LMFDB - Embedded newform 4560.2.d.l.2431.8

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 35x^{10} + 202x^{8} + 362x^{6} + 245x^{4} + 63x^{2} + 4 $ x^12 + 35x^10 + 202x^8 + 362x^6 + 245x^4 + 63*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2431.8
Root			$-1.19434i$ of defining polynomial
Character	$\chi$	$=$	4560.2431
Dual form			4560.2.d.l.2431.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.25182i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.25182i q^{7} +1.00000 q^{9} +0.630063i q^{11} -3.96496i q^{13} +1.00000 q^{15} -6.90373 q^{17} +(-2.11425 - 3.81182i) q^{19} +1.25182i q^{21} -6.49272i q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.83404i q^{29} -7.48802 q^{31} +0.630063i q^{33} +1.25182i q^{35} +9.57487i q^{37} -3.96496i q^{39} -7.74454i q^{41} -7.37089i q^{43} +1.00000 q^{45} -3.33490i q^{47} +5.43294 q^{49} -6.90373 q^{51} -9.95682i q^{53} +0.630063i q^{55} +(-2.11425 - 3.81182i) q^{57} -9.92095 q^{59} -12.4329 q^{61} +1.25182i q^{63} -3.96496i q^{65} -7.82681 q^{67} -6.49272i q^{69} +1.41571 q^{71} +7.64421 q^{73} +1.00000 q^{75} -0.788728 q^{77} -13.8485 q^{79} +1.00000 q^{81} +9.45397i q^{83} -6.90373 q^{85} +2.83404i q^{87} -4.52855i q^{89} +4.96343 q^{91} -7.48802 q^{93} +(-2.11425 - 3.81182i) q^{95} +3.90679i q^{97} +0.630063i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{3} + 12 q^{5} + 12 q^{9}+O(q^{10}) $ 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9	$ 12 q + 12 q^{3} + 12 q^{5} + 12 q^{9} + 12 q^{15} + 4 q^{17} + 12 q^{25} + 12 q^{27} - 12 q^{31} + 12 q^{45} - 28 q^{49} + 4 q^{51} + 52 q^{59} - 56 q^{61} - 32 q^{67} + 8 q^{71} + 32 q^{73} + 12 q^{75} + 24 q^{77} - 28 q^{79} + 12 q^{81} + 4 q^{85} + 32 q^{91} - 12 q^{93}+O(q^{100}) $ 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9 + 12 * q^15 + 4 * q^17 + 12 * q^25 + 12 * q^27 - 12 * q^31 + 12 * q^45 - 28 * q^49 + 4 * q^51 + 52 * q^59 - 56 * q^61 - 32 * q^67 + 8 * q^71 + 32 * q^73 + 12 * q^75 + 24 * q^77 - 28 * q^79 + 12 * q^81 + 4 * q^85 + 32 * q^91 - 12 * q^93

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
$7$			1.25182i			0.473145i	0.971614	+	0.236573i	$0.0760241\pi$
$7$			1.25182i			0.473145i	−0.971614	+	0.236573i	$0.923976\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			0.630063i			0.189971i	0.995479	+	0.0949855i	$0.0302805\pi$
$11$			0.630063i			0.189971i	−0.995479	+	0.0949855i	$0.969720\pi$
$12$		0			0
$13$		−	3.96496i		−	1.09968i	−0.835269	−	0.549841i	$-0.814688\pi$
$13$		−	3.96496i		−	1.09968i	0.835269	−	0.549841i	$-0.185312\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	−6.90373			−1.67440			−0.837200	−	0.546896i	$-0.815809\pi$
$17$	−6.90373			−1.67440			−0.837200	+	0.546896i	$0.815809\pi$
$18$		0			0
$19$	−2.11425	−	3.81182i	−0.485041	−	0.874491i
$19$	−2.11425	−	3.81182i	−0.485041	−	0.874491i
$20$		0			0
$21$			1.25182i			0.273170i
$22$		0			0
$23$		−	6.49272i		−	1.35382i	−0.736064	−	0.676912i	$-0.763318\pi$
$23$		−	6.49272i		−	1.35382i	0.736064	−	0.676912i	$-0.236682\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$			2.83404i			0.526268i	0.964759	+	0.263134i	$0.0847561\pi$
$29$			2.83404i			0.526268i	−0.964759	+	0.263134i	$0.915244\pi$
$30$		0			0
$31$	−7.48802			−1.34489			−0.672444	−	0.740148i	$-0.734755\pi$
$31$	−7.48802			−1.34489			−0.672444	+	0.740148i	$0.734755\pi$
$32$		0			0
$33$			0.630063i			0.109680i
$34$		0			0
$35$			1.25182i			0.211597i
$36$		0			0
$37$			9.57487i			1.57410i	0.616890	+	0.787049i	$0.288392\pi$
$37$			9.57487i			1.57410i	−0.616890	+	0.787049i	$0.711608\pi$
$38$		0			0
$39$		−	3.96496i		−	0.634902i
$40$		0			0
$41$		−	7.74454i		−	1.20949i	−0.796418	−	0.604747i	$-0.793274\pi$
$41$		−	7.74454i		−	1.20949i	0.796418	−	0.604747i	$-0.206726\pi$
$42$		0			0
$43$		−	7.37089i		−	1.12405i	−0.827120	−	0.562025i	$-0.810022\pi$
$43$		−	7.37089i		−	1.12405i	0.827120	−	0.562025i	$-0.189978\pi$
$44$		0			0
$45$	1.00000			0.149071
$46$		0			0
$47$		−	3.33490i		−	0.486445i	−0.969971	−	0.243222i	$-0.921796\pi$
$47$		−	3.33490i		−	0.486445i	0.969971	−	0.243222i	$-0.0782045\pi$
$48$		0			0
$49$	5.43294			0.776134
$50$		0			0
$51$	−6.90373			−0.966716
$52$		0			0
$53$		−	9.95682i		−	1.36767i	−0.729635	−	0.683837i	$-0.760310\pi$
$53$		−	9.95682i		−	1.36767i	0.729635	−	0.683837i	$-0.239690\pi$
$54$		0			0
$55$			0.630063i			0.0849576i
$56$		0			0
$57$	−2.11425	−	3.81182i	−0.280039	−	0.504888i
$58$		0			0
$59$	−9.92095			−1.29160			−0.645799	−	0.763507i	$-0.723476\pi$
$59$	−9.92095			−1.29160			−0.645799	+	0.763507i	$0.723476\pi$
$60$		0			0
$61$	−12.4329			−1.59187			−0.795937	−	0.605379i	$-0.793021\pi$
$61$	−12.4329			−1.59187			−0.795937	+	0.605379i	$0.793021\pi$
$62$		0			0
$63$			1.25182i			0.157715i
$64$		0			0
$65$		−	3.96496i		−	0.491793i
$66$		0			0
$67$	−7.82681			−0.956197			−0.478098	−	0.878306i	$-0.658674\pi$
$67$	−7.82681			−0.956197			−0.478098	+	0.878306i	$0.658674\pi$
$68$		0			0
$69$		−	6.49272i		−	0.781631i
$70$		0			0
$71$	1.41571			0.168014			0.0840072	−	0.996465i	$-0.473228\pi$
$71$	1.41571			0.168014			0.0840072	+	0.996465i	$0.473228\pi$
$72$		0			0
$73$	7.64421			0.894687			0.447343	−	0.894362i	$-0.352370\pi$
$73$	7.64421			0.894687			0.447343	+	0.894362i	$0.352370\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	−0.788728			−0.0898839
$78$		0			0
$79$	−13.8485			−1.55808			−0.779040	−	0.626974i	$-0.784293\pi$
$79$	−13.8485			−1.55808			−0.779040	+	0.626974i	$0.784293\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			9.45397i			1.03771i	0.854863	+	0.518854i	$0.173641\pi$
$83$			9.45397i			1.03771i	−0.854863	+	0.518854i	$0.826359\pi$
$84$		0			0
$85$	−6.90373			−0.748815
$86$		0			0
$87$			2.83404i			0.303841i
$88$		0			0
$89$		−	4.52855i		−	0.480026i	−0.970770	−	0.240013i	$-0.922848\pi$
$89$		−	4.52855i		−	0.480026i	0.970770	−	0.240013i	$-0.0771516\pi$
$90$		0			0
$91$	4.96343			0.520309
$92$		0			0
$93$	−7.48802			−0.776471
$94$		0			0
$95$	−2.11425	−	3.81182i	−0.216917	−	0.391084i
$96$		0			0
$97$			3.90679i			0.396674i	0.980134	+	0.198337i	$0.0635541\pi$
$97$			3.90679i			0.396674i	−0.980134	+	0.198337i	$0.936446\pi$
$98$		0			0
$99$			0.630063i			0.0633237i
$100$		0			0
$101$	14.7934			1.47200			0.736001	−	0.676981i	$-0.236712\pi$
$101$	14.7934			1.47200			0.736001	+	0.676981i	$0.236712\pi$
$102$		0			0
$103$	7.82681			0.771198			0.385599	−	0.922666i	$-0.373995\pi$
$103$	7.82681			0.771198			0.385599	+	0.922666i	$0.373995\pi$
$104$		0			0
$105$			1.25182i			0.122166i
$106$		0			0
$107$	−1.24252			−0.120119			−0.0600596	−	0.998195i	$-0.519129\pi$
$107$	−1.24252			−0.120119			−0.0600596	+	0.998195i	$0.519129\pi$
$108$		0			0
$109$		−	17.2068i		−	1.64811i	−0.566507	−	0.824057i	$-0.691706\pi$
$109$		−	17.2068i		−	1.64811i	0.566507	−	0.824057i	$-0.308294\pi$
$110$		0			0
$111$			9.57487i			0.908806i
$112$		0			0
$113$			9.29604i			0.874498i	0.899341	+	0.437249i	$0.144047\pi$
$113$			9.29604i			0.874498i	−0.899341	+	0.437249i	$0.855953\pi$
$114$		0			0
$115$		−	6.49272i		−	0.605449i
$116$		0			0
$117$		−	3.96496i		−	0.366561i
$118$		0			0
$119$		−	8.64226i		−	0.792235i
$120$		0			0
$121$	10.6030			0.963911
$122$		0			0
$123$		−	7.74454i		−	0.698302i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	13.4710			1.19536			0.597680	−	0.801735i	$-0.296089\pi$
$127$	13.4710			1.19536			0.597680	+	0.801735i	$0.296089\pi$
$128$		0			0
$129$		−	7.37089i		−	0.648971i
$130$		0			0
$131$		−	8.50842i		−	0.743385i	−0.928356	−	0.371692i	$-0.878778\pi$
$131$		−	8.50842i		−	0.743385i	0.928356	−	0.371692i	$-0.121222\pi$
$132$		0			0
$133$	4.77173	−	2.64667i	0.413761	−	0.229495i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	−0.608472			−0.0519853			−0.0259926	−	0.999662i	$-0.508275\pi$
$137$	−0.608472			−0.0519853			−0.0259926	+	0.999662i	$0.508275\pi$
$138$		0			0
$139$			15.1578i			1.28567i	0.766006	+	0.642834i	$0.222241\pi$
$139$			15.1578i			1.28567i	−0.766006	+	0.642834i	$0.777759\pi$
$140$		0			0
$141$		−	3.33490i		−	0.280849i
$142$		0			0
$143$	2.49817			0.208908
$144$		0			0
$145$			2.83404i			0.235354i
$146$		0			0
$147$	5.43294			0.448101
$148$		0			0
$149$	−15.5603			−1.27475			−0.637376	−	0.770553i	$-0.719980\pi$
$149$	−15.5603			−1.27475			−0.637376	+	0.770553i	$0.719980\pi$
$150$		0			0
$151$	6.31793			0.514146			0.257073	−	0.966392i	$-0.417242\pi$
$151$	6.31793			0.514146			0.257073	+	0.966392i	$0.417242\pi$
$152$		0			0
$153$	−6.90373			−0.558134
$154$		0			0
$155$	−7.48802			−0.601452
$156$		0			0
$157$	14.9069			1.18970			0.594851	−	0.803836i	$-0.297211\pi$
$157$	14.9069			1.18970			0.594851	+	0.803836i	$0.297211\pi$
$158$		0			0
$159$		−	9.95682i		−	0.789627i
$160$		0			0
$161$	8.12774			0.640556
$162$		0			0
$163$		−	11.8536i		−	0.928447i	−0.885718	−	0.464223i	$-0.846333\pi$
$163$		−	11.8536i		−	0.928447i	0.885718	−	0.464223i	$-0.153667\pi$
$164$		0			0
$165$			0.630063i			0.0490503i
$166$		0			0
$167$	−13.4742			−1.04267			−0.521333	−	0.853353i	$-0.674565\pi$
$167$	−13.4742			−1.04267			−0.521333	+	0.853353i	$0.674565\pi$
$168$		0			0
$169$	−2.72091			−0.209301
$170$		0			0
$171$	−2.11425	−	3.81182i	−0.161680	−	0.291497i
$172$		0			0
$173$		−	8.69008i		−	0.660695i	−0.943859	−	0.330347i	$-0.892834\pi$
$173$		−	8.69008i		−	0.660695i	0.943859	−	0.330347i	$-0.107166\pi$
$174$		0			0
$175$			1.25182i			0.0946290i
$176$		0			0
$177$	−9.92095			−0.745704
$178$		0			0
$179$	3.95895			0.295906			0.147953	−	0.988994i	$-0.452732\pi$
$179$	3.95895			0.295906			0.147953	+	0.988994i	$0.452732\pi$
$180$		0			0
$181$		−	11.1264i		−	0.827016i	−0.910500	−	0.413508i	$-0.864303\pi$
$181$		−	11.1264i		−	0.827016i	0.910500	−	0.413508i	$-0.135697\pi$
$182$		0			0
$183$	−12.4329			−0.919069
$184$		0			0
$185$			9.57487i			0.703958i
$186$		0			0
$187$		−	4.34978i		−	0.318088i
$188$		0			0
$189$			1.25182i			0.0910568i
$190$		0			0
$191$			9.75195i			0.705626i	0.935694	+	0.352813i	$0.114775\pi$
$191$			9.75195i			0.705626i	−0.935694	+	0.352813i	$0.885225\pi$
$192$		0			0
$193$			17.5048i			1.26002i	0.776586	+	0.630011i	$0.216950\pi$
$193$			17.5048i			1.26002i	−0.776586	+	0.630011i	$0.783050\pi$
$194$		0			0
$195$		−	3.96496i		−	0.283937i
$196$		0			0
$197$	21.6568			1.54298			0.771492	−	0.636239i	$-0.219511\pi$
$197$	21.6568			1.54298			0.771492	+	0.636239i	$0.219511\pi$
$198$		0			0
$199$			4.70101i			0.333246i	0.986021	+	0.166623i	$0.0532863\pi$
$199$			4.70101i			0.333246i	−0.986021	+	0.166623i	$0.946714\pi$
$200$		0			0
$201$	−7.82681			−0.552061
$202$		0			0
$203$	−3.54772			−0.249001
$204$		0			0
$205$		−	7.74454i		−	0.540902i
$206$		0			0
$207$		−	6.49272i		−	0.451275i
$208$		0			0
$209$	2.40168	−	1.33211i	0.166128	−	0.0921438i
$210$		0			0
$211$	−28.4183			−1.95640			−0.978198	−	0.207674i	$-0.933411\pi$
$211$	−28.4183			−1.95640			−0.978198	+	0.207674i	$0.933411\pi$
$212$		0			0
$213$	1.41571			0.0970032
$214$		0			0
$215$		−	7.37089i		−	0.502691i
$216$		0			0
$217$		−	9.37368i		−	0.636327i
$218$		0			0
$219$	7.64421			0.516548
$220$		0			0
$221$			27.3730i			1.84131i
$222$		0			0
$223$	0.509638			0.0341279			0.0170639	−	0.999854i	$-0.494568\pi$
$223$	0.509638			0.0341279			0.0170639	+	0.999854i	$0.494568\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−16.3056			−1.08224			−0.541122	−	0.840944i	$-0.682000\pi$
$227$	−16.3056			−1.08224			−0.541122	+	0.840944i	$0.682000\pi$
$228$		0			0
$229$	19.6375			1.29768			0.648840	−	0.760925i	$-0.275254\pi$
$229$	19.6375			1.29768			0.648840	+	0.760925i	$0.275254\pi$
$230$		0			0
$231$	−0.788728			−0.0518945
$232$		0			0
$233$	−17.0910			−1.11967			−0.559836	−	0.828604i	$-0.689136\pi$
$233$	−17.0910			−1.11967			−0.559836	+	0.828604i	$0.689136\pi$
$234$		0			0
$235$		−	3.33490i		−	0.217545i
$236$		0			0
$237$	−13.8485			−0.899558
$238$		0			0
$239$		−	19.0418i		−	1.23171i	−0.787860	−	0.615855i	$-0.788811\pi$
$239$		−	19.0418i		−	1.23171i	0.787860	−	0.615855i	$-0.211189\pi$
$240$		0			0
$241$			3.56019i			0.229332i	0.993404	+	0.114666i	$0.0365798\pi$
$241$			3.56019i			0.229332i	−0.993404	+	0.114666i	$0.963420\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	5.43294			0.347098
$246$		0			0
$247$	−15.1137	+	8.38290i	−0.961662	+	0.533391i
$248$		0			0
$249$			9.45397i			0.599121i
$250$		0			0
$251$			15.5397i			0.980860i	0.871481	+	0.490430i	$0.163160\pi$
$251$			15.5397i			0.980860i	−0.871481	+	0.490430i	$0.836840\pi$
$252$		0			0
$253$	4.09082			0.257187
$254$		0			0
$255$	−6.90373			−0.432328
$256$		0			0
$257$		−	1.13092i		−	0.0705449i	−0.999378	−	0.0352725i	$-0.988770\pi$
$257$		−	1.13092i		−	0.0705449i	0.999378	−	0.0352725i	$-0.0112299\pi$
$258$		0			0
$259$	−11.9861			−0.744777
$260$		0			0
$261$			2.83404i			0.175423i
$262$		0			0
$263$		−	4.33000i		−	0.266999i	−0.991049	−	0.133500i	$-0.957378\pi$
$263$		−	4.33000i		−	0.266999i	0.991049	−	0.133500i	$-0.0426215\pi$
$264$		0			0
$265$		−	9.95682i		−	0.611642i
$266$		0			0
$267$		−	4.52855i		−	0.277143i
$268$		0			0
$269$		−	2.64379i		−	0.161195i	−0.996747	−	0.0805974i	$-0.974317\pi$
$269$		−	2.64379i		−	0.161195i	0.996747	−	0.0805974i	$-0.0256828\pi$
$270$		0			0
$271$		−	25.2272i		−	1.53244i	−0.642576	−	0.766222i	$-0.722134\pi$
$271$		−	25.2272i		−	1.53244i	0.642576	−	0.766222i	$-0.277866\pi$
$272$		0			0
$273$	4.96343			0.300401
$274$		0			0
$275$			0.630063i			0.0379942i
$276$		0			0
$277$	23.6753			1.42251			0.711256	−	0.702933i	$-0.248127\pi$
$277$	23.6753			1.42251			0.711256	+	0.702933i	$0.248127\pi$
$278$		0			0
$279$	−7.48802			−0.448296
$280$		0			0
$281$		−	25.0714i		−	1.49563i	−0.663906	−	0.747816i	$-0.731103\pi$
$281$		−	25.0714i		−	1.49563i	0.663906	−	0.747816i	$-0.268897\pi$
$282$		0			0
$283$			10.4772i			0.622802i	0.950279	+	0.311401i	$0.100798\pi$
$283$			10.4772i			0.622802i	−0.950279	+	0.311401i	$0.899202\pi$
$284$		0			0
$285$	−2.11425	−	3.81182i	−0.125237	−	0.225793i
$286$		0			0
$287$	9.69480			0.572266
$288$		0			0
$289$	30.6615			1.80362
$290$		0			0
$291$			3.90679i			0.229020i
$292$		0			0
$293$		−	29.4541i		−	1.72073i	−0.509681	−	0.860364i	$-0.670237\pi$
$293$		−	29.4541i		−	1.72073i	0.509681	−	0.860364i	$-0.329763\pi$
$294$		0			0
$295$	−9.92095			−0.577620
$296$		0			0
$297$			0.630063i			0.0365599i
$298$		0			0
$299$	−25.7434			−1.48878
$300$		0			0
$301$	9.22706			0.531839
$302$		0			0
$303$	14.7934			0.849861
$304$		0			0
$305$	−12.4329			−0.711908
$306$		0			0
$307$	−30.3610			−1.73279			−0.866396	−	0.499358i	$-0.833569\pi$
$307$	−30.3610			−1.73279			−0.866396	+	0.499358i	$0.833569\pi$
$308$		0			0
$309$	7.82681			0.445252
$310$		0			0
$311$		−	19.7798i		−	1.12161i	−0.827948	−	0.560805i	$-0.810492\pi$
$311$		−	19.7798i		−	1.12161i	0.827948	−	0.560805i	$-0.189508\pi$
$312$		0			0
$313$	−21.1152			−1.19350			−0.596752	−	0.802426i	$-0.703542\pi$
$313$	−21.1152			−1.19350			−0.596752	+	0.802426i	$0.703542\pi$
$314$		0			0
$315$			1.25182i			0.0705323i
$316$		0			0
$317$		−	14.7057i		−	0.825955i	−0.910741	−	0.412977i	$-0.864489\pi$
$317$		−	14.7057i		−	0.825955i	0.910741	−	0.412977i	$-0.135511\pi$
$318$		0			0
$319$	−1.78562			−0.0999756
$320$		0			0
$321$	−1.24252			−0.0693509
$322$		0			0
$323$	14.5962	+	26.3158i	0.812154	+	1.46425i
$324$		0			0
$325$		−	3.96496i		−	0.219936i
$326$		0			0
$327$		−	17.2068i		−	0.951539i
$328$		0			0
$329$	4.17471			0.230159
$330$		0			0
$331$	12.8795			0.707924			0.353962	−	0.935260i	$-0.384834\pi$
$331$	12.8795			0.707924			0.353962	+	0.935260i	$0.384834\pi$
$332$		0			0
$333$			9.57487i			0.524700i
$334$		0			0
$335$	−7.82681			−0.427624
$336$		0			0
$337$			6.25842i			0.340918i	0.985365	+	0.170459i	$0.0545250\pi$
$337$			6.25842i			0.340918i	−0.985365	+	0.170459i	$0.945475\pi$
$338$		0			0
$339$			9.29604i			0.504891i
$340$		0			0
$341$		−	4.71792i		−	0.255490i
$342$		0			0
$343$			15.5639i			0.840369i
$344$		0			0
$345$		−	6.49272i		−	0.349556i
$346$		0			0
$347$			31.2171i			1.67582i	0.545808	+	0.837910i	$0.316223\pi$
$347$			31.2171i			1.67582i	−0.545808	+	0.837910i	$0.683777\pi$
$348$		0			0
$349$	11.8911			0.636515			0.318257	−	0.948004i	$-0.396902\pi$
$349$	11.8911			0.636515			0.318257	+	0.948004i	$0.396902\pi$
$350$		0			0
$351$		−	3.96496i		−	0.211634i
$352$		0			0
$353$	−30.7603			−1.63720			−0.818602	−	0.574361i	$-0.805251\pi$
$353$	−30.7603			−1.63720			−0.818602	+	0.574361i	$0.805251\pi$
$354$		0			0
$355$	1.41571			0.0751383
$356$		0			0
$357$		−	8.64226i		−	0.457397i
$358$		0			0
$359$		−	24.6285i		−	1.29984i	−0.760002	−	0.649920i	$-0.774802\pi$
$359$		−	24.6285i		−	1.29984i	0.760002	−	0.649920i	$-0.225198\pi$
$360$		0			0
$361$	−10.0599	+	16.1182i	−0.529470	+	0.848329i
$362$		0			0
$363$	10.6030			0.556514
$364$		0			0
$365$	7.64421			0.400116
$366$		0			0
$367$			9.12358i			0.476247i	0.971235	+	0.238123i	$0.0765323\pi$
$367$			9.12358i			0.476247i	−0.971235	+	0.238123i	$0.923468\pi$
$368$		0			0
$369$		−	7.74454i		−	0.403165i
$370$		0			0
$371$	12.4642			0.647108
$372$		0			0
$373$			26.8785i			1.39171i	0.718180	+	0.695857i	$0.244975\pi$
$373$			26.8785i			1.39171i	−0.718180	+	0.695857i	$0.755025\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	11.2369			0.578727
$378$		0			0
$379$	8.96192			0.460343			0.230171	−	0.973150i	$-0.426071\pi$
$379$	8.96192			0.460343			0.230171	+	0.973150i	$0.426071\pi$
$380$		0			0
$381$	13.4710			0.690141
$382$		0			0
$383$	−3.84395			−0.196416			−0.0982082	−	0.995166i	$-0.531311\pi$
$383$	−3.84395			−0.196416			−0.0982082	+	0.995166i	$0.531311\pi$
$384$		0			0
$385$	−0.788728			−0.0401973
$386$		0			0
$387$		−	7.37089i		−	0.374684i
$388$		0			0
$389$	−3.35048			−0.169876			−0.0849380	−	0.996386i	$-0.527069\pi$
$389$	−3.35048			−0.169876			−0.0849380	+	0.996386i	$0.527069\pi$
$390$		0			0
$391$			44.8240i			2.26685i
$392$		0			0
$393$		−	8.50842i		−	0.429193i
$394$		0			0
$395$	−13.8485			−0.696794
$396$		0			0
$397$	3.42125			0.171708			0.0858538	−	0.996308i	$-0.472638\pi$
$397$	3.42125			0.171708			0.0858538	+	0.996308i	$0.472638\pi$
$398$		0			0
$399$	4.77173	−	2.64667i	0.238885	−	0.132499i
$400$		0			0
$401$			30.4328i			1.51974i	0.650073	+	0.759872i	$0.274738\pi$
$401$			30.4328i			1.51974i	−0.650073	+	0.759872i	$0.725262\pi$
$402$		0			0
$403$			29.6897i			1.47895i
$404$		0			0
$405$	1.00000			0.0496904
$406$		0			0
$407$	−6.03277			−0.299033
$408$		0			0
$409$			14.1192i			0.698151i	0.937095	+	0.349075i	$0.113504\pi$
$409$			14.1192i			0.698151i	−0.937095	+	0.349075i	$0.886496\pi$
$410$		0			0
$411$	−0.608472			−0.0300137
$412$		0			0
$413$		−	12.4193i		−	0.611113i
$414$		0			0
$415$			9.45397i			0.464077i
$416$		0			0
$417$			15.1578i			0.742280i
$418$		0			0
$419$		−	19.0252i		−	0.929440i	−0.885458	−	0.464720i	$-0.846155\pi$
$419$		−	19.0252i		−	0.929440i	0.885458	−	0.464720i	$-0.153845\pi$
$420$		0			0
$421$			11.9451i			0.582168i	0.956698	+	0.291084i	$0.0940159\pi$
$421$			11.9451i			0.582168i	−0.956698	+	0.291084i	$0.905984\pi$
$422$		0			0
$423$		−	3.33490i		−	0.162148i
$424$		0			0
$425$	−6.90373			−0.334880
$426$		0			0
$427$		−	15.5639i		−	0.753188i
$428$		0			0
$429$	2.49817			0.120613
$430$		0			0
$431$	2.95197			0.142191			0.0710957	−	0.997470i	$-0.477350\pi$
$431$	2.95197			0.142191			0.0710957	+	0.997470i	$0.477350\pi$
$432$		0			0
$433$			29.6192i			1.42341i	0.702478	+	0.711705i	$0.252077\pi$
$433$			29.6192i			1.42341i	−0.702478	+	0.711705i	$0.747923\pi$
$434$		0			0
$435$			2.83404i			0.135882i
$436$		0			0
$437$	−24.7491	+	13.7272i	−1.18391	+	0.656661i
$438$		0			0
$439$	−37.6919			−1.79894			−0.899469	−	0.436985i	$-0.856046\pi$
$439$	−37.6919			−1.79894			−0.899469	+	0.436985i	$0.856046\pi$
$440$		0			0
$441$	5.43294			0.258711
$442$		0			0
$443$		−	25.8140i		−	1.22646i	−0.789904	−	0.613231i	$-0.789870\pi$
$443$		−	25.8140i		−	1.22646i	0.789904	−	0.613231i	$-0.210130\pi$
$444$		0			0
$445$		−	4.52855i		−	0.214674i
$446$		0			0
$447$	−15.5603			−0.735978
$448$		0			0
$449$		−	11.2815i		−	0.532408i	−0.963917	−	0.266204i	$-0.914231\pi$
$449$		−	11.2815i		−	0.532408i	0.963917	−	0.266204i	$-0.0857694\pi$
$450$		0			0
$451$	4.87955			0.229769
$452$		0			0
$453$	6.31793			0.296842
$454$		0			0
$455$	4.96343			0.232689
$456$		0			0
$457$	24.7642			1.15842			0.579209	−	0.815179i	$-0.303361\pi$
$457$	24.7642			1.15842			0.579209	+	0.815179i	$0.303361\pi$
$458$		0			0
$459$	−6.90373			−0.322239
$460$		0			0
$461$	−17.7142			−0.825031			−0.412515	−	0.910951i	$-0.635350\pi$
$461$	−17.7142			−0.825031			−0.412515	+	0.910951i	$0.635350\pi$
$462$		0			0
$463$		−	0.191937i		−	0.00892008i	−0.999990	−	0.00446004i	$-0.998580\pi$
$463$		−	0.191937i		−	0.00892008i	0.999990	−	0.00446004i	$-0.00141968\pi$
$464$		0			0
$465$	−7.48802			−0.347248
$466$		0			0
$467$		−	31.4589i		−	1.45574i	−0.685713	−	0.727872i	$-0.740510\pi$
$467$		−	31.4589i		−	1.45574i	0.685713	−	0.727872i	$-0.259490\pi$
$468$		0			0
$469$		−	9.79779i		−	0.452420i
$470$		0			0
$471$	14.9069			0.686875
$472$		0			0
$473$	4.64412			0.213537
$474$		0			0
$475$	−2.11425	−	3.81182i	−0.0970083	−	0.174898i
$476$		0			0
$477$		−	9.95682i		−	0.455891i
$478$		0			0
$479$			23.7854i			1.08678i	0.839480	+	0.543391i	$0.182860\pi$
$479$			23.7854i			1.08678i	−0.839480	+	0.543391i	$0.817140\pi$
$480$		0			0
$481$	37.9640			1.73101
$482$		0			0
$483$	8.12774			0.369825
$484$		0			0
$485$			3.90679i			0.177398i
$486$		0			0
$487$	4.55349			0.206338			0.103169	−	0.994664i	$-0.467102\pi$
$487$	4.55349			0.206338			0.103169	+	0.994664i	$0.467102\pi$
$488$		0			0
$489$		−	11.8536i		−	0.536039i
$490$		0			0
$491$		−	36.5290i		−	1.64853i	−0.566204	−	0.824265i	$-0.691588\pi$
$491$		−	36.5290i		−	1.64853i	0.566204	−	0.824265i	$-0.308412\pi$
$492$		0			0
$493$		−	19.5654i		−	0.881183i
$494$		0			0
$495$			0.630063i			0.0283192i
$496$		0			0
$497$			1.77223i			0.0794952i
$498$		0			0
$499$			0.0346536i			0.00155131i	1.00000		0.000775653i	$0.000246898\pi$
$499$			0.0346536i			0.00155131i	−1.00000		0.000775653i	$0.999753\pi$
$500$		0			0
$501$	−13.4742			−0.601983
$502$		0			0
$503$		−	14.6538i		−	0.653383i	−0.945131	−	0.326691i	$-0.894066\pi$
$503$		−	14.6538i		−	0.653383i	0.945131	−	0.326691i	$-0.105934\pi$
$504$		0			0
$505$	14.7934			0.658299
$506$		0			0
$507$	−2.72091			−0.120840
$508$		0			0
$509$		−	34.1830i		−	1.51513i	−0.652758	−	0.757567i	$-0.726388\pi$
$509$		−	34.1830i		−	1.51513i	0.652758	−	0.757567i	$-0.273612\pi$
$510$		0			0
$511$			9.56921i			0.423317i
$512$		0			0
$513$	−2.11425	−	3.81182i	−0.0933463	−	0.168296i
$514$		0			0
$515$	7.82681			0.344890
$516$		0			0
$517$	2.10119			0.0924104
$518$		0			0
$519$		−	8.69008i		−	0.381452i
$520$		0			0
$521$			21.6319i			0.947712i	0.880602	+	0.473856i	$0.157138\pi$
$521$			21.6319i			0.947712i	−0.880602	+	0.473856i	$0.842862\pi$
$522$		0			0
$523$	−26.3171			−1.15077			−0.575383	−	0.817884i	$-0.695147\pi$
$523$	−26.3171			−1.15077			−0.575383	+	0.817884i	$0.695147\pi$
$524$		0			0
$525$			1.25182i			0.0546341i
$526$		0			0
$527$	51.6953			2.25188
$528$		0			0
$529$	−19.1554			−0.832842
$530$		0			0
$531$	−9.92095			−0.430533
$532$		0			0
$533$	−30.7068			−1.33006
$534$		0			0
$535$	−1.24252			−0.0537190
$536$		0			0
$537$	3.95895			0.170841
$538$		0			0
$539$			3.42309i			0.147443i
$540$		0			0
$541$	16.6108			0.714155			0.357078	−	0.934075i	$-0.383773\pi$
$541$	16.6108			0.714155			0.357078	+	0.934075i	$0.383773\pi$
$542$		0			0
$543$		−	11.1264i		−	0.477478i
$544$		0			0
$545$		−	17.2068i		−	0.737059i
$546$		0			0
$547$	37.7975			1.61611			0.808053	−	0.589110i	$-0.200522\pi$
$547$	37.7975			1.61611			0.808053	+	0.589110i	$0.200522\pi$
$548$		0			0
$549$	−12.4329			−0.530625
$550$		0			0
$551$	10.8028	−	5.99186i	0.460217	−	0.255262i
$552$		0			0
$553$		−	17.3359i		−	0.737198i
$554$		0			0
$555$			9.57487i			0.406431i
$556$		0			0
$557$	9.58935			0.406314			0.203157	−	0.979146i	$-0.434880\pi$
$557$	9.58935			0.406314			0.203157	+	0.979146i	$0.434880\pi$
$558$		0			0
$559$	−29.2253			−1.23610
$560$		0			0
$561$		−	4.34978i		−	0.183648i
$562$		0			0
$563$	46.0307			1.93996			0.969982	−	0.243177i	$-0.0781896\pi$
$563$	46.0307			1.93996			0.969982	+	0.243177i	$0.0781896\pi$
$564$		0			0
$565$			9.29604i			0.391087i
$566$		0			0
$567$			1.25182i			0.0525717i
$568$		0			0
$569$			23.2020i			0.972679i	0.873770	+	0.486339i	$0.161668\pi$
$569$			23.2020i			0.972679i	−0.873770	+	0.486339i	$0.838332\pi$
$570$		0			0
$571$			20.4551i			0.856020i	0.903774	+	0.428010i	$0.140785\pi$
$571$			20.4551i			0.856020i	−0.903774	+	0.428010i	$0.859215\pi$
$572$		0			0
$573$			9.75195i			0.407393i
$574$		0			0
$575$		−	6.49272i		−	0.270765i
$576$		0			0
$577$	−32.0188			−1.33296			−0.666480	−	0.745522i	$-0.732200\pi$
$577$	−32.0188			−1.33296			−0.666480	+	0.745522i	$0.732200\pi$
$578$		0			0
$579$			17.5048i			0.727474i
$580$		0			0
$581$	−11.8347			−0.490986
$582$		0			0
$583$	6.27342			0.259818
$584$		0			0
$585$		−	3.96496i		−	0.163931i
$586$		0			0
$587$		−	13.9566i		−	0.576052i	−0.957623	−	0.288026i	$-0.907001\pi$
$587$		−	13.9566i		−	0.576052i	0.957623	−	0.288026i	$-0.0929990\pi$
$588$		0			0
$589$	15.8315	+	28.5430i	0.652326	+	1.17609i
$590$		0			0
$591$	21.6568			0.890842
$592$		0			0
$593$	30.2855			1.24367			0.621837	−	0.783146i	$-0.286386\pi$
$593$	30.2855			1.24367			0.621837	+	0.783146i	$0.286386\pi$
$594$		0			0
$595$		−	8.64226i		−	0.354298i
$596$		0			0
$597$			4.70101i			0.192400i
$598$		0			0
$599$	−35.4402			−1.44805			−0.724024	−	0.689775i	$-0.757710\pi$
$599$	−35.4402			−1.44805			−0.724024	+	0.689775i	$0.757710\pi$
$600$		0			0
$601$			13.2505i			0.540497i	0.962791	+	0.270249i	$0.0871059\pi$
$601$			13.2505i			0.540497i	−0.962791	+	0.270249i	$0.912894\pi$
$602$		0			0
$603$	−7.82681			−0.318732
$604$		0			0
$605$	10.6030			0.431074
$606$		0			0
$607$	−28.2364			−1.14608			−0.573040	−	0.819528i	$-0.694236\pi$
$607$	−28.2364			−1.14608			−0.573040	+	0.819528i	$0.694236\pi$
$608$		0			0
$609$	−3.54772			−0.143761
$610$		0			0
$611$	−13.2227			−0.534935
$612$		0			0
$613$	15.1176			0.610593			0.305297	−	0.952257i	$-0.401244\pi$
$613$	15.1176			0.610593			0.305297	+	0.952257i	$0.401244\pi$
$614$		0			0
$615$		−	7.74454i		−	0.312290i
$616$		0			0
$617$	−5.32628			−0.214428			−0.107214	−	0.994236i	$-0.534193\pi$
$617$	−5.32628			−0.214428			−0.107214	+	0.994236i	$0.534193\pi$
$618$		0			0
$619$		−	37.8063i		−	1.51956i	−0.650178	−	0.759782i	$-0.725306\pi$
$619$		−	37.8063i		−	1.51956i	0.650178	−	0.759782i	$-0.274694\pi$
$620$		0			0
$621$		−	6.49272i		−	0.260544i
$622$		0			0
$623$	5.66895			0.227122
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	2.40168	−	1.33211i	0.0959140	−	0.0531993i
$628$		0			0
$629$		−	66.1023i		−	2.63567i
$630$		0			0
$631$			6.40333i			0.254912i	0.991844	+	0.127456i	$0.0406812\pi$
$631$			6.40333i			0.254912i	−0.991844	+	0.127456i	$0.959319\pi$
$632$		0			0
$633$	−28.4183			−1.12953
$634$		0			0
$635$	13.4710			0.534581
$636$		0			0
$637$		−	21.5414i		−	0.853500i
$638$		0			0
$639$	1.41571			0.0560048
$640$		0			0
$641$		−	33.4809i		−	1.32242i	−0.750203	−	0.661208i	$-0.770044\pi$
$641$		−	33.4809i		−	1.32242i	0.750203	−	0.661208i	$-0.229956\pi$
$642$		0			0
$643$			24.5225i			0.967072i	0.875325	+	0.483536i	$0.160648\pi$
$643$			24.5225i			0.967072i	−0.875325	+	0.483536i	$0.839352\pi$
$644$		0			0
$645$		−	7.37089i		−	0.290229i
$646$		0			0
$647$			41.4957i			1.63136i	0.578501	+	0.815682i	$0.303638\pi$
$647$			41.4957i			1.63136i	−0.578501	+	0.815682i	$0.696362\pi$
$648$		0			0
$649$		−	6.25082i		−	0.245366i
$650$		0			0
$651$		−	9.37368i		−	0.367384i
$652$		0			0
$653$	6.80018			0.266112			0.133056	−	0.991109i	$-0.457521\pi$
$653$	6.80018			0.266112			0.133056	+	0.991109i	$0.457521\pi$
$654$		0			0
$655$		−	8.50842i		−	0.332452i
$656$		0			0
$657$	7.64421			0.298229
$658$		0			0
$659$	−36.8583			−1.43580			−0.717898	−	0.696149i	$-0.754895\pi$
$659$	−36.8583			−1.43580			−0.717898	+	0.696149i	$0.754895\pi$
$660$		0			0
$661$			6.97211i			0.271183i	0.990765	+	0.135592i	$0.0432936\pi$
$661$			6.97211i			0.271183i	−0.990765	+	0.135592i	$0.956706\pi$
$662$		0			0
$663$			27.3730i			1.06308i
$664$		0			0
$665$	4.77173	−	2.64667i	0.185040	−	0.102633i
$666$		0			0
$667$	18.4006			0.712474
$668$		0			0
$669$	0.509638			0.0197037
$670$		0			0
$671$		−	7.83353i		−	0.302410i
$672$		0			0
$673$		−	16.1423i		−	0.622239i	−0.950371	−	0.311119i	$-0.899296\pi$
$673$		−	16.1423i		−	0.622239i	0.950371	−	0.311119i	$-0.100704\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$			34.9941i			1.34493i	0.740128	+	0.672466i	$0.234765\pi$
$677$			34.9941i			1.34493i	−0.740128	+	0.672466i	$0.765235\pi$
$678$		0			0
$679$	−4.89062			−0.187685
$680$		0			0
$681$	−16.3056			−0.624833
$682$		0			0
$683$	22.5170			0.861589			0.430795	−	0.902450i	$-0.358233\pi$
$683$	22.5170			0.861589			0.430795	+	0.902450i	$0.358233\pi$
$684$		0			0
$685$	−0.608472			−0.0232485
$686$		0			0
$687$	19.6375			0.749216
$688$		0			0
$689$	−39.4784			−1.50401
$690$		0			0
$691$			39.5236i			1.50355i	0.659420	+	0.751775i	$0.270802\pi$
$691$			39.5236i			1.50355i	−0.659420	+	0.751775i	$0.729198\pi$
$692$		0			0
$693$	−0.788728			−0.0299613
$694$		0			0
$695$			15.1578i			0.574968i
$696$		0			0
$697$			53.4662i			2.02518i
$698$		0			0
$699$	−17.0910			−0.646442
$700$		0			0
$701$	33.7767			1.27573			0.637864	−	0.770149i	$-0.279818\pi$
$701$	33.7767			1.27573			0.637864	+	0.770149i	$0.279818\pi$
$702$		0			0
$703$	36.4977	−	20.2436i	1.37654	−	0.763503i
$704$		0			0
$705$		−	3.33490i		−	0.125600i
$706$		0			0
$707$			18.5188i			0.696470i
$708$		0			0
$709$	1.19262			0.0447898			0.0223949	−	0.999749i	$-0.492871\pi$
$709$	1.19262			0.0447898			0.0223949	+	0.999749i	$0.492871\pi$
$710$		0			0
$711$	−13.8485			−0.519360
$712$		0			0
$713$			48.6176i			1.82074i
$714$		0			0
$715$	2.49817			0.0934264
$716$		0			0
$717$		−	19.0418i		−	0.711128i
$718$		0			0
$719$		−	40.1949i		−	1.49902i	−0.661994	−	0.749509i	$-0.730290\pi$
$719$		−	40.1949i		−	1.49902i	0.661994	−	0.749509i	$-0.269710\pi$
$720$		0			0
$721$			9.79779i			0.364889i
$722$		0			0
$723$			3.56019i			0.132405i
$724$		0			0
$725$			2.83404i			0.105254i
$726$		0			0
$727$			2.40487i			0.0891918i	0.999005	+	0.0445959i	$0.0142000\pi$
$727$			2.40487i			0.0891918i	−0.999005	+	0.0445959i	$0.985800\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			50.8867i			1.88211i
$732$		0			0
$733$	−17.5401			−0.647859			−0.323930	−	0.946081i	$-0.605004\pi$
$733$	−17.5401			−0.647859			−0.323930	+	0.946081i	$0.605004\pi$
$734$		0			0
$735$	5.43294			0.200397
$736$		0			0
$737$		−	4.93138i		−	0.181650i
$738$		0			0
$739$		−	16.4628i		−	0.605592i	−0.953055	−	0.302796i	$-0.902080\pi$
$739$		−	16.4628i		−	0.605592i	0.953055	−	0.302796i	$-0.0979201\pi$
$740$		0			0
$741$	−15.1137	+	8.38290i	−0.555216	+	0.307954i
$742$		0			0
$743$	37.3965			1.37195			0.685973	−	0.727627i	$-0.259377\pi$
$743$	37.3965			1.37195			0.685973	+	0.727627i	$0.259377\pi$
$744$		0			0
$745$	−15.5603			−0.570086
$746$		0			0
$747$			9.45397i			0.345903i
$748$		0			0
$749$		−	1.55542i		−	0.0568339i
$750$		0			0
$751$	33.2116			1.21191			0.605954	−	0.795500i	$-0.292792\pi$
$751$	33.2116			1.21191			0.605954	+	0.795500i	$0.292792\pi$
$752$		0			0
$753$			15.5397i			0.566300i
$754$		0			0
$755$	6.31793			0.229933
$756$		0			0
$757$	−12.3936			−0.450452			−0.225226	−	0.974307i	$-0.572312\pi$
$757$	−12.3936			−0.450452			−0.225226	+	0.974307i	$0.572312\pi$
$758$		0			0
$759$	4.09082			0.148487
$760$		0			0
$761$	−20.0173			−0.725627			−0.362813	−	0.931862i	$-0.618184\pi$
$761$	−20.0173			−0.725627			−0.362813	+	0.931862i	$0.618184\pi$
$762$		0			0
$763$	21.5399			0.779797
$764$		0			0
$765$	−6.90373			−0.249605
$766$		0			0
$767$			39.3362i			1.42035i
$768$		0			0
$769$	43.6864			1.57537			0.787686	−	0.616077i	$-0.211279\pi$
$769$	43.6864			1.57537			0.787686	+	0.616077i	$0.211279\pi$
$770$		0			0
$771$		−	1.13092i		−	0.0407291i
$772$		0			0
$773$			32.6219i			1.17333i	0.809830	+	0.586664i	$0.199559\pi$
$773$			32.6219i			1.17333i	−0.809830	+	0.586664i	$0.800441\pi$
$774$		0			0
$775$	−7.48802			−0.268978
$776$		0			0
$777$	−11.9861			−0.429997
$778$		0			0
$779$	−29.5208	+	16.3739i	−1.05769	+	0.586655i
$780$		0			0
$781$			0.891989i			0.0319179i
$782$		0			0
$783$			2.83404i			0.101280i
$784$		0			0
$785$	14.9069			0.532051
$786$		0			0
$787$	44.0146			1.56895			0.784475	−	0.620160i	$-0.212932\pi$
$787$	44.0146			1.56895			0.784475	+	0.620160i	$0.212932\pi$
$788$		0			0
$789$		−	4.33000i		−	0.154152i
$790$		0			0
$791$	−11.6370			−0.413764
$792$		0			0
$793$			49.2961i			1.75056i
$794$		0			0
$795$		−	9.95682i		−	0.353132i
$796$		0			0
$797$		−	46.1194i		−	1.63363i	−0.576896	−	0.816817i	$-0.695736\pi$
$797$		−	46.1194i		−	1.63363i	0.576896	−	0.816817i	$-0.304264\pi$
$798$		0			0
$799$			23.0232i			0.814504i
$800$		0			0
$801$		−	4.52855i		−	0.160009i
$802$		0			0
$803$			4.81633i			0.169965i
$804$		0			0
$805$	8.12774			0.286465
$806$		0			0
$807$		−	2.64379i		−	0.0930659i
$808$		0			0
$809$	−15.5769			−0.547656			−0.273828	−	0.961779i	$-0.588290\pi$
$809$	−15.5769			−0.547656			−0.273828	+	0.961779i	$0.588290\pi$
$810$		0			0
$811$	−6.69944			−0.235249			−0.117625	−	0.993058i	$-0.537528\pi$
$811$	−6.69944			−0.235249			−0.117625	+	0.993058i	$0.537528\pi$
$812$		0			0
$813$		−	25.2272i		−	0.884757i
$814$		0			0
$815$		−	11.8536i		−	0.415214i
$816$		0			0
$817$	−28.0965	+	15.5839i	−0.982972	+	0.545211i
$818$		0			0
$819$	4.96343			0.173436
$820$		0			0
$821$	−50.1444			−1.75005			−0.875025	−	0.484078i	$-0.839155\pi$
$821$	−50.1444			−1.75005			−0.875025	+	0.484078i	$0.839155\pi$
$822$		0			0
$823$			22.6443i			0.789331i	0.918825	+	0.394666i	$0.129140\pi$
$823$			22.6443i			0.789331i	−0.918825	+	0.394666i	$0.870860\pi$
$824$		0			0
$825$			0.630063i			0.0219360i
$826$		0			0
$827$	9.28426			0.322845			0.161423	−	0.986885i	$-0.448392\pi$
$827$	9.28426			0.322845			0.161423	+	0.986885i	$0.448392\pi$
$828$		0			0
$829$			5.47550i			0.190172i	0.995469	+	0.0950859i	$0.0303126\pi$
$829$			5.47550i			0.190172i	−0.995469	+	0.0950859i	$0.969687\pi$
$830$		0			0
$831$	23.6753			0.821288
$832$		0			0
$833$	−37.5075			−1.29956
$834$		0			0
$835$	−13.4742			−0.466294
$836$		0			0
$837$	−7.48802			−0.258824
$838$		0			0
$839$	55.8731			1.92896			0.964478	−	0.264165i	$-0.0850963\pi$
$839$	55.8731			1.92896			0.964478	+	0.264165i	$0.0850963\pi$
$840$		0			0
$841$	20.9682			0.723042
$842$		0			0
$843$		−	25.0714i		−	0.863503i
$844$		0			0
$845$	−2.72091			−0.0936022
$846$		0			0
$847$			13.2731i			0.456070i
$848$		0			0
$849$			10.4772i			0.359575i
$850$		0			0
$851$	62.1669			2.13105
$852$		0			0
$853$	23.6753			0.810628			0.405314	−	0.914178i	$-0.367162\pi$
$853$	23.6753			0.810628			0.405314	+	0.914178i	$0.367162\pi$
$854$		0			0
$855$	−2.11425	−	3.81182i	−0.0723057	−	0.130361i
$856$		0			0
$857$			22.8312i			0.779898i	0.920836	+	0.389949i	$0.127507\pi$
$857$			22.8312i			0.779898i	−0.920836	+	0.389949i	$0.872493\pi$
$858$		0			0
$859$		−	35.8305i		−	1.22252i	−0.791430	−	0.611260i	$-0.790663\pi$
$859$		−	35.8305i		−	1.22252i	0.791430	−	0.611260i	$-0.209337\pi$
$860$		0			0
$861$	9.69480			0.330398
$862$		0			0
$863$	−0.528289			−0.0179832			−0.00899159	−	0.999960i	$-0.502862\pi$
$863$	−0.528289			−0.0179832			−0.00899159	+	0.999960i	$0.502862\pi$
$864$		0			0
$865$		−	8.69008i		−	0.295472i
$866$		0			0
$867$	30.6615			1.04132
$868$		0			0
$869$		−	8.72543i		−	0.295990i
$870$		0			0
$871$			31.0330i			1.05151i
$872$		0			0
$873$			3.90679i			0.132225i
$874$		0			0
$875$			1.25182i			0.0423194i
$876$		0			0
$877$		−	40.2630i		−	1.35959i	−0.733404	−	0.679793i	$-0.762070\pi$
$877$		−	40.2630i		−	1.35959i	0.733404	−	0.679793i	$-0.237930\pi$
$878$		0			0
$879$		−	29.4541i		−	0.993462i
$880$		0			0
$881$	−28.3270			−0.954359			−0.477180	−	0.878806i	$-0.658341\pi$
$881$	−28.3270			−0.954359			−0.477180	+	0.878806i	$0.658341\pi$
$882$		0			0
$883$			43.9458i			1.47890i	0.673214	+	0.739448i	$0.264913\pi$
$883$			43.9458i			1.47890i	−0.673214	+	0.739448i	$0.735087\pi$
$884$		0			0
$885$	−9.92095			−0.333489
$886$		0			0
$887$	−47.8147			−1.60546			−0.802730	−	0.596343i	$-0.796620\pi$
$887$	−47.8147			−1.60546			−0.802730	+	0.596343i	$0.796620\pi$
$888$		0			0
$889$			16.8633i			0.565579i
$890$		0			0
$891$			0.630063i			0.0211079i
$892$		0			0
$893$	−12.7120	+	7.05080i	−0.425392	+	0.235946i
$894$		0			0
$895$	3.95895			0.132333
$896$		0			0
$897$	−25.7434			−0.859546
$898$		0			0
$899$		−	21.2213i		−	0.707771i
$900$		0			0
$901$			68.7392i			2.29003i
$902$		0			0
$903$	9.22706			0.307057
$904$		0			0
$905$		−	11.1264i		−	0.369853i
$906$		0			0
$907$	33.3042			1.10585			0.552924	−	0.833232i	$-0.313512\pi$
$907$	33.3042			1.10585			0.552924	+	0.833232i	$0.313512\pi$
$908$		0			0
$909$	14.7934			0.490667
$910$		0			0
$911$	10.9036			0.361252			0.180626	−	0.983552i	$-0.442188\pi$
$911$	10.9036			0.361252			0.180626	+	0.983552i	$0.442188\pi$
$912$		0			0
$913$	−5.95659			−0.197134
$914$		0			0
$915$	−12.4329			−0.411020
$916$		0			0
$917$	10.6511			0.351729
$918$		0			0
$919$		−	27.0384i		−	0.891914i	−0.895054	−	0.445957i	$-0.852863\pi$
$919$		−	27.0384i		−	0.891914i	0.895054	−	0.445957i	$-0.147137\pi$
$920$		0			0
$921$	−30.3610			−1.00043
$922$		0			0
$923$		−	5.61325i		−	0.184762i
$924$		0			0
$925$			9.57487i			0.314820i
$926$		0			0
$927$	7.82681			0.257066
$928$		0			0
$929$	6.19877			0.203375			0.101688	−	0.994816i	$-0.467576\pi$
$929$	6.19877			0.203375			0.101688	+	0.994816i	$0.467576\pi$
$930$		0			0
$931$	−11.4866	−	20.7094i	−0.376457	−	0.678722i
$932$		0			0
$933$		−	19.7798i		−	0.647562i
$934$		0			0
$935$		−	4.34978i		−	0.142253i
$936$		0			0
$937$	−46.4565			−1.51767			−0.758833	−	0.651285i	$-0.774230\pi$
$937$	−46.4565			−1.51767			−0.758833	+	0.651285i	$0.774230\pi$
$938$		0			0
$939$	−21.1152			−0.689069
$940$		0			0
$941$		−	54.2829i		−	1.76957i	−0.465999	−	0.884785i	$-0.654305\pi$
$941$		−	54.2829i		−	1.76957i	0.465999	−	0.884785i	$-0.345695\pi$
$942$		0			0
$943$	−50.2831			−1.63744
$944$		0			0
$945$			1.25182i			0.0407219i
$946$		0			0
$947$			14.7316i			0.478711i	0.970932	+	0.239356i	$0.0769362\pi$
$947$			14.7316i			0.478711i	−0.970932	+	0.239356i	$0.923064\pi$
$948$		0			0
$949$		−	30.3090i		−	0.983871i
$950$		0			0
$951$		−	14.7057i		−	0.476865i
$952$		0			0
$953$			53.1515i			1.72175i	0.508820	+	0.860873i	$0.330082\pi$
$953$			53.1515i			1.72175i	−0.508820	+	0.860873i	$0.669918\pi$
$954$		0			0
$955$			9.75195i			0.315566i
$956$		0			0
$957$	−1.78562			−0.0577210
$958$		0			0
$959$		−	0.761700i		−	0.0245966i
$960$		0			0
$961$	25.0704			0.808723
$962$		0			0
$963$	−1.24252			−0.0400398
$964$		0			0
$965$			17.5048i			0.563499i
$966$		0			0
$967$		−	14.1838i		−	0.456119i	−0.973647	−	0.228060i	$-0.926762\pi$
$967$		−	14.1838i		−	0.456119i	0.973647	−	0.228060i	$-0.0732381\pi$
$968$		0			0
$969$	14.5962	+	26.3158i	0.468897	+	0.845384i
$970$		0			0
$971$	42.2231			1.35500			0.677502	−	0.735521i	$-0.263062\pi$
$971$	42.2231			1.35500			0.677502	+	0.735521i	$0.263062\pi$
$972$		0			0
$973$	−18.9749			−0.608307
$974$		0			0
$975$		−	3.96496i		−	0.126980i
$976$		0			0
$977$			53.1731i			1.70116i	0.525847	+	0.850579i	$0.323749\pi$
$977$			53.1731i			1.70116i	−0.525847	+	0.850579i	$0.676251\pi$
$978$		0			0
$979$	2.85327			0.0911910
$980$		0			0
$981$		−	17.2068i		−	0.549371i
$982$		0			0
$983$	27.9449			0.891303			0.445652	−	0.895206i	$-0.352972\pi$
$983$	27.9449			0.891303			0.445652	+	0.895206i	$0.352972\pi$
$984$		0			0
$985$	21.6568			0.690043
$986$		0			0
$987$	4.17471			0.132882
$988$		0			0
$989$	−47.8571			−1.52177
$990$		0			0
$991$	31.2741			0.993454			0.496727	−	0.867907i	$-0.334535\pi$
$991$	31.2741			0.993454			0.496727	+	0.867907i	$0.334535\pi$
$992$		0			0
$993$	12.8795			0.408720
$994$		0			0
$995$			4.70101i			0.149032i
$996$		0			0
$997$	19.8729			0.629380			0.314690	−	0.949195i	$-0.398100\pi$
$997$	19.8729			0.629380			0.314690	+	0.949195i	$0.398100\pi$
$998$		0			0
$999$			9.57487i			0.302935i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.d.l.2431.8	yes	12
4.3	odd	2		4560.2.d.j.2431.5	✓	12
19.18	odd	2		4560.2.d.j.2431.8	yes	12
76.75	even	2	inner	4560.2.d.l.2431.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4560.2.d.j.2431.5	✓	12	4.3	odd	2
4560.2.d.j.2431.8	yes	12	19.18	odd	2
4560.2.d.l.2431.5	yes	12	76.75	even	2	inner
4560.2.d.l.2431.8	yes	12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.25182i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.25182i q^{7} +1.00000 q^{9} +0.630063i q^{11} -3.96496i q^{13} +1.00000 q^{15} -6.90373 q^{17} +(-2.11425 - 3.81182i) q^{19} +1.25182i q^{21} -6.49272i q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.83404i q^{29} -7.48802 q^{31} +0.630063i q^{33} +1.25182i q^{35} +9.57487i q^{37} -3.96496i q^{39} -7.74454i q^{41} -7.37089i q^{43} +1.00000 q^{45} -3.33490i q^{47} +5.43294 q^{49} -6.90373 q^{51} -9.95682i q^{53} +0.630063i q^{55} +(-2.11425 - 3.81182i) q^{57} -9.92095 q^{59} -12.4329 q^{61} +1.25182i q^{63} -3.96496i q^{65} -7.82681 q^{67} -6.49272i q^{69} +1.41571 q^{71} +7.64421 q^{73} +1.00000 q^{75} -0.788728 q^{77} -13.8485 q^{79} +1.00000 q^{81} +9.45397i q^{83} -6.90373 q^{85} +2.83404i q^{87} -4.52855i q^{89} +4.96343 q^{91} -7.48802 q^{93} +(-2.11425 - 3.81182i) q^{95} +3.90679i q^{97} +0.630063i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{3} + 12 q^{5} + 12 q^{9}+O(q^{10}) \) 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9	\( 12 q + 12 q^{3} + 12 q^{5} + 12 q^{9} + 12 q^{15} + 4 q^{17} + 12 q^{25} + 12 q^{27} - 12 q^{31} + 12 q^{45} - 28 q^{49} + 4 q^{51} + 52 q^{59} - 56 q^{61} - 32 q^{67} + 8 q^{71} + 32 q^{73} + 12 q^{75} + 24 q^{77} - 28 q^{79} + 12 q^{81} + 4 q^{85} + 32 q^{91} - 12 q^{93}+O(q^{100}) \) 12 * q + 12 * q^3 + 12 * q^5 + 12 * q^9 + 12 * q^15 + 4 * q^17 + 12 * q^25 + 12 * q^27 - 12 * q^31 + 12 * q^45 - 28 * q^49 + 4 * q^51 + 52 * q^59 - 56 * q^61 - 32 * q^67 + 8 * q^71 + 32 * q^73 + 12 * q^75 + 24 * q^77 - 28 * q^79 + 12 * q^81 + 4 * q^85 + 32 * q^91 - 12 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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