LMFDB - Embedded newform 1280.3.e.l.639.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,3,Mod(639,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1280.639");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1280.e (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:	$34.8774738381$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			639.13
Character	$\chi$	$=$	1280.639
Dual form			1280.3.e.l.639.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00470i q^{3} +(4.99385 + 0.247851i) q^{5} -4.85433 q^{7} +4.98119 q^{9} +O(q^{10})$	$q-2.00470i q^{3} +(4.99385 + 0.247851i) q^{5} -4.85433 q^{7} +4.98119 q^{9} -0.0522635 q^{11} -8.88065 q^{13} +(0.496866 - 10.0112i) q^{15} -3.74825i q^{17} -20.3394 q^{19} +9.73146i q^{21} -15.3665 q^{23} +(24.8771 + 2.47546i) q^{25} -28.0281i q^{27} -38.3095i q^{29} -34.5984i q^{31} +0.104773i q^{33} +(-24.2418 - 1.20315i) q^{35} -13.5464 q^{37} +17.8030i q^{39} +30.8458 q^{41} -5.41787i q^{43} +(24.8753 + 1.23459i) q^{45} -59.9167 q^{47} -25.4355 q^{49} -7.51411 q^{51} +5.15237 q^{53} +(-0.260996 - 0.0129536i) q^{55} +40.7743i q^{57} +88.9020 q^{59} -90.8553i q^{61} -24.1803 q^{63} +(-44.3487 - 2.20108i) q^{65} -71.2737i q^{67} +30.8052i q^{69} +125.923i q^{71} -113.127i q^{73} +(4.96256 - 49.8712i) q^{75} +0.253704 q^{77} -1.80121i q^{79} -11.3571 q^{81} -135.565i q^{83} +(0.929007 - 18.7182i) q^{85} -76.7989 q^{87} +70.2086 q^{89} +43.1096 q^{91} -69.3593 q^{93} +(-101.572 - 5.04114i) q^{95} +106.921i q^{97} -0.260334 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 72 q^{9}+O(q^{10}) $ 24 * q - 72 * q^9	$ 24 q - 72 q^{9} + 16 q^{11} - 48 q^{19} - 24 q^{25} + 112 q^{35} - 80 q^{41} + 168 q^{49} + 576 q^{51} - 496 q^{59} + 32 q^{65} - 224 q^{75} + 184 q^{81} - 144 q^{89} + 864 q^{91} - 368 q^{99}+O(q^{100}) $ 24 * q - 72 * q^9 + 16 * q^11 - 48 * q^19 - 24 * q^25 + 112 * q^35 - 80 * q^41 + 168 * q^49 + 576 * q^51 - 496 * q^59 + 32 * q^65 - 224 * q^75 + 184 * q^81 - 144 * q^89 + 864 * q^91 - 368 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	2.00470i		−	0.668233i	−0.942532	−	0.334116i	$-0.891562\pi$
$3$		−	2.00470i		−	0.668233i	0.942532	−	0.334116i	$-0.108438\pi$
$4$		0			0
$5$	4.99385	+	0.247851i	0.998771	+	0.0495702i
$5$	4.99385	+	0.247851i	0.998771	+	0.0495702i
$6$		0			0
$7$	−4.85433			−0.693475			−0.346738	−	0.937962i	$-0.612711\pi$
$7$	−4.85433			−0.693475			−0.346738	+	0.937962i	$0.612711\pi$
$8$		0			0
$9$	4.98119			0.553465
$10$		0			0
$11$	−0.0522635			−0.00475123			−0.00237561	−	0.999997i	$-0.500756\pi$
$11$	−0.0522635			−0.00475123			−0.00237561	+	0.999997i	$0.500756\pi$
$12$		0			0
$13$	−8.88065			−0.683127			−0.341563	−	0.939859i	$-0.610956\pi$
$13$	−8.88065			−0.683127			−0.341563	+	0.939859i	$0.610956\pi$
$14$		0			0
$15$	0.496866	−	10.0112i	0.0331244	−	0.667411i
$16$		0			0
$17$		−	3.74825i		−	0.220485i	−0.993905	−	0.110243i	$-0.964837\pi$
$17$		−	3.74825i		−	0.220485i	0.993905	−	0.110243i	$-0.0351628\pi$
$18$		0			0
$19$	−20.3394			−1.07049			−0.535247	−	0.844695i	$-0.679782\pi$
$19$	−20.3394			−1.07049			−0.535247	+	0.844695i	$0.679782\pi$
$20$		0			0
$21$			9.73146i			0.463403i
$22$		0			0
$23$	−15.3665			−0.668108			−0.334054	−	0.942554i	$-0.608417\pi$
$23$	−15.3665			−0.668108			−0.334054	+	0.942554i	$0.608417\pi$
$24$		0			0
$25$	24.8771	+	2.47546i	0.995086	+	0.0990185i
$26$		0			0
$27$		−	28.0281i		−	1.03808i
$28$		0			0
$29$		−	38.3095i		−	1.32102i	−0.750819	−	0.660508i	$-0.770341\pi$
$29$		−	38.3095i		−	1.32102i	0.750819	−	0.660508i	$-0.229659\pi$
$30$		0			0
$31$		−	34.5984i		−	1.11608i	−0.829815	−	0.558038i	$-0.811554\pi$
$31$		−	34.5984i		−	1.11608i	0.829815	−	0.558038i	$-0.188446\pi$
$32$		0			0
$33$			0.104773i			0.00317493i
$34$		0			0
$35$	−24.2418	−	1.20315i	−0.692623	−	0.0343757i
$36$		0			0
$37$	−13.5464			−0.366118			−0.183059	−	0.983102i	$-0.558600\pi$
$37$	−13.5464			−0.366118			−0.183059	+	0.983102i	$0.558600\pi$
$38$		0			0
$39$			17.8030i			0.456488i
$40$		0			0
$41$	30.8458			0.752338			0.376169	−	0.926551i	$-0.377241\pi$
$41$	30.8458			0.752338			0.376169	+	0.926551i	$0.377241\pi$
$42$		0			0
$43$		−	5.41787i		−	0.125997i	−0.998014	−	0.0629985i	$-0.979934\pi$
$43$		−	5.41787i		−	0.125997i	0.998014	−	0.0629985i	$-0.0200663\pi$
$44$		0			0
$45$	24.8753	+	1.23459i	0.552785	+	0.0274354i
$46$		0			0
$47$	−59.9167			−1.27482			−0.637412	−	0.770523i	$-0.719995\pi$
$47$	−59.9167			−1.27482			−0.637412	+	0.770523i	$0.719995\pi$
$48$		0			0
$49$	−25.4355			−0.519092
$50$		0			0
$51$	−7.51411			−0.147335
$52$		0			0
$53$	5.15237			0.0972145			0.0486072	−	0.998818i	$-0.484522\pi$
$53$	5.15237			0.0972145			0.0486072	+	0.998818i	$0.484522\pi$
$54$		0			0
$55$	−0.260996	−	0.0129536i	−0.00474539	−	0.000235519i
$56$		0			0
$57$			40.7743i			0.715339i
$58$		0			0
$59$	88.9020			1.50681			0.753407	−	0.657554i	$-0.228409\pi$
$59$	88.9020			1.50681			0.753407	+	0.657554i	$0.228409\pi$
$60$		0			0
$61$		−	90.8553i		−	1.48943i	−0.667382	−	0.744716i	$-0.732585\pi$
$61$		−	90.8553i		−	1.48943i	0.667382	−	0.744716i	$-0.267415\pi$
$62$		0			0
$63$	−24.1803			−0.383814
$64$		0			0
$65$	−44.3487	−	2.20108i	−0.682287	−	0.0338627i
$66$		0			0
$67$		−	71.2737i		−	1.06379i	−0.846812	−	0.531893i	$-0.821481\pi$
$67$		−	71.2737i		−	1.06379i	0.846812	−	0.531893i	$-0.178519\pi$
$68$		0			0
$69$			30.8052i			0.446452i
$70$		0			0
$71$			125.923i			1.77357i	0.462184	+	0.886784i	$0.347066\pi$
$71$			125.923i			1.77357i	−0.462184	+	0.886784i	$0.652934\pi$
$72$		0			0
$73$		−	113.127i		−	1.54968i	−0.632154	−	0.774842i	$-0.717829\pi$
$73$		−	113.127i		−	1.54968i	0.632154	−	0.774842i	$-0.282171\pi$
$74$		0			0
$75$	4.96256	−	49.8712i	0.0661674	−	0.664949i
$76$		0			0
$77$	0.253704			0.00329486
$78$		0			0
$79$		−	1.80121i		−	0.0228001i	−0.999935	−	0.0114001i	$-0.996371\pi$
$79$		−	1.80121i		−	0.0228001i	0.999935	−	0.0114001i	$-0.00362883\pi$
$80$		0			0
$81$	−11.3571			−0.140211
$82$		0			0
$83$		−	135.565i		−	1.63332i	−0.577121	−	0.816659i	$-0.695824\pi$
$83$		−	135.565i		−	1.63332i	0.577121	−	0.816659i	$-0.304176\pi$
$84$		0			0
$85$	0.929007	−	18.7182i	0.0109295	−	0.220214i
$86$		0			0
$87$	−76.7989			−0.882746
$88$		0			0
$89$	70.2086			0.788861			0.394430	−	0.918926i	$-0.370942\pi$
$89$	70.2086			0.788861			0.394430	+	0.918926i	$0.370942\pi$
$90$		0			0
$91$	43.1096			0.473732
$92$		0			0
$93$	−69.3593			−0.745799
$94$		0			0
$95$	−101.572	−	5.04114i	−1.06918	−	0.0530646i
$96$		0			0
$97$			106.921i			1.10228i	0.834413	+	0.551140i	$0.185807\pi$
$97$			106.921i			1.10228i	−0.834413	+	0.551140i	$0.814193\pi$
$98$		0			0
$99$	−0.260334			−0.00262964
$100$		0			0
$101$			107.595i			1.06530i	0.846335	+	0.532650i	$0.178804\pi$
$101$			107.595i			1.06530i	−0.846335	+	0.532650i	$0.821196\pi$
$102$		0			0
$103$	−166.182			−1.61342			−0.806711	−	0.590946i	$-0.798755\pi$
$103$	−166.182			−1.61342			−0.806711	+	0.590946i	$0.798755\pi$
$104$		0			0
$105$	−2.41195	+	48.5975i	−0.0229710	+	0.462833i
$106$		0			0
$107$		−	115.967i		−	1.08380i	−0.840443	−	0.541901i	$-0.817705\pi$
$107$		−	115.967i		−	1.08380i	0.840443	−	0.541901i	$-0.182295\pi$
$108$		0			0
$109$			87.0881i			0.798974i	0.916739	+	0.399487i	$0.130812\pi$
$109$			87.0881i			0.798974i	−0.916739	+	0.399487i	$0.869188\pi$
$110$		0			0
$111$			27.1564i			0.244652i
$112$		0			0
$113$		−	31.3690i		−	0.277602i	−0.990320	−	0.138801i	$-0.955675\pi$
$113$		−	31.3690i		−	0.277602i	0.990320	−	0.138801i	$-0.0443248\pi$
$114$		0			0
$115$	−76.7380	−	3.80860i	−0.667287	−	0.0331183i
$116$		0			0
$117$	−44.2362			−0.378087
$118$		0			0
$119$			18.1952i			0.152901i
$120$		0			0
$121$	−120.997			−0.999977
$122$		0			0
$123$		−	61.8366i		−	0.502737i
$124$		0			0
$125$	123.619	+	18.5279i	0.988954	+	0.148223i
$126$		0			0
$127$	−68.8645			−0.542240			−0.271120	−	0.962546i	$-0.587394\pi$
$127$	−68.8645			−0.542240			−0.271120	+	0.962546i	$0.587394\pi$
$128$		0			0
$129$	−10.8612			−0.0841953
$130$		0			0
$131$	125.101			0.954968			0.477484	−	0.878640i	$-0.341549\pi$
$131$	125.101			0.954968			0.477484	+	0.878640i	$0.341549\pi$
$132$		0			0
$133$	98.7341			0.742361
$134$		0			0
$135$	6.94678	−	139.968i	0.0514576	−	1.03680i
$136$		0			0
$137$			101.124i			0.738131i	0.929403	+	0.369065i	$0.120322\pi$
$137$			101.124i			0.738131i	−0.929403	+	0.369065i	$0.879678\pi$
$138$		0			0
$139$	−180.726			−1.30019			−0.650094	−	0.759854i	$-0.725271\pi$
$139$	−180.726			−1.30019			−0.650094	+	0.759854i	$0.725271\pi$
$140$		0			0
$141$			120.115i			0.851879i
$142$		0			0
$143$	0.464134			0.00324569
$144$		0			0
$145$	9.49504	−	191.312i	0.0654830	−	1.31939i
$146$		0			0
$147$			50.9905i			0.346874i
$148$		0			0
$149$		−	18.0144i		−	0.120902i	−0.998171	−	0.0604510i	$-0.980746\pi$
$149$		−	18.0144i		−	0.120902i	0.998171	−	0.0604510i	$-0.0192539\pi$
$150$		0			0
$151$		−	47.4348i		−	0.314138i	−0.987588	−	0.157069i	$-0.949795\pi$
$151$		−	47.4348i		−	0.314138i	0.987588	−	0.157069i	$-0.0502045\pi$
$152$		0			0
$153$		−	18.6707i		−	0.122031i
$154$		0			0
$155$	8.57524	−	172.779i	0.0553241	−	1.11470i
$156$		0			0
$157$	−68.0919			−0.433707			−0.216853	−	0.976204i	$-0.569579\pi$
$157$	−68.0919			−0.433707			−0.216853	+	0.976204i	$0.569579\pi$
$158$		0			0
$159$		−	10.3289i		−	0.0649619i
$160$		0			0
$161$	74.5940			0.463316
$162$		0			0
$163$		−	167.395i		−	1.02696i	−0.858100	−	0.513482i	$-0.828355\pi$
$163$		−	167.395i		−	1.02696i	0.858100	−	0.513482i	$-0.171645\pi$
$164$		0			0
$165$	−0.0259680	+	0.523219i	−0.000157382	+	0.00317102i
$166$		0			0
$167$	18.0753			0.108235			0.0541176	−	0.998535i	$-0.482765\pi$
$167$	18.0753			0.108235			0.0541176	+	0.998535i	$0.482765\pi$
$168$		0			0
$169$	−90.1341			−0.533338
$170$		0			0
$171$	−101.314			−0.592481
$172$		0			0
$173$	−117.236			−0.677663			−0.338832	−	0.940847i	$-0.610032\pi$
$173$	−117.236			−0.677663			−0.338832	+	0.940847i	$0.610032\pi$
$174$		0			0
$175$	−120.762	−	12.0167i	−0.690067	−	0.0686669i
$176$		0			0
$177$		−	178.222i		−	1.00690i
$178$		0			0
$179$	−59.2931			−0.331246			−0.165623	−	0.986189i	$-0.552964\pi$
$179$	−59.2931			−0.331246			−0.165623	+	0.986189i	$0.552964\pi$
$180$		0			0
$181$		−	132.103i		−	0.729849i	−0.931037	−	0.364925i	$-0.881095\pi$
$181$		−	132.103i		−	0.729849i	0.931037	−	0.364925i	$-0.118905\pi$
$182$		0			0
$183$	−182.137			−0.995287
$184$		0			0
$185$	−67.6486	−	3.35748i	−0.365668	−	0.0181486i
$186$		0			0
$187$			0.195897i			0.00104758i
$188$		0			0
$189$			136.057i			0.719880i
$190$		0			0
$191$			204.069i			1.06842i	0.845351	+	0.534212i	$0.179391\pi$
$191$			204.069i			1.06842i	−0.845351	+	0.534212i	$0.820609\pi$
$192$		0			0
$193$			337.234i			1.74732i	0.486533	+	0.873662i	$0.338261\pi$
$193$			337.234i			1.74732i	−0.486533	+	0.873662i	$0.661739\pi$
$194$		0			0
$195$	−4.41250	+	88.9057i	−0.0226282	+	0.455927i
$196$		0			0
$197$	−210.797			−1.07003			−0.535017	−	0.844841i	$-0.679695\pi$
$197$	−210.797			−1.07003			−0.535017	+	0.844841i	$0.679695\pi$
$198$		0			0
$199$		−	261.667i		−	1.31491i	−0.753494	−	0.657455i	$-0.771633\pi$
$199$		−	261.667i		−	1.31491i	0.753494	−	0.657455i	$-0.228367\pi$
$200$		0			0
$201$	−142.882			−0.710857
$202$		0			0
$203$			185.967i			0.916092i
$204$		0			0
$205$	154.040	+	7.64517i	0.751413	+	0.0372935i
$206$		0			0
$207$	−76.5433			−0.369774
$208$		0			0
$209$	1.06301			0.00508616
$210$		0			0
$211$	232.011			1.09958			0.549789	−	0.835304i	$-0.314708\pi$
$211$	232.011			1.09958			0.549789	+	0.835304i	$0.314708\pi$
$212$		0			0
$213$	252.438			1.18516
$214$		0			0
$215$	1.34282	−	27.0561i	0.00624570	−	0.125842i
$216$		0			0
$217$			167.952i			0.773972i
$218$		0			0
$219$	−226.785			−1.03555
$220$		0			0
$221$			33.2869i			0.150619i
$222$		0			0
$223$	234.832			1.05306			0.526530	−	0.850157i	$-0.323493\pi$
$223$	234.832			1.05306			0.526530	+	0.850157i	$0.323493\pi$
$224$		0			0
$225$	123.918	+	12.3307i	0.550745	+	0.0548033i
$226$		0			0
$227$		−	52.3990i		−	0.230833i	−0.993317	−	0.115416i	$-0.963180\pi$
$227$		−	52.3990i		−	0.230833i	0.993317	−	0.115416i	$-0.0368202\pi$
$228$		0			0
$229$		−	325.834i		−	1.42286i	−0.702759	−	0.711428i	$-0.748049\pi$
$229$		−	325.834i		−	1.42286i	0.702759	−	0.711428i	$-0.251951\pi$
$230$		0			0
$231$		−	0.508600i		−	0.00220173i
$232$		0			0
$233$		−	384.230i		−	1.64906i	−0.565821	−	0.824528i	$-0.691440\pi$
$233$		−	384.230i		−	1.64906i	0.565821	−	0.824528i	$-0.308560\pi$
$234$		0			0
$235$	−299.215	−	14.8504i	−1.27326	−	0.0631933i
$236$		0			0
$237$	−3.61088			−0.0152358
$238$		0			0
$239$		−	84.2197i		−	0.352384i	−0.984356	−	0.176192i	$-0.943622\pi$
$239$		−	84.2197i		−	0.352384i	0.984356	−	0.176192i	$-0.0563779\pi$
$240$		0			0
$241$	163.620			0.678920			0.339460	−	0.940621i	$-0.389756\pi$
$241$	163.620			0.678920			0.339460	+	0.940621i	$0.389756\pi$
$242$		0			0
$243$		−	229.485i		−	0.944382i
$244$		0			0
$245$	−127.021	−	6.30422i	−0.518454	−	0.0257315i
$246$		0			0
$247$	180.627			0.731283
$248$		0			0
$249$	−271.768			−1.09144
$250$		0			0
$251$	358.000			1.42630			0.713148	−	0.701014i	$-0.247269\pi$
$251$	358.000			1.42630			0.713148	+	0.701014i	$0.247269\pi$
$252$		0			0
$253$	0.803107			0.00317433
$254$		0			0
$255$	−37.5243	−	1.86238i	−0.147154	−	0.00730345i
$256$		0			0
$257$			289.464i			1.12632i	0.826349	+	0.563159i	$0.190414\pi$
$257$			289.464i			1.12632i	−0.826349	+	0.563159i	$0.809586\pi$
$258$		0			0
$259$	65.7585			0.253894
$260$		0			0
$261$		−	190.827i		−	0.731136i
$262$		0			0
$263$	−221.486			−0.842151			−0.421076	−	0.907025i	$-0.638347\pi$
$263$	−221.486			−0.842151			−0.421076	+	0.907025i	$0.638347\pi$
$264$		0			0
$265$	25.7302	+	1.27702i	0.0970950	+	0.00481894i
$266$		0			0
$267$		−	140.747i		−	0.527143i
$268$		0			0
$269$		−	282.618i		−	1.05062i	−0.850910	−	0.525312i	$-0.823949\pi$
$269$		−	282.618i		−	1.05062i	0.850910	−	0.525312i	$-0.176051\pi$
$270$		0			0
$271$			421.440i			1.55513i	0.628804	+	0.777564i	$0.283545\pi$
$271$			421.440i			1.55513i	−0.628804	+	0.777564i	$0.716455\pi$
$272$		0			0
$273$		−	86.4217i		−	0.316563i
$274$		0			0
$275$	−1.30017	−	0.129376i	−0.00472788	−	0.000470460i
$276$		0			0
$277$	−259.738			−0.937682			−0.468841	−	0.883283i	$-0.655328\pi$
$277$	−259.738			−0.937682			−0.468841	+	0.883283i	$0.655328\pi$
$278$		0			0
$279$		−	172.341i		−	0.617709i
$280$		0			0
$281$	425.546			1.51440			0.757199	−	0.653184i	$-0.226567\pi$
$281$	425.546			1.51440			0.757199	+	0.653184i	$0.226567\pi$
$282$		0			0
$283$			417.501i			1.47527i	0.675200	+	0.737634i	$0.264057\pi$
$283$			417.501i			1.47527i	−0.675200	+	0.737634i	$0.735943\pi$
$284$		0			0
$285$	−10.1060	+	203.621i	−0.0354595	+	0.714460i
$286$		0			0
$287$	−149.736			−0.521727
$288$		0			0
$289$	274.951			0.951386
$290$		0			0
$291$	214.345			0.736579
$292$		0			0
$293$	334.004			1.13995			0.569973	−	0.821664i	$-0.306954\pi$
$293$	334.004			1.13995			0.569973	+	0.821664i	$0.306954\pi$
$294$		0			0
$295$	443.964	+	22.0345i	1.50496	+	0.0746931i
$296$		0			0
$297$			1.46484i			0.00493214i
$298$		0			0
$299$	136.464			0.456403
$300$		0			0
$301$			26.3001i			0.0873758i
$302$		0			0
$303$	215.696			0.711869
$304$		0			0
$305$	22.5186	−	453.718i	0.0738314	−	1.48760i
$306$		0			0
$307$			426.544i			1.38939i	0.719303	+	0.694697i	$0.244461\pi$
$307$			426.544i			1.38939i	−0.719303	+	0.694697i	$0.755539\pi$
$308$		0			0
$309$			333.146i			1.07814i
$310$		0			0
$311$			559.575i			1.79927i	0.436638	+	0.899637i	$0.356169\pi$
$311$			559.575i			1.79927i	−0.436638	+	0.899637i	$0.643831\pi$
$312$		0			0
$313$			125.597i			0.401270i	0.979666	+	0.200635i	$0.0643004\pi$
$313$			125.597i			0.401270i	−0.979666	+	0.200635i	$0.935700\pi$
$314$		0			0
$315$	−120.753	−	5.99311i	−0.383342	−	0.0190257i
$316$		0			0
$317$	137.398			0.433433			0.216716	−	0.976235i	$-0.430465\pi$
$317$	137.398			0.433433			0.216716	+	0.976235i	$0.430465\pi$
$318$		0			0
$319$			2.00219i			0.00627645i
$320$		0			0
$321$	−232.478			−0.724232
$322$		0			0
$323$			76.2371i			0.236028i
$324$		0			0
$325$	−220.925	−	21.9837i	−0.679770	−	0.0676422i
$326$		0			0
$327$	174.585			0.533900
$328$		0			0
$329$	290.855			0.884059
$330$		0			0
$331$	463.536			1.40041			0.700206	−	0.713941i	$-0.253092\pi$
$331$	463.536			1.40041			0.700206	+	0.713941i	$0.253092\pi$
$332$		0			0
$333$	−67.4770			−0.202634
$334$		0			0
$335$	17.6652	−	355.930i	0.0527321	−	1.06248i
$336$		0			0
$337$		−	478.938i		−	1.42118i	−0.703605	−	0.710591i	$-0.748428\pi$
$337$		−	478.938i		−	1.42118i	0.703605	−	0.710591i	$-0.251572\pi$
$338$		0			0
$339$	−62.8854			−0.185503
$340$		0			0
$341$			1.80823i			0.00530274i
$342$		0			0
$343$	361.334			1.05345
$344$		0			0
$345$	−7.63509	+	153.836i	−0.0221307	+	0.445903i
$346$		0			0
$347$			44.7114i			0.128851i	0.997923	+	0.0644257i	$0.0205215\pi$
$347$			44.7114i			0.128851i	−0.997923	+	0.0644257i	$0.979478\pi$
$348$		0			0
$349$		−	32.2561i		−	0.0924245i	−0.998932	−	0.0462122i	$-0.985285\pi$
$349$		−	32.2561i		−	0.0924245i	0.998932	−	0.0462122i	$-0.0147151\pi$
$350$		0			0
$351$			248.907i			0.709138i
$352$		0			0
$353$			661.810i			1.87482i	0.348233	+	0.937408i	$0.386782\pi$
$353$			661.810i			1.87482i	−0.348233	+	0.937408i	$0.613218\pi$
$354$		0			0
$355$	−31.2102	+	628.843i	−0.0879161	+	1.77139i
$356$		0			0
$357$	36.4759			0.102173
$358$		0			0
$359$			220.336i			0.613750i	0.951750	+	0.306875i	$0.0992833\pi$
$359$			220.336i			0.613750i	−0.951750	+	0.306875i	$0.900717\pi$
$360$		0			0
$361$	52.6909			0.145958
$362$		0			0
$363$			242.563i			0.668218i
$364$		0			0
$365$	28.0386	−	564.940i	0.0768182	−	1.54778i
$366$		0			0
$367$	469.269			1.27866			0.639332	−	0.768931i	$-0.279211\pi$
$367$	469.269			1.27866			0.639332	+	0.768931i	$0.279211\pi$
$368$		0			0
$369$	153.649			0.416393
$370$		0			0
$371$	−25.0113			−0.0674158
$372$		0			0
$373$	229.799			0.616083			0.308042	−	0.951373i	$-0.400326\pi$
$373$	229.799			0.616083			0.308042	+	0.951373i	$0.400326\pi$
$374$		0			0
$375$	37.1429	−	247.819i	0.0990477	−	0.660851i
$376$		0			0
$377$			340.213i			0.902422i
$378$		0			0
$379$	−35.7198			−0.0942475			−0.0471237	−	0.998889i	$-0.515006\pi$
$379$	−35.7198			−0.0942475			−0.0471237	+	0.998889i	$0.515006\pi$
$380$		0			0
$381$			138.053i			0.362343i
$382$		0			0
$383$	662.223			1.72904			0.864521	−	0.502597i	$-0.167622\pi$
$383$	662.223			1.72904			0.864521	+	0.502597i	$0.167622\pi$
$384$		0			0
$385$	1.26696	+	0.0628808i	0.00329081	+	0.000163327i
$386$		0			0
$387$		−	26.9874i		−	0.0697349i
$388$		0			0
$389$			213.828i			0.549687i	0.961489	+	0.274843i	$0.0886260\pi$
$389$			213.828i			0.549687i	−0.961489	+	0.274843i	$0.911374\pi$
$390$		0			0
$391$			57.5974i			0.147308i
$392$		0			0
$393$		−	250.789i		−	0.638141i
$394$		0			0
$395$	0.446431	−	8.99497i	0.00113021	−	0.0227721i
$396$		0			0
$397$	−158.309			−0.398762			−0.199381	−	0.979922i	$-0.563893\pi$
$397$	−158.309			−0.398762			−0.199381	+	0.979922i	$0.563893\pi$
$398$		0			0
$399$		−	197.932i		−	0.496070i
$400$		0			0
$401$	−506.079			−1.26204			−0.631021	−	0.775766i	$-0.717364\pi$
$401$	−506.079			−1.26204			−0.631021	+	0.775766i	$0.717364\pi$
$402$		0			0
$403$			307.256i			0.762422i
$404$		0			0
$405$	−56.7158	−	2.81488i	−0.140039	−	0.00695031i
$406$		0			0
$407$	0.707981			0.00173951
$408$		0			0
$409$	290.255			0.709670			0.354835	−	0.934929i	$-0.384537\pi$
$409$	290.255			0.709670			0.354835	+	0.934929i	$0.384537\pi$
$410$		0			0
$411$	202.723			0.493243
$412$		0			0
$413$	−431.560			−1.04494
$414$		0			0
$415$	33.6000	−	676.993i	0.0809639	−	1.63131i
$416$		0			0
$417$			362.301i			0.868828i
$418$		0			0
$419$	−702.070			−1.67558			−0.837792	−	0.545989i	$-0.816154\pi$
$419$	−702.070			−1.67558			−0.837792	+	0.545989i	$0.816154\pi$
$420$		0			0
$421$			181.963i			0.432215i	0.976370	+	0.216108i	$0.0693362\pi$
$421$			181.963i			0.432215i	−0.976370	+	0.216108i	$0.930664\pi$
$422$		0			0
$423$	−298.456			−0.705570
$424$		0			0
$425$	9.27865	−	93.2457i	0.0218321	−	0.219402i
$426$		0			0
$427$			441.041i			1.03288i
$428$		0			0
$429$		−	0.930449i		−	0.00216888i
$430$		0			0
$431$		−	166.957i		−	0.387370i	−0.981064	−	0.193685i	$-0.937956\pi$
$431$		−	166.957i		−	0.387370i	0.981064	−	0.193685i	$-0.0620440\pi$
$432$		0			0
$433$		−	323.197i		−	0.746414i	−0.927748	−	0.373207i	$-0.878258\pi$
$433$		−	323.197i		−	0.746414i	0.927748	−	0.373207i	$-0.121742\pi$
$434$		0			0
$435$	−383.523	−	19.0347i	−0.881661	−	0.0437579i
$436$		0			0
$437$	312.545			0.715206
$438$		0			0
$439$			27.9377i			0.0636395i	0.999494	+	0.0318197i	$0.0101302\pi$
$439$			27.9377i			0.0636395i	−0.999494	+	0.0318197i	$0.989870\pi$
$440$		0			0
$441$	−126.699			−0.287299
$442$		0			0
$443$			279.321i			0.630522i	0.949005	+	0.315261i	$0.102092\pi$
$443$			279.321i			0.630522i	−0.949005	+	0.315261i	$0.897908\pi$
$444$		0			0
$445$	350.611	+	17.4013i	0.787891	+	0.0391040i
$446$		0			0
$447$	−36.1135			−0.0807907
$448$		0			0
$449$	−709.613			−1.58043			−0.790215	−	0.612830i	$-0.790031\pi$
$449$	−709.613			−1.58043			−0.790215	+	0.612830i	$0.790031\pi$
$450$		0			0
$451$	−1.61211			−0.00357453
$452$		0			0
$453$	−95.0926			−0.209917
$454$		0			0
$455$	215.283	+	10.6847i	0.473149	+	0.0234830i
$456$		0			0
$457$		−	620.812i		−	1.35845i	−0.733930	−	0.679225i	$-0.762316\pi$
$457$		−	620.812i		−	1.35845i	0.733930	−	0.679225i	$-0.237684\pi$
$458$		0			0
$459$	−105.056			−0.228880
$460$		0			0
$461$		−	387.594i		−	0.840767i	−0.907347	−	0.420383i	$-0.861896\pi$
$461$		−	387.594i		−	0.840767i	0.907347	−	0.420383i	$-0.138104\pi$
$462$		0			0
$463$	224.135			0.484092			0.242046	−	0.970265i	$-0.422182\pi$
$463$	224.135			0.484092			0.242046	+	0.970265i	$0.422182\pi$
$464$		0			0
$465$	−346.370	−	17.1908i	−0.744882	−	0.0369694i
$466$		0			0
$467$			294.857i			0.631386i	0.948861	+	0.315693i	$0.102237\pi$
$467$			294.857i			0.631386i	−0.948861	+	0.315693i	$0.897763\pi$
$468$		0			0
$469$			345.986i			0.737709i
$470$		0			0
$471$			136.504i			0.289817i
$472$		0			0
$473$			0.283157i			0.000598641i
$474$		0			0
$475$	−505.986	−	50.3494i	−1.06523	−	0.105999i
$476$		0			0
$477$	25.6649			0.0538048
$478$		0			0
$479$		−	809.238i		−	1.68943i	−0.535215	−	0.844716i	$-0.679770\pi$
$479$		−	809.238i		−	1.68943i	0.535215	−	0.844716i	$-0.320230\pi$
$480$		0			0
$481$	120.301			0.250105
$482$		0			0
$483$		−	149.538i		−	0.309603i
$484$		0			0
$485$	−26.5005	+	533.948i	−0.0546402	+	1.10092i
$486$		0			0
$487$	578.125			1.18712			0.593558	−	0.804791i	$-0.297723\pi$
$487$	578.125			1.18712			0.593558	+	0.804791i	$0.297723\pi$
$488$		0			0
$489$	−335.577			−0.686251
$490$		0			0
$491$	880.655			1.79360			0.896798	−	0.442441i	$-0.145887\pi$
$491$	880.655			1.79360			0.896798	+	0.442441i	$0.145887\pi$
$492$		0			0
$493$	−143.593			−0.291264
$494$		0			0
$495$	−1.30007	−	0.0645241i	−0.00262641	−	0.000130352i
$496$		0			0
$497$		−	611.273i		−	1.22993i
$498$		0			0
$499$	90.6206			0.181604			0.0908022	−	0.995869i	$-0.471057\pi$
$499$	90.6206			0.181604			0.0908022	+	0.995869i	$0.471057\pi$
$500$		0			0
$501$		−	36.2355i		−	0.0723262i
$502$		0			0
$503$	407.230			0.809603			0.404802	−	0.914405i	$-0.367341\pi$
$503$	407.230			0.809603			0.404802	+	0.914405i	$0.367341\pi$
$504$		0			0
$505$	−26.6676	+	537.315i	−0.0528072	+	1.06399i
$506$		0			0
$507$			180.692i			0.356394i
$508$		0			0
$509$		−	244.966i		−	0.481269i	−0.970616	−	0.240634i	$-0.922644\pi$
$509$		−	244.966i		−	0.481269i	0.970616	−	0.240634i	$-0.0773555\pi$
$510$		0			0
$511$			549.155i			1.07467i
$512$		0			0
$513$			570.074i			1.11125i
$514$		0			0
$515$	−829.891	−	41.1885i	−1.61144	−	0.0799776i
$516$		0			0
$517$	3.13146			0.00605698
$518$		0			0
$519$			235.022i			0.452837i
$520$		0			0
$521$	346.677			0.665407			0.332703	−	0.943032i	$-0.392039\pi$
$521$	346.677			0.665407			0.332703	+	0.943032i	$0.392039\pi$
$522$		0			0
$523$			358.764i			0.685973i	0.939340	+	0.342987i	$0.111439\pi$
$523$			358.764i			0.685973i	−0.939340	+	0.342987i	$0.888561\pi$
$524$		0			0
$525$	−24.0899	+	242.091i	−0.0458855	+	0.461126i
$526$		0			0
$527$	−129.683			−0.246078
$528$		0			0
$529$	−292.871			−0.553632
$530$		0			0
$531$	442.838			0.833969
$532$		0			0
$533$	−273.931			−0.513942
$534$		0			0
$535$	28.7425	−	579.121i	0.0537242	−	1.08247i
$536$		0			0
$537$			118.865i			0.221350i
$538$		0			0
$539$	1.32935			0.00246633
$540$		0			0
$541$		−	45.9247i		−	0.0848886i	−0.999099	−	0.0424443i	$-0.986485\pi$
$541$		−	45.9247i		−	0.0848886i	0.999099	−	0.0424443i	$-0.0135145\pi$
$542$		0			0
$543$	−264.826			−0.487709
$544$		0			0
$545$	−21.5849	+	434.905i	−0.0396053	+	0.797991i
$546$		0			0
$547$			351.059i			0.641790i	0.947115	+	0.320895i	$0.103984\pi$
$547$			351.059i			0.641790i	−0.947115	+	0.320895i	$0.896016\pi$
$548$		0			0
$549$		−	452.567i		−	0.824348i
$550$		0			0
$551$			779.191i			1.41414i
$552$		0			0
$553$			8.74365i			0.0158113i
$554$		0			0
$555$	−6.73074	+	135.615i	−0.0121275	+	0.244351i
$556$		0			0
$557$	678.109			1.21743			0.608716	−	0.793388i	$-0.291685\pi$
$557$	678.109			1.21743			0.608716	+	0.793388i	$0.291685\pi$
$558$		0			0
$559$			48.1142i			0.0860720i
$560$		0			0
$561$	0.392714			0.000700024
$562$		0			0
$563$		−	224.849i		−	0.399376i	−0.979860	−	0.199688i	$-0.936007\pi$
$563$		−	224.849i		−	0.399376i	0.979860	−	0.199688i	$-0.0639929\pi$
$564$		0			0
$565$	7.77484	−	156.652i	0.0137608	−	0.277261i
$566$		0			0
$567$	55.1312			0.0972332
$568$		0			0
$569$	217.927			0.383000			0.191500	−	0.981493i	$-0.438665\pi$
$569$	217.927			0.383000			0.191500	+	0.981493i	$0.438665\pi$
$570$		0			0
$571$	−26.8930			−0.0470981			−0.0235490	−	0.999723i	$-0.507497\pi$
$571$	−26.8930			−0.0470981			−0.0235490	+	0.999723i	$0.507497\pi$
$572$		0			0
$573$	409.096			0.713955
$574$		0			0
$575$	−382.274	−	38.0392i	−0.664825	−	0.0661551i
$576$		0			0
$577$			434.655i			0.753301i	0.926355	+	0.376651i	$0.122924\pi$
$577$			434.655i			0.753301i	−0.926355	+	0.376651i	$0.877076\pi$
$578$		0			0
$579$	676.051			1.16762
$580$		0			0
$581$			658.079i			1.13267i
$582$		0			0
$583$	−0.269281			−0.000461888
$584$		0			0
$585$	−220.909	−	10.9640i	−0.377622	−	0.0187418i
$586$		0			0
$587$		−	393.225i		−	0.669889i	−0.942238	−	0.334944i	$-0.891282\pi$
$587$		−	393.225i		−	0.669889i	0.942238	−	0.334944i	$-0.108718\pi$
$588$		0			0
$589$			703.710i			1.19475i
$590$		0			0
$591$			422.584i			0.715032i
$592$		0			0
$593$		−	787.343i		−	1.32773i	−0.747853	−	0.663865i	$-0.768915\pi$
$593$		−	787.343i		−	1.32773i	0.747853	−	0.663865i	$-0.231085\pi$
$594$		0			0
$595$	−4.50970	+	90.8643i	−0.00757933	+	0.152713i
$596$		0			0
$597$	−524.563			−0.878666
$598$		0			0
$599$		−	479.835i		−	0.801061i	−0.916283	−	0.400530i	$-0.868826\pi$
$599$		−	479.835i		−	0.801061i	0.916283	−	0.400530i	$-0.131174\pi$
$600$		0			0
$601$	−426.464			−0.709591			−0.354796	−	0.934944i	$-0.615450\pi$
$601$	−426.464			−0.709591			−0.354796	+	0.934944i	$0.615450\pi$
$602$		0			0
$603$		−	355.027i		−	0.588768i
$604$		0			0
$605$	−604.243	−	29.9893i	−0.998748	−	0.0495691i
$606$		0			0
$607$	834.209			1.37431			0.687157	−	0.726509i	$-0.258858\pi$
$607$	834.209			1.37431			0.687157	+	0.726509i	$0.258858\pi$
$608$		0			0
$609$	372.807			0.612163
$610$		0			0
$611$	532.099			0.870866
$612$		0			0
$613$	−109.201			−0.178141			−0.0890706	−	0.996025i	$-0.528390\pi$
$613$	−109.201			−0.178141			−0.0890706	+	0.996025i	$0.528390\pi$
$614$		0			0
$615$	15.3263	−	308.803i	0.0249208	−	0.502119i
$616$		0			0
$617$		−	41.6758i		−	0.0675458i	−0.999430	−	0.0337729i	$-0.989248\pi$
$617$		−	41.6758i		−	0.0675458i	0.999430	−	0.0337729i	$-0.0107523\pi$
$618$		0			0
$619$	136.579			0.220644			0.110322	−	0.993896i	$-0.464812\pi$
$619$	136.579			0.220644			0.110322	+	0.993896i	$0.464812\pi$
$620$		0			0
$621$			430.693i			0.693547i
$622$		0			0
$623$	−340.816			−0.547055
$624$		0			0
$625$	612.744	+	123.165i	0.980391	+	0.197064i
$626$		0			0
$627$		−	2.13101i		−	0.00339874i
$628$		0			0
$629$			50.7752i			0.0807237i
$630$		0			0
$631$			680.088i			1.07779i	0.842372	+	0.538897i	$0.181159\pi$
$631$			680.088i			1.07779i	−0.842372	+	0.538897i	$0.818841\pi$
$632$		0			0
$633$		−	465.112i		−	0.734774i
$634$		0			0
$635$	−343.899	−	17.0681i	−0.541574	−	0.0268790i
$636$		0			0
$637$	225.884			0.354606
$638$		0			0
$639$			627.248i			0.981608i
$640$		0			0
$641$	319.731			0.498801			0.249400	−	0.968400i	$-0.419766\pi$
$641$	319.731			0.498801			0.249400	+	0.968400i	$0.419766\pi$
$642$		0			0
$643$			523.796i			0.814613i	0.913292	+	0.407306i	$0.133532\pi$
$643$			523.796i			0.814613i	−0.913292	+	0.407306i	$0.866468\pi$
$644$		0			0
$645$	−54.2392	−	2.69196i	−0.0840918	−	0.00417358i
$646$		0			0
$647$	−17.3046			−0.0267460			−0.0133730	−	0.999911i	$-0.504257\pi$
$647$	−17.3046			−0.0267460			−0.0133730	+	0.999911i	$0.504257\pi$
$648$		0			0
$649$	−4.64633			−0.00715922
$650$		0			0
$651$	336.693			0.517193
$652$		0			0
$653$	−366.938			−0.561927			−0.280963	−	0.959719i	$-0.590654\pi$
$653$	−366.938			−0.561927			−0.280963	+	0.959719i	$0.590654\pi$
$654$		0			0
$655$	624.735	+	31.0064i	0.953794	+	0.0473380i
$656$		0			0
$657$		−	563.506i		−	0.857696i
$658$		0			0
$659$	−539.045			−0.817974			−0.408987	−	0.912540i	$-0.634118\pi$
$659$	−539.045			−0.817974			−0.408987	+	0.912540i	$0.634118\pi$
$660$		0			0
$661$		−	219.459i		−	0.332010i	−0.986125	−	0.166005i	$-0.946913\pi$
$661$		−	219.459i		−	0.332010i	0.986125	−	0.166005i	$-0.0530868\pi$
$662$		0			0
$663$	66.7301			0.100649
$664$		0			0
$665$	493.063	+	24.4713i	0.741449	+	0.0367990i
$666$		0			0
$667$			588.682i			0.882582i
$668$		0			0
$669$		−	470.768i		−	0.703689i
$670$		0			0
$671$			4.74842i			0.00707663i
$672$		0			0
$673$			126.916i			0.188583i	0.995545	+	0.0942914i	$0.0300585\pi$
$673$			126.916i			0.188583i	−0.995545	+	0.0942914i	$0.969941\pi$
$674$		0			0
$675$	69.3824	−	697.258i	0.102789	−	1.03297i
$676$		0			0
$677$	1013.26			1.49670			0.748349	−	0.663305i	$-0.230847\pi$
$677$	1013.26			1.49670			0.748349	+	0.663305i	$0.230847\pi$
$678$		0			0
$679$		−	519.030i		−	0.764404i
$680$		0			0
$681$	−105.044			−0.154250
$682$		0			0
$683$		−	543.680i		−	0.796018i	−0.917382	−	0.398009i	$-0.869701\pi$
$683$		−	543.680i		−	0.796018i	0.917382	−	0.398009i	$-0.130299\pi$
$684$		0			0
$685$	−25.0637	+	504.998i	−0.0365893	+	0.737223i
$686$		0			0
$687$	−653.199			−0.950799
$688$		0			0
$689$	−45.7564			−0.0664098
$690$		0			0
$691$	1307.88			1.89273			0.946364	−	0.323102i	$-0.104726\pi$
$691$	1307.88			1.89273			0.946364	+	0.323102i	$0.104726\pi$
$692$		0			0
$693$	1.26375			0.00182359
$694$		0			0
$695$	−902.520	−	44.7932i	−1.29859	−	0.0644506i
$696$		0			0
$697$		−	115.618i		−	0.165879i
$698$		0			0
$699$	−770.266			−1.10195
$700$		0			0
$701$		−	351.710i		−	0.501726i	−0.968023	−	0.250863i	$-0.919286\pi$
$701$		−	351.710i		−	0.501726i	0.968023	−	0.250863i	$-0.0807144\pi$
$702$		0			0
$703$	275.525			0.391928
$704$		0			0
$705$	−29.7706	+	599.836i	−0.0422278	+	0.850832i
$706$		0			0
$707$		−	522.303i		−	0.738760i
$708$		0			0
$709$		−	1195.09i		−	1.68560i	−0.538225	−	0.842801i	$-0.680905\pi$
$709$		−	1195.09i		−	1.68560i	0.538225	−	0.842801i	$-0.319095\pi$
$710$		0			0
$711$		−	8.97215i		−	0.0126191i
$712$		0			0
$713$			531.656i			0.745660i
$714$		0			0
$715$	2.31782	+	0.115036i	0.00324170	+	0.000160890i
$716$		0			0
$717$	−168.835			−0.235474
$718$		0			0
$719$			76.7870i			0.106797i	0.998573	+	0.0533985i	$0.0170053\pi$
$719$			76.7870i			0.106797i	−0.998573	+	0.0533985i	$0.982995\pi$
$720$		0			0
$721$	806.704			1.11887
$722$		0			0
$723$		−	328.008i		−	0.453676i
$724$		0			0
$725$	94.8337	−	953.030i	0.130805	−	1.31452i
$726$		0			0
$727$	−697.902			−0.959976			−0.479988	−	0.877275i	$-0.659359\pi$
$727$	−697.902			−0.959976			−0.479988	+	0.877275i	$0.659359\pi$
$728$		0			0
$729$	−562.262			−0.771279
$730$		0			0
$731$	−20.3075			−0.0277805
$732$		0			0
$733$	−694.828			−0.947923			−0.473962	−	0.880546i	$-0.657176\pi$
$733$	−694.828			−0.947923			−0.473962	+	0.880546i	$0.657176\pi$
$734$		0			0
$735$	−12.6381	+	254.639i	−0.0171946	+	0.346448i
$736$		0			0
$737$			3.72501i			0.00505429i
$738$		0			0
$739$	−370.791			−0.501746			−0.250873	−	0.968020i	$-0.580718\pi$
$739$	−370.791			−0.501746			−0.250873	+	0.968020i	$0.580718\pi$
$740$		0			0
$741$		−	362.103i		−	0.488668i
$742$		0			0
$743$	−1148.87			−1.54626			−0.773129	−	0.634249i	$-0.781310\pi$
$743$	−1148.87			−1.54626			−0.773129	+	0.634249i	$0.781310\pi$
$744$		0			0
$745$	4.46489	−	89.9613i	0.00599314	−	0.120753i
$746$		0			0
$747$		−	675.276i		−	0.903984i
$748$		0			0
$749$			562.941i			0.751589i
$750$		0			0
$751$			1040.20i			1.38509i	0.721376	+	0.692544i	$0.243510\pi$
$751$			1040.20i			1.38509i	−0.721376	+	0.692544i	$0.756490\pi$
$752$		0			0
$753$		−	717.682i		−	0.953097i
$754$		0			0
$755$	11.7568	−	236.883i	0.0155719	−	0.313752i
$756$		0			0
$757$	1456.91			1.92458			0.962290	−	0.272026i	$-0.0876938\pi$
$757$	1456.91			1.92458			0.962290	+	0.272026i	$0.0876938\pi$
$758$		0			0
$759$		−	1.60999i		−	0.00212119i
$760$		0			0
$761$	−575.305			−0.755986			−0.377993	−	0.925808i	$-0.623386\pi$
$761$	−575.305			−0.755986			−0.377993	+	0.925808i	$0.623386\pi$
$762$		0			0
$763$		−	422.754i		−	0.554068i
$764$		0			0
$765$	4.62756	−	93.2388i	0.00604909	−	0.121881i
$766$		0			0
$767$	−789.508			−1.02935
$768$		0			0
$769$	712.792			0.926908			0.463454	−	0.886121i	$-0.346610\pi$
$769$	712.792			0.926908			0.463454	+	0.886121i	$0.346610\pi$
$770$		0			0
$771$	580.287			0.752642
$772$		0			0
$773$	−1126.56			−1.45739			−0.728695	−	0.684839i	$-0.759873\pi$
$773$	−1126.56			−1.45739			−0.728695	+	0.684839i	$0.759873\pi$
$774$		0			0
$775$	85.6470	−	860.709i	0.110512	−	1.11059i
$776$		0			0
$777$		−	131.826i		−	0.169660i
$778$		0			0
$779$	−627.386			−0.805373
$780$		0			0
$781$		−	6.58120i		−	0.00842663i
$782$		0			0
$783$	−1073.74			−1.37132
$784$		0			0
$785$	−340.041	−	16.8767i	−0.433173	−	0.0214989i
$786$		0			0
$787$		−	900.518i		−	1.14424i	−0.820169	−	0.572121i	$-0.806121\pi$
$787$		−	900.518i		−	1.14424i	0.820169	−	0.572121i	$-0.193879\pi$
$788$		0			0
$789$			444.012i			0.562753i
$790$		0			0
$791$			152.275i			0.192510i
$792$		0			0
$793$			806.854i			1.01747i
$794$		0			0
$795$	2.56004	−	51.5812i	0.00322017	−	0.0648820i
$796$		0			0
$797$	157.087			0.197098			0.0985489	−	0.995132i	$-0.468580\pi$
$797$	157.087			0.197098			0.0985489	+	0.995132i	$0.468580\pi$
$798$		0			0
$799$			224.583i			0.281080i
$800$		0			0
$801$	349.722			0.436607
$802$		0			0
$803$			5.91241i			0.00736291i
$804$		0			0
$805$	372.511	+	18.4882i	0.462747	+	0.0229667i
$806$		0			0
$807$	−566.563			−0.702061
$808$		0			0
$809$	64.8889			0.0802088			0.0401044	−	0.999195i	$-0.487231\pi$
$809$	64.8889			0.0802088			0.0401044	+	0.999195i	$0.487231\pi$
$810$		0			0
$811$	−682.468			−0.841515			−0.420757	−	0.907173i	$-0.638236\pi$
$811$	−682.468			−0.841515			−0.420757	+	0.907173i	$0.638236\pi$
$812$		0			0
$813$	844.859			1.03919
$814$		0			0
$815$	41.4890	−	835.946i	0.0509068	−	1.02570i
$816$		0			0
$817$			110.196i			0.134879i
$818$		0			0
$819$	214.737			0.262194
$820$		0			0
$821$		−	766.593i		−	0.933731i	−0.884328	−	0.466865i	$-0.845383\pi$
$821$		−	766.593i		−	0.933731i	0.884328	−	0.466865i	$-0.154617\pi$
$822$		0			0
$823$	1241.47			1.50847			0.754233	−	0.656607i	$-0.228009\pi$
$823$	1241.47			1.50847			0.754233	+	0.656607i	$0.228009\pi$
$824$		0			0
$825$	−0.259361	+	2.60644i	−0.000314377	+	0.00315932i
$826$		0			0
$827$		−	1206.45i		−	1.45883i	−0.684073	−	0.729414i	$-0.739793\pi$
$827$		−	1206.45i		−	1.45883i	0.684073	−	0.729414i	$-0.260207\pi$
$828$		0			0
$829$		−	391.321i		−	0.472040i	−0.971748	−	0.236020i	$-0.924157\pi$
$829$		−	391.321i		−	0.472040i	0.971748	−	0.236020i	$-0.0758431\pi$
$830$		0			0
$831$			520.696i			0.626590i
$832$		0			0
$833$			95.3386i			0.114452i
$834$		0			0
$835$	90.2652	+	4.47997i	0.108102	+	0.00536524i
$836$		0			0
$837$	−969.725			−1.15857
$838$		0			0
$839$		−	1418.91i		−	1.69120i	−0.533821	−	0.845598i	$-0.679244\pi$
$839$		−	1418.91i		−	1.69120i	0.533821	−	0.845598i	$-0.320756\pi$
$840$		0			0
$841$	−626.615			−0.745084
$842$		0			0
$843$		−	853.091i		−	1.01197i
$844$		0			0
$845$	−450.116	−	22.3398i	−0.532682	−	0.0264377i
$846$		0			0
$847$	587.360			0.693460
$848$		0			0
$849$	836.964			0.985823
$850$		0			0
$851$	208.160			0.244607
$852$		0			0
$853$	−1295.85			−1.51917			−0.759585	−	0.650408i	$-0.774598\pi$
$853$	−1295.85			−1.51917			−0.759585	+	0.650408i	$0.774598\pi$
$854$		0			0
$855$	−505.949	−	25.1108i	−0.591753	−	0.0293694i
$856$		0			0
$857$		−	427.770i		−	0.499148i	−0.968356	−	0.249574i	$-0.919709\pi$
$857$		−	427.770i		−	0.499148i	0.968356	−	0.249574i	$-0.0802907\pi$
$858$		0			0
$859$	−63.2699			−0.0736553			−0.0368277	−	0.999322i	$-0.511725\pi$
$859$	−63.2699			−0.0736553			−0.0368277	+	0.999322i	$0.511725\pi$
$860$		0			0
$861$			300.175i			0.348635i
$862$		0			0
$863$	−1387.34			−1.60758			−0.803790	−	0.594914i	$-0.797186\pi$
$863$	−1387.34			−1.60758			−0.803790	+	0.594914i	$0.797186\pi$
$864$		0			0
$865$	−585.458	−	29.0570i	−0.676830	−	0.0335919i
$866$		0			0
$867$		−	551.193i		−	0.635747i
$868$		0			0
$869$			0.0941375i			0.000108329i
$870$		0			0
$871$			632.956i			0.726701i
$872$		0			0
$873$			532.594i			0.610073i
$874$		0			0
$875$	−600.088	−	89.9406i	−0.685815	−	0.102789i
$876$		0			0
$877$	−235.329			−0.268334			−0.134167	−	0.990959i	$-0.542836\pi$
$877$	−235.329			−0.268334			−0.134167	+	0.990959i	$0.542836\pi$
$878$		0			0
$879$		−	669.577i		−	0.761749i
$880$		0			0
$881$	−190.504			−0.216236			−0.108118	−	0.994138i	$-0.534482\pi$
$881$	−190.504			−0.216236			−0.108118	+	0.994138i	$0.534482\pi$
$882$		0			0
$883$			596.863i			0.675950i	0.941155	+	0.337975i	$0.109742\pi$
$883$			596.863i			0.675950i	−0.941155	+	0.337975i	$0.890258\pi$
$884$		0			0
$885$	44.1724	−	890.013i	0.0499124	−	1.00566i
$886$		0			0
$887$	−1132.56			−1.27684			−0.638420	−	0.769688i	$-0.720412\pi$
$887$	−1132.56			−1.27684			−0.638420	+	0.769688i	$0.720412\pi$
$888$		0			0
$889$	334.291			0.376030
$890$		0			0
$891$	0.593564			0.000666177
$892$		0			0
$893$	1218.67			1.36469
$894$		0			0
$895$	−296.101	−	14.6959i	−0.330839	−	0.0164199i
$896$		0			0
$897$		−	273.570i		−	0.304983i
$898$		0			0
$899$	−1325.45			−1.47436
$900$		0			0
$901$		−	19.3123i		−	0.0214344i
$902$		0			0
$903$	52.7238			0.0583874
$904$		0			0
$905$	32.7418	−	659.702i	0.0361788	−	0.728952i
$906$		0			0
$907$			1272.37i			1.40283i	0.712752	+	0.701416i	$0.247448\pi$
$907$			1272.37i			1.40283i	−0.712752	+	0.701416i	$0.752552\pi$
$908$		0			0
$909$			535.952i			0.589607i
$910$		0			0
$911$		−	241.993i		−	0.265635i	−0.991141	−	0.132817i	$-0.957598\pi$
$911$		−	241.993i		−	0.265635i	0.991141	−	0.132817i	$-0.0424024\pi$
$912$		0			0
$913$			7.08512i			0.00776027i
$914$		0			0
$915$	−909.568	−	45.1429i	−0.994063	−	0.0493366i
$916$		0			0
$917$	−607.280			−0.662247
$918$		0			0
$919$		−	340.754i		−	0.370787i	−0.982664	−	0.185394i	$-0.940644\pi$
$919$		−	340.754i		−	0.370787i	0.982664	−	0.185394i	$-0.0593560\pi$
$920$		0			0
$921$	855.092			0.928438
$922$		0			0
$923$		−	1118.28i		−	1.21157i
$924$		0			0
$925$	−336.995	−	33.5335i	−0.364319	−	0.0362525i
$926$		0			0
$927$	−827.786			−0.892973
$928$		0			0
$929$	−103.798			−0.111731			−0.0558653	−	0.998438i	$-0.517792\pi$
$929$	−103.798			−0.111731			−0.0558653	+	0.998438i	$0.517792\pi$
$930$		0			0
$931$	517.343			0.555685
$932$		0			0
$933$	1121.78			1.20233
$934$		0			0
$935$	−0.0485532	+	0.978279i	−5.19285e−5	+	0.00104629i
$936$		0			0
$937$		−	1428.78i		−	1.52485i	−0.647079	−	0.762423i	$-0.724010\pi$
$937$		−	1428.78i		−	1.52485i	0.647079	−	0.762423i	$-0.275990\pi$
$938$		0			0
$939$	251.785			0.268141
$940$		0			0
$941$		−	978.179i		−	1.03951i	−0.854315	−	0.519755i	$-0.826023\pi$
$941$		−	978.179i		−	1.03951i	0.854315	−	0.519755i	$-0.173977\pi$
$942$		0			0
$943$	−473.992			−0.502643
$944$		0			0
$945$	−33.7219	+	679.450i	−0.0356846	+	0.718995i
$946$		0			0
$947$		−	789.655i		−	0.833849i	−0.908941	−	0.416924i	$-0.863108\pi$
$947$		−	789.655i		−	0.833849i	0.908941	−	0.416924i	$-0.136892\pi$
$948$		0			0
$949$			1004.64i			1.05863i
$950$		0			0
$951$		−	275.442i		−	0.289634i
$952$		0			0
$953$			814.080i			0.854229i	0.904198	+	0.427114i	$0.140470\pi$
$953$			814.080i			0.854229i	−0.904198	+	0.427114i	$0.859530\pi$
$954$		0			0
$955$	−50.5787	+	1019.09i	−0.0529619	+	1.06711i
$956$		0			0
$957$	4.01378			0.00419413
$958$		0			0
$959$		−	490.889i		−	0.511875i
$960$		0			0
$961$	−236.048			−0.245627
$962$		0			0
$963$		−	577.652i		−	0.599846i
$964$		0			0
$965$	−83.5837	+	1684.09i	−0.0866152	+	1.74518i
$966$		0			0
$967$	−498.366			−0.515373			−0.257687	−	0.966229i	$-0.582960\pi$
$967$	−498.366			−0.515373			−0.257687	+	0.966229i	$0.582960\pi$
$968$		0			0
$969$	152.832			0.157722
$970$		0			0
$971$	175.760			0.181009			0.0905047	−	0.995896i	$-0.471152\pi$
$971$	175.760			0.181009			0.0905047	+	0.995896i	$0.471152\pi$
$972$		0			0
$973$	877.304			0.901648
$974$		0			0
$975$	−44.0707	+	442.888i	−0.0452007	+	0.454244i
$976$		0			0
$977$			808.069i			0.827092i	0.910483	+	0.413546i	$0.135710\pi$
$977$			808.069i			0.827092i	−0.910483	+	0.413546i	$0.864290\pi$
$978$		0			0
$979$	−3.66935			−0.00374806
$980$		0			0
$981$			433.802i			0.442204i
$982$		0			0
$983$	330.736			0.336455			0.168228	−	0.985748i	$-0.446196\pi$
$983$	330.736			0.336455			0.168228	+	0.985748i	$0.446196\pi$
$984$		0			0
$985$	−1052.69	−	52.2462i	−1.06872	−	0.0530418i
$986$		0			0
$987$		−	583.077i		−	0.590757i
$988$		0			0
$989$			83.2537i			0.0841796i
$990$		0			0
$991$		−	582.843i		−	0.588136i	−0.955784	−	0.294068i	$-0.904991\pi$
$991$		−	582.843i		−	0.588136i	0.955784	−	0.294068i	$-0.0950092\pi$
$992$		0			0
$993$		−	929.250i		−	0.935801i
$994$		0			0
$995$	64.8544	−	1306.73i	0.0651803	−	1.31329i
$996$		0			0
$997$	995.382			0.998377			0.499189	−	0.866493i	$-0.333631\pi$
$997$	995.382			0.998377			0.499189	+	0.866493i	$0.333631\pi$
$998$		0			0
$999$			379.679i			0.380059i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.3.e.l.639.13		24
4.3	odd	2		1280.3.e.k.639.12		24
5.4	even	2	inner	1280.3.e.l.639.12		24
8.3	odd	2	inner	1280.3.e.l.639.11		24
8.5	even	2		1280.3.e.k.639.14		24
16.3	odd	4		640.3.h.a.639.7	✓	24
16.5	even	4		640.3.h.b.639.7	yes	24
16.11	odd	4		640.3.h.b.639.18	yes	24
16.13	even	4		640.3.h.a.639.18	yes	24
20.19	odd	2		1280.3.e.k.639.13		24
40.19	odd	2	inner	1280.3.e.l.639.14		24
40.29	even	2		1280.3.e.k.639.11		24
80.19	odd	4		640.3.h.a.639.17	yes	24
80.29	even	4		640.3.h.a.639.8	yes	24
80.59	odd	4		640.3.h.b.639.8	yes	24
80.69	even	4		640.3.h.b.639.17	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.3.h.a.639.7	✓	24	16.3	odd	4
640.3.h.a.639.8	yes	24	80.29	even	4
640.3.h.a.639.17	yes	24	80.19	odd	4
640.3.h.a.639.18	yes	24	16.13	even	4
640.3.h.b.639.7	yes	24	16.5	even	4
640.3.h.b.639.8	yes	24	80.59	odd	4
640.3.h.b.639.17	yes	24	80.69	even	4
640.3.h.b.639.18	yes	24	16.11	odd	4
1280.3.e.k.639.11		24	40.29	even	2
1280.3.e.k.639.12		24	4.3	odd	2
1280.3.e.k.639.13		24	20.19	odd	2
1280.3.e.k.639.14		24	8.5	even	2
1280.3.e.l.639.11		24	8.3	odd	2	inner
1280.3.e.l.639.12		24	5.4	even	2	inner
1280.3.e.l.639.13		24	1.1	even	1	trivial
1280.3.e.l.639.14		24	40.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1280.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00470i q^{3} +(4.99385 + 0.247851i) q^{5} -4.85433 q^{7} +4.98119 q^{9} +O(q^{10})\)	\(q-2.00470i q^{3} +(4.99385 + 0.247851i) q^{5} -4.85433 q^{7} +4.98119 q^{9} -0.0522635 q^{11} -8.88065 q^{13} +(0.496866 - 10.0112i) q^{15} -3.74825i q^{17} -20.3394 q^{19} +9.73146i q^{21} -15.3665 q^{23} +(24.8771 + 2.47546i) q^{25} -28.0281i q^{27} -38.3095i q^{29} -34.5984i q^{31} +0.104773i q^{33} +(-24.2418 - 1.20315i) q^{35} -13.5464 q^{37} +17.8030i q^{39} +30.8458 q^{41} -5.41787i q^{43} +(24.8753 + 1.23459i) q^{45} -59.9167 q^{47} -25.4355 q^{49} -7.51411 q^{51} +5.15237 q^{53} +(-0.260996 - 0.0129536i) q^{55} +40.7743i q^{57} +88.9020 q^{59} -90.8553i q^{61} -24.1803 q^{63} +(-44.3487 - 2.20108i) q^{65} -71.2737i q^{67} +30.8052i q^{69} +125.923i q^{71} -113.127i q^{73} +(4.96256 - 49.8712i) q^{75} +0.253704 q^{77} -1.80121i q^{79} -11.3571 q^{81} -135.565i q^{83} +(0.929007 - 18.7182i) q^{85} -76.7989 q^{87} +70.2086 q^{89} +43.1096 q^{91} -69.3593 q^{93} +(-101.572 - 5.04114i) q^{95} +106.921i q^{97} -0.260334 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{9}+O(q^{10}) \) 24 * q - 72 * q^9	\( 24 q - 72 q^{9} + 16 q^{11} - 48 q^{19} - 24 q^{25} + 112 q^{35} - 80 q^{41} + 168 q^{49} + 576 q^{51} - 496 q^{59} + 32 q^{65} - 224 q^{75} + 184 q^{81} - 144 q^{89} + 864 q^{91} - 368 q^{99}+O(q^{100}) \) 24 * q - 72 * q^9 + 16 * q^11 - 48 * q^19 - 24 * q^25 + 112 * q^35 - 80 * q^41 + 168 * q^49 + 576 * q^51 - 496 * q^59 + 32 * q^65 - 224 * q^75 + 184 * q^81 - 144 * q^89 + 864 * q^91 - 368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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