	1.06907i
$796$		0			0
$797$	−227.620	+	227.620i	−0.285596	+	0.285596i	−0.835336	−	0.549740i	$-0.814727\pi$
$797$	−227.620	+	227.620i	−0.285596	+	0.285596i	0.549740	+	0.835336i	$0.314727\pi$
$798$		0			0
$799$		−	1154.29i		−	1.44467i
$800$		0			0
$801$	−186.207			−0.232469
$802$		0			0
$803$	−983.932	−	983.932i	−1.22532	−	1.22532i
$804$		0			0
$805$	−128.288	−	128.288i	−0.159364	−	0.159364i
$806$		0			0
$807$	458.512			0.568168
$808$		0			0
$809$			24.1128i			0.0298057i	0.999889	+	0.0149029i	$0.00474391\pi$
$809$			24.1128i			0.0298057i	−0.999889	+	0.0149029i	$0.995256\pi$
$810$		0			0
$811$	199.450	−	199.450i	0.245931	−	0.245931i	−0.573367	−	0.819299i	$-0.694363\pi$
$811$	199.450	−	199.450i	0.245931	−	0.245931i	0.819299	+	0.573367i	$0.194363\pi$
$812$		0			0
$813$	88.0815	−	88.0815i	0.108341	−	0.108341i
$814$		0			0
$815$			2727.58i			3.34672i
$816$		0			0
$817$	−201.005			−0.246028
$818$		0			0
$819$	−241.038	−	241.038i	−0.294308	−	0.294308i
$820$		0			0
$821$	−7.50885	−	7.50885i	−0.00914598	−	0.00914598i	0.702519	−	0.711665i	$-0.252059\pi$
$821$	−7.50885	−	7.50885i	−0.00914598	−	0.00914598i	−0.711665	+	0.702519i	$0.752059\pi$
$822$		0			0
$823$	−1221.45			−1.48414			−0.742072	−	0.670320i	$-0.766157\pi$
$823$	−1221.45			−1.48414			−0.742072	+	0.670320i	$0.766157\pi$
$824$		0			0
$825$		−	1301.97i		−	1.57815i
$826$		0			0
$827$	−513.090	+	513.090i	−0.620423	+	0.620423i	−0.945640	−	0.325216i	$-0.894563\pi$
$827$	−513.090	+	513.090i	−0.620423	+	0.620423i	0.325216	+	0.945640i	$0.394563\pi$
$828$		0			0
$829$	−142.936	+	142.936i	−0.172420	+	0.172420i	−0.788042	−	0.615622i	$-0.788905\pi$
$829$	−142.936	+	142.936i	−0.172420	+	0.172420i	0.615622	+	0.788042i	$0.288905\pi$
$830$		0			0
$831$		−	74.5206i		−	0.0896758i
$832$		0			0
$833$	519.519			0.623672
$834$		0			0
$835$	−178.519	−	178.519i	−0.213795	−	0.213795i
$836$		0			0
$837$	187.051	+	187.051i	0.223478	+	0.223478i
$838$		0			0
$839$	358.691			0.427523			0.213761	−	0.976886i	$-0.431429\pi$
$839$	358.691			0.427523			0.213761	+	0.976886i	$0.431429\pi$
$840$		0			0
$841$			603.968i			0.718155i
$842$		0			0
$843$	339.696	−	339.696i	0.402961	−	0.402961i
$844$		0			0
$845$	−1.22728	+	1.22728i	−0.00145240	+	0.00145240i
$846$		0			0
$847$		−	397.983i		−	0.469873i
$848$		0			0
$849$	−329.276			−0.387840
$850$		0			0
$851$	107.318	+	107.318i	0.126108	+	0.126108i
$852$		0			0
$853$	−108.362	−	108.362i	−0.127037	−	0.127037i	0.640730	−	0.767767i	$-0.278632\pi$
$853$	−108.362	−	108.362i	−0.127037	−	0.127037i	−0.767767	+	0.640730i	$0.778632\pi$
$854$		0			0
$855$	108.334			0.126707
$856$		0			0
$857$			332.547i			0.388037i	0.980998	+	0.194018i	$0.0621521\pi$
$857$			332.547i			0.388037i	−0.980998	+	0.194018i	$0.937848\pi$
$858$		0			0
$859$	−727.423	+	727.423i	−0.846826	+	0.846826i	−0.989736	−	0.142910i	$-0.954354\pi$
$859$	−727.423	+	727.423i	−0.846826	+	0.846826i	0.142910	+	0.989736i	$0.454354\pi$
$860$		0			0
$861$	295.661	−	295.661i	0.343393	−	0.343393i
$862$		0			0
$863$			818.786i			0.948767i	0.880318	+	0.474384i	$0.157329\pi$
$863$			818.786i			0.948767i	−0.880318	+	0.474384i	$0.842671\pi$
$864$		0			0
$865$	2884.56			3.33475
$866$		0			0
$867$	83.7149	+	83.7149i	0.0965570	+	0.0965570i
$868$		0			0
$869$	455.563	+	455.563i	0.524238	+	0.524238i
$870$		0			0
$871$	−803.511			−0.922515
$872$		0			0
$873$		−	68.2554i		−	0.0781849i
$874$		0			0
$875$	−1876.33	+	1876.33i	−2.14437	+	2.14437i
$876$		0			0
$877$	−148.834	+	148.834i	−0.169709	+	0.169709i	−0.786851	−	0.617143i	$-0.788290\pi$
$877$	−148.834	+	148.834i	−0.169709	+	0.169709i	0.617143	+	0.786851i	$0.288290\pi$
$878$		0			0
$879$			107.737i			0.122567i
$880$		0			0
$881$	971.130			1.10230			0.551152	−	0.834405i	$-0.314188\pi$
$881$	971.130			1.10230			0.551152	+	0.834405i	$0.314188\pi$
$882$		0			0
$883$	−724.455	−	724.455i	−0.820447	−	0.820447i	0.165725	−	0.986172i	$-0.447004\pi$
$883$	−724.455	−	724.455i	−0.820447	−	0.820447i	−0.986172	+	0.165725i	$0.947004\pi$
$884$		0			0
$885$	325.277	+	325.277i	0.367545	+	0.367545i
$886$		0			0
$887$	1575.17			1.77584			0.887920	−	0.459999i	$-0.152150\pi$
$887$	1575.17			1.77584			0.887920	+	0.459999i	$0.152150\pi$
$888$		0			0
$889$			513.867i			0.578028i
$890$		0			0
$891$	82.1192	−	82.1192i	0.0921652	−	0.0921652i
$892$		0			0
$893$	−170.881	+	170.881i	−0.191356	+	0.191356i
$894$		0			0
$895$			1819.18i			2.03260i
$896$		0			0
$897$	51.1656			0.0570408
$898$		0			0
$899$	554.220	+	554.220i	0.616485	+	0.616485i
$900$		0			0
$901$	1016.63	+	1016.63i	1.12833	+	1.12833i
$902$		0			0
$903$	769.315			0.851954
$904$		0			0
$905$		−	1608.83i		−	1.77772i
$906$		0			0
$907$	713.184	−	713.184i	0.786310	−	0.786310i	−0.194577	−	0.980887i	$-0.562333\pi$
$907$	713.184	−	713.184i	0.786310	−	0.786310i	0.980887	+	0.194577i	$0.0623334\pi$
$908$		0			0
$909$	−220.794	+	220.794i	−0.242897	+	0.242897i
$910$		0			0
$911$			1386.77i			1.52225i	0.648604	+	0.761126i	$0.275353\pi$
$911$			1386.77i			1.52225i	−0.648604	+	0.761126i	$0.724647\pi$
$912$		0			0
$913$	−273.905			−0.300005
$914$		0			0
$915$	−200.608	−	200.608i	−0.219244	−	0.219244i
$916$		0			0
$917$	−1119.34	−	1119.34i	−1.22065	−	1.22065i
$918$		0			0
$919$	1717.04			1.86838			0.934191	−	0.356774i	$-0.116123\pi$
$919$	1717.04			1.86838			0.934191	+	0.356774i	$0.116123\pi$
$920$		0			0
$921$		−	95.7296i		−	0.103941i
$922$		0			0
$923$	188.543	−	188.543i	0.204272	−	0.204272i
$924$		0			0
$925$	2749.65	−	2749.65i	2.97259	−	2.97259i
$926$		0			0
$927$			234.179i			0.252620i
$928$		0			0
$929$	−926.210			−0.996997			−0.498499	−	0.866890i	$-0.666115\pi$
$929$	−926.210			−0.996997			−0.498499	+	0.866890i	$0.666115\pi$
$930$		0			0
$931$	−76.9096	−	76.9096i	−0.0826096	−	0.0826096i
$932$		0			0
$933$	−411.068	−	411.068i	−0.440587	−	0.440587i
$934$		0			0
$935$	−2225.71			−2.38043
$936$		0			0
$937$		−	1149.08i		−	1.22634i	−0.789952	−	0.613168i	$-0.789895\pi$
$937$		−	1149.08i		−	1.22634i	0.789952	−	0.613168i	$-0.210105\pi$
$938$		0			0
$939$	−306.844	+	306.844i	−0.326777	+	0.326777i
$940$		0			0
$941$	552.794	−	552.794i	0.587454	−	0.587454i	−0.349487	−	0.936941i	$-0.613644\pi$
$941$	552.794	−	552.794i	0.587454	−	0.587454i	0.936941	+	0.349487i	$0.113644\pi$
$942$		0			0
$943$			62.7606i			0.0665542i
$944$		0			0
$945$	−414.633			−0.438765
$946$		0			0
$947$	−494.177	−	494.177i	−0.521834	−	0.521834i	0.396291	−	0.918125i	$-0.370297\pi$
$947$	−494.177	−	494.177i	−0.521834	−	0.521834i	−0.918125	+	0.396291i	$0.870297\pi$
$948$		0			0
$949$	−990.711	−	990.711i	−1.04395	−	1.04395i
$950$		0			0
$951$	64.0310			0.0673302
$952$		0			0
$953$			350.020i			0.367282i	0.982993	+	0.183641i	$0.0587884\pi$
$953$			350.020i			0.367282i	−0.982993	+	0.183641i	$0.941212\pi$
$954$		0			0
$955$	370.067	−	370.067i	0.387505	−	0.387505i
$956$		0			0
$957$	243.313	−	243.313i	0.254246	−	0.254246i
$958$		0			0
$959$		−	171.037i		−	0.178349i
$960$		0			0
$961$	−1630.72			−1.69690
$962$		0			0
$963$	212.426	+	212.426i	0.220588	+	0.220588i
$964$		0			0
$965$	−761.318	−	761.318i	−0.788930	−	0.788930i
$966$		0			0
$967$	302.823			0.313157			0.156578	−	0.987666i	$-0.449954\pi$
$967$	302.823			0.313157			0.156578	+	0.987666i	$0.449954\pi$
$968$		0			0
$969$		−	129.584i		−	0.133730i
$970$		0			0
$971$	675.049	−	675.049i	0.695210	−	0.695210i	−0.268164	−	0.963373i	$-0.586417\pi$
$971$	675.049	−	675.049i	0.695210	−	0.695210i	0.963373	+	0.268164i	$0.0864167\pi$
$972$		0			0
$973$	−1457.14	+	1457.14i	−1.49758	+	1.49758i
$974$		0			0
$975$		−	1310.94i		−	1.34456i
$976$		0			0
$977$	−1603.49			−1.64124			−0.820621	−	0.571473i	$-0.806372\pi$
$977$	−1603.49			−1.64124			−0.820621	+	0.571473i	$0.806372\pi$
$978$		0			0
$979$	−566.341	−	566.341i	−0.578489	−	0.578489i
$980$		0			0
$981$	146.700	+	146.700i	0.149541	+	0.149541i
$982$		0			0
$983$	−202.760			−0.206267			−0.103133	−	0.994668i	$-0.532887\pi$
$983$	−202.760			−0.206267			−0.103133	+	0.994668i	$0.532887\pi$
$984$		0			0
$985$		−	902.497i		−	0.916240i
$986$		0			0
$987$	654.022	−	654.022i	0.662637	−	0.662637i
$988$		0			0
$989$	−81.6520	+	81.6520i	−0.0825601	+	0.0825601i
$990$		0			0
$991$			646.058i			0.651925i	0.945383	+	0.325963i	$0.105688\pi$
$991$			646.058i			0.651925i	−0.945383	+	0.325963i	$0.894312\pi$
$992$		0			0
$993$	167.240			0.168419
$994$		0			0
$995$	−221.551	−	221.551i	−0.222665	−	0.222665i
$996$		0			0
$997$	−869.330	−	869.330i	−0.871946	−	0.871946i	0.120738	−	0.992684i	$-0.461474\pi$
$997$	−869.330	−	869.330i	−0.871946	−	0.871946i	−0.992684	+	0.120738i	$0.961474\pi$
$998$		0			0
$999$	346.856			0.347204

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.l.f.319.8	yes	16
4.3	odd	2	inner	768.3.l.f.319.4	yes	16
8.3	odd	2		768.3.l.e.319.5	yes	16
8.5	even	2		768.3.l.e.319.1	✓	16
16.3	odd	4	inner	768.3.l.f.703.8	yes	16
16.5	even	4		768.3.l.e.703.5	yes	16
16.11	odd	4		768.3.l.e.703.1	yes	16
16.13	even	4	inner	768.3.l.f.703.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.3.l.e.319.1	✓	16	8.5	even	2
768.3.l.e.319.5	yes	16	8.3	odd	2
768.3.l.e.703.1	yes	16	16.11	odd	4
768.3.l.e.703.5	yes	16	16.5	even	4
768.3.l.f.319.4	yes	16	4.3	odd	2	inner
768.3.l.f.319.8	yes	16	1.1	even	1	trivial
768.3.l.f.703.4	yes	16	16.13	even	4	inner
768.3.l.f.703.8	yes	16	16.3	odd	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)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} +(6.45189 + 6.45189i) q^{5} +8.74541 q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 + 1.22474i) q^{3} +(6.45189 + 6.45189i) q^{5} +8.74541 q^{7} +3.00000i q^{9} +(-9.12435 + 9.12435i) q^{11} +(-9.18721 + 9.18721i) q^{13} +15.8038i q^{15} +18.9038 q^{17} +(-2.79852 - 2.79852i) q^{19} +(10.7109 + 10.7109i) q^{21} -2.27362 q^{23} +58.2538i q^{25} +(-3.67423 + 3.67423i) q^{27} +(-10.8865 + 10.8865i) q^{29} -50.9089i q^{31} -22.3500 q^{33} +(56.4245 + 56.4245i) q^{35} +(-47.2012 - 47.2012i) q^{37} -22.5040 q^{39} -27.6038i q^{41} +(35.9127 - 35.9127i) q^{43} +(-19.3557 + 19.3557i) q^{45} -61.0614i q^{47} +27.4823 q^{49} +(23.1523 + 23.1523i) q^{51} +(53.7789 + 53.7789i) q^{53} -117.739 q^{55} -6.85494i q^{57} +(20.5821 - 20.5821i) q^{59} +(-12.6936 + 12.6936i) q^{61} +26.2362i q^{63} -118.550 q^{65} +(43.7298 + 43.7298i) q^{67} +(-2.78461 - 2.78461i) q^{69} -20.5223 q^{71} +107.836i q^{73} +(-71.3460 + 71.3460i) q^{75} +(-79.7962 + 79.7962i) q^{77} -49.9282i q^{79} -9.00000 q^{81} +(15.0096 + 15.0096i) q^{83} +(121.965 + 121.965i) q^{85} -26.6664 q^{87} +62.0691i q^{89} +(-80.3460 + 80.3460i) q^{91} +(62.3504 - 62.3504i) q^{93} -36.1114i q^{95} -22.7518 q^{97} +(-27.3731 - 27.3731i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{5}+O(q^{10}) \) 16 * q + 8 * q^5	\( 16 q + 8 q^{5} - 32 q^{13} + 112 q^{17} + 72 q^{21} - 56 q^{29} - 48 q^{33} + 272 q^{37} - 24 q^{45} + 304 q^{49} + 504 q^{53} - 176 q^{61} + 624 q^{65} + 288 q^{69} - 848 q^{77} - 144 q^{81} + 880 q^{85} + 216 q^{93} - 160 q^{97}+O(q^{100}) \) 16 * q + 8 * q^5 - 32 * q^13 + 112 * q^17 + 72 * q^21 - 56 * q^29 - 48 * q^33 + 272 * q^37 - 24 * q^45 + 304 * q^49 + 504 * q^53 - 176 * q^61 + 624 * q^65 + 288 * q^69 - 848 * q^77 - 144 * q^81 + 880 * q^85 + 216 * q^93 - 160 * q^97

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