LMFDB - Embedded newform 2700.2.s.d.2449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(1549,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2700.1549");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.s (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:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 3x^{14} + 5x^{12} + 15x^{10} + 45x^{8} + 60x^{6} + 80x^{4} + 192x^{2} + 256 $ x^16 + 3x^14 + 5x^12 + 15x^10 + 45x^8 + 60x^6 + 80x^4 + 192*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{10} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2449.4
Root			$0.263711 - 1.38941i$ of defining polynomial
Character	$\chi$	$=$	2700.2449
Dual form			2700.2.s.d.1549.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0748933 - 0.0432397i) q^{7} +O(q^{10})$	$q+(-0.0748933 - 0.0432397i) q^{7} +(-0.456760 + 0.791132i) q^{11} +(-2.27331 + 1.31249i) q^{13} -2.08648i q^{17} -4.93847 q^{19} +(7.34128 - 4.23849i) q^{23} +(-1.19899 + 2.07671i) q^{29} +(-1.81249 - 3.13933i) q^{31} -5.85199i q^{37} +(3.32497 + 5.75902i) q^{41} +(-7.14469 - 4.12499i) q^{43} +(2.32831 + 1.34425i) q^{47} +(-3.49626 - 6.05570i) q^{49} -5.73642i q^{53} +(-6.16922 - 10.6854i) q^{59} +(3.16823 - 5.48753i) q^{61} +(5.33946 - 3.08274i) q^{67} +12.3905 q^{71} -5.31349i q^{73} +(0.0684166 - 0.0395003i) q^{77} +(-6.72394 + 11.6462i) q^{79} +(5.26457 + 3.03950i) q^{83} -8.13440 q^{89} +0.227007 q^{91} +(-9.61459 - 5.55098i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 6 q^{11} + 16 q^{19} - 18 q^{29} - 4 q^{31} - 18 q^{41} + 18 q^{49} - 30 q^{59} + 2 q^{61} + 48 q^{71} - 14 q^{79} + 12 q^{89} - 44 q^{91}+O(q^{100}) $ 16 * q - 6 * q^11 + 16 * q^19 - 18 * q^29 - 4 * q^31 - 18 * q^41 + 18 * q^49 - 30 * q^59 + 2 * q^61 + 48 * q^71 - 14 * q^79 + 12 * q^89 - 44 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$-1$

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
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.0748933	−	0.0432397i	−0.0283070	−	0.0163431i	0.485780	−	0.874081i	$-0.338536\pi$
$7$	−0.0748933	−	0.0432397i	−0.0283070	−	0.0163431i	−0.514087	+	0.857738i	$0.671869\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.456760	+	0.791132i	−0.137718	+	0.238535i	−0.926633	−	0.375968i	$-0.877310\pi$
$11$	−0.456760	+	0.791132i	−0.137718	+	0.238535i	0.788914	+	0.614503i	$0.210644\pi$
$12$		0			0
$13$	−2.27331	+	1.31249i	−0.630501	+	0.364020i	−0.780946	−	0.624598i	$-0.785263\pi$
$13$	−2.27331	+	1.31249i	−0.630501	+	0.364020i	0.150445	+	0.988618i	$0.451929\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2.08648i		−	0.506046i	−0.967460	−	0.253023i	$-0.918575\pi$
$17$		−	2.08648i		−	0.506046i	0.967460	−	0.253023i	$-0.0814248\pi$
$18$		0			0
$19$	−4.93847			−1.13296			−0.566482	−	0.824074i	$-0.691696\pi$
$19$	−4.93847			−1.13296			−0.566482	+	0.824074i	$0.691696\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.34128	−	4.23849i	1.53076	−	0.883786i	0.531436	−	0.847098i	$-0.321653\pi$
$23$	7.34128	−	4.23849i	1.53076	−	0.883786i	0.999327	−	0.0366878i	$-0.0116807\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.19899	+	2.07671i	−0.222647	+	0.385636i	−0.955611	−	0.294632i	$-0.904803\pi$
$29$	−1.19899	+	2.07671i	−0.222647	+	0.385636i	0.732964	+	0.680267i	$0.238136\pi$
$30$		0			0
$31$	−1.81249	−	3.13933i	−0.325533	−	0.563840i	0.656087	−	0.754685i	$-0.272211\pi$
$31$	−1.81249	−	3.13933i	−0.325533	−	0.563840i	−0.981620	+	0.190845i	$0.938877\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	5.85199i		−	0.962062i	−0.876704	−	0.481031i	$-0.840262\pi$
$37$		−	5.85199i		−	0.962062i	0.876704	−	0.481031i	$-0.159738\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.32497	+	5.75902i	0.519273	+	0.899407i	0.999749	+	0.0223994i	$0.00713054\pi$
$41$	3.32497	+	5.75902i	0.519273	+	0.899407i	−0.480476	+	0.877008i	$0.659536\pi$
$42$		0			0
$43$	−7.14469	−	4.12499i	−1.08955	−	0.629055i	−0.156097	−	0.987742i	$-0.549891\pi$
$43$	−7.14469	−	4.12499i	−1.08955	−	0.629055i	−0.933458	+	0.358687i	$0.883224\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.32831	+	1.34425i	0.339619	+	0.196079i	0.660103	−	0.751175i	$-0.270512\pi$
$47$	2.32831	+	1.34425i	0.339619	+	0.196079i	−0.320485	+	0.947254i	$0.603846\pi$
$48$		0			0
$49$	−3.49626	−	6.05570i	−0.499466	−	0.865100i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5.73642i		−	0.787958i	−0.919120	−	0.393979i	$-0.871098\pi$
$53$		−	5.73642i		−	0.787958i	0.919120	−	0.393979i	$-0.128902\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.16922	−	10.6854i	−0.803164	−	1.39112i	−0.917524	−	0.397681i	$-0.869815\pi$
$59$	−6.16922	−	10.6854i	−0.803164	−	1.39112i	0.114360	−	0.993439i	$-0.463518\pi$
$60$		0			0
$61$	3.16823	−	5.48753i	0.405650	−	0.702606i	−0.588747	−	0.808317i	$-0.700379\pi$
$61$	3.16823	−	5.48753i	0.405650	−	0.702606i	0.994397	+	0.105711i	$0.0337119\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	5.33946	−	3.08274i	0.652319	−	0.376617i	−0.137025	−	0.990568i	$-0.543754\pi$
$67$	5.33946	−	3.08274i	0.652319	−	0.376617i	0.789344	+	0.613951i	$0.210421\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.3905			1.47048			0.735241	−	0.677806i	$-0.237069\pi$
$71$	12.3905			1.47048			0.735241	+	0.677806i	$0.237069\pi$
$72$		0			0
$73$		−	5.31349i		−	0.621897i	−0.950427	−	0.310948i	$-0.899353\pi$
$73$		−	5.31349i		−	0.621897i	0.950427	−	0.310948i	$-0.100647\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.0684166	−	0.0395003i	0.00779679	−	0.00450148i
$78$		0			0
$79$	−6.72394	+	11.6462i	−0.756503	+	1.31030i	0.188121	+	0.982146i	$0.439760\pi$
$79$	−6.72394	+	11.6462i	−0.756503	+	1.31030i	−0.944624	+	0.328155i	$0.893573\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.26457	+	3.03950i	0.577862	+	0.333629i	0.760283	−	0.649592i	$-0.225060\pi$
$83$	5.26457	+	3.03950i	0.577862	+	0.333629i	−0.182422	+	0.983220i	$0.558394\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.13440			−0.862244			−0.431122	−	0.902294i	$-0.641882\pi$
$89$	−8.13440			−0.862244			−0.431122	+	0.902294i	$0.641882\pi$
$90$		0			0
$91$	0.227007			0.0237968
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−9.61459	−	5.55098i	−0.976213	−	0.563617i	−0.0750885	−	0.997177i	$-0.523924\pi$
$97$	−9.61459	−	5.55098i	−0.976213	−	0.563617i	−0.901125	+	0.433560i	$0.857257\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−6.03950	+	10.4607i	−0.600953	+	1.04088i	0.391724	+	0.920083i	$0.371879\pi$
$101$	−6.03950	+	10.4607i	−0.600953	+	1.04088i	−0.992677	+	0.120798i	$0.961455\pi$
$102$		0			0
$103$	3.78240	−	2.18377i	0.372691	−	0.215173i	−0.301943	−	0.953326i	$-0.597635\pi$
$103$	3.78240	−	2.18377i	0.372691	−	0.215173i	0.674633	+	0.738153i	$0.264302\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	18.9614i		−	1.83307i	−0.399953	−	0.916536i	$-0.630973\pi$
$107$		−	18.9614i		−	1.83307i	0.399953	−	0.916536i	$-0.369027\pi$
$108$		0			0
$109$	−15.1904			−1.45498			−0.727490	−	0.686119i	$-0.759313\pi$
$109$	−15.1904			−1.45498			−0.727490	+	0.686119i	$0.759313\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.71752	−	3.30101i	0.537859	−	0.310533i	−0.206352	−	0.978478i	$-0.566159\pi$
$113$	5.71752	−	3.30101i	0.537859	−	0.310533i	0.744211	+	0.667945i	$0.232826\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.0902186	+	0.156263i	−0.00827033	+	0.0143246i
$120$		0			0
$121$	5.08274	+	8.80356i	0.462067	+	0.800324i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	7.42492i		−	0.658855i	−0.944181	−	0.329427i	$-0.893144\pi$
$127$		−	7.42492i		−	0.658855i	0.944181	−	0.329427i	$-0.106856\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−7.98072	−	13.8230i	−0.697279	−	1.20772i	−0.969406	−	0.245461i	$-0.921061\pi$
$131$	−7.98072	−	13.8230i	−0.697279	−	1.20772i	0.272128	−	0.962261i	$-0.412273\pi$
$132$		0			0
$133$	0.369858	+	0.213538i	0.0320708	+	0.0185161i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.9266	+	9.19525i	1.36070	+	0.785603i	0.989718	−	0.143035i	$-0.0456861\pi$
$137$	15.9266	+	9.19525i	1.36070	+	0.785603i	0.370987	+	0.928638i	$0.379019\pi$
$138$		0			0
$139$	2.78074	+	4.81638i	0.235859	+	0.408520i	0.959522	−	0.281634i	$-0.0908763\pi$
$139$	2.78074	+	4.81638i	0.235859	+	0.408520i	−0.723663	+	0.690154i	$0.757543\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2.39798i		−	0.200529i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.7780	−	20.4001i	−0.964891	−	1.67124i	−0.709909	−	0.704294i	$-0.751264\pi$
$149$	−11.7780	−	20.4001i	−0.964891	−	1.67124i	−0.254982	−	0.966946i	$-0.582069\pi$
$150$		0			0
$151$	−0.0114831	+	0.0198893i	−0.000934479	+	0.00161856i	−0.866492	−	0.499190i	$-0.833631\pi$
$151$	−0.0114831	+	0.0198893i	−0.000934479	+	0.00161856i	0.865558	+	0.500809i	$0.166964\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−10.5303	+	6.07967i	−0.840410	+	0.485211i	−0.857404	−	0.514645i	$-0.827924\pi$
$157$	−10.5303	+	6.07967i	−0.840410	+	0.485211i	0.0169936	+	0.999856i	$0.494590\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.733083			−0.0577751
$162$		0			0
$163$		−	19.0654i		−	1.49332i	−0.665208	−	0.746658i	$-0.731657\pi$
$163$		−	19.0654i		−	1.49332i	0.665208	−	0.746658i	$-0.268343\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.57868	−	0.911449i	0.122162	−	0.0705300i	−0.437674	−	0.899134i	$-0.644198\pi$
$167$	1.57868	−	0.911449i	0.122162	−	0.0705300i	0.559836	+	0.828604i	$0.310864\pi$
$168$		0			0
$169$	−3.05472	+	5.29093i	−0.234979	+	0.406995i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−18.0034	−	10.3942i	−1.36877	−	0.790259i	−0.377999	−	0.925806i	$-0.623388\pi$
$173$	−18.0034	−	10.3942i	−1.36877	−	0.790259i	−0.990771	+	0.135547i	$0.956721\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3.12692			−0.233717			−0.116858	−	0.993149i	$-0.537282\pi$
$179$	−3.12692			−0.233717			−0.116858	+	0.993149i	$0.537282\pi$
$180$		0			0
$181$	22.9249			1.70399			0.851996	−	0.523549i	$-0.175392\pi$
$181$	22.9249			1.70399			0.851996	+	0.523549i	$0.175392\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.65068	+	0.953021i	0.120710	+	0.0696918i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.97698	−	6.88833i	0.287764	−	0.498422i	−0.685512	−	0.728062i	$-0.740421\pi$
$191$	3.97698	−	6.88833i	0.287764	−	0.498422i	0.973276	+	0.229640i	$0.0737547\pi$
$192$		0			0
$193$	4.49333	−	2.59422i	0.323437	−	0.186736i	−0.329487	−	0.944160i	$-0.606876\pi$
$193$	4.49333	−	2.59422i	0.323437	−	0.186736i	0.652923	+	0.757424i	$0.273542\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	21.3384i		−	1.52030i	−0.649747	−	0.760150i	$-0.725125\pi$
$197$		−	21.3384i		−	1.52030i	0.649747	−	0.760150i	$-0.274875\pi$
$198$		0			0
$199$	−9.50608			−0.673868			−0.336934	−	0.941528i	$-0.609390\pi$
$199$	−9.50608			−0.673868			−0.336934	+	0.941528i	$0.609390\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.179593	−	0.103688i	0.0126049	−	0.00727746i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.25570	−	3.90698i	0.156030	−	0.270252i
$210$		0			0
$211$	0.295285	+	0.511448i	0.0203282	+	0.0352096i	0.876011	−	0.482292i	$-0.160196\pi$
$211$	0.295285	+	0.511448i	0.0203282	+	0.0352096i	−0.855682	+	0.517501i	$0.826862\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			0.313486i			0.0212808i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.73849	+	4.74320i	0.184211	+	0.319062i
$222$		0			0
$223$	4.99308	+	2.88276i	0.334361	+	0.193044i	0.657776	−	0.753214i	$-0.271497\pi$
$223$	4.99308	+	2.88276i	0.334361	+	0.193044i	−0.323414	+	0.946257i	$0.604831\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.4801	+	6.62805i	0.761962	+	0.439919i	0.830000	−	0.557764i	$-0.188340\pi$
$227$	11.4801	+	6.62805i	0.761962	+	0.439919i	−0.0680374	+	0.997683i	$0.521674\pi$
$228$		0			0
$229$	3.92500	+	6.79831i	0.259372	+	0.449245i	0.966074	−	0.258266i	$-0.0831512\pi$
$229$	3.92500	+	6.79831i	0.259372	+	0.449245i	−0.706702	+	0.707511i	$0.749818\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			15.9094i			1.04226i	0.853478	+	0.521129i	$0.174489\pi$
$233$			15.9094i			1.04226i	−0.853478	+	0.521129i	$0.825511\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−7.23849	−	12.5374i	−0.468219	−	0.810979i	0.531121	−	0.847296i	$-0.321771\pi$
$239$	−7.23849	−	12.5374i	−0.468219	−	0.810979i	−0.999340	+	0.0363167i	$0.988438\pi$
$240$		0			0
$241$	−3.32803	+	5.76432i	−0.214378	+	0.371313i	−0.953080	−	0.302719i	$-0.902106\pi$
$241$	−3.32803	+	5.76432i	−0.214378	+	0.371313i	0.738702	+	0.674032i	$0.235439\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	11.2267	−	6.48171i	0.714335	−	0.412421i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	14.6885			0.927130			0.463565	−	0.886063i	$-0.346570\pi$
$251$	14.6885			0.927130			0.463565	+	0.886063i	$0.346570\pi$
$252$		0			0
$253$			7.74390i			0.486855i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.7973	+	8.54324i	−0.923032	+	0.532913i	−0.884601	−	0.466349i	$-0.845569\pi$
$257$	−14.7973	+	8.54324i	−0.923032	+	0.532913i	−0.0384308	+	0.999261i	$0.512236\pi$
$258$		0			0
$259$	−0.253038	+	0.438275i	−0.0157230	+	0.0272331i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.89476	+	1.67129i	0.178499	+	0.103056i	0.586587	−	0.809886i	$-0.300471\pi$
$263$	2.89476	+	1.67129i	0.178499	+	0.103056i	−0.408088	+	0.912942i	$0.633805\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.82704			0.111397			0.0556983	−	0.998448i	$-0.482261\pi$
$269$	1.82704			0.111397			0.0556983	+	0.998448i	$0.482261\pi$
$270$		0			0
$271$	−24.0634			−1.46175			−0.730874	−	0.682513i	$-0.760887\pi$
$271$	−24.0634			−1.46175			−0.730874	+	0.682513i	$0.760887\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−12.0646	−	6.96550i	−0.724891	−	0.418516i	0.0916590	−	0.995790i	$-0.470783\pi$
$277$	−12.0646	−	6.96550i	−0.724891	−	0.418516i	−0.816550	+	0.577274i	$0.804116\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.62391	+	16.6691i	−0.574114	+	0.994395i	0.422023	+	0.906585i	$0.361320\pi$
$281$	−9.62391	+	16.6691i	−0.574114	+	0.994395i	−0.996137	+	0.0878099i	$0.972013\pi$
$282$		0			0
$283$	19.2907	−	11.1375i	1.14671	−	0.662053i	0.198627	−	0.980075i	$-0.436352\pi$
$283$	19.2907	−	11.1375i	1.14671	−	0.662053i	0.948083	+	0.318022i	$0.103019\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	0.575082i		−	0.0339460i
$288$		0			0
$289$	12.6466			0.743918
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.81053	−	1.04531i	0.105772	−	0.0610678i	−0.446181	−	0.894943i	$-0.647216\pi$
$293$	1.81053	−	1.04531i	0.105772	−	0.0610678i	0.551953	+	0.833875i	$0.313883\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−11.1260	+	19.2708i	−0.643432	+	1.11446i
$300$		0			0
$301$	0.356726	+	0.617868i	0.0205613	+	0.0356133i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	1.74056i		−	0.0993391i	−0.998766	−	0.0496696i	$-0.984183\pi$
$307$		−	1.74056i		−	0.0993391i	0.998766	−	0.0496696i	$-0.0158168\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.8682	−	20.5563i	−0.672984	−	1.16564i	−0.977054	−	0.212993i	$-0.931679\pi$
$311$	−11.8682	−	20.5563i	−0.672984	−	1.16564i	0.304069	−	0.952650i	$-0.401655\pi$
$312$		0			0
$313$	−9.42330	−	5.44054i	−0.532636	−	0.307518i	0.209453	−	0.977819i	$-0.432832\pi$
$313$	−9.42330	−	5.44054i	−0.532636	−	0.307518i	−0.742089	+	0.670301i	$0.766165\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.93468	+	4.00374i	0.389490	+	0.224872i	0.681939	−	0.731409i	$-0.261137\pi$
$317$	6.93468	+	4.00374i	0.389490	+	0.224872i	−0.292449	+	0.956281i	$0.594470\pi$
$318$		0			0
$319$	−1.09530	−	1.89712i	−0.0613251	−	0.106218i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			10.3040i			0.573331i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.116250	−	0.201351i	−0.00640906	−	0.0111008i
$330$		0			0
$331$	2.10976	−	3.65422i	0.115963	−	0.200854i	−0.802201	−	0.597054i	$-0.796338\pi$
$331$	2.10976	−	3.65422i	0.115963	−	0.200854i	0.918164	+	0.396200i	$0.129671\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−5.53964	+	3.19832i	−0.301764	+	0.174223i	−0.643235	−	0.765669i	$-0.722408\pi$
$337$	−5.53964	+	3.19832i	−0.301764	+	0.174223i	0.341471	+	0.939892i	$0.389075\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.31150			0.179328
$342$		0			0
$343$			1.21006i			0.0653373i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.7048	+	6.75777i	−0.628347	+	0.362776i	−0.780112	−	0.625641i	$-0.784838\pi$
$347$	−11.7048	+	6.75777i	−0.628347	+	0.362776i	0.151765	+	0.988417i	$0.451504\pi$
$348$		0			0
$349$	0.408788	−	0.708042i	0.0218819	−	0.0379006i	−0.854877	−	0.518831i	$-0.826368\pi$
$349$	0.408788	−	0.708042i	0.0218819	−	0.0379006i	0.876759	+	0.480930i	$0.159701\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−16.4029	−	9.47023i	−0.873039	−	0.504049i	−0.00468224	−	0.999989i	$-0.501490\pi$
$353$	−16.4029	−	9.47023i	−0.873039	−	0.504049i	−0.868357	+	0.495940i	$0.834824\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	36.4863			1.92568			0.962838	−	0.270081i	$-0.0870505\pi$
$359$	36.4863			1.92568			0.962838	+	0.270081i	$0.0870505\pi$
$360$		0			0
$361$	5.38851			0.283606
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	16.8225	+	9.71246i	0.878126	+	0.506986i	0.870040	−	0.492981i	$-0.164093\pi$
$367$	16.8225	+	9.71246i	0.878126	+	0.506986i	0.00808588	+	0.999967i	$0.497426\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.248041	+	0.429619i	−0.0128776	+	0.0223047i
$372$		0			0
$373$	−25.2679	+	14.5884i	−1.30832	+	0.755359i	−0.981816	−	0.189836i	$-0.939204\pi$
$373$	−25.2679	+	14.5884i	−1.30832	+	0.755359i	−0.326505	+	0.945195i	$0.605871\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	6.29466i		−	0.324192i
$378$		0			0
$379$	21.8060			1.12010			0.560048	−	0.828460i	$-0.310783\pi$
$379$	21.8060			1.12010			0.560048	+	0.828460i	$0.310783\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	8.37526	−	4.83546i	0.427956	−	0.247080i	−0.270520	−	0.962714i	$-0.587195\pi$
$383$	8.37526	−	4.83546i	0.427956	−	0.247080i	0.698475	+	0.715634i	$0.253862\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.86447	+	13.6217i	−0.398744	+	0.690646i	−0.993571	−	0.113208i	$-0.963887\pi$
$389$	−7.86447	+	13.6217i	−0.398744	+	0.690646i	0.594827	+	0.803854i	$0.297221\pi$
$390$		0			0
$391$	−8.84352	−	15.3174i	−0.447236	−	0.774636i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			22.8999i			1.14931i	0.818394	+	0.574657i	$0.194864\pi$
$397$			22.8999i			1.14931i	−0.818394	+	0.574657i	$0.805136\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.04999	+	5.28274i	0.152309	+	0.263807i	0.932076	−	0.362263i	$-0.117996\pi$
$401$	3.04999	+	5.28274i	0.152309	+	0.263807i	−0.779767	+	0.626070i	$0.784662\pi$
$402$		0			0
$403$	8.24070	+	4.75777i	0.410499	+	0.237001i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.62970	+	2.67296i	0.229486	+	0.132494i
$408$		0			0
$409$	−14.4344	−	25.0011i	−0.713736	−	1.23623i	−0.963445	−	0.267906i	$-0.913668\pi$
$409$	−14.4344	−	25.0011i	−0.713736	−	1.23623i	0.249709	−	0.968321i	$-0.419665\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1.06702i			0.0525046i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	8.97117	+	15.5385i	0.438270	+	0.759106i	0.997556	−	0.0698684i	$-0.0222579\pi$
$419$	8.97117	+	15.5385i	0.438270	+	0.759106i	−0.559286	+	0.828975i	$0.688925\pi$
$420$		0			0
$421$	0.361544	−	0.626213i	0.0176206	−	0.0305198i	−0.857081	−	0.515182i	$-0.827724\pi$
$421$	0.361544	−	0.626213i	0.0176206	−	0.0305198i	0.874701	+	0.484662i	$0.161058\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.474558	+	0.273986i	−0.0229655	+	0.0132591i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.19656			0.394815			0.197407	−	0.980322i	$-0.436748\pi$
$431$	8.19656			0.394815			0.197407	+	0.980322i	$0.436748\pi$
$432$		0			0
$433$			35.7464i			1.71786i	0.512091	+	0.858931i	$0.328871\pi$
$433$			35.7464i			1.71786i	−0.512091	+	0.858931i	$0.671129\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−36.2547	+	20.9317i	−1.73430	+	1.00130i
$438$		0			0
$439$	9.13440	−	15.8212i	0.435961	−	0.755107i	−0.561413	−	0.827536i	$-0.689742\pi$
$439$	9.13440	−	15.8212i	0.435961	−	0.755107i	0.997374	+	0.0724295i	$0.0230752\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	25.8355	+	14.9161i	1.22748	+	0.708687i	0.966502	−	0.256658i	$-0.0826214\pi$
$443$	25.8355	+	14.9161i	1.22748	+	0.708687i	0.260979	+	0.965345i	$0.415955\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	38.9423			1.83780			0.918901	−	0.394488i	$-0.129078\pi$
$449$	38.9423			1.83780			0.918901	+	0.394488i	$0.129078\pi$
$450$		0			0
$451$	−6.07486			−0.286054
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	17.7453	+	10.2452i	0.830089	+	0.479252i	0.853883	−	0.520465i	$-0.174241\pi$
$457$	17.7453	+	10.2452i	0.830089	+	0.479252i	−0.0237941	+	0.999717i	$0.507575\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14.0858	+	24.3972i	−0.656039	+	1.13629i	0.325593	+	0.945510i	$0.394436\pi$
$461$	−14.0858	+	24.3972i	−0.656039	+	1.13629i	−0.981632	+	0.190783i	$0.938897\pi$
$462$		0			0
$463$	−29.7216	+	17.1598i	−1.38128	+	0.797481i	−0.992311	−	0.123770i	$-0.960501\pi$
$463$	−29.7216	+	17.1598i	−1.38128	+	0.797481i	−0.388967	+	0.921252i	$0.627168\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			4.33096i			0.200413i	0.994967	+	0.100206i	$0.0319503\pi$
$467$			4.33096i			0.200413i	−0.994967	+	0.100206i	$0.968050\pi$
$468$		0			0
$469$	−0.533186			−0.0246203
$470$		0			0
$471$		0			0
$472$		0			0
$473$	6.52682	−	3.76826i	0.300103	−	0.173265i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.4925	+	25.1018i	−0.662180	+	1.14693i	0.317861	+	0.948137i	$0.397035\pi$
$479$	−14.4925	+	25.1018i	−0.662180	+	1.14693i	−0.980042	+	0.198793i	$0.936298\pi$
$480$		0			0
$481$	7.68070	+	13.3034i	0.350210	+	0.606581i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	11.2189i		−	0.508376i	−0.967155	−	0.254188i	$-0.918192\pi$
$487$		−	11.2189i		−	0.508376i	0.967155	−	0.254188i	$-0.0818083\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.7817	−	22.1386i	−0.576831	−	0.999101i	−0.995840	−	0.0911190i	$-0.970956\pi$
$491$	−12.7817	−	22.1386i	−0.576831	−	0.999101i	0.419009	−	0.907982i	$-0.362378\pi$
$492$		0			0
$493$	4.33301	+	2.50167i	0.195149	+	0.112669i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.927965	−	0.535761i	−0.0416249	−	0.0240322i
$498$		0			0
$499$	9.46856	+	16.4000i	0.423871	+	0.734166i	0.996314	−	0.0857782i	$-0.0273376\pi$
$499$	9.46856	+	16.4000i	0.423871	+	0.734166i	−0.572443	+	0.819944i	$0.694004\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	21.1790i		−	0.944324i	−0.881512	−	0.472162i	$-0.843474\pi$
$503$		−	21.1790i		−	0.944324i	0.881512	−	0.472162i	$-0.156526\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.3280	+	28.2809i	0.723725	+	1.25353i	0.959497	+	0.281720i	$0.0909049\pi$
$509$	16.3280	+	28.2809i	0.723725	+	1.25353i	−0.235772	+	0.971808i	$0.575762\pi$
$510$		0			0
$511$	−0.229753	+	0.397944i	−0.0101637	+	0.0176040i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−2.12696	+	1.22800i	−0.0935435	+	0.0540074i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.06550			−0.0904912			−0.0452456	−	0.998976i	$-0.514407\pi$
$521$	−2.06550			−0.0904912			−0.0452456	+	0.998976i	$0.514407\pi$
$522$		0			0
$523$			8.34790i			0.365028i	0.983203	+	0.182514i	$0.0584235\pi$
$523$			8.34790i			0.365028i	−0.983203	+	0.182514i	$0.941576\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.55015	+	3.78173i	−0.285329	+	0.164735i
$528$		0			0
$529$	24.4296	−	42.3133i	1.06216	−	1.83971i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−15.1173	−	8.72800i	−0.654805	−	0.378052i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.38781			0.275143
$540$		0			0
$541$	−12.2175			−0.525273			−0.262637	−	0.964895i	$-0.584592\pi$
$541$	−12.2175			−0.525273			−0.262637	+	0.964895i	$0.584592\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.22891	+	4.75097i	0.351843	+	0.203137i	0.665497	−	0.746401i	$-0.268220\pi$
$547$	8.22891	+	4.75097i	0.351843	+	0.203137i	−0.313654	+	0.949537i	$0.601553\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.92118	−	10.2558i	0.252251	−	0.436911i
$552$		0			0
$553$	1.00716	−	0.581482i	0.0428286	−	0.0247271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	19.1999i		−	0.813526i	−0.913534	−	0.406763i	$-0.866658\pi$
$557$		−	19.1999i		−	0.813526i	0.913534	−	0.406763i	$-0.133342\pi$
$558$		0			0
$559$	21.6561			0.915954
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−14.0477	+	8.11044i	−0.592040	+	0.341814i	−0.765904	−	0.642955i	$-0.777708\pi$
$563$	−14.0477	+	8.11044i	−0.592040	+	0.341814i	0.173864	+	0.984770i	$0.444375\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	14.0286	−	24.2983i	0.588111	−	1.01864i	−0.406369	−	0.913709i	$-0.633205\pi$
$569$	14.0286	−	24.2983i	0.588111	−	1.01864i	0.994480	−	0.104929i	$-0.0334616\pi$
$570$		0			0
$571$	−20.7722	−	35.9785i	−0.869289	−	1.50565i	−0.862725	−	0.505674i	$-0.831244\pi$
$571$	−20.7722	−	35.9785i	−0.869289	−	1.50565i	−0.00656379	−	0.999978i	$-0.502089\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$			18.4113i			0.766473i	0.923650	+	0.383236i	$0.125191\pi$
$577$			18.4113i			0.766473i	−0.923650	+	0.383236i	$0.874809\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.262854	−	0.455276i	−0.0109050	−	0.0188880i
$582$		0			0
$583$	4.53826	+	2.62017i	0.187956	+	0.108516i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.0767	+	17.9422i	1.28267	+	0.740552i	0.977336	−	0.211692i	$-0.0678974\pi$
$587$	31.0767	+	17.9422i	1.28267	+	0.740552i	0.305338	+	0.952244i	$0.401231\pi$
$588$		0			0
$589$	8.95095	+	15.5035i	0.368817	+	0.638811i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			47.6039i			1.95486i	0.211265	+	0.977429i	$0.432242\pi$
$593$			47.6039i			1.95486i	−0.211265	+	0.977429i	$0.567758\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.4279	+	21.5258i	0.507791	+	0.879521i	0.999959	+	0.00902025i	$0.00287127\pi$
$599$	12.4279	+	21.5258i	0.507791	+	0.879521i	−0.492168	+	0.870500i	$0.663795\pi$
$600$		0			0
$601$	−11.5482	+	20.0021i	−0.471062	+	0.815904i	−0.999452	−	0.0330979i	$-0.989463\pi$
$601$	−11.5482	+	20.0021i	−0.471062	+	0.815904i	0.528390	+	0.849002i	$0.322796\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−4.00708	+	2.31349i	−0.162642	+	0.0939015i	−0.579112	−	0.815248i	$-0.696601\pi$
$607$	−4.00708	+	2.31349i	−0.162642	+	0.0939015i	0.416470	+	0.909150i	$0.363267\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.05728			−0.285507
$612$		0			0
$613$		−	12.1791i		−	0.491909i	−0.969281	−	0.245954i	$-0.920899\pi$
$613$		−	12.1791i		−	0.491909i	0.969281	−	0.245954i	$-0.0791013\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−40.3021	+	23.2684i	−1.62250	+	0.936752i	−0.636254	+	0.771479i	$0.719517\pi$
$617$	−40.3021	+	23.2684i	−1.62250	+	0.936752i	−0.986248	+	0.165273i	$0.947150\pi$
$618$		0			0
$619$	7.90198	−	13.6866i	0.317608	−	0.550112i	−0.662381	−	0.749167i	$-0.730454\pi$
$619$	7.90198	−	13.6866i	0.317608	−	0.550112i	0.979988	+	0.199055i	$0.0637872\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.609212	+	0.351729i	0.0244076	+	0.0140917i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.2101			−0.486847
$630$		0			0
$631$	−24.6749			−0.982292			−0.491146	−	0.871077i	$-0.663422\pi$
$631$	−24.6749			−0.982292			−0.491146	+	0.871077i	$0.663422\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	15.8961	+	9.17764i	0.629828	+	0.363631i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.757771	−	1.31250i	0.0299301	−	0.0518405i	−0.850672	−	0.525696i	$-0.823805\pi$
$641$	0.757771	−	1.31250i	0.0299301	−	0.0518405i	0.880602	+	0.473856i	$0.157138\pi$
$642$		0			0
$643$	21.1309	−	12.1999i	0.833321	−	0.481118i	−0.0216672	−	0.999765i	$-0.506897\pi$
$643$	21.1309	−	12.1999i	0.833321	−	0.481118i	0.854988	+	0.518647i	$0.173564\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			26.9195i			1.05832i	0.848523	+	0.529158i	$0.177492\pi$
$647$			26.9195i			1.05832i	−0.848523	+	0.529158i	$0.822508\pi$
$648$		0			0
$649$	11.2714			0.442442
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.08609	+	4.09116i	−0.277300	+	0.160099i	−0.632201	−	0.774805i	$-0.717848\pi$
$653$	−7.08609	+	4.09116i	−0.277300	+	0.160099i	0.354900	+	0.934904i	$0.384515\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−7.55504	+	13.0857i	−0.294303	+	0.509747i	−0.974822	−	0.222983i	$-0.928421\pi$
$659$	−7.55504	+	13.0857i	−0.294303	+	0.509747i	0.680520	+	0.732730i	$0.261754\pi$
$660$		0			0
$661$	17.0300	+	29.4969i	0.662392	+	1.14730i	0.979985	+	0.199069i	$0.0637918\pi$
$661$	17.0300	+	29.4969i	0.662392	+	1.14730i	−0.317594	+	0.948227i	$0.602875\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			20.3276i			0.787089i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.89424	+	5.01297i	0.111731	+	0.193524i
$672$		0			0
$673$	25.4728	+	14.7067i	0.981905	+	0.566903i	0.902845	−	0.429967i	$-0.141475\pi$
$673$	25.4728	+	14.7067i	0.981905	+	0.566903i	0.0790600	+	0.996870i	$0.474808\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−29.3615	−	16.9519i	−1.12846	−	0.651514i	−0.184909	−	0.982756i	$-0.559199\pi$
$677$	−29.3615	−	16.9519i	−1.12846	−	0.651514i	−0.943546	+	0.331242i	$0.892532\pi$
$678$		0			0
$679$	0.480045	+	0.831463i	0.0184224	+	0.0319086i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			29.1078i			1.11378i	0.830586	+	0.556890i	$0.188005\pi$
$683$			29.1078i			1.11378i	−0.830586	+	0.556890i	$0.811995\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.52901	+	13.0406i	0.286832	+	0.496808i
$690$		0			0
$691$	5.21079	−	9.02536i	0.198228	−	0.343341i	−0.749726	−	0.661748i	$-0.769815\pi$
$691$	5.21079	−	9.02536i	0.198228	−	0.343341i	0.947954	+	0.318408i	$0.103148\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	12.0161	−	6.93748i	0.455141	−	0.262776i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	9.75406			0.368406			0.184203	−	0.982888i	$-0.441030\pi$
$701$	9.75406			0.368406			0.184203	+	0.982888i	$0.441030\pi$
$702$		0			0
$703$			28.8999i			1.08998i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.904636	−	0.522292i	0.0340223	−	0.0196428i
$708$		0			0
$709$	−16.1539	+	27.9795i	−0.606674	+	1.05079i	0.385110	+	0.922870i	$0.374163\pi$
$709$	−16.1539	+	27.9795i	−0.606674	+	1.05079i	−0.991785	+	0.127920i	$0.959170\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−26.6120	−	15.3645i	−0.996629	−	0.575404i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	18.4905			0.689579			0.344789	−	0.938680i	$-0.387950\pi$
$719$	18.4905			0.689579			0.344789	+	0.938680i	$0.387950\pi$
$720$		0			0
$721$	−0.377701			−0.0140663
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−44.5755	−	25.7357i	−1.65321	−	0.954484i	−0.975739	−	0.218939i	$-0.929740\pi$
$727$	−44.5755	−	25.7357i	−1.65321	−	0.954484i	−0.677476	−	0.735545i	$-0.736926\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.60670	+	14.9072i	−0.318330	+	0.551364i
$732$		0			0
$733$	−5.75091	+	3.32029i	−0.212415	+	0.122638i	−0.602433	−	0.798169i	$-0.705802\pi$
$733$	−5.75091	+	3.32029i	−0.212415	+	0.122638i	0.390018	+	0.920807i	$0.372469\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			5.63229i			0.207468i
$738$		0			0
$739$	38.6058			1.42014			0.710068	−	0.704133i	$-0.248664\pi$
$739$	38.6058			1.42014			0.710068	+	0.704133i	$0.248664\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−22.1619	+	12.7952i	−0.813043	+	0.469410i	−0.848011	−	0.529978i	$-0.822200\pi$
$743$	−22.1619	+	12.7952i	−0.813043	+	0.469410i	0.0349688	+	0.999388i	$0.488867\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.819886	+	1.42008i	−0.0299580	+	0.0518888i
$750$		0			0
$751$	−7.55572	−	13.0869i	−0.275712	−	0.477547i	0.694603	−	0.719394i	$-0.255580\pi$
$751$	−7.55572	−	13.0869i	−0.275712	−	0.477547i	−0.970315	+	0.241847i	$0.922247\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			13.2710i			0.482341i	0.970483	+	0.241170i	$0.0775313\pi$
$757$			13.2710i			0.482341i	−0.970483	+	0.241170i	$0.922469\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.25871	−	16.0366i	−0.335628	−	0.581325i	0.647977	−	0.761660i	$-0.275615\pi$
$761$	−9.25871	−	16.0366i	−0.335628	−	0.581325i	−0.983605	+	0.180335i	$0.942282\pi$
$762$		0			0
$763$	1.13766	+	0.656829i	0.0411861	+	0.0237788i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	28.0490	+	16.1941i	1.01279	+	0.584736i
$768$		0			0
$769$	−1.00941	−	1.74835i	−0.0364003	−	0.0630472i	0.847251	−	0.531192i	$-0.178256\pi$
$769$	−1.00941	−	1.74835i	−0.0364003	−	0.0630472i	−0.883652	+	0.468145i	$0.844922\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			5.76154i			0.207228i	0.994618	+	0.103614i	$0.0330407\pi$
$773$			5.76154i			0.207228i	−0.994618	+	0.103614i	$0.966959\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16.4203	−	28.4407i	−0.588317	−	1.01900i
$780$		0			0
$781$	−5.65949	+	9.80252i	−0.202512	+	0.350762i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	6.34834	−	3.66521i	0.226294	−	0.130651i	−0.382567	−	0.923928i	$-0.624960\pi$
$787$	6.34834	−	3.66521i	0.226294	−	0.130651i	0.608861	+	0.793277i	$0.291627\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.570938			−0.0203002
$792$		0			0
$793$			16.6331i			0.590659i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.16920	−	3.56179i	0.218524	−	0.126165i	−0.386742	−	0.922188i	$-0.626400\pi$
$797$	6.16920	−	3.56179i	0.218524	−	0.126165i	0.605267	+	0.796023i	$0.293066\pi$
$798$		0			0
$799$	2.80475	−	4.85797i	0.0992249	−	0.171863i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.20367	+	2.42699i	0.148344	+	0.0856466i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	29.0809			1.02243			0.511215	−	0.859453i	$-0.329196\pi$
$809$	29.0809			1.02243			0.511215	+	0.859453i	$0.329196\pi$
$810$		0			0
$811$	34.3214			1.20519			0.602593	−	0.798048i	$-0.294134\pi$
$811$	34.3214			1.20519			0.602593	+	0.798048i	$0.294134\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	35.2838	+	20.3711i	1.23443	+	0.712696i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.4384	−	37.1324i	0.748206	−	1.29593i	−0.200476	−	0.979699i	$-0.564249\pi$
$821$	21.4384	−	37.1324i	0.748206	−	1.29593i	0.948682	−	0.316232i	$-0.102418\pi$
$822$		0			0
$823$	40.0371	−	23.1154i	1.39561	−	0.805753i	0.401677	−	0.915781i	$-0.368427\pi$
$823$	40.0371	−	23.1154i	1.39561	−	0.805753i	0.993929	+	0.110028i	$0.0350940\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	23.6350i		−	0.821869i	−0.911665	−	0.410934i	$-0.865203\pi$
$827$		−	23.6350i		−	0.821869i	0.911665	−	0.410934i	$-0.134797\pi$
$828$		0			0
$829$	18.8979			0.656352			0.328176	−	0.944617i	$-0.393566\pi$
$829$	18.8979			0.656352			0.328176	+	0.944617i	$0.393566\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12.6351	+	7.29488i	−0.437780	+	0.252752i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.21547	+	9.03346i	−0.180058	+	0.311870i	−0.941900	−	0.335893i	$-0.890962\pi$
$839$	−5.21547	+	9.03346i	−0.180058	+	0.311870i	0.761842	+	0.647763i	$0.224295\pi$
$840$		0			0
$841$	11.6248	+	20.1348i	0.400857	+	0.694304i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	0.879104i		−	0.0302064i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.8036	−	42.9611i	−0.850257	−	1.47269i
$852$		0			0
$853$	−34.5379	−	19.9405i	−1.18255	−	0.682748i	−0.225951	−	0.974139i	$-0.572549\pi$
$853$	−34.5379	−	19.9405i	−1.18255	−	0.682748i	−0.956604	+	0.291390i	$0.905882\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−15.6984	−	9.06346i	−0.536246	−	0.309602i	0.207310	−	0.978275i	$-0.433529\pi$
$857$	−15.6984	−	9.06346i	−0.536246	−	0.309602i	−0.743556	+	0.668673i	$0.766862\pi$
$858$		0			0
$859$	2.01040	+	3.48212i	0.0685941	+	0.118808i	0.898283	−	0.439418i	$-0.144815\pi$
$859$	2.01040	+	3.48212i	0.0685941	+	0.118808i	−0.829689	+	0.558227i	$0.811482\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	43.9540i		−	1.49621i	−0.663580	−	0.748105i	$-0.730964\pi$
$863$		−	43.9540i		−	1.49621i	0.663580	−	0.748105i	$-0.269036\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.14246	−	10.6391i	−0.208369	−	0.360905i
$870$		0			0
$871$	−8.09215	+	14.0160i	−0.274192	+	0.474915i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−42.9313	+	24.7864i	−1.44969	+	0.836978i	−0.998462	−	0.0554326i	$-0.982346\pi$
$877$	−42.9313	+	24.7864i	−1.44969	+	0.836978i	−0.451225	+	0.892410i	$0.649013\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−51.0775			−1.72085			−0.860423	−	0.509580i	$-0.829801\pi$
$881$	−51.0775			−1.72085			−0.860423	+	0.509580i	$0.829801\pi$
$882$		0			0
$883$		−	51.1113i		−	1.72003i	−0.510266	−	0.860017i	$-0.670453\pi$
$883$		−	51.1113i		−	1.72003i	0.510266	−	0.860017i	$-0.329547\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	29.4079	−	16.9786i	0.987420	−	0.570087i	0.0829179	−	0.996556i	$-0.473576\pi$
$887$	29.4079	−	16.9786i	0.987420	−	0.570087i	0.904502	+	0.426469i	$0.140243\pi$
$888$		0			0
$889$	−0.321051	+	0.556076i	−0.0107677	+	0.0186502i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−11.4983	−	6.63854i	−0.384776	−	0.222150i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.69264			0.289916
$900$		0			0
$901$	−11.9689			−0.398742
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−5.02638	−	2.90198i	−0.166898	−	0.0963588i	0.414224	−	0.910175i	$-0.364053\pi$
$907$	−5.02638	−	2.90198i	−0.166898	−	0.0963588i	−0.581122	+	0.813816i	$0.697386\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−2.09995	+	3.63722i	−0.0695744	+	0.120506i	−0.898714	−	0.438535i	$-0.855497\pi$
$911$	−2.09995	+	3.63722i	−0.0695744	+	0.120506i	0.829140	+	0.559042i	$0.188831\pi$
$912$		0			0
$913$	−4.80929	+	2.77665i	−0.159164	+	0.0918936i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1.38033i			0.0455827i
$918$		0			0
$919$	−25.2655			−0.833431			−0.416715	−	0.909037i	$-0.636819\pi$
$919$	−25.2655			−0.833431			−0.416715	+	0.909037i	$0.636819\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−28.1674	+	16.2624i	−0.927141	+	0.535285i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.1932	−	31.5115i	0.596899	−	1.03386i	−0.396377	−	0.918088i	$-0.629733\pi$
$929$	18.1932	−	31.5115i	0.596899	−	1.03386i	0.993276	−	0.115771i	$-0.0369340\pi$
$930$		0			0
$931$	17.2662	+	29.9059i	0.565876	+	0.980127i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			17.5425i			0.573088i	0.958067	+	0.286544i	$0.0925064\pi$
$937$			17.5425i			0.573088i	−0.958067	+	0.286544i	$0.907494\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.1692	+	21.0777i	0.396705	+	0.687114i	0.993317	−	0.115416i	$-0.0368202\pi$
$941$	12.1692	+	21.0777i	0.396705	+	0.687114i	−0.596612	+	0.802530i	$0.703487\pi$
$942$		0			0
$943$	48.8191	+	28.1857i	1.58977	+	0.917853i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.64316	−	1.52603i	−0.0858913	−	0.0495893i	0.456439	−	0.889755i	$-0.349125\pi$
$947$	−2.64316	−	1.52603i	−0.0858913	−	0.0495893i	−0.542330	+	0.840165i	$0.682458\pi$
$948$		0			0
$949$	6.97392	+	12.0792i	0.226383	+	0.392107i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			11.6054i			0.375934i	0.982175	+	0.187967i	$0.0601899\pi$
$953$			11.6054i			0.375934i	−0.982175	+	0.187967i	$0.939810\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.795199	−	1.37732i	−0.0256783	−	0.0444761i
$960$		0			0
$961$	8.92974	−	15.4668i	0.288056	−	0.498928i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	14.5825	−	8.41919i	0.468941	−	0.270743i	−0.246856	−	0.969052i	$-0.579397\pi$
$967$	14.5825	−	8.41919i	0.468941	−	0.270743i	0.715796	+	0.698309i	$0.246064\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	18.3864			0.590046			0.295023	−	0.955490i	$-0.404673\pi$
$971$	18.3864			0.590046			0.295023	+	0.955490i	$0.404673\pi$
$972$		0			0
$973$		−	0.480952i		−	0.0154186i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25.5206	−	14.7343i	0.816477	−	0.471393i	−0.0327228	−	0.999464i	$-0.510418\pi$
$977$	25.5206	−	14.7343i	0.816477	−	0.471393i	0.849200	+	0.528071i	$0.177085\pi$
$978$		0			0
$979$	3.71547	−	6.43538i	0.118747	−	0.205676i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.6664	+	10.7771i	0.595366	+	0.343735i	0.767216	−	0.641388i	$-0.221641\pi$
$983$	18.6664	+	10.7771i	0.595366	+	0.343735i	−0.171851	+	0.985123i	$0.554975\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−69.9349			−2.22380
$990$		0			0
$991$	9.01027			0.286221			0.143110	−	0.989707i	$-0.454290\pi$
$991$	9.01027			0.286221			0.143110	+	0.989707i	$0.454290\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−26.4136	−	15.2499i	−0.836526	−	0.482969i	0.0195556	−	0.999809i	$-0.493775\pi$
$997$	−26.4136	−	15.2499i	−0.836526	−	0.482969i	−0.856082	+	0.516840i	$0.827108\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.s.d.2449.4		16
3.2	odd	2		900.2.s.d.349.3		16
5.2	odd	4		2700.2.i.e.1801.3		8
5.3	odd	4		2700.2.i.d.1801.2		8
5.4	even	2	inner	2700.2.s.d.2449.5		16
9.2	odd	6		8100.2.d.q.649.5		8
9.4	even	3	inner	2700.2.s.d.1549.5		16
9.5	odd	6		900.2.s.d.49.6		16
9.7	even	3		8100.2.d.s.649.5		8
15.2	even	4		900.2.i.d.601.4	yes	8
15.8	even	4		900.2.i.e.601.1	yes	8
15.14	odd	2		900.2.s.d.349.6		16
45.2	even	12		8100.2.a.x.1.2		4
45.4	even	6	inner	2700.2.s.d.1549.4		16
45.7	odd	12		8100.2.a.y.1.2		4
45.13	odd	12		2700.2.i.d.901.2		8
45.14	odd	6		900.2.s.d.49.3		16
45.22	odd	12		2700.2.i.e.901.3		8
45.23	even	12		900.2.i.e.301.1	yes	8
45.29	odd	6		8100.2.d.q.649.4		8
45.32	even	12		900.2.i.d.301.4	✓	8
45.34	even	6		8100.2.d.s.649.4		8
45.38	even	12		8100.2.a.z.1.3		4
45.43	odd	12		8100.2.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.2.i.d.301.4	✓	8	45.32	even	12
900.2.i.d.601.4	yes	8	15.2	even	4
900.2.i.e.301.1	yes	8	45.23	even	12
900.2.i.e.601.1	yes	8	15.8	even	4
900.2.s.d.49.3		16	45.14	odd	6
900.2.s.d.49.6		16	9.5	odd	6
900.2.s.d.349.3		16	3.2	odd	2
900.2.s.d.349.6		16	15.14	odd	2
2700.2.i.d.901.2		8	45.13	odd	12
2700.2.i.d.1801.2		8	5.3	odd	4
2700.2.i.e.901.3		8	45.22	odd	12
2700.2.i.e.1801.3		8	5.2	odd	4
2700.2.s.d.1549.4		16	45.4	even	6	inner
2700.2.s.d.1549.5		16	9.4	even	3	inner
2700.2.s.d.2449.4		16	1.1	even	1	trivial
2700.2.s.d.2449.5		16	5.4	even	2	inner
8100.2.a.x.1.2		4	45.2	even	12
8100.2.a.y.1.2		4	45.7	odd	12
8100.2.a.z.1.3		4	45.38	even	12
8100.2.a.ba.1.3		4	45.43	odd	12
8100.2.d.q.649.4		8	45.29	odd	6
8100.2.d.q.649.5		8	9.2	odd	6
8100.2.d.s.649.4		8	45.34	even	6
8100.2.d.s.649.5		8	9.7	even	3

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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.0748933 - 0.0432397i) q^{7} +O(q^{10})\)	\(q+(-0.0748933 - 0.0432397i) q^{7} +(-0.456760 + 0.791132i) q^{11} +(-2.27331 + 1.31249i) q^{13} -2.08648i q^{17} -4.93847 q^{19} +(7.34128 - 4.23849i) q^{23} +(-1.19899 + 2.07671i) q^{29} +(-1.81249 - 3.13933i) q^{31} -5.85199i q^{37} +(3.32497 + 5.75902i) q^{41} +(-7.14469 - 4.12499i) q^{43} +(2.32831 + 1.34425i) q^{47} +(-3.49626 - 6.05570i) q^{49} -5.73642i q^{53} +(-6.16922 - 10.6854i) q^{59} +(3.16823 - 5.48753i) q^{61} +(5.33946 - 3.08274i) q^{67} +12.3905 q^{71} -5.31349i q^{73} +(0.0684166 - 0.0395003i) q^{77} +(-6.72394 + 11.6462i) q^{79} +(5.26457 + 3.03950i) q^{83} -8.13440 q^{89} +0.227007 q^{91} +(-9.61459 - 5.55098i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 6 q^{11} + 16 q^{19} - 18 q^{29} - 4 q^{31} - 18 q^{41} + 18 q^{49} - 30 q^{59} + 2 q^{61} + 48 q^{71} - 14 q^{79} + 12 q^{89} - 44 q^{91}+O(q^{100}) \) 16 * q - 6 * q^11 + 16 * q^19 - 18 * q^29 - 4 * q^31 - 18 * q^41 + 18 * q^49 - 30 * q^59 + 2 * q^61 + 48 * q^71 - 14 * q^79 + 12 * q^89 - 44 * q^91

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