LMFDB - Embedded newform 560.2.k.b.111.7

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			111.7
Root			$0.337637 - 0.337637i$ of defining polynomial
Character	$\chi$	$=$	560.111
Dual form			560.2.k.b.111.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.96176 q^{3} -1.00000i q^{5} +(-0.337637 + 2.62412i) q^{7} +5.77200 q^{9} +O(q^{10})$	$q+2.96176 q^{3} -1.00000i q^{5} +(-0.337637 + 2.62412i) q^{7} +5.77200 q^{9} -3.63703i q^{11} -4.77200i q^{13} -2.96176i q^{15} +4.77200i q^{17} +4.57296 q^{19} +(-1.00000 + 7.77200i) q^{21} +5.24824i q^{23} -1.00000 q^{25} +8.20999 q^{27} -4.77200 q^{29} +5.92351 q^{31} -10.7720i q^{33} +(2.62412 + 0.337637i) q^{35} -11.5440 q^{37} -14.1335i q^{39} +6.00000i q^{41} -2.02582i q^{43} -5.77200i q^{45} -1.61121 q^{47} +(-6.77200 - 1.77200i) q^{49} +14.1335i q^{51} -6.00000 q^{53} -3.63703 q^{55} +13.5440 q^{57} -7.27406 q^{59} +3.54400i q^{61} +(-1.94884 + 15.1464i) q^{63} -4.77200 q^{65} -12.5223i q^{67} +15.5440i q^{69} +7.27406i q^{71} -6.00000i q^{73} -2.96176 q^{75} +(9.54400 + 1.22800i) q^{77} -6.85944i q^{79} +7.00000 q^{81} +2.02582 q^{83} +4.77200 q^{85} -14.1335 q^{87} -12.0000i q^{89} +(12.5223 + 1.61121i) q^{91} +17.5440 q^{93} -4.57296i q^{95} +16.7720i q^{97} -20.9930i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.96176			1.70997			0.854985	−	0.518652i	$-0.173566\pi$
$3$	2.96176			1.70997			0.854985	+	0.518652i	$0.173566\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	−0.337637	+	2.62412i	−0.127615	+	0.991824i
$7$	−0.337637	+	2.62412i	−0.127615	+	0.991824i
$8$		0			0
$9$	5.77200			1.92400
$10$		0			0
$11$		−	3.63703i		−	1.09661i	−0.836280	−	0.548303i	$-0.815274\pi$
$11$		−	3.63703i		−	1.09661i	0.836280	−	0.548303i	$-0.184726\pi$
$12$		0			0
$13$		−	4.77200i		−	1.32352i	−0.749718	−	0.661758i	$-0.769811\pi$
$13$		−	4.77200i		−	1.32352i	0.749718	−	0.661758i	$-0.230189\pi$
$14$		0			0
$15$		−	2.96176i		−	0.764722i
$16$		0			0
$17$			4.77200i			1.15738i	0.815547	+	0.578690i	$0.196436\pi$
$17$			4.77200i			1.15738i	−0.815547	+	0.578690i	$0.803564\pi$
$18$		0			0
$19$	4.57296			1.04911			0.524555	−	0.851377i	$-0.324232\pi$
$19$	4.57296			1.04911			0.524555	+	0.851377i	$0.324232\pi$
$20$		0			0
$21$	−1.00000	+	7.77200i	−0.218218	+	1.69599i
$22$		0			0
$23$			5.24824i			1.09433i	0.837024	+	0.547167i	$0.184294\pi$
$23$			5.24824i			1.09433i	−0.837024	+	0.547167i	$0.815706\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	8.20999			1.58001
$28$		0			0
$29$	−4.77200			−0.886139			−0.443069	−	0.896487i	$-0.646110\pi$
$29$	−4.77200			−0.886139			−0.443069	+	0.896487i	$0.646110\pi$
$30$		0			0
$31$	5.92351			1.06389			0.531947	−	0.846778i	$-0.321460\pi$
$31$	5.92351			1.06389			0.531947	+	0.846778i	$0.321460\pi$
$32$		0			0
$33$		−	10.7720i		−	1.87516i
$34$		0			0
$35$	2.62412	+	0.337637i	0.443557	+	0.0570711i
$36$		0			0
$37$	−11.5440			−1.89782			−0.948911	−	0.315543i	$-0.897813\pi$
$37$	−11.5440			−1.89782			−0.948911	+	0.315543i	$0.897813\pi$
$38$		0			0
$39$		−	14.1335i		−	2.26317i
$40$		0			0
$41$			6.00000i			0.937043i	0.883452	+	0.468521i	$0.155213\pi$
$41$			6.00000i			0.937043i	−0.883452	+	0.468521i	$0.844787\pi$
$42$		0			0
$43$		−	2.02582i		−	0.308935i	−0.987998	−	0.154468i	$-0.950634\pi$
$43$		−	2.02582i		−	0.308935i	0.987998	−	0.154468i	$-0.0493663\pi$
$44$		0			0
$45$		−	5.77200i		−	0.860439i
$46$		0			0
$47$	−1.61121			−0.235019			−0.117509	−	0.993072i	$-0.537491\pi$
$47$	−1.61121			−0.235019			−0.117509	+	0.993072i	$0.537491\pi$
$48$		0			0
$49$	−6.77200	−	1.77200i	−0.967429	−	0.253143i
$50$		0			0
$51$			14.1335i			1.97909i
$52$		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
$55$	−3.63703			−0.490417
$56$		0			0
$57$	13.5440			1.79395
$58$		0			0
$59$	−7.27406			−0.947002			−0.473501	−	0.880793i	$-0.657010\pi$
$59$	−7.27406			−0.947002			−0.473501	+	0.880793i	$0.657010\pi$
$60$		0			0
$61$			3.54400i			0.453763i	0.973922	+	0.226882i	$0.0728530\pi$
$61$			3.54400i			0.453763i	−0.973922	+	0.226882i	$0.927147\pi$
$62$		0			0
$63$	−1.94884	+	15.1464i	−0.245531	+	1.90827i
$64$		0			0
$65$	−4.77200			−0.591894
$66$		0			0
$67$		−	12.5223i		−	1.52984i	−0.644124	−	0.764921i	$-0.722778\pi$
$67$		−	12.5223i		−	1.52984i	0.644124	−	0.764921i	$-0.277222\pi$
$68$		0			0
$69$			15.5440i			1.87128i
$70$		0			0
$71$			7.27406i			0.863272i	0.902048	+	0.431636i	$0.142064\pi$
$71$			7.27406i			0.863272i	−0.902048	+	0.431636i	$0.857936\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$	−2.96176			−0.341994
$76$		0			0
$77$	9.54400	+	1.22800i	1.08764	+	0.139943i
$78$		0			0
$79$		−	6.85944i		−	0.771748i	−0.922552	−	0.385874i	$-0.873900\pi$
$79$		−	6.85944i		−	0.771748i	0.922552	−	0.385874i	$-0.126100\pi$
$80$		0			0
$81$	7.00000			0.777778
$82$		0			0
$83$	2.02582			0.222363			0.111182	−	0.993800i	$-0.464536\pi$
$83$	2.02582			0.222363			0.111182	+	0.993800i	$0.464536\pi$
$84$		0			0
$85$	4.77200			0.517596
$86$		0			0
$87$	−14.1335			−1.51527
$88$		0			0
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$	12.5223	+	1.61121i	1.31269	+	0.168900i
$92$		0			0
$93$	17.5440			1.81923
$94$		0			0
$95$		−	4.57296i		−	0.469176i
$96$		0			0
$97$			16.7720i			1.70294i	0.524404	+	0.851469i	$0.324288\pi$
$97$			16.7720i			1.70294i	−0.524404	+	0.851469i	$0.675712\pi$
$98$		0			0
$99$		−	20.9930i		−	2.10987i
$100$		0			0
$101$		−	9.54400i		−	0.949664i	−0.880076	−	0.474832i	$-0.842509\pi$
$101$		−	9.54400i		−	0.949664i	0.880076	−	0.474832i	$-0.157491\pi$
$102$		0			0
$103$	4.31231			0.424904			0.212452	−	0.977171i	$-0.431855\pi$
$103$	4.31231			0.424904			0.212452	+	0.977171i	$0.431855\pi$
$104$		0			0
$105$	7.77200	+	1.00000i	0.758470	+	0.0975900i
$106$		0			0
$107$			12.5223i			1.21058i	0.796006	+	0.605288i	$0.206942\pi$
$107$			12.5223i			1.21058i	−0.796006	+	0.605288i	$0.793058\pi$
$108$		0			0
$109$	6.77200			0.648640			0.324320	−	0.945947i	$-0.394865\pi$
$109$	6.77200			0.648640			0.324320	+	0.945947i	$0.394865\pi$
$110$		0			0
$111$	−34.1905			−3.24522
$112$		0			0
$113$	−15.5440			−1.46226			−0.731128	−	0.682240i	$-0.761006\pi$
$113$	−15.5440			−1.46226			−0.731128	+	0.682240i	$0.761006\pi$
$114$		0			0
$115$	5.24824			0.489401
$116$		0			0
$117$		−	27.5440i		−	2.54644i
$118$		0			0
$119$	−12.5223	−	1.61121i	−1.14792	−	0.147699i
$120$		0			0
$121$	−2.22800			−0.202545
$122$		0			0
$123$			17.7705i			1.60232i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			12.5223i			1.11117i	0.831458	+	0.555587i	$0.187507\pi$
$127$			12.5223i			1.11117i	−0.831458	+	0.555587i	$0.812493\pi$
$128$		0			0
$129$		−	6.00000i		−	0.528271i
$130$		0			0
$131$	−10.4965			−0.917081			−0.458541	−	0.888673i	$-0.651628\pi$
$131$	−10.4965			−0.917081			−0.458541	+	0.888673i	$0.651628\pi$
$132$		0			0
$133$	−1.54400	+	12.0000i	−0.133882	+	1.04053i
$134$		0			0
$135$		−	8.20999i		−	0.706604i
$136$		0			0
$137$	−3.54400			−0.302785			−0.151392	−	0.988474i	$-0.548376\pi$
$137$	−3.54400			−0.302785			−0.151392	+	0.988474i	$0.548376\pi$
$138$		0			0
$139$	9.14593			0.775747			0.387874	−	0.921713i	$-0.373210\pi$
$139$	9.14593			0.775747			0.387874	+	0.921713i	$0.373210\pi$
$140$		0			0
$141$	−4.77200			−0.401875
$142$		0			0
$143$	−17.3559			−1.45138
$144$		0			0
$145$			4.77200i			0.396293i
$146$		0			0
$147$	−20.0570	−	5.24824i	−1.65428	−	0.432867i
$148$		0			0
$149$	9.54400			0.781875			0.390938	−	0.920417i	$-0.372151\pi$
$149$	9.54400			0.781875			0.390938	+	0.920417i	$0.372151\pi$
$150$		0			0
$151$		−	3.63703i		−	0.295977i	−0.988989	−	0.147989i	$-0.952720\pi$
$151$		−	3.63703i		−	0.295977i	0.988989	−	0.147989i	$-0.0472799\pi$
$152$		0			0
$153$			27.5440i			2.22680i
$154$		0			0
$155$		−	5.92351i		−	0.475788i
$156$		0			0
$157$		−	6.00000i		−	0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$		−	6.00000i		−	0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$		0			0
$159$	−17.7705			−1.40930
$160$		0			0
$161$	−13.7720	−	1.77200i	−1.08539	−	0.139653i
$162$		0			0
$163$			15.7447i			1.23322i	0.787268	+	0.616611i	$0.211495\pi$
$163$			15.7447i			1.23322i	−0.787268	+	0.616611i	$0.788505\pi$
$164$		0			0
$165$	−10.7720			−0.838599
$166$		0			0
$167$	23.4334			1.81333			0.906665	−	0.421851i	$-0.138619\pi$
$167$	23.4334			1.81333			0.906665	+	0.421851i	$0.138619\pi$
$168$		0			0
$169$	−9.77200			−0.751692
$170$		0			0
$171$	26.3952			2.01849
$172$		0			0
$173$			2.31601i			0.176083i	0.996117	+	0.0880413i	$0.0280608\pi$
$173$			2.31601i			0.176083i	−0.996117	+	0.0880413i	$0.971939\pi$
$174$		0			0
$175$	0.337637	−	2.62412i	0.0255230	−	0.198365i
$176$		0			0
$177$	−21.5440			−1.61935
$178$		0			0
$179$		−	3.22241i		−	0.240854i	−0.992722	−	0.120427i	$-0.961574\pi$
$179$		−	3.22241i		−	0.240854i	0.992722	−	0.120427i	$-0.0384264\pi$
$180$		0			0
$181$		−	9.54400i		−	0.709400i	−0.934980	−	0.354700i	$-0.884583\pi$
$181$		−	9.54400i		−	0.709400i	0.934980	−	0.354700i	$-0.115417\pi$
$182$		0			0
$183$			10.4965i			0.775922i
$184$		0			0
$185$			11.5440i			0.848732i
$186$		0			0
$187$	17.3559			1.26919
$188$		0			0
$189$	−2.77200	+	21.5440i	−0.201633	+	1.56710i
$190$		0			0
$191$		−	24.6300i		−	1.78216i	−0.453843	−	0.891082i	$-0.649947\pi$
$191$		−	24.6300i		−	1.78216i	0.453843	−	0.891082i	$-0.350053\pi$
$192$		0			0
$193$	−4.45600			−0.320750			−0.160375	−	0.987056i	$-0.551270\pi$
$193$	−4.45600			−0.320750			−0.160375	+	0.987056i	$0.551270\pi$
$194$		0			0
$195$	−14.1335			−1.01212
$196$		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
$199$	4.57296			0.324169			0.162084	−	0.986777i	$-0.448178\pi$
$199$	4.57296			0.324169			0.162084	+	0.986777i	$0.448178\pi$
$200$		0			0
$201$		−	37.0880i		−	2.61599i
$202$		0			0
$203$	1.61121	−	12.5223i	0.113085	−	0.878893i
$204$		0			0
$205$	6.00000			0.419058
$206$		0			0
$207$			30.2928i			2.10550i
$208$		0			0
$209$		−	16.6320i		−	1.15046i
$210$		0			0
$211$			17.3559i			1.19483i	0.801932	+	0.597415i	$0.203806\pi$
$211$			17.3559i			1.19483i	−0.801932	+	0.597415i	$0.796194\pi$
$212$		0			0
$213$			21.5440i			1.47617i
$214$		0			0
$215$	−2.02582			−0.138160
$216$		0			0
$217$	−2.00000	+	15.5440i	−0.135769	+	1.05520i
$218$		0			0
$219$		−	17.7705i		−	1.20082i
$220$		0			0
$221$	22.7720			1.53181
$222$		0			0
$223$	13.4582			0.901230			0.450615	−	0.892718i	$-0.351205\pi$
$223$	13.4582			0.901230			0.450615	+	0.892718i	$0.351205\pi$
$224$		0			0
$225$	−5.77200			−0.384800
$226$		0			0
$227$	−16.1593			−1.07253			−0.536266	−	0.844049i	$-0.680166\pi$
$227$	−16.1593			−1.07253			−0.536266	+	0.844049i	$0.680166\pi$
$228$		0			0
$229$		−	21.5440i		−	1.42367i	−0.702348	−	0.711834i	$-0.747865\pi$
$229$		−	21.5440i		−	1.42367i	0.702348	−	0.711834i	$-0.252135\pi$
$230$		0			0
$231$	28.2670	+	3.63703i	1.85983	+	0.239299i
$232$		0			0
$233$	8.45600			0.553971			0.276985	−	0.960874i	$-0.410665\pi$
$233$	8.45600			0.553971			0.276985	+	0.960874i	$0.410665\pi$
$234$		0			0
$235$			1.61121i			0.105104i
$236$		0			0
$237$		−	20.3160i		−	1.31967i
$238$		0			0
$239$			14.1335i			0.914221i	0.889410	+	0.457110i	$0.151116\pi$
$239$			14.1335i			0.914221i	−0.889410	+	0.457110i	$0.848884\pi$
$240$		0			0
$241$		−	8.45600i		−	0.544699i	−0.962198	−	0.272349i	$-0.912199\pi$
$241$		−	8.45600i		−	0.544699i	0.962198	−	0.272349i	$-0.0878006\pi$
$242$		0			0
$243$	−3.89769			−0.250037
$244$		0			0
$245$	−1.77200	+	6.77200i	−0.113209	+	0.432647i
$246$		0			0
$247$		−	21.8222i		−	1.38851i
$248$		0			0
$249$	6.00000			0.380235
$250$		0			0
$251$	−25.0446			−1.58080			−0.790401	−	0.612590i	$-0.790128\pi$
$251$	−25.0446			−1.58080			−0.790401	+	0.612590i	$0.790128\pi$
$252$		0			0
$253$	19.0880			1.20005
$254$		0			0
$255$	14.1335			0.885075
$256$		0			0
$257$			13.0880i			0.816407i	0.912891	+	0.408204i	$0.133845\pi$
$257$			13.0880i			0.816407i	−0.912891	+	0.408204i	$0.866155\pi$
$258$		0			0
$259$	3.89769	−	30.2928i	0.242191	−	1.88231i
$260$		0			0
$261$	−27.5440			−1.70493
$262$		0			0
$263$			8.47065i			0.522323i	0.965295	+	0.261161i	$0.0841055\pi$
$263$			8.47065i			0.522323i	−0.965295	+	0.261161i	$0.915895\pi$
$264$		0			0
$265$			6.00000i			0.368577i
$266$		0			0
$267$		−	35.5411i		−	2.17508i
$268$		0			0
$269$			15.5440i			0.947735i	0.880596	+	0.473867i	$0.157142\pi$
$269$			15.5440i			0.947735i	−0.880596	+	0.473867i	$0.842858\pi$
$270$		0			0
$271$	−1.35055			−0.0820401			−0.0410200	−	0.999158i	$-0.513061\pi$
$271$	−1.35055			−0.0820401			−0.0410200	+	0.999158i	$0.513061\pi$
$272$		0			0
$273$	37.0880	+	4.77200i	2.24467	+	0.288815i
$274$		0			0
$275$			3.63703i			0.219321i
$276$		0			0
$277$	19.5440			1.17429			0.587143	−	0.809483i	$-0.300253\pi$
$277$	19.5440			1.17429			0.587143	+	0.809483i	$0.300253\pi$
$278$		0			0
$279$	34.1905			2.04693
$280$		0			0
$281$	20.3160			1.21195			0.605976	−	0.795483i	$-0.292783\pi$
$281$	20.3160			1.21195			0.605976	+	0.795483i	$0.292783\pi$
$282$		0			0
$283$	23.9547			1.42396			0.711980	−	0.702200i	$-0.247799\pi$
$283$	23.9547			1.42396			0.711980	+	0.702200i	$0.247799\pi$
$284$		0			0
$285$		−	13.5440i		−	0.802278i
$286$		0			0
$287$	−15.7447	−	2.02582i	−0.929381	−	0.119581i
$288$		0			0
$289$	−5.77200			−0.339530
$290$		0			0
$291$			49.6746i			2.91198i
$292$		0			0
$293$			14.3160i			0.836350i	0.908366	+	0.418175i	$0.137330\pi$
$293$			14.3160i			0.836350i	−0.908366	+	0.418175i	$0.862670\pi$
$294$		0			0
$295$			7.27406i			0.423512i
$296$		0			0
$297$		−	29.8600i		−	1.73265i
$298$		0			0
$299$	25.0446			1.44837
$300$		0			0
$301$	5.31601	+	0.683994i	0.306409	+	0.0394248i
$302$		0			0
$303$		−	28.2670i		−	1.62390i
$304$		0			0
$305$	3.54400			0.202929
$306$		0			0
$307$	31.2288			1.78232			0.891160	−	0.453689i	$-0.149892\pi$
$307$	31.2288			1.78232			0.891160	+	0.453689i	$0.149892\pi$
$308$		0			0
$309$	12.7720			0.726574
$310$		0			0
$311$	24.2154			1.37313			0.686564	−	0.727070i	$-0.259118\pi$
$311$	24.2154			1.37313			0.686564	+	0.727070i	$0.259118\pi$
$312$		0			0
$313$		−	4.77200i		−	0.269729i	−0.990864	−	0.134865i	$-0.956940\pi$
$313$		−	4.77200i		−	0.269729i	0.990864	−	0.134865i	$-0.0430600\pi$
$314$		0			0
$315$	15.1464	+	1.94884i	0.853404	+	0.109805i
$316$		0			0
$317$	25.0880			1.40908			0.704541	−	0.709663i	$-0.251153\pi$
$317$	25.0880			1.40908			0.704541	+	0.709663i	$0.251153\pi$
$318$		0			0
$319$			17.3559i			0.971745i
$320$		0			0
$321$			37.0880i			2.07005i
$322$		0			0
$323$			21.8222i			1.21422i
$324$		0			0
$325$			4.77200i			0.264703i
$326$		0			0
$327$	20.0570			1.10916
$328$		0			0
$329$	0.544004	−	4.22800i	0.0299919	−	0.233097i
$330$		0			0
$331$			13.7189i			0.754058i	0.926201	+	0.377029i	$0.123054\pi$
$331$			13.7189i			0.754058i	−0.926201	+	0.377029i	$0.876946\pi$
$332$		0			0
$333$	−66.6320			−3.65141
$334$		0			0
$335$	−12.5223			−0.684166
$336$		0			0
$337$	4.45600			0.242734			0.121367	−	0.992608i	$-0.461272\pi$
$337$	4.45600			0.242734			0.121367	+	0.992608i	$0.461272\pi$
$338$		0			0
$339$	−46.0376			−2.50042
$340$		0			0
$341$		−	21.5440i		−	1.16667i
$342$		0			0
$343$	6.93643	−	17.1722i	0.374532	−	0.927214i
$344$		0			0
$345$	15.5440			0.836861
$346$		0			0
$347$			5.24824i			0.281740i	0.990028	+	0.140870i	$0.0449900\pi$
$347$			5.24824i			0.281740i	−0.990028	+	0.140870i	$0.955010\pi$
$348$		0			0
$349$		−	2.45600i		−	0.131466i	−0.997837	−	0.0657332i	$-0.979061\pi$
$349$		−	2.45600i		−	0.131466i	0.997837	−	0.0657332i	$-0.0209386\pi$
$350$		0			0
$351$		−	39.1781i		−	2.09117i
$352$		0			0
$353$		−	35.8600i		−	1.90864i	−0.298793	−	0.954318i	$-0.596584\pi$
$353$		−	35.8600i		−	1.90864i	0.298793	−	0.954318i	$-0.403416\pi$
$354$		0			0
$355$	7.27406			0.386067
$356$		0			0
$357$	−37.0880	−	4.77200i	−1.96291	−	0.252561i
$358$		0			0
$359$			3.22241i			0.170072i	0.996378	+	0.0850362i	$0.0271006\pi$
$359$			3.22241i			0.170072i	−0.996378	+	0.0850362i	$0.972899\pi$
$360$		0			0
$361$	1.91199			0.100631
$362$		0			0
$363$	−6.59879			−0.346347
$364$		0			0
$365$	−6.00000			−0.314054
$366$		0			0
$367$	−14.8088			−0.773012			−0.386506	−	0.922287i	$-0.626318\pi$
$367$	−14.8088			−0.773012			−0.386506	+	0.922287i	$0.626318\pi$
$368$		0			0
$369$			34.6320i			1.80287i
$370$		0			0
$371$	2.02582	−	15.7447i	0.105176	−	0.817425i
$372$		0			0
$373$	4.45600			0.230723			0.115361	−	0.993324i	$-0.463197\pi$
$373$	4.45600			0.230723			0.115361	+	0.993324i	$0.463197\pi$
$374$		0			0
$375$			2.96176i			0.152944i
$376$		0			0
$377$			22.7720i			1.17282i
$378$		0			0
$379$			3.22241i			0.165524i	0.996569	+	0.0827621i	$0.0263742\pi$
$379$			3.22241i			0.165524i	−0.996569	+	0.0827621i	$0.973626\pi$
$380$		0			0
$381$			37.0880i			1.90008i
$382$		0			0
$383$	8.47065			0.432830			0.216415	−	0.976301i	$-0.430564\pi$
$383$	8.47065			0.432830			0.216415	+	0.976301i	$0.430564\pi$
$384$		0			0
$385$	1.22800	−	9.54400i	0.0625846	−	0.486407i
$386$		0			0
$387$		−	11.6931i		−	0.594392i
$388$		0			0
$389$	−20.3160			−1.03006			−0.515031	−	0.857171i	$-0.672220\pi$
$389$	−20.3160			−1.03006			−0.515031	+	0.857171i	$0.672220\pi$
$390$		0			0
$391$	−25.0446			−1.26656
$392$		0			0
$393$	−31.0880			−1.56818
$394$		0			0
$395$	−6.85944			−0.345136
$396$		0			0
$397$			16.7720i			0.841763i	0.907116	+	0.420881i	$0.138279\pi$
$397$			16.7720i			0.841763i	−0.907116	+	0.420881i	$0.861721\pi$
$398$		0			0
$399$	−4.57296	+	35.5411i	−0.228935	+	1.77928i
$400$		0			0
$401$	−7.22800			−0.360949			−0.180475	−	0.983580i	$-0.557763\pi$
$401$	−7.22800			−0.360949			−0.180475	+	0.983580i	$0.557763\pi$
$402$		0			0
$403$		−	28.2670i		−	1.40808i
$404$		0			0
$405$		−	7.00000i		−	0.347833i
$406$		0			0
$407$			41.9859i			2.08116i
$408$		0			0
$409$		−	37.0880i		−	1.83388i	−0.399021	−	0.916942i	$-0.630650\pi$
$409$		−	37.0880i		−	1.83388i	0.399021	−	0.916942i	$-0.369350\pi$
$410$		0			0
$411$	−10.4965			−0.517753
$412$		0			0
$413$	2.45600	−	19.0880i	0.120852	−	0.939259i
$414$		0			0
$415$		−	2.02582i		−	0.0994438i
$416$		0			0
$417$	27.0880			1.32651
$418$		0			0
$419$	−13.7189			−0.670212			−0.335106	−	0.942181i	$-0.608772\pi$
$419$	−13.7189			−0.670212			−0.335106	+	0.942181i	$0.608772\pi$
$420$		0			0
$421$	−13.8600			−0.675496			−0.337748	−	0.941237i	$-0.609665\pi$
$421$	−13.8600			−0.675496			−0.337748	+	0.941237i	$0.609665\pi$
$422$		0			0
$423$	−9.29989			−0.452176
$424$		0			0
$425$		−	4.77200i		−	0.231476i
$426$		0			0
$427$	−9.29989	−	1.19659i	−0.450053	−	0.0579070i
$428$		0			0
$429$	−51.4040			−2.48181
$430$		0			0
$431$		−	31.0748i		−	1.49682i	−0.663236	−	0.748410i	$-0.730817\pi$
$431$		−	31.0748i		−	1.49682i	0.663236	−	0.748410i	$-0.269183\pi$
$432$		0			0
$433$		−	25.0880i		−	1.20565i	−0.797872	−	0.602826i	$-0.794041\pi$
$433$		−	25.0880i		−	1.20565i	0.797872	−	0.602826i	$-0.205959\pi$
$434$		0			0
$435$			14.1335i			0.677650i
$436$		0			0
$437$			24.0000i			1.14808i
$438$		0			0
$439$	11.8470			0.565428			0.282714	−	0.959204i	$-0.408765\pi$
$439$	11.8470			0.565428			0.282714	+	0.959204i	$0.408765\pi$
$440$		0			0
$441$	−39.0880	−	10.2280i	−1.86133	−	0.487048i
$442$		0			0
$443$		−	8.47065i		−	0.402453i	−0.979545	−	0.201226i	$-0.935507\pi$
$443$		−	8.47065i		−	0.402453i	0.979545	−	0.201226i	$-0.0644927\pi$
$444$		0			0
$445$	−12.0000			−0.568855
$446$		0			0
$447$	28.2670			1.33698
$448$		0			0
$449$	29.8600			1.40918			0.704590	−	0.709614i	$-0.251131\pi$
$449$	29.8600			1.40918			0.704590	+	0.709614i	$0.251131\pi$
$450$		0			0
$451$	21.8222			1.02757
$452$		0			0
$453$		−	10.7720i		−	0.506113i
$454$		0			0
$455$	1.61121	−	12.5223i	0.0755345	−	0.587055i
$456$		0			0
$457$	28.1760			1.31802			0.659009	−	0.752135i	$-0.270976\pi$
$457$	28.1760			1.31802			0.659009	+	0.752135i	$0.270976\pi$
$458$		0			0
$459$			39.1781i			1.82868i
$460$		0			0
$461$			7.08801i			0.330121i	0.986283	+	0.165061i	$0.0527820\pi$
$461$			7.08801i			0.330121i	−0.986283	+	0.165061i	$0.947218\pi$
$462$		0			0
$463$		−	1.19659i		−	0.0556102i	−0.999613	−	0.0278051i	$-0.991148\pi$
$463$		−	1.19659i		−	0.0556102i	0.999613	−	0.0278051i	$-0.00885178\pi$
$464$		0			0
$465$		−	17.5440i		−	0.813584i
$466$		0			0
$467$	−33.1006			−1.53171			−0.765857	−	0.643010i	$-0.777685\pi$
$467$	−33.1006			−1.53171			−0.765857	+	0.643010i	$0.777685\pi$
$468$		0			0
$469$	32.8600	+	4.22800i	1.51733	+	0.195231i
$470$		0			0
$471$		−	17.7705i		−	0.818823i
$472$		0			0
$473$	−7.36799			−0.338780
$474$		0			0
$475$	−4.57296			−0.209822
$476$		0			0
$477$	−34.6320			−1.58569
$478$		0			0
$479$	29.0963			1.32944			0.664721	−	0.747092i	$-0.268550\pi$
$479$	29.0963			1.32944			0.664721	+	0.747092i	$0.268550\pi$
$480$		0			0
$481$			55.0880i			2.51180i
$482$		0			0
$483$	−40.7893	−	5.24824i	−1.85598	−	0.238803i
$484$		0			0
$485$	16.7720			0.761577
$486$		0			0
$487$		−	15.7447i		−	0.713461i	−0.934207	−	0.356731i	$-0.883891\pi$
$487$		−	15.7447i		−	0.713461i	0.934207	−	0.356731i	$-0.116109\pi$
$488$		0			0
$489$			46.6320i			2.10877i
$490$		0			0
$491$		−	24.6300i		−	1.11154i	−0.831338	−	0.555768i	$-0.812424\pi$
$491$		−	24.6300i		−	1.11154i	0.831338	−	0.555768i	$-0.187576\pi$
$492$		0			0
$493$		−	22.7720i		−	1.02560i
$494$		0			0
$495$	−20.9930			−0.943563
$496$		0			0
$497$	−19.0880	−	2.45600i	−0.856214	−	0.110166i
$498$		0			0
$499$			28.6816i			1.28397i	0.766719	+	0.641983i	$0.221888\pi$
$499$			28.6816i			1.28397i	−0.766719	+	0.641983i	$0.778112\pi$
$500$		0			0
$501$	69.4040			3.10074
$502$		0			0
$503$	−8.88527			−0.396175			−0.198087	−	0.980184i	$-0.563473\pi$
$503$	−8.88527			−0.396175			−0.198087	+	0.980184i	$0.563473\pi$
$504$		0			0
$505$	−9.54400			−0.424703
$506$		0			0
$507$	−28.9423			−1.28537
$508$		0			0
$509$		−	19.0880i		−	0.846061i	−0.906115	−	0.423031i	$-0.860966\pi$
$509$		−	19.0880i		−	0.846061i	0.906115	−	0.423031i	$-0.139034\pi$
$510$		0			0
$511$	15.7447	+	2.02582i	0.696505	+	0.0896172i
$512$		0			0
$513$	37.5440			1.65761
$514$		0			0
$515$		−	4.31231i		−	0.190023i
$516$		0			0
$517$			5.86001i			0.257723i
$518$		0			0
$519$			6.85944i			0.301096i
$520$		0			0
$521$			21.5440i			0.943860i	0.881636	+	0.471930i	$0.156442\pi$
$521$			21.5440i			0.943860i	−0.881636	+	0.471930i	$0.843558\pi$
$522$		0			0
$523$	−38.9175			−1.70174			−0.850871	−	0.525375i	$-0.823925\pi$
$523$	−38.9175			−1.70174			−0.850871	+	0.525375i	$0.823925\pi$
$524$		0			0
$525$	1.00000	−	7.77200i	0.0436436	−	0.339198i
$526$		0			0
$527$			28.2670i			1.23133i
$528$		0			0
$529$	−4.54400			−0.197565
$530$		0			0
$531$	−41.9859			−1.82203
$532$		0			0
$533$	28.6320			1.24019
$534$		0			0
$535$	12.5223			0.541386
$536$		0			0
$537$		−	9.54400i		−	0.411854i
$538$		0			0
$539$	−6.44483	+	24.6300i	−0.277598	+	1.06089i
$540$		0			0
$541$	−15.2280			−0.654703			−0.327351	−	0.944903i	$-0.606156\pi$
$541$	−15.2280			−0.654703			−0.327351	+	0.944903i	$0.606156\pi$
$542$		0			0
$543$		−	28.2670i		−	1.21305i
$544$		0			0
$545$		−	6.77200i		−	0.290081i
$546$		0			0
$547$		−	1.19659i		−	0.0511624i	−0.999673	−	0.0255812i	$-0.991856\pi$
$547$		−	1.19659i		−	0.0511624i	0.999673	−	0.0255812i	$-0.00814364\pi$
$548$		0			0
$549$			20.4560i			0.873041i
$550$		0			0
$551$	−21.8222			−0.929657
$552$		0			0
$553$	18.0000	+	2.31601i	0.765438	+	0.0984866i
$554$		0			0
$555$			34.1905i			1.45131i
$556$		0			0
$557$	20.4560			0.866748			0.433374	−	0.901214i	$-0.357323\pi$
$557$	20.4560			0.866748			0.433374	+	0.901214i	$0.357323\pi$
$558$		0			0
$559$	−9.66724			−0.408881
$560$		0			0
$561$	51.4040			2.17028
$562$		0			0
$563$	−8.47065			−0.356995			−0.178498	−	0.983940i	$-0.557124\pi$
$563$	−8.47065			−0.356995			−0.178498	+	0.983940i	$0.557124\pi$
$564$		0			0
$565$			15.5440i			0.653941i
$566$		0			0
$567$	−2.36346	+	18.3688i	−0.0992561	+	0.771418i
$568$		0			0
$569$	−16.6320			−0.697250			−0.348625	−	0.937262i	$-0.613351\pi$
$569$	−16.6320			−0.697250			−0.348625	+	0.937262i	$0.613351\pi$
$570$		0			0
$571$		−	7.27406i		−	0.304410i	−0.988349	−	0.152205i	$-0.951363\pi$
$571$		−	7.27406i		−	0.304410i	0.988349	−	0.152205i	$-0.0486374\pi$
$572$		0			0
$573$		−	72.9480i		−	3.04745i
$574$		0			0
$575$		−	5.24824i		−	0.218867i
$576$		0			0
$577$		−	0.139991i		−	0.00582789i	−0.999996	−	0.00291394i	$-0.999072\pi$
$577$		−	0.139991i		−	0.00582789i	0.999996	−	0.00291394i	$-0.000927538\pi$
$578$		0			0
$579$	−13.1976			−0.548473
$580$		0			0
$581$	−0.683994	+	5.31601i	−0.0283769	+	0.220545i
$582$		0			0
$583$			21.8222i			0.903783i
$584$		0			0
$585$	−27.5440			−1.13880
$586$		0			0
$587$	−2.02582			−0.0836147			−0.0418074	−	0.999126i	$-0.513312\pi$
$587$	−2.02582			−0.0836147			−0.0418074	+	0.999126i	$0.513312\pi$
$588$		0			0
$589$	27.0880			1.11614
$590$		0			0
$591$	−17.7705			−0.730982
$592$		0			0
$593$		−	11.8600i		−	0.487032i	−0.969897	−	0.243516i	$-0.921699\pi$
$593$		−	11.8600i		−	0.487032i	0.969897	−	0.243516i	$-0.0783009\pi$
$594$		0			0
$595$	−1.61121	+	12.5223i	−0.0660530	+	0.513364i
$596$		0			0
$597$	13.5440			0.554319
$598$		0			0
$599$		−	14.9627i		−	0.611361i	−0.952134	−	0.305681i	$-0.901116\pi$
$599$		−	14.9627i		−	0.611361i	0.952134	−	0.305681i	$-0.0988840\pi$
$600$		0			0
$601$		−	44.1760i		−	1.80198i	−0.433843	−	0.900989i	$-0.642843\pi$
$601$		−	44.1760i		−	1.80198i	0.433843	−	0.900989i	$-0.357157\pi$
$602$		0			0
$603$		−	72.2787i		−	2.94342i
$604$		0			0
$605$			2.22800i			0.0905810i
$606$		0			0
$607$	−10.7571			−0.436619			−0.218309	−	0.975880i	$-0.570054\pi$
$607$	−10.7571			−0.436619			−0.218309	+	0.975880i	$0.570054\pi$
$608$		0			0
$609$	4.77200	−	37.0880i	0.193371	−	1.50288i
$610$		0			0
$611$			7.68868i			0.311051i
$612$		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
$615$	17.7705			0.716577
$616$		0			0
$617$	−15.5440			−0.625778			−0.312889	−	0.949790i	$-0.601297\pi$
$617$	−15.5440			−0.625778			−0.312889	+	0.949790i	$0.601297\pi$
$618$		0			0
$619$	−11.8470			−0.476172			−0.238086	−	0.971244i	$-0.576520\pi$
$619$	−11.8470			−0.476172			−0.238086	+	0.971244i	$0.576520\pi$
$620$		0			0
$621$			43.0880i			1.72906i
$622$		0			0
$623$	31.4894	+	4.05165i	1.26160	+	0.162326i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	49.2600i		−	1.96725i
$628$		0			0
$629$		−	55.0880i		−	2.19650i
$630$		0			0
$631$			10.9111i			0.434364i	0.976131	+	0.217182i	$0.0696865\pi$
$631$			10.9111i			0.434364i	−0.976131	+	0.217182i	$0.930314\pi$
$632$		0			0
$633$			51.4040i			2.04313i
$634$		0			0
$635$	12.5223			0.496932
$636$		0			0
$637$	−8.45600	+	32.3160i	−0.335039	+	1.28041i
$638$		0			0
$639$			41.9859i			1.66094i
$640$		0			0
$641$	−4.63201			−0.182953			−0.0914767	−	0.995807i	$-0.529159\pi$
$641$	−4.63201			−0.182953			−0.0914767	+	0.995807i	$0.529159\pi$
$642$		0			0
$643$	23.9547			0.944682			0.472341	−	0.881416i	$-0.343409\pi$
$643$	23.9547			0.944682			0.472341	+	0.881416i	$0.343409\pi$
$644$		0			0
$645$	−6.00000			−0.236250
$646$		0			0
$647$	16.5740			0.651589			0.325795	−	0.945441i	$-0.394368\pi$
$647$	16.5740			0.651589			0.325795	+	0.945441i	$0.394368\pi$
$648$		0			0
$649$			26.4560i			1.03849i
$650$		0			0
$651$	−5.92351	+	46.0376i	−0.232161	+	1.80435i
$652$		0			0
$653$	−1.08801			−0.0425770			−0.0212885	−	0.999773i	$-0.506777\pi$
$653$	−1.08801			−0.0425770			−0.0212885	+	0.999773i	$0.506777\pi$
$654$		0			0
$655$			10.4965i			0.410131i
$656$		0			0
$657$		−	34.6320i		−	1.35112i
$658$		0			0
$659$			49.6746i			1.93505i	0.252781	+	0.967524i	$0.418655\pi$
$659$			49.6746i			1.93505i	−0.252781	+	0.967524i	$0.581345\pi$
$660$		0			0
$661$		−	15.5440i		−	0.604592i	−0.953214	−	0.302296i	$-0.902247\pi$
$661$		−	15.5440i		−	0.604592i	0.953214	−	0.302296i	$-0.0977530\pi$
$662$		0			0
$663$	67.4451			2.61935
$664$		0			0
$665$	12.0000	+	1.54400i	0.465340	+	0.0598739i
$666$		0			0
$667$		−	25.0446i		−	0.969731i
$668$		0			0
$669$	39.8600			1.54108
$670$		0			0
$671$	12.8897			0.497600
$672$		0			0
$673$	−35.5440			−1.37012			−0.685060	−	0.728486i	$-0.740224\pi$
$673$	−35.5440			−1.37012			−0.685060	+	0.728486i	$0.740224\pi$
$674$		0			0
$675$	−8.20999			−0.316003
$676$		0			0
$677$			16.7720i			0.644600i	0.946638	+	0.322300i	$0.104456\pi$
$677$			16.7720i			0.644600i	−0.946638	+	0.322300i	$0.895544\pi$
$678$		0			0
$679$	−44.0117	−	5.66286i	−1.68902	−	0.217320i
$680$		0			0
$681$	−47.8600			−1.83400
$682$		0			0
$683$		−	36.7377i		−	1.40573i	−0.711324	−	0.702864i	$-0.751904\pi$
$683$		−	36.7377i		−	1.40573i	0.711324	−	0.702864i	$-0.248096\pi$
$684$		0			0
$685$			3.54400i			0.135409i
$686$		0			0
$687$		−	63.8081i		−	2.43443i
$688$		0			0
$689$			28.6320i			1.09079i
$690$		0			0
$691$	13.1976			0.502059			0.251030	−	0.967979i	$-0.419231\pi$
$691$	13.1976			0.502059			0.251030	+	0.967979i	$0.419231\pi$
$692$		0			0
$693$	55.0880	+	7.08801i	2.09262	+	0.269251i
$694$		0			0
$695$		−	9.14593i		−	0.346925i
$696$		0			0
$697$	−28.6320			−1.08451
$698$		0			0
$699$	25.0446			0.947274
$700$		0			0
$701$	−29.8600			−1.12780			−0.563898	−	0.825844i	$-0.690699\pi$
$701$	−29.8600			−1.12780			−0.563898	+	0.825844i	$0.690699\pi$
$702$		0			0
$703$	−52.7903			−1.99102
$704$		0			0
$705$			4.77200i			0.179724i
$706$		0			0
$707$	25.0446	+	3.22241i	0.941899	+	0.121191i
$708$		0			0
$709$	−13.6840			−0.513913			−0.256957	−	0.966423i	$-0.582720\pi$
$709$	−13.6840			−0.513913			−0.256957	+	0.966423i	$0.582720\pi$
$710$		0			0
$711$		−	39.5927i		−	1.48484i
$712$		0			0
$713$			31.0880i			1.16426i
$714$		0			0
$715$			17.3559i			0.649075i
$716$		0			0
$717$			41.8600i			1.56329i
$718$		0			0
$719$	−6.44483			−0.240351			−0.120176	−	0.992753i	$-0.538346\pi$
$719$	−6.44483			−0.240351			−0.120176	+	0.992753i	$0.538346\pi$
$720$		0			0
$721$	−1.45600	+	11.3160i	−0.0542241	+	0.421430i
$722$		0			0
$723$		−	25.0446i		−	0.931419i
$724$		0			0
$725$	4.77200			0.177228
$726$		0			0
$727$	−24.3693			−0.903808			−0.451904	−	0.892066i	$-0.649255\pi$
$727$	−24.3693			−0.903808			−0.451904	+	0.892066i	$0.649255\pi$
$728$		0			0
$729$	−32.5440			−1.20533
$730$		0			0
$731$	9.66724			0.357556
$732$		0			0
$733$			31.2280i			1.15343i	0.816945	+	0.576716i	$0.195666\pi$
$733$			31.2280i			1.15343i	−0.816945	+	0.576716i	$0.804334\pi$
$734$		0			0
$735$	−5.24824	+	20.0570i	−0.193584	+	0.739814i
$736$		0			0
$737$	−45.5440			−1.67763
$738$		0			0
$739$		−	42.4005i		−	1.55973i	−0.625949	−	0.779864i	$-0.715288\pi$
$739$		−	42.4005i		−	1.55973i	0.625949	−	0.779864i	$-0.284712\pi$
$740$		0			0
$741$		−	64.6320i		−	2.37432i
$742$		0			0
$743$			4.41900i			0.162117i	0.996709	+	0.0810587i	$0.0258301\pi$
$743$			4.41900i			0.162117i	−0.996709	+	0.0810587i	$0.974170\pi$
$744$		0			0
$745$		−	9.54400i		−	0.349665i
$746$		0			0
$747$	11.6931			0.427827
$748$		0			0
$749$	−32.8600	−	4.22800i	−1.20068	−	0.154488i
$750$		0			0
$751$			25.4592i			0.929020i	0.885568	+	0.464510i	$0.153770\pi$
$751$			25.4592i			0.929020i	−0.885568	+	0.464510i	$0.846230\pi$
$752$		0			0
$753$	−74.1760			−2.70312
$754$		0			0
$755$	−3.63703			−0.132365
$756$		0			0
$757$	11.5440			0.419574			0.209787	−	0.977747i	$-0.432723\pi$
$757$	11.5440			0.419574			0.209787	+	0.977747i	$0.432723\pi$
$758$		0			0
$759$	56.5340			2.05206
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−2.28648	+	17.7705i	−0.0827762	+	0.643337i
$764$		0			0
$765$	27.5440			0.995856
$766$		0			0
$767$			34.7118i			1.25337i
$768$		0			0
$769$			28.6320i			1.03250i	0.856439	+	0.516248i	$0.172672\pi$
$769$			28.6320i			1.03250i	−0.856439	+	0.516248i	$0.827328\pi$
$770$		0			0
$771$			38.7635i			1.39603i
$772$		0			0
$773$			43.2280i			1.55480i	0.629005	+	0.777402i	$0.283463\pi$
$773$			43.2280i			1.55480i	−0.629005	+	0.777402i	$0.716537\pi$
$774$		0			0
$775$	−5.92351			−0.212779
$776$		0			0
$777$	11.5440	−	89.7200i	0.414139	−	3.21869i
$778$		0			0
$779$			27.4378i			0.983060i
$780$		0			0
$781$	26.4560			0.946670
$782$		0			0
$783$	−39.1781			−1.40011
$784$		0			0
$785$	−6.00000			−0.214149
$786$		0			0
$787$	13.9795			0.498317			0.249159	−	0.968463i	$-0.419846\pi$
$787$	13.9795			0.498317			0.249159	+	0.968463i	$0.419846\pi$
$788$		0			0
$789$			25.0880i			0.893157i
$790$		0			0
$791$	5.24824	−	40.7893i	0.186606	−	1.45030i
$792$		0			0
$793$	16.9120			0.600562
$794$		0			0
$795$			17.7705i			0.630256i
$796$		0			0
$797$		−	19.2280i		−	0.681091i	−0.940228	−	0.340545i	$-0.889388\pi$
$797$		−	19.2280i		−	0.681091i	0.940228	−	0.340545i	$-0.110612\pi$
$798$		0			0
$799$		−	7.68868i		−	0.272006i
$800$		0			0
$801$		−	69.2640i		−	2.44732i
$802$		0			0
$803$	−21.8222			−0.770088
$804$		0			0
$805$	−1.77200	+	13.7720i	−0.0624549	+	0.485399i
$806$		0			0
$807$			46.0376i			1.62060i
$808$		0			0
$809$	14.3160			0.503324			0.251662	−	0.967815i	$-0.419023\pi$
$809$	14.3160			0.503324			0.251662	+	0.967815i	$0.419023\pi$
$810$		0			0
$811$	42.2938			1.48514			0.742569	−	0.669770i	$-0.233607\pi$
$811$	42.2938			1.48514			0.742569	+	0.669770i	$0.233607\pi$
$812$		0			0
$813$	−4.00000			−0.140286
$814$		0			0
$815$	15.7447			0.551513
$816$		0			0
$817$		−	9.26402i		−	0.324107i
$818$		0			0
$819$	72.2787	+	9.29989i	2.52562	+	0.324964i
$820$		0			0
$821$	−15.6840			−0.547375			−0.273688	−	0.961819i	$-0.588243\pi$
$821$	−15.6840			−0.547375			−0.273688	+	0.961819i	$0.588243\pi$
$822$		0			0
$823$		−	18.9671i		−	0.661153i	−0.943779	−	0.330576i	$-0.892757\pi$
$823$		−	18.9671i		−	0.661153i	0.943779	−	0.330576i	$-0.107243\pi$
$824$		0			0
$825$			10.7720i			0.375033i
$826$		0			0
$827$			5.24824i			0.182499i	0.995828	+	0.0912496i	$0.0290861\pi$
$827$			5.24824i			0.182499i	−0.995828	+	0.0912496i	$0.970914\pi$
$828$		0			0
$829$		−	49.0880i		−	1.70490i	−0.522811	−	0.852448i	$-0.675117\pi$
$829$		−	49.0880i		−	1.70490i	0.522811	−	0.852448i	$-0.324883\pi$
$830$		0			0
$831$	57.8846			2.00799
$832$		0			0
$833$	8.45600	−	32.3160i	0.292983	−	1.11968i
$834$		0			0
$835$		−	23.4334i		−	0.810946i
$836$		0			0
$837$	48.6320			1.68097
$838$		0			0
$839$	6.44483			0.222500			0.111250	−	0.993792i	$-0.464515\pi$
$839$	6.44483			0.222500			0.111250	+	0.993792i	$0.464515\pi$
$840$		0			0
$841$	−6.22800			−0.214759
$842$		0			0
$843$	60.1711			2.07240
$844$		0			0
$845$			9.77200i			0.336167i
$846$		0			0
$847$	0.752256	−	5.84653i	0.0258478	−	0.200889i
$848$		0			0
$849$	70.9480			2.43493
$850$		0			0
$851$		−	60.5857i		−	2.07685i
$852$		0			0
$853$			20.1760i			0.690814i	0.938453	+	0.345407i	$0.112259\pi$
$853$			20.1760i			0.690814i	−0.938453	+	0.345407i	$0.887741\pi$
$854$		0			0
$855$		−	26.3952i		−	0.902695i
$856$		0			0
$857$			6.00000i			0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$			6.00000i			0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$		0			0
$859$	40.9433			1.39697			0.698483	−	0.715626i	$-0.253859\pi$
$859$	40.9433			1.39697			0.698483	+	0.715626i	$0.253859\pi$
$860$		0			0
$861$	−46.6320	−	6.00000i	−1.58921	−	0.204479i
$862$		0			0
$863$		−	6.07747i		−	0.206880i	−0.994636	−	0.103440i	$-0.967015\pi$
$863$		−	6.07747i		−	0.206880i	0.994636	−	0.103440i	$-0.0329849\pi$
$864$		0			0
$865$	2.31601			0.0787466
$866$		0			0
$867$	−17.0953			−0.580586
$868$		0			0
$869$	−24.9480			−0.846304
$870$		0			0
$871$	−59.7564			−2.02477
$872$		0			0
$873$			96.8080i			3.27646i
$874$		0			0
$875$	−2.62412	−	0.337637i	−0.0887114	−	0.0114142i
$876$		0			0
$877$	−5.08801			−0.171810			−0.0859049	−	0.996303i	$-0.527378\pi$
$877$	−5.08801			−0.171810			−0.0859049	+	0.996303i	$0.527378\pi$
$878$		0			0
$879$			42.4005i			1.43013i
$880$		0			0
$881$			34.6320i			1.16678i	0.812191	+	0.583391i	$0.198274\pi$
$881$			34.6320i			1.16678i	−0.812191	+	0.583391i	$0.801726\pi$
$882$		0			0
$883$			11.6931i			0.393503i	0.980453	+	0.196751i	$0.0630392\pi$
$883$			11.6931i			0.393503i	−0.980453	+	0.196751i	$0.936961\pi$
$884$		0			0
$885$			21.5440i			0.724194i
$886$		0			0
$887$	54.5082			1.83021			0.915103	−	0.403220i	$-0.132109\pi$
$887$	54.5082			1.83021			0.915103	+	0.403220i	$0.132109\pi$
$888$		0			0
$889$	−32.8600	−	4.22800i	−1.10209	−	0.141803i
$890$		0			0
$891$		−	25.4592i		−	0.852916i
$892$		0			0
$893$	−7.36799			−0.246560
$894$		0			0
$895$	−3.22241			−0.107713
$896$		0			0
$897$	74.1760			2.47667
$898$		0			0
$899$	−28.2670			−0.942758
$900$		0			0
$901$		−	28.6320i		−	0.953871i
$902$		0			0
$903$	15.7447	+	2.02582i	0.523951	+	0.0674152i
$904$		0			0
$905$	−9.54400			−0.317253
$906$		0			0
$907$		−	19.7964i		−	0.657327i	−0.944447	−	0.328664i	$-0.893402\pi$
$907$		−	19.7964i		−	0.657327i	0.944447	−	0.328664i	$-0.106598\pi$
$908$		0			0
$909$		−	55.0880i		−	1.82715i
$910$		0			0
$911$		−	34.7118i		−	1.15005i	−0.818134	−	0.575027i	$-0.804991\pi$
$911$		−	34.7118i		−	1.15005i	0.818134	−	0.575027i	$-0.195009\pi$
$912$		0			0
$913$		−	7.36799i		−	0.243845i
$914$		0			0
$915$	10.4965			0.347003
$916$		0			0
$917$	3.54400	−	27.5440i	0.117033	−	0.909583i
$918$		0			0
$919$			22.2368i			0.733525i	0.930315	+	0.366762i	$0.119534\pi$
$919$			22.2368i			0.733525i	−0.930315	+	0.366762i	$0.880466\pi$
$920$		0			0
$921$	92.4920			3.04772
$922$		0			0
$923$	34.7118			1.14255
$924$		0			0
$925$	11.5440			0.379565
$926$		0			0
$927$	24.8906			0.817516
$928$		0			0
$929$		−	1.08801i		−	0.0356964i	−0.999841	−	0.0178482i	$-0.994318\pi$
$929$		−	1.08801i		−	0.0356964i	0.999841	−	0.0178482i	$-0.00568156\pi$
$930$		0			0
$931$	−30.9681	−	8.10330i	−1.01494	−	0.265575i
$932$		0			0
$933$	71.7200			2.34801
$934$		0			0
$935$		−	17.3559i		−	0.567599i
$936$		0			0
$937$		−	19.2280i		−	0.628151i	−0.949398	−	0.314076i	$-0.898305\pi$
$937$		−	19.2280i		−	0.628151i	0.949398	−	0.314076i	$-0.101695\pi$
$938$		0			0
$939$		−	14.1335i		−	0.461230i
$940$		0			0
$941$		−	26.1760i		−	0.853314i	−0.904414	−	0.426657i	$-0.859691\pi$
$941$		−	26.1760i		−	0.853314i	0.904414	−	0.426657i	$-0.140309\pi$
$942$		0			0
$943$	−31.4894			−1.02544
$944$		0			0
$945$	21.5440	+	2.77200i	0.700826	+	0.0901732i
$946$		0			0
$947$			33.5153i			1.08910i	0.838729	+	0.544550i	$0.183299\pi$
$947$			33.5153i			1.08910i	−0.838729	+	0.544550i	$0.816701\pi$
$948$		0			0
$949$	−28.6320			−0.929434
$950$		0			0
$951$	74.3046			2.40949
$952$		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
$955$	−24.6300			−0.797008
$956$		0			0
$957$			51.4040i			1.66166i
$958$		0			0
$959$	1.19659	−	9.29989i	0.0386399	−	0.300309i
$960$		0			0
$961$	4.08801			0.131871
$962$		0			0
$963$			72.2787i			2.32915i
$964$		0			0
$965$			4.45600i			0.143444i
$966$		0			0
$967$			41.6186i			1.33836i	0.743099	+	0.669181i	$0.233355\pi$
$967$			41.6186i			1.33836i	−0.743099	+	0.669181i	$0.766645\pi$
$968$		0			0
$969$			64.6320i			2.07628i
$970$		0			0
$971$	−32.3187			−1.03716			−0.518578	−	0.855031i	$-0.673538\pi$
$971$	−32.3187			−1.03716			−0.518578	+	0.855031i	$0.673538\pi$
$972$		0			0
$973$	−3.08801	+	24.0000i	−0.0989970	+	0.769405i
$974$		0			0
$975$			14.1335i			0.452635i
$976$		0			0
$977$	37.0880			1.18655			0.593275	−	0.805000i	$-0.297835\pi$
$977$	37.0880			1.18655			0.593275	+	0.805000i	$0.297835\pi$
$978$		0			0
$979$	−43.6444			−1.39488
$980$		0			0
$981$	39.0880			1.24798
$982$		0			0
$983$	16.1593			0.515403			0.257701	−	0.966225i	$-0.417035\pi$
$983$	16.1593			0.515403			0.257701	+	0.966225i	$0.417035\pi$
$984$		0			0
$985$			6.00000i			0.191176i
$986$		0			0
$987$	1.61121	−	12.5223i	0.0512853	−	0.398589i
$988$		0			0
$989$	10.6320			0.338078
$990$		0			0
$991$			42.8151i			1.36007i	0.733181	+	0.680034i	$0.238035\pi$
$991$			42.8151i			1.36007i	−0.733181	+	0.680034i	$0.761965\pi$
$992$		0			0
$993$			40.6320i			1.28942i
$994$		0			0
$995$		−	4.57296i		−	0.144973i
$996$		0			0
$997$		−	33.4040i		−	1.05792i	−0.848648	−	0.528958i	$-0.822583\pi$
$997$		−	33.4040i		−	1.05792i	0.848648	−	0.528958i	$-0.177417\pi$
$998$		0			0
$999$	−94.7762			−2.99859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.2.k.b.111.7	yes	8
3.2	odd	2		5040.2.d.d.4591.6		8
4.3	odd	2	inner	560.2.k.b.111.1	✓	8
5.2	odd	4		2800.2.e.g.2799.1		8
5.3	odd	4		2800.2.e.h.2799.8		8
5.4	even	2		2800.2.k.m.2351.1		8
7.6	odd	2	inner	560.2.k.b.111.2	yes	8
8.3	odd	2		2240.2.k.d.1791.8		8
8.5	even	2		2240.2.k.d.1791.2		8
12.11	even	2		5040.2.d.d.4591.7		8
20.3	even	4		2800.2.e.h.2799.1		8
20.7	even	4		2800.2.e.g.2799.8		8
20.19	odd	2		2800.2.k.m.2351.8		8
21.20	even	2		5040.2.d.d.4591.3		8
28.27	even	2	inner	560.2.k.b.111.8	yes	8
35.13	even	4		2800.2.e.g.2799.2		8
35.27	even	4		2800.2.e.h.2799.7		8
35.34	odd	2		2800.2.k.m.2351.7		8
56.13	odd	2		2240.2.k.d.1791.7		8
56.27	even	2		2240.2.k.d.1791.1		8
84.83	odd	2		5040.2.d.d.4591.2		8
140.27	odd	4		2800.2.e.h.2799.2		8
140.83	odd	4		2800.2.e.g.2799.7		8
140.139	even	2		2800.2.k.m.2351.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.k.b.111.1	✓	8	4.3	odd	2	inner
560.2.k.b.111.2	yes	8	7.6	odd	2	inner
560.2.k.b.111.7	yes	8	1.1	even	1	trivial
560.2.k.b.111.8	yes	8	28.27	even	2	inner
2240.2.k.d.1791.1		8	56.27	even	2
2240.2.k.d.1791.2		8	8.5	even	2
2240.2.k.d.1791.7		8	56.13	odd	2
2240.2.k.d.1791.8		8	8.3	odd	2
2800.2.e.g.2799.1		8	5.2	odd	4
2800.2.e.g.2799.2		8	35.13	even	4
2800.2.e.g.2799.7		8	140.83	odd	4
2800.2.e.g.2799.8		8	20.7	even	4
2800.2.e.h.2799.1		8	20.3	even	4
2800.2.e.h.2799.2		8	140.27	odd	4
2800.2.e.h.2799.7		8	35.27	even	4
2800.2.e.h.2799.8		8	5.3	odd	4
2800.2.k.m.2351.1		8	5.4	even	2
2800.2.k.m.2351.2		8	140.139	even	2
2800.2.k.m.2351.7		8	35.34	odd	2
2800.2.k.m.2351.8		8	20.19	odd	2
5040.2.d.d.4591.2		8	84.83	odd	2
5040.2.d.d.4591.3		8	21.20	even	2
5040.2.d.d.4591.6		8	3.2	odd	2
5040.2.d.d.4591.7		8	12.11	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)

\(f(q)\)	\(=\)	\(q+2.96176 q^{3} -1.00000i q^{5} +(-0.337637 + 2.62412i) q^{7} +5.77200 q^{9} +O(q^{10})\)	\(q+2.96176 q^{3} -1.00000i q^{5} +(-0.337637 + 2.62412i) q^{7} +5.77200 q^{9} -3.63703i q^{11} -4.77200i q^{13} -2.96176i q^{15} +4.77200i q^{17} +4.57296 q^{19} +(-1.00000 + 7.77200i) q^{21} +5.24824i q^{23} -1.00000 q^{25} +8.20999 q^{27} -4.77200 q^{29} +5.92351 q^{31} -10.7720i q^{33} +(2.62412 + 0.337637i) q^{35} -11.5440 q^{37} -14.1335i q^{39} +6.00000i q^{41} -2.02582i q^{43} -5.77200i q^{45} -1.61121 q^{47} +(-6.77200 - 1.77200i) q^{49} +14.1335i q^{51} -6.00000 q^{53} -3.63703 q^{55} +13.5440 q^{57} -7.27406 q^{59} +3.54400i q^{61} +(-1.94884 + 15.1464i) q^{63} -4.77200 q^{65} -12.5223i q^{67} +15.5440i q^{69} +7.27406i q^{71} -6.00000i q^{73} -2.96176 q^{75} +(9.54400 + 1.22800i) q^{77} -6.85944i q^{79} +7.00000 q^{81} +2.02582 q^{83} +4.77200 q^{85} -14.1335 q^{87} -12.0000i q^{89} +(12.5223 + 1.61121i) q^{91} +17.5440 q^{93} -4.57296i q^{95} +16.7720i q^{97} -20.9930i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{9}+O(q^{10}) \) 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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Modular forms

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Representations

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Database

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