LMFDB - Embedded newform 1100.2.k.a.857.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1100,2,Mod(593,1100)] 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("1100.593"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1100.k (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.78354422234$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 9 $ x^4 - 5*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			857.2
Root			$1.65831 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1100.857
Dual form			1100.2.k.a.593.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.00000i) q^{3} +(3.31662 - 3.31662i) q^{7} +1.00000i q^{9} +3.31662i q^{11} +(3.31662 + 3.31662i) q^{13} +(3.31662 - 3.31662i) q^{17} -6.63325i q^{21} +(-3.00000 + 3.00000i) q^{23} +(4.00000 + 4.00000i) q^{27} +6.63325 q^{29} -4.00000 q^{31} +(3.31662 + 3.31662i) q^{33} +(-5.00000 - 5.00000i) q^{37} +6.63325 q^{39} +(-3.31662 - 3.31662i) q^{43} +(-5.00000 - 5.00000i) q^{47} -15.0000i q^{49} -6.63325i q^{51} +(3.00000 - 3.00000i) q^{53} +10.0000i q^{59} -13.2665i q^{61} +(3.31662 + 3.31662i) q^{63} +(3.00000 + 3.00000i) q^{67} +6.00000i q^{69} -4.00000 q^{71} +(3.31662 + 3.31662i) q^{73} +(11.0000 + 11.0000i) q^{77} -13.2665 q^{79} +5.00000 q^{81} +(-3.31662 - 3.31662i) q^{83} +(6.63325 - 6.63325i) q^{87} +12.0000i q^{89} +22.0000 q^{91} +(-4.00000 + 4.00000i) q^{93} +(-5.00000 - 5.00000i) q^{97} -3.31662 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 12 q^{23} + 16 q^{27} - 16 q^{31} - 20 q^{37} - 20 q^{47} + 12 q^{53} + 12 q^{67} - 16 q^{71} + 44 q^{77} + 20 q^{81} + 88 q^{91} - 16 q^{93} - 20 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 12 * q^23 + 16 * q^27 - 16 * q^31 - 20 * q^37 - 20 * q^47 + 12 * q^53 + 12 * q^67 - 16 * q^71 + 44 * q^77 + 20 * q^81 + 88 * q^91 - 16 * q^93 - 20 * q^97

Character values

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

$n$	$101$	$177$	$551$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$

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
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	1.00000	−	1.00000i	0.577350	−	0.577350i	0.934172	+	0.356822i	$0.116140\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.31662	−	3.31662i	1.25357	−	1.25357i	0.299456	−	0.954110i	$-0.403195\pi$
$7$	3.31662	−	3.31662i	1.25357	−	1.25357i	0.954110	−	0.299456i	$-0.0968053\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$			3.31662i			1.00000i
$11$			3.31662i			1.00000i
$12$		0			0
$13$	3.31662	+	3.31662i	0.919866	+	0.919866i	0.997019	−	0.0771531i	$-0.0245830\pi$
$13$	3.31662	+	3.31662i	0.919866	+	0.919866i	−0.0771531	+	0.997019i	$0.524583\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.31662	−	3.31662i	0.804400	−	0.804400i	−0.179380	−	0.983780i	$-0.557409\pi$
$17$	3.31662	−	3.31662i	0.804400	−	0.804400i	0.983780	+	0.179380i	$0.0574092\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		−	6.63325i		−	1.44749i
$22$		0			0
$23$	−3.00000	+	3.00000i	−0.625543	+	0.625543i	−0.946943	−	0.321400i	$-0.895847\pi$
$23$	−3.00000	+	3.00000i	−0.625543	+	0.625543i	0.321400	+	0.946943i	$0.395847\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.00000	+	4.00000i	0.769800	+	0.769800i
$28$		0			0
$29$	6.63325			1.23176			0.615882	−	0.787839i	$-0.288800\pi$
$29$	6.63325			1.23176			0.615882	+	0.787839i	$0.288800\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$	3.31662	+	3.31662i	0.577350	+	0.577350i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	0.164399	−	0.986394i	$-0.447432\pi$
$37$	−5.00000	−	5.00000i	−0.821995	−	0.821995i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$	6.63325			1.06217
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−3.31662	−	3.31662i	−0.505781	−	0.505781i	0.407448	−	0.913228i	$-0.366419\pi$
$43$	−3.31662	−	3.31662i	−0.505781	−	0.505781i	−0.913228	+	0.407448i	$0.866419\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.00000	−	5.00000i	−0.729325	−	0.729325i	0.241160	−	0.970485i	$-0.422472\pi$
$47$	−5.00000	−	5.00000i	−0.729325	−	0.729325i	−0.970485	+	0.241160i	$0.922472\pi$
$48$		0			0
$49$		−	15.0000i		−	2.14286i
$50$		0			0
$51$		−	6.63325i		−	0.928841i
$52$		0			0
$53$	3.00000	−	3.00000i	0.412082	−	0.412082i	−0.470381	−	0.882463i	$-0.655884\pi$
$53$	3.00000	−	3.00000i	0.412082	−	0.412082i	0.882463	+	0.470381i	$0.155884\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			10.0000i			1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$			10.0000i			1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$		0			0
$61$		−	13.2665i		−	1.69860i	−0.527910	−	0.849301i	$-0.677024\pi$
$61$		−	13.2665i		−	1.69860i	0.527910	−	0.849301i	$-0.322976\pi$
$62$		0			0
$63$	3.31662	+	3.31662i	0.417855	+	0.417855i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.00000	+	3.00000i	0.366508	+	0.366508i	0.866202	−	0.499694i	$-0.166554\pi$
$67$	3.00000	+	3.00000i	0.366508	+	0.366508i	−0.499694	+	0.866202i	$0.666554\pi$
$68$		0			0
$69$			6.00000i			0.722315i
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$	3.31662	+	3.31662i	0.388182	+	0.388182i	0.874038	−	0.485857i	$-0.161492\pi$
$73$	3.31662	+	3.31662i	0.388182	+	0.388182i	−0.485857	+	0.874038i	$0.661492\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	11.0000	+	11.0000i	1.25357	+	1.25357i
$78$		0			0
$79$	−13.2665			−1.49260			−0.746299	−	0.665611i	$-0.768171\pi$
$79$	−13.2665			−1.49260			−0.746299	+	0.665611i	$0.768171\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	−3.31662	−	3.31662i	−0.364047	−	0.364047i	0.501254	−	0.865300i	$-0.332872\pi$
$83$	−3.31662	−	3.31662i	−0.364047	−	0.364047i	−0.865300	+	0.501254i	$0.832872\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	6.63325	−	6.63325i	0.711159	−	0.711159i
$88$		0			0
$89$			12.0000i			1.27200i	0.771690	+	0.635999i	$0.219412\pi$
$89$			12.0000i			1.27200i	−0.771690	+	0.635999i	$0.780588\pi$
$90$		0			0
$91$	22.0000			2.30623
$92$		0			0
$93$	−4.00000	+	4.00000i	−0.414781	+	0.414781i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.00000	−	5.00000i	−0.507673	−	0.507673i	0.406138	−	0.913812i	$-0.366875\pi$
$97$	−5.00000	−	5.00000i	−0.507673	−	0.507673i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$	−3.31662			−0.333333
$100$		0			0
$101$			13.2665i			1.32007i	0.751237	+	0.660033i	$0.229458\pi$
$101$			13.2665i			1.32007i	−0.751237	+	0.660033i	$0.770542\pi$
$102$		0			0
$103$	9.00000	−	9.00000i	0.886796	−	0.886796i	−0.107418	−	0.994214i	$-0.534258\pi$
$103$	9.00000	−	9.00000i	0.886796	−	0.886796i	0.994214	+	0.107418i	$0.0342582\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.31662	−	3.31662i	0.320630	−	0.320630i	−0.528379	−	0.849009i	$-0.677200\pi$
$107$	3.31662	−	3.31662i	0.320630	−	0.320630i	0.849009	+	0.528379i	$0.177200\pi$
$108$		0			0
$109$	6.63325			0.635350			0.317675	−	0.948200i	$-0.397098\pi$
$109$	6.63325			0.635350			0.317675	+	0.948200i	$0.397098\pi$
$110$		0			0
$111$	−10.0000			−0.949158
$112$		0			0
$113$	−1.00000	+	1.00000i	−0.0940721	+	0.0940721i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	−1.00000	+	1.00000i	−0.0940721	+	0.0940721i	0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.31662	+	3.31662i	−0.306622	+	0.306622i
$118$		0			0
$119$		−	22.0000i		−	2.01674i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	3.31662	−	3.31662i	0.294303	−	0.294303i	−0.544475	−	0.838777i	$-0.683271\pi$
$127$	3.31662	−	3.31662i	0.294303	−	0.294303i	0.838777	+	0.544475i	$0.183271\pi$
$128$		0			0
$129$	−6.63325			−0.584025
$130$		0			0
$131$		−	6.63325i		−	0.579550i	−0.957095	−	0.289775i	$-0.906420\pi$
$131$		−	6.63325i		−	0.579550i	0.957095	−	0.289775i	$-0.0935804\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	−	1.00000i	−0.0854358	−	0.0854358i	0.663097	−	0.748533i	$-0.269242\pi$
$137$	−1.00000	−	1.00000i	−0.0854358	−	0.0854358i	−0.748533	+	0.663097i	$0.769242\pi$
$138$		0			0
$139$	−13.2665			−1.12525			−0.562625	−	0.826712i	$-0.690208\pi$
$139$	−13.2665			−1.12525			−0.562625	+	0.826712i	$0.690208\pi$
$140$		0			0
$141$	−10.0000			−0.842152
$142$		0			0
$143$	−11.0000	+	11.0000i	−0.919866	+	0.919866i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−15.0000	−	15.0000i	−1.23718	−	1.23718i
$148$		0			0
$149$	6.63325			0.543417			0.271708	−	0.962380i	$-0.412411\pi$
$149$	6.63325			0.543417			0.271708	+	0.962380i	$0.412411\pi$
$150$		0			0
$151$			6.63325i			0.539806i	0.962887	+	0.269903i	$0.0869917\pi$
$151$			6.63325i			0.539806i	−0.962887	+	0.269903i	$0.913008\pi$
$152$		0			0
$153$	3.31662	+	3.31662i	0.268133	+	0.268133i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.00000	−	1.00000i	−0.0798087	−	0.0798087i	0.666076	−	0.745884i	$-0.267973\pi$
$157$	−1.00000	−	1.00000i	−0.0798087	−	0.0798087i	−0.745884	+	0.666076i	$0.767973\pi$
$158$		0			0
$159$		−	6.00000i		−	0.475831i
$160$		0			0
$161$			19.8997i			1.56832i
$162$		0			0
$163$	−3.00000	+	3.00000i	−0.234978	+	0.234978i	−0.814767	−	0.579789i	$-0.803135\pi$
$163$	−3.00000	+	3.00000i	−0.234978	+	0.234978i	0.579789	+	0.814767i	$0.303135\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−9.94987	+	9.94987i	−0.769944	+	0.769944i	−0.978096	−	0.208152i	$-0.933255\pi$
$167$	−9.94987	+	9.94987i	−0.769944	+	0.769944i	0.208152	+	0.978096i	$0.433255\pi$
$168$		0			0
$169$			9.00000i			0.692308i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	3.31662	+	3.31662i	0.252158	+	0.252158i	0.821855	−	0.569697i	$-0.192939\pi$
$173$	3.31662	+	3.31662i	0.252158	+	0.252158i	−0.569697	+	0.821855i	$0.692939\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	10.0000	+	10.0000i	0.751646	+	0.751646i
$178$		0			0
$179$		−	6.00000i		−	0.448461i	−0.974536	−	0.224231i	$-0.928013\pi$
$179$		−	6.00000i		−	0.448461i	0.974536	−	0.224231i	$-0.0719869\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$	−13.2665	−	13.2665i	−0.980688	−	0.980688i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	11.0000	+	11.0000i	0.804400	+	0.804400i
$188$		0			0
$189$	26.5330			1.92999
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$	−9.94987	−	9.94987i	−0.716208	−	0.716208i	0.251619	−	0.967826i	$-0.419037\pi$
$193$	−9.94987	−	9.94987i	−0.716208	−	0.716208i	−0.967826	+	0.251619i	$0.919037\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−9.94987	+	9.94987i	−0.708899	+	0.708899i	−0.966304	−	0.257405i	$-0.917133\pi$
$197$	−9.94987	+	9.94987i	−0.708899	+	0.708899i	0.257405	+	0.966304i	$0.417133\pi$
$198$		0			0
$199$			22.0000i			1.55954i	0.626067	+	0.779769i	$0.284664\pi$
$199$			22.0000i			1.55954i	−0.626067	+	0.779769i	$0.715336\pi$
$200$		0			0
$201$	6.00000			0.423207
$202$		0			0
$203$	22.0000	−	22.0000i	1.54410	−	1.54410i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−3.00000	−	3.00000i	−0.208514	−	0.208514i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	6.63325i		−	0.456652i	−0.973585	−	0.228326i	$-0.926675\pi$
$211$		−	6.63325i		−	0.456652i	0.973585	−	0.228326i	$-0.0733252\pi$
$212$		0			0
$213$	−4.00000	+	4.00000i	−0.274075	+	0.274075i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−13.2665	+	13.2665i	−0.900589	+	0.900589i
$218$		0			0
$219$	6.63325			0.448233
$220$		0			0
$221$	22.0000			1.47988
$222$		0			0
$223$	−7.00000	+	7.00000i	−0.468755	+	0.468755i	−0.901511	−	0.432756i	$-0.857541\pi$
$223$	−7.00000	+	7.00000i	−0.468755	+	0.468755i	0.432756	+	0.901511i	$0.357541\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.94987	+	9.94987i	−0.660396	+	0.660396i	−0.955473	−	0.295077i	$-0.904655\pi$
$227$	−9.94987	+	9.94987i	−0.660396	+	0.660396i	0.295077	+	0.955473i	$0.404655\pi$
$228$		0			0
$229$		−	4.00000i		−	0.264327i	−0.991228	−	0.132164i	$-0.957808\pi$
$229$		−	4.00000i		−	0.264327i	0.991228	−	0.132164i	$-0.0421925\pi$
$230$		0			0
$231$	22.0000			1.44749
$232$		0			0
$233$	3.31662	+	3.31662i	0.217279	+	0.217279i	0.807351	−	0.590072i	$-0.200900\pi$
$233$	3.31662	+	3.31662i	0.217279	+	0.217279i	−0.590072	+	0.807351i	$0.700900\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−13.2665	+	13.2665i	−0.861752	+	0.861752i
$238$		0			0
$239$	13.2665			0.858138			0.429069	−	0.903272i	$-0.358842\pi$
$239$	13.2665			0.858138			0.429069	+	0.903272i	$0.358842\pi$
$240$		0			0
$241$			26.5330i			1.70914i	0.519336	+	0.854570i	$0.326179\pi$
$241$			26.5330i			1.70914i	−0.519336	+	0.854570i	$0.673821\pi$
$242$		0			0
$243$	−7.00000	+	7.00000i	−0.449050	+	0.449050i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−6.63325			−0.420365
$250$		0			0
$251$	−8.00000			−0.504956			−0.252478	−	0.967603i	$-0.581245\pi$
$251$	−8.00000			−0.504956			−0.252478	+	0.967603i	$0.581245\pi$
$252$		0			0
$253$	−9.94987	−	9.94987i	−0.625543	−	0.625543i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.00000	−	9.00000i	−0.561405	−	0.561405i	0.368302	−	0.929706i	$-0.379939\pi$
$257$	−9.00000	−	9.00000i	−0.561405	−	0.561405i	−0.929706	+	0.368302i	$0.879939\pi$
$258$		0			0
$259$	−33.1662			−2.06085
$260$		0			0
$261$			6.63325i			0.410588i
$262$		0			0
$263$	−3.31662	−	3.31662i	−0.204512	−	0.204512i	0.597418	−	0.801930i	$-0.296193\pi$
$263$	−3.31662	−	3.31662i	−0.204512	−	0.204512i	−0.801930	+	0.597418i	$0.796193\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	12.0000	+	12.0000i	0.734388	+	0.734388i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		−	19.8997i		−	1.20882i	−0.796672	−	0.604412i	$-0.793408\pi$
$271$		−	19.8997i		−	1.20882i	0.796672	−	0.604412i	$-0.206592\pi$
$272$		0			0
$273$	22.0000	−	22.0000i	1.33150	−	1.33150i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	16.5831	−	16.5831i	0.996383	−	0.996383i	−0.00361013	−	0.999993i	$-0.501149\pi$
$277$	16.5831	−	16.5831i	0.996383	−	0.996383i	0.999993	+	0.00361013i	$0.00114914\pi$
$278$		0			0
$279$		−	4.00000i		−	0.239474i
$280$		0			0
$281$			26.5330i			1.58283i	0.611282	+	0.791413i	$0.290654\pi$
$281$			26.5330i			1.58283i	−0.611282	+	0.791413i	$0.709346\pi$
$282$		0			0
$283$	−16.5831	−	16.5831i	−0.985764	−	0.985764i	0.0141357	−	0.999900i	$-0.495500\pi$
$283$	−16.5831	−	16.5831i	−0.985764	−	0.985764i	−0.999900	+	0.0141357i	$0.995500\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	5.00000i		−	0.294118i
$290$		0			0
$291$	−10.0000			−0.586210
$292$		0			0
$293$	16.5831	+	16.5831i	0.968796	+	0.968796i	0.999528	−	0.0307312i	$-0.00978360\pi$
$293$	16.5831	+	16.5831i	0.968796	+	0.968796i	−0.0307312	+	0.999528i	$0.509784\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−13.2665	+	13.2665i	−0.769800	+	0.769800i
$298$		0			0
$299$	−19.8997			−1.15083
$300$		0			0
$301$	−22.0000			−1.26806
$302$		0			0
$303$	13.2665	+	13.2665i	0.762140	+	0.762140i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	3.31662	−	3.31662i	0.189290	−	0.189290i	−0.606099	−	0.795389i	$-0.707267\pi$
$307$	3.31662	−	3.31662i	0.189290	−	0.189290i	0.795389	+	0.606099i	$0.207267\pi$
$308$		0			0
$309$		−	18.0000i		−	1.02398i
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$	−21.0000	+	21.0000i	−1.18699	+	1.18699i	−0.209095	+	0.977895i	$0.567052\pi$
$313$	−21.0000	+	21.0000i	−1.18699	+	1.18699i	−0.977895	+	0.209095i	$0.932948\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.0000	−	17.0000i	−0.954815	−	0.954815i	0.0442073	−	0.999022i	$-0.485924\pi$
$317$	−17.0000	−	17.0000i	−0.954815	−	0.954815i	−0.999022	+	0.0442073i	$0.985924\pi$
$318$		0			0
$319$			22.0000i			1.23176i
$320$		0			0
$321$		−	6.63325i		−	0.370232i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	6.63325	−	6.63325i	0.366820	−	0.366820i
$328$		0			0
$329$	−33.1662			−1.82851
$330$		0			0
$331$	−16.0000			−0.879440			−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000			−0.879440			−0.439720	+	0.898135i	$0.644922\pi$
$332$		0			0
$333$	5.00000	−	5.00000i	0.273998	−	0.273998i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	3.31662	−	3.31662i	0.180668	−	0.180668i	−0.610979	−	0.791647i	$-0.709224\pi$
$337$	3.31662	−	3.31662i	0.180668	−	0.180668i	0.791647	+	0.610979i	$0.209224\pi$
$338$		0			0
$339$			2.00000i			0.108625i
$340$		0			0
$341$		−	13.2665i		−	0.718421i
$342$		0			0
$343$	−26.5330	−	26.5330i	−1.43265	−	1.43265i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−9.94987	+	9.94987i	−0.534137	+	0.534137i	−0.921801	−	0.387664i	$-0.873282\pi$
$347$	−9.94987	+	9.94987i	−0.534137	+	0.534137i	0.387664	+	0.921801i	$0.373282\pi$
$348$		0			0
$349$	−19.8997			−1.06521			−0.532605	−	0.846364i	$-0.678787\pi$
$349$	−19.8997			−1.06521			−0.532605	+	0.846364i	$0.678787\pi$
$350$		0			0
$351$			26.5330i			1.41623i
$352$		0			0
$353$	−9.00000	+	9.00000i	−0.479022	+	0.479022i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−9.00000	+	9.00000i	−0.479022	+	0.479022i	0.425797	+	0.904819i	$0.359994\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−22.0000	−	22.0000i	−1.16436	−	1.16436i
$358$		0			0
$359$	13.2665			0.700179			0.350090	−	0.936716i	$-0.386151\pi$
$359$	13.2665			0.700179			0.350090	+	0.936716i	$0.386151\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$	−11.0000	+	11.0000i	−0.577350	+	0.577350i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	15.0000	+	15.0000i	0.782994	+	0.782994i	0.980335	−	0.197341i	$-0.0632307\pi$
$367$	15.0000	+	15.0000i	0.782994	+	0.782994i	−0.197341	+	0.980335i	$0.563231\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	19.8997i		−	1.03314i
$372$		0			0
$373$	3.31662	+	3.31662i	0.171728	+	0.171728i	0.787738	−	0.616010i	$-0.211252\pi$
$373$	3.31662	+	3.31662i	0.171728	+	0.171728i	−0.616010	+	0.787738i	$0.711252\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	22.0000	+	22.0000i	1.13306	+	1.13306i
$378$		0			0
$379$			2.00000i			0.102733i	0.998680	+	0.0513665i	$0.0163577\pi$
$379$			2.00000i			0.102733i	−0.998680	+	0.0513665i	$0.983642\pi$
$380$		0			0
$381$		−	6.63325i		−	0.339832i
$382$		0			0
$383$	5.00000	−	5.00000i	0.255488	−	0.255488i	−0.567728	−	0.823216i	$-0.692177\pi$
$383$	5.00000	−	5.00000i	0.255488	−	0.255488i	0.823216	+	0.567728i	$0.192177\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	3.31662	−	3.31662i	0.168594	−	0.168594i
$388$		0			0
$389$		−	32.0000i		−	1.62246i	−0.584724	−	0.811232i	$-0.698797\pi$
$389$		−	32.0000i		−	1.62246i	0.584724	−	0.811232i	$-0.301203\pi$
$390$		0			0
$391$			19.8997i			1.00637i
$392$		0			0
$393$	−6.63325	−	6.63325i	−0.334603	−	0.334603i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	23.0000	+	23.0000i	1.15434	+	1.15434i	0.985674	+	0.168663i	$0.0539450\pi$
$397$	23.0000	+	23.0000i	1.15434	+	1.15434i	0.168663	+	0.985674i	$0.446055\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.0000			1.09863			0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000			1.09863			0.549314	+	0.835616i	$0.314889\pi$
$402$		0			0
$403$	−13.2665	−	13.2665i	−0.660851	−	0.660851i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	16.5831	−	16.5831i	0.821995	−	0.821995i
$408$		0			0
$409$	19.8997			0.983979			0.491990	−	0.870601i	$-0.336270\pi$
$409$	19.8997			0.983979			0.491990	+	0.870601i	$0.336270\pi$
$410$		0			0
$411$	−2.00000			−0.0986527
$412$		0			0
$413$	33.1662	+	33.1662i	1.63200	+	1.63200i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−13.2665	+	13.2665i	−0.649663	+	0.649663i
$418$		0			0
$419$			2.00000i			0.0977064i	0.998806	+	0.0488532i	$0.0155566\pi$
$419$			2.00000i			0.0977064i	−0.998806	+	0.0488532i	$0.984443\pi$
$420$		0			0
$421$	22.0000			1.07221			0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000			1.07221			0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$	5.00000	−	5.00000i	0.243108	−	0.243108i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−44.0000	−	44.0000i	−2.12931	−	2.12931i
$428$		0			0
$429$			22.0000i			1.06217i
$430$		0			0
$431$		−	33.1662i		−	1.59756i	−0.601622	−	0.798781i	$-0.705479\pi$
$431$		−	33.1662i		−	1.59756i	0.601622	−	0.798781i	$-0.294521\pi$
$432$		0			0
$433$	15.0000	−	15.0000i	0.720854	−	0.720854i	−0.247925	−	0.968779i	$-0.579749\pi$
$433$	15.0000	−	15.0000i	0.720854	−	0.720854i	0.968779	+	0.247925i	$0.0797487\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	15.0000			0.714286
$442$		0			0
$443$	−15.0000	+	15.0000i	−0.712672	+	0.712672i	−0.967093	−	0.254422i	$-0.918115\pi$
$443$	−15.0000	+	15.0000i	−0.712672	+	0.712672i	0.254422	+	0.967093i	$0.418115\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	6.63325	−	6.63325i	0.313742	−	0.313742i
$448$		0			0
$449$			8.00000i			0.377543i	0.982021	+	0.188772i	$0.0604506\pi$
$449$			8.00000i			0.377543i	−0.982021	+	0.188772i	$0.939549\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	6.63325	+	6.63325i	0.311657	+	0.311657i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.31662	−	3.31662i	0.155145	−	0.155145i	−0.625266	−	0.780411i	$-0.715010\pi$
$457$	3.31662	−	3.31662i	0.155145	−	0.155145i	0.780411	+	0.625266i	$0.215010\pi$
$458$		0			0
$459$	26.5330			1.23845
$460$		0			0
$461$		−	39.7995i		−	1.85365i	−0.375497	−	0.926824i	$-0.622528\pi$
$461$		−	39.7995i		−	1.85365i	0.375497	−	0.926824i	$-0.377472\pi$
$462$		0			0
$463$	9.00000	−	9.00000i	0.418265	−	0.418265i	−0.466340	−	0.884606i	$-0.654428\pi$
$463$	9.00000	−	9.00000i	0.418265	−	0.418265i	0.884606	+	0.466340i	$0.154428\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.00000	−	1.00000i	−0.0462745	−	0.0462745i	0.683591	−	0.729865i	$-0.260417\pi$
$467$	−1.00000	−	1.00000i	−0.0462745	−	0.0462745i	−0.729865	+	0.683591i	$0.760417\pi$
$468$		0			0
$469$	19.8997			0.918885
$470$		0			0
$471$	−2.00000			−0.0921551
$472$		0			0
$473$	11.0000	−	11.0000i	0.505781	−	0.505781i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	3.00000	+	3.00000i	0.137361	+	0.137361i
$478$		0			0
$479$	13.2665			0.606162			0.303081	−	0.952965i	$-0.401985\pi$
$479$	13.2665			0.606162			0.303081	+	0.952965i	$0.401985\pi$
$480$		0			0
$481$		−	33.1662i		−	1.51225i
$482$		0			0
$483$	19.8997	+	19.8997i	0.905470	+	0.905470i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	23.0000	+	23.0000i	1.04223	+	1.04223i	0.999068	+	0.0431614i	$0.0137430\pi$
$487$	23.0000	+	23.0000i	1.04223	+	1.04223i	0.0431614	+	0.999068i	$0.486257\pi$
$488$		0			0
$489$			6.00000i			0.271329i
$490$		0			0
$491$			33.1662i			1.49677i	0.663264	+	0.748386i	$0.269171\pi$
$491$			33.1662i			1.49677i	−0.663264	+	0.748386i	$0.730829\pi$
$492$		0			0
$493$	22.0000	−	22.0000i	0.990830	−	0.990830i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−13.2665	+	13.2665i	−0.595084	+	0.595084i
$498$		0			0
$499$		−	22.0000i		−	0.984855i	−0.870353	−	0.492428i	$-0.836110\pi$
$499$		−	22.0000i		−	0.984855i	0.870353	−	0.492428i	$-0.163890\pi$
$500$		0			0
$501$			19.8997i			0.889055i
$502$		0			0
$503$	9.94987	+	9.94987i	0.443643	+	0.443643i	0.893234	−	0.449591i	$-0.148430\pi$
$503$	9.94987	+	9.94987i	0.443643	+	0.443643i	−0.449591	+	0.893234i	$0.648430\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	9.00000	+	9.00000i	0.399704	+	0.399704i
$508$		0			0
$509$			36.0000i			1.59567i	0.602875	+	0.797836i	$0.294022\pi$
$509$			36.0000i			1.59567i	−0.602875	+	0.797836i	$0.705978\pi$
$510$		0			0
$511$	22.0000			0.973223
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	16.5831	−	16.5831i	0.729325	−	0.729325i
$518$		0			0
$519$	6.63325			0.291167
$520$		0			0
$521$	−34.0000			−1.48957			−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000			−1.48957			−0.744784	+	0.667306i	$0.767447\pi$
$522$		0			0
$523$	−16.5831	−	16.5831i	−0.725129	−	0.725129i	0.244516	−	0.969645i	$-0.421371\pi$
$523$	−16.5831	−	16.5831i	−0.725129	−	0.725129i	−0.969645	+	0.244516i	$0.921371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.2665	+	13.2665i	−0.577898	+	0.577898i
$528$		0			0
$529$			5.00000i			0.217391i
$530$		0			0
$531$	−10.0000			−0.433963
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−6.00000	−	6.00000i	−0.258919	−	0.258919i
$538$		0			0
$539$	49.7494			2.14286
$540$		0			0
$541$		−	13.2665i		−	0.570371i	−0.958472	−	0.285186i	$-0.907945\pi$
$541$		−	13.2665i		−	0.570371i	0.958472	−	0.285186i	$-0.0920553\pi$
$542$		0			0
$543$	2.00000	−	2.00000i	0.0858282	−	0.0858282i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	29.8496	−	29.8496i	1.27628	−	1.27628i	0.333543	−	0.942735i	$-0.391756\pi$
$547$	29.8496	−	29.8496i	1.27628	−	1.27628i	0.942735	−	0.333543i	$-0.108244\pi$
$548$		0			0
$549$	13.2665			0.566200
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−44.0000	+	44.0000i	−1.87107	+	1.87107i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	29.8496	−	29.8496i	1.26477	−	1.26477i	0.316015	−	0.948754i	$-0.397655\pi$
$557$	29.8496	−	29.8496i	1.26477	−	1.26477i	0.948754	−	0.316015i	$-0.102345\pi$
$558$		0			0
$559$		−	22.0000i		−	0.930501i
$560$		0			0
$561$	22.0000			0.928841
$562$		0			0
$563$	9.94987	+	9.94987i	0.419337	+	0.419337i	0.884975	−	0.465638i	$-0.154175\pi$
$563$	9.94987	+	9.94987i	0.419337	+	0.419337i	−0.465638	+	0.884975i	$0.654175\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	16.5831	−	16.5831i	0.696426	−	0.696426i
$568$		0			0
$569$	19.8997			0.834241			0.417120	−	0.908851i	$-0.363039\pi$
$569$	19.8997			0.834241			0.417120	+	0.908851i	$0.363039\pi$
$570$		0			0
$571$		−	19.8997i		−	0.832779i	−0.909186	−	0.416389i	$-0.863295\pi$
$571$		−	19.8997i		−	0.832779i	0.909186	−	0.416389i	$-0.136705\pi$
$572$		0			0
$573$	−12.0000	+	12.0000i	−0.501307	+	0.501307i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−21.0000	−	21.0000i	−0.874241	−	0.874241i	0.118690	−	0.992931i	$-0.462131\pi$
$577$	−21.0000	−	21.0000i	−0.874241	−	0.874241i	−0.992931	+	0.118690i	$0.962131\pi$
$578$		0			0
$579$	−19.8997			−0.827005
$580$		0			0
$581$	−22.0000			−0.912714
$582$		0			0
$583$	9.94987	+	9.94987i	0.412082	+	0.412082i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−13.0000	−	13.0000i	−0.536567	−	0.536567i	0.385952	−	0.922519i	$-0.373873\pi$
$587$	−13.0000	−	13.0000i	−0.536567	−	0.536567i	−0.922519	+	0.385952i	$0.873873\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$			19.8997i			0.818566i
$592$		0			0
$593$	−9.94987	−	9.94987i	−0.408592	−	0.408592i	0.472655	−	0.881247i	$-0.343296\pi$
$593$	−9.94987	−	9.94987i	−0.408592	−	0.408592i	−0.881247	+	0.472655i	$0.843296\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	22.0000	+	22.0000i	0.900400	+	0.900400i
$598$		0			0
$599$			30.0000i			1.22577i	0.790173	+	0.612883i	$0.209990\pi$
$599$			30.0000i			1.22577i	−0.790173	+	0.612883i	$0.790010\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−3.00000	+	3.00000i	−0.122169	+	0.122169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	3.31662	−	3.31662i	0.134618	−	0.134618i	−0.636587	−	0.771205i	$-0.719654\pi$
$607$	3.31662	−	3.31662i	0.134618	−	0.134618i	0.771205	+	0.636587i	$0.219654\pi$
$608$		0			0
$609$		−	44.0000i		−	1.78297i
$610$		0			0
$611$		−	33.1662i		−	1.34176i
$612$		0			0
$613$	16.5831	+	16.5831i	0.669786	+	0.669786i	0.957666	−	0.287880i	$-0.0929505\pi$
$613$	16.5831	+	16.5831i	0.669786	+	0.669786i	−0.287880	+	0.957666i	$0.592951\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.00000	−	1.00000i	−0.0402585	−	0.0402585i	0.686691	−	0.726949i	$-0.259063\pi$
$617$	−1.00000	−	1.00000i	−0.0402585	−	0.0402585i	−0.726949	+	0.686691i	$0.759063\pi$
$618$		0			0
$619$		−	38.0000i		−	1.52735i	−0.645601	−	0.763674i	$-0.723393\pi$
$619$		−	38.0000i		−	1.52735i	0.645601	−	0.763674i	$-0.276607\pi$
$620$		0			0
$621$	−24.0000			−0.963087
$622$		0			0
$623$	39.7995	+	39.7995i	1.59453	+	1.59453i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−33.1662			−1.32242
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		0			0
$633$	−6.63325	−	6.63325i	−0.263648	−	0.263648i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	49.7494	−	49.7494i	1.97114	−	1.97114i
$638$		0			0
$639$		−	4.00000i		−	0.158238i
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$	21.0000	−	21.0000i	0.828159	−	0.828159i	−0.159103	−	0.987262i	$-0.550860\pi$
$643$	21.0000	−	21.0000i	0.828159	−	0.828159i	0.987262	+	0.159103i	$0.0508601\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17.0000	−	17.0000i	−0.668339	−	0.668339i	0.288992	−	0.957331i	$-0.406680\pi$
$647$	−17.0000	−	17.0000i	−0.668339	−	0.668339i	−0.957331	+	0.288992i	$0.906680\pi$
$648$		0			0
$649$	−33.1662			−1.30189
$650$		0			0
$651$			26.5330i			1.03991i
$652$		0			0
$653$	−5.00000	+	5.00000i	−0.195665	+	0.195665i	−0.798139	−	0.602474i	$-0.794182\pi$
$653$	−5.00000	+	5.00000i	−0.195665	+	0.195665i	0.602474	+	0.798139i	$0.294182\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−3.31662	+	3.31662i	−0.129394	+	0.129394i
$658$		0			0
$659$	−13.2665			−0.516789			−0.258395	−	0.966039i	$-0.583194\pi$
$659$	−13.2665			−0.516789			−0.258395	+	0.966039i	$0.583194\pi$
$660$		0			0
$661$	34.0000			1.32245			0.661223	−	0.750189i	$-0.270038\pi$
$661$	34.0000			1.32245			0.661223	+	0.750189i	$0.270038\pi$
$662$		0			0
$663$	22.0000	−	22.0000i	0.854409	−	0.854409i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−19.8997	+	19.8997i	−0.770521	+	0.770521i
$668$		0			0
$669$			14.0000i			0.541271i
$670$		0			0
$671$	44.0000			1.69860
$672$		0			0
$673$	−9.94987	−	9.94987i	−0.383539	−	0.383539i	0.488836	−	0.872376i	$-0.337422\pi$
$673$	−9.94987	−	9.94987i	−0.383539	−	0.383539i	−0.872376	+	0.488836i	$0.837422\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	29.8496	−	29.8496i	1.14721	−	1.14721i	0.160116	−	0.987098i	$-0.448813\pi$
$677$	29.8496	−	29.8496i	1.14721	−	1.14721i	0.987098	−	0.160116i	$-0.0511868\pi$
$678$		0			0
$679$	−33.1662			−1.27280
$680$		0			0
$681$			19.8997i			0.762560i
$682$		0			0
$683$	−23.0000	+	23.0000i	−0.880071	+	0.880071i	−0.993541	−	0.113471i	$-0.963803\pi$
$683$	−23.0000	+	23.0000i	−0.880071	+	0.880071i	0.113471	+	0.993541i	$0.463803\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−4.00000	−	4.00000i	−0.152610	−	0.152610i
$688$		0			0
$689$	19.8997			0.758120
$690$		0			0
$691$	8.00000			0.304334			0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000			0.304334			0.152167	+	0.988355i	$0.451375\pi$
$692$		0			0
$693$	−11.0000	+	11.0000i	−0.417855	+	0.417855i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	6.63325			0.250893
$700$		0			0
$701$			13.2665i			0.501069i	0.968108	+	0.250534i	$0.0806063\pi$
$701$			13.2665i			0.501069i	−0.968108	+	0.250534i	$0.919394\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	44.0000	+	44.0000i	1.65479	+	1.65479i
$708$		0			0
$709$			40.0000i			1.50223i	0.660171	+	0.751116i	$0.270484\pi$
$709$			40.0000i			1.50223i	−0.660171	+	0.751116i	$0.729516\pi$
$710$		0			0
$711$		−	13.2665i		−	0.497533i
$712$		0			0
$713$	12.0000	−	12.0000i	0.449404	−	0.449404i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	13.2665	−	13.2665i	0.495446	−	0.495446i
$718$		0			0
$719$		−	2.00000i		−	0.0745874i	−0.999304	−	0.0372937i	$-0.988126\pi$
$719$		−	2.00000i		−	0.0745874i	0.999304	−	0.0372937i	$-0.0118737\pi$
$720$		0			0
$721$		−	59.6992i		−	2.22332i
$722$		0			0
$723$	26.5330	+	26.5330i	0.986773	+	0.986773i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−25.0000	−	25.0000i	−0.927199	−	0.927199i	0.0703254	−	0.997524i	$-0.477596\pi$
$727$	−25.0000	−	25.0000i	−0.927199	−	0.927199i	−0.997524	+	0.0703254i	$0.977596\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$	−22.0000			−0.813699
$732$		0			0
$733$	−36.4829	−	36.4829i	−1.34753	−	1.34753i	−0.888341	−	0.459185i	$-0.848142\pi$
$733$	−36.4829	−	36.4829i	−1.34753	−	1.34753i	−0.459185	−	0.888341i	$-0.651858\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−9.94987	+	9.94987i	−0.366508	+	0.366508i
$738$		0			0
$739$	39.7995			1.46405			0.732024	−	0.681279i	$-0.238576\pi$
$739$	39.7995			1.46405			0.732024	+	0.681279i	$0.238576\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.5831	−	16.5831i	−0.608376	−	0.608376i	0.334146	−	0.942521i	$-0.391552\pi$
$743$	−16.5831	−	16.5831i	−0.608376	−	0.608376i	−0.942521	+	0.334146i	$0.891552\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	3.31662	−	3.31662i	0.121349	−	0.121349i
$748$		0			0
$749$		−	22.0000i		−	0.803863i
$750$		0			0
$751$	28.0000			1.02173			0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000			1.02173			0.510867	+	0.859660i	$0.329324\pi$
$752$		0			0
$753$	−8.00000	+	8.00000i	−0.291536	+	0.291536i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	3.00000	+	3.00000i	0.109037	+	0.109037i	0.759520	−	0.650484i	$-0.225434\pi$
$757$	3.00000	+	3.00000i	0.109037	+	0.109037i	−0.650484	+	0.759520i	$0.725434\pi$
$758$		0			0
$759$	−19.8997			−0.722315
$760$		0			0
$761$			26.5330i			0.961820i	0.876770	+	0.480910i	$0.159694\pi$
$761$			26.5330i			0.961820i	−0.876770	+	0.480910i	$0.840306\pi$
$762$		0			0
$763$	22.0000	−	22.0000i	0.796453	−	0.796453i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−33.1662	+	33.1662i	−1.19756	+	1.19756i
$768$		0			0
$769$	19.8997			0.717603			0.358802	−	0.933414i	$-0.383185\pi$
$769$	19.8997			0.717603			0.358802	+	0.933414i	$0.383185\pi$
$770$		0			0
$771$	−18.0000			−0.648254
$772$		0			0
$773$	−5.00000	+	5.00000i	−0.179838	+	0.179838i	−0.791285	−	0.611448i	$-0.790588\pi$
$773$	−5.00000	+	5.00000i	−0.179838	+	0.179838i	0.611448	+	0.791285i	$0.290588\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−33.1662	+	33.1662i	−1.18983	+	1.18983i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		−	13.2665i		−	0.474713i
$782$		0			0
$783$	26.5330	+	26.5330i	0.948212	+	0.948212i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−23.2164	+	23.2164i	−0.827574	+	0.827574i	−0.987181	−	0.159606i	$-0.948977\pi$
$787$	−23.2164	+	23.2164i	−0.827574	+	0.827574i	0.159606	+	0.987181i	$0.448977\pi$
$788$		0			0
$789$	−6.63325			−0.236150
$790$		0			0
$791$			6.63325i			0.235851i
$792$		0			0
$793$	44.0000	−	44.0000i	1.56249	−	1.56249i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19.0000	+	19.0000i	0.673015	+	0.673015i	0.958410	−	0.285395i	$-0.0921249\pi$
$797$	19.0000	+	19.0000i	0.673015	+	0.673015i	−0.285395	+	0.958410i	$0.592125\pi$
$798$		0			0
$799$	−33.1662			−1.17334
$800$		0			0
$801$	−12.0000			−0.423999
$802$		0			0
$803$	−11.0000	+	11.0000i	−0.388182	+	0.388182i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	19.8997			0.699638			0.349819	−	0.936817i	$-0.386243\pi$
$809$	19.8997			0.699638			0.349819	+	0.936817i	$0.386243\pi$
$810$		0			0
$811$			33.1662i			1.16462i	0.812965	+	0.582312i	$0.197852\pi$
$811$			33.1662i			1.16462i	−0.812965	+	0.582312i	$0.802148\pi$
$812$		0			0
$813$	−19.8997	−	19.8997i	−0.697915	−	0.697915i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$			22.0000i			0.768742i
$820$		0			0
$821$		−	39.7995i		−	1.38901i	−0.719487	−	0.694506i	$-0.755623\pi$
$821$		−	39.7995i		−	1.38901i	0.719487	−	0.694506i	$-0.244377\pi$
$822$		0			0
$823$	−27.0000	+	27.0000i	−0.941161	+	0.941161i	−0.998363	−	0.0572018i	$-0.981782\pi$
$823$	−27.0000	+	27.0000i	−0.941161	+	0.941161i	0.0572018	+	0.998363i	$0.481782\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.31662	−	3.31662i	0.115330	−	0.115330i	−0.647086	−	0.762417i	$-0.724013\pi$
$827$	3.31662	−	3.31662i	0.115330	−	0.115330i	0.762417	+	0.647086i	$0.224013\pi$
$828$		0			0
$829$		−	20.0000i		−	0.694629i	−0.937749	−	0.347314i	$-0.887094\pi$
$829$		−	20.0000i		−	0.694629i	0.937749	−	0.347314i	$-0.112906\pi$
$830$		0			0
$831$		−	33.1662i		−	1.15052i
$832$		0			0
$833$	−49.7494	−	49.7494i	−1.72371	−	1.72371i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−16.0000	−	16.0000i	−0.553041	−	0.553041i
$838$		0			0
$839$			30.0000i			1.03572i	0.855467	+	0.517858i	$0.173270\pi$
$839$			30.0000i			1.03572i	−0.855467	+	0.517858i	$0.826730\pi$
$840$		0			0
$841$	15.0000			0.517241
$842$		0			0
$843$	26.5330	+	26.5330i	0.913845	+	0.913845i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−36.4829	+	36.4829i	−1.25357	+	1.25357i
$848$		0			0
$849$	−33.1662			−1.13826
$850$		0			0
$851$	30.0000			1.02839
$852$		0			0
$853$	−9.94987	−	9.94987i	−0.340677	−	0.340677i	0.515945	−	0.856622i	$-0.327441\pi$
$853$	−9.94987	−	9.94987i	−0.340677	−	0.340677i	−0.856622	+	0.515945i	$0.827441\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−23.2164	+	23.2164i	−0.793056	+	0.793056i	−0.981990	−	0.188934i	$-0.939497\pi$
$857$	−23.2164	+	23.2164i	−0.793056	+	0.793056i	0.188934	+	0.981990i	$0.439497\pi$
$858$		0			0
$859$			10.0000i			0.341196i	0.985341	+	0.170598i	$0.0545699\pi$
$859$			10.0000i			0.341196i	−0.985341	+	0.170598i	$0.945430\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	29.0000	−	29.0000i	0.987171	−	0.987171i	−0.0127473	−	0.999919i	$-0.504058\pi$
$863$	29.0000	−	29.0000i	0.987171	−	0.987171i	0.999919	+	0.0127473i	$0.00405769\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−5.00000	−	5.00000i	−0.169809	−	0.169809i
$868$		0			0
$869$		−	44.0000i		−	1.49260i
$870$		0			0
$871$			19.8997i			0.674277i
$872$		0			0
$873$	5.00000	−	5.00000i	0.169224	−	0.169224i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3.31662	−	3.31662i	0.111994	−	0.111994i	−0.648889	−	0.760883i	$-0.724766\pi$
$877$	3.31662	−	3.31662i	0.111994	−	0.111994i	0.760883	+	0.648889i	$0.224766\pi$
$878$		0			0
$879$	33.1662			1.11867
$880$		0			0
$881$	54.0000			1.81931			0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000			1.81931			0.909653	+	0.415369i	$0.136347\pi$
$882$		0			0
$883$	−27.0000	+	27.0000i	−0.908622	+	0.908622i	−0.996161	−	0.0875388i	$-0.972100\pi$
$883$	−27.0000	+	27.0000i	−0.908622	+	0.908622i	0.0875388	+	0.996161i	$0.472100\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16.5831	−	16.5831i	0.556807	−	0.556807i	−0.371590	−	0.928397i	$-0.621187\pi$
$887$	16.5831	−	16.5831i	0.556807	−	0.556807i	0.928397	+	0.371590i	$0.121187\pi$
$888$		0			0
$889$		−	22.0000i		−	0.737856i
$890$		0			0
$891$			16.5831i			0.555556i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−19.8997	+	19.8997i	−0.664433	+	0.664433i
$898$		0			0
$899$	−26.5330			−0.884925
$900$		0			0
$901$		−	19.8997i		−	0.662957i
$902$		0			0
$903$	−22.0000	+	22.0000i	−0.732114	+	0.732114i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	7.00000	+	7.00000i	0.232431	+	0.232431i	0.813707	−	0.581276i	$-0.197446\pi$
$907$	7.00000	+	7.00000i	0.232431	+	0.232431i	−0.581276	+	0.813707i	$0.697446\pi$
$908$		0			0
$909$	−13.2665			−0.440022
$910$		0			0
$911$	−44.0000			−1.45779			−0.728893	−	0.684628i	$-0.759965\pi$
$911$	−44.0000			−1.45779			−0.728893	+	0.684628i	$0.759965\pi$
$912$		0			0
$913$	11.0000	−	11.0000i	0.364047	−	0.364047i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.0000	−	22.0000i	−0.726504	−	0.726504i
$918$		0			0
$919$	26.5330			0.875243			0.437621	−	0.899159i	$-0.355821\pi$
$919$	26.5330			0.875243			0.437621	+	0.899159i	$0.355821\pi$
$920$		0			0
$921$		−	6.63325i		−	0.218573i
$922$		0			0
$923$	−13.2665	−	13.2665i	−0.436672	−	0.436672i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	9.00000	+	9.00000i	0.295599	+	0.295599i
$928$		0			0
$929$		−	44.0000i		−	1.44359i	−0.692105	−	0.721797i	$-0.743317\pi$
$929$		−	44.0000i		−	1.44359i	0.692105	−	0.721797i	$-0.256683\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	12.0000	−	12.0000i	0.392862	−	0.392862i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−9.94987	+	9.94987i	−0.325048	+	0.325048i	−0.850700	−	0.525652i	$-0.823822\pi$
$937$	−9.94987	+	9.94987i	−0.325048	+	0.325048i	0.525652	+	0.850700i	$0.323822\pi$
$938$		0			0
$939$			42.0000i			1.37062i
$940$		0			0
$941$			39.7995i			1.29743i	0.761033	+	0.648713i	$0.224693\pi$
$941$			39.7995i			1.29743i	−0.761033	+	0.648713i	$0.775307\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.0000	+	19.0000i	0.617417	+	0.617417i	0.944868	−	0.327451i	$-0.106190\pi$
$947$	19.0000	+	19.0000i	0.617417	+	0.617417i	−0.327451	+	0.944868i	$0.606190\pi$
$948$		0			0
$949$			22.0000i			0.714150i
$950$		0			0
$951$	−34.0000			−1.10253
$952$		0			0
$953$	−36.4829	−	36.4829i	−1.18180	−	1.18180i	−0.979279	−	0.202518i	$-0.935088\pi$
$953$	−36.4829	−	36.4829i	−1.18180	−	1.18180i	−0.202518	−	0.979279i	$-0.564912\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	22.0000	+	22.0000i	0.711159	+	0.711159i
$958$		0			0
$959$	−6.63325			−0.214199
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$	3.31662	+	3.31662i	0.106877	+	0.106877i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−9.94987	+	9.94987i	−0.319966	+	0.319966i	−0.848754	−	0.528788i	$-0.822647\pi$
$967$	−9.94987	+	9.94987i	−0.319966	+	0.319966i	0.528788	+	0.848754i	$0.322647\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8.00000			−0.256732			−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000			−0.256732			−0.128366	+	0.991727i	$0.540973\pi$
$972$		0			0
$973$	−44.0000	+	44.0000i	−1.41058	+	1.41058i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−13.0000	−	13.0000i	−0.415907	−	0.415907i	0.467883	−	0.883790i	$-0.345017\pi$
$977$	−13.0000	−	13.0000i	−0.415907	−	0.415907i	−0.883790	+	0.467883i	$0.845017\pi$
$978$		0			0
$979$	−39.7995			−1.27200
$980$		0			0
$981$			6.63325i			0.211783i
$982$		0			0
$983$	−15.0000	+	15.0000i	−0.478426	+	0.478426i	−0.904628	−	0.426202i	$-0.859851\pi$
$983$	−15.0000	+	15.0000i	−0.478426	+	0.478426i	0.426202	+	0.904628i	$0.359851\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−33.1662	+	33.1662i	−1.05569	+	1.05569i
$988$		0			0
$989$	19.8997			0.632775
$990$		0			0
$991$	−44.0000			−1.39771			−0.698853	−	0.715265i	$-0.746306\pi$
$991$	−44.0000			−1.39771			−0.698853	+	0.715265i	$0.746306\pi$
$992$		0			0
$993$	−16.0000	+	16.0000i	−0.507745	+	0.507745i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	16.5831	−	16.5831i	0.525193	−	0.525193i	−0.393942	−	0.919135i	$-0.628889\pi$
$997$	16.5831	−	16.5831i	0.525193	−	0.525193i	0.919135	+	0.393942i	$0.128889\pi$
$998$		0			0
$999$		−	40.0000i		−	1.26554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.2.k.a.857.2		4
5.2	odd	4		220.2.k.a.153.2	yes	4
5.3	odd	4	inner	1100.2.k.a.593.1		4
5.4	even	2		220.2.k.a.197.1	yes	4
11.10	odd	2	inner	1100.2.k.a.857.1		4
15.2	even	4		1980.2.y.a.1693.2		4
15.14	odd	2		1980.2.y.a.1297.1		4
20.7	even	4		880.2.bd.f.593.1		4
20.19	odd	2		880.2.bd.f.417.2		4
55.32	even	4		220.2.k.a.153.1	✓	4
55.43	even	4	inner	1100.2.k.a.593.2		4
55.54	odd	2		220.2.k.a.197.2	yes	4
165.32	odd	4		1980.2.y.a.1693.1		4
165.164	even	2		1980.2.y.a.1297.2		4
220.87	odd	4		880.2.bd.f.593.2		4
220.219	even	2		880.2.bd.f.417.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.k.a.153.1	✓	4	55.32	even	4
220.2.k.a.153.2	yes	4	5.2	odd	4
220.2.k.a.197.1	yes	4	5.4	even	2
220.2.k.a.197.2	yes	4	55.54	odd	2
880.2.bd.f.417.1		4	220.219	even	2
880.2.bd.f.417.2		4	20.19	odd	2
880.2.bd.f.593.1		4	20.7	even	4
880.2.bd.f.593.2		4	220.87	odd	4
1100.2.k.a.593.1		4	5.3	odd	4	inner
1100.2.k.a.593.2		4	55.43	even	4	inner
1100.2.k.a.857.1		4	11.10	odd	2	inner
1100.2.k.a.857.2		4	1.1	even	1	trivial
1980.2.y.a.1297.1		4	15.14	odd	2
1980.2.y.a.1297.2		4	165.164	even	2
1980.2.y.a.1693.1		4	165.32	odd	4
1980.2.y.a.1693.2		4	15.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1100.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.00000i) q^{3} +(3.31662 - 3.31662i) q^{7} +1.00000i q^{9} +3.31662i q^{11} +(3.31662 + 3.31662i) q^{13} +(3.31662 - 3.31662i) q^{17} -6.63325i q^{21} +(-3.00000 + 3.00000i) q^{23} +(4.00000 + 4.00000i) q^{27} +6.63325 q^{29} -4.00000 q^{31} +(3.31662 + 3.31662i) q^{33} +(-5.00000 - 5.00000i) q^{37} +6.63325 q^{39} +(-3.31662 - 3.31662i) q^{43} +(-5.00000 - 5.00000i) q^{47} -15.0000i q^{49} -6.63325i q^{51} +(3.00000 - 3.00000i) q^{53} +10.0000i q^{59} -13.2665i q^{61} +(3.31662 + 3.31662i) q^{63} +(3.00000 + 3.00000i) q^{67} +6.00000i q^{69} -4.00000 q^{71} +(3.31662 + 3.31662i) q^{73} +(11.0000 + 11.0000i) q^{77} -13.2665 q^{79} +5.00000 q^{81} +(-3.31662 - 3.31662i) q^{83} +(6.63325 - 6.63325i) q^{87} +12.0000i q^{89} +22.0000 q^{91} +(-4.00000 + 4.00000i) q^{93} +(-5.00000 - 5.00000i) q^{97} -3.31662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 12 q^{23} + 16 q^{27} - 16 q^{31} - 20 q^{37} - 20 q^{47} + 12 q^{53} + 12 q^{67} - 16 q^{71} + 44 q^{77} + 20 q^{81} + 88 q^{91} - 16 q^{93} - 20 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 12 * q^23 + 16 * q^27 - 16 * q^31 - 20 * q^37 - 20 * q^47 + 12 * q^53 + 12 * q^67 - 16 * q^71 + 44 * q^77 + 20 * q^81 + 88 * q^91 - 16 * q^93 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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