LMFDB - Embedded newform 1008.3.f.h.433.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,3,Mod(433,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.433"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1008, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,-4,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			433.4
Root			$0.765367i$ of defining polynomial
Character	$\chi$	$=$	1008.433
Dual form			1008.3.f.h.433.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.92177i q^{5} +(-3.82843 - 5.86030i) q^{7} -9.31371 q^{11} +0.262632i q^{13} -14.7821i q^{17} -25.4972i q^{19} +16.6274 q^{23} -54.5980 q^{25} -14.6863 q^{29} +8.65914i q^{31} +(52.2843 - 34.1563i) q^{35} +43.9411 q^{37} -22.9159i q^{41} +46.0000 q^{43} -86.1562i q^{47} +(-19.6863 + 44.8715i) q^{49} +60.6274 q^{53} -83.0948i q^{55} +58.1228i q^{59} +9.97230i q^{61} -2.34315 q^{65} +20.6274 q^{67} -78.9706 q^{71} +21.4303i q^{73} +(35.6569 + 54.5811i) q^{77} +90.2843 q^{79} -71.8544i q^{83} +131.882 q^{85} +2.01094i q^{89} +(1.53911 - 1.00547i) q^{91} +227.480 q^{95} -158.581i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} + 8 q^{11} - 24 q^{23} - 60 q^{25} - 104 q^{29} + 96 q^{35} + 40 q^{37} + 184 q^{43} - 124 q^{49} + 152 q^{53} - 32 q^{65} - 8 q^{67} - 248 q^{71} + 120 q^{77} + 248 q^{79} + 256 q^{85} - 288 q^{91}+ \cdots + 480 q^{95}+O(q^{100}) $ 4 * q - 4 * q^7 + 8 * q^11 - 24 * q^23 - 60 * q^25 - 104 * q^29 + 96 * q^35 + 40 * q^37 + 184 * q^43 - 124 * q^49 + 152 * q^53 - 32 * q^65 - 8 * q^67 - 248 * q^71 + 120 * q^77 + 248 * q^79 + 256 * q^85 - 288 * q^91 + 480 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$			8.92177i			1.78435i	0.451686	+	0.892177i	$0.350823\pi$
$5$			8.92177i			1.78435i	−0.451686	+	0.892177i	$0.649177\pi$
$6$		0			0
$7$	−3.82843	−	5.86030i	−0.546918	−	0.837186i
$7$	−3.82843	−	5.86030i	−0.546918	−	0.837186i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.31371			−0.846701			−0.423350	−	0.905966i	$-0.639146\pi$
$11$	−9.31371			−0.846701			−0.423350	+	0.905966i	$0.639146\pi$
$12$		0			0
$13$			0.262632i			0.0202025i	0.999949	+	0.0101012i	$0.00321538\pi$
$13$			0.262632i			0.0202025i	−0.999949	+	0.0101012i	$0.996785\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	14.7821i		−	0.869534i	−0.900543	−	0.434767i	$-0.856831\pi$
$17$		−	14.7821i		−	0.869534i	0.900543	−	0.434767i	$-0.143169\pi$
$18$		0			0
$19$		−	25.4972i		−	1.34196i	−0.741476	−	0.670979i	$-0.765874\pi$
$19$		−	25.4972i		−	1.34196i	0.741476	−	0.670979i	$-0.234126\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	16.6274			0.722931			0.361466	−	0.932385i	$-0.382276\pi$
$23$	16.6274			0.722931			0.361466	+	0.932385i	$0.382276\pi$
$24$		0			0
$25$	−54.5980			−2.18392
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−14.6863			−0.506424			−0.253212	−	0.967411i	$-0.581487\pi$
$29$	−14.6863			−0.506424			−0.253212	+	0.967411i	$0.581487\pi$
$30$		0			0
$31$			8.65914i			0.279327i	0.990199	+	0.139664i	$0.0446021\pi$
$31$			8.65914i			0.279327i	−0.990199	+	0.139664i	$0.955398\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	52.2843	−	34.1563i	1.49384	−	0.975896i
$36$		0			0
$37$	43.9411			1.18760			0.593799	−	0.804613i	$-0.297627\pi$
$37$	43.9411			1.18760			0.593799	+	0.804613i	$0.297627\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	22.9159i		−	0.558925i	−0.960157	−	0.279463i	$-0.909844\pi$
$41$		−	22.9159i		−	0.558925i	0.960157	−	0.279463i	$-0.0901563\pi$
$42$		0			0
$43$	46.0000			1.06977			0.534884	−	0.844926i	$-0.320355\pi$
$43$	46.0000			1.06977			0.534884	+	0.844926i	$0.320355\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	86.1562i		−	1.83311i	−0.399907	−	0.916556i	$-0.630958\pi$
$47$		−	86.1562i		−	1.83311i	0.399907	−	0.916556i	$-0.369042\pi$
$48$		0			0
$49$	−19.6863	+	44.8715i	−0.401761	+	0.915745i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	60.6274			1.14391			0.571957	−	0.820284i	$-0.306185\pi$
$53$	60.6274			1.14391			0.571957	+	0.820284i	$0.306185\pi$
$54$		0			0
$55$		−	83.0948i		−	1.51081i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			58.1228i			0.985133i	0.870275	+	0.492566i	$0.163941\pi$
$59$			58.1228i			0.985133i	−0.870275	+	0.492566i	$0.836059\pi$
$60$		0			0
$61$			9.97230i			0.163480i	0.996654	+	0.0817402i	$0.0260478\pi$
$61$			9.97230i			0.163480i	−0.996654	+	0.0817402i	$0.973952\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.34315			−0.0360484
$66$		0			0
$67$	20.6274			0.307872			0.153936	−	0.988081i	$-0.450805\pi$
$67$	20.6274			0.307872			0.153936	+	0.988081i	$0.450805\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−78.9706			−1.11226			−0.556131	−	0.831095i	$-0.687715\pi$
$71$	−78.9706			−1.11226			−0.556131	+	0.831095i	$0.687715\pi$
$72$		0			0
$73$			21.4303i			0.293565i	0.989169	+	0.146783i	$0.0468918\pi$
$73$			21.4303i			0.293565i	−0.989169	+	0.146783i	$0.953108\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	35.6569	+	54.5811i	0.463076	+	0.708846i
$78$		0			0
$79$	90.2843			1.14284			0.571419	−	0.820658i	$-0.306393\pi$
$79$	90.2843			1.14284			0.571419	+	0.820658i	$0.306393\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	71.8544i		−	0.865715i	−0.901462	−	0.432858i	$-0.857505\pi$
$83$		−	71.8544i		−	0.865715i	0.901462	−	0.432858i	$-0.142495\pi$
$84$		0			0
$85$	131.882			1.55156
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2.01094i			0.0225948i	0.999936	+	0.0112974i	$0.00359615\pi$
$89$			2.01094i			0.0225948i	−0.999936	+	0.0112974i	$0.996404\pi$
$90$		0			0
$91$	1.53911	−	1.00547i	0.0169132	−	0.0110491i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	227.480			2.39453
$96$		0			0
$97$		−	158.581i		−	1.63485i	−0.576032	−	0.817427i	$-0.695399\pi$
$97$		−	158.581i		−	1.63485i	0.576032	−	0.817427i	$-0.304601\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	93.5022i		−	0.925764i	−0.886420	−	0.462882i	$-0.846815\pi$
$101$		−	93.5022i		−	0.925764i	0.886420	−	0.462882i	$-0.153185\pi$
$102$		0			0
$103$		−	20.3797i		−	0.197862i	−0.995094	−	0.0989308i	$-0.968458\pi$
$103$		−	20.3797i		−	0.197862i	0.995094	−	0.0989308i	$-0.0315422\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	14.1177			0.131942			0.0659708	−	0.997822i	$-0.478986\pi$
$107$	14.1177			0.131942			0.0659708	+	0.997822i	$0.478986\pi$
$108$		0			0
$109$	23.2548			0.213347			0.106674	−	0.994294i	$-0.465980\pi$
$109$	23.2548			0.213347			0.106674	+	0.994294i	$0.465980\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−13.7157			−0.121378			−0.0606891	−	0.998157i	$-0.519330\pi$
$113$	−13.7157			−0.121378			−0.0606891	+	0.998157i	$0.519330\pi$
$114$		0			0
$115$			148.346i			1.28997i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−86.6274	+	56.5921i	−0.727961	+	0.475564i
$120$		0			0
$121$	−34.2548			−0.283098
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	264.066i		−	2.11253i
$126$		0			0
$127$	−177.765			−1.39972			−0.699860	−	0.714280i	$-0.746754\pi$
$127$	−177.765			−1.39972			−0.699860	+	0.714280i	$0.746754\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	112.614i		−	0.859648i	−0.902913	−	0.429824i	$-0.858576\pi$
$131$		−	112.614i		−	0.859648i	0.902913	−	0.429824i	$-0.141424\pi$
$132$		0			0
$133$	−149.421	+	97.6142i	−1.12347	+	0.733941i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−121.765			−0.888792			−0.444396	−	0.895830i	$-0.646582\pi$
$137$	−121.765			−0.888792			−0.444396	+	0.895830i	$0.646582\pi$
$138$		0			0
$139$		−	175.329i		−	1.26136i	−0.776043	−	0.630679i	$-0.782776\pi$
$139$		−	175.329i		−	1.26136i	0.776043	−	0.630679i	$-0.217224\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2.44608i		−	0.0171055i
$144$		0			0
$145$		−	131.028i		−	0.903639i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	237.765			1.59573			0.797867	−	0.602833i	$-0.205961\pi$
$149$	237.765			1.59573			0.797867	+	0.602833i	$0.205961\pi$
$150$		0			0
$151$	−99.3726			−0.658097			−0.329048	−	0.944313i	$-0.606728\pi$
$151$	−99.3726			−0.658097			−0.329048	+	0.944313i	$0.606728\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−77.2548			−0.498418
$156$		0			0
$157$		−	96.6538i		−	0.615629i	−0.951446	−	0.307815i	$-0.900402\pi$
$157$		−	96.6538i		−	0.615629i	0.951446	−	0.307815i	$-0.0995977\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−63.6569	−	97.4417i	−0.395384	−	0.605228i
$162$		0			0
$163$	−170.569			−1.04643			−0.523216	−	0.852200i	$-0.675268\pi$
$163$	−170.569			−1.04643			−0.523216	+	0.852200i	$0.675268\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	94.9055i		−	0.568296i	−0.958780	−	0.284148i	$-0.908289\pi$
$167$		−	94.9055i		−	0.568296i	0.958780	−	0.284148i	$-0.0917108\pi$
$168$		0			0
$169$	168.931			0.999592
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	182.720i		−	1.05618i	−0.849187	−	0.528092i	$-0.822908\pi$
$173$		−	182.720i		−	1.05618i	0.849187	−	0.528092i	$-0.177092\pi$
$174$		0			0
$175$	209.024	+	319.961i	1.19443	+	1.82835i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−204.059			−1.13999			−0.569997	−	0.821647i	$-0.693056\pi$
$179$	−204.059			−1.13999			−0.569997	+	0.821647i	$0.693056\pi$
$180$		0			0
$181$		−	229.167i		−	1.26612i	−0.774104	−	0.633059i	$-0.781799\pi$
$181$		−	229.167i		−	1.26612i	0.774104	−	0.633059i	$-0.218201\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			392.033i			2.11910i
$186$		0			0
$187$			137.676i			0.736235i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.59798			0.0502512			0.0251256	−	0.999684i	$-0.492001\pi$
$191$	9.59798			0.0502512			0.0251256	+	0.999684i	$0.492001\pi$
$192$		0			0
$193$	−3.42136			−0.0177272			−0.00886362	−	0.999961i	$-0.502821\pi$
$193$	−3.42136			−0.0177272			−0.00886362	+	0.999961i	$0.502821\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	146.451			0.743405			0.371703	−	0.928352i	$-0.378774\pi$
$197$	146.451			0.743405			0.371703	+	0.928352i	$0.378774\pi$
$198$		0			0
$199$		−	244.647i		−	1.22938i	−0.788768	−	0.614691i	$-0.789281\pi$
$199$		−	244.647i		−	1.22938i	0.788768	−	0.614691i	$-0.210719\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	56.2254	+	86.0661i	0.276972	+	0.423971i
$204$		0			0
$205$	204.451			0.997321
$206$		0			0
$207$		0			0
$208$		0			0
$209$			237.474i			1.13624i
$210$		0			0
$211$	−311.019			−1.47403			−0.737013	−	0.675879i	$-0.763764\pi$
$211$	−311.019			−1.47403			−0.737013	+	0.675879i	$0.763764\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			410.401i			1.90884i
$216$		0			0
$217$	50.7452	−	33.1509i	0.233849	−	0.152769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.88225			0.0175667
$222$		0			0
$223$			116.156i			0.520877i	0.965490	+	0.260438i	$0.0838671\pi$
$223$			116.156i			0.520877i	−0.965490	+	0.260438i	$0.916133\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	59.1734i		−	0.260676i	−0.991470	−	0.130338i	$-0.958394\pi$
$227$		−	59.1734i		−	0.260676i	0.991470	−	0.130338i	$-0.0416062\pi$
$228$		0			0
$229$		−	128.664i		−	0.561852i	−0.959729	−	0.280926i	$-0.909359\pi$
$229$		−	128.664i		−	0.561852i	0.959729	−	0.280926i	$-0.0906415\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	88.9807			0.381891			0.190946	−	0.981601i	$-0.438845\pi$
$233$	88.9807			0.381891			0.190946	+	0.981601i	$0.438845\pi$
$234$		0			0
$235$	768.666			3.27092
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.13708			−0.0298623			−0.0149311	−	0.999889i	$-0.504753\pi$
$239$	−7.13708			−0.0298623			−0.0149311	+	0.999889i	$0.504753\pi$
$240$		0			0
$241$			367.451i			1.52469i	0.647170	+	0.762346i	$0.275953\pi$
$241$			367.451i			1.52469i	−0.647170	+	0.762346i	$0.724047\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−400.333	−	175.637i	−1.63401	−	0.716884i
$246$		0			0
$247$	6.69639			0.0271109
$248$		0			0
$249$		0			0
$250$		0			0
$251$			176.995i			0.705159i	0.935782	+	0.352579i	$0.114695\pi$
$251$			176.995i			0.705159i	−0.935782	+	0.352579i	$0.885305\pi$
$252$		0			0
$253$	−154.863			−0.612106
$254$		0			0
$255$		0			0
$256$		0			0
$257$			27.4631i			0.106860i	0.998572	+	0.0534301i	$0.0170154\pi$
$257$			27.4631i			0.106860i	−0.998572	+	0.0534301i	$0.982985\pi$
$258$		0			0
$259$	−168.225	−	257.508i	−0.649519	−	0.994240i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	77.1472			0.293335			0.146668	−	0.989186i	$-0.453145\pi$
$263$	77.1472			0.293335			0.146668	+	0.989186i	$0.453145\pi$
$264$		0			0
$265$			540.904i			2.04115i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			34.3739i			0.127784i	0.997957	+	0.0638920i	$0.0203513\pi$
$269$			34.3739i			0.127784i	−0.997957	+	0.0638920i	$0.979649\pi$
$270$		0			0
$271$			202.927i			0.748809i	0.927266	+	0.374404i	$0.122153\pi$
$271$			202.927i			0.748809i	−0.927266	+	0.374404i	$0.877847\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	508.510			1.84913
$276$		0			0
$277$	−254.098			−0.917320			−0.458660	−	0.888612i	$-0.651670\pi$
$277$	−254.098			−0.917320			−0.458660	+	0.888612i	$0.651670\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	221.529			0.788359			0.394180	−	0.919033i	$-0.371029\pi$
$281$	221.529			0.788359			0.394180	+	0.919033i	$0.371029\pi$
$282$		0			0
$283$			283.786i			1.00278i	0.865223	+	0.501388i	$0.167177\pi$
$283$			283.786i			1.00278i	−0.865223	+	0.501388i	$0.832823\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−134.294	+	87.7320i	−0.467925	+	0.305687i
$288$		0			0
$289$	70.4903			0.243911
$290$		0			0
$291$		0			0
$292$		0			0
$293$			398.073i			1.35861i	0.733855	+	0.679306i	$0.237719\pi$
$293$			398.073i			1.35861i	−0.733855	+	0.679306i	$0.762281\pi$
$294$		0			0
$295$	−518.558			−1.75783
$296$		0			0
$297$		0			0
$298$		0			0
$299$			4.36690i			0.0146050i
$300$		0			0
$301$	−176.108	−	269.574i	−0.585075	−	0.895594i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−88.9706			−0.291707
$306$		0			0
$307$			167.720i			0.546320i	0.961969	+	0.273160i	$0.0880689\pi$
$307$			167.720i			0.546320i	−0.961969	+	0.273160i	$0.911931\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	284.101i		−	0.913508i	−0.889593	−	0.456754i	$-0.849012\pi$
$311$		−	284.101i		−	0.913508i	0.889593	−	0.456754i	$-0.150988\pi$
$312$		0			0
$313$			514.311i			1.64317i	0.570089	+	0.821583i	$0.306909\pi$
$313$			514.311i			1.64317i	−0.570089	+	0.821583i	$0.693091\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−18.5685			−0.0585758			−0.0292879	−	0.999571i	$-0.509324\pi$
$317$	−18.5685			−0.0585758			−0.0292879	+	0.999571i	$0.509324\pi$
$318$		0			0
$319$	136.784			0.428789
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−376.902			−1.16688
$324$		0			0
$325$		−	14.3392i		−	0.0441206i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−504.902	+	329.843i	−1.53466	+	1.00256i
$330$		0			0
$331$	−194.333			−0.587109			−0.293554	−	0.955942i	$-0.594838\pi$
$331$	−194.333			−0.587109			−0.293554	+	0.955942i	$0.594838\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			184.033i			0.549352i
$336$		0			0
$337$	−125.265			−0.371706			−0.185853	−	0.982578i	$-0.559505\pi$
$337$	−125.265			−0.371706			−0.185853	+	0.982578i	$0.559505\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	80.6487i		−	0.236506i
$342$		0			0
$343$	338.328	−	56.4196i	0.986379	−	0.164489i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−160.745			−0.463243			−0.231621	−	0.972806i	$-0.574403\pi$
$347$	−160.745			−0.463243			−0.231621	+	0.972806i	$0.574403\pi$
$348$		0			0
$349$		−	93.0671i		−	0.266668i	−0.991071	−	0.133334i	$-0.957432\pi$
$349$		−	93.0671i		−	0.266668i	0.991071	−	0.133334i	$-0.0425683\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			189.631i			0.537198i	0.963252	+	0.268599i	$0.0865606\pi$
$353$			189.631i			0.537198i	−0.963252	+	0.268599i	$0.913439\pi$
$354$		0			0
$355$		−	704.557i		−	1.98467i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−494.431			−1.37724			−0.688622	−	0.725121i	$-0.741784\pi$
$359$	−494.431			−1.37724			−0.688622	+	0.725121i	$0.741784\pi$
$360$		0			0
$361$	−289.108			−0.800852
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−191.196			−0.523825
$366$		0			0
$367$		−	718.994i		−	1.95911i	−0.201171	−	0.979556i	$-0.564475\pi$
$367$		−	718.994i		−	1.95911i	0.201171	−	0.979556i	$-0.435525\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−232.108	−	355.295i	−0.625627	−	0.957668i
$372$		0			0
$373$	132.843			0.356147			0.178073	−	0.984017i	$-0.443014\pi$
$373$	132.843			0.356147			0.178073	+	0.984017i	$0.443014\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	3.85710i		−	0.0102310i
$378$		0			0
$379$	−324.607			−0.856483			−0.428242	−	0.903664i	$-0.640867\pi$
$379$	−324.607			−0.856483			−0.428242	+	0.903664i	$0.640867\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	391.252i		−	1.02155i	−0.859715	−	0.510773i	$-0.829359\pi$
$383$		−	391.252i		−	1.02155i	0.859715	−	0.510773i	$-0.170641\pi$
$384$		0			0
$385$	−486.960	+	318.122i	−1.26483	+	0.826292i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−223.352			−0.574171			−0.287085	−	0.957905i	$-0.592686\pi$
$389$	−223.352			−0.574171			−0.287085	+	0.957905i	$0.592686\pi$
$390$		0			0
$391$		−	245.788i		−	0.628613i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			805.496i			2.03923i
$396$		0			0
$397$			223.044i			0.561824i	0.959733	+	0.280912i	$0.0906370\pi$
$397$			223.044i			0.561824i	−0.959733	+	0.280912i	$0.909363\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−136.225			−0.339714			−0.169857	−	0.985469i	$-0.554331\pi$
$401$	−136.225			−0.339714			−0.169857	+	0.985469i	$0.554331\pi$
$402$		0			0
$403$	−2.27417			−0.00564310
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−409.255			−1.00554
$408$		0			0
$409$		−	671.587i		−	1.64202i	−0.570913	−	0.821010i	$-0.693411\pi$
$409$		−	671.587i		−	1.64202i	0.570913	−	0.821010i	$-0.306589\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	340.617	−	222.519i	0.824739	−	0.538787i
$414$		0			0
$415$	641.068			1.54474
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	10.8053i		−	0.0257882i	−0.999917	−	0.0128941i	$-0.995896\pi$
$419$		−	10.8053i		−	0.0257882i	0.999917	−	0.0128941i	$-0.00410443\pi$
$420$		0			0
$421$	421.647			1.00154			0.500768	−	0.865581i	$-0.333051\pi$
$421$	421.647			1.00154			0.500768	+	0.865581i	$0.333051\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			807.071i			1.89899i
$426$		0			0
$427$	58.4407	−	38.1782i	0.136863	−	0.0894104i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	793.529			1.84113			0.920567	−	0.390584i	$-0.127727\pi$
$431$	793.529			1.84113			0.920567	+	0.390584i	$0.127727\pi$
$432$		0			0
$433$			254.627i			0.588053i	0.955797	+	0.294027i	$0.0949954\pi$
$433$			254.627i			0.588053i	−0.955797	+	0.294027i	$0.905005\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	423.953i		−	0.970144i
$438$		0			0
$439$			839.517i			1.91234i	0.292814	+	0.956169i	$0.405408\pi$
$439$			839.517i			1.91234i	−0.292814	+	0.956169i	$0.594592\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	774.215			1.74766			0.873832	−	0.486228i	$-0.161627\pi$
$443$	774.215			1.74766			0.873832	+	0.486228i	$0.161627\pi$
$444$		0			0
$445$	−17.9411			−0.0403171
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−751.921			−1.67466			−0.837328	−	0.546700i	$-0.815884\pi$
$449$	−751.921			−1.67466			−0.837328	+	0.546700i	$0.815884\pi$
$450$		0			0
$451$			213.432i			0.473243i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	8.97056	+	13.7315i	0.0197155	+	0.0301792i
$456$		0			0
$457$	−251.990			−0.551400			−0.275700	−	0.961244i	$-0.588910\pi$
$457$	−251.990			−0.551400			−0.275700	+	0.961244i	$0.588910\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	93.4121i		−	0.202629i	−0.994854	−	0.101315i	$-0.967695\pi$
$461$		−	93.4121i		−	0.202629i	0.994854	−	0.101315i	$-0.0323049\pi$
$462$		0			0
$463$	−167.990			−0.362829			−0.181415	−	0.983407i	$-0.558068\pi$
$463$	−167.990			−0.362829			−0.181415	+	0.983407i	$0.558068\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			364.179i			0.779828i	0.920851	+	0.389914i	$0.127495\pi$
$467$			364.179i			0.779828i	−0.920851	+	0.389914i	$0.872505\pi$
$468$		0			0
$469$	−78.9706	−	120.883i	−0.168381	−	0.257746i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−428.431			−0.905773
$474$		0			0
$475$			1392.10i			2.93073i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	137.061i		−	0.286139i	−0.989713	−	0.143069i	$-0.954303\pi$
$479$		−	137.061i		−	0.286139i	0.989713	−	0.143069i	$-0.0456972\pi$
$480$		0			0
$481$			11.5404i			0.0239924i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1414.82			2.91716
$486$		0			0
$487$	−557.411			−1.14458			−0.572291	−	0.820051i	$-0.693945\pi$
$487$	−557.411			−1.14458			−0.572291	+	0.820051i	$0.693945\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−537.647			−1.09500			−0.547502	−	0.836805i	$-0.684421\pi$
$491$	−537.647			−1.09500			−0.547502	+	0.836805i	$0.684421\pi$
$492$		0			0
$493$			217.094i			0.440353i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	302.333	+	462.791i	0.608316	+	0.931170i
$498$		0			0
$499$	633.980			1.27050			0.635250	−	0.772306i	$-0.280897\pi$
$499$	633.980			1.27050			0.635250	+	0.772306i	$0.280897\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	91.1385i		−	0.181190i	−0.995888	−	0.0905949i	$-0.971123\pi$
$503$		−	91.1385i		−	0.181190i	0.995888	−	0.0905949i	$-0.0288769\pi$
$504$		0			0
$505$	834.205			1.65189
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	353.366i		−	0.694237i	−0.937821	−	0.347118i	$-0.887160\pi$
$509$		−	353.366i		−	0.694237i	0.937821	−	0.347118i	$-0.112840\pi$
$510$		0			0
$511$	125.588	−	82.0442i	0.245769	−	0.160556i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	181.823			0.353055
$516$		0			0
$517$			802.434i			1.55210i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	949.039i		−	1.82157i	−0.412878	−	0.910786i	$-0.635476\pi$
$521$		−	949.039i		−	1.82157i	0.412878	−	0.910786i	$-0.364524\pi$
$522$		0			0
$523$		−	662.702i		−	1.26712i	−0.773695	−	0.633558i	$-0.781594\pi$
$523$		−	662.702i		−	1.26712i	0.773695	−	0.633558i	$-0.218406\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	128.000			0.242884
$528$		0			0
$529$	−252.529			−0.477371
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.01847			0.0112917
$534$		0			0
$535$			125.955i			0.235430i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	183.352	−	417.920i	0.340171	−	0.775362i
$540$		0			0
$541$	1026.90			1.89815			0.949077	−	0.315043i	$-0.102019\pi$
$541$	1026.90			1.89815			0.949077	+	0.315043i	$0.102019\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			207.474i			0.380687i
$546$		0			0
$547$	258.686			0.472918			0.236459	−	0.971641i	$-0.424013\pi$
$547$	258.686			0.472918			0.236459	+	0.971641i	$0.424013\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			374.459i			0.679600i
$552$		0			0
$553$	−345.647	−	529.093i	−0.625039	−	0.956769i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	741.529			1.33129			0.665645	−	0.746268i	$-0.268156\pi$
$557$	741.529			1.33129			0.665645	+	0.746268i	$0.268156\pi$
$558$		0			0
$559$			12.0811i			0.0216120i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	80.5135i		−	0.143008i	−0.997440	−	0.0715040i	$-0.977220\pi$
$563$		−	80.5135i		−	0.143008i	0.997440	−	0.0715040i	$-0.0227799\pi$
$564$		0			0
$565$		−	122.369i		−	0.216582i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	742.971			1.30575			0.652874	−	0.757467i	$-0.273563\pi$
$569$	742.971			1.30575			0.652874	+	0.757467i	$0.273563\pi$
$570$		0			0
$571$	−597.882			−1.04708			−0.523540	−	0.852001i	$-0.675389\pi$
$571$	−597.882			−1.04708			−0.523540	+	0.852001i	$0.675389\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−907.823			−1.57882
$576$		0			0
$577$		−	830.767i		−	1.43980i	−0.694075	−	0.719902i	$-0.744187\pi$
$577$		−	830.767i		−	1.43980i	0.694075	−	0.719902i	$-0.255813\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−421.088	+	275.089i	−0.724765	+	0.473475i
$582$		0			0
$583$	−564.666			−0.968552
$584$		0			0
$585$		0			0
$586$		0			0
$587$			700.310i			1.19303i	0.802601	+	0.596516i	$0.203449\pi$
$587$			700.310i			1.19303i	−0.802601	+	0.596516i	$0.796551\pi$
$588$		0			0
$589$	220.784			0.374845
$590$		0			0
$591$		0			0
$592$		0			0
$593$			417.050i			0.703288i	0.936134	+	0.351644i	$0.114377\pi$
$593$			417.050i			0.703288i	−0.936134	+	0.351644i	$0.885623\pi$
$594$		0			0
$595$	−504.902	−	772.870i	−0.848574	−	1.29894i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	223.892			0.373777			0.186888	−	0.982381i	$-0.440160\pi$
$599$	223.892			0.373777			0.186888	+	0.982381i	$0.440160\pi$
$600$		0			0
$601$		−	721.185i		−	1.19998i	−0.800009	−	0.599988i	$-0.795172\pi$
$601$		−	721.185i		−	1.19998i	0.800009	−	0.599988i	$-0.204828\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	305.614i		−	0.505147i
$606$		0			0
$607$			320.494i			0.527996i	0.964523	+	0.263998i	$0.0850413\pi$
$607$			320.494i			0.527996i	−0.964523	+	0.263998i	$0.914959\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	22.6274			0.0370334
$612$		0			0
$613$	195.373			0.318715			0.159358	−	0.987221i	$-0.449058\pi$
$613$	195.373			0.318715			0.159358	+	0.987221i	$0.449058\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−838.284			−1.35865			−0.679323	−	0.733840i	$-0.737726\pi$
$617$	−838.284			−1.35865			−0.679323	+	0.733840i	$0.737726\pi$
$618$		0			0
$619$			515.407i			0.832644i	0.909217	+	0.416322i	$0.136681\pi$
$619$			515.407i			0.832644i	−0.909217	+	0.416322i	$0.863319\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	11.7847	−	7.69873i	0.0189161	−	0.0123575i
$624$		0			0
$625$	990.990			1.58558
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	649.541i		−	1.03266i
$630$		0			0
$631$	392.538			0.622089			0.311045	−	0.950395i	$-0.399321\pi$
$631$	392.538			0.622089			0.311045	+	0.950395i	$0.399321\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	1585.97i		−	2.49760i
$636$		0			0
$637$	−11.7847	−	5.17026i	−0.0185003	−	0.00811657i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	813.460			1.26905			0.634524	−	0.772903i	$-0.281196\pi$
$641$	813.460			1.26905			0.634524	+	0.772903i	$0.281196\pi$
$642$		0			0
$643$		−	906.299i		−	1.40948i	−0.709463	−	0.704742i	$-0.751063\pi$
$643$		−	906.299i		−	1.40948i	0.709463	−	0.704742i	$-0.248937\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	114.325i		−	0.176700i	−0.996089	−	0.0883499i	$-0.971841\pi$
$647$		−	114.325i		−	0.176700i	0.996089	−	0.0883499i	$-0.0281594\pi$
$648$		0			0
$649$		−	541.339i		−	0.834113i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−415.685			−0.636578			−0.318289	−	0.947994i	$-0.603108\pi$
$653$	−415.685			−0.636578			−0.318289	+	0.947994i	$0.603108\pi$
$654$		0			0
$655$	1004.71			1.53392
$656$		0			0
$657$		0			0
$658$		0			0
$659$	650.431			0.986996			0.493498	−	0.869747i	$-0.335718\pi$
$659$	650.431			0.986996			0.493498	+	0.869747i	$0.335718\pi$
$660$		0			0
$661$		−	1023.81i		−	1.54888i	−0.632645	−	0.774442i	$-0.718031\pi$
$661$		−	1023.81i		−	1.54888i	0.632645	−	0.774442i	$-0.281969\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−870.891	−	1333.10i	−1.30961	−	2.00467i
$666$		0			0
$667$	−244.195			−0.366110
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	92.8791i		−	0.138419i
$672$		0			0
$673$	−1187.21			−1.76406			−0.882032	−	0.471190i	$-0.843824\pi$
$673$	−1187.21			−1.76406			−0.882032	+	0.471190i	$0.843824\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	905.826i		−	1.33800i	−0.743262	−	0.669000i	$-0.766723\pi$
$677$		−	905.826i		−	1.33800i	0.743262	−	0.669000i	$-0.233277\pi$
$678$		0			0
$679$	−929.332	+	607.116i	−1.36868	+	0.894132i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−315.255			−0.461574			−0.230787	−	0.973004i	$-0.574130\pi$
$683$	−315.255			−0.461574			−0.230787	+	0.973004i	$0.574130\pi$
$684$		0			0
$685$		−	1086.35i		−	1.58592i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			15.9227i			0.0231099i
$690$		0			0
$691$			334.960i			0.484747i	0.970183	+	0.242374i	$0.0779260\pi$
$691$			334.960i			0.484747i	−0.970183	+	0.242374i	$0.922074\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1564.24			2.25071
$696$		0			0
$697$	−338.745			−0.486005
$698$		0			0
$699$		0			0
$700$		0			0
$701$	362.353			0.516909			0.258455	−	0.966023i	$-0.416787\pi$
$701$	362.353			0.516909			0.258455	+	0.966023i	$0.416787\pi$
$702$		0			0
$703$		−	1120.38i		−	1.59371i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−547.951	+	357.966i	−0.775037	+	0.506317i
$708$		0			0
$709$	−1117.96			−1.57681			−0.788406	−	0.615155i	$-0.789093\pi$
$709$	−1117.96			−1.57681			−0.788406	+	0.615155i	$0.789093\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			143.979i			0.201934i
$714$		0			0
$715$	21.8234			0.0305222
$716$		0			0
$717$		0			0
$718$		0			0
$719$			972.571i			1.35267i	0.736593	+	0.676336i	$0.236433\pi$
$719$			972.571i			1.35267i	−0.736593	+	0.676336i	$0.763567\pi$
$720$		0			0
$721$	−119.431	+	78.0224i	−0.165647	+	0.108214i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	801.842			1.10599
$726$		0			0
$727$			660.646i			0.908729i	0.890816	+	0.454365i	$0.150134\pi$
$727$			660.646i			0.908729i	−0.890816	+	0.454365i	$0.849866\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	679.975i		−	0.930199i
$732$		0			0
$733$			258.222i			0.352280i	0.984365	+	0.176140i	$0.0563612\pi$
$733$			258.222i			0.352280i	−0.984365	+	0.176140i	$0.943639\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−192.118			−0.260675
$738$		0			0
$739$	333.391			0.451138			0.225569	−	0.974227i	$-0.427576\pi$
$739$	333.391			0.451138			0.225569	+	0.974227i	$0.427576\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−610.118			−0.821154			−0.410577	−	0.911826i	$-0.634673\pi$
$743$	−610.118			−0.821154			−0.410577	+	0.911826i	$0.634673\pi$
$744$		0			0
$745$			2121.28i			2.84736i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−54.0488	−	82.7343i	−0.0721612	−	0.110460i
$750$		0			0
$751$	18.1177			0.0241248			0.0120624	−	0.999927i	$-0.496160\pi$
$751$	18.1177			0.0241248			0.0120624	+	0.999927i	$0.496160\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	886.579i		−	1.17428i
$756$		0			0
$757$	107.137			0.141529			0.0707643	−	0.997493i	$-0.477456\pi$
$757$	107.137			0.141529			0.0707643	+	0.997493i	$0.477456\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	898.284i		−	1.18040i	−0.807257	−	0.590200i	$-0.799049\pi$
$761$		−	898.284i		−	1.18040i	0.807257	−	0.590200i	$-0.200951\pi$
$762$		0			0
$763$	−89.0294	−	136.280i	−0.116683	−	0.178611i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15.2649			−0.0199021
$768$		0			0
$769$			929.350i			1.20852i	0.796788	+	0.604259i	$0.206531\pi$
$769$			929.350i			1.20852i	−0.796788	+	0.604259i	$0.793469\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			914.395i			1.18292i	0.806335	+	0.591459i	$0.201448\pi$
$773$			914.395i			1.18292i	−0.806335	+	0.591459i	$0.798552\pi$
$774$		0			0
$775$		−	472.771i		−	0.610028i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−584.293			−0.750055
$780$		0			0
$781$	735.509			0.941753
$782$		0			0
$783$		0			0
$784$		0			0
$785$	862.323			1.09850
$786$		0			0
$787$			761.720i			0.967878i	0.875102	+	0.483939i	$0.160794\pi$
$787$			761.720i			0.967878i	−0.875102	+	0.483939i	$0.839206\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	52.5097	+	80.3783i	0.0663839	+	0.101616i
$792$		0			0
$793$	−2.61905			−0.00330271
$794$		0			0
$795$		0			0
$796$		0			0
$797$			395.267i			0.495943i	0.968767	+	0.247972i	$0.0797639\pi$
$797$			395.267i			0.495943i	−0.968767	+	0.247972i	$0.920236\pi$
$798$		0			0
$799$	−1273.57			−1.59395
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	199.595i		−	0.248562i
$804$		0			0
$805$	869.352	−	567.932i	1.07994	−	0.705505i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	877.793			1.08503			0.542517	−	0.840045i	$-0.317471\pi$
$809$	877.793			1.08503			0.542517	+	0.840045i	$0.317471\pi$
$810$		0			0
$811$		−	851.717i		−	1.05021i	−0.851039	−	0.525103i	$-0.824027\pi$
$811$		−	851.717i		−	1.05021i	0.851039	−	0.525103i	$-0.175973\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	1521.77i		−	1.86721i
$816$		0			0
$817$		−	1172.87i		−	1.43558i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−437.038			−0.532324			−0.266162	−	0.963928i	$-0.585756\pi$
$821$	−437.038			−0.532324			−0.266162	+	0.963928i	$0.585756\pi$
$822$		0			0
$823$	1592.34			1.93480			0.967402	−	0.253247i	$-0.0814985\pi$
$823$	1592.34			1.93480			0.967402	+	0.253247i	$0.0814985\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1182.45			1.42981			0.714904	−	0.699223i	$-0.246470\pi$
$827$	1182.45			1.42981			0.714904	+	0.699223i	$0.246470\pi$
$828$		0			0
$829$			1048.05i			1.26423i	0.774873	+	0.632117i	$0.217814\pi$
$829$			1048.05i			1.26423i	−0.774873	+	0.632117i	$0.782186\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	663.294	+	291.004i	0.796271	+	0.349345i
$834$		0			0
$835$	846.725			1.01404
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	585.611i		−	0.697986i	−0.937125	−	0.348993i	$-0.886524\pi$
$839$		−	585.611i		−	0.697986i	0.937125	−	0.348993i	$-0.113476\pi$
$840$		0			0
$841$	−625.313			−0.743535
$842$		0			0
$843$		0			0
$844$		0			0
$845$			1507.16i			1.78363i
$846$		0			0
$847$	131.142	+	200.744i	0.154831	+	0.237006i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	730.627			0.858552
$852$		0			0
$853$			901.189i			1.05649i	0.849091	+	0.528247i	$0.177150\pi$
$853$			901.189i			1.05649i	−0.849091	+	0.528247i	$0.822850\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			965.997i			1.12718i	0.826053	+	0.563592i	$0.190581\pi$
$857$			965.997i			1.12718i	−0.826053	+	0.563592i	$0.809419\pi$
$858$		0			0
$859$		−	250.830i		−	0.292003i	−0.989284	−	0.146001i	$-0.953360\pi$
$859$		−	250.830i		−	0.292003i	0.989284	−	0.146001i	$-0.0466404\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1497.13			1.73479			0.867397	−	0.497617i	$-0.165791\pi$
$863$	1497.13			1.73479			0.867397	+	0.497617i	$0.165791\pi$
$864$		0			0
$865$	1630.18			1.88461
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−840.881			−0.967643
$870$		0			0
$871$			5.41743i			0.00621978i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1547.51	+	1010.96i	−1.76858	+	1.15538i
$876$		0			0
$877$	141.354			0.161179			0.0805896	−	0.996747i	$-0.474320\pi$
$877$	141.354			0.161179			0.0805896	+	0.996747i	$0.474320\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1413.83i		−	1.60480i	−0.596788	−	0.802399i	$-0.703557\pi$
$881$		−	1413.83i		−	1.60480i	0.596788	−	0.802399i	$-0.296443\pi$
$882$		0			0
$883$	1386.04			1.56969			0.784845	−	0.619692i	$-0.212742\pi$
$883$	1386.04			1.56969			0.784845	+	0.619692i	$0.212742\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			303.595i			0.342272i	0.985247	+	0.171136i	$0.0547437\pi$
$887$			303.595i			0.342272i	−0.985247	+	0.171136i	$0.945256\pi$
$888$		0			0
$889$	680.558	+	1041.75i	0.765533	+	1.17183i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2196.74			−2.45996
$894$		0			0
$895$		−	1820.57i		−	2.03415i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	127.171i		−	0.141458i
$900$		0			0
$901$		−	896.199i		−	0.994671i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2044.58			2.25920
$906$		0			0
$907$	−1304.02			−1.43773			−0.718864	−	0.695151i	$-0.755337\pi$
$907$	−1304.02			−1.43773			−0.718864	+	0.695151i	$0.755337\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−466.118			−0.511655			−0.255828	−	0.966722i	$-0.582348\pi$
$911$	−466.118			−0.511655			−0.255828	+	0.966722i	$0.582348\pi$
$912$		0			0
$913$			669.231i			0.733002i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−659.951	+	431.134i	−0.719685	+	0.470157i
$918$		0			0
$919$	1178.34			1.28220			0.641100	−	0.767458i	$-0.278478\pi$
$919$	1178.34			1.28220			0.641100	+	0.767458i	$0.278478\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	20.7402i		−	0.0224705i
$924$		0			0
$925$	−2399.10			−2.59362
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	911.161i		−	0.980798i	−0.871498	−	0.490399i	$-0.836851\pi$
$929$		−	911.161i		−	0.980798i	0.871498	−	0.490399i	$-0.163149\pi$
$930$		0			0
$931$	1144.10	+	501.945i	1.22889	+	0.539147i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1228.31			−1.31370
$936$		0			0
$937$		−	1035.35i		−	1.10496i	−0.833527	−	0.552479i	$-0.813682\pi$
$937$		−	1035.35i		−	1.10496i	0.833527	−	0.552479i	$-0.186318\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			64.4479i			0.0684887i	0.999413	+	0.0342443i	$0.0109024\pi$
$941$			64.4479i			0.0684887i	−0.999413	+	0.0342443i	$0.989098\pi$
$942$		0			0
$943$		−	381.033i		−	0.404065i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	132.274			0.139677			0.0698385	−	0.997558i	$-0.477752\pi$
$947$	132.274			0.139677			0.0698385	+	0.997558i	$0.477752\pi$
$948$		0			0
$949$	−5.62828			−0.00593075
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1346.86			−1.41329			−0.706643	−	0.707570i	$-0.749791\pi$
$953$	−1346.86			−1.41329			−0.706643	+	0.707570i	$0.749791\pi$
$954$		0			0
$955$			85.6310i			0.0896659i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	466.167	+	713.577i	0.486096	+	0.744084i
$960$		0			0
$961$	886.019			0.921976
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	30.5246i		−	0.0316317i
$966$		0			0
$967$	498.979			0.516007			0.258004	−	0.966144i	$-0.416935\pi$
$967$	498.979			0.516007			0.258004	+	0.966144i	$0.416935\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1799.26i			1.85299i	0.376304	+	0.926496i	$0.377195\pi$
$971$			1799.26i			1.85299i	−0.376304	+	0.926496i	$0.622805\pi$
$972$		0			0
$973$	−1027.48	+	671.234i	−1.05599	+	0.689860i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1381.61			−1.41413			−0.707066	−	0.707148i	$-0.749982\pi$
$977$	−1381.61			−1.41413			−0.707066	+	0.707148i	$0.749982\pi$
$978$		0			0
$979$		−	18.7293i		−	0.0191310i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1306.67i			1.32927i	0.747167	+	0.664636i	$0.231413\pi$
$983$			1306.67i			1.32927i	−0.747167	+	0.664636i	$0.768587\pi$
$984$		0			0
$985$			1306.60i			1.32650i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	764.861			0.773368
$990$		0			0
$991$	−1033.99			−1.04338			−0.521689	−	0.853136i	$-0.674698\pi$
$991$	−1033.99			−1.04338			−0.521689	+	0.853136i	$0.674698\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2182.68			2.19365
$996$		0			0
$997$			391.500i			0.392678i	0.980536	+	0.196339i	$0.0629052\pi$
$997$			391.500i			0.392678i	−0.980536	+	0.196339i	$0.937095\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.f.h.433.4		4
3.2	odd	2		112.3.c.c.97.3		4
4.3	odd	2		504.3.f.a.433.4		4
7.6	odd	2	inner	1008.3.f.h.433.1		4
12.11	even	2		56.3.c.a.41.2	✓	4
21.2	odd	6		784.3.s.f.129.2		8
21.5	even	6		784.3.s.f.129.3		8
21.11	odd	6		784.3.s.f.705.3		8
21.17	even	6		784.3.s.f.705.2		8
21.20	even	2		112.3.c.c.97.2		4
24.5	odd	2		448.3.c.e.321.2		4
24.11	even	2		448.3.c.f.321.3		4
28.27	even	2		504.3.f.a.433.1		4
60.23	odd	4		1400.3.p.a.1049.5		8
60.47	odd	4		1400.3.p.a.1049.4		8
60.59	even	2		1400.3.f.a.601.3		4
84.11	even	6		392.3.o.b.313.2		8
84.23	even	6		392.3.o.b.129.3		8
84.47	odd	6		392.3.o.b.129.2		8
84.59	odd	6		392.3.o.b.313.3		8
84.83	odd	2		56.3.c.a.41.3	yes	4
168.83	odd	2		448.3.c.f.321.2		4
168.125	even	2		448.3.c.e.321.3		4
420.83	even	4		1400.3.p.a.1049.3		8
420.167	even	4		1400.3.p.a.1049.6		8
420.419	odd	2		1400.3.f.a.601.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.c.a.41.2	✓	4	12.11	even	2
56.3.c.a.41.3	yes	4	84.83	odd	2
112.3.c.c.97.2		4	21.20	even	2
112.3.c.c.97.3		4	3.2	odd	2
392.3.o.b.129.2		8	84.47	odd	6
392.3.o.b.129.3		8	84.23	even	6
392.3.o.b.313.2		8	84.11	even	6
392.3.o.b.313.3		8	84.59	odd	6
448.3.c.e.321.2		4	24.5	odd	2
448.3.c.e.321.3		4	168.125	even	2
448.3.c.f.321.2		4	168.83	odd	2
448.3.c.f.321.3		4	24.11	even	2
504.3.f.a.433.1		4	28.27	even	2
504.3.f.a.433.4		4	4.3	odd	2
784.3.s.f.129.2		8	21.2	odd	6
784.3.s.f.129.3		8	21.5	even	6
784.3.s.f.705.2		8	21.17	even	6
784.3.s.f.705.3		8	21.11	odd	6
1008.3.f.h.433.1		4	7.6	odd	2	inner
1008.3.f.h.433.4		4	1.1	even	1	trivial
1400.3.f.a.601.2		4	420.419	odd	2
1400.3.f.a.601.3		4	60.59	even	2
1400.3.p.a.1049.3		8	420.83	even	4
1400.3.p.a.1049.4		8	60.47	odd	4
1400.3.p.a.1049.5		8	60.23	odd	4
1400.3.p.a.1049.6		8	420.167	even	4

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.92177i q^{5} +(-3.82843 - 5.86030i) q^{7} -9.31371 q^{11} +0.262632i q^{13} -14.7821i q^{17} -25.4972i q^{19} +16.6274 q^{23} -54.5980 q^{25} -14.6863 q^{29} +8.65914i q^{31} +(52.2843 - 34.1563i) q^{35} +43.9411 q^{37} -22.9159i q^{41} +46.0000 q^{43} -86.1562i q^{47} +(-19.6863 + 44.8715i) q^{49} +60.6274 q^{53} -83.0948i q^{55} +58.1228i q^{59} +9.97230i q^{61} -2.34315 q^{65} +20.6274 q^{67} -78.9706 q^{71} +21.4303i q^{73} +(35.6569 + 54.5811i) q^{77} +90.2843 q^{79} -71.8544i q^{83} +131.882 q^{85} +2.01094i q^{89} +(1.53911 - 1.00547i) q^{91} +227.480 q^{95} -158.581i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} + 8 q^{11} - 24 q^{23} - 60 q^{25} - 104 q^{29} + 96 q^{35} + 40 q^{37} + 184 q^{43} - 124 q^{49} + 152 q^{53} - 32 q^{65} - 8 q^{67} - 248 q^{71} + 120 q^{77} + 248 q^{79} + 256 q^{85} - 288 q^{91}+ \cdots + 480 q^{95}+O(q^{100}) \) 4 * q - 4 * q^7 + 8 * q^11 - 24 * q^23 - 60 * q^25 - 104 * q^29 + 96 * q^35 + 40 * q^37 + 184 * q^43 - 124 * q^49 + 152 * q^53 - 32 * q^65 - 8 * q^67 - 248 * q^71 + 120 * q^77 + 248 * q^79 + 256 * q^85 - 288 * q^91 + 480 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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