LMFDB - Embedded newform 2016.3.e.c.1007.28

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2016,3,Mod(1007,2016)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2016, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2016.1007"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-336] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$54.9320212950$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1007.28
Character	$\chi$	$=$	2016.1007
Dual form			2016.3.e.c.1007.27

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.51431i q^{5} +(6.42922 - 2.76859i) q^{7} -5.26675i q^{11} -9.36128 q^{13} +12.3664 q^{17} +8.84770i q^{19} -4.29743 q^{23} +12.6496 q^{25} -18.6786 q^{29} +52.1525 q^{31} +(9.72968 + 22.5943i) q^{35} -45.4695i q^{37} -42.0916 q^{41} +23.2557 q^{43} +28.5041i q^{47} +(33.6698 - 35.5998i) q^{49} +18.1585 q^{53} +18.5090 q^{55} +76.3857 q^{59} +3.84787 q^{61} -32.8984i q^{65} -4.55636 q^{67} -40.3858 q^{71} -123.009i q^{73} +(-14.5815 - 33.8611i) q^{77} +107.545i q^{79} +92.3016 q^{83} +43.4593i q^{85} -1.80053 q^{89} +(-60.1857 + 25.9176i) q^{91} -31.0935 q^{95} -122.204i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 336 q^{25} + 384 q^{43} + 368 q^{49} + 896 q^{67} + 576 q^{91}+O(q^{100}) $ 48 * q - 336 * q^25 + 384 * q^43 + 368 * q^49 + 896 * q^67 + 576 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$5$			3.51431i			0.702861i	0.936214	+	0.351431i	$0.114305\pi$
$5$			3.51431i			0.702861i	−0.936214	+	0.351431i	$0.885695\pi$
$6$		0			0
$7$	6.42922	−	2.76859i	0.918460	−	0.395513i
$7$	6.42922	−	2.76859i	0.918460	−	0.395513i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	5.26675i		−	0.478795i	−0.970922	−	0.239398i	$-0.923050\pi$
$11$		−	5.26675i		−	0.478795i	0.970922	−	0.239398i	$-0.0769499\pi$
$12$		0			0
$13$	−9.36128			−0.720098			−0.360049	−	0.932933i	$-0.617240\pi$
$13$	−9.36128			−0.720098			−0.360049	+	0.932933i	$0.617240\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	12.3664			0.727435			0.363718	−	0.931509i	$-0.381507\pi$
$17$	12.3664			0.727435			0.363718	+	0.931509i	$0.381507\pi$
$18$		0			0
$19$			8.84770i			0.465669i	0.972516	+	0.232834i	$0.0748000\pi$
$19$			8.84770i			0.465669i	−0.972516	+	0.232834i	$0.925200\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.29743			−0.186845			−0.0934223	−	0.995627i	$-0.529781\pi$
$23$	−4.29743			−0.186845			−0.0934223	+	0.995627i	$0.529781\pi$
$24$		0			0
$25$	12.6496			0.505986
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−18.6786			−0.644088			−0.322044	−	0.946725i	$-0.604370\pi$
$29$	−18.6786			−0.644088			−0.322044	+	0.946725i	$0.604370\pi$
$30$		0			0
$31$	52.1525			1.68234			0.841170	−	0.540771i	$-0.181867\pi$
$31$	52.1525			1.68234			0.841170	+	0.540771i	$0.181867\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9.72968	+	22.5943i	0.277991	+	0.645550i
$36$		0			0
$37$		−	45.4695i		−	1.22891i	−0.788954	−	0.614453i	$-0.789377\pi$
$37$		−	45.4695i		−	1.22891i	0.788954	−	0.614453i	$-0.210623\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−42.0916			−1.02662			−0.513312	−	0.858202i	$-0.671582\pi$
$41$	−42.0916			−1.02662			−0.513312	+	0.858202i	$0.671582\pi$
$42$		0			0
$43$	23.2557			0.540830			0.270415	−	0.962744i	$-0.412839\pi$
$43$	23.2557			0.540830			0.270415	+	0.962744i	$0.412839\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			28.5041i			0.606469i	0.952916	+	0.303235i	$0.0980666\pi$
$47$			28.5041i			0.606469i	−0.952916	+	0.303235i	$0.901933\pi$
$48$		0			0
$49$	33.6698	−	35.5998i	0.687139	−	0.726526i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	18.1585			0.342614			0.171307	−	0.985218i	$-0.445201\pi$
$53$	18.1585			0.342614			0.171307	+	0.985218i	$0.445201\pi$
$54$		0			0
$55$	18.5090			0.336527
$56$		0			0
$57$		0			0
$58$		0			0
$59$	76.3857			1.29467			0.647336	−	0.762205i	$-0.275883\pi$
$59$	76.3857			1.29467			0.647336	+	0.762205i	$0.275883\pi$
$60$		0			0
$61$	3.84787			0.0630799			0.0315400	−	0.999502i	$-0.489959\pi$
$61$	3.84787			0.0630799			0.0315400	+	0.999502i	$0.489959\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	32.8984i		−	0.506129i
$66$		0			0
$67$	−4.55636			−0.0680054			−0.0340027	−	0.999422i	$-0.510825\pi$
$67$	−4.55636			−0.0680054			−0.0340027	+	0.999422i	$0.510825\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−40.3858			−0.568814			−0.284407	−	0.958704i	$-0.591797\pi$
$71$	−40.3858			−0.568814			−0.284407	+	0.958704i	$0.591797\pi$
$72$		0			0
$73$		−	123.009i		−	1.68505i	−0.538656	−	0.842526i	$-0.681068\pi$
$73$		−	123.009i		−	1.68505i	0.538656	−	0.842526i	$-0.318932\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−14.5815	−	33.8611i	−0.189370	−	0.439754i
$78$		0			0
$79$			107.545i			1.36133i	0.732596	+	0.680663i	$0.238308\pi$
$79$			107.545i			1.36133i	−0.732596	+	0.680663i	$0.761692\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	92.3016			1.11207			0.556034	−	0.831160i	$-0.312323\pi$
$83$	92.3016			1.11207			0.556034	+	0.831160i	$0.312323\pi$
$84$		0			0
$85$			43.4593i			0.511286i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.80053			−0.0202306			−0.0101153	−	0.999949i	$-0.503220\pi$
$89$	−1.80053			−0.0202306			−0.0101153	+	0.999949i	$0.503220\pi$
$90$		0			0
$91$	−60.1857	+	25.9176i	−0.661382	+	0.284808i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−31.0935			−0.327300
$96$		0			0
$97$		−	122.204i		−	1.25984i	−0.776660	−	0.629919i	$-0.783088\pi$
$97$		−	122.204i		−	1.25984i	0.776660	−	0.629919i	$-0.216912\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			85.2602i			0.844160i	0.906559	+	0.422080i	$0.138700\pi$
$101$			85.2602i			0.844160i	−0.906559	+	0.422080i	$0.861300\pi$
$102$		0			0
$103$	27.4148			0.266164			0.133082	−	0.991105i	$-0.457513\pi$
$103$	27.4148			0.266164			0.133082	+	0.991105i	$0.457513\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			116.898i			1.09250i	0.837622	+	0.546250i	$0.183945\pi$
$107$			116.898i			1.09250i	−0.837622	+	0.546250i	$0.816055\pi$
$108$		0			0
$109$			109.910i			1.00835i	0.863602	+	0.504175i	$0.168203\pi$
$109$			109.910i			1.00835i	−0.863602	+	0.504175i	$0.831797\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	91.9016i		−	0.813288i	−0.913587	−	0.406644i	$-0.866699\pi$
$113$		−	91.9016i		−	0.813288i	0.913587	−	0.406644i	$-0.133301\pi$
$114$		0			0
$115$		−	15.1025i		−	0.131326i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	79.5063	−	34.2375i	0.668120	−	0.287710i
$120$		0			0
$121$	93.2614			0.770755
$122$		0			0
$123$		0			0
$124$		0			0
$125$			132.312i			1.05850i
$126$		0			0
$127$		−	54.4707i		−	0.428903i	−0.976735	−	0.214452i	$-0.931204\pi$
$127$		−	54.4707i		−	0.428903i	0.976735	−	0.214452i	$-0.0687964\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	229.881			1.75481			0.877407	−	0.479746i	$-0.159271\pi$
$131$	229.881			1.75481			0.877407	+	0.479746i	$0.159271\pi$
$132$		0			0
$133$	24.4957	+	56.8838i	0.184178	+	0.427698i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			218.060i			1.59168i	0.605507	+	0.795840i	$0.292970\pi$
$137$			218.060i			1.59168i	−0.605507	+	0.795840i	$0.707030\pi$
$138$		0			0
$139$		−	275.165i		−	1.97960i	−0.142458	−	0.989801i	$-0.545501\pi$
$139$		−	275.165i		−	1.97960i	0.142458	−	0.989801i	$-0.454499\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			49.3035i			0.344780i
$144$		0			0
$145$		−	65.6422i		−	0.452705i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−158.666			−1.06487			−0.532437	−	0.846469i	$-0.678724\pi$
$149$	−158.666			−1.06487			−0.532437	+	0.846469i	$0.678724\pi$
$150$		0			0
$151$			62.5692i			0.414366i	0.978302	+	0.207183i	$0.0664295\pi$
$151$			62.5692i			0.414366i	−0.978302	+	0.207183i	$0.933570\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			183.280i			1.18245i
$156$		0			0
$157$	119.454			0.760853			0.380427	−	0.924811i	$-0.375777\pi$
$157$	119.454			0.760853			0.380427	+	0.924811i	$0.375777\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−27.6291	+	11.8978i	−0.171609	+	0.0738995i
$162$		0			0
$163$	114.091			0.699945			0.349972	−	0.936760i	$-0.386191\pi$
$163$	114.091			0.699945			0.349972	+	0.936760i	$0.386191\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	7.52860i		−	0.0450814i	−0.999746	−	0.0225407i	$-0.992824\pi$
$167$		−	7.52860i		−	0.0450814i	0.999746	−	0.0225407i	$-0.00717554\pi$
$168$		0			0
$169$	−81.3664			−0.481458
$170$		0			0
$171$		0			0
$172$		0			0
$173$			247.124i			1.42846i	0.699910	+	0.714231i	$0.253223\pi$
$173$			247.124i			1.42846i	−0.699910	+	0.714231i	$0.746777\pi$
$174$		0			0
$175$	81.3274	−	35.0217i	0.464728	−	0.200124i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	185.560i		−	1.03665i	−0.855184	−	0.518324i	$-0.826556\pi$
$179$		−	185.560i		−	1.03665i	0.855184	−	0.518324i	$-0.173444\pi$
$180$		0			0
$181$	322.522			1.78189			0.890944	−	0.454113i	$-0.150044\pi$
$181$	322.522			1.78189			0.890944	+	0.454113i	$0.150044\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	159.794			0.863750
$186$		0			0
$187$		−	65.1307i		−	0.348292i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	301.533			1.57871			0.789355	−	0.613937i	$-0.210415\pi$
$191$	301.533			1.57871			0.789355	+	0.613937i	$0.210415\pi$
$192$		0			0
$193$	−11.9190			−0.0617567			−0.0308783	−	0.999523i	$-0.509830\pi$
$193$	−11.9190			−0.0617567			−0.0308783	+	0.999523i	$0.509830\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	149.961			0.761224			0.380612	−	0.924735i	$-0.375713\pi$
$197$	149.961			0.761224			0.380612	+	0.924735i	$0.375713\pi$
$198$		0			0
$199$	17.5356			0.0881185			0.0440593	−	0.999029i	$-0.485971\pi$
$199$	17.5356			0.0881185			0.0440593	+	0.999029i	$0.485971\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−120.089	+	51.7133i	−0.591570	+	0.254745i
$204$		0			0
$205$		−	147.923i		−	0.721574i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	46.5986			0.222960
$210$		0			0
$211$	377.299			1.78815			0.894074	−	0.447919i	$-0.147835\pi$
$211$	377.299			1.78815			0.894074	+	0.447919i	$0.147835\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			81.7276i			0.380128i
$216$		0			0
$217$	335.300	−	144.389i	1.54516	−	0.665388i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−115.765			−0.523825
$222$		0			0
$223$	191.043			0.856695			0.428348	−	0.903614i	$-0.359096\pi$
$223$	191.043			0.856695			0.428348	+	0.903614i	$0.359096\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	54.0537			0.238122			0.119061	−	0.992887i	$-0.462012\pi$
$227$	54.0537			0.238122			0.119061	+	0.992887i	$0.462012\pi$
$228$		0			0
$229$	63.8040			0.278620			0.139310	−	0.990249i	$-0.455512\pi$
$229$	63.8040			0.278620			0.139310	+	0.990249i	$0.455512\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			91.0939i			0.390961i	0.980708	+	0.195480i	$0.0626266\pi$
$233$			91.0939i			0.390961i	−0.980708	+	0.195480i	$0.937373\pi$
$234$		0			0
$235$	−100.172			−0.426264
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−390.181			−1.63256			−0.816279	−	0.577658i	$-0.803967\pi$
$239$	−390.181			−1.63256			−0.816279	+	0.577658i	$0.803967\pi$
$240$		0			0
$241$		−	66.2992i		−	0.275100i	−0.990495	−	0.137550i	$-0.956077\pi$
$241$		−	66.2992i		−	0.275100i	0.990495	−	0.137550i	$-0.0439228\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	125.109	+	118.326i	0.510647	+	0.482963i
$246$		0			0
$247$		−	82.8258i		−	0.335327i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−72.6421			−0.289411			−0.144705	−	0.989475i	$-0.546223\pi$
$251$	−72.6421			−0.289411			−0.144705	+	0.989475i	$0.546223\pi$
$252$		0			0
$253$			22.6335i			0.0894603i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	480.730			1.87055			0.935273	−	0.353927i	$-0.115154\pi$
$257$	480.730			1.87055			0.935273	+	0.353927i	$0.115154\pi$
$258$		0			0
$259$	−125.887	−	292.334i	−0.486048	−	1.12870i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−221.781			−0.843274			−0.421637	−	0.906765i	$-0.638544\pi$
$263$	−221.781			−0.843274			−0.421637	+	0.906765i	$0.638544\pi$
$264$		0			0
$265$			63.8147i			0.240810i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	65.8358i		−	0.244743i	−0.992484	−	0.122371i	$-0.960950\pi$
$269$		−	65.8358i		−	0.244743i	0.992484	−	0.122371i	$-0.0390499\pi$
$270$		0			0
$271$	−63.0008			−0.232475			−0.116238	−	0.993221i	$-0.537083\pi$
$271$	−63.0008			−0.232475			−0.116238	+	0.993221i	$0.537083\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	66.6225i		−	0.242264i
$276$		0			0
$277$			274.997i			0.992769i	0.868103	+	0.496385i	$0.165339\pi$
$277$			274.997i			0.992769i	−0.868103	+	0.496385i	$0.834661\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	148.229i		−	0.527504i	−0.964591	−	0.263752i	$-0.915040\pi$
$281$		−	148.229i		−	0.527504i	0.964591	−	0.263752i	$-0.0849600\pi$
$282$		0			0
$283$		−	265.522i		−	0.938240i	−0.883134	−	0.469120i	$-0.844571\pi$
$283$		−	265.522i		−	0.938240i	0.883134	−	0.469120i	$-0.155429\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−270.616	+	116.534i	−0.942913	+	0.406043i
$288$		0			0
$289$	−136.072			−0.470838
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	521.117i		−	1.77855i	−0.457368	−	0.889277i	$-0.651208\pi$
$293$		−	521.117i		−	1.77855i	0.457368	−	0.889277i	$-0.348792\pi$
$294$		0			0
$295$			268.443i			0.909975i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	40.2294			0.134546
$300$		0			0
$301$	149.516	−	64.3855i	0.496731	−	0.213905i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			13.5226i			0.0443364i
$306$		0			0
$307$			120.087i			0.391162i	0.980688	+	0.195581i	$0.0626592\pi$
$307$			120.087i			0.391162i	−0.980688	+	0.195581i	$0.937341\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			351.784i			1.13114i	0.824701	+	0.565570i	$0.191344\pi$
$311$			351.784i			1.13114i	−0.824701	+	0.565570i	$0.808656\pi$
$312$		0			0
$313$			55.1841i			0.176307i	0.996107	+	0.0881536i	$0.0280966\pi$
$313$			55.1841i			0.176307i	−0.996107	+	0.0881536i	$0.971903\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−410.736			−1.29570			−0.647849	−	0.761769i	$-0.724331\pi$
$317$	−410.736			−1.29570			−0.647849	+	0.761769i	$0.724331\pi$
$318$		0			0
$319$			98.3753i			0.308386i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			109.414i			0.338744i
$324$		0			0
$325$	−118.417			−0.364360
$326$		0			0
$327$		0			0
$328$		0			0
$329$	78.9162	+	183.259i	0.239867	+	0.557018i
$330$		0			0
$331$	284.206			0.858629			0.429314	−	0.903155i	$-0.358755\pi$
$331$	284.206			0.858629			0.429314	+	0.903155i	$0.358755\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	16.0125i		−	0.0477984i
$336$		0			0
$337$	−459.374			−1.36313			−0.681564	−	0.731758i	$-0.738700\pi$
$337$	−459.374			−1.36313			−0.681564	+	0.731758i	$0.738700\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	274.674i		−	0.805496i
$342$		0			0
$343$	117.909	−	322.097i	0.343759	−	0.939058i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	460.926i		−	1.32832i	−0.747592	−	0.664159i	$-0.768790\pi$
$347$		−	460.926i		−	1.32832i	0.747592	−	0.664159i	$-0.231210\pi$
$348$		0			0
$349$	−359.787			−1.03091			−0.515454	−	0.856917i	$-0.672377\pi$
$349$	−359.787			−1.03091			−0.515454	+	0.856917i	$0.672377\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	152.580			0.432238			0.216119	−	0.976367i	$-0.430660\pi$
$353$	152.580			0.432238			0.216119	+	0.976367i	$0.430660\pi$
$354$		0			0
$355$		−	141.928i		−	0.399797i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	405.919			1.13069			0.565347	−	0.824853i	$-0.308742\pi$
$359$	405.919			1.13069			0.565347	+	0.824853i	$0.308742\pi$
$360$		0			0
$361$	282.718			0.783153
$362$		0			0
$363$		0			0
$364$		0			0
$365$	432.291			1.18436
$366$		0			0
$367$	−70.1579			−0.191166			−0.0955830	−	0.995421i	$-0.530472\pi$
$367$	−70.1579			−0.191166			−0.0955830	+	0.995421i	$0.530472\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	116.745	−	50.2736i	0.314677	−	0.135508i
$372$		0			0
$373$		−	330.205i		−	0.885268i	−0.896702	−	0.442634i	$-0.854044\pi$
$373$		−	330.205i		−	0.885268i	0.896702	−	0.442634i	$-0.145956\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	174.855			0.463807
$378$		0			0
$379$	29.3602			0.0774675			0.0387337	−	0.999250i	$-0.487668\pi$
$379$	29.3602			0.0774675			0.0387337	+	0.999250i	$0.487668\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			328.601i			0.857966i	0.903313	+	0.428983i	$0.141128\pi$
$383$			328.601i			0.857966i	−0.903313	+	0.428983i	$0.858872\pi$
$384$		0			0
$385$	118.998	−	51.2438i	0.309086	−	0.133101i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−397.533			−1.02194			−0.510968	−	0.859599i	$-0.670713\pi$
$389$	−397.533			−1.02194			−0.510968	+	0.859599i	$0.670713\pi$
$390$		0			0
$391$	−53.1437			−0.135917
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−377.945			−0.956824
$396$		0			0
$397$	−192.561			−0.485041			−0.242520	−	0.970146i	$-0.577974\pi$
$397$	−192.561			−0.485041			−0.242520	+	0.970146i	$0.577974\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			447.317i			1.11550i	0.830008	+	0.557752i	$0.188336\pi$
$401$			447.317i			1.11550i	−0.830008	+	0.557752i	$0.811664\pi$
$402$		0			0
$403$	−488.215			−1.21145
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−239.476			−0.588394
$408$		0			0
$409$			556.925i			1.36167i	0.732435	+	0.680837i	$0.238384\pi$
$409$			556.925i			1.36167i	−0.732435	+	0.680837i	$0.761616\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	491.100	−	211.481i	1.18911	−	0.512060i
$414$		0			0
$415$			324.376i			0.781629i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−655.843			−1.56526			−0.782628	−	0.622489i	$-0.786121\pi$
$419$	−655.843			−1.56526			−0.782628	+	0.622489i	$0.786121\pi$
$420$		0			0
$421$			358.380i			0.851258i	0.904898	+	0.425629i	$0.139947\pi$
$421$			358.380i			0.851258i	−0.904898	+	0.425629i	$0.860053\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	156.431			0.368072
$426$		0			0
$427$	24.7388	−	10.6532i	0.0579364	−	0.0249489i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	340.972			0.791117			0.395559	−	0.918441i	$-0.370551\pi$
$431$	340.972			0.791117			0.395559	+	0.918441i	$0.370551\pi$
$432$		0			0
$433$		−	213.233i		−	0.492454i	−0.969212	−	0.246227i	$-0.920809\pi$
$433$		−	213.233i		−	0.492454i	0.969212	−	0.246227i	$-0.0791909\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	38.0223i		−	0.0870077i
$438$		0			0
$439$	238.439			0.543140			0.271570	−	0.962419i	$-0.412457\pi$
$439$	238.439			0.543140			0.271570	+	0.962419i	$0.412457\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	112.783i		−	0.254589i	−0.991865	−	0.127295i	$-0.959371\pi$
$443$		−	112.783i		−	0.254589i	0.991865	−	0.127295i	$-0.0406294\pi$
$444$		0			0
$445$		−	6.32760i		−	0.0142193i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	366.262i		−	0.815729i	−0.913043	−	0.407864i	$-0.866274\pi$
$449$		−	366.262i		−	0.815729i	0.913043	−	0.407864i	$-0.133726\pi$
$450$		0			0
$451$			221.686i			0.491543i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−91.0823	−	211.511i	−0.200181	−	0.464860i
$456$		0			0
$457$	−127.312			−0.278582			−0.139291	−	0.990252i	$-0.544482\pi$
$457$	−127.312			−0.278582			−0.139291	+	0.990252i	$0.544482\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			588.619i			1.27683i	0.769692	+	0.638415i	$0.220410\pi$
$461$			588.619i			1.27683i	−0.769692	+	0.638415i	$0.779590\pi$
$462$		0			0
$463$		−	269.281i		−	0.581601i	−0.956784	−	0.290801i	$-0.906078\pi$
$463$		−	269.281i		−	0.581601i	0.956784	−	0.290801i	$-0.0939217\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−357.434			−0.765383			−0.382692	−	0.923876i	$-0.625003\pi$
$467$	−357.434			−0.765383			−0.382692	+	0.923876i	$0.625003\pi$
$468$		0			0
$469$	−29.2939	+	12.6147i	−0.0624603	+	0.0268970i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	122.482i		−	0.258947i
$474$		0			0
$475$			111.920i			0.235622i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	758.388i		−	1.58327i	−0.610992	−	0.791637i	$-0.709229\pi$
$479$		−	758.388i		−	1.58327i	0.610992	−	0.791637i	$-0.290771\pi$
$480$		0			0
$481$			425.653i			0.884933i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	429.464			0.885492
$486$		0			0
$487$			616.289i			1.26548i	0.774364	+	0.632741i	$0.218070\pi$
$487$			616.289i			1.26548i	−0.774364	+	0.632741i	$0.781930\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			574.215i			1.16948i	0.811221	+	0.584740i	$0.198803\pi$
$491$			574.215i			1.16948i	−0.811221	+	0.584740i	$0.801197\pi$
$492$		0			0
$493$	−230.987			−0.468532
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−259.649	+	111.812i	−0.522433	+	0.224973i
$498$		0			0
$499$	613.646			1.22975			0.614875	−	0.788624i	$-0.289206\pi$
$499$	613.646			1.22975			0.614875	+	0.788624i	$0.289206\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	55.3348i		−	0.110010i	−0.998486	−	0.0550048i	$-0.982483\pi$
$503$		−	55.3348i		−	0.110010i	0.998486	−	0.0550048i	$-0.0175174\pi$
$504$		0			0
$505$	−299.630			−0.593328
$506$		0			0
$507$		0			0
$508$		0			0
$509$			310.802i			0.610613i	0.952254	+	0.305307i	$0.0987589\pi$
$509$			310.802i			0.610613i	−0.952254	+	0.305307i	$0.901241\pi$
$510$		0			0
$511$	−340.561	−	790.851i	−0.666460	−	1.54765i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			96.3442i			0.187076i
$516$		0			0
$517$	150.124			0.290375
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−774.806			−1.48715			−0.743576	−	0.668652i	$-0.766872\pi$
$521$	−774.806			−1.48715			−0.743576	+	0.668652i	$0.766872\pi$
$522$		0			0
$523$			440.022i			0.841342i	0.907213	+	0.420671i	$0.138205\pi$
$523$			440.022i			0.841342i	−0.907213	+	0.420671i	$0.861795\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	644.939			1.22379
$528$		0			0
$529$	−510.532			−0.965089
$530$		0			0
$531$		0			0
$532$		0			0
$533$	394.031			0.739270
$534$		0			0
$535$	−410.814			−0.767876
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−187.495	−	177.330i	−0.347857	−	0.328999i
$540$		0			0
$541$		−	1012.28i		−	1.87113i	−0.353159	−	0.935563i	$-0.614893\pi$
$541$		−	1012.28i		−	1.87113i	0.353159	−	0.935563i	$-0.385107\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−386.258			−0.708730
$546$		0			0
$547$	−588.492			−1.07585			−0.537927	−	0.842991i	$-0.680792\pi$
$547$	−588.492			−1.07585			−0.537927	+	0.842991i	$0.680792\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	165.262i		−	0.299932i
$552$		0			0
$553$	297.748	+	691.430i	0.538423	+	1.25032i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1101.14			−1.97691			−0.988454	−	0.151520i	$-0.951583\pi$
$557$	−1101.14			−1.97691			−0.988454	+	0.151520i	$0.951583\pi$
$558$		0			0
$559$	−217.703			−0.389451
$560$		0			0
$561$		0			0
$562$		0			0
$563$	394.840			0.701315			0.350658	−	0.936504i	$-0.385958\pi$
$563$	394.840			0.701315			0.350658	+	0.936504i	$0.385958\pi$
$564$		0			0
$565$	322.970			0.571629
$566$		0			0
$567$		0			0
$568$		0			0
$569$			314.095i			0.552013i	0.961156	+	0.276006i	$0.0890111\pi$
$569$			314.095i			0.552013i	−0.961156	+	0.276006i	$0.910989\pi$
$570$		0			0
$571$	−715.232			−1.25260			−0.626298	−	0.779584i	$-0.715431\pi$
$571$	−715.232			−1.25260			−0.626298	+	0.779584i	$0.715431\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−54.3609			−0.0945407
$576$		0			0
$577$			250.055i			0.433371i	0.976241	+	0.216685i	$0.0695246\pi$
$577$			250.055i			0.433371i	−0.976241	+	0.216685i	$0.930475\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	593.428	−	255.546i	1.02139	−	0.439837i
$582$		0			0
$583$		−	95.6365i		−	0.164042i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	608.274			1.03624			0.518121	−	0.855308i	$-0.326632\pi$
$587$	608.274			1.03624			0.518121	+	0.855308i	$0.326632\pi$
$588$		0			0
$589$			461.430i			0.783413i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1006.48			−1.69727			−0.848634	−	0.528980i	$-0.822575\pi$
$593$	−1006.48			−1.69727			−0.848634	+	0.528980i	$0.822575\pi$
$594$		0			0
$595$	120.321	+	279.410i	0.202220	+	0.469596i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−81.6309			−0.136279			−0.0681393	−	0.997676i	$-0.521706\pi$
$599$	−81.6309			−0.136279			−0.0681393	+	0.997676i	$0.521706\pi$
$600$		0			0
$601$		−	84.9314i		−	0.141317i	−0.997501	−	0.0706584i	$-0.977490\pi$
$601$		−	84.9314i		−	0.141317i	0.997501	−	0.0706584i	$-0.0225100\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			327.749i			0.541734i
$606$		0			0
$607$	−964.122			−1.58834			−0.794170	−	0.607696i	$-0.792094\pi$
$607$	−964.122			−1.58834			−0.794170	+	0.607696i	$0.792094\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	266.835i		−	0.436718i
$612$		0			0
$613$			282.857i			0.461430i	0.973021	+	0.230715i	$0.0741065\pi$
$613$			282.857i			0.461430i	−0.973021	+	0.230715i	$0.925894\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	805.890i		−	1.30614i	−0.757296	−	0.653071i	$-0.773480\pi$
$617$		−	805.890i		−	1.30614i	0.757296	−	0.653071i	$-0.226520\pi$
$618$		0			0
$619$			970.057i			1.56714i	0.621306	+	0.783568i	$0.286602\pi$
$619$			970.057i			1.56714i	−0.621306	+	0.783568i	$0.713398\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.5760	+	4.98492i	−0.0185810	+	0.00800148i
$624$		0			0
$625$	−148.745			−0.237992
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	562.294i		−	0.893949i
$630$		0			0
$631$		−	750.402i		−	1.18923i	−0.804012	−	0.594613i	$-0.797305\pi$
$631$		−	750.402i		−	1.18923i	0.804012	−	0.594613i	$-0.202695\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	191.427			0.301459
$636$		0			0
$637$	−315.192	+	333.260i	−0.494807	+	0.523171i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	1031.35i		−	1.60897i	−0.593974	−	0.804484i	$-0.702442\pi$
$641$		−	1031.35i		−	1.60897i	0.593974	−	0.804484i	$-0.297558\pi$
$642$		0			0
$643$			545.671i			0.848634i	0.905514	+	0.424317i	$0.139486\pi$
$643$			545.671i			0.848634i	−0.905514	+	0.424317i	$0.860514\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	291.352i		−	0.450313i	−0.974323	−	0.225156i	$-0.927711\pi$
$647$		−	291.352i		−	0.450313i	0.974323	−	0.225156i	$-0.0722893\pi$
$648$		0			0
$649$		−	402.304i		−	0.619883i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−383.246			−0.586900			−0.293450	−	0.955974i	$-0.594803\pi$
$653$	−383.246			−0.586900			−0.293450	+	0.955974i	$0.594803\pi$
$654$		0			0
$655$			807.871i			1.23339i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			60.8502i			0.0923372i	0.998934	+	0.0461686i	$0.0147011\pi$
$659$			60.8502i			0.0923372i	−0.998934	+	0.0461686i	$0.985299\pi$
$660$		0			0
$661$	519.891			0.786523			0.393261	−	0.919427i	$-0.371347\pi$
$661$	519.891			0.786523			0.393261	+	0.919427i	$0.371347\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−199.907	+	86.0854i	−0.300612	+	0.129452i
$666$		0			0
$667$	80.2697			0.120344
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	20.2658i		−	0.0302024i
$672$		0			0
$673$	871.592			1.29508			0.647542	−	0.762029i	$-0.275797\pi$
$673$	871.592			1.29508			0.647542	+	0.762029i	$0.275797\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	458.477i		−	0.677219i	−0.940927	−	0.338610i	$-0.890043\pi$
$677$		−	458.477i		−	0.677219i	0.940927	−	0.338610i	$-0.109957\pi$
$678$		0			0
$679$	−338.334	−	785.679i	−0.498283	−	1.15711i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	270.189i		−	0.395591i	−0.980243	−	0.197796i	$-0.936622\pi$
$683$		−	270.189i		−	0.395591i	0.980243	−	0.197796i	$-0.0633783\pi$
$684$		0			0
$685$	−766.330			−1.11873
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−169.987			−0.246716
$690$		0			0
$691$			931.061i			1.34741i	0.739000	+	0.673705i	$0.235298\pi$
$691$			931.061i			1.34741i	−0.739000	+	0.673705i	$0.764702\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	967.013			1.39139
$696$		0			0
$697$	−520.521			−0.746802
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−421.119			−0.600740			−0.300370	−	0.953823i	$-0.597110\pi$
$701$	−421.119			−0.600740			−0.300370	+	0.953823i	$0.597110\pi$
$702$		0			0
$703$	402.301			0.572263
$704$		0			0
$705$		0			0
$706$		0			0
$707$	236.051	+	548.157i	0.333877	+	0.775328i
$708$		0			0
$709$		−	1227.26i		−	1.73097i	−0.500931	−	0.865487i	$-0.667009\pi$
$709$		−	1227.26i		−	1.73097i	0.500931	−	0.865487i	$-0.332991\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−224.122			−0.314336
$714$		0			0
$715$	−173.268			−0.242332
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1202.04i		−	1.67182i	−0.548869	−	0.835908i	$-0.684941\pi$
$719$		−	1202.04i		−	1.67182i	0.548869	−	0.835908i	$-0.315059\pi$
$720$		0			0
$721$	176.256	−	75.9005i	0.244461	−	0.105271i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−236.277			−0.325900
$726$		0			0
$727$	−725.480			−0.997910			−0.498955	−	0.866628i	$-0.666283\pi$
$727$	−725.480			−0.997910			−0.498955	+	0.866628i	$0.666283\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	287.589			0.393418
$732$		0			0
$733$	−1032.80			−1.40900			−0.704499	−	0.709705i	$-0.748828\pi$
$733$	−1032.80			−1.40900			−0.704499	+	0.709705i	$0.748828\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			23.9972i			0.0325607i
$738$		0			0
$739$	−1062.50			−1.43775			−0.718877	−	0.695137i	$-0.755344\pi$
$739$	−1062.50			−1.43775			−0.718877	+	0.695137i	$0.755344\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−684.488			−0.921249			−0.460625	−	0.887595i	$-0.652375\pi$
$743$	−684.488			−0.921249			−0.460625	+	0.887595i	$0.652375\pi$
$744$		0			0
$745$		−	557.602i		−	0.748459i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	323.642	+	751.560i	0.432098	+	1.00342i
$750$		0			0
$751$		−	565.603i		−	0.753133i	−0.926390	−	0.376567i	$-0.877105\pi$
$751$		−	565.603i		−	0.753133i	0.926390	−	0.376567i	$-0.122895\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−219.887			−0.291242
$756$		0			0
$757$			383.583i			0.506714i	0.967373	+	0.253357i	$0.0815348\pi$
$757$			383.583i			0.506714i	−0.967373	+	0.253357i	$0.918465\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1312.55			−1.72476			−0.862382	−	0.506258i	$-0.831028\pi$
$761$	−1312.55			−1.72476			−0.862382	+	0.506258i	$0.831028\pi$
$762$		0			0
$763$	304.296	+	706.636i	0.398816	+	0.926129i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−715.068			−0.932292
$768$		0			0
$769$			1180.37i			1.53494i	0.641086	+	0.767469i	$0.278484\pi$
$769$			1180.37i			1.53494i	−0.641086	+	0.767469i	$0.721516\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1269.43i		−	1.64221i	−0.570780	−	0.821103i	$-0.693359\pi$
$773$		−	1269.43i		−	1.64221i	0.570780	−	0.821103i	$-0.306641\pi$
$774$		0			0
$775$	659.711			0.851240
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	372.414i		−	0.478066i
$780$		0			0
$781$			212.702i			0.272345i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			419.798i			0.534774i
$786$		0			0
$787$		−	1159.82i		−	1.47372i	−0.676044	−	0.736861i	$-0.736307\pi$
$787$		−	1159.82i		−	1.47372i	0.676044	−	0.736861i	$-0.263693\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−254.438	−	590.856i	−0.321666	−	0.746973i
$792$		0			0
$793$	−36.0210			−0.0454237
$794$		0			0
$795$		0			0
$796$		0			0
$797$			478.221i			0.600026i	0.953935	+	0.300013i	$0.0969910\pi$
$797$			478.221i			0.600026i	−0.953935	+	0.300013i	$0.903009\pi$
$798$		0			0
$799$			352.493i			0.441167i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−647.856			−0.806795
$804$		0			0
$805$	−41.8126	−	97.0971i	−0.0519411	−	0.120618i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1061.21i		−	1.31176i	−0.754865	−	0.655881i	$-0.772297\pi$
$809$		−	1061.21i		−	1.31176i	0.754865	−	0.655881i	$-0.227703\pi$
$810$		0			0
$811$			738.364i			0.910437i	0.890380	+	0.455218i	$0.150439\pi$
$811$			738.364i			0.910437i	−0.890380	+	0.455218i	$0.849561\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			400.951i			0.491964i
$816$		0			0
$817$			205.759i			0.251847i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−448.627			−0.546440			−0.273220	−	0.961952i	$-0.588089\pi$
$821$	−448.627			−0.546440			−0.273220	+	0.961952i	$0.588089\pi$
$822$		0			0
$823$			1327.80i			1.61337i	0.590980	+	0.806686i	$0.298741\pi$
$823$			1327.80i			1.61337i	−0.590980	+	0.806686i	$0.701259\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1315.08i		−	1.59018i	−0.606493	−	0.795089i	$-0.707424\pi$
$827$		−	1315.08i		−	1.59018i	0.606493	−	0.795089i	$-0.292576\pi$
$828$		0			0
$829$	−1191.63			−1.43744			−0.718718	−	0.695302i	$-0.755271\pi$
$829$	−1191.63			−1.43744			−0.718718	+	0.695302i	$0.755271\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	416.374	−	440.241i	0.499849	−	0.528501i
$834$		0			0
$835$	26.4578			0.0316860
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	192.974i		−	0.230005i	−0.993365	−	0.115002i	$-0.963312\pi$
$839$		−	192.974i		−	0.230005i	0.993365	−	0.115002i	$-0.0366876\pi$
$840$		0			0
$841$	−492.111			−0.585150
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	285.947i		−	0.338398i
$846$		0			0
$847$	599.598	−	258.203i	0.707908	−	0.304844i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			195.402i			0.229614i
$852$		0			0
$853$	−236.364			−0.277097			−0.138549	−	0.990356i	$-0.544244\pi$
$853$	−236.364			−0.277097			−0.138549	+	0.990356i	$0.544244\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	464.302			0.541775			0.270888	−	0.962611i	$-0.412683\pi$
$857$	464.302			0.541775			0.270888	+	0.962611i	$0.412683\pi$
$858$		0			0
$859$		−	567.819i		−	0.661023i	−0.943802	−	0.330511i	$-0.892779\pi$
$859$		−	567.819i		−	0.661023i	0.943802	−	0.330511i	$-0.107221\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	610.907			0.707888			0.353944	−	0.935267i	$-0.384840\pi$
$863$	610.907			0.707888			0.353944	+	0.935267i	$0.384840\pi$
$864$		0			0
$865$	−868.469			−1.00401
$866$		0			0
$867$		0			0
$868$		0			0
$869$	566.411			0.651797
$870$		0			0
$871$	42.6534			0.0489706
$872$		0			0
$873$		0			0
$874$		0			0
$875$	366.319	+	850.666i	0.418651	+	0.972190i
$876$		0			0
$877$		−	780.619i		−	0.890101i	−0.895505	−	0.445050i	$-0.853186\pi$
$877$		−	780.619i		−	0.890101i	0.895505	−	0.445050i	$-0.146814\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−539.433			−0.612296			−0.306148	−	0.951984i	$-0.599040\pi$
$881$	−539.433			−0.612296			−0.306148	+	0.951984i	$0.599040\pi$
$882$		0			0
$883$	−797.997			−0.903734			−0.451867	−	0.892085i	$-0.649242\pi$
$883$	−797.997			−0.903734			−0.451867	+	0.892085i	$0.649242\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	435.503i		−	0.490985i	−0.969399	−	0.245492i	$-0.921050\pi$
$887$		−	435.503i		−	0.490985i	0.969399	−	0.245492i	$-0.0789496\pi$
$888$		0			0
$889$	−150.807	−	350.204i	−0.169637	−	0.393931i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−252.196			−0.282414
$894$		0			0
$895$	652.115			0.728620
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−974.134			−1.08358
$900$		0			0
$901$	224.556			0.249230
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1133.44i			1.25242i
$906$		0			0
$907$	834.673			0.920257			0.460128	−	0.887852i	$-0.347803\pi$
$907$	834.673			0.920257			0.460128	+	0.887852i	$0.347803\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−543.809			−0.596937			−0.298468	−	0.954420i	$-0.596476\pi$
$911$	−543.809			−0.596937			−0.298468	+	0.954420i	$0.596476\pi$
$912$		0			0
$913$		−	486.129i		−	0.532453i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1477.95	−	636.446i	1.61173	−	0.694052i
$918$		0			0
$919$			257.470i			0.280163i	0.990140	+	0.140082i	$0.0447365\pi$
$919$			257.470i			0.280163i	−0.990140	+	0.140082i	$0.955264\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	378.063			0.409602
$924$		0			0
$925$		−	575.173i		−	0.621809i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	943.723			1.01585			0.507924	−	0.861402i	$-0.330413\pi$
$929$	943.723			1.01585			0.507924	+	0.861402i	$0.330413\pi$
$930$		0			0
$931$	314.976	+	297.900i	0.338321	+	0.319979i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	228.889			0.244801
$936$		0			0
$937$			468.548i			0.500052i	0.968239	+	0.250026i	$0.0804391\pi$
$937$			468.548i			0.500052i	−0.968239	+	0.250026i	$0.919561\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1430.10i			1.51977i	0.650058	+	0.759885i	$0.274745\pi$
$941$			1430.10i			1.51977i	−0.650058	+	0.759885i	$0.725255\pi$
$942$		0			0
$943$	180.885			0.191819
$944$		0			0
$945$		0			0
$946$		0			0
$947$			782.150i			0.825924i	0.910748	+	0.412962i	$0.135506\pi$
$947$			782.150i			0.825924i	−0.910748	+	0.412962i	$0.864494\pi$
$948$		0			0
$949$			1151.52i			1.21340i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	200.954i		−	0.210864i	−0.994427	−	0.105432i	$-0.966377\pi$
$953$		−	200.954i		−	0.210864i	0.994427	−	0.105432i	$-0.0336226\pi$
$954$		0			0
$955$			1059.68i			1.10961i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	603.720	+	1401.96i	0.629530	+	1.46189i
$960$		0			0
$961$	1758.89			1.83027
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	41.8872i		−	0.0434064i
$966$		0			0
$967$		−	297.101i		−	0.307240i	−0.988130	−	0.153620i	$-0.950907\pi$
$967$		−	297.101i		−	0.307240i	0.988130	−	0.153620i	$-0.0490931\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−534.138			−0.550090			−0.275045	−	0.961431i	$-0.588693\pi$
$971$	−534.138			−0.550090			−0.275045	+	0.961431i	$0.588693\pi$
$972$		0			0
$973$	−761.819	−	1769.09i	−0.782959	−	1.81819i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			631.207i			0.646066i	0.946388	+	0.323033i	$0.104702\pi$
$977$			631.207i			0.646066i	−0.946388	+	0.323033i	$0.895298\pi$
$978$		0			0
$979$			9.48292i			0.00968633i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1569.91i			1.59706i	0.601955	+	0.798530i	$0.294389\pi$
$983$			1569.91i			1.59706i	−0.601955	+	0.798530i	$0.705611\pi$
$984$		0			0
$985$			527.009i			0.535035i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−99.9395			−0.101051
$990$		0			0
$991$			890.354i			0.898440i	0.893421	+	0.449220i	$0.148298\pi$
$991$			890.354i			0.898440i	−0.893421	+	0.449220i	$0.851702\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			61.6254i			0.0619351i
$996$		0			0
$997$	918.467			0.921230			0.460615	−	0.887600i	$-0.347629\pi$
$997$	918.467			0.921230			0.460615	+	0.887600i	$0.347629\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.e.c.1007.28		48
3.2	odd	2	inner	2016.3.e.c.1007.17		48
4.3	odd	2		504.3.e.c.251.11	yes	48
7.6	odd	2	inner	2016.3.e.c.1007.29		48
8.3	odd	2	inner	2016.3.e.c.1007.15		48
8.5	even	2		504.3.e.c.251.39	yes	48
12.11	even	2		504.3.e.c.251.38	yes	48
21.20	even	2	inner	2016.3.e.c.1007.16		48
24.5	odd	2		504.3.e.c.251.10	yes	48
24.11	even	2	inner	2016.3.e.c.1007.30		48
28.27	even	2		504.3.e.c.251.12	yes	48
56.13	odd	2		504.3.e.c.251.40	yes	48
56.27	even	2	inner	2016.3.e.c.1007.18		48
84.83	odd	2		504.3.e.c.251.37	yes	48
168.83	odd	2	inner	2016.3.e.c.1007.27		48
168.125	even	2		504.3.e.c.251.9	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.3.e.c.251.9	✓	48	168.125	even	2
504.3.e.c.251.10	yes	48	24.5	odd	2
504.3.e.c.251.11	yes	48	4.3	odd	2
504.3.e.c.251.12	yes	48	28.27	even	2
504.3.e.c.251.37	yes	48	84.83	odd	2
504.3.e.c.251.38	yes	48	12.11	even	2
504.3.e.c.251.39	yes	48	8.5	even	2
504.3.e.c.251.40	yes	48	56.13	odd	2
2016.3.e.c.1007.15		48	8.3	odd	2	inner
2016.3.e.c.1007.16		48	21.20	even	2	inner
2016.3.e.c.1007.17		48	3.2	odd	2	inner
2016.3.e.c.1007.18		48	56.27	even	2	inner
2016.3.e.c.1007.27		48	168.83	odd	2	inner
2016.3.e.c.1007.28		48	1.1	even	1	trivial
2016.3.e.c.1007.29		48	7.6	odd	2	inner
2016.3.e.c.1007.30		48	24.11	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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2016.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.51431i q^{5} +(6.42922 - 2.76859i) q^{7} -5.26675i q^{11} -9.36128 q^{13} +12.3664 q^{17} +8.84770i q^{19} -4.29743 q^{23} +12.6496 q^{25} -18.6786 q^{29} +52.1525 q^{31} +(9.72968 + 22.5943i) q^{35} -45.4695i q^{37} -42.0916 q^{41} +23.2557 q^{43} +28.5041i q^{47} +(33.6698 - 35.5998i) q^{49} +18.1585 q^{53} +18.5090 q^{55} +76.3857 q^{59} +3.84787 q^{61} -32.8984i q^{65} -4.55636 q^{67} -40.3858 q^{71} -123.009i q^{73} +(-14.5815 - 33.8611i) q^{77} +107.545i q^{79} +92.3016 q^{83} +43.4593i q^{85} -1.80053 q^{89} +(-60.1857 + 25.9176i) q^{91} -31.0935 q^{95} -122.204i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 336 q^{25} + 384 q^{43} + 368 q^{49} + 896 q^{67} + 576 q^{91}+O(q^{100}) \) 48 * q - 336 * q^25 + 384 * q^43 + 368 * q^49 + 896 * q^67 + 576 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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