LMFDB - Embedded newform 1100.3.e.c.549.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1100,3,Mod(549,1100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [16,0,0,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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} + 174 x^{14} + 10969 x^{12} + 318076 x^{10} + 4442560 x^{8} + 28982576 x^{6} + 77210944 x^{4} + \cdots + 26790976 $ x^16 + 174x^14 + 10969x^12 + 318076x^10 + 4442560x^8 + 28982576x^6 + 77210944x^4 + 82074624*x^2 + 26790976
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			549.12
Root			$-6.54086i$ of defining polynomial
Character	$\chi$	$=$	1100.549
Dual form			1100.3.e.c.549.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.09157i q^{3} +6.85033 q^{7} +4.62533 q^{9} +(-10.3907 + 3.61010i) q^{11} -8.70538 q^{13} -30.8540 q^{17} -13.4701i q^{19} +14.3280i q^{21} +8.15908i q^{23} +28.4984i q^{27} +32.9665i q^{29} -33.0321 q^{31} +(-7.55078 - 21.7330i) q^{33} +58.3473i q^{37} -18.2079i q^{39} -80.8006i q^{41} +38.7283 q^{43} +40.3954i q^{47} -2.07294 q^{49} -64.5333i q^{51} -0.654819i q^{53} +28.1736 q^{57} -33.8941 q^{59} +111.376i q^{61} +31.6850 q^{63} +23.2936i q^{67} -17.0653 q^{69} -70.1769 q^{71} +56.6520 q^{73} +(-71.1799 + 24.7304i) q^{77} -46.2341i q^{79} -17.9784 q^{81} -151.786 q^{83} -68.9518 q^{87} -137.804 q^{89} -59.6348 q^{91} -69.0890i q^{93} +154.530i q^{97} +(-48.0605 + 16.6979i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9} + 6 q^{11} + 28 q^{31} - 28 q^{49} - 256 q^{59} + 352 q^{69} - 68 q^{71} - 256 q^{81} + 292 q^{89} + 228 q^{91} - 16 q^{99}+O(q^{100}) $ 16 * q - 32 * q^9 + 6 * q^11 + 28 * q^31 - 28 * q^49 - 256 * q^59 + 352 * q^69 - 68 * q^71 - 256 * q^81 + 292 * q^89 + 228 * q^91 - 16 * 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
$3$			2.09157i			0.697191i	0.937273	+	0.348595i	$0.113341\pi$
$3$			2.09157i			0.697191i	−0.937273	+	0.348595i	$0.886659\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	6.85033			0.978619			0.489309	−	0.872110i	$-0.337249\pi$
$7$	6.85033			0.978619			0.489309	+	0.872110i	$0.337249\pi$
$8$		0			0
$9$	4.62533			0.513925
$10$		0			0
$11$	−10.3907	+	3.61010i	−0.944611	+	0.328191i
$11$	−10.3907	+	3.61010i	−0.944611	+	0.328191i
$12$		0			0
$13$	−8.70538			−0.669645			−0.334822	−	0.942281i	$-0.608676\pi$
$13$	−8.70538			−0.669645			−0.334822	+	0.942281i	$0.608676\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−30.8540			−1.81494			−0.907470	−	0.420116i	$-0.861989\pi$
$17$	−30.8540			−1.81494			−0.907470	+	0.420116i	$0.861989\pi$
$18$		0			0
$19$		−	13.4701i		−	0.708951i	−0.935065	−	0.354476i	$-0.884659\pi$
$19$		−	13.4701i		−	0.708951i	0.935065	−	0.354476i	$-0.115341\pi$
$20$		0			0
$21$			14.3280i			0.682284i
$22$		0			0
$23$			8.15908i			0.354743i	0.984144	+	0.177371i	$0.0567594\pi$
$23$			8.15908i			0.354743i	−0.984144	+	0.177371i	$0.943241\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			28.4984i			1.05549i
$28$		0			0
$29$			32.9665i			1.13677i	0.822761	+	0.568387i	$0.192433\pi$
$29$			32.9665i			1.13677i	−0.822761	+	0.568387i	$0.807567\pi$
$30$		0			0
$31$	−33.0321			−1.06555			−0.532776	−	0.846256i	$-0.678851\pi$
$31$	−33.0321			−1.06555			−0.532776	+	0.846256i	$0.678851\pi$
$32$		0			0
$33$	−7.55078	−	21.7330i	−0.228811	−	0.658574i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			58.3473i			1.57695i	0.615064	+	0.788477i	$0.289130\pi$
$37$			58.3473i			1.57695i	−0.615064	+	0.788477i	$0.710870\pi$
$38$		0			0
$39$		−	18.2079i		−	0.466870i
$40$		0			0
$41$		−	80.8006i		−	1.97075i	−0.170407	−	0.985374i	$-0.554508\pi$
$41$		−	80.8006i		−	1.97075i	0.170407	−	0.985374i	$-0.445492\pi$
$42$		0			0
$43$	38.7283			0.900659			0.450329	−	0.892862i	$-0.351307\pi$
$43$	38.7283			0.900659			0.450329	+	0.892862i	$0.351307\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			40.3954i			0.859477i	0.902953	+	0.429739i	$0.141394\pi$
$47$			40.3954i			0.859477i	−0.902953	+	0.429739i	$0.858606\pi$
$48$		0			0
$49$	−2.07294			−0.0423049
$50$		0			0
$51$		−	64.5333i		−	1.26536i
$52$		0			0
$53$		−	0.654819i		−	0.0123551i	−0.999981	−	0.00617754i	$-0.998034\pi$
$53$		−	0.654819i		−	0.0123551i	0.999981	−	0.00617754i	$-0.00196638\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	28.1736			0.494274
$58$		0			0
$59$	−33.8941			−0.574477			−0.287238	−	0.957859i	$-0.592737\pi$
$59$	−33.8941			−0.574477			−0.287238	+	0.957859i	$0.592737\pi$
$60$		0			0
$61$			111.376i			1.82584i	0.408141	+	0.912919i	$0.366177\pi$
$61$			111.376i			1.82584i	−0.408141	+	0.912919i	$0.633823\pi$
$62$		0			0
$63$	31.6850			0.502937
$64$		0			0
$65$		0			0
$66$		0			0
$67$			23.2936i			0.347666i	0.984775	+	0.173833i	$0.0556153\pi$
$67$			23.2936i			0.347666i	−0.984775	+	0.173833i	$0.944385\pi$
$68$		0			0
$69$	−17.0653			−0.247323
$70$		0			0
$71$	−70.1769			−0.988407			−0.494203	−	0.869346i	$-0.664540\pi$
$71$	−70.1769			−0.988407			−0.494203	+	0.869346i	$0.664540\pi$
$72$		0			0
$73$	56.6520			0.776054			0.388027	−	0.921648i	$-0.373157\pi$
$73$	56.6520			0.776054			0.388027	+	0.921648i	$0.373157\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−71.1799	+	24.7304i	−0.924415	+	0.321174i
$78$		0			0
$79$		−	46.2341i		−	0.585242i	−0.956229	−	0.292621i	$-0.905473\pi$
$79$		−	46.2341i		−	0.585242i	0.956229	−	0.292621i	$-0.0945274\pi$
$80$		0			0
$81$	−17.9784			−0.221956
$82$		0			0
$83$	−151.786			−1.82875			−0.914375	−	0.404867i	$-0.867318\pi$
$83$	−151.786			−1.82875			−0.914375	+	0.404867i	$0.867318\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−68.9518			−0.792549
$88$		0			0
$89$	−137.804			−1.54836			−0.774178	−	0.632968i	$-0.781837\pi$
$89$	−137.804			−1.54836			−0.774178	+	0.632968i	$0.781837\pi$
$90$		0			0
$91$	−59.6348			−0.655327
$92$		0			0
$93$		−	69.0890i		−	0.742893i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			154.530i			1.59309i	0.604580	+	0.796545i	$0.293341\pi$
$97$			154.530i			1.59309i	−0.604580	+	0.796545i	$0.706659\pi$
$98$		0			0
$99$	−48.0605	+	16.6979i	−0.485460	+	0.168665i
$100$		0			0
$101$		−	43.1844i		−	0.427568i	−0.976881	−	0.213784i	$-0.931421\pi$
$101$		−	43.1844i		−	0.427568i	0.976881	−	0.213784i	$-0.0685789\pi$
$102$		0			0
$103$		−	132.608i		−	1.28746i	−0.765255	−	0.643728i	$-0.777387\pi$
$103$		−	132.608i		−	1.28746i	0.765255	−	0.643728i	$-0.222613\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	75.4144			0.704807			0.352404	−	0.935848i	$-0.385364\pi$
$107$	75.4144			0.704807			0.352404	+	0.935848i	$0.385364\pi$
$108$		0			0
$109$		−	145.652i		−	1.33626i	−0.744045	−	0.668129i	$-0.767095\pi$
$109$		−	145.652i		−	1.33626i	0.744045	−	0.668129i	$-0.232905\pi$
$110$		0			0
$111$	−122.038			−1.09944
$112$		0			0
$113$			5.20398i			0.0460529i	0.999735	+	0.0230265i	$0.00733020\pi$
$113$			5.20398i			0.0460529i	−0.999735	+	0.0230265i	$0.992670\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−40.2652			−0.344147
$118$		0			0
$119$	−211.360			−1.77614
$120$		0			0
$121$	94.9344	−	75.0231i	0.784582	−	0.620025i
$122$		0			0
$123$	169.000			1.37399
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−145.449			−1.14527			−0.572633	−	0.819812i	$-0.694078\pi$
$127$	−145.449			−1.14527			−0.572633	+	0.819812i	$0.694078\pi$
$128$		0			0
$129$			81.0031i			0.627931i
$130$		0			0
$131$			67.5119i			0.515358i	0.966231	+	0.257679i	$0.0829577\pi$
$131$			67.5119i			0.515358i	−0.966231	+	0.257679i	$0.917042\pi$
$132$		0			0
$133$		−	92.2745i		−	0.693793i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			146.229i			1.06737i	0.845685	+	0.533683i	$0.179192\pi$
$137$			146.229i			1.06737i	−0.845685	+	0.533683i	$0.820808\pi$
$138$		0			0
$139$		−	15.4107i		−	0.110868i	−0.998462	−	0.0554342i	$-0.982346\pi$
$139$		−	15.4107i		−	0.110868i	0.998462	−	0.0554342i	$-0.0176543\pi$
$140$		0			0
$141$	−84.4900			−0.599220
$142$		0			0
$143$	90.4552	−	31.4273i	0.632554	−	0.219771i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	4.33570i		−	0.0294946i
$148$		0			0
$149$		−	152.533i		−	1.02371i	−0.859071	−	0.511856i	$-0.828958\pi$
$149$		−	152.533i		−	1.02371i	0.859071	−	0.511856i	$-0.171042\pi$
$150$		0			0
$151$			40.4135i			0.267639i	0.991006	+	0.133820i	$0.0427243\pi$
$151$			40.4135i			0.267639i	−0.991006	+	0.133820i	$0.957276\pi$
$152$		0			0
$153$	−142.710			−0.932744
$154$		0			0
$155$		0			0
$156$		0			0
$157$			85.6731i			0.545688i	0.962058	+	0.272844i	$0.0879643\pi$
$157$			85.6731i			0.545688i	−0.962058	+	0.272844i	$0.912036\pi$
$158$		0			0
$159$	1.36960			0.00861384
$160$		0			0
$161$			55.8924i			0.347158i
$162$		0			0
$163$		−	6.20830i		−	0.0380877i	−0.999819	−	0.0190439i	$-0.993938\pi$
$163$		−	6.20830i		−	0.0380877i	0.999819	−	0.0190439i	$-0.00606222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	15.9363			0.0954272			0.0477136	−	0.998861i	$-0.484807\pi$
$167$	15.9363			0.0954272			0.0477136	+	0.998861i	$0.484807\pi$
$168$		0			0
$169$	−93.2164			−0.551576
$170$		0			0
$171$		−	62.3035i		−	0.364348i
$172$		0			0
$173$	104.481			0.603937			0.301969	−	0.953318i	$-0.402356\pi$
$173$	104.481			0.603937			0.301969	+	0.953318i	$0.402356\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	70.8920i		−	0.400520i
$178$		0			0
$179$	221.475			1.23729			0.618645	−	0.785670i	$-0.287682\pi$
$179$	221.475			1.23729			0.618645	+	0.785670i	$0.287682\pi$
$180$		0			0
$181$	122.870			0.678841			0.339420	−	0.940635i	$-0.389769\pi$
$181$	122.870			0.678841			0.339420	+	0.940635i	$0.389769\pi$
$182$		0			0
$183$	−232.951			−1.27296
$184$		0			0
$185$		0			0
$186$		0			0
$187$	320.595	−	111.386i	1.71441	−	0.595647i
$188$		0			0
$189$			195.223i			1.03293i
$190$		0			0
$191$	−95.4979			−0.499989			−0.249994	−	0.968247i	$-0.580429\pi$
$191$	−95.4979			−0.499989			−0.249994	+	0.968247i	$0.580429\pi$
$192$		0			0
$193$	182.564			0.945929			0.472965	−	0.881081i	$-0.343184\pi$
$193$	182.564			0.945929			0.472965	+	0.881081i	$0.343184\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	79.1315			0.401683			0.200841	−	0.979624i	$-0.435632\pi$
$197$	79.1315			0.401683			0.200841	+	0.979624i	$0.435632\pi$
$198$		0			0
$199$	−121.450			−0.610300			−0.305150	−	0.952304i	$-0.598707\pi$
$199$	−121.450			−0.610300			−0.305150	+	0.952304i	$0.598707\pi$
$200$		0			0
$201$	−48.7203			−0.242389
$202$		0			0
$203$			225.831i			1.11247i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			37.7384i			0.182311i
$208$		0			0
$209$	48.6283	+	139.964i	0.232671	+	0.669684i
$210$		0			0
$211$		−	394.485i		−	1.86960i	−0.355177	−	0.934799i	$-0.615579\pi$
$211$		−	394.485i		−	1.86960i	0.355177	−	0.934799i	$-0.384421\pi$
$212$		0			0
$213$		−	146.780i		−	0.689108i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−226.281			−1.04277
$218$		0			0
$219$			118.492i			0.541058i
$220$		0			0
$221$	268.596			1.21537
$222$		0			0
$223$			287.156i			1.28770i	0.765153	+	0.643849i	$0.222663\pi$
$223$			287.156i			1.28770i	−0.765153	+	0.643849i	$0.777337\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−223.474			−0.984467			−0.492234	−	0.870463i	$-0.663819\pi$
$227$	−223.474			−0.984467			−0.492234	+	0.870463i	$0.663819\pi$
$228$		0			0
$229$	195.387			0.853217			0.426608	−	0.904437i	$-0.359708\pi$
$229$	195.387			0.853217			0.426608	+	0.904437i	$0.359708\pi$
$230$		0			0
$231$	−51.7254	−	148.878i	−0.223919	−	0.644493i
$232$		0			0
$233$	43.4960			0.186678			0.0933390	−	0.995634i	$-0.470246\pi$
$233$	43.4960			0.186678			0.0933390	+	0.995634i	$0.470246\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	96.7019			0.408025
$238$		0			0
$239$		−	210.701i		−	0.881593i	−0.897607	−	0.440797i	$-0.854696\pi$
$239$		−	210.701i		−	0.881593i	0.897607	−	0.440797i	$-0.145304\pi$
$240$		0			0
$241$			171.506i			0.711641i	0.934554	+	0.355821i	$0.115799\pi$
$241$			171.506i			0.711641i	−0.934554	+	0.355821i	$0.884201\pi$
$242$		0			0
$243$			218.882i			0.900749i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			117.262i			0.474746i
$248$		0			0
$249$		−	317.472i		−	1.27499i
$250$		0			0
$251$	−188.857			−0.752418			−0.376209	−	0.926535i	$-0.622773\pi$
$251$	−188.857			−0.752418			−0.376209	+	0.926535i	$0.622773\pi$
$252$		0			0
$253$	−29.4551	−	84.7788i	−0.116423	−	0.335094i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			225.478i			0.877347i	0.898647	+	0.438673i	$0.144552\pi$
$257$			225.478i			0.877347i	−0.898647	+	0.438673i	$0.855448\pi$
$258$		0			0
$259$			399.699i			1.54324i
$260$		0			0
$261$			152.481i			0.584217i
$262$		0			0
$263$	−236.256			−0.898313			−0.449156	−	0.893453i	$-0.648275\pi$
$263$	−236.256			−0.898313			−0.449156	+	0.893453i	$0.648275\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	288.226i		−	1.07950i
$268$		0			0
$269$	350.154			1.30169			0.650844	−	0.759211i	$-0.274415\pi$
$269$	350.154			1.30169			0.650844	+	0.759211i	$0.274415\pi$
$270$		0			0
$271$			456.267i			1.68364i	0.539757	+	0.841821i	$0.318516\pi$
$271$			456.267i			1.68364i	−0.539757	+	0.841821i	$0.681484\pi$
$272$		0			0
$273$		−	124.730i		−	0.456888i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	103.023			0.371925			0.185963	−	0.982557i	$-0.440460\pi$
$277$	103.023			0.371925			0.185963	+	0.982557i	$0.440460\pi$
$278$		0			0
$279$	−152.784			−0.547614
$280$		0			0
$281$			205.264i			0.730476i	0.930914	+	0.365238i	$0.119012\pi$
$281$			205.264i			0.730476i	−0.930914	+	0.365238i	$0.880988\pi$
$282$		0			0
$283$	−255.134			−0.901535			−0.450767	−	0.892641i	$-0.648850\pi$
$283$	−255.134			−0.901535			−0.450767	+	0.892641i	$0.648850\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	553.511i		−	1.92861i
$288$		0			0
$289$	662.969			2.29401
$290$		0			0
$291$	−323.210			−1.11069
$292$		0			0
$293$	64.7857			0.221112			0.110556	−	0.993870i	$-0.464737\pi$
$293$	64.7857			0.221112			0.110556	+	0.993870i	$0.464737\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−102.882	−	296.119i	−0.346403	−	0.997032i
$298$		0			0
$299$		−	71.0279i		−	0.237552i
$300$		0			0
$301$	265.302			0.881402
$302$		0			0
$303$	90.3232			0.298096
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−277.572			−0.904145			−0.452072	−	0.891981i	$-0.649315\pi$
$307$	−277.572			−0.904145			−0.452072	+	0.891981i	$0.649315\pi$
$308$		0			0
$309$	277.359			0.897602
$310$		0			0
$311$	345.088			1.10961			0.554805	−	0.831981i	$-0.312793\pi$
$311$	345.088			1.10961			0.554805	+	0.831981i	$0.312793\pi$
$312$		0			0
$313$		−	39.5116i		−	0.126235i	−0.998006	−	0.0631176i	$-0.979896\pi$
$313$		−	39.5116i		−	0.126235i	0.998006	−	0.0631176i	$-0.0201043\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			32.7689i			0.103372i	0.998663	+	0.0516860i	$0.0164595\pi$
$317$			32.7689i			0.103372i	−0.998663	+	0.0516860i	$0.983540\pi$
$318$		0			0
$319$	−119.012	−	342.546i	−0.373079	−	1.07381i
$320$		0			0
$321$			157.735i			0.491385i
$322$		0			0
$323$			415.606i			1.28670i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	304.642			0.931627
$328$		0			0
$329$			276.722i			0.841101i
$330$		0			0
$331$	475.800			1.43746			0.718731	−	0.695288i	$-0.244723\pi$
$331$	475.800			1.43746			0.718731	+	0.695288i	$0.244723\pi$
$332$		0			0
$333$			269.875i			0.810437i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	422.230			1.25291			0.626455	−	0.779458i	$-0.284505\pi$
$337$	422.230			1.25291			0.626455	+	0.779458i	$0.284505\pi$
$338$		0			0
$339$	−10.8845			−0.0321077
$340$		0			0
$341$	343.228	−	119.249i	1.00653	−	0.349704i
$342$		0			0
$343$	−349.867			−1.02002
$344$		0			0
$345$		0			0
$346$		0			0
$347$	334.106			0.962842			0.481421	−	0.876490i	$-0.340121\pi$
$347$	334.106			0.962842			0.481421	+	0.876490i	$0.340121\pi$
$348$		0			0
$349$			628.513i			1.80090i	0.434964	+	0.900448i	$0.356761\pi$
$349$			628.513i			1.80090i	−0.434964	+	0.900448i	$0.643239\pi$
$350$		0			0
$351$		−	248.089i		−	0.706806i
$352$		0			0
$353$		−	490.528i		−	1.38960i	−0.719204	−	0.694799i	$-0.755493\pi$
$353$		−	490.528i		−	1.38960i	0.719204	−	0.694799i	$-0.244507\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	442.075i		−	1.23830i
$358$		0			0
$359$			305.050i			0.849723i	0.905258	+	0.424861i	$0.139677\pi$
$359$			305.050i			0.849723i	−0.905258	+	0.424861i	$0.860323\pi$
$360$		0			0
$361$	179.557			0.497388
$362$		0			0
$363$	156.916	+	198.562i	0.432276	+	0.547003i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	315.354i		−	0.859274i	−0.903002	−	0.429637i	$-0.858642\pi$
$367$		−	315.354i		−	0.859274i	0.903002	−	0.429637i	$-0.141358\pi$
$368$		0			0
$369$		−	373.729i		−	1.01282i
$370$		0			0
$371$		−	4.48573i		−	0.0120909i
$372$		0			0
$373$	498.605			1.33674			0.668372	−	0.743827i	$-0.266991\pi$
$373$	498.605			1.33674			0.668372	+	0.743827i	$0.266991\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	286.986i		−	0.761235i
$378$		0			0
$379$	267.849			0.706725			0.353362	−	0.935487i	$-0.385038\pi$
$379$	267.849			0.706725			0.353362	+	0.935487i	$0.385038\pi$
$380$		0			0
$381$		−	304.217i		−	0.798469i
$382$		0			0
$383$			263.703i			0.688519i	0.938874	+	0.344260i	$0.111870\pi$
$383$			263.703i			0.688519i	−0.938874	+	0.344260i	$0.888130\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	179.131			0.462871
$388$		0			0
$389$	−246.626			−0.634001			−0.317001	−	0.948425i	$-0.602676\pi$
$389$	−246.626			−0.634001			−0.317001	+	0.948425i	$0.602676\pi$
$390$		0			0
$391$		−	251.740i		−	0.643837i
$392$		0			0
$393$	−141.206			−0.359303
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	469.581i		−	1.18282i	−0.806369	−	0.591412i	$-0.798571\pi$
$397$		−	469.581i		−	1.18282i	0.806369	−	0.591412i	$-0.201429\pi$
$398$		0			0
$399$	192.999			0.483706
$400$		0			0
$401$	8.43435			0.0210333			0.0105167	−	0.999945i	$-0.496652\pi$
$401$	8.43435			0.0210333			0.0105167	+	0.999945i	$0.496652\pi$
$402$		0			0
$403$	287.557			0.713541
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−210.640	−	606.271i	−0.517542	−	1.48961i
$408$		0			0
$409$			121.636i			0.297398i	0.988882	+	0.148699i	$0.0475086\pi$
$409$			121.636i			0.297398i	−0.988882	+	0.148699i	$0.952491\pi$
$410$		0			0
$411$	−305.848			−0.744157
$412$		0			0
$413$	−232.186			−0.562194
$414$		0			0
$415$		0			0
$416$		0			0
$417$	32.2326			0.0772964
$418$		0			0
$419$	−434.497			−1.03698			−0.518492	−	0.855082i	$-0.673507\pi$
$419$	−434.497			−1.03698			−0.518492	+	0.855082i	$0.673507\pi$
$420$		0			0
$421$	458.288			1.08857			0.544285	−	0.838900i	$-0.316801\pi$
$421$	458.288			1.08857			0.544285	+	0.838900i	$0.316801\pi$
$422$		0			0
$423$			186.842i			0.441707i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			762.963i			1.78680i
$428$		0			0
$429$	65.7324	+	189.194i	0.153222	+	0.441011i
$430$		0			0
$431$			710.617i			1.64876i	0.566034	+	0.824382i	$0.308477\pi$
$431$			710.617i			1.64876i	−0.566034	+	0.824382i	$0.691523\pi$
$432$		0			0
$433$		−	602.203i		−	1.39077i	−0.718638	−	0.695384i	$-0.755234\pi$
$433$		−	602.203i		−	1.39077i	0.718638	−	0.695384i	$-0.244766\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	109.903			0.251495
$438$		0			0
$439$		−	262.319i		−	0.597537i	−0.954326	−	0.298769i	$-0.903424\pi$
$439$		−	262.319i		−	0.597537i	0.954326	−	0.298769i	$-0.0965759\pi$
$440$		0			0
$441$	−9.58802			−0.0217415
$442$		0			0
$443$			559.726i			1.26349i	0.775176	+	0.631745i	$0.217661\pi$
$443$			559.726i			1.26349i	−0.775176	+	0.631745i	$0.782339\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	319.034			0.713722
$448$		0			0
$449$	−391.702			−0.872388			−0.436194	−	0.899853i	$-0.643674\pi$
$449$	−391.702			−0.872388			−0.436194	+	0.899853i	$0.643674\pi$
$450$		0			0
$451$	291.698	+	839.577i	0.646781	+	1.86159i
$452$		0			0
$453$	−84.5278			−0.186596
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−123.702			−0.270682			−0.135341	−	0.990799i	$-0.543213\pi$
$457$	−123.702			−0.270682			−0.135341	+	0.990799i	$0.543213\pi$
$458$		0			0
$459$		−	879.288i		−	1.91566i
$460$		0			0
$461$		−	188.305i		−	0.408470i	−0.978922	−	0.204235i	$-0.934529\pi$
$461$		−	188.305i		−	0.408470i	0.978922	−	0.204235i	$-0.0654707\pi$
$462$		0			0
$463$		−	116.141i		−	0.250844i	−0.992103	−	0.125422i	$-0.959972\pi$
$463$		−	116.141i		−	0.250844i	0.992103	−	0.125422i	$-0.0400285\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			54.3031i			0.116281i	0.998308	+	0.0581403i	$0.0185171\pi$
$467$			54.3031i			0.116281i	−0.998308	+	0.0581403i	$0.981483\pi$
$468$		0			0
$469$			159.569i			0.340233i
$470$		0			0
$471$	−179.191			−0.380449
$472$		0			0
$473$	−402.416	+	139.813i	−0.850773	+	0.295588i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	3.02875i		−	0.00634958i
$478$		0			0
$479$		−	103.374i		−	0.215812i	−0.994161	−	0.107906i	$-0.965586\pi$
$479$		−	103.374i		−	0.215812i	0.994161	−	0.107906i	$-0.0344145\pi$
$480$		0			0
$481$		−	507.936i		−	1.05600i
$482$		0			0
$483$	−116.903			−0.242035
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	587.899i		−	1.20719i	−0.797293	−	0.603593i	$-0.793735\pi$
$487$		−	587.899i		−	1.20719i	0.797293	−	0.603593i	$-0.206265\pi$
$488$		0			0
$489$	12.9851			0.0265544
$490$		0			0
$491$		−	586.225i		−	1.19394i	−0.802263	−	0.596970i	$-0.796371\pi$
$491$		−	586.225i		−	1.19394i	0.802263	−	0.596970i	$-0.203629\pi$
$492$		0			0
$493$		−	1017.15i		−	2.06318i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−480.735			−0.967274
$498$		0			0
$499$	−280.363			−0.561850			−0.280925	−	0.959730i	$-0.590641\pi$
$499$	−280.363			−0.561850			−0.280925	+	0.959730i	$0.590641\pi$
$500$		0			0
$501$			33.3320i			0.0665310i
$502$		0			0
$503$	687.382			1.36656			0.683282	−	0.730154i	$-0.260552\pi$
$503$	687.382			1.36656			0.683282	+	0.730154i	$0.260552\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	194.969i		−	0.384554i
$508$		0			0
$509$	780.381			1.53317			0.766583	−	0.642146i	$-0.221956\pi$
$509$	780.381			1.53317			0.766583	+	0.642146i	$0.221956\pi$
$510$		0			0
$511$	388.085			0.759462
$512$		0			0
$513$	383.875			0.748294
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−145.831	−	419.738i	−0.282073	−	0.811872i
$518$		0			0
$519$			218.530i			0.421059i
$520$		0			0
$521$	−306.042			−0.587413			−0.293707	−	0.955896i	$-0.594889\pi$
$521$	−306.042			−0.587413			−0.293707	+	0.955896i	$0.594889\pi$
$522$		0			0
$523$	−482.421			−0.922411			−0.461206	−	0.887293i	$-0.652583\pi$
$523$	−482.421			−0.922411			−0.461206	+	0.887293i	$0.652583\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1019.17			1.93391
$528$		0			0
$529$	462.429			0.874158
$530$		0			0
$531$	−156.771			−0.295238
$532$		0			0
$533$			703.400i			1.31970i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			463.231i			0.862628i
$538$		0			0
$539$	21.5393	−	7.48351i	0.0399617	−	0.0138841i
$540$		0			0
$541$			630.921i			1.16621i	0.812396	+	0.583106i	$0.198163\pi$
$541$			630.921i			1.16621i	−0.812396	+	0.583106i	$0.801837\pi$
$542$		0			0
$543$			256.992i			0.473281i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−519.754			−0.950190			−0.475095	−	0.879934i	$-0.657586\pi$
$547$	−519.754			−0.950190			−0.475095	+	0.879934i	$0.657586\pi$
$548$		0			0
$549$			515.151i			0.938344i
$550$		0			0
$551$	444.061			0.805918
$552$		0			0
$553$		−	316.719i		−	0.572729i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−92.6779			−0.166388			−0.0831938	−	0.996533i	$-0.526512\pi$
$557$	−92.6779			−0.166388			−0.0831938	+	0.996533i	$0.526512\pi$
$558$		0			0
$559$	−337.145			−0.603121
$560$		0			0
$561$	232.972	+	670.548i	0.415279	+	1.19527i
$562$		0			0
$563$	−312.203			−0.554534			−0.277267	−	0.960793i	$-0.589429\pi$
$563$	−312.203			−0.554534			−0.277267	+	0.960793i	$0.589429\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−123.158			−0.217210
$568$		0			0
$569$			1.95145i			0.00342961i	0.999999	+	0.00171481i	$0.000545840\pi$
$569$			1.95145i			0.00342961i	−0.999999	+	0.00171481i	$0.999454\pi$
$570$		0			0
$571$		−	19.4418i		−	0.0340486i	−0.999855	−	0.0170243i	$-0.994581\pi$
$571$		−	19.4418i		−	0.0340486i	0.999855	−	0.0170243i	$-0.00541927\pi$
$572$		0			0
$573$		−	199.741i		−	0.348588i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			1090.83i			1.89052i	0.326325	+	0.945258i	$0.394190\pi$
$577$			1090.83i			1.89052i	−0.326325	+	0.945258i	$0.605810\pi$
$578$		0			0
$579$			381.847i			0.659493i
$580$		0			0
$581$	−1039.79			−1.78965
$582$		0			0
$583$	2.36396	+	6.80404i	0.00405482	+	0.0116707i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			389.066i			0.662805i	0.943490	+	0.331402i	$0.107522\pi$
$587$			389.066i			0.662805i	−0.943490	+	0.331402i	$0.892478\pi$
$588$		0			0
$589$			444.945i			0.755425i
$590$		0			0
$591$			165.509i			0.280050i
$592$		0			0
$593$	189.048			0.318799			0.159400	−	0.987214i	$-0.449044\pi$
$593$	189.048			0.318799			0.159400	+	0.987214i	$0.449044\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	254.021i		−	0.425496i
$598$		0			0
$599$	−613.899			−1.02487			−0.512437	−	0.858725i	$-0.671257\pi$
$599$	−613.899			−1.02487			−0.512437	+	0.858725i	$0.671257\pi$
$600$		0			0
$601$		−	188.463i		−	0.313583i	−0.987632	−	0.156791i	$-0.949885\pi$
$601$		−	188.463i		−	0.313583i	0.987632	−	0.156791i	$-0.0501150\pi$
$602$		0			0
$603$			107.741i			0.178674i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−816.608			−1.34532			−0.672659	−	0.739953i	$-0.734848\pi$
$607$	−816.608			−1.34532			−0.672659	+	0.739953i	$0.734848\pi$
$608$		0			0
$609$	−472.342			−0.775603
$610$		0			0
$611$		−	351.658i		−	0.575544i
$612$		0			0
$613$	−539.684			−0.880398			−0.440199	−	0.897900i	$-0.645092\pi$
$613$	−539.684			−0.880398			−0.440199	+	0.897900i	$0.645092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			54.2911i			0.0879921i	0.999032	+	0.0439961i	$0.0140089\pi$
$617$			54.2911i			0.0879921i	−0.999032	+	0.0439961i	$0.985991\pi$
$618$		0			0
$619$	1012.42			1.63557			0.817786	−	0.575522i	$-0.195201\pi$
$619$	1012.42			1.63557			0.817786	+	0.575522i	$0.195201\pi$
$620$		0			0
$621$	−232.520			−0.374429
$622$		0			0
$623$	−944.001			−1.51525
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−292.745	+	101.710i	−0.466897	+	0.162216i
$628$		0			0
$629$		−	1800.25i		−	2.86208i
$630$		0			0
$631$	−810.399			−1.28431			−0.642154	−	0.766576i	$-0.721959\pi$
$631$	−810.399			−1.28431			−0.642154	+	0.766576i	$0.721959\pi$
$632$		0			0
$633$	825.094			1.30347
$634$		0			0
$635$		0			0
$636$		0			0
$637$	18.0457			0.0283292
$638$		0			0
$639$	−324.591			−0.507967
$640$		0			0
$641$	−276.256			−0.430977			−0.215488	−	0.976506i	$-0.569134\pi$
$641$	−276.256			−0.430977			−0.215488	+	0.976506i	$0.569134\pi$
$642$		0			0
$643$		−	666.135i		−	1.03598i	−0.855387	−	0.517990i	$-0.826680\pi$
$643$		−	666.135i		−	1.03598i	0.855387	−	0.517990i	$-0.173320\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1196.84i			1.84983i	0.380170	+	0.924916i	$0.375865\pi$
$647$			1196.84i			1.84983i	−0.380170	+	0.924916i	$0.624135\pi$
$648$		0			0
$649$	352.185	−	122.361i	0.542657	−	0.188538i
$650$		0			0
$651$		−	473.283i		−	0.727009i
$652$		0			0
$653$		−	588.543i		−	0.901291i	−0.892703	−	0.450645i	$-0.851194\pi$
$653$		−	588.543i		−	0.901291i	0.892703	−	0.450645i	$-0.148806\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	262.034			0.398834
$658$		0			0
$659$		−	578.998i		−	0.878601i	−0.898340	−	0.439301i	$-0.855226\pi$
$659$		−	578.998i		−	0.878601i	0.898340	−	0.439301i	$-0.144774\pi$
$660$		0			0
$661$	404.959			0.612646			0.306323	−	0.951928i	$-0.400901\pi$
$661$	404.959			0.612646			0.306323	+	0.951928i	$0.400901\pi$
$662$		0			0
$663$			561.787i			0.847341i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−268.976			−0.403263
$668$		0			0
$669$	−600.608			−0.897771
$670$		0			0
$671$	−402.079	−	1157.28i	−0.599223	−	1.72471i
$672$		0			0
$673$	756.306			1.12378			0.561891	−	0.827211i	$-0.310074\pi$
$673$	756.306			1.12378			0.561891	+	0.827211i	$0.310074\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	240.947			0.355904			0.177952	−	0.984039i	$-0.443053\pi$
$677$	240.947			0.355904			0.177952	+	0.984039i	$0.443053\pi$
$678$		0			0
$679$			1058.58i			1.55903i
$680$		0			0
$681$		−	467.412i		−	0.686361i
$682$		0			0
$683$			1245.15i			1.82307i	0.411226	+	0.911533i	$0.365101\pi$
$683$			1245.15i			1.82307i	−0.411226	+	0.911533i	$0.634899\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			408.665i			0.594855i
$688$		0			0
$689$			5.70045i			0.00827351i
$690$		0			0
$691$	−604.155			−0.874320			−0.437160	−	0.899384i	$-0.644016\pi$
$691$	−604.155			−0.874320			−0.437160	+	0.899384i	$0.644016\pi$
$692$		0			0
$693$	−329.230	+	114.386i	−0.475080	+	0.165059i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			2493.02i			3.57679i
$698$		0			0
$699$			90.9749i			0.130150i
$700$		0			0
$701$			611.107i			0.871764i	0.900004	+	0.435882i	$0.143564\pi$
$701$			611.107i			0.871764i	−0.900004	+	0.435882i	$0.856436\pi$
$702$		0			0
$703$	785.943			1.11798
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	295.827i		−	0.418426i
$708$		0			0
$709$	438.658			0.618699			0.309350	−	0.950948i	$-0.399889\pi$
$709$	438.658			0.618699			0.309350	+	0.950948i	$0.399889\pi$
$710$		0			0
$711$		−	213.848i		−	0.300770i
$712$		0			0
$713$		−	269.512i		−	0.377997i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	440.696			0.614639
$718$		0			0
$719$	−1129.43			−1.57084			−0.785419	−	0.618964i	$-0.787552\pi$
$719$	−1129.43			−1.57084			−0.785419	+	0.618964i	$0.787552\pi$
$720$		0			0
$721$		−	908.408i		−	1.25993i
$722$		0			0
$723$	−358.716			−0.496150
$724$		0			0
$725$		0			0
$726$		0			0
$727$			27.2412i			0.0374707i	0.999824	+	0.0187354i	$0.00596400\pi$
$727$			27.2412i			0.0374707i	−0.999824	+	0.0187354i	$0.994036\pi$
$728$		0			0
$729$	−619.613			−0.849950
$730$		0			0
$731$	−1194.92			−1.63464
$732$		0			0
$733$	757.221			1.03304			0.516522	−	0.856274i	$-0.327226\pi$
$733$	757.221			1.03304			0.516522	+	0.856274i	$0.327226\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−84.0922	−	242.038i	−0.114101	−	0.328409i
$738$		0			0
$739$			984.769i			1.33257i	0.745697	+	0.666285i	$0.232117\pi$
$739$			984.769i			1.33257i	−0.745697	+	0.666285i	$0.767883\pi$
$740$		0			0
$741$	−245.262			−0.330988
$742$		0			0
$743$	1166.00			1.56931			0.784654	−	0.619935i	$-0.212841\pi$
$743$	1166.00			1.56931			0.784654	+	0.619935i	$0.212841\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−702.061			−0.939841
$748$		0			0
$749$	516.614			0.689738
$750$		0			0
$751$	966.474			1.28692			0.643458	−	0.765481i	$-0.277499\pi$
$751$	966.474			1.28692			0.643458	+	0.765481i	$0.277499\pi$
$752$		0			0
$753$		−	395.008i		−	0.524579i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			1191.59i			1.57409i	0.616894	+	0.787046i	$0.288391\pi$
$757$			1191.59i			1.57409i	−0.616894	+	0.787046i	$0.711609\pi$
$758$		0			0
$759$	177.321	−	61.6074i	0.233624	−	0.0811692i
$760$		0			0
$761$		−	251.478i		−	0.330457i	−0.986255	−	0.165229i	$-0.947164\pi$
$761$		−	251.478i		−	0.330457i	0.986255	−	0.165229i	$-0.0528362\pi$
$762$		0			0
$763$		−	997.766i		−	1.30769i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	295.061			0.384695
$768$		0			0
$769$		−	510.406i		−	0.663727i	−0.943327	−	0.331864i	$-0.892323\pi$
$769$		−	510.406i		−	0.663727i	0.943327	−	0.331864i	$-0.107677\pi$
$770$		0			0
$771$	−471.604			−0.611678
$772$		0			0
$773$			838.431i			1.08465i	0.840170	+	0.542323i	$0.182455\pi$
$773$			838.431i			1.08465i	−0.840170	+	0.542323i	$0.817545\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−835.998			−1.07593
$778$		0			0
$779$	−1088.39			−1.39716
$780$		0			0
$781$	729.189	−	253.345i	0.933661	−	0.324386i
$782$		0			0
$783$	−939.490			−1.19986
$784$		0			0
$785$		0			0
$786$		0			0
$787$	86.4925			0.109902			0.0549508	−	0.998489i	$-0.482500\pi$
$787$	86.4925			0.109902			0.0549508	+	0.998489i	$0.482500\pi$
$788$		0			0
$789$		−	494.147i		−	0.626295i
$790$		0			0
$791$			35.6490i			0.0450683i
$792$		0			0
$793$		−	969.571i		−	1.22266i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	154.435i		−	0.193770i	−0.995296	−	0.0968849i	$-0.969112\pi$
$797$		−	154.435i		−	0.193770i	0.995296	−	0.0968849i	$-0.0308879\pi$
$798$		0			0
$799$		−	1246.36i		−	1.55990i
$800$		0			0
$801$	−637.387			−0.795739
$802$		0			0
$803$	−588.655	+	204.519i	−0.733070	+	0.254694i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			732.373i			0.907525i
$808$		0			0
$809$			1219.75i			1.50772i	0.657033	+	0.753862i	$0.271811\pi$
$809$			1219.75i			1.50772i	−0.657033	+	0.753862i	$0.728189\pi$
$810$		0			0
$811$			1277.26i			1.57491i	0.616369	+	0.787457i	$0.288603\pi$
$811$			1277.26i			1.57491i	−0.616369	+	0.787457i	$0.711397\pi$
$812$		0			0
$813$	−954.315			−1.17382
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	521.674i		−	0.638523i
$818$		0			0
$819$	−275.830			−0.336789
$820$		0			0
$821$			1435.21i			1.74812i	0.485817	+	0.874061i	$0.338522\pi$
$821$			1435.21i			1.74812i	−0.485817	+	0.874061i	$0.661478\pi$
$822$		0			0
$823$		−	1039.17i		−	1.26266i	−0.775516	−	0.631328i	$-0.782510\pi$
$823$		−	1039.17i		−	1.26266i	0.775516	−	0.631328i	$-0.217490\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	376.097			0.454773			0.227386	−	0.973805i	$-0.426982\pi$
$827$	376.097			0.454773			0.227386	+	0.973805i	$0.426982\pi$
$828$		0			0
$829$	−87.4548			−0.105494			−0.0527472	−	0.998608i	$-0.516798\pi$
$829$	−87.4548			−0.105494			−0.0527472	+	0.998608i	$0.516798\pi$
$830$		0			0
$831$			215.481i			0.259303i
$832$		0			0
$833$	63.9585			0.0767809
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	941.361i		−	1.12468i
$838$		0			0
$839$	93.2621			0.111159			0.0555793	−	0.998454i	$-0.482299\pi$
$839$	93.2621			0.111159			0.0555793	+	0.998454i	$0.482299\pi$
$840$		0			0
$841$	−245.788			−0.292257
$842$		0			0
$843$	−429.324			−0.509281
$844$		0			0
$845$		0			0
$846$		0			0
$847$	650.332	−	513.933i	0.767807	−	0.606769i
$848$		0			0
$849$		−	533.632i		−	0.628542i
$850$		0			0
$851$	−476.060			−0.559413
$852$		0			0
$853$	−425.799			−0.499178			−0.249589	−	0.968352i	$-0.580295\pi$
$853$	−425.799			−0.499178			−0.249589	+	0.968352i	$0.580295\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−869.217			−1.01426			−0.507128	−	0.861871i	$-0.669293\pi$
$857$	−869.217			−1.01426			−0.507128	+	0.861871i	$0.669293\pi$
$858$		0			0
$859$	233.966			0.272370			0.136185	−	0.990683i	$-0.456516\pi$
$859$	233.966			0.272370			0.136185	+	0.990683i	$0.456516\pi$
$860$		0			0
$861$	1157.71			1.34461
$862$		0			0
$863$		−	972.669i		−	1.12708i	−0.826089	−	0.563539i	$-0.809439\pi$
$863$		−	972.669i		−	1.12708i	0.826089	−	0.563539i	$-0.190561\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			1386.65i			1.59936i
$868$		0			0
$869$	166.910	+	480.406i	0.192071	+	0.552826i
$870$		0			0
$871$		−	202.780i		−	0.232813i
$872$		0			0
$873$			714.750i			0.818729i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1259.13			−1.43573			−0.717864	−	0.696183i	$-0.754880\pi$
$877$	−1259.13			−1.43573			−0.717864	+	0.696183i	$0.754880\pi$
$878$		0			0
$879$			135.504i			0.154157i
$880$		0			0
$881$	−1162.94			−1.32003			−0.660013	−	0.751255i	$-0.729449\pi$
$881$	−1162.94			−1.32003			−0.660013	+	0.751255i	$0.729449\pi$
$882$		0			0
$883$		−	1072.88i		−	1.21504i	−0.794305	−	0.607519i	$-0.792165\pi$
$883$		−	1072.88i		−	1.21504i	0.794305	−	0.607519i	$-0.207835\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1335.00			−1.50508			−0.752539	−	0.658548i	$-0.771171\pi$
$887$	−1335.00			−1.50508			−0.752539	+	0.658548i	$0.771171\pi$
$888$		0			0
$889$	−996.373			−1.12078
$890$		0			0
$891$	186.809	−	64.9038i	0.209662	−	0.0728438i
$892$		0			0
$893$	544.130			0.609328
$894$		0			0
$895$		0			0
$896$		0			0
$897$	148.560			0.165619
$898$		0			0
$899$		−	1088.95i		−	1.21129i
$900$		0			0
$901$			20.2038i			0.0224237i
$902$		0			0
$903$			554.898i			0.614505i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			16.1018i			0.0177529i	0.999961	+	0.00887643i	$0.00282549\pi$
$907$			16.1018i			0.0177529i	−0.999961	+	0.00887643i	$0.997175\pi$
$908$		0			0
$909$		−	199.742i		−	0.219738i
$910$		0			0
$911$	−977.120			−1.07258			−0.536290	−	0.844034i	$-0.680175\pi$
$911$	−977.120			−1.07258			−0.536290	+	0.844034i	$0.680175\pi$
$912$		0			0
$913$	1577.17	−	547.963i	1.72746	−	0.600179i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			462.479i			0.504339i
$918$		0			0
$919$		−	1286.07i		−	1.39942i	−0.714427	−	0.699710i	$-0.753313\pi$
$919$		−	1286.07i		−	1.39942i	0.714427	−	0.699710i	$-0.246687\pi$
$920$		0			0
$921$		−	580.563i		−	0.630361i
$922$		0			0
$923$	610.917			0.661881
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	613.355i		−	0.661656i
$928$		0			0
$929$	−1596.59			−1.71861			−0.859305	−	0.511463i	$-0.829104\pi$
$929$	−1596.59			−1.71861			−0.859305	+	0.511463i	$0.829104\pi$
$930$		0			0
$931$			27.9227i			0.0299921i
$932$		0			0
$933$			721.777i			0.773609i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	653.371			0.697301			0.348650	−	0.937253i	$-0.386640\pi$
$937$	653.371			0.697301			0.348650	+	0.937253i	$0.386640\pi$
$938$		0			0
$939$	82.6414			0.0880100
$940$		0			0
$941$		−	430.775i		−	0.457784i	−0.973452	−	0.228892i	$-0.926490\pi$
$941$		−	430.775i		−	0.457784i	0.973452	−	0.228892i	$-0.0735103\pi$
$942$		0			0
$943$	659.259			0.699108
$944$		0			0
$945$		0			0
$946$		0			0
$947$			81.6426i			0.0862118i	0.999071	+	0.0431059i	$0.0137253\pi$
$947$			81.6426i			0.0862118i	−0.999071	+	0.0431059i	$0.986275\pi$
$948$		0			0
$949$	−493.177			−0.519681
$950$		0			0
$951$	−68.5386			−0.0720700
$952$		0			0
$953$	−1569.16			−1.64654			−0.823272	−	0.567647i	$-0.807854\pi$
$953$	−1569.16			−1.64654			−0.823272	+	0.567647i	$0.807854\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	716.459	−	248.923i	0.748651	−	0.260107i
$958$		0			0
$959$			1001.72i			1.04454i
$960$		0			0
$961$	130.120			0.135401
$962$		0			0
$963$	348.816			0.362218
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1468.26			1.51837			0.759184	−	0.650876i	$-0.225598\pi$
$967$	1468.26			1.51837			0.759184	+	0.650876i	$0.225598\pi$
$968$		0			0
$969$	−869.269			−0.897079
$970$		0			0
$971$	882.221			0.908569			0.454285	−	0.890857i	$-0.349895\pi$
$971$	882.221			0.908569			0.454285	+	0.890857i	$0.349895\pi$
$972$		0			0
$973$		−	105.568i		−	0.108498i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1552.16i			1.58870i	0.607462	+	0.794349i	$0.292188\pi$
$977$			1552.16i			1.58870i	−0.607462	+	0.794349i	$0.707812\pi$
$978$		0			0
$979$	1431.88	−	497.485i	1.46260	−	0.508156i
$980$		0			0
$981$		−	673.689i		−	0.686737i
$982$		0			0
$983$		−	1497.65i		−	1.52355i	−0.647839	−	0.761777i	$-0.724327\pi$
$983$		−	1497.65i		−	1.52355i	0.647839	−	0.761777i	$-0.275673\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−578.784			−0.586408
$988$		0			0
$989$			315.988i			0.319502i
$990$		0			0
$991$	983.889			0.992824			0.496412	−	0.868087i	$-0.334650\pi$
$991$	983.889			0.992824			0.496412	+	0.868087i	$0.334650\pi$
$992$		0			0
$993$			995.170i			1.00219i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1568.77			1.57350			0.786748	−	0.617275i	$-0.211763\pi$
$997$	1568.77			1.57350			0.786748	+	0.617275i	$0.211763\pi$
$998$		0			0
$999$	−1662.80			−1.66447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.3.e.c.549.12		16
5.2	odd	4		1100.3.f.e.901.6	yes	8
5.3	odd	4		1100.3.f.c.901.3	✓	8
5.4	even	2	inner	1100.3.e.c.549.5		16
11.10	odd	2	inner	1100.3.e.c.549.11		16
55.32	even	4		1100.3.f.e.901.5	yes	8
55.43	even	4		1100.3.f.c.901.4	yes	8
55.54	odd	2	inner	1100.3.e.c.549.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.3.e.c.549.5		16	5.4	even	2	inner
1100.3.e.c.549.6		16	55.54	odd	2	inner
1100.3.e.c.549.11		16	11.10	odd	2	inner
1100.3.e.c.549.12		16	1.1	even	1	trivial
1100.3.f.c.901.3	✓	8	5.3	odd	4
1100.3.f.c.901.4	yes	8	55.43	even	4
1100.3.f.e.901.5	yes	8	55.32	even	4
1100.3.f.e.901.6	yes	8	5.2	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+2.09157i q^{3} +6.85033 q^{7} +4.62533 q^{9} +(-10.3907 + 3.61010i) q^{11} -8.70538 q^{13} -30.8540 q^{17} -13.4701i q^{19} +14.3280i q^{21} +8.15908i q^{23} +28.4984i q^{27} +32.9665i q^{29} -33.0321 q^{31} +(-7.55078 - 21.7330i) q^{33} +58.3473i q^{37} -18.2079i q^{39} -80.8006i q^{41} +38.7283 q^{43} +40.3954i q^{47} -2.07294 q^{49} -64.5333i q^{51} -0.654819i q^{53} +28.1736 q^{57} -33.8941 q^{59} +111.376i q^{61} +31.6850 q^{63} +23.2936i q^{67} -17.0653 q^{69} -70.1769 q^{71} +56.6520 q^{73} +(-71.1799 + 24.7304i) q^{77} -46.2341i q^{79} -17.9784 q^{81} -151.786 q^{83} -68.9518 q^{87} -137.804 q^{89} -59.6348 q^{91} -69.0890i q^{93} +154.530i q^{97} +(-48.0605 + 16.6979i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9} + 6 q^{11} + 28 q^{31} - 28 q^{49} - 256 q^{59} + 352 q^{69} - 68 q^{71} - 256 q^{81} + 292 q^{89} + 228 q^{91} - 16 q^{99}+O(q^{100}) \) 16 * q - 32 * q^9 + 6 * q^11 + 28 * q^31 - 28 * q^49 - 256 * q^59 + 352 * q^69 - 68 * q^71 - 256 * q^81 + 292 * q^89 + 228 * q^91 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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