LMFDB - Embedded newform 720.6.f.a.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,6,Mod(289,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("720.289");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	720.f (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:	$115.476350265$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	720.289
Dual form			720.6.f.a.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-55.0000 + 10.0000i) q^{5} -158.000i q^{7} -148.000 q^{11} -684.000i q^{13} -2048.00i q^{17} +2220.00 q^{19} +1246.00i q^{23} +(2925.00 - 1100.00i) q^{25} -270.000 q^{29} +2048.00 q^{31} +(1580.00 + 8690.00i) q^{35} -4372.00i q^{37} +2398.00 q^{41} +2294.00i q^{43} -10682.0i q^{47} -8157.00 q^{49} +2964.00i q^{53} +(8140.00 - 1480.00i) q^{55} +39740.0 q^{59} -42298.0 q^{61} +(6840.00 + 37620.0i) q^{65} -32098.0i q^{67} -4248.00 q^{71} -30104.0i q^{73} +23384.0i q^{77} +35280.0 q^{79} +27826.0i q^{83} +(20480.0 + 112640. i) q^{85} -85210.0 q^{89} -108072. q^{91} +(-122100. + 22200.0i) q^{95} -97232.0i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 110 q^{5} - 296 q^{11} + 4440 q^{19} + 5850 q^{25} - 540 q^{29} + 4096 q^{31} + 3160 q^{35} + 4796 q^{41} - 16314 q^{49} + 16280 q^{55} + 79480 q^{59} - 84596 q^{61} + 13680 q^{65} - 8496 q^{71} + 70560 q^{79}+ \cdots - 244200 q^{95}+O(q^{100}) $ 2 * q - 110 * q^5 - 296 * q^11 + 4440 * q^19 + 5850 * q^25 - 540 * q^29 + 4096 * q^31 + 3160 * q^35 + 4796 * q^41 - 16314 * q^49 + 16280 * q^55 + 79480 * q^59 - 84596 * q^61 + 13680 * q^65 - 8496 * q^71 + 70560 * q^79 + 40960 * q^85 - 170420 * q^89 - 216144 * q^91 - 244200 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−55.0000	+	10.0000i	−0.983870	+	0.178885i
$5$	−55.0000	+	10.0000i	−0.983870	+	0.178885i
$6$		0			0
$7$		−	158.000i		−	1.21874i	−0.792885	−	0.609371i	$-0.791422\pi$
$7$		−	158.000i		−	1.21874i	0.792885	−	0.609371i	$-0.208578\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−148.000			−0.368791			−0.184395	−	0.982852i	$-0.559033\pi$
$11$	−148.000			−0.368791			−0.184395	+	0.982852i	$0.559033\pi$
$12$		0			0
$13$		−	684.000i		−	1.12253i	−0.827636	−	0.561265i	$-0.810315\pi$
$13$		−	684.000i		−	1.12253i	0.827636	−	0.561265i	$-0.189685\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	2048.00i		−	1.71873i	−0.511363	−	0.859365i	$-0.670859\pi$
$17$		−	2048.00i		−	1.71873i	0.511363	−	0.859365i	$-0.329141\pi$
$18$		0			0
$19$	2220.00			1.41081			0.705406	−	0.708804i	$-0.250765\pi$
$19$	2220.00			1.41081			0.705406	+	0.708804i	$0.250765\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1246.00i			0.491132i	0.969380	+	0.245566i	$0.0789738\pi$
$23$			1246.00i			0.491132i	−0.969380	+	0.245566i	$0.921026\pi$
$24$		0			0
$25$	2925.00	−	1100.00i	0.936000	−	0.352000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−270.000			−0.0596168			−0.0298084	−	0.999556i	$-0.509490\pi$
$29$	−270.000			−0.0596168			−0.0298084	+	0.999556i	$0.509490\pi$
$30$		0			0
$31$	2048.00			0.382759			0.191380	−	0.981516i	$-0.438704\pi$
$31$	2048.00			0.382759			0.191380	+	0.981516i	$0.438704\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1580.00	+	8690.00i	0.218015	+	1.19908i
$36$		0			0
$37$		−	4372.00i		−	0.525020i	−0.964929	−	0.262510i	$-0.915450\pi$
$37$		−	4372.00i		−	0.525020i	0.964929	−	0.262510i	$-0.0845503\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2398.00			0.222787			0.111393	−	0.993776i	$-0.464469\pi$
$41$	2398.00			0.222787			0.111393	+	0.993776i	$0.464469\pi$
$42$		0			0
$43$			2294.00i			0.189200i	0.995515	+	0.0946002i	$0.0301573\pi$
$43$			2294.00i			0.189200i	−0.995515	+	0.0946002i	$0.969843\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10682.0i		−	0.705355i	−0.935745	−	0.352678i	$-0.885271\pi$
$47$		−	10682.0i		−	0.705355i	0.935745	−	0.352678i	$-0.114729\pi$
$48$		0			0
$49$	−8157.00			−0.485333
$50$		0			0
$51$		0			0
$52$		0			0
$53$			2964.00i			0.144940i	0.997371	+	0.0724700i	$0.0230882\pi$
$53$			2964.00i			0.144940i	−0.997371	+	0.0724700i	$0.976912\pi$
$54$		0			0
$55$	8140.00	−	1480.00i	0.362842	−	0.0659713i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	39740.0			1.48627			0.743135	−	0.669141i	$-0.233338\pi$
$59$	39740.0			1.48627			0.743135	+	0.669141i	$0.233338\pi$
$60$		0			0
$61$	−42298.0			−1.45544			−0.727722	−	0.685873i	$-0.759421\pi$
$61$	−42298.0			−1.45544			−0.727722	+	0.685873i	$0.759421\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6840.00	+	37620.0i	0.200804	+	1.10442i
$66$		0			0
$67$		−	32098.0i		−	0.873556i	−0.899569	−	0.436778i	$-0.856119\pi$
$67$		−	32098.0i		−	0.873556i	0.899569	−	0.436778i	$-0.143881\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4248.00			−0.100009			−0.0500044	−	0.998749i	$-0.515924\pi$
$71$	−4248.00			−0.100009			−0.0500044	+	0.998749i	$0.515924\pi$
$72$		0			0
$73$		−	30104.0i		−	0.661176i	−0.943775	−	0.330588i	$-0.892753\pi$
$73$		−	30104.0i		−	0.661176i	0.943775	−	0.330588i	$-0.107247\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			23384.0i			0.449461i
$78$		0			0
$79$	35280.0			0.636005			0.318003	−	0.948090i	$-0.396988\pi$
$79$	35280.0			0.636005			0.318003	+	0.948090i	$0.396988\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			27826.0i			0.443359i	0.975120	+	0.221680i	$0.0711539\pi$
$83$			27826.0i			0.443359i	−0.975120	+	0.221680i	$0.928846\pi$
$84$		0			0
$85$	20480.0	+	112640.i	0.307456	+	1.69101i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−85210.0			−1.14029			−0.570145	−	0.821544i	$-0.693113\pi$
$89$	−85210.0			−1.14029			−0.570145	+	0.821544i	$0.693113\pi$
$90$		0			0
$91$	−108072.			−1.36807
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−122100.	+	22200.0i	−1.38805	+	0.252374i
$96$		0			0
$97$		−	97232.0i		−	1.04925i	−0.851333	−	0.524626i	$-0.824205\pi$
$97$		−	97232.0i		−	1.04925i	0.851333	−	0.524626i	$-0.175795\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4298.00			0.0419240			0.0209620	−	0.999780i	$-0.493327\pi$
$101$	4298.00			0.0419240			0.0209620	+	0.999780i	$0.493327\pi$
$102$		0			0
$103$			124114.i			1.15273i	0.817192	+	0.576365i	$0.195529\pi$
$103$			124114.i			1.15273i	−0.817192	+	0.576365i	$0.804471\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	42342.0i		−	0.357530i	−0.983892	−	0.178765i	$-0.942790\pi$
$107$		−	42342.0i		−	0.357530i	0.983892	−	0.178765i	$-0.0572101\pi$
$108$		0			0
$109$	35990.0			0.290145			0.145073	−	0.989421i	$-0.453658\pi$
$109$	35990.0			0.290145			0.145073	+	0.989421i	$0.453658\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	228816.i		−	1.68574i	−0.538118	−	0.842869i	$-0.680865\pi$
$113$		−	228816.i		−	1.68574i	0.538118	−	0.842869i	$-0.319135\pi$
$114$		0			0
$115$	−12460.0	−	68530.0i	−0.0878564	−	0.483210i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−323584.			−2.09469
$120$		0			0
$121$	−139147.			−0.863993
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−149875.	+	89750.0i	−0.857935	+	0.513759i
$126$		0			0
$127$		−	175238.i		−	0.964093i	−0.876146	−	0.482047i	$-0.839894\pi$
$127$		−	175238.i		−	0.964093i	0.876146	−	0.482047i	$-0.160106\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	299652.			1.52559			0.762797	−	0.646638i	$-0.223826\pi$
$131$	299652.			1.52559			0.762797	+	0.646638i	$0.223826\pi$
$132$		0			0
$133$		−	350760.i		−	1.71942i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	107928.i		−	0.491284i	−0.969361	−	0.245642i	$-0.921001\pi$
$137$		−	107928.i		−	0.491284i	0.969361	−	0.245642i	$-0.0789988\pi$
$138$		0			0
$139$	−196460.			−0.862456			−0.431228	−	0.902243i	$-0.641920\pi$
$139$	−196460.			−0.862456			−0.431228	+	0.902243i	$0.641920\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			101232.i			0.413978i
$144$		0			0
$145$	14850.0	−	2700.00i	0.0586552	−	0.0106646i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	138850.			0.512366			0.256183	−	0.966628i	$-0.417535\pi$
$149$	138850.			0.512366			0.256183	+	0.966628i	$0.417535\pi$
$150$		0			0
$151$	−416152.			−1.48528			−0.742642	−	0.669688i	$-0.766428\pi$
$151$	−416152.			−1.48528			−0.742642	+	0.669688i	$0.766428\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−112640.	+	20480.0i	−0.376585	+	0.0684701i
$156$		0			0
$157$			433108.i			1.40232i	0.713004	+	0.701160i	$0.247334\pi$
$157$			433108.i			1.40232i	−0.713004	+	0.701160i	$0.752666\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	196868.			0.598564
$162$		0			0
$163$			149134.i			0.439651i	0.975539	+	0.219825i	$0.0705487\pi$
$163$			149134.i			0.439651i	−0.975539	+	0.219825i	$0.929451\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	559602.i		−	1.55270i	−0.630301	−	0.776351i	$-0.717068\pi$
$167$		−	559602.i		−	1.55270i	0.630301	−	0.776351i	$-0.282932\pi$
$168$		0			0
$169$	−96563.0			−0.260072
$170$		0			0
$171$		0			0
$172$		0			0
$173$			343804.i			0.873365i	0.899616	+	0.436682i	$0.143847\pi$
$173$			343804.i			0.873365i	−0.899616	+	0.436682i	$0.856153\pi$
$174$		0			0
$175$	−173800.	−	462150.i	−0.428997	−	1.14074i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−23980.0			−0.0559392			−0.0279696	−	0.999609i	$-0.508904\pi$
$179$	−23980.0			−0.0559392			−0.0279696	+	0.999609i	$0.508904\pi$
$180$		0			0
$181$	−651898.			−1.47905			−0.739526	−	0.673128i	$-0.764950\pi$
$181$	−651898.			−1.47905			−0.739526	+	0.673128i	$0.764950\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	43720.0	+	240460.i	0.0939184	+	0.516551i
$186$		0			0
$187$			303104.i			0.633852i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	202752.			0.402144			0.201072	−	0.979576i	$-0.435557\pi$
$191$	202752.			0.402144			0.201072	+	0.979576i	$0.435557\pi$
$192$		0			0
$193$			452656.i			0.874732i	0.899284	+	0.437366i	$0.144089\pi$
$193$			452656.i			0.874732i	−0.899284	+	0.437366i	$0.855911\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	337468.i		−	0.619537i	−0.950812	−	0.309768i	$-0.899748\pi$
$197$		−	337468.i		−	0.619537i	0.950812	−	0.309768i	$-0.100252\pi$
$198$		0			0
$199$	−561000.			−1.00422			−0.502112	−	0.864803i	$-0.667443\pi$
$199$	−561000.			−1.00422			−0.502112	+	0.864803i	$0.667443\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			42660.0i			0.0726576i
$204$		0			0
$205$	−131890.	+	23980.0i	−0.219193	+	0.0398533i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−328560.			−0.520294
$210$		0			0
$211$	805548.			1.24562			0.622810	−	0.782373i	$-0.285991\pi$
$211$	805548.			1.24562			0.622810	+	0.782373i	$0.285991\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−22940.0	−	126170.i	−0.0338452	−	0.186149i
$216$		0			0
$217$		−	323584.i		−	0.466485i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.40083e6			−1.92932
$222$		0			0
$223$			1.21855e6i			1.64090i	0.571717	+	0.820451i	$0.306278\pi$
$223$			1.21855e6i			1.64090i	−0.571717	+	0.820451i	$0.693722\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			564338.i			0.726900i	0.931614	+	0.363450i	$0.118401\pi$
$227$			564338.i			0.726900i	−0.931614	+	0.363450i	$0.881599\pi$
$228$		0			0
$229$	−560330.			−0.706082			−0.353041	−	0.935608i	$-0.614852\pi$
$229$	−560330.			−0.706082			−0.353041	+	0.935608i	$0.614852\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	293576.i		−	0.354267i	−0.984187	−	0.177134i	$-0.943318\pi$
$233$		−	293576.i		−	0.354267i	0.984187	−	0.177134i	$-0.0566824\pi$
$234$		0			0
$235$	106820.	+	587510.i	0.126178	+	0.693978i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−584240.			−0.661602			−0.330801	−	0.943701i	$-0.607319\pi$
$239$	−584240.			−0.661602			−0.330801	+	0.943701i	$0.607319\pi$
$240$		0			0
$241$	−563798.			−0.625289			−0.312645	−	0.949870i	$-0.601215\pi$
$241$	−563798.			−0.625289			−0.312645	+	0.949870i	$0.601215\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	448635.	−	81570.0i	0.477505	−	0.0868191i
$246$		0			0
$247$		−	1.51848e6i		−	1.58368i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.01975e6			−1.02167			−0.510833	−	0.859680i	$-0.670663\pi$
$251$	−1.01975e6			−1.02167			−0.510833	+	0.859680i	$0.670663\pi$
$252$		0			0
$253$		−	184408.i		−	0.181125i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	657408.i		−	0.620872i	−0.950594	−	0.310436i	$-0.899525\pi$
$257$		−	657408.i		−	0.620872i	0.950594	−	0.310436i	$-0.100475\pi$
$258$		0			0
$259$	−690776.			−0.639864
$260$		0			0
$261$		0			0
$262$		0			0
$263$			562366.i			0.501337i	0.968073	+	0.250668i	$0.0806504\pi$
$263$			562366.i			0.501337i	−0.968073	+	0.250668i	$0.919350\pi$
$264$		0			0
$265$	−29640.0	−	163020.i	−0.0259277	−	0.142602i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	366570.			0.308870			0.154435	−	0.988003i	$-0.450644\pi$
$269$	366570.			0.308870			0.154435	+	0.988003i	$0.450644\pi$
$270$		0			0
$271$	−1.16075e6			−0.960099			−0.480050	−	0.877241i	$-0.659381\pi$
$271$	−1.16075e6			−0.960099			−0.480050	+	0.877241i	$0.659381\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−432900.	+	162800.i	−0.345188	+	0.129814i
$276$		0			0
$277$		−	2.51501e6i		−	1.96943i	−0.174172	−	0.984715i	$-0.555725\pi$
$277$		−	2.51501e6i		−	1.96943i	0.174172	−	0.984715i	$-0.444275\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.08600e6			−1.57597			−0.787987	−	0.615692i	$-0.788876\pi$
$281$	−2.08600e6			−1.57597			−0.787987	+	0.615692i	$0.788876\pi$
$282$		0			0
$283$		−	2.23803e6i		−	1.66111i	−0.556935	−	0.830556i	$-0.688023\pi$
$283$		−	2.23803e6i		−	1.66111i	0.556935	−	0.830556i	$-0.311977\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	378884.i		−	0.271520i
$288$		0			0
$289$	−2.77445e6			−1.95403
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	975756.i		−	0.664006i	−0.943278	−	0.332003i	$-0.892276\pi$
$293$		−	975756.i		−	0.664006i	0.943278	−	0.332003i	$-0.107724\pi$
$294$		0			0
$295$	−2.18570e6	+	397400.i	−1.46230	+	0.265872i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	852264.			0.551310
$300$		0			0
$301$	362452.			0.230587
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.32639e6	−	422980.i	1.43197	−	0.260358i
$306$		0			0
$307$		−	87858.0i		−	0.0532029i	−0.999646	−	0.0266015i	$-0.991531\pi$
$307$		−	87858.0i		−	0.0532029i	0.999646	−	0.0266015i	$-0.00846850\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	599352.			0.351383			0.175692	−	0.984445i	$-0.443784\pi$
$311$	599352.			0.351383			0.175692	+	0.984445i	$0.443784\pi$
$312$		0			0
$313$			2.09342e6i			1.20780i	0.797060	+	0.603900i	$0.206387\pi$
$313$			2.09342e6i			1.20780i	−0.797060	+	0.603900i	$0.793613\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			2.41625e6i			1.35050i	0.737590	+	0.675249i	$0.235964\pi$
$317$			2.41625e6i			1.35050i	−0.737590	+	0.675249i	$0.764036\pi$
$318$		0			0
$319$	39960.0			0.0219861
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4.54656e6i		−	2.42480i
$324$		0			0
$325$	−752400.	−	2.00070e6i	−0.395130	−	1.05069i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.68776e6			−0.859647
$330$		0			0
$331$	1.64095e6			0.823237			0.411618	−	0.911356i	$-0.364964\pi$
$331$	1.64095e6			0.823237			0.411618	+	0.911356i	$0.364964\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	320980.	+	1.76539e6i	0.156267	+	0.859466i
$336$		0			0
$337$			2.18773e6i			1.04935i	0.851304	+	0.524673i	$0.175812\pi$
$337$			2.18773e6i			1.04935i	−0.851304	+	0.524673i	$0.824188\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−303104.			−0.141158
$342$		0			0
$343$		−	1.36670e6i		−	0.627246i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			2.74502e6i			1.22383i	0.790923	+	0.611916i	$0.209601\pi$
$347$			2.74502e6i			1.22383i	−0.790923	+	0.611916i	$0.790399\pi$
$348$		0			0
$349$	2.65115e6			1.16512			0.582560	−	0.812788i	$-0.302051\pi$
$349$	2.65115e6			1.16512			0.582560	+	0.812788i	$0.302051\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			3.05766e6i			1.30603i	0.757345	+	0.653015i	$0.226496\pi$
$353$			3.05766e6i			1.30603i	−0.757345	+	0.653015i	$0.773504\pi$
$354$		0			0
$355$	233640.	−	42480.0i	0.0983957	−	0.0178901i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.79356e6			−1.55350			−0.776749	−	0.629810i	$-0.783133\pi$
$359$	−3.79356e6			−1.55350			−0.776749	+	0.629810i	$0.783133\pi$
$360$		0			0
$361$	2.45230e6			0.990389
$362$		0			0
$363$		0			0
$364$		0			0
$365$	301040.	+	1.65572e6i	0.118275	+	0.650511i
$366$		0			0
$367$			3.11060e6i			1.20553i	0.797917	+	0.602767i	$0.205935\pi$
$367$			3.11060e6i			1.20553i	−0.797917	+	0.602767i	$0.794065\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	468312.			0.176645
$372$		0			0
$373$			1.41520e6i			0.526677i	0.964703	+	0.263339i	$0.0848236\pi$
$373$			1.41520e6i			0.526677i	−0.964703	+	0.263339i	$0.915176\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			184680.i			0.0669216i
$378$		0			0
$379$	−3.90262e6			−1.39559			−0.697796	−	0.716297i	$-0.745836\pi$
$379$	−3.90262e6			−1.39559			−0.697796	+	0.716297i	$0.745836\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	695674.i		−	0.242331i	−0.992632	−	0.121165i	$-0.961337\pi$
$383$		−	695674.i		−	0.242331i	0.992632	−	0.121165i	$-0.0386632\pi$
$384$		0			0
$385$	−233840.	−	1.28612e6i	−0.0804020	−	0.442211i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	498290.			0.166958			0.0834792	−	0.996510i	$-0.473397\pi$
$389$	498290.			0.166958			0.0834792	+	0.996510i	$0.473397\pi$
$390$		0			0
$391$	2.55181e6			0.844124
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1.94040e6	+	352800.i	−0.625747	+	0.113772i
$396$		0			0
$397$			1.09567e6i			0.348901i	0.984666	+	0.174451i	$0.0558150\pi$
$397$			1.09567e6i			0.348901i	−0.984666	+	0.174451i	$0.944185\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.49160e6			0.773779			0.386890	−	0.922126i	$-0.373549\pi$
$401$	2.49160e6			0.773779			0.386890	+	0.922126i	$0.373549\pi$
$402$		0			0
$403$		−	1.40083e6i		−	0.429659i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			647056.i			0.193623i
$408$		0			0
$409$	3.63349e6			1.07403			0.537014	−	0.843573i	$-0.319552\pi$
$409$	3.63349e6			1.07403			0.537014	+	0.843573i	$0.319552\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	6.27892e6i		−	1.81138i
$414$		0			0
$415$	−278260.	−	1.53043e6i	−0.0793105	−	0.436208i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.64378e6			1.01395			0.506976	−	0.861960i	$-0.330763\pi$
$419$	3.64378e6			1.01395			0.506976	+	0.861960i	$0.330763\pi$
$420$		0			0
$421$	−1.82530e6			−0.501913			−0.250957	−	0.967998i	$-0.580745\pi$
$421$	−1.82530e6			−0.501913			−0.250957	+	0.967998i	$0.580745\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.25280e6	−	5.99040e6i	−0.604993	−	1.60873i
$426$		0			0
$427$			6.68308e6i			1.77381i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.85435e6			0.740141			0.370070	−	0.929004i	$-0.379334\pi$
$431$	2.85435e6			0.740141			0.370070	+	0.929004i	$0.379334\pi$
$432$		0			0
$433$			587776.i			0.150658i	0.997159	+	0.0753290i	$0.0240007\pi$
$433$			587776.i			0.150658i	−0.997159	+	0.0753290i	$0.975999\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			2.76612e6i			0.692895i
$438$		0			0
$439$	6.11604e6			1.51464			0.757319	−	0.653045i	$-0.226509\pi$
$439$	6.11604e6			1.51464			0.757319	+	0.653045i	$0.226509\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			2.35771e6i			0.570795i	0.958409	+	0.285398i	$0.0921257\pi$
$443$			2.35771e6i			0.570795i	−0.958409	+	0.285398i	$0.907874\pi$
$444$		0			0
$445$	4.68655e6	−	852100.i	1.12190	−	0.203981i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.49735e6			1.28688			0.643439	−	0.765497i	$-0.277507\pi$
$449$	5.49735e6			1.28688			0.643439	+	0.765497i	$0.277507\pi$
$450$		0			0
$451$	−354904.			−0.0821617
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.94396e6	−	1.08072e6i	1.34601	−	0.244729i
$456$		0			0
$457$		−	1.16039e6i		−	0.259905i	−0.991520	−	0.129952i	$-0.958518\pi$
$457$		−	1.16039e6i		−	0.259905i	0.991520	−	0.129952i	$-0.0414824\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.30330e6			0.504775			0.252387	−	0.967626i	$-0.418784\pi$
$461$	2.30330e6			0.504775			0.252387	+	0.967626i	$0.418784\pi$
$462$		0			0
$463$			2.71343e6i			0.588257i	0.955766	+	0.294128i	$0.0950293\pi$
$463$			2.71343e6i			0.588257i	−0.955766	+	0.294128i	$0.904971\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			4.05050e6i			0.859441i	0.902962	+	0.429721i	$0.141388\pi$
$467$			4.05050e6i			0.859441i	−0.902962	+	0.429721i	$0.858612\pi$
$468$		0			0
$469$	−5.07148e6			−1.06464
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	339512.i		−	0.0697754i
$474$		0			0
$475$	6.49350e6	−	2.44200e6i	1.32052	−	0.496606i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.60528e6			−1.11624			−0.558121	−	0.829759i	$-0.688478\pi$
$479$	−5.60528e6			−1.11624			−0.558121	+	0.829759i	$0.688478\pi$
$480$		0			0
$481$	−2.99045e6			−0.589350
$482$		0			0
$483$		0			0
$484$		0			0
$485$	972320.	+	5.34776e6i	0.187696	+	1.03233i
$486$		0			0
$487$		−	7.13168e6i		−	1.36260i	−0.732003	−	0.681301i	$-0.761414\pi$
$487$		−	7.13168e6i		−	1.36260i	0.732003	−	0.681301i	$-0.238586\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.88145e6			1.10098			0.550492	−	0.834841i	$-0.314440\pi$
$491$	5.88145e6			1.10098			0.550492	+	0.834841i	$0.314440\pi$
$492$		0			0
$493$			552960.i			0.102465i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			671184.i			0.121885i
$498$		0			0
$499$	1.75710e6			0.315897			0.157948	−	0.987447i	$-0.449512\pi$
$499$	1.75710e6			0.315897			0.157948	+	0.987447i	$0.449512\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	4.91411e6i		−	0.866015i	−0.901390	−	0.433007i	$-0.857452\pi$
$503$		−	4.91411e6i		−	0.866015i	0.901390	−	0.433007i	$-0.142548\pi$
$504$		0			0
$505$	−236390.	+	42980.0i	−0.0412478	+	0.00749960i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.75499e6			−0.984578			−0.492289	−	0.870432i	$-0.663840\pi$
$509$	−5.75499e6			−0.984578			−0.492289	+	0.870432i	$0.663840\pi$
$510$		0			0
$511$	−4.75643e6			−0.805803
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.24114e6	−	6.82627e6i	−0.206207	−	1.13414i
$516$		0			0
$517$			1.58094e6i			0.260128i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.61980e6			0.261437			0.130718	−	0.991420i	$-0.458272\pi$
$521$	1.61980e6			0.261437			0.130718	+	0.991420i	$0.458272\pi$
$522$		0			0
$523$			1.19117e7i			1.90422i	0.305751	+	0.952112i	$0.401093\pi$
$523$			1.19117e7i			1.90422i	−0.305751	+	0.952112i	$0.598907\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	4.19430e6i		−	0.657860i
$528$		0			0
$529$	4.88383e6			0.758789
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	1.64023e6i		−	0.250085i
$534$		0			0
$535$	423420.	+	2.32881e6i	0.0639568	+	0.351763i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.20724e6			0.178986
$540$		0			0
$541$	4.07630e6			0.598788			0.299394	−	0.954130i	$-0.403215\pi$
$541$	4.07630e6			0.598788			0.299394	+	0.954130i	$0.403215\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.97945e6	+	359900.i	−0.285465	+	0.0519028i
$546$		0			0
$547$		−	1.23680e7i		−	1.76739i	−0.468065	−	0.883694i	$-0.655049\pi$
$547$		−	1.23680e7i		−	1.76739i	0.468065	−	0.883694i	$-0.344951\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−599400.			−0.0841081
$552$		0			0
$553$		−	5.57424e6i		−	0.775127i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	130308.i		−	0.0177964i	−0.999960	−	0.00889822i	$-0.997168\pi$
$557$		−	130308.i		−	0.0177964i	0.999960	−	0.00889822i	$-0.00283243\pi$
$558$		0			0
$559$	1.56910e6			0.212383
$560$		0			0
$561$		0			0
$562$		0			0
$563$			5.91687e6i			0.786721i	0.919384	+	0.393361i	$0.128688\pi$
$563$			5.91687e6i			0.786721i	−0.919384	+	0.393361i	$0.871312\pi$
$564$		0			0
$565$	2.28816e6	+	1.25849e7i	0.301554	+	1.65855i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.03013e6			−1.16927			−0.584633	−	0.811298i	$-0.698761\pi$
$569$	−9.03013e6			−1.16927			−0.584633	+	0.811298i	$0.698761\pi$
$570$		0			0
$571$	1.07093e7			1.37459			0.687294	−	0.726379i	$-0.258798\pi$
$571$	1.07093e7			1.37459			0.687294	+	0.726379i	$0.258798\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.37060e6	+	3.64455e6i	0.172879	+	0.459700i
$576$		0			0
$577$		−	1.22051e6i		−	0.152617i	−0.997084	−	0.0763084i	$-0.975687\pi$
$577$		−	1.22051e6i		−	0.152617i	0.997084	−	0.0763084i	$-0.0243134\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.39651e6			0.540341
$582$		0			0
$583$		−	438672.i		−	0.0534526i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.47104e7i		−	1.76210i	−0.473026	−	0.881049i	$-0.656838\pi$
$587$		−	1.47104e7i		−	1.76210i	0.473026	−	0.881049i	$-0.343162\pi$
$588$		0			0
$589$	4.54656e6			0.540001
$590$		0			0
$591$		0			0
$592$		0			0
$593$			8.52014e6i			0.994970i	0.867472	+	0.497485i	$0.165743\pi$
$593$			8.52014e6i			0.994970i	−0.867472	+	0.497485i	$0.834257\pi$
$594$		0			0
$595$	1.77971e7	−	3.23584e6i	2.06090	−	0.374709i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.90100e6			−0.330355			−0.165177	−	0.986264i	$-0.552820\pi$
$599$	−2.90100e6			−0.330355			−0.165177	+	0.986264i	$0.552820\pi$
$600$		0			0
$601$	5.72760e6			0.646825			0.323412	−	0.946258i	$-0.395170\pi$
$601$	5.72760e6			0.646825			0.323412	+	0.946258i	$0.395170\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.65308e6	−	1.39147e6i	0.850057	−	0.154556i
$606$		0			0
$607$			8.79924e6i			0.969334i	0.874699	+	0.484667i	$0.161059\pi$
$607$			8.79924e6i			0.969334i	−0.874699	+	0.484667i	$0.838941\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.30649e6			−0.791782
$612$		0			0
$613$		−	1.03408e6i		−	0.111149i	−0.998455	−	0.0555744i	$-0.982301\pi$
$613$		−	1.03408e6i		−	0.111149i	0.998455	−	0.0555744i	$-0.0176990\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.29854e7i		−	1.37323i	−0.727020	−	0.686616i	$-0.759095\pi$
$617$		−	1.29854e7i		−	1.37323i	0.727020	−	0.686616i	$-0.240905\pi$
$618$		0			0
$619$	7.92002e6			0.830806			0.415403	−	0.909637i	$-0.363641\pi$
$619$	7.92002e6			0.830806			0.415403	+	0.909637i	$0.363641\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			1.34632e7i			1.38972i
$624$		0			0
$625$	7.34562e6	−	6.43500e6i	0.752192	−	0.658944i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8.95386e6			−0.902368
$630$		0			0
$631$	−1.68218e7			−1.68189			−0.840945	−	0.541120i	$-0.818001\pi$
$631$	−1.68218e7			−1.68189			−0.840945	+	0.541120i	$0.818001\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.75238e6	+	9.63809e6i	0.172462	+	0.948542i
$636$		0			0
$637$			5.57939e6i			0.544801i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.55154e7			1.49148			0.745741	−	0.666236i	$-0.232096\pi$
$641$	1.55154e7			1.49148			0.745741	+	0.666236i	$0.232096\pi$
$642$		0			0
$643$			1.05801e7i			1.00916i	0.863364	+	0.504582i	$0.168354\pi$
$643$			1.05801e7i			1.00916i	−0.863364	+	0.504582i	$0.831646\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.37883e7i			1.29494i	0.762090	+	0.647471i	$0.224173\pi$
$647$			1.37883e7i			1.29494i	−0.762090	+	0.647471i	$0.775827\pi$
$648$		0			0
$649$	−5.88152e6			−0.548123
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	1.58924e6i		−	0.145850i	−0.997337	−	0.0729248i	$-0.976767\pi$
$653$		−	1.58924e6i		−	0.145850i	0.997337	−	0.0729248i	$-0.0232333\pi$
$654$		0			0
$655$	−1.64809e7	+	2.99652e6i	−1.50099	+	0.272907i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.12434e6			0.818442			0.409221	−	0.912435i	$-0.365801\pi$
$659$	9.12434e6			0.818442			0.409221	+	0.912435i	$0.365801\pi$
$660$		0			0
$661$	6.50310e6			0.578918			0.289459	−	0.957190i	$-0.406525\pi$
$661$	6.50310e6			0.578918			0.289459	+	0.957190i	$0.406525\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.50760e6	+	1.92918e7i	0.307578	+	1.69168i
$666$		0			0
$667$		−	336420.i		−	0.0292797i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6.26010e6			0.536754
$672$		0			0
$673$			2.17810e6i			0.185370i	0.995695	+	0.0926850i	$0.0295449\pi$
$673$			2.17810e6i			0.185370i	−0.995695	+	0.0926850i	$0.970455\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	3.98419e6i		−	0.334094i	−0.985949	−	0.167047i	$-0.946577\pi$
$677$		−	3.98419e6i		−	0.334094i	0.985949	−	0.167047i	$-0.0534231\pi$
$678$		0			0
$679$	−1.53627e7			−1.27877
$680$		0			0
$681$		0			0
$682$		0			0
$683$			5.91563e6i			0.485231i	0.970122	+	0.242616i	$0.0780054\pi$
$683$			5.91563e6i			0.485231i	−0.970122	+	0.242616i	$0.921995\pi$
$684$		0			0
$685$	1.07928e6	+	5.93604e6i	0.0878836	+	0.483360i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.02738e6			0.162700
$690$		0			0
$691$	1.55471e7			1.23867			0.619335	−	0.785127i	$-0.287402\pi$
$691$	1.55471e7			1.23867			0.619335	+	0.785127i	$0.287402\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.08053e7	−	1.96460e6i	0.848545	−	0.154281i
$696$		0			0
$697$		−	4.91110e6i		−	0.382910i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2.27103e7			1.74553			0.872766	−	0.488139i	$-0.162324\pi$
$701$	2.27103e7			1.74553			0.872766	+	0.488139i	$0.162324\pi$
$702$		0			0
$703$		−	9.70584e6i		−	0.740704i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	679084.i		−	0.0510946i
$708$		0			0
$709$	−6.29841e6			−0.470560			−0.235280	−	0.971928i	$-0.575601\pi$
$709$	−6.29841e6			−0.470560			−0.235280	+	0.971928i	$0.575601\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			2.55181e6i			0.187985i
$714$		0			0
$715$	−1.01232e6	−	5.56776e6i	−0.0740547	−	0.407301i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.11911e7			−1.52873			−0.764367	−	0.644782i	$-0.776948\pi$
$719$	−2.11911e7			−1.52873			−0.764367	+	0.644782i	$0.776948\pi$
$720$		0			0
$721$	1.96100e7			1.40488
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−789750.	+	297000.i	−0.0558013	+	0.0209851i
$726$		0			0
$727$		−	1.35610e7i		−	0.951605i	−0.879552	−	0.475803i	$-0.842158\pi$
$727$		−	1.35610e7i		−	0.951605i	0.879552	−	0.475803i	$-0.157842\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.69811e6			0.325185
$732$		0			0
$733$		−	2.69413e7i		−	1.85208i	−0.377429	−	0.926038i	$-0.623192\pi$
$733$		−	2.69413e7i		−	1.85208i	0.377429	−	0.926038i	$-0.376808\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			4.75050e6i			0.322160i
$738$		0			0
$739$	2.77414e6			0.186860			0.0934302	−	0.995626i	$-0.470217\pi$
$739$	2.77414e6			0.186860			0.0934302	+	0.995626i	$0.470217\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.85538e7i		−	1.23299i	−0.787358	−	0.616497i	$-0.788551\pi$
$743$		−	1.85538e7i		−	1.23299i	0.787358	−	0.616497i	$-0.211449\pi$
$744$		0			0
$745$	−7.63675e6	+	1.38850e6i	−0.504101	+	0.0916548i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.69004e6			−0.435736
$750$		0			0
$751$	2.19285e6			0.141876			0.0709380	−	0.997481i	$-0.477401\pi$
$751$	2.19285e6			0.141876			0.0709380	+	0.997481i	$0.477401\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2.28884e7	−	4.16152e6i	1.46133	−	0.265696i
$756$		0			0
$757$		−	9.48749e6i		−	0.601744i	−0.953665	−	0.300872i	$-0.902722\pi$
$757$		−	9.48749e6i		−	0.601744i	0.953665	−	0.300872i	$-0.0972777\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.69580e6			−0.606907			−0.303453	−	0.952846i	$-0.598140\pi$
$761$	−9.69580e6			−0.606907			−0.303453	+	0.952846i	$0.598140\pi$
$762$		0			0
$763$		−	5.68642e6i		−	0.353612i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	2.71822e7i		−	1.66838i
$768$		0			0
$769$	−9.32787e6			−0.568809			−0.284405	−	0.958704i	$-0.591796\pi$
$769$	−9.32787e6			−0.568809			−0.284405	+	0.958704i	$0.591796\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	9.68080e6i		−	0.582723i	−0.956613	−	0.291362i	$-0.905892\pi$
$773$		−	9.68080e6i		−	0.582723i	0.956613	−	0.291362i	$-0.0941083\pi$
$774$		0			0
$775$	5.99040e6	−	2.25280e6i	0.358263	−	0.134731i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.32356e6			0.314310
$780$		0			0
$781$	628704.			0.0368824
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.33108e6	−	2.38209e7i	−0.250855	−	1.37970i
$786$		0			0
$787$			5.52302e6i			0.317863i	0.987290	+	0.158931i	$0.0508049\pi$
$787$			5.52302e6i			0.317863i	−0.987290	+	0.158931i	$0.949195\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.61529e7			−2.05448
$792$		0			0
$793$			2.89318e7i			1.63378i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1.71119e7i			0.954230i	0.878841	+	0.477115i	$0.158318\pi$
$797$			1.71119e7i			0.954230i	−0.878841	+	0.477115i	$0.841682\pi$
$798$		0			0
$799$	−2.18767e7			−1.21232
$800$		0			0
$801$		0			0
$802$		0			0
$803$			4.45539e6i			0.243836i
$804$		0			0
$805$	−1.08277e7	+	1.96868e6i	−0.588909	+	0.107074i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1.45309e7			−0.780586			−0.390293	−	0.920691i	$-0.627626\pi$
$809$	−1.45309e7			−0.780586			−0.390293	+	0.920691i	$0.627626\pi$
$810$		0			0
$811$	2.13545e7			1.14009			0.570044	−	0.821614i	$-0.306926\pi$
$811$	2.13545e7			1.14009			0.570044	+	0.821614i	$0.306926\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.49134e6	−	8.20237e6i	−0.0786471	−	0.432559i
$816$		0			0
$817$			5.09268e6i			0.266926i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.67967e7			−1.90525			−0.952623	−	0.304154i	$-0.901626\pi$
$821$	−3.67967e7			−1.90525			−0.952623	+	0.304154i	$0.901626\pi$
$822$		0			0
$823$		−	3.30668e7i		−	1.70174i	−0.525376	−	0.850870i	$-0.676075\pi$
$823$		−	3.30668e7i		−	1.70174i	0.525376	−	0.850870i	$-0.323925\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1.77309e7i			0.901505i	0.892649	+	0.450752i	$0.148844\pi$
$827$			1.77309e7i			0.901505i	−0.892649	+	0.450752i	$0.851156\pi$
$828$		0			0
$829$	−1.29375e7			−0.653830			−0.326915	−	0.945054i	$-0.606009\pi$
$829$	−1.29375e7			−0.653830			−0.326915	+	0.945054i	$0.606009\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			1.67055e7i			0.834157i
$834$		0			0
$835$	5.59602e6	+	3.07781e7i	0.277756	+	1.52766i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.31812e7			−1.62738			−0.813688	−	0.581302i	$-0.802543\pi$
$839$	−3.31812e7			−1.62738			−0.813688	+	0.581302i	$0.802543\pi$
$840$		0			0
$841$	−2.04382e7			−0.996446
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5.31096e6	−	965630.i	0.255877	−	0.0465231i
$846$		0			0
$847$			2.19852e7i			1.05299i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.44751e6			0.257854
$852$		0			0
$853$			5.17224e6i			0.243392i	0.992567	+	0.121696i	$0.0388332\pi$
$853$			5.17224e6i			0.243392i	−0.992567	+	0.121696i	$0.961167\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1.05320e7i			0.489845i	0.969543	+	0.244922i	$0.0787625\pi$
$857$			1.05320e7i			0.489845i	−0.969543	+	0.244922i	$0.921238\pi$
$858$		0			0
$859$	−1.14741e7			−0.530563			−0.265282	−	0.964171i	$-0.585465\pi$
$859$	−1.14741e7			−0.530563			−0.265282	+	0.964171i	$0.585465\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.92722e7i		−	0.880856i	−0.897788	−	0.440428i	$-0.854827\pi$
$863$		−	1.92722e7i		−	0.880856i	0.897788	−	0.440428i	$-0.145173\pi$
$864$		0			0
$865$	−3.43804e6	−	1.89092e7i	−0.156232	−	0.859277i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.22144e6			−0.234553
$870$		0			0
$871$	−2.19550e7			−0.980593
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.41805e7	+	2.36802e7i	0.626140	+	1.04560i
$876$		0			0
$877$			2.30524e7i			1.01208i	0.862509	+	0.506042i	$0.168892\pi$
$877$			2.30524e7i			1.01208i	−0.862509	+	0.506042i	$0.831108\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.26690e7			−0.983994			−0.491997	−	0.870597i	$-0.663733\pi$
$881$	−2.26690e7			−0.983994			−0.491997	+	0.870597i	$0.663733\pi$
$882$		0			0
$883$			3.67337e6i			0.158549i	0.996853	+	0.0792745i	$0.0252604\pi$
$883$			3.67337e6i			0.158549i	−0.996853	+	0.0792745i	$0.974740\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			3.39649e7i			1.44951i	0.689007	+	0.724755i	$0.258047\pi$
$887$			3.39649e7i			1.44951i	−0.689007	+	0.724755i	$0.741953\pi$
$888$		0			0
$889$	−2.76876e7			−1.17498
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	2.37140e7i		−	0.995123i
$894$		0			0
$895$	1.31890e6	−	239800.i	0.0550369	−	0.0100067i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−552960.			−0.0228189
$900$		0			0
$901$	6.07027e6			0.249113
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.58544e7	−	6.51898e6i	1.45519	−	0.264581i
$906$		0			0
$907$			2.13327e7i			0.861050i	0.902579	+	0.430525i	$0.141672\pi$
$907$			2.13327e7i			0.861050i	−0.902579	+	0.430525i	$0.858328\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.03512e7			−0.413235			−0.206617	−	0.978422i	$-0.566246\pi$
$911$	−1.03512e7			−0.413235			−0.206617	+	0.978422i	$0.566246\pi$
$912$		0			0
$913$		−	4.11825e6i		−	0.163507i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	4.73450e7i		−	1.85931i
$918$		0			0
$919$	−2.59019e7			−1.01168			−0.505839	−	0.862628i	$-0.668817\pi$
$919$	−2.59019e7			−1.01168			−0.505839	+	0.862628i	$0.668817\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.90563e6i			0.112263i
$924$		0			0
$925$	−4.80920e6	−	1.27881e7i	−0.184807	−	0.491419i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.13230e7			−1.19076			−0.595379	−	0.803445i	$-0.702998\pi$
$929$	−3.13230e7			−1.19076			−0.595379	+	0.803445i	$0.702998\pi$
$930$		0			0
$931$	−1.81085e7			−0.684714
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.03104e6	−	1.66707e7i	−0.113387	−	0.623628i
$936$		0			0
$937$		−	2.08461e7i		−	0.775667i	−0.921729	−	0.387833i	$-0.873224\pi$
$937$		−	2.08461e7i		−	0.775667i	0.921729	−	0.387833i	$-0.126776\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3.82929e7			1.40976			0.704878	−	0.709328i	$-0.251002\pi$
$941$	3.82929e7			1.40976			0.704878	+	0.709328i	$0.251002\pi$
$942$		0			0
$943$			2.98791e6i			0.109418i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	4.25088e7i		−	1.54029i	−0.637866	−	0.770147i	$-0.720183\pi$
$947$		−	4.25088e7i		−	1.54029i	0.637866	−	0.770147i	$-0.279817\pi$
$948$		0			0
$949$	−2.05911e7			−0.742189
$950$		0			0
$951$		0			0
$952$		0			0
$953$			3.91855e7i			1.39763i	0.715302	+	0.698816i	$0.246289\pi$
$953$			3.91855e7i			1.39763i	−0.715302	+	0.698816i	$0.753711\pi$
$954$		0			0
$955$	−1.11514e7	+	2.02752e6i	−0.395658	+	0.0719377i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.70526e7			−0.598749
$960$		0			0
$961$	−2.44348e7			−0.853495
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.52656e6	−	2.48961e7i	−0.156477	−	0.860622i
$966$		0			0
$967$			1.84836e7i			0.635653i	0.948149	+	0.317827i	$0.102953\pi$
$967$			1.84836e7i			0.635653i	−0.948149	+	0.317827i	$0.897047\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3.95031e7			1.34457			0.672284	−	0.740294i	$-0.265314\pi$
$971$	3.95031e7			1.34457			0.672284	+	0.740294i	$0.265314\pi$
$972$		0			0
$973$			3.10407e7i			1.05111i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	3.29043e7i		−	1.10285i	−0.834225	−	0.551425i	$-0.814084\pi$
$977$		−	3.29043e7i		−	1.10285i	0.834225	−	0.551425i	$-0.185916\pi$
$978$		0			0
$979$	1.26111e7			0.420529
$980$		0			0
$981$		0			0
$982$		0			0
$983$			2.65797e7i			0.877338i	0.898649	+	0.438669i	$0.144550\pi$
$983$			2.65797e7i			0.877338i	−0.898649	+	0.438669i	$0.855450\pi$
$984$		0			0
$985$	3.37468e6	+	1.85607e7i	0.110826	+	0.609544i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2.85832e6			−0.0929225
$990$		0			0
$991$	−1.92964e7			−0.624153			−0.312077	−	0.950057i	$-0.601025\pi$
$991$	−1.92964e7			−0.624153			−0.312077	+	0.950057i	$0.601025\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.08550e7	−	5.61000e6i	0.988025	−	0.179641i
$996$		0			0
$997$		−	5.12017e7i		−	1.63135i	−0.578511	−	0.815674i	$-0.696366\pi$
$997$		−	5.12017e7i		−	1.63135i	0.578511	−	0.815674i	$-0.303634\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.6.f.a.289.2		2
3.2	odd	2		80.6.c.c.49.2		2
4.3	odd	2		90.6.c.a.19.2		2
5.4	even	2	inner	720.6.f.a.289.1		2
12.11	even	2		10.6.b.a.9.1	✓	2
15.2	even	4		400.6.a.k.1.1		1
15.8	even	4		400.6.a.c.1.1		1
15.14	odd	2		80.6.c.c.49.1		2
20.3	even	4		450.6.a.w.1.1		1
20.7	even	4		450.6.a.c.1.1		1
20.19	odd	2		90.6.c.a.19.1		2
24.5	odd	2		320.6.c.a.129.1		2
24.11	even	2		320.6.c.b.129.2		2
60.23	odd	4		50.6.a.c.1.1		1
60.47	odd	4		50.6.a.e.1.1		1
60.59	even	2		10.6.b.a.9.2	yes	2
120.29	odd	2		320.6.c.a.129.2		2
120.59	even	2		320.6.c.b.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.b.a.9.1	✓	2	12.11	even	2
10.6.b.a.9.2	yes	2	60.59	even	2
50.6.a.c.1.1		1	60.23	odd	4
50.6.a.e.1.1		1	60.47	odd	4
80.6.c.c.49.1		2	15.14	odd	2
80.6.c.c.49.2		2	3.2	odd	2
90.6.c.a.19.1		2	20.19	odd	2
90.6.c.a.19.2		2	4.3	odd	2
320.6.c.a.129.1		2	24.5	odd	2
320.6.c.a.129.2		2	120.29	odd	2
320.6.c.b.129.1		2	120.59	even	2
320.6.c.b.129.2		2	24.11	even	2
400.6.a.c.1.1		1	15.8	even	4
400.6.a.k.1.1		1	15.2	even	4
450.6.a.c.1.1		1	20.7	even	4
450.6.a.w.1.1		1	20.3	even	4
720.6.f.a.289.1		2	5.4	even	2	inner
720.6.f.a.289.2		2	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-55.0000 + 10.0000i) q^{5} -158.000i q^{7} -148.000 q^{11} -684.000i q^{13} -2048.00i q^{17} +2220.00 q^{19} +1246.00i q^{23} +(2925.00 - 1100.00i) q^{25} -270.000 q^{29} +2048.00 q^{31} +(1580.00 + 8690.00i) q^{35} -4372.00i q^{37} +2398.00 q^{41} +2294.00i q^{43} -10682.0i q^{47} -8157.00 q^{49} +2964.00i q^{53} +(8140.00 - 1480.00i) q^{55} +39740.0 q^{59} -42298.0 q^{61} +(6840.00 + 37620.0i) q^{65} -32098.0i q^{67} -4248.00 q^{71} -30104.0i q^{73} +23384.0i q^{77} +35280.0 q^{79} +27826.0i q^{83} +(20480.0 + 112640. i) q^{85} -85210.0 q^{89} -108072. q^{91} +(-122100. + 22200.0i) q^{95} -97232.0i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 110 q^{5} - 296 q^{11} + 4440 q^{19} + 5850 q^{25} - 540 q^{29} + 4096 q^{31} + 3160 q^{35} + 4796 q^{41} - 16314 q^{49} + 16280 q^{55} + 79480 q^{59} - 84596 q^{61} + 13680 q^{65} - 8496 q^{71} + 70560 q^{79}+ \cdots - 244200 q^{95}+O(q^{100}) \) 2 * q - 110 * q^5 - 296 * q^11 + 4440 * q^19 + 5850 * q^25 - 540 * q^29 + 4096 * q^31 + 3160 * q^35 + 4796 * q^41 - 16314 * q^49 + 16280 * q^55 + 79480 * q^59 - 84596 * q^61 + 13680 * q^65 - 8496 * q^71 + 70560 * q^79 + 40960 * q^85 - 170420 * q^89 - 216144 * q^91 - 244200 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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