LMFDB - Embedded newform 2016.2.cr.a.1873.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(1297,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))

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

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

chi := DirichletCharacter("2016.1297");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.cr (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1873.4
Root			$1.40126 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	2016.1873
Dual form			2016.2.cr.a.1297.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.93649 - 1.11803i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})$	$q+(1.93649 - 1.11803i) q^{5} +(0.500000 + 2.59808i) q^{7} +(1.93649 + 1.11803i) q^{11} +(1.73205 - 3.00000i) q^{17} +(6.70820 - 3.87298i) q^{19} +(-3.46410 - 6.00000i) q^{23} +2.23607i q^{29} +(-0.500000 + 0.866025i) q^{31} +(3.87298 + 4.47214i) q^{35} +(-6.70820 + 3.87298i) q^{37} +10.3923 q^{41} +(1.73205 + 3.00000i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(9.68246 + 5.59017i) q^{53} +5.00000 q^{55} +(1.93649 + 1.11803i) q^{59} +(-6.70820 + 3.87298i) q^{61} +(6.70820 + 3.87298i) q^{67} +10.3923 q^{71} +(5.00000 - 8.66025i) q^{73} +(-1.93649 + 5.59017i) q^{77} +(-6.50000 - 11.2583i) q^{79} +11.1803i q^{83} -7.74597i q^{85} +(-6.92820 - 12.0000i) q^{89} +(8.66025 - 15.0000i) q^{95} -1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7}+O(q^{10}) $ 8 * q + 4 * q^7	$ 8 q + 4 q^{7} - 4 q^{31} - 52 q^{49} + 40 q^{55} + 40 q^{73} - 52 q^{79} - 8 q^{97}+O(q^{100}) $ 8 * q + 4 * q^7 - 4 * q^31 - 52 * q^49 + 40 * q^55 + 40 * q^73 - 52 * q^79 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$-1$	$1$

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
$5$	1.93649	−	1.11803i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$5$	1.93649	−	1.11803i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.93649	+	1.11803i	0.583874	+	0.337100i	0.762672	−	0.646786i	$-0.223887\pi$
$11$	1.93649	+	1.11803i	0.583874	+	0.337100i	−0.178797	+	0.983886i	$0.557221\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.73205	−	3.00000i	0.420084	−	0.727607i	−0.575863	−	0.817546i	$-0.695334\pi$
$17$	1.73205	−	3.00000i	0.420084	−	0.727607i	0.995947	+	0.0899392i	$0.0286673\pi$
$18$		0			0
$19$	6.70820	−	3.87298i	1.53897	−	0.888523i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	6.70820	−	3.87298i	1.53897	−	0.888523i	0.998899	−	0.0469020i	$-0.0149348\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.46410	−	6.00000i	−0.722315	−	1.25109i	−0.960070	−	0.279761i	$-0.909745\pi$
$23$	−3.46410	−	6.00000i	−0.722315	−	1.25109i	0.237754	−	0.971325i	$-0.423589\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.23607i			0.415227i	0.978211	+	0.207614i	$0.0665697\pi$
$29$			2.23607i			0.415227i	−0.978211	+	0.207614i	$0.933430\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	−0.907428	−	0.420208i	$-0.861957\pi$
$31$	−0.500000	+	0.866025i	−0.0898027	+	0.155543i	0.817625	+	0.575751i	$0.195290\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.87298	+	4.47214i	0.654654	+	0.755929i
$36$		0			0
$37$	−6.70820	+	3.87298i	−1.10282	+	0.636715i	−0.936961	−	0.349435i	$-0.886374\pi$
$37$	−6.70820	+	3.87298i	−1.10282	+	0.636715i	−0.165861	+	0.986149i	$0.553040\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.3923			1.62301			0.811503	−	0.584349i	$-0.198650\pi$
$41$	10.3923			1.62301			0.811503	+	0.584349i	$0.198650\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.73205	+	3.00000i	0.252646	+	0.437595i	0.964253	−	0.264982i	$-0.0853660\pi$
$47$	1.73205	+	3.00000i	0.252646	+	0.437595i	−0.711608	+	0.702577i	$0.752033\pi$
$48$		0			0
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.68246	+	5.59017i	1.32999	+	0.767869i	0.985297	−	0.170848i	$-0.0546505\pi$
$53$	9.68246	+	5.59017i	1.32999	+	0.767869i	0.344690	+	0.938716i	$0.387984\pi$
$54$		0			0
$55$	5.00000			0.674200
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.93649	+	1.11803i	0.252110	+	0.145556i	0.620730	−	0.784024i	$-0.286836\pi$
$59$	1.93649	+	1.11803i	0.252110	+	0.145556i	−0.368620	+	0.929580i	$0.620170\pi$
$60$		0			0
$61$	−6.70820	+	3.87298i	−0.858898	+	0.495885i	−0.863643	−	0.504104i	$-0.831823\pi$
$61$	−6.70820	+	3.87298i	−0.858898	+	0.495885i	0.00474543	+	0.999989i	$0.498489\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.70820	+	3.87298i	0.819538	+	0.473160i	0.850257	−	0.526368i	$-0.176447\pi$
$67$	6.70820	+	3.87298i	0.819538	+	0.473160i	−0.0307194	+	0.999528i	$0.509780\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.3923			1.23334			0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923			1.23334			0.616670	+	0.787222i	$0.288481\pi$
$72$		0			0
$73$	5.00000	−	8.66025i	0.585206	−	1.01361i	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	5.00000	−	8.66025i	0.585206	−	1.01361i	0.994850	−	0.101361i	$-0.0323196\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.93649	+	5.59017i	−0.220684	+	0.637059i
$78$		0			0
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	0.225018	−	0.974355i	$-0.427756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			11.1803i			1.22720i	0.789616	+	0.613601i	$0.210280\pi$
$83$			11.1803i			1.22720i	−0.789616	+	0.613601i	$0.789720\pi$
$84$		0			0
$85$		−	7.74597i		−	0.840168i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.92820	−	12.0000i	−0.734388	−	1.27200i	−0.954991	−	0.296634i	$-0.904136\pi$
$89$	−6.92820	−	12.0000i	−0.734388	−	1.27200i	0.220603	−	0.975364i	$-0.429197\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.66025	−	15.0000i	0.888523	−	1.53897i
$96$		0			0
$97$	−1.00000			−0.101535			−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000			−0.101535			−0.0507673	+	0.998711i	$0.516167\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.87298	+	2.23607i	0.385376	+	0.222497i	0.680155	−	0.733069i	$-0.261913\pi$
$101$	3.87298	+	2.23607i	0.385376	+	0.222497i	−0.294779	+	0.955566i	$0.595246\pi$
$102$		0			0
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	0.992990	−	0.118199i	$-0.0377120\pi$
$103$	4.00000	+	6.92820i	0.394132	+	0.682656i	−0.598858	+	0.800855i	$0.704379\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.68246	−	5.59017i	0.936039	−	0.540422i	0.0473223	−	0.998880i	$-0.484931\pi$
$107$	9.68246	−	5.59017i	0.936039	−	0.540422i	0.888716	+	0.458458i	$0.151598\pi$
$108$		0			0
$109$	−6.70820	−	3.87298i	−0.642529	−	0.370965i	0.143059	−	0.989714i	$-0.454306\pi$
$109$	−6.70820	−	3.87298i	−0.642529	−	0.370965i	−0.785588	+	0.618750i	$0.787640\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.3923			−0.977626			−0.488813	−	0.872389i	$-0.662570\pi$
$113$	−10.3923			−0.977626			−0.488813	+	0.872389i	$0.662570\pi$
$114$		0			0
$115$	−13.4164	−	7.74597i	−1.25109	−	0.722315i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.66025	+	3.00000i	0.793884	+	0.275010i
$120$		0			0
$121$	−3.00000	−	5.19615i	−0.272727	−	0.472377i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			1.00000i
$126$		0			0
$127$	−5.00000			−0.443678			−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000			−0.443678			−0.221839	+	0.975083i	$0.571206\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.68246	−	5.59017i	0.845960	−	0.488415i	−0.0133255	−	0.999911i	$-0.504242\pi$
$131$	9.68246	−	5.59017i	0.845960	−	0.488415i	0.859286	+	0.511496i	$0.170908\pi$
$132$		0			0
$133$	13.4164	+	15.4919i	1.16335	+	1.34332i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.73205	−	3.00000i	0.147979	−	0.256307i	−0.782501	−	0.622649i	$-0.786057\pi$
$137$	1.73205	−	3.00000i	0.147979	−	0.256307i	0.930480	+	0.366342i	$0.119390\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	2.50000	+	4.33013i	0.207614	+	0.359597i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	19.3649	−	11.1803i	1.58644	−	0.915929i	0.592548	−	0.805535i	$-0.298122\pi$
$149$	19.3649	−	11.1803i	1.58644	−	0.915929i	0.993888	−	0.110394i	$-0.0352112\pi$
$150$		0			0
$151$	−3.50000	+	6.06218i	−0.284826	+	0.493333i	−0.972567	−	0.232623i	$-0.925269\pi$
$151$	−3.50000	+	6.06218i	−0.284826	+	0.493333i	0.687741	+	0.725956i	$0.258602\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			2.23607i			0.179605i
$156$		0			0
$157$	13.4164	+	7.74597i	1.07075	+	0.618195i	0.928385	−	0.371619i	$-0.121197\pi$
$157$	13.4164	+	7.74597i	1.07075	+	0.618195i	0.142361	+	0.989815i	$0.454531\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	13.8564	−	12.0000i	1.09204	−	0.945732i
$162$		0			0
$163$	−13.4164	+	7.74597i	−1.05085	+	0.606711i	−0.922888	−	0.385068i	$-0.874178\pi$
$163$	−13.4164	+	7.74597i	−1.05085	+	0.606711i	−0.127966	+	0.991779i	$0.540845\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.3923			−0.804181			−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923			−0.804181			−0.402090	+	0.915600i	$0.631716\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.87298	+	2.23607i	−0.294457	+	0.170005i	−0.639950	−	0.768416i	$-0.721045\pi$
$173$	−3.87298	+	2.23607i	−0.294457	+	0.170005i	0.345493	+	0.938421i	$0.387712\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.74597	+	4.47214i	0.578961	+	0.334263i	0.760720	−	0.649080i	$-0.224846\pi$
$179$	7.74597	+	4.47214i	0.578961	+	0.334263i	−0.181760	+	0.983343i	$0.558179\pi$
$180$		0			0
$181$		−	23.2379i		−	1.72726i	−0.504127	−	0.863630i	$-0.668186\pi$
$181$		−	23.2379i		−	1.72726i	0.504127	−	0.863630i	$-0.331814\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.66025	+	15.0000i	−0.636715	+	1.10282i
$186$		0			0
$187$	6.70820	−	3.87298i	0.490552	−	0.283221i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.46410	−	6.00000i	−0.250654	−	0.434145i	0.713052	−	0.701111i	$-0.247312\pi$
$191$	−3.46410	−	6.00000i	−0.250654	−	0.434145i	−0.963706	+	0.266966i	$0.913979\pi$
$192$		0			0
$193$	−8.50000	+	14.7224i	−0.611843	+	1.05974i	0.379086	+	0.925361i	$0.376238\pi$
$193$	−8.50000	+	14.7224i	−0.611843	+	1.05974i	−0.990930	+	0.134382i	$0.957095\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4.47214i		−	0.318626i	−0.987228	−	0.159313i	$-0.949072\pi$
$197$		−	4.47214i		−	0.318626i	0.987228	−	0.159313i	$-0.0509280\pi$
$198$		0			0
$199$	4.00000	−	6.92820i	0.283552	−	0.491127i	−0.688705	−	0.725042i	$-0.741820\pi$
$199$	4.00000	−	6.92820i	0.283552	−	0.491127i	0.972257	+	0.233915i	$0.0751537\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.80948	+	1.11803i	−0.407745	+	0.0784706i
$204$		0			0
$205$	20.1246	−	11.6190i	1.40556	−	0.811503i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	17.3205			1.19808
$210$		0			0
$211$			23.2379i			1.59976i	0.600158	+	0.799882i	$0.295104\pi$
$211$			23.2379i			1.59976i	−0.600158	+	0.799882i	$0.704896\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.50000	−	0.866025i	−0.169711	−	0.0587896i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	7.00000			0.468755			0.234377	−	0.972146i	$-0.424695\pi$
$223$	7.00000			0.468755			0.234377	+	0.972146i	$0.424695\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−21.3014	−	12.2984i	−1.41382	−	0.816272i	−0.418078	−	0.908411i	$-0.637296\pi$
$227$	−21.3014	−	12.2984i	−1.41382	−	0.816272i	−0.995746	+	0.0921394i	$0.970629\pi$
$228$		0			0
$229$	−6.70820	+	3.87298i	−0.443291	+	0.255934i	−0.704992	−	0.709215i	$-0.749050\pi$
$229$	−6.70820	+	3.87298i	−0.443291	+	0.255934i	0.261702	+	0.965149i	$0.415716\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.73205	−	3.00000i	−0.113470	−	0.196537i	0.803697	−	0.595039i	$-0.202863\pi$
$233$	−1.73205	−	3.00000i	−0.113470	−	0.196537i	−0.917167	+	0.398502i	$0.869530\pi$
$234$		0			0
$235$	6.70820	+	3.87298i	0.437595	+	0.252646i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−20.7846			−1.34444			−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846			−1.34444			−0.672222	+	0.740349i	$0.734660\pi$
$240$		0			0
$241$	−11.5000	+	19.9186i	−0.740780	+	1.28307i	0.211360	+	0.977408i	$0.432211\pi$
$241$	−11.5000	+	19.9186i	−0.740780	+	1.28307i	−0.952141	+	0.305661i	$0.901123\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.68246	+	12.2984i	−0.618590	+	0.785714i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			24.5967i			1.55253i	0.630405	+	0.776266i	$0.282889\pi$
$251$			24.5967i			1.55253i	−0.630405	+	0.776266i	$0.717111\pi$
$252$		0			0
$253$		−	15.4919i		−	0.973970i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.46410	+	6.00000i	0.216085	+	0.374270i	0.953608	−	0.301052i	$-0.0973379\pi$
$257$	3.46410	+	6.00000i	0.216085	+	0.374270i	−0.737523	+	0.675322i	$0.764005\pi$
$258$		0			0
$259$	−13.4164	−	15.4919i	−0.833655	−	0.962622i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.46410	−	6.00000i	0.213606	−	0.369976i	−0.739235	−	0.673448i	$-0.764813\pi$
$263$	3.46410	−	6.00000i	0.213606	−	0.369976i	0.952840	+	0.303472i	$0.0981459\pi$
$264$		0			0
$265$	25.0000			1.53574
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−25.1744	−	14.5344i	−1.53491	−	0.886181i	−0.999125	−	0.0418260i	$-0.986682\pi$
$269$	−25.1744	−	14.5344i	−1.53491	−	0.886181i	−0.535785	−	0.844355i	$-0.679984\pi$
$270$		0			0
$271$	−0.500000	−	0.866025i	−0.0303728	−	0.0526073i	0.850439	−	0.526073i	$-0.176336\pi$
$271$	−0.500000	−	0.866025i	−0.0303728	−	0.0526073i	−0.880812	+	0.473466i	$0.843003\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	13.4164	+	7.74597i	0.806114	+	0.465410i	0.845605	−	0.533810i	$-0.179240\pi$
$277$	13.4164	+	7.74597i	0.806114	+	0.465410i	−0.0394907	+	0.999220i	$0.512574\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.3923			0.619953			0.309976	−	0.950744i	$-0.399679\pi$
$281$	10.3923			0.619953			0.309976	+	0.950744i	$0.399679\pi$
$282$		0			0
$283$	−13.4164	−	7.74597i	−0.797523	−	0.460450i	0.0450815	−	0.998983i	$-0.485645\pi$
$283$	−13.4164	−	7.74597i	−0.797523	−	0.460450i	−0.842604	+	0.538533i	$0.818979\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.19615	+	27.0000i	0.306719	+	1.59376i
$288$		0			0
$289$	2.50000	+	4.33013i	0.147059	+	0.254713i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	11.1803i		−	0.653162i	−0.945169	−	0.326581i	$-0.894103\pi$
$293$		−	11.1803i		−	0.653162i	0.945169	−	0.326581i	$-0.105897\pi$
$294$		0			0
$295$	5.00000			0.291111
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.66025	+	15.0000i	−0.495885	+	0.858898i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.1244	+	21.0000i	−0.687509	+	1.19080i	0.285132	+	0.958488i	$0.407963\pi$
$311$	−12.1244	+	21.0000i	−0.687509	+	1.19080i	−0.972641	+	0.232313i	$0.925371\pi$
$312$		0			0
$313$	−11.5000	−	19.9186i	−0.650018	−	1.12586i	−0.983118	−	0.182973i	$-0.941428\pi$
$313$	−11.5000	−	19.9186i	−0.650018	−	1.12586i	0.333099	−	0.942892i	$-0.391906\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.93649	−	1.11803i	0.108764	−	0.0627950i	−0.444631	−	0.895714i	$-0.646665\pi$
$317$	1.93649	−	1.11803i	0.108764	−	0.0627950i	0.553395	+	0.832919i	$0.313332\pi$
$318$		0			0
$319$	−2.50000	+	4.33013i	−0.139973	+	0.242441i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	26.8328i		−	1.49302i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.92820	+	6.00000i	−0.381964	+	0.330791i
$330$		0			0
$331$	−13.4164	+	7.74597i	−0.737432	+	0.425757i	−0.821135	−	0.570734i	$-0.806659\pi$
$331$	−13.4164	+	7.74597i	−0.737432	+	0.425757i	0.0837026	+	0.996491i	$0.473325\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	17.3205			0.946320
$336$		0			0
$337$	−19.0000			−1.03500			−0.517498	−	0.855684i	$-0.673136\pi$
$337$	−19.0000			−1.03500			−0.517498	+	0.855684i	$0.673136\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.93649	+	1.11803i	−0.104867	+	0.0605449i
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.4919	−	8.94427i	−0.831651	−	0.480154i	0.0227669	−	0.999741i	$-0.492752\pi$
$347$	−15.4919	−	8.94427i	−0.831651	−	0.480154i	−0.854417	+	0.519587i	$0.826086\pi$
$348$		0			0
$349$			23.2379i			1.24390i	0.783058	+	0.621948i	$0.213659\pi$
$349$			23.2379i			1.24390i	−0.783058	+	0.621948i	$0.786341\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.8564	+	24.0000i	−0.737502	+	1.27739i	0.216115	+	0.976368i	$0.430661\pi$
$353$	−13.8564	+	24.0000i	−0.737502	+	1.27739i	−0.953617	+	0.301023i	$0.902672\pi$
$354$		0			0
$355$	20.1246	−	11.6190i	1.06810	−	0.616670i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.66025	−	15.0000i	−0.457071	−	0.791670i	0.541734	−	0.840550i	$-0.317768\pi$
$359$	−8.66025	−	15.0000i	−0.457071	−	0.791670i	−0.998805	+	0.0488803i	$0.984435\pi$
$360$		0			0
$361$	20.5000	−	35.5070i	1.07895	−	1.86879i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	22.3607i		−	1.17041i
$366$		0			0
$367$	2.50000	−	4.33013i	0.130499	−	0.226031i	−0.793370	−	0.608740i	$-0.791675\pi$
$367$	2.50000	−	4.33013i	0.130499	−	0.226031i	0.923869	+	0.382709i	$0.125009\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.68246	+	27.9508i	−0.502688	+	1.45114i
$372$		0			0
$373$	−6.70820	+	3.87298i	−0.347338	+	0.200535i	−0.663512	−	0.748166i	$-0.730935\pi$
$373$	−6.70820	+	3.87298i	−0.347338	+	0.200535i	0.316174	+	0.948701i	$0.397602\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			23.2379i			1.19365i	0.802371	+	0.596825i	$0.203571\pi$
$379$			23.2379i			1.19365i	−0.802371	+	0.596825i	$0.796429\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.46410	−	6.00000i	−0.177007	−	0.306586i	0.763847	−	0.645398i	$-0.223308\pi$
$383$	−3.46410	−	6.00000i	−0.177007	−	0.306586i	−0.940854	+	0.338812i	$0.889975\pi$
$384$		0			0
$385$	2.50000	+	12.9904i	0.127412	+	0.662051i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.87298	+	2.23607i	0.196368	+	0.113373i	0.594960	−	0.803755i	$-0.297168\pi$
$389$	3.87298	+	2.23607i	0.196368	+	0.113373i	−0.398592	+	0.917128i	$0.630501\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−25.1744	−	14.5344i	−1.26666	−	0.731307i
$396$		0			0
$397$	−26.8328	+	15.4919i	−1.34670	+	0.777518i	−0.987781	−	0.155851i	$-0.950188\pi$
$397$	−26.8328	+	15.4919i	−1.34670	+	0.777518i	−0.358920	+	0.933368i	$0.616855\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−17.3205	−	30.0000i	−0.864945	−	1.49813i	−0.867102	−	0.498131i	$-0.834020\pi$
$401$	−17.3205	−	30.0000i	−0.864945	−	1.49813i	0.00215698	−	0.999998i	$-0.499313\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−17.3205			−0.858546
$408$		0			0
$409$	3.50000	−	6.06218i	0.173064	−	0.299755i	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	3.50000	−	6.06218i	0.173064	−	0.299755i	0.939490	+	0.342578i	$0.111300\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.93649	+	5.59017i	−0.0952885	+	0.275074i
$414$		0			0
$415$	12.5000	+	21.6506i	0.613601	+	1.06279i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	35.7771i		−	1.74783i	−0.486083	−	0.873913i	$-0.661575\pi$
$419$		−	35.7771i		−	1.74783i	0.486083	−	0.873913i	$-0.338425\pi$
$420$		0			0
$421$		−	23.2379i		−	1.13255i	−0.824218	−	0.566273i	$-0.808385\pi$
$421$		−	23.2379i		−	1.13255i	0.824218	−	0.566273i	$-0.191615\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−13.4164	−	15.4919i	−0.649265	−	0.749707i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.1244	+	21.0000i	−0.584010	+	1.01153i	0.410988	+	0.911641i	$0.365184\pi$
$431$	−12.1244	+	21.0000i	−0.584010	+	1.01153i	−0.994998	+	0.0998939i	$0.968150\pi$
$432$		0			0
$433$	−22.0000			−1.05725			−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000			−1.05725			−0.528626	+	0.848855i	$0.677293\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−46.4758	−	26.8328i	−2.22324	−	1.28359i
$438$		0			0
$439$	−3.50000	−	6.06218i	−0.167046	−	0.289332i	0.770334	−	0.637641i	$-0.220089\pi$
$439$	−3.50000	−	6.06218i	−0.167046	−	0.289332i	−0.937380	+	0.348309i	$0.886756\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	21.3014	−	12.2984i	1.01206	−	0.584313i	0.100266	−	0.994961i	$-0.468031\pi$
$443$	21.3014	−	12.2984i	1.01206	−	0.584313i	0.911794	+	0.410647i	$0.134697\pi$
$444$		0			0
$445$	−26.8328	−	15.4919i	−1.27200	−	0.734388i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−20.7846			−0.980886			−0.490443	−	0.871473i	$-0.663165\pi$
$449$	−20.7846			−0.980886			−0.490443	+	0.871473i	$0.663165\pi$
$450$		0			0
$451$	20.1246	+	11.6190i	0.947631	+	0.547115i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2.50000	−	4.33013i	−0.116945	−	0.202555i	0.801611	−	0.597847i	$-0.203977\pi$
$457$	−2.50000	−	4.33013i	−0.116945	−	0.202555i	−0.918556	+	0.395292i	$0.870643\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			22.3607i			1.04144i	0.853727	+	0.520720i	$0.174337\pi$
$461$			22.3607i			1.04144i	−0.853727	+	0.520720i	$0.825663\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.74597	+	4.47214i	−0.358441	+	0.206946i	−0.668397	−	0.743805i	$-0.733019\pi$
$467$	−7.74597	+	4.47214i	−0.358441	+	0.206946i	0.309956	+	0.950751i	$0.399686\pi$
$468$		0			0
$469$	−6.70820	+	19.3649i	−0.309756	+	0.894189i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.92820	+	12.0000i	−0.316558	+	0.548294i	−0.979767	−	0.200140i	$-0.935860\pi$
$479$	−6.92820	+	12.0000i	−0.316558	+	0.548294i	0.663210	+	0.748434i	$0.269194\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.93649	+	1.11803i	−0.0879316	+	0.0507673i
$486$		0			0
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	−0.877132	−	0.480250i	$-0.840546\pi$
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	0.854475	+	0.519493i	$0.173879\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	29.0689i		−	1.31186i	−0.754822	−	0.655930i	$-0.772277\pi$
$491$		−	29.0689i		−	1.31186i	0.754822	−	0.655930i	$-0.227723\pi$
$492$		0			0
$493$	6.70820	+	3.87298i	0.302122	+	0.174430i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.19615	+	27.0000i	0.233079	+	1.21112i
$498$		0			0
$499$	26.8328	−	15.4919i	1.20120	−	0.693514i	0.240379	−	0.970679i	$-0.422728\pi$
$499$	26.8328	−	15.4919i	1.20120	−	0.693514i	0.960822	+	0.277165i	$0.0893948\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	31.1769			1.39011			0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769			1.39011			0.695055	+	0.718957i	$0.255380\pi$
$504$		0			0
$505$	10.0000			0.444994
$506$		0			0
$507$		0			0
$508$		0			0
$509$	25.1744	−	14.5344i	1.11584	−	0.644228i	0.175501	−	0.984479i	$-0.443846\pi$
$509$	25.1744	−	14.5344i	1.11584	−	0.644228i	0.940334	+	0.340251i	$0.110512\pi$
$510$		0			0
$511$	25.0000	+	8.66025i	1.10593	+	0.383107i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	15.4919	+	8.94427i	0.682656	+	0.394132i
$516$		0			0
$517$			7.74597i			0.340667i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−13.8564	+	24.0000i	−0.607060	+	1.05146i	0.384662	+	0.923057i	$0.374318\pi$
$521$	−13.8564	+	24.0000i	−0.607060	+	1.05146i	−0.991722	+	0.128402i	$0.959015\pi$
$522$		0			0
$523$	26.8328	−	15.4919i	1.17332	−	0.677415i	0.218858	−	0.975757i	$-0.429767\pi$
$523$	26.8328	−	15.4919i	1.17332	−	0.677415i	0.954459	+	0.298342i	$0.0964335\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.73205	+	3.00000i	0.0754493	+	0.130682i
$528$		0			0
$529$	−12.5000	+	21.6506i	−0.543478	+	0.941332i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	12.5000	−	21.6506i	0.540422	−	0.936039i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−15.4919	−	2.23607i	−0.667285	−	0.0963143i
$540$		0			0
$541$	13.4164	−	7.74597i	0.576816	−	0.333025i	−0.183051	−	0.983103i	$-0.558597\pi$
$541$	13.4164	−	7.74597i	0.576816	−	0.333025i	0.759867	+	0.650078i	$0.225264\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−17.3205			−0.741929
$546$		0			0
$547$		−	23.2379i		−	0.993581i	−0.867871	−	0.496790i	$-0.834512\pi$
$547$		−	23.2379i		−	0.993581i	0.867871	−	0.496790i	$-0.165488\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.66025	+	15.0000i	0.368939	+	0.639021i
$552$		0			0
$553$	26.0000	−	22.5167i	1.10563	−	0.957506i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.5554	−	7.82624i	−0.574362	−	0.331608i	0.184527	−	0.982827i	$-0.440925\pi$
$557$	−13.5554	−	7.82624i	−0.574362	−	0.331608i	−0.758890	+	0.651219i	$0.774258\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−32.9204	−	19.0066i	−1.38743	−	0.801032i	−0.394403	−	0.918938i	$-0.629049\pi$
$563$	−32.9204	−	19.0066i	−1.38743	−	0.801032i	−0.993025	+	0.117906i	$0.962382\pi$
$564$		0			0
$565$	−20.1246	+	11.6190i	−0.846649	+	0.488813i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.73205	−	3.00000i	−0.0726113	−	0.125767i	0.827434	−	0.561563i	$-0.189800\pi$
$569$	−1.73205	−	3.00000i	−0.0726113	−	0.125767i	−0.900045	+	0.435797i	$0.856467\pi$
$570$		0			0
$571$	6.70820	+	3.87298i	0.280730	+	0.162079i	0.633754	−	0.773535i	$-0.281513\pi$
$571$	6.70820	+	3.87298i	0.280730	+	0.162079i	−0.353024	+	0.935614i	$0.614847\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−17.5000	+	30.3109i	−0.728535	+	1.26186i	0.228968	+	0.973434i	$0.426465\pi$
$577$	−17.5000	+	30.3109i	−0.728535	+	1.26186i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−29.0474	+	5.59017i	−1.20509	+	0.231919i
$582$		0			0
$583$	12.5000	+	21.6506i	0.517697	+	0.896678i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			38.0132i			1.56897i	0.620147	+	0.784485i	$0.287073\pi$
$587$			38.0132i			1.56897i	−0.620147	+	0.784485i	$0.712927\pi$
$588$		0			0
$589$			7.74597i			0.319167i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3.46410	+	6.00000i	0.142254	+	0.246390i	0.928345	−	0.371720i	$-0.121232\pi$
$593$	3.46410	+	6.00000i	0.142254	+	0.246390i	−0.786091	+	0.618110i	$0.787898\pi$
$594$		0			0
$595$	20.1246	−	3.87298i	0.825029	−	0.158777i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.73205	+	3.00000i	−0.0707697	+	0.122577i	−0.899239	−	0.437458i	$-0.855879\pi$
$599$	−1.73205	+	3.00000i	−0.0707697	+	0.122577i	0.828469	+	0.560035i	$0.189212\pi$
$600$		0			0
$601$	−7.00000			−0.285536			−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000			−0.285536			−0.142768	+	0.989756i	$0.545600\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.6190	−	6.70820i	−0.472377	−	0.272727i
$606$		0			0
$607$	14.5000	+	25.1147i	0.588537	+	1.01938i	0.994424	+	0.105453i	$0.0336291\pi$
$607$	14.5000	+	25.1147i	0.588537	+	1.01938i	−0.405887	+	0.913923i	$0.633038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−26.8328	−	15.4919i	−1.08377	−	0.625713i	−0.151857	−	0.988402i	$-0.548525\pi$
$613$	−26.8328	−	15.4919i	−1.08377	−	0.625713i	−0.931910	+	0.362689i	$0.881859\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−31.1769			−1.25514			−0.627568	−	0.778562i	$-0.715949\pi$
$617$	−31.1769			−1.25514			−0.627568	+	0.778562i	$0.715949\pi$
$618$		0			0
$619$	6.70820	+	3.87298i	0.269625	+	0.155668i	0.628717	−	0.777634i	$-0.283580\pi$
$619$	6.70820	+	3.87298i	0.269625	+	0.155668i	−0.359092	+	0.933302i	$0.616914\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	27.7128	−	24.0000i	1.11029	−	0.961540i
$624$		0			0
$625$	12.5000	+	21.6506i	0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			26.8328i			1.06989i
$630$		0			0
$631$	−11.0000			−0.437903			−0.218952	−	0.975736i	$-0.570264\pi$
$631$	−11.0000			−0.437903			−0.218952	+	0.975736i	$0.570264\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9.68246	+	5.59017i	−0.384237	+	0.221839i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	22.5167	−	39.0000i	0.889355	−	1.54041i	0.0487148	−	0.998813i	$-0.484487\pi$
$641$	22.5167	−	39.0000i	0.889355	−	1.54041i	0.840640	−	0.541595i	$-0.182179\pi$
$642$		0			0
$643$			23.2379i			0.916413i	0.888846	+	0.458207i	$0.151508\pi$
$643$			23.2379i			0.916413i	−0.888846	+	0.458207i	$0.848492\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−22.5167	+	39.0000i	−0.885221	+	1.53325i	−0.0397614	+	0.999209i	$0.512660\pi$
$647$	−22.5167	+	39.0000i	−0.885221	+	1.53325i	−0.845460	+	0.534039i	$0.820674\pi$
$648$		0			0
$649$	2.50000	+	4.33013i	0.0981336	+	0.169972i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	36.7933	−	21.2426i	1.43983	−	0.831289i	0.441997	−	0.897016i	$-0.354270\pi$
$653$	36.7933	−	21.2426i	1.43983	−	0.831289i	0.997838	+	0.0657275i	$0.0209368\pi$
$654$		0			0
$655$	12.5000	−	21.6506i	0.488415	−	0.845960i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			17.8885i			0.696839i	0.937339	+	0.348419i	$0.113281\pi$
$659$			17.8885i			0.696839i	−0.937339	+	0.348419i	$0.886719\pi$
$660$		0			0
$661$	13.4164	+	7.74597i	0.521838	+	0.301283i	0.737686	−	0.675144i	$-0.235918\pi$
$661$	13.4164	+	7.74597i	0.521838	+	0.301283i	−0.215848	+	0.976427i	$0.569252\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	43.3013	+	15.0000i	1.67915	+	0.581675i
$666$		0			0
$667$	13.4164	−	7.74597i	0.519485	−	0.299925i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.3205			−0.668651
$672$		0			0
$673$	−7.00000			−0.269830			−0.134915	−	0.990857i	$-0.543076\pi$
$673$	−7.00000			−0.269830			−0.134915	+	0.990857i	$0.543076\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−21.3014	+	12.2984i	−0.818680	+	0.472665i	−0.849961	−	0.526846i	$-0.823375\pi$
$677$	−21.3014	+	12.2984i	−0.818680	+	0.472665i	0.0312813	+	0.999511i	$0.490041\pi$
$678$		0			0
$679$	−0.500000	−	2.59808i	−0.0191882	−	0.0997050i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−9.68246	−	5.59017i	−0.370489	−	0.213902i	0.303183	−	0.952932i	$-0.401951\pi$
$683$	−9.68246	−	5.59017i	−0.370489	−	0.213902i	−0.673672	+	0.739030i	$0.735284\pi$
$684$		0			0
$685$		−	7.74597i		−	0.295958i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−33.5410	+	19.3649i	−1.27596	+	0.736676i	−0.976103	−	0.217308i	$-0.930272\pi$
$691$	−33.5410	+	19.3649i	−1.27596	+	0.736676i	−0.299857	+	0.953984i	$0.596939\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	18.0000	−	31.1769i	0.681799	−	1.18091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	24.5967i		−	0.929006i	−0.885571	−	0.464503i	$-0.846233\pi$
$701$		−	24.5967i		−	0.929006i	0.885571	−	0.464503i	$-0.153767\pi$
$702$		0			0
$703$	−30.0000	+	51.9615i	−1.13147	+	1.95977i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.87298	+	11.1803i	−0.145659	+	0.420480i
$708$		0			0
$709$	13.4164	−	7.74597i	0.503864	−	0.290906i	−0.226444	−	0.974024i	$-0.572710\pi$
$709$	13.4164	−	7.74597i	0.503864	−	0.290906i	0.730308	+	0.683118i	$0.239377\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.92820			0.259463
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.2487	−	42.0000i	−0.904324	−	1.56634i	−0.821822	−	0.569745i	$-0.807042\pi$
$719$	−24.2487	−	42.0000i	−0.904324	−	1.56634i	−0.0825027	−	0.996591i	$-0.526291\pi$
$720$		0			0
$721$	−16.0000	+	13.8564i	−0.595871	+	0.516040i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−29.0000			−1.07555			−0.537775	−	0.843088i	$-0.680735\pi$
$727$	−29.0000			−1.07555			−0.537775	+	0.843088i	$0.680735\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	13.4164	−	7.74597i	0.495546	−	0.286104i	−0.231326	−	0.972876i	$-0.574306\pi$
$733$	13.4164	−	7.74597i	0.495546	−	0.286104i	0.726872	+	0.686772i	$0.240973\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8.66025	+	15.0000i	0.319005	+	0.552532i
$738$		0			0
$739$	−13.4164	−	7.74597i	−0.493531	−	0.284940i	0.232507	−	0.972595i	$-0.425307\pi$
$739$	−13.4164	−	7.74597i	−0.493531	−	0.284940i	−0.726038	+	0.687655i	$0.758640\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.3923			−0.381257			−0.190628	−	0.981662i	$-0.561053\pi$
$743$	−10.3923			−0.381257			−0.190628	+	0.981662i	$0.561053\pi$
$744$		0			0
$745$	25.0000	−	43.3013i	0.915929	−	1.58644i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	19.3649	+	22.3607i	0.707579	+	0.817041i
$750$		0			0
$751$	−21.5000	−	37.2391i	−0.784546	−	1.35887i	−0.929270	−	0.369402i	$-0.879563\pi$
$751$	−21.5000	−	37.2391i	−0.784546	−	1.35887i	0.144724	−	0.989472i	$-0.453771\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			15.6525i			0.569652i
$756$		0			0
$757$			23.2379i			0.844596i	0.906457	+	0.422298i	$0.138776\pi$
$757$			23.2379i			0.844596i	−0.906457	+	0.422298i	$0.861224\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.46410	+	6.00000i	0.125574	+	0.217500i	0.921957	−	0.387292i	$-0.126590\pi$
$761$	3.46410	+	6.00000i	0.125574	+	0.217500i	−0.796383	+	0.604792i	$0.793256\pi$
$762$		0			0
$763$	6.70820	−	19.3649i	0.242853	−	0.701057i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−31.0000			−1.11789			−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000			−1.11789			−0.558944	+	0.829205i	$0.688793\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.87298	+	2.23607i	0.139302	+	0.0804258i	0.568031	−	0.823007i	$-0.307705\pi$
$773$	3.87298	+	2.23607i	0.139302	+	0.0804258i	−0.428730	+	0.903433i	$0.641039\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	69.7137	−	40.2492i	2.49775	−	1.44208i
$780$		0			0
$781$	20.1246	+	11.6190i	0.720115	+	0.415759i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	34.6410			1.23639
$786$		0			0
$787$	−33.5410	−	19.3649i	−1.19561	−	0.690285i	−0.236035	−	0.971745i	$-0.575848\pi$
$787$	−33.5410	−	19.3649i	−1.19561	−	0.690285i	−0.959573	+	0.281460i	$0.909181\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5.19615	−	27.0000i	−0.184754	−	0.960009i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$			15.6525i			0.554439i	0.960807	+	0.277220i	$0.0894129\pi$
$797$			15.6525i			0.554439i	−0.960807	+	0.277220i	$0.910587\pi$
$798$		0			0
$799$	12.0000			0.424529
$800$		0			0
$801$		0			0
$802$		0			0
$803$	19.3649	−	11.1803i	0.683373	−	0.394546i
$804$		0			0
$805$	13.4164	−	38.7298i	0.472866	−	1.36505i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	17.3205	−	30.0000i	0.608957	−	1.05474i	−0.382456	−	0.923974i	$-0.624922\pi$
$809$	17.3205	−	30.0000i	0.608957	−	1.05474i	0.991413	−	0.130770i	$-0.0417450\pi$
$810$		0			0
$811$			23.2379i			0.815993i	0.912983	+	0.407997i	$0.133772\pi$
$811$			23.2379i			0.815993i	−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−17.3205	+	30.0000i	−0.606711	+	1.05085i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	13.5554	−	7.82624i	0.473088	−	0.273138i	−0.244443	−	0.969664i	$-0.578605\pi$
$821$	13.5554	−	7.82624i	0.473088	−	0.273138i	0.717532	+	0.696526i	$0.245272\pi$
$822$		0			0
$823$	10.0000	−	17.3205i	0.348578	−	0.603755i	−0.637419	−	0.770517i	$-0.719998\pi$
$823$	10.0000	−	17.3205i	0.348578	−	0.603755i	0.985997	+	0.166762i	$0.0533313\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			24.5967i			0.855313i	0.903941	+	0.427656i	$0.140661\pi$
$827$			24.5967i			0.855313i	−0.903941	+	0.427656i	$0.859339\pi$
$828$		0			0
$829$	−6.70820	−	3.87298i	−0.232986	−	0.134514i	0.378963	−	0.925412i	$-0.376281\pi$
$829$	−6.70820	−	3.87298i	−0.232986	−	0.134514i	−0.611949	+	0.790898i	$0.709614\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.46410	+	24.0000i	−0.120024	+	0.831551i
$834$		0			0
$835$	−20.1246	+	11.6190i	−0.696441	+	0.402090i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	31.1769			1.07635			0.538173	−	0.842834i	$-0.319115\pi$
$839$	31.1769			1.07635			0.538173	+	0.842834i	$0.319115\pi$
$840$		0			0
$841$	24.0000			0.827586
$842$		0			0
$843$		0			0
$844$		0			0
$845$	25.1744	−	14.5344i	0.866025	−	0.500000i
$846$		0			0
$847$	12.0000	−	10.3923i	0.412325	−	0.357084i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	46.4758	+	26.8328i	1.59317	+	0.919817i
$852$		0			0
$853$			23.2379i			0.795651i	0.917461	+	0.397825i	$0.130235\pi$
$853$			23.2379i			0.795651i	−0.917461	+	0.397825i	$0.869765\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.1244	−	21.0000i	0.414160	−	0.717346i	−0.581180	−	0.813775i	$-0.697409\pi$
$857$	12.1244	−	21.0000i	0.414160	−	0.717346i	0.995340	+	0.0964289i	$0.0307420\pi$
$858$		0			0
$859$	−13.4164	+	7.74597i	−0.457762	+	0.264289i	−0.711103	−	0.703088i	$-0.751804\pi$
$859$	−13.4164	+	7.74597i	−0.457762	+	0.264289i	0.253341	+	0.967377i	$0.418471\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13.8564	−	24.0000i	−0.471678	−	0.816970i	0.527797	−	0.849370i	$-0.323018\pi$
$863$	−13.8564	−	24.0000i	−0.471678	−	0.816970i	−0.999475	+	0.0324008i	$0.989685\pi$
$864$		0			0
$865$	−5.00000	+	8.66025i	−0.170005	+	0.294457i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	29.0689i		−	0.986094i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−29.0474	+	5.59017i	−0.981981	+	0.188982i
$876$		0			0
$877$	−26.8328	+	15.4919i	−0.906080	+	0.523125i	−0.879168	−	0.476512i	$-0.841901\pi$
$877$	−26.8328	+	15.4919i	−0.906080	+	0.523125i	−0.0269120	+	0.999638i	$0.508567\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	10.3923			0.350126			0.175063	−	0.984557i	$-0.443987\pi$
$881$	10.3923			0.350126			0.175063	+	0.984557i	$0.443987\pi$
$882$		0			0
$883$			23.2379i			0.782018i	0.920387	+	0.391009i	$0.127874\pi$
$883$			23.2379i			0.782018i	−0.920387	+	0.391009i	$0.872126\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−13.8564	−	24.0000i	−0.465253	−	0.805841i	0.533960	−	0.845510i	$-0.320703\pi$
$887$	−13.8564	−	24.0000i	−0.465253	−	0.805841i	−0.999213	+	0.0396684i	$0.987370\pi$
$888$		0			0
$889$	−2.50000	−	12.9904i	−0.0838473	−	0.435683i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	23.2379	+	13.4164i	0.777627	+	0.448963i
$894$		0			0
$895$	20.0000			0.668526
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.93649	−	1.11803i	−0.0645856	−	0.0372885i
$900$		0			0
$901$	33.5410	−	19.3649i	1.11741	−	0.645139i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−25.9808	−	45.0000i	−0.863630	−	1.49585i
$906$		0			0
$907$	−13.4164	−	7.74597i	−0.445485	−	0.257201i	0.260437	−	0.965491i	$-0.416133\pi$
$907$	−13.4164	−	7.74597i	−0.445485	−	0.257201i	−0.705921	+	0.708290i	$0.749467\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	10.3923			0.344312			0.172156	−	0.985070i	$-0.444927\pi$
$911$	10.3923			0.344312			0.172156	+	0.985070i	$0.444927\pi$
$912$		0			0
$913$	−12.5000	+	21.6506i	−0.413690	+	0.716531i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	19.3649	+	22.3607i	0.639486	+	0.738415i
$918$		0			0
$919$	4.00000	+	6.92820i	0.131948	+	0.228540i	0.924427	−	0.381358i	$-0.124544\pi$
$919$	4.00000	+	6.92820i	0.131948	+	0.228540i	−0.792480	+	0.609898i	$0.791210\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.3205	−	30.0000i	−0.568267	−	0.984268i	−0.996737	−	0.0807121i	$-0.974281\pi$
$929$	−17.3205	−	30.0000i	−0.568267	−	0.984268i	0.428470	−	0.903556i	$-0.359053\pi$
$930$		0			0
$931$	−33.5410	+	42.6028i	−1.09926	+	1.39625i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	8.66025	−	15.0000i	0.283221	−	0.490552i
$936$		0			0
$937$	5.00000			0.163343			0.0816714	−	0.996659i	$-0.473974\pi$
$937$	5.00000			0.163343			0.0816714	+	0.996659i	$0.473974\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9.68246	+	5.59017i	0.315639	+	0.182234i	0.649447	−	0.760407i	$-0.275000\pi$
$941$	9.68246	+	5.59017i	0.315639	+	0.182234i	−0.333808	+	0.942641i	$0.608334\pi$
$942$		0			0
$943$	−36.0000	−	62.3538i	−1.17232	−	2.03052i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−30.9839	+	17.8885i	−1.00684	+	0.581300i	−0.910265	−	0.414026i	$-0.864122\pi$
$947$	−30.9839	+	17.8885i	−1.00684	+	0.581300i	−0.0965754	+	0.995326i	$0.530789\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−20.7846			−0.673280			−0.336640	−	0.941634i	$-0.609290\pi$
$953$	−20.7846			−0.673280			−0.336640	+	0.941634i	$0.609290\pi$
$954$		0			0
$955$	−13.4164	−	7.74597i	−0.434145	−	0.250654i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	8.66025	+	3.00000i	0.279654	+	0.0968751i
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			38.0132i			1.22369i
$966$		0			0
$967$	19.0000			0.610999			0.305499	−	0.952192i	$-0.401177\pi$
$967$	19.0000			0.610999			0.305499	+	0.952192i	$0.401177\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	9.68246	−	5.59017i	0.310725	−	0.179397i	−0.336526	−	0.941674i	$-0.609252\pi$
$971$	9.68246	−	5.59017i	0.310725	−	0.179397i	0.647251	+	0.762277i	$0.275919\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6.92820	−	12.0000i	0.221653	−	0.383914i	−0.733657	−	0.679520i	$-0.762188\pi$
$977$	6.92820	−	12.0000i	0.221653	−	0.383914i	0.955310	+	0.295606i	$0.0955215\pi$
$978$		0			0
$979$		−	30.9839i		−	0.990249i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	19.0526	−	33.0000i	0.607682	−	1.05254i	−0.383939	−	0.923358i	$-0.625433\pi$
$983$	19.0526	−	33.0000i	0.607682	−	1.05254i	0.991621	−	0.129178i	$-0.0412339\pi$
$984$		0			0
$985$	−5.00000	−	8.66025i	−0.159313	−	0.275939i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.500000	+	0.866025i	−0.0158830	+	0.0275102i	−0.873858	−	0.486182i	$-0.838389\pi$
$991$	−0.500000	+	0.866025i	−0.0158830	+	0.0275102i	0.857975	+	0.513692i	$0.171723\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	17.8885i		−	0.567105i
$996$		0			0
$997$	−6.70820	−	3.87298i	−0.212451	−	0.122659i	0.389999	−	0.920815i	$-0.372475\pi$
$997$	−6.70820	−	3.87298i	−0.212451	−	0.122659i	−0.602450	+	0.798157i	$0.705809\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.cr.a.1873.4		8
3.2	odd	2	inner	2016.2.cr.a.1873.1		8
4.3	odd	2		504.2.cj.a.109.4	yes	8
7.2	even	3	inner	2016.2.cr.a.1297.2		8
8.3	odd	2		504.2.cj.a.109.2	yes	8
8.5	even	2	inner	2016.2.cr.a.1873.2		8
12.11	even	2		504.2.cj.a.109.1	yes	8
21.2	odd	6	inner	2016.2.cr.a.1297.3		8
24.5	odd	2	inner	2016.2.cr.a.1873.3		8
24.11	even	2		504.2.cj.a.109.3	yes	8
28.23	odd	6		504.2.cj.a.37.2	yes	8
56.37	even	6	inner	2016.2.cr.a.1297.4		8
56.51	odd	6		504.2.cj.a.37.4	yes	8
84.23	even	6		504.2.cj.a.37.3	yes	8
168.107	even	6		504.2.cj.a.37.1	✓	8
168.149	odd	6	inner	2016.2.cr.a.1297.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.cj.a.37.1	✓	8	168.107	even	6
504.2.cj.a.37.2	yes	8	28.23	odd	6
504.2.cj.a.37.3	yes	8	84.23	even	6
504.2.cj.a.37.4	yes	8	56.51	odd	6
504.2.cj.a.109.1	yes	8	12.11	even	2
504.2.cj.a.109.2	yes	8	8.3	odd	2
504.2.cj.a.109.3	yes	8	24.11	even	2
504.2.cj.a.109.4	yes	8	4.3	odd	2
2016.2.cr.a.1297.1		8	168.149	odd	6	inner
2016.2.cr.a.1297.2		8	7.2	even	3	inner
2016.2.cr.a.1297.3		8	21.2	odd	6	inner
2016.2.cr.a.1297.4		8	56.37	even	6	inner
2016.2.cr.a.1873.1		8	3.2	odd	2	inner
2016.2.cr.a.1873.2		8	8.5	even	2	inner
2016.2.cr.a.1873.3		8	24.5	odd	2	inner
2016.2.cr.a.1873.4		8	1.1	even	1	trivial

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.cr (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.93649 - 1.11803i) q^{5} +(0.500000 + 2.59808i) q^{7} +O(q^{10})\)	\(q+(1.93649 - 1.11803i) q^{5} +(0.500000 + 2.59808i) q^{7} +(1.93649 + 1.11803i) q^{11} +(1.73205 - 3.00000i) q^{17} +(6.70820 - 3.87298i) q^{19} +(-3.46410 - 6.00000i) q^{23} +2.23607i q^{29} +(-0.500000 + 0.866025i) q^{31} +(3.87298 + 4.47214i) q^{35} +(-6.70820 + 3.87298i) q^{37} +10.3923 q^{41} +(1.73205 + 3.00000i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(9.68246 + 5.59017i) q^{53} +5.00000 q^{55} +(1.93649 + 1.11803i) q^{59} +(-6.70820 + 3.87298i) q^{61} +(6.70820 + 3.87298i) q^{67} +10.3923 q^{71} +(5.00000 - 8.66025i) q^{73} +(-1.93649 + 5.59017i) q^{77} +(-6.50000 - 11.2583i) q^{79} +11.1803i q^{83} -7.74597i q^{85} +(-6.92820 - 12.0000i) q^{89} +(8.66025 - 15.0000i) q^{95} -1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7}+O(q^{10}) \) 8 * q + 4 * q^7	\( 8 q + 4 q^{7} - 4 q^{31} - 52 q^{49} + 40 q^{55} + 40 q^{73} - 52 q^{79} - 8 q^{97}+O(q^{100}) \) 8 * q + 4 * q^7 - 4 * q^31 - 52 * q^49 + 40 * q^55 + 40 * q^73 - 52 * q^79 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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