LMFDB - Embedded newform 768.3.h.f.641.8

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			641.8
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	768.641
Dual form			768.3.h.f.641.6

$q$-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	$-0.766269\pi$
$7$	−10.3923			−1.48461			−0.742307	+	0.670059i	$0.766269\pi$
$8$		0			0
$9$	3.00000	+	8.48528i	0.333333	+	0.942809i
$10$		0			0
$11$	−14.6969			−1.33609			−0.668043	−	0.744123i	$-0.732868\pi$
$11$	−14.6969			−1.33609			−0.668043	+	0.744123i	$0.732868\pi$
$12$		0			0
$13$		−	6.00000i		−	0.461538i	−0.973009	−	0.230769i	$-0.925876\pi$
$13$		−	6.00000i		−	0.461538i	0.973009	−	0.230769i	$-0.0741242\pi$
$14$		0			0
$15$	6.92820	+	4.89898i	0.461880	+	0.326599i
$16$		0			0
$17$			22.6274i			1.33102i	0.746387	+	0.665512i	$0.231787\pi$
$17$			22.6274i			1.33102i	−0.746387	+	0.665512i	$0.768213\pi$
$18$		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	$-0.667716\pi$
$31$	−31.1769			−1.00571			−0.502853	+	0.864372i	$0.667716\pi$
$32$		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	$0.171656\pi$
$37$			38.0000i			1.02703i	−0.858082	+	0.513514i	$0.828344\pi$
$38$		0			0
$39$	10.3923	−	14.6969i	0.266469	−	0.376845i
$40$		0			0
$41$			5.65685i			0.137972i	0.997618	+	0.0689860i	$0.0219764\pi$
$41$			5.65685i			0.137972i	−0.997618	+	0.0689860i	$0.978024\pi$
$42$		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	$-0.539749\pi$
$59$	−14.6969			−0.249101			−0.124550	+	0.992213i	$0.539749\pi$
$60$		0			0
$61$		−	22.0000i		−	0.360656i	−0.983607	−	0.180328i	$-0.942284\pi$
$61$		−	22.0000i		−	0.360656i	0.983607	−	0.180328i	$-0.0577159\pi$
$62$		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	$-0.563225\pi$
$79$	−31.1769			−0.394644			−0.197322	+	0.980339i	$0.563225\pi$
$80$		0			0
$81$	−63.0000	+	50.9117i	−0.777778	+	0.628539i
$82$		0			0
$83$	−73.4847			−0.885358			−0.442679	−	0.896680i	$-0.645972\pi$
$83$	−73.4847			−0.885358			−0.442679	+	0.896680i	$0.645972\pi$
$84$		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	$0.0101176\pi$
$89$			5.65685i			0.0635602i	−0.999495	+	0.0317801i	$0.989882\pi$
$90$		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	$-0.385115\pi$
$103$	72.7461			0.706273			0.353137	+	0.935572i	$0.385115\pi$
$104$		0			0
$105$	−72.0000	−	50.9117i	−0.685714	−	0.484873i
$106$		0			0
$107$	−73.4847			−0.686773			−0.343386	−	0.939194i	$-0.611574\pi$
$107$	−73.4847			−0.686773			−0.343386	+	0.939194i	$0.611574\pi$
$108$		0			0
$109$			138.000i			1.26606i	0.774129	+	0.633028i	$0.218188\pi$
$109$			138.000i			1.26606i	−0.774129	+	0.633028i	$0.781812\pi$
$110$		0			0
$111$	−65.8179	+	93.0806i	−0.592954	+	0.838564i
$112$		0			0
$113$		−	192.333i		−	1.70206i	−0.525115	−	0.851031i	$-0.675978\pi$
$113$		−	192.333i		−	1.70206i	0.525115	−	0.851031i	$-0.324022\pi$
$114$		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	$-0.434419\pi$
$127$	51.9615			0.409146			0.204573	+	0.978851i	$0.434419\pi$
$128$		0			0
$129$	−18.0000	+	25.4558i	−0.139535	+	0.197332i
$130$		0			0
$131$	44.0908			0.336571			0.168286	−	0.985738i	$-0.446177\pi$
$131$	44.0908			0.336571			0.168286	+	0.985738i	$0.446177\pi$
$132$		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	$-0.953837\pi$
$137$		−	39.5980i		−	0.289036i	0.989502	−	0.144518i	$-0.0461632\pi$
$138$		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	$-0.698904\pi$
$151$	−176.669			−1.16999			−0.584997	+	0.811035i	$0.698904\pi$
$152$		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	$0.163166\pi$
$157$			154.000i			0.980892i	−0.871472	+	0.490446i	$0.836834\pi$
$158$		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	$-0.407219\pi$
$179$	102.879			0.574741			0.287370	+	0.957820i	$0.407219\pi$
$180$		0			0
$181$		−	234.000i		−	1.29282i	−0.762991	−	0.646409i	$-0.776270\pi$
$181$		−	234.000i		−	1.29282i	0.762991	−	0.646409i	$-0.223730\pi$
$182$		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	$-0.199765\pi$
$199$	322.161			1.61890			0.809451	+	0.587188i	$0.199765\pi$
$200$		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	$-0.803560\pi$
$223$	−363.731			−1.63108			−0.815540	+	0.578701i	$0.803560\pi$
$224$		0			0
$225$	−51.0000	−	144.250i	−0.226667	−	0.641110i
$226$		0			0
$227$	−14.6969			−0.0647442			−0.0323721	−	0.999476i	$-0.510306\pi$
$227$	−14.6969			−0.0647442			−0.0323721	+	0.999476i	$0.510306\pi$
$228$		0			0
$229$			246.000i			1.07424i	0.843507	+	0.537118i	$0.180487\pi$
$229$			246.000i			1.07424i	−0.843507	+	0.537118i	$0.819513\pi$
$230$		0			0
$231$	374.123	+	264.545i	1.61958	+	1.14522i
$232$		0			0
$233$		−	209.304i		−	0.898299i	−0.893457	−	0.449149i	$-0.851727\pi$
$233$		−	209.304i		−	0.898299i	0.893457	−	0.449149i	$-0.148273\pi$
$234$		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	$-0.209835\pi$
$251$	396.817			1.58095			0.790473	+	0.612497i	$0.209835\pi$
$252$		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	$-0.730833\pi$
$257$		−	384.666i		−	1.49676i	0.663273	−	0.748378i	$-0.269167\pi$
$258$		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	$-0.419812\pi$
$271$	135.100			0.498524			0.249262	+	0.968436i	$0.419812\pi$
$272$		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	$0.116337\pi$
$277$			198.000i			0.714801i	−0.933951	+	0.357401i	$0.883663\pi$
$278$		0			0
$279$	−93.5307	−	264.545i	−0.335236	−	0.948190i
$280$		0			0
$281$		−	197.990i		−	0.704590i	−0.935889	−	0.352295i	$-0.885401\pi$
$281$		−	197.990i		−	0.704590i	0.935889	−	0.352295i	$-0.114599\pi$
$282$		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	$-0.710496\pi$
$347$	−426.211			−1.22827			−0.614137	+	0.789199i	$0.710496\pi$
$348$		0			0
$349$		−	166.000i		−	0.475645i	−0.971309	−	0.237822i	$-0.923566\pi$
$349$		−	166.000i		−	0.475645i	0.971309	−	0.237822i	$-0.0764336\pi$
$350$		0			0
$351$	155.885	+	44.0908i	0.444116	+	0.125615i
$352$		0			0
$353$		−	22.6274i		−	0.0641003i	−0.999486	−	0.0320502i	$-0.989796\pi$
$353$		−	22.6274i		−	0.0641003i	0.999486	−	0.0320502i	$-0.0102036\pi$
$354$		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	$-0.441076\pi$
$367$	135.100			0.368120			0.184060	+	0.982915i	$0.441076\pi$
$368$		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	$0.0505616\pi$
$373$			118.000i			0.316354i	−0.987411	+	0.158177i	$0.949438\pi$
$374$		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	$-0.913135\pi$
$397$		−	214.000i		−	0.539043i	0.962994	−	0.269521i	$-0.0868655\pi$
$398$		0			0
$399$	−187.061	+	264.545i	−0.468826	+	0.663020i
$400$		0			0
$401$			226.274i			0.564275i	0.959374	+	0.282137i	$0.0910434\pi$
$401$			226.274i			0.564275i	−0.959374	+	0.282137i	$0.908957\pi$
$402$		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	$-0.644473\pi$
$419$	−367.423			−0.876906			−0.438453	+	0.898754i	$0.644473\pi$
$420$		0			0
$421$			534.000i			1.26841i	0.773166	+	0.634204i	$0.218672\pi$
$421$			534.000i			1.26841i	−0.773166	+	0.634204i	$0.781328\pi$
$422$		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	$-0.627732\pi$
$439$	−342.946			−0.781198			−0.390599	+	0.920561i	$0.627732\pi$
$440$		0			0
$441$	177.000	+	500.632i	0.401361	+	1.13522i
$442$		0			0
$443$	514.393			1.16116			0.580579	−	0.814204i	$-0.302826\pi$
$443$	514.393			1.16116			0.580579	+	0.814204i	$0.302826\pi$
$444$		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	$0.0769432\pi$
$449$			214.960i			0.478754i	−0.970927	+	0.239377i	$0.923057\pi$
$450$		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	$-0.660300\pi$
$463$	−446.869			−0.965160			−0.482580	+	0.875852i	$0.660300\pi$
$464$		0			0
$465$	−216.000	−	152.735i	−0.464516	−	0.328463i
$466$		0			0
$467$	−426.211			−0.912658			−0.456329	−	0.889811i	$-0.650836\pi$
$467$	−426.211			−0.912658			−0.456329	+	0.889811i	$0.650836\pi$
$468$		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	$-0.644121\pi$
$487$	−426.084			−0.874917			−0.437458	+	0.899239i	$0.644121\pi$
$488$		0			0
$489$	−126.000	+	178.191i	−0.257669	+	0.364399i
$490$		0			0
$491$	−191.060			−0.389125			−0.194562	−	0.980890i	$-0.562329\pi$
$491$	−191.060			−0.389125			−0.194562	+	0.980890i	$0.562329\pi$
$492$		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	$-0.987901\pi$
$521$		−	39.5980i		−	0.0760038i	0.999278	−	0.0380019i	$-0.0120993\pi$
$522$		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	$0.0986534\pi$
$541$			330.000i			0.609982i	−0.952355	+	0.304991i	$0.901347\pi$
$542$		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	$-0.660430\pi$
$563$	−543.787			−0.965873			−0.482937	+	0.875655i	$0.660430\pi$
$564$		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	$-0.998418\pi$
$569$		−	5.65685i		−	0.00994175i	0.999988	−	0.00497087i	$-0.00158228\pi$
$570$		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	$-0.488043\pi$
$587$	44.0908			0.0751121			0.0375561	+	0.999295i	$0.488043\pi$
$588$		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	$0.0704119\pi$
$593$			260.215i			0.438812i	−0.975634	+	0.219406i	$0.929588\pi$
$594$		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	$-0.668531\pi$
$607$	−613.146			−1.01013			−0.505063	+	0.863083i	$0.668531\pi$
$608$		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	$-0.775758\pi$
$613$		−	794.000i		−	1.29527i	0.761951	−	0.647635i	$-0.224242\pi$
$614$		0			0
$615$	−27.7128	+	39.1918i	−0.0450615	+	0.0637266i
$616$		0			0
$617$		−	639.225i		−	1.03602i	−0.855374	−	0.518010i	$-0.826673\pi$
$617$		−	639.225i		−	1.03602i	0.855374	−	0.518010i	$-0.173327\pi$
$618$		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	$-0.523613\pi$
$631$	−93.5307			−0.148226			−0.0741131	+	0.997250i	$0.523613\pi$
$632$		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	$-0.838682\pi$
$641$		−	622.254i		−	0.970755i	0.874305	−	0.485378i	$-0.161318\pi$
$642$		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	$-0.156922\pi$
$659$	1161.06			1.76185			0.880924	+	0.473257i	$0.156922\pi$
$660$		0			0
$661$		−	442.000i		−	0.668684i	−0.942452	−	0.334342i	$-0.891486\pi$
$661$		−	442.000i		−	0.668684i	0.942452	−	0.334342i	$-0.108514\pi$
$662$		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	$-0.558550\pi$
$683$	−249.848			−0.365810			−0.182905	+	0.983131i	$0.558550\pi$
$684$		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	$0.0229166\pi$
$709$			102.000i			0.143865i	−0.997410	+	0.0719323i	$0.977083\pi$
$710$		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	$-0.236627\pi$
$727$	1070.41			1.47236			0.736181	+	0.676785i	$0.236627\pi$
$728$		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	$0.0830122\pi$
$733$			378.000i			0.515689i	−0.966186	+	0.257844i	$0.916988\pi$
$734$		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	$-0.453586\pi$
$751$	218.238			0.290597			0.145299	+	0.989388i	$0.453586\pi$
$752$		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	$-0.741733\pi$
$757$		−	1098.00i		−	1.45046i	0.688505	−	0.725231i	$-0.258267\pi$
$758$		0			0
$759$	−748.246	+	1058.18i	−0.985831	+	1.39418i
$760$		0			0
$761$		−	1250.16i		−	1.64279i	−0.570358	−	0.821396i	$-0.693196\pi$
$761$		−	1250.16i		−	1.64279i	0.570358	−	0.821396i	$-0.306804\pi$
$762$		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	$0.274153\pi$
$809$			1227.54i			1.51735i	−0.651468	+	0.758676i	$0.725847\pi$
$810$		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	$-0.420807\pi$
$823$	405.300			0.492466			0.246233	+	0.969211i	$0.420807\pi$
$824$		0			0
$825$	612.000	+	432.749i	0.741818	+	0.524545i
$826$		0			0
$827$	1043.48			1.26177			0.630884	−	0.775877i	$-0.282692\pi$
$827$	1043.48			1.26177			0.630884	+	0.775877i	$0.282692\pi$
$828$		0			0
$829$		−	486.000i		−	0.586248i	−0.956074	−	0.293124i	$-0.905305\pi$
$829$		−	486.000i		−	0.586248i	0.956074	−	0.293124i	$-0.0946950\pi$
$830$		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	$-0.587465\pi$
$853$		−	1642.00i		−	1.92497i	0.271334	−	0.962485i	$-0.412535\pi$
$854$		0			0
$855$	249.415	−	88.1816i	0.291714	−	0.103136i
$856$		0			0
$857$			1057.83i			1.23434i	0.786829	+	0.617171i	$0.211721\pi$
$857$			1057.83i			1.23434i	−0.786829	+	0.617171i	$0.788279\pi$
$858$		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	$-0.898475\pi$
$877$		−	550.000i		−	0.627138i	0.949565	−	0.313569i	$-0.101525\pi$
$878$		0			0
$879$	−879.882	−	622.170i	−1.00100	−	0.707816i
$880$		0			0
$881$			1210.57i			1.37408i	0.726618	+	0.687041i	$0.241091\pi$
$881$			1210.57i			1.37408i	−0.726618	+	0.687041i	$0.758909\pi$
$882$		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	$-0.302123\pi$
$919$	1070.41			1.16475			0.582376	+	0.812919i	$0.302123\pi$
$920$		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	$0.116972\pi$
$929$			667.509i			0.718524i	−0.933237	+	0.359262i	$0.883028\pi$
$930$		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	$-0.716347\pi$
$947$	−1190.45			−1.25708			−0.628539	+	0.777778i	$0.716347\pi$
$948$		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	$0.0675816\pi$
$953$			401.637i			0.421445i	−0.977546	+	0.210722i	$0.932418\pi$
$954$		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	$-0.763366\pi$
$967$	−1423.75			−1.47233			−0.736166	+	0.676801i	$0.763366\pi$
$968$		0			0
$969$	576.000	+	407.294i	0.594427	+	0.420324i
$970$		0			0
$971$	808.332			0.832473			0.416237	−	0.909256i	$-0.363349\pi$
$971$	808.332			0.832473			0.416237	+	0.909256i	$0.363349\pi$
$972$		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	$-0.906511\pi$
$977$		−	565.685i		−	0.579002i	0.957178	−	0.289501i	$-0.0934894\pi$
$978$		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	$-0.410363\pi$
$991$	550.792			0.555794			0.277897	+	0.960611i	$0.410363\pi$
$992$		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	$0.298058\pi$
$997$			1606.00i			1.61083i	−0.592710	+	0.805416i	$0.701942\pi$
$998$		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.641.8		8
3.2	odd	2	inner	768.3.h.f.641.3		8
4.3	odd	2	inner	768.3.h.f.641.2		8
8.3	odd	2	inner	768.3.h.f.641.7		8
8.5	even	2	inner	768.3.h.f.641.1		8
12.11	even	2	inner	768.3.h.f.641.5		8
16.3	odd	4		192.3.e.e.65.3		4
16.5	even	4		96.3.e.a.65.3	yes	4
16.11	odd	4		96.3.e.a.65.2	yes	4
16.13	even	4		192.3.e.e.65.2		4
24.5	odd	2	inner	768.3.h.f.641.6		8
24.11	even	2	inner	768.3.h.f.641.4		8
48.5	odd	4		96.3.e.a.65.4	yes	4
48.11	even	4		96.3.e.a.65.1	✓	4
48.29	odd	4		192.3.e.e.65.1		4
48.35	even	4		192.3.e.e.65.4		4

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

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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