LMFDB - Embedded newform 2700.3.p.b.1601.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,3,Mod(1601,2700)]

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, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2700.1601");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2700.p (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:	$73.5696713773$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1601.1
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	2700.1601
Dual form			2700.3.p.b.2501.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.05842 + 7.02939i) q^{7} +O(q^{10})$	$q+(-4.05842 + 7.02939i) q^{7} +(17.6168 + 10.1711i) q^{11} +(3.05842 + 5.29734i) q^{13} +17.9653i q^{17} +9.11684 q^{19} +(29.0584 - 16.7769i) q^{23} +(-14.4090 - 8.31901i) q^{29} +(11.1753 + 19.3561i) q^{31} +50.4674 q^{37} +(-29.9674 + 17.3017i) q^{41} +(11.5000 - 19.9186i) q^{43} +(-33.1753 - 19.1537i) q^{47} +(-8.44158 - 14.6212i) q^{49} +19.0149i q^{53} +(2.96738 - 1.71322i) q^{59} +(23.1753 - 40.1407i) q^{61} +(-3.14947 - 5.45504i) q^{67} +35.9306i q^{71} -47.3505 q^{73} +(-142.993 + 82.5571i) q^{77} +(42.2921 - 73.2521i) q^{79} +(-33.1753 - 19.1537i) q^{83} +143.723i q^{89} -49.6495 q^{91} +(40.3832 - 69.9457i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{7}+O(q^{10}) $ 4 * q + q^7	$ 4 q + q^{7} + 36 q^{11} - 5 q^{13} + 2 q^{19} + 99 q^{23} + 63 q^{29} - 7 q^{31} + 64 q^{37} + 18 q^{41} + 46 q^{43} - 81 q^{47} - 51 q^{49} - 126 q^{59} + 41 q^{61} - 116 q^{67} - 86 q^{73} - 279 q^{77} + 83 q^{79} - 81 q^{83} - 302 q^{91} + 196 q^{97}+O(q^{100}) $ 4 * q + q^7 + 36 * q^11 - 5 * q^13 + 2 * q^19 + 99 * q^23 + 63 * q^29 - 7 * q^31 + 64 * q^37 + 18 * q^41 + 46 * q^43 - 81 * q^47 - 51 * q^49 - 126 * q^59 + 41 * q^61 - 116 * q^67 - 86 * q^73 - 279 * q^77 + 83 * q^79 - 81 * q^83 - 302 * q^91 + 196 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

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
$7$	−4.05842	+	7.02939i	−0.579775	+	1.00420i	0.415730	+	0.909488i	$0.363526\pi$
$7$	−4.05842	+	7.02939i	−0.579775	+	1.00420i	−0.995505	+	0.0947110i	$0.969807\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	17.6168	+	10.1711i	1.60153	+	0.924645i	0.991181	+	0.132513i	$0.0423045\pi$
$11$	17.6168	+	10.1711i	1.60153	+	0.924645i	0.610350	+	0.792132i	$0.291029\pi$
$12$		0			0
$13$	3.05842	+	5.29734i	0.235263	+	0.407488i	0.959349	−	0.282222i	$-0.0910714\pi$
$13$	3.05842	+	5.29734i	0.235263	+	0.407488i	−0.724086	+	0.689710i	$0.757738\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			17.9653i			1.05678i	0.849001	+	0.528392i	$0.177205\pi$
$17$			17.9653i			1.05678i	−0.849001	+	0.528392i	$0.822795\pi$
$18$		0			0
$19$	9.11684			0.479834			0.239917	−	0.970793i	$-0.422880\pi$
$19$	9.11684			0.479834			0.239917	+	0.970793i	$0.422880\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	29.0584	−	16.7769i	1.26341	−	0.729430i	0.289677	−	0.957124i	$-0.406452\pi$
$23$	29.0584	−	16.7769i	1.26341	−	0.729430i	0.973733	+	0.227695i	$0.0731188\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−14.4090	−	8.31901i	−0.496860	−	0.286863i	0.230556	−	0.973059i	$-0.425946\pi$
$29$	−14.4090	−	8.31901i	−0.496860	−	0.286863i	−0.727416	+	0.686197i	$0.759279\pi$
$30$		0			0
$31$	11.1753	+	19.3561i	0.360492	+	0.624391i	0.988042	−	0.154185i	$-0.0492753\pi$
$31$	11.1753	+	19.3561i	0.360492	+	0.624391i	−0.627549	+	0.778577i	$0.715942\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	50.4674			1.36398			0.681992	−	0.731360i	$-0.261114\pi$
$37$	50.4674			1.36398			0.681992	+	0.731360i	$0.261114\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−29.9674	+	17.3017i	−0.730912	+	0.421992i	−0.818756	−	0.574142i	$-0.805336\pi$
$41$	−29.9674	+	17.3017i	−0.730912	+	0.421992i	0.0878440	+	0.996134i	$0.472002\pi$
$42$		0			0
$43$	11.5000	−	19.9186i	0.267442	−	0.463223i	−0.700759	−	0.713398i	$-0.747155\pi$
$43$	11.5000	−	19.9186i	0.267442	−	0.463223i	0.968200	+	0.250176i	$0.0804883\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−33.1753	−	19.1537i	−0.705857	−	0.407527i	0.103668	−	0.994612i	$-0.466942\pi$
$47$	−33.1753	−	19.1537i	−0.705857	−	0.407527i	−0.809525	+	0.587085i	$0.800275\pi$
$48$		0			0
$49$	−8.44158	−	14.6212i	−0.172277	−	0.298393i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			19.0149i			0.358771i	0.983779	+	0.179386i	$0.0574110\pi$
$53$			19.0149i			0.358771i	−0.983779	+	0.179386i	$0.942589\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.96738	−	1.71322i	0.0502945	−	0.0290375i	−0.474642	−	0.880179i	$-0.657422\pi$
$59$	2.96738	−	1.71322i	0.0502945	−	0.0290375i	0.524936	+	0.851141i	$0.324089\pi$
$60$		0			0
$61$	23.1753	−	40.1407i	0.379922	−	0.658045i	−0.611128	−	0.791532i	$-0.709284\pi$
$61$	23.1753	−	40.1407i	0.379922	−	0.658045i	0.991051	+	0.133487i	$0.0426174\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−3.14947	−	5.45504i	−0.0470070	−	0.0814185i	0.841565	−	0.540157i	$-0.181635\pi$
$67$	−3.14947	−	5.45504i	−0.0470070	−	0.0814185i	−0.888572	+	0.458738i	$0.848302\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			35.9306i			0.506065i	0.967458	+	0.253033i	$0.0814280\pi$
$71$			35.9306i			0.506065i	−0.967458	+	0.253033i	$0.918572\pi$
$72$		0			0
$73$	−47.3505			−0.648637			−0.324319	−	0.945948i	$-0.605135\pi$
$73$	−47.3505			−0.648637			−0.324319	+	0.945948i	$0.605135\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−142.993	+	82.5571i	−1.85705	+	1.07217i
$78$		0			0
$79$	42.2921	−	73.2521i	0.535343	−	0.927242i	−0.463803	−	0.885938i	$-0.653516\pi$
$79$	42.2921	−	73.2521i	0.535343	−	0.927242i	0.999147	−	0.0413035i	$-0.0131510\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−33.1753	−	19.1537i	−0.399702	−	0.230768i	0.286653	−	0.958034i	$-0.407457\pi$
$83$	−33.1753	−	19.1537i	−0.399702	−	0.230768i	−0.686355	+	0.727266i	$0.740791\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			143.723i			1.61486i	0.589963	+	0.807430i	$0.299142\pi$
$89$			143.723i			1.61486i	−0.589963	+	0.807430i	$0.700858\pi$
$90$		0			0
$91$	−49.6495			−0.545599
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	40.3832	−	69.9457i	0.416321	−	0.721089i	−0.579245	−	0.815154i	$-0.696653\pi$
$97$	40.3832	−	69.9457i	0.416321	−	0.721089i	0.995566	+	0.0940641i	$0.0299859\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−105.942	−	61.1654i	−1.04893	−	0.605598i	−0.126578	−	0.991957i	$-0.540399\pi$
$101$	−105.942	−	61.1654i	−1.04893	−	0.605598i	−0.922349	+	0.386359i	$0.873733\pi$
$102$		0			0
$103$	36.8247	+	63.7823i	0.357522	+	0.619246i	0.987546	−	0.157330i	$-0.0502885\pi$
$103$	36.8247	+	63.7823i	0.357522	+	0.619246i	−0.630024	+	0.776575i	$0.716955\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			72.9108i			0.681410i	0.940170	+	0.340705i	$0.110666\pi$
$107$			72.9108i			0.681410i	−0.940170	+	0.340705i	$0.889334\pi$
$108$		0			0
$109$	31.2989			0.287146			0.143573	−	0.989640i	$-0.454141\pi$
$109$	31.2989			0.287146			0.143573	+	0.989640i	$0.454141\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	16.2269	−	9.36858i	0.143601	−	0.0829078i	−0.426478	−	0.904498i	$-0.640246\pi$
$113$	16.2269	−	9.36858i	0.143601	−	0.0829078i	0.570079	+	0.821590i	$0.306913\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−126.285	−	72.9108i	−1.06122	−	0.612696i
$120$		0			0
$121$	146.402	+	253.576i	1.20993	+	2.09567i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	126.103			0.992939			0.496469	−	0.868054i	$-0.334630\pi$
$127$	126.103			0.992939			0.496469	+	0.868054i	$0.334630\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−140.694	+	81.2299i	−1.07400	+	0.620075i	−0.929272	−	0.369396i	$-0.879565\pi$
$131$	−140.694	+	81.2299i	−1.07400	+	0.620075i	−0.144730	+	0.989471i	$0.546231\pi$
$132$		0			0
$133$	−37.0000	+	64.0859i	−0.278195	+	0.481849i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	90.3832	+	52.1827i	0.659731	+	0.380896i	0.792174	−	0.610295i	$-0.208949\pi$
$137$	90.3832	+	52.1827i	0.659731	+	0.380896i	−0.132443	+	0.991191i	$0.542282\pi$
$138$		0			0
$139$	−30.6168	−	53.0299i	−0.220265	−	0.381510i	0.734623	−	0.678475i	$-0.237359\pi$
$139$	−30.6168	−	53.0299i	−0.220265	−	0.381510i	−0.954888	+	0.296965i	$0.904026\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			124.430i			0.870139i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	128.344	−	74.0993i	0.861367	−	0.497311i	−0.00310272	−	0.999995i	$-0.500988\pi$
$149$	128.344	−	74.0993i	0.861367	−	0.497311i	0.864470	+	0.502685i	$0.167654\pi$
$150$		0			0
$151$	−127.526	+	220.881i	−0.844542	+	1.46279i	0.0414769	+	0.999139i	$0.486794\pi$
$151$	−127.526	+	220.881i	−0.844542	+	1.46279i	−0.886019	+	0.463650i	$0.846540\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	146.227	+	253.272i	0.931381	+	1.61320i	0.780963	+	0.624577i	$0.214729\pi$
$157$	146.227	+	253.272i	0.931381	+	1.61320i	0.150418	+	0.988622i	$0.451938\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			272.351i			1.69162i
$162$		0			0
$163$	−93.5326			−0.573820			−0.286910	−	0.957958i	$-0.592628\pi$
$163$	−93.5326			−0.573820			−0.286910	+	0.957958i	$0.592628\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−97.2269	+	56.1340i	−0.582197	+	0.336131i	−0.762006	−	0.647570i	$-0.775785\pi$
$167$	−97.2269	+	56.1340i	−0.582197	+	0.336131i	0.179809	+	0.983702i	$0.442452\pi$
$168$		0			0
$169$	65.7921	−	113.955i	0.389302	−	0.674292i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−205.227	−	118.488i	−1.18628	−	0.684900i	−0.228823	−	0.973468i	$-0.573488\pi$
$173$	−205.227	−	118.488i	−1.18628	−	0.684900i	−0.957460	+	0.288568i	$0.906821\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	234.599i		−	1.31061i	−0.755366	−	0.655304i	$-0.772541\pi$
$179$		−	234.599i		−	1.31061i	0.755366	−	0.655304i	$-0.227459\pi$
$180$		0			0
$181$	221.636			1.22451			0.612254	−	0.790661i	$-0.290263\pi$
$181$	221.636			1.22451			0.612254	+	0.790661i	$0.290263\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−182.727	+	316.492i	−0.977149	+	1.69247i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	130.162	+	75.1488i	0.681474	+	0.393449i	0.800410	−	0.599452i	$-0.204615\pi$
$191$	130.162	+	75.1488i	0.681474	+	0.393449i	−0.118936	+	0.992902i	$0.537948\pi$
$192$		0			0
$193$	−24.5000	−	42.4352i	−0.126943	−	0.219872i	0.795548	−	0.605891i	$-0.207183\pi$
$193$	−24.5000	−	42.4352i	−0.126943	−	0.219872i	−0.922491	+	0.386019i	$0.873850\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			276.827i			1.40521i	0.711579	+	0.702606i	$0.247980\pi$
$197$			276.827i			1.40521i	−0.711579	+	0.702606i	$0.752020\pi$
$198$		0			0
$199$	−198.935			−0.999672			−0.499836	−	0.866120i	$-0.666606\pi$
$199$	−198.935			−0.999672			−0.499836	+	0.866120i	$0.666606\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	116.955	−	67.5241i	0.576134	−	0.332631i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	160.610	+	92.7282i	0.768469	+	0.443676i
$210$		0			0
$211$	−47.0068	−	81.4182i	−0.222781	−	0.385868i	0.732870	−	0.680368i	$-0.238180\pi$
$211$	−47.0068	−	81.4182i	−0.222781	−	0.385868i	−0.955651	+	0.294500i	$0.904847\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−181.416			−0.836017
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−95.1684	+	54.9455i	−0.430626	+	0.248622i
$222$		0			0
$223$	−77.8763	+	134.886i	−0.349221	+	0.604869i	−0.986111	−	0.166086i	$-0.946887\pi$
$223$	−77.8763	+	134.886i	−0.349221	+	0.604869i	0.636890	+	0.770955i	$0.280220\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−138.448	−	79.9332i	−0.609905	−	0.352129i	0.163023	−	0.986622i	$-0.447875\pi$
$227$	−138.448	−	79.9332i	−0.609905	−	0.352129i	−0.772928	+	0.634493i	$0.781209\pi$
$228$		0			0
$229$	19.1237	+	33.1232i	0.0835095	+	0.144643i	0.904755	−	0.425932i	$-0.140054\pi$
$229$	19.1237	+	33.1232i	0.0835095	+	0.144643i	−0.821246	+	0.570575i	$0.806720\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			157.490i			0.675921i	0.941160	+	0.337960i	$0.109737\pi$
$233$			157.490i			0.675921i	−0.941160	+	0.337960i	$0.890263\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	62.4742	−	36.0695i	0.261398	−	0.150918i	−0.363574	−	0.931565i	$-0.618444\pi$
$239$	62.4742	−	36.0695i	0.261398	−	0.150918i	0.624972	+	0.780647i	$0.285110\pi$
$240$		0			0
$241$	−113.370	+	196.362i	−0.470413	+	0.814779i	−0.999427	−	0.0338337i	$-0.989228\pi$
$241$	−113.370	+	196.362i	−0.470413	+	0.814779i	0.529015	+	0.848613i	$0.322562\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	27.8832	+	48.2950i	0.112887	+	0.195526i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			222.931i			0.888171i	0.895985	+	0.444085i	$0.146471\pi$
$251$			222.931i			0.888171i	−0.895985	+	0.444085i	$0.853529\pi$
$252$		0			0
$253$	682.557			2.69785
$254$		0			0
$255$		0			0
$256$		0			0
$257$	92.2011	−	53.2323i	0.358759	−	0.207130i	−0.309777	−	0.950809i	$-0.600254\pi$
$257$	92.2011	−	53.2323i	0.358759	−	0.207130i	0.668536	+	0.743680i	$0.266921\pi$
$258$		0			0
$259$	−204.818	+	354.755i	−0.790803	+	1.36971i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	155.344	+	89.6877i	0.590660	+	0.341018i	0.765359	−	0.643604i	$-0.222562\pi$
$263$	155.344	+	89.6877i	0.590660	+	0.341018i	−0.174698	+	0.984622i	$0.555895\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	416.351i		−	1.54777i	−0.633324	−	0.773887i	$-0.718310\pi$
$269$		−	416.351i		−	1.54777i	0.633324	−	0.773887i	$-0.281690\pi$
$270$		0			0
$271$	396.907			1.46460			0.732302	−	0.680980i	$-0.238446\pi$
$271$	396.907			1.46460			0.732302	+	0.680980i	$0.238446\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−57.7731	+	100.066i	−0.208567	+	0.361249i	−0.951263	−	0.308379i	$-0.900213\pi$
$277$	−57.7731	+	100.066i	−0.208567	+	0.361249i	0.742696	+	0.669629i	$0.233547\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−422.564	−	243.967i	−1.50379	−	0.868211i	−0.999990	−	0.00438786i	$-0.998603\pi$
$281$	−422.564	−	243.967i	−1.50379	−	0.868211i	−0.503795	−	0.863823i	$-0.668063\pi$
$282$		0			0
$283$	−169.825	−	294.145i	−0.600087	−	1.03938i	−0.992807	−	0.119724i	$-0.961799\pi$
$283$	−169.825	−	294.145i	−0.600087	−	1.03938i	0.392720	−	0.919658i	$-0.371534\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	280.870i		−	0.978641i
$288$		0			0
$289$	−33.7527			−0.116791
$290$		0			0
$291$		0			0
$292$		0			0
$293$	122.409	−	70.6728i	0.417778	−	0.241204i	−0.276348	−	0.961058i	$-0.589124\pi$
$293$	122.409	−	70.6728i	0.417778	−	0.241204i	0.694126	+	0.719853i	$0.255791\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	177.746	+	102.622i	0.594468	+	0.343216i
$300$		0			0
$301$	93.3437	+	161.676i	0.310112	+	0.537130i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	120.649			0.392995			0.196498	−	0.980504i	$-0.437043\pi$
$307$	120.649			0.392995			0.196498	+	0.980504i	$0.437043\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	119.254	−	68.8514i	0.383454	−	0.221387i	−0.295866	−	0.955229i	$-0.595608\pi$
$311$	119.254	−	68.8514i	0.383454	−	0.221387i	0.679320	+	0.733842i	$0.262275\pi$
$312$		0			0
$313$	129.266	−	223.896i	0.412991	−	0.715322i	−0.582224	−	0.813029i	$-0.697817\pi$
$313$	129.266	−	223.896i	0.412991	−	0.715322i	0.995215	+	0.0977064i	$0.0311506\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−16.7079	−	9.64630i	−0.0527063	−	0.0304300i	0.473415	−	0.880839i	$-0.343021\pi$
$317$	−16.7079	−	9.64630i	−0.0527063	−	0.0304300i	−0.526122	+	0.850409i	$0.676354\pi$
$318$		0			0
$319$	−169.227	−	293.110i	−0.530492	−	0.918839i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			163.787i			0.507081i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	269.278	−	155.468i	0.818476	−	0.472547i
$330$		0			0
$331$	98.3953	−	170.426i	0.297267	−	0.514881i	−0.678243	−	0.734838i	$-0.737258\pi$
$331$	98.3953	−	170.426i	0.297267	−	0.514881i	0.975510	+	0.219957i	$0.0705916\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	158.720	+	274.911i	0.470979	+	0.815760i	0.999449	−	0.0331921i	$-0.0105673\pi$
$337$	158.720	+	274.911i	0.470979	+	0.815760i	−0.528470	+	0.848952i	$0.677234\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			454.659i			1.33331i
$342$		0			0
$343$	−260.687			−0.760022
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−537.407	+	310.272i	−1.54872	+	0.894157i	−0.550485	+	0.834845i	$0.685557\pi$
$347$	−537.407	+	310.272i	−1.54872	+	0.894157i	−0.998240	+	0.0593116i	$0.981109\pi$
$348$		0			0
$349$	189.512	−	328.245i	0.543015	−	0.940529i	−0.455714	−	0.890126i	$-0.650616\pi$
$349$	189.512	−	328.245i	0.543015	−	0.940529i	0.998729	−	0.0504030i	$-0.0160506\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−213.514	−	123.272i	−0.604855	−	0.349213i	0.166094	−	0.986110i	$-0.446884\pi$
$353$	−213.514	−	123.272i	−0.604855	−	0.349213i	−0.770949	+	0.636897i	$0.780218\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			572.791i			1.59552i	0.602976	+	0.797759i	$0.293981\pi$
$359$			572.791i			1.59552i	−0.602976	+	0.797759i	$0.706019\pi$
$360$		0			0
$361$	−277.883			−0.769759
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−93.9279	+	162.688i	−0.255934	+	0.443291i	−0.965149	−	0.261701i	$-0.915716\pi$
$367$	−93.9279	+	162.688i	−0.255934	+	0.443291i	0.709214	+	0.704993i	$0.249050\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−133.663	−	77.1704i	−0.360278	−	0.208007i
$372$		0			0
$373$	75.0584	+	130.005i	0.201229	+	0.348539i	0.948925	−	0.315503i	$-0.102173\pi$
$373$	75.0584	+	130.005i	0.201229	+	0.348539i	−0.747696	+	0.664042i	$0.768840\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	101.772i		−	0.269953i
$378$		0			0
$379$	−26.6222			−0.0702432			−0.0351216	−	0.999383i	$-0.511182\pi$
$379$	−26.6222			−0.0702432			−0.0351216	+	0.999383i	$0.511182\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−444.966	+	256.901i	−1.16179	+	0.670760i	−0.951733	−	0.306929i	$-0.900699\pi$
$383$	−444.966	+	256.901i	−1.16179	+	0.670760i	−0.210058	+	0.977689i	$0.567365\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	22.1616	+	12.7950i	0.0569707	+	0.0328921i	0.528215	−	0.849111i	$-0.322862\pi$
$389$	22.1616	+	12.7950i	0.0569707	+	0.0328921i	−0.471244	+	0.882003i	$0.656195\pi$
$390$		0			0
$391$	301.402	+	522.044i	0.770849	+	1.33515i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−388.804			−0.979356			−0.489678	−	0.871903i	$-0.662886\pi$
$397$	−388.804			−0.979356			−0.489678	+	0.871903i	$0.662886\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	34.0842	−	19.6785i	0.0849981	−	0.0490736i	−0.456899	−	0.889519i	$-0.651040\pi$
$401$	34.0842	−	19.6785i	0.0849981	−	0.0490736i	0.541897	+	0.840445i	$0.317706\pi$
$402$		0			0
$403$	−68.3574	+	118.398i	−0.169621	+	0.293793i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	889.076	+	513.308i	2.18446	+	1.26120i
$408$		0			0
$409$	−86.7200	−	150.204i	−0.212029	−	0.367246i	0.740320	−	0.672255i	$-0.234674\pi$
$409$	−86.7200	−	150.204i	−0.212029	−	0.367246i	−0.952350	+	0.305009i	$0.901341\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			27.8118i			0.0673409i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−115.031	+	66.4132i	−0.274537	+	0.158504i	−0.630948	−	0.775825i	$-0.717334\pi$
$419$	−115.031	+	66.4132i	−0.274537	+	0.158504i	0.356411	+	0.934329i	$0.384000\pi$
$420$		0			0
$421$	−317.447	+	549.834i	−0.754031	+	1.30602i	0.191824	+	0.981429i	$0.438560\pi$
$421$	−317.447	+	549.834i	−0.754031	+	1.30602i	−0.945855	+	0.324590i	$0.894774\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	188.110	+	325.816i	0.440539	+	0.763035i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			602.424i			1.39774i	0.715251	+	0.698868i	$0.246313\pi$
$431$			602.424i			1.39774i	−0.715251	+	0.698868i	$0.753687\pi$
$432$		0			0
$433$	−266.155			−0.614676			−0.307338	−	0.951600i	$-0.599438\pi$
$433$	−266.155			−0.614676			−0.307338	+	0.951600i	$0.599438\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	264.921	−	152.952i	0.606227	−	0.350005i
$438$		0			0
$439$	−250.330	+	433.584i	−0.570228	+	0.987664i	0.426314	+	0.904575i	$0.359812\pi$
$439$	−250.330	+	433.584i	−0.570228	+	0.987664i	−0.996542	+	0.0830886i	$0.973522\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−261.098	−	150.745i	−0.589386	−	0.340282i	0.175469	−	0.984485i	$-0.443856\pi$
$443$	−261.098	−	150.745i	−0.589386	−	0.340282i	−0.764855	+	0.644203i	$0.777189\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	565.321i		−	1.25907i	−0.776973	−	0.629534i	$-0.783246\pi$
$449$		−	565.321i		−	1.25907i	0.776973	−	0.629534i	$-0.216754\pi$
$450$		0			0
$451$	−703.907			−1.56077
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	26.1495	−	45.2922i	0.0572198	−	0.0991077i	−0.835997	−	0.548735i	$-0.815110\pi$
$457$	26.1495	−	45.2922i	0.0572198	−	0.0991077i	0.893216	+	0.449627i	$0.148443\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−166.357	−	96.0465i	−0.360862	−	0.208344i	0.308597	−	0.951193i	$-0.400141\pi$
$461$	−166.357	−	96.0465i	−0.360862	−	0.208344i	−0.669459	+	0.742849i	$0.733474\pi$
$462$		0			0
$463$	283.110	+	490.361i	0.611469	+	1.05909i	0.990993	+	0.133913i	$0.0427543\pi$
$463$	283.110	+	490.361i	0.611469	+	1.05909i	−0.379524	+	0.925182i	$0.623912\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			174.405i			0.373459i	0.982411	+	0.186729i	$0.0597888\pi$
$467$			174.405i			0.373459i	−0.982411	+	0.186729i	$0.940211\pi$
$468$		0			0
$469$	51.1275			0.109014
$470$		0			0
$471$		0			0
$472$		0			0
$473$	405.187	−	233.935i	0.856633	−	0.494577i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−473.784	−	273.539i	−0.989110	−	0.571063i	−0.0841020	−	0.996457i	$-0.526802\pi$
$479$	−473.784	−	273.539i	−0.989110	−	0.571063i	−0.905008	+	0.425394i	$0.860135\pi$
$480$		0			0
$481$	154.351	+	267.343i	0.320895	+	0.555807i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	769.945			1.58100			0.790498	−	0.612464i	$-0.209822\pi$
$487$	769.945			1.58100			0.790498	+	0.612464i	$0.209822\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	154.916	−	89.4407i	0.315511	−	0.182160i	−0.333879	−	0.942616i	$-0.608358\pi$
$491$	154.916	−	89.4407i	0.315511	−	0.182160i	0.649390	+	0.760456i	$0.275024\pi$
$492$		0			0
$493$	149.454	−	258.861i	0.303152	−	0.525074i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−252.571	−	145.822i	−0.508190	−	0.293404i
$498$		0			0
$499$	192.655	+	333.688i	0.386082	+	0.668713i	0.991919	−	0.126876i	$-0.0404949\pi$
$499$	192.655	+	333.688i	0.386082	+	0.668713i	−0.605837	+	0.795589i	$0.707162\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			67.6630i			0.134519i	0.997736	+	0.0672594i	$0.0214255\pi$
$503$			67.6630i			0.134519i	−0.997736	+	0.0672594i	$0.978574\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−523.292	+	302.123i	−1.02808	+	0.593562i	−0.916434	−	0.400187i	$-0.868945\pi$
$509$	−523.292	+	302.123i	−1.02808	+	0.593562i	−0.111645	+	0.993748i	$0.535612\pi$
$510$		0			0
$511$	192.168	−	332.846i	0.376063	−	0.651361i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−389.629	−	674.857i	−0.753634	−	1.30533i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	273.678i		−	0.525294i	−0.964892	−	0.262647i	$-0.915405\pi$
$521$		−	273.678i		−	0.525294i	0.964892	−	0.262647i	$-0.0845954\pi$
$522$		0			0
$523$	−687.402			−1.31434			−0.657172	−	0.753740i	$-0.728248\pi$
$523$	−687.402			−1.31434			−0.657172	+	0.753740i	$0.728248\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−347.739	+	200.767i	−0.659846	+	0.380962i
$528$		0			0
$529$	298.428	−	516.892i	0.564136	−	0.977112i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−183.306	−	105.832i	−0.343913	−	0.198558i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	343.440i		−	0.637180i
$540$		0			0
$541$	664.543			1.22836			0.614180	−	0.789166i	$-0.289487\pi$
$541$	664.543			1.22836			0.614180	+	0.789166i	$0.289487\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	259.603	−	449.646i	0.474594	−	0.822022i	−0.524982	−	0.851113i	$-0.675928\pi$
$547$	259.603	−	449.646i	0.474594	−	0.822022i	0.999577	+	0.0290914i	$0.00926138\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−131.364	−	75.8431i	−0.238410	−	0.137646i
$552$		0			0
$553$	343.278	+	594.576i	0.620757	+	1.07518i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	422.648i		−	0.758794i	−0.925234	−	0.379397i	$-0.876131\pi$
$557$		−	422.648i		−	0.758794i	0.925234	−	0.379397i	$-0.123869\pi$
$558$		0			0
$559$	140.687			0.251677
$560$		0			0
$561$		0			0
$562$		0			0
$563$	798.799	−	461.187i	1.41883	−	0.819159i	0.422630	−	0.906302i	$-0.361107\pi$
$563$	798.799	−	461.187i	1.41883	−	0.819159i	0.996196	+	0.0871428i	$0.0277737\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	914.445	+	527.955i	1.60711	+	0.927865i	0.990013	+	0.140978i	$0.0450247\pi$
$569$	914.445	+	527.955i	1.60711	+	0.927865i	0.617097	+	0.786887i	$0.288309\pi$
$570$		0			0
$571$	−401.524	−	695.460i	−0.703195	−	1.21797i	−0.967339	−	0.253486i	$-0.918423\pi$
$571$	−401.524	−	695.460i	−0.703195	−	1.21797i	0.264144	−	0.964483i	$-0.414911\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	96.6495			0.167503			0.0837517	−	0.996487i	$-0.473310\pi$
$577$	96.6495			0.167503			0.0837517	+	0.996487i	$0.473310\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	269.278	−	155.468i	0.463474	−	0.267587i
$582$		0			0
$583$	−193.402	+	334.982i	−0.331736	+	0.574584i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−870.497	−	502.582i	−1.48296	−	0.856187i	−0.483146	−	0.875540i	$-0.660506\pi$
$587$	−870.497	−	502.582i	−1.48296	−	0.856187i	−0.999813	+	0.0193528i	$0.993839\pi$
$588$		0			0
$589$	101.883	+	176.467i	0.172976	+	0.299604i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			752.444i			1.26888i	0.772973	+	0.634439i	$0.218769\pi$
$593$			752.444i			1.26888i	−0.772973	+	0.634439i	$0.781231\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	24.0857	−	13.9059i	0.0402099	−	0.0232152i	−0.479760	−	0.877400i	$-0.659276\pi$
$599$	24.0857	−	13.9059i	0.0402099	−	0.0232152i	0.519970	+	0.854184i	$0.325943\pi$
$600$		0			0
$601$	475.356	−	823.340i	0.790942	−	1.36995i	−0.134443	−	0.990921i	$-0.542925\pi$
$601$	475.356	−	823.340i	0.790942	−	1.36995i	0.925385	−	0.379030i	$-0.123742\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−161.306	−	279.390i	−0.265743	−	0.460280i	0.702015	−	0.712162i	$-0.252284\pi$
$607$	−161.306	−	279.390i	−0.265743	−	0.460280i	−0.967758	+	0.251882i	$0.918950\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	234.321i		−	0.383504i
$612$		0			0
$613$	−138.206			−0.225459			−0.112730	−	0.993626i	$-0.535959\pi$
$613$	−138.206			−0.225459			−0.112730	+	0.993626i	$0.535959\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−682.084	+	393.802i	−1.10548	+	0.638252i	−0.937656	−	0.347564i	$-0.887009\pi$
$617$	−682.084	+	393.802i	−1.10548	+	0.638252i	−0.167829	+	0.985816i	$0.553676\pi$
$618$		0			0
$619$	121.747	−	210.873i	0.196684	−	0.340667i	−0.750767	−	0.660567i	$-0.770316\pi$
$619$	121.747	−	210.873i	0.196684	−	0.340667i	0.947451	+	0.319900i	$0.103649\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1010.28	−	583.287i	−1.62164	−	0.936255i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$			906.662i			1.44143i
$630$		0			0
$631$	111.924			0.177376			0.0886879	−	0.996059i	$-0.471733\pi$
$631$	111.924			0.177376			0.0886879	+	0.996059i	$0.471733\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	51.6358	−	89.4359i	0.0810609	−	0.140402i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−632.095	−	364.940i	−0.986107	−	0.569329i	−0.0819990	−	0.996632i	$-0.526130\pi$
$641$	−632.095	−	364.940i	−0.986107	−	0.569329i	−0.904108	+	0.427303i	$0.859464\pi$
$642$		0			0
$643$	−288.500	−	499.697i	−0.448678	−	0.777133i	0.549622	−	0.835413i	$-0.314772\pi$
$643$	−288.500	−	499.697i	−0.448678	−	0.777133i	−0.998300	+	0.0582801i	$0.981438\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			129.029i			0.199426i	0.995016	+	0.0997130i	$0.0317925\pi$
$647$			129.029i			0.199426i	−0.995016	+	0.0997130i	$0.968208\pi$
$648$		0			0
$649$	69.7011			0.107398
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1026.62	−	592.717i	1.57215	−	0.907682i	0.576247	−	0.817275i	$-0.304517\pi$
$653$	1026.62	−	592.717i	1.57215	−	0.907682i	0.995905	−	0.0904070i	$-0.0288168\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−947.808	−	547.217i	−1.43825	−	0.830375i	−0.440524	−	0.897741i	$-0.645207\pi$
$659$	−947.808	−	547.217i	−1.43825	−	0.830375i	−0.997728	+	0.0673658i	$0.978541\pi$
$660$		0			0
$661$	−604.876	−	1047.68i	−0.915093	−	1.58499i	−0.806765	−	0.590872i	$-0.798784\pi$
$661$	−604.876	−	1047.68i	−0.915093	−	1.58499i	−0.108327	−	0.994115i	$-0.534549\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−558.269			−0.836984
$668$		0			0
$669$		0			0
$670$		0			0
$671$	816.550	−	471.435i	1.21692	−	0.702586i
$672$		0			0
$673$	508.615	−	880.948i	0.755743	−	1.30899i	−0.189260	−	0.981927i	$-0.560609\pi$
$673$	508.615	−	880.948i	0.755743	−	1.30899i	0.945004	−	0.327059i	$-0.106058\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	689.890	+	398.308i	1.01904	+	0.588343i	0.913826	−	0.406107i	$-0.133114\pi$
$677$	689.890	+	398.308i	1.01904	+	0.588343i	0.105214	+	0.994450i	$0.466447\pi$
$678$		0			0
$679$	327.784	+	567.738i	0.482745	+	0.836139i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			400.485i			0.586361i	0.956057	+	0.293181i	$0.0947138\pi$
$683$			400.485i			0.586361i	−0.956057	+	0.293181i	$0.905286\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−100.728	+	58.1556i	−0.146195	+	0.0844057i
$690$		0			0
$691$	216.423	−	374.855i	0.313202	−	0.542482i	−0.665852	−	0.746084i	$-0.731932\pi$
$691$	216.423	−	374.855i	0.313202	−	0.542482i	0.979054	+	0.203602i	$0.0652650\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−310.830	−	538.373i	−0.445954	−	0.772415i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	65.4412i		−	0.0933541i	−0.998910	−	0.0466770i	$-0.985137\pi$
$701$		−	65.4412i		−	0.0933541i	0.998910	−	0.0466770i	$-0.0148632\pi$
$702$		0			0
$703$	460.103			0.654485
$704$		0			0
$705$		0			0
$706$		0			0
$707$	859.911	−	496.470i	1.21628	−	0.702221i
$708$		0			0
$709$	−100.461	+	174.003i	−0.141693	+	0.245420i	−0.928134	−	0.372245i	$-0.878588\pi$
$709$	−100.461	+	174.003i	−0.141693	+	0.245420i	0.786441	+	0.617665i	$0.211921\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	649.471	+	374.972i	0.910899	+	0.525908i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$			1062.98i			1.47841i	0.673478	+	0.739207i	$0.264800\pi$
$719$			1062.98i			1.47841i	−0.673478	+	0.739207i	$0.735200\pi$
$720$		0			0
$721$	−597.801			−0.829128
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	495.629	−	858.455i	0.681746	−	1.18082i	−0.292702	−	0.956204i	$-0.594554\pi$
$727$	495.629	−	858.455i	0.681746	−	1.18082i	0.974448	−	0.224614i	$-0.0721122\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	357.844	+	206.601i	0.489526	+	0.282628i
$732$		0			0
$733$	−590.134	−	1022.14i	−0.805095	−	1.39446i	−0.916227	−	0.400659i	$-0.868781\pi$
$733$	−590.134	−	1022.14i	−0.805095	−	1.39446i	0.111133	−	0.993806i	$-0.464552\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	128.134i		−	0.173859i
$738$		0			0
$739$	599.351			0.811029			0.405515	−	0.914089i	$-0.367092\pi$
$739$	599.351			0.811029			0.405515	+	0.914089i	$0.367092\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	287.083	−	165.747i	0.386383	−	0.223078i	−0.294209	−	0.955741i	$-0.595056\pi$
$743$	287.083	−	165.747i	0.386383	−	0.223078i	0.680592	+	0.732663i	$0.261723\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−512.519	−	295.903i	−0.684271	−	0.395064i
$750$		0			0
$751$	−76.0448	−	131.713i	−0.101258	−	0.175384i	0.810945	−	0.585122i	$-0.198953\pi$
$751$	−76.0448	−	131.713i	−0.101258	−	0.175384i	−0.912203	+	0.409738i	$0.865620\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1179.61			1.55827			0.779134	−	0.626858i	$-0.215659\pi$
$757$	1179.61			1.55827			0.779134	+	0.626858i	$0.215659\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1162.58	−	671.214i	1.52770	−	0.882016i	0.528239	−	0.849096i	$-0.322853\pi$
$761$	1162.58	−	671.214i	1.52770	−	0.882016i	0.999458	−	0.0329205i	$-0.0104808\pi$
$762$		0			0
$763$	−127.024	+	220.013i	−0.166480	+	0.288352i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	18.1510	+	10.4795i	0.0236649	+	0.0136629i
$768$		0			0
$769$	−548.512	−	950.051i	−0.713280	−	1.23544i	−0.963619	−	0.267279i	$-0.913876\pi$
$769$	−548.512	−	950.051i	−0.713280	−	1.23544i	0.250339	−	0.968158i	$-0.419458\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1181.39i			1.52832i	0.645028	+	0.764159i	$0.276846\pi$
$773$			1181.39i			1.52832i	−0.645028	+	0.764159i	$0.723154\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−273.208	+	157.737i	−0.350716	+	0.202486i
$780$		0			0
$781$	−365.454	+	632.984i	−0.467931	+	0.810479i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−18.0311	−	31.2308i	−0.0229112	−	0.0396834i	0.854342	−	0.519710i	$-0.173960\pi$
$787$	−18.0311	−	31.2308i	−0.0229112	−	0.0396834i	−0.877254	+	0.480027i	$0.840627\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			152.087i			0.192271i
$792$		0			0
$793$	283.519			0.357527
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−115.618	+	66.7523i	−0.145067	+	0.0837544i	−0.570777	−	0.821105i	$-0.693358\pi$
$797$	−115.618	+	66.7523i	−0.145067	+	0.0837544i	0.425710	+	0.904860i	$0.360024\pi$
$798$		0			0
$799$	344.103	−	596.004i	0.430667	−	0.745938i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−834.167	−	481.607i	−1.03881	−	0.599759i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1053.66i			1.30242i	0.758898	+	0.651209i	$0.225738\pi$
$809$			1053.66i			1.30242i	−0.758898	+	0.651209i	$0.774262\pi$
$810$		0			0
$811$	434.464			0.535714			0.267857	−	0.963459i	$-0.413684\pi$
$811$	434.464			0.535714			0.267857	+	0.963459i	$0.413684\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	104.844	−	181.595i	0.128328	−	0.222270i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−252.436	−	145.744i	−0.307474	−	0.177520i	0.338322	−	0.941031i	$-0.390141\pi$
$821$	−252.436	−	145.744i	−0.307474	−	0.177520i	−0.645796	+	0.763510i	$0.723474\pi$
$822$		0			0
$823$	168.409	+	291.693i	0.204628	+	0.354426i	0.950014	−	0.312207i	$-0.101068\pi$
$823$	168.409	+	291.693i	0.204628	+	0.354426i	−0.745386	+	0.666633i	$0.767735\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1029.27i		−	1.24458i	−0.782785	−	0.622292i	$-0.786202\pi$
$827$		−	1029.27i		−	1.24458i	0.782785	−	0.622292i	$-0.213798\pi$
$828$		0			0
$829$	790.674			0.953768			0.476884	−	0.878966i	$-0.341766\pi$
$829$	790.674			0.953768			0.476884	+	0.878966i	$0.341766\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	262.675	−	151.656i	0.315336	−	0.182060i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−655.031	−	378.182i	−0.780728	−	0.450754i	0.0559600	−	0.998433i	$-0.482178\pi$
$839$	−655.031	−	378.182i	−0.780728	−	0.450754i	−0.836688	+	0.547679i	$0.815511\pi$
$840$		0			0
$841$	−282.088	−	488.591i	−0.335420	−	0.580964i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−2376.65			−2.80596
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1466.50	−	846.686i	1.72327	−	0.994930i
$852$		0			0
$853$	−599.175	+	1037.80i	−0.702433	+	1.21665i	0.265177	+	0.964200i	$0.414570\pi$
$853$	−599.175	+	1037.80i	−0.702433	+	1.21665i	−0.967610	+	0.252450i	$0.918764\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	605.629	+	349.660i	0.706685	+	0.408005i	0.809832	−	0.586661i	$-0.199558\pi$
$857$	605.629	+	349.660i	0.706685	+	0.408005i	−0.103147	+	0.994666i	$0.532891\pi$
$858$		0			0
$859$	278.734	+	482.781i	0.324486	+	0.562027i	0.981408	−	0.191932i	$-0.0614753\pi$
$859$	278.734	+	482.781i	0.324486	+	0.562027i	−0.656922	+	0.753959i	$0.728142\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			99.3954i			0.115174i	0.998340	+	0.0575871i	$0.0183407\pi$
$863$			99.3954i			0.115174i	−0.998340	+	0.0575871i	$0.981659\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1490.11	−	860.314i	1.71474	−	0.990004i
$870$		0			0
$871$	19.2648	−	33.3676i	0.0221180	−	0.0383096i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	401.292	+	695.058i	0.457574	+	0.792541i	0.998832	−	0.0483154i	$-0.0153853\pi$
$877$	401.292	+	695.058i	0.457574	+	0.792541i	−0.541258	+	0.840856i	$0.682052\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			524.266i			0.595080i	0.954709	+	0.297540i	$0.0961662\pi$
$881$			524.266i			0.595080i	−0.954709	+	0.297540i	$0.903834\pi$
$882$		0			0
$883$	993.894			1.12559			0.562794	−	0.826597i	$-0.309727\pi$
$883$	993.894			1.12559			0.562794	+	0.826597i	$0.309727\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	616.643	−	356.019i	0.695200	−	0.401374i	−0.110357	−	0.993892i	$-0.535199\pi$
$887$	616.643	−	356.019i	0.695200	−	0.401374i	0.805557	+	0.592518i	$0.201866\pi$
$888$		0			0
$889$	−511.780	+	886.429i	−0.575681	+	0.997108i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−302.454	−	174.622i	−0.338694	−	0.195545i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	371.869i		−	0.413647i
$900$		0			0
$901$	−341.609			−0.379144
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−374.473	+	648.606i	−0.412870	+	0.715111i	−0.995202	−	0.0978396i	$-0.968807\pi$
$907$	−374.473	+	648.606i	−0.412870	+	0.715111i	0.582333	+	0.812951i	$0.302140\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	3.87633	+	2.23800i	0.00425503	+	0.00245664i	0.502126	−	0.864794i	$-0.332551\pi$
$911$	3.87633	+	2.23800i	0.00425503	+	0.00245664i	−0.497871	+	0.867251i	$0.665885\pi$
$912$		0			0
$913$	−389.629	−	674.857i	−0.426757	−	0.739165i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1318.66i		−	1.43802i
$918$		0			0
$919$	−1592.91			−1.73331			−0.866653	−	0.498912i	$-0.833733\pi$
$919$	−1592.91			−1.73331			−0.866653	+	0.498912i	$0.833733\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−190.337	+	109.891i	−0.206215	+	0.119059i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	770.784	+	445.012i	0.829692	+	0.479023i	0.853747	−	0.520688i	$-0.174324\pi$
$929$	770.784	+	445.012i	0.829692	+	0.479023i	−0.0240553	+	0.999711i	$0.507658\pi$
$930$		0			0
$931$	−76.9605	−	133.300i	−0.0826644	−	0.143179i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−443.554			−0.473377			−0.236688	−	0.971586i	$-0.576062\pi$
$937$	−443.554			−0.473377			−0.236688	+	0.971586i	$0.576062\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	69.7458	−	40.2678i	0.0741188	−	0.0427925i	−0.462483	−	0.886628i	$-0.653041\pi$
$941$	69.7458	−	40.2678i	0.0741188	−	0.0427925i	0.536601	+	0.843836i	$0.319708\pi$
$942$		0			0
$943$	−580.536	+	1005.52i	−0.615627	+	1.06630i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	245.861	+	141.948i	0.259621	+	0.149892i	0.624162	−	0.781295i	$-0.285441\pi$
$947$	245.861	+	141.948i	0.259621	+	0.149892i	−0.364541	+	0.931187i	$0.618774\pi$
$948$		0			0
$949$	−144.818	−	250.832i	−0.152601	−	0.264312i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1123.17i		−	1.17857i	−0.807927	−	0.589283i	$-0.799410\pi$
$953$		−	1123.17i		−	1.17857i	0.807927	−	0.589283i	$-0.200590\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−733.626	+	423.559i	−0.764991	+	0.441668i
$960$		0			0
$961$	230.727	−	399.631i	0.240090	−	0.415849i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−699.536	−	1211.63i	−0.723409	−	1.25298i	−0.959626	−	0.281281i	$-0.909241\pi$
$967$	−699.536	−	1211.63i	−0.723409	−	1.25298i	0.236217	−	0.971700i	$-0.424093\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1705.41i		−	1.75634i	−0.478345	−	0.878172i	$-0.658763\pi$
$971$		−	1705.41i		−	1.75634i	0.478345	−	0.878172i	$-0.341237\pi$
$972$		0			0
$973$	497.024			0.510816
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1005.50	−	580.524i	1.02917	−	0.594190i	0.112422	−	0.993661i	$-0.464139\pi$
$977$	1005.50	−	580.524i	1.02917	−	0.594190i	0.916746	+	0.399470i	$0.130806\pi$
$978$		0			0
$979$	−1461.81	+	2531.94i	−1.49317	+	2.58625i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1023.46	−	590.895i	−1.04116	−	0.601114i	−0.120999	−	0.992653i	$-0.538610\pi$
$983$	−1023.46	−	590.895i	−1.04116	−	0.601114i	−0.920162	+	0.391539i	$0.871943\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	771.737i		−	0.780320i
$990$		0			0
$991$	−969.527			−0.978332			−0.489166	−	0.872191i	$-0.662699\pi$
$991$	−969.527			−0.978332			−0.489166	+	0.872191i	$0.662699\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	390.124	−	675.714i	0.391298	−	0.677747i	−0.601323	−	0.799006i	$-0.705360\pi$
$997$	390.124	−	675.714i	0.391298	−	0.677747i	0.992621	+	0.121258i	$0.0386930\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.3.p.b.1601.1		4
3.2	odd	2		900.3.p.a.101.1		4
5.2	odd	4		2700.3.u.b.2249.2		8
5.3	odd	4		2700.3.u.b.2249.3		8
5.4	even	2		108.3.g.a.89.2		4
9.4	even	3		900.3.p.a.401.1		4
9.5	odd	6	inner	2700.3.p.b.2501.1		4
15.2	even	4		900.3.u.a.749.2		8
15.8	even	4		900.3.u.a.749.3		8
15.14	odd	2		36.3.g.a.29.2	yes	4
20.19	odd	2		432.3.q.b.305.2		4
40.19	odd	2		1728.3.q.h.1601.1		4
40.29	even	2		1728.3.q.g.1601.1		4
45.4	even	6		36.3.g.a.5.2	✓	4
45.13	odd	12		900.3.u.a.149.2		8
45.14	odd	6		108.3.g.a.17.2		4
45.22	odd	12		900.3.u.a.149.3		8
45.23	even	12		2700.3.u.b.449.2		8
45.29	odd	6		324.3.c.b.161.2		4
45.32	even	12		2700.3.u.b.449.3		8
45.34	even	6		324.3.c.b.161.3		4
60.59	even	2		144.3.q.b.65.1		4
120.29	odd	2		576.3.q.d.65.1		4
120.59	even	2		576.3.q.g.65.2		4
180.59	even	6		432.3.q.b.17.2		4
180.79	odd	6		1296.3.e.e.161.3		4
180.119	even	6		1296.3.e.e.161.2		4
180.139	odd	6		144.3.q.b.113.1		4
360.59	even	6		1728.3.q.h.449.1		4
360.139	odd	6		576.3.q.g.257.2		4
360.149	odd	6		1728.3.q.g.449.1		4
360.229	even	6		576.3.q.d.257.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.3.g.a.5.2	✓	4	45.4	even	6
36.3.g.a.29.2	yes	4	15.14	odd	2
108.3.g.a.17.2		4	45.14	odd	6
108.3.g.a.89.2		4	5.4	even	2
144.3.q.b.65.1		4	60.59	even	2
144.3.q.b.113.1		4	180.139	odd	6
324.3.c.b.161.2		4	45.29	odd	6
324.3.c.b.161.3		4	45.34	even	6
432.3.q.b.17.2		4	180.59	even	6
432.3.q.b.305.2		4	20.19	odd	2
576.3.q.d.65.1		4	120.29	odd	2
576.3.q.d.257.1		4	360.229	even	6
576.3.q.g.65.2		4	120.59	even	2
576.3.q.g.257.2		4	360.139	odd	6
900.3.p.a.101.1		4	3.2	odd	2
900.3.p.a.401.1		4	9.4	even	3
900.3.u.a.149.2		8	45.13	odd	12
900.3.u.a.149.3		8	45.22	odd	12
900.3.u.a.749.2		8	15.2	even	4
900.3.u.a.749.3		8	15.8	even	4
1296.3.e.e.161.2		4	180.119	even	6
1296.3.e.e.161.3		4	180.79	odd	6
1728.3.q.g.449.1		4	360.149	odd	6
1728.3.q.g.1601.1		4	40.29	even	2
1728.3.q.h.449.1		4	360.59	even	6
1728.3.q.h.1601.1		4	40.19	odd	2
2700.3.p.b.1601.1		4	1.1	even	1	trivial
2700.3.p.b.2501.1		4	9.5	odd	6	inner
2700.3.u.b.449.2		8	45.23	even	12
2700.3.u.b.449.3		8	45.32	even	12
2700.3.u.b.2249.2		8	5.2	odd	4
2700.3.u.b.2249.3		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-4.05842 + 7.02939i) q^{7} +O(q^{10})\)	\(q+(-4.05842 + 7.02939i) q^{7} +(17.6168 + 10.1711i) q^{11} +(3.05842 + 5.29734i) q^{13} +17.9653i q^{17} +9.11684 q^{19} +(29.0584 - 16.7769i) q^{23} +(-14.4090 - 8.31901i) q^{29} +(11.1753 + 19.3561i) q^{31} +50.4674 q^{37} +(-29.9674 + 17.3017i) q^{41} +(11.5000 - 19.9186i) q^{43} +(-33.1753 - 19.1537i) q^{47} +(-8.44158 - 14.6212i) q^{49} +19.0149i q^{53} +(2.96738 - 1.71322i) q^{59} +(23.1753 - 40.1407i) q^{61} +(-3.14947 - 5.45504i) q^{67} +35.9306i q^{71} -47.3505 q^{73} +(-142.993 + 82.5571i) q^{77} +(42.2921 - 73.2521i) q^{79} +(-33.1753 - 19.1537i) q^{83} +143.723i q^{89} -49.6495 q^{91} +(40.3832 - 69.9457i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{7}+O(q^{10}) \) 4 * q + q^7	\( 4 q + q^{7} + 36 q^{11} - 5 q^{13} + 2 q^{19} + 99 q^{23} + 63 q^{29} - 7 q^{31} + 64 q^{37} + 18 q^{41} + 46 q^{43} - 81 q^{47} - 51 q^{49} - 126 q^{59} + 41 q^{61} - 116 q^{67} - 86 q^{73} - 279 q^{77} + 83 q^{79} - 81 q^{83} - 302 q^{91} + 196 q^{97}+O(q^{100}) \) 4 * q + q^7 + 36 * q^11 - 5 * q^13 + 2 * q^19 + 99 * q^23 + 63 * q^29 - 7 * q^31 + 64 * q^37 + 18 * q^41 + 46 * q^43 - 81 * q^47 - 51 * q^49 - 126 * q^59 + 41 * q^61 - 116 * q^67 - 86 * q^73 - 279 * q^77 + 83 * q^79 - 81 * q^83 - 302 * q^91 + 196 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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