LMFDB - Embedded newform 1100.3.e.b.549.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,3,Mod(549,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1100.549");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1100.e (of order $2$, degree $1$, 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:	$29.9728290796$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 85x^{14} + 2456x^{12} + 32605x^{10} + 215801x^{8} + 712960x^{6} + 1098976x^{4} + 633600x^{2} + 92416 $ x^16 + 85x^14 + 2456x^12 + 32605x^10 + 215801x^8 + 712960x^6 + 1098976x^4 + 633600*x^2 + 92416
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{25}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			549.15
Root			$-6.55851i$ of defining polynomial
Character	$\chi$	$=$	1100.549
Dual form			1100.3.e.b.549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.94892i q^{3} -13.1585 q^{7} -15.4918 q^{9} +(-3.18499 - 10.5288i) q^{11} -14.6396 q^{13} +9.90801 q^{17} +11.5333i q^{19} -65.1206i q^{21} +14.6044i q^{23} -32.1276i q^{27} +8.28275i q^{29} +53.2285 q^{31} +(52.1063 - 15.7623i) q^{33} +41.1146i q^{37} -72.4502i q^{39} -74.6449i q^{41} +30.4377 q^{43} +0.697244i q^{47} +124.147 q^{49} +49.0340i q^{51} -17.0571i q^{53} -57.0773 q^{57} +6.29242 q^{59} -50.6864i q^{61} +203.850 q^{63} -81.8378i q^{67} -72.2760 q^{69} -62.4365 q^{71} -11.5551 q^{73} +(41.9098 + 138.544i) q^{77} +79.2397i q^{79} +19.5705 q^{81} -81.8105 q^{83} -40.9907 q^{87} -136.935 q^{89} +192.636 q^{91} +263.424i q^{93} -80.2686i q^{97} +(49.3414 + 163.111i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 56 q^{9} + 48 q^{11} + 40 q^{31} + 488 q^{49} + 32 q^{59} + 112 q^{69} - 440 q^{71} - 448 q^{81} - 440 q^{89} - 144 q^{91} + 80 q^{99}+O(q^{100}) $ 16 * q - 56 * q^9 + 48 * q^11 + 40 * q^31 + 488 * q^49 + 32 * q^59 + 112 * q^69 - 440 * q^71 - 448 * q^81 - 440 * q^89 - 144 * q^91 + 80 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			4.94892i			1.64964i	0.565395	+	0.824820i	$0.308724\pi$
$3$			4.94892i			1.64964i	−0.565395	+	0.824820i	$0.691276\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−13.1585			−1.87979			−0.939896	−	0.341462i	$-0.889078\pi$
$7$	−13.1585			−1.87979			−0.939896	+	0.341462i	$0.889078\pi$
$8$		0			0
$9$	−15.4918			−1.72132
$10$		0			0
$11$	−3.18499	−	10.5288i	−0.289545	−	0.957165i
$11$	−3.18499	−	10.5288i	−0.289545	−	0.957165i
$12$		0			0
$13$	−14.6396			−1.12612			−0.563061	−	0.826415i	$-0.690376\pi$
$13$	−14.6396			−1.12612			−0.563061	+	0.826415i	$0.690376\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	9.90801			0.582824			0.291412	−	0.956598i	$-0.405875\pi$
$17$	9.90801			0.582824			0.291412	+	0.956598i	$0.405875\pi$
$18$		0			0
$19$			11.5333i			0.607015i	0.952829	+	0.303507i	$0.0981577\pi$
$19$			11.5333i			0.607015i	−0.952829	+	0.303507i	$0.901842\pi$
$20$		0			0
$21$		−	65.1206i		−	3.10098i
$22$		0			0
$23$			14.6044i			0.634974i	0.948263	+	0.317487i	$0.102839\pi$
$23$			14.6044i			0.634974i	−0.948263	+	0.317487i	$0.897161\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	32.1276i		−	1.18991i
$28$		0			0
$29$			8.28275i			0.285612i	0.989751	+	0.142806i	$0.0456125\pi$
$29$			8.28275i			0.285612i	−0.989751	+	0.142806i	$0.954387\pi$
$30$		0			0
$31$	53.2285			1.71705			0.858524	−	0.512773i	$-0.171382\pi$
$31$	53.2285			1.71705			0.858524	+	0.512773i	$0.171382\pi$
$32$		0			0
$33$	52.1063	−	15.7623i	1.57898	−	0.477645i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			41.1146i			1.11121i	0.831448	+	0.555603i	$0.187512\pi$
$37$			41.1146i			1.11121i	−0.831448	+	0.555603i	$0.812488\pi$
$38$		0			0
$39$		−	72.4502i		−	1.85770i
$40$		0			0
$41$		−	74.6449i		−	1.82061i	−0.413940	−	0.910304i	$-0.635848\pi$
$41$		−	74.6449i		−	1.82061i	0.413940	−	0.910304i	$-0.364152\pi$
$42$		0			0
$43$	30.4377			0.707855			0.353927	−	0.935273i	$-0.384846\pi$
$43$	30.4377			0.707855			0.353927	+	0.935273i	$0.384846\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			0.697244i			0.0148350i	0.999972	+	0.00741749i	$0.00236108\pi$
$47$			0.697244i			0.0148350i	−0.999972	+	0.00741749i	$0.997639\pi$
$48$		0			0
$49$	124.147			2.53361
$50$		0			0
$51$			49.0340i			0.961451i
$52$		0			0
$53$		−	17.0571i		−	0.321831i	−0.986968	−	0.160916i	$-0.948555\pi$
$53$		−	17.0571i		−	0.321831i	0.986968	−	0.160916i	$-0.0514447\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−57.0773			−1.00136
$58$		0			0
$59$	6.29242			0.106651			0.0533256	−	0.998577i	$-0.483018\pi$
$59$	6.29242			0.106651			0.0533256	+	0.998577i	$0.483018\pi$
$60$		0			0
$61$		−	50.6864i		−	0.830925i	−0.909610	−	0.415462i	$-0.863620\pi$
$61$		−	50.6864i		−	0.830925i	0.909610	−	0.415462i	$-0.136380\pi$
$62$		0			0
$63$	203.850			3.23571
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	81.8378i		−	1.22146i	−0.791839	−	0.610730i	$-0.790876\pi$
$67$		−	81.8378i		−	1.22146i	0.791839	−	0.610730i	$-0.209124\pi$
$68$		0			0
$69$	−72.2760			−1.04748
$70$		0			0
$71$	−62.4365			−0.879387			−0.439694	−	0.898148i	$-0.644913\pi$
$71$	−62.4365			−0.879387			−0.439694	+	0.898148i	$0.644913\pi$
$72$		0			0
$73$	−11.5551			−0.158289			−0.0791445	−	0.996863i	$-0.525219\pi$
$73$	−11.5551			−0.158289			−0.0791445	+	0.996863i	$0.525219\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	41.9098	+	138.544i	0.544283	+	1.79927i
$78$		0			0
$79$			79.2397i			1.00303i	0.865148	+	0.501517i	$0.167224\pi$
$79$			79.2397i			1.00303i	−0.865148	+	0.501517i	$0.832776\pi$
$80$		0			0
$81$	19.5705			0.241611
$82$		0			0
$83$	−81.8105			−0.985669			−0.492834	−	0.870123i	$-0.664039\pi$
$83$	−81.8105			−0.985669			−0.492834	+	0.870123i	$0.664039\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−40.9907			−0.471158
$88$		0			0
$89$	−136.935			−1.53860			−0.769300	−	0.638888i	$-0.779395\pi$
$89$	−136.935			−1.53860			−0.769300	+	0.638888i	$0.779395\pi$
$90$		0			0
$91$	192.636			2.11687
$92$		0			0
$93$			263.424i			2.83251i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	80.2686i		−	0.827511i	−0.910388	−	0.413756i	$-0.864217\pi$
$97$		−	80.2686i		−	0.827511i	0.910388	−	0.413756i	$-0.135783\pi$
$98$		0			0
$99$	49.3414	+	163.111i	0.498398	+	1.64758i
$100$		0			0
$101$			19.3173i			0.191261i	0.995417	+	0.0956303i	$0.0304867\pi$
$101$			19.3173i			0.191261i	−0.995417	+	0.0956303i	$0.969513\pi$
$102$		0			0
$103$			71.4905i			0.694082i	0.937850	+	0.347041i	$0.112814\pi$
$103$			71.4905i			0.694082i	−0.937850	+	0.347041i	$0.887186\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	148.254			1.38555			0.692774	−	0.721154i	$-0.256388\pi$
$107$	148.254			1.38555			0.692774	+	0.721154i	$0.256388\pi$
$108$		0			0
$109$		−	201.904i		−	1.85233i	−0.377116	−	0.926166i	$-0.623084\pi$
$109$		−	201.904i		−	1.85233i	0.377116	−	0.926166i	$-0.376916\pi$
$110$		0			0
$111$	−203.473			−1.83309
$112$		0			0
$113$		−	179.659i		−	1.58990i	−0.606673	−	0.794951i	$-0.707496\pi$
$113$		−	179.659i		−	1.58990i	0.606673	−	0.794951i	$-0.292504\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	226.794			1.93841
$118$		0			0
$119$	−130.375			−1.09559
$120$		0			0
$121$	−100.712	+	67.0683i	−0.832328	+	0.554284i
$122$		0			0
$123$	369.412			3.00335
$124$		0			0
$125$		0			0
$126$		0			0
$127$	173.681			1.36757			0.683784	−	0.729685i	$-0.260333\pi$
$127$	173.681			1.36757			0.683784	+	0.729685i	$0.260333\pi$
$128$		0			0
$129$			150.634i			1.16771i
$130$		0			0
$131$		−	99.4932i		−	0.759490i	−0.925091	−	0.379745i	$-0.876012\pi$
$131$		−	99.4932i		−	0.759490i	0.925091	−	0.379745i	$-0.123988\pi$
$132$		0			0
$133$		−	151.761i		−	1.14106i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	155.034i		−	1.13163i	−0.824531	−	0.565816i	$-0.808561\pi$
$137$		−	155.034i		−	1.13163i	0.824531	−	0.565816i	$-0.191439\pi$
$138$		0			0
$139$			62.7620i			0.451525i	0.974182	+	0.225763i	$0.0724874\pi$
$139$			62.7620i			0.451525i	−0.974182	+	0.225763i	$0.927513\pi$
$140$		0			0
$141$	−3.45061			−0.0244724
$142$		0			0
$143$	46.6269	+	154.137i	0.326063	+	1.07788i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			614.395i			4.17955i
$148$		0			0
$149$		−	137.649i		−	0.923818i	−0.886927	−	0.461909i	$-0.847165\pi$
$149$		−	137.649i		−	0.923818i	0.886927	−	0.461909i	$-0.152835\pi$
$150$		0			0
$151$			229.174i			1.51771i	0.651259	+	0.758856i	$0.274241\pi$
$151$			229.174i			1.51771i	−0.651259	+	0.758856i	$0.725759\pi$
$152$		0			0
$153$	−153.493			−1.00322
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	137.708i		−	0.877118i	−0.898702	−	0.438559i	$-0.855489\pi$
$157$		−	137.708i		−	0.877118i	0.898702	−	0.438559i	$-0.144511\pi$
$158$		0			0
$159$	84.4141			0.530906
$160$		0			0
$161$		−	192.172i		−	1.19362i
$162$		0			0
$163$			38.9324i			0.238849i	0.992843	+	0.119425i	$0.0381049\pi$
$163$			38.9324i			0.238849i	−0.992843	+	0.119425i	$0.961895\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	165.244			0.989488			0.494744	−	0.869039i	$-0.335262\pi$
$167$	165.244			0.989488			0.494744	+	0.869039i	$0.335262\pi$
$168$		0			0
$169$	45.3175			0.268151
$170$		0			0
$171$		−	178.672i		−	1.04486i
$172$		0			0
$173$	−297.612			−1.72030			−0.860149	−	0.510042i	$-0.829630\pi$
$173$	−297.612			−1.72030			−0.860149	+	0.510042i	$0.829630\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			31.1407i			0.175936i
$178$		0			0
$179$	208.567			1.16518			0.582588	−	0.812767i	$-0.302040\pi$
$179$	208.567			1.16518			0.582588	+	0.812767i	$0.302040\pi$
$180$		0			0
$181$	234.341			1.29470			0.647351	−	0.762192i	$-0.275877\pi$
$181$	234.341			1.29470			0.647351	+	0.762192i	$0.275877\pi$
$182$		0			0
$183$	250.843			1.37073
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−31.5569	−	104.320i	−0.168754	−	0.557859i
$188$		0			0
$189$			422.752i			2.23678i
$190$		0			0
$191$	186.250			0.975128			0.487564	−	0.873087i	$-0.337885\pi$
$191$	186.250			0.975128			0.487564	+	0.873087i	$0.337885\pi$
$192$		0			0
$193$	−144.553			−0.748982			−0.374491	−	0.927231i	$-0.622182\pi$
$193$	−144.553			−0.748982			−0.374491	+	0.927231i	$0.622182\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	159.129			0.807762			0.403881	−	0.914812i	$-0.367661\pi$
$197$	159.129			0.807762			0.403881	+	0.914812i	$0.367661\pi$
$198$		0			0
$199$	−208.591			−1.04820			−0.524098	−	0.851658i	$-0.675598\pi$
$199$	−208.591			−1.04820			−0.524098	+	0.851658i	$0.675598\pi$
$200$		0			0
$201$	405.009			2.01497
$202$		0			0
$203$		−	108.989i		−	0.536891i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	226.249i		−	1.09299i
$208$		0			0
$209$	121.432	−	36.7334i	0.581013	−	0.175758i
$210$		0			0
$211$		−	249.692i		−	1.18337i	−0.806168	−	0.591687i	$-0.798462\pi$
$211$		−	249.692i		−	1.18337i	0.806168	−	0.591687i	$-0.201538\pi$
$212$		0			0
$213$		−	308.993i		−	1.45067i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−700.409			−3.22769
$218$		0			0
$219$		−	57.1852i		−	0.261120i
$220$		0			0
$221$	−145.049			−0.656331
$222$		0			0
$223$			94.8922i			0.425526i	0.977104	+	0.212763i	$0.0682461\pi$
$223$			94.8922i			0.425526i	−0.977104	+	0.212763i	$0.931754\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−365.237			−1.60897			−0.804487	−	0.593970i	$-0.797560\pi$
$227$	−365.237			−1.60897			−0.804487	+	0.593970i	$0.797560\pi$
$228$		0			0
$229$	181.047			0.790600			0.395300	−	0.918552i	$-0.370641\pi$
$229$	181.047			0.790600			0.395300	+	0.918552i	$0.370641\pi$
$230$		0			0
$231$	−685.642	+	207.408i	−2.96815	+	0.897872i
$232$		0			0
$233$	291.907			1.25282			0.626410	−	0.779494i	$-0.284524\pi$
$233$	291.907			1.25282			0.626410	+	0.779494i	$0.284524\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−392.151			−1.65465
$238$		0			0
$239$		−	43.0882i		−	0.180285i	−0.995929	−	0.0901427i	$-0.971268\pi$
$239$		−	43.0882i		−	0.180285i	0.995929	−	0.0901427i	$-0.0287323\pi$
$240$		0			0
$241$			77.0546i			0.319729i	0.987139	+	0.159864i	$0.0511057\pi$
$241$			77.0546i			0.319729i	−0.987139	+	0.159864i	$0.948894\pi$
$242$		0			0
$243$		−	192.296i		−	0.791340i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	168.842i		−	0.683572i
$248$		0			0
$249$		−	404.874i		−	1.62600i
$250$		0			0
$251$	7.50457			0.0298987			0.0149494	−	0.999888i	$-0.495241\pi$
$251$	7.50457			0.0298987			0.0149494	+	0.999888i	$0.495241\pi$
$252$		0			0
$253$	153.767	−	46.5149i	0.607774	−	0.183853i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	86.1842i		−	0.335347i	−0.985843	−	0.167674i	$-0.946375\pi$
$257$		−	86.1842i		−	0.335347i	0.985843	−	0.167674i	$-0.0536255\pi$
$258$		0			0
$259$		−	541.008i		−	2.08883i
$260$		0			0
$261$		−	128.315i		−	0.491629i
$262$		0			0
$263$	−328.758			−1.25003			−0.625016	−	0.780612i	$-0.714907\pi$
$263$	−328.758			−1.25003			−0.625016	+	0.780612i	$0.714907\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	677.683i		−	2.53814i
$268$		0			0
$269$	358.896			1.33418			0.667092	−	0.744975i	$-0.267539\pi$
$269$	358.896			1.33418			0.667092	+	0.744975i	$0.267539\pi$
$270$		0			0
$271$		−	223.030i		−	0.822989i	−0.911412	−	0.411494i	$-0.865007\pi$
$271$		−	223.030i		−	0.822989i	0.911412	−	0.411494i	$-0.134993\pi$
$272$		0			0
$273$			953.339i			3.49208i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−81.1615			−0.293002			−0.146501	−	0.989211i	$-0.546801\pi$
$277$	−81.1615			−0.293002			−0.146501	+	0.989211i	$0.546801\pi$
$278$		0			0
$279$	−824.607			−2.95558
$280$		0			0
$281$			91.9342i			0.327168i	0.986529	+	0.163584i	$0.0523054\pi$
$281$			91.9342i			0.327168i	−0.986529	+	0.163584i	$0.947695\pi$
$282$		0			0
$283$	1.55872			0.00550784			0.00275392	−	0.999996i	$-0.499123\pi$
$283$	1.55872			0.00550784			0.00275392	+	0.999996i	$0.499123\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			982.218i			3.42236i
$288$		0			0
$289$	−190.831			−0.660316
$290$		0			0
$291$	397.243			1.36510
$292$		0			0
$293$	−361.504			−1.23380			−0.616900	−	0.787041i	$-0.711612\pi$
$293$	−361.504			−1.23380			−0.616900	+	0.787041i	$0.711612\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−338.265	+	102.326i	−1.13894	+	0.344532i
$298$		0			0
$299$		−	213.802i		−	0.715058i
$300$		0			0
$301$	−400.516			−1.33062
$302$		0			0
$303$	−95.5999			−0.315511
$304$		0			0
$305$		0			0
$306$		0			0
$307$	51.3776			0.167354			0.0836769	−	0.996493i	$-0.473334\pi$
$307$	51.3776			0.167354			0.0836769	+	0.996493i	$0.473334\pi$
$308$		0			0
$309$	−353.801			−1.14499
$310$		0			0
$311$	198.075			0.636898			0.318449	−	0.947940i	$-0.396838\pi$
$311$	198.075			0.636898			0.318449	+	0.947940i	$0.396838\pi$
$312$		0			0
$313$			22.2970i			0.0712365i	0.999365	+	0.0356183i	$0.0113400\pi$
$313$			22.2970i			0.0712365i	−0.999365	+	0.0356183i	$0.988660\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	102.256i		−	0.322574i	−0.986908	−	0.161287i	$-0.948436\pi$
$317$		−	102.256i		−	0.322574i	0.986908	−	0.161287i	$-0.0515644\pi$
$318$		0			0
$319$	87.2075	−	26.3805i	0.273378	−	0.0826975i
$320$		0			0
$321$			733.696i			2.28566i
$322$		0			0
$323$			114.272i			0.353783i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	999.208			3.05568
$328$		0			0
$329$		−	9.17471i		−	0.0278867i
$330$		0			0
$331$	−353.647			−1.06842			−0.534211	−	0.845351i	$-0.679391\pi$
$331$	−353.647			−1.06842			−0.534211	+	0.845351i	$0.679391\pi$
$332$		0			0
$333$		−	636.941i		−	1.91273i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−237.983			−0.706182			−0.353091	−	0.935589i	$-0.614869\pi$
$337$	−237.983			−0.706182			−0.353091	+	0.935589i	$0.614869\pi$
$338$		0			0
$339$	889.119			2.62277
$340$		0			0
$341$	−169.532	−	560.433i	−0.497162	−	1.64350i
$342$		0			0
$343$	−988.826			−2.88288
$344$		0			0
$345$		0			0
$346$		0			0
$347$	142.124			0.409580			0.204790	−	0.978806i	$-0.434349\pi$
$347$	142.124			0.409580			0.204790	+	0.978806i	$0.434349\pi$
$348$		0			0
$349$			25.4867i			0.0730278i	0.999333	+	0.0365139i	$0.0116253\pi$
$349$			25.4867i			0.0730278i	−0.999333	+	0.0365139i	$0.988375\pi$
$350$		0			0
$351$			470.335i			1.33999i
$352$		0			0
$353$		−	155.144i		−	0.439502i	−0.975556	−	0.219751i	$-0.929476\pi$
$353$		−	155.144i		−	0.439502i	0.975556	−	0.219751i	$-0.0705245\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	645.216i		−	1.80733i
$358$		0			0
$359$			6.51482i			0.0181471i	0.999959	+	0.00907357i	$0.00288825\pi$
$359$			6.51482i			0.0181471i	−0.999959	+	0.00907357i	$0.997112\pi$
$360$		0			0
$361$	227.984			0.631533
$362$		0			0
$363$	−331.916	−	498.414i	−0.914369	−	1.37304i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			102.889i			0.280353i	0.990127	+	0.140176i	$0.0447670\pi$
$367$			102.889i			0.280353i	−0.990127	+	0.140176i	$0.955233\pi$
$368$		0			0
$369$			1156.39i			3.13384i
$370$		0			0
$371$			224.446i			0.604976i
$372$		0			0
$373$	119.790			0.321152			0.160576	−	0.987024i	$-0.448665\pi$
$373$	119.790			0.321152			0.160576	+	0.987024i	$0.448665\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	121.256i		−	0.321634i
$378$		0			0
$379$	−175.693			−0.463570			−0.231785	−	0.972767i	$-0.574457\pi$
$379$	−175.693			−0.463570			−0.231785	+	0.972767i	$0.574457\pi$
$380$		0			0
$381$			859.534i			2.25599i
$382$		0			0
$383$		−	225.422i		−	0.588570i	−0.955718	−	0.294285i	$-0.904918\pi$
$383$		−	225.422i		−	0.588570i	0.955718	−	0.294285i	$-0.0950815\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−471.537			−1.21844
$388$		0			0
$389$	−91.2920			−0.234684			−0.117342	−	0.993092i	$-0.537437\pi$
$389$	−91.2920			−0.234684			−0.117342	+	0.993092i	$0.537437\pi$
$390$		0			0
$391$			144.701i			0.370078i
$392$		0			0
$393$	492.384			1.25289
$394$		0			0
$395$		0			0
$396$		0			0
$397$			602.789i			1.51836i	0.650880	+	0.759181i	$0.274400\pi$
$397$			602.789i			1.51836i	−0.650880	+	0.759181i	$0.725600\pi$
$398$		0			0
$399$	751.054			1.88234
$400$		0			0
$401$	277.075			0.690961			0.345480	−	0.938426i	$-0.387716\pi$
$401$	277.075			0.690961			0.345480	+	0.938426i	$0.387716\pi$
$402$		0			0
$403$	−779.243			−1.93361
$404$		0			0
$405$		0			0
$406$		0			0
$407$	432.888	−	130.950i	1.06361	−	0.321744i
$408$		0			0
$409$		−	165.156i		−	0.403804i	−0.979406	−	0.201902i	$-0.935288\pi$
$409$		−	165.156i		−	0.403804i	0.979406	−	0.201902i	$-0.0647122\pi$
$410$		0			0
$411$	767.249			1.86679
$412$		0			0
$413$	−82.7990			−0.200482
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−310.604			−0.744855
$418$		0			0
$419$	619.744			1.47910			0.739551	−	0.673100i	$-0.235038\pi$
$419$	619.744			1.47910			0.739551	+	0.673100i	$0.235038\pi$
$420$		0			0
$421$	232.190			0.551520			0.275760	−	0.961226i	$-0.411070\pi$
$421$	232.190			0.551520			0.275760	+	0.961226i	$0.411070\pi$
$422$		0			0
$423$		−	10.8016i		−	0.0255357i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			666.959i			1.56197i
$428$		0			0
$429$	−762.814	+	230.753i	−1.77812	+	0.537886i
$430$		0			0
$431$			606.608i			1.40744i	0.710476	+	0.703721i	$0.248480\pi$
$431$			606.608i			1.40744i	−0.710476	+	0.703721i	$0.751520\pi$
$432$		0			0
$433$		−	85.1841i		−	0.196730i	−0.995150	−	0.0983650i	$-0.968639\pi$
$433$		−	85.1841i		−	0.196730i	0.995150	−	0.0983650i	$-0.0313613\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−168.436			−0.385438
$438$		0			0
$439$			559.497i			1.27448i	0.770665	+	0.637240i	$0.219924\pi$
$439$			559.497i			1.27448i	−0.770665	+	0.637240i	$0.780076\pi$
$440$		0			0
$441$	−1923.27			−4.36115
$442$		0			0
$443$			171.266i			0.386606i	0.981139	+	0.193303i	$0.0619200\pi$
$443$			171.266i			0.386606i	−0.981139	+	0.193303i	$0.938080\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	681.214			1.52397
$448$		0			0
$449$	−701.180			−1.56165			−0.780824	−	0.624751i	$-0.785200\pi$
$449$	−701.180			−1.56165			−0.780824	+	0.624751i	$0.785200\pi$
$450$		0			0
$451$	−785.922	+	237.743i	−1.74262	+	0.527147i
$452$		0			0
$453$	−1134.17			−2.50368
$454$		0			0
$455$		0			0
$456$		0			0
$457$	94.6651			0.207145			0.103572	−	0.994622i	$-0.466973\pi$
$457$	94.6651			0.207145			0.103572	+	0.994622i	$0.466973\pi$
$458$		0			0
$459$		−	318.321i		−	0.693509i
$460$		0			0
$461$			386.694i			0.838815i	0.907798	+	0.419408i	$0.137762\pi$
$461$			386.694i			0.838815i	−0.907798	+	0.419408i	$0.862238\pi$
$462$		0			0
$463$		−	364.997i		−	0.788330i	−0.919040	−	0.394165i	$-0.871034\pi$
$463$		−	364.997i		−	0.788330i	0.919040	−	0.394165i	$-0.128966\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	719.317i		−	1.54029i	−0.637867	−	0.770147i	$-0.720183\pi$
$467$		−	719.317i		−	1.54029i	0.637867	−	0.770147i	$-0.279817\pi$
$468$		0			0
$469$			1076.87i			2.29609i
$470$		0			0
$471$	681.504			1.44693
$472$		0			0
$473$	−96.9440	−	320.473i	−0.204955	−	0.677533i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			264.245i			0.553973i
$478$		0			0
$479$		−	373.191i		−	0.779104i	−0.921004	−	0.389552i	$-0.872630\pi$
$479$		−	373.191i		−	0.779104i	0.921004	−	0.389552i	$-0.127370\pi$
$480$		0			0
$481$		−	601.901i		−	1.25135i
$482$		0			0
$483$	951.047			1.96904
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	480.297i		−	0.986237i	−0.869962	−	0.493118i	$-0.835857\pi$
$487$		−	480.297i		−	0.986237i	0.869962	−	0.493118i	$-0.164143\pi$
$488$		0			0
$489$	−192.673			−0.394015
$490$		0			0
$491$		−	107.055i		−	0.218035i	−0.994040	−	0.109017i	$-0.965230\pi$
$491$		−	107.055i		−	0.218035i	0.994040	−	0.109017i	$-0.0347704\pi$
$492$		0			0
$493$			82.0656i			0.166462i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	821.573			1.65306
$498$		0			0
$499$	−296.804			−0.594798			−0.297399	−	0.954753i	$-0.596119\pi$
$499$	−296.804			−0.594798			−0.297399	+	0.954753i	$0.596119\pi$
$500$		0			0
$501$			817.782i			1.63230i
$502$		0			0
$503$	−538.023			−1.06963			−0.534815	−	0.844969i	$-0.679619\pi$
$503$	−538.023			−1.06963			−0.534815	+	0.844969i	$0.679619\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			224.273i			0.442353i
$508$		0			0
$509$	383.073			0.752599			0.376299	−	0.926498i	$-0.377196\pi$
$509$	383.073			0.752599			0.376299	+	0.926498i	$0.377196\pi$
$510$		0			0
$511$	152.048			0.297550
$512$		0			0
$513$	370.537			0.722293
$514$		0			0
$515$		0			0
$516$		0			0
$517$	7.34115	−	2.22072i	0.0141995	−	0.00429539i
$518$		0			0
$519$		−	1472.86i		−	2.83787i
$520$		0			0
$521$	−484.670			−0.930270			−0.465135	−	0.885240i	$-0.653994\pi$
$521$	−484.670			−0.930270			−0.465135	+	0.885240i	$0.653994\pi$
$522$		0			0
$523$	481.382			0.920424			0.460212	−	0.887809i	$-0.347773\pi$
$523$	481.382			0.920424			0.460212	+	0.887809i	$0.347773\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	527.389			1.00074
$528$		0			0
$529$	315.712			0.596809
$530$		0			0
$531$	−97.4811			−0.183580
$532$		0			0
$533$			1092.77i			2.05023i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			1032.18i			1.92212i
$538$		0			0
$539$	−395.407	−	1307.12i	−0.733595	−	2.42509i
$540$		0			0
$541$		−	534.638i		−	0.988241i	−0.869393	−	0.494121i	$-0.835490\pi$
$541$		−	534.638i		−	0.988241i	0.869393	−	0.494121i	$-0.164510\pi$
$542$		0			0
$543$			1159.74i			2.13580i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−601.686			−1.09997			−0.549987	−	0.835173i	$-0.685367\pi$
$547$	−601.686			−1.09997			−0.549987	+	0.835173i	$0.685367\pi$
$548$		0			0
$549$			785.226i			1.43028i
$550$		0			0
$551$	−95.5273			−0.173371
$552$		0			0
$553$		−	1042.68i		−	1.88549i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−802.187			−1.44019			−0.720096	−	0.693874i	$-0.755902\pi$
$557$	−802.187			−1.44019			−0.720096	+	0.693874i	$0.755902\pi$
$558$		0			0
$559$	−445.596			−0.797131
$560$		0			0
$561$	516.270	−	156.173i	0.920267	−	0.278383i
$562$		0			0
$563$	729.072			1.29498			0.647489	−	0.762075i	$-0.275819\pi$
$563$	729.072			1.29498			0.647489	+	0.762075i	$0.275819\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−257.519			−0.454178
$568$		0			0
$569$		−	629.473i		−	1.10628i	−0.833089	−	0.553140i	$-0.813430\pi$
$569$		−	629.473i		−	1.10628i	0.833089	−	0.553140i	$-0.186570\pi$
$570$		0			0
$571$		−	615.487i		−	1.07791i	−0.842335	−	0.538955i	$-0.818819\pi$
$571$		−	615.487i		−	1.07791i	0.842335	−	0.538955i	$-0.181181\pi$
$572$		0			0
$573$			921.735i			1.60861i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	162.104i		−	0.280942i	−0.990085	−	0.140471i	$-0.955138\pi$
$577$		−	162.104i		−	0.280942i	0.990085	−	0.140471i	$-0.0448617\pi$
$578$		0			0
$579$		−	715.384i		−	1.23555i
$580$		0			0
$581$	1076.51			1.85285
$582$		0			0
$583$	−179.591	+	54.3266i	−0.308046	+	0.0931845i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	661.750i		−	1.12734i	−0.825999	−	0.563671i	$-0.809389\pi$
$587$		−	661.750i		−	1.12734i	0.825999	−	0.563671i	$-0.190611\pi$
$588$		0			0
$589$			613.899i			1.04227i
$590$		0			0
$591$			787.518i			1.33252i
$592$		0			0
$593$	−77.0827			−0.129988			−0.0649939	−	0.997886i	$-0.520703\pi$
$593$	−77.0827			−0.129988			−0.0649939	+	0.997886i	$0.520703\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	1032.30i		−	1.72915i
$598$		0			0
$599$	414.443			0.691891			0.345945	−	0.938255i	$-0.387558\pi$
$599$	414.443			0.691891			0.345945	+	0.938255i	$0.387558\pi$
$600$		0			0
$601$			258.961i			0.430883i	0.976517	+	0.215441i	$0.0691191\pi$
$601$			258.961i			0.430883i	−0.976517	+	0.215441i	$0.930881\pi$
$602$		0			0
$603$			1267.82i			2.10252i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−447.958			−0.737986			−0.368993	−	0.929432i	$-0.620297\pi$
$607$	−447.958			−0.737986			−0.368993	+	0.929432i	$0.620297\pi$
$608$		0			0
$609$	539.378			0.885678
$610$		0			0
$611$		−	10.2074i		−	0.0167060i
$612$		0			0
$613$	−762.804			−1.24438			−0.622190	−	0.782867i	$-0.713757\pi$
$613$	−762.804			−1.24438			−0.622190	+	0.782867i	$0.713757\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	531.407i		−	0.861276i	−0.902525	−	0.430638i	$-0.858289\pi$
$617$		−	531.407i		−	0.861276i	0.902525	−	0.430638i	$-0.141711\pi$
$618$		0			0
$619$	−648.349			−1.04741			−0.523707	−	0.851898i	$-0.675451\pi$
$619$	−648.349			−1.04741			−0.523707	+	0.851898i	$0.675451\pi$
$620$		0			0
$621$	469.204			0.755562
$622$		0			0
$623$	1801.87			2.89225
$624$		0			0
$625$		0			0
$626$		0			0
$627$	181.791	+	600.956i	0.289937	+	0.958463i
$628$		0			0
$629$			407.364i			0.647638i
$630$		0			0
$631$	−500.262			−0.792808			−0.396404	−	0.918076i	$-0.629742\pi$
$631$	−500.262			−0.792808			−0.396404	+	0.918076i	$0.629742\pi$
$632$		0			0
$633$	1235.71			1.95214
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−1817.46			−2.85316
$638$		0			0
$639$	967.256			1.51370
$640$		0			0
$641$	195.726			0.305345			0.152673	−	0.988277i	$-0.451212\pi$
$641$	195.726			0.305345			0.152673	+	0.988277i	$0.451212\pi$
$642$		0			0
$643$		−	1016.58i		−	1.58100i	−0.612465	−	0.790498i	$-0.709822\pi$
$643$		−	1016.58i		−	1.58100i	0.612465	−	0.790498i	$-0.290178\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			489.683i			0.756851i	0.925632	+	0.378426i	$0.123534\pi$
$647$			489.683i			0.756851i	−0.925632	+	0.378426i	$0.876466\pi$
$648$		0			0
$649$	−20.0413	−	66.2517i	−0.0308803	−	0.102083i
$650$		0			0
$651$		−	3466.27i		−	5.32453i
$652$		0			0
$653$		−	293.173i		−	0.448964i	−0.974478	−	0.224482i	$-0.927931\pi$
$653$		−	293.173i		−	0.448964i	0.974478	−	0.224482i	$-0.0720690\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	179.010			0.272465
$658$		0			0
$659$		−	590.485i		−	0.896032i	−0.894026	−	0.448016i	$-0.852131\pi$
$659$		−	590.485i		−	0.896032i	0.894026	−	0.448016i	$-0.147869\pi$
$660$		0			0
$661$	43.8951			0.0664072			0.0332036	−	0.999449i	$-0.489429\pi$
$661$	43.8951			0.0664072			0.0332036	+	0.999449i	$0.489429\pi$
$662$		0			0
$663$		−	717.837i		−	1.08271i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−120.965			−0.181356
$668$		0			0
$669$	−469.614			−0.701964
$670$		0			0
$671$	−533.668	+	161.436i	−0.795332	+	0.240590i
$672$		0			0
$673$	869.115			1.29140			0.645702	−	0.763590i	$-0.276565\pi$
$673$	869.115			1.29140			0.645702	+	0.763590i	$0.276565\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	480.512			0.709766			0.354883	−	0.934911i	$-0.384521\pi$
$677$	480.512			0.709766			0.354883	+	0.934911i	$0.384521\pi$
$678$		0			0
$679$			1056.22i			1.55555i
$680$		0			0
$681$		−	1807.53i		−	2.65423i
$682$		0			0
$683$		−	1111.42i		−	1.62726i	−0.581380	−	0.813632i	$-0.697487\pi$
$683$		−	1111.42i		−	1.62726i	0.581380	−	0.813632i	$-0.302513\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			895.989i			1.30421i
$688$		0			0
$689$			249.708i			0.362421i
$690$		0			0
$691$	373.681			0.540783			0.270391	−	0.962750i	$-0.412847\pi$
$691$	373.681			0.540783			0.270391	+	0.962750i	$0.412847\pi$
$692$		0			0
$693$	−649.260	−	2146.30i	−0.936883	−	3.09711i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	739.583i		−	1.06109i
$698$		0			0
$699$			1444.63i			2.06670i
$700$		0			0
$701$		−	732.819i		−	1.04539i	−0.852520	−	0.522695i	$-0.824927\pi$
$701$		−	732.819i		−	1.04539i	0.852520	−	0.522695i	$-0.175073\pi$
$702$		0			0
$703$	−474.186			−0.674518
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	254.188i		−	0.359530i
$708$		0			0
$709$	799.109			1.12709			0.563547	−	0.826084i	$-0.309437\pi$
$709$	799.109			1.12709			0.563547	+	0.826084i	$0.309437\pi$
$710$		0			0
$711$		−	1227.57i		−	1.72654i
$712$		0			0
$713$			777.370i			1.09028i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	213.240			0.297406
$718$		0			0
$719$	375.329			0.522015			0.261008	−	0.965337i	$-0.415945\pi$
$719$	375.329			0.522015			0.261008	+	0.965337i	$0.415945\pi$
$720$		0			0
$721$		−	940.710i		−	1.30473i
$722$		0			0
$723$	−381.337			−0.527438
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	21.7939i		−	0.0299779i	−0.999888	−	0.0149889i	$-0.995229\pi$
$727$		−	21.7939i		−	0.0299779i	0.999888	−	0.0149889i	$-0.00477130\pi$
$728$		0			0
$729$	1127.79			1.54704
$730$		0			0
$731$	301.578			0.412555
$732$		0			0
$733$	34.7783			0.0474466			0.0237233	−	0.999719i	$-0.492448\pi$
$733$	34.7783			0.0474466			0.0237233	+	0.999719i	$0.492448\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−861.654	+	260.653i	−1.16914	+	0.353667i
$738$		0			0
$739$		−	332.608i		−	0.450079i	−0.974350	−	0.225039i	$-0.927749\pi$
$739$		−	332.608i		−	0.450079i	0.974350	−	0.225039i	$-0.0722511\pi$
$740$		0			0
$741$	835.588			1.12765
$742$		0			0
$743$	999.033			1.34459			0.672297	−	0.740282i	$-0.265308\pi$
$743$	999.033			1.34459			0.672297	+	0.740282i	$0.265308\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1267.40			1.69665
$748$		0			0
$749$	−1950.80			−2.60454
$750$		0			0
$751$	−218.656			−0.291153			−0.145576	−	0.989347i	$-0.546504\pi$
$751$	−218.656			−0.291153			−0.145576	+	0.989347i	$0.546504\pi$
$752$		0			0
$753$			37.1396i			0.0493221i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	158.061i		−	0.208800i	−0.994535	−	0.104400i	$-0.966708\pi$
$757$		−	158.061i		−	0.208800i	0.994535	−	0.104400i	$-0.0332922\pi$
$758$		0			0
$759$	230.198	+	760.980i	0.303292	+	1.00261i
$760$		0			0
$761$			389.588i			0.511942i	0.966684	+	0.255971i	$0.0823952\pi$
$761$			389.588i			0.511942i	−0.966684	+	0.255971i	$0.917605\pi$
$762$		0			0
$763$			2656.76i			3.48200i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−92.1184			−0.120102
$768$		0			0
$769$			687.354i			0.893828i	0.894577	+	0.446914i	$0.147477\pi$
$769$			687.354i			0.893828i	−0.894577	+	0.446914i	$0.852523\pi$
$770$		0			0
$771$	426.519			0.553202
$772$		0			0
$773$			934.266i			1.20862i	0.796748	+	0.604312i	$0.206552\pi$
$773$			934.266i			1.20862i	−0.796748	+	0.604312i	$0.793448\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2677.41			3.44583
$778$		0			0
$779$	860.901			1.10514
$780$		0			0
$781$	198.860	+	657.382i	0.254622	+	0.841718i
$782$		0			0
$783$	266.105			0.339853
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−793.478			−1.00823			−0.504116	−	0.863636i	$-0.668181\pi$
$787$	−793.478			−1.00823			−0.504116	+	0.863636i	$0.668181\pi$
$788$		0			0
$789$		−	1627.00i		−	2.06210i
$790$		0			0
$791$			2364.05i			2.98869i
$792$		0			0
$793$			742.028i			0.935723i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1015.43i			1.27407i	0.770837	+	0.637033i	$0.219838\pi$
$797$			1015.43i			1.27407i	−0.770837	+	0.637033i	$0.780162\pi$
$798$		0			0
$799$			6.90831i			0.00864619i
$800$		0			0
$801$	2121.38			2.64842
$802$		0			0
$803$	36.8029	+	121.661i	0.0458317	+	0.151509i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			1776.15i			2.20093i
$808$		0			0
$809$		−	759.906i		−	0.939316i	−0.882849	−	0.469658i	$-0.844377\pi$
$809$		−	759.906i		−	0.939316i	0.882849	−	0.469658i	$-0.155623\pi$
$810$		0			0
$811$		−	626.134i		−	0.772052i	−0.922488	−	0.386026i	$-0.873848\pi$
$811$		−	626.134i		−	0.772052i	0.922488	−	0.386026i	$-0.126152\pi$
$812$		0			0
$813$	1103.76			1.35764
$814$		0			0
$815$		0			0
$816$		0			0
$817$			351.047i			0.429678i
$818$		0			0
$819$	−2984.28			−3.64381
$820$		0			0
$821$		−	677.557i		−	0.825283i	−0.910894	−	0.412641i	$-0.864606\pi$
$821$		−	677.557i		−	0.825283i	0.910894	−	0.412641i	$-0.135394\pi$
$822$		0			0
$823$		−	55.0463i		−	0.0668850i	−0.999441	−	0.0334425i	$-0.989353\pi$
$823$		−	55.0463i		−	0.0668850i	0.999441	−	0.0334425i	$-0.0106471\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	198.508			0.240034			0.120017	−	0.992772i	$-0.461705\pi$
$827$	198.508			0.240034			0.120017	+	0.992772i	$0.461705\pi$
$828$		0			0
$829$	1191.24			1.43696			0.718481	−	0.695547i	$-0.244838\pi$
$829$	1191.24			1.43696			0.718481	+	0.695547i	$0.244838\pi$
$830$		0			0
$831$		−	401.662i		−	0.483348i
$832$		0			0
$833$	1230.05			1.47665
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	1710.10i		−	2.04313i
$838$		0			0
$839$	−159.788			−0.190450			−0.0952252	−	0.995456i	$-0.530357\pi$
$839$	−159.788			−0.190450			−0.0952252	+	0.995456i	$0.530357\pi$
$840$		0			0
$841$	772.396			0.918426
$842$		0			0
$843$	−454.975			−0.539709
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1325.22	−	882.521i	1.56460	−	1.04194i
$848$		0			0
$849$			7.71398i			0.00908596i
$850$		0			0
$851$	−600.454			−0.705586
$852$		0			0
$853$	−214.501			−0.251467			−0.125733	−	0.992064i	$-0.540128\pi$
$853$	−214.501			−0.251467			−0.125733	+	0.992064i	$0.540128\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	773.417			0.902471			0.451235	−	0.892405i	$-0.350983\pi$
$857$	773.417			0.902471			0.451235	+	0.892405i	$0.350983\pi$
$858$		0			0
$859$	−497.526			−0.579193			−0.289596	−	0.957149i	$-0.593521\pi$
$859$	−497.526			−0.579193			−0.289596	+	0.957149i	$0.593521\pi$
$860$		0			0
$861$	−4860.92			−5.64567
$862$		0			0
$863$		−	476.209i		−	0.551807i	−0.961185	−	0.275903i	$-0.911023\pi$
$863$		−	476.209i		−	0.551807i	0.961185	−	0.275903i	$-0.0889770\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	944.409i		−	1.08928i
$868$		0			0
$869$	834.299	−	252.378i	0.960068	−	0.290423i
$870$		0			0
$871$			1198.07i			1.37551i
$872$		0			0
$873$			1243.51i			1.42441i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	329.549			0.375768			0.187884	−	0.982191i	$-0.439837\pi$
$877$	329.549			0.375768			0.187884	+	0.982191i	$0.439837\pi$
$878$		0			0
$879$		−	1789.05i		−	2.03533i
$880$		0			0
$881$	4.25551			0.00483031			0.00241516	−	0.999997i	$-0.499231\pi$
$881$	4.25551			0.00483031			0.00241516	+	0.999997i	$0.499231\pi$
$882$		0			0
$883$		−	288.392i		−	0.326605i	−0.986576	−	0.163302i	$-0.947785\pi$
$883$		−	288.392i		−	0.326605i	0.986576	−	0.163302i	$-0.0522146\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−437.564			−0.493308			−0.246654	−	0.969104i	$-0.579331\pi$
$887$	−437.564			−0.493308			−0.246654	+	0.969104i	$0.579331\pi$
$888$		0			0
$889$	−2285.39			−2.57074
$890$		0			0
$891$	−62.3319	−	206.054i	−0.0699572	−	0.231262i
$892$		0			0
$893$	−8.04151			−0.00900505
$894$		0			0
$895$		0			0
$896$		0			0
$897$	1058.09			1.17959
$898$		0			0
$899$			440.878i			0.490410i
$900$		0			0
$901$		−	169.002i		−	0.187571i
$902$		0			0
$903$		−	1982.12i		−	2.19504i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1071.31i			1.18115i	0.806982	+	0.590576i	$0.201100\pi$
$907$			1071.31i			1.18115i	−0.806982	+	0.590576i	$0.798900\pi$
$908$		0			0
$909$		−	299.261i		−	0.329220i
$910$		0			0
$911$	−503.288			−0.552457			−0.276228	−	0.961092i	$-0.589085\pi$
$911$	−503.288			−0.552457			−0.276228	+	0.961092i	$0.589085\pi$
$912$		0			0
$913$	260.566	+	861.367i	0.285395	+	0.943447i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1309.19i			1.42768i
$918$		0			0
$919$			1781.27i			1.93827i	0.246534	+	0.969134i	$0.420708\pi$
$919$			1781.27i			1.93827i	−0.246534	+	0.969134i	$0.579292\pi$
$920$		0			0
$921$			254.264i			0.276074i
$922$		0			0
$923$	914.045			0.990297
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	1107.52i		−	1.19473i
$928$		0			0
$929$	879.394			0.946603			0.473301	−	0.880901i	$-0.343062\pi$
$929$	879.394			0.946603			0.473301	+	0.880901i	$0.343062\pi$
$930$		0			0
$931$			1431.82i			1.53794i
$932$		0			0
$933$			980.260i			1.05065i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−446.198			−0.476198			−0.238099	−	0.971241i	$-0.576524\pi$
$937$	−446.198			−0.476198			−0.238099	+	0.971241i	$0.576524\pi$
$938$		0			0
$939$	−110.346			−0.117515
$940$		0			0
$941$			245.275i			0.260653i	0.991471	+	0.130327i	$0.0416026\pi$
$941$			245.275i			0.260653i	−0.991471	+	0.130327i	$0.958397\pi$
$942$		0			0
$943$	1090.14			1.15604
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	236.400i		−	0.249631i	−0.992180	−	0.124815i	$-0.960166\pi$
$947$		−	236.400i		−	0.249631i	0.992180	−	0.124815i	$-0.0398339\pi$
$948$		0			0
$949$	169.162			0.178253
$950$		0			0
$951$	506.056			0.532131
$952$		0			0
$953$	−1049.09			−1.10083			−0.550414	−	0.834892i	$-0.685530\pi$
$953$	−1049.09			−1.10083			−0.550414	+	0.834892i	$0.685530\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	130.555	+	431.583i	0.136421	+	0.450975i
$958$		0			0
$959$			2040.02i			2.12723i
$960$		0			0
$961$	1872.27			1.94825
$962$		0			0
$963$	−2296.72			−2.38497
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−523.118			−0.540970			−0.270485	−	0.962724i	$-0.587184\pi$
$967$	−523.118			−0.540970			−0.270485	+	0.962724i	$0.587184\pi$
$968$		0			0
$969$	−565.523			−0.583615
$970$		0			0
$971$	−1692.15			−1.74268			−0.871341	−	0.490677i	$-0.836749\pi$
$971$	−1692.15			−1.74268			−0.871341	+	0.490677i	$0.836749\pi$
$972$		0			0
$973$		−	825.857i		−	0.848773i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			998.995i			1.02251i	0.859428	+	0.511257i	$0.170820\pi$
$977$			998.995i			1.02251i	−0.859428	+	0.511257i	$0.829180\pi$
$978$		0			0
$979$	436.138	+	1441.77i	0.445493	+	1.47269i
$980$		0			0
$981$			3127.87i			3.18845i
$982$		0			0
$983$			1768.79i			1.79938i	0.436528	+	0.899691i	$0.356208\pi$
$983$			1768.79i			1.79938i	−0.436528	+	0.899691i	$0.643792\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	45.4050			0.0460030
$988$		0			0
$989$			444.525i			0.449469i
$990$		0			0
$991$	854.135			0.861892			0.430946	−	0.902378i	$-0.358180\pi$
$991$	854.135			0.861892			0.430946	+	0.902378i	$0.358180\pi$
$992$		0			0
$993$		−	1750.17i		−	1.76251i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1362.72			−1.36682			−0.683408	−	0.730036i	$-0.739503\pi$
$997$	−1362.72			−1.36682			−0.683408	+	0.730036i	$0.739503\pi$
$998$		0			0
$999$	1320.91			1.32224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.3.e.b.549.15		16
5.2	odd	4		1100.3.f.f.901.7		8
5.3	odd	4		220.3.f.a.21.2	yes	8
5.4	even	2	inner	1100.3.e.b.549.2		16
11.10	odd	2	inner	1100.3.e.b.549.16		16
15.8	even	4		1980.3.b.a.901.8		8
20.3	even	4		880.3.j.b.241.7		8
55.32	even	4		1100.3.f.f.901.8		8
55.43	even	4		220.3.f.a.21.1	✓	8
55.54	odd	2	inner	1100.3.e.b.549.1		16
165.98	odd	4		1980.3.b.a.901.5		8
220.43	odd	4		880.3.j.b.241.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.3.f.a.21.1	✓	8	55.43	even	4
220.3.f.a.21.2	yes	8	5.3	odd	4
880.3.j.b.241.7		8	20.3	even	4
880.3.j.b.241.8		8	220.43	odd	4
1100.3.e.b.549.1		16	55.54	odd	2	inner
1100.3.e.b.549.2		16	5.4	even	2	inner
1100.3.e.b.549.15		16	1.1	even	1	trivial
1100.3.e.b.549.16		16	11.10	odd	2	inner
1100.3.f.f.901.7		8	5.2	odd	4
1100.3.f.f.901.8		8	55.32	even	4
1980.3.b.a.901.5		8	165.98	odd	4
1980.3.b.a.901.8		8	15.8	even	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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1100.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.94892i q^{3} -13.1585 q^{7} -15.4918 q^{9} +(-3.18499 - 10.5288i) q^{11} -14.6396 q^{13} +9.90801 q^{17} +11.5333i q^{19} -65.1206i q^{21} +14.6044i q^{23} -32.1276i q^{27} +8.28275i q^{29} +53.2285 q^{31} +(52.1063 - 15.7623i) q^{33} +41.1146i q^{37} -72.4502i q^{39} -74.6449i q^{41} +30.4377 q^{43} +0.697244i q^{47} +124.147 q^{49} +49.0340i q^{51} -17.0571i q^{53} -57.0773 q^{57} +6.29242 q^{59} -50.6864i q^{61} +203.850 q^{63} -81.8378i q^{67} -72.2760 q^{69} -62.4365 q^{71} -11.5551 q^{73} +(41.9098 + 138.544i) q^{77} +79.2397i q^{79} +19.5705 q^{81} -81.8105 q^{83} -40.9907 q^{87} -136.935 q^{89} +192.636 q^{91} +263.424i q^{93} -80.2686i q^{97} +(49.3414 + 163.111i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 56 q^{9} + 48 q^{11} + 40 q^{31} + 488 q^{49} + 32 q^{59} + 112 q^{69} - 440 q^{71} - 448 q^{81} - 440 q^{89} - 144 q^{91} + 80 q^{99}+O(q^{100}) \) 16 * q - 56 * q^9 + 48 * q^11 + 40 * q^31 + 488 * q^49 + 32 * q^59 + 112 * q^69 - 440 * q^71 - 448 * q^81 - 440 * q^89 - 144 * q^91 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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