LMFDB - Embedded newform 320.6.c.l.129.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(129,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("320.129");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.c (of order $2$, degree $1$, not 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:	$51.3228223402$
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} - 6 x^{11} + 18 x^{10} + 348 x^{9} + 21226 x^{8} - 87824 x^{7} + 205428 x^{6} + 2113880 x^{5} + \cdots + 1072562500 $ x^12 - 6x^11 + 18x^10 + 348x^9 + 21226x^8 - 87824x^7 + 205428x^6 + 2113880x^5 + 27827225x^4 - 59798850x^3 + 89646050x^2 + 438522500*x + 1072562500
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{46}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.11
Root			$9.04538 + 9.04538i$ of defining polynomial
Character	$\chi$	$=$	320.129
Dual form			320.6.c.l.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+21.9637i q^{3} +(-18.7585 - 52.6604i) q^{5} -29.6983i q^{7} -239.406 q^{9} +O(q^{10})$	$q+21.9637i q^{3} +(-18.7585 - 52.6604i) q^{5} -29.6983i q^{7} -239.406 q^{9} -227.001 q^{11} -1067.66i q^{13} +(1156.62 - 412.007i) q^{15} +686.258i q^{17} +1309.11 q^{19} +652.286 q^{21} +4076.26i q^{23} +(-2421.24 + 1975.66i) q^{25} +78.9360i q^{27} -4535.21 q^{29} +7691.77 q^{31} -4985.79i q^{33} +(-1563.92 + 557.096i) q^{35} +9329.38i q^{37} +23449.8 q^{39} +3753.20 q^{41} -2934.28i q^{43} +(4490.90 + 12607.2i) q^{45} +3690.55i q^{47} +15925.0 q^{49} -15072.8 q^{51} +4816.68i q^{53} +(4258.20 + 11954.0i) q^{55} +28752.9i q^{57} -31168.2 q^{59} +33809.4 q^{61} +7109.96i q^{63} +(-56223.3 + 20027.7i) q^{65} -24844.0i q^{67} -89529.9 q^{69} +14153.6 q^{71} +73107.8i q^{73} +(-43392.9 - 53179.4i) q^{75} +6741.55i q^{77} -73646.1 q^{79} -59909.4 q^{81} +67353.7i q^{83} +(36138.6 - 12873.2i) q^{85} -99610.3i q^{87} +60277.5 q^{89} -31707.6 q^{91} +168940. i q^{93} +(-24556.9 - 68938.2i) q^{95} -15478.8i q^{97} +54345.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 60 q^{5} - 268 q^{9}+O(q^{10}) $ 12 * q + 60 * q^5 - 268 * q^9	$ 12 q + 60 q^{5} - 268 q^{9} - 12080 q^{21} - 16420 q^{25} + 7864 q^{29} + 65560 q^{41} + 60420 q^{45} + 17956 q^{49} + 118232 q^{61} + 11360 q^{65} - 204464 q^{69} - 346436 q^{81} - 96320 q^{85} + 354936 q^{89}+O(q^{100}) $ 12 * q + 60 * q^5 - 268 * q^9 - 12080 * q^21 - 16420 * q^25 + 7864 * q^29 + 65560 * q^41 + 60420 * q^45 + 17956 * q^49 + 118232 * q^61 + 11360 * q^65 - 204464 * q^69 - 346436 * q^81 - 96320 * q^85 + 354936 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			21.9637i			1.40897i	0.709716	+	0.704487i	$0.248823\pi$
$3$			21.9637i			1.40897i	−0.709716	+	0.704487i	$0.751177\pi$
$4$		0			0
$5$	−18.7585	−	52.6604i	−0.335563	−	0.942018i
$5$	−18.7585	−	52.6604i	−0.335563	−	0.942018i
$6$		0			0
$7$		−	29.6983i		−	0.229080i	−0.993419	−	0.114540i	$-0.963461\pi$
$7$		−	29.6983i		−	0.229080i	0.993419	−	0.114540i	$-0.0365394\pi$
$8$		0			0
$9$	−239.406			−0.985210
$10$		0			0
$11$	−227.001			−0.565648			−0.282824	−	0.959172i	$-0.591271\pi$
$11$	−227.001			−0.565648			−0.282824	+	0.959172i	$0.591271\pi$
$12$		0			0
$13$		−	1067.66i		−	1.75216i	−0.482165	−	0.876080i	$-0.660150\pi$
$13$		−	1067.66i		−	1.75216i	0.482165	−	0.876080i	$-0.339850\pi$
$14$		0			0
$15$	1156.62	−	412.007i	1.32728	−	0.472799i
$16$		0			0
$17$			686.258i			0.575924i	0.957642	+	0.287962i	$0.0929777\pi$
$17$			686.258i			0.575924i	−0.957642	+	0.287962i	$0.907022\pi$
$18$		0			0
$19$	1309.11			0.831940			0.415970	−	0.909378i	$-0.363442\pi$
$19$	1309.11			0.831940			0.415970	+	0.909378i	$0.363442\pi$
$20$		0			0
$21$	652.286			0.322768
$22$		0			0
$23$			4076.26i			1.60673i	0.595489	+	0.803364i	$0.296958\pi$
$23$			4076.26i			1.60673i	−0.595489	+	0.803364i	$0.703042\pi$
$24$		0			0
$25$	−2421.24	+	1975.66i	−0.774796	+	0.632212i
$26$		0			0
$27$			78.9360i			0.0208385i
$28$		0			0
$29$	−4535.21			−1.00139			−0.500694	−	0.865624i	$-0.666922\pi$
$29$	−4535.21			−1.00139			−0.500694	+	0.865624i	$0.666922\pi$
$30$		0			0
$31$	7691.77			1.43755			0.718773	−	0.695245i	$-0.244704\pi$
$31$	7691.77			1.43755			0.718773	+	0.695245i	$0.244704\pi$
$32$		0			0
$33$		−	4985.79i		−	0.796983i
$34$		0			0
$35$	−1563.92	+	557.096i	−0.215797	+	0.0768706i
$36$		0			0
$37$			9329.38i			1.12034i	0.828379	+	0.560168i	$0.189263\pi$
$37$			9329.38i			1.12034i	−0.828379	+	0.560168i	$0.810737\pi$
$38$		0			0
$39$	23449.8			2.46875
$40$		0			0
$41$	3753.20			0.348692			0.174346	−	0.984684i	$-0.444219\pi$
$41$	3753.20			0.348692			0.174346	+	0.984684i	$0.444219\pi$
$42$		0			0
$43$		−	2934.28i		−	0.242008i	−0.992652	−	0.121004i	$-0.961389\pi$
$43$		−	2934.28i		−	0.242008i	0.992652	−	0.121004i	$-0.0386114\pi$
$44$		0			0
$45$	4490.90	+	12607.2i	0.330600	+	0.928086i
$46$		0			0
$47$			3690.55i			0.243695i	0.992549	+	0.121847i	$0.0388819\pi$
$47$			3690.55i			0.243695i	−0.992549	+	0.121847i	$0.961118\pi$
$48$		0			0
$49$	15925.0			0.947522
$50$		0			0
$51$	−15072.8			−0.811463
$52$		0			0
$53$			4816.68i			0.235536i	0.993041	+	0.117768i	$0.0375740\pi$
$53$			4816.68i			0.235536i	−0.993041	+	0.117768i	$0.962426\pi$
$54$		0			0
$55$	4258.20	+	11954.0i	0.189810	+	0.532850i
$56$		0			0
$57$			28752.9i			1.17218i
$58$		0			0
$59$	−31168.2			−1.16569			−0.582843	−	0.812585i	$-0.698060\pi$
$59$	−31168.2			−1.16569			−0.582843	+	0.812585i	$0.698060\pi$
$60$		0			0
$61$	33809.4			1.16336			0.581678	−	0.813419i	$-0.302396\pi$
$61$	33809.4			1.16336			0.581678	+	0.813419i	$0.302396\pi$
$62$		0			0
$63$			7109.96i			0.225692i
$64$		0			0
$65$	−56223.3	+	20027.7i	−1.65057	+	0.587959i
$66$		0			0
$67$		−	24844.0i		−	0.676136i	−0.941122	−	0.338068i	$-0.890227\pi$
$67$		−	24844.0i		−	0.676136i	0.941122	−	0.338068i	$-0.109773\pi$
$68$		0			0
$69$	−89529.9			−2.26384
$70$		0			0
$71$	14153.6			0.333212			0.166606	−	0.986024i	$-0.446719\pi$
$71$	14153.6			0.333212			0.166606	+	0.986024i	$0.446719\pi$
$72$		0			0
$73$			73107.8i			1.60567i	0.596200	+	0.802836i	$0.296677\pi$
$73$			73107.8i			1.60567i	−0.596200	+	0.802836i	$0.703323\pi$
$74$		0			0
$75$	−43392.9	−	53179.4i	−0.890771	−	1.09167i
$76$		0			0
$77$			6741.55i			0.129578i
$78$		0			0
$79$	−73646.1			−1.32764			−0.663822	−	0.747890i	$-0.731067\pi$
$79$	−73646.1			−1.32764			−0.663822	+	0.747890i	$0.731067\pi$
$80$		0			0
$81$	−59909.4			−1.01457
$82$		0			0
$83$			67353.7i			1.07316i	0.843848	+	0.536582i	$0.180285\pi$
$83$			67353.7i			1.07316i	−0.843848	+	0.536582i	$0.819715\pi$
$84$		0			0
$85$	36138.6	−	12873.2i	0.542531	−	0.193259i
$86$		0			0
$87$		−	99610.3i		−	1.41093i
$88$		0			0
$89$	60277.5			0.806641			0.403320	−	0.915059i	$-0.367856\pi$
$89$	60277.5			0.806641			0.403320	+	0.915059i	$0.367856\pi$
$90$		0			0
$91$	−31707.6			−0.401384
$92$		0			0
$93$			168940.i			2.02547i
$94$		0			0
$95$	−24556.9	−	68938.2i	−0.279168	−	0.783702i
$96$		0			0
$97$		−	15478.8i		−	0.167035i	−0.996506	−	0.0835175i	$-0.973385\pi$
$97$		−	15478.8i		−	0.167035i	0.996506	−	0.0835175i	$-0.0266155\pi$
$98$		0			0
$99$	54345.4			0.557282
$100$		0			0
$101$	−37535.9			−0.366137			−0.183069	−	0.983100i	$-0.558603\pi$
$101$	−37535.9			−0.366137			−0.183069	+	0.983100i	$0.558603\pi$
$102$		0			0
$103$			183229.i			1.70177i	0.525354	+	0.850884i	$0.323933\pi$
$103$			183229.i			1.70177i	−0.525354	+	0.850884i	$0.676067\pi$
$104$		0			0
$105$	−12235.9	−	34349.6i	−0.108309	−	0.304053i
$106$		0			0
$107$			72550.4i			0.612605i	0.951934	+	0.306302i	$0.0990919\pi$
$107$			72550.4i			0.612605i	−0.951934	+	0.306302i	$0.900908\pi$
$108$		0			0
$109$	−159664.			−1.28719			−0.643593	−	0.765368i	$-0.722557\pi$
$109$	−159664.			−1.28719			−0.643593	+	0.765368i	$0.722557\pi$
$110$		0			0
$111$	−204908.			−1.57853
$112$		0			0
$113$			130294.i			0.959905i	0.877294	+	0.479953i	$0.159346\pi$
$113$			130294.i			0.959905i	−0.877294	+	0.479953i	$0.840654\pi$
$114$		0			0
$115$	214657.	−	76464.6i	1.51357	−	0.539158i
$116$		0			0
$117$			255604.i			1.72625i
$118$		0			0
$119$	20380.7			0.131933
$120$		0			0
$121$	−109522.			−0.680043
$122$		0			0
$123$			82434.4i			0.491299i
$124$		0			0
$125$	149458.	+	90442.8i	0.855547	+	0.517725i
$126$		0			0
$127$		−	240270.i		−	1.32187i	−0.750442	−	0.660936i	$-0.770159\pi$
$127$		−	240270.i		−	1.32187i	0.750442	−	0.660936i	$-0.229841\pi$
$128$		0			0
$129$	64447.7			0.340984
$130$		0			0
$131$	389775.			1.98443			0.992214	−	0.124546i	$-0.0397475\pi$
$131$	389775.			1.98443			0.992214	+	0.124546i	$0.0397475\pi$
$132$		0			0
$133$		−	38878.3i		−	0.190581i
$134$		0			0
$135$	4156.80	−	1480.72i	0.0196302	−	0.00699261i
$136$		0			0
$137$			434535.i			1.97799i	0.147962	+	0.988993i	$0.452729\pi$
$137$			434535.i			1.97799i	−0.147962	+	0.988993i	$0.547271\pi$
$138$		0			0
$139$	114647.			0.503298			0.251649	−	0.967819i	$-0.419027\pi$
$139$	114647.			0.503298			0.251649	+	0.967819i	$0.419027\pi$
$140$		0			0
$141$	−81058.3			−0.343360
$142$		0			0
$143$			242359.i			0.991105i
$144$		0			0
$145$	85073.9	+	238826.i	0.336029	+	0.943326i
$146$		0			0
$147$			349773.i			1.33504i
$148$		0			0
$149$	294922.			1.08828			0.544141	−	0.838994i	$-0.316856\pi$
$149$	294922.			1.08828			0.544141	+	0.838994i	$0.316856\pi$
$150$		0			0
$151$	404524.			1.44378			0.721892	−	0.692006i	$-0.243273\pi$
$151$	404524.			1.44378			0.721892	+	0.692006i	$0.243273\pi$
$152$		0			0
$153$		−	164294.i		−	0.567406i
$154$		0			0
$155$	−144286.	−	405051.i	−0.482387	−	1.35419i
$156$		0			0
$157$		−	377457.i		−	1.22213i	−0.791579	−	0.611067i	$-0.790741\pi$
$157$		−	377457.i		−	1.22213i	0.791579	−	0.611067i	$-0.209259\pi$
$158$		0			0
$159$	−105792.			−0.331865
$160$		0			0
$161$	121058.			0.368069
$162$		0			0
$163$			422416.i			1.24529i	0.782503	+	0.622647i	$0.213943\pi$
$163$			422416.i			1.24529i	−0.782503	+	0.622647i	$0.786057\pi$
$164$		0			0
$165$	−262554.	+	93526.1i	−0.750773	+	0.267438i
$166$		0			0
$167$		−	360875.i		−	1.00130i	−0.865649	−	0.500652i	$-0.833094\pi$
$167$		−	360875.i		−	1.00130i	0.865649	−	0.500652i	$-0.166906\pi$
$168$		0			0
$169$	−768601.			−2.07007
$170$		0			0
$171$	−313409.			−0.819636
$172$		0			0
$173$		−	405196.i		−	1.02932i	−0.857395	−	0.514659i	$-0.827919\pi$
$173$		−	405196.i		−	1.02932i	0.857395	−	0.514659i	$-0.172081\pi$
$174$		0			0
$175$	58673.8	+	71906.6i	0.144827	+	0.177490i
$176$		0			0
$177$		−	684570.i		−	1.64242i
$178$		0			0
$179$	198995.			0.464205			0.232102	−	0.972691i	$-0.425440\pi$
$179$	198995.			0.464205			0.232102	+	0.972691i	$0.425440\pi$
$180$		0			0
$181$	−243402.			−0.552240			−0.276120	−	0.961123i	$-0.589049\pi$
$181$	−243402.			−0.552240			−0.276120	+	0.961123i	$0.589049\pi$
$182$		0			0
$183$			742581.i			1.63914i
$184$		0			0
$185$	491289.	−	175005.i	1.05538	−	0.375943i
$186$		0			0
$187$		−	155781.i		−	0.325770i
$188$		0			0
$189$	2344.27			0.00477367
$190$		0			0
$191$	−239201.			−0.474438			−0.237219	−	0.971456i	$-0.576236\pi$
$191$	−239201.			−0.474438			−0.237219	+	0.971456i	$0.576236\pi$
$192$		0			0
$193$		−	506878.i		−	0.979513i	−0.871859	−	0.489757i	$-0.837086\pi$
$193$		−	506878.i		−	0.979513i	0.871859	−	0.489757i	$-0.162914\pi$
$194$		0			0
$195$	−439883.	−	1.23487e6i	−0.828420	−	2.32561i
$196$		0			0
$197$			493617.i			0.906201i	0.891460	+	0.453100i	$0.149682\pi$
$197$			493617.i			0.906201i	−0.891460	+	0.453100i	$0.850318\pi$
$198$		0			0
$199$	489526.			0.876281			0.438140	−	0.898907i	$-0.355637\pi$
$199$	489526.			0.876281			0.438140	+	0.898907i	$0.355637\pi$
$200$		0			0
$201$	545666.			0.952658
$202$		0			0
$203$			134688.i			0.229398i
$204$		0			0
$205$	−70404.5	−	197645.i	−0.117008	−	0.328474i
$206$		0			0
$207$		−	975881.i		−	1.58296i
$208$		0			0
$209$	−297169.			−0.470585
$210$		0			0
$211$	−1.19087e6			−1.84145			−0.920723	−	0.390217i	$-0.872400\pi$
$211$	−1.19087e6			−1.84145			−0.920723	+	0.390217i	$0.872400\pi$
$212$		0			0
$213$			310866.i			0.469487i
$214$		0			0
$215$	−154520.	+	55042.7i	−0.227976	+	0.0812089i
$216$		0			0
$217$		−	228432.i		−	0.329313i
$218$		0			0
$219$	−1.60572e6			−2.26235
$220$		0			0
$221$	732689.			1.00911
$222$		0			0
$223$			876410.i			1.18017i	0.807341	+	0.590086i	$0.200906\pi$
$223$			876410.i			1.18017i	−0.807341	+	0.590086i	$0.799094\pi$
$224$		0			0
$225$	579659.	−	472986.i	0.763336	−	0.622862i
$226$		0			0
$227$			1.06583e6i			1.37285i	0.727200	+	0.686425i	$0.240821\pi$
$227$			1.06583e6i			1.37285i	−0.727200	+	0.686425i	$0.759179\pi$
$228$		0			0
$229$	657279.			0.828250			0.414125	−	0.910220i	$-0.364088\pi$
$229$	657279.			0.828250			0.414125	+	0.910220i	$0.364088\pi$
$230$		0			0
$231$	−148070.			−0.182573
$232$		0			0
$233$			549169.i			0.662699i	0.943508	+	0.331349i	$0.107504\pi$
$233$			549169.i			0.662699i	−0.943508	+	0.331349i	$0.892496\pi$
$234$		0			0
$235$	194346.	−	69229.3i	0.229565	−	0.0817749i
$236$		0			0
$237$		−	1.61754e6i		−	1.87062i
$238$		0			0
$239$	453609.			0.513673			0.256836	−	0.966455i	$-0.417320\pi$
$239$	453609.			0.513673			0.256836	+	0.966455i	$0.417320\pi$
$240$		0			0
$241$	1.08405e6			1.20229			0.601143	−	0.799142i	$-0.294712\pi$
$241$	1.08405e6			1.20229			0.601143	+	0.799142i	$0.294712\pi$
$242$		0			0
$243$		−	1.29665e6i		−	1.40867i
$244$		0			0
$245$	−298730.	−	838617.i	−0.317953	−	0.892583i
$246$		0			0
$247$		−	1.39768e6i		−	1.45769i
$248$		0			0
$249$	−1.47934e6			−1.51206
$250$		0			0
$251$	1.10334e6			1.10541			0.552705	−	0.833377i	$-0.313595\pi$
$251$	1.10334e6			1.10541			0.552705	+	0.833377i	$0.313595\pi$
$252$		0			0
$253$		−	925315.i		−	0.908842i
$254$		0			0
$255$	282743.	+	793740.i	0.272297	+	0.764412i
$256$		0			0
$257$		−	360858.i		−	0.340803i	−0.985375	−	0.170402i	$-0.945493\pi$
$257$		−	360858.i		−	0.340803i	0.985375	−	0.170402i	$-0.0545065\pi$
$258$		0			0
$259$	277067.			0.256646
$260$		0			0
$261$	1.08576e6			0.986578
$262$		0			0
$263$			492090.i			0.438687i	0.975648	+	0.219344i	$0.0703916\pi$
$263$			492090.i			0.438687i	−0.975648	+	0.219344i	$0.929608\pi$
$264$		0			0
$265$	253648.	−	90353.8i	0.221880	−	0.0790372i
$266$		0			0
$267$			1.32392e6i			1.13654i
$268$		0			0
$269$	−809822.			−0.682352			−0.341176	−	0.939999i	$-0.610825\pi$
$269$	−809822.			−0.682352			−0.341176	+	0.939999i	$0.610825\pi$
$270$		0			0
$271$	1.07473e6			0.888949			0.444475	−	0.895791i	$-0.353390\pi$
$271$	1.07473e6			0.888949			0.444475	+	0.895791i	$0.353390\pi$
$272$		0			0
$273$		−	696419.i		−	0.565541i
$274$		0			0
$275$	549623.	−	448477.i	0.438261	−	0.357609i
$276$		0			0
$277$		−	625727.i		−	0.489988i	−0.969525	−	0.244994i	$-0.921214\pi$
$277$		−	625727.i		−	0.489988i	0.969525	−	0.244994i	$-0.0787859\pi$
$278$		0			0
$279$	−1.84146e6			−1.41629
$280$		0			0
$281$	239071.			0.180618			0.0903091	−	0.995914i	$-0.471215\pi$
$281$	239071.			0.180618			0.0903091	+	0.995914i	$0.471215\pi$
$282$		0			0
$283$		−	96643.3i		−	0.0717308i	−0.999357	−	0.0358654i	$-0.988581\pi$
$283$		−	96643.3i		−	0.0717308i	0.999357	−	0.0358654i	$-0.0114188\pi$
$284$		0			0
$285$	1.51414e6	−	539363.i	1.10422	−	0.393340i
$286$		0			0
$287$		−	111464.i		−	0.0798783i
$288$		0			0
$289$	948906.			0.668311
$290$		0			0
$291$	339972.			0.235348
$292$		0			0
$293$			695903.i			0.473565i	0.971563	+	0.236783i	$0.0760929\pi$
$293$			695903.i			0.473565i	−0.971563	+	0.236783i	$0.923907\pi$
$294$		0			0
$295$	584669.	+	1.64133e6i	0.391160	+	1.09810i
$296$		0			0
$297$		−	17918.5i		−	0.0117872i
$298$		0			0
$299$	4.35205e6			2.81524
$300$		0			0
$301$	−87143.1			−0.0554392
$302$		0			0
$303$		−	824430.i		−	0.515878i
$304$		0			0
$305$	−634214.	−	1.78042e6i	−0.390379	−	1.09590i
$306$		0			0
$307$			1.89838e6i			1.14958i	0.818302	+	0.574788i	$0.194916\pi$
$307$			1.89838e6i			1.14958i	−0.818302	+	0.574788i	$0.805084\pi$
$308$		0			0
$309$	−4.02439e6			−2.39775
$310$		0			0
$311$	−1.32482e6			−0.776706			−0.388353	−	0.921511i	$-0.626956\pi$
$311$	−1.32482e6			−0.776706			−0.388353	+	0.921511i	$0.626956\pi$
$312$		0			0
$313$		−	671307.i		−	0.387311i	−0.981070	−	0.193656i	$-0.937966\pi$
$313$		−	671307.i		−	0.387311i	0.981070	−	0.193656i	$-0.0620345\pi$
$314$		0			0
$315$	374413.	−	133372.i	0.212606	−	0.0757337i
$316$		0			0
$317$			2.05383e6i			1.14793i	0.818879	+	0.573967i	$0.194596\pi$
$317$			2.05383e6i			1.14793i	−0.818879	+	0.573967i	$0.805404\pi$
$318$		0			0
$319$	1.02950e6			0.566433
$320$		0			0
$321$	−1.59348e6			−0.863145
$322$		0			0
$323$			898387.i			0.479134i
$324$		0			0
$325$	2.10933e6	+	2.58505e6i	1.10774	+	1.35757i
$326$		0			0
$327$		−	3.50683e6i		−	1.81361i
$328$		0			0
$329$	109603.			0.0558256
$330$		0			0
$331$	2.74106e6			1.37515			0.687573	−	0.726115i	$-0.258676\pi$
$331$	2.74106e6			1.37515			0.687573	+	0.726115i	$0.258676\pi$
$332$		0			0
$333$		−	2.23351e6i		−	1.10377i
$334$		0			0
$335$	−1.30829e6	+	466036.i	−0.636932	+	0.226886i
$336$		0			0
$337$		−	1.46812e6i		−	0.704183i	−0.935966	−	0.352092i	$-0.885471\pi$
$337$		−	1.46812e6i		−	0.704183i	0.935966	−	0.352092i	$-0.114529\pi$
$338$		0			0
$339$	−2.86175e6			−1.35248
$340$		0			0
$341$	−1.74604e6			−0.813145
$342$		0			0
$343$		−	972085.i		−	0.446138i
$344$		0			0
$345$	1.67945e6	+	4.71468e6i	0.759659	+	2.13258i
$346$		0			0
$347$			1.12356e6i			0.500926i	0.968126	+	0.250463i	$0.0805828\pi$
$347$			1.12356e6i			0.500926i	−0.968126	+	0.250463i	$0.919417\pi$
$348$		0			0
$349$	−1.71043e6			−0.751693			−0.375846	−	0.926682i	$-0.622648\pi$
$349$	−1.71043e6			−0.751693			−0.375846	+	0.926682i	$0.622648\pi$
$350$		0			0
$351$	84276.6			0.0365123
$352$		0			0
$353$		−	2.43615e6i		−	1.04056i	−0.853996	−	0.520280i	$-0.825827\pi$
$353$		−	2.43615e6i		−	1.04056i	0.853996	−	0.520280i	$-0.174173\pi$
$354$		0			0
$355$	−265500.	−	745334.i	−0.111813	−	0.313892i
$356$		0			0
$357$			447637.i			0.185890i
$358$		0			0
$359$	−2.00932e6			−0.822837			−0.411418	−	0.911447i	$-0.634966\pi$
$359$	−2.00932e6			−0.822837			−0.411418	+	0.911447i	$0.634966\pi$
$360$		0			0
$361$	−762332.			−0.307876
$362$		0			0
$363$		−	2.40550e6i		−	0.958163i
$364$		0			0
$365$	3.84989e6	−	1.37139e6i	1.51257	−	0.538803i
$366$		0			0
$367$			3.79537e6i			1.47092i	0.677567	+	0.735461i	$0.263034\pi$
$367$			3.79537e6i			1.47092i	−0.677567	+	0.735461i	$0.736966\pi$
$368$		0			0
$369$	−898539.			−0.343535
$370$		0			0
$371$	143047.			0.0539566
$372$		0			0
$373$			2.29348e6i			0.853537i	0.904361	+	0.426769i	$0.140348\pi$
$373$			2.29348e6i			0.853537i	−0.904361	+	0.426769i	$0.859652\pi$
$374$		0			0
$375$	−1.98646e6	+	3.28266e6i	−0.729461	+	1.20544i
$376$		0			0
$377$			4.84206e6i			1.75459i
$378$		0			0
$379$	−2.13249e6			−0.762588			−0.381294	−	0.924454i	$-0.624521\pi$
$379$	−2.13249e6			−0.762588			−0.381294	+	0.924454i	$0.624521\pi$
$380$		0			0
$381$	5.27722e6			1.86248
$382$		0			0
$383$		−	4.10113e6i		−	1.42859i	−0.699847	−	0.714293i	$-0.746749\pi$
$383$		−	4.10113e6i		−	1.42859i	0.699847	−	0.714293i	$-0.253251\pi$
$384$		0			0
$385$	355012.	−	126461.i	0.122065	−	0.0434817i
$386$		0			0
$387$			702484.i			0.238429i
$388$		0			0
$389$	−4.88764e6			−1.63766			−0.818832	−	0.574033i	$-0.805378\pi$
$389$	−4.88764e6			−1.63766			−0.818832	+	0.574033i	$0.805378\pi$
$390$		0			0
$391$	−2.79737e6			−0.925353
$392$		0			0
$393$			8.56091e6i			2.79601i
$394$		0			0
$395$	1.38149e6	+	3.87823e6i	0.445508	+	1.25067i
$396$		0			0
$397$		−	674221.i		−	0.214697i	−0.994221	−	0.107348i	$-0.965764\pi$
$397$		−	674221.i		−	0.214697i	0.994221	−	0.107348i	$-0.0342360\pi$
$398$		0			0
$399$	853914.			0.268523
$400$		0			0
$401$	2.13743e6			0.663789			0.331894	−	0.943317i	$-0.392312\pi$
$401$	2.13743e6			0.663789			0.331894	+	0.943317i	$0.392312\pi$
$402$		0			0
$403$		−	8.21218e6i		−	2.51881i
$404$		0			0
$405$	1.12381e6	+	3.15485e6i	0.340452	+	0.955744i
$406$		0			0
$407$		−	2.11778e6i		−	0.633716i
$408$		0			0
$409$	−1.64322e6			−0.485722			−0.242861	−	0.970061i	$-0.578086\pi$
$409$	−1.64322e6			−0.485722			−0.242861	+	0.970061i	$0.578086\pi$
$410$		0			0
$411$	−9.54401e6			−2.78693
$412$		0			0
$413$			925642.i			0.267035i
$414$		0			0
$415$	3.54687e6	−	1.26346e6i	1.01094	−	0.360114i
$416$		0			0
$417$			2.51807e6i			0.709134i
$418$		0			0
$419$	1.29320e6			0.359859			0.179929	−	0.983680i	$-0.442413\pi$
$419$	1.29320e6			0.359859			0.179929	+	0.983680i	$0.442413\pi$
$420$		0			0
$421$	−131852.			−0.0362560			−0.0181280	−	0.999836i	$-0.505771\pi$
$421$	−131852.			−0.0362560			−0.0181280	+	0.999836i	$0.505771\pi$
$422$		0			0
$423$		−	883540.i		−	0.240091i
$424$		0			0
$425$	−1.35581e6	−	1.66159e6i	−0.364106	−	0.446224i
$426$		0			0
$427$		−	1.00408e6i		−	0.266501i
$428$		0			0
$429$	−5.32312e6			−1.39644
$430$		0			0
$431$	1.09123e6			0.282960			0.141480	−	0.989941i	$-0.454814\pi$
$431$	1.09123e6			0.282960			0.141480	+	0.989941i	$0.454814\pi$
$432$		0			0
$433$		−	1.10549e6i		−	0.283357i	−0.989913	−	0.141679i	$-0.954750\pi$
$433$		−	1.10549e6i		−	0.283357i	0.989913	−	0.141679i	$-0.0452499\pi$
$434$		0			0
$435$	−5.24552e6	+	1.86854e6i	−1.32912	+	0.473456i
$436$		0			0
$437$			5.33627e6i			1.33670i
$438$		0			0
$439$	5.82681e6			1.44301			0.721505	−	0.692409i	$-0.243451\pi$
$439$	5.82681e6			1.44301			0.721505	+	0.692409i	$0.243451\pi$
$440$		0			0
$441$	−3.81254e6			−0.933509
$442$		0			0
$443$		−	1.88621e6i		−	0.456647i	−0.973585	−	0.228324i	$-0.926676\pi$
$443$		−	1.88621e6i		−	0.456647i	0.973585	−	0.228324i	$-0.0733245\pi$
$444$		0			0
$445$	−1.13072e6	−	3.17424e6i	−0.270678	−	0.759870i
$446$		0			0
$447$			6.47760e6i			1.53336i
$448$		0			0
$449$	−8.06463e6			−1.88786			−0.943928	−	0.330152i	$-0.892900\pi$
$449$	−8.06463e6			−1.88786			−0.943928	+	0.330152i	$0.892900\pi$
$450$		0			0
$451$	−851980.			−0.197237
$452$		0			0
$453$			8.88487e6i			2.03426i
$454$		0			0
$455$	594788.	+	1.66974e6i	0.134690	+	0.378111i
$456$		0			0
$457$		−	3.33016e6i		−	0.745889i	−0.927854	−	0.372945i	$-0.878348\pi$
$457$		−	3.33016e6i		−	0.745889i	0.927854	−	0.372945i	$-0.121652\pi$
$458$		0			0
$459$	−54170.5			−0.0120014
$460$		0			0
$461$	−4.99208e6			−1.09403			−0.547015	−	0.837123i	$-0.684236\pi$
$461$	−4.99208e6			−1.09403			−0.547015	+	0.837123i	$0.684236\pi$
$462$		0			0
$463$		−	1.89423e6i		−	0.410659i	−0.978693	−	0.205329i	$-0.934173\pi$
$463$		−	1.89423e6i		−	0.410659i	0.978693	−	0.205329i	$-0.0658265\pi$
$464$		0			0
$465$	8.89645e6	−	3.16906e6i	1.90803	−	0.679671i
$466$		0			0
$467$			3.60284e6i			0.764455i	0.924068	+	0.382228i	$0.124843\pi$
$467$			3.60284e6i			0.764455i	−0.924068	+	0.382228i	$0.875157\pi$
$468$		0			0
$469$	−737824.			−0.154889
$470$		0			0
$471$	8.29037e6			1.72196
$472$		0			0
$473$			666084.i			0.136891i
$474$		0			0
$475$	−3.16966e6	+	2.58636e6i	−0.644583	+	0.525962i
$476$		0			0
$477$		−	1.15314e6i		−	0.232053i
$478$		0			0
$479$	−5.65624e6			−1.12639			−0.563196	−	0.826324i	$-0.690428\pi$
$479$	−5.65624e6			−1.12639			−0.563196	+	0.826324i	$0.690428\pi$
$480$		0			0
$481$	9.96059e6			1.96301
$482$		0			0
$483$			2.65889e6i			0.518600i
$484$		0			0
$485$	−815119.	+	290359.i	−0.157350	+	0.0560507i
$486$		0			0
$487$			5.12375e6i			0.978961i	0.872014	+	0.489480i	$0.162813\pi$
$487$			5.12375e6i			0.978961i	−0.872014	+	0.489480i	$0.837187\pi$
$488$		0			0
$489$	−9.27784e6			−1.75459
$490$		0			0
$491$	8.93216e6			1.67206			0.836032	−	0.548681i	$-0.184870\pi$
$491$	8.93216e6			1.67206			0.836032	+	0.548681i	$0.184870\pi$
$492$		0			0
$493$		−	3.11233e6i		−	0.576724i
$494$		0			0
$495$	−1.01944e6	−	2.86185e6i	−0.187003	−	0.524970i
$496$		0			0
$497$		−	420338.i		−	0.0763321i
$498$		0			0
$499$	−1.78948e6			−0.321719			−0.160859	−	0.986977i	$-0.551427\pi$
$499$	−1.78948e6			−0.321719			−0.160859	+	0.986977i	$0.551427\pi$
$500$		0			0
$501$	7.92617e6			1.41081
$502$		0			0
$503$			296978.i			0.0523365i	0.999658	+	0.0261683i	$0.00833057\pi$
$503$			296978.i			0.0523365i	−0.999658	+	0.0261683i	$0.991669\pi$
$504$		0			0
$505$	704118.	+	1.97666e6i	0.122862	+	0.344908i
$506$		0			0
$507$		−	1.68814e7i		−	2.91667i
$508$		0			0
$509$	3.91977e6			0.670603			0.335302	−	0.942111i	$-0.391162\pi$
$509$	3.91977e6			0.670603			0.335302	+	0.942111i	$0.391162\pi$
$510$		0			0
$511$	2.17118e6			0.367827
$512$		0			0
$513$			103336.i			0.0173363i
$514$		0			0
$515$	9.64889e6	−	3.43710e6i	1.60310	−	0.571049i
$516$		0			0
$517$		−	837759.i		−	0.137845i
$518$		0			0
$519$	8.89962e6			1.45028
$520$		0			0
$521$	4.74987e6			0.766633			0.383317	−	0.923617i	$-0.374782\pi$
$521$	4.74987e6			0.766633			0.383317	+	0.923617i	$0.374782\pi$
$522$		0			0
$523$			8.15275e6i			1.30332i	0.758513	+	0.651658i	$0.225926\pi$
$523$			8.15275e6i			1.30332i	−0.758513	+	0.651658i	$0.774074\pi$
$524$		0			0
$525$	−1.57934e6	+	1.28870e6i	−0.250079	+	0.204057i
$526$		0			0
$527$			5.27854e6i			0.827918i
$528$		0			0
$529$	−1.01795e7			−1.58157
$530$		0			0
$531$	7.46185e6			1.14844
$532$		0			0
$533$		−	4.00714e6i		−	0.610965i
$534$		0			0
$535$	3.82053e6	−	1.36094e6i	0.577085	−	0.205567i
$536$		0			0
$537$			4.37068e6i			0.654053i
$538$		0			0
$539$	−3.61499e6			−0.535964
$540$		0			0
$541$	835785.			0.122773			0.0613863	−	0.998114i	$-0.480448\pi$
$541$	835785.			0.122773			0.0613863	+	0.998114i	$0.480448\pi$
$542$		0			0
$543$		−	5.34602e6i		−	0.778092i
$544$		0			0
$545$	2.99507e6	+	8.40799e6i	0.431932	+	1.21255i
$546$		0			0
$547$			7.88859e6i			1.12728i	0.826021	+	0.563639i	$0.190599\pi$
$547$			7.88859e6i			1.12728i	−0.826021	+	0.563639i	$0.809401\pi$
$548$		0			0
$549$	−8.09418e6			−1.14615
$550$		0			0
$551$	−5.93709e6			−0.833095
$552$		0			0
$553$			2.18716e6i			0.304137i
$554$		0			0
$555$	3.84377e6	+	1.07905e7i	0.529694	+	1.48700i
$556$		0			0
$557$		−	7.42914e6i		−	1.01461i	−0.861765	−	0.507307i	$-0.830641\pi$
$557$		−	7.42914e6i		−	1.01461i	0.861765	−	0.507307i	$-0.169359\pi$
$558$		0			0
$559$	−3.13281e6			−0.424037
$560$		0			0
$561$	3.42154e6			0.459002
$562$		0			0
$563$			5.23381e6i			0.695901i	0.937513	+	0.347950i	$0.113122\pi$
$563$			5.23381e6i			0.695901i	−0.937513	+	0.347950i	$0.886878\pi$
$564$		0			0
$565$	6.86134e6	−	2.44412e6i	0.904248	−	0.322108i
$566$		0			0
$567$			1.77921e6i			0.232418i
$568$		0			0
$569$	−7.64257e6			−0.989598			−0.494799	−	0.869008i	$-0.664758\pi$
$569$	−7.64257e6			−0.989598			−0.494799	+	0.869008i	$0.664758\pi$
$570$		0			0
$571$	−1.06264e7			−1.36394			−0.681971	−	0.731379i	$-0.738877\pi$
$571$	−1.06264e7			−1.36394			−0.681971	+	0.731379i	$0.738877\pi$
$572$		0			0
$573$		−	5.25375e6i		−	0.668471i
$574$		0			0
$575$	−8.05331e6	−	9.86959e6i	−1.01579	−	1.24489i
$576$		0			0
$577$		−	7.92313e6i		−	0.990735i	−0.868684	−	0.495367i	$-0.835033\pi$
$577$		−	7.92313e6i		−	0.990735i	0.868684	−	0.495367i	$-0.164967\pi$
$578$		0			0
$579$	1.11329e7			1.38011
$580$		0			0
$581$	2.00029e6			0.245840
$582$		0			0
$583$		−	1.09339e6i		−	0.133231i
$584$		0			0
$585$	1.34602e7	−	4.79475e6i	1.62615	−	0.579264i
$586$		0			0
$587$		−	1.41402e7i		−	1.69380i	−0.531755	−	0.846898i	$-0.678467\pi$
$587$		−	1.41402e7i		−	1.69380i	0.531755	−	0.846898i	$-0.321533\pi$
$588$		0			0
$589$	1.00694e7			1.19595
$590$		0			0
$591$	−1.08417e7			−1.27681
$592$		0			0
$593$		−	6.60665e6i		−	0.771516i	−0.922600	−	0.385758i	$-0.873940\pi$
$593$		−	6.60665e6i		−	0.771516i	0.922600	−	0.385758i	$-0.126060\pi$
$594$		0			0
$595$	−382312.	−	1.07326e6i	−0.0442716	−	0.124283i
$596$		0			0
$597$			1.07518e7i			1.23466i
$598$		0			0
$599$	−1.73390e7			−1.97450			−0.987252	−	0.159166i	$-0.949120\pi$
$599$	−1.73390e7			−1.97450			−0.987252	+	0.159166i	$0.949120\pi$
$600$		0			0
$601$	6.83953e6			0.772397			0.386198	−	0.922416i	$-0.373788\pi$
$601$	6.83953e6			0.772397			0.386198	+	0.922416i	$0.373788\pi$
$602$		0			0
$603$			5.94780e6i			0.666136i
$604$		0			0
$605$	2.05446e6	+	5.76745e6i	0.228197	+	0.640612i
$606$		0			0
$607$			5.78383e6i			0.637153i	0.947897	+	0.318576i	$0.103205\pi$
$607$			5.78383e6i			0.637153i	−0.947897	+	0.318576i	$0.896795\pi$
$608$		0			0
$609$	−2.95826e6			−0.323216
$610$		0			0
$611$	3.94025e6			0.426993
$612$		0			0
$613$		−	1.29204e7i		−	1.38875i	−0.719614	−	0.694374i	$-0.755681\pi$
$613$		−	1.29204e7i		−	1.38875i	0.719614	−	0.694374i	$-0.244319\pi$
$614$		0			0
$615$	4.34103e6	−	1.54635e6i	0.462812	−	0.164861i
$616$		0			0
$617$			3.21433e6i			0.339920i	0.985451	+	0.169960i	$0.0543639\pi$
$617$			3.21433e6i			0.339920i	−0.985451	+	0.169960i	$0.945636\pi$
$618$		0			0
$619$	3.18358e6			0.333956			0.166978	−	0.985961i	$-0.446599\pi$
$619$	3.18358e6			0.333956			0.166978	+	0.985961i	$0.446599\pi$
$620$		0			0
$621$	−321764.			−0.0334817
$622$		0			0
$623$		−	1.79014e6i		−	0.184785i
$624$		0			0
$625$	1.95914e6	−	9.56709e6i	0.200616	−	0.979670i
$626$		0			0
$627$		−	6.52695e6i		−	0.663042i
$628$		0			0
$629$	−6.40236e6			−0.645229
$630$		0			0
$631$	−2.08225e6			−0.208190			−0.104095	−	0.994567i	$-0.533195\pi$
$631$	−2.08225e6			−0.208190			−0.104095	+	0.994567i	$0.533195\pi$
$632$		0			0
$633$		−	2.61560e7i		−	2.59455i
$634$		0			0
$635$	−1.26527e7	+	4.50710e6i	−1.24523	+	0.443571i
$636$		0			0
$637$		−	1.70025e7i		−	1.66021i
$638$		0			0
$639$	−3.38845e6			−0.328284
$640$		0			0
$641$	−1.96753e6			−0.189137			−0.0945687	−	0.995518i	$-0.530147\pi$
$641$	−1.96753e6			−0.189137			−0.0945687	+	0.995518i	$0.530147\pi$
$642$		0			0
$643$		−	6.15416e6i		−	0.587004i	−0.955958	−	0.293502i	$-0.905179\pi$
$643$		−	6.15416e6i		−	0.587004i	0.955958	−	0.293502i	$-0.0948208\pi$
$644$		0			0
$645$	−1.20894e6	−	3.39384e6i	−0.114421	−	0.321213i
$646$		0			0
$647$		−	8.21959e6i		−	0.771951i	−0.922509	−	0.385975i	$-0.873865\pi$
$647$		−	8.21959e6i		−	0.771951i	0.922509	−	0.385975i	$-0.126135\pi$
$648$		0			0
$649$	7.07520e6			0.659367
$650$		0			0
$651$	5.01723e6			0.463993
$652$		0			0
$653$			7.81613e6i			0.717314i	0.933470	+	0.358657i	$0.116765\pi$
$653$			7.81613e6i			0.717314i	−0.933470	+	0.358657i	$0.883235\pi$
$654$		0			0
$655$	−7.31159e6	−	2.05257e7i	−0.665900	−	1.86937i
$656$		0			0
$657$		−	1.75025e7i		−	1.58192i
$658$		0			0
$659$	100743.			0.00903649			0.00451824	−	0.999990i	$-0.498562\pi$
$659$	100743.			0.00903649			0.00451824	+	0.999990i	$0.498562\pi$
$660$		0			0
$661$	−5.45043e6			−0.485207			−0.242603	−	0.970126i	$-0.578001\pi$
$661$	−5.45043e6			−0.485207			−0.242603	+	0.970126i	$0.578001\pi$
$662$		0			0
$663$			1.60926e7i			1.42181i
$664$		0			0
$665$	−2.04735e6	+	729300.i	−0.179530	+	0.0639517i
$666$		0			0
$667$		−	1.84867e7i		−	1.60896i
$668$		0			0
$669$	−1.92492e7			−1.66283
$670$		0			0
$671$	−7.67477e6			−0.658050
$672$		0			0
$673$		−	3.25735e6i		−	0.277221i	−0.990347	−	0.138611i	$-0.955736\pi$
$673$		−	3.25735e6i		−	0.277221i	0.990347	−	0.138611i	$-0.0442637\pi$
$674$		0			0
$675$	−155951.	−	191123.i	−0.0131743	−	0.0161455i
$676$		0			0
$677$		−	5.82826e6i		−	0.488728i	−0.969684	−	0.244364i	$-0.921421\pi$
$677$		−	5.82826e6i		−	0.488728i	0.969684	−	0.244364i	$-0.0785792\pi$
$678$		0			0
$679$	−459694.			−0.0382643
$680$		0			0
$681$	−2.34096e7			−1.93431
$682$		0			0
$683$			2.13611e7i			1.75215i	0.482174	+	0.876076i	$0.339847\pi$
$683$			2.13611e7i			1.75215i	−0.482174	+	0.876076i	$0.660153\pi$
$684$		0			0
$685$	2.28828e7	−	8.15123e6i	1.86330	−	0.663738i
$686$		0			0
$687$			1.44363e7i			1.16698i
$688$		0			0
$689$	5.14257e6			0.412698
$690$		0			0
$691$	−1.58254e7			−1.26083			−0.630417	−	0.776256i	$-0.717116\pi$
$691$	−1.58254e7			−1.26083			−0.630417	+	0.776256i	$0.717116\pi$
$692$		0			0
$693$		−	1.61397e6i		−	0.127662i
$694$		0			0
$695$	−2.15060e6	−	6.03735e6i	−0.168888	−	0.474115i
$696$		0			0
$697$			2.57567e6i			0.200820i
$698$		0			0
$699$	−1.20618e7			−0.933726
$700$		0			0
$701$	2.75175e6			0.211502			0.105751	−	0.994393i	$-0.466275\pi$
$701$	2.75175e6			0.211502			0.105751	+	0.994393i	$0.466275\pi$
$702$		0			0
$703$			1.22132e7i			0.932052i
$704$		0			0
$705$	1.52053e6	+	4.26856e6i	0.115219	+	0.323451i
$706$		0			0
$707$			1.11475e6i			0.0838746i
$708$		0			0
$709$	−1.10623e7			−0.826474			−0.413237	−	0.910624i	$-0.635602\pi$
$709$	−1.10623e7			−0.826474			−0.413237	+	0.910624i	$0.635602\pi$
$710$		0			0
$711$	1.76313e7			1.30801
$712$		0			0
$713$			3.13536e7i			2.30974i
$714$		0			0
$715$	1.27627e7	−	4.54630e6i	0.933639	−	0.332578i
$716$		0			0
$717$			9.96294e6i			0.723752i
$718$		0			0
$719$	2.57064e7			1.85447			0.927234	−	0.374483i	$-0.122180\pi$
$719$	2.57064e7			1.85447			0.927234	+	0.374483i	$0.122180\pi$
$720$		0			0
$721$	5.44158e6			0.389840
$722$		0			0
$723$			2.38099e7i			1.69399i
$724$		0			0
$725$	1.09808e7	−	8.96005e6i	0.775872	−	0.633090i
$726$		0			0
$727$		−	1.44657e7i		−	1.01509i	−0.861626	−	0.507544i	$-0.830553\pi$
$727$		−	1.44657e7i		−	1.01509i	0.861626	−	0.507544i	$-0.169447\pi$
$728$		0			0
$729$	1.39214e7			0.970205
$730$		0			0
$731$	2.01367e6			0.139378
$732$		0			0
$733$		−	6.59256e6i		−	0.453204i	−0.973987	−	0.226602i	$-0.927238\pi$
$733$		−	6.59256e6i		−	0.453204i	0.973987	−	0.226602i	$-0.0727617\pi$
$734$		0			0
$735$	1.84192e7	−	6.56122e6i	1.25763	−	0.447988i
$736$		0			0
$737$			5.63960e6i			0.382455i
$738$		0			0
$739$	−8.77086e6			−0.590788			−0.295394	−	0.955376i	$-0.595451\pi$
$739$	−8.77086e6			−0.590788			−0.295394	+	0.955376i	$0.595451\pi$
$740$		0			0
$741$	3.06983e7			2.05385
$742$		0			0
$743$			5.10837e6i			0.339477i	0.985489	+	0.169738i	$0.0542923\pi$
$743$			5.10837e6i			0.339477i	−0.985489	+	0.169738i	$0.945708\pi$
$744$		0			0
$745$	−5.53230e6	−	1.55307e7i	−0.365187	−	1.02518i
$746$		0			0
$747$		−	1.61249e7i		−	1.05729i
$748$		0			0
$749$	2.15462e6			0.140335
$750$		0			0
$751$	1.65673e7			1.07190			0.535948	−	0.844251i	$-0.319954\pi$
$751$	1.65673e7			1.07190			0.535948	+	0.844251i	$0.319954\pi$
$752$		0			0
$753$			2.42334e7i			1.55750i
$754$		0			0
$755$	−7.58828e6	−	2.13024e7i	−0.484480	−	1.36007i
$756$		0			0
$757$		−	1.57190e7i		−	0.996978i	−0.866896	−	0.498489i	$-0.833888\pi$
$757$		−	1.57190e7i		−	0.996978i	0.866896	−	0.498489i	$-0.166112\pi$
$758$		0			0
$759$	2.03234e7			1.28054
$760$		0			0
$761$	2.04524e7			1.28022			0.640108	−	0.768285i	$-0.278890\pi$
$761$	2.04524e7			1.28022			0.640108	+	0.768285i	$0.278890\pi$
$762$		0			0
$763$			4.74176e6i			0.294868i
$764$		0			0
$765$	−8.65181e6	+	3.08192e6i	−0.534507	+	0.190400i
$766$		0			0
$767$			3.32769e7i			2.04247i
$768$		0			0
$769$	−3.29593e6			−0.200984			−0.100492	−	0.994938i	$-0.532042\pi$
$769$	−3.29593e6			−0.200984			−0.100492	+	0.994938i	$0.532042\pi$
$770$		0			0
$771$	7.92580e6			0.480183
$772$		0			0
$773$			1.32963e7i			0.800354i	0.916438	+	0.400177i	$0.131051\pi$
$773$			1.32963e7i			0.800354i	−0.916438	+	0.400177i	$0.868949\pi$
$774$		0			0
$775$	−1.86236e7	+	1.51963e7i	−1.11380	+	0.908834i
$776$		0			0
$777$			6.08542e6i			0.361608i
$778$		0			0
$779$	4.91335e6			0.290091
$780$		0			0
$781$	−3.21288e6			−0.188481
$782$		0			0
$783$		−	357991.i		−	0.0208674i
$784$		0			0
$785$	−1.98770e7	+	7.08054e6i	−1.15127	+	0.410102i
$786$		0			0
$787$		−	1.98876e6i		−	0.114458i	−0.998361	−	0.0572290i	$-0.981773\pi$
$787$		−	1.98876e6i		−	0.114458i	0.998361	−	0.0572290i	$-0.0182265\pi$
$788$		0			0
$789$	−1.08081e7			−0.618099
$790$		0			0
$791$	3.86951e6			0.219895
$792$		0			0
$793$		−	3.60969e7i		−	2.03839i
$794$		0			0
$795$	1.98451e6	+	5.57107e6i	0.111361	+	0.312623i
$796$		0			0
$797$			2.20524e7i			1.22973i	0.788632	+	0.614866i	$0.210790\pi$
$797$			2.20524e7i			1.22973i	−0.788632	+	0.614866i	$0.789210\pi$
$798$		0			0
$799$	−2.53267e6			−0.140350
$800$		0			0
$801$	−1.44308e7			−0.794710
$802$		0			0
$803$		−	1.65956e7i		−	0.908245i
$804$		0			0
$805$	−2.27087e6	−	6.37496e6i	−0.123510	−	0.346727i
$806$		0			0
$807$		−	1.77867e7i		−	0.961417i
$808$		0			0
$809$	5.20415e6			0.279562			0.139781	−	0.990182i	$-0.455360\pi$
$809$	5.20415e6			0.279562			0.139781	+	0.990182i	$0.455360\pi$
$810$		0			0
$811$	1.00506e7			0.536585			0.268293	−	0.963337i	$-0.413541\pi$
$811$	1.00506e7			0.536585			0.268293	+	0.963337i	$0.413541\pi$
$812$		0			0
$813$			2.36051e7i			1.25251i
$814$		0			0
$815$	2.22446e7	−	7.92390e6i	1.17309	−	0.417874i
$816$		0			0
$817$		−	3.84129e6i		−	0.201336i
$818$		0			0
$819$	7.59100e6			0.395448
$820$		0			0
$821$	−1.48945e7			−0.771204			−0.385602	−	0.922665i	$-0.626006\pi$
$821$	−1.48945e7			−0.771204			−0.385602	+	0.922665i	$0.626006\pi$
$822$		0			0
$823$			4.97038e6i			0.255794i	0.991787	+	0.127897i	$0.0408227\pi$
$823$			4.97038e6i			0.255794i	−0.991787	+	0.127897i	$0.959177\pi$
$824$		0			0
$825$	9.85024e6	+	1.20718e7i	0.503862	+	0.617499i
$826$		0			0
$827$			1.50182e7i			0.763579i	0.924249	+	0.381790i	$0.124692\pi$
$827$			1.50182e7i			0.763579i	−0.924249	+	0.381790i	$0.875308\pi$
$828$		0			0
$829$	−2.69399e7			−1.36148			−0.680738	−	0.732527i	$-0.738341\pi$
$829$	−2.69399e7			−1.36148			−0.680738	+	0.732527i	$0.738341\pi$
$830$		0			0
$831$	1.37433e7			0.690380
$832$		0			0
$833$			1.09287e7i			0.545701i
$834$		0			0
$835$	−1.90038e7	+	6.76948e6i	−0.943246	+	0.336000i
$836$		0			0
$837$			607157.i			0.0299562i
$838$		0			0
$839$	−1.35679e7			−0.665441			−0.332720	−	0.943026i	$-0.607966\pi$
$839$	−1.35679e7			−0.665441			−0.332720	+	0.943026i	$0.607966\pi$
$840$		0			0
$841$	57008.6			0.00277939
$842$		0			0
$843$			5.25090e6i			0.254487i
$844$		0			0
$845$	1.44178e7	+	4.04748e7i	0.694637	+	1.95004i
$846$		0			0
$847$			3.25261e6i			0.155784i
$848$		0			0
$849$	2.12265e6			0.101067
$850$		0			0
$851$	−3.80290e7			−1.80007
$852$		0			0
$853$		−	8.75046e6i		−	0.411773i	−0.978576	−	0.205887i	$-0.933992\pi$
$853$		−	8.75046e6i		−	0.411773i	0.978576	−	0.205887i	$-0.0660078\pi$
$854$		0			0
$855$	5.87908e6	+	1.65042e7i	0.275039	+	0.772111i
$856$		0			0
$857$		−	2.85540e7i		−	1.32805i	−0.747710	−	0.664025i	$-0.768847\pi$
$857$		−	2.85540e7i		−	1.32805i	0.747710	−	0.664025i	$-0.231153\pi$
$858$		0			0
$859$	1.50591e7			0.696333			0.348167	−	0.937433i	$-0.386804\pi$
$859$	1.50591e7			0.696333			0.348167	+	0.937433i	$0.386804\pi$
$860$		0			0
$861$	2.44816e6			0.112547
$862$		0			0
$863$		−	1.17155e7i		−	0.535467i	−0.963493	−	0.267734i	$-0.913725\pi$
$863$		−	1.17155e7i		−	0.535467i	0.963493	−	0.267734i	$-0.0862747\pi$
$864$		0			0
$865$	−2.13378e7	+	7.60087e6i	−0.969636	+	0.345401i
$866$		0			0
$867$			2.08415e7i			0.941634i
$868$		0			0
$869$	1.67177e7			0.750979
$870$		0			0
$871$	−2.65249e7			−1.18470
$872$		0			0
$873$			3.70572e6i			0.164565i
$874$		0			0
$875$	2.68600e6	−	4.43865e6i	0.118600	−	0.195989i
$876$		0			0
$877$			2.27293e7i			0.997901i	0.866630	+	0.498951i	$0.166281\pi$
$877$			2.27293e7i			0.997901i	−0.866630	+	0.498951i	$0.833719\pi$
$878$		0			0
$879$	−1.52846e7			−0.667241
$880$		0			0
$881$	1.47515e7			0.640317			0.320159	−	0.947364i	$-0.396264\pi$
$881$	1.47515e7			0.640317			0.320159	+	0.947364i	$0.396264\pi$
$882$		0			0
$883$		−	2.41244e7i		−	1.04125i	−0.853786	−	0.520625i	$-0.825699\pi$
$883$		−	2.41244e7i		−	1.04125i	0.853786	−	0.520625i	$-0.174301\pi$
$884$		0			0
$885$	−3.60497e7	+	1.28415e7i	−1.54719	+	0.551135i
$886$		0			0
$887$		−	1.83011e7i		−	0.781031i	−0.920596	−	0.390515i	$-0.872297\pi$
$887$		−	1.83011e7i		−	0.781031i	0.920596	−	0.390515i	$-0.127703\pi$
$888$		0			0
$889$	−7.13560e6			−0.302814
$890$		0			0
$891$	1.35995e7			0.573890
$892$		0			0
$893$			4.83133e6i			0.202740i
$894$		0			0
$895$	−3.73285e6	−	1.04792e7i	−0.155770	−	0.437289i
$896$		0			0
$897$			9.55874e7i			3.96661i
$898$		0			0
$899$	−3.48838e7			−1.43954
$900$		0			0
$901$	−3.30549e6			−0.135651
$902$		0			0
$903$		−	1.91399e6i		−	0.0781124i
$904$		0			0
$905$	4.56586e6	+	1.28176e7i	0.185311	+	0.520220i
$906$		0			0
$907$			1.52809e7i			0.616782i	0.951260	+	0.308391i	$0.0997905\pi$
$907$			1.52809e7i			0.616782i	−0.951260	+	0.308391i	$0.900210\pi$
$908$		0			0
$909$	8.98633e6			0.360722
$910$		0			0
$911$	−1.77783e7			−0.709732			−0.354866	−	0.934917i	$-0.615473\pi$
$911$	−1.77783e7			−0.709732			−0.354866	+	0.934917i	$0.615473\pi$
$912$		0			0
$913$		−	1.52894e7i		−	0.607033i
$914$		0			0
$915$	3.91046e7	−	1.39297e7i	1.54410	−	0.550034i
$916$		0			0
$917$		−	1.15756e7i		−	0.454592i
$918$		0			0
$919$	−4.10179e7			−1.60208			−0.801040	−	0.598611i	$-0.795720\pi$
$919$	−4.10179e7			−1.60208			−0.801040	+	0.598611i	$0.795720\pi$
$920$		0			0
$921$	−4.16956e7			−1.61972
$922$		0			0
$923$		−	1.51112e7i		−	0.583841i
$924$		0			0
$925$	−1.84317e7	−	2.25886e7i	−0.708290	−	0.868031i
$926$		0			0
$927$		−	4.38660e7i		−	1.67660i
$928$		0			0
$929$	−2.93427e7			−1.11548			−0.557739	−	0.830016i	$-0.688331\pi$
$929$	−2.93427e7			−1.11548			−0.557739	+	0.830016i	$0.688331\pi$
$930$		0			0
$931$	2.08476e7			0.788282
$932$		0			0
$933$		−	2.90981e7i		−	1.09436i
$934$		0			0
$935$	−8.20351e6	+	2.92223e6i	−0.306881	+	0.109316i
$936$		0			0
$937$		−	2.04953e7i		−	0.762614i	−0.924448	−	0.381307i	$-0.875474\pi$
$937$		−	2.04953e7i		−	0.762614i	0.924448	−	0.381307i	$-0.124526\pi$
$938$		0			0
$939$	1.47444e7			0.545712
$940$		0			0
$941$	−4.61324e7			−1.69837			−0.849184	−	0.528097i	$-0.822906\pi$
$941$	−4.61324e7			−1.69837			−0.849184	+	0.528097i	$0.822906\pi$
$942$		0			0
$943$			1.52990e7i			0.560253i
$944$		0			0
$945$	−43974.9	−	123450.i	−0.00160186	−	0.00449688i
$946$		0			0
$947$			2.69893e7i			0.977949i	0.872298	+	0.488974i	$0.162629\pi$
$947$			2.69893e7i			0.977949i	−0.872298	+	0.488974i	$0.837371\pi$
$948$		0			0
$949$	7.80542e7			2.81339
$950$		0			0
$951$	−4.51098e7			−1.61741
$952$		0			0
$953$			2.07300e7i			0.739377i	0.929156	+	0.369689i	$0.120536\pi$
$953$			2.07300e7i			0.739377i	−0.929156	+	0.369689i	$0.879464\pi$
$954$		0			0
$955$	4.48705e6	+	1.25964e7i	0.159204	+	0.446929i
$956$		0			0
$957$			2.26116e7i			0.798090i
$958$		0			0
$959$	1.29050e7			0.453116
$960$		0			0
$961$	3.05341e7			1.06654
$962$		0			0
$963$		−	1.73690e7i		−	0.603544i
$964$		0			0
$965$	−2.66924e7	+	9.50828e6i	−0.922719	+	0.328688i
$966$		0			0
$967$		−	1.26875e7i		−	0.436324i	−0.975913	−	0.218162i	$-0.929994\pi$
$967$		−	1.26875e7i		−	0.436324i	0.975913	−	0.218162i	$-0.0700062\pi$
$968$		0			0
$969$	−1.97319e7			−0.675088
$970$		0			0
$971$	−1.10734e7			−0.376905			−0.188452	−	0.982082i	$-0.560347\pi$
$971$	−1.10734e7			−0.376905			−0.188452	+	0.982082i	$0.560347\pi$
$972$		0			0
$973$		−	3.40482e6i		−	0.115295i
$974$		0			0
$975$	−5.67774e7	+	4.63288e7i	−1.91278	+	1.56077i
$976$		0			0
$977$			1.14221e7i			0.382833i	0.981509	+	0.191417i	$0.0613082\pi$
$977$			1.14221e7i			0.382833i	−0.981509	+	0.191417i	$0.938692\pi$
$978$		0			0
$979$	−1.36830e7			−0.456274
$980$		0			0
$981$	3.82246e7			1.26815
$982$		0			0
$983$		−	8.53757e6i		−	0.281806i	−0.990023	−	0.140903i	$-0.954999\pi$
$983$		−	8.53757e6i		−	0.281806i	0.990023	−	0.140903i	$-0.0450006\pi$
$984$		0			0
$985$	2.59941e7	−	9.25952e6i	0.853657	−	0.304087i
$986$		0			0
$987$			2.40730e6i			0.0786568i
$988$		0			0
$989$	1.19609e7			0.388841
$990$		0			0
$991$	8.29735e6			0.268383			0.134192	−	0.990955i	$-0.457156\pi$
$991$	8.29735e6			0.268383			0.134192	+	0.990955i	$0.457156\pi$
$992$		0			0
$993$			6.02040e7i			1.93755i
$994$		0			0
$995$	−9.18278e6	−	2.57786e7i	−0.294047	−	0.825472i
$996$		0			0
$997$			1.92324e6i			0.0612767i	0.999531	+	0.0306384i	$0.00975402\pi$
$997$			1.92324e6i			0.0612767i	−0.999531	+	0.0306384i	$0.990246\pi$
$998$		0			0
$999$	−736424.			−0.0233461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.c.l.129.11		12
4.3	odd	2	inner	320.6.c.l.129.1		12
5.4	even	2	inner	320.6.c.l.129.2		12
8.3	odd	2		160.6.c.d.129.12	yes	12
8.5	even	2		160.6.c.d.129.2	yes	12
20.19	odd	2	inner	320.6.c.l.129.12		12
40.3	even	4		800.6.a.w.1.1		6
40.13	odd	4		800.6.a.bb.1.6		6
40.19	odd	2		160.6.c.d.129.1	✓	12
40.27	even	4		800.6.a.bb.1.5		6
40.29	even	2		160.6.c.d.129.11	yes	12
40.37	odd	4		800.6.a.w.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.c.d.129.1	✓	12	40.19	odd	2
160.6.c.d.129.2	yes	12	8.5	even	2
160.6.c.d.129.11	yes	12	40.29	even	2
160.6.c.d.129.12	yes	12	8.3	odd	2
320.6.c.l.129.1		12	4.3	odd	2	inner
320.6.c.l.129.2		12	5.4	even	2	inner
320.6.c.l.129.11		12	1.1	even	1	trivial
320.6.c.l.129.12		12	20.19	odd	2	inner
800.6.a.w.1.1		6	40.3	even	4
800.6.a.w.1.2		6	40.37	odd	4
800.6.a.bb.1.5		6	40.27	even	4
800.6.a.bb.1.6		6	40.13	odd	4

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+21.9637i q^{3} +(-18.7585 - 52.6604i) q^{5} -29.6983i q^{7} -239.406 q^{9} +O(q^{10})\)	\(q+21.9637i q^{3} +(-18.7585 - 52.6604i) q^{5} -29.6983i q^{7} -239.406 q^{9} -227.001 q^{11} -1067.66i q^{13} +(1156.62 - 412.007i) q^{15} +686.258i q^{17} +1309.11 q^{19} +652.286 q^{21} +4076.26i q^{23} +(-2421.24 + 1975.66i) q^{25} +78.9360i q^{27} -4535.21 q^{29} +7691.77 q^{31} -4985.79i q^{33} +(-1563.92 + 557.096i) q^{35} +9329.38i q^{37} +23449.8 q^{39} +3753.20 q^{41} -2934.28i q^{43} +(4490.90 + 12607.2i) q^{45} +3690.55i q^{47} +15925.0 q^{49} -15072.8 q^{51} +4816.68i q^{53} +(4258.20 + 11954.0i) q^{55} +28752.9i q^{57} -31168.2 q^{59} +33809.4 q^{61} +7109.96i q^{63} +(-56223.3 + 20027.7i) q^{65} -24844.0i q^{67} -89529.9 q^{69} +14153.6 q^{71} +73107.8i q^{73} +(-43392.9 - 53179.4i) q^{75} +6741.55i q^{77} -73646.1 q^{79} -59909.4 q^{81} +67353.7i q^{83} +(36138.6 - 12873.2i) q^{85} -99610.3i q^{87} +60277.5 q^{89} -31707.6 q^{91} +168940. i q^{93} +(-24556.9 - 68938.2i) q^{95} -15478.8i q^{97} +54345.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 60 q^{5} - 268 q^{9}+O(q^{10}) \) 12 * q + 60 * q^5 - 268 * q^9	\( 12 q + 60 q^{5} - 268 q^{9} - 12080 q^{21} - 16420 q^{25} + 7864 q^{29} + 65560 q^{41} + 60420 q^{45} + 17956 q^{49} + 118232 q^{61} + 11360 q^{65} - 204464 q^{69} - 346436 q^{81} - 96320 q^{85} + 354936 q^{89}+O(q^{100}) \) 12 * q + 60 * q^5 - 268 * q^9 - 12080 * q^21 - 16420 * q^25 + 7864 * q^29 + 65560 * q^41 + 60420 * q^45 + 17956 * q^49 + 118232 * q^61 + 11360 * q^65 - 204464 * q^69 - 346436 * q^81 - 96320 * q^85 + 354936 * q^89

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