LMFDB - Embedded newform 768.3.h.e.641.1

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

$q$-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	$0.125666\pi$
$13$			10.0000i			0.769231i	−0.923077	+	0.384615i	$0.874334\pi$
$14$		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	$-0.754850\pi$
$19$		−	26.4575i		−	1.39250i	0.717799	−	0.696250i	$-0.245150\pi$
$20$		0			0
$21$	7.48331	+	14.0000i	0.356348	+	0.666667i
$22$		0			0
$23$			16.9706i			0.737851i	0.929459	+	0.368925i	$0.120274\pi$
$23$			16.9706i			0.737851i	−0.929459	+	0.368925i	$0.879726\pi$
$24$		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	$-0.956853\pi$
$37$		−	10.0000i		−	0.270270i	0.990827	−	0.135135i	$-0.0431469\pi$
$38$		0			0
$39$	26.4575	−	14.1421i	0.678398	−	0.362619i
$40$		0			0
$41$		−	14.9666i		−	0.365040i	−0.983202	−	0.182520i	$-0.941575\pi$
$41$		−	14.9666i		−	0.365040i	0.983202	−	0.182520i	$-0.0584254\pi$
$42$		0			0
$43$		−	58.2065i		−	1.35364i	−0.736148	−	0.676820i	$-0.763357\pi$
$43$		−	58.2065i		−	1.35364i	0.736148	−	0.676820i	$-0.236643\pi$
$44$		0			0
$45$	37.4166	−	56.0000i	0.831479	−	1.24444i
$46$		0			0
$47$		−	11.3137i		−	0.240717i	−0.992730	−	0.120359i	$-0.961596\pi$
$47$		−	11.3137i		−	0.240717i	0.992730	−	0.120359i	$-0.0384044\pi$
$48$		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	$0.264090\pi$
$61$			90.0000i			1.47541i	−0.675123	+	0.737705i	$0.735910\pi$
$62$		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	$-0.987427\pi$
$67$		−	5.29150i		−	0.0789777i	0.999220	−	0.0394888i	$-0.0125730\pi$
$68$		0			0
$69$	44.8999	−	24.0000i	0.650723	−	0.347826i
$70$		0			0
$71$			28.2843i			0.398370i	0.979962	+	0.199185i	$0.0638295\pi$
$71$			28.2843i			0.398370i	−0.979962	+	0.199185i	$0.936171\pi$
$72$		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	$-0.861886\pi$
$89$		−	74.8331i		−	0.840822i	0.907334	−	0.420411i	$-0.138114\pi$
$90$		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	$-0.895947\pi$
$109$		−	70.0000i		−	0.642202i	0.947045	−	0.321101i	$-0.104053\pi$
$110$		0			0
$111$	−26.4575	+	14.1421i	−0.238356	+	0.127407i
$112$		0			0
$113$			149.666i			1.32448i	0.749292	+	0.662240i	$0.230394\pi$
$113$			149.666i			1.32448i	−0.749292	+	0.662240i	$0.769606\pi$
$114$		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	$0.305660\pi$
$137$			224.499i			1.63868i	−0.573306	+	0.819341i	$0.694340\pi$
$138$		0			0
$139$			132.288i			0.951709i	0.879524	+	0.475855i	$0.157861\pi$
$139$			132.288i			0.951709i	−0.879524	+	0.475855i	$0.842139\pi$
$140$		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	$0.182106\pi$
$157$			170.000i			1.08280i	−0.840764	+	0.541401i	$0.817894\pi$
$158$		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	$0.0997937\pi$
$163$			100.539i			0.616801i	−0.951257	+	0.308400i	$0.900206\pi$
$164$		0			0
$165$	−149.666	−	280.000i	−0.907068	−	1.69697i
$166$		0			0
$167$		−	243.245i		−	1.45656i	−0.685282	−	0.728278i	$-0.740321\pi$
$167$		−	243.245i		−	1.45656i	0.685282	−	0.728278i	$-0.259679\pi$
$168$		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	$0.257586\pi$
$181$			262.000i			1.44751i	−0.690055	+	0.723757i	$0.742414\pi$
$182$		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	$-0.651740\pi$
$191$		−	339.411i		−	1.77702i	0.458855	−	0.888511i	$-0.348260\pi$
$192$		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	$0.242236\pi$
$211$			291.033i			1.37930i	−0.724142	+	0.689651i	$0.757764\pi$
$212$		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	$0.285626\pi$
$229$			358.000i			1.56332i	−0.623706	+	0.781659i	$0.714374\pi$
$230$		0			0
$231$	−105.830	−	197.990i	−0.458139	−	0.857099i
$232$		0			0
$233$		−	224.499i		−	0.963517i	−0.876304	−	0.481758i	$-0.839998\pi$
$233$		−	224.499i		−	0.963517i	0.876304	−	0.481758i	$-0.160002\pi$
$234$		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	$0.115530\pi$
$239$			169.706i			0.710065i	−0.934854	+	0.355033i	$0.884470\pi$
$240$		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	$-0.802130\pi$
$257$		−	299.333i		−	1.16472i	0.812932	−	0.582359i	$-0.197870\pi$
$258$		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	$-0.906221\pi$
$263$		−	152.735i		−	0.580742i	0.956914	−	0.290371i	$-0.0937787\pi$
$264$		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	$-0.654516\pi$
$277$		−	490.000i		−	1.76895i	0.466585	−	0.884477i	$-0.345484\pi$
$278$		0			0
$279$	−132.288	+	197.990i	−0.474149	+	0.709641i
$280$		0			0
$281$		−	134.700i		−	0.479358i	−0.970852	−	0.239679i	$-0.922958\pi$
$281$		−	134.700i		−	0.479358i	0.970852	−	0.239679i	$-0.0770422\pi$
$282$		0			0
$283$			5.29150i			0.0186979i	0.999956	+	0.00934894i	$0.00297590\pi$
$283$			5.29150i			0.0186979i	−0.999956	+	0.00934894i	$0.997024\pi$
$284$		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	$-0.975286\pi$
$307$		−	47.6235i		−	0.155125i	0.996987	−	0.0775627i	$-0.0247138\pi$
$308$		0			0
$309$	67.3498	+	126.000i	0.217961	+	0.407767i
$310$		0			0
$311$		−	254.558i		−	0.818516i	−0.912419	−	0.409258i	$-0.865788\pi$
$311$		−	254.558i		−	0.818516i	0.912419	−	0.409258i	$-0.134212\pi$
$312$		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	$0.0640390\pi$
$331$			132.288i			0.399660i	−0.979831	+	0.199830i	$0.935961\pi$
$332$		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	$0.140406\pi$
$349$			298.000i			0.853868i	−0.904283	+	0.426934i	$0.859594\pi$
$350$		0			0
$351$	−26.4575	+	268.701i	−0.0753775	+	0.765529i
$352$		0			0
$353$			299.333i			0.847968i	0.905670	+	0.423984i	$0.139369\pi$
$353$			299.333i			0.847968i	−0.905670	+	0.423984i	$0.860631\pi$
$354$		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	$-0.730789\pi$
$359$		−	537.401i		−	1.49694i	0.663169	−	0.748470i	$-0.269211\pi$
$360$		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	$-0.814781\pi$
$373$		−	410.000i		−	1.09920i	0.835429	−	0.549598i	$-0.185219\pi$
$374$		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	$0.337574\pi$
$379$			661.438i			1.74522i	−0.488419	+	0.872609i	$0.662426\pi$
$380$		0			0
$381$	−217.016	−	406.000i	−0.569596	−	1.06562i
$382$		0			0
$383$			158.392i			0.413556i	0.978388	+	0.206778i	$0.0662978\pi$
$383$			158.392i			0.413556i	−0.978388	+	0.206778i	$0.933702\pi$
$384$		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	$-0.906454\pi$
$397$		−	230.000i		−	0.579345i	0.957126	−	0.289673i	$-0.0935464\pi$
$398$		0			0
$399$	370.405	−	197.990i	0.928334	−	0.496215i
$400$		0			0
$401$			299.333i			0.746465i	0.927738	+	0.373233i	$0.121751\pi$
$401$			299.333i			0.746465i	−0.927738	+	0.373233i	$0.878249\pi$
$402$		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	$0.0264934\pi$
$421$			70.0000i			0.166271i	−0.996538	+	0.0831354i	$0.973507\pi$
$422$		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	$-0.798882\pi$
$431$		−	509.117i		−	1.18125i	0.806948	−	0.590623i	$-0.201118\pi$
$432$		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	$0.0749631\pi$
$449$			209.533i			0.466666i	−0.972397	+	0.233333i	$0.925037\pi$
$450$		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	$0.115278\pi$
$479$			339.411i			0.708583i	−0.935135	+	0.354291i	$0.884722\pi$
$480$		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	$-0.812059\pi$
$499$		−	555.608i		−	1.11344i	0.830699	−	0.556721i	$-0.187941\pi$
$500$		0			0
$501$	−643.565	+	344.000i	−1.28456	+	0.686627i
$502$		0			0
$503$			831.558i			1.65320i	0.562793	+	0.826598i	$0.309727\pi$
$503$			831.558i			1.65320i	−0.562793	+	0.826598i	$0.690273\pi$
$504$		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	$0.0691220\pi$
$521$			224.499i			0.430901i	−0.976515	+	0.215451i	$0.930878\pi$
$522$		0			0
$523$		−	904.847i		−	1.73011i	−0.501678	−	0.865054i	$-0.667284\pi$
$523$		−	904.847i		−	1.73011i	0.501678	−	0.865054i	$-0.332716\pi$
$524$		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	$-0.841093\pi$
$541$		−	518.000i		−	0.957486i	0.877955	−	0.478743i	$-0.158907\pi$
$542$		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	$-0.632849\pi$
$547$		−	1000.09i		−	1.82833i	0.405347	−	0.914163i	$-0.367151\pi$
$548$		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	$0.181281\pi$
$569$			613.632i			1.07844i	−0.842165	+	0.539220i	$0.818719\pi$
$570$		0			0
$571$			343.948i			0.602360i	0.953567	+	0.301180i	$0.0973805\pi$
$571$			343.948i			0.602360i	−0.953567	+	0.301180i	$0.902619\pi$
$572$		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	$-0.876412\pi$
$593$		−	448.999i		−	0.757165i	0.925568	−	0.378583i	$-0.123588\pi$
$594$		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	$-0.868537\pi$
$599$		−	480.833i		−	0.802726i	0.915919	−	0.401363i	$-0.131463\pi$
$600$		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	$0.171787\pi$
$613$			630.000i			1.02773i	−0.857870	+	0.513866i	$0.828213\pi$
$614$		0			0
$615$	296.324	−	158.392i	0.481828	−	0.257548i
$616$		0			0
$617$		−	1122.50i		−	1.81928i	−0.415395	−	0.909641i	$-0.636357\pi$
$617$		−	1122.50i		−	1.81928i	0.415395	−	0.909641i	$-0.363643\pi$
$618$		0			0
$619$			555.608i			0.897589i	0.893635	+	0.448795i	$0.148147\pi$
$619$			555.608i			0.897589i	−0.893635	+	0.448795i	$0.851853\pi$
$620$		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	$0.0981671\pi$
$641$			389.132i			0.607071i	−0.952820	+	0.303535i	$0.901833\pi$
$642$		0			0
$643$			269.867i			0.419699i	0.977734	+	0.209850i	$0.0672974\pi$
$643$			269.867i			0.419699i	−0.977734	+	0.209850i	$0.932703\pi$
$644$		0			0
$645$	1152.43	−	616.000i	1.78671	−	0.955039i
$646$		0			0
$647$		−	1170.97i		−	1.80984i	−0.425578	−	0.904922i	$-0.639929\pi$
$647$		−	1170.97i		−	1.80984i	0.425578	−	0.904922i	$-0.360071\pi$
$648$		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	$-0.958954\pi$
$661$		−	170.000i		−	0.257186i	0.991697	−	0.128593i	$-0.0410461\pi$
$662$		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	$-0.993906\pi$
$691$		−	26.4575i		−	0.0382887i	0.999817	−	0.0191444i	$-0.00609421\pi$
$692$		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	$0.0409680\pi$
$709$			182.000i			0.256700i	−0.991729	+	0.128350i	$0.959032\pi$
$710$		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	$0.0887998\pi$
$719$			395.980i			0.550737i	−0.961339	+	0.275368i	$0.911200\pi$
$720$		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	$0.146222\pi$
$733$			650.000i			0.886767i	−0.896332	+	0.443383i	$0.853778\pi$
$734$		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	$-0.925240\pi$
$739$		−	343.948i		−	0.465423i	0.972546	−	0.232712i	$-0.0747598\pi$
$740$		0			0
$741$	−374.166	−	700.000i	−0.504947	−	0.944669i
$742$		0			0
$743$			661.852i			0.890783i	0.895336	+	0.445392i	$0.146936\pi$
$743$			661.852i			0.890783i	−0.895336	+	0.445392i	$0.853064\pi$
$744$		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	$-0.756056\pi$
$757$		−	1050.00i		−	1.38705i	0.720431	−	0.693527i	$-0.243944\pi$
$758$		0			0
$759$	−634.980	+	339.411i	−0.836601	+	0.447182i
$760$		0			0
$761$			673.498i			0.885018i	0.896764	+	0.442509i	$0.145911\pi$
$761$			673.498i			0.885018i	−0.896764	+	0.442509i	$0.854089\pi$
$762$		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	$0.194948\pi$
$787$			904.847i			1.14974i	−0.818244	+	0.574871i	$0.805052\pi$
$788$		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	$0.169894\pi$
$809$			823.165i			1.01751i	−0.860912	+	0.508754i	$0.830106\pi$
$810$		0			0
$811$			26.4575i			0.0326233i	0.999867	+	0.0163117i	$0.00519239\pi$
$811$			26.4575i			0.0326233i	−0.999867	+	0.0163117i	$0.994808\pi$
$812$		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	$0.0326948\pi$
$829$			170.000i			0.205066i	−0.994730	+	0.102533i	$0.967305\pi$
$830$		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	$0.138461\pi$
$839$			707.107i			0.842797i	−0.906876	+	0.421399i	$0.861539\pi$
$840$		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	$-0.891563\pi$
$853$		−	570.000i		−	0.668230i	0.942532	−	0.334115i	$-0.108437\pi$
$854$		0			0
$855$	−1481.62	−	989.949i	−1.73289	−	1.15784i
$856$		0			0
$857$		−	523.832i		−	0.611239i	−0.952154	−	0.305620i	$-0.901136\pi$
$857$		−	523.832i		−	0.611239i	0.952154	−	0.305620i	$-0.0988636\pi$
$858$		0			0
$859$		−	1455.16i		−	1.69402i	−0.531577	−	0.847010i	$-0.678400\pi$
$859$		−	1455.16i		−	1.69402i	0.531577	−	0.847010i	$-0.321600\pi$
$860$		0			0
$861$	209.533	−	112.000i	0.243360	−	0.130081i
$862$		0			0
$863$		−	181.019i		−	0.209756i	−0.994485	−	0.104878i	$-0.966555\pi$
$863$		−	181.019i		−	0.209756i	0.994485	−	0.104878i	$-0.0334452\pi$
$864$		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	$0.0901228\pi$
$877$			490.000i			0.558723i	−0.960186	+	0.279361i	$0.909877\pi$
$878$		0			0
$879$	−158.745	−	296.985i	−0.180597	−	0.337867i
$880$		0			0
$881$			987.798i			1.12122i	0.828079	+	0.560612i	$0.189434\pi$
$881$			987.798i			1.12122i	−0.828079	+	0.560612i	$0.810566\pi$
$882$		0			0
$883$		−	957.762i		−	1.08467i	−0.840163	−	0.542334i	$-0.817541\pi$
$883$		−	957.762i		−	1.08467i	0.840163	−	0.542334i	$-0.182459\pi$
$884$		0			0
$885$	1047.66	+	1960.00i	1.18380	+	2.21469i
$886$		0			0
$887$			1374.62i			1.54974i	0.632123	+	0.774868i	$0.282184\pi$
$887$			1374.62i			1.54974i	−0.632123	+	0.774868i	$0.717816\pi$
$888$		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	$0.0951933\pi$
$907$			534.442i			0.589241i	−0.955614	+	0.294621i	$0.904807\pi$
$908$		0			0
$909$	−561.249	+	840.000i	−0.617435	+	0.924092i
$910$		0			0
$911$		−	169.706i		−	0.186285i	−0.995653	−	0.0931425i	$-0.970309\pi$
$911$		−	169.706i		−	0.186285i	0.995653	−	0.0931425i	$-0.0296912\pi$
$912$		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	$0.0153904\pi$
$929$			89.7998i			0.0966628i	−0.998831	+	0.0483314i	$0.984610\pi$
$930$		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	$-0.962420\pi$
$953$		−	224.499i		−	0.235571i	0.993039	−	0.117786i	$-0.0375796\pi$
$954$		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	$0.0991176\pi$
$977$			598.665i			0.612759i	−0.951910	+	0.306379i	$0.900882\pi$
$978$		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	$-0.712398\pi$
$983$		−	1544.32i		−	1.57103i	0.618843	−	0.785514i	$-0.287602\pi$
$984$		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	$-0.866847\pi$
$997$		−	810.000i		−	0.812437i	0.913776	−	0.406219i	$-0.133153\pi$
$998$		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.641.1		8
3.2	odd	2	inner	768.3.h.e.641.6		8
4.3	odd	2	inner	768.3.h.e.641.7		8
8.3	odd	2	inner	768.3.h.e.641.2		8
8.5	even	2	inner	768.3.h.e.641.8		8
12.11	even	2	inner	768.3.h.e.641.4		8
16.3	odd	4		192.3.e.f.65.2		4
16.5	even	4		96.3.e.b.65.2	yes	4
16.11	odd	4		96.3.e.b.65.3	yes	4
16.13	even	4		192.3.e.f.65.3		4
24.5	odd	2	inner	768.3.h.e.641.3		8
24.11	even	2	inner	768.3.h.e.641.5		8
48.5	odd	4		96.3.e.b.65.1	✓	4
48.11	even	4		96.3.e.b.65.4	yes	4
48.29	odd	4		192.3.e.f.65.4		4
48.35	even	4		192.3.e.f.65.1		4

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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