LMFDB - Embedded newform 768.3.h.f.641.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(641,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.641");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.4
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	768.641
Dual form			768.3.h.f.641.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.44949 + 1.73205i) q^{3} +2.82843 q^{5} +10.3923 q^{7} +(3.00000 - 8.48528i) q^{9} +O(q^{10})$	\(q+(-2.44949 + 1.73205i) q^{3} +2.82843 q^{5} +10.3923 q^{7} +(3.00000 - 8.48528i) q^{9} +14.6969 q^{11} +6.00000i q^{13} +(-6.92820 + 4.89898i) q^{15} -22.6274i q^{17} -10.3923i q^{19} +(-25.4558 + 18.0000i) q^{21} -29.3939i q^{23} -17.0000 q^{25} +(7.34847 + 25.9808i) q^{27} -31.1127 q^{29} +31.1769 q^{31} +(-36.0000 + 25.4558i) q^{33} +29.3939 q^{35} -38.0000i q^{37} +(-10.3923 - 14.6969i) q^{39} -5.65685i q^{41} +10.3923i q^{43} +(8.48528 - 24.0000i) q^{45} -58.7878i q^{47} +59.0000 q^{49} +(39.1918 + 55.4256i) q^{51} +14.1421 q^{53} +41.5692 q^{55} +(18.0000 + 25.4558i) q^{57} +14.6969 q^{59} +22.0000i q^{61} +(31.1769 - 88.1816i) q^{63} +16.9706i q^{65} +114.315i q^{67} +(50.9117 + 72.0000i) q^{69} +29.3939i q^{71} +30.0000 q^{73} +(41.6413 - 29.4449i) q^{75} +152.735 q^{77} +31.1769 q^{79} +(-63.0000 - 50.9117i) q^{81} +73.4847 q^{83} -64.0000i q^{85} +(76.2102 - 53.8888i) q^{87} -5.65685i q^{89} +62.3538i q^{91} +(-76.3675 + 54.0000i) q^{93} -29.3939i q^{95} +90.0000 q^{97} +(44.0908 - 124.708i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} - 136 q^{25} - 288 q^{33} + 472 q^{49} + 144 q^{57} + 240 q^{73} - 504 q^{81} + 720 q^{97}+O(q^{100}) $ 8 * q + 24 * q^9 - 136 * q^25 - 288 * q^33 + 472 * q^49 + 144 * q^57 + 240 * q^73 - 504 * q^81 + 720 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−2.44949	+	1.73205i	−0.816497	+	0.577350i
$3$	−2.44949	+	1.73205i	−0.816497	+	0.577350i
$4$		0			0
$5$	2.82843			0.565685			0.282843	−	0.959166i	$-0.408723\pi$
$5$	2.82843			0.565685			0.282843	+	0.959166i	$0.408723\pi$
$6$		0			0
$7$	10.3923			1.48461			0.742307	−	0.670059i	$-0.233731\pi$
$7$	10.3923			1.48461			0.742307	+	0.670059i	$0.233731\pi$
$8$		0			0
$9$	3.00000	−	8.48528i	0.333333	−	0.942809i
$10$		0			0
$11$	14.6969			1.33609			0.668043	−	0.744123i	$-0.267132\pi$
$11$	14.6969			1.33609			0.668043	+	0.744123i	$0.267132\pi$
$12$		0			0
$13$			6.00000i			0.461538i	0.973009	+	0.230769i	$0.0741242\pi$
$13$			6.00000i			0.461538i	−0.973009	+	0.230769i	$0.925876\pi$
$14$		0			0
$15$	−6.92820	+	4.89898i	−0.461880	+	0.326599i
$16$		0			0
$17$		−	22.6274i		−	1.33102i	−0.746387	−	0.665512i	$-0.768213\pi$
$17$		−	22.6274i		−	1.33102i	0.746387	−	0.665512i	$-0.231787\pi$
$18$		0			0
$19$		−	10.3923i		−	0.546963i	−0.961877	−	0.273482i	$-0.911825\pi$
$19$		−	10.3923i		−	0.546963i	0.961877	−	0.273482i	$-0.0881753\pi$
$20$		0			0
$21$	−25.4558	+	18.0000i	−1.21218	+	0.857143i
$22$		0			0
$23$		−	29.3939i		−	1.27799i	−0.769209	−	0.638997i	$-0.779349\pi$
$23$		−	29.3939i		−	1.27799i	0.769209	−	0.638997i	$-0.220651\pi$
$24$		0			0
$25$	−17.0000			−0.680000
$26$		0			0
$27$	7.34847	+	25.9808i	0.272166	+	0.962250i
$28$		0			0
$29$	−31.1127			−1.07285			−0.536426	−	0.843947i	$-0.680226\pi$
$29$	−31.1127			−1.07285			−0.536426	+	0.843947i	$0.680226\pi$
$30$		0			0
$31$	31.1769			1.00571			0.502853	−	0.864372i	$-0.332284\pi$
$31$	31.1769			1.00571			0.502853	+	0.864372i	$0.332284\pi$
$32$		0			0
$33$	−36.0000	+	25.4558i	−1.09091	+	0.771389i
$34$		0			0
$35$	29.3939			0.839825
$36$		0			0
$37$		−	38.0000i		−	1.02703i	−0.858082	−	0.513514i	$-0.828344\pi$
$37$		−	38.0000i		−	1.02703i	0.858082	−	0.513514i	$-0.171656\pi$
$38$		0			0
$39$	−10.3923	−	14.6969i	−0.266469	−	0.376845i
$40$		0			0
$41$		−	5.65685i		−	0.137972i	−0.997618	−	0.0689860i	$-0.978024\pi$
$41$		−	5.65685i		−	0.137972i	0.997618	−	0.0689860i	$-0.0219764\pi$
$42$		0			0
$43$			10.3923i			0.241682i	0.992672	+	0.120841i	$0.0385590\pi$
$43$			10.3923i			0.241682i	−0.992672	+	0.120841i	$0.961441\pi$
$44$		0			0
$45$	8.48528	−	24.0000i	0.188562	−	0.533333i
$46$		0			0
$47$		−	58.7878i		−	1.25080i	−0.780303	−	0.625402i	$-0.784935\pi$
$47$		−	58.7878i		−	1.25080i	0.780303	−	0.625402i	$-0.215065\pi$
$48$		0			0
$49$	59.0000			1.20408
$50$		0			0
$51$	39.1918	+	55.4256i	0.768467	+	1.08678i
$52$		0			0
$53$	14.1421			0.266833			0.133416	−	0.991060i	$-0.457405\pi$
$53$	14.1421			0.266833			0.133416	+	0.991060i	$0.457405\pi$
$54$		0			0
$55$	41.5692			0.755804
$56$		0			0
$57$	18.0000	+	25.4558i	0.315789	+	0.446594i
$58$		0			0
$59$	14.6969			0.249101			0.124550	−	0.992213i	$-0.460251\pi$
$59$	14.6969			0.249101			0.124550	+	0.992213i	$0.460251\pi$
$60$		0			0
$61$			22.0000i			0.360656i	0.983607	+	0.180328i	$0.0577159\pi$
$61$			22.0000i			0.360656i	−0.983607	+	0.180328i	$0.942284\pi$
$62$		0			0
$63$	31.1769	−	88.1816i	0.494872	−	1.39971i
$64$		0			0
$65$			16.9706i			0.261086i
$66$		0			0
$67$			114.315i			1.70620i	0.521748	+	0.853100i	$0.325280\pi$
$67$			114.315i			1.70620i	−0.521748	+	0.853100i	$0.674720\pi$
$68$		0			0
$69$	50.9117	+	72.0000i	0.737851	+	1.04348i
$70$		0			0
$71$			29.3939i			0.413998i	0.978341	+	0.206999i	$0.0663697\pi$
$71$			29.3939i			0.413998i	−0.978341	+	0.206999i	$0.933630\pi$
$72$		0			0
$73$	30.0000			0.410959			0.205479	−	0.978661i	$-0.434125\pi$
$73$	30.0000			0.410959			0.205479	+	0.978661i	$0.434125\pi$
$74$		0			0
$75$	41.6413	−	29.4449i	0.555218	−	0.392598i
$76$		0			0
$77$	152.735			1.98357
$78$		0			0
$79$	31.1769			0.394644			0.197322	−	0.980339i	$-0.436775\pi$
$79$	31.1769			0.394644			0.197322	+	0.980339i	$0.436775\pi$
$80$		0			0
$81$	−63.0000	−	50.9117i	−0.777778	−	0.628539i
$82$		0			0
$83$	73.4847			0.885358			0.442679	−	0.896680i	$-0.354028\pi$
$83$	73.4847			0.885358			0.442679	+	0.896680i	$0.354028\pi$
$84$		0			0
$85$		−	64.0000i		−	0.752941i
$86$		0			0
$87$	76.2102	−	53.8888i	0.875980	−	0.619411i
$88$		0			0
$89$		−	5.65685i		−	0.0635602i	−0.999495	−	0.0317801i	$-0.989882\pi$
$89$		−	5.65685i		−	0.0635602i	0.999495	−	0.0317801i	$-0.0101176\pi$
$90$		0			0
$91$			62.3538i			0.685207i
$92$		0			0
$93$	−76.3675	+	54.0000i	−0.821156	+	0.580645i
$94$		0			0
$95$		−	29.3939i		−	0.309409i
$96$		0			0
$97$	90.0000			0.927835			0.463918	−	0.885878i	$-0.346443\pi$
$97$	90.0000			0.927835			0.463918	+	0.885878i	$0.346443\pi$
$98$		0			0
$99$	44.0908	−	124.708i	0.445362	−	1.25967i
$100$		0			0
$101$	2.82843			0.0280042			0.0140021	−	0.999902i	$-0.495543\pi$
$101$	2.82843			0.0280042			0.0140021	+	0.999902i	$0.495543\pi$
$102$		0			0
$103$	−72.7461			−0.706273			−0.353137	−	0.935572i	$-0.614885\pi$
$103$	−72.7461			−0.706273			−0.353137	+	0.935572i	$0.614885\pi$
$104$		0			0
$105$	−72.0000	+	50.9117i	−0.685714	+	0.484873i
$106$		0			0
$107$	73.4847			0.686773			0.343386	−	0.939194i	$-0.388426\pi$
$107$	73.4847			0.686773			0.343386	+	0.939194i	$0.388426\pi$
$108$		0			0
$109$		−	138.000i		−	1.26606i	−0.774129	−	0.633028i	$-0.781812\pi$
$109$		−	138.000i		−	1.26606i	0.774129	−	0.633028i	$-0.218188\pi$
$110$		0			0
$111$	65.8179	+	93.0806i	0.592954	+	0.838564i
$112$		0			0
$113$			192.333i			1.70206i	0.525115	+	0.851031i	$0.324022\pi$
$113$			192.333i			1.70206i	−0.525115	+	0.851031i	$0.675978\pi$
$114$		0			0
$115$		−	83.1384i		−	0.722943i
$116$		0			0
$117$	50.9117	+	18.0000i	0.435143	+	0.153846i
$118$		0			0
$119$		−	235.151i		−	1.97606i
$120$		0			0
$121$	95.0000			0.785124
$122$		0			0
$123$	9.79796	+	13.8564i	0.0796582	+	0.112654i
$124$		0			0
$125$	−118.794			−0.950352
$126$		0			0
$127$	−51.9615			−0.409146			−0.204573	−	0.978851i	$-0.565581\pi$
$127$	−51.9615			−0.409146			−0.204573	+	0.978851i	$0.565581\pi$
$128$		0			0
$129$	−18.0000	−	25.4558i	−0.139535	−	0.197332i
$130$		0			0
$131$	−44.0908			−0.336571			−0.168286	−	0.985738i	$-0.553823\pi$
$131$	−44.0908			−0.336571			−0.168286	+	0.985738i	$0.553823\pi$
$132$		0			0
$133$		−	108.000i		−	0.812030i
$134$		0			0
$135$	20.7846	+	73.4847i	0.153960	+	0.544331i
$136$		0			0
$137$			39.5980i			0.289036i	0.989502	+	0.144518i	$0.0461632\pi$
$137$			39.5980i			0.289036i	−0.989502	+	0.144518i	$0.953837\pi$
$138$		0			0
$139$			135.100i			0.971942i	0.873975	+	0.485971i	$0.161534\pi$
$139$			135.100i			0.971942i	−0.873975	+	0.485971i	$0.838466\pi$
$140$		0			0
$141$	101.823	+	144.000i	0.722152	+	1.02128i
$142$		0			0
$143$			88.1816i			0.616655i
$144$		0			0
$145$	−88.0000			−0.606897
$146$		0			0
$147$	−144.520	+	102.191i	−0.983129	+	0.695177i
$148$		0			0
$149$	−189.505			−1.27184			−0.635922	−	0.771754i	$-0.719380\pi$
$149$	−189.505			−1.27184			−0.635922	+	0.771754i	$0.719380\pi$
$150$		0			0
$151$	176.669			1.16999			0.584997	−	0.811035i	$-0.301096\pi$
$151$	176.669			1.16999			0.584997	+	0.811035i	$0.301096\pi$
$152$		0			0
$153$	−192.000	−	67.8823i	−1.25490	−	0.443675i
$154$		0			0
$155$	88.1816			0.568914
$156$		0			0
$157$		−	154.000i		−	0.980892i	−0.871472	−	0.490446i	$-0.836834\pi$
$157$		−	154.000i		−	0.980892i	0.871472	−	0.490446i	$-0.163166\pi$
$158$		0			0
$159$	−34.6410	+	24.4949i	−0.217868	+	0.154056i
$160$		0			0
$161$		−	305.470i		−	1.89733i
$162$		0			0
$163$			72.7461i			0.446295i	0.974785	+	0.223148i	$0.0716332\pi$
$163$			72.7461i			0.446295i	−0.974785	+	0.223148i	$0.928367\pi$
$164$		0			0
$165$	−101.823	+	72.0000i	−0.617111	+	0.436364i
$166$		0			0
$167$			264.545i			1.58410i	0.610455	+	0.792051i	$0.290986\pi$
$167$			264.545i			1.58410i	−0.610455	+	0.792051i	$0.709014\pi$
$168$		0			0
$169$	133.000			0.786982
$170$		0			0
$171$	−88.1816	−	31.1769i	−0.515682	−	0.182321i
$172$		0			0
$173$	−223.446			−1.29159			−0.645797	−	0.763509i	$-0.723475\pi$
$173$	−223.446			−1.29159			−0.645797	+	0.763509i	$0.723475\pi$
$174$		0			0
$175$	−176.669			−1.00954
$176$		0			0
$177$	−36.0000	+	25.4558i	−0.203390	+	0.143818i
$178$		0			0
$179$	−102.879			−0.574741			−0.287370	−	0.957820i	$-0.592781\pi$
$179$	−102.879			−0.574741			−0.287370	+	0.957820i	$0.592781\pi$
$180$		0			0
$181$			234.000i			1.29282i	0.762991	+	0.646409i	$0.223730\pi$
$181$			234.000i			1.29282i	−0.762991	+	0.646409i	$0.776270\pi$
$182$		0			0
$183$	−38.1051	−	53.8888i	−0.208225	−	0.294474i
$184$		0			0
$185$		−	107.480i		−	0.580974i
$186$		0			0
$187$		−	332.554i		−	1.77836i
$188$		0			0
$189$	76.3675	+	270.000i	0.404061	+	1.42857i
$190$		0			0
$191$		−	117.576i		−	0.615579i	−0.951455	−	0.307789i	$-0.900411\pi$
$191$		−	117.576i		−	0.615579i	0.951455	−	0.307789i	$-0.0995892\pi$
$192$		0			0
$193$	82.0000			0.424870			0.212435	−	0.977175i	$-0.431861\pi$
$193$	82.0000			0.424870			0.212435	+	0.977175i	$0.431861\pi$
$194$		0			0
$195$	−29.3939	−	41.5692i	−0.150738	−	0.213175i
$196$		0			0
$197$	206.475			1.04810			0.524049	−	0.851688i	$-0.324421\pi$
$197$	206.475			1.04810			0.524049	+	0.851688i	$0.324421\pi$
$198$		0			0
$199$	−322.161			−1.61890			−0.809451	−	0.587188i	$-0.800235\pi$
$199$	−322.161			−1.61890			−0.809451	+	0.587188i	$0.800235\pi$
$200$		0			0
$201$	−198.000	−	280.014i	−0.985075	−	1.39311i
$202$		0			0
$203$	−323.333			−1.59277
$204$		0			0
$205$		−	16.0000i		−	0.0780488i
$206$		0			0
$207$	−249.415	−	88.1816i	−1.20490	−	0.425998i
$208$		0			0
$209$		−	152.735i		−	0.730790i
$210$		0			0
$211$		−	135.100i		−	0.640284i	−0.947370	−	0.320142i	$-0.896269\pi$
$211$		−	135.100i		−	0.640284i	0.947370	−	0.320142i	$-0.103731\pi$
$212$		0			0
$213$	−50.9117	−	72.0000i	−0.239022	−	0.338028i
$214$		0			0
$215$			29.3939i			0.136716i
$216$		0			0
$217$	324.000			1.49309
$218$		0			0
$219$	−73.4847	+	51.9615i	−0.335547	+	0.237267i
$220$		0			0
$221$	135.765			0.614319
$222$		0			0
$223$	363.731			1.63108			0.815540	−	0.578701i	$-0.196440\pi$
$223$	363.731			1.63108			0.815540	+	0.578701i	$0.196440\pi$
$224$		0			0
$225$	−51.0000	+	144.250i	−0.226667	+	0.641110i
$226$		0			0
$227$	14.6969			0.0647442			0.0323721	−	0.999476i	$-0.489694\pi$
$227$	14.6969			0.0647442			0.0323721	+	0.999476i	$0.489694\pi$
$228$		0			0
$229$		−	246.000i		−	1.07424i	−0.843507	−	0.537118i	$-0.819513\pi$
$229$		−	246.000i		−	1.07424i	0.843507	−	0.537118i	$-0.180487\pi$
$230$		0			0
$231$	−374.123	+	264.545i	−1.61958	+	1.14522i
$232$		0			0
$233$			209.304i			0.898299i	0.893457	+	0.449149i	$0.148273\pi$
$233$			209.304i			0.898299i	−0.893457	+	0.449149i	$0.851727\pi$
$234$		0			0
$235$		−	166.277i		−	0.707561i
$236$		0			0
$237$	−76.3675	+	54.0000i	−0.322226	+	0.227848i
$238$		0			0
$239$			411.514i			1.72182i	0.508760	+	0.860909i	$0.330104\pi$
$239$			411.514i			1.72182i	−0.508760	+	0.860909i	$0.669896\pi$
$240$		0			0
$241$	210.000			0.871369			0.435685	−	0.900099i	$-0.356506\pi$
$241$	210.000			0.871369			0.435685	+	0.900099i	$0.356506\pi$
$242$		0			0
$243$	242.499	+	15.5885i	0.997940	+	0.0641500i
$244$		0			0
$245$	166.877			0.681131
$246$		0			0
$247$	62.3538			0.252445
$248$		0			0
$249$	−180.000	+	127.279i	−0.722892	+	0.511162i
$250$		0			0
$251$	−396.817			−1.58095			−0.790473	−	0.612497i	$-0.790165\pi$
$251$	−396.817			−1.58095			−0.790473	+	0.612497i	$0.790165\pi$
$252$		0			0
$253$		−	432.000i		−	1.70751i
$254$		0			0
$255$	110.851	+	156.767i	0.434711	+	0.614774i
$256$		0			0
$257$			384.666i			1.49676i	0.663273	+	0.748378i	$0.269167\pi$
$257$			384.666i			1.49676i	−0.663273	+	0.748378i	$0.730833\pi$
$258$		0			0
$259$		−	394.908i		−	1.52474i
$260$		0			0
$261$	−93.3381	+	264.000i	−0.357617	+	1.01149i
$262$		0			0
$263$			499.696i			1.89998i	0.312274	+	0.949992i	$0.398909\pi$
$263$			499.696i			1.89998i	−0.312274	+	0.949992i	$0.601091\pi$
$264$		0			0
$265$	40.0000			0.150943
$266$		0			0
$267$	9.79796	+	13.8564i	0.0366965	+	0.0518967i
$268$		0			0
$269$	−19.7990			−0.0736022			−0.0368011	−	0.999323i	$-0.511717\pi$
$269$	−19.7990			−0.0736022			−0.0368011	+	0.999323i	$0.511717\pi$
$270$		0			0
$271$	−135.100			−0.498524			−0.249262	−	0.968436i	$-0.580188\pi$
$271$	−135.100			−0.498524			−0.249262	+	0.968436i	$0.580188\pi$
$272$		0			0
$273$	−108.000	−	152.735i	−0.395604	−	0.559469i
$274$		0			0
$275$	−249.848			−0.908538
$276$		0			0
$277$		−	198.000i		−	0.714801i	−0.933951	−	0.357401i	$-0.883663\pi$
$277$		−	198.000i		−	0.714801i	0.933951	−	0.357401i	$-0.116337\pi$
$278$		0			0
$279$	93.5307	−	264.545i	0.335236	−	0.948190i
$280$		0			0
$281$			197.990i			0.704590i	0.935889	+	0.352295i	$0.114599\pi$
$281$			197.990i			0.704590i	−0.935889	+	0.352295i	$0.885401\pi$
$282$		0			0
$283$			51.9615i			0.183610i	0.995777	+	0.0918048i	$0.0292636\pi$
$283$			51.9615i			0.183610i	−0.995777	+	0.0918048i	$0.970736\pi$
$284$		0			0
$285$	50.9117	+	72.0000i	0.178638	+	0.252632i
$286$		0			0
$287$		−	58.7878i		−	0.204835i
$288$		0			0
$289$	−223.000			−0.771626
$290$		0			0
$291$	−220.454	+	155.885i	−0.757574	+	0.535686i
$292$		0			0
$293$	−359.210			−1.22597			−0.612987	−	0.790093i	$-0.710032\pi$
$293$	−359.210			−1.22597			−0.612987	+	0.790093i	$0.710032\pi$
$294$		0			0
$295$	41.5692			0.140913
$296$		0			0
$297$	108.000	+	381.838i	0.363636	+	1.28565i
$298$		0			0
$299$	176.363			0.589844
$300$		0			0
$301$			108.000i			0.358804i
$302$		0			0
$303$	−6.92820	+	4.89898i	−0.0228654	+	0.0161682i
$304$		0			0
$305$			62.2254i			0.204018i
$306$		0			0
$307$			197.454i			0.643172i	0.946880	+	0.321586i	$0.104216\pi$
$307$			197.454i			0.643172i	−0.946880	+	0.321586i	$0.895784\pi$
$308$		0			0
$309$	178.191	−	126.000i	0.576670	−	0.407767i
$310$		0			0
$311$		−	264.545i		−	0.850627i	−0.905046	−	0.425313i	$-0.860164\pi$
$311$		−	264.545i		−	0.850627i	0.905046	−	0.425313i	$-0.139836\pi$
$312$		0			0
$313$	−266.000			−0.849840			−0.424920	−	0.905231i	$-0.639698\pi$
$313$	−266.000			−0.849840			−0.424920	+	0.905231i	$0.639698\pi$
$314$		0			0
$315$	88.1816	−	249.415i	0.279942	−	0.791795i
$316$		0			0
$317$	421.436			1.32945			0.664725	−	0.747088i	$-0.268549\pi$
$317$	421.436			1.32945			0.664725	+	0.747088i	$0.268549\pi$
$318$		0			0
$319$	−457.261			−1.43342
$320$		0			0
$321$	−180.000	+	127.279i	−0.560748	+	0.396508i
$322$		0			0
$323$	−235.151			−0.728022
$324$		0			0
$325$		−	102.000i		−	0.313846i
$326$		0			0
$327$	239.023	+	338.030i	0.730957	+	1.03373i
$328$		0			0
$329$		−	610.940i		−	1.85696i
$330$		0			0
$331$		−	197.454i		−	0.596537i	−0.954482	−	0.298269i	$-0.903591\pi$
$331$		−	197.454i		−	0.596537i	0.954482	−	0.298269i	$-0.0964091\pi$
$332$		0			0
$333$	−322.441	−	114.000i	−0.968290	−	0.342342i
$334$		0			0
$335$			323.333i			0.965172i
$336$		0			0
$337$	522.000			1.54896			0.774481	−	0.632598i	$-0.218011\pi$
$337$	522.000			1.54896			0.774481	+	0.632598i	$0.218011\pi$
$338$		0			0
$339$	−333.131	−	471.118i	−0.982686	−	1.38973i
$340$		0			0
$341$	458.205			1.34371
$342$		0			0
$343$	103.923			0.302983
$344$		0			0
$345$	144.000	+	203.647i	0.417391	+	0.590280i
$346$		0			0
$347$	426.211			1.22827			0.614137	−	0.789199i	$-0.289504\pi$
$347$	426.211			1.22827			0.614137	+	0.789199i	$0.289504\pi$
$348$		0			0
$349$			166.000i			0.475645i	0.971309	+	0.237822i	$0.0764336\pi$
$349$			166.000i			0.475645i	−0.971309	+	0.237822i	$0.923566\pi$
$350$		0			0
$351$	−155.885	+	44.0908i	−0.444116	+	0.125615i
$352$		0			0
$353$			22.6274i			0.0641003i	0.999486	+	0.0320502i	$0.0102036\pi$
$353$			22.6274i			0.0641003i	−0.999486	+	0.0320502i	$0.989796\pi$
$354$		0			0
$355$			83.1384i			0.234193i
$356$		0			0
$357$	407.294	+	576.000i	1.14088	+	1.61345i
$358$		0			0
$359$		−	88.1816i		−	0.245631i	−0.992430	−	0.122816i	$-0.960808\pi$
$359$		−	88.1816i		−	0.245631i	0.992430	−	0.122816i	$-0.0391924\pi$
$360$		0			0
$361$	253.000			0.700831
$362$		0			0
$363$	−232.702	+	164.545i	−0.641051	+	0.453292i
$364$		0			0
$365$	84.8528			0.232473
$366$		0			0
$367$	−135.100			−0.368120			−0.184060	−	0.982915i	$-0.558924\pi$
$367$	−135.100			−0.368120			−0.184060	+	0.982915i	$0.558924\pi$
$368$		0			0
$369$	−48.0000	−	16.9706i	−0.130081	−	0.0459907i
$370$		0			0
$371$	146.969			0.396144
$372$		0			0
$373$		−	118.000i		−	0.316354i	−0.987411	−	0.158177i	$-0.949438\pi$
$373$		−	118.000i		−	0.316354i	0.987411	−	0.158177i	$-0.0505616\pi$
$374$		0			0
$375$	290.985	−	205.757i	0.775959	−	0.548686i
$376$		0			0
$377$		−	186.676i		−	0.495162i
$378$		0			0
$379$			592.361i			1.56296i	0.623931	+	0.781479i	$0.285535\pi$
$379$			592.361i			1.56296i	−0.623931	+	0.781479i	$0.714465\pi$
$380$		0			0
$381$	127.279	−	90.0000i	0.334066	−	0.236220i
$382$		0			0
$383$		−	352.727i		−	0.920957i	−0.887671	−	0.460478i	$-0.847678\pi$
$383$		−	352.727i		−	0.920957i	0.887671	−	0.460478i	$-0.152322\pi$
$384$		0			0
$385$	432.000			1.12208
$386$		0			0
$387$	88.1816	+	31.1769i	0.227860	+	0.0805605i
$388$		0			0
$389$	−359.210			−0.923420			−0.461710	−	0.887031i	$-0.652764\pi$
$389$	−359.210			−0.923420			−0.461710	+	0.887031i	$0.652764\pi$
$390$		0			0
$391$	−665.108			−1.70104
$392$		0			0
$393$	108.000	−	76.3675i	0.274809	−	0.194319i
$394$		0			0
$395$	88.1816			0.223245
$396$		0			0
$397$			214.000i			0.539043i	0.962994	+	0.269521i	$0.0868655\pi$
$397$			214.000i			0.539043i	−0.962994	+	0.269521i	$0.913135\pi$
$398$		0			0
$399$	187.061	+	264.545i	0.468826	+	0.663020i
$400$		0			0
$401$		−	226.274i		−	0.564275i	−0.959374	−	0.282137i	$-0.908957\pi$
$401$		−	226.274i		−	0.564275i	0.959374	−	0.282137i	$-0.0910434\pi$
$402$		0			0
$403$			187.061i			0.464172i
$404$		0			0
$405$	−178.191	−	144.000i	−0.439978	−	0.355556i
$406$		0			0
$407$		−	558.484i		−	1.37220i
$408$		0			0
$409$	−114.000			−0.278729			−0.139364	−	0.990241i	$-0.544506\pi$
$409$	−114.000			−0.278729			−0.139364	+	0.990241i	$0.544506\pi$
$410$		0			0
$411$	−68.5857	−	96.9948i	−0.166875	−	0.235997i
$412$		0			0
$413$	152.735			0.369819
$414$		0			0
$415$	207.846			0.500834
$416$		0			0
$417$	−234.000	−	330.926i	−0.561151	−	0.793587i
$418$		0			0
$419$	367.423			0.876906			0.438453	−	0.898754i	$-0.355527\pi$
$419$	367.423			0.876906			0.438453	+	0.898754i	$0.355527\pi$
$420$		0			0
$421$		−	534.000i		−	1.26841i	−0.773166	−	0.634204i	$-0.781328\pi$
$421$		−	534.000i		−	1.26841i	0.773166	−	0.634204i	$-0.218672\pi$
$422$		0			0
$423$	−498.831	−	176.363i	−1.17927	−	0.416934i
$424$		0			0
$425$			384.666i			0.905097i
$426$		0			0
$427$			228.631i			0.535435i
$428$		0			0
$429$	−152.735	−	216.000i	−0.356026	−	0.503497i
$430$		0			0
$431$		−	293.939i		−	0.681993i	−0.940065	−	0.340996i	$-0.889236\pi$
$431$		−	293.939i		−	0.681993i	0.940065	−	0.340996i	$-0.110764\pi$
$432$		0			0
$433$	−614.000			−1.41801			−0.709007	−	0.705202i	$-0.750857\pi$
$433$	−614.000			−1.41801			−0.709007	+	0.705202i	$0.750857\pi$
$434$		0			0
$435$	215.555	−	152.420i	0.495529	−	0.350392i
$436$		0			0
$437$	−305.470			−0.699016
$438$		0			0
$439$	342.946			0.781198			0.390599	−	0.920561i	$-0.372268\pi$
$439$	342.946			0.781198			0.390599	+	0.920561i	$0.372268\pi$
$440$		0			0
$441$	177.000	−	500.632i	0.401361	−	1.13522i
$442$		0			0
$443$	−514.393			−1.16116			−0.580579	−	0.814204i	$-0.697174\pi$
$443$	−514.393			−1.16116			−0.580579	+	0.814204i	$0.697174\pi$
$444$		0			0
$445$		−	16.0000i		−	0.0359551i
$446$		0			0
$447$	464.190	−	328.232i	1.03846	−	0.734299i
$448$		0			0
$449$		−	214.960i		−	0.478754i	−0.970927	−	0.239377i	$-0.923057\pi$
$449$		−	214.960i		−	0.478754i	0.970927	−	0.239377i	$-0.0769432\pi$
$450$		0			0
$451$		−	83.1384i		−	0.184342i
$452$		0			0
$453$	−432.749	+	306.000i	−0.955297	+	0.675497i
$454$		0			0
$455$			176.363i			0.387612i
$456$		0			0
$457$	−786.000			−1.71991			−0.859956	−	0.510368i	$-0.829509\pi$
$457$	−786.000			−1.71991			−0.859956	+	0.510368i	$0.829509\pi$
$458$		0			0
$459$	587.878	−	166.277i	1.28078	−	0.362259i
$460$		0			0
$461$	−811.759			−1.76086			−0.880432	−	0.474172i	$-0.842748\pi$
$461$	−811.759			−1.76086			−0.880432	+	0.474172i	$0.842748\pi$
$462$		0			0
$463$	446.869			0.965160			0.482580	−	0.875852i	$-0.339700\pi$
$463$	446.869			0.965160			0.482580	+	0.875852i	$0.339700\pi$
$464$		0			0
$465$	−216.000	+	152.735i	−0.464516	+	0.328463i
$466$		0			0
$467$	426.211			0.912658			0.456329	−	0.889811i	$-0.349164\pi$
$467$	426.211			0.912658			0.456329	+	0.889811i	$0.349164\pi$
$468$		0			0
$469$			1188.00i			2.53305i
$470$		0			0
$471$	266.736	+	377.221i	0.566318	+	0.800895i
$472$		0			0
$473$			152.735i			0.322907i
$474$		0			0
$475$			176.669i			0.371935i
$476$		0			0
$477$	42.4264	−	120.000i	0.0889442	−	0.251572i
$478$		0			0
$479$		−	117.576i		−	0.245460i	−0.992440	−	0.122730i	$-0.960835\pi$
$479$		−	117.576i		−	0.245460i	0.992440	−	0.122730i	$-0.0391650\pi$
$480$		0			0
$481$	228.000			0.474012
$482$		0			0
$483$	529.090	+	748.246i	1.09542	+	1.54916i
$484$		0			0
$485$	254.558			0.524863
$486$		0			0
$487$	426.084			0.874917			0.437458	−	0.899239i	$-0.355879\pi$
$487$	426.084			0.874917			0.437458	+	0.899239i	$0.355879\pi$
$488$		0			0
$489$	−126.000	−	178.191i	−0.257669	−	0.364399i
$490$		0			0
$491$	191.060			0.389125			0.194562	−	0.980890i	$-0.437671\pi$
$491$	191.060			0.389125			0.194562	+	0.980890i	$0.437671\pi$
$492$		0			0
$493$			704.000i			1.42799i
$494$		0			0
$495$	124.708	−	352.727i	0.251935	−	0.712579i
$496$		0			0
$497$			305.470i			0.614628i
$498$		0			0
$499$			862.561i			1.72858i	0.502994	+	0.864290i	$0.332232\pi$
$499$			862.561i			1.72858i	−0.502994	+	0.864290i	$0.667768\pi$
$500$		0			0
$501$	−458.205	−	648.000i	−0.914581	−	1.29341i
$502$		0			0
$503$			205.757i			0.409060i	0.978860	+	0.204530i	$0.0655666\pi$
$503$			205.757i			0.409060i	−0.978860	+	0.204530i	$0.934433\pi$
$504$		0			0
$505$	8.00000			0.0158416
$506$		0			0
$507$	−325.782	+	230.363i	−0.642568	+	0.454364i
$508$		0			0
$509$	−845.700			−1.66149			−0.830746	−	0.556651i	$-0.812086\pi$
$509$	−845.700			−1.66149			−0.830746	+	0.556651i	$0.812086\pi$
$510$		0			0
$511$	311.769			0.610116
$512$		0			0
$513$	270.000	−	76.3675i	0.526316	−	0.148865i
$514$		0			0
$515$	−205.757			−0.399528
$516$		0			0
$517$		−	864.000i		−	1.67118i
$518$		0			0
$519$	547.328	−	387.019i	1.05458	−	0.745702i
$520$		0			0
$521$			39.5980i			0.0760038i	0.999278	+	0.0380019i	$0.0120993\pi$
$521$			39.5980i			0.0760038i	−0.999278	+	0.0380019i	$0.987901\pi$
$522$		0			0
$523$		−	654.715i		−	1.25185i	−0.779885	−	0.625923i	$-0.784723\pi$
$523$		−	654.715i		−	1.25185i	0.779885	−	0.625923i	$-0.215277\pi$
$524$		0			0
$525$	432.749	−	306.000i	0.824284	−	0.582857i
$526$		0			0
$527$		−	705.453i		−	1.33862i
$528$		0			0
$529$	−335.000			−0.633270
$530$		0			0
$531$	44.0908	−	124.708i	0.0830336	−	0.234854i
$532$		0			0
$533$	33.9411			0.0636794
$534$		0			0
$535$	207.846			0.388497
$536$		0			0
$537$	252.000	−	178.191i	0.469274	−	0.331827i
$538$		0			0
$539$	867.119			1.60876
$540$		0			0
$541$		−	330.000i		−	0.609982i	−0.952355	−	0.304991i	$-0.901347\pi$
$541$		−	330.000i		−	0.609982i	0.952355	−	0.304991i	$-0.0986534\pi$
$542$		0			0
$543$	−405.300	−	573.181i	−0.746409	−	1.05558i
$544$		0			0
$545$		−	390.323i		−	0.716189i
$546$		0			0
$547$		−	93.5307i		−	0.170989i	−0.996339	−	0.0854943i	$-0.972753\pi$
$547$		−	93.5307i		−	0.170989i	0.996339	−	0.0854943i	$-0.0272469\pi$
$548$		0			0
$549$	186.676	+	66.0000i	0.340029	+	0.120219i
$550$		0			0
$551$			323.333i			0.586811i
$552$		0			0
$553$	324.000			0.585895
$554$		0			0
$555$	186.161	+	263.272i	0.335426	+	0.474363i
$556$		0			0
$557$	613.769			1.10192			0.550959	−	0.834532i	$-0.314262\pi$
$557$	613.769			1.10192			0.550959	+	0.834532i	$0.314262\pi$
$558$		0			0
$559$	−62.3538			−0.111545
$560$		0			0
$561$	576.000	+	814.587i	1.02674	+	1.45203i
$562$		0			0
$563$	543.787			0.965873			0.482937	−	0.875655i	$-0.339570\pi$
$563$	543.787			0.965873			0.482937	+	0.875655i	$0.339570\pi$
$564$		0			0
$565$			544.000i			0.962832i
$566$		0			0
$567$	−654.715	−	529.090i	−1.15470	−	0.933139i
$568$		0			0
$569$			5.65685i			0.00994175i	0.999988	+	0.00497087i	$0.00158228\pi$
$569$			5.65685i			0.00994175i	−0.999988	+	0.00497087i	$0.998418\pi$
$570$		0			0
$571$		−	613.146i		−	1.07381i	−0.843642	−	0.536905i	$-0.819593\pi$
$571$		−	613.146i		−	1.07381i	0.843642	−	0.536905i	$-0.180407\pi$
$572$		0			0
$573$	203.647	+	288.000i	0.355404	+	0.502618i
$574$		0			0
$575$			499.696i			0.869036i
$576$		0			0
$577$	−502.000			−0.870017			−0.435009	−	0.900426i	$-0.643255\pi$
$577$	−502.000			−0.870017			−0.435009	+	0.900426i	$0.643255\pi$
$578$		0			0
$579$	−200.858	+	142.028i	−0.346905	+	0.245299i
$580$		0			0
$581$	763.675			1.31442
$582$		0			0
$583$	207.846			0.356511
$584$		0			0
$585$	144.000	+	50.9117i	0.246154	+	0.0870285i
$586$		0			0
$587$	−44.0908			−0.0751121			−0.0375561	−	0.999295i	$-0.511957\pi$
$587$	−44.0908			−0.0751121			−0.0375561	+	0.999295i	$0.511957\pi$
$588$		0			0
$589$		−	324.000i		−	0.550085i
$590$		0			0
$591$	−505.759	+	357.626i	−0.855768	+	0.605119i
$592$		0			0
$593$		−	260.215i		−	0.438812i	−0.975634	−	0.219406i	$-0.929588\pi$
$593$		−	260.215i		−	0.438812i	0.975634	−	0.219406i	$-0.0704119\pi$
$594$		0			0
$595$		−	665.108i		−	1.11783i
$596$		0			0
$597$	789.131	−	558.000i	1.32183	−	0.934673i
$598$		0			0
$599$		−	734.847i		−	1.22679i	−0.789776	−	0.613395i	$-0.789803\pi$
$599$		−	734.847i		−	1.22679i	0.789776	−	0.613395i	$-0.210197\pi$
$600$		0			0
$601$	−130.000			−0.216306			−0.108153	−	0.994134i	$-0.534494\pi$
$601$	−130.000			−0.216306			−0.108153	+	0.994134i	$0.534494\pi$
$602$		0			0
$603$	969.998	+	342.946i	1.60862	+	0.568733i
$604$		0			0
$605$	268.701			0.444133
$606$		0			0
$607$	613.146			1.01013			0.505063	−	0.863083i	$-0.331469\pi$
$607$	613.146			1.01013			0.505063	+	0.863083i	$0.331469\pi$
$608$		0			0
$609$	792.000	−	560.029i	1.30049	−	0.919587i
$610$		0			0
$611$	352.727			0.577294
$612$		0			0
$613$			794.000i			1.29527i	0.761951	+	0.647635i	$0.224242\pi$
$613$			794.000i			1.29527i	−0.761951	+	0.647635i	$0.775758\pi$
$614$		0			0
$615$	27.7128	+	39.1918i	0.0450615	+	0.0637266i
$616$		0			0
$617$			639.225i			1.03602i	0.855374	+	0.518010i	$0.173327\pi$
$617$			639.225i			1.03602i	−0.855374	+	0.518010i	$0.826673\pi$
$618$		0			0
$619$			966.484i			1.56136i	0.624928	+	0.780682i	$0.285128\pi$
$619$			966.484i			1.56136i	−0.624928	+	0.780682i	$0.714872\pi$
$620$		0			0
$621$	763.675	−	216.000i	1.22975	−	0.347826i
$622$		0			0
$623$		−	58.7878i		−	0.0943624i
$624$		0			0
$625$	89.0000			0.142400
$626$		0			0
$627$	264.545	+	374.123i	0.421922	+	0.596687i
$628$		0			0
$629$	−859.842			−1.36700
$630$		0			0
$631$	93.5307			0.148226			0.0741131	−	0.997250i	$-0.476387\pi$
$631$	93.5307			0.148226			0.0741131	+	0.997250i	$0.476387\pi$
$632$		0			0
$633$	234.000	+	330.926i	0.369668	+	0.522790i
$634$		0			0
$635$	−146.969			−0.231448
$636$		0			0
$637$			354.000i			0.555730i
$638$		0			0
$639$	249.415	+	88.1816i	0.390321	+	0.137999i
$640$		0			0
$641$			622.254i			0.970755i	0.874305	+	0.485378i	$0.161318\pi$
$641$			622.254i			0.970755i	−0.874305	+	0.485378i	$0.838682\pi$
$642$		0			0
$643$		−	592.361i		−	0.921246i	−0.887596	−	0.460623i	$-0.847626\pi$
$643$		−	592.361i		−	0.921246i	0.887596	−	0.460623i	$-0.152374\pi$
$644$		0			0
$645$	−50.9117	−	72.0000i	−0.0789328	−	0.111628i
$646$		0			0
$647$		−	1263.94i		−	1.95353i	−0.214303	−	0.976767i	$-0.568748\pi$
$647$		−	1263.94i		−	1.95353i	0.214303	−	0.976767i	$-0.431252\pi$
$648$		0			0
$649$	216.000			0.332820
$650$		0			0
$651$	−793.635	+	561.184i	−1.21910	+	0.862035i
$652$		0			0
$653$	591.141			0.905270			0.452635	−	0.891696i	$-0.350484\pi$
$653$	591.141			0.905270			0.452635	+	0.891696i	$0.350484\pi$
$654$		0			0
$655$	−124.708			−0.190393
$656$		0			0
$657$	90.0000	−	254.558i	0.136986	−	0.387456i
$658$		0			0
$659$	−1161.06			−1.76185			−0.880924	−	0.473257i	$-0.843078\pi$
$659$	−1161.06			−1.76185			−0.880924	+	0.473257i	$0.843078\pi$
$660$		0			0
$661$			442.000i			0.668684i	0.942452	+	0.334342i	$0.108514\pi$
$661$			442.000i			0.668684i	−0.942452	+	0.334342i	$0.891486\pi$
$662$		0			0
$663$	−332.554	+	235.151i	−0.501589	+	0.354677i
$664$		0			0
$665$		−	305.470i		−	0.459354i
$666$		0			0
$667$			914.523i			1.37110i
$668$		0			0
$669$	−890.955	+	630.000i	−1.33177	+	0.941704i
$670$		0			0
$671$			323.333i			0.481867i
$672$		0			0
$673$	−742.000			−1.10253			−0.551263	−	0.834332i	$-0.685854\pi$
$673$	−742.000			−1.10253			−0.551263	+	0.834332i	$0.685854\pi$
$674$		0			0
$675$	−124.924	−	441.673i	−0.185073	−	0.654330i
$676$		0			0
$677$	2.82843			0.00417788			0.00208894	−	0.999998i	$-0.499335\pi$
$677$	2.82843			0.00417788			0.00208894	+	0.999998i	$0.499335\pi$
$678$		0			0
$679$	935.307			1.37748
$680$		0			0
$681$	−36.0000	+	25.4558i	−0.0528634	+	0.0373801i
$682$		0			0
$683$	249.848			0.365810			0.182905	−	0.983131i	$-0.441450\pi$
$683$	249.848			0.365810			0.182905	+	0.983131i	$0.441450\pi$
$684$		0			0
$685$			112.000i			0.163504i
$686$		0			0
$687$	426.084	+	602.574i	0.620210	+	0.877110i
$688$		0			0
$689$			84.8528i			0.123154i
$690$		0			0
$691$			322.161i			0.466225i	0.972450	+	0.233112i	$0.0748910\pi$
$691$			322.161i			0.466225i	−0.972450	+	0.233112i	$0.925109\pi$
$692$		0			0
$693$	458.205	−	1296.00i	0.661191	−	1.87013i
$694$		0			0
$695$			382.120i			0.549814i
$696$		0			0
$697$	−128.000			−0.183644
$698$		0			0
$699$	−362.524	−	512.687i	−0.518633	−	0.733458i
$700$		0			0
$701$	−234.759			−0.334892			−0.167446	−	0.985881i	$-0.553552\pi$
$701$	−234.759			−0.334892			−0.167446	+	0.985881i	$0.553552\pi$
$702$		0			0
$703$	−394.908			−0.561746
$704$		0			0
$705$	288.000	+	407.294i	0.408511	+	0.577721i
$706$		0			0
$707$	29.3939			0.0415755
$708$		0			0
$709$		−	102.000i		−	0.143865i	−0.997410	−	0.0719323i	$-0.977083\pi$
$709$		−	102.000i		−	0.143865i	0.997410	−	0.0719323i	$-0.0229166\pi$
$710$		0			0
$711$	93.5307	−	264.545i	0.131548	−	0.372074i
$712$		0			0
$713$		−	916.410i		−	1.28529i
$714$		0			0
$715$			249.415i			0.348833i
$716$		0			0
$717$	−712.764	−	1008.00i	−0.994092	−	1.40586i
$718$		0			0
$719$		−	529.090i		−	0.735869i	−0.929852	−	0.367934i	$-0.880065\pi$
$719$		−	529.090i		−	0.735869i	0.929852	−	0.367934i	$-0.119935\pi$
$720$		0			0
$721$	−756.000			−1.04854
$722$		0			0
$723$	−514.393	+	363.731i	−0.711470	+	0.503085i
$724$		0			0
$725$	528.916			0.729539
$726$		0			0
$727$	−1070.41			−1.47236			−0.736181	−	0.676785i	$-0.763373\pi$
$727$	−1070.41			−1.47236			−0.736181	+	0.676785i	$0.763373\pi$
$728$		0			0
$729$	−621.000	+	381.838i	−0.851852	+	0.523783i
$730$		0			0
$731$	235.151			0.321684
$732$		0			0
$733$		−	378.000i		−	0.515689i	−0.966186	−	0.257844i	$-0.916988\pi$
$733$		−	378.000i		−	0.515689i	0.966186	−	0.257844i	$-0.0830122\pi$
$734$		0			0
$735$	−408.764	+	289.040i	−0.556141	+	0.393251i
$736$		0			0
$737$			1680.09i			2.27963i
$738$		0			0
$739$			1111.98i			1.50470i	0.658761	+	0.752352i	$0.271081\pi$
$739$			1111.98i			1.50470i	−0.658761	+	0.752352i	$0.728919\pi$
$740$		0			0
$741$	−152.735	+	108.000i	−0.206120	+	0.145749i
$742$		0			0
$743$		−	205.757i		−	0.276928i	−0.990368	−	0.138464i	$-0.955784\pi$
$743$		−	205.757i		−	0.276928i	0.990368	−	0.138464i	$-0.0442164\pi$
$744$		0			0
$745$	−536.000			−0.719463
$746$		0			0
$747$	220.454	−	623.538i	0.295119	−	0.834723i
$748$		0			0
$749$	763.675			1.01959
$750$		0			0
$751$	−218.238			−0.290597			−0.145299	−	0.989388i	$-0.546414\pi$
$751$	−218.238			−0.290597			−0.145299	+	0.989388i	$0.546414\pi$
$752$		0			0
$753$	972.000	−	687.308i	1.29084	−	0.912759i
$754$		0			0
$755$	499.696			0.661849
$756$		0			0
$757$			1098.00i			1.45046i	0.688505	+	0.725231i	$0.258267\pi$
$757$			1098.00i			1.45046i	−0.688505	+	0.725231i	$0.741733\pi$
$758$		0			0
$759$	748.246	+	1058.18i	0.985831	+	1.39418i
$760$		0			0
$761$			1250.16i			1.64279i	0.570358	+	0.821396i	$0.306804\pi$
$761$			1250.16i			1.64279i	−0.570358	+	0.821396i	$0.693196\pi$
$762$		0			0
$763$		−	1434.14i		−	1.87960i
$764$		0			0
$765$	−543.058	−	192.000i	−0.709880	−	0.250980i
$766$		0			0
$767$			88.1816i			0.114970i
$768$		0			0
$769$	−334.000			−0.434330			−0.217165	−	0.976135i	$-0.569681\pi$
$769$	−334.000			−0.434330			−0.217165	+	0.976135i	$0.569681\pi$
$770$		0			0
$771$	−666.261	−	942.236i	−0.864152	−	1.22210i
$772$		0			0
$773$	−562.857			−0.728146			−0.364073	−	0.931370i	$-0.618614\pi$
$773$	−562.857			−0.728146			−0.364073	+	0.931370i	$0.618614\pi$
$774$		0			0
$775$	−530.008			−0.683881
$776$		0			0
$777$	684.000	+	967.322i	0.880309	+	1.24494i
$778$		0			0
$779$	−58.7878			−0.0754657
$780$		0			0
$781$			432.000i			0.553137i
$782$		0			0
$783$	−228.631	−	808.332i	−0.291993	−	1.03235i
$784$		0			0
$785$		−	435.578i		−	0.554876i
$786$		0			0
$787$		−	841.777i		−	1.06960i	−0.844978	−	0.534801i	$-0.820387\pi$
$787$		−	841.777i		−	1.06960i	0.844978	−	0.534801i	$-0.179613\pi$
$788$		0			0
$789$	−865.499	−	1224.00i	−1.09696	−	1.55133i
$790$		0			0
$791$			1998.78i			2.52691i
$792$		0			0
$793$	−132.000			−0.166456
$794$		0			0
$795$	−97.9796	+	69.2820i	−0.123245	+	0.0871472i
$796$		0			0
$797$	−393.151			−0.493289			−0.246645	−	0.969106i	$-0.579328\pi$
$797$	−393.151			−0.493289			−0.246645	+	0.969106i	$0.579328\pi$
$798$		0			0
$799$	−1330.22			−1.66485
$800$		0			0
$801$	−48.0000	−	16.9706i	−0.0599251	−	0.0211867i
$802$		0			0
$803$	440.908			0.549076
$804$		0			0
$805$		−	864.000i		−	1.07329i
$806$		0			0
$807$	48.4974	−	34.2929i	0.0600959	−	0.0424942i
$808$		0			0
$809$		−	1227.54i		−	1.51735i	−0.651468	−	0.758676i	$-0.725847\pi$
$809$		−	1227.54i		−	1.51735i	0.651468	−	0.758676i	$-0.274153\pi$
$810$		0			0
$811$		−	1153.55i		−	1.42237i	−0.703003	−	0.711187i	$-0.748158\pi$
$811$		−	1153.55i		−	1.42237i	0.703003	−	0.711187i	$-0.251842\pi$
$812$		0			0
$813$	330.926	−	234.000i	0.407043	−	0.287823i
$814$		0			0
$815$			205.757i			0.252463i
$816$		0			0
$817$	108.000			0.132191
$818$		0			0
$819$	529.090	+	187.061i	0.646019	+	0.228402i
$820$		0			0
$821$	−370.524			−0.451308			−0.225654	−	0.974207i	$-0.572452\pi$
$821$	−370.524			−0.451308			−0.225654	+	0.974207i	$0.572452\pi$
$822$		0			0
$823$	−405.300			−0.492466			−0.246233	−	0.969211i	$-0.579193\pi$
$823$	−405.300			−0.492466			−0.246233	+	0.969211i	$0.579193\pi$
$824$		0			0
$825$	612.000	−	432.749i	0.741818	−	0.524545i
$826$		0			0
$827$	−1043.48			−1.26177			−0.630884	−	0.775877i	$-0.717308\pi$
$827$	−1043.48			−1.26177			−0.630884	+	0.775877i	$0.717308\pi$
$828$		0			0
$829$			486.000i			0.586248i	0.956074	+	0.293124i	$0.0946950\pi$
$829$			486.000i			0.586248i	−0.956074	+	0.293124i	$0.905305\pi$
$830$		0			0
$831$	342.946	+	484.999i	0.412691	+	0.583633i
$832$		0			0
$833$		−	1335.02i		−	1.60266i
$834$		0			0
$835$			748.246i			0.896103i
$836$		0			0
$837$	229.103	+	810.000i	0.273719	+	0.967742i
$838$		0			0
$839$			264.545i			0.315310i	0.987494	+	0.157655i	$0.0503934\pi$
$839$			264.545i			0.315310i	−0.987494	+	0.157655i	$0.949607\pi$
$840$		0			0
$841$	127.000			0.151011
$842$		0			0
$843$	−342.929	−	484.974i	−0.406795	−	0.575296i
$844$		0			0
$845$	376.181			0.445184
$846$		0			0
$847$	987.269			1.16561
$848$		0			0
$849$	−90.0000	−	127.279i	−0.106007	−	0.149917i
$850$		0			0
$851$	−1116.97			−1.31254
$852$		0			0
$853$			1642.00i			1.92497i	0.271334	+	0.962485i	$0.412535\pi$
$853$			1642.00i			1.92497i	−0.271334	+	0.962485i	$0.587465\pi$
$854$		0			0
$855$	−249.415	−	88.1816i	−0.291714	−	0.103136i
$856$		0			0
$857$		−	1057.83i		−	1.23434i	−0.786829	−	0.617171i	$-0.788279\pi$
$857$		−	1057.83i		−	1.23434i	0.786829	−	0.617171i	$-0.211721\pi$
$858$		0			0
$859$			93.5307i			0.108883i	0.998517	+	0.0544416i	$0.0173379\pi$
$859$			93.5307i			0.108883i	−0.998517	+	0.0544416i	$0.982662\pi$
$860$		0			0
$861$	101.823	+	144.000i	0.118262	+	0.167247i
$862$		0			0
$863$			235.151i			0.272481i	0.990676	+	0.136240i	$0.0435020\pi$
$863$			235.151i			0.272481i	−0.990676	+	0.136240i	$0.956498\pi$
$864$		0			0
$865$	−632.000			−0.730636
$866$		0			0
$867$	546.236	−	386.247i	0.630030	−	0.445499i
$868$		0			0
$869$	458.205			0.527279
$870$		0			0
$871$	−685.892			−0.787477
$872$		0			0
$873$	270.000	−	763.675i	0.309278	−	0.874771i
$874$		0			0
$875$	−1234.54			−1.41091
$876$		0			0
$877$			550.000i			0.627138i	0.949565	+	0.313569i	$0.101525\pi$
$877$			550.000i			0.627138i	−0.949565	+	0.313569i	$0.898475\pi$
$878$		0			0
$879$	879.882	−	622.170i	1.00100	−	0.707816i
$880$		0			0
$881$		−	1210.57i		−	1.37408i	−0.726618	−	0.687041i	$-0.758909\pi$
$881$		−	1210.57i		−	1.37408i	0.726618	−	0.687041i	$-0.241091\pi$
$882$		0			0
$883$		−	841.777i		−	0.953314i	−0.879089	−	0.476657i	$-0.841848\pi$
$883$		−	841.777i		−	0.953314i	0.879089	−	0.476657i	$-0.158152\pi$
$884$		0			0
$885$	−101.823	+	72.0000i	−0.115055	+	0.0813559i
$886$		0			0
$887$			1146.36i			1.29240i	0.763167	+	0.646201i	$0.223643\pi$
$887$			1146.36i			1.29240i	−0.763167	+	0.646201i	$0.776357\pi$
$888$		0			0
$889$	−540.000			−0.607424
$890$		0			0
$891$	−925.907	−	748.246i	−1.03918	−	0.839782i
$892$		0			0
$893$	−610.940			−0.684144
$894$		0			0
$895$	−290.985			−0.325122
$896$		0			0
$897$	−432.000	+	305.470i	−0.481605	+	0.340546i
$898$		0			0
$899$	−969.998			−1.07897
$900$		0			0
$901$		−	320.000i		−	0.355161i
$902$		0			0
$903$	−187.061	−	264.545i	−0.207156	−	0.292962i
$904$		0			0
$905$			661.852i			0.731328i
$906$		0			0
$907$			1506.88i			1.66139i	0.556725	+	0.830697i	$0.312058\pi$
$907$			1506.88i			1.66139i	−0.556725	+	0.830697i	$0.687942\pi$
$908$		0			0
$909$	8.48528	−	24.0000i	0.00933474	−	0.0264026i
$910$		0			0
$911$			1469.69i			1.61328i	0.591046	+	0.806638i	$0.298715\pi$
$911$			1469.69i			1.61328i	−0.591046	+	0.806638i	$0.701285\pi$
$912$		0			0
$913$	1080.00			1.18291
$914$		0			0
$915$	−107.778	−	152.420i	−0.117790	−	0.166580i
$916$		0			0
$917$	−458.205			−0.499679
$918$		0			0
$919$	−1070.41			−1.16475			−0.582376	−	0.812919i	$-0.697877\pi$
$919$	−1070.41			−1.16475			−0.582376	+	0.812919i	$0.697877\pi$
$920$		0			0
$921$	−342.000	−	483.661i	−0.371336	−	0.525148i
$922$		0			0
$923$	−176.363			−0.191076
$924$		0			0
$925$			646.000i			0.698378i
$926$		0			0
$927$	−218.238	+	617.271i	−0.235424	+	0.665881i
$928$		0			0
$929$		−	667.509i		−	0.718524i	−0.933237	−	0.359262i	$-0.883028\pi$
$929$		−	667.509i		−	0.718524i	0.933237	−	0.359262i	$-0.116972\pi$
$930$		0			0
$931$		−	613.146i		−	0.658589i
$932$		0			0
$933$	458.205	+	648.000i	0.491110	+	0.694534i
$934$		0			0
$935$		−	940.604i		−	1.00599i
$936$		0			0
$937$	−1402.00			−1.49626			−0.748132	−	0.663550i	$-0.769049\pi$
$937$	−1402.00			−1.49626			−0.748132	+	0.663550i	$0.769049\pi$
$938$		0			0
$939$	651.564	−	460.726i	0.693892	−	0.490656i
$940$		0			0
$941$	−1694.23			−1.80045			−0.900227	−	0.435420i	$-0.856600\pi$
$941$	−1694.23			−1.80045			−0.900227	+	0.435420i	$0.856600\pi$
$942$		0			0
$943$	−166.277			−0.176328
$944$		0			0
$945$	216.000	+	763.675i	0.228571	+	0.808122i
$946$		0			0
$947$	1190.45			1.25708			0.628539	−	0.777778i	$-0.283653\pi$
$947$	1190.45			1.25708			0.628539	+	0.777778i	$0.283653\pi$
$948$		0			0
$949$			180.000i			0.189673i
$950$		0			0
$951$	−1032.30	+	729.948i	−1.08549	+	0.767558i
$952$		0			0
$953$		−	401.637i		−	0.421445i	−0.977546	−	0.210722i	$-0.932418\pi$
$953$		−	401.637i		−	0.421445i	0.977546	−	0.210722i	$-0.0675816\pi$
$954$		0			0
$955$		−	332.554i		−	0.348224i
$956$		0			0
$957$	1120.06	−	792.000i	1.17038	−	0.827586i
$958$		0			0
$959$			411.514i			0.429108i
$960$		0			0
$961$	11.0000			0.0114464
$962$		0			0
$963$	220.454	−	623.538i	0.228924	−	0.647496i
$964$		0			0
$965$	231.931			0.240343
$966$		0			0
$967$	1423.75			1.47233			0.736166	−	0.676801i	$-0.236634\pi$
$967$	1423.75			1.47233			0.736166	+	0.676801i	$0.236634\pi$
$968$		0			0
$969$	576.000	−	407.294i	0.594427	−	0.420324i
$970$		0			0
$971$	−808.332			−0.832473			−0.416237	−	0.909256i	$-0.636651\pi$
$971$	−808.332			−0.832473			−0.416237	+	0.909256i	$0.636651\pi$
$972$		0			0
$973$			1404.00i			1.44296i
$974$		0			0
$975$	176.669	+	249.848i	0.181199	+	0.256254i
$976$		0			0
$977$			565.685i			0.579002i	0.957178	+	0.289501i	$0.0934894\pi$
$977$			565.685i			0.579002i	−0.957178	+	0.289501i	$0.906511\pi$
$978$		0			0
$979$		−	83.1384i		−	0.0849218i
$980$		0			0
$981$	−1170.97	−	414.000i	−1.19365	−	0.422018i
$982$		0			0
$983$			558.484i			0.568142i	0.958803	+	0.284071i	$0.0916852\pi$
$983$			558.484i			0.568142i	−0.958803	+	0.284071i	$0.908315\pi$
$984$		0			0
$985$	584.000			0.592893
$986$		0			0
$987$	1058.18	+	1496.49i	1.07212	+	1.51620i
$988$		0			0
$989$	305.470			0.308868
$990$		0			0
$991$	−550.792			−0.555794			−0.277897	−	0.960611i	$-0.589637\pi$
$991$	−550.792			−0.555794			−0.277897	+	0.960611i	$0.589637\pi$
$992$		0			0
$993$	342.000	+	483.661i	0.344411	+	0.487071i
$994$		0			0
$995$	−911.210			−0.915789
$996$		0			0
$997$		−	1606.00i		−	1.61083i	−0.592710	−	0.805416i	$-0.701942\pi$
$997$		−	1606.00i		−	1.61083i	0.592710	−	0.805416i	$-0.298058\pi$
$998$		0			0
$999$	987.269	−	279.242i	0.988257	−	0.279521i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.h.f.641.4		8
3.2	odd	2	inner	768.3.h.f.641.7		8
4.3	odd	2	inner	768.3.h.f.641.6		8
8.3	odd	2	inner	768.3.h.f.641.3		8
8.5	even	2	inner	768.3.h.f.641.5		8
12.11	even	2	inner	768.3.h.f.641.1		8
16.3	odd	4		96.3.e.a.65.4	yes	4
16.5	even	4		192.3.e.e.65.4		4
16.11	odd	4		192.3.e.e.65.1		4
16.13	even	4		96.3.e.a.65.1	✓	4
24.5	odd	2	inner	768.3.h.f.641.2		8
24.11	even	2	inner	768.3.h.f.641.8		8
48.5	odd	4		192.3.e.e.65.3		4
48.11	even	4		192.3.e.e.65.2		4
48.29	odd	4		96.3.e.a.65.2	yes	4
48.35	even	4		96.3.e.a.65.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.e.a.65.1	✓	4	16.13	even	4
96.3.e.a.65.2	yes	4	48.29	odd	4
96.3.e.a.65.3	yes	4	48.35	even	4
96.3.e.a.65.4	yes	4	16.3	odd	4
192.3.e.e.65.1		4	16.11	odd	4
192.3.e.e.65.2		4	48.11	even	4
192.3.e.e.65.3		4	48.5	odd	4
192.3.e.e.65.4		4	16.5	even	4
768.3.h.f.641.1		8	12.11	even	2	inner
768.3.h.f.641.2		8	24.5	odd	2	inner
768.3.h.f.641.3		8	8.3	odd	2	inner
768.3.h.f.641.4		8	1.1	even	1	trivial
768.3.h.f.641.5		8	8.5	even	2	inner
768.3.h.f.641.6		8	4.3	odd	2	inner
768.3.h.f.641.7		8	3.2	odd	2	inner
768.3.h.f.641.8		8	24.11	even	2	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.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.44949 + 1.73205i) q^{3} +2.82843 q^{5} +10.3923 q^{7} +(3.00000 - 8.48528i) q^{9} +O(q^{10})\)	\(q+(-2.44949 + 1.73205i) q^{3} +2.82843 q^{5} +10.3923 q^{7} +(3.00000 - 8.48528i) q^{9} +14.6969 q^{11} +6.00000i q^{13} +(-6.92820 + 4.89898i) q^{15} -22.6274i q^{17} -10.3923i q^{19} +(-25.4558 + 18.0000i) q^{21} -29.3939i q^{23} -17.0000 q^{25} +(7.34847 + 25.9808i) q^{27} -31.1127 q^{29} +31.1769 q^{31} +(-36.0000 + 25.4558i) q^{33} +29.3939 q^{35} -38.0000i q^{37} +(-10.3923 - 14.6969i) q^{39} -5.65685i q^{41} +10.3923i q^{43} +(8.48528 - 24.0000i) q^{45} -58.7878i q^{47} +59.0000 q^{49} +(39.1918 + 55.4256i) q^{51} +14.1421 q^{53} +41.5692 q^{55} +(18.0000 + 25.4558i) q^{57} +14.6969 q^{59} +22.0000i q^{61} +(31.1769 - 88.1816i) q^{63} +16.9706i q^{65} +114.315i q^{67} +(50.9117 + 72.0000i) q^{69} +29.3939i q^{71} +30.0000 q^{73} +(41.6413 - 29.4449i) q^{75} +152.735 q^{77} +31.1769 q^{79} +(-63.0000 - 50.9117i) q^{81} +73.4847 q^{83} -64.0000i q^{85} +(76.2102 - 53.8888i) q^{87} -5.65685i q^{89} +62.3538i q^{91} +(-76.3675 + 54.0000i) q^{93} -29.3939i q^{95} +90.0000 q^{97} +(44.0908 - 124.708i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} - 136 q^{25} - 288 q^{33} + 472 q^{49} + 144 q^{57} + 240 q^{73} - 504 q^{81} + 720 q^{97}+O(q^{100}) \) 8 * q + 24 * q^9 - 136 * q^25 - 288 * q^33 + 472 * q^49 + 144 * q^57 + 240 * q^73 - 504 * q^81 + 720 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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