LMFDB - Embedded newform 560.2.k.b.111.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,2,Mod(111,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("560.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	560.k (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:	$4.47162251319$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.116319195136.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 77x^{4} + 4 $ x^8 + 77*x^4 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.6
Root			$2.09428 + 2.09428i$ of defining polynomial
Character	$\chi$	$=$	560.111
Dual form			560.2.k.b.111.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.477491 q^{3} +1.00000i q^{5} +(-2.09428 + 1.61679i) q^{7} -2.77200 q^{9} +O(q^{10})$	$q+0.477491 q^{3} +1.00000i q^{5} +(-2.09428 + 1.61679i) q^{7} -2.77200 q^{9} +4.66605i q^{11} -3.77200i q^{13} +0.477491i q^{15} +3.77200i q^{17} -7.42213 q^{19} +(-1.00000 + 0.772002i) q^{21} +3.23357i q^{23} -1.00000 q^{25} -2.75608 q^{27} +3.77200 q^{29} +0.954983 q^{31} +2.22800i q^{33} +(-1.61679 - 2.09428i) q^{35} +5.54400 q^{37} -1.80110i q^{39} -6.00000i q^{41} +12.5657i q^{43} -2.77200i q^{45} +7.89962 q^{47} +(1.77200 - 6.77200i) q^{49} +1.80110i q^{51} -6.00000 q^{53} -4.66605 q^{55} -3.54400 q^{57} -9.33210 q^{59} +13.5440i q^{61} +(5.80534 - 4.48174i) q^{63} +3.77200 q^{65} +6.09852i q^{67} +1.54400i q^{69} -9.33210i q^{71} +6.00000i q^{73} -0.477491 q^{75} +(-7.54400 - 9.77200i) q^{77} -11.1332i q^{79} +7.00000 q^{81} +12.5657 q^{83} -3.77200 q^{85} +1.80110 q^{87} +12.0000i q^{89} +(6.09852 + 7.89962i) q^{91} +0.455996 q^{93} -7.42213i q^{95} -8.22800i q^{97} -12.9343i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{9}+O(q^{10}) $ 8 * q + 12 * q^9	$ 8 q + 12 q^{9} - 8 q^{21} - 8 q^{25} - 4 q^{29} - 24 q^{37} - 20 q^{49} - 48 q^{53} + 40 q^{57} - 4 q^{65} + 8 q^{77} + 56 q^{81} + 4 q^{85} + 72 q^{93}+O(q^{100}) $ 8 * q + 12 * q^9 - 8 * q^21 - 8 * q^25 - 4 * q^29 - 24 * q^37 - 20 * q^49 - 48 * q^53 + 40 * q^57 - 4 * q^65 + 8 * q^77 + 56 * q^81 + 4 * q^85 + 72 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$351$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	0.477491			0.275680			0.137840	−	0.990455i	$-0.455984\pi$
$3$	0.477491			0.275680			0.137840	+	0.990455i	$0.455984\pi$
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−2.09428	+	1.61679i	−0.791563	+	0.611088i
$7$	−2.09428	+	1.61679i	−0.791563	+	0.611088i
$8$		0			0
$9$	−2.77200			−0.924001
$10$		0			0
$11$			4.66605i			1.40687i	0.710761	+	0.703433i	$0.248351\pi$
$11$			4.66605i			1.40687i	−0.710761	+	0.703433i	$0.751649\pi$
$12$		0			0
$13$		−	3.77200i		−	1.04617i	−0.852282	−	0.523083i	$-0.824782\pi$
$13$		−	3.77200i		−	1.04617i	0.852282	−	0.523083i	$-0.175218\pi$
$14$		0			0
$15$			0.477491i			0.123288i
$16$		0			0
$17$			3.77200i			0.914845i	0.889250	+	0.457422i	$0.151227\pi$
$17$			3.77200i			0.914845i	−0.889250	+	0.457422i	$0.848773\pi$
$18$		0			0
$19$	−7.42213			−1.70275			−0.851377	−	0.524555i	$-0.824232\pi$
$19$	−7.42213			−1.70275			−0.851377	+	0.524555i	$0.824232\pi$
$20$		0			0
$21$	−1.00000	+	0.772002i	−0.218218	+	0.168465i
$22$		0			0
$23$			3.23357i			0.674247i	0.941460	+	0.337123i	$0.109454\pi$
$23$			3.23357i			0.674247i	−0.941460	+	0.337123i	$0.890546\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−2.75608			−0.530408
$28$		0			0
$29$	3.77200			0.700443			0.350222	−	0.936667i	$-0.386106\pi$
$29$	3.77200			0.700443			0.350222	+	0.936667i	$0.386106\pi$
$30$		0			0
$31$	0.954983			0.171520			0.0857600	−	0.996316i	$-0.472668\pi$
$31$	0.954983			0.171520			0.0857600	+	0.996316i	$0.472668\pi$
$32$		0			0
$33$			2.22800i			0.387845i
$34$		0			0
$35$	−1.61679	−	2.09428i	−0.273287	−	0.353998i
$36$		0			0
$37$	5.54400			0.911429			0.455714	−	0.890126i	$-0.349384\pi$
$37$	5.54400			0.911429			0.455714	+	0.890126i	$0.349384\pi$
$38$		0			0
$39$		−	1.80110i		−	0.288407i
$40$		0			0
$41$		−	6.00000i		−	0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$		−	6.00000i		−	0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$		0			0
$43$			12.5657i			1.91625i	0.286356	+	0.958123i	$0.407556\pi$
$43$			12.5657i			1.91625i	−0.286356	+	0.958123i	$0.592444\pi$
$44$		0			0
$45$		−	2.77200i		−	0.413226i
$46$		0			0
$47$	7.89962			1.15228			0.576139	−	0.817352i	$-0.304559\pi$
$47$	7.89962			1.15228			0.576139	+	0.817352i	$0.304559\pi$
$48$		0			0
$49$	1.77200	−	6.77200i	0.253143	−	0.967429i
$50$		0			0
$51$			1.80110i			0.252204i
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−4.66605			−0.629170
$56$		0			0
$57$	−3.54400			−0.469415
$58$		0			0
$59$	−9.33210			−1.21494			−0.607468	−	0.794344i	$-0.707815\pi$
$59$	−9.33210			−1.21494			−0.607468	+	0.794344i	$0.707815\pi$
$60$		0			0
$61$			13.5440i			1.73413i	0.498193	+	0.867066i	$0.333997\pi$
$61$			13.5440i			1.73413i	−0.498193	+	0.867066i	$0.666003\pi$
$62$		0			0
$63$	5.80534	−	4.48174i	0.731404	−	0.564646i
$64$		0			0
$65$	3.77200			0.467859
$66$		0			0
$67$			6.09852i			0.745053i	0.928022	+	0.372527i	$0.121508\pi$
$67$			6.09852i			0.745053i	−0.928022	+	0.372527i	$0.878492\pi$
$68$		0			0
$69$			1.54400i			0.185876i
$70$		0			0
$71$		−	9.33210i		−	1.10752i	−0.832678	−	0.553758i	$-0.813193\pi$
$71$		−	9.33210i		−	1.10752i	0.832678	−	0.553758i	$-0.186807\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$	−0.477491			−0.0551360
$76$		0			0
$77$	−7.54400	−	9.77200i	−0.859719	−	1.11362i
$78$		0			0
$79$		−	11.1332i		−	1.25258i	−0.779589	−	0.626291i	$-0.784572\pi$
$79$		−	11.1332i		−	1.25258i	0.779589	−	0.626291i	$-0.215428\pi$
$80$		0			0
$81$	7.00000			0.777778
$82$		0			0
$83$	12.5657			1.37926			0.689631	−	0.724161i	$-0.257773\pi$
$83$	12.5657			1.37926			0.689631	+	0.724161i	$0.257773\pi$
$84$		0			0
$85$	−3.77200			−0.409131
$86$		0			0
$87$	1.80110			0.193098
$88$		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
$91$	6.09852	+	7.89962i	0.639299	+	0.828105i
$92$		0			0
$93$	0.455996			0.0472846
$94$		0			0
$95$		−	7.42213i		−	0.761494i
$96$		0			0
$97$		−	8.22800i		−	0.835427i	−0.908579	−	0.417713i	$-0.862832\pi$
$97$		−	8.22800i		−	0.835427i	0.908579	−	0.417713i	$-0.137168\pi$
$98$		0			0
$99$		−	12.9343i		−	1.29995i
$100$		0			0
$101$		−	7.54400i		−	0.750656i	−0.926892	−	0.375328i	$-0.877530\pi$
$101$		−	7.54400i		−	0.750656i	0.926892	−	0.375328i	$-0.122470\pi$
$102$		0			0
$103$	8.85460			0.872470			0.436235	−	0.899833i	$-0.356312\pi$
$103$	8.85460			0.872470			0.436235	+	0.899833i	$0.356312\pi$
$104$		0			0
$105$	−0.772002	−	1.00000i	−0.0753397	−	0.0975900i
$106$		0			0
$107$		−	6.09852i		−	0.589566i	−0.955564	−	0.294783i	$-0.904753\pi$
$107$		−	6.09852i		−	0.589566i	0.955564	−	0.294783i	$-0.0952474\pi$
$108$		0			0
$109$	−1.77200			−0.169727			−0.0848635	−	0.996393i	$-0.527045\pi$
$109$	−1.77200			−0.169727			−0.0848635	+	0.996393i	$0.527045\pi$
$110$		0			0
$111$	2.64721			0.251262
$112$		0			0
$113$	1.54400			0.145248			0.0726238	−	0.997359i	$-0.476863\pi$
$113$	1.54400			0.145248			0.0726238	+	0.997359i	$0.476863\pi$
$114$		0			0
$115$	−3.23357			−0.301532
$116$		0			0
$117$			10.4560i			0.966657i
$118$		0			0
$119$	−6.09852	−	7.89962i	−0.559051	−	0.724157i
$120$		0			0
$121$	−10.7720			−0.979273
$122$		0			0
$123$		−	2.86495i		−	0.258324i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	6.09852i		−	0.541156i	−0.962698	−	0.270578i	$-0.912785\pi$
$127$		−	6.09852i		−	0.541156i	0.962698	−	0.270578i	$-0.0872149\pi$
$128$		0			0
$129$			6.00000i			0.528271i
$130$		0			0
$131$	6.46715			0.565037			0.282519	−	0.959262i	$-0.408830\pi$
$131$	6.46715			0.565037			0.282519	+	0.959262i	$0.408830\pi$
$132$		0			0
$133$	15.5440	−	12.0000i	1.34784	−	1.04053i
$134$		0			0
$135$		−	2.75608i		−	0.237206i
$136$		0			0
$137$	13.5440			1.15714			0.578571	−	0.815632i	$-0.303611\pi$
$137$	13.5440			1.15714			0.578571	+	0.815632i	$0.303611\pi$
$138$		0			0
$139$	−14.8443			−1.25907			−0.629537	−	0.776971i	$-0.716755\pi$
$139$	−14.8443			−1.25907			−0.629537	+	0.776971i	$0.716755\pi$
$140$		0			0
$141$	3.77200			0.317660
$142$		0			0
$143$	17.6003			1.47181
$144$		0			0
$145$			3.77200i			0.313248i
$146$		0			0
$147$	0.846116	−	3.23357i	0.0697865	−	0.266701i
$148$		0			0
$149$	−7.54400			−0.618029			−0.309014	−	0.951057i	$-0.599999\pi$
$149$	−7.54400			−0.618029			−0.309014	+	0.951057i	$0.599999\pi$
$150$		0			0
$151$			4.66605i			0.379718i	0.981811	+	0.189859i	$0.0608030\pi$
$151$			4.66605i			0.379718i	−0.981811	+	0.189859i	$0.939197\pi$
$152$		0			0
$153$		−	10.4560i		−	0.845317i
$154$		0			0
$155$			0.954983i			0.0767061i
$156$		0			0
$157$			6.00000i			0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$			6.00000i			0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$		0			0
$159$	−2.86495			−0.227205
$160$		0			0
$161$	−5.22800	−	6.77200i	−0.412024	−	0.533709i
$162$		0			0
$163$			9.70072i			0.759819i	0.925024	+	0.379910i	$0.124045\pi$
$163$			9.70072i			0.759819i	−0.925024	+	0.379910i	$0.875955\pi$
$164$		0			0
$165$	−2.22800			−0.173449
$166$		0			0
$167$	20.0967			1.55513			0.777563	−	0.628805i	$-0.216455\pi$
$167$	20.0967			1.55513			0.777563	+	0.628805i	$0.216455\pi$
$168$		0			0
$169$	−1.22800			−0.0944614
$170$		0			0
$171$	20.5742			1.57335
$172$		0			0
$173$			23.3160i			1.77268i	0.463032	+	0.886342i	$0.346761\pi$
$173$			23.3160i			1.77268i	−0.463032	+	0.886342i	$0.653239\pi$
$174$		0			0
$175$	2.09428	−	1.61679i	0.158313	−	0.122218i
$176$		0			0
$177$	−4.45600			−0.334933
$178$		0			0
$179$		−	15.7992i		−	1.18089i	−0.807078	−	0.590445i	$-0.798952\pi$
$179$		−	15.7992i		−	1.18089i	0.807078	−	0.590445i	$-0.201048\pi$
$180$		0			0
$181$		−	7.54400i		−	0.560741i	−0.959892	−	0.280371i	$-0.909543\pi$
$181$		−	7.54400i		−	0.560741i	0.959892	−	0.280371i	$-0.0904574\pi$
$182$		0			0
$183$			6.46715i			0.478065i
$184$		0			0
$185$			5.54400i			0.407603i
$186$		0			0
$187$	−17.6003			−1.28706
$188$		0			0
$189$	5.77200	−	4.45600i	0.419851	−	0.324126i
$190$		0			0
$191$		−	8.26825i		−	0.598269i	−0.954211	−	0.299135i	$-0.903302\pi$
$191$		−	8.26825i		−	0.598269i	0.954211	−	0.299135i	$-0.0966980\pi$
$192$		0			0
$193$	−21.5440			−1.55077			−0.775386	−	0.631488i	$-0.782445\pi$
$193$	−21.5440			−1.55077			−0.775386	+	0.631488i	$0.782445\pi$
$194$		0			0
$195$	1.80110			0.128979
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	−7.42213			−0.526141			−0.263070	−	0.964777i	$-0.584735\pi$
$199$	−7.42213			−0.526141			−0.263070	+	0.964777i	$0.584735\pi$
$200$		0			0
$201$			2.91199i			0.205396i
$202$		0			0
$203$	−7.89962	+	6.09852i	−0.554445	+	0.428032i
$204$		0			0
$205$	6.00000			0.419058
$206$		0			0
$207$		−	8.96347i		−	0.623004i
$208$		0			0
$209$		−	34.6320i		−	2.39555i
$210$		0			0
$211$			17.6003i			1.21166i	0.795595	+	0.605829i	$0.207158\pi$
$211$			17.6003i			1.21166i	−0.795595	+	0.605829i	$0.792842\pi$
$212$		0			0
$213$		−	4.45600i		−	0.305320i
$214$		0			0
$215$	−12.5657			−0.856972
$216$		0			0
$217$	−2.00000	+	1.54400i	−0.135769	+	0.104814i
$218$		0			0
$219$			2.86495i			0.193595i
$220$		0			0
$221$	14.2280			0.957079
$222$		0			0
$223$	−5.98966			−0.401097			−0.200548	−	0.979684i	$-0.564272\pi$
$223$	−5.98966			−0.401097			−0.200548	+	0.979684i	$0.564272\pi$
$224$		0			0
$225$	2.77200			0.184800
$226$		0			0
$227$	−10.7646			−0.714470			−0.357235	−	0.934015i	$-0.616280\pi$
$227$	−10.7646			−0.714470			−0.357235	+	0.934015i	$0.616280\pi$
$228$		0			0
$229$			4.45600i			0.294461i	0.989102	+	0.147230i	$0.0470358\pi$
$229$			4.45600i			0.294461i	−0.989102	+	0.147230i	$0.952964\pi$
$230$		0			0
$231$	−3.60220	−	4.66605i	−0.237007	−	0.307003i
$232$		0			0
$233$	25.5440			1.67344			0.836722	−	0.547628i	$-0.184469\pi$
$233$	25.5440			1.67344			0.836722	+	0.547628i	$0.184469\pi$
$234$		0			0
$235$			7.89962i			0.515314i
$236$		0			0
$237$		−	5.31601i		−	0.345312i
$238$		0			0
$239$			1.80110i			0.116503i	0.998302	+	0.0582517i	$0.0185526\pi$
$239$			1.80110i			0.116503i	−0.998302	+	0.0582517i	$0.981447\pi$
$240$		0			0
$241$			25.5440i			1.64543i	0.568451	+	0.822717i	$0.307543\pi$
$241$			25.5440i			1.64543i	−0.568451	+	0.822717i	$0.692457\pi$
$242$		0			0
$243$	11.6107			0.744826
$244$		0			0
$245$	6.77200	+	1.77200i	0.432647	+	0.113209i
$246$		0			0
$247$			27.9963i			1.78136i
$248$		0			0
$249$	6.00000			0.380235
$250$		0			0
$251$	−12.1970			−0.769871			−0.384935	−	0.922944i	$-0.625776\pi$
$251$	−12.1970			−0.769871			−0.384935	+	0.922944i	$0.625776\pi$
$252$		0			0
$253$	−15.0880			−0.948575
$254$		0			0
$255$	−1.80110			−0.112789
$256$		0			0
$257$			21.0880i			1.31543i	0.753265	+	0.657717i	$0.228478\pi$
$257$			21.0880i			1.31543i	−0.753265	+	0.657717i	$0.771522\pi$
$258$		0			0
$259$	−11.6107	+	8.96347i	−0.721453	+	0.556963i
$260$		0			0
$261$	−10.4560			−0.647210
$262$		0			0
$263$			19.0328i			1.17361i	0.809727	+	0.586807i	$0.199615\pi$
$263$			19.0328i			1.17361i	−0.809727	+	0.586807i	$0.800385\pi$
$264$		0			0
$265$		−	6.00000i		−	0.368577i
$266$		0			0
$267$			5.72990i			0.350664i
$268$		0			0
$269$			1.54400i			0.0941396i	0.998892	+	0.0470698i	$0.0149883\pi$
$269$			1.54400i			0.0941396i	−0.998892	+	0.0470698i	$0.985012\pi$
$270$		0			0
$271$	−8.37711			−0.508873			−0.254437	−	0.967089i	$-0.581890\pi$
$271$	−8.37711			−0.508873			−0.254437	+	0.967089i	$0.581890\pi$
$272$		0			0
$273$	2.91199	+	3.77200i	0.176242	+	0.228292i
$274$		0			0
$275$		−	4.66605i		−	0.281373i
$276$		0			0
$277$	2.45600			0.147567			0.0737833	−	0.997274i	$-0.476493\pi$
$277$	2.45600			0.147567			0.0737833	+	0.997274i	$0.476493\pi$
$278$		0			0
$279$	−2.64721			−0.158485
$280$		0			0
$281$	−5.31601			−0.317126			−0.158563	−	0.987349i	$-0.550686\pi$
$281$	−5.31601			−0.317126			−0.158563	+	0.987349i	$0.550686\pi$
$282$		0			0
$283$	−12.4568			−0.740480			−0.370240	−	0.928936i	$-0.620725\pi$
$283$	−12.4568			−0.740480			−0.370240	+	0.928936i	$0.620725\pi$
$284$		0			0
$285$		−	3.54400i		−	0.209929i
$286$		0			0
$287$	9.70072	+	12.5657i	0.572615	+	0.741728i
$288$		0			0
$289$	2.77200			0.163059
$290$		0			0
$291$		−	3.92880i		−	0.230310i
$292$		0			0
$293$			11.3160i			0.661088i	0.943791	+	0.330544i	$0.107232\pi$
$293$			11.3160i			0.661088i	−0.943791	+	0.330544i	$0.892768\pi$
$294$		0			0
$295$		−	9.33210i		−	0.543336i
$296$		0			0
$297$		−	12.8600i		−	0.746213i
$298$		0			0
$299$	12.1970			0.705373
$300$		0			0
$301$	−20.3160	−	26.3160i	−1.17100	−	1.51683i
$302$		0			0
$303$		−	3.60220i		−	0.206941i
$304$		0			0
$305$	−13.5440			−0.775527
$306$		0			0
$307$	−3.12471			−0.178336			−0.0891682	−	0.996017i	$-0.528421\pi$
$307$	−3.12471			−0.178336			−0.0891682	+	0.996017i	$0.528421\pi$
$308$		0			0
$309$	4.22800			0.240522
$310$		0			0
$311$	−28.7335			−1.62933			−0.814665	−	0.579932i	$-0.803079\pi$
$311$	−28.7335			−1.62933			−0.814665	+	0.579932i	$0.803079\pi$
$312$		0			0
$313$		−	3.77200i		−	0.213206i	−0.994302	−	0.106603i	$-0.966003\pi$
$313$		−	3.77200i		−	0.213206i	0.994302	−	0.106603i	$-0.0339974\pi$
$314$		0			0
$315$	4.48174	+	5.80534i	0.252517	+	0.327094i
$316$		0			0
$317$	−9.08801			−0.510433			−0.255217	−	0.966884i	$-0.582147\pi$
$317$	−9.08801			−0.510433			−0.255217	+	0.966884i	$0.582147\pi$
$318$		0			0
$319$			17.6003i			0.985430i
$320$		0			0
$321$		−	2.91199i		−	0.162532i
$322$		0			0
$323$		−	27.9963i		−	1.55776i
$324$		0			0
$325$			3.77200i			0.209233i
$326$		0			0
$327$	−0.846116			−0.0467903
$328$		0			0
$329$	−16.5440	+	12.7720i	−0.912100	+	0.704143i
$330$		0			0
$331$			22.2664i			1.22387i	0.790908	+	0.611936i	$0.209609\pi$
$331$			22.2664i			1.22387i	−0.790908	+	0.611936i	$0.790391\pi$
$332$		0			0
$333$	−15.3680			−0.842161
$334$		0			0
$335$	−6.09852			−0.333198
$336$		0			0
$337$	21.5440			1.17358			0.586788	−	0.809740i	$-0.300392\pi$
$337$	21.5440			1.17358			0.586788	+	0.809740i	$0.300392\pi$
$338$		0			0
$339$	0.737249			0.0400419
$340$		0			0
$341$			4.45600i			0.241306i
$342$		0			0
$343$	7.23782	+	17.0474i	0.390805	+	0.920473i
$344$		0			0
$345$	−1.54400			−0.0831264
$346$		0			0
$347$			3.23357i			0.173587i	0.996226	+	0.0867937i	$0.0276621\pi$
$347$			3.23357i			0.173587i	−0.996226	+	0.0867937i	$0.972338\pi$
$348$		0			0
$349$			19.5440i			1.04617i	0.852282	+	0.523083i	$0.175218\pi$
$349$			19.5440i			1.04617i	−0.852282	+	0.523083i	$0.824782\pi$
$350$		0			0
$351$			10.3959i			0.554895i
$352$		0			0
$353$		−	6.86001i		−	0.365121i	−0.983195	−	0.182561i	$-0.941561\pi$
$353$		−	6.86001i		−	0.365121i	0.983195	−	0.182561i	$-0.0584386\pi$
$354$		0			0
$355$	9.33210			0.495296
$356$		0			0
$357$	−2.91199	−	3.77200i	−0.154119	−	0.199636i
$358$		0			0
$359$			15.7992i			0.833852i	0.908940	+	0.416926i	$0.136893\pi$
$359$			15.7992i			0.833852i	−0.908940	+	0.416926i	$0.863107\pi$
$360$		0			0
$361$	36.0880			1.89937
$362$		0			0
$363$	−5.14354			−0.269966
$364$		0			0
$365$	−6.00000			−0.314054
$366$		0			0
$367$	−2.38746			−0.124624			−0.0623121	−	0.998057i	$-0.519847\pi$
$367$	−2.38746			−0.124624			−0.0623121	+	0.998057i	$0.519847\pi$
$368$		0			0
$369$			16.6320i			0.865828i
$370$		0			0
$371$	12.5657	−	9.70072i	0.652377	−	0.503636i
$372$		0			0
$373$	21.5440			1.11551			0.557753	−	0.830007i	$-0.311664\pi$
$373$	21.5440			1.11551			0.557753	+	0.830007i	$0.311664\pi$
$374$		0			0
$375$		−	0.477491i		−	0.0246576i
$376$		0			0
$377$		−	14.2280i		−	0.732779i
$378$		0			0
$379$			15.7992i			0.811553i	0.913972	+	0.405776i	$0.132999\pi$
$379$			15.7992i			0.811553i	−0.913972	+	0.405776i	$0.867001\pi$
$380$		0			0
$381$		−	2.91199i		−	0.149186i
$382$		0			0
$383$	−19.0328			−0.972531			−0.486266	−	0.873811i	$-0.661641\pi$
$383$	−19.0328			−0.972531			−0.486266	+	0.873811i	$0.661641\pi$
$384$		0			0
$385$	9.77200	−	7.54400i	0.498027	−	0.384478i
$386$		0			0
$387$		−	34.8321i		−	1.77061i
$388$		0			0
$389$	5.31601			0.269532			0.134766	−	0.990877i	$-0.456972\pi$
$389$	5.31601			0.269532			0.134766	+	0.990877i	$0.456972\pi$
$390$		0			0
$391$	−12.1970			−0.616831
$392$		0			0
$393$	3.08801			0.155769
$394$		0			0
$395$	11.1332			0.560172
$396$		0			0
$397$		−	8.22800i		−	0.412951i	−0.978452	−	0.206476i	$-0.933801\pi$
$397$		−	8.22800i		−	0.412951i	0.978452	−	0.206476i	$-0.0661994\pi$
$398$		0			0
$399$	7.42213	−	5.72990i	0.371571	−	0.286854i
$400$		0			0
$401$	−15.7720			−0.787616			−0.393808	−	0.919193i	$-0.628843\pi$
$401$	−15.7720			−0.787616			−0.393808	+	0.919193i	$0.628843\pi$
$402$		0			0
$403$		−	3.60220i		−	0.179438i
$404$		0			0
$405$			7.00000i			0.347833i
$406$		0			0
$407$			25.8686i			1.28226i
$408$		0			0
$409$			2.91199i			0.143989i	0.997405	+	0.0719944i	$0.0229364\pi$
$409$			2.91199i			0.143989i	−0.997405	+	0.0719944i	$0.977064\pi$
$410$		0			0
$411$	6.46715			0.319001
$412$		0			0
$413$	19.5440	−	15.0880i	0.961698	−	0.742432i
$414$		0			0
$415$			12.5657i			0.616824i
$416$		0			0
$417$	−7.08801			−0.347101
$418$		0			0
$419$	22.2664			1.08778			0.543892	−	0.839155i	$-0.316950\pi$
$419$	22.2664			1.08778			0.543892	+	0.839155i	$0.316950\pi$
$420$		0			0
$421$	28.8600			1.40655			0.703275	−	0.710918i	$-0.251720\pi$
$421$	28.8600			1.40655			0.703275	+	0.710918i	$0.251720\pi$
$422$		0			0
$423$	−21.8978			−1.06471
$424$		0			0
$425$		−	3.77200i		−	0.182969i
$426$		0			0
$427$	−21.8978	−	28.3649i	−1.05971	−	1.37267i
$428$		0			0
$429$	8.40401			0.405750
$430$		0			0
$431$		−	39.8667i		−	1.92031i	−0.279463	−	0.960156i	$-0.590157\pi$
$431$		−	39.8667i		−	1.92031i	0.279463	−	0.960156i	$-0.409843\pi$
$432$		0			0
$433$		−	9.08801i		−	0.436742i	−0.975866	−	0.218371i	$-0.929926\pi$
$433$		−	9.08801i		−	0.436742i	0.975866	−	0.218371i	$-0.0700743\pi$
$434$		0			0
$435$			1.80110i			0.0863561i
$436$		0			0
$437$		−	24.0000i		−	1.14808i
$438$		0			0
$439$	1.90997			0.0911577			0.0455789	−	0.998961i	$-0.485487\pi$
$439$	1.90997			0.0911577			0.0455789	+	0.998961i	$0.485487\pi$
$440$		0			0
$441$	−4.91199	+	18.7720i	−0.233904	+	0.893905i
$442$		0			0
$443$		−	19.0328i		−	0.904276i	−0.891948	−	0.452138i	$-0.850661\pi$
$443$		−	19.0328i		−	0.904276i	0.891948	−	0.452138i	$-0.149339\pi$
$444$		0			0
$445$	−12.0000			−0.568855
$446$		0			0
$447$	−3.60220			−0.170378
$448$		0			0
$449$	−12.8600			−0.606901			−0.303451	−	0.952847i	$-0.598139\pi$
$449$	−12.8600			−0.606901			−0.303451	+	0.952847i	$0.598139\pi$
$450$		0			0
$451$	27.9963			1.31829
$452$		0			0
$453$			2.22800i			0.104680i
$454$		0			0
$455$	−7.89962	+	6.09852i	−0.370340	+	0.285903i
$456$		0			0
$457$	−40.1760			−1.87935			−0.939677	−	0.342062i	$-0.888875\pi$
$457$	−40.1760			−1.87935			−0.939677	+	0.342062i	$0.888875\pi$
$458$		0			0
$459$		−	10.3959i		−	0.485241i
$460$		0			0
$461$			27.0880i			1.26161i	0.775940	+	0.630807i	$0.217276\pi$
$461$			27.0880i			1.26161i	−0.775940	+	0.630807i	$0.782724\pi$
$462$		0			0
$463$		−	28.3649i		−	1.31823i	−0.752043	−	0.659115i	$-0.770931\pi$
$463$		−	28.3649i		−	1.31823i	0.752043	−	0.659115i	$-0.229069\pi$
$464$		0			0
$465$			0.455996i			0.0211463i
$466$		0			0
$467$	27.3011			1.26334			0.631671	−	0.775236i	$-0.282369\pi$
$467$	27.3011			1.26334			0.631671	+	0.775236i	$0.282369\pi$
$468$		0			0
$469$	−9.86001	−	12.7720i	−0.455293	−	0.589756i
$470$		0			0
$471$			2.86495i			0.132010i
$472$		0			0
$473$	−58.6320			−2.69590
$474$		0			0
$475$	7.42213			0.340551
$476$		0			0
$477$	16.6320			0.761527
$478$		0			0
$479$	37.3284			1.70558			0.852789	−	0.522256i	$-0.174910\pi$
$479$	37.3284			1.70558			0.852789	+	0.522256i	$0.174910\pi$
$480$		0			0
$481$		−	20.9120i		−	0.953505i
$482$		0			0
$483$	−2.49632	−	3.23357i	−0.113587	−	0.147133i
$484$		0			0
$485$	8.22800			0.373614
$486$		0			0
$487$		−	9.70072i		−	0.439582i	−0.975547	−	0.219791i	$-0.929463\pi$
$487$		−	9.70072i		−	0.439582i	0.975547	−	0.219791i	$-0.0705375\pi$
$488$		0			0
$489$			4.63201i			0.209467i
$490$		0			0
$491$		−	8.26825i		−	0.373141i	−0.982442	−	0.186570i	$-0.940263\pi$
$491$		−	8.26825i		−	0.373141i	0.982442	−	0.186570i	$-0.0597372\pi$
$492$		0			0
$493$			14.2280i			0.640797i
$494$		0			0
$495$	12.9343			0.581353
$496$		0			0
$497$	15.0880	+	19.5440i	0.676790	+	0.876668i
$498$		0			0
$499$		−	16.8631i		−	0.754896i	−0.926031	−	0.377448i	$-0.876802\pi$
$499$		−	16.8631i		−	0.754896i	0.926031	−	0.377448i	$-0.123198\pi$
$500$		0			0
$501$	9.59599			0.428717
$502$		0			0
$503$	−1.43247			−0.0638709			−0.0319354	−	0.999490i	$-0.510167\pi$
$503$	−1.43247			−0.0638709			−0.0319354	+	0.999490i	$0.510167\pi$
$504$		0			0
$505$	7.54400			0.335704
$506$		0			0
$507$	−0.586359			−0.0260411
$508$		0			0
$509$		−	15.0880i		−	0.668764i	−0.942438	−	0.334382i	$-0.891472\pi$
$509$		−	15.0880i		−	0.668764i	0.942438	−	0.334382i	$-0.108528\pi$
$510$		0			0
$511$	−9.70072	−	12.5657i	−0.429135	−	0.555872i
$512$		0			0
$513$	20.4560			0.903154
$514$		0			0
$515$			8.85460i			0.390180i
$516$		0			0
$517$			36.8600i			1.62110i
$518$		0			0
$519$			11.1332i			0.488693i
$520$		0			0
$521$		−	4.45600i		−	0.195221i	−0.995225	−	0.0976104i	$-0.968880\pi$
$521$		−	4.45600i		−	0.195221i	0.995225	−	0.0976104i	$-0.0311199\pi$
$522$		0			0
$523$	−26.6727			−1.16631			−0.583157	−	0.812359i	$-0.698183\pi$
$523$	−26.6727			−1.16631			−0.583157	+	0.812359i	$0.698183\pi$
$524$		0			0
$525$	1.00000	−	0.772002i	0.0436436	−	0.0336929i
$526$		0			0
$527$			3.60220i			0.156914i
$528$		0			0
$529$	12.5440			0.545391
$530$		0			0
$531$	25.8686			1.12260
$532$		0			0
$533$	−22.6320			−0.980301
$534$		0			0
$535$	6.09852			0.263662
$536$		0			0
$537$		−	7.54400i		−	0.325548i
$538$		0			0
$539$	31.5985	+	8.26825i	1.36104	+	0.356139i
$540$		0			0
$541$	−23.7720			−1.02204			−0.511019	−	0.859569i	$-0.670732\pi$
$541$	−23.7720			−1.02204			−0.511019	+	0.859569i	$0.670732\pi$
$542$		0			0
$543$		−	3.60220i		−	0.154585i
$544$		0			0
$545$		−	1.77200i		−	0.0759042i
$546$		0			0
$547$		−	28.3649i		−	1.21280i	−0.795161	−	0.606398i	$-0.792614\pi$
$547$		−	28.3649i		−	1.21280i	0.795161	−	0.606398i	$-0.207386\pi$
$548$		0			0
$549$		−	37.5440i		−	1.60234i
$550$		0			0
$551$	−27.9963			−1.19268
$552$		0			0
$553$	18.0000	+	23.3160i	0.765438	+	0.991497i
$554$		0			0
$555$			2.64721i			0.112368i
$556$		0			0
$557$	37.5440			1.59079			0.795395	−	0.606091i	$-0.207263\pi$
$557$	37.5440			1.59079			0.795395	+	0.606091i	$0.207263\pi$
$558$		0			0
$559$	47.3977			2.00471
$560$		0			0
$561$	−8.40401			−0.354818
$562$		0			0
$563$	19.0328			0.802138			0.401069	−	0.916048i	$-0.368639\pi$
$563$	19.0328			0.802138			0.401069	+	0.916048i	$0.368639\pi$
$564$		0			0
$565$			1.54400i			0.0649567i
$566$		0			0
$567$	−14.6599	+	11.3175i	−0.615660	+	0.475291i
$568$		0			0
$569$	34.6320			1.45185			0.725925	−	0.687774i	$-0.241412\pi$
$569$	34.6320			1.45185			0.725925	+	0.687774i	$0.241412\pi$
$570$		0			0
$571$			9.33210i			0.390536i	0.980750	+	0.195268i	$0.0625577\pi$
$571$			9.33210i			0.390536i	−0.980750	+	0.195268i	$0.937442\pi$
$572$		0			0
$573$		−	3.94802i		−	0.164931i
$574$		0			0
$575$		−	3.23357i		−	0.134849i
$576$		0			0
$577$			42.8600i			1.78429i	0.451754	+	0.892143i	$0.350799\pi$
$577$			42.8600i			1.78429i	−0.451754	+	0.892143i	$0.649201\pi$
$578$		0			0
$579$	−10.2871			−0.427516
$580$		0			0
$581$	−26.3160	+	20.3160i	−1.09177	+	0.842850i
$582$		0			0
$583$		−	27.9963i		−	1.15949i
$584$		0			0
$585$	−10.4560			−0.432302
$586$		0			0
$587$	−12.5657			−0.518641			−0.259320	−	0.965791i	$-0.583499\pi$
$587$	−12.5657			−0.518641			−0.259320	+	0.965791i	$0.583499\pi$
$588$		0			0
$589$	−7.08801			−0.292056
$590$		0			0
$591$	−2.86495			−0.117848
$592$		0			0
$593$		−	30.8600i		−	1.26727i	−0.773633	−	0.633634i	$-0.781562\pi$
$593$		−	30.8600i		−	1.26727i	0.773633	−	0.633634i	$-0.218438\pi$
$594$		0			0
$595$	7.89962	−	6.09852i	0.323853	−	0.250015i
$596$		0			0
$597$	−3.54400			−0.145046
$598$		0			0
$599$			39.1295i			1.59879i	0.600808	+	0.799394i	$0.294846\pi$
$599$			39.1295i			1.59879i	−0.600808	+	0.799394i	$0.705154\pi$
$600$		0			0
$601$		−	24.1760i		−	0.986160i	−0.869984	−	0.493080i	$-0.835871\pi$
$601$		−	24.1760i		−	0.986160i	0.869984	−	0.493080i	$-0.164129\pi$
$602$		0			0
$603$		−	16.9051i		−	0.688430i
$604$		0			0
$605$		−	10.7720i		−	0.437944i
$606$		0			0
$607$	22.7439			0.923146			0.461573	−	0.887102i	$-0.347285\pi$
$607$	22.7439			0.923146			0.461573	+	0.887102i	$0.347285\pi$
$608$		0			0
$609$	−3.77200	+	2.91199i	−0.152849	+	0.118000i
$610$		0			0
$611$		−	29.7974i		−	1.20547i
$612$		0			0
$613$	−34.0000			−1.37325			−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000			−1.37325			−0.686624	+	0.727013i	$0.740908\pi$
$614$		0			0
$615$	2.86495			0.115526
$616$		0			0
$617$	1.54400			0.0621593			0.0310796	−	0.999517i	$-0.490105\pi$
$617$	1.54400			0.0621593			0.0310796	+	0.999517i	$0.490105\pi$
$618$		0			0
$619$	−1.90997			−0.0767680			−0.0383840	−	0.999263i	$-0.512221\pi$
$619$	−1.90997			−0.0767680			−0.0383840	+	0.999263i	$0.512221\pi$
$620$		0			0
$621$		−	8.91199i		−	0.357626i
$622$		0			0
$623$	−19.4014	−	25.1313i	−0.777302	−	1.00687i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	16.5365i		−	0.660404i
$628$		0			0
$629$			20.9120i			0.833816i
$630$		0			0
$631$		−	13.9981i		−	0.557257i	−0.960399	−	0.278629i	$-0.910120\pi$
$631$		−	13.9981i		−	0.557257i	0.960399	−	0.278629i	$-0.0898799\pi$
$632$		0			0
$633$			8.40401i			0.334030i
$634$		0			0
$635$	6.09852			0.242012
$636$		0			0
$637$	−25.5440	−	6.68399i	−1.01209	−	0.264829i
$638$		0			0
$639$			25.8686i			1.02335i
$640$		0			0
$641$	46.6320			1.84185			0.920927	−	0.389735i	$-0.127434\pi$
$641$	46.6320			1.84185			0.920927	+	0.389735i	$0.127434\pi$
$642$		0			0
$643$	−12.4568			−0.491248			−0.245624	−	0.969365i	$-0.578993\pi$
$643$	−12.4568			−0.491248			−0.245624	+	0.969365i	$0.578993\pi$
$644$		0			0
$645$	−6.00000			−0.236250
$646$		0			0
$647$	31.2299			1.22777			0.613886	−	0.789395i	$-0.289605\pi$
$647$	31.2299			1.22777			0.613886	+	0.789395i	$0.289605\pi$
$648$		0			0
$649$		−	43.5440i		−	1.70925i
$650$		0			0
$651$	−0.954983	+	0.737249i	−0.0374287	+	0.0288951i
$652$		0			0
$653$	33.0880			1.29483			0.647417	−	0.762136i	$-0.275849\pi$
$653$	33.0880			1.29483			0.647417	+	0.762136i	$0.275849\pi$
$654$		0			0
$655$			6.46715i			0.252692i
$656$		0			0
$657$		−	16.6320i		−	0.648877i
$658$		0			0
$659$		−	3.92880i		−	0.153044i	−0.997068	−	0.0765221i	$-0.975618\pi$
$659$		−	3.92880i		−	0.153044i	0.997068	−	0.0765221i	$-0.0243816\pi$
$660$		0			0
$661$		−	1.54400i		−	0.0600548i	−0.999549	−	0.0300274i	$-0.990441\pi$
$661$		−	1.54400i		−	0.0600548i	0.999549	−	0.0300274i	$-0.00955945\pi$
$662$		0			0
$663$	6.79375			0.263847
$664$		0			0
$665$	12.0000	+	15.5440i	0.465340	+	0.602771i
$666$		0			0
$667$			12.1970i			0.472271i
$668$		0			0
$669$	−2.86001			−0.110574
$670$		0			0
$671$	−63.1970			−2.43969
$672$		0			0
$673$	−18.4560			−0.711426			−0.355713	−	0.934595i	$-0.615762\pi$
$673$	−18.4560			−0.711426			−0.355713	+	0.934595i	$0.615762\pi$
$674$		0			0
$675$	2.75608			0.106082
$676$		0			0
$677$		−	8.22800i		−	0.316228i	−0.987421	−	0.158114i	$-0.949459\pi$
$677$		−	8.22800i		−	0.316228i	0.987421	−	0.158114i	$-0.0505413\pi$
$678$		0			0
$679$	13.3029	+	17.2317i	0.510519	+	0.661293i
$680$		0			0
$681$	−5.13999			−0.196965
$682$		0			0
$683$		−	22.6350i		−	0.866105i	−0.901369	−	0.433052i	$-0.857437\pi$
$683$		−	22.6350i		−	0.866105i	0.901369	−	0.433052i	$-0.142563\pi$
$684$		0			0
$685$			13.5440i			0.517490i
$686$		0			0
$687$			2.12770i			0.0811768i
$688$		0			0
$689$			22.6320i			0.862211i
$690$		0			0
$691$	10.2871			0.391339			0.195669	−	0.980670i	$-0.437312\pi$
$691$	10.2871			0.391339			0.195669	+	0.980670i	$0.437312\pi$
$692$		0			0
$693$	20.9120	+	27.0880i	0.794381	+	1.02899i
$694$		0			0
$695$		−	14.8443i		−	0.563075i
$696$		0			0
$697$	22.6320			0.857249
$698$		0			0
$699$	12.1970			0.461335
$700$		0			0
$701$	12.8600			0.485716			0.242858	−	0.970062i	$-0.421915\pi$
$701$	12.8600			0.485716			0.242858	+	0.970062i	$0.421915\pi$
$702$		0			0
$703$	−41.1483			−1.55194
$704$		0			0
$705$			3.77200i			0.142062i
$706$		0			0
$707$	12.1970	+	15.7992i	0.458717	+	0.594192i
$708$		0			0
$709$	−39.3160			−1.47654			−0.738272	−	0.674503i	$-0.764358\pi$
$709$	−39.3160			−1.47654			−0.738272	+	0.674503i	$0.764358\pi$
$710$		0			0
$711$			30.8612i			1.15739i
$712$		0			0
$713$			3.08801i			0.115647i
$714$		0			0
$715$			17.6003i			0.658215i
$716$		0			0
$717$			0.860009i			0.0321176i
$718$		0			0
$719$	31.5985			1.17842			0.589212	−	0.807978i	$-0.299438\pi$
$719$	31.5985			1.17842			0.589212	+	0.807978i	$0.299438\pi$
$720$		0			0
$721$	−18.5440	+	14.3160i	−0.690615	+	0.533156i
$722$		0			0
$723$			12.1970i			0.453613i
$724$		0			0
$725$	−3.77200			−0.140089
$726$		0			0
$727$	−8.00849			−0.297018			−0.148509	−	0.988911i	$-0.547447\pi$
$727$	−8.00849			−0.297018			−0.148509	+	0.988911i	$0.547447\pi$
$728$		0			0
$729$	−15.4560			−0.572444
$730$		0			0
$731$	−47.3977			−1.75307
$732$		0			0
$733$		−	39.7720i		−	1.46901i	−0.678602	−	0.734506i	$-0.737414\pi$
$733$		−	39.7720i		−	1.46901i	0.678602	−	0.734506i	$-0.262586\pi$
$734$		0			0
$735$	3.23357	+	0.846116i	0.119272	+	0.0312095i
$736$		0			0
$737$	−28.4560			−1.04819
$738$		0			0
$739$		−	5.40330i		−	0.198763i	−0.995049	−	0.0993817i	$-0.968314\pi$
$739$		−	5.40330i		−	0.198763i	0.995049	−	0.0993817i	$-0.0316865\pi$
$740$		0			0
$741$			13.3680i			0.491085i
$742$		0			0
$743$			44.1642i			1.62023i	0.586274	+	0.810113i	$0.300594\pi$
$743$			44.1642i			1.62023i	−0.586274	+	0.810113i	$0.699406\pi$
$744$		0			0
$745$		−	7.54400i		−	0.276391i
$746$		0			0
$747$	−34.8321			−1.27444
$748$		0			0
$749$	9.86001	+	12.7720i	0.360277	+	0.466679i
$750$		0			0
$751$		−	32.6623i		−	1.19187i	−0.803034	−	0.595933i	$-0.796782\pi$
$751$		−	32.6623i		−	1.19187i	0.803034	−	0.595933i	$-0.203218\pi$
$752$		0			0
$753$	−5.82399			−0.212238
$754$		0			0
$755$	−4.66605			−0.169815
$756$		0			0
$757$	−5.54400			−0.201500			−0.100750	−	0.994912i	$-0.532124\pi$
$757$	−5.54400			−0.201500			−0.100750	+	0.994912i	$0.532124\pi$
$758$		0			0
$759$	−7.20440			−0.261503
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	3.71106	−	2.86495i	0.134350	−	0.103718i
$764$		0			0
$765$	10.4560			0.378037
$766$		0			0
$767$			35.2007i			1.27102i
$768$		0			0
$769$			22.6320i			0.816131i	0.912953	+	0.408066i	$0.133797\pi$
$769$			22.6320i			0.816131i	−0.912953	+	0.408066i	$0.866203\pi$
$770$		0			0
$771$			10.0693i			0.362639i
$772$		0			0
$773$		−	51.7720i		−	1.86211i	−0.364880	−	0.931055i	$-0.618890\pi$
$773$		−	51.7720i		−	1.86211i	0.364880	−	0.931055i	$-0.381110\pi$
$774$		0			0
$775$	−0.954983			−0.0343040
$776$		0			0
$777$	−5.54400	+	4.27998i	−0.198890	+	0.153543i
$778$		0			0
$779$			44.5328i			1.59555i
$780$		0			0
$781$	43.5440			1.55813
$782$		0			0
$783$	−10.3959			−0.371521
$784$		0			0
$785$	−6.00000			−0.214149
$786$		0			0
$787$	−38.5431			−1.37391			−0.686957	−	0.726698i	$-0.741054\pi$
$787$	−38.5431			−1.37391			−0.686957	+	0.726698i	$0.741054\pi$
$788$		0			0
$789$			9.08801i			0.323542i
$790$		0			0
$791$	−3.23357	+	2.49632i	−0.114973	+	0.0887591i
$792$		0			0
$793$	51.0880			1.81419
$794$		0			0
$795$		−	2.86495i		−	0.101609i
$796$		0			0
$797$			27.7720i			0.983735i	0.870670	+	0.491867i	$0.163686\pi$
$797$			27.7720i			0.983735i	−0.870670	+	0.491867i	$0.836314\pi$
$798$		0			0
$799$			29.7974i			1.05416i
$800$		0			0
$801$		−	33.2640i		−	1.17533i
$802$		0			0
$803$	−27.9963			−0.987968
$804$		0			0
$805$	6.77200	−	5.22800i	0.238682	−	0.184263i
$806$		0			0
$807$			0.737249i			0.0259524i
$808$		0			0
$809$	−11.3160			−0.397850			−0.198925	−	0.980015i	$-0.563745\pi$
$809$	−11.3160			−0.397850			−0.198925	+	0.980015i	$0.563745\pi$
$810$		0			0
$811$	47.6155			1.67200			0.836002	−	0.548726i	$-0.184887\pi$
$811$	47.6155			1.67200			0.836002	+	0.548726i	$0.184887\pi$
$812$		0			0
$813$	−4.00000			−0.140286
$814$		0			0
$815$	−9.70072			−0.339801
$816$		0			0
$817$		−	93.2640i		−	3.26290i
$818$		0			0
$819$	−16.9051	−	21.8978i	−0.590713	−	0.765170i
$820$		0			0
$821$	−41.3160			−1.44194			−0.720969	−	0.692967i	$-0.756303\pi$
$821$	−41.3160			−1.44194			−0.720969	+	0.692967i	$0.756303\pi$
$822$		0			0
$823$		−	25.5000i		−	0.888873i	−0.895810	−	0.444436i	$-0.853404\pi$
$823$		−	25.5000i		−	0.888873i	0.895810	−	0.444436i	$-0.146596\pi$
$824$		0			0
$825$		−	2.22800i		−	0.0775689i
$826$		0			0
$827$			3.23357i			0.112442i	0.998418	+	0.0562212i	$0.0179052\pi$
$827$			3.23357i			0.112442i	−0.998418	+	0.0562212i	$0.982095\pi$
$828$		0			0
$829$			14.9120i			0.517915i	0.965889	+	0.258957i	$0.0833789\pi$
$829$			14.9120i			0.517915i	−0.965889	+	0.258957i	$0.916621\pi$
$830$		0			0
$831$	1.17272			0.0406811
$832$		0			0
$833$	25.5440	+	6.68399i	0.885047	+	0.231587i
$834$		0			0
$835$			20.0967i			0.695474i
$836$		0			0
$837$	−2.63201			−0.0909756
$838$		0			0
$839$	−31.5985			−1.09090			−0.545450	−	0.838143i	$-0.683641\pi$
$839$	−31.5985			−1.09090			−0.545450	+	0.838143i	$0.683641\pi$
$840$		0			0
$841$	−14.7720			−0.509379
$842$		0			0
$843$	−2.53835			−0.0874253
$844$		0			0
$845$		−	1.22800i		−	0.0422444i
$846$		0			0
$847$	22.5596	−	17.4160i	0.775156	−	0.598422i
$848$		0			0
$849$	−5.94802			−0.204135
$850$		0			0
$851$			17.9269i			0.614528i
$852$		0			0
$853$			48.1760i			1.64952i	0.565486	+	0.824758i	$0.308689\pi$
$853$			48.1760i			1.64952i	−0.565486	+	0.824758i	$0.691311\pi$
$854$		0			0
$855$			20.5742i			0.703621i
$856$		0			0
$857$		−	6.00000i		−	0.204956i	−0.994735	−	0.102478i	$-0.967323\pi$
$857$		−	6.00000i		−	0.204956i	0.994735	−	0.102478i	$-0.0326771\pi$
$858$		0			0
$859$	39.2383			1.33880			0.669398	−	0.742904i	$-0.266552\pi$
$859$	39.2383			1.33880			0.669398	+	0.742904i	$0.266552\pi$
$860$		0			0
$861$	4.63201	+	6.00000i	0.157859	+	0.204479i
$862$		0			0
$863$			37.6970i			1.28322i	0.767031	+	0.641611i	$0.221733\pi$
$863$			37.6970i			1.28322i	−0.767031	+	0.641611i	$0.778267\pi$
$864$		0			0
$865$	−23.3160			−0.792768
$866$		0			0
$867$	1.32361			0.0449521
$868$		0			0
$869$	51.9480			1.76222
$870$		0			0
$871$	23.0036			0.779449
$872$		0			0
$873$			22.8080i			0.771935i
$874$		0			0
$875$	1.61679	+	2.09428i	0.0546574	+	0.0707995i
$876$		0			0
$877$	29.0880			0.982232			0.491116	−	0.871094i	$-0.336589\pi$
$877$	29.0880			0.982232			0.491116	+	0.871094i	$0.336589\pi$
$878$		0			0
$879$			5.40330i			0.182249i
$880$		0			0
$881$			16.6320i			0.560347i	0.959949	+	0.280173i	$0.0903919\pi$
$881$			16.6320i			0.560347i	−0.959949	+	0.280173i	$0.909608\pi$
$882$		0			0
$883$			34.8321i			1.17219i	0.810242	+	0.586096i	$0.199336\pi$
$883$			34.8321i			1.17219i	−0.810242	+	0.586096i	$0.800664\pi$
$884$		0			0
$885$		−	4.45600i		−	0.149787i
$886$		0			0
$887$	−19.7701			−0.663814			−0.331907	−	0.943312i	$-0.607692\pi$
$887$	−19.7701			−0.663814			−0.331907	+	0.943312i	$0.607692\pi$
$888$		0			0
$889$	9.86001	+	12.7720i	0.330694	+	0.428359i
$890$		0			0
$891$			32.6623i			1.09423i
$892$		0			0
$893$	−58.6320			−1.96205
$894$		0			0
$895$	15.7992			0.528110
$896$		0			0
$897$	5.82399			0.194457
$898$		0			0
$899$	3.60220			0.120140
$900$		0			0
$901$		−	22.6320i		−	0.753982i
$902$		0			0
$903$	−9.70072	−	12.5657i	−0.322820	−	0.418159i
$904$		0			0
$905$	7.54400			0.250771
$906$		0			0
$907$			15.4306i			0.512365i	0.966628	+	0.256183i	$0.0824648\pi$
$907$			15.4306i			0.512365i	−0.966628	+	0.256183i	$0.917535\pi$
$908$		0			0
$909$			20.9120i			0.693607i
$910$		0			0
$911$		−	35.2007i		−	1.16625i	−0.812382	−	0.583125i	$-0.801830\pi$
$911$		−	35.2007i		−	1.16625i	0.812382	−	0.583125i	$-0.198170\pi$
$912$		0			0
$913$			58.6320i			1.94044i
$914$		0			0
$915$	−6.46715			−0.213797
$916$		0			0
$917$	−13.5440	+	10.4560i	−0.447262	+	0.345287i
$918$		0			0
$919$		−	48.4616i		−	1.59860i	−0.600932	−	0.799300i	$-0.705204\pi$
$919$		−	48.4616i		−	1.59860i	0.600932	−	0.799300i	$-0.294796\pi$
$920$		0			0
$921$	−1.49202			−0.0491638
$922$		0			0
$923$	−35.2007			−1.15864
$924$		0			0
$925$	−5.54400			−0.182286
$926$		0			0
$927$	−24.5450			−0.806163
$928$		0			0
$929$		−	33.0880i		−	1.08558i	−0.839868	−	0.542791i	$-0.817368\pi$
$929$		−	33.0880i		−	1.08558i	0.839868	−	0.542791i	$-0.182632\pi$
$930$		0			0
$931$	−13.1520	+	50.2627i	−0.431040	+	1.64729i
$932$		0			0
$933$	−13.7200			−0.449173
$934$		0			0
$935$		−	17.6003i		−	0.575593i
$936$		0			0
$937$			27.7720i			0.907272i	0.891187	+	0.453636i	$0.149873\pi$
$937$			27.7720i			0.907272i	−0.891187	+	0.453636i	$0.850127\pi$
$938$		0			0
$939$		−	1.80110i		−	0.0587766i
$940$		0			0
$941$		−	42.1760i		−	1.37490i	−0.726232	−	0.687449i	$-0.758730\pi$
$941$		−	42.1760i		−	1.37490i	0.726232	−	0.687449i	$-0.241270\pi$
$942$		0			0
$943$	19.4014			0.631798
$944$		0			0
$945$	4.45600	+	5.77200i	0.144954	+	0.187763i
$946$		0			0
$947$			6.83577i			0.222133i	0.993813	+	0.111066i	$0.0354266\pi$
$947$			6.83577i			0.222133i	−0.993813	+	0.111066i	$0.964573\pi$
$948$		0			0
$949$	22.6320			0.734666
$950$		0			0
$951$	−4.33945			−0.140716
$952$		0			0
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$		0			0
$955$	8.26825			0.267554
$956$		0			0
$957$			8.40401i			0.271663i
$958$		0			0
$959$	−28.3649	+	21.8978i	−0.915951	+	0.707116i
$960$		0			0
$961$	−30.0880			−0.970581
$962$		0			0
$963$			16.9051i			0.544760i
$964$		0			0
$965$		−	21.5440i		−	0.693526i
$966$		0			0
$967$		−	43.4269i		−	1.39652i	−0.715847	−	0.698258i	$-0.753959\pi$
$967$		−	43.4269i		−	1.39652i	0.715847	−	0.698258i	$-0.246041\pi$
$968$		0			0
$969$		−	13.3680i		−	0.429442i
$970$		0			0
$971$	−21.5291			−0.690903			−0.345451	−	0.938437i	$-0.612274\pi$
$971$	−21.5291			−0.690903			−0.345451	+	0.938437i	$0.612274\pi$
$972$		0			0
$973$	31.0880	−	24.0000i	0.996636	−	0.769405i
$974$		0			0
$975$			1.80110i			0.0576813i
$976$		0			0
$977$	2.91199			0.0931629			0.0465814	−	0.998914i	$-0.485167\pi$
$977$	2.91199			0.0931629			0.0465814	+	0.998914i	$0.485167\pi$
$978$		0			0
$979$	−55.9926			−1.78953
$980$		0			0
$981$	4.91199			0.156828
$982$		0			0
$983$	10.7646			0.343336			0.171668	−	0.985155i	$-0.445084\pi$
$983$	10.7646			0.343336			0.171668	+	0.985155i	$0.445084\pi$
$984$		0			0
$985$		−	6.00000i		−	0.191176i
$986$		0			0
$987$	−7.89962	+	6.09852i	−0.251448	+	0.194118i
$988$		0			0
$989$	−40.6320			−1.29202
$990$		0			0
$991$		−	15.0620i		−	0.478460i	−0.970963	−	0.239230i	$-0.923105\pi$
$991$		−	15.0620i		−	0.478460i	0.970963	−	0.239230i	$-0.0768950\pi$
$992$		0			0
$993$			10.6320i			0.337397i
$994$		0			0
$995$		−	7.42213i		−	0.235297i
$996$		0			0
$997$		−	26.4040i		−	0.836223i	−0.908396	−	0.418112i	$-0.862692\pi$
$997$		−	26.4040i		−	0.836223i	0.908396	−	0.418112i	$-0.137308\pi$
$998$		0			0
$999$	−15.2797			−0.483429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.k.b.111.6	yes	8
3.2	odd	2		5040.2.d.d.4591.1		8
4.3	odd	2	inner	560.2.k.b.111.4	yes	8
5.2	odd	4		2800.2.e.h.2799.3		8
5.3	odd	4		2800.2.e.g.2799.6		8
5.4	even	2		2800.2.k.m.2351.3		8
7.6	odd	2	inner	560.2.k.b.111.3	✓	8
8.3	odd	2		2240.2.k.d.1791.5		8
8.5	even	2		2240.2.k.d.1791.3		8
12.11	even	2		5040.2.d.d.4591.4		8
20.3	even	4		2800.2.e.g.2799.3		8
20.7	even	4		2800.2.e.h.2799.6		8
20.19	odd	2		2800.2.k.m.2351.6		8
21.20	even	2		5040.2.d.d.4591.8		8
28.27	even	2	inner	560.2.k.b.111.5	yes	8
35.13	even	4		2800.2.e.h.2799.4		8
35.27	even	4		2800.2.e.g.2799.5		8
35.34	odd	2		2800.2.k.m.2351.5		8
56.13	odd	2		2240.2.k.d.1791.6		8
56.27	even	2		2240.2.k.d.1791.4		8
84.83	odd	2		5040.2.d.d.4591.5		8
140.27	odd	4		2800.2.e.g.2799.4		8
140.83	odd	4		2800.2.e.h.2799.5		8
140.139	even	2		2800.2.k.m.2351.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.k.b.111.3	✓	8	7.6	odd	2	inner
560.2.k.b.111.4	yes	8	4.3	odd	2	inner
560.2.k.b.111.5	yes	8	28.27	even	2	inner
560.2.k.b.111.6	yes	8	1.1	even	1	trivial
2240.2.k.d.1791.3		8	8.5	even	2
2240.2.k.d.1791.4		8	56.27	even	2
2240.2.k.d.1791.5		8	8.3	odd	2
2240.2.k.d.1791.6		8	56.13	odd	2
2800.2.e.g.2799.3		8	20.3	even	4
2800.2.e.g.2799.4		8	140.27	odd	4
2800.2.e.g.2799.5		8	35.27	even	4
2800.2.e.g.2799.6		8	5.3	odd	4
2800.2.e.h.2799.3		8	5.2	odd	4
2800.2.e.h.2799.4		8	35.13	even	4
2800.2.e.h.2799.5		8	140.83	odd	4
2800.2.e.h.2799.6		8	20.7	even	4
2800.2.k.m.2351.3		8	5.4	even	2
2800.2.k.m.2351.4		8	140.139	even	2
2800.2.k.m.2351.5		8	35.34	odd	2
2800.2.k.m.2351.6		8	20.19	odd	2
5040.2.d.d.4591.1		8	3.2	odd	2
5040.2.d.d.4591.4		8	12.11	even	2
5040.2.d.d.4591.5		8	84.83	odd	2
5040.2.d.d.4591.8		8	21.20	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	560.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.477491 q^{3} +1.00000i q^{5} +(-2.09428 + 1.61679i) q^{7} -2.77200 q^{9} +O(q^{10})\)	\(q+0.477491 q^{3} +1.00000i q^{5} +(-2.09428 + 1.61679i) q^{7} -2.77200 q^{9} +4.66605i q^{11} -3.77200i q^{13} +0.477491i q^{15} +3.77200i q^{17} -7.42213 q^{19} +(-1.00000 + 0.772002i) q^{21} +3.23357i q^{23} -1.00000 q^{25} -2.75608 q^{27} +3.77200 q^{29} +0.954983 q^{31} +2.22800i q^{33} +(-1.61679 - 2.09428i) q^{35} +5.54400 q^{37} -1.80110i q^{39} -6.00000i q^{41} +12.5657i q^{43} -2.77200i q^{45} +7.89962 q^{47} +(1.77200 - 6.77200i) q^{49} +1.80110i q^{51} -6.00000 q^{53} -4.66605 q^{55} -3.54400 q^{57} -9.33210 q^{59} +13.5440i q^{61} +(5.80534 - 4.48174i) q^{63} +3.77200 q^{65} +6.09852i q^{67} +1.54400i q^{69} -9.33210i q^{71} +6.00000i q^{73} -0.477491 q^{75} +(-7.54400 - 9.77200i) q^{77} -11.1332i q^{79} +7.00000 q^{81} +12.5657 q^{83} -3.77200 q^{85} +1.80110 q^{87} +12.0000i q^{89} +(6.09852 + 7.89962i) q^{91} +0.455996 q^{93} -7.42213i q^{95} -8.22800i q^{97} -12.9343i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{9}+O(q^{10}) \) 8 * q + 12 * q^9	\( 8 q + 12 q^{9} - 8 q^{21} - 8 q^{25} - 4 q^{29} - 24 q^{37} - 20 q^{49} - 48 q^{53} + 40 q^{57} - 4 q^{65} + 8 q^{77} + 56 q^{81} + 4 q^{85} + 72 q^{93}+O(q^{100}) \) 8 * q + 12 * q^9 - 8 * q^21 - 8 * q^25 - 4 * q^29 - 24 * q^37 - 20 * q^49 - 48 * q^53 + 40 * q^57 - 4 * q^65 + 8 * q^77 + 56 * q^81 + 4 * q^85 + 72 * q^93

Introduction

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