LMFDB - Embedded newform 2016.2.k.b.1889.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(1889,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2016.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.k (of order $2$, degree $1$, 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:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.101415451701035401216.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 18x^{12} + 145x^{8} - 72x^{4} + 16 $ x^16 - 18x^12 + 145x^8 - 72*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.6
Root			$1.81768 - 0.332046i$ of defining polynomial
Character	$\chi$	$=$	2016.1889
Dual form			2016.2.k.b.1889.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.841723 q^{5} +(-1.16372 + 2.37608i) q^{7} +O(q^{10})$	$q-0.841723 q^{5} +(-1.16372 + 2.37608i) q^{7} +3.64575i q^{11} -5.53019i q^{13} -0.841723 q^{17} +3.06871i q^{19} -3.64575i q^{23} -4.29150 q^{25} +8.89753i q^{29} -7.82087i q^{31} +(0.979531 - 2.00000i) q^{35} -3.29150 q^{37} -8.66259 q^{41} +2.32744 q^{43} +9.10132 q^{47} +(-4.29150 - 5.53019i) q^{49} -4.24264i q^{53} -3.06871i q^{55} -11.0604 q^{59} -9.10132i q^{61} +4.65489i q^{65} -8.48528 q^{67} +0.354249i q^{71} -14.6315i q^{73} +(-8.66259 - 4.24264i) q^{77} +8.48528 q^{79} -9.10132 q^{83} +0.708497 q^{85} -6.97915 q^{89} +(13.1402 + 6.43560i) q^{91} -2.58301i q^{95} -3.57113i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{25} + 32 q^{37} + 16 q^{49} + 96 q^{85}+O(q^{100}) $ 16 * q + 16 * q^25 + 32 * q^37 + 16 * q^49 + 96 * q^85

Display 100 coefficients 10 coefficients

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
$5$	−0.841723			−0.376430			−0.188215	−	0.982128i	$-0.560270\pi$
$5$	−0.841723			−0.376430			−0.188215	+	0.982128i	$0.560270\pi$
$6$		0			0
$7$	−1.16372	+	2.37608i	−0.439846	+	0.898073i
$7$	−1.16372	+	2.37608i	−0.439846	+	0.898073i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.64575i			1.09924i	0.835416	+	0.549618i	$0.185227\pi$
$11$			3.64575i			1.09924i	−0.835416	+	0.549618i	$0.814773\pi$
$12$		0			0
$13$		−	5.53019i		−	1.53380i	−0.641767	−	0.766899i	$-0.721799\pi$
$13$		−	5.53019i		−	1.53380i	0.641767	−	0.766899i	$-0.278201\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.841723			−0.204148			−0.102074	−	0.994777i	$-0.532548\pi$
$17$	−0.841723			−0.204148			−0.102074	+	0.994777i	$0.532548\pi$
$18$		0			0
$19$			3.06871i			0.704011i	0.935998	+	0.352005i	$0.114500\pi$
$19$			3.06871i			0.704011i	−0.935998	+	0.352005i	$0.885500\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.64575i		−	0.760192i	−0.924947	−	0.380096i	$-0.875891\pi$
$23$		−	3.64575i		−	0.760192i	0.924947	−	0.380096i	$-0.124109\pi$
$24$		0			0
$25$	−4.29150			−0.858301
$26$		0			0
$27$		0			0
$28$		0			0
$29$			8.89753i			1.65223i	0.563502	+	0.826115i	$0.309454\pi$
$29$			8.89753i			1.65223i	−0.563502	+	0.826115i	$0.690546\pi$
$30$		0			0
$31$		−	7.82087i		−	1.40467i	−0.711847	−	0.702335i	$-0.752141\pi$
$31$		−	7.82087i		−	1.40467i	0.711847	−	0.702335i	$-0.247859\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.979531	−	2.00000i	0.165571	−	0.338062i
$36$		0			0
$37$	−3.29150			−0.541120			−0.270560	−	0.962703i	$-0.587209\pi$
$37$	−3.29150			−0.541120			−0.270560	+	0.962703i	$0.587209\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.66259			−1.35287			−0.676435	−	0.736502i	$-0.736476\pi$
$41$	−8.66259			−1.35287			−0.676435	+	0.736502i	$0.736476\pi$
$42$		0			0
$43$	2.32744			0.354932			0.177466	−	0.984127i	$-0.443210\pi$
$43$	2.32744			0.354932			0.177466	+	0.984127i	$0.443210\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.10132			1.32756			0.663782	−	0.747926i	$-0.268950\pi$
$47$	9.10132			1.32756			0.663782	+	0.747926i	$0.268950\pi$
$48$		0			0
$49$	−4.29150	−	5.53019i	−0.613072	−	0.790027i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4.24264i		−	0.582772i	−0.956606	−	0.291386i	$-0.905884\pi$
$53$		−	4.24264i		−	0.582772i	0.956606	−	0.291386i	$-0.0941163\pi$
$54$		0			0
$55$		−	3.06871i		−	0.413785i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.0604			−1.43994			−0.719969	−	0.694006i	$-0.755844\pi$
$59$	−11.0604			−1.43994			−0.719969	+	0.694006i	$0.755844\pi$
$60$		0			0
$61$		−	9.10132i		−	1.16530i	−0.812722	−	0.582652i	$-0.802015\pi$
$61$		−	9.10132i		−	1.16530i	0.812722	−	0.582652i	$-0.197985\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			4.65489i			0.577368i
$66$		0			0
$67$	−8.48528			−1.03664			−0.518321	−	0.855186i	$-0.673443\pi$
$67$	−8.48528			−1.03664			−0.518321	+	0.855186i	$0.673443\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			0.354249i			0.0420416i	0.999779	+	0.0210208i	$0.00669162\pi$
$71$			0.354249i			0.0420416i	−0.999779	+	0.0210208i	$0.993308\pi$
$72$		0			0
$73$		−	14.6315i		−	1.71249i	−0.516571	−	0.856244i	$-0.672792\pi$
$73$		−	14.6315i		−	1.71249i	0.516571	−	0.856244i	$-0.327208\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−8.66259	−	4.24264i	−0.987194	−	0.483494i
$78$		0			0
$79$	8.48528			0.954669			0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528			0.954669			0.477334	+	0.878722i	$0.341603\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.10132			−0.998999			−0.499500	−	0.866314i	$-0.666483\pi$
$83$	−9.10132			−0.998999			−0.499500	+	0.866314i	$0.666483\pi$
$84$		0			0
$85$	0.708497			0.0768473
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.97915			−0.739788			−0.369894	−	0.929074i	$-0.620606\pi$
$89$	−6.97915			−0.739788			−0.369894	+	0.929074i	$0.620606\pi$
$90$		0			0
$91$	13.1402	+	6.43560i	1.37746	+	0.674635i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	2.58301i		−	0.265011i
$96$		0			0
$97$		−	3.57113i		−	0.362593i	−0.983428	−	0.181297i	$-0.941971\pi$
$97$		−	3.57113i		−	0.362593i	0.983428	−	0.181297i	$-0.0580294\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−14.8000			−1.47266			−0.736328	−	0.676624i	$-0.763442\pi$
$101$	−14.8000			−1.47266			−0.736328	+	0.676624i	$0.763442\pi$
$102$		0			0
$103$		−	5.05034i		−	0.497625i	−0.968552	−	0.248812i	$-0.919960\pi$
$103$		−	5.05034i		−	0.497625i	0.968552	−	0.248812i	$-0.0800402\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			6.22876i			0.602157i	0.953599	+	0.301078i	$0.0973466\pi$
$107$			6.22876i			0.602157i	−0.953599	+	0.301078i	$0.902653\pi$
$108$		0			0
$109$	−10.5830			−1.01367			−0.506834	−	0.862044i	$-0.669184\pi$
$109$	−10.5830			−1.01367			−0.506834	+	0.862044i	$0.669184\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			0.412247i			0.0387809i	0.999812	+	0.0193905i	$0.00617256\pi$
$113$			0.412247i			0.0387809i	−0.999812	+	0.0193905i	$0.993827\pi$
$114$		0			0
$115$			3.06871i			0.286159i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.979531	−	2.00000i	0.0897935	−	0.183340i
$120$		0			0
$121$	−2.29150			−0.208318
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.82087			0.699520
$126$		0			0
$127$	−17.7951			−1.57906			−0.789528	−	0.613715i	$-0.789675\pi$
$127$	−17.7951			−1.57906			−0.789528	+	0.613715i	$0.789675\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.95906			−0.171164			−0.0855821	−	0.996331i	$-0.527275\pi$
$131$	−1.95906			−0.171164			−0.0855821	+	0.996331i	$0.527275\pi$
$132$		0			0
$133$	−7.29150	−	3.57113i	−0.632253	−	0.309656i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			8.89753i			0.760167i	0.924952	+	0.380084i	$0.124105\pi$
$137$			8.89753i			0.760167i	−0.924952	+	0.380084i	$0.875895\pi$
$138$		0			0
$139$			12.8712i			1.09172i	0.837876	+	0.545861i	$0.183797\pi$
$139$			12.8712i			1.09172i	−0.837876	+	0.545861i	$0.816203\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	20.1617			1.68601
$144$		0			0
$145$		−	7.48925i		−	0.621949i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			8.89753i			0.728914i	0.931220	+	0.364457i	$0.118745\pi$
$149$			8.89753i			0.728914i	−0.931220	+	0.364457i	$0.881255\pi$
$150$		0			0
$151$	−19.2980			−1.57045			−0.785225	−	0.619211i	$-0.787453\pi$
$151$	−19.2980			−1.57045			−0.785225	+	0.619211i	$0.787453\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			6.58301i			0.528760i
$156$		0			0
$157$		−	1.95906i		−	0.156350i	−0.996940	−	0.0781751i	$-0.975091\pi$
$157$		−	1.95906i		−	0.156350i	0.996940	−	0.0781751i	$-0.0249093\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.66259	+	4.24264i	0.682708	+	0.334367i
$162$		0			0
$163$	8.48528			0.664619			0.332309	−	0.943170i	$-0.392172\pi$
$163$	8.48528			0.664619			0.332309	+	0.943170i	$0.392172\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.0194			1.00747			0.503737	−	0.863857i	$-0.331958\pi$
$167$	13.0194			1.00747			0.503737	+	0.863857i	$0.331958\pi$
$168$		0			0
$169$	−17.5830			−1.35254
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.0295			−0.914585			−0.457292	−	0.889316i	$-0.651181\pi$
$173$	−12.0295			−0.914585			−0.457292	+	0.889316i	$0.651181\pi$
$174$		0			0
$175$	4.99412	−	10.1969i	0.377520	−	0.770817i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			18.9373i			1.41544i	0.706495	+	0.707718i	$0.250275\pi$
$179$			18.9373i			1.41544i	−0.706495	+	0.707718i	$0.749725\pi$
$180$		0			0
$181$		−	5.53019i		−	0.411056i	−0.978651	−	0.205528i	$-0.934109\pi$
$181$		−	5.53019i		−	0.411056i	0.978651	−	0.205528i	$-0.0658911\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.77053			0.203694
$186$		0			0
$187$		−	3.06871i		−	0.224406i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	25.5203i		−	1.84658i	−0.384102	−	0.923291i	$-0.625489\pi$
$191$		−	25.5203i		−	1.84658i	0.384102	−	0.923291i	$-0.374511\pi$
$192$		0			0
$193$	15.2915			1.10071			0.550353	−	0.834932i	$-0.314493\pi$
$193$	15.2915			1.10071			0.550353	+	0.834932i	$0.314493\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	5.06713i		−	0.361018i	−0.983573	−	0.180509i	$-0.942225\pi$
$197$		−	5.06713i		−	0.361018i	0.983573	−	0.180509i	$-0.0577746\pi$
$198$		0			0
$199$			7.52269i			0.533269i	0.963798	+	0.266635i	$0.0859117\pi$
$199$			7.52269i			0.533269i	−0.963798	+	0.266635i	$0.914088\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−21.1412	−	10.3542i	−1.48382	−	0.726726i
$204$		0			0
$205$	7.29150			0.509261
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−11.1878			−0.773874
$210$		0			0
$211$	6.98233			0.480684			0.240342	−	0.970688i	$-0.422740\pi$
$211$	6.98233			0.480684			0.240342	+	0.970688i	$0.422740\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.95906			−0.133607
$216$		0			0
$217$	18.5830	+	9.10132i	1.26150	+	0.617838i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			4.65489i			0.313122i
$222$		0			0
$223$		−	4.75216i		−	0.318228i	−0.987260	−	0.159114i	$-0.949136\pi$
$223$		−	4.75216i		−	0.318228i	0.987260	−	0.159114i	$-0.0508637\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.0604			0.734103			0.367052	−	0.930201i	$-0.380367\pi$
$227$	11.0604			0.734103			0.367052	+	0.930201i	$0.380367\pi$
$228$		0			0
$229$		−	5.53019i		−	0.365445i	−0.983165	−	0.182723i	$-0.941509\pi$
$229$		−	5.53019i		−	0.365445i	0.983165	−	0.182723i	$-0.0584910\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	13.5524i		−	0.887848i	−0.896064	−	0.443924i	$-0.853586\pi$
$233$		−	13.5524i		−	0.887848i	0.896064	−	0.443924i	$-0.146414\pi$
$234$		0			0
$235$	−7.66079			−0.499735
$236$		0			0
$237$		0			0
$238$		0			0
$239$			5.06275i			0.327482i	0.986503	+	0.163741i	$0.0523561\pi$
$239$			5.06275i			0.327482i	−0.986503	+	0.163741i	$0.947644\pi$
$240$		0			0
$241$			7.48925i			0.482425i	0.970472	+	0.241213i	$0.0775451\pi$
$241$			7.48925i			0.482425i	−0.970472	+	0.241213i	$0.922455\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	3.61226	+	4.65489i	0.230779	+	0.297390i
$246$		0			0
$247$	16.9706			1.07981
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−20.1617			−1.27260			−0.636298	−	0.771444i	$-0.719535\pi$
$251$	−20.1617			−1.27260			−0.636298	+	0.771444i	$0.719535\pi$
$252$		0			0
$253$	13.2915			0.835630
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.0295			0.750379			0.375189	−	0.926948i	$-0.377578\pi$
$257$	12.0295			0.750379			0.375189	+	0.926948i	$0.377578\pi$
$258$		0			0
$259$	3.83039	−	7.82087i	0.238009	−	0.485965i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			30.9373i			1.90767i	0.300327	+	0.953836i	$0.402904\pi$
$263$			30.9373i			1.90767i	−0.300327	+	0.953836i	$0.597096\pi$
$264$		0			0
$265$			3.57113i			0.219373i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	29.3547			1.78979			0.894893	−	0.446281i	$-0.147252\pi$
$269$	29.3547			1.78979			0.894893	+	0.446281i	$0.147252\pi$
$270$		0			0
$271$		−	23.4626i		−	1.42525i	−0.701544	−	0.712626i	$-0.747506\pi$
$271$		−	23.4626i		−	1.42525i	0.701544	−	0.712626i	$-0.252494\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	15.6458i		−	0.943474i
$276$		0			0
$277$	3.87451			0.232797			0.116398	−	0.993203i	$-0.462865\pi$
$277$	3.87451			0.232797			0.116398	+	0.993203i	$0.462865\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	25.8681i		−	1.54316i	−0.636132	−	0.771580i	$-0.719467\pi$
$281$		−	25.8681i		−	1.54316i	0.636132	−	0.771580i	$-0.280533\pi$
$282$		0			0
$283$			25.4442i			1.51250i	0.654281	+	0.756251i	$0.272971\pi$
$283$			25.4442i			1.51250i	−0.654281	+	0.756251i	$0.727029\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.0808	−	20.5830i	0.595054	−	1.21498i
$288$		0			0
$289$	−16.2915			−0.958324
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6.97915			−0.407726			−0.203863	−	0.978999i	$-0.565350\pi$
$293$	−6.97915			−0.407726			−0.203863	+	0.978999i	$0.565350\pi$
$294$		0			0
$295$	9.30978			0.542036
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−20.1617			−1.16598
$300$		0			0
$301$	−2.70850	+	5.53019i	−0.156115	+	0.318755i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			7.66079i			0.438655i
$306$		0			0
$307$			3.06871i			0.175141i	0.996158	+	0.0875703i	$0.0279103\pi$
$307$			3.06871i			0.175141i	−0.996158	+	0.0875703i	$0.972090\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.0604			−0.627177			−0.313588	−	0.949559i	$-0.601531\pi$
$311$	−11.0604			−0.627177			−0.313588	+	0.949559i	$0.601531\pi$
$312$		0			0
$313$			22.1208i			1.25034i	0.780489	+	0.625170i	$0.214970\pi$
$313$			22.1208i			1.25034i	−0.780489	+	0.625170i	$0.785030\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	8.89753i		−	0.499735i	−0.968280	−	0.249867i	$-0.919613\pi$
$317$		−	8.89753i		−	0.499735i	0.968280	−	0.249867i	$-0.0803871\pi$
$318$		0			0
$319$	−32.4382			−1.81619
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2.58301i		−	0.143722i
$324$		0			0
$325$			23.7328i			1.31646i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−10.5914	+	21.6255i	−0.583923	+	1.19225i
$330$		0			0
$331$	19.2980			1.06071			0.530357	−	0.847774i	$-0.322058\pi$
$331$	19.2980			1.06071			0.530357	+	0.847774i	$0.322058\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.14226			0.390223
$336$		0			0
$337$	−9.41699			−0.512976			−0.256488	−	0.966547i	$-0.582565\pi$
$337$	−9.41699			−0.512976			−0.256488	+	0.966547i	$0.582565\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	28.5129			1.54406
$342$		0			0
$343$	18.1343	−	3.76135i	0.979159	−	0.203094i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			18.9373i			1.01661i	0.861179	+	0.508303i	$0.169727\pi$
$347$			18.9373i			1.01661i	−0.861179	+	0.508303i	$0.830273\pi$
$348$		0			0
$349$			20.1617i			1.07923i	0.841912	+	0.539615i	$0.181430\pi$
$349$			20.1617i			1.07923i	−0.841912	+	0.539615i	$0.818570\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−22.6209			−1.20399			−0.601994	−	0.798500i	$-0.705627\pi$
$353$	−22.6209			−1.20399			−0.601994	+	0.798500i	$0.705627\pi$
$354$		0			0
$355$		−	0.298179i		−	0.0158257i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	1.77124i		−	0.0934827i	−0.998907	−	0.0467413i	$-0.985116\pi$
$359$		−	1.77124i		−	0.0934827i	0.998907	−	0.0467413i	$-0.0148836\pi$
$360$		0			0
$361$	9.58301			0.504369
$362$		0			0
$363$		0			0
$364$		0			0
$365$			12.3157i			0.644632i
$366$		0			0
$367$		−	4.75216i		−	0.248061i	−0.992278	−	0.124030i	$-0.960418\pi$
$367$		−	4.75216i		−	0.248061i	0.992278	−	0.124030i	$-0.0395820\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.0808	+	4.93725i	0.523372	+	0.256329i
$372$		0			0
$373$	8.58301			0.444411			0.222206	−	0.975000i	$-0.428674\pi$
$373$	8.58301			0.444411			0.222206	+	0.975000i	$0.428674\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	49.2050			2.53419
$378$		0			0
$379$	−19.2980			−0.991272			−0.495636	−	0.868530i	$-0.665065\pi$
$379$	−19.2980			−0.991272			−0.495636	+	0.868530i	$0.665065\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	24.0798			1.23042			0.615211	−	0.788363i	$-0.289071\pi$
$383$	24.0798			1.23042			0.615211	+	0.788363i	$0.289071\pi$
$384$		0			0
$385$	7.29150	+	3.57113i	0.371609	+	0.182002i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	21.2132i		−	1.07555i	−0.843088	−	0.537776i	$-0.819265\pi$
$389$		−	21.2132i		−	1.07555i	0.843088	−	0.537776i	$-0.180735\pi$
$390$		0			0
$391$			3.06871i			0.155191i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.14226			−0.359366
$396$		0			0
$397$			35.1402i			1.76364i	0.471590	+	0.881818i	$0.343680\pi$
$397$			35.1402i			1.76364i	−0.471590	+	0.881818i	$0.656320\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	17.3828i		−	0.868056i	−0.900899	−	0.434028i	$-0.857092\pi$
$401$		−	17.3828i		−	0.868056i	0.900899	−	0.434028i	$-0.142908\pi$
$402$		0			0
$403$	−43.2509			−2.15448
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	12.0000i		−	0.594818i
$408$		0			0
$409$			7.48925i			0.370320i	0.982708	+	0.185160i	$0.0592803\pi$
$409$			7.48925i			0.370320i	−0.982708	+	0.185160i	$0.940720\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.8712	−	26.2803i	0.633351	−	1.29317i
$414$		0			0
$415$	7.66079			0.376053
$416$		0			0
$417$		0			0
$418$		0			0
$419$	40.3234			1.96993			0.984963	−	0.172763i	$-0.0552694\pi$
$419$	40.3234			1.96993			0.984963	+	0.172763i	$0.0552694\pi$
$420$		0			0
$421$	9.29150			0.452840			0.226420	−	0.974030i	$-0.427298\pi$
$421$	9.29150			0.452840			0.226420	+	0.974030i	$0.427298\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.61226			0.175220
$426$		0			0
$427$	21.6255	+	10.5914i	1.04653	+	0.512554i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1.52026i		−	0.0732283i	−0.999329	−	0.0366142i	$-0.988343\pi$
$431$		−	1.52026i		−	0.0732283i	0.999329	−	0.0366142i	$-0.0116573\pi$
$432$		0			0
$433$			11.0604i			0.531528i	0.964038	+	0.265764i	$0.0856242\pi$
$433$			11.0604i			0.531528i	−0.964038	+	0.265764i	$0.914376\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	11.1878			0.535183
$438$		0			0
$439$			30.4946i			1.45543i	0.685881	+	0.727713i	$0.259417\pi$
$439$			30.4946i			1.45543i	−0.685881	+	0.727713i	$0.740583\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	9.06275i		−	0.430584i	−0.976550	−	0.215292i	$-0.930930\pi$
$443$		−	9.06275i		−	0.430584i	0.976550	−	0.215292i	$-0.0690703\pi$
$444$		0			0
$445$	5.87451			0.278478
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	35.1779i		−	1.66015i	−0.557655	−	0.830073i	$-0.688299\pi$
$449$		−	35.1779i		−	1.66015i	0.557655	−	0.830073i	$-0.311701\pi$
$450$		0			0
$451$		−	31.5817i		−	1.48712i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−11.0604	−	5.41699i	−0.518519	−	0.253953i
$456$		0			0
$457$	27.7490			1.29804			0.649022	−	0.760770i	$-0.275178\pi$
$457$	27.7490			1.29804			0.649022	+	0.760770i	$0.275178\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11.4331			−0.532494			−0.266247	−	0.963905i	$-0.585784\pi$
$461$	−11.4331			−0.532494			−0.266247	+	0.963905i	$0.585784\pi$
$462$		0			0
$463$	−17.7951			−0.827006			−0.413503	−	0.910503i	$-0.635695\pi$
$463$	−17.7951			−0.827006			−0.413503	+	0.910503i	$0.635695\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.95906			0.0906546			0.0453273	−	0.998972i	$-0.485567\pi$
$467$	1.95906			0.0906546			0.0453273	+	0.998972i	$0.485567\pi$
$468$		0			0
$469$	9.87451	−	20.1617i	0.455962	−	0.930981i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			8.48528i			0.390154i
$474$		0			0
$475$		−	13.1694i		−	0.604253i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$			18.2026i			0.829969i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			3.00590i			0.136491i
$486$		0			0
$487$	−31.6137			−1.43255			−0.716276	−	0.697817i	$-0.754155\pi$
$487$	−31.6137			−1.43255			−0.716276	+	0.697817i	$0.754155\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			36.8118i			1.66129i	0.556801	+	0.830646i	$0.312029\pi$
$491$			36.8118i			1.66129i	−0.556801	+	0.830646i	$0.687971\pi$
$492$		0			0
$493$		−	7.48925i		−	0.337299i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.841723	−	0.412247i	−0.0377564	−	0.0184918i
$498$		0			0
$499$	40.9235			1.83199			0.915993	−	0.401195i	$-0.131405\pi$
$499$	40.9235			1.83199			0.915993	+	0.401195i	$0.131405\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−35.1402			−1.56682			−0.783412	−	0.621503i	$-0.786523\pi$
$503$	−35.1402			−1.56682			−0.783412	+	0.621503i	$0.786523\pi$
$504$		0			0
$505$	12.4575			0.554352
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−19.8504			−0.879852			−0.439926	−	0.898034i	$-0.644995\pi$
$509$	−19.8504			−0.879852			−0.439926	+	0.898034i	$0.644995\pi$
$510$		0			0
$511$	34.7656	+	17.0270i	1.53794	+	0.753230i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			4.25098i			0.187321i
$516$		0			0
$517$			33.1811i			1.45930i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	32.1252			1.40743			0.703715	−	0.710482i	$-0.251523\pi$
$521$	32.1252			1.40743			0.703715	+	0.710482i	$0.251523\pi$
$522$		0			0
$523$			41.3842i			1.80960i	0.425834	+	0.904801i	$0.359981\pi$
$523$			41.3842i			1.80960i	−0.425834	+	0.904801i	$0.640019\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			6.58301i			0.286760i
$528$		0			0
$529$	9.70850			0.422109
$530$		0			0
$531$		0			0
$532$		0			0
$533$			47.9058i			2.07503i
$534$		0			0
$535$		−	5.24289i		−	0.226670i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	20.1617	−	15.6458i	0.868426	−	0.673910i
$540$		0			0
$541$	−9.29150			−0.399473			−0.199736	−	0.979850i	$-0.564009\pi$
$541$	−9.29150			−0.399473			−0.199736	+	0.979850i	$0.564009\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.90796			0.381575
$546$		0			0
$547$	−27.1048			−1.15892			−0.579459	−	0.815001i	$-0.696736\pi$
$547$	−27.1048			−1.15892			−0.579459	+	0.815001i	$0.696736\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−27.3040			−1.16319
$552$		0			0
$553$	−9.87451	+	20.1617i	−0.419907	+	0.857363i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	36.0024i		−	1.52547i	−0.646712	−	0.762734i	$-0.723856\pi$
$557$		−	36.0024i		−	1.52547i	0.646712	−	0.762734i	$-0.276144\pi$
$558$		0			0
$559$		−	12.8712i		−	0.544394i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−13.0194			−0.548704			−0.274352	−	0.961629i	$-0.588463\pi$
$563$	−13.0194			−0.548704			−0.274352	+	0.961629i	$0.588463\pi$
$564$		0			0
$565$		−	0.346998i		−	0.0145983i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			31.3475i			1.31415i	0.753823	+	0.657077i	$0.228207\pi$
$569$			31.3475i			1.31415i	−0.753823	+	0.657077i	$0.771793\pi$
$570$		0			0
$571$	−16.1461			−0.675692			−0.337846	−	0.941201i	$-0.609698\pi$
$571$	−16.1461			−0.675692			−0.337846	+	0.941201i	$0.609698\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			15.6458i			0.652473i
$576$		0			0
$577$			29.2630i			1.21824i	0.793080	+	0.609118i	$0.208476\pi$
$577$			29.2630i			1.21824i	−0.793080	+	0.609118i	$0.791524\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10.5914	−	21.6255i	0.439405	−	0.897175i
$582$		0			0
$583$	15.4676			0.640603
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.0194			−0.537370			−0.268685	−	0.963228i	$-0.586589\pi$
$587$	−13.0194			−0.537370			−0.268685	+	0.963228i	$0.586589\pi$
$588$		0			0
$589$	24.0000			0.988903
$590$		0			0
$591$		0			0
$592$		0			0
$593$	40.5424			1.66488			0.832439	−	0.554117i	$-0.186944\pi$
$593$	40.5424			1.66488			0.832439	+	0.554117i	$0.186944\pi$
$594$		0			0
$595$	−0.824494	+	1.68345i	−0.0338010	+	0.0690145i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			28.8118i			1.17722i	0.808418	+	0.588608i	$0.200324\pi$
$599$			28.8118i			1.17722i	−0.808418	+	0.588608i	$0.799676\pi$
$600$		0			0
$601$		−	14.9785i		−	0.610986i	−0.952194	−	0.305493i	$-0.901179\pi$
$601$		−	14.9785i		−	0.610986i	0.952194	−	0.305493i	$-0.0988213\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.92881			0.0784173
$606$		0			0
$607$		−	1.98162i		−	0.0804317i	−0.999191	−	0.0402158i	$-0.987195\pi$
$607$		−	1.98162i		−	0.0804317i	0.999191	−	0.0402158i	$-0.0128046\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	50.3320i		−	2.03622i
$612$		0			0
$613$	−32.5830			−1.31602			−0.658008	−	0.753011i	$-0.728601\pi$
$613$	−32.5830			−1.31602			−0.658008	+	0.753011i	$0.728601\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	16.5583i		−	0.666613i	−0.942819	−	0.333306i	$-0.891836\pi$
$617$		−	16.5583i		−	0.666613i	0.942819	−	0.333306i	$-0.108164\pi$
$618$		0			0
$619$		−	28.5129i		−	1.14603i	−0.819544	−	0.573016i	$-0.805773\pi$
$619$		−	28.5129i		−	1.14603i	0.819544	−	0.573016i	$-0.194227\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.12179	−	16.5830i	0.325393	−	0.664384i
$624$		0			0
$625$	14.8745			0.594980
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.77053			0.110468
$630$		0			0
$631$	27.1048			1.07903			0.539513	−	0.841977i	$-0.318608\pi$
$631$	27.1048			1.07903			0.539513	+	0.841977i	$0.318608\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.9785			0.594404
$636$		0			0
$637$	−30.5830	+	23.7328i	−1.21174	+	0.940329i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			39.0083i			1.54073i	0.637601	+	0.770367i	$0.279927\pi$
$641$			39.0083i			1.54073i	−0.637601	+	0.770367i	$0.720073\pi$
$642$		0			0
$643$		−	34.3522i		−	1.35472i	−0.735653	−	0.677359i	$-0.763124\pi$
$643$		−	34.3522i		−	1.35472i	0.735653	−	0.677359i	$-0.236876\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−29.2630			−1.15045			−0.575224	−	0.817996i	$-0.695085\pi$
$647$	−29.2630			−1.15045			−0.575224	+	0.817996i	$0.695085\pi$
$648$		0			0
$649$		−	40.3234i		−	1.58283i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			13.5524i			0.530347i	0.964201	+	0.265174i	$0.0854292\pi$
$653$			13.5524i			0.530347i	−0.964201	+	0.265174i	$0.914571\pi$
$654$		0			0
$655$	1.64899			0.0644313
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	28.8118i		−	1.12235i	−0.827698	−	0.561173i	$-0.810350\pi$
$659$		−	28.8118i		−	1.12235i	0.827698	−	0.561173i	$-0.189650\pi$
$660$		0			0
$661$			1.95906i			0.0761987i	0.999274	+	0.0380994i	$0.0121303\pi$
$661$			1.95906i			0.0761987i	−0.999274	+	0.0380994i	$0.987870\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.13742	+	3.00590i	0.237999	+	0.116564i
$666$		0			0
$667$	32.4382			1.25601
$668$		0			0
$669$		0			0
$670$		0			0
$671$	33.1811			1.28094
$672$		0			0
$673$	−15.2915			−0.589444			−0.294722	−	0.955583i	$-0.595227\pi$
$673$	−15.2915			−0.589444			−0.294722	+	0.955583i	$0.595227\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.61226			−0.138830			−0.0694151	−	0.997588i	$-0.522113\pi$
$677$	−3.61226			−0.138830			−0.0694151	+	0.997588i	$0.522113\pi$
$678$		0			0
$679$	8.48528	+	4.15580i	0.325635	+	0.159485i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			30.2288i			1.15667i	0.815799	+	0.578336i	$0.196298\pi$
$683$			30.2288i			1.15667i	−0.815799	+	0.578336i	$0.803702\pi$
$684$		0			0
$685$		−	7.48925i		−	0.286150i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−23.4626			−0.893854
$690$		0			0
$691$		−	34.0540i		−	1.29548i	−0.761863	−	0.647738i	$-0.775715\pi$
$691$		−	34.0540i		−	1.29548i	0.761863	−	0.647738i	$-0.224285\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	10.8340i		−	0.410957i
$696$		0			0
$697$	7.29150			0.276185
$698$		0			0
$699$		0			0
$700$		0			0
$701$			33.5289i			1.26637i	0.774001	+	0.633184i	$0.218252\pi$
$701$			33.5289i			1.26637i	−0.774001	+	0.633184i	$0.781748\pi$
$702$		0			0
$703$		−	10.1007i		−	0.380954i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	17.2231	−	35.1660i	0.647742	−	1.32255i
$708$		0			0
$709$	28.0000			1.05156			0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000			1.05156			0.525781	+	0.850620i	$0.323773\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−28.5129			−1.06782
$714$		0			0
$715$	−16.9706			−0.634663
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−18.2026			−0.678844			−0.339422	−	0.940634i	$-0.610231\pi$
$719$	−18.2026			−0.678844			−0.339422	+	0.940634i	$0.610231\pi$
$720$		0			0
$721$	12.0000	+	5.87719i	0.446903	+	0.218878i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	38.1838i		−	1.41811i
$726$		0			0
$727$			36.3338i			1.34755i	0.738938	+	0.673773i	$0.235328\pi$
$727$			36.3338i			1.34755i	−0.738938	+	0.673773i	$0.764672\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.95906			−0.0724586
$732$		0			0
$733$		−	23.7328i		−	0.876592i	−0.898831	−	0.438296i	$-0.855582\pi$
$733$		−	23.7328i		−	0.876592i	0.898831	−	0.438296i	$-0.144418\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	30.9352i		−	1.13951i
$738$		0			0
$739$	−44.0754			−1.62134			−0.810670	−	0.585504i	$-0.800897\pi$
$739$	−44.0754			−1.62134			−0.810670	+	0.585504i	$0.800897\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			18.2288i			0.668748i	0.942440	+	0.334374i	$0.108525\pi$
$743$			18.2288i			0.668748i	−0.942440	+	0.334374i	$0.891475\pi$
$744$		0			0
$745$		−	7.48925i		−	0.274385i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−14.8000	−	7.24854i	−0.540781	−	0.264856i
$750$		0			0
$751$	−16.2921			−0.594507			−0.297254	−	0.954799i	$-0.596071\pi$
$751$	−16.2921			−0.594507			−0.297254	+	0.954799i	$0.596071\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.2436			0.591164
$756$		0			0
$757$	41.1660			1.49620			0.748102	−	0.663584i	$-0.230965\pi$
$757$	41.1660			1.49620			0.748102	+	0.663584i	$0.230965\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−52.8173			−1.91462			−0.957312	−	0.289055i	$-0.906659\pi$
$761$	−52.8173			−1.91462			−0.957312	+	0.289055i	$0.906659\pi$
$762$		0			0
$763$	12.3157	−	25.1461i	0.445857	−	0.910348i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			61.1660i			2.20858i
$768$		0			0
$769$		−	25.3449i		−	0.913960i	−0.889477	−	0.456980i	$-0.848931\pi$
$769$		−	25.3449i		−	0.913960i	0.889477	−	0.456980i	$-0.151069\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.20861			0.151373			0.0756867	−	0.997132i	$-0.475885\pi$
$773$	4.20861			0.151373			0.0756867	+	0.997132i	$0.475885\pi$
$774$		0			0
$775$			33.5633i			1.20563i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	26.5830i		−	0.952435i
$780$		0			0
$781$	−1.29150			−0.0462136
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1.64899i			0.0588549i
$786$		0			0
$787$			25.7424i			0.917618i	0.888535	+	0.458809i	$0.151724\pi$
$787$			25.7424i			0.917618i	−0.888535	+	0.458809i	$0.848276\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.979531	−	0.479741i	−0.0348281	−	0.0170576i
$792$		0			0
$793$	−50.3320			−1.78734
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.0295			0.426106			0.213053	−	0.977041i	$-0.431659\pi$
$797$	12.0295			0.426106			0.213053	+	0.977041i	$0.431659\pi$
$798$		0			0
$799$	−7.66079			−0.271019
$800$		0			0
$801$		0			0
$802$		0			0
$803$	53.3428			1.88243
$804$		0			0
$805$	−7.29150	−	3.57113i	−0.256992	−	0.125866i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	31.3475i		−	1.10212i	−0.834466	−	0.551059i	$-0.814224\pi$
$809$		−	31.3475i		−	1.10212i	0.834466	−	0.551059i	$-0.185776\pi$
$810$		0			0
$811$			31.2835i			1.09851i	0.835654	+	0.549256i	$0.185089\pi$
$811$			31.2835i			1.09851i	−0.835654	+	0.549256i	$0.814911\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.14226			−0.250182
$816$		0			0
$817$			7.14226i			0.249876i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			13.5524i			0.472983i	0.971634	+	0.236491i	$0.0759975\pi$
$821$			13.5524i			0.472983i	−0.971634	+	0.236491i	$0.924003\pi$
$822$		0			0
$823$	−25.4558			−0.887335			−0.443667	−	0.896191i	$-0.646323\pi$
$823$	−25.4558			−0.887335			−0.443667	+	0.896191i	$0.646323\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12.3542i			0.429599i	0.976658	+	0.214800i	$0.0689099\pi$
$827$			12.3542i			0.429599i	−0.976658	+	0.214800i	$0.931090\pi$
$828$		0			0
$829$		−	30.8751i		−	1.07234i	−0.844111	−	0.536168i	$-0.819871\pi$
$829$		−	30.8751i		−	1.07234i	0.844111	−	0.536168i	$-0.180129\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.61226	+	4.65489i	0.125157	+	0.161282i
$834$		0			0
$835$	−10.9588			−0.379244
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−38.3643			−1.32448			−0.662242	−	0.749290i	$-0.730395\pi$
$839$	−38.3643			−1.32448			−0.662242	+	0.749290i	$0.730395\pi$
$840$		0			0
$841$	−50.1660			−1.72986
$842$		0			0
$843$		0			0
$844$		0			0
$845$	14.8000			0.509136
$846$		0			0
$847$	2.66667	−	5.44479i	0.0916279	−	0.187085i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			12.0000i			0.411355i
$852$		0			0
$853$		−	27.3040i		−	0.934870i	−0.884028	−	0.467435i	$-0.845178\pi$
$853$		−	27.3040i		−	0.934870i	0.884028	−	0.467435i	$-0.154822\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−11.4331			−0.390548			−0.195274	−	0.980749i	$-0.562560\pi$
$857$	−11.4331			−0.390548			−0.195274	+	0.980749i	$0.562560\pi$
$858$		0			0
$859$		−	25.4442i		−	0.868146i	−0.900878	−	0.434073i	$-0.857076\pi$
$859$		−	25.4442i		−	0.868146i	0.900878	−	0.434073i	$-0.142924\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	3.64575i		−	0.124103i	−0.998073	−	0.0620514i	$-0.980236\pi$
$863$		−	3.64575i		−	0.124103i	0.998073	−	0.0620514i	$-0.0197643\pi$
$864$		0			0
$865$	10.1255			0.344277
$866$		0			0
$867$		0			0
$868$		0			0
$869$			30.9352i			1.04941i
$870$		0			0
$871$			46.9252i			1.59000i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9.10132	+	18.5830i	−0.307681	+	0.628220i
$876$		0			0
$877$	−14.0000			−0.472746			−0.236373	−	0.971662i	$-0.575959\pi$
$877$	−14.0000			−0.472746			−0.236373	+	0.971662i	$0.575959\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.69934			−0.158325			−0.0791624	−	0.996862i	$-0.525225\pi$
$881$	−4.69934			−0.158325			−0.0791624	+	0.996862i	$0.525225\pi$
$882$		0			0
$883$	28.6078			0.962729			0.481364	−	0.876521i	$-0.340141\pi$
$883$	28.6078			0.962729			0.481364	+	0.876521i	$0.340141\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−53.3428			−1.79108			−0.895539	−	0.444984i	$-0.853209\pi$
$887$	−53.3428			−1.79108			−0.895539	+	0.444984i	$0.853209\pi$
$888$		0			0
$889$	20.7085	−	42.2825i	0.694541	−	1.41811i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			27.9293i			0.934619i
$894$		0			0
$895$		−	15.9399i		−	0.532813i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	69.5864			2.32084
$900$		0			0
$901$			3.57113i			0.118972i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			4.65489i			0.154734i
$906$		0			0
$907$	−23.9529			−0.795343			−0.397671	−	0.917528i	$-0.630182\pi$
$907$	−23.9529			−0.795343			−0.397671	+	0.917528i	$0.630182\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	23.6458i		−	0.783419i	−0.920089	−	0.391709i	$-0.871884\pi$
$911$		−	23.6458i		−	0.783419i	0.920089	−	0.391709i	$-0.128116\pi$
$912$		0			0
$913$		−	33.1811i		−	1.09814i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2.27980	−	4.65489i	0.0752858	−	0.153718i
$918$		0			0
$919$	54.8881			1.81059			0.905296	−	0.424781i	$-0.139649\pi$
$919$	54.8881			1.81059			0.905296	+	0.424781i	$0.139649\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.95906			0.0644833
$924$		0			0
$925$	14.1255			0.464443
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−44.9964			−1.47628			−0.738142	−	0.674645i	$-0.764297\pi$
$929$	−44.9964			−1.47628			−0.738142	+	0.674645i	$0.764297\pi$
$930$		0			0
$931$	16.9706	−	13.1694i	0.556188	−	0.431609i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			2.58301i			0.0844733i
$936$		0			0
$937$			44.2415i			1.44531i	0.691210	+	0.722654i	$0.257078\pi$
$937$			44.2415i			1.44531i	−0.691210	+	0.722654i	$0.742922\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−50.0467			−1.63148			−0.815739	−	0.578421i	$-0.803669\pi$
$941$	−50.0467			−1.63148			−0.815739	+	0.578421i	$0.803669\pi$
$942$		0			0
$943$			31.5817i			1.02844i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			10.2288i			0.332390i	0.986093	+	0.166195i	$0.0531481\pi$
$947$			10.2288i			0.332390i	−0.986093	+	0.166195i	$0.946852\pi$
$948$		0			0
$949$	−80.9150			−2.62661
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	25.0436i		−	0.811242i	−0.914041	−	0.405621i	$-0.867055\pi$
$953$		−	25.0436i		−	0.811242i	0.914041	−	0.405621i	$-0.132945\pi$
$954$		0			0
$955$			21.4810i			0.695108i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−21.1412	−	10.3542i	−0.682686	−	0.334356i
$960$		0			0
$961$	−30.1660			−0.973097
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−12.8712			−0.414339
$966$		0			0
$967$	−8.48528			−0.272868			−0.136434	−	0.990649i	$-0.543564\pi$
$967$	−8.48528			−0.272868			−0.136434	+	0.990649i	$0.543564\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−33.1811			−1.06483			−0.532417	−	0.846482i	$-0.678716\pi$
$971$	−33.1811			−1.06483			−0.532417	+	0.846482i	$0.678716\pi$
$972$		0			0
$973$	−30.5830	−	14.9785i	−0.980446	−	0.480189i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			60.6337i			1.93984i	0.243415	+	0.969922i	$0.421732\pi$
$977$			60.6337i			1.93984i	−0.243415	+	0.969922i	$0.578268\pi$
$978$		0			0
$979$		−	25.4442i		−	0.813201i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	13.0194			0.415256			0.207628	−	0.978208i	$-0.433426\pi$
$983$	13.0194			0.415256			0.207628	+	0.978208i	$0.433426\pi$
$984$		0			0
$985$			4.26512i			0.135898i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	8.48528i		−	0.269816i
$990$		0			0
$991$	−54.8881			−1.74358			−0.871789	−	0.489881i	$-0.837040\pi$
$991$	−54.8881			−1.74358			−0.871789	+	0.489881i	$0.837040\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	6.33202i		−	0.200739i
$996$		0			0
$997$		−	9.10132i		−	0.288242i	−0.989560	−	0.144121i	$-0.953965\pi$
$997$		−	9.10132i		−	0.288242i	0.989560	−	0.144121i	$-0.0460354\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.k.b.1889.6	yes	16
3.2	odd	2	inner	2016.2.k.b.1889.10	yes	16
4.3	odd	2	inner	2016.2.k.b.1889.7	yes	16
7.6	odd	2	inner	2016.2.k.b.1889.9	yes	16
8.3	odd	2		4032.2.k.h.3905.11		16
8.5	even	2		4032.2.k.h.3905.10		16
12.11	even	2	inner	2016.2.k.b.1889.11	yes	16
21.20	even	2	inner	2016.2.k.b.1889.5	✓	16
24.5	odd	2		4032.2.k.h.3905.6		16
24.11	even	2		4032.2.k.h.3905.7		16
28.27	even	2	inner	2016.2.k.b.1889.12	yes	16
56.13	odd	2		4032.2.k.h.3905.5		16
56.27	even	2		4032.2.k.h.3905.8		16
84.83	odd	2	inner	2016.2.k.b.1889.8	yes	16
168.83	odd	2		4032.2.k.h.3905.12		16
168.125	even	2		4032.2.k.h.3905.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2016.2.k.b.1889.5	✓	16	21.20	even	2	inner
2016.2.k.b.1889.6	yes	16	1.1	even	1	trivial
2016.2.k.b.1889.7	yes	16	4.3	odd	2	inner
2016.2.k.b.1889.8	yes	16	84.83	odd	2	inner
2016.2.k.b.1889.9	yes	16	7.6	odd	2	inner
2016.2.k.b.1889.10	yes	16	3.2	odd	2	inner
2016.2.k.b.1889.11	yes	16	12.11	even	2	inner
2016.2.k.b.1889.12	yes	16	28.27	even	2	inner
4032.2.k.h.3905.5		16	56.13	odd	2
4032.2.k.h.3905.6		16	24.5	odd	2
4032.2.k.h.3905.7		16	24.11	even	2
4032.2.k.h.3905.8		16	56.27	even	2
4032.2.k.h.3905.9		16	168.125	even	2
4032.2.k.h.3905.10		16	8.5	even	2
4032.2.k.h.3905.11		16	8.3	odd	2
4032.2.k.h.3905.12		16	168.83	odd	2

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:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.841723 q^{5} +(-1.16372 + 2.37608i) q^{7} +O(q^{10})\)	\(q-0.841723 q^{5} +(-1.16372 + 2.37608i) q^{7} +3.64575i q^{11} -5.53019i q^{13} -0.841723 q^{17} +3.06871i q^{19} -3.64575i q^{23} -4.29150 q^{25} +8.89753i q^{29} -7.82087i q^{31} +(0.979531 - 2.00000i) q^{35} -3.29150 q^{37} -8.66259 q^{41} +2.32744 q^{43} +9.10132 q^{47} +(-4.29150 - 5.53019i) q^{49} -4.24264i q^{53} -3.06871i q^{55} -11.0604 q^{59} -9.10132i q^{61} +4.65489i q^{65} -8.48528 q^{67} +0.354249i q^{71} -14.6315i q^{73} +(-8.66259 - 4.24264i) q^{77} +8.48528 q^{79} -9.10132 q^{83} +0.708497 q^{85} -6.97915 q^{89} +(13.1402 + 6.43560i) q^{91} -2.58301i q^{95} -3.57113i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{25} + 32 q^{37} + 16 q^{49} + 96 q^{85}+O(q^{100}) \) 16 * q + 16 * q^25 + 32 * q^37 + 16 * q^49 + 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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