LMFDB - Embedded newform 768.3.l.e.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:	$16$
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			319.1
Root			$2.27793 + 1.31516i$ of defining polynomial
Character	$\chi$	$=$	768.319
Dual form			768.3.l.e.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(-6.45189 - 6.45189i) q^{5} +8.74541 q^{7} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.22474 - 1.22474i) q^{3} +(-6.45189 - 6.45189i) q^{5} +8.74541 q^{7} +3.00000i q^{9} +(9.12435 - 9.12435i) q^{11} +(9.18721 - 9.18721i) q^{13} +15.8038i q^{15} +18.9038 q^{17} +(2.79852 + 2.79852i) q^{19} +(-10.7109 - 10.7109i) q^{21} -2.27362 q^{23} +58.2538i q^{25} +(3.67423 - 3.67423i) q^{27} +(10.8865 - 10.8865i) q^{29} -50.9089i q^{31} -22.3500 q^{33} +(-56.4245 - 56.4245i) q^{35} +(47.2012 + 47.2012i) q^{37} -22.5040 q^{39} -27.6038i q^{41} +(-35.9127 + 35.9127i) q^{43} +(19.3557 - 19.3557i) q^{45} -61.0614i q^{47} +27.4823 q^{49} +(-23.1523 - 23.1523i) q^{51} +(-53.7789 - 53.7789i) q^{53} -117.739 q^{55} -6.85494i q^{57} +(-20.5821 + 20.5821i) q^{59} +(12.6936 - 12.6936i) q^{61} +26.2362i q^{63} -118.550 q^{65} +(-43.7298 - 43.7298i) q^{67} +(2.78461 + 2.78461i) q^{69} -20.5223 q^{71} +107.836i q^{73} +(71.3460 - 71.3460i) q^{75} +(79.7962 - 79.7962i) q^{77} -49.9282i q^{79} -9.00000 q^{81} +(-15.0096 - 15.0096i) q^{83} +(-121.965 - 121.965i) q^{85} -26.6664 q^{87} +62.0691i q^{89} +(80.3460 - 80.3460i) q^{91} +(-62.3504 + 62.3504i) q^{93} -36.1114i q^{95} -22.7518 q^{97} +(27.3731 + 27.3731i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.884200\pi$
$5$	−6.45189	−	6.45189i	−1.29038	−	1.29038i	−0.355826	−	0.934552i	$-0.615800\pi$
$6$		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	−0.157959	−	0.987446i	$-0.550491\pi$
$11$	9.12435	−	9.12435i	0.829486	−	0.829486i	0.987446	+	0.157959i	$0.0504914\pi$
$12$		0			0
$13$	9.18721	−	9.18721i	0.706709	−	0.706709i	−0.259133	−	0.965842i	$-0.583437\pi$
$13$	9.18721	−	9.18721i	0.706709	−	0.706709i	0.965842	+	0.259133i	$0.0834368\pi$
$14$		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	0.776906	−	0.629616i	$-0.216788\pi$
$19$	2.79852	+	2.79852i	0.147290	+	0.147290i	−0.629616	+	0.776906i	$0.716788\pi$
$20$		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	−0.494042	−	0.869438i	$-0.664481\pi$
$29$	10.8865	−	10.8865i	0.375396	−	0.375396i	0.869438	+	0.494042i	$0.164481\pi$
$30$		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	$0.107947\pi$
$37$	47.2012	+	47.2012i	1.27571	+	1.27571i	0.332661	+	0.943046i	$0.392053\pi$
$38$		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	−0.988220	−	0.153041i	$-0.951093\pi$
$43$	−35.9127	+	35.9127i	−0.835179	+	0.835179i	0.153041	+	0.988220i	$0.451093\pi$
$44$		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	$-0.995287\pi$
$53$	−53.7789	−	53.7789i	−1.01470	−	1.01470i	−0.0148067	−	0.999890i	$-0.504713\pi$
$54$		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	−0.859681	−	0.510831i	$-0.829338\pi$
$59$	−20.5821	+	20.5821i	−0.348850	+	0.348850i	0.510831	+	0.859681i	$0.329338\pi$
$60$		0			0
$61$	12.6936	−	12.6936i	0.208092	−	0.208092i	−0.595364	−	0.803456i	$-0.702992\pi$
$61$	12.6936	−	12.6936i	0.208092	−	0.208092i	0.803456	+	0.595364i	$0.202992\pi$
$62$		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	0.300954	−	0.953639i	$-0.402695\pi$
$67$	−43.7298	−	43.7298i	−0.652684	−	0.652684i	−0.953639	+	0.300954i	$0.902695\pi$
$68$		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	0.610883	−	0.791721i	$-0.290815\pi$
$83$	−15.0096	−	15.0096i	−0.180838	−	0.180838i	−0.791721	+	0.610883i	$0.790815\pi$
$84$		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	0.241668	−	0.970359i	$-0.422306\pi$
$101$	−73.5978	−	73.5978i	−0.728691	−	0.728691i	−0.970359	+	0.241668i	$0.922306\pi$
$102$		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	−0.955796	−	0.294032i	$-0.905003\pi$
$107$	−70.8088	+	70.8088i	−0.661764	+	0.661764i	0.294032	+	0.955796i	$0.405003\pi$
$108$		0			0
$109$	−48.9000	+	48.9000i	−0.448624	+	0.448624i	−0.894897	−	0.446273i	$-0.852751\pi$
$109$	−48.9000	+	48.9000i	−0.448624	+	0.448624i	0.446273	+	0.894897i	$0.352751\pi$
$110$		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	0.999742	−	0.0227077i	$-0.00722870\pi$
$131$	127.992	+	127.992i	0.977034	+	0.977034i	−0.0227077	+	0.999742i	$0.507229\pi$
$132$		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	$-0.428045\pi$
$139$	166.618	−	166.618i	1.19869	−	1.19869i	0.974559	−	0.224132i	$-0.0719547\pi$
$140$		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	0.861657	−	0.507491i	$-0.169427\pi$
$149$	52.7707	+	52.7707i	0.354166	+	0.354166i	−0.507491	+	0.861657i	$0.669427\pi$
$150$		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	−0.623170	−	0.782086i	$-0.714156\pi$
$157$	24.9498	−	24.9498i	0.158916	−	0.158916i	0.782086	+	0.623170i	$0.214156\pi$
$158$		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	$-0.880623\pi$
$163$	−211.378	−	211.378i	−1.29680	−	1.29680i	−0.366304	−	0.930495i	$-0.619377\pi$
$164$		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	$0.616784\pi$
$173$	−223.544	+	223.544i	−1.29216	+	1.29216i	−0.933448	+	0.358713i	$0.883216\pi$
$174$		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	0.193501	−	0.981100i	$-0.438016\pi$
$179$	−140.980	−	140.980i	−0.787599	−	0.787599i	−0.981100	+	0.193501i	$0.938016\pi$
$180$		0			0
$181$	124.679	+	124.679i	0.688836	+	0.688836i	0.961975	−	0.273139i	$-0.0880617\pi$
$181$	124.679	+	124.679i	0.688836	+	0.688836i	−0.273139	+	0.961975i	$0.588062\pi$
$182$		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	0.861976	−	0.506948i	$-0.169227\pi$
$197$	69.9405	+	69.9405i	0.355028	+	0.355028i	−0.506948	+	0.861976i	$0.669227\pi$
$198$		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	$0.0636505\pi$
$211$	248.707	+	248.707i	1.17871	+	1.17871i	0.198634	+	0.980074i	$0.436349\pi$
$212$		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	0.456839	−	0.889549i	$-0.348982\pi$
$227$	−98.2253	−	98.2253i	−0.432711	−	0.432711i	−0.889549	+	0.456839i	$0.848982\pi$
$228$		0			0
$229$	21.3388	+	21.3388i	0.0931823	+	0.0931823i	0.752161	−	0.658979i	$-0.229011\pi$
$229$	21.3388	+	21.3388i	0.0931823	+	0.0931823i	−0.658979	+	0.752161i	$0.729011\pi$
$230$		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	−0.816021	−	0.578022i	$-0.803825\pi$
$251$	−59.7377	+	59.7377i	−0.237999	+	0.237999i	0.578022	+	0.816021i	$0.303825\pi$
$252$		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	−0.963515	−	0.267654i	$-0.913752\pi$
$269$	−187.187	+	187.187i	−0.695861	+	0.695861i	0.267654	+	0.963515i	$0.413752\pi$
$270$		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	0.759886	−	0.650056i	$-0.225255\pi$
$277$	30.4229	+	30.4229i	0.109830	+	0.109830i	−0.650056	+	0.759886i	$0.725255\pi$
$278$		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	−0.428525	−	0.903530i	$-0.640966\pi$
$283$	134.427	−	134.427i	0.475005	−	0.475005i	0.903530	+	0.428525i	$0.140966\pi$
$284$		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	0.628055	−	0.778169i	$-0.283851\pi$
$293$	−43.9833	−	43.9833i	−0.150114	−	0.150114i	−0.778169	+	0.628055i	$0.783851\pi$
$294$		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	0.767887	−	0.640586i	$-0.221308\pi$
$307$	39.0814	+	39.0814i	0.127301	+	0.127301i	−0.640586	+	0.767887i	$0.721308\pi$
$308$		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	−0.747135	−	0.664673i	$-0.768571\pi$
$317$	−26.1406	+	26.1406i	−0.0824623	+	0.0824623i	0.664673	+	0.747135i	$0.268571\pi$
$318$		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	−0.802680	−	0.596410i	$-0.796593\pi$
$331$	−68.2754	+	68.2754i	−0.206270	+	0.206270i	0.596410	+	0.802680i	$0.296593\pi$
$332$		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	−0.787669	−	0.616099i	$-0.788712\pi$
$347$	−59.5347	+	59.5347i	−0.171570	+	0.171570i	0.616099	+	0.787669i	$0.288712\pi$
$348$		0			0
$349$	−148.260	+	148.260i	−0.424813	+	0.424813i	−0.886857	−	0.462044i	$-0.847116\pi$
$349$	−148.260	+	148.260i	−0.424813	+	0.424813i	0.462044	+	0.886857i	$0.347116\pi$
$350$		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	0.855350	−	0.518050i	$-0.173342\pi$
$373$	125.813	+	125.813i	0.337301	+	0.337301i	−0.518050	+	0.855350i	$0.673342\pi$
$374$		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	−0.569615	−	0.821912i	$-0.692908\pi$
$379$	95.6205	−	95.6205i	0.252297	−	0.252297i	0.821912	+	0.569615i	$0.192908\pi$
$380$		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	$0.0834564\pi$
$389$	476.532	+	476.532i	1.22502	+	1.22502i	0.259192	+	0.965826i	$0.416544\pi$
$390$		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	−0.419344	−	0.907827i	$-0.637740\pi$
$397$	193.928	−	193.928i	0.488483	−	0.488483i	0.907827	+	0.419344i	$0.137740\pi$
$398$		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	0.318154	−	0.948039i	$-0.396937\pi$
$419$	−263.922	−	263.922i	−0.629885	−	0.629885i	−0.948039	+	0.318154i	$0.896937\pi$
$420$		0			0
$421$	−180.379	−	180.379i	−0.428455	−	0.428455i	0.459647	−	0.888102i	$-0.347976\pi$
$421$	−180.379	−	180.379i	−0.428455	−	0.428455i	−0.888102	+	0.459647i	$0.847976\pi$
$422$		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	−0.861705	−	0.507409i	$-0.830603\pi$
$443$	−156.953	+	156.953i	−0.354297	+	0.354297i	0.507409	+	0.861705i	$0.330603\pi$
$444$		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	−0.915052	−	0.403335i	$-0.867851\pi$
$461$	−235.902	+	235.902i	−0.511717	+	0.511717i	0.403335	+	0.915052i	$0.367851\pi$
$462$		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	0.984906	−	0.173088i	$-0.0553746\pi$
$467$	379.119	+	379.119i	0.811818	+	0.811818i	−0.173088	+	0.984906i	$0.555375\pi$
$468$		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	$-0.361905\pi$
$491$	651.908	−	651.908i	1.32772	−	1.32772i	0.907359	−	0.420356i	$-0.138095\pi$
$492$		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	0.158156	−	0.987414i	$-0.449445\pi$
$499$	−413.800	−	413.800i	−0.829258	−	0.829258i	−0.987414	+	0.158156i	$0.949445\pi$
$500$		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	−0.497390	−	0.867527i	$-0.665708\pi$
$509$	188.400	−	188.400i	0.370137	−	0.370137i	0.867527	+	0.497390i	$0.165708\pi$
$510$		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	−0.0162219	−	0.999868i	$-0.505164\pi$
$523$	514.447	−	514.447i	0.983647	−	0.983647i	0.999868	+	0.0162219i	$0.00516381\pi$
$524$		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	−0.478927	−	0.877855i	$-0.658974\pi$
$541$	215.820	−	215.820i	0.398927	−	0.398927i	0.877855	+	0.478927i	$0.158974\pi$
$542$		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	0.988477	−	0.151371i	$-0.0483687\pi$
$547$	457.897	+	457.897i	0.837107	+	0.837107i	−0.151371	+	0.988477i	$0.548369\pi$
$548$		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	−0.714834	−	0.699294i	$-0.753498\pi$
$557$	−8.65563	+	8.65563i	−0.0155397	+	0.0155397i	0.699294	+	0.714834i	$0.253498\pi$
$558$		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	0.994050	−	0.108921i	$-0.0347397\pi$
$563$	498.328	+	498.328i	0.885129	+	0.885129i	−0.108921	+	0.994050i	$0.534740\pi$
$564$		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	−0.999253	−	0.0386402i	$-0.987697\pi$
$571$	−548.510	+	548.510i	−0.960613	+	0.960613i	0.0386402	+	0.999253i	$0.487697\pi$
$572$		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	−0.463994	−	0.885839i	$-0.653584\pi$
$587$	247.623	−	247.623i	0.421845	−	0.421845i	0.885839	+	0.463994i	$0.153584\pi$
$588$		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	0.724833	−	0.688924i	$-0.241917\pi$
$613$	22.0125	+	22.0125i	0.0359094	+	0.0359094i	−0.688924	+	0.724833i	$0.741917\pi$
$614$		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	−0.290711	−	0.956811i	$-0.593892\pi$
$619$	412.316	−	412.316i	0.666100	−	0.666100i	0.956811	+	0.290711i	$0.0938917\pi$
$620$		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	0.624391	−	0.781112i	$-0.285347\pi$
$643$	−100.772	−	100.772i	−0.156722	−	0.156722i	−0.781112	+	0.624391i	$0.785347\pi$
$644$		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	−0.847402	−	0.530951i	$-0.821835\pi$
$653$	−206.642	+	206.642i	−0.316451	+	0.316451i	0.530951	+	0.847402i	$0.321835\pi$
$654$		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	0.748793	−	0.662804i	$-0.230634\pi$
$659$	56.6668	+	56.6668i	0.0859891	+	0.0859891i	−0.662804	+	0.748793i	$0.730634\pi$
$660$		0			0
$661$	781.642	+	781.642i	1.18251	+	1.18251i	0.979091	+	0.203424i	$0.0652070\pi$
$661$	781.642	+	781.642i	1.18251	+	1.18251i	0.203424	+	0.979091i	$0.434793\pi$
$662$		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	0.443017	−	0.896513i	$-0.353908\pi$
$677$	−307.017	−	307.017i	−0.453497	−	0.453497i	−0.896513	+	0.443017i	$0.853908\pi$
$678$		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	−0.116425	−	0.993200i	$-0.537143\pi$
$683$	598.837	−	598.837i	0.876775	−	0.876775i	0.993200	+	0.116425i	$0.0371434\pi$
$684$		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	0.990669	−	0.136288i	$-0.0435174\pi$
$691$	590.377	+	590.377i	0.854381	+	0.854381i	−0.136288	+	0.990669i	$0.543517\pi$
$692$		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	$0.540148\pi$
$701$	−783.614	+	783.614i	−1.11785	+	1.11785i	−0.992056	+	0.125795i	$0.959852\pi$
$702$		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	0.671593	−	0.740920i	$-0.265610\pi$
$709$	−49.1534	−	49.1534i	−0.0693277	−	0.0693277i	−0.740920	+	0.671593i	$0.765610\pi$
$710$		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	−0.501061	−	0.865412i	$-0.667057\pi$
$733$	267.069	−	267.069i	0.364351	−	0.364351i	0.865412	+	0.501061i	$0.167057\pi$
$734$		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	0.983235	−	0.182345i	$-0.0583688\pi$
$739$	591.857	+	591.857i	0.800889	+	0.800889i	−0.182345	+	0.983235i	$0.558369\pi$
$740$		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	$0.0825192\pi$
$757$	925.760	+	925.760i	1.22293	+	1.22293i	0.256348	+	0.966585i	$0.417481\pi$
$758$		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	0.594806	−	0.803869i	$-0.297229\pi$
$773$	−161.605	−	161.605i	−0.209063	−	0.209063i	−0.803869	+	0.594806i	$0.797229\pi$
$774$		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	0.808173	−	0.588945i	$-0.200456\pi$
$787$	172.533	+	172.533i	0.219229	+	0.219229i	−0.588945	+	0.808173i	$0.700456\pi$
$788$		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	−0.549740	−	0.835336i	$-0.685273\pi$
$797$	227.620	−	227.620i	0.285596	−	0.285596i	0.835336	+	0.549740i	$0.185273\pi$
$798$		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	−0.819299	−	0.573367i	$-0.805637\pi$
$811$	−199.450	+	199.450i	−0.245931	+	0.245931i	0.573367	+	0.819299i	$0.305637\pi$
$812$		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	0.711665	−	0.702519i	$-0.247941\pi$
$821$	7.50885	+	7.50885i	0.00914598	+	0.00914598i	−0.702519	+	0.711665i	$0.747941\pi$
$822$		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	−0.325216	−	0.945640i	$-0.605437\pi$
$827$	513.090	−	513.090i	0.620423	−	0.620423i	0.945640	+	0.325216i	$0.105437\pi$
$828$		0			0
$829$	142.936	−	142.936i	0.172420	−	0.172420i	−0.615622	−	0.788042i	$-0.711095\pi$
$829$	142.936	−	142.936i	0.172420	−	0.172420i	0.788042	+	0.615622i	$0.211095\pi$
$830$		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	0.767767	−	0.640730i	$-0.221368\pi$
$853$	108.362	+	108.362i	0.127037	+	0.127037i	−0.640730	+	0.767767i	$0.721368\pi$
$854$		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	−0.142910	−	0.989736i	$-0.545646\pi$
$859$	727.423	−	727.423i	0.846826	−	0.846826i	0.989736	+	0.142910i	$0.0456459\pi$
$860$		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	−0.617143	−	0.786851i	$-0.711710\pi$
$877$	148.834	−	148.834i	0.169709	−	0.169709i	0.786851	+	0.617143i	$0.211710\pi$
$878$		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	0.986172	−	0.165725i	$-0.0529964\pi$
$883$	724.455	+	724.455i	0.820447	+	0.820447i	−0.165725	+	0.986172i	$0.552996\pi$
$884$		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	−0.980887	−	0.194577i	$-0.937667\pi$
$907$	−713.184	+	713.184i	−0.786310	+	0.786310i	0.194577	+	0.980887i	$0.437667\pi$
$908$		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	−0.936941	−	0.349487i	$-0.886356\pi$
$941$	−552.794	+	552.794i	−0.587454	+	0.587454i	0.349487	+	0.936941i	$0.386356\pi$
$942$		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	0.918125	−	0.396291i	$-0.129703\pi$
$947$	494.177	+	494.177i	0.521834	+	0.521834i	−0.396291	+	0.918125i	$0.629703\pi$
$948$		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	−0.963373	−	0.268164i	$-0.913583\pi$
$971$	−675.049	+	675.049i	−0.695210	+	0.695210i	0.268164	+	0.963373i	$0.413583\pi$
$972$		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	0.992684	−	0.120738i	$-0.0385262\pi$
$997$	869.330	+	869.330i	0.871946	+	0.871946i	−0.120738	+	0.992684i	$0.538526\pi$
$998$		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.e.319.1	✓	16
4.3	odd	2	inner	768.3.l.e.319.5	yes	16
8.3	odd	2		768.3.l.f.319.4	yes	16
8.5	even	2		768.3.l.f.319.8	yes	16
16.3	odd	4	inner	768.3.l.e.703.1	yes	16
16.5	even	4		768.3.l.f.703.4	yes	16
16.11	odd	4		768.3.l.f.703.8	yes	16
16.13	even	4	inner	768.3.l.e.703.5	yes	16

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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