LMFDB - Embedded newform 1980.2.y.a.1693.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1980,2,Mod(1297,1980)] 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("1980.1297"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1980, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1980.y (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$15.8103796002$
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			1693.2
Root			$-1.65831 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1980.1693
Dual form			1980.2.y.a.1297.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 2.00000i) q^{5} +(3.31662 + 3.31662i) q^{7} -3.31662i q^{11} +(3.31662 - 3.31662i) q^{13} +(-3.31662 - 3.31662i) q^{17} +(-3.00000 - 3.00000i) q^{23} +(-3.00000 - 4.00000i) q^{25} +6.63325 q^{29} -4.00000 q^{31} +(9.94987 - 3.31662i) q^{35} +(5.00000 - 5.00000i) q^{37} +(-3.31662 + 3.31662i) q^{43} +(-5.00000 + 5.00000i) q^{47} +15.0000i q^{49} +(3.00000 + 3.00000i) q^{53} +(-6.63325 - 3.31662i) q^{55} +10.0000i q^{59} -13.2665i q^{61} +(-3.31662 - 9.94987i) q^{65} +(-3.00000 + 3.00000i) q^{67} +4.00000 q^{71} +(3.31662 - 3.31662i) q^{73} +(11.0000 - 11.0000i) q^{77} +13.2665 q^{79} +(3.31662 - 3.31662i) q^{83} +(-9.94987 + 3.31662i) q^{85} +12.0000i q^{89} +22.0000 q^{91} +(5.00000 - 5.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 12 q^{23} - 12 q^{25} - 16 q^{31} + 20 q^{37} - 20 q^{47} + 12 q^{53} - 12 q^{67} + 16 q^{71} + 44 q^{77} + 88 q^{91} + 20 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 12 * q^23 - 12 * q^25 - 16 * q^31 + 20 * q^37 - 20 * q^47 + 12 * q^53 - 12 * q^67 + 16 * q^71 + 44 * q^77 + 88 * q^91 + 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1980\mathbb{Z}\right)^\times$.

$n$	$397$	$541$	$991$	$1541$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$	3.31662	+	3.31662i	1.25357	+	1.25357i	0.954110	+	0.299456i	$0.0968053\pi$
$7$	3.31662	+	3.31662i	1.25357	+	1.25357i	0.299456	+	0.954110i	$0.403195\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.31662i		−	1.00000i
$11$		−	3.31662i		−	1.00000i
$12$		0			0
$13$	3.31662	−	3.31662i	0.919866	−	0.919866i	−0.0771531	−	0.997019i	$-0.524583\pi$
$13$	3.31662	−	3.31662i	0.919866	−	0.919866i	0.997019	+	0.0771531i	$0.0245830\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.31662	−	3.31662i	−0.804400	−	0.804400i	0.179380	−	0.983780i	$-0.442591\pi$
$17$	−3.31662	−	3.31662i	−0.804400	−	0.804400i	−0.983780	+	0.179380i	$0.942591\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.00000	−	3.00000i	−0.625543	−	0.625543i	0.321400	−	0.946943i	$-0.395847\pi$
$23$	−3.00000	−	3.00000i	−0.625543	−	0.625543i	−0.946943	+	0.321400i	$0.895847\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		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
$33$		0			0
$34$		0			0
$35$	9.94987	−	3.31662i	1.68184	−	0.560612i
$36$		0			0
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	5.00000	−	5.00000i	0.821995	−	0.821995i	0.986394	+	0.164399i	$0.0525685\pi$
$38$		0			0
$39$		0			0
$40$		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	−0.913228	−	0.407448i	$-0.866419\pi$
$43$	−3.31662	+	3.31662i	−0.505781	+	0.505781i	0.407448	+	0.913228i	$0.366419\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−5.00000	+	5.00000i	−0.729325	+	0.729325i	−0.970485	−	0.241160i	$-0.922472\pi$
$47$	−5.00000	+	5.00000i	−0.729325	+	0.729325i	0.241160	+	0.970485i	$0.422472\pi$
$48$		0			0
$49$			15.0000i			2.14286i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.00000	+	3.00000i	0.412082	+	0.412082i	0.882463	−	0.470381i	$-0.155884\pi$
$53$	3.00000	+	3.00000i	0.412082	+	0.412082i	−0.470381	+	0.882463i	$0.655884\pi$
$54$		0			0
$55$	−6.63325	−	3.31662i	−0.894427	−	0.447214i
$56$		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
$63$		0			0
$64$		0			0
$65$	−3.31662	−	9.94987i	−0.411377	−	1.23413i
$66$		0			0
$67$	−3.00000	+	3.00000i	−0.366508	+	0.366508i	−0.866202	−	0.499694i	$-0.833446\pi$
$67$	−3.00000	+	3.00000i	−0.366508	+	0.366508i	0.499694	+	0.866202i	$0.333446\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$	3.31662	−	3.31662i	0.388182	−	0.388182i	−0.485857	−	0.874038i	$-0.661492\pi$
$73$	3.31662	−	3.31662i	0.388182	−	0.388182i	0.874038	+	0.485857i	$0.161492\pi$
$74$		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	$-0.231829\pi$
$79$	13.2665			1.49260			0.746299	+	0.665611i	$0.231829\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.31662	−	3.31662i	0.364047	−	0.364047i	−0.501254	−	0.865300i	$-0.667128\pi$
$83$	3.31662	−	3.31662i	0.364047	−	0.364047i	0.865300	+	0.501254i	$0.167128\pi$
$84$		0			0
$85$	−9.94987	+	3.31662i	−1.07922	+	0.359738i
$86$		0			0
$87$		0			0
$88$		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
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	5.00000	−	5.00000i	0.507673	−	0.507673i	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	5.00000	−	5.00000i	0.507673	−	0.507673i	0.913812	+	0.406138i	$0.133125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.2665i		−	1.32007i	−0.751237	−	0.660033i	$-0.770542\pi$
$101$		−	13.2665i		−	1.32007i	0.751237	−	0.660033i	$-0.229458\pi$
$102$		0			0
$103$	−9.00000	−	9.00000i	−0.886796	−	0.886796i	0.107418	−	0.994214i	$-0.465742\pi$
$103$	−9.00000	−	9.00000i	−0.886796	−	0.886796i	−0.994214	+	0.107418i	$0.965742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.31662	−	3.31662i	−0.320630	−	0.320630i	0.528379	−	0.849009i	$-0.322800\pi$
$107$	−3.31662	−	3.31662i	−0.320630	−	0.320630i	−0.849009	+	0.528379i	$0.822800\pi$
$108$		0			0
$109$	−6.63325			−0.635350			−0.317675	−	0.948200i	$-0.602902\pi$
$109$	−6.63325			−0.635350			−0.317675	+	0.948200i	$0.602902\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.00000	−	1.00000i	−0.0940721	−	0.0940721i	0.658505	−	0.752577i	$-0.271189\pi$
$113$	−1.00000	−	1.00000i	−0.0940721	−	0.0940721i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$	−9.00000	+	3.00000i	−0.839254	+	0.279751i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	22.0000i		−	2.01674i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$	3.31662	+	3.31662i	0.294303	+	0.294303i	0.838777	−	0.544475i	$-0.183271\pi$
$127$	3.31662	+	3.31662i	0.294303	+	0.294303i	−0.544475	+	0.838777i	$0.683271\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			6.63325i			0.579550i	0.957095	+	0.289775i	$0.0935804\pi$
$131$			6.63325i			0.579550i	−0.957095	+	0.289775i	$0.906420\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	+	1.00000i	−0.0854358	+	0.0854358i	−0.748533	−	0.663097i	$-0.769242\pi$
$137$	−1.00000	+	1.00000i	−0.0854358	+	0.0854358i	0.663097	+	0.748533i	$0.269242\pi$
$138$		0			0
$139$	13.2665			1.12525			0.562625	−	0.826712i	$-0.309792\pi$
$139$	13.2665			1.12525			0.562625	+	0.826712i	$0.309792\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−11.0000	−	11.0000i	−0.919866	−	0.919866i
$144$		0			0
$145$	6.63325	−	13.2665i	0.550861	−	1.10172i
$146$		0			0
$147$		0			0
$148$		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
$153$		0			0
$154$		0			0
$155$	−4.00000	+	8.00000i	−0.321288	+	0.642575i
$156$		0			0
$157$	1.00000	−	1.00000i	0.0798087	−	0.0798087i	−0.666076	−	0.745884i	$-0.732027\pi$
$157$	1.00000	−	1.00000i	0.0798087	−	0.0798087i	0.745884	+	0.666076i	$0.232027\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	19.8997i		−	1.56832i
$162$		0			0
$163$	3.00000	+	3.00000i	0.234978	+	0.234978i	0.814767	−	0.579789i	$-0.196865\pi$
$163$	3.00000	+	3.00000i	0.234978	+	0.234978i	−0.579789	+	0.814767i	$0.696865\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.94987	+	9.94987i	0.769944	+	0.769944i	0.978096	−	0.208152i	$-0.0667449\pi$
$167$	9.94987	+	9.94987i	0.769944	+	0.769944i	−0.208152	+	0.978096i	$0.566745\pi$
$168$		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	$-0.807061\pi$
$173$	−3.31662	+	3.31662i	−0.252158	+	0.252158i	0.569697	+	0.821855i	$0.307061\pi$
$174$		0			0
$175$	3.31662	−	23.2164i	0.250713	−	1.75499i
$176$		0			0
$177$		0			0
$178$		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
$183$		0			0
$184$		0			0
$185$	−5.00000	−	15.0000i	−0.367607	−	1.10282i
$186$		0			0
$187$	−11.0000	+	11.0000i	−0.804400	+	0.804400i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$	−9.94987	+	9.94987i	−0.716208	+	0.716208i	−0.967826	−	0.251619i	$-0.919037\pi$
$193$	−9.94987	+	9.94987i	−0.716208	+	0.716208i	0.251619	+	0.967826i	$0.419037\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.94987	+	9.94987i	0.708899	+	0.708899i	0.966304	−	0.257405i	$-0.0828673\pi$
$197$	9.94987	+	9.94987i	0.708899	+	0.708899i	−0.257405	+	0.966304i	$0.582867\pi$
$198$		0			0
$199$		−	22.0000i		−	1.55954i	−0.626067	−	0.779769i	$-0.715336\pi$
$199$		−	22.0000i		−	1.55954i	0.626067	−	0.779769i	$-0.284664\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	22.0000	+	22.0000i	1.54410	+	1.54410i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		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
$213$		0			0
$214$		0			0
$215$	3.31662	+	9.94987i	0.226192	+	0.678576i
$216$		0			0
$217$	−13.2665	−	13.2665i	−0.900589	−	0.900589i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−22.0000			−1.47988
$222$		0			0
$223$	7.00000	+	7.00000i	0.468755	+	0.468755i	0.901511	−	0.432756i	$-0.142459\pi$
$223$	7.00000	+	7.00000i	0.468755	+	0.468755i	−0.432756	+	0.901511i	$0.642459\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.94987	+	9.94987i	0.660396	+	0.660396i	0.955473	−	0.295077i	$-0.0953453\pi$
$227$	9.94987	+	9.94987i	0.660396	+	0.660396i	−0.295077	+	0.955473i	$0.595345\pi$
$228$		0			0
$229$			4.00000i			0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$			4.00000i			0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.31662	+	3.31662i	−0.217279	+	0.217279i	−0.807351	−	0.590072i	$-0.799100\pi$
$233$	−3.31662	+	3.31662i	−0.217279	+	0.217279i	0.590072	+	0.807351i	$0.299100\pi$
$234$		0			0
$235$	5.00000	+	15.0000i	0.326164	+	0.978492i
$236$		0			0
$237$		0			0
$238$		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
$243$		0			0
$244$		0			0
$245$	30.0000	+	15.0000i	1.91663	+	0.958315i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.00000			0.504956			0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000			0.504956			0.252478	+	0.967603i	$0.418755\pi$
$252$		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	−0.929706	−	0.368302i	$-0.879939\pi$
$257$	−9.00000	+	9.00000i	−0.561405	+	0.561405i	0.368302	+	0.929706i	$0.379939\pi$
$258$		0			0
$259$	33.1662			2.06085
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.31662	−	3.31662i	0.204512	−	0.204512i	−0.597418	−	0.801930i	$-0.703807\pi$
$263$	3.31662	−	3.31662i	0.204512	−	0.204512i	0.801930	+	0.597418i	$0.203807\pi$
$264$		0			0
$265$	9.00000	−	3.00000i	0.552866	−	0.184289i
$266$		0			0
$267$		0			0
$268$		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
$273$		0			0
$274$		0			0
$275$	−13.2665	+	9.94987i	−0.800000	+	0.600000i
$276$		0			0
$277$	16.5831	+	16.5831i	0.996383	+	0.996383i	0.999993	−	0.00361013i	$-0.00114914\pi$
$277$	16.5831	+	16.5831i	0.996383	+	0.996383i	−0.00361013	+	0.999993i	$0.501149\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	26.5330i		−	1.58283i	−0.611282	−	0.791413i	$-0.709346\pi$
$281$		−	26.5330i		−	1.58283i	0.611282	−	0.791413i	$-0.290654\pi$
$282$		0			0
$283$	−16.5831	+	16.5831i	−0.985764	+	0.985764i	−0.999900	−	0.0141357i	$-0.995500\pi$
$283$	−16.5831	+	16.5831i	−0.985764	+	0.985764i	0.0141357	+	0.999900i	$0.495500\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			5.00000i			0.294118i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.5831	+	16.5831i	−0.968796	+	0.968796i	−0.999528	−	0.0307312i	$-0.990216\pi$
$293$	−16.5831	+	16.5831i	−0.968796	+	0.968796i	0.0307312	+	0.999528i	$0.490216\pi$
$294$		0			0
$295$	20.0000	+	10.0000i	1.16445	+	0.582223i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−19.8997			−1.15083
$300$		0			0
$301$	−22.0000			−1.26806
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−26.5330	−	13.2665i	−1.51928	−	0.759638i
$306$		0			0
$307$	3.31662	+	3.31662i	0.189290	+	0.189290i	0.795389	−	0.606099i	$-0.207267\pi$
$307$	3.31662	+	3.31662i	0.189290	+	0.189290i	−0.606099	+	0.795389i	$0.707267\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$	21.0000	+	21.0000i	1.18699	+	1.18699i	0.977895	+	0.209095i	$0.0670517\pi$
$313$	21.0000	+	21.0000i	1.18699	+	1.18699i	0.209095	+	0.977895i	$0.432948\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.0000	+	17.0000i	−0.954815	+	0.954815i	−0.999022	−	0.0442073i	$-0.985924\pi$
$317$	−17.0000	+	17.0000i	−0.954815	+	0.954815i	0.0442073	+	0.999022i	$0.485924\pi$
$318$		0			0
$319$		−	22.0000i		−	1.23176i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−23.2164	−	3.31662i	−1.28781	−	0.183973i
$326$		0			0
$327$		0			0
$328$		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
$333$		0			0
$334$		0			0
$335$	3.00000	+	9.00000i	0.163908	+	0.491723i
$336$		0			0
$337$	3.31662	+	3.31662i	0.180668	+	0.180668i	0.791647	−	0.610979i	$-0.209224\pi$
$337$	3.31662	+	3.31662i	0.180668	+	0.180668i	−0.610979	+	0.791647i	$0.709224\pi$
$338$		0			0
$339$		0			0
$340$		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	$-0.126718\pi$
$347$	9.94987	+	9.94987i	0.534137	+	0.534137i	−0.387664	+	0.921801i	$0.626718\pi$
$348$		0			0
$349$	19.8997			1.06521			0.532605	−	0.846364i	$-0.321213\pi$
$349$	19.8997			1.06521			0.532605	+	0.846364i	$0.321213\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.00000	−	9.00000i	−0.479022	−	0.479022i	0.425797	−	0.904819i	$-0.359994\pi$
$353$	−9.00000	−	9.00000i	−0.479022	−	0.479022i	−0.904819	+	0.425797i	$0.859994\pi$
$354$		0			0
$355$	4.00000	−	8.00000i	0.212298	−	0.424596i
$356$		0			0
$357$		0			0
$358$		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
$363$		0			0
$364$		0			0
$365$	−3.31662	−	9.94987i	−0.173600	−	0.520800i
$366$		0			0
$367$	−15.0000	+	15.0000i	−0.782994	+	0.782994i	−0.980335	−	0.197341i	$-0.936769\pi$
$367$	−15.0000	+	15.0000i	−0.782994	+	0.782994i	0.197341	+	0.980335i	$0.436769\pi$
$368$		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	−0.616010	−	0.787738i	$-0.711252\pi$
$373$	3.31662	−	3.31662i	0.171728	−	0.171728i	0.787738	+	0.616010i	$0.211252\pi$
$374$		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	$-0.983642\pi$
$379$		−	2.00000i		−	0.102733i	0.998680	−	0.0513665i	$-0.0163577\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.00000	+	5.00000i	0.255488	+	0.255488i	0.823216	−	0.567728i	$-0.192177\pi$
$383$	5.00000	+	5.00000i	0.255488	+	0.255488i	−0.567728	+	0.823216i	$0.692177\pi$
$384$		0			0
$385$	−11.0000	−	33.0000i	−0.560612	−	1.68184i
$386$		0			0
$387$		0			0
$388$		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
$393$		0			0
$394$		0			0
$395$	13.2665	−	26.5330i	0.667510	−	1.33502i
$396$		0			0
$397$	−23.0000	+	23.0000i	−1.15434	+	1.15434i	−0.168663	+	0.985674i	$0.553945\pi$
$397$	−23.0000	+	23.0000i	−1.15434	+	1.15434i	−0.985674	+	0.168663i	$0.946055\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−22.0000			−1.09863			−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000			−1.09863			−0.549314	+	0.835616i	$0.685111\pi$
$402$		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	$-0.663730\pi$
$409$	−19.8997			−0.983979			−0.491990	+	0.870601i	$0.663730\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−33.1662	+	33.1662i	−1.63200	+	1.63200i
$414$		0			0
$415$	−3.31662	−	9.94987i	−0.162807	−	0.488420i
$416$		0			0
$417$		0			0
$418$		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
$423$		0			0
$424$		0			0
$425$	−3.31662	+	23.2164i	−0.160880	+	1.12616i
$426$		0			0
$427$	44.0000	−	44.0000i	2.12931	−	2.12931i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			33.1662i			1.59756i	0.601622	+	0.798781i	$0.294521\pi$
$431$			33.1662i			1.59756i	−0.601622	+	0.798781i	$0.705479\pi$
$432$		0			0
$433$	−15.0000	−	15.0000i	−0.720854	−	0.720854i	0.247925	−	0.968779i	$-0.420251\pi$
$433$	−15.0000	−	15.0000i	−0.720854	−	0.720854i	−0.968779	+	0.247925i	$0.920251\pi$
$434$		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
$441$		0			0
$442$		0			0
$443$	−15.0000	−	15.0000i	−0.712672	−	0.712672i	0.254422	−	0.967093i	$-0.418115\pi$
$443$	−15.0000	−	15.0000i	−0.712672	−	0.712672i	−0.967093	+	0.254422i	$0.918115\pi$
$444$		0			0
$445$	24.0000	+	12.0000i	1.13771	+	0.568855i
$446$		0			0
$447$		0			0
$448$		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
$453$		0			0
$454$		0			0
$455$	22.0000	−	44.0000i	1.03138	−	2.06275i
$456$		0			0
$457$	3.31662	+	3.31662i	0.155145	+	0.155145i	0.780411	−	0.625266i	$-0.215010\pi$
$457$	3.31662	+	3.31662i	0.155145	+	0.155145i	−0.625266	+	0.780411i	$0.715010\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			39.7995i			1.85365i	0.375497	+	0.926824i	$0.377472\pi$
$461$			39.7995i			1.85365i	−0.375497	+	0.926824i	$0.622528\pi$
$462$		0			0
$463$	−9.00000	−	9.00000i	−0.418265	−	0.418265i	0.466340	−	0.884606i	$-0.345572\pi$
$463$	−9.00000	−	9.00000i	−0.418265	−	0.418265i	−0.884606	+	0.466340i	$0.845572\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.00000	+	1.00000i	−0.0462745	+	0.0462745i	−0.729865	−	0.683591i	$-0.760417\pi$
$467$	−1.00000	+	1.00000i	−0.0462745	+	0.0462745i	0.683591	+	0.729865i	$0.260417\pi$
$468$		0			0
$469$	−19.8997			−0.918885
$470$		0			0
$471$		0			0
$472$		0			0
$473$	11.0000	+	11.0000i	0.505781	+	0.505781i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		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
$483$		0			0
$484$		0			0
$485$	−5.00000	−	15.0000i	−0.227038	−	0.681115i
$486$		0			0
$487$	−23.0000	+	23.0000i	−1.04223	+	1.04223i	−0.0431614	+	0.999068i	$0.513743\pi$
$487$	−23.0000	+	23.0000i	−1.04223	+	1.04223i	−0.999068	+	0.0431614i	$0.986257\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	33.1662i		−	1.49677i	−0.663264	−	0.748386i	$-0.730829\pi$
$491$		−	33.1662i		−	1.49677i	0.663264	−	0.748386i	$-0.269171\pi$
$492$		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	$0.163890\pi$
$499$			22.0000i			0.984855i	−0.870353	+	0.492428i	$0.836110\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−9.94987	+	9.94987i	−0.443643	+	0.443643i	−0.893234	−	0.449591i	$-0.851570\pi$
$503$	−9.94987	+	9.94987i	−0.443643	+	0.443643i	0.449591	+	0.893234i	$0.351570\pi$
$504$		0			0
$505$	−26.5330	−	13.2665i	−1.18070	−	0.590351i
$506$		0			0
$507$		0			0
$508$		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
$515$	−27.0000	+	9.00000i	−1.18976	+	0.396587i
$516$		0			0
$517$	16.5831	+	16.5831i	0.729325	+	0.729325i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	34.0000			1.48957			0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000			1.48957			0.744784	+	0.667306i	$0.232553\pi$
$522$		0			0
$523$	−16.5831	+	16.5831i	−0.725129	+	0.725129i	−0.969645	−	0.244516i	$-0.921371\pi$
$523$	−16.5831	+	16.5831i	−0.725129	+	0.725129i	0.244516	+	0.969645i	$0.421371\pi$
$524$		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
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−9.94987	+	3.31662i	−0.430171	+	0.143390i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		0			0
$545$	−6.63325	+	13.2665i	−0.284137	+	0.568274i
$546$		0			0
$547$	29.8496	+	29.8496i	1.27628	+	1.27628i	0.942735	+	0.333543i	$0.108244\pi$
$547$	29.8496	+	29.8496i	1.27628	+	1.27628i	0.333543	+	0.942735i	$0.391756\pi$
$548$		0			0
$549$		0			0
$550$		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	−0.948754	−	0.316015i	$-0.897655\pi$
$557$	−29.8496	−	29.8496i	−1.26477	−	1.26477i	−0.316015	−	0.948754i	$-0.602345\pi$
$558$		0			0
$559$			22.0000i			0.930501i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9.94987	+	9.94987i	−0.419337	+	0.419337i	−0.884975	−	0.465638i	$-0.845825\pi$
$563$	−9.94987	+	9.94987i	−0.419337	+	0.419337i	0.465638	+	0.884975i	$0.345825\pi$
$564$		0			0
$565$	−3.00000	+	1.00000i	−0.126211	+	0.0420703i
$566$		0			0
$567$		0			0
$568$		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
$573$		0			0
$574$		0			0
$575$	−3.00000	+	21.0000i	−0.125109	+	0.875761i
$576$		0			0
$577$	21.0000	−	21.0000i	0.874241	−	0.874241i	−0.118690	−	0.992931i	$-0.537869\pi$
$577$	21.0000	−	21.0000i	0.874241	−	0.874241i	0.992931	+	0.118690i	$0.0378694\pi$
$578$		0			0
$579$		0			0
$580$		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	−0.922519	−	0.385952i	$-0.873873\pi$
$587$	−13.0000	+	13.0000i	−0.536567	+	0.536567i	0.385952	+	0.922519i	$0.373873\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.94987	−	9.94987i	0.408592	−	0.408592i	−0.472655	−	0.881247i	$-0.656704\pi$
$593$	9.94987	−	9.94987i	0.408592	−	0.408592i	0.881247	+	0.472655i	$0.156704\pi$
$594$		0			0
$595$	−44.0000	−	22.0000i	−1.80382	−	0.901912i
$596$		0			0
$597$		0			0
$598$		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
$603$		0			0
$604$		0			0
$605$	−11.0000	+	22.0000i	−0.447214	+	0.894427i
$606$		0			0
$607$	3.31662	+	3.31662i	0.134618	+	0.134618i	0.771205	−	0.636587i	$-0.219654\pi$
$607$	3.31662	+	3.31662i	0.134618	+	0.134618i	−0.636587	+	0.771205i	$0.719654\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			33.1662i			1.34176i
$612$		0			0
$613$	16.5831	−	16.5831i	0.669786	−	0.669786i	−0.287880	−	0.957666i	$-0.592951\pi$
$613$	16.5831	−	16.5831i	0.669786	−	0.669786i	0.957666	+	0.287880i	$0.0929505\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.00000	+	1.00000i	−0.0402585	+	0.0402585i	−0.726949	−	0.686691i	$-0.759063\pi$
$617$	−1.00000	+	1.00000i	−0.0402585	+	0.0402585i	0.686691	+	0.726949i	$0.259063\pi$
$618$		0			0
$619$			38.0000i			1.52735i	0.645601	+	0.763674i	$0.276607\pi$
$619$			38.0000i			1.52735i	−0.645601	+	0.763674i	$0.723393\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−39.7995	+	39.7995i	−1.59453	+	1.59453i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		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
$633$		0			0
$634$		0			0
$635$	9.94987	−	3.31662i	0.394849	−	0.131616i
$636$		0			0
$637$	49.7494	+	49.7494i	1.97114	+	1.97114i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$	−21.0000	−	21.0000i	−0.828159	−	0.828159i	0.159103	−	0.987262i	$-0.449140\pi$
$643$	−21.0000	−	21.0000i	−0.828159	−	0.828159i	−0.987262	+	0.159103i	$0.949140\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17.0000	+	17.0000i	−0.668339	+	0.668339i	−0.957331	−	0.288992i	$-0.906680\pi$
$647$	−17.0000	+	17.0000i	−0.668339	+	0.668339i	0.288992	+	0.957331i	$0.406680\pi$
$648$		0			0
$649$	33.1662			1.30189
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.00000	−	5.00000i	−0.195665	−	0.195665i	0.602474	−	0.798139i	$-0.294182\pi$
$653$	−5.00000	−	5.00000i	−0.195665	−	0.195665i	−0.798139	+	0.602474i	$0.794182\pi$
$654$		0			0
$655$	13.2665	+	6.63325i	0.518365	+	0.259183i
$656$		0			0
$657$		0			0
$658$		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
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−19.8997	−	19.8997i	−0.770521	−	0.770521i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−44.0000			−1.69860
$672$		0			0
$673$	−9.94987	+	9.94987i	−0.383539	+	0.383539i	−0.872376	−	0.488836i	$-0.837422\pi$
$673$	−9.94987	+	9.94987i	−0.383539	+	0.383539i	0.488836	+	0.872376i	$0.337422\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−29.8496	−	29.8496i	−1.14721	−	1.14721i	−0.987098	−	0.160116i	$-0.948813\pi$
$677$	−29.8496	−	29.8496i	−1.14721	−	1.14721i	−0.160116	−	0.987098i	$-0.551187\pi$
$678$		0			0
$679$	33.1662			1.27280
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−23.0000	−	23.0000i	−0.880071	−	0.880071i	0.113471	−	0.993541i	$-0.463803\pi$
$683$	−23.0000	−	23.0000i	−0.880071	−	0.880071i	−0.993541	+	0.113471i	$0.963803\pi$
$684$		0			0
$685$	1.00000	+	3.00000i	0.0382080	+	0.114624i
$686$		0			0
$687$		0			0
$688$		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
$693$		0			0
$694$		0			0
$695$	13.2665	−	26.5330i	0.503227	−	1.00645i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	13.2665i		−	0.501069i	−0.968108	−	0.250534i	$-0.919394\pi$
$701$		−	13.2665i		−	0.501069i	0.968108	−	0.250534i	$-0.0806063\pi$
$702$		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	$-0.729516\pi$
$709$		−	40.0000i		−	1.50223i	0.660171	−	0.751116i	$-0.270484\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12.0000	+	12.0000i	0.449404	+	0.449404i
$714$		0			0
$715$	−33.0000	+	11.0000i	−1.23413	+	0.411377i
$716$		0			0
$717$		0			0
$718$		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
$723$		0			0
$724$		0			0
$725$	−19.8997	−	26.5330i	−0.739058	−	0.985411i
$726$		0			0
$727$	25.0000	−	25.0000i	0.927199	−	0.927199i	−0.0703254	−	0.997524i	$-0.522404\pi$
$727$	25.0000	−	25.0000i	0.927199	−	0.927199i	0.997524	+	0.0703254i	$0.0224038\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	22.0000			0.813699
$732$		0			0
$733$	−36.4829	+	36.4829i	−1.34753	+	1.34753i	−0.459185	+	0.888341i	$0.651858\pi$
$733$	−36.4829	+	36.4829i	−1.34753	+	1.34753i	−0.888341	+	0.459185i	$0.848142\pi$
$734$		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	$-0.761424\pi$
$739$	−39.7995			−1.46405			−0.732024	+	0.681279i	$0.761424\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	16.5831	−	16.5831i	0.608376	−	0.608376i	−0.334146	−	0.942521i	$-0.608448\pi$
$743$	16.5831	−	16.5831i	0.608376	−	0.608376i	0.942521	+	0.334146i	$0.108448\pi$
$744$		0			0
$745$	6.63325	−	13.2665i	0.243023	−	0.486047i
$746$		0			0
$747$		0			0
$748$		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
$753$		0			0
$754$		0			0
$755$	13.2665	+	6.63325i	0.482817	+	0.241409i
$756$		0			0
$757$	−3.00000	+	3.00000i	−0.109037	+	0.109037i	−0.759520	−	0.650484i	$-0.774566\pi$
$757$	−3.00000	+	3.00000i	−0.109037	+	0.109037i	0.650484	+	0.759520i	$0.274566\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	26.5330i		−	0.961820i	−0.876770	−	0.480910i	$-0.840306\pi$
$761$		−	26.5330i		−	0.961820i	0.876770	−	0.480910i	$-0.159694\pi$
$762$		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	$-0.616815\pi$
$769$	−19.8997			−0.717603			−0.358802	+	0.933414i	$0.616815\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.00000	−	5.00000i	−0.179838	−	0.179838i	0.611448	−	0.791285i	$-0.290588\pi$
$773$	−5.00000	−	5.00000i	−0.179838	−	0.179838i	−0.791285	+	0.611448i	$0.790588\pi$
$774$		0			0
$775$	12.0000	+	16.0000i	0.431053	+	0.574737i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		−	13.2665i		−	0.474713i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.00000	−	3.00000i	−0.0356915	−	0.107075i
$786$		0			0
$787$	−23.2164	−	23.2164i	−0.827574	−	0.827574i	0.159606	−	0.987181i	$-0.448977\pi$
$787$	−23.2164	−	23.2164i	−0.827574	−	0.827574i	−0.987181	+	0.159606i	$0.948977\pi$
$788$		0			0
$789$		0			0
$790$		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	−0.285395	−	0.958410i	$-0.592125\pi$
$797$	19.0000	−	19.0000i	0.673015	−	0.673015i	0.958410	+	0.285395i	$0.0921249\pi$
$798$		0			0
$799$	33.1662			1.17334
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.0000	−	11.0000i	−0.388182	−	0.388182i
$804$		0			0
$805$	−39.7995	−	19.8997i	−1.40275	−	0.701374i
$806$		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
$813$		0			0
$814$		0			0
$815$	9.00000	−	3.00000i	0.315256	−	0.105085i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			39.7995i			1.38901i	0.719487	+	0.694506i	$0.244377\pi$
$821$			39.7995i			1.38901i	−0.719487	+	0.694506i	$0.755623\pi$
$822$		0			0
$823$	27.0000	+	27.0000i	0.941161	+	0.941161i	0.998363	−	0.0572018i	$-0.0182178\pi$
$823$	27.0000	+	27.0000i	0.941161	+	0.941161i	−0.0572018	+	0.998363i	$0.518218\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.31662	−	3.31662i	−0.115330	−	0.115330i	0.647086	−	0.762417i	$-0.275987\pi$
$827$	−3.31662	−	3.31662i	−0.115330	−	0.115330i	−0.762417	+	0.647086i	$0.775987\pi$
$828$		0			0
$829$			20.0000i			0.694629i	0.937749	+	0.347314i	$0.112906\pi$
$829$			20.0000i			0.694629i	−0.937749	+	0.347314i	$0.887094\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	49.7494	−	49.7494i	1.72371	−	1.72371i
$834$		0			0
$835$	29.8496	−	9.94987i	1.03299	−	0.344330i
$836$		0			0
$837$		0			0
$838$		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
$843$		0			0
$844$		0			0
$845$	−18.0000	−	9.00000i	−0.619219	−	0.309609i
$846$		0			0
$847$	−36.4829	−	36.4829i	−1.25357	−	1.25357i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−30.0000			−1.02839
$852$		0			0
$853$	−9.94987	+	9.94987i	−0.340677	+	0.340677i	−0.856622	−	0.515945i	$-0.827441\pi$
$853$	−9.94987	+	9.94987i	−0.340677	+	0.340677i	0.515945	+	0.856622i	$0.327441\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	23.2164	+	23.2164i	0.793056	+	0.793056i	0.981990	−	0.188934i	$-0.0605031\pi$
$857$	23.2164	+	23.2164i	0.793056	+	0.793056i	−0.188934	+	0.981990i	$0.560503\pi$
$858$		0			0
$859$		−	10.0000i		−	0.341196i	−0.985341	−	0.170598i	$-0.945430\pi$
$859$		−	10.0000i		−	0.341196i	0.985341	−	0.170598i	$-0.0545699\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	29.0000	+	29.0000i	0.987171	+	0.987171i	0.999919	−	0.0127473i	$-0.00405769\pi$
$863$	29.0000	+	29.0000i	0.987171	+	0.987171i	−0.0127473	+	0.999919i	$0.504058\pi$
$864$		0			0
$865$	3.31662	+	9.94987i	0.112769	+	0.338306i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	44.0000i		−	1.49260i
$870$		0			0
$871$			19.8997i			0.674277i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−43.1161	−	29.8496i	−1.45759	−	1.00910i
$876$		0			0
$877$	3.31662	+	3.31662i	0.111994	+	0.111994i	0.760883	−	0.648889i	$-0.224766\pi$
$877$	3.31662	+	3.31662i	0.111994	+	0.111994i	−0.648889	+	0.760883i	$0.724766\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−54.0000			−1.81931			−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000			−1.81931			−0.909653	+	0.415369i	$0.863653\pi$
$882$		0			0
$883$	27.0000	+	27.0000i	0.908622	+	0.908622i	0.996161	−	0.0875388i	$-0.0279002\pi$
$883$	27.0000	+	27.0000i	0.908622	+	0.908622i	−0.0875388	+	0.996161i	$0.527900\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−16.5831	−	16.5831i	−0.556807	−	0.556807i	0.371590	−	0.928397i	$-0.378813\pi$
$887$	−16.5831	−	16.5831i	−0.556807	−	0.556807i	−0.928397	+	0.371590i	$0.878813\pi$
$888$		0			0
$889$			22.0000i			0.737856i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−12.0000	−	6.00000i	−0.401116	−	0.200558i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−26.5330			−0.884925
$900$		0			0
$901$		−	19.8997i		−	0.662957i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.00000	−	4.00000i	0.0664822	−	0.132964i
$906$		0			0
$907$	−7.00000	+	7.00000i	−0.232431	+	0.232431i	−0.813707	−	0.581276i	$-0.802554\pi$
$907$	−7.00000	+	7.00000i	−0.232431	+	0.232431i	0.581276	+	0.813707i	$0.302554\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	44.0000			1.45779			0.728893	−	0.684628i	$-0.240035\pi$
$911$	44.0000			1.45779			0.728893	+	0.684628i	$0.240035\pi$
$912$		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	$-0.644179\pi$
$919$	−26.5330			−0.875243			−0.437621	+	0.899159i	$0.644179\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	13.2665	−	13.2665i	0.436672	−	0.436672i
$924$		0			0
$925$	−35.0000	−	5.00000i	−1.15079	−	0.164399i
$926$		0			0
$927$		0			0
$928$		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
$933$		0			0
$934$		0			0
$935$	11.0000	+	33.0000i	0.359738	+	1.07922i
$936$		0			0
$937$	−9.94987	−	9.94987i	−0.325048	−	0.325048i	0.525652	−	0.850700i	$-0.323822\pi$
$937$	−9.94987	−	9.94987i	−0.325048	−	0.325048i	−0.850700	+	0.525652i	$0.823822\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	39.7995i		−	1.29743i	−0.761033	−	0.648713i	$-0.775307\pi$
$941$		−	39.7995i		−	1.29743i	0.761033	−	0.648713i	$-0.224693\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.0000	−	19.0000i	0.617417	−	0.617417i	−0.327451	−	0.944868i	$-0.606190\pi$
$947$	19.0000	−	19.0000i	0.617417	−	0.617417i	0.944868	+	0.327451i	$0.106190\pi$
$948$		0			0
$949$		−	22.0000i		−	0.714150i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	36.4829	−	36.4829i	1.18180	−	1.18180i	0.202518	−	0.979279i	$-0.435088\pi$
$953$	36.4829	−	36.4829i	1.18180	−	1.18180i	0.979279	−	0.202518i	$-0.0649123\pi$
$954$		0			0
$955$	12.0000	−	24.0000i	0.388311	−	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.63325			−0.214199
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	9.94987	+	29.8496i	0.320298	+	0.960893i
$966$		0			0
$967$	−9.94987	−	9.94987i	−0.319966	−	0.319966i	0.528788	−	0.848754i	$-0.322647\pi$
$967$	−9.94987	−	9.94987i	−0.319966	−	0.319966i	−0.848754	+	0.528788i	$0.822647\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.00000			0.256732			0.128366	−	0.991727i	$-0.459027\pi$
$971$	8.00000			0.256732			0.128366	+	0.991727i	$0.459027\pi$
$972$		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	−0.883790	−	0.467883i	$-0.845017\pi$
$977$	−13.0000	+	13.0000i	−0.415907	+	0.415907i	0.467883	+	0.883790i	$0.345017\pi$
$978$		0			0
$979$	39.7995			1.27200
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−15.0000	−	15.0000i	−0.478426	−	0.478426i	0.426202	−	0.904628i	$-0.359851\pi$
$983$	−15.0000	−	15.0000i	−0.478426	−	0.478426i	−0.904628	+	0.426202i	$0.859851\pi$
$984$		0			0
$985$	29.8496	−	9.94987i	0.951088	−	0.317029i
$986$		0			0
$987$		0			0
$988$		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
$993$		0			0
$994$		0			0
$995$	−44.0000	−	22.0000i	−1.39489	−	0.697447i
$996$		0			0
$997$	16.5831	+	16.5831i	0.525193	+	0.525193i	0.919135	−	0.393942i	$-0.128889\pi$
$997$	16.5831	+	16.5831i	0.525193	+	0.525193i	−0.393942	+	0.919135i	$0.628889\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1980.2.y.a.1693.2		4
3.2	odd	2		220.2.k.a.153.2	yes	4
5.2	odd	4	inner	1980.2.y.a.1297.1		4
11.10	odd	2	inner	1980.2.y.a.1693.1		4
12.11	even	2		880.2.bd.f.593.1		4
15.2	even	4		220.2.k.a.197.1	yes	4
15.8	even	4		1100.2.k.a.857.2		4
15.14	odd	2		1100.2.k.a.593.1		4
33.32	even	2		220.2.k.a.153.1	✓	4
55.32	even	4	inner	1980.2.y.a.1297.2		4
60.47	odd	4		880.2.bd.f.417.2		4
132.131	odd	2		880.2.bd.f.593.2		4
165.32	odd	4		220.2.k.a.197.2	yes	4
165.98	odd	4		1100.2.k.a.857.1		4
165.164	even	2		1100.2.k.a.593.2		4
660.527	even	4		880.2.bd.f.417.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.k.a.153.1	✓	4	33.32	even	2
220.2.k.a.153.2	yes	4	3.2	odd	2
220.2.k.a.197.1	yes	4	15.2	even	4
220.2.k.a.197.2	yes	4	165.32	odd	4
880.2.bd.f.417.1		4	660.527	even	4
880.2.bd.f.417.2		4	60.47	odd	4
880.2.bd.f.593.1		4	12.11	even	2
880.2.bd.f.593.2		4	132.131	odd	2
1100.2.k.a.593.1		4	15.14	odd	2
1100.2.k.a.593.2		4	165.164	even	2
1100.2.k.a.857.1		4	165.98	odd	4
1100.2.k.a.857.2		4	15.8	even	4
1980.2.y.a.1297.1		4	5.2	odd	4	inner
1980.2.y.a.1297.2		4	55.32	even	4	inner
1980.2.y.a.1693.1		4	11.10	odd	2	inner
1980.2.y.a.1693.2		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1980.y (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 2.00000i) q^{5} +(3.31662 + 3.31662i) q^{7} -3.31662i q^{11} +(3.31662 - 3.31662i) q^{13} +(-3.31662 - 3.31662i) q^{17} +(-3.00000 - 3.00000i) q^{23} +(-3.00000 - 4.00000i) q^{25} +6.63325 q^{29} -4.00000 q^{31} +(9.94987 - 3.31662i) q^{35} +(5.00000 - 5.00000i) q^{37} +(-3.31662 + 3.31662i) q^{43} +(-5.00000 + 5.00000i) q^{47} +15.0000i q^{49} +(3.00000 + 3.00000i) q^{53} +(-6.63325 - 3.31662i) q^{55} +10.0000i q^{59} -13.2665i q^{61} +(-3.31662 - 9.94987i) q^{65} +(-3.00000 + 3.00000i) q^{67} +4.00000 q^{71} +(3.31662 - 3.31662i) q^{73} +(11.0000 - 11.0000i) q^{77} +13.2665 q^{79} +(3.31662 - 3.31662i) q^{83} +(-9.94987 + 3.31662i) q^{85} +12.0000i q^{89} +22.0000 q^{91} +(5.00000 - 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 12 q^{23} - 12 q^{25} - 16 q^{31} + 20 q^{37} - 20 q^{47} + 12 q^{53} - 12 q^{67} + 16 q^{71} + 44 q^{77} + 88 q^{91} + 20 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 12 * q^23 - 12 * q^25 - 16 * q^31 + 20 * q^37 - 20 * q^47 + 12 * q^53 - 12 * q^67 + 16 * q^71 + 44 * q^77 + 88 * q^91 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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