LMFDB - Embedded newform 4560.2.d.f.2431.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(2431,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("4560.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.d (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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.792772608.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 22x^{4} + 62x^{2} + 12 $ x^6 + 22x^4 + 62x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.4
Root			$-4.32698i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.f.2431.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +0.624069i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +0.624069i q^{7} +1.00000 q^{9} -4.32698i q^{11} -0.624069i q^{13} -1.00000 q^{15} -4.41188 q^{17} +(-2.29966 + 3.70291i) q^{19} -0.624069i q^{21} -1.92071i q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.16840i q^{29} -0.700336 q^{31} +4.32698i q^{33} +0.624069i q^{35} +2.40627i q^{37} +0.624069i q^{39} +7.89138i q^{41} -2.40627i q^{43} +1.00000 q^{45} +1.64369i q^{47} +6.61054 q^{49} +4.41188 q^{51} +1.92071i q^{53} -4.32698i q^{55} +(2.29966 - 3.70291i) q^{57} -10.0224 q^{59} +11.3221 q^{61} +0.624069i q^{63} -0.624069i q^{65} +4.31087 q^{67} +1.92071i q^{69} +10.0224 q^{71} +6.31087 q^{73} -1.00000 q^{75} +2.70034 q^{77} -3.71155 q^{79} +1.00000 q^{81} -4.23697i q^{83} -4.41188 q^{85} -8.16840i q^{87} +3.61290i q^{89} +0.389463 q^{91} +0.700336 q^{93} +(-2.29966 + 3.70291i) q^{95} +2.94033i q^{97} -4.32698i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9} - 6 q^{15} - 4 q^{17} - 6 q^{19} + 6 q^{25} - 6 q^{27} - 12 q^{31} + 6 q^{45} - 14 q^{49} + 4 q^{51} + 6 q^{57} + 16 q^{59} - 16 q^{61} - 20 q^{67} - 16 q^{71} - 8 q^{73} - 6 q^{75} + 24 q^{77} + 8 q^{79} + 6 q^{81} - 4 q^{85} + 56 q^{91} + 12 q^{93} - 6 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9 - 6 * q^15 - 4 * q^17 - 6 * q^19 + 6 * q^25 - 6 * q^27 - 12 * q^31 + 6 * q^45 - 14 * q^49 + 4 * q^51 + 6 * q^57 + 16 * q^59 - 16 * q^61 - 20 * q^67 - 16 * q^71 - 8 * q^73 - 6 * q^75 + 24 * q^77 + 8 * q^79 + 6 * q^81 - 4 * q^85 + 56 * q^91 + 12 * q^93 - 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$.

$n$	$1141$	$1711$	$1921$	$2737$	$3041$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$			0.624069i			0.235876i	0.993021	+	0.117938i	$0.0376284\pi$
$7$			0.624069i			0.235876i	−0.993021	+	0.117938i	$0.962372\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	4.32698i		−	1.30463i	−0.757946	−	0.652317i	$-0.773797\pi$
$11$		−	4.32698i		−	1.30463i	0.757946	−	0.652317i	$-0.226203\pi$
$12$		0			0
$13$		−	0.624069i		−	0.173086i	−0.996248	−	0.0865429i	$-0.972418\pi$
$13$		−	0.624069i		−	0.173086i	0.996248	−	0.0865429i	$-0.0275820\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$	−4.41188			−1.07004			−0.535019	−	0.844840i	$-0.679696\pi$
$17$	−4.41188			−1.07004			−0.535019	+	0.844840i	$0.679696\pi$
$18$		0			0
$19$	−2.29966	+	3.70291i	−0.527579	+	0.849506i
$19$	−2.29966	+	3.70291i	−0.527579	+	0.849506i
$20$		0			0
$21$		−	0.624069i		−	0.136183i
$22$		0			0
$23$		−	1.92071i		−	0.400496i	−0.979745	−	0.200248i	$-0.935825\pi$
$23$		−	1.92071i		−	0.400496i	0.979745	−	0.200248i	$-0.0641747\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$			8.16840i			1.51683i	0.651770	+	0.758417i	$0.274027\pi$
$29$			8.16840i			1.51683i	−0.651770	+	0.758417i	$0.725973\pi$
$30$		0			0
$31$	−0.700336			−0.125784			−0.0628920	−	0.998020i	$-0.520032\pi$
$31$	−0.700336			−0.125784			−0.0628920	+	0.998020i	$0.520032\pi$
$32$		0			0
$33$			4.32698i			0.753230i
$34$		0			0
$35$			0.624069i			0.105487i
$36$		0			0
$37$			2.40627i			0.395588i	0.980244	+	0.197794i	$0.0633778\pi$
$37$			2.40627i			0.395588i	−0.980244	+	0.197794i	$0.936622\pi$
$38$		0			0
$39$			0.624069i			0.0999311i
$40$		0			0
$41$			7.89138i			1.23243i	0.787579	+	0.616213i	$0.211334\pi$
$41$			7.89138i			1.23243i	−0.787579	+	0.616213i	$0.788666\pi$
$42$		0			0
$43$		−	2.40627i		−	0.366953i	−0.983024	−	0.183476i	$-0.941265\pi$
$43$		−	2.40627i		−	0.366953i	0.983024	−	0.183476i	$-0.0587351\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$			1.64369i			0.239757i	0.992789	+	0.119878i	$0.0382505\pi$
$47$			1.64369i			0.239757i	−0.992789	+	0.119878i	$0.961750\pi$
$48$		0			0
$49$	6.61054			0.944362
$50$		0			0
$51$	4.41188			0.617787
$52$		0			0
$53$			1.92071i			0.263830i	0.991261	+	0.131915i	$0.0421126\pi$
$53$			1.92071i			0.263830i	−0.991261	+	0.131915i	$0.957887\pi$
$54$		0			0
$55$		−	4.32698i		−	0.583450i
$56$		0			0
$57$	2.29966	−	3.70291i	0.304598	−	0.490462i
$58$		0			0
$59$	−10.0224			−1.30481			−0.652404	−	0.757871i	$-0.726239\pi$
$59$	−10.0224			−1.30481			−0.652404	+	0.757871i	$0.726239\pi$
$60$		0			0
$61$	11.3221			1.44964			0.724822	−	0.688936i	$-0.241922\pi$
$61$	11.3221			1.44964			0.724822	+	0.688936i	$0.241922\pi$
$62$		0			0
$63$			0.624069i			0.0786254i
$64$		0			0
$65$		−	0.624069i		−	0.0774063i
$66$		0			0
$67$	4.31087			0.526657			0.263328	−	0.964706i	$-0.415180\pi$
$67$	4.31087			0.526657			0.263328	+	0.964706i	$0.415180\pi$
$68$		0			0
$69$			1.92071i			0.231226i
$70$		0			0
$71$	10.0224			1.18944			0.594721	−	0.803932i	$-0.297262\pi$
$71$	10.0224			1.18944			0.594721	+	0.803932i	$0.297262\pi$
$72$		0			0
$73$	6.31087			0.738632			0.369316	−	0.929304i	$-0.379592\pi$
$73$	6.31087			0.738632			0.369316	+	0.929304i	$0.379592\pi$
$74$		0			0
$75$	−1.00000			−0.115470
$76$		0			0
$77$	2.70034			0.307732
$78$		0			0
$79$	−3.71155			−0.417581			−0.208791	−	0.977960i	$-0.566953\pi$
$79$	−3.71155			−0.417581			−0.208791	+	0.977960i	$0.566953\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	4.23697i		−	0.465068i	−0.972588	−	0.232534i	$-0.925298\pi$
$83$		−	4.23697i		−	0.465068i	0.972588	−	0.232534i	$-0.0747017\pi$
$84$		0			0
$85$	−4.41188			−0.478536
$86$		0			0
$87$		−	8.16840i		−	0.875744i
$88$		0			0
$89$			3.61290i			0.382967i	0.981496	+	0.191483i	$0.0613298\pi$
$89$			3.61290i			0.382967i	−0.981496	+	0.191483i	$0.938670\pi$
$90$		0			0
$91$	0.389463			0.0408268
$92$		0			0
$93$	0.700336			0.0726214
$94$		0			0
$95$	−2.29966	+	3.70291i	−0.235941	+	0.379911i
$96$		0			0
$97$			2.94033i			0.298545i	0.988796	+	0.149273i	$0.0476932\pi$
$97$			2.94033i			0.298545i	−0.988796	+	0.149273i	$0.952307\pi$
$98$		0			0
$99$		−	4.32698i		−	0.434878i
$100$		0			0
$101$	0.599328			0.0596354			0.0298177	−	0.999555i	$-0.490507\pi$
$101$	0.599328			0.0596354			0.0298177	+	0.999555i	$0.490507\pi$
$102$		0			0
$103$	12.8238			1.26356			0.631781	−	0.775147i	$-0.282324\pi$
$103$	12.8238			1.26356			0.631781	+	0.775147i	$0.282324\pi$
$104$		0			0
$105$		−	0.624069i		−	0.0609029i
$106$		0			0
$107$	−7.71155			−0.745503			−0.372752	−	0.927931i	$-0.621586\pi$
$107$	−7.71155			−0.745503			−0.372752	+	0.927931i	$0.621586\pi$
$108$		0			0
$109$			12.2184i			1.17031i	0.810923	+	0.585153i	$0.198966\pi$
$109$			12.2184i			1.17031i	−0.810923	+	0.585153i	$0.801034\pi$
$110$		0			0
$111$		−	2.40627i		−	0.228393i
$112$		0			0
$113$			7.01027i			0.659471i	0.944073	+	0.329735i	$0.106960\pi$
$113$			7.01027i			0.659471i	−0.944073	+	0.329735i	$0.893040\pi$
$114$		0			0
$115$		−	1.92071i		−	0.179107i
$116$		0			0
$117$		−	0.624069i		−	0.0576952i
$118$		0			0
$119$		−	2.75332i		−	0.252396i
$120$		0			0
$121$	−7.72275			−0.702069
$122$		0			0
$123$		−	7.89138i		−	0.711542i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	4.31087			0.382528			0.191264	−	0.981539i	$-0.438741\pi$
$127$	4.31087			0.382528			0.191264	+	0.981539i	$0.438741\pi$
$128$		0			0
$129$			2.40627i			0.211860i
$130$		0			0
$131$		−	2.26776i		−	0.198135i	−0.995081	−	0.0990676i	$-0.968414\pi$
$131$		−	2.26776i		−	0.198135i	0.995081	−	0.0990676i	$-0.0315860\pi$
$132$		0			0
$133$	−2.31087	−	1.43515i	−0.200378	−	0.124443i
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	3.68913			0.315183			0.157592	−	0.987504i	$-0.449627\pi$
$137$	3.68913			0.315183			0.157592	+	0.987504i	$0.449627\pi$
$138$		0			0
$139$			16.3368i			1.38567i	0.721097	+	0.692835i	$0.243638\pi$
$139$			16.3368i			1.38567i	−0.721097	+	0.692835i	$0.756362\pi$
$140$		0			0
$141$		−	1.64369i		−	0.138424i
$142$		0			0
$143$	−2.70034			−0.225813
$144$		0			0
$145$			8.16840i			0.678349i
$146$		0			0
$147$	−6.61054			−0.545228
$148$		0			0
$149$	−2.82376			−0.231332			−0.115666	−	0.993288i	$-0.536900\pi$
$149$	−2.82376			−0.231332			−0.115666	+	0.993288i	$0.536900\pi$
$150$		0			0
$151$	−9.83497			−0.800359			−0.400179	−	0.916437i	$-0.631052\pi$
$151$	−9.83497			−0.800359			−0.400179	+	0.916437i	$0.631052\pi$
$152$		0			0
$153$	−4.41188			−0.356679
$154$		0			0
$155$	−0.700336			−0.0562523
$156$		0			0
$157$	−9.11222			−0.727234			−0.363617	−	0.931549i	$-0.618458\pi$
$157$	−9.11222			−0.727234			−0.363617	+	0.931549i	$0.618458\pi$
$158$		0			0
$159$		−	1.92071i		−	0.152322i
$160$		0			0
$161$	1.19866			0.0944674
$162$		0			0
$163$		−	14.3676i		−	1.12536i	−0.826676	−	0.562678i	$-0.809771\pi$
$163$		−	14.3676i		−	1.12536i	0.826676	−	0.562678i	$-0.190229\pi$
$164$		0			0
$165$			4.32698i			0.336855i
$166$		0			0
$167$	20.4343			1.58125			0.790627	−	0.612298i	$-0.209755\pi$
$167$	20.4343			1.58125			0.790627	+	0.612298i	$0.209755\pi$
$168$		0			0
$169$	12.6105			0.970041
$170$		0			0
$171$	−2.29966	+	3.70291i	−0.175860	+	0.283169i
$172$		0			0
$173$		−	20.1997i		−	1.53576i	−0.640595	−	0.767879i	$-0.721312\pi$
$173$		−	20.1997i		−	1.53576i	0.640595	−	0.767879i	$-0.278688\pi$
$174$		0			0
$175$			0.624069i			0.0471752i
$176$		0			0
$177$	10.0224			0.753331
$178$		0			0
$179$	−1.19866			−0.0895918			−0.0447959	−	0.998996i	$-0.514264\pi$
$179$	−1.19866			−0.0895918			−0.0447959	+	0.998996i	$0.514264\pi$
$180$		0			0
$181$			8.65396i			0.643244i	0.946868	+	0.321622i	$0.104228\pi$
$181$			8.65396i			0.643244i	−0.946868	+	0.321622i	$0.895772\pi$
$182$		0			0
$183$	−11.3221			−0.836952
$184$		0			0
$185$			2.40627i			0.176912i
$186$		0			0
$187$			19.0901i			1.39601i
$188$		0			0
$189$		−	0.624069i		−	0.0453944i
$190$		0			0
$191$			16.5453i			1.19718i	0.801056	+	0.598589i	$0.204272\pi$
$191$			16.5453i			1.19718i	−0.801056	+	0.598589i	$0.795728\pi$
$192$		0			0
$193$			8.02989i			0.578004i	0.957328	+	0.289002i	$0.0933234\pi$
$193$			8.02989i			0.578004i	−0.957328	+	0.289002i	$0.906677\pi$
$194$		0			0
$195$			0.624069i			0.0446905i
$196$		0			0
$197$	14.4343			1.02840			0.514201	−	0.857670i	$-0.328089\pi$
$197$	14.4343			1.02840			0.514201	+	0.857670i	$0.328089\pi$
$198$		0			0
$199$			2.31626i			0.164195i	0.996624	+	0.0820977i	$0.0261620\pi$
$199$			2.31626i			0.164195i	−0.996624	+	0.0820977i	$0.973838\pi$
$200$		0			0
$201$	−4.31087			−0.304066
$202$		0			0
$203$	−5.09765			−0.357785
$204$		0			0
$205$			7.89138i			0.551158i
$206$		0			0
$207$		−	1.92071i		−	0.133499i
$208$		0			0
$209$	16.0224	+	9.95060i	1.10829	+	0.688297i
$210$		0			0
$211$	2.08644			0.143636			0.0718182	−	0.997418i	$-0.477120\pi$
$211$	2.08644			0.143636			0.0718182	+	0.997418i	$0.477120\pi$
$212$		0			0
$213$	−10.0224			−0.686725
$214$		0			0
$215$		−	2.40627i		−	0.164106i
$216$		0			0
$217$		−	0.437058i		−	0.0296694i
$218$		0			0
$219$	−6.31087			−0.426449
$220$		0			0
$221$			2.75332i			0.185208i
$222$		0			0
$223$	−4.82376			−0.323023			−0.161511	−	0.986871i	$-0.551637\pi$
$223$	−4.82376			−0.323023			−0.161511	+	0.986871i	$0.551637\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−3.81255			−0.253048			−0.126524	−	0.991964i	$-0.540382\pi$
$227$	−3.81255			−0.253048			−0.126524	+	0.991964i	$0.540382\pi$
$228$		0			0
$229$	13.6105			0.899410			0.449705	−	0.893177i	$-0.351529\pi$
$229$	13.6105			0.899410			0.449705	+	0.893177i	$0.351529\pi$
$230$		0			0
$231$	−2.70034			−0.177669
$232$		0			0
$233$	9.42309			0.617327			0.308664	−	0.951171i	$-0.400118\pi$
$233$	9.42309			0.617327			0.308664	+	0.951171i	$0.400118\pi$
$234$		0			0
$235$			1.64369i			0.107223i
$236$		0			0
$237$	3.71155			0.241091
$238$		0			0
$239$			6.38620i			0.413089i	0.978437	+	0.206544i	$0.0662218\pi$
$239$			6.38620i			0.413089i	−0.978437	+	0.206544i	$0.933778\pi$
$240$		0			0
$241$			1.34514i			0.0866482i	0.999061	+	0.0433241i	$0.0137948\pi$
$241$			1.34514i			0.0866482i	−0.999061	+	0.0433241i	$0.986205\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	6.61054			0.422332
$246$		0			0
$247$	2.31087	+	1.43515i	0.147037	+	0.0913164i
$248$		0			0
$249$			4.23697i			0.268507i
$250$		0			0
$251$			1.01962i			0.0643579i	0.999482	+	0.0321790i	$0.0102447\pi$
$251$			1.01962i			0.0643579i	−0.999482	+	0.0321790i	$0.989755\pi$
$252$		0			0
$253$	−8.31087			−0.522500
$254$		0			0
$255$	4.41188			0.276283
$256$		0			0
$257$			25.3863i			1.58355i	0.610810	+	0.791777i	$0.290844\pi$
$257$			25.3863i			1.58355i	−0.610810	+	0.791777i	$0.709156\pi$
$258$		0			0
$259$	−1.50168			−0.0933098
$260$		0			0
$261$			8.16840i			0.505611i
$262$		0			0
$263$		−	6.73325i		−	0.415190i	−0.978215	−	0.207595i	$-0.933436\pi$
$263$		−	6.73325i		−	0.415190i	0.978215	−	0.207595i	$-0.0665636\pi$
$264$		0			0
$265$			1.92071i			0.117988i
$266$		0			0
$267$		−	3.61290i		−	0.221106i
$268$		0			0
$269$			12.2669i			0.747924i	0.927444	+	0.373962i	$0.122001\pi$
$269$			12.2669i			0.747924i	−0.927444	+	0.373962i	$0.877999\pi$
$270$		0			0
$271$			23.1886i			1.40861i	0.709899	+	0.704303i	$0.248740\pi$
$271$			23.1886i			1.40861i	−0.709899	+	0.704303i	$0.751260\pi$
$272$		0			0
$273$	−0.389463			−0.0235714
$274$		0			0
$275$		−	4.32698i		−	0.260927i
$276$		0			0
$277$	−3.40067			−0.204327			−0.102163	−	0.994768i	$-0.532576\pi$
$277$	−3.40067			−0.204327			−0.102163	+	0.994768i	$0.532576\pi$
$278$		0			0
$279$	−0.700336			−0.0419280
$280$		0			0
$281$			12.4469i			0.742518i	0.928529	+	0.371259i	$0.121074\pi$
$281$			12.4469i			0.742518i	−0.928529	+	0.371259i	$0.878926\pi$
$282$		0			0
$283$			25.3378i			1.50618i	0.657920	+	0.753088i	$0.271437\pi$
$283$			25.3378i			1.50618i	−0.657920	+	0.753088i	$0.728563\pi$
$284$		0			0
$285$	2.29966	−	3.70291i	0.136220	−	0.219341i
$286$		0			0
$287$	−4.92477			−0.290700
$288$		0			0
$289$	2.46469			0.144982
$290$		0			0
$291$		−	2.94033i		−	0.172365i
$292$		0			0
$293$		−	7.28729i		−	0.425728i	−0.977082	−	0.212864i	$-0.931721\pi$
$293$		−	7.28729i		−	0.425728i	0.977082	−	0.212864i	$-0.0682791\pi$
$294$		0			0
$295$	−10.0224			−0.583528
$296$		0			0
$297$			4.32698i			0.251077i
$298$		0			0
$299$	−1.19866			−0.0693201
$300$		0			0
$301$	1.50168			0.0865554
$302$		0			0
$303$	−0.599328			−0.0344305
$304$		0			0
$305$	11.3221			0.648301
$306$		0			0
$307$	14.0224			0.800302			0.400151	−	0.916449i	$-0.368958\pi$
$307$	14.0224			0.800302			0.400151	+	0.916449i	$0.368958\pi$
$308$		0			0
$309$	−12.8238			−0.729518
$310$		0			0
$311$			17.2594i			0.978692i	0.872090	+	0.489346i	$0.162764\pi$
$311$			17.2594i			0.978692i	−0.872090	+	0.489346i	$0.837236\pi$
$312$		0			0
$313$	15.1346			0.855460			0.427730	−	0.903907i	$-0.359313\pi$
$313$	15.1346			0.855460			0.427730	+	0.903907i	$0.359313\pi$
$314$		0			0
$315$			0.624069i			0.0351623i
$316$		0			0
$317$		−	15.1102i		−	0.848673i	−0.905505	−	0.424336i	$-0.860507\pi$
$317$		−	15.1102i		−	0.848673i	0.905505	−	0.424336i	$-0.139493\pi$
$318$		0			0
$319$	35.3445			1.97891
$320$		0			0
$321$	7.71155			0.430416
$322$		0			0
$323$	10.1458	−	16.3368i	0.564530	−	0.909004i
$324$		0			0
$325$		−	0.624069i		−	0.0346171i
$326$		0			0
$327$		−	12.2184i		−	0.675677i
$328$		0			0
$329$	−1.02578			−0.0565529
$330$		0			0
$331$	16.8092			0.923917			0.461958	−	0.886902i	$-0.347147\pi$
$331$	16.8092			0.923917			0.461958	+	0.886902i	$0.347147\pi$
$332$		0			0
$333$			2.40627i			0.131863i
$334$		0			0
$335$	4.31087			0.235528
$336$		0			0
$337$			34.9829i			1.90564i	0.303539	+	0.952819i	$0.401832\pi$
$337$			34.9829i			1.90564i	−0.303539	+	0.952819i	$0.598168\pi$
$338$		0			0
$339$		−	7.01027i		−	0.380746i
$340$		0			0
$341$			3.03034i			0.164102i
$342$		0			0
$343$			8.49392i			0.458629i
$344$		0			0
$345$			1.92071i			0.103408i
$346$		0			0
$347$		−	20.4768i		−	1.09925i	−0.835411	−	0.549625i	$-0.814771\pi$
$347$		−	20.4768i		−	1.09925i	0.835411	−	0.549625i	$-0.185229\pi$
$348$		0			0
$349$	5.50953			0.294918			0.147459	−	0.989068i	$-0.452890\pi$
$349$	5.50953			0.294918			0.147459	+	0.989068i	$0.452890\pi$
$350$		0			0
$351$			0.624069i			0.0333104i
$352$		0			0
$353$	−26.0448			−1.38623			−0.693113	−	0.720829i	$-0.743761\pi$
$353$	−26.0448			−1.38623			−0.693113	+	0.720829i	$0.743761\pi$
$354$		0			0
$355$	10.0224			0.531935
$356$		0			0
$357$			2.75332i			0.145721i
$358$		0			0
$359$		−	17.0994i		−	0.902471i	−0.892405	−	0.451235i	$-0.850984\pi$
$359$		−	17.0994i		−	0.902471i	0.892405	−	0.451235i	$-0.149016\pi$
$360$		0			0
$361$	−8.42309	−	17.0309i	−0.443321	−	0.896363i
$362$		0			0
$363$	7.72275			0.405339
$364$		0			0
$365$	6.31087			0.330326
$366$		0			0
$367$			13.0995i			0.683787i	0.939739	+	0.341893i	$0.111068\pi$
$367$			13.0995i			0.683787i	−0.939739	+	0.341893i	$0.888932\pi$
$368$		0			0
$369$			7.89138i			0.410809i
$370$		0			0
$371$	−1.19866			−0.0622311
$372$		0			0
$373$		−	10.3462i		−	0.535703i	−0.963460	−	0.267852i	$-0.913686\pi$
$373$		−	10.3462i		−	0.535703i	0.963460	−	0.267852i	$-0.0863137\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	5.09765			0.262542
$378$		0			0
$379$	16.8092			0.863430			0.431715	−	0.902010i	$-0.357909\pi$
$379$	16.8092			0.863430			0.431715	+	0.902010i	$0.357909\pi$
$380$		0			0
$381$	−4.31087			−0.220853
$382$		0			0
$383$	−30.8462			−1.57617			−0.788083	−	0.615569i	$-0.788926\pi$
$383$	−30.8462			−1.57617			−0.788083	+	0.615569i	$0.788926\pi$
$384$		0			0
$385$	2.70034			0.137622
$386$		0			0
$387$		−	2.40627i		−	0.122318i
$388$		0			0
$389$	7.97758			0.404479			0.202240	−	0.979336i	$-0.435178\pi$
$389$	7.97758			0.404479			0.202240	+	0.979336i	$0.435178\pi$
$390$		0			0
$391$			8.47394i			0.428546i
$392$		0			0
$393$			2.26776i			0.114393i
$394$		0			0
$395$	−3.71155			−0.186748
$396$		0			0
$397$	−11.1987			−0.562044			−0.281022	−	0.959701i	$-0.590673\pi$
$397$	−11.1987			−0.562044			−0.281022	+	0.959701i	$0.590673\pi$
$398$		0			0
$399$	2.31087	+	1.43515i	0.115688	+	0.0718474i
$400$		0			0
$401$		−	24.4082i		−	1.21889i	−0.792829	−	0.609444i	$-0.791393\pi$
$401$		−	24.4082i		−	1.21889i	0.792829	−	0.609444i	$-0.208607\pi$
$402$		0			0
$403$			0.437058i			0.0217714i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	10.4119			0.516098
$408$		0			0
$409$			24.5337i			1.21311i	0.795040	+	0.606557i	$0.207450\pi$
$409$			24.5337i			1.21311i	−0.795040	+	0.606557i	$0.792550\pi$
$410$		0			0
$411$	−3.68913			−0.181971
$412$		0			0
$413$		−	6.25469i		−	0.307773i
$414$		0			0
$415$		−	4.23697i		−	0.207985i
$416$		0			0
$417$		−	16.3368i		−	0.800017i
$418$		0			0
$419$		−	37.7917i		−	1.84624i	−0.384507	−	0.923122i	$-0.625628\pi$
$419$		−	37.7917i		−	1.84624i	0.384507	−	0.923122i	$-0.374372\pi$
$420$		0			0
$421$		−	15.9628i		−	0.777978i	−0.921242	−	0.388989i	$-0.872824\pi$
$421$		−	15.9628i		−	0.777978i	0.921242	−	0.388989i	$-0.127176\pi$
$422$		0			0
$423$			1.64369i			0.0799190i
$424$		0			0
$425$	−4.41188			−0.214008
$426$		0			0
$427$			7.06577i			0.341936i
$428$		0			0
$429$	2.70034			0.130373
$430$		0			0
$431$	−8.82376			−0.425026			−0.212513	−	0.977158i	$-0.568165\pi$
$431$	−8.82376			−0.425026			−0.212513	+	0.977158i	$0.568165\pi$
$432$		0			0
$433$		−	3.39737i		−	0.163267i	−0.996662	−	0.0816335i	$-0.973986\pi$
$433$		−	3.39737i		−	0.163267i	0.996662	−	0.0816335i	$-0.0260137\pi$
$434$		0			0
$435$		−	8.16840i		−	0.391645i
$436$		0			0
$437$	7.11222	+	4.41699i	0.340223	+	0.211293i
$438$		0			0
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$	6.61054			0.314787
$442$		0			0
$443$		−	27.9796i		−	1.32935i	−0.747132	−	0.664675i	$-0.768570\pi$
$443$		−	27.9796i		−	1.32935i	0.747132	−	0.664675i	$-0.231430\pi$
$444$		0			0
$445$			3.61290i			0.171268i
$446$		0			0
$447$	2.82376			0.133559
$448$		0			0
$449$			17.5165i			0.826653i	0.910583	+	0.413326i	$0.135633\pi$
$449$			17.5165i			0.826653i	−0.910583	+	0.413326i	$0.864367\pi$
$450$		0			0
$451$	34.1458			1.60787
$452$		0			0
$453$	9.83497			0.462087
$454$		0			0
$455$	0.389463			0.0182583
$456$		0			0
$457$	−0.535307			−0.0250406			−0.0125203	−	0.999922i	$-0.503985\pi$
$457$	−0.535307			−0.0250406			−0.0125203	+	0.999922i	$0.503985\pi$
$458$		0			0
$459$	4.41188			0.205929
$460$		0			0
$461$	11.6475			0.542479			0.271240	−	0.962512i	$-0.412566\pi$
$461$	11.6475			0.542479			0.271240	+	0.962512i	$0.412566\pi$
$462$		0			0
$463$			23.8326i			1.10760i	0.832651	+	0.553798i	$0.186822\pi$
$463$			23.8326i			1.10760i	−0.832651	+	0.553798i	$0.813178\pi$
$464$		0			0
$465$	0.700336			0.0324773
$466$		0			0
$467$			39.0328i			1.80622i	0.429405	+	0.903112i	$0.358723\pi$
$467$			39.0328i			1.80622i	−0.429405	+	0.903112i	$0.641277\pi$
$468$		0			0
$469$			2.69028i			0.124226i
$470$		0			0
$471$	9.11222			0.419869
$472$		0			0
$473$	−10.4119			−0.478739
$474$		0			0
$475$	−2.29966	+	3.70291i	−0.105516	+	0.169901i
$476$		0			0
$477$			1.92071i			0.0879433i
$478$		0			0
$479$			24.9423i			1.13964i	0.821769	+	0.569820i	$0.192987\pi$
$479$			24.9423i			1.13964i	−0.821769	+	0.569820i	$0.807013\pi$
$480$		0			0
$481$	1.50168			0.0684707
$482$		0			0
$483$	−1.19866			−0.0545408
$484$		0			0
$485$			2.94033i			0.133514i
$486$		0			0
$487$	−22.3333			−1.01202			−0.506009	−	0.862528i	$-0.668880\pi$
$487$	−22.3333			−1.01202			−0.506009	+	0.862528i	$0.668880\pi$
$488$		0			0
$489$			14.3676i			0.649725i
$490$		0			0
$491$		−	12.4269i		−	0.560818i	−0.959881	−	0.280409i	$-0.909530\pi$
$491$		−	12.4269i		−	0.560818i	0.959881	−	0.280409i	$-0.0904701\pi$
$492$		0			0
$493$		−	36.0380i		−	1.62307i
$494$		0			0
$495$		−	4.32698i		−	0.194483i
$496$		0			0
$497$			6.25469i			0.280561i
$498$		0			0
$499$			19.3472i			0.866098i	0.901370	+	0.433049i	$0.142562\pi$
$499$			19.3472i			0.866098i	−0.901370	+	0.433049i	$0.857438\pi$
$500$		0			0
$501$	−20.4343			−0.912937
$502$		0			0
$503$		−	0.395552i		−	0.0176368i	−0.999961	−	0.00881839i	$-0.997193\pi$
$503$		−	0.395552i		−	0.0176368i	0.999961	−	0.00881839i	$-0.00280702\pi$
$504$		0			0
$505$	0.599328			0.0266698
$506$		0			0
$507$	−12.6105			−0.560054
$508$		0			0
$509$			22.1889i			0.983507i	0.870734	+	0.491754i	$0.163644\pi$
$509$			22.1889i			0.983507i	−0.870734	+	0.491754i	$0.836356\pi$
$510$		0			0
$511$			3.93842i			0.174226i
$512$		0			0
$513$	2.29966	−	3.70291i	0.101533	−	0.163487i
$514$		0			0
$515$	12.8238			0.565082
$516$		0			0
$517$	7.11222			0.312795
$518$		0			0
$519$			20.1997i			0.886671i
$520$		0			0
$521$		−	23.5141i		−	1.03017i	−0.857139	−	0.515086i	$-0.827760\pi$
$521$		−	23.5141i		−	1.03017i	0.857139	−	0.515086i	$-0.172240\pi$
$522$		0			0
$523$	−22.1604			−0.969007			−0.484504	−	0.874789i	$-0.661000\pi$
$523$	−22.1604			−0.969007			−0.484504	+	0.874789i	$0.661000\pi$
$524$		0			0
$525$		−	0.624069i		−	0.0272366i
$526$		0			0
$527$	3.08980			0.134594
$528$		0			0
$529$	19.3109			0.839603
$530$		0			0
$531$	−10.0224			−0.434936
$532$		0			0
$533$	4.92477			0.213315
$534$		0			0
$535$	−7.71155			−0.333399
$536$		0			0
$537$	1.19866			0.0517258
$538$		0			0
$539$		−	28.6037i		−	1.23205i
$540$		0			0
$541$	−9.19866			−0.395481			−0.197741	−	0.980254i	$-0.563360\pi$
$541$	−9.19866			−0.395481			−0.197741	+	0.980254i	$0.563360\pi$
$542$		0			0
$543$		−	8.65396i		−	0.371377i
$544$		0			0
$545$			12.2184i			0.523377i
$546$		0			0
$547$	2.55449			0.109222			0.0546111	−	0.998508i	$-0.482608\pi$
$547$	2.55449			0.109222			0.0546111	+	0.998508i	$0.482608\pi$
$548$		0			0
$549$	11.3221			0.483215
$550$		0			0
$551$	−30.2469	−	18.7846i	−1.28856	−	0.800250i
$552$		0			0
$553$		−	2.31626i		−	0.0984975i
$554$		0			0
$555$		−	2.40627i		−	0.102140i
$556$		0			0
$557$	−40.3557			−1.70993			−0.854963	−	0.518689i	$-0.826420\pi$
$557$	−40.3557			−1.70993			−0.854963	+	0.518689i	$0.826420\pi$
$558$		0			0
$559$	−1.50168			−0.0635143
$560$		0			0
$561$		−	19.0901i		−	0.805985i
$562$		0			0
$563$	−41.0112			−1.72842			−0.864208	−	0.503134i	$-0.832180\pi$
$563$	−41.0112			−1.72842			−0.864208	+	0.503134i	$0.832180\pi$
$564$		0			0
$565$			7.01027i			0.294924i
$566$		0			0
$567$			0.624069i			0.0262085i
$568$		0			0
$569$			29.4147i			1.23313i	0.787304	+	0.616565i	$0.211476\pi$
$569$			29.4147i			1.23313i	−0.787304	+	0.616565i	$0.788524\pi$
$570$		0			0
$571$			13.5635i			0.567615i	0.958881	+	0.283807i	$0.0915977\pi$
$571$			13.5635i			0.567615i	−0.958881	+	0.283807i	$0.908402\pi$
$572$		0			0
$573$		−	16.5453i		−	0.691191i
$574$		0			0
$575$		−	1.92071i		−	0.0800991i
$576$		0			0
$577$	−4.91020			−0.204414			−0.102207	−	0.994763i	$-0.532590\pi$
$577$	−4.91020			−0.204414			−0.102207	+	0.994763i	$0.532590\pi$
$578$		0			0
$579$		−	8.02989i		−	0.333711i
$580$		0			0
$581$	2.64416			0.109698
$582$		0			0
$583$	8.31087			0.344201
$584$		0			0
$585$		−	0.624069i		−	0.0258021i
$586$		0			0
$587$			20.2968i			0.837737i	0.908047	+	0.418868i	$0.137573\pi$
$587$			20.2968i			0.837737i	−0.908047	+	0.418868i	$0.862427\pi$
$588$		0			0
$589$	1.61054	−	2.59328i	0.0663610	−	0.106854i
$590$		0			0
$591$	−14.4343			−0.593748
$592$		0			0
$593$	27.1571			1.11521			0.557603	−	0.830108i	$-0.311721\pi$
$593$	27.1571			1.11521			0.557603	+	0.830108i	$0.311721\pi$
$594$		0			0
$595$		−	2.75332i		−	0.112875i
$596$		0			0
$597$		−	2.31626i		−	0.0947983i
$598$		0			0
$599$	−19.7980			−0.808924			−0.404462	−	0.914555i	$-0.632541\pi$
$599$	−19.7980			−0.808924			−0.404462	+	0.914555i	$0.632541\pi$
$600$		0			0
$601$			31.2315i			1.27396i	0.770881	+	0.636979i	$0.219816\pi$
$601$			31.2315i			1.27396i	−0.770881	+	0.636979i	$0.780184\pi$
$602$		0			0
$603$	4.31087			0.175552
$604$		0			0
$605$	−7.72275			−0.313975
$606$		0			0
$607$	4.31087			0.174973			0.0874865	−	0.996166i	$-0.472117\pi$
$607$	4.31087			0.174973			0.0874865	+	0.996166i	$0.472117\pi$
$608$		0			0
$609$	5.09765			0.206567
$610$		0			0
$611$	1.02578			0.0414985
$612$		0			0
$613$	−43.1346			−1.74219			−0.871096	−	0.491113i	$-0.836590\pi$
$613$	−43.1346			−1.74219			−0.871096	+	0.491113i	$0.836590\pi$
$614$		0			0
$615$		−	7.89138i		−	0.318211i
$616$		0			0
$617$	6.38946			0.257230			0.128615	−	0.991695i	$-0.458947\pi$
$617$	6.38946			0.257230			0.128615	+	0.991695i	$0.458947\pi$
$618$		0			0
$619$			25.0878i			1.00836i	0.863598	+	0.504181i	$0.168206\pi$
$619$			25.0878i			1.00836i	−0.863598	+	0.504181i	$0.831794\pi$
$620$		0			0
$621$			1.92071i			0.0770754i
$622$		0			0
$623$	−2.25470			−0.0903327
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−16.0224	−	9.95060i	−0.639874	−	0.397389i
$628$		0			0
$629$		−	10.6162i		−	0.423295i
$630$		0			0
$631$			12.9524i			0.515627i	0.966195	+	0.257814i	$0.0830021\pi$
$631$			12.9524i			0.515627i	−0.966195	+	0.257814i	$0.916998\pi$
$632$		0			0
$633$	−2.08644			−0.0829285
$634$		0			0
$635$	4.31087			0.171072
$636$		0			0
$637$		−	4.12543i		−	0.163456i
$638$		0			0
$639$	10.0224			0.396481
$640$		0			0
$641$		−	22.2660i		−	0.879453i	−0.898132	−	0.439726i	$-0.855075\pi$
$641$		−	22.2660i		−	0.879453i	0.898132	−	0.439726i	$-0.144925\pi$
$642$		0			0
$643$			26.8430i			1.05858i	0.848440	+	0.529292i	$0.177542\pi$
$643$			26.8430i			1.05858i	−0.848440	+	0.529292i	$0.822458\pi$
$644$		0			0
$645$			2.40627i			0.0947468i
$646$		0			0
$647$			9.78357i			0.384632i	0.981333	+	0.192316i	$0.0615998\pi$
$647$			9.78357i			0.384632i	−0.981333	+	0.192316i	$0.938400\pi$
$648$		0			0
$649$			43.3668i			1.70230i
$650$		0			0
$651$			0.437058i			0.0171297i
$652$		0			0
$653$	−47.4309			−1.85612			−0.928058	−	0.372436i	$-0.878523\pi$
$653$	−47.4309			−1.85612			−0.928058	+	0.372436i	$0.878523\pi$
$654$		0			0
$655$		−	2.26776i		−	0.0886087i
$656$		0			0
$657$	6.31087			0.246211
$658$		0			0
$659$	11.4679			0.446727			0.223363	−	0.974735i	$-0.428296\pi$
$659$	11.4679			0.446727			0.223363	+	0.974735i	$0.428296\pi$
$660$		0			0
$661$		−	3.38438i		−	0.131637i	−0.997832	−	0.0658187i	$-0.979034\pi$
$661$		−	3.38438i		−	0.131637i	0.997832	−	0.0658187i	$-0.0209659\pi$
$662$		0			0
$663$		−	2.75332i		−	0.106930i
$664$		0			0
$665$	−2.31087	−	1.43515i	−0.0896118	−	0.0556527i
$666$		0			0
$667$	15.6891			0.607485
$668$		0			0
$669$	4.82376			0.186497
$670$		0			0
$671$		−	48.9904i		−	1.89125i
$672$		0			0
$673$		−	38.2702i		−	1.47521i	−0.675233	−	0.737605i	$-0.735957\pi$
$673$		−	38.2702i		−	1.47521i	0.675233	−	0.737605i	$-0.264043\pi$
$674$		0			0
$675$	−1.00000			−0.0384900
$676$		0			0
$677$		−	39.8239i		−	1.53056i	−0.643699	−	0.765279i	$-0.722601\pi$
$677$		−	39.8239i		−	1.53056i	0.643699	−	0.765279i	$-0.277399\pi$
$678$		0			0
$679$	−1.83497			−0.0704197
$680$		0			0
$681$	3.81255			0.146097
$682$		0			0
$683$	8.73732			0.334324			0.167162	−	0.985929i	$-0.446540\pi$
$683$	8.73732			0.334324			0.167162	+	0.985929i	$0.446540\pi$
$684$		0			0
$685$	3.68913			0.140954
$686$		0			0
$687$	−13.6105			−0.519274
$688$		0			0
$689$	1.19866			0.0456652
$690$		0			0
$691$			7.50282i			0.285421i	0.989764	+	0.142710i	$0.0455818\pi$
$691$			7.50282i			0.285421i	−0.989764	+	0.142710i	$0.954418\pi$
$692$		0			0
$693$	2.70034			0.102577
$694$		0			0
$695$			16.3368i			0.619690i
$696$		0			0
$697$		−	34.8158i		−	1.31874i
$698$		0			0
$699$	−9.42309			−0.356414
$700$		0			0
$701$	−48.0673			−1.81548			−0.907738	−	0.419538i	$-0.862192\pi$
$701$	−48.0673			−1.81548			−0.907738	+	0.419538i	$0.862192\pi$
$702$		0			0
$703$	−8.91020	−	5.53361i	−0.336055	−	0.208704i
$704$		0			0
$705$		−	1.64369i		−	0.0619050i
$706$		0			0
$707$			0.374022i			0.0140666i
$708$		0			0
$709$	46.3703			1.74147			0.870736	−	0.491750i	$-0.163643\pi$
$709$	46.3703			1.74147			0.870736	+	0.491750i	$0.163643\pi$
$710$		0			0
$711$	−3.71155			−0.139194
$712$		0			0
$713$			1.34514i			0.0503760i
$714$		0			0
$715$	−2.70034			−0.100987
$716$		0			0
$717$		−	6.38620i		−	0.238497i
$718$		0			0
$719$		−	31.2800i		−	1.16655i	−0.812276	−	0.583273i	$-0.801772\pi$
$719$		−	31.2800i		−	1.16655i	0.812276	−	0.583273i	$-0.198228\pi$
$720$		0			0
$721$			8.00292i			0.298044i
$722$		0			0
$723$		−	1.34514i		−	0.0500264i
$724$		0			0
$725$			8.16840i			0.303367i
$726$		0			0
$727$		−	18.7231i		−	0.694401i	−0.937791	−	0.347201i	$-0.887132\pi$
$727$		−	18.7231i		−	0.694401i	0.937791	−	0.347201i	$-0.112868\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			10.6162i			0.392653i
$732$		0			0
$733$	−5.86201			−0.216518			−0.108259	−	0.994123i	$-0.534528\pi$
$733$	−5.86201			−0.216518			−0.108259	+	0.994123i	$0.534528\pi$
$734$		0			0
$735$	−6.61054			−0.243833
$736$		0			0
$737$		−	18.6531i		−	0.687094i
$738$		0			0
$739$			4.71554i			0.173464i	0.996232	+	0.0867319i	$0.0276424\pi$
$739$			4.71554i			0.173464i	−0.996232	+	0.0867319i	$0.972358\pi$
$740$		0			0
$741$	−2.31087	−	1.43515i	−0.0848920	−	0.0527216i
$742$		0			0
$743$	31.8720			1.16927			0.584634	−	0.811297i	$-0.301238\pi$
$743$	31.8720			1.16927			0.584634	+	0.811297i	$0.301238\pi$
$744$		0			0
$745$	−2.82376			−0.103455
$746$		0			0
$747$		−	4.23697i		−	0.155023i
$748$		0			0
$749$		−	4.81254i		−	0.175846i
$750$		0			0
$751$	47.9022			1.74798			0.873989	−	0.485947i	$-0.161525\pi$
$751$	47.9022			1.74798			0.873989	+	0.485947i	$0.161525\pi$
$752$		0			0
$753$		−	1.01962i		−	0.0371571i
$754$		0			0
$755$	−9.83497			−0.357931
$756$		0			0
$757$	−17.7980			−0.646879			−0.323439	−	0.946249i	$-0.604839\pi$
$757$	−17.7980			−0.646879			−0.323439	+	0.946249i	$0.604839\pi$
$758$		0			0
$759$	8.31087			0.301666
$760$		0			0
$761$	17.0482			0.617997			0.308998	−	0.951063i	$-0.400006\pi$
$761$	17.0482			0.617997			0.308998	+	0.951063i	$0.400006\pi$
$762$		0			0
$763$	−7.62511			−0.276047
$764$		0			0
$765$	−4.41188			−0.159512
$766$		0			0
$767$			6.25469i			0.225844i
$768$		0			0
$769$	−9.02578			−0.325478			−0.162739	−	0.986669i	$-0.552033\pi$
$769$	−9.02578			−0.325478			−0.162739	+	0.986669i	$0.552033\pi$
$770$		0			0
$771$		−	25.3863i		−	0.914266i
$772$		0			0
$773$		−	21.9189i		−	0.788369i	−0.919031	−	0.394184i	$-0.871027\pi$
$773$		−	21.9189i		−	0.788369i	0.919031	−	0.394184i	$-0.128973\pi$
$774$		0			0
$775$	−0.700336			−0.0251568
$776$		0			0
$777$	1.50168			0.0538725
$778$		0			0
$779$	−29.2211	−	18.1475i	−1.04695	−	0.650203i
$780$		0			0
$781$		−	43.3668i		−	1.55179i
$782$		0			0
$783$		−	8.16840i		−	0.291915i
$784$		0			0
$785$	−9.11222			−0.325229
$786$		0			0
$787$	−43.4039			−1.54718			−0.773591	−	0.633685i	$-0.781541\pi$
$787$	−43.4039			−1.54718			−0.773591	+	0.633685i	$0.781541\pi$
$788$		0			0
$789$			6.73325i			0.239710i
$790$		0			0
$791$	−4.37489			−0.155553
$792$		0			0
$793$		−	7.06577i		−	0.250913i
$794$		0			0
$795$		−	1.92071i		−	0.0681206i
$796$		0			0
$797$		−	44.0394i		−	1.55995i	−0.625808	−	0.779977i	$-0.715231\pi$
$797$		−	44.0394i		−	1.55995i	0.625808	−	0.779977i	$-0.284769\pi$
$798$		0			0
$799$		−	7.25177i		−	0.256549i
$800$		0			0
$801$			3.61290i			0.127656i
$802$		0			0
$803$		−	27.3070i		−	0.963644i
$804$		0			0
$805$	1.19866			0.0422471
$806$		0			0
$807$		−	12.2669i		−	0.431814i
$808$		0			0
$809$	21.8428			0.767953			0.383976	−	0.923343i	$-0.374554\pi$
$809$	21.8428			0.767953			0.383976	+	0.923343i	$0.374554\pi$
$810$		0			0
$811$	−2.84618			−0.0999429			−0.0499714	−	0.998751i	$-0.515913\pi$
$811$	−2.84618			−0.0999429			−0.0499714	+	0.998751i	$0.515913\pi$
$812$		0			0
$813$		−	23.1886i		−	0.813259i
$814$		0			0
$815$		−	14.3676i		−	0.503275i
$816$		0			0
$817$	8.91020	+	5.53361i	0.311728	+	0.193597i
$818$		0			0
$819$	0.389463			0.0136089
$820$		0			0
$821$	−30.6666			−1.07027			−0.535136	−	0.844766i	$-0.679739\pi$
$821$	−30.6666			−1.07027			−0.535136	+	0.844766i	$0.679739\pi$
$822$		0			0
$823$		−	15.3587i		−	0.535370i	−0.963506	−	0.267685i	$-0.913741\pi$
$823$		−	15.3587i		−	0.535370i	0.963506	−	0.267685i	$-0.0862587\pi$
$824$		0			0
$825$			4.32698i			0.150646i
$826$		0			0
$827$	−0.389463			−0.0135429			−0.00677147	−	0.999977i	$-0.502155\pi$
$827$	−0.389463			−0.0135429			−0.00677147	+	0.999977i	$0.502155\pi$
$828$		0			0
$829$		−	24.0196i		−	0.834237i	−0.908852	−	0.417118i	$-0.863040\pi$
$829$		−	24.0196i		−	0.834237i	0.908852	−	0.417118i	$-0.136960\pi$
$830$		0			0
$831$	3.40067			0.117968
$832$		0			0
$833$	−29.1649			−1.01050
$834$		0			0
$835$	20.4343			0.707158
$836$		0			0
$837$	0.700336			0.0242071
$838$		0			0
$839$	1.44551			0.0499045			0.0249522	−	0.999689i	$-0.492057\pi$
$839$	1.44551			0.0499045			0.0249522	+	0.999689i	$0.492057\pi$
$840$		0			0
$841$	−37.7228			−1.30078
$842$		0			0
$843$		−	12.4469i		−	0.428693i
$844$		0			0
$845$	12.6105			0.433816
$846$		0			0
$847$		−	4.81953i		−	0.165601i
$848$		0			0
$849$		−	25.3378i		−	0.869591i
$850$		0			0
$851$	4.62175			0.158431
$852$		0			0
$853$	−18.9966			−0.650433			−0.325216	−	0.945640i	$-0.605437\pi$
$853$	−18.9966			−0.650433			−0.325216	+	0.945640i	$0.605437\pi$
$854$		0			0
$855$	−2.29966	+	3.70291i	−0.0786469	+	0.126637i
$856$		0			0
$857$			36.5365i			1.24806i	0.781399	+	0.624032i	$0.214507\pi$
$857$			36.5365i			1.24806i	−0.781399	+	0.624032i	$0.785493\pi$
$858$		0			0
$859$			35.9610i			1.22697i	0.789705	+	0.613486i	$0.210233\pi$
$859$			35.9610i			1.22697i	−0.789705	+	0.613486i	$0.789767\pi$
$860$		0			0
$861$	4.92477			0.167836
$862$		0			0
$863$	14.5577			0.495551			0.247775	−	0.968818i	$-0.420301\pi$
$863$	14.5577			0.495551			0.247775	+	0.968818i	$0.420301\pi$
$864$		0			0
$865$		−	20.1997i		−	0.686812i
$866$		0			0
$867$	−2.46469			−0.0837054
$868$		0			0
$869$			16.0598i			0.544791i
$870$		0			0
$871$		−	2.69028i		−	0.0911568i
$872$		0			0
$873$			2.94033i			0.0995151i
$874$		0			0
$875$			0.624069i			0.0210974i
$876$		0			0
$877$		−	52.2078i		−	1.76293i	−0.472248	−	0.881466i	$-0.656557\pi$
$877$		−	52.2078i		−	1.76293i	0.472248	−	0.881466i	$-0.343443\pi$
$878$		0			0
$879$			7.28729i			0.245794i
$880$		0			0
$881$	5.46793			0.184219			0.0921096	−	0.995749i	$-0.470639\pi$
$881$	5.46793			0.184219			0.0921096	+	0.995749i	$0.470639\pi$
$882$		0			0
$883$			5.07658i			0.170840i	0.996345	+	0.0854202i	$0.0272233\pi$
$883$			5.07658i			0.170840i	−0.996345	+	0.0854202i	$0.972777\pi$
$884$		0			0
$885$	10.0224			0.336900
$886$		0			0
$887$	−3.42309			−0.114936			−0.0574681	−	0.998347i	$-0.518303\pi$
$887$	−3.42309			−0.114936			−0.0574681	+	0.998347i	$0.518303\pi$
$888$		0			0
$889$			2.69028i			0.0902292i
$890$		0			0
$891$		−	4.32698i		−	0.144959i
$892$		0			0
$893$	−6.08644	−	3.77994i	−0.203675	−	0.126491i
$894$		0			0
$895$	−1.19866			−0.0400667
$896$		0			0
$897$	1.19866			0.0400220
$898$		0			0
$899$		−	5.72062i		−	0.190793i
$900$		0			0
$901$		−	8.47394i		−	0.282308i
$902$		0			0
$903$	−1.50168			−0.0499728
$904$		0			0
$905$			8.65396i			0.287667i
$906$		0			0
$907$	51.7788			1.71929			0.859643	−	0.510895i	$-0.170686\pi$
$907$	51.7788			1.71929			0.859643	+	0.510895i	$0.170686\pi$
$908$		0			0
$909$	0.599328			0.0198785
$910$		0			0
$911$	22.0224			0.729635			0.364818	−	0.931079i	$-0.381131\pi$
$911$	22.0224			0.729635			0.364818	+	0.931079i	$0.381131\pi$
$912$		0			0
$913$	−18.3333			−0.606743
$914$		0			0
$915$	−11.3221			−0.374296
$916$		0			0
$917$	1.41524			0.0467353
$918$		0			0
$919$		−	16.1568i		−	0.532963i	−0.963840	−	0.266482i	$-0.914139\pi$
$919$		−	16.1568i		−	0.532963i	0.963840	−	0.266482i	$-0.0858611\pi$
$920$		0			0
$921$	−14.0224			−0.462054
$922$		0			0
$923$		−	6.25469i		−	0.205875i
$924$		0			0
$925$			2.40627i			0.0791177i
$926$		0			0
$927$	12.8238			0.421188
$928$		0			0
$929$	−12.8462			−0.421469			−0.210735	−	0.977543i	$-0.567586\pi$
$929$	−12.8462			−0.421469			−0.210735	+	0.977543i	$0.567586\pi$
$930$		0			0
$931$	−15.2020	+	24.4782i	−0.498226	+	0.802241i
$932$		0			0
$933$		−	17.2594i		−	0.565048i
$934$		0			0
$935$			19.0901i			0.624314i
$936$		0			0
$937$	17.8428			0.582900			0.291450	−	0.956586i	$-0.405862\pi$
$937$	17.8428			0.582900			0.291450	+	0.956586i	$0.405862\pi$
$938$		0			0
$939$	−15.1346			−0.493900
$940$		0			0
$941$		−	57.9499i		−	1.88911i	−0.328348	−	0.944557i	$-0.606492\pi$
$941$		−	57.9499i		−	1.88911i	0.328348	−	0.944557i	$-0.393508\pi$
$942$		0			0
$943$	15.1571			0.493582
$944$		0			0
$945$		−	0.624069i		−	0.0203010i
$946$		0			0
$947$			17.7035i			0.575286i	0.957738	+	0.287643i	$0.0928716\pi$
$947$			17.7035i			0.575286i	−0.957738	+	0.287643i	$0.907128\pi$
$948$		0			0
$949$		−	3.93842i		−	0.127847i
$950$		0			0
$951$			15.1102i			0.489981i
$952$		0			0
$953$		−	30.0188i		−	0.972405i	−0.873846	−	0.486203i	$-0.838382\pi$
$953$		−	30.0188i		−	0.972405i	0.873846	−	0.486203i	$-0.161618\pi$
$954$		0			0
$955$			16.5453i			0.535395i
$956$		0			0
$957$	−35.3445			−1.14253
$958$		0			0
$959$			2.30227i			0.0743442i
$960$		0			0
$961$	−30.5095			−0.984178
$962$		0			0
$963$	−7.71155			−0.248501
$964$		0			0
$965$			8.02989i			0.258491i
$966$		0			0
$967$		−	30.2474i		−	0.972689i	−0.873767	−	0.486345i	$-0.838330\pi$
$967$		−	30.2474i		−	0.972689i	0.873767	−	0.486345i	$-0.161670\pi$
$968$		0			0
$969$	−10.1458	+	16.3368i	−0.325931	+	0.524814i
$970$		0			0
$971$	−2.22443			−0.0713855			−0.0356927	−	0.999363i	$-0.511364\pi$
$971$	−2.22443			−0.0713855			−0.0356927	+	0.999363i	$0.511364\pi$
$972$		0			0
$973$	−10.1953			−0.326846
$974$		0			0
$975$			0.624069i			0.0199862i
$976$		0			0
$977$			6.12216i			0.195865i	0.995193	+	0.0979327i	$0.0312230\pi$
$977$			6.12216i			0.195865i	−0.995193	+	0.0979327i	$0.968777\pi$
$978$		0			0
$979$	15.6330			0.499631
$980$		0			0
$981$			12.2184i			0.390102i
$982$		0			0
$983$	−8.26142			−0.263498			−0.131749	−	0.991283i	$-0.542059\pi$
$983$	−8.26142			−0.263498			−0.131749	+	0.991283i	$0.542059\pi$
$984$		0			0
$985$	14.4343			0.459915
$986$		0			0
$987$	1.02578			0.0326508
$988$		0			0
$989$	−4.62175			−0.146963
$990$		0			0
$991$	−8.55449			−0.271742			−0.135871	−	0.990727i	$-0.543383\pi$
$991$	−8.55449			−0.271742			−0.135871	+	0.990727i	$0.543383\pi$
$992$		0			0
$993$	−16.8092			−0.533424
$994$		0			0
$995$			2.31626i			0.0734304i
$996$		0			0
$997$	−4.66335			−0.147690			−0.0738449	−	0.997270i	$-0.523527\pi$
$997$	−4.66335			−0.147690			−0.0738449	+	0.997270i	$0.523527\pi$
$998$		0			0
$999$		−	2.40627i		−	0.0761310i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.f.2431.4	yes	6
4.3	odd	2		4560.2.d.h.2431.3	yes	6
19.18	odd	2		4560.2.d.h.2431.4	yes	6
76.75	even	2	inner	4560.2.d.f.2431.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.f.2431.3	✓	6	76.75	even	2	inner
4560.2.d.f.2431.4	yes	6	1.1	even	1	trivial
4560.2.d.h.2431.3	yes	6	4.3	odd	2
4560.2.d.h.2431.4	yes	6	19.18	odd	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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +0.624069i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +0.624069i q^{7} +1.00000 q^{9} -4.32698i q^{11} -0.624069i q^{13} -1.00000 q^{15} -4.41188 q^{17} +(-2.29966 + 3.70291i) q^{19} -0.624069i q^{21} -1.92071i q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.16840i q^{29} -0.700336 q^{31} +4.32698i q^{33} +0.624069i q^{35} +2.40627i q^{37} +0.624069i q^{39} +7.89138i q^{41} -2.40627i q^{43} +1.00000 q^{45} +1.64369i q^{47} +6.61054 q^{49} +4.41188 q^{51} +1.92071i q^{53} -4.32698i q^{55} +(2.29966 - 3.70291i) q^{57} -10.0224 q^{59} +11.3221 q^{61} +0.624069i q^{63} -0.624069i q^{65} +4.31087 q^{67} +1.92071i q^{69} +10.0224 q^{71} +6.31087 q^{73} -1.00000 q^{75} +2.70034 q^{77} -3.71155 q^{79} +1.00000 q^{81} -4.23697i q^{83} -4.41188 q^{85} -8.16840i q^{87} +3.61290i q^{89} +0.389463 q^{91} +0.700336 q^{93} +(-2.29966 + 3.70291i) q^{95} +2.94033i q^{97} -4.32698i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{5} + 6 q^{9} - 6 q^{15} - 4 q^{17} - 6 q^{19} + 6 q^{25} - 6 q^{27} - 12 q^{31} + 6 q^{45} - 14 q^{49} + 4 q^{51} + 6 q^{57} + 16 q^{59} - 16 q^{61} - 20 q^{67} - 16 q^{71} - 8 q^{73} - 6 q^{75} + 24 q^{77} + 8 q^{79} + 6 q^{81} - 4 q^{85} + 56 q^{91} + 12 q^{93} - 6 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^5 + 6 * q^9 - 6 * q^15 - 4 * q^17 - 6 * q^19 + 6 * q^25 - 6 * q^27 - 12 * q^31 + 6 * q^45 - 14 * q^49 + 4 * q^51 + 6 * q^57 + 16 * q^59 - 16 * q^61 - 20 * q^67 - 16 * q^71 - 8 * q^73 - 6 * q^75 + 24 * q^77 + 8 * q^79 + 6 * q^81 - 4 * q^85 + 56 * q^91 + 12 * q^93 - 6 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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