LMFDB - Embedded newform 280.2.n.a.139.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,2,Mod(139,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("280.139");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	280.n (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$2.23581125660$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 9 $ x^4 + 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			139.4
Root			$0.707107 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	280.139
Dual form			280.2.n.a.139.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +(1.41421 + 2.23607i) q^{7} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +(1.41421 + 2.23607i) q^{7} +2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} -2.00000 q^{11} -4.47214i q^{13} +(2.00000 + 3.16228i) q^{14} +4.00000 q^{16} -4.24264 q^{18} -6.32456i q^{19} +4.47214i q^{20} -2.82843 q^{22} +8.48528 q^{23} -5.00000 q^{25} -6.32456i q^{26} +(2.82843 + 4.47214i) q^{28} +5.65685 q^{32} +(-5.00000 + 3.16228i) q^{35} -6.00000 q^{36} -11.3137 q^{37} -8.94427i q^{38} +6.32456i q^{40} -12.6491i q^{41} -4.00000 q^{44} -6.70820i q^{45} +12.0000 q^{46} +13.4164i q^{47} +(-3.00000 + 6.32456i) q^{49} -7.07107 q^{50} -8.94427i q^{52} -5.65685 q^{53} -4.47214i q^{55} +(4.00000 + 6.32456i) q^{56} +6.32456i q^{59} +(-4.24264 - 6.70820i) q^{63} +8.00000 q^{64} +10.0000 q^{65} +(-7.07107 + 4.47214i) q^{70} -8.48528 q^{72} -16.0000 q^{74} -12.6491i q^{76} +(-2.82843 - 4.47214i) q^{77} +8.94427i q^{80} +9.00000 q^{81} -17.8885i q^{82} -5.65685 q^{88} +12.6491i q^{89} -9.48683i q^{90} +(10.0000 - 6.32456i) q^{91} +16.9706 q^{92} +18.9737i q^{94} +14.1421 q^{95} +(-4.24264 + 8.94427i) q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 12 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 - 12 * q^9	$ 4 q + 8 q^{4} - 12 q^{9} - 8 q^{11} + 8 q^{14} + 16 q^{16} - 20 q^{25} - 20 q^{35} - 24 q^{36} - 16 q^{44} + 48 q^{46} - 12 q^{49} + 16 q^{56} + 32 q^{64} + 40 q^{65} - 64 q^{74} + 36 q^{81} + 40 q^{91} + 24 q^{99}+O(q^{100}) $ 4 * q + 8 * q^4 - 12 * q^9 - 8 * q^11 + 8 * q^14 + 16 * q^16 - 20 * q^25 - 20 * q^35 - 24 * q^36 - 16 * q^44 + 48 * q^46 - 12 * q^49 + 16 * q^56 + 32 * q^64 + 40 * q^65 - 64 * q^74 + 36 * q^81 + 40 * q^91 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/280\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$141$	$241$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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 $

$2$	1.41421			1.00000
$2$	1.41421			1.00000
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	2.00000			1.00000
$5$			2.23607i			1.00000i
$5$			2.23607i			1.00000i
$6$		0			0
$7$	1.41421	+	2.23607i	0.534522	+	0.845154i
$7$	1.41421	+	2.23607i	0.534522	+	0.845154i
$8$	2.82843			1.00000
$9$	−3.00000			−1.00000
$10$			3.16228i			1.00000i
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$		−	4.47214i		−	1.24035i	−0.784465	−	0.620174i	$-0.787062\pi$
$13$		−	4.47214i		−	1.24035i	0.784465	−	0.620174i	$-0.212938\pi$
$14$	2.00000	+	3.16228i	0.534522	+	0.845154i
$15$		0			0
$16$	4.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−4.24264			−1.00000
$19$		−	6.32456i		−	1.45095i	−0.688247	−	0.725476i	$-0.741620\pi$
$19$		−	6.32456i		−	1.45095i	0.688247	−	0.725476i	$-0.258380\pi$
$20$			4.47214i			1.00000i
$21$		0			0
$22$	−2.82843			−0.603023
$23$	8.48528			1.76930			0.884652	−	0.466252i	$-0.154396\pi$
$23$	8.48528			1.76930			0.884652	+	0.466252i	$0.154396\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		−	6.32456i		−	1.24035i
$27$		0			0
$28$	2.82843	+	4.47214i	0.534522	+	0.845154i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	5.65685			1.00000
$33$		0			0
$34$		0			0
$35$	−5.00000	+	3.16228i	−0.845154	+	0.534522i
$36$	−6.00000			−1.00000
$37$	−11.3137			−1.85996			−0.929981	−	0.367607i	$-0.880177\pi$
$37$	−11.3137			−1.85996			−0.929981	+	0.367607i	$0.880177\pi$
$38$		−	8.94427i		−	1.45095i
$39$		0			0
$40$			6.32456i			1.00000i
$41$		−	12.6491i		−	1.97546i	−0.156174	−	0.987730i	$-0.549916\pi$
$41$		−	12.6491i		−	1.97546i	0.156174	−	0.987730i	$-0.450084\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−4.00000			−0.603023
$45$		−	6.70820i		−	1.00000i
$46$	12.0000			1.76930
$47$			13.4164i			1.95698i	0.206284	+	0.978492i	$0.433863\pi$
$47$			13.4164i			1.95698i	−0.206284	+	0.978492i	$0.566137\pi$
$48$		0			0
$49$	−3.00000	+	6.32456i	−0.428571	+	0.903508i
$50$	−7.07107			−1.00000
$51$		0			0
$52$		−	8.94427i		−	1.24035i
$53$	−5.65685			−0.777029			−0.388514	−	0.921443i	$-0.627012\pi$
$53$	−5.65685			−0.777029			−0.388514	+	0.921443i	$0.627012\pi$
$54$		0			0
$55$		−	4.47214i		−	0.603023i
$56$	4.00000	+	6.32456i	0.534522	+	0.845154i
$57$		0			0
$58$		0			0
$59$			6.32456i			0.823387i	0.911322	+	0.411693i	$0.135063\pi$
$59$			6.32456i			0.823387i	−0.911322	+	0.411693i	$0.864937\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$	−4.24264	−	6.70820i	−0.534522	−	0.845154i
$64$	8.00000			1.00000
$65$	10.0000			1.24035
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$	−7.07107	+	4.47214i	−0.845154	+	0.534522i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−8.48528			−1.00000
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	−16.0000			−1.85996
$75$		0			0
$76$		−	12.6491i		−	1.45095i
$77$	−2.82843	−	4.47214i	−0.322329	−	0.509647i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$			8.94427i			1.00000i
$81$	9.00000			1.00000
$82$		−	17.8885i		−	1.97546i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−5.65685			−0.603023
$89$			12.6491i			1.34080i	0.741999	+	0.670402i	$0.233878\pi$
$89$			12.6491i			1.34080i	−0.741999	+	0.670402i	$0.766122\pi$
$90$		−	9.48683i		−	1.00000i
$91$	10.0000	−	6.32456i	1.04828	−	0.662994i
$92$	16.9706			1.76930
$93$		0			0
$94$			18.9737i			1.95698i
$95$	14.1421			1.45095
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−4.24264	+	8.94427i	−0.428571	+	0.903508i
$99$	6.00000			0.603023
$100$	−10.0000			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			4.47214i			0.440653i	0.975426	+	0.220326i	$0.0707122\pi$
$103$			4.47214i			0.440653i	−0.975426	+	0.220326i	$0.929288\pi$
$104$		−	12.6491i		−	1.24035i
$105$		0			0
$106$	−8.00000			−0.777029
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		−	6.32456i		−	0.603023i
$111$		0			0
$112$	5.65685	+	8.94427i	0.534522	+	0.845154i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$			18.9737i			1.76930i
$116$		0			0
$117$			13.4164i			1.24035i
$118$			8.94427i			0.823387i
$119$		0			0
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	11.1803i		−	1.00000i
$126$	−6.00000	−	9.48683i	−0.534522	−	0.845154i
$127$	−2.82843			−0.250982			−0.125491	−	0.992095i	$-0.540051\pi$
$127$	−2.82843			−0.250982			−0.125491	+	0.992095i	$0.540051\pi$
$128$	11.3137			1.00000
$129$		0			0
$130$	14.1421			1.24035
$131$		−	6.32456i		−	0.552579i	−0.961074	−	0.276289i	$-0.910895\pi$
$131$		−	6.32456i		−	0.552579i	0.961074	−	0.276289i	$-0.0891049\pi$
$132$		0			0
$133$	14.1421	−	8.94427i	1.22628	−	0.775567i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		−	18.9737i		−	1.60933i	−0.593732	−	0.804663i	$-0.702346\pi$
$139$		−	18.9737i		−	1.60933i	0.593732	−	0.804663i	$-0.297654\pi$
$140$	−10.0000	+	6.32456i	−0.845154	+	0.534522i
$141$		0			0
$142$		0			0
$143$			8.94427i			0.747958i
$144$	−12.0000			−1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−22.6274			−1.85996
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		−	17.8885i		−	1.45095i
$153$		0			0
$154$	−4.00000	−	6.32456i	−0.322329	−	0.509647i
$155$		0			0
$156$		0			0
$157$			22.3607i			1.78458i	0.451466	+	0.892288i	$0.350901\pi$
$157$			22.3607i			1.78458i	−0.451466	+	0.892288i	$0.649099\pi$
$158$		0			0
$159$		0			0
$160$			12.6491i			1.00000i
$161$	12.0000	+	18.9737i	0.945732	+	1.49533i
$162$	12.7279			1.00000
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		−	25.2982i		−	1.97546i
$165$		0			0
$166$		0			0
$167$		−	4.47214i		−	0.346064i	−0.984916	−	0.173032i	$-0.944644\pi$
$167$		−	4.47214i		−	0.346064i	0.984916	−	0.173032i	$-0.0553564\pi$
$168$		0			0
$169$	−7.00000			−0.538462
$170$		0			0
$171$			18.9737i			1.45095i
$172$		0			0
$173$			13.4164i			1.02003i	0.860165	+	0.510015i	$0.170360\pi$
$173$			13.4164i			1.02003i	−0.860165	+	0.510015i	$0.829640\pi$
$174$		0			0
$175$	−7.07107	−	11.1803i	−0.534522	−	0.845154i
$176$	−8.00000			−0.603023
$177$		0			0
$178$			17.8885i			1.34080i
$179$	26.0000			1.94333			0.971666	−	0.236360i	$-0.0759544\pi$
$179$	26.0000			1.94333			0.971666	+	0.236360i	$0.0759544\pi$
$180$		−	13.4164i		−	1.00000i
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$	14.1421	−	8.94427i	1.04828	−	0.662994i
$183$		0			0
$184$	24.0000			1.76930
$185$		−	25.2982i		−	1.85996i
$186$		0			0
$187$		0			0
$188$			26.8328i			1.95698i
$189$		0			0
$190$	20.0000			1.45095
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	−6.00000	+	12.6491i	−0.428571	+	0.903508i
$197$	16.9706			1.20910			0.604551	−	0.796566i	$-0.293352\pi$
$197$	16.9706			1.20910			0.604551	+	0.796566i	$0.293352\pi$
$198$	8.48528			0.603023
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−14.1421			−1.00000
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	28.2843			1.97546
$206$			6.32456i			0.440653i
$207$	−25.4558			−1.76930
$208$		−	17.8885i		−	1.24035i
$209$			12.6491i			0.874957i
$210$		0			0
$211$	−22.0000			−1.51454			−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000			−1.51454			−0.757271	+	0.653101i	$0.773468\pi$
$212$	−11.3137			−0.777029
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		−	8.94427i		−	0.603023i
$221$		0			0
$222$		0			0
$223$		−	22.3607i		−	1.49738i	−0.662919	−	0.748691i	$-0.730683\pi$
$223$		−	22.3607i		−	1.49738i	0.662919	−	0.748691i	$-0.269317\pi$
$224$	8.00000	+	12.6491i	0.534522	+	0.845154i
$225$	15.0000			1.00000
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$			26.8328i			1.76930i
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$			18.9737i			1.24035i
$235$	−30.0000			−1.95698
$236$			12.6491i			0.823387i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	25.2982i		−	1.62960i	−0.579741	−	0.814801i	$-0.696846\pi$
$241$		−	25.2982i		−	1.62960i	0.579741	−	0.814801i	$-0.303154\pi$
$242$	−9.89949			−0.636364
$243$		0			0
$244$		0			0
$245$	−14.1421	−	6.70820i	−0.903508	−	0.428571i
$246$		0			0
$247$	−28.2843			−1.79969
$248$		0			0
$249$		0			0
$250$		−	15.8114i		−	1.00000i
$251$			31.6228i			1.99601i	0.0631194	+	0.998006i	$0.479895\pi$
$251$			31.6228i			1.99601i	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	−8.48528	−	13.4164i	−0.534522	−	0.845154i
$253$	−16.9706			−1.06693
$254$	−4.00000			−0.250982
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	−16.0000	−	25.2982i	−0.994192	−	1.57195i
$260$	20.0000			1.24035
$261$		0			0
$262$		−	8.94427i		−	0.552579i
$263$	−8.48528			−0.523225			−0.261612	−	0.965173i	$-0.584254\pi$
$263$	−8.48528			−0.523225			−0.261612	+	0.965173i	$0.584254\pi$
$264$		0			0
$265$		−	12.6491i		−	0.777029i
$266$	20.0000	−	12.6491i	1.22628	−	0.775567i
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	10.0000			0.603023
$276$		0			0
$277$	11.3137			0.679775			0.339887	−	0.940466i	$-0.389611\pi$
$277$	11.3137			0.679775			0.339887	+	0.940466i	$0.389611\pi$
$278$		−	26.8328i		−	1.60933i
$279$		0			0
$280$	−14.1421	+	8.94427i	−0.845154	+	0.534522i
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$			12.6491i			0.747958i
$287$	28.2843	−	17.8885i	1.66957	−	1.05593i
$288$	−16.9706			−1.00000
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	4.47214i		−	0.261265i	−0.991431	−	0.130632i	$-0.958299\pi$
$293$		−	4.47214i		−	0.261265i	0.991431	−	0.130632i	$-0.0417008\pi$
$294$		0			0
$295$	−14.1421			−0.823387
$296$	−32.0000			−1.85996
$297$		0			0
$298$		0			0
$299$		−	37.9473i		−	2.19455i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		−	25.2982i		−	1.45095i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−5.65685	−	8.94427i	−0.322329	−	0.509647i
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$			31.6228i			1.78458i
$315$	15.0000	−	9.48683i	0.845154	−	0.534522i
$316$		0			0
$317$	−16.9706			−0.953162			−0.476581	−	0.879131i	$-0.658124\pi$
$317$	−16.9706			−0.953162			−0.476581	+	0.879131i	$0.658124\pi$
$318$		0			0
$319$		0			0
$320$			17.8885i			1.00000i
$321$		0			0
$322$	16.9706	+	26.8328i	0.945732	+	1.49533i
$323$		0			0
$324$	18.0000			1.00000
$325$			22.3607i			1.24035i
$326$		0			0
$327$		0			0
$328$		−	35.7771i		−	1.97546i
$329$	−30.0000	+	18.9737i	−1.65395	+	1.04605i
$330$		0			0
$331$	−18.0000			−0.989369			−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000			−0.989369			−0.494685	+	0.869072i	$0.664716\pi$
$332$		0			0
$333$	33.9411			1.85996
$334$		−	6.32456i		−	0.346064i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−9.89949			−0.538462
$339$		0			0
$340$		0			0
$341$		0			0
$342$			26.8328i			1.45095i
$343$	−18.3848	+	2.23607i	−0.992685	+	0.120736i
$344$		0			0
$345$		0			0
$346$			18.9737i			1.02003i
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$	−10.0000	−	15.8114i	−0.534522	−	0.845154i
$351$		0			0
$352$	−11.3137			−0.603023
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$			25.2982i			1.34080i
$357$		0			0
$358$	36.7696			1.94333
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		−	18.9737i		−	1.00000i
$361$	−21.0000			−1.10526
$362$		0			0
$363$		0			0
$364$	20.0000	−	12.6491i	1.04828	−	0.662994i
$365$		0			0
$366$		0			0
$367$			22.3607i			1.16722i	0.812035	+	0.583609i	$0.198360\pi$
$367$			22.3607i			1.16722i	−0.812035	+	0.583609i	$0.801640\pi$
$368$	33.9411			1.76930
$369$			37.9473i			1.97546i
$370$		−	35.7771i		−	1.85996i
$371$	−8.00000	−	12.6491i	−0.415339	−	0.656709i
$372$		0			0
$373$	22.6274			1.17160			0.585802	−	0.810454i	$-0.300780\pi$
$373$	22.6274			1.17160			0.585802	+	0.810454i	$0.300780\pi$
$374$		0			0
$375$		0			0
$376$			37.9473i			1.95698i
$377$		0			0
$378$		0			0
$379$	34.0000			1.74646			0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000			1.74646			0.873231	+	0.487306i	$0.162020\pi$
$380$	28.2843			1.45095
$381$		0			0
$382$		0			0
$383$			13.4164i			0.685546i	0.939418	+	0.342773i	$0.111366\pi$
$383$			13.4164i			0.685546i	−0.939418	+	0.342773i	$0.888634\pi$
$384$		0			0
$385$	10.0000	−	6.32456i	0.509647	−	0.322329i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−8.48528	+	17.8885i	−0.428571	+	0.903508i
$393$		0			0
$394$	24.0000			1.20910
$395$		0			0
$396$	12.0000			0.603023
$397$			4.47214i			0.224450i	0.993683	+	0.112225i	$0.0357978\pi$
$397$			4.47214i			0.224450i	−0.993683	+	0.112225i	$0.964202\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$	38.0000			1.89763			0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000			1.89763			0.948815	+	0.315833i	$0.102284\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$			20.1246i			1.00000i
$406$		0			0
$407$	22.6274			1.12160
$408$		0			0
$409$		−	37.9473i		−	1.87637i	−0.346128	−	0.938187i	$-0.612504\pi$
$409$		−	37.9473i		−	1.87637i	0.346128	−	0.938187i	$-0.387496\pi$
$410$	40.0000			1.97546
$411$		0			0
$412$			8.94427i			0.440653i
$413$	−14.1421	+	8.94427i	−0.695889	+	0.440119i
$414$	−36.0000			−1.76930
$415$		0			0
$416$		−	25.2982i		−	1.24035i
$417$		0			0
$418$			17.8885i			0.874957i
$419$			31.6228i			1.54487i	0.635092	+	0.772437i	$0.280962\pi$
$419$			31.6228i			1.54487i	−0.635092	+	0.772437i	$0.719038\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−31.1127			−1.51454
$423$		−	40.2492i		−	1.95698i
$424$	−16.0000			−0.777029
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	53.6656i		−	2.56718i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		−	12.6491i		−	0.603023i
$441$	9.00000	−	18.9737i	0.428571	−	0.903508i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−28.2843			−1.34080
$446$		−	31.6228i		−	1.49738i
$447$		0			0
$448$	11.3137	+	17.8885i	0.534522	+	0.845154i
$449$	34.0000			1.60456			0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000			1.60456			0.802280	+	0.596948i	$0.203620\pi$
$450$	21.2132			1.00000
$451$			25.2982i			1.19125i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	14.1421	+	22.3607i	0.662994	+	1.04828i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$			37.9473i			1.76930i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	36.7696			1.70883			0.854413	−	0.519594i	$-0.173917\pi$
$463$	36.7696			1.70883			0.854413	+	0.519594i	$0.173917\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$			26.8328i			1.24035i
$469$		0			0
$470$	−42.4264			−1.95698
$471$		0			0
$472$			17.8885i			0.823387i
$473$		0			0
$474$		0			0
$475$			31.6228i			1.45095i
$476$		0			0
$477$	16.9706			0.777029
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$			50.5964i			2.30700i
$482$		−	35.7771i		−	1.62960i
$483$		0			0
$484$	−14.0000			−0.636364
$485$		0			0
$486$		0			0
$487$	31.1127			1.40985			0.704925	−	0.709281i	$-0.250980\pi$
$487$	31.1127			1.40985			0.704925	+	0.709281i	$0.250980\pi$
$488$		0			0
$489$		0			0
$490$	−20.0000	−	9.48683i	−0.903508	−	0.428571i
$491$	2.00000			0.0902587			0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000			0.0902587			0.0451294	+	0.998981i	$0.485630\pi$
$492$		0			0
$493$		0			0
$494$	−40.0000			−1.79969
$495$			13.4164i			0.603023i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6.00000			−0.268597			−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000			−0.268597			−0.134298	+	0.990941i	$0.542878\pi$
$500$		−	22.3607i		−	1.00000i
$501$		0			0
$502$			44.7214i			1.99601i
$503$			40.2492i			1.79462i	0.441397	+	0.897312i	$0.354483\pi$
$503$			40.2492i			1.79462i	−0.441397	+	0.897312i	$0.645517\pi$
$504$	−12.0000	−	18.9737i	−0.534522	−	0.845154i
$505$		0			0
$506$	−24.0000			−1.06693
$507$		0			0
$508$	−5.65685			−0.250982
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	22.6274			1.00000
$513$		0			0
$514$		0			0
$515$	−10.0000			−0.440653
$516$		0			0
$517$		−	26.8328i		−	1.18011i
$518$	−22.6274	−	35.7771i	−0.994192	−	1.57195i
$519$		0			0
$520$	28.2843			1.24035
$521$		−	25.2982i		−	1.10834i	−0.832405	−	0.554168i	$-0.813037\pi$
$521$		−	25.2982i		−	1.10834i	0.832405	−	0.554168i	$-0.186963\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		−	12.6491i		−	0.552579i
$525$		0			0
$526$	−12.0000			−0.523225
$527$		0			0
$528$		0			0
$529$	49.0000			2.13043
$530$		−	17.8885i		−	0.777029i
$531$		−	18.9737i		−	0.823387i
$532$	28.2843	−	17.8885i	1.22628	−	0.775567i
$533$	−56.5685			−2.45026
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.00000	−	12.6491i	0.258438	−	0.544836i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	14.1421			0.603023
$551$		0			0
$552$		0			0
$553$		0			0
$554$	16.0000			0.679775
$555$		0			0
$556$		−	37.9473i		−	1.60933i
$557$	−45.2548			−1.91751			−0.958754	−	0.284236i	$-0.908260\pi$
$557$	−45.2548			−1.91751			−0.958754	+	0.284236i	$0.908260\pi$
$558$		0			0
$559$		0			0
$560$	−20.0000	+	12.6491i	−0.845154	+	0.534522i
$561$		0			0
$562$	−31.1127			−1.31241
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	12.7279	+	20.1246i	0.534522	+	0.845154i
$568$		0			0
$569$	−46.0000			−1.92842			−0.964210	−	0.265139i	$-0.914582\pi$
$569$	−46.0000			−1.92842			−0.964210	+	0.265139i	$0.914582\pi$
$570$		0			0
$571$	18.0000			0.753277			0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000			0.753277			0.376638	+	0.926360i	$0.377080\pi$
$572$			17.8885i			0.747958i
$573$		0			0
$574$	40.0000	−	25.2982i	1.66957	−	1.05593i
$575$	−42.4264			−1.76930
$576$	−24.0000			−1.00000
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−24.0416			−1.00000
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	11.3137			0.468566
$584$		0			0
$585$	−30.0000			−1.24035
$586$		−	6.32456i		−	0.261265i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$	−20.0000			−0.823387
$591$		0			0
$592$	−45.2548			−1.85996
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		−	53.6656i		−	2.19455i
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			25.2982i			1.03194i	0.856608	+	0.515968i	$0.172568\pi$
$601$			25.2982i			1.03194i	−0.856608	+	0.515968i	$0.827432\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	15.6525i		−	0.636364i
$606$		0			0
$607$		−	49.1935i		−	1.99670i	−0.0574012	−	0.998351i	$-0.518281\pi$
$607$		−	49.1935i		−	1.99670i	0.0574012	−	0.998351i	$-0.481719\pi$
$608$		−	35.7771i		−	1.45095i
$609$		0			0
$610$		0			0
$611$	60.0000			2.42734
$612$		0			0
$613$	−5.65685			−0.228478			−0.114239	−	0.993453i	$-0.536443\pi$
$613$	−5.65685			−0.228478			−0.114239	+	0.993453i	$0.536443\pi$
$614$		0			0
$615$		0			0
$616$	−8.00000	−	12.6491i	−0.322329	−	0.509647i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$			18.9737i			0.762616i	0.924448	+	0.381308i	$0.124526\pi$
$619$			18.9737i			0.762616i	−0.924448	+	0.381308i	$0.875474\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−28.2843	+	17.8885i	−1.13319	+	0.716689i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$			44.7214i			1.78458i
$629$		0			0
$630$	21.2132	−	13.4164i	0.845154	−	0.534522i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	−24.0000			−0.953162
$635$		−	6.32456i		−	0.250982i
$636$		0			0
$637$	28.2843	+	13.4164i	1.12066	+	0.531577i
$638$		0			0
$639$		0			0
$640$			25.2982i			1.00000i
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	24.0000	+	37.9473i	0.945732	+	1.49533i
$645$		0			0
$646$		0			0
$647$		−	40.2492i		−	1.58236i	−0.611583	−	0.791180i	$-0.709467\pi$
$647$		−	40.2492i		−	1.58236i	0.611583	−	0.791180i	$-0.290533\pi$
$648$	25.4558			1.00000
$649$		−	12.6491i		−	0.496521i
$650$			31.6228i			1.24035i
$651$		0			0
$652$		0			0
$653$	−50.9117			−1.99233			−0.996164	−	0.0875041i	$-0.972111\pi$
$653$	−50.9117			−1.99233			−0.996164	+	0.0875041i	$0.972111\pi$
$654$		0			0
$655$	14.1421			0.552579
$656$		−	50.5964i		−	1.97546i
$657$		0			0
$658$	−42.4264	+	26.8328i	−1.65395	+	1.04605i
$659$	−26.0000			−1.01282			−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000			−1.01282			−0.506408	+	0.862294i	$0.669027\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−25.4558			−0.989369
$663$		0			0
$664$		0			0
$665$	20.0000	+	31.6228i	0.775567	+	1.22628i
$666$	48.0000			1.85996
$667$		0			0
$668$		−	8.94427i		−	0.346064i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	−14.0000			−0.538462
$677$		−	49.1935i		−	1.89066i	−0.326116	−	0.945330i	$-0.605740\pi$
$677$		−	49.1935i		−	1.89066i	0.326116	−	0.945330i	$-0.394260\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$			37.9473i			1.45095i
$685$		0			0
$686$	−26.0000	+	3.16228i	−0.992685	+	0.120736i
$687$		0			0
$688$		0			0
$689$			25.2982i			0.963785i
$690$		0			0
$691$		−	31.6228i		−	1.20299i	−0.798878	−	0.601494i	$-0.794573\pi$
$691$		−	31.6228i		−	1.20299i	0.798878	−	0.601494i	$-0.205427\pi$
$692$			26.8328i			1.02003i
$693$	8.48528	+	13.4164i	0.322329	+	0.509647i
$694$		0			0
$695$	42.4264			1.60933
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−14.1421	−	22.3607i	−0.534522	−	0.845154i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$			71.5542i			2.69872i
$704$	−16.0000			−0.603023
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$			35.7771i			1.34080i
$713$		0			0
$714$		0			0
$715$	−20.0000			−0.747958
$716$	52.0000			1.94333
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		−	26.8328i		−	1.00000i
$721$	−10.0000	+	6.32456i	−0.372419	+	0.235539i
$722$	−29.6985			−1.10526
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	4.47214i		−	0.165862i	−0.996555	−	0.0829312i	$-0.973572\pi$
$727$		−	4.47214i		−	0.165862i	0.996555	−	0.0829312i	$-0.0264282\pi$
$728$	28.2843	−	17.8885i	1.04828	−	0.662994i
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	49.1935i		−	1.81700i	−0.417881	−	0.908502i	$-0.637227\pi$
$733$		−	49.1935i		−	1.81700i	0.417881	−	0.908502i	$-0.362773\pi$
$734$			31.6228i			1.16722i
$735$		0			0
$736$	48.0000			1.76930
$737$		0			0
$738$			53.6656i			1.97546i
$739$	54.0000			1.98642			0.993211	−	0.116326i	$-0.0371118\pi$
$739$	54.0000			1.98642			0.993211	+	0.116326i	$0.0371118\pi$
$740$		−	50.5964i		−	1.85996i
$741$		0			0
$742$	−11.3137	−	17.8885i	−0.415339	−	0.656709i
$743$	36.7696			1.34894			0.674472	−	0.738300i	$-0.264371\pi$
$743$	36.7696			1.34894			0.674472	+	0.738300i	$0.264371\pi$
$744$		0			0
$745$		0			0
$746$	32.0000			1.17160
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			53.6656i			1.95698i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	45.2548			1.64481			0.822407	−	0.568899i	$-0.192630\pi$
$757$	45.2548			1.64481			0.822407	+	0.568899i	$0.192630\pi$
$758$	48.0833			1.74646
$759$		0			0
$760$	40.0000			1.45095
$761$		−	50.5964i		−	1.83412i	−0.398750	−	0.917060i	$-0.630556\pi$
$761$		−	50.5964i		−	1.83412i	0.398750	−	0.917060i	$-0.369444\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$			18.9737i			0.685546i
$767$	28.2843			1.02129
$768$		0			0
$769$		−	12.6491i		−	0.456139i	−0.973645	−	0.228069i	$-0.926759\pi$
$769$		−	12.6491i		−	0.456139i	0.973645	−	0.228069i	$-0.0732413\pi$
$770$	14.1421	−	8.94427i	0.509647	−	0.322329i
$771$		0			0
$772$		0			0
$773$			22.3607i			0.804258i	0.915583	+	0.402129i	$0.131730\pi$
$773$			22.3607i			0.804258i	−0.915583	+	0.402129i	$0.868270\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−80.0000			−2.86630
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−12.0000	+	25.2982i	−0.428571	+	0.903508i
$785$	−50.0000			−1.78458
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	33.9411			1.20910
$789$		0			0
$790$		0			0
$791$		0			0
$792$	16.9706			0.603023
$793$		0			0
$794$			6.32456i			0.224450i
$795$		0			0
$796$		0			0
$797$			40.2492i			1.42570i	0.701316	+	0.712850i	$0.252596\pi$
$797$			40.2492i			1.42570i	−0.701316	+	0.712850i	$0.747404\pi$
$798$		0			0
$799$		0			0
$800$	−28.2843			−1.00000
$801$		−	37.9473i		−	1.34080i
$802$	53.7401			1.89763
$803$		0			0
$804$		0			0
$805$	−42.4264	+	26.8328i	−1.49533	+	0.945732i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	26.0000			0.914111			0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000			0.914111			0.457056	+	0.889438i	$0.348904\pi$
$810$			28.4605i			1.00000i
$811$		−	56.9210i		−	1.99877i	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$		−	56.9210i		−	1.99877i	0.0351147	−	0.999383i	$-0.488820\pi$
$812$		0			0
$813$		0			0
$814$	32.0000			1.12160
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	53.6656i		−	1.87637i
$819$	−30.0000	+	18.9737i	−1.04828	+	0.662994i
$820$	56.5685			1.97546
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−48.0833			−1.67608			−0.838039	−	0.545611i	$-0.816298\pi$
$823$	−48.0833			−1.67608			−0.838039	+	0.545611i	$0.816298\pi$
$824$			12.6491i			0.440653i
$825$		0			0
$826$	−20.0000	+	12.6491i	−0.695889	+	0.440119i
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	−50.9117			−1.76930
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		−	35.7771i		−	1.24035i
$833$		0			0
$834$		0			0
$835$	10.0000			0.346064
$836$			25.2982i			0.874957i
$837$		0			0
$838$			44.7214i			1.54487i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$	−44.0000			−1.51454
$845$		−	15.6525i		−	0.538462i
$846$		−	56.9210i		−	1.95698i
$847$	−9.89949	−	15.6525i	−0.340151	−	0.537825i
$848$	−22.6274			−0.777029
$849$		0			0
$850$		0			0
$851$	−96.0000			−3.29084
$852$		0			0
$853$			58.1378i			1.99060i	0.0968435	+	0.995300i	$0.469125\pi$
$853$			58.1378i			1.99060i	−0.0968435	+	0.995300i	$0.530875\pi$
$854$		0			0
$855$	−42.4264			−1.45095
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$			56.9210i			1.94212i	0.238837	+	0.971060i	$0.423234\pi$
$859$			56.9210i			1.94212i	−0.238837	+	0.971060i	$0.576766\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.48528			0.288842			0.144421	−	0.989516i	$-0.453868\pi$
$863$	8.48528			0.288842			0.144421	+	0.989516i	$0.453868\pi$
$864$		0			0
$865$	−30.0000			−1.02003
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		−	75.8947i		−	2.56718i
$875$	25.0000	−	15.8114i	0.845154	−	0.534522i
$876$		0			0
$877$	−11.3137			−0.382037			−0.191018	−	0.981586i	$-0.561179\pi$
$877$	−11.3137			−0.382037			−0.191018	+	0.981586i	$0.561179\pi$
$878$		0			0
$879$		0			0
$880$		−	17.8885i		−	0.603023i
$881$		−	12.6491i		−	0.426159i	−0.977035	−	0.213080i	$-0.931651\pi$
$881$		−	12.6491i		−	0.426159i	0.977035	−	0.213080i	$-0.0683494\pi$
$882$	12.7279	−	26.8328i	0.428571	−	0.903508i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			4.47214i			0.150160i	0.997178	+	0.0750798i	$0.0239212\pi$
$887$			4.47214i			0.150160i	−0.997178	+	0.0750798i	$0.976079\pi$
$888$		0			0
$889$	−4.00000	−	6.32456i	−0.134156	−	0.212119i
$890$	−40.0000			−1.34080
$891$	−18.0000			−0.603023
$892$		−	44.7214i		−	1.49738i
$893$	84.8528			2.83949
$894$		0			0
$895$			58.1378i			1.94333i
$896$	16.0000	+	25.2982i	0.534522	+	0.845154i
$897$		0			0
$898$	48.0833			1.60456
$899$		0			0
$900$	30.0000			1.00000
$901$		0			0
$902$			35.7771i			1.19125i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$		0			0
$910$	20.0000	+	31.6228i	0.662994	+	1.04828i
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.1421	−	8.94427i	0.467014	−	0.295366i
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$			53.6656i			1.76930i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	56.5685			1.85996
$926$	52.0000			1.70883
$927$		−	13.4164i		−	0.440653i
$928$		0			0
$929$		−	50.5964i		−	1.66002i	−0.557752	−	0.830008i	$-0.688336\pi$
$929$		−	50.5964i		−	1.66002i	0.557752	−	0.830008i	$-0.311664\pi$
$930$		0			0
$931$	40.0000	+	18.9737i	1.31095	+	0.621837i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$			37.9473i			1.24035i
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$	−60.0000			−1.95698
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		−	107.331i		−	3.49519i
$944$			25.2982i			0.823387i
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$			44.7214i			1.45095i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$	24.0000			0.777029
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$			71.5542i			2.30700i
$963$		0			0
$964$		−	50.5964i		−	1.62960i
$965$		0			0
$966$		0			0
$967$	−53.7401			−1.72817			−0.864083	−	0.503350i	$-0.832101\pi$
$967$	−53.7401			−1.72817			−0.864083	+	0.503350i	$0.832101\pi$
$968$	−19.7990			−0.636364
$969$		0			0
$970$		0			0
$971$		−	6.32456i		−	0.202965i	−0.994837	−	0.101482i	$-0.967641\pi$
$971$		−	6.32456i		−	0.202965i	0.994837	−	0.101482i	$-0.0323585\pi$
$972$		0			0
$973$	42.4264	−	26.8328i	1.36013	−	0.860221i
$974$	44.0000			1.40985
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		−	25.2982i		−	0.808535i
$980$	−28.2843	−	13.4164i	−0.903508	−	0.428571i
$981$		0			0
$982$	2.82843			0.0902587
$983$		−	40.2492i		−	1.28375i	−0.766809	−	0.641875i	$-0.778157\pi$
$983$		−	40.2492i		−	1.28375i	0.766809	−	0.641875i	$-0.221843\pi$
$984$		0			0
$985$			37.9473i			1.20910i
$986$		0			0
$987$		0			0
$988$	−56.5685			−1.79969
$989$		0			0
$990$			18.9737i			0.603023i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$			49.1935i			1.55797i	0.627040	+	0.778987i	$0.284266\pi$
$997$			49.1935i			1.55797i	−0.627040	+	0.778987i	$0.715734\pi$
$998$	−8.48528			−0.268597
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.2.n.a.139.4	yes	4
4.3	odd	2		1120.2.n.a.559.3		4
5.4	even	2	inner	280.2.n.a.139.1	✓	4
7.6	odd	2	inner	280.2.n.a.139.3	yes	4
8.3	odd	2	inner	280.2.n.a.139.1	✓	4
8.5	even	2		1120.2.n.a.559.2		4
20.19	odd	2		1120.2.n.a.559.2		4
28.27	even	2		1120.2.n.a.559.1		4
35.34	odd	2	inner	280.2.n.a.139.2	yes	4
40.19	odd	2	CM	280.2.n.a.139.4	yes	4
40.29	even	2		1120.2.n.a.559.3		4
56.13	odd	2		1120.2.n.a.559.4		4
56.27	even	2	inner	280.2.n.a.139.2	yes	4
140.139	even	2		1120.2.n.a.559.4		4
280.69	odd	2		1120.2.n.a.559.1		4
280.139	even	2	inner	280.2.n.a.139.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.n.a.139.1	✓	4	5.4	even	2	inner
280.2.n.a.139.1	✓	4	8.3	odd	2	inner
280.2.n.a.139.2	yes	4	35.34	odd	2	inner
280.2.n.a.139.2	yes	4	56.27	even	2	inner
280.2.n.a.139.3	yes	4	7.6	odd	2	inner
280.2.n.a.139.3	yes	4	280.139	even	2	inner
280.2.n.a.139.4	yes	4	1.1	even	1	trivial
280.2.n.a.139.4	yes	4	40.19	odd	2	CM
1120.2.n.a.559.1		4	28.27	even	2
1120.2.n.a.559.1		4	280.69	odd	2
1120.2.n.a.559.2		4	8.5	even	2
1120.2.n.a.559.2		4	20.19	odd	2
1120.2.n.a.559.3		4	4.3	odd	2
1120.2.n.a.559.3		4	40.29	even	2
1120.2.n.a.559.4		4	56.13	odd	2
1120.2.n.a.559.4		4	140.139	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	280.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +(1.41421 + 2.23607i) q^{7} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +2.23607i q^{5} +(1.41421 + 2.23607i) q^{7} +2.82843 q^{8} -3.00000 q^{9} +3.16228i q^{10} -2.00000 q^{11} -4.47214i q^{13} +(2.00000 + 3.16228i) q^{14} +4.00000 q^{16} -4.24264 q^{18} -6.32456i q^{19} +4.47214i q^{20} -2.82843 q^{22} +8.48528 q^{23} -5.00000 q^{25} -6.32456i q^{26} +(2.82843 + 4.47214i) q^{28} +5.65685 q^{32} +(-5.00000 + 3.16228i) q^{35} -6.00000 q^{36} -11.3137 q^{37} -8.94427i q^{38} +6.32456i q^{40} -12.6491i q^{41} -4.00000 q^{44} -6.70820i q^{45} +12.0000 q^{46} +13.4164i q^{47} +(-3.00000 + 6.32456i) q^{49} -7.07107 q^{50} -8.94427i q^{52} -5.65685 q^{53} -4.47214i q^{55} +(4.00000 + 6.32456i) q^{56} +6.32456i q^{59} +(-4.24264 - 6.70820i) q^{63} +8.00000 q^{64} +10.0000 q^{65} +(-7.07107 + 4.47214i) q^{70} -8.48528 q^{72} -16.0000 q^{74} -12.6491i q^{76} +(-2.82843 - 4.47214i) q^{77} +8.94427i q^{80} +9.00000 q^{81} -17.8885i q^{82} -5.65685 q^{88} +12.6491i q^{89} -9.48683i q^{90} +(10.0000 - 6.32456i) q^{91} +16.9706 q^{92} +18.9737i q^{94} +14.1421 q^{95} +(-4.24264 + 8.94427i) q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 12 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 - 12 * q^9	\( 4 q + 8 q^{4} - 12 q^{9} - 8 q^{11} + 8 q^{14} + 16 q^{16} - 20 q^{25} - 20 q^{35} - 24 q^{36} - 16 q^{44} + 48 q^{46} - 12 q^{49} + 16 q^{56} + 32 q^{64} + 40 q^{65} - 64 q^{74} + 36 q^{81} + 40 q^{91} + 24 q^{99}+O(q^{100}) \) 4 * q + 8 * q^4 - 12 * q^9 - 8 * q^11 + 8 * q^14 + 16 * q^16 - 20 * q^25 - 20 * q^35 - 24 * q^36 - 16 * q^44 + 48 * q^46 - 12 * q^49 + 16 * q^56 + 32 * q^64 + 40 * q^65 - 64 * q^74 + 36 * q^81 + 40 * q^91 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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