LMFDB - Embedded newform 2700.2.i.e.1801.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(901,2700)]

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, 0]))

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

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

chi := DirichletCharacter("2700.901");

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.i (of order $3$, 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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.142635249.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 3x^{6} + 3x^{5} - 11x^{4} + 6x^{3} + 12x^{2} - 24x + 16 $ x^8 - 3x^7 + 3x^6 + 3x^5 - 11x^4 + 6x^3 + 12x^2 - 24*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{5} $
Twist minimal:	no (minimal twist has level 900)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1801.3
Root			$1.38941 + 0.263711i$ of defining polynomial
Character	$\chi$	$=$	2700.1801
Dual form			2700.2.i.e.901.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0432397 - 0.0748933i) q^{7} +O(q^{10})$	$q+(0.0432397 - 0.0748933i) q^{7} +(-0.456760 + 0.791132i) q^{11} +(1.31249 + 2.27331i) q^{13} +2.08648 q^{17} +4.93847 q^{19} +(-4.23849 - 7.34128i) q^{23} +(1.19899 - 2.07671i) q^{29} +(-1.81249 - 3.13933i) q^{31} +5.85199 q^{37} +(3.32497 + 5.75902i) q^{41} +(-4.12499 + 7.14469i) q^{43} +(-1.34425 + 2.32831i) q^{47} +(3.49626 + 6.05570i) q^{49} -5.73642 q^{53} +(6.16922 + 10.6854i) q^{59} +(3.16823 - 5.48753i) q^{61} +(3.08274 + 5.33946i) q^{67} +12.3905 q^{71} -5.31349 q^{73} +(0.0395003 + 0.0684166i) q^{77} +(6.72394 - 11.6462i) q^{79} +(3.03950 - 5.26457i) q^{83} +8.13440 q^{89} +0.227007 q^{91} +(5.55098 - 9.61459i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{7}+O(q^{10}) $ 8 * q + q^7	$ 8 q + q^{7} - 3 q^{11} - 2 q^{13} + 18 q^{17} - 8 q^{19} + 3 q^{23} + 9 q^{29} - 2 q^{31} - 2 q^{37} - 9 q^{41} - 8 q^{43} - 12 q^{47} - 9 q^{49} + 24 q^{53} + 15 q^{59} + q^{61} - 11 q^{67} + 24 q^{71} - 20 q^{73} - 36 q^{77} + 7 q^{79} - 12 q^{83} - 6 q^{89} - 22 q^{91} - 5 q^{97}+O(q^{100}) $ 8 * q + q^7 - 3 * q^11 - 2 * q^13 + 18 * q^17 - 8 * q^19 + 3 * q^23 + 9 * q^29 - 2 * q^31 - 2 * q^37 - 9 * q^41 - 8 * q^43 - 12 * q^47 - 9 * q^49 + 24 * q^53 + 15 * q^59 + q^61 - 11 * q^67 + 24 * q^71 - 20 * q^73 - 36 * q^77 + 7 * q^79 - 12 * q^83 - 6 * q^89 - 22 * q^91 - 5 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$7$	0.0432397	−	0.0748933i	0.0163431	−	0.0283070i	−0.857738	−	0.514087i	$-0.828131\pi$
$7$	0.0432397	−	0.0748933i	0.0163431	−	0.0283070i	0.874081	+	0.485780i	$0.161464\pi$
$8$		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
$13$	1.31249	+	2.27331i	0.364020	+	0.630501i	0.988618	−	0.150445i	$-0.0480706\pi$
$13$	1.31249	+	2.27331i	0.364020	+	0.630501i	−0.624598	+	0.780946i	$0.714737\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.08648			0.506046			0.253023	−	0.967460i	$-0.418575\pi$
$17$	2.08648			0.506046			0.253023	+	0.967460i	$0.418575\pi$
$18$		0			0
$19$	4.93847			1.13296			0.566482	−	0.824074i	$-0.308304\pi$
$19$	4.93847			1.13296			0.566482	+	0.824074i	$0.308304\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.23849	−	7.34128i	−0.883786	−	1.53076i	−0.847098	−	0.531436i	$-0.821653\pi$
$23$	−4.23849	−	7.34128i	−0.883786	−	1.53076i	−0.0366878	−	0.999327i	$-0.511681\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.19899	−	2.07671i	0.222647	−	0.385636i	−0.732964	−	0.680267i	$-0.761864\pi$
$29$	1.19899	−	2.07671i	0.222647	−	0.385636i	0.955611	+	0.294632i	$0.0951970\pi$
$30$		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
$37$	5.85199			0.962062			0.481031	−	0.876704i	$-0.340262\pi$
$37$	5.85199			0.962062			0.481031	+	0.876704i	$0.340262\pi$
$38$		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
$43$	−4.12499	+	7.14469i	−0.629055	+	1.08955i	0.358687	+	0.933458i	$0.383224\pi$
$43$	−4.12499	+	7.14469i	−0.629055	+	1.08955i	−0.987742	+	0.156097i	$0.950109\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.34425	+	2.32831i	−0.196079	+	0.339619i	−0.947254	−	0.320485i	$-0.896154\pi$
$47$	−1.34425	+	2.32831i	−0.196079	+	0.339619i	0.751175	+	0.660103i	$0.229488\pi$
$48$		0			0
$49$	3.49626	+	6.05570i	0.499466	+	0.865100i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.73642			−0.787958			−0.393979	−	0.919120i	$-0.628902\pi$
$53$	−5.73642			−0.787958			−0.393979	+	0.919120i	$0.628902\pi$
$54$		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	$0.130185\pi$
$59$	6.16922	+	10.6854i	0.803164	+	1.39112i	−0.114360	+	0.993439i	$0.536482\pi$
$60$		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
$67$	3.08274	+	5.33946i	0.376617	+	0.652319i	0.990568	−	0.137025i	$-0.0437541\pi$
$67$	3.08274	+	5.33946i	0.376617	+	0.652319i	−0.613951	+	0.789344i	$0.710421\pi$
$68$		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
$73$	−5.31349			−0.621897			−0.310948	−	0.950427i	$-0.600647\pi$
$73$	−5.31349			−0.621897			−0.310948	+	0.950427i	$0.600647\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.0395003	+	0.0684166i	0.00450148	+	0.00779679i
$78$		0			0
$79$	6.72394	−	11.6462i	0.756503	−	1.31030i	−0.188121	−	0.982146i	$-0.560240\pi$
$79$	6.72394	−	11.6462i	0.756503	−	1.31030i	0.944624	−	0.328155i	$-0.106427\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.03950	−	5.26457i	0.333629	−	0.577862i	−0.649592	−	0.760283i	$-0.725060\pi$
$83$	3.03950	−	5.26457i	0.333629	−	0.577862i	0.983220	+	0.182422i	$0.0583936\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.13440			0.862244			0.431122	−	0.902294i	$-0.358118\pi$
$89$	8.13440			0.862244			0.431122	+	0.902294i	$0.358118\pi$
$90$		0			0
$91$	0.227007			0.0237968
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.55098	−	9.61459i	0.563617	−	0.976213i	−0.433560	−	0.901125i	$-0.642743\pi$
$97$	5.55098	−	9.61459i	0.563617	−	0.976213i	0.997177	−	0.0750885i	$-0.0239239\pi$
$98$		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
$103$	−2.18377	−	3.78240i	−0.215173	−	0.372691i	0.738153	−	0.674633i	$-0.235698\pi$
$103$	−2.18377	−	3.78240i	−0.215173	−	0.372691i	−0.953326	+	0.301943i	$0.902365\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.9614			1.83307			0.916536	−	0.399953i	$-0.130973\pi$
$107$	18.9614			1.83307			0.916536	+	0.399953i	$0.130973\pi$
$108$		0			0
$109$	15.1904			1.45498			0.727490	−	0.686119i	$-0.240687\pi$
$109$	15.1904			1.45498			0.727490	+	0.686119i	$0.240687\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.30101	−	5.71752i	−0.310533	−	0.537859i	0.667945	−	0.744211i	$-0.267174\pi$
$113$	−3.30101	−	5.71752i	−0.310533	−	0.537859i	−0.978478	+	0.206352i	$0.933841\pi$
$114$		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
$127$	7.42492			0.658855			0.329427	−	0.944181i	$-0.393144\pi$
$127$	7.42492			0.658855			0.329427	+	0.944181i	$0.393144\pi$
$128$		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
$133$	0.213538	−	0.369858i	0.0185161	−	0.0320708i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.19525	+	15.9266i	−0.785603	+	1.36070i	0.143035	+	0.989718i	$0.454314\pi$
$137$	−9.19525	+	15.9266i	−0.785603	+	1.36070i	−0.928638	+	0.370987i	$0.879019\pi$
$138$		0			0
$139$	−2.78074	−	4.81638i	−0.235859	−	0.408520i	0.723663	−	0.690154i	$-0.242457\pi$
$139$	−2.78074	−	4.81638i	−0.235859	−	0.408520i	−0.959522	+	0.281634i	$0.909124\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.39798			−0.200529
$144$		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	$0.248736\pi$
$149$	11.7780	+	20.4001i	0.964891	+	1.67124i	0.254982	+	0.966946i	$0.417931\pi$
$150$		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
$157$	−6.07967	−	10.5303i	−0.485211	−	0.840410i	0.514645	−	0.857404i	$-0.327924\pi$
$157$	−6.07967	−	10.5303i	−0.485211	−	0.840410i	−0.999856	+	0.0169936i	$0.994590\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−0.733083			−0.0577751
$162$		0			0
$163$	−19.0654			−1.49332			−0.746658	−	0.665208i	$-0.768343\pi$
$163$	−19.0654			−1.49332			−0.746658	+	0.665208i	$0.768343\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.911449	+	1.57868i	0.0705300	+	0.122162i	0.899134	−	0.437674i	$-0.144198\pi$
$167$	0.911449	+	1.57868i	0.0705300	+	0.122162i	−0.828604	+	0.559836i	$0.810864\pi$
$168$		0			0
$169$	3.05472	−	5.29093i	0.234979	−	0.406995i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.3942	+	18.0034i	−0.790259	+	1.36877i	0.135547	+	0.990771i	$0.456721\pi$
$173$	−10.3942	+	18.0034i	−0.790259	+	1.36877i	−0.925806	+	0.377999i	$0.876612\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.12692			0.233717			0.116858	−	0.993149i	$-0.462718\pi$
$179$	3.12692			0.233717			0.116858	+	0.993149i	$0.462718\pi$
$180$		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
$187$	−0.953021	+	1.65068i	−0.0696918	+	0.120710i
$188$		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
$193$	−2.59422	−	4.49333i	−0.186736	−	0.323437i	0.757424	−	0.652923i	$-0.226458\pi$
$193$	−2.59422	−	4.49333i	−0.186736	−	0.323437i	−0.944160	+	0.329487i	$0.893124\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	21.3384			1.52030			0.760150	−	0.649747i	$-0.225125\pi$
$197$	21.3384			1.52030			0.760150	+	0.649747i	$0.225125\pi$
$198$		0			0
$199$	9.50608			0.673868			0.336934	−	0.941528i	$-0.390610\pi$
$199$	9.50608			0.673868			0.336934	+	0.941528i	$0.390610\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.103688	−	0.179593i	−0.00727746	−	0.0126049i
$204$		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
$217$	−0.313486			−0.0212808
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.73849	+	4.74320i	0.184211	+	0.319062i
$222$		0			0
$223$	2.88276	−	4.99308i	0.193044	−	0.334361i	−0.753214	−	0.657776i	$-0.771497\pi$
$223$	2.88276	−	4.99308i	0.193044	−	0.334361i	0.946257	+	0.323414i	$0.104831\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.62805	+	11.4801i	−0.439919	+	0.761962i	−0.997683	−	0.0680374i	$-0.978326\pi$
$227$	−6.62805	+	11.4801i	−0.439919	+	0.761962i	0.557764	+	0.830000i	$0.311660\pi$
$228$		0			0
$229$	−3.92500	−	6.79831i	−0.259372	−	0.449245i	0.706702	−	0.707511i	$-0.250182\pi$
$229$	−3.92500	−	6.79831i	−0.259372	−	0.449245i	−0.966074	+	0.258266i	$0.916849\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.9094			1.04226			0.521129	−	0.853478i	$-0.325511\pi$
$233$	15.9094			1.04226			0.521129	+	0.853478i	$0.325511\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.23849	+	12.5374i	0.468219	+	0.810979i	0.999340	−	0.0363167i	$-0.0115625\pi$
$239$	7.23849	+	12.5374i	0.468219	+	0.810979i	−0.531121	+	0.847296i	$0.678229\pi$
$240$		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
$247$	6.48171	+	11.2267i	0.412421	+	0.714335i
$248$		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
$253$	7.74390			0.486855
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.54324	−	14.7973i	−0.532913	−	0.923032i	−0.999261	−	0.0384308i	$-0.987764\pi$
$257$	−8.54324	−	14.7973i	−0.532913	−	0.923032i	0.466349	−	0.884601i	$-0.345569\pi$
$258$		0			0
$259$	0.253038	−	0.438275i	0.0157230	−	0.0272331i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.67129	−	2.89476i	0.103056	−	0.178499i	−0.809886	−	0.586587i	$-0.800471\pi$
$263$	1.67129	−	2.89476i	0.103056	−	0.178499i	0.912942	+	0.408088i	$0.133805\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.82704			−0.111397			−0.0556983	−	0.998448i	$-0.517739\pi$
$269$	−1.82704			−0.111397			−0.0556983	+	0.998448i	$0.517739\pi$
$270$		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
$277$	6.96550	−	12.0646i	0.418516	−	0.724891i	−0.577274	−	0.816550i	$-0.695884\pi$
$277$	6.96550	−	12.0646i	0.418516	−	0.724891i	0.995790	+	0.0916590i	$0.0292170\pi$
$278$		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
$283$	−11.1375	−	19.2907i	−0.662053	−	1.14671i	−0.980075	−	0.198627i	$-0.936352\pi$
$283$	−11.1375	−	19.2907i	−0.662053	−	1.14671i	0.318022	−	0.948083i	$-0.396981\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	0.575082			0.0339460
$288$		0			0
$289$	−12.6466			−0.743918
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.04531	−	1.81053i	−0.0610678	−	0.105772i	0.833875	−	0.551953i	$-0.186117\pi$
$293$	−1.04531	−	1.81053i	−0.0610678	−	0.105772i	−0.894943	+	0.446181i	$0.852784\pi$
$294$		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
$307$	1.74056			0.0993391			0.0496696	−	0.998766i	$-0.484183\pi$
$307$	1.74056			0.0993391			0.0496696	+	0.998766i	$0.484183\pi$
$308$		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
$313$	−5.44054	+	9.42330i	−0.307518	+	0.532636i	−0.977819	−	0.209453i	$-0.932832\pi$
$313$	−5.44054	+	9.42330i	−0.307518	+	0.532636i	0.670301	+	0.742089i	$0.266165\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.00374	+	6.93468i	−0.224872	+	0.389490i	−0.956281	−	0.292449i	$-0.905530\pi$
$317$	−4.00374	+	6.93468i	−0.224872	+	0.389490i	0.731409	+	0.681939i	$0.238863\pi$
$318$		0			0
$319$	1.09530	+	1.89712i	0.0613251	+	0.106218i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	10.3040			0.573331
$324$		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
$337$	−3.19832	−	5.53964i	−0.174223	−	0.301764i	0.765669	−	0.643235i	$-0.222408\pi$
$337$	−3.19832	−	5.53964i	−0.174223	−	0.301764i	−0.939892	+	0.341471i	$0.889075\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.31150			0.179328
$342$		0			0
$343$	1.21006			0.0653373
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.75777	−	11.7048i	−0.362776	−	0.628347i	0.625641	−	0.780112i	$-0.284838\pi$
$347$	−6.75777	−	11.7048i	−0.362776	−	0.628347i	−0.988417	+	0.151765i	$0.951504\pi$
$348$		0			0
$349$	−0.408788	+	0.708042i	−0.0218819	+	0.0379006i	−0.876759	−	0.480930i	$-0.840299\pi$
$349$	−0.408788	+	0.708042i	−0.0218819	+	0.0379006i	0.854877	+	0.518831i	$0.173632\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.47023	+	16.4029i	−0.504049	+	0.873039i	0.495940	+	0.868357i	$0.334824\pi$
$353$	−9.47023	+	16.4029i	−0.504049	+	0.873039i	−0.999989	+	0.00468224i	$0.998510\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−36.4863			−1.92568			−0.962838	−	0.270081i	$-0.912950\pi$
$359$	−36.4863			−1.92568			−0.962838	+	0.270081i	$0.912950\pi$
$360$		0			0
$361$	5.38851			0.283606
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−9.71246	+	16.8225i	−0.506986	+	0.878126i	0.492981	+	0.870040i	$0.335907\pi$
$367$	−9.71246	+	16.8225i	−0.506986	+	0.878126i	−0.999967	+	0.00808588i	$0.997426\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.248041	+	0.429619i	−0.0128776	+	0.0223047i
$372$		0			0
$373$	14.5884	+	25.2679i	0.755359	+	1.30832i	0.945195	+	0.326505i	$0.105871\pi$
$373$	14.5884	+	25.2679i	0.755359	+	1.30832i	−0.189836	+	0.981816i	$0.560796\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.29466			0.324192
$378$		0			0
$379$	−21.8060			−1.12010			−0.560048	−	0.828460i	$-0.689217\pi$
$379$	−21.8060			−1.12010			−0.560048	+	0.828460i	$0.689217\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.83546	−	8.37526i	−0.247080	−	0.427956i	0.715634	−	0.698475i	$-0.246138\pi$
$383$	−4.83546	−	8.37526i	−0.247080	−	0.427956i	−0.962714	+	0.270520i	$0.912805\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.86447	−	13.6217i	0.398744	−	0.690646i	−0.594827	−	0.803854i	$-0.702779\pi$
$389$	7.86447	−	13.6217i	0.398744	−	0.690646i	0.993571	+	0.113208i	$0.0361127\pi$
$390$		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
$397$	−22.8999			−1.14931			−0.574657	−	0.818394i	$-0.694864\pi$
$397$	−22.8999			−1.14931			−0.574657	+	0.818394i	$0.694864\pi$
$398$		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
$403$	4.75777	−	8.24070i	0.237001	−	0.410499i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.67296	+	4.62970i	−0.132494	+	0.229486i
$408$		0			0
$409$	14.4344	+	25.0011i	0.713736	+	1.23623i	0.963445	+	0.267906i	$0.0863316\pi$
$409$	14.4344	+	25.0011i	0.713736	+	1.23623i	−0.249709	+	0.968321i	$0.580335\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.06702			0.0525046
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.97117	−	15.5385i	−0.438270	−	0.759106i	0.559286	−	0.828975i	$-0.311075\pi$
$419$	−8.97117	−	15.5385i	−0.438270	−	0.759106i	−0.997556	+	0.0698684i	$0.977742\pi$
$420$		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
$427$	−0.273986	−	0.474558i	−0.0132591	−	0.0229655i
$428$		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
$433$	35.7464			1.71786			0.858931	−	0.512091i	$-0.171129\pi$
$433$	35.7464			1.71786			0.858931	+	0.512091i	$0.171129\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−20.9317	−	36.2547i	−1.00130	−	1.73430i
$438$		0			0
$439$	−9.13440	+	15.8212i	−0.435961	+	0.755107i	−0.997374	−	0.0724295i	$-0.976925\pi$
$439$	−9.13440	+	15.8212i	−0.435961	+	0.755107i	0.561413	+	0.827536i	$0.310258\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	14.9161	−	25.8355i	0.708687	−	1.22748i	−0.256658	−	0.966502i	$-0.582621\pi$
$443$	14.9161	−	25.8355i	0.708687	−	1.22748i	0.965345	−	0.260979i	$-0.0840453\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−38.9423			−1.83780			−0.918901	−	0.394488i	$-0.870922\pi$
$449$	−38.9423			−1.83780			−0.918901	+	0.394488i	$0.870922\pi$
$450$		0			0
$451$	−6.07486			−0.286054
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−10.2452	+	17.7453i	−0.479252	+	0.830089i	−0.999717	−	0.0237941i	$-0.992425\pi$
$457$	−10.2452	+	17.7453i	−0.479252	+	0.830089i	0.520465	+	0.853883i	$0.325759\pi$
$458$		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
$463$	17.1598	+	29.7216i	0.797481	+	1.38128i	0.921252	+	0.388967i	$0.127168\pi$
$463$	17.1598	+	29.7216i	0.797481	+	1.38128i	−0.123770	+	0.992311i	$0.539499\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.33096			−0.200413			−0.100206	−	0.994967i	$-0.531950\pi$
$467$	−4.33096			−0.200413			−0.100206	+	0.994967i	$0.531950\pi$
$468$		0			0
$469$	0.533186			0.0246203
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.76826	−	6.52682i	−0.173265	−	0.300103i
$474$		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	$-0.602965\pi$
$479$	14.4925	−	25.1018i	0.662180	−	1.14693i	0.980042	−	0.198793i	$-0.0637020\pi$
$480$		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
$487$	11.2189			0.508376			0.254188	−	0.967155i	$-0.418192\pi$
$487$	11.2189			0.508376			0.254188	+	0.967155i	$0.418192\pi$
$488$		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
$493$	2.50167	−	4.33301i	0.112669	−	0.195149i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.535761	−	0.927965i	0.0240322	−	0.0416249i
$498$		0			0
$499$	−9.46856	−	16.4000i	−0.423871	−	0.734166i	0.572443	−	0.819944i	$-0.305996\pi$
$499$	−9.46856	−	16.4000i	−0.423871	−	0.734166i	−0.996314	+	0.0857782i	$0.972662\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−21.1790			−0.944324			−0.472162	−	0.881512i	$-0.656526\pi$
$503$	−21.1790			−0.944324			−0.472162	+	0.881512i	$0.656526\pi$
$504$		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	$-0.909095\pi$
$509$	−16.3280	−	28.2809i	−0.723725	−	1.25353i	0.235772	−	0.971808i	$-0.424238\pi$
$510$		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
$517$	−1.22800	−	2.12696i	−0.0540074	−	0.0935435i
$518$		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
$523$	8.34790			0.365028			0.182514	−	0.983203i	$-0.441576\pi$
$523$	8.34790			0.365028			0.182514	+	0.983203i	$0.441576\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.78173	−	6.55015i	−0.164735	−	0.285329i
$528$		0			0
$529$	−24.4296	+	42.3133i	−1.06216	+	1.83971i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−8.72800	+	15.1173i	−0.378052	+	0.654805i
$534$		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
$547$	−4.75097	+	8.22891i	−0.203137	+	0.351843i	−0.949537	−	0.313654i	$-0.898447\pi$
$547$	−4.75097	+	8.22891i	−0.203137	+	0.351843i	0.746401	+	0.665497i	$0.231780\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.92118	−	10.2558i	0.252251	−	0.436911i
$552$		0			0
$553$	−0.581482	−	1.00716i	−0.0247271	−	0.0428286i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	19.1999			0.813526			0.406763	−	0.913534i	$-0.366658\pi$
$557$	19.1999			0.813526			0.406763	+	0.913534i	$0.366658\pi$
$558$		0			0
$559$	−21.6561			−0.915954
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.11044	+	14.0477i	0.341814	+	0.592040i	0.984770	−	0.173864i	$-0.0556253\pi$
$563$	8.11044	+	14.0477i	0.341814	+	0.592040i	−0.642955	+	0.765904i	$0.722292\pi$
$564$		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	$0.366795\pi$
$569$	−14.0286	+	24.2983i	−0.588111	+	1.01864i	−0.994480	+	0.104929i	$0.966538\pi$
$570$		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
$577$	−18.4113			−0.766473			−0.383236	−	0.923650i	$-0.625191\pi$
$577$	−18.4113			−0.766473			−0.383236	+	0.923650i	$0.625191\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.262854	−	0.455276i	−0.0109050	−	0.0188880i
$582$		0			0
$583$	2.62017	−	4.53826i	0.108516	−	0.187956i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−17.9422	+	31.0767i	−0.740552	+	1.28267i	0.211692	+	0.977336i	$0.432103\pi$
$587$	−17.9422	+	31.0767i	−0.740552	+	1.28267i	−0.952244	+	0.305338i	$0.901231\pi$
$588$		0			0
$589$	−8.95095	−	15.5035i	−0.368817	−	0.638811i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	47.6039			1.95486			0.977429	−	0.211265i	$-0.0677582\pi$
$593$	47.6039			1.95486			0.977429	+	0.211265i	$0.0677582\pi$
$594$		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	$-0.997129\pi$
$599$	−12.4279	−	21.5258i	−0.507791	−	0.879521i	0.492168	−	0.870500i	$-0.336205\pi$
$600$		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
$607$	−2.31349	−	4.00708i	−0.0939015	−	0.162642i	0.815248	−	0.579112i	$-0.196601\pi$
$607$	−2.31349	−	4.00708i	−0.0939015	−	0.162642i	−0.909150	+	0.416470i	$0.863267\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.05728			−0.285507
$612$		0			0
$613$	−12.1791			−0.491909			−0.245954	−	0.969281i	$-0.579101\pi$
$613$	−12.1791			−0.491909			−0.245954	+	0.969281i	$0.579101\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−23.2684	−	40.3021i	−0.936752	−	1.62250i	−0.771479	−	0.636254i	$-0.780483\pi$
$617$	−23.2684	−	40.3021i	−0.936752	−	1.62250i	−0.165273	−	0.986248i	$-0.552850\pi$
$618$		0			0
$619$	−7.90198	+	13.6866i	−0.317608	+	0.550112i	−0.979988	−	0.199055i	$-0.936213\pi$
$619$	−7.90198	+	13.6866i	−0.317608	+	0.550112i	0.662381	+	0.749167i	$0.269546\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.351729	−	0.609212i	0.0140917	−	0.0244076i
$624$		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
$637$	−9.17764	+	15.8961i	−0.363631	+	0.629828i
$638$		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
$643$	−12.1999	−	21.1309i	−0.481118	−	0.833321i	0.518647	−	0.854988i	$-0.326436\pi$
$643$	−12.1999	−	21.1309i	−0.481118	−	0.833321i	−0.999765	+	0.0216672i	$0.993103\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−26.9195			−1.05832			−0.529158	−	0.848523i	$-0.677492\pi$
$647$	−26.9195			−1.05832			−0.529158	+	0.848523i	$0.677492\pi$
$648$		0			0
$649$	−11.2714			−0.442442
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.09116	+	7.08609i	0.160099	+	0.277300i	0.934904	−	0.354900i	$-0.115485\pi$
$653$	4.09116	+	7.08609i	0.160099	+	0.277300i	−0.774805	+	0.632201i	$0.782152\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.55504	−	13.0857i	0.294303	−	0.509747i	−0.680520	−	0.732730i	$-0.738246\pi$
$659$	7.55504	−	13.0857i	0.294303	−	0.509747i	0.974822	+	0.222983i	$0.0715793\pi$
$660$		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
$667$	−20.3276			−0.787089
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.89424	+	5.01297i	0.111731	+	0.193524i
$672$		0			0
$673$	14.7067	−	25.4728i	0.566903	−	0.981905i	−0.429967	−	0.902845i	$-0.641475\pi$
$673$	14.7067	−	25.4728i	0.566903	−	0.981905i	0.996870	−	0.0790600i	$-0.0251919\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16.9519	−	29.3615i	0.651514	−	1.12846i	−0.331242	−	0.943546i	$-0.607468\pi$
$677$	16.9519	−	29.3615i	0.651514	−	1.12846i	0.982756	−	0.184909i	$-0.0591992\pi$
$678$		0			0
$679$	−0.480045	−	0.831463i	−0.0184224	−	0.0319086i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	29.1078			1.11378			0.556890	−	0.830586i	$-0.311995\pi$
$683$	29.1078			1.11378			0.556890	+	0.830586i	$0.311995\pi$
$684$		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
$697$	6.93748	+	12.0161i	0.262776	+	0.455141i
$698$		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
$703$	28.8999			1.08998
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.522292	+	0.904636i	0.0196428	+	0.0340223i
$708$		0			0
$709$	16.1539	−	27.9795i	0.606674	−	1.05079i	−0.385110	−	0.922870i	$-0.625837\pi$
$709$	16.1539	−	27.9795i	0.606674	−	1.05079i	0.991785	−	0.127920i	$-0.0408300\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−15.3645	+	26.6120i	−0.575404	+	0.996629i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−18.4905			−0.689579			−0.344789	−	0.938680i	$-0.612050\pi$
$719$	−18.4905			−0.689579			−0.344789	+	0.938680i	$0.612050\pi$
$720$		0			0
$721$	−0.377701			−0.0140663
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	25.7357	−	44.5755i	0.954484	−	1.65321i	0.218939	−	0.975739i	$-0.429740\pi$
$727$	25.7357	−	44.5755i	0.954484	−	1.65321i	0.735545	−	0.677476i	$-0.236926\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.60670	+	14.9072i	−0.318330	+	0.551364i
$732$		0			0
$733$	3.32029	+	5.75091i	0.122638	+	0.212415i	0.920807	−	0.390018i	$-0.127531\pi$
$733$	3.32029	+	5.75091i	0.122638	+	0.212415i	−0.798169	+	0.602433i	$0.794198\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.63229			−0.207468
$738$		0			0
$739$	−38.6058			−1.42014			−0.710068	−	0.704133i	$-0.751336\pi$
$739$	−38.6058			−1.42014			−0.710068	+	0.704133i	$0.751336\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.7952	+	22.1619i	0.469410	+	0.813043i	0.999388	−	0.0349688i	$-0.0111332\pi$
$743$	12.7952	+	22.1619i	0.469410	+	0.813043i	−0.529978	+	0.848011i	$0.677800\pi$
$744$		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
$757$	−13.2710			−0.482341			−0.241170	−	0.970483i	$-0.577531\pi$
$757$	−13.2710			−0.482341			−0.241170	+	0.970483i	$0.577531\pi$
$758$		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
$763$	0.656829	−	1.13766i	0.0237788	−	0.0411861i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−16.1941	+	28.0490i	−0.584736	+	1.01279i
$768$		0			0
$769$	1.00941	+	1.74835i	0.0364003	+	0.0630472i	0.883652	−	0.468145i	$-0.155078\pi$
$769$	1.00941	+	1.74835i	0.0364003	+	0.0630472i	−0.847251	+	0.531192i	$0.821744\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.76154			0.207228			0.103614	−	0.994618i	$-0.466959\pi$
$773$	5.76154			0.207228			0.103614	+	0.994618i	$0.466959\pi$
$774$		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
$787$	3.66521	+	6.34834i	0.130651	+	0.226294i	0.923928	−	0.382567i	$-0.124960\pi$
$787$	3.66521	+	6.34834i	0.130651	+	0.226294i	−0.793277	+	0.608861i	$0.791627\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−0.570938			−0.0203002
$792$		0			0
$793$	16.6331			0.590659
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.56179	+	6.16920i	0.126165	+	0.218524i	0.922188	−	0.386742i	$-0.126400\pi$
$797$	3.56179	+	6.16920i	0.126165	+	0.218524i	−0.796023	+	0.605267i	$0.793066\pi$
$798$		0			0
$799$	−2.80475	+	4.85797i	−0.0992249	+	0.171863i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.42699	−	4.20367i	0.0856466	−	0.148344i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−29.0809			−1.02243			−0.511215	−	0.859453i	$-0.670804\pi$
$809$	−29.0809			−1.02243			−0.511215	+	0.859453i	$0.670804\pi$
$810$		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
$817$	−20.3711	+	35.2838i	−0.712696	+	1.23443i
$818$		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
$823$	−23.1154	−	40.0371i	−0.805753	−	1.39561i	−0.915781	−	0.401677i	$-0.868427\pi$
$823$	−23.1154	−	40.0371i	−0.805753	−	1.39561i	0.110028	−	0.993929i	$-0.464906\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	23.6350			0.821869			0.410934	−	0.911665i	$-0.365203\pi$
$827$	23.6350			0.821869			0.410934	+	0.911665i	$0.365203\pi$
$828$		0			0
$829$	−18.8979			−0.656352			−0.328176	−	0.944617i	$-0.606434\pi$
$829$	−18.8979			−0.656352			−0.328176	+	0.944617i	$0.606434\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7.29488	+	12.6351i	0.252752	+	0.437780i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5.21547	−	9.03346i	0.180058	−	0.311870i	−0.761842	−	0.647763i	$-0.775705\pi$
$839$	5.21547	−	9.03346i	0.180058	−	0.311870i	0.941900	+	0.335893i	$0.109038\pi$
$840$		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
$847$	0.879104			0.0302064
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−24.8036	−	42.9611i	−0.850257	−	1.47269i
$852$		0			0
$853$	−19.9405	+	34.5379i	−0.682748	+	1.18255i	0.291390	+	0.956604i	$0.405882\pi$
$853$	−19.9405	+	34.5379i	−0.682748	+	1.18255i	−0.974139	+	0.225951i	$0.927451\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.06346	−	15.6984i	0.309602	−	0.536246i	−0.668673	−	0.743556i	$-0.733138\pi$
$857$	9.06346	−	15.6984i	0.309602	−	0.536246i	0.978275	+	0.207310i	$0.0664709\pi$
$858$		0			0
$859$	−2.01040	−	3.48212i	−0.0685941	−	0.118808i	0.829689	−	0.558227i	$-0.188518\pi$
$859$	−2.01040	−	3.48212i	−0.0685941	−	0.118808i	−0.898283	+	0.439418i	$0.855185\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−43.9540			−1.49621			−0.748105	−	0.663580i	$-0.769036\pi$
$863$	−43.9540			−1.49621			−0.748105	+	0.663580i	$0.769036\pi$
$864$		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
$877$	−24.7864	−	42.9313i	−0.836978	−	1.44969i	−0.892410	−	0.451225i	$-0.850987\pi$
$877$	−24.7864	−	42.9313i	−0.836978	−	1.44969i	0.0554326	−	0.998462i	$-0.482346\pi$
$878$		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
$883$	−51.1113			−1.72003			−0.860017	−	0.510266i	$-0.829547\pi$
$883$	−51.1113			−1.72003			−0.860017	+	0.510266i	$0.829547\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.9786	+	29.4079i	0.570087	+	0.987420i	0.996556	+	0.0829179i	$0.0264239\pi$
$887$	16.9786	+	29.4079i	0.570087	+	0.987420i	−0.426469	+	0.904502i	$0.640243\pi$
$888$		0			0
$889$	0.321051	−	0.556076i	0.0107677	−	0.0186502i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.63854	+	11.4983i	−0.222150	+	0.384776i
$894$		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
$907$	2.90198	−	5.02638i	0.0963588	−	0.166898i	−0.813816	−	0.581122i	$-0.802614\pi$
$907$	2.90198	−	5.02638i	0.0963588	−	0.166898i	0.910175	+	0.414224i	$0.135947\pi$
$908$		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
$913$	2.77665	+	4.80929i	0.0918936	+	0.159164i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.38033			−0.0455827
$918$		0			0
$919$	25.2655			0.833431			0.416715	−	0.909037i	$-0.363181\pi$
$919$	25.2655			0.833431			0.416715	+	0.909037i	$0.363181\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	16.2624	+	28.1674i	0.535285	+	0.927141i
$924$		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	$0.370267\pi$
$929$	−18.1932	+	31.5115i	−0.596899	+	1.03386i	−0.993276	+	0.115771i	$0.963066\pi$
$930$		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
$937$	−17.5425			−0.573088			−0.286544	−	0.958067i	$-0.592506\pi$
$937$	−17.5425			−0.573088			−0.286544	+	0.958067i	$0.592506\pi$
$938$		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
$943$	28.1857	−	48.8191i	0.917853	−	1.58977i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.52603	−	2.64316i	0.0495893	−	0.0858913i	−0.840165	−	0.542330i	$-0.817542\pi$
$947$	1.52603	−	2.64316i	0.0495893	−	0.0858913i	0.889755	+	0.456439i	$0.150875\pi$
$948$		0			0
$949$	−6.97392	−	12.0792i	−0.226383	−	0.392107i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	11.6054			0.375934			0.187967	−	0.982175i	$-0.439810\pi$
$953$	11.6054			0.375934			0.187967	+	0.982175i	$0.439810\pi$
$954$		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
$967$	8.41919	+	14.5825i	0.270743	+	0.468941i	0.969052	−	0.246856i	$-0.0793973\pi$
$967$	8.41919	+	14.5825i	0.270743	+	0.468941i	−0.698309	+	0.715796i	$0.746064\pi$
$968$		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
$973$	−0.480952			−0.0154186
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.7343	+	25.5206i	0.471393	+	0.816477i	0.999464	−	0.0327228i	$-0.0104178\pi$
$977$	14.7343	+	25.5206i	0.471393	+	0.816477i	−0.528071	+	0.849200i	$0.677085\pi$
$978$		0			0
$979$	−3.71547	+	6.43538i	−0.118747	+	0.205676i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	10.7771	−	18.6664i	0.343735	−	0.595366i	−0.641388	−	0.767216i	$-0.721641\pi$
$983$	10.7771	−	18.6664i	0.343735	−	0.595366i	0.985123	+	0.171851i	$0.0549746\pi$
$984$		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
$997$	15.2499	−	26.4136i	0.482969	−	0.836526i	−0.516840	−	0.856082i	$-0.672892\pi$
$997$	15.2499	−	26.4136i	0.482969	−	0.836526i	0.999809	+	0.0195556i	$0.00622512\pi$
$998$		0			0
$999$		0			0

Twists

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

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

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.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.0432397 - 0.0748933i) q^{7} +O(q^{10})\)	\(q+(0.0432397 - 0.0748933i) q^{7} +(-0.456760 + 0.791132i) q^{11} +(1.31249 + 2.27331i) q^{13} +2.08648 q^{17} +4.93847 q^{19} +(-4.23849 - 7.34128i) q^{23} +(1.19899 - 2.07671i) q^{29} +(-1.81249 - 3.13933i) q^{31} +5.85199 q^{37} +(3.32497 + 5.75902i) q^{41} +(-4.12499 + 7.14469i) q^{43} +(-1.34425 + 2.32831i) q^{47} +(3.49626 + 6.05570i) q^{49} -5.73642 q^{53} +(6.16922 + 10.6854i) q^{59} +(3.16823 - 5.48753i) q^{61} +(3.08274 + 5.33946i) q^{67} +12.3905 q^{71} -5.31349 q^{73} +(0.0395003 + 0.0684166i) q^{77} +(6.72394 - 11.6462i) q^{79} +(3.03950 - 5.26457i) q^{83} +8.13440 q^{89} +0.227007 q^{91} +(5.55098 - 9.61459i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{7}+O(q^{10}) \) 8 * q + q^7	\( 8 q + q^{7} - 3 q^{11} - 2 q^{13} + 18 q^{17} - 8 q^{19} + 3 q^{23} + 9 q^{29} - 2 q^{31} - 2 q^{37} - 9 q^{41} - 8 q^{43} - 12 q^{47} - 9 q^{49} + 24 q^{53} + 15 q^{59} + q^{61} - 11 q^{67} + 24 q^{71} - 20 q^{73} - 36 q^{77} + 7 q^{79} - 12 q^{83} - 6 q^{89} - 22 q^{91} - 5 q^{97}+O(q^{100}) \) 8 * q + q^7 - 3 * q^11 - 2 * q^13 + 18 * q^17 - 8 * q^19 + 3 * q^23 + 9 * q^29 - 2 * q^31 - 2 * q^37 - 9 * q^41 - 8 * q^43 - 12 * q^47 - 9 * q^49 + 24 * q^53 + 15 * q^59 + q^61 - 11 * q^67 + 24 * q^71 - 20 * q^73 - 36 * q^77 + 7 * q^79 - 12 * q^83 - 6 * q^89 - 22 * q^91 - 5 * q^97

Introduction

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