LMFDB - Embedded newform 720.4.f.e.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,4,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, 4, names="a")

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

chi := DirichletCharacter("720.289");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
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:	$42.4813752041$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
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			720.4.f.e.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 + 11.0000i) q^{5} +10.0000i q^{7} +O(q^{10})$	$q+(2.00000 + 11.0000i) q^{5} +10.0000i q^{7} -14.0000 q^{11} -82.0000i q^{13} +18.0000i q^{17} -136.000 q^{19} -140.000i q^{23} +(-117.000 + 44.0000i) q^{25} +112.000 q^{29} -72.0000 q^{31} +(-110.000 + 20.0000i) q^{35} -26.0000i q^{37} +446.000 q^{41} -396.000i q^{43} +144.000i q^{47} +243.000 q^{49} -158.000i q^{53} +(-28.0000 - 154.000i) q^{55} +342.000 q^{59} +314.000 q^{61} +(902.000 - 164.000i) q^{65} -152.000i q^{67} -932.000 q^{71} -548.000i q^{73} -140.000i q^{77} -512.000 q^{79} +284.000i q^{83} +(-198.000 + 36.0000i) q^{85} -810.000 q^{89} +820.000 q^{91} +(-272.000 - 1496.00i) q^{95} -1304.00i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5}+O(q^{10}) $ 2 * q + 4 * q^5	$ 2 q + 4 q^{5} - 28 q^{11} - 272 q^{19} - 234 q^{25} + 224 q^{29} - 144 q^{31} - 220 q^{35} + 892 q^{41} + 486 q^{49} - 56 q^{55} + 684 q^{59} + 628 q^{61} + 1804 q^{65} - 1864 q^{71} - 1024 q^{79} - 396 q^{85} - 1620 q^{89} + 1640 q^{91} - 544 q^{95}+O(q^{100}) $ 2 * q + 4 * q^5 - 28 * q^11 - 272 * q^19 - 234 * q^25 + 224 * q^29 - 144 * q^31 - 220 * q^35 + 892 * q^41 + 486 * q^49 - 56 * q^55 + 684 * q^59 + 628 * q^61 + 1804 * q^65 - 1864 * q^71 - 1024 * q^79 - 396 * q^85 - 1620 * q^89 + 1640 * q^91 - 544 * q^95

Display 100 coefficients 10 coefficients

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
$5$	2.00000	+	11.0000i	0.178885	+	0.983870i
$5$	2.00000	+	11.0000i	0.178885	+	0.983870i
$6$		0			0
$7$			10.0000i			0.539949i	0.962867	+	0.269975i	$0.0870153\pi$
$7$			10.0000i			0.539949i	−0.962867	+	0.269975i	$0.912985\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−14.0000			−0.383742			−0.191871	−	0.981420i	$-0.561455\pi$
$11$	−14.0000			−0.383742			−0.191871	+	0.981420i	$0.561455\pi$
$12$		0			0
$13$		−	82.0000i		−	1.74944i	−0.484629	−	0.874720i	$-0.661046\pi$
$13$		−	82.0000i		−	1.74944i	0.484629	−	0.874720i	$-0.338954\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			18.0000i			0.256802i	0.991722	+	0.128401i	$0.0409845\pi$
$17$			18.0000i			0.256802i	−0.991722	+	0.128401i	$0.959015\pi$
$18$		0			0
$19$	−136.000			−1.64213			−0.821067	−	0.570832i	$-0.806621\pi$
$19$	−136.000			−1.64213			−0.821067	+	0.570832i	$0.806621\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	140.000i		−	1.26922i	−0.772833	−	0.634609i	$-0.781161\pi$
$23$		−	140.000i		−	1.26922i	0.772833	−	0.634609i	$-0.218839\pi$
$24$		0			0
$25$	−117.000	+	44.0000i	−0.936000	+	0.352000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	112.000			0.717168			0.358584	−	0.933497i	$-0.383260\pi$
$29$	112.000			0.717168			0.358584	+	0.933497i	$0.383260\pi$
$30$		0			0
$31$	−72.0000			−0.417148			−0.208574	−	0.978007i	$-0.566882\pi$
$31$	−72.0000			−0.417148			−0.208574	+	0.978007i	$0.566882\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−110.000	+	20.0000i	−0.531240	+	0.0965891i
$36$		0			0
$37$		−	26.0000i		−	0.115524i	−0.998330	−	0.0577618i	$-0.981604\pi$
$37$		−	26.0000i		−	0.115524i	0.998330	−	0.0577618i	$-0.0183964\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	446.000			1.69887			0.849433	−	0.527697i	$-0.176944\pi$
$41$	446.000			1.69887			0.849433	+	0.527697i	$0.176944\pi$
$42$		0			0
$43$		−	396.000i		−	1.40441i	−0.711977	−	0.702203i	$-0.752200\pi$
$43$		−	396.000i		−	1.40441i	0.711977	−	0.702203i	$-0.247800\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			144.000i			0.446906i	0.974715	+	0.223453i	$0.0717328\pi$
$47$			144.000i			0.446906i	−0.974715	+	0.223453i	$0.928267\pi$
$48$		0			0
$49$	243.000			0.708455
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	158.000i		−	0.409490i	−0.978815	−	0.204745i	$-0.934363\pi$
$53$		−	158.000i		−	0.409490i	0.978815	−	0.204745i	$-0.0656365\pi$
$54$		0			0
$55$	−28.0000	−	154.000i	−0.0686458	−	0.377552i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	342.000			0.754654			0.377327	−	0.926080i	$-0.376843\pi$
$59$	342.000			0.754654			0.377327	+	0.926080i	$0.376843\pi$
$60$		0			0
$61$	314.000			0.659075			0.329538	−	0.944142i	$-0.393107\pi$
$61$	314.000			0.659075			0.329538	+	0.944142i	$0.393107\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	902.000	−	164.000i	1.72122	−	0.312949i
$66$		0			0
$67$		−	152.000i		−	0.277161i	−0.990351	−	0.138580i	$-0.955746\pi$
$67$		−	152.000i		−	0.277161i	0.990351	−	0.138580i	$-0.0442539\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−932.000			−1.55786			−0.778930	−	0.627111i	$-0.784237\pi$
$71$	−932.000			−1.55786			−0.778930	+	0.627111i	$0.784237\pi$
$72$		0			0
$73$		−	548.000i		−	0.878610i	−0.898338	−	0.439305i	$-0.855225\pi$
$73$		−	548.000i		−	0.878610i	0.898338	−	0.439305i	$-0.144775\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	140.000i		−	0.207201i
$78$		0			0
$79$	−512.000			−0.729171			−0.364585	−	0.931170i	$-0.618789\pi$
$79$	−512.000			−0.729171			−0.364585	+	0.931170i	$0.618789\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			284.000i			0.375579i	0.982209	+	0.187789i	$0.0601323\pi$
$83$			284.000i			0.375579i	−0.982209	+	0.187789i	$0.939868\pi$
$84$		0			0
$85$	−198.000	+	36.0000i	−0.252660	+	0.0459382i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−810.000			−0.964717			−0.482359	−	0.875974i	$-0.660220\pi$
$89$	−810.000			−0.964717			−0.482359	+	0.875974i	$0.660220\pi$
$90$		0			0
$91$	820.000			0.944608
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−272.000	−	1496.00i	−0.293754	−	1.61565i
$96$		0			0
$97$		−	1304.00i		−	1.36496i	−0.730904	−	0.682480i	$-0.760901\pi$
$97$		−	1304.00i		−	1.36496i	0.730904	−	0.682480i	$-0.239099\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−936.000			−0.922133			−0.461067	−	0.887365i	$-0.652533\pi$
$101$	−936.000			−0.922133			−0.461067	+	0.887365i	$0.652533\pi$
$102$		0			0
$103$		−	1450.00i		−	1.38711i	−0.720402	−	0.693557i	$-0.756043\pi$
$103$		−	1450.00i		−	1.38711i	0.720402	−	0.693557i	$-0.243957\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1292.00i		−	1.16731i	−0.812001	−	0.583656i	$-0.801622\pi$
$107$		−	1292.00i		−	1.16731i	0.812001	−	0.583656i	$-0.198378\pi$
$108$		0			0
$109$	2142.00			1.88226			0.941130	−	0.338044i	$-0.109765\pi$
$109$	2142.00			1.88226			0.941130	+	0.338044i	$0.109765\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1418.00i			1.18048i	0.807228	+	0.590240i	$0.200967\pi$
$113$			1418.00i			1.18048i	−0.807228	+	0.590240i	$0.799033\pi$
$114$		0			0
$115$	1540.00	−	280.000i	1.24875	−	0.227045i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−180.000			−0.138660
$120$		0			0
$121$	−1135.00			−0.852742
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−718.000	−	1199.00i	−0.513759	−	0.857935i
$126$		0			0
$127$			1674.00i			1.16963i	0.811165	+	0.584817i	$0.198834\pi$
$127$			1674.00i			1.16963i	−0.811165	+	0.584817i	$0.801166\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1134.00			0.756321			0.378160	−	0.925740i	$-0.376557\pi$
$131$	1134.00			0.756321			0.378160	+	0.925740i	$0.376557\pi$
$132$		0			0
$133$		−	1360.00i		−	0.886669i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2866.00i		−	1.78729i	−0.448773	−	0.893646i	$-0.648139\pi$
$137$		−	2866.00i		−	1.78729i	0.448773	−	0.893646i	$-0.351861\pi$
$138$		0			0
$139$	−764.000			−0.466199			−0.233099	−	0.972453i	$-0.574887\pi$
$139$	−764.000			−0.466199			−0.233099	+	0.972453i	$0.574887\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1148.00i			0.671333i
$144$		0			0
$145$	224.000	+	1232.00i	0.128291	+	0.705600i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3060.00			−1.68245			−0.841225	−	0.540686i	$-0.818165\pi$
$149$	−3060.00			−1.68245			−0.841225	+	0.540686i	$0.818165\pi$
$150$		0			0
$151$	1304.00			0.702768			0.351384	−	0.936231i	$-0.385711\pi$
$151$	1304.00			0.702768			0.351384	+	0.936231i	$0.385711\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−144.000	−	792.000i	−0.0746217	−	0.410419i
$156$		0			0
$157$		−	1582.00i		−	0.804187i	−0.915599	−	0.402093i	$-0.868283\pi$
$157$		−	1582.00i		−	0.804187i	0.915599	−	0.402093i	$-0.131717\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1400.00			0.685313
$162$		0			0
$163$		−	3232.00i		−	1.55307i	−0.630077	−	0.776533i	$-0.716976\pi$
$163$		−	3232.00i		−	1.55307i	0.630077	−	0.776533i	$-0.283024\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2988.00i			1.38454i	0.721638	+	0.692271i	$0.243390\pi$
$167$			2988.00i			1.38454i	−0.721638	+	0.692271i	$0.756610\pi$
$168$		0			0
$169$	−4527.00			−2.06054
$170$		0			0
$171$		0			0
$172$		0			0
$173$			918.000i			0.403435i	0.979444	+	0.201717i	$0.0646523\pi$
$173$			918.000i			0.403435i	−0.979444	+	0.201717i	$0.935348\pi$
$174$		0			0
$175$	−440.000	−	1170.00i	−0.190062	−	0.505392i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1238.00			0.516941			0.258471	−	0.966019i	$-0.416781\pi$
$179$	1238.00			0.516941			0.258471	+	0.966019i	$0.416781\pi$
$180$		0			0
$181$	1450.00			0.595457			0.297728	−	0.954651i	$-0.403771\pi$
$181$	1450.00			0.595457			0.297728	+	0.954651i	$0.403771\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	286.000	−	52.0000i	0.113660	−	0.0206655i
$186$		0			0
$187$		−	252.000i		−	0.0985458i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4860.00			1.84114			0.920569	−	0.390581i	$-0.127726\pi$
$191$	4860.00			1.84114			0.920569	+	0.390581i	$0.127726\pi$
$192$		0			0
$193$		−	1412.00i		−	0.526622i	−0.964711	−	0.263311i	$-0.915186\pi$
$193$		−	1412.00i		−	0.526622i	0.964711	−	0.263311i	$-0.0848145\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1170.00i		−	0.423142i	−0.977363	−	0.211571i	$-0.932142\pi$
$197$		−	1170.00i		−	0.423142i	0.977363	−	0.211571i	$-0.0678580\pi$
$198$		0			0
$199$	2080.00			0.740941			0.370471	−	0.928844i	$-0.379196\pi$
$199$	2080.00			0.740941			0.370471	+	0.928844i	$0.379196\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1120.00i			0.387234i
$204$		0			0
$205$	892.000	+	4906.00i	0.303902	+	1.67146i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1904.00			0.630155
$210$		0			0
$211$	−2692.00			−0.878317			−0.439159	−	0.898410i	$-0.644723\pi$
$211$	−2692.00			−0.878317			−0.439159	+	0.898410i	$0.644723\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4356.00	−	792.000i	1.38175	−	0.251228i
$216$		0			0
$217$		−	720.000i		−	0.225239i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1476.00			0.449260
$222$		0			0
$223$			846.000i			0.254046i	0.991900	+	0.127023i	$0.0405423\pi$
$223$			846.000i			0.254046i	−0.991900	+	0.127023i	$0.959458\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	2884.00i		−	0.843250i	−0.906770	−	0.421625i	$-0.861460\pi$
$227$		−	2884.00i		−	0.843250i	0.906770	−	0.421625i	$-0.138540\pi$
$228$		0			0
$229$	−3150.00			−0.908986			−0.454493	−	0.890750i	$-0.650180\pi$
$229$	−3150.00			−0.908986			−0.454493	+	0.890750i	$0.650180\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4014.00i			1.12861i	0.825567	+	0.564304i	$0.190856\pi$
$233$			4014.00i			1.12861i	−0.825567	+	0.564304i	$0.809144\pi$
$234$		0			0
$235$	−1584.00	+	288.000i	−0.439697	+	0.0799449i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4900.00			−1.32617			−0.663085	−	0.748544i	$-0.730753\pi$
$239$	−4900.00			−1.32617			−0.663085	+	0.748544i	$0.730753\pi$
$240$		0			0
$241$	−2314.00			−0.618497			−0.309249	−	0.950981i	$-0.600078\pi$
$241$	−2314.00			−0.618497			−0.309249	+	0.950981i	$0.600078\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	486.000	+	2673.00i	0.126732	+	0.697027i
$246$		0			0
$247$			11152.0i			2.87281i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2002.00			−0.503447			−0.251723	−	0.967799i	$-0.580997\pi$
$251$	−2002.00			−0.503447			−0.251723	+	0.967799i	$0.580997\pi$
$252$		0			0
$253$			1960.00i			0.487052i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	450.000i		−	0.109223i	−0.998508	−	0.0546113i	$-0.982608\pi$
$257$		−	450.000i		−	0.109223i	0.998508	−	0.0546113i	$-0.0173920\pi$
$258$		0			0
$259$	260.000			0.0623769
$260$		0			0
$261$		0			0
$262$		0			0
$263$			180.000i			0.0422026i	0.999777	+	0.0211013i	$0.00671725\pi$
$263$			180.000i			0.0422026i	−0.999777	+	0.0211013i	$0.993283\pi$
$264$		0			0
$265$	1738.00	−	316.000i	0.402885	−	0.0732518i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2448.00			0.554859			0.277430	−	0.960746i	$-0.410517\pi$
$269$	2448.00			0.554859			0.277430	+	0.960746i	$0.410517\pi$
$270$		0			0
$271$	−6776.00			−1.51887			−0.759433	−	0.650586i	$-0.774524\pi$
$271$	−6776.00			−1.51887			−0.759433	+	0.650586i	$0.774524\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1638.00	−	616.000i	0.359182	−	0.135077i
$276$		0			0
$277$		−	6426.00i		−	1.39387i	−0.717136	−	0.696933i	$-0.754547\pi$
$277$		−	6426.00i		−	1.39387i	0.717136	−	0.696933i	$-0.245453\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2718.00			−0.577019			−0.288509	−	0.957477i	$-0.593160\pi$
$281$	−2718.00			−0.577019			−0.288509	+	0.957477i	$0.593160\pi$
$282$		0			0
$283$		−	6048.00i		−	1.27038i	−0.772358	−	0.635188i	$-0.780923\pi$
$283$		−	6048.00i		−	1.27038i	0.772358	−	0.635188i	$-0.219077\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4460.00i			0.917301i
$288$		0			0
$289$	4589.00			0.934053
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1246.00i		−	0.248437i	−0.992255	−	0.124219i	$-0.960358\pi$
$293$		−	1246.00i		−	0.248437i	0.992255	−	0.124219i	$-0.0396424\pi$
$294$		0			0
$295$	684.000	+	3762.00i	0.134997	+	0.742482i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−11480.0			−2.22042
$300$		0			0
$301$	3960.00			0.758308
$302$		0			0
$303$		0			0
$304$		0			0
$305$	628.000	+	3454.00i	0.117899	+	0.648444i
$306$		0			0
$307$			1244.00i			0.231267i	0.993292	+	0.115633i	$0.0368897\pi$
$307$			1244.00i			0.231267i	−0.993292	+	0.115633i	$0.963110\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6372.00			−1.16181			−0.580905	−	0.813971i	$-0.697301\pi$
$311$	−6372.00			−1.16181			−0.580905	+	0.813971i	$0.697301\pi$
$312$		0			0
$313$			5500.00i			0.993222i	0.867973	+	0.496611i	$0.165422\pi$
$313$			5500.00i			0.993222i	−0.867973	+	0.496611i	$0.834578\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			378.000i			0.0669735i	0.999439	+	0.0334867i	$0.0106612\pi$
$317$			378.000i			0.0669735i	−0.999439	+	0.0334867i	$0.989339\pi$
$318$		0			0
$319$	−1568.00			−0.275207
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	2448.00i		−	0.421704i
$324$		0			0
$325$	3608.00	+	9594.00i	0.615803	+	1.63747i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1440.00			−0.241306
$330$		0			0
$331$	11888.0			1.97409			0.987045	−	0.160446i	$-0.0512931\pi$
$331$	11888.0			1.97409			0.987045	+	0.160446i	$0.0512931\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1672.00	−	304.000i	0.272690	−	0.0495800i
$336$		0			0
$337$			9116.00i			1.47353i	0.676148	+	0.736766i	$0.263648\pi$
$337$			9116.00i			1.47353i	−0.676148	+	0.736766i	$0.736352\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1008.00			0.160077
$342$		0			0
$343$			5860.00i			0.922479i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			4676.00i			0.723403i	0.932294	+	0.361701i	$0.117804\pi$
$347$			4676.00i			0.723403i	−0.932294	+	0.361701i	$0.882196\pi$
$348$		0			0
$349$	−11906.0			−1.82611			−0.913057	−	0.407833i	$-0.866285\pi$
$349$	−11906.0			−1.82611			−0.913057	+	0.407833i	$0.866285\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			2142.00i			0.322966i	0.986875	+	0.161483i	$0.0516278\pi$
$353$			2142.00i			0.322966i	−0.986875	+	0.161483i	$0.948372\pi$
$354$		0			0
$355$	−1864.00	−	10252.0i	−0.278678	−	1.53273i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9824.00			−1.44426			−0.722132	−	0.691755i	$-0.756838\pi$
$359$	−9824.00			−1.44426			−0.722132	+	0.691755i	$0.756838\pi$
$360$		0			0
$361$	11637.0			1.69660
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6028.00	−	1096.00i	0.864438	−	0.157171i
$366$		0			0
$367$			5354.00i			0.761516i	0.924675	+	0.380758i	$0.124337\pi$
$367$			5354.00i			0.761516i	−0.924675	+	0.380758i	$0.875663\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1580.00			0.221104
$372$		0			0
$373$			7694.00i			1.06804i	0.845471	+	0.534022i	$0.179320\pi$
$373$			7694.00i			1.06804i	−0.845471	+	0.534022i	$0.820680\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	9184.00i		−	1.25464i
$378$		0			0
$379$	−6004.00			−0.813733			−0.406866	−	0.913488i	$-0.633379\pi$
$379$	−6004.00			−0.813733			−0.406866	+	0.913488i	$0.633379\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			9432.00i			1.25836i	0.777259	+	0.629181i	$0.216610\pi$
$383$			9432.00i			1.25836i	−0.777259	+	0.629181i	$0.783390\pi$
$384$		0			0
$385$	1540.00	−	280.000i	0.203859	−	0.0370653i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−6156.00			−0.802369			−0.401185	−	0.915997i	$-0.631401\pi$
$389$	−6156.00			−0.802369			−0.401185	+	0.915997i	$0.631401\pi$
$390$		0			0
$391$	2520.00			0.325938
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1024.00	−	5632.00i	−0.130438	−	0.717409i
$396$		0			0
$397$		−	7866.00i		−	0.994416i	−0.867631	−	0.497208i	$-0.834359\pi$
$397$		−	7866.00i		−	0.994416i	0.867631	−	0.497208i	$-0.165641\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2074.00			0.258281			0.129140	−	0.991626i	$-0.458778\pi$
$401$	2074.00			0.258281			0.129140	+	0.991626i	$0.458778\pi$
$402$		0			0
$403$			5904.00i			0.729775i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			364.000i			0.0443312i
$408$		0			0
$409$	2746.00			0.331983			0.165991	−	0.986127i	$-0.446918\pi$
$409$	2746.00			0.331983			0.165991	+	0.986127i	$0.446918\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			3420.00i			0.407475i
$414$		0			0
$415$	−3124.00	+	568.000i	−0.369521	+	0.0671856i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6426.00			0.749238			0.374619	−	0.927179i	$-0.377774\pi$
$419$	6426.00			0.749238			0.374619	+	0.927179i	$0.377774\pi$
$420$		0			0
$421$	−10610.0			−1.22827			−0.614133	−	0.789203i	$-0.710494\pi$
$421$	−10610.0			−1.22827			−0.614133	+	0.789203i	$0.710494\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−792.000	−	2106.00i	−0.0903945	−	0.240367i
$426$		0			0
$427$			3140.00i			0.355867i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8384.00			0.936991			0.468495	−	0.883466i	$-0.344796\pi$
$431$	8384.00			0.936991			0.468495	+	0.883466i	$0.344796\pi$
$432$		0			0
$433$			1980.00i			0.219752i	0.993945	+	0.109876i	$0.0350454\pi$
$433$			1980.00i			0.219752i	−0.993945	+	0.109876i	$0.964955\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			19040.0i			2.08423i
$438$		0			0
$439$	864.000			0.0939327			0.0469664	−	0.998896i	$-0.485045\pi$
$439$	864.000			0.0939327			0.0469664	+	0.998896i	$0.485045\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	13212.0i		−	1.41698i	−0.705722	−	0.708489i	$-0.749377\pi$
$443$		−	13212.0i		−	1.41698i	0.705722	−	0.708489i	$-0.250623\pi$
$444$		0			0
$445$	−1620.00	−	8910.00i	−0.172574	−	0.949156i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−16290.0			−1.71219			−0.856094	−	0.516820i	$-0.827116\pi$
$449$	−16290.0			−1.71219			−0.856094	+	0.516820i	$0.827116\pi$
$450$		0			0
$451$	−6244.00			−0.651926
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1640.00	+	9020.00i	0.168977	+	0.929372i
$456$		0			0
$457$		−	6336.00i		−	0.648546i	−0.945964	−	0.324273i	$-0.894880\pi$
$457$		−	6336.00i		−	0.648546i	0.945964	−	0.324273i	$-0.105120\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13716.0			1.38572			0.692861	−	0.721071i	$-0.256350\pi$
$461$	13716.0			1.38572			0.692861	+	0.721071i	$0.256350\pi$
$462$		0			0
$463$			14626.0i			1.46809i	0.679098	+	0.734047i	$0.262371\pi$
$463$			14626.0i			1.46809i	−0.679098	+	0.734047i	$0.737629\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			5796.00i			0.574319i	0.957883	+	0.287159i	$0.0927109\pi$
$467$			5796.00i			0.574319i	−0.957883	+	0.287159i	$0.907289\pi$
$468$		0			0
$469$	1520.00			0.149653
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5544.00i			0.538929i
$474$		0			0
$475$	15912.0	−	5984.00i	1.53704	−	0.578031i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8348.00			0.796305			0.398152	−	0.917319i	$-0.369652\pi$
$479$	8348.00			0.796305			0.398152	+	0.917319i	$0.369652\pi$
$480$		0			0
$481$	−2132.00			−0.202102
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14344.0	−	2608.00i	1.34294	−	0.244172i
$486$		0			0
$487$		−	1898.00i		−	0.176605i	−0.996094	−	0.0883025i	$-0.971856\pi$
$487$		−	1898.00i		−	0.176605i	0.996094	−	0.0883025i	$-0.0281442\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	16654.0			1.53072			0.765361	−	0.643601i	$-0.222560\pi$
$491$	16654.0			1.53072			0.765361	+	0.643601i	$0.222560\pi$
$492$		0			0
$493$			2016.00i			0.184171i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	9320.00i		−	0.841165i
$498$		0			0
$499$	−10600.0			−0.950944			−0.475472	−	0.879731i	$-0.657723\pi$
$499$	−10600.0			−0.950944			−0.475472	+	0.879731i	$0.657723\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	10048.0i		−	0.890692i	−0.895359	−	0.445346i	$-0.853081\pi$
$503$		−	10048.0i		−	0.890692i	0.895359	−	0.445346i	$-0.146919\pi$
$504$		0			0
$505$	−1872.00	−	10296.0i	−0.164956	−	0.907259i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−15088.0			−1.31388			−0.656939	−	0.753944i	$-0.728149\pi$
$509$	−15088.0			−1.31388			−0.656939	+	0.753944i	$0.728149\pi$
$510$		0			0
$511$	5480.00			0.474405
$512$		0			0
$513$		0			0
$514$		0			0
$515$	15950.0	−	2900.00i	1.36474	−	0.248135i
$516$		0			0
$517$		−	2016.00i		−	0.171496i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8582.00			−0.721659			−0.360829	−	0.932632i	$-0.617506\pi$
$521$	−8582.00			−0.721659			−0.360829	+	0.932632i	$0.617506\pi$
$522$		0			0
$523$		−	16928.0i		−	1.41532i	−0.706556	−	0.707658i	$-0.749752\pi$
$523$		−	16928.0i		−	1.41532i	0.706556	−	0.707658i	$-0.250248\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1296.00i		−	0.107125i
$528$		0			0
$529$	−7433.00			−0.610915
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	36572.0i		−	2.97206i
$534$		0			0
$535$	14212.0	−	2584.00i	1.14848	−	0.208815i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3402.00			−0.271864
$540$		0			0
$541$	−11150.0			−0.886092			−0.443046	−	0.896499i	$-0.646102\pi$
$541$	−11150.0			−0.886092			−0.443046	+	0.896499i	$0.646102\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4284.00	+	23562.0i	0.336709	+	1.85190i
$546$		0			0
$547$		−	19628.0i		−	1.53425i	−0.641500	−	0.767123i	$-0.721688\pi$
$547$		−	19628.0i		−	1.53425i	0.641500	−	0.767123i	$-0.278312\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−15232.0			−1.17769
$552$		0			0
$553$		−	5120.00i		−	0.393715i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			8694.00i			0.661358i	0.943743	+	0.330679i	$0.107278\pi$
$557$			8694.00i			0.661358i	−0.943743	+	0.330679i	$0.892722\pi$
$558$		0			0
$559$	−32472.0			−2.45692
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	828.000i		−	0.0619823i	−0.999520	−	0.0309912i	$-0.990134\pi$
$563$		−	828.000i		−	0.0619823i	0.999520	−	0.0309912i	$-0.00986637\pi$
$564$		0			0
$565$	−15598.0	+	2836.00i	−1.16144	+	0.211171i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−17514.0			−1.29038			−0.645189	−	0.764023i	$-0.723221\pi$
$569$	−17514.0			−1.29038			−0.645189	+	0.764023i	$0.723221\pi$
$570$		0			0
$571$	5552.00			0.406907			0.203454	−	0.979085i	$-0.434783\pi$
$571$	5552.00			0.406907			0.203454	+	0.979085i	$0.434783\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	6160.00	+	16380.0i	0.446765	+	1.18799i
$576$		0			0
$577$			12896.0i			0.930446i	0.885193	+	0.465223i	$0.154026\pi$
$577$			12896.0i			0.930446i	−0.885193	+	0.465223i	$0.845974\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2840.00			−0.202794
$582$		0			0
$583$			2212.00i			0.157138i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1588.00i			0.111659i	0.998440	+	0.0558295i	$0.0177803\pi$
$587$			1588.00i			0.111659i	−0.998440	+	0.0558295i	$0.982220\pi$
$588$		0			0
$589$	9792.00			0.685012
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	22262.0i		−	1.54164i	−0.637055	−	0.770819i	$-0.719848\pi$
$593$		−	22262.0i		−	1.54164i	0.637055	−	0.770819i	$-0.280152\pi$
$594$		0			0
$595$	−360.000	−	1980.00i	−0.0248043	−	0.136424i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4640.00			−0.316503			−0.158251	−	0.987399i	$-0.550586\pi$
$599$	−4640.00			−0.316503			−0.158251	+	0.987399i	$0.550586\pi$
$600$		0			0
$601$	−2574.00			−0.174702			−0.0873508	−	0.996178i	$-0.527840\pi$
$601$	−2574.00			−0.174702			−0.0873508	+	0.996178i	$0.527840\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2270.00	−	12485.0i	−0.152543	−	0.838987i
$606$		0			0
$607$		−	11170.0i		−	0.746913i	−0.927648	−	0.373457i	$-0.878172\pi$
$607$		−	11170.0i		−	0.746913i	0.927648	−	0.373457i	$-0.121828\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	11808.0			0.781834
$612$		0			0
$613$		−	3646.00i		−	0.240229i	−0.992760	−	0.120115i	$-0.961674\pi$
$613$		−	3646.00i		−	0.240229i	0.992760	−	0.120115i	$-0.0383262\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	7646.00i		−	0.498892i	−0.968389	−	0.249446i	$-0.919751\pi$
$617$		−	7646.00i		−	0.498892i	0.968389	−	0.249446i	$-0.0802485\pi$
$618$		0			0
$619$	26668.0			1.73163			0.865814	−	0.500366i	$-0.166801\pi$
$619$	26668.0			1.73163			0.865814	+	0.500366i	$0.166801\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	8100.00i		−	0.520898i
$624$		0			0
$625$	11753.0	−	10296.0i	0.752192	−	0.658944i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	468.000			0.0296667
$630$		0			0
$631$	−7712.00			−0.486545			−0.243272	−	0.969958i	$-0.578221\pi$
$631$	−7712.00			−0.486545			−0.243272	+	0.969958i	$0.578221\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−18414.0	+	3348.00i	−1.15077	+	0.209230i
$636$		0			0
$637$		−	19926.0i		−	1.23940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	22302.0			1.37422			0.687111	−	0.726553i	$-0.258879\pi$
$641$	22302.0			1.37422			0.687111	+	0.726553i	$0.258879\pi$
$642$		0			0
$643$		−	2232.00i		−	0.136892i	−0.997655	−	0.0684459i	$-0.978196\pi$
$643$		−	2232.00i		−	0.136892i	0.997655	−	0.0684459i	$-0.0218041\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	9464.00i		−	0.575067i	−0.957771	−	0.287533i	$-0.907165\pi$
$647$		−	9464.00i		−	0.575067i	0.957771	−	0.287533i	$-0.0928352\pi$
$648$		0			0
$649$	−4788.00			−0.289592
$650$		0			0
$651$		0			0
$652$		0			0
$653$			4878.00i			0.292329i	0.989260	+	0.146165i	$0.0466929\pi$
$653$			4878.00i			0.292329i	−0.989260	+	0.146165i	$0.953307\pi$
$654$		0			0
$655$	2268.00	+	12474.0i	0.135295	+	0.744121i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10206.0			−0.603292			−0.301646	−	0.953420i	$-0.597536\pi$
$659$	−10206.0			−0.603292			−0.301646	+	0.953420i	$0.597536\pi$
$660$		0			0
$661$	20906.0			1.23018			0.615090	−	0.788457i	$-0.289120\pi$
$661$	20906.0			1.23018			0.615090	+	0.788457i	$0.289120\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	14960.0	−	2720.00i	0.872367	−	0.158612i
$666$		0			0
$667$		−	15680.0i		−	0.910243i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4396.00			−0.252915
$672$		0			0
$673$		−	6812.00i		−	0.390168i	−0.980787	−	0.195084i	$-0.937502\pi$
$673$		−	6812.00i		−	0.390168i	0.980787	−	0.195084i	$-0.0624980\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			10026.0i			0.569174i	0.958650	+	0.284587i	$0.0918564\pi$
$677$			10026.0i			0.569174i	−0.958650	+	0.284587i	$0.908144\pi$
$678$		0			0
$679$	13040.0			0.737009
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	21236.0i		−	1.18971i	−0.803832	−	0.594856i	$-0.797209\pi$
$683$		−	21236.0i		−	1.18971i	0.803832	−	0.594856i	$-0.202791\pi$
$684$		0			0
$685$	31526.0	−	5732.00i	1.75846	−	0.319720i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12956.0			−0.716378
$690$		0			0
$691$	11520.0			0.634213			0.317107	−	0.948390i	$-0.397289\pi$
$691$	11520.0			0.634213			0.317107	+	0.948390i	$0.397289\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1528.00	−	8404.00i	−0.0833962	−	0.458679i
$696$		0			0
$697$			8028.00i			0.436273i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	400.000			0.0215518			0.0107759	−	0.999942i	$-0.496570\pi$
$701$	400.000			0.0215518			0.0107759	+	0.999942i	$0.496570\pi$
$702$		0			0
$703$			3536.00i			0.189705i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	9360.00i		−	0.497905i
$708$		0			0
$709$	5930.00			0.314113			0.157056	−	0.987590i	$-0.449800\pi$
$709$	5930.00			0.314113			0.157056	+	0.987590i	$0.449800\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			10080.0i			0.529452i
$714$		0			0
$715$	−12628.0	+	2296.00i	−0.660504	+	0.120092i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7160.00			−0.371381			−0.185691	−	0.982608i	$-0.559452\pi$
$719$	−7160.00			−0.371381			−0.185691	+	0.982608i	$0.559452\pi$
$720$		0			0
$721$	14500.0			0.748971
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13104.0	+	4928.00i	−0.671269	+	0.252443i
$726$		0			0
$727$		−	8874.00i		−	0.452708i	−0.974045	−	0.226354i	$-0.927319\pi$
$727$		−	8874.00i		−	0.452708i	0.974045	−	0.226354i	$-0.0726806\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7128.00			0.360655
$732$		0			0
$733$			5562.00i			0.280269i	0.990132	+	0.140134i	$0.0447535\pi$
$733$			5562.00i			0.280269i	−0.990132	+	0.140134i	$0.955247\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			2128.00i			0.106358i
$738$		0			0
$739$	−12096.0			−0.602109			−0.301055	−	0.953607i	$-0.597339\pi$
$739$	−12096.0			−0.602109			−0.301055	+	0.953607i	$0.597339\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			21312.0i			1.05230i	0.850391	+	0.526152i	$0.176366\pi$
$743$			21312.0i			1.05230i	−0.850391	+	0.526152i	$0.823634\pi$
$744$		0			0
$745$	−6120.00	−	33660.0i	−0.300966	−	1.65531i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12920.0			0.630289
$750$		0			0
$751$	6832.00			0.331962			0.165981	−	0.986129i	$-0.446921\pi$
$751$	6832.00			0.331962			0.165981	+	0.986129i	$0.446921\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2608.00	+	14344.0i	0.125715	+	0.691433i
$756$		0			0
$757$			7174.00i			0.344443i	0.985058	+	0.172222i	$0.0550945\pi$
$757$			7174.00i			0.344443i	−0.985058	+	0.172222i	$0.944905\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	15394.0			0.733288			0.366644	−	0.930361i	$-0.380507\pi$
$761$	15394.0			0.733288			0.366644	+	0.930361i	$0.380507\pi$
$762$		0			0
$763$			21420.0i			1.01633i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	28044.0i		−	1.32022i
$768$		0			0
$769$	−9730.00			−0.456271			−0.228136	−	0.973629i	$-0.573263\pi$
$769$	−9730.00			−0.456271			−0.228136	+	0.973629i	$0.573263\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			24206.0i			1.12630i	0.826355	+	0.563150i	$0.190411\pi$
$773$			24206.0i			1.12630i	−0.826355	+	0.563150i	$0.809589\pi$
$774$		0			0
$775$	8424.00	−	3168.00i	0.390450	−	0.146836i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−60656.0			−2.78976
$780$		0			0
$781$	13048.0			0.597816
$782$		0			0
$783$		0			0
$784$		0			0
$785$	17402.0	−	3164.00i	0.791215	−	0.143857i
$786$		0			0
$787$		−	31312.0i		−	1.41824i	−0.705089	−	0.709118i	$-0.749093\pi$
$787$		−	31312.0i		−	1.41824i	0.705089	−	0.709118i	$-0.250907\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14180.0			−0.637399
$792$		0			0
$793$		−	25748.0i		−	1.15301i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			10746.0i			0.477595i	0.971069	+	0.238797i	$0.0767531\pi$
$797$			10746.0i			0.477595i	−0.971069	+	0.238797i	$0.923247\pi$
$798$		0			0
$799$	−2592.00			−0.114766
$800$		0			0
$801$		0			0
$802$		0			0
$803$			7672.00i			0.337159i
$804$		0			0
$805$	2800.00	+	15400.0i	0.122593	+	0.674259i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30546.0			1.32749			0.663745	−	0.747959i	$-0.268966\pi$
$809$	30546.0			1.32749			0.663745	+	0.747959i	$0.268966\pi$
$810$		0			0
$811$	2628.00			0.113787			0.0568937	−	0.998380i	$-0.481880\pi$
$811$	2628.00			0.113787			0.0568937	+	0.998380i	$0.481880\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	35552.0	−	6464.00i	1.52802	−	0.277821i
$816$		0			0
$817$			53856.0i			2.30622i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−26280.0			−1.11715			−0.558574	−	0.829455i	$-0.688651\pi$
$821$	−26280.0			−1.11715			−0.558574	+	0.829455i	$0.688651\pi$
$822$		0			0
$823$			26146.0i			1.10740i	0.832715	+	0.553701i	$0.186785\pi$
$823$			26146.0i			1.10740i	−0.832715	+	0.553701i	$0.813215\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	29268.0i		−	1.23065i	−0.788273	−	0.615325i	$-0.789025\pi$
$827$		−	29268.0i		−	1.23065i	0.788273	−	0.615325i	$-0.210975\pi$
$828$		0			0
$829$	−6202.00			−0.259836			−0.129918	−	0.991525i	$-0.541471\pi$
$829$	−6202.00			−0.259836			−0.129918	+	0.991525i	$0.541471\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			4374.00i			0.181933i
$834$		0			0
$835$	−32868.0	+	5976.00i	−1.36221	+	0.247674i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4680.00			−0.192576			−0.0962882	−	0.995353i	$-0.530697\pi$
$839$	−4680.00			−0.192576			−0.0962882	+	0.995353i	$0.530697\pi$
$840$		0			0
$841$	−11845.0			−0.485670
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9054.00	−	49797.0i	−0.368600	−	2.02730i
$846$		0			0
$847$		−	11350.0i		−	0.460438i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3640.00			−0.146625
$852$		0			0
$853$			12122.0i			0.486576i	0.969954	+	0.243288i	$0.0782260\pi$
$853$			12122.0i			0.486576i	−0.969954	+	0.243288i	$0.921774\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	26006.0i		−	1.03658i	−0.855205	−	0.518289i	$-0.826569\pi$
$857$		−	26006.0i		−	1.03658i	0.855205	−	0.518289i	$-0.173431\pi$
$858$		0			0
$859$	−17684.0			−0.702410			−0.351205	−	0.936299i	$-0.614228\pi$
$859$	−17684.0			−0.702410			−0.351205	+	0.936299i	$0.614228\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			15084.0i			0.594977i	0.954725	+	0.297489i	$0.0961490\pi$
$863$			15084.0i			0.594977i	−0.954725	+	0.297489i	$0.903851\pi$
$864$		0			0
$865$	−10098.0	+	1836.00i	−0.396928	+	0.0721686i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7168.00			0.279813
$870$		0			0
$871$	−12464.0			−0.484875
$872$		0			0
$873$		0			0
$874$		0			0
$875$	11990.0	−	7180.00i	0.463241	−	0.277404i
$876$		0			0
$877$		−	17614.0i		−	0.678201i	−0.940750	−	0.339101i	$-0.889877\pi$
$877$		−	17614.0i		−	0.678201i	0.940750	−	0.339101i	$-0.110123\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3298.00			0.126121			0.0630604	−	0.998010i	$-0.479914\pi$
$881$	3298.00			0.126121			0.0630604	+	0.998010i	$0.479914\pi$
$882$		0			0
$883$		−	9496.00i		−	0.361909i	−0.983491	−	0.180955i	$-0.942081\pi$
$883$		−	9496.00i		−	0.361909i	0.983491	−	0.180955i	$-0.0579187\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			50220.0i			1.90104i	0.310663	+	0.950520i	$0.399449\pi$
$887$			50220.0i			1.90104i	−0.310663	+	0.950520i	$0.600551\pi$
$888$		0			0
$889$	−16740.0			−0.631543
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	19584.0i		−	0.733879i
$894$		0			0
$895$	2476.00	+	13618.0i	0.0924732	+	0.508603i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−8064.00			−0.299165
$900$		0			0
$901$	2844.00			0.105158
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2900.00	+	15950.0i	0.106519	+	0.585852i
$906$		0			0
$907$			20636.0i			0.755465i	0.925915	+	0.377733i	$0.123296\pi$
$907$			20636.0i			0.755465i	−0.925915	+	0.377733i	$0.876704\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	13680.0			0.497518			0.248759	−	0.968565i	$-0.419977\pi$
$911$	13680.0			0.497518			0.248759	+	0.968565i	$0.419977\pi$
$912$		0			0
$913$		−	3976.00i		−	0.144125i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			11340.0i			0.408375i
$918$		0			0
$919$	21456.0			0.770150			0.385075	−	0.922885i	$-0.374176\pi$
$919$	21456.0			0.770150			0.385075	+	0.922885i	$0.374176\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			76424.0i			2.72538i
$924$		0			0
$925$	1144.00	+	3042.00i	0.0406643	+	0.108130i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16510.0			0.583074			0.291537	−	0.956560i	$-0.405833\pi$
$929$	16510.0			0.583074			0.291537	+	0.956560i	$0.405833\pi$
$930$		0			0
$931$	−33048.0			−1.16338
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2772.00	−	504.000i	0.0969562	−	0.0176284i
$936$		0			0
$937$			36296.0i			1.26546i	0.774371	+	0.632731i	$0.218066\pi$
$937$			36296.0i			1.26546i	−0.774371	+	0.632731i	$0.781934\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13540.0			−0.469066			−0.234533	−	0.972108i	$-0.575356\pi$
$941$	−13540.0			−0.469066			−0.234533	+	0.972108i	$0.575356\pi$
$942$		0			0
$943$		−	62440.0i		−	2.15623i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	32940.0i		−	1.13031i	−0.824984	−	0.565156i	$-0.808816\pi$
$947$		−	32940.0i		−	1.13031i	0.824984	−	0.565156i	$-0.191184\pi$
$948$		0			0
$949$	−44936.0			−1.53708
$950$		0			0
$951$		0			0
$952$		0			0
$953$			22482.0i			0.764180i	0.924125	+	0.382090i	$0.124796\pi$
$953$			22482.0i			0.764180i	−0.924125	+	0.382090i	$0.875204\pi$
$954$		0			0
$955$	9720.00	+	53460.0i	0.329353	+	1.81144i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	28660.0			0.965047
$960$		0			0
$961$	−24607.0			−0.825988
$962$		0			0
$963$		0			0
$964$		0			0
$965$	15532.0	−	2824.00i	0.518127	−	0.0942049i
$966$		0			0
$967$		−	9566.00i		−	0.318120i	−0.987269	−	0.159060i	$-0.949154\pi$
$967$		−	9566.00i		−	0.318120i	0.987269	−	0.159060i	$-0.0508463\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	10062.0			0.332549			0.166274	−	0.986080i	$-0.446826\pi$
$971$	10062.0			0.332549			0.166274	+	0.986080i	$0.446826\pi$
$972$		0			0
$973$		−	7640.00i		−	0.251724i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			48506.0i			1.58838i	0.607671	+	0.794189i	$0.292104\pi$
$977$			48506.0i			1.58838i	−0.607671	+	0.794189i	$0.707896\pi$
$978$		0			0
$979$	11340.0			0.370202
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	41144.0i		−	1.33498i	−0.744617	−	0.667492i	$-0.767368\pi$
$983$		−	41144.0i		−	1.33498i	0.744617	−	0.667492i	$-0.232632\pi$
$984$		0			0
$985$	12870.0	−	2340.00i	0.416317	−	0.0756940i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−55440.0			−1.78250
$990$		0			0
$991$	−16120.0			−0.516719			−0.258360	−	0.966049i	$-0.583182\pi$
$991$	−16120.0			−0.516719			−0.258360	+	0.966049i	$0.583182\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4160.00	+	22880.0i	0.132544	+	0.728990i
$996$		0			0
$997$			36666.0i			1.16472i	0.812932	+	0.582359i	$0.197870\pi$
$997$			36666.0i			1.16472i	−0.812932	+	0.582359i	$0.802130\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.4.f.e.289.2		2
3.2	odd	2		240.4.f.c.49.1		2
4.3	odd	2		360.4.f.b.289.2		2
5.4	even	2	inner	720.4.f.e.289.1		2
12.11	even	2		120.4.f.b.49.2	yes	2
15.2	even	4		1200.4.a.e.1.1		1
15.8	even	4		1200.4.a.bg.1.1		1
15.14	odd	2		240.4.f.c.49.2		2
20.3	even	4		1800.4.a.i.1.1		1
20.7	even	4		1800.4.a.x.1.1		1
20.19	odd	2		360.4.f.b.289.1		2
24.5	odd	2		960.4.f.e.769.2		2
24.11	even	2		960.4.f.f.769.1		2
60.23	odd	4		600.4.a.c.1.1		1
60.47	odd	4		600.4.a.p.1.1		1
60.59	even	2		120.4.f.b.49.1	✓	2
120.29	odd	2		960.4.f.e.769.1		2
120.59	even	2		960.4.f.f.769.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.b.49.1	✓	2	60.59	even	2
120.4.f.b.49.2	yes	2	12.11	even	2
240.4.f.c.49.1		2	3.2	odd	2
240.4.f.c.49.2		2	15.14	odd	2
360.4.f.b.289.1		2	20.19	odd	2
360.4.f.b.289.2		2	4.3	odd	2
600.4.a.c.1.1		1	60.23	odd	4
600.4.a.p.1.1		1	60.47	odd	4
720.4.f.e.289.1		2	5.4	even	2	inner
720.4.f.e.289.2		2	1.1	even	1	trivial
960.4.f.e.769.1		2	120.29	odd	2
960.4.f.e.769.2		2	24.5	odd	2
960.4.f.f.769.1		2	24.11	even	2
960.4.f.f.769.2		2	120.59	even	2
1200.4.a.e.1.1		1	15.2	even	4
1200.4.a.bg.1.1		1	15.8	even	4
1800.4.a.i.1.1		1	20.3	even	4
1800.4.a.x.1.1		1	20.7	even	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 + 11.0000i) q^{5} +10.0000i q^{7} +O(q^{10})\)	\(q+(2.00000 + 11.0000i) q^{5} +10.0000i q^{7} -14.0000 q^{11} -82.0000i q^{13} +18.0000i q^{17} -136.000 q^{19} -140.000i q^{23} +(-117.000 + 44.0000i) q^{25} +112.000 q^{29} -72.0000 q^{31} +(-110.000 + 20.0000i) q^{35} -26.0000i q^{37} +446.000 q^{41} -396.000i q^{43} +144.000i q^{47} +243.000 q^{49} -158.000i q^{53} +(-28.0000 - 154.000i) q^{55} +342.000 q^{59} +314.000 q^{61} +(902.000 - 164.000i) q^{65} -152.000i q^{67} -932.000 q^{71} -548.000i q^{73} -140.000i q^{77} -512.000 q^{79} +284.000i q^{83} +(-198.000 + 36.0000i) q^{85} -810.000 q^{89} +820.000 q^{91} +(-272.000 - 1496.00i) q^{95} -1304.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5}+O(q^{10}) \) 2 * q + 4 * q^5	\( 2 q + 4 q^{5} - 28 q^{11} - 272 q^{19} - 234 q^{25} + 224 q^{29} - 144 q^{31} - 220 q^{35} + 892 q^{41} + 486 q^{49} - 56 q^{55} + 684 q^{59} + 628 q^{61} + 1804 q^{65} - 1864 q^{71} - 1024 q^{79} - 396 q^{85} - 1620 q^{89} + 1640 q^{91} - 544 q^{95}+O(q^{100}) \) 2 * q + 4 * q^5 - 28 * q^11 - 272 * q^19 - 234 * q^25 + 224 * q^29 - 144 * q^31 - 220 * q^35 + 892 * q^41 + 486 * q^49 - 56 * q^55 + 684 * q^59 + 628 * q^61 + 1804 * q^65 - 1864 * q^71 - 1024 * q^79 - 396 * q^85 - 1620 * q^89 + 1640 * q^91 - 544 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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