LMFDB - Embedded newform 768.3.h.e.641.3

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:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.3
Root			$-1.28897 - 0.581861i$ of defining polynomial
Character	$\chi$	$=$	768.641
Dual form			768.3.h.e.641.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 2.64575i) q^{3} -7.48331 q^{5} -5.29150 q^{7} +(-5.00000 - 7.48331i) q^{9} -14.1421 q^{11} -10.0000i q^{13} +(10.5830 - 19.7990i) q^{15} +26.4575i q^{19} +(7.48331 - 14.0000i) q^{21} -16.9706i q^{23} +31.0000 q^{25} +(26.8701 - 2.64575i) q^{27} -37.4166 q^{29} +26.4575 q^{31} +(20.0000 - 37.4166i) q^{33} +39.5980 q^{35} +10.0000i q^{37} +(26.4575 + 14.1421i) q^{39} +14.9666i q^{41} +58.2065i q^{43} +(37.4166 + 56.0000i) q^{45} +11.3137i q^{47} -21.0000 q^{49} -37.4166 q^{53} +105.830 q^{55} +(-70.0000 - 37.4166i) q^{57} +98.9949 q^{59} -90.0000i q^{61} +(26.4575 + 39.5980i) q^{63} +74.8331i q^{65} +5.29150i q^{67} +(44.8999 + 24.0000i) q^{69} -28.2843i q^{71} +30.0000 q^{73} +(-43.8406 + 82.0183i) q^{75} +74.8331 q^{77} +26.4575 q^{79} +(-31.0000 + 74.8331i) q^{81} -25.4558 q^{83} +(52.9150 - 98.9949i) q^{87} +74.8331i q^{89} +52.9150i q^{91} +(-37.4166 + 70.0000i) q^{93} -197.990i q^{95} +10.0000 q^{97} +(70.7107 + 105.830i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{9} + 248 q^{25} + 160 q^{33} - 168 q^{49} - 560 q^{57} + 240 q^{73} - 248 q^{81} + 80 q^{97}+O(q^{100}) $ 8 * q - 40 * q^9 + 248 * q^25 + 160 * q^33 - 168 * q^49 - 560 * q^57 + 240 * q^73 - 248 * q^81 + 80 * q^97

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
$3$	−1.41421	+	2.64575i	−0.471405	+	0.881917i
$3$	−1.41421	+	2.64575i	−0.471405	+	0.881917i
$4$		0			0
$5$	−7.48331			−1.49666			−0.748331	−	0.663325i	$-0.769145\pi$
$5$	−7.48331			−1.49666			−0.748331	+	0.663325i	$0.769145\pi$
$6$		0			0
$7$	−5.29150			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−5.29150			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$		0			0
$9$	−5.00000	−	7.48331i	−0.555556	−	0.831479i
$10$		0			0
$11$	−14.1421			−1.28565			−0.642824	−	0.766014i	$-0.722237\pi$
$11$	−14.1421			−1.28565			−0.642824	+	0.766014i	$0.722237\pi$
$12$		0			0
$13$		−	10.0000i		−	0.769231i	−0.923077	−	0.384615i	$-0.874334\pi$
$13$		−	10.0000i		−	0.769231i	0.923077	−	0.384615i	$-0.125666\pi$
$14$		0			0
$15$	10.5830	−	19.7990i	0.705534	−	1.31993i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$			26.4575i			1.39250i	0.717799	+	0.696250i	$0.245150\pi$
$19$			26.4575i			1.39250i	−0.717799	+	0.696250i	$0.754850\pi$
$20$		0			0
$21$	7.48331	−	14.0000i	0.356348	−	0.666667i
$22$		0			0
$23$		−	16.9706i		−	0.737851i	−0.929459	−	0.368925i	$-0.879726\pi$
$23$		−	16.9706i		−	0.737851i	0.929459	−	0.368925i	$-0.120274\pi$
$24$		0			0
$25$	31.0000			1.24000
$26$		0			0
$27$	26.8701	−	2.64575i	0.995187	−	0.0979908i
$28$		0			0
$29$	−37.4166			−1.29023			−0.645113	−	0.764087i	$-0.723190\pi$
$29$	−37.4166			−1.29023			−0.645113	+	0.764087i	$0.723190\pi$
$30$		0			0
$31$	26.4575			0.853468			0.426734	−	0.904377i	$-0.359664\pi$
$31$	26.4575			0.853468			0.426734	+	0.904377i	$0.359664\pi$
$32$		0			0
$33$	20.0000	−	37.4166i	0.606061	−	1.13384i
$34$		0			0
$35$	39.5980			1.13137
$36$		0			0
$37$			10.0000i			0.270270i	0.990827	+	0.135135i	$0.0431469\pi$
$37$			10.0000i			0.270270i	−0.990827	+	0.135135i	$0.956853\pi$
$38$		0			0
$39$	26.4575	+	14.1421i	0.678398	+	0.362619i
$40$		0			0
$41$			14.9666i			0.365040i	0.983202	+	0.182520i	$0.0584254\pi$
$41$			14.9666i			0.365040i	−0.983202	+	0.182520i	$0.941575\pi$
$42$		0			0
$43$			58.2065i			1.35364i	0.736148	+	0.676820i	$0.236643\pi$
$43$			58.2065i			1.35364i	−0.736148	+	0.676820i	$0.763357\pi$
$44$		0			0
$45$	37.4166	+	56.0000i	0.831479	+	1.24444i
$46$		0			0
$47$			11.3137i			0.240717i	0.992730	+	0.120359i	$0.0384044\pi$
$47$			11.3137i			0.240717i	−0.992730	+	0.120359i	$0.961596\pi$
$48$		0			0
$49$	−21.0000			−0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−37.4166			−0.705973			−0.352987	−	0.935628i	$-0.614834\pi$
$53$	−37.4166			−0.705973			−0.352987	+	0.935628i	$0.614834\pi$
$54$		0			0
$55$	105.830			1.92418
$56$		0			0
$57$	−70.0000	−	37.4166i	−1.22807	−	0.656431i
$58$		0			0
$59$	98.9949			1.67788			0.838940	−	0.544224i	$-0.183176\pi$
$59$	98.9949			1.67788			0.838940	+	0.544224i	$0.183176\pi$
$60$		0			0
$61$		−	90.0000i		−	1.47541i	−0.675123	−	0.737705i	$-0.735910\pi$
$61$		−	90.0000i		−	1.47541i	0.675123	−	0.737705i	$-0.264090\pi$
$62$		0			0
$63$	26.4575	+	39.5980i	0.419961	+	0.628539i
$64$		0			0
$65$			74.8331i			1.15128i
$66$		0			0
$67$			5.29150i			0.0789777i	0.999220	+	0.0394888i	$0.0125730\pi$
$67$			5.29150i			0.0789777i	−0.999220	+	0.0394888i	$0.987427\pi$
$68$		0			0
$69$	44.8999	+	24.0000i	0.650723	+	0.347826i
$70$		0			0
$71$		−	28.2843i		−	0.398370i	−0.979962	−	0.199185i	$-0.936171\pi$
$71$		−	28.2843i		−	0.398370i	0.979962	−	0.199185i	$-0.0638295\pi$
$72$		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
$75$	−43.8406	+	82.0183i	−0.584542	+	1.09358i
$76$		0			0
$77$	74.8331			0.971859
$78$		0			0
$79$	26.4575			0.334905			0.167453	−	0.985880i	$-0.446446\pi$
$79$	26.4575			0.334905			0.167453	+	0.985880i	$0.446446\pi$
$80$		0			0
$81$	−31.0000	+	74.8331i	−0.382716	+	0.923866i
$82$		0			0
$83$	−25.4558			−0.306697			−0.153348	−	0.988172i	$-0.549006\pi$
$83$	−25.4558			−0.306697			−0.153348	+	0.988172i	$0.549006\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	52.9150	−	98.9949i	0.608219	−	1.13787i
$88$		0			0
$89$			74.8331i			0.840822i	0.907334	+	0.420411i	$0.138114\pi$
$89$			74.8331i			0.840822i	−0.907334	+	0.420411i	$0.861886\pi$
$90$		0			0
$91$			52.9150i			0.581484i
$92$		0			0
$93$	−37.4166	+	70.0000i	−0.402329	+	0.752688i
$94$		0			0
$95$		−	197.990i		−	2.08410i
$96$		0			0
$97$	10.0000			0.103093			0.0515464	−	0.998671i	$-0.483585\pi$
$97$	10.0000			0.103093			0.0515464	+	0.998671i	$0.483585\pi$
$98$		0			0
$99$	70.7107	+	105.830i	0.714249	+	1.06899i
$100$		0			0
$101$	112.250			1.11138			0.555692	−	0.831388i	$-0.312454\pi$
$101$	112.250			1.11138			0.555692	+	0.831388i	$0.312454\pi$
$102$		0			0
$103$	−47.6235			−0.462364			−0.231182	−	0.972910i	$-0.574259\pi$
$103$	−47.6235			−0.462364			−0.231182	+	0.972910i	$0.574259\pi$
$104$		0			0
$105$	−56.0000	+	104.766i	−0.533333	+	0.997775i
$106$		0			0
$107$	−2.82843			−0.0264339			−0.0132169	−	0.999913i	$-0.504207\pi$
$107$	−2.82843			−0.0264339			−0.0132169	+	0.999913i	$0.504207\pi$
$108$		0			0
$109$			70.0000i			0.642202i	0.947045	+	0.321101i	$0.104053\pi$
$109$			70.0000i			0.642202i	−0.947045	+	0.321101i	$0.895947\pi$
$110$		0			0
$111$	−26.4575	−	14.1421i	−0.238356	−	0.127407i
$112$		0			0
$113$		−	149.666i		−	1.32448i	−0.749292	−	0.662240i	$-0.769606\pi$
$113$		−	149.666i		−	1.32448i	0.749292	−	0.662240i	$-0.230394\pi$
$114$		0			0
$115$			126.996i			1.10431i
$116$		0			0
$117$	−74.8331	+	50.0000i	−0.639600	+	0.427350i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	79.0000			0.652893
$122$		0			0
$123$	−39.5980	−	21.1660i	−0.321935	−	0.172081i
$124$		0			0
$125$	−44.8999			−0.359199
$126$		0			0
$127$	153.454			1.20830			0.604148	−	0.796872i	$-0.293514\pi$
$127$	153.454			1.20830			0.604148	+	0.796872i	$0.293514\pi$
$128$		0			0
$129$	−154.000	−	82.3165i	−1.19380	−	0.638112i
$130$		0			0
$131$	−70.7107			−0.539776			−0.269888	−	0.962892i	$-0.586987\pi$
$131$	−70.7107			−0.539776			−0.269888	+	0.962892i	$0.586987\pi$
$132$		0			0
$133$		−	140.000i		−	1.05263i
$134$		0			0
$135$	−201.077	+	19.7990i	−1.48946	+	0.146659i
$136$		0			0
$137$		−	224.499i		−	1.63868i	−0.573306	−	0.819341i	$-0.694340\pi$
$137$		−	224.499i		−	1.63868i	0.573306	−	0.819341i	$-0.305660\pi$
$138$		0			0
$139$		−	132.288i		−	0.951709i	−0.879524	−	0.475855i	$-0.842139\pi$
$139$		−	132.288i		−	0.951709i	0.879524	−	0.475855i	$-0.157861\pi$
$140$		0			0
$141$	−29.9333	−	16.0000i	−0.212293	−	0.113475i
$142$		0			0
$143$			141.421i			0.988961i
$144$		0			0
$145$	280.000			1.93103
$146$		0			0
$147$	29.6985	−	55.5608i	0.202031	−	0.377964i
$148$		0			0
$149$	22.4499			0.150671			0.0753354	−	0.997158i	$-0.475997\pi$
$149$	22.4499			0.150671			0.0753354	+	0.997158i	$0.475997\pi$
$150$		0			0
$151$	79.3725			0.525646			0.262823	−	0.964844i	$-0.415346\pi$
$151$	79.3725			0.525646			0.262823	+	0.964844i	$0.415346\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−197.990			−1.27735
$156$		0			0
$157$		−	170.000i		−	1.08280i	−0.840764	−	0.541401i	$-0.817894\pi$
$157$		−	170.000i		−	1.08280i	0.840764	−	0.541401i	$-0.182106\pi$
$158$		0			0
$159$	52.9150	−	98.9949i	0.332799	−	0.622610i
$160$		0			0
$161$			89.7998i			0.557763i
$162$		0			0
$163$		−	100.539i		−	0.616801i	−0.951257	−	0.308400i	$-0.900206\pi$
$163$		−	100.539i		−	0.616801i	0.951257	−	0.308400i	$-0.0997937\pi$
$164$		0			0
$165$	−149.666	+	280.000i	−0.907068	+	1.69697i
$166$		0			0
$167$			243.245i			1.45656i	0.685282	+	0.728278i	$0.259679\pi$
$167$			243.245i			1.45656i	−0.685282	+	0.728278i	$0.740321\pi$
$168$		0			0
$169$	69.0000			0.408284
$170$		0			0
$171$	197.990	−	132.288i	1.15784	−	0.773611i
$172$		0			0
$173$	112.250			0.648842			0.324421	−	0.945913i	$-0.394831\pi$
$173$	112.250			0.648842			0.324421	+	0.945913i	$0.394831\pi$
$174$		0			0
$175$	−164.037			−0.937352
$176$		0			0
$177$	−140.000	+	261.916i	−0.790960	+	1.47975i
$178$		0			0
$179$	212.132			1.18510			0.592548	−	0.805535i	$-0.298122\pi$
$179$	212.132			1.18510			0.592548	+	0.805535i	$0.298122\pi$
$180$		0			0
$181$		−	262.000i		−	1.44751i	−0.690055	−	0.723757i	$-0.742414\pi$
$181$		−	262.000i		−	1.44751i	0.690055	−	0.723757i	$-0.257586\pi$
$182$		0			0
$183$	238.118	+	127.279i	1.30119	+	0.695515i
$184$		0			0
$185$		−	74.8331i		−	0.404504i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−142.183	+	14.0000i	−0.752291	+	0.0740741i
$190$		0			0
$191$			339.411i			1.77702i	0.458855	+	0.888511i	$0.348260\pi$
$191$			339.411i			1.77702i	−0.458855	+	0.888511i	$0.651740\pi$
$192$		0			0
$193$	210.000			1.08808			0.544041	−	0.839058i	$-0.316893\pi$
$193$	210.000			1.08808			0.544041	+	0.839058i	$0.316893\pi$
$194$		0			0
$195$	−197.990	−	105.830i	−1.01533	−	0.542718i
$196$		0			0
$197$	−187.083			−0.949659			−0.474830	−	0.880078i	$-0.657490\pi$
$197$	−187.083			−0.949659			−0.474830	+	0.880078i	$0.657490\pi$
$198$		0			0
$199$	−343.948			−1.72838			−0.864190	−	0.503165i	$-0.832169\pi$
$199$	−343.948			−1.72838			−0.864190	+	0.503165i	$0.832169\pi$
$200$		0			0
$201$	−14.0000	−	7.48331i	−0.0696517	−	0.0372304i
$202$		0			0
$203$	197.990			0.975320
$204$		0			0
$205$		−	112.000i		−	0.546341i
$206$		0			0
$207$	−126.996	+	84.8528i	−0.613508	+	0.409917i
$208$		0			0
$209$		−	374.166i		−	1.79027i
$210$		0			0
$211$		−	291.033i		−	1.37930i	−0.724142	−	0.689651i	$-0.757764\pi$
$211$		−	291.033i		−	1.37930i	0.724142	−	0.689651i	$-0.242236\pi$
$212$		0			0
$213$	74.8331	+	40.0000i	0.351329	+	0.187793i
$214$		0			0
$215$		−	435.578i		−	2.02594i
$216$		0			0
$217$	−140.000			−0.645161
$218$		0			0
$219$	−42.4264	+	79.3725i	−0.193728	+	0.362432i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−312.199			−1.39999			−0.699997	−	0.714146i	$-0.746815\pi$
$223$	−312.199			−1.39999			−0.699997	+	0.714146i	$0.746815\pi$
$224$		0			0
$225$	−155.000	−	231.983i	−0.688889	−	1.03103i
$226$		0			0
$227$	−59.3970			−0.261661			−0.130830	−	0.991405i	$-0.541764\pi$
$227$	−59.3970			−0.261661			−0.130830	+	0.991405i	$0.541764\pi$
$228$		0			0
$229$		−	358.000i		−	1.56332i	−0.623706	−	0.781659i	$-0.714374\pi$
$229$		−	358.000i		−	1.56332i	0.623706	−	0.781659i	$-0.285626\pi$
$230$		0			0
$231$	−105.830	+	197.990i	−0.458139	+	0.857099i
$232$		0			0
$233$			224.499i			0.963517i	0.876304	+	0.481758i	$0.160002\pi$
$233$			224.499i			0.963517i	−0.876304	+	0.481758i	$0.839998\pi$
$234$		0			0
$235$		−	84.6640i		−	0.360273i
$236$		0			0
$237$	−37.4166	+	70.0000i	−0.157876	+	0.295359i
$238$		0			0
$239$		−	169.706i		−	0.710065i	−0.934854	−	0.355033i	$-0.884470\pi$
$239$		−	169.706i		−	0.710065i	0.934854	−	0.355033i	$-0.115530\pi$
$240$		0			0
$241$	−110.000			−0.456432			−0.228216	−	0.973611i	$-0.573289\pi$
$241$	−110.000			−0.456432			−0.228216	+	0.973611i	$0.573289\pi$
$242$		0			0
$243$	−154.149	−	187.848i	−0.634359	−	0.773038i
$244$		0			0
$245$	157.150			0.641427
$246$		0			0
$247$	264.575			1.07115
$248$		0			0
$249$	36.0000	−	67.3498i	0.144578	−	0.270481i
$250$		0			0
$251$	−296.985			−1.18321			−0.591603	−	0.806229i	$-0.701505\pi$
$251$	−296.985			−1.18321			−0.591603	+	0.806229i	$0.701505\pi$
$252$		0			0
$253$			240.000i			0.948617i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			299.333i			1.16472i	0.812932	+	0.582359i	$0.197870\pi$
$257$			299.333i			1.16472i	−0.812932	+	0.582359i	$0.802130\pi$
$258$		0			0
$259$		−	52.9150i		−	0.204305i
$260$		0			0
$261$	187.083	+	280.000i	0.716793	+	1.07280i
$262$		0			0
$263$			152.735i			0.580742i	0.956914	+	0.290371i	$0.0937787\pi$
$263$			152.735i			0.580742i	−0.956914	+	0.290371i	$0.906221\pi$
$264$		0			0
$265$	280.000			1.05660
$266$		0			0
$267$	−197.990	−	105.830i	−0.741535	−	0.396367i
$268$		0			0
$269$	−426.549			−1.58568			−0.792842	−	0.609427i	$-0.791399\pi$
$269$	−426.549			−1.58568			−0.792842	+	0.609427i	$0.791399\pi$
$270$		0			0
$271$	−396.863			−1.46444			−0.732219	−	0.681069i	$-0.761515\pi$
$271$	−396.863			−1.46444			−0.732219	+	0.681069i	$0.761515\pi$
$272$		0			0
$273$	−140.000	−	74.8331i	−0.512821	−	0.274114i
$274$		0			0
$275$	−438.406			−1.59420
$276$		0			0
$277$			490.000i			1.76895i	0.466585	+	0.884477i	$0.345484\pi$
$277$			490.000i			1.76895i	−0.466585	+	0.884477i	$0.654516\pi$
$278$		0			0
$279$	−132.288	−	197.990i	−0.474149	−	0.709641i
$280$		0			0
$281$			134.700i			0.479358i	0.970852	+	0.239679i	$0.0770422\pi$
$281$			134.700i			0.479358i	−0.970852	+	0.239679i	$0.922958\pi$
$282$		0			0
$283$		−	5.29150i		−	0.0186979i	−0.999956	−	0.00934894i	$-0.997024\pi$
$283$		−	5.29150i		−	0.0186979i	0.999956	−	0.00934894i	$-0.00297590\pi$
$284$		0			0
$285$	523.832	+	280.000i	1.83801	+	0.982456i
$286$		0			0
$287$		−	79.1960i		−	0.275944i
$288$		0			0
$289$	289.000			1.00000
$290$		0			0
$291$	−14.1421	+	26.4575i	−0.0485984	+	0.0909193i
$292$		0			0
$293$	112.250			0.383105			0.191552	−	0.981482i	$-0.438648\pi$
$293$	112.250			0.383105			0.191552	+	0.981482i	$0.438648\pi$
$294$		0			0
$295$	−740.810			−2.51122
$296$		0			0
$297$	−380.000	+	37.4166i	−1.27946	+	0.125982i
$298$		0			0
$299$	−169.706			−0.567577
$300$		0			0
$301$		−	308.000i		−	1.02326i
$302$		0			0
$303$	−158.745	+	296.985i	−0.523911	+	0.980148i
$304$		0			0
$305$			673.498i			2.20819i
$306$		0			0
$307$			47.6235i			0.155125i	0.996987	+	0.0775627i	$0.0247138\pi$
$307$			47.6235i			0.155125i	−0.996987	+	0.0775627i	$0.975286\pi$
$308$		0			0
$309$	67.3498	−	126.000i	0.217961	−	0.407767i
$310$		0			0
$311$			254.558i			0.818516i	0.912419	+	0.409258i	$0.134212\pi$
$311$			254.558i			0.818516i	−0.912419	+	0.409258i	$0.865788\pi$
$312$		0			0
$313$	230.000			0.734824			0.367412	−	0.930058i	$-0.380244\pi$
$313$	230.000			0.734824			0.367412	+	0.930058i	$0.380244\pi$
$314$		0			0
$315$	−197.990	−	296.324i	−0.628539	−	0.940712i
$316$		0			0
$317$	561.249			1.77050			0.885250	−	0.465115i	$-0.153987\pi$
$317$	561.249			1.77050			0.885250	+	0.465115i	$0.153987\pi$
$318$		0			0
$319$	529.150			1.65878
$320$		0			0
$321$	4.00000	−	7.48331i	0.0124611	−	0.0233125i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	310.000i		−	0.953846i
$326$		0			0
$327$	−185.203	−	98.9949i	−0.566369	−	0.302737i
$328$		0			0
$329$		−	59.8665i		−	0.181965i
$330$		0			0
$331$		−	132.288i		−	0.399660i	−0.979831	−	0.199830i	$-0.935961\pi$
$331$		−	132.288i		−	0.399660i	0.979831	−	0.199830i	$-0.0640390\pi$
$332$		0			0
$333$	74.8331	−	50.0000i	0.224724	−	0.150150i
$334$		0			0
$335$		−	39.5980i		−	0.118203i
$336$		0			0
$337$	−70.0000			−0.207715			−0.103858	−	0.994592i	$-0.533119\pi$
$337$	−70.0000			−0.207715			−0.103858	+	0.994592i	$0.533119\pi$
$338$		0			0
$339$	395.980	+	211.660i	1.16808	+	0.624366i
$340$		0			0
$341$	−374.166			−1.09726
$342$		0			0
$343$	370.405			1.07990
$344$		0			0
$345$	−336.000	−	179.600i	−0.973913	−	0.520578i
$346$		0			0
$347$	540.230			1.55686			0.778429	−	0.627733i	$-0.216017\pi$
$347$	540.230			1.55686			0.778429	+	0.627733i	$0.216017\pi$
$348$		0			0
$349$		−	298.000i		−	0.853868i	−0.904283	−	0.426934i	$-0.859594\pi$
$349$		−	298.000i		−	0.853868i	0.904283	−	0.426934i	$-0.140406\pi$
$350$		0			0
$351$	−26.4575	−	268.701i	−0.0753775	−	0.765529i
$352$		0			0
$353$		−	299.333i		−	0.847968i	−0.905670	−	0.423984i	$-0.860631\pi$
$353$		−	299.333i		−	0.847968i	0.905670	−	0.423984i	$-0.139369\pi$
$354$		0			0
$355$			211.660i			0.596226i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			537.401i			1.49694i	0.663169	+	0.748470i	$0.269211\pi$
$359$			537.401i			1.49694i	−0.663169	+	0.748470i	$0.730789\pi$
$360$		0			0
$361$	−339.000			−0.939058
$362$		0			0
$363$	−111.723	+	209.014i	−0.307777	+	0.575797i
$364$		0			0
$365$	−224.499			−0.615067
$366$		0			0
$367$	111.122			0.302784			0.151392	−	0.988474i	$-0.451625\pi$
$367$	111.122			0.302784			0.151392	+	0.988474i	$0.451625\pi$
$368$		0			0
$369$	112.000	−	74.8331i	0.303523	−	0.202800i
$370$		0			0
$371$	197.990			0.533665
$372$		0			0
$373$			410.000i			1.09920i	0.835429	+	0.549598i	$0.185219\pi$
$373$			410.000i			1.09920i	−0.835429	+	0.549598i	$0.814781\pi$
$374$		0			0
$375$	63.4980	−	118.794i	0.169328	−	0.316784i
$376$		0			0
$377$			374.166i			0.992482i
$378$		0			0
$379$		−	661.438i		−	1.74522i	−0.488419	−	0.872609i	$-0.662426\pi$
$379$		−	661.438i		−	1.74522i	0.488419	−	0.872609i	$-0.337574\pi$
$380$		0			0
$381$	−217.016	+	406.000i	−0.569596	+	1.06562i
$382$		0			0
$383$		−	158.392i		−	0.413556i	−0.978388	−	0.206778i	$-0.933702\pi$
$383$		−	158.392i		−	0.413556i	0.978388	−	0.206778i	$-0.0662978\pi$
$384$		0			0
$385$	−560.000			−1.45455
$386$		0			0
$387$	435.578	−	291.033i	1.12552	−	0.752022i
$388$		0			0
$389$	−246.949			−0.634831			−0.317416	−	0.948287i	$-0.602815\pi$
$389$	−246.949			−0.634831			−0.317416	+	0.948287i	$0.602815\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	100.000	−	187.083i	0.254453	−	0.476038i
$394$		0			0
$395$	−197.990			−0.501240
$396$		0			0
$397$			230.000i			0.579345i	0.957126	+	0.289673i	$0.0935464\pi$
$397$			230.000i			0.579345i	−0.957126	+	0.289673i	$0.906454\pi$
$398$		0			0
$399$	370.405	+	197.990i	0.928334	+	0.496215i
$400$		0			0
$401$		−	299.333i		−	0.746465i	−0.927738	−	0.373233i	$-0.878249\pi$
$401$		−	299.333i		−	0.746465i	0.927738	−	0.373233i	$-0.121751\pi$
$402$		0			0
$403$		−	264.575i		−	0.656514i
$404$		0			0
$405$	231.983	−	560.000i	0.572797	−	1.38272i
$406$		0			0
$407$		−	141.421i		−	0.347473i
$408$		0			0
$409$	590.000			1.44254			0.721271	−	0.692653i	$-0.243558\pi$
$409$	590.000			1.44254			0.721271	+	0.692653i	$0.243558\pi$
$410$		0			0
$411$	593.970	+	317.490i	1.44518	+	0.772482i
$412$		0			0
$413$	−523.832			−1.26836
$414$		0			0
$415$	190.494			0.459022
$416$		0			0
$417$	350.000	+	187.083i	0.839329	+	0.448640i
$418$		0			0
$419$	98.9949			0.236265			0.118132	−	0.992998i	$-0.462309\pi$
$419$	98.9949			0.236265			0.118132	+	0.992998i	$0.462309\pi$
$420$		0			0
$421$		−	70.0000i		−	0.166271i	−0.996538	−	0.0831354i	$-0.973507\pi$
$421$		−	70.0000i		−	0.166271i	0.996538	−	0.0831354i	$-0.0264934\pi$
$422$		0			0
$423$	84.6640	−	56.5685i	0.200151	−	0.133732i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			476.235i			1.11531i
$428$		0			0
$429$	−374.166	−	200.000i	−0.872181	−	0.466200i
$430$		0			0
$431$			509.117i			1.18125i	0.806948	+	0.590623i	$0.201118\pi$
$431$			509.117i			1.18125i	−0.806948	+	0.590623i	$0.798882\pi$
$432$		0			0
$433$	10.0000			0.0230947			0.0115473	−	0.999933i	$-0.496324\pi$
$433$	10.0000			0.0230947			0.0115473	+	0.999933i	$0.496324\pi$
$434$		0			0
$435$	−395.980	+	740.810i	−0.910298	+	1.70301i
$436$		0			0
$437$	448.999			1.02746
$438$		0			0
$439$	502.693			1.14509			0.572543	−	0.819875i	$-0.305957\pi$
$439$	502.693			1.14509			0.572543	+	0.819875i	$0.305957\pi$
$440$		0			0
$441$	105.000	+	157.150i	0.238095	+	0.356348i
$442$		0			0
$443$	313.955			0.708703			0.354351	−	0.935112i	$-0.384702\pi$
$443$	313.955			0.708703			0.354351	+	0.935112i	$0.384702\pi$
$444$		0			0
$445$		−	560.000i		−	1.25843i
$446$		0			0
$447$	−31.7490	+	59.3970i	−0.0710269	+	0.132879i
$448$		0			0
$449$		−	209.533i		−	0.466666i	−0.972397	−	0.233333i	$-0.925037\pi$
$449$		−	209.533i		−	0.466666i	0.972397	−	0.233333i	$-0.0749631\pi$
$450$		0			0
$451$		−	211.660i		−	0.469313i
$452$		0			0
$453$	−112.250	+	210.000i	−0.247792	+	0.463576i
$454$		0			0
$455$		−	395.980i		−	0.870285i
$456$		0			0
$457$	430.000			0.940919			0.470460	−	0.882422i	$-0.344088\pi$
$457$	430.000			0.940919			0.470460	+	0.882422i	$0.344088\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	112.250			0.243492			0.121746	−	0.992561i	$-0.461151\pi$
$461$	112.250			0.243492			0.121746	+	0.992561i	$0.461151\pi$
$462$		0			0
$463$	−100.539			−0.217146			−0.108573	−	0.994088i	$-0.534628\pi$
$463$	−100.539			−0.217146			−0.108573	+	0.994088i	$0.534628\pi$
$464$		0			0
$465$	280.000	−	523.832i	0.602151	−	1.12652i
$466$		0			0
$467$	−138.593			−0.296773			−0.148386	−	0.988929i	$-0.547408\pi$
$467$	−138.593			−0.296773			−0.148386	+	0.988929i	$0.547408\pi$
$468$		0			0
$469$		−	28.0000i		−	0.0597015i
$470$		0			0
$471$	449.778	+	240.416i	0.954942	+	0.510438i
$472$		0			0
$473$		−	823.165i		−	1.74031i
$474$		0			0
$475$			820.183i			1.72670i
$476$		0			0
$477$	187.083	+	280.000i	0.392207	+	0.587002i
$478$		0			0
$479$		−	339.411i		−	0.708583i	−0.935135	−	0.354291i	$-0.884722\pi$
$479$		−	339.411i		−	0.708583i	0.935135	−	0.354291i	$-0.115278\pi$
$480$		0			0
$481$	100.000			0.207900
$482$		0			0
$483$	−237.588	−	126.996i	−0.491900	−	0.262932i
$484$		0			0
$485$	−74.8331			−0.154295
$486$		0			0
$487$	375.697			0.771451			0.385726	−	0.922614i	$-0.373951\pi$
$487$	375.697			0.771451			0.385726	+	0.922614i	$0.373951\pi$
$488$		0			0
$489$	266.000	+	142.183i	0.543967	+	0.290763i
$490$		0			0
$491$	−523.259			−1.06570			−0.532850	−	0.846210i	$-0.678879\pi$
$491$	−523.259			−1.06570			−0.532850	+	0.846210i	$0.678879\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−529.150	−	791.960i	−1.06899	−	1.59992i
$496$		0			0
$497$			149.666i			0.301139i
$498$		0			0
$499$			555.608i			1.11344i	0.830699	+	0.556721i	$0.187941\pi$
$499$			555.608i			1.11344i	−0.830699	+	0.556721i	$0.812059\pi$
$500$		0			0
$501$	−643.565	−	344.000i	−1.28456	−	0.686627i
$502$		0			0
$503$		−	831.558i		−	1.65320i	−0.562793	−	0.826598i	$-0.690273\pi$
$503$		−	831.558i		−	1.65320i	0.562793	−	0.826598i	$-0.309727\pi$
$504$		0			0
$505$	−840.000			−1.66337
$506$		0			0
$507$	−97.5807	+	182.557i	−0.192467	+	0.360073i
$508$		0			0
$509$	561.249			1.10265			0.551325	−	0.834291i	$-0.314123\pi$
$509$	561.249			1.10265			0.551325	+	0.834291i	$0.314123\pi$
$510$		0			0
$511$	−158.745			−0.310656
$512$		0			0
$513$	70.0000	+	710.915i	0.136452	+	1.38580i
$514$		0			0
$515$	356.382			0.692004
$516$		0			0
$517$		−	160.000i		−	0.309478i
$518$		0			0
$519$	−158.745	+	296.985i	−0.305867	+	0.572225i
$520$		0			0
$521$		−	224.499i		−	0.430901i	−0.976515	−	0.215451i	$-0.930878\pi$
$521$		−	224.499i		−	0.430901i	0.976515	−	0.215451i	$-0.0691220\pi$
$522$		0			0
$523$			904.847i			1.73011i	0.501678	+	0.865054i	$0.332716\pi$
$523$			904.847i			1.73011i	−0.501678	+	0.865054i	$0.667284\pi$
$524$		0			0
$525$	231.983	−	434.000i	0.441872	−	0.826667i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	241.000			0.455577
$530$		0			0
$531$	−494.975	−	740.810i	−0.932156	−	1.39512i
$532$		0			0
$533$	149.666			0.280800
$534$		0			0
$535$	21.1660			0.0395626
$536$		0			0
$537$	−300.000	+	561.249i	−0.558659	+	1.04516i
$538$		0			0
$539$	296.985			0.550992
$540$		0			0
$541$			518.000i			0.957486i	0.877955	+	0.478743i	$0.158907\pi$
$541$			518.000i			0.957486i	−0.877955	+	0.478743i	$0.841093\pi$
$542$		0			0
$543$	693.187	+	370.524i	1.27659	+	0.682365i
$544$		0			0
$545$		−	523.832i		−	0.961160i
$546$		0			0
$547$			1000.09i			1.82833i	0.405347	+	0.914163i	$0.367151\pi$
$547$			1000.09i			1.82833i	−0.405347	+	0.914163i	$0.632849\pi$
$548$		0			0
$549$	−673.498	+	450.000i	−1.22677	+	0.819672i
$550$		0			0
$551$		−	989.949i		−	1.79664i
$552$		0			0
$553$	−140.000			−0.253165
$554$		0			0
$555$	197.990	+	105.830i	0.356739	+	0.190685i
$556$		0			0
$557$	411.582			0.738927			0.369463	−	0.929245i	$-0.379541\pi$
$557$	411.582			0.738927			0.369463	+	0.929245i	$0.379541\pi$
$558$		0			0
$559$	582.065			1.04126
$560$		0			0
$561$		0			0
$562$		0			0
$563$	653.367			1.16051			0.580255	−	0.814435i	$-0.302953\pi$
$563$	653.367			1.16051			0.580255	+	0.814435i	$0.302953\pi$
$564$		0			0
$565$			1120.00i			1.98230i
$566$		0			0
$567$	164.037	−	395.980i	0.289306	−	0.698377i
$568$		0			0
$569$		−	613.632i		−	1.07844i	−0.842165	−	0.539220i	$-0.818719\pi$
$569$		−	613.632i		−	1.07844i	0.842165	−	0.539220i	$-0.181281\pi$
$570$		0			0
$571$		−	343.948i		−	0.602360i	−0.953567	−	0.301180i	$-0.902619\pi$
$571$		−	343.948i		−	0.602360i	0.953567	−	0.301180i	$-0.0973805\pi$
$572$		0			0
$573$	−897.998	−	480.000i	−1.56719	−	0.837696i
$574$		0			0
$575$		−	526.087i		−	0.914935i
$576$		0			0
$577$	−390.000			−0.675910			−0.337955	−	0.941162i	$-0.609735\pi$
$577$	−390.000			−0.675910			−0.337955	+	0.941162i	$0.609735\pi$
$578$		0			0
$579$	−296.985	+	555.608i	−0.512927	+	0.959599i
$580$		0			0
$581$	134.700			0.231841
$582$		0			0
$583$	529.150			0.907633
$584$		0			0
$585$	560.000	−	374.166i	0.957265	−	0.639600i
$586$		0			0
$587$	653.367			1.11306			0.556530	−	0.830827i	$-0.312132\pi$
$587$	653.367			1.11306			0.556530	+	0.830827i	$0.312132\pi$
$588$		0			0
$589$			700.000i			1.18846i
$590$		0			0
$591$	264.575	−	494.975i	0.447674	−	0.837521i
$592$		0			0
$593$			448.999i			0.757165i	0.925568	+	0.378583i	$0.123588\pi$
$593$			448.999i			0.757165i	−0.925568	+	0.378583i	$0.876412\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	486.415	−	910.000i	0.814766	−	1.52429i
$598$		0			0
$599$			480.833i			0.802726i	0.915919	+	0.401363i	$0.131463\pi$
$599$			480.833i			0.802726i	−0.915919	+	0.401363i	$0.868537\pi$
$600$		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
$603$	39.5980	−	26.4575i	0.0656683	−	0.0438765i
$604$		0			0
$605$	−591.182			−0.977160
$606$		0			0
$607$	153.454			0.252807			0.126403	−	0.991979i	$-0.459657\pi$
$607$	153.454			0.252807			0.126403	+	0.991979i	$0.459657\pi$
$608$		0			0
$609$	−280.000	+	523.832i	−0.459770	+	0.860151i
$610$		0			0
$611$	113.137			0.185167
$612$		0			0
$613$		−	630.000i		−	1.02773i	−0.857870	−	0.513866i	$-0.828213\pi$
$613$		−	630.000i		−	1.02773i	0.857870	−	0.513866i	$-0.171787\pi$
$614$		0			0
$615$	296.324	+	158.392i	0.481828	+	0.257548i
$616$		0			0
$617$			1122.50i			1.81928i	0.415395	+	0.909641i	$0.363643\pi$
$617$			1122.50i			1.81928i	−0.415395	+	0.909641i	$0.636357\pi$
$618$		0			0
$619$		−	555.608i		−	0.897589i	−0.893635	−	0.448795i	$-0.851853\pi$
$619$		−	555.608i		−	0.897589i	0.893635	−	0.448795i	$-0.148147\pi$
$620$		0			0
$621$	−44.8999	−	456.000i	−0.0723026	−	0.734300i
$622$		0			0
$623$		−	395.980i		−	0.635602i
$624$		0			0
$625$	−439.000			−0.702400
$626$		0			0
$627$	989.949	+	529.150i	1.57887	+	0.843940i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−978.928			−1.55139			−0.775696	−	0.631107i	$-0.782601\pi$
$631$	−978.928			−1.55139			−0.775696	+	0.631107i	$0.782601\pi$
$632$		0			0
$633$	770.000	+	411.582i	1.21643	+	0.650209i
$634$		0			0
$635$	−1148.34			−1.80841
$636$		0			0
$637$			210.000i			0.329670i
$638$		0			0
$639$	−211.660	+	141.421i	−0.331236	+	0.221317i
$640$		0			0
$641$		−	389.132i		−	0.607071i	−0.952820	−	0.303535i	$-0.901833\pi$
$641$		−	389.132i		−	0.607071i	0.952820	−	0.303535i	$-0.0981671\pi$
$642$		0			0
$643$		−	269.867i		−	0.419699i	−0.977734	−	0.209850i	$-0.932703\pi$
$643$		−	269.867i		−	0.419699i	0.977734	−	0.209850i	$-0.0672974\pi$
$644$		0			0
$645$	1152.43	+	616.000i	1.78671	+	0.955039i
$646$		0			0
$647$			1170.97i			1.80984i	0.425578	+	0.904922i	$0.360071\pi$
$647$			1170.97i			1.80984i	−0.425578	+	0.904922i	$0.639929\pi$
$648$		0			0
$649$	−1400.00			−2.15716
$650$		0			0
$651$	197.990	−	370.405i	0.304132	−	0.568979i
$652$		0			0
$653$	710.915			1.08869			0.544345	−	0.838861i	$-0.316778\pi$
$653$	710.915			1.08869			0.544345	+	0.838861i	$0.316778\pi$
$654$		0			0
$655$	529.150			0.807863
$656$		0			0
$657$	−150.000	−	224.499i	−0.228311	−	0.341704i
$658$		0			0
$659$	−127.279			−0.193140			−0.0965700	−	0.995326i	$-0.530787\pi$
$659$	−127.279			−0.193140			−0.0965700	+	0.995326i	$0.530787\pi$
$660$		0			0
$661$			170.000i			0.257186i	0.991697	+	0.128593i	$0.0410461\pi$
$661$			170.000i			0.257186i	−0.991697	+	0.128593i	$0.958954\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			1047.66i			1.57543i
$666$		0			0
$667$			634.980i			0.951994i
$668$		0			0
$669$	441.516	−	826.000i	0.659963	−	1.23468i
$670$		0			0
$671$			1272.79i			1.89686i
$672$		0			0
$673$	−310.000			−0.460624			−0.230312	−	0.973117i	$-0.573975\pi$
$673$	−310.000			−0.460624			−0.230312	+	0.973117i	$0.573975\pi$
$674$		0			0
$675$	832.972	−	82.0183i	1.23403	−	0.121509i
$676$		0			0
$677$	1309.58			1.93439			0.967194	−	0.254041i	$-0.0817597\pi$
$677$	1309.58			1.93439			0.967194	+	0.254041i	$0.0817597\pi$
$678$		0			0
$679$	−52.9150			−0.0779308
$680$		0			0
$681$	84.0000	−	157.150i	0.123348	−	0.230763i
$682$		0			0
$683$	1162.48			1.70203			0.851013	−	0.525145i	$-0.175989\pi$
$683$	1162.48			1.70203			0.851013	+	0.525145i	$0.175989\pi$
$684$		0			0
$685$			1680.00i			2.45255i
$686$		0			0
$687$	947.179	+	506.288i	1.37872	+	0.736956i
$688$		0			0
$689$			374.166i			0.543056i
$690$		0			0
$691$			26.4575i			0.0382887i	0.999817	+	0.0191444i	$0.00609421\pi$
$691$			26.4575i			0.0382887i	−0.999817	+	0.0191444i	$0.993906\pi$
$692$		0			0
$693$	−374.166	−	560.000i	−0.539922	−	0.808081i
$694$		0			0
$695$			989.949i			1.42439i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−593.970	−	317.490i	−0.849742	−	0.454206i
$700$		0			0
$701$	22.4499			0.0320256			0.0160128	−	0.999872i	$-0.494903\pi$
$701$	22.4499			0.0320256			0.0160128	+	0.999872i	$0.494903\pi$
$702$		0			0
$703$	−264.575			−0.376352
$704$		0			0
$705$	224.000	+	119.733i	0.317730	+	0.169834i
$706$		0			0
$707$	−593.970			−0.840127
$708$		0			0
$709$		−	182.000i		−	0.256700i	−0.991729	−	0.128350i	$-0.959032\pi$
$709$		−	182.000i		−	0.256700i	0.991729	−	0.128350i	$-0.0409680\pi$
$710$		0			0
$711$	−132.288	−	197.990i	−0.186058	−	0.278467i
$712$		0			0
$713$		−	448.999i		−	0.629732i
$714$		0			0
$715$		−	1058.30i		−	1.48014i
$716$		0			0
$717$	448.999	+	240.000i	0.626219	+	0.334728i
$718$		0			0
$719$		−	395.980i		−	0.550737i	−0.961339	−	0.275368i	$-0.911200\pi$
$719$		−	395.980i		−	0.550737i	0.961339	−	0.275368i	$-0.0887998\pi$
$720$		0			0
$721$	252.000			0.349515
$722$		0			0
$723$	155.563	−	291.033i	0.215164	−	0.402535i
$724$		0			0
$725$	−1159.91			−1.59988
$726$		0			0
$727$	−47.6235			−0.0655069			−0.0327535	−	0.999463i	$-0.510428\pi$
$727$	−47.6235			−0.0655069			−0.0327535	+	0.999463i	$0.510428\pi$
$728$		0			0
$729$	715.000	−	142.183i	0.980796	−	0.195038i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	650.000i		−	0.886767i	−0.896332	−	0.443383i	$-0.853778\pi$
$733$		−	650.000i		−	0.886767i	0.896332	−	0.443383i	$-0.146222\pi$
$734$		0			0
$735$	−222.243	+	415.779i	−0.302372	+	0.565685i
$736$		0			0
$737$		−	74.8331i		−	0.101538i
$738$		0			0
$739$			343.948i			0.465423i	0.972546	+	0.232712i	$0.0747598\pi$
$739$			343.948i			0.465423i	−0.972546	+	0.232712i	$0.925240\pi$
$740$		0			0
$741$	−374.166	+	700.000i	−0.504947	+	0.944669i
$742$		0			0
$743$		−	661.852i		−	0.890783i	−0.895336	−	0.445392i	$-0.853064\pi$
$743$		−	661.852i		−	0.890783i	0.895336	−	0.445392i	$-0.146936\pi$
$744$		0			0
$745$	−168.000			−0.225503
$746$		0			0
$747$	127.279	+	190.494i	0.170387	+	0.255012i
$748$		0			0
$749$	14.9666			0.0199821
$750$		0			0
$751$	−608.523			−0.810283			−0.405142	−	0.914254i	$-0.632778\pi$
$751$	−608.523			−0.810283			−0.405142	+	0.914254i	$0.632778\pi$
$752$		0			0
$753$	420.000	−	785.748i	0.557769	−	1.04349i
$754$		0			0
$755$	−593.970			−0.786715
$756$		0			0
$757$			1050.00i			1.38705i	0.720431	+	0.693527i	$0.243944\pi$
$757$			1050.00i			1.38705i	−0.720431	+	0.693527i	$0.756056\pi$
$758$		0			0
$759$	−634.980	−	339.411i	−0.836601	−	0.447182i
$760$		0			0
$761$		−	673.498i		−	0.885018i	−0.896764	−	0.442509i	$-0.854089\pi$
$761$		−	673.498i		−	0.885018i	0.896764	−	0.442509i	$-0.145911\pi$
$762$		0			0
$763$		−	370.405i		−	0.485459i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	989.949i		−	1.29068i
$768$		0			0
$769$	1138.00			1.47984			0.739922	−	0.672693i	$-0.234862\pi$
$769$	1138.00			1.47984			0.739922	+	0.672693i	$0.234862\pi$
$770$		0			0
$771$	−791.960	−	423.320i	−1.02718	−	0.549053i
$772$		0			0
$773$	−785.748			−1.01649			−0.508246	−	0.861212i	$-0.669706\pi$
$773$	−785.748			−1.01649			−0.508246	+	0.861212i	$0.669706\pi$
$774$		0			0
$775$	820.183			1.05830
$776$		0			0
$777$	140.000	+	74.8331i	0.180180	+	0.0963104i
$778$		0			0
$779$	−395.980			−0.508318
$780$		0			0
$781$			400.000i			0.512164i
$782$		0			0
$783$	−1005.39	+	98.9949i	−1.28402	+	0.126430i
$784$		0			0
$785$			1272.16i			1.62059i
$786$		0			0
$787$		−	904.847i		−	1.14974i	−0.818244	−	0.574871i	$-0.805052\pi$
$787$		−	904.847i		−	1.14974i	0.818244	−	0.574871i	$-0.194948\pi$
$788$		0			0
$789$	−404.099	−	216.000i	−0.512166	−	0.273764i
$790$		0			0
$791$			791.960i			1.00121i
$792$		0			0
$793$	−900.000			−1.13493
$794$		0			0
$795$	−395.980	+	740.810i	−0.498088	+	0.931837i
$796$		0			0
$797$	−1234.75			−1.54924			−0.774622	−	0.632425i	$-0.782060\pi$
$797$	−1234.75			−1.54924			−0.774622	+	0.632425i	$0.782060\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	560.000	−	374.166i	0.699126	−	0.467123i
$802$		0			0
$803$	−424.264			−0.528349
$804$		0			0
$805$		−	672.000i		−	0.834783i
$806$		0			0
$807$	603.231	−	1128.54i	0.747499	−	1.39844i
$808$		0			0
$809$		−	823.165i		−	1.01751i	−0.860912	−	0.508754i	$-0.830106\pi$
$809$		−	823.165i		−	1.01751i	0.860912	−	0.508754i	$-0.169894\pi$
$810$		0			0
$811$		−	26.4575i		−	0.0326233i	−0.999867	−	0.0163117i	$-0.994808\pi$
$811$		−	26.4575i		−	0.0326233i	0.999867	−	0.0163117i	$-0.00519239\pi$
$812$		0			0
$813$	561.249	−	1050.00i	0.690343	−	1.29151i
$814$		0			0
$815$			752.362i			0.923143i
$816$		0			0
$817$	−1540.00			−1.88494
$818$		0			0
$819$	395.980	−	264.575i	0.483492	−	0.323047i
$820$		0			0
$821$	−97.2831			−0.118493			−0.0592467	−	0.998243i	$-0.518870\pi$
$821$	−97.2831			−0.118493			−0.0592467	+	0.998243i	$0.518870\pi$
$822$		0			0
$823$	−894.264			−1.08659			−0.543295	−	0.839542i	$-0.682824\pi$
$823$	−894.264			−1.08659			−0.543295	+	0.839542i	$0.682824\pi$
$824$		0			0
$825$	620.000	−	1159.91i	0.751515	−	1.40596i
$826$		0			0
$827$	−285.671			−0.345431			−0.172715	−	0.984972i	$-0.555254\pi$
$827$	−285.671			−0.345431			−0.172715	+	0.984972i	$0.555254\pi$
$828$		0			0
$829$		−	170.000i		−	0.205066i	−0.994730	−	0.102533i	$-0.967305\pi$
$829$		−	170.000i		−	0.205066i	0.994730	−	0.102533i	$-0.0326948\pi$
$830$		0			0
$831$	−1296.42	−	692.965i	−1.56007	−	0.833892i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		−	1820.28i		−	2.17997i
$836$		0			0
$837$	710.915	−	70.0000i	0.849361	−	0.0836320i
$838$		0			0
$839$		−	707.107i		−	0.842797i	−0.906876	−	0.421399i	$-0.861539\pi$
$839$		−	707.107i		−	0.842797i	0.906876	−	0.421399i	$-0.138461\pi$
$840$		0			0
$841$	559.000			0.664685
$842$		0			0
$843$	−356.382	−	190.494i	−0.422754	−	0.225972i
$844$		0			0
$845$	−516.349			−0.611064
$846$		0			0
$847$	−418.029			−0.493540
$848$		0			0
$849$	14.0000	+	7.48331i	0.0164900	+	0.00881427i
$850$		0			0
$851$	169.706			0.199419
$852$		0			0
$853$			570.000i			0.668230i	0.942532	+	0.334115i	$0.108437\pi$
$853$			570.000i			0.668230i	−0.942532	+	0.334115i	$0.891563\pi$
$854$		0			0
$855$	−1481.62	+	989.949i	−1.73289	+	1.15784i
$856$		0			0
$857$			523.832i			0.611239i	0.952154	+	0.305620i	$0.0988636\pi$
$857$			523.832i			0.611239i	−0.952154	+	0.305620i	$0.901136\pi$
$858$		0			0
$859$			1455.16i			1.69402i	0.531577	+	0.847010i	$0.321600\pi$
$859$			1455.16i			1.69402i	−0.531577	+	0.847010i	$0.678400\pi$
$860$		0			0
$861$	209.533	+	112.000i	0.243360	+	0.130081i
$862$		0			0
$863$			181.019i			0.209756i	0.994485	+	0.104878i	$0.0334452\pi$
$863$			181.019i			0.209756i	−0.994485	+	0.104878i	$0.966555\pi$
$864$		0			0
$865$	−840.000			−0.971098
$866$		0			0
$867$	−408.708	+	764.622i	−0.471405	+	0.881917i
$868$		0			0
$869$	−374.166			−0.430570
$870$		0			0
$871$	52.9150			0.0607520
$872$		0			0
$873$	−50.0000	−	74.8331i	−0.0572738	−	0.0857195i
$874$		0			0
$875$	237.588			0.271529
$876$		0			0
$877$		−	490.000i		−	0.558723i	−0.960186	−	0.279361i	$-0.909877\pi$
$877$		−	490.000i		−	0.558723i	0.960186	−	0.279361i	$-0.0901228\pi$
$878$		0			0
$879$	−158.745	+	296.985i	−0.180597	+	0.337867i
$880$		0			0
$881$		−	987.798i		−	1.12122i	−0.828079	−	0.560612i	$-0.810566\pi$
$881$		−	987.798i		−	1.12122i	0.828079	−	0.560612i	$-0.189434\pi$
$882$		0			0
$883$			957.762i			1.08467i	0.840163	+	0.542334i	$0.182459\pi$
$883$			957.762i			1.08467i	−0.840163	+	0.542334i	$0.817541\pi$
$884$		0			0
$885$	1047.66	−	1960.00i	1.18380	−	2.21469i
$886$		0			0
$887$		−	1374.62i		−	1.54974i	−0.632123	−	0.774868i	$-0.717816\pi$
$887$		−	1374.62i		−	1.54974i	0.632123	−	0.774868i	$-0.282184\pi$
$888$		0			0
$889$	−812.000			−0.913386
$890$		0			0
$891$	438.406	−	1058.30i	0.492038	−	1.18777i
$892$		0			0
$893$	−299.333			−0.335199
$894$		0			0
$895$	−1587.45			−1.77369
$896$		0			0
$897$	240.000	−	448.999i	0.267559	−	0.500556i
$898$		0			0
$899$	−989.949			−1.10117
$900$		0			0
$901$		0			0
$902$		0			0
$903$	814.891	+	435.578i	0.902427	+	0.482367i
$904$		0			0
$905$			1960.63i			2.16644i
$906$		0			0
$907$		−	534.442i		−	0.589241i	−0.955614	−	0.294621i	$-0.904807\pi$
$907$		−	534.442i		−	0.589241i	0.955614	−	0.294621i	$-0.0951933\pi$
$908$		0			0
$909$	−561.249	−	840.000i	−0.617435	−	0.924092i
$910$		0			0
$911$			169.706i			0.186285i	0.995653	+	0.0931425i	$0.0296912\pi$
$911$			169.706i			0.186285i	−0.995653	+	0.0931425i	$0.970309\pi$
$912$		0			0
$913$	360.000			0.394304
$914$		0			0
$915$	−1781.91	−	952.470i	−1.94744	−	1.04095i
$916$		0			0
$917$	374.166			0.408032
$918$		0			0
$919$	−555.608			−0.604579			−0.302289	−	0.953216i	$-0.597751\pi$
$919$	−555.608			−0.604579			−0.302289	+	0.953216i	$0.597751\pi$
$920$		0			0
$921$	−126.000	−	67.3498i	−0.136808	−	0.0731269i
$922$		0			0
$923$	−282.843			−0.306438
$924$		0			0
$925$			310.000i			0.335135i
$926$		0			0
$927$	238.118	+	356.382i	0.256869	+	0.384446i
$928$		0			0
$929$		−	89.7998i		−	0.0966628i	−0.998831	−	0.0483314i	$-0.984610\pi$
$929$		−	89.7998i		−	0.0966628i	0.998831	−	0.0483314i	$-0.0153904\pi$
$930$		0			0
$931$		−	555.608i		−	0.596786i
$932$		0			0
$933$	−673.498	−	360.000i	−0.721863	−	0.385852i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1270.00			1.35539			0.677695	−	0.735343i	$-0.262979\pi$
$937$	1270.00			1.35539			0.677695	+	0.735343i	$0.262979\pi$
$938$		0			0
$939$	−325.269	+	608.523i	−0.346399	+	0.648054i
$940$		0			0
$941$	−187.083			−0.198813			−0.0994064	−	0.995047i	$-0.531694\pi$
$941$	−187.083			−0.198813			−0.0994064	+	0.995047i	$0.531694\pi$
$942$		0			0
$943$	253.992			0.269345
$944$		0			0
$945$	1064.00	−	104.766i	1.12593	−	0.110864i
$946$		0			0
$947$	280.014			0.295686			0.147843	−	0.989011i	$-0.452767\pi$
$947$	280.014			0.295686			0.147843	+	0.989011i	$0.452767\pi$
$948$		0			0
$949$		−	300.000i		−	0.316122i
$950$		0			0
$951$	−793.725	+	1484.92i	−0.834622	+	1.56143i
$952$		0			0
$953$			224.499i			0.235571i	0.993039	+	0.117786i	$0.0375796\pi$
$953$			224.499i			0.235571i	−0.993039	+	0.117786i	$0.962420\pi$
$954$		0			0
$955$		−	2539.92i		−	2.65960i
$956$		0			0
$957$	−748.331	+	1400.00i	−0.781956	+	1.46290i
$958$		0			0
$959$			1187.94i			1.23873i
$960$		0			0
$961$	−261.000			−0.271592
$962$		0			0
$963$	14.1421	+	21.1660i	0.0146855	+	0.0219792i
$964$		0			0
$965$	−1571.50			−1.62849
$966$		0			0
$967$	206.369			0.213411			0.106706	−	0.994291i	$-0.465970\pi$
$967$	206.369			0.213411			0.106706	+	0.994291i	$0.465970\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1711.20			−1.76231			−0.881153	−	0.472832i	$-0.843232\pi$
$971$	−1711.20			−1.76231			−0.881153	+	0.472832i	$0.843232\pi$
$972$		0			0
$973$			700.000i			0.719424i
$974$		0			0
$975$	820.183	+	438.406i	0.841213	+	0.449647i
$976$		0			0
$977$		−	598.665i		−	0.612759i	−0.951910	−	0.306379i	$-0.900882\pi$
$977$		−	598.665i		−	0.612759i	0.951910	−	0.306379i	$-0.0991176\pi$
$978$		0			0
$979$		−	1058.30i		−	1.08100i
$980$		0			0
$981$	523.832	−	350.000i	0.533978	−	0.356779i
$982$		0			0
$983$			1544.32i			1.57103i	0.618843	+	0.785514i	$0.287602\pi$
$983$			1544.32i			1.57103i	−0.618843	+	0.785514i	$0.712398\pi$
$984$		0			0
$985$	1400.00			1.42132
$986$		0			0
$987$	158.392	+	84.6640i	0.160478	+	0.0857792i
$988$		0			0
$989$	987.798			0.998784
$990$		0			0
$991$	−608.523			−0.614049			−0.307025	−	0.951702i	$-0.599333\pi$
$991$	−608.523			−0.614049			−0.307025	+	0.951702i	$0.599333\pi$
$992$		0			0
$993$	350.000	+	187.083i	0.352467	+	0.188402i
$994$		0			0
$995$	2573.87			2.58680
$996$		0			0
$997$			810.000i			0.812437i	0.913776	+	0.406219i	$0.133153\pi$
$997$			810.000i			0.812437i	−0.913776	+	0.406219i	$0.866847\pi$
$998$		0			0
$999$	26.4575	+	268.701i	0.0264840	+	0.268970i

Twists

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

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

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

\(f(q)\)	\(=\)	\(q+(-1.41421 + 2.64575i) q^{3} -7.48331 q^{5} -5.29150 q^{7} +(-5.00000 - 7.48331i) q^{9} -14.1421 q^{11} -10.0000i q^{13} +(10.5830 - 19.7990i) q^{15} +26.4575i q^{19} +(7.48331 - 14.0000i) q^{21} -16.9706i q^{23} +31.0000 q^{25} +(26.8701 - 2.64575i) q^{27} -37.4166 q^{29} +26.4575 q^{31} +(20.0000 - 37.4166i) q^{33} +39.5980 q^{35} +10.0000i q^{37} +(26.4575 + 14.1421i) q^{39} +14.9666i q^{41} +58.2065i q^{43} +(37.4166 + 56.0000i) q^{45} +11.3137i q^{47} -21.0000 q^{49} -37.4166 q^{53} +105.830 q^{55} +(-70.0000 - 37.4166i) q^{57} +98.9949 q^{59} -90.0000i q^{61} +(26.4575 + 39.5980i) q^{63} +74.8331i q^{65} +5.29150i q^{67} +(44.8999 + 24.0000i) q^{69} -28.2843i q^{71} +30.0000 q^{73} +(-43.8406 + 82.0183i) q^{75} +74.8331 q^{77} +26.4575 q^{79} +(-31.0000 + 74.8331i) q^{81} -25.4558 q^{83} +(52.9150 - 98.9949i) q^{87} +74.8331i q^{89} +52.9150i q^{91} +(-37.4166 + 70.0000i) q^{93} -197.990i q^{95} +10.0000 q^{97} +(70.7107 + 105.830i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{9} + 248 q^{25} + 160 q^{33} - 168 q^{49} - 560 q^{57} + 240 q^{73} - 248 q^{81} + 80 q^{97}+O(q^{100}) \) 8 * q - 40 * q^9 + 248 * q^25 + 160 * q^33 - 168 * q^49 - 560 * q^57 + 240 * q^73 - 248 * q^81 + 80 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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