LMFDB - Embedded newform 153.4.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,4,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("153.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	153.a (trivial)

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:	yes
Analytic conductor:	$9.02729223088$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.5912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x - 10 $ x^3 - x^2 - 14*x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.55528$ of defining polynomial
Character	$\chi$	$=$	153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.55528 q^{2} -1.47057 q^{4} -8.47057 q^{5} +3.66117 q^{7} -24.1999 q^{8} +O(q^{10})$	\(q+2.55528 q^{2} -1.47057 q^{4} -8.47057 q^{5} +3.66117 q^{7} -24.1999 q^{8} -21.6446 q^{10} -61.4727 q^{11} +20.9881 q^{13} +9.35531 q^{14} -50.0729 q^{16} -17.0000 q^{17} -102.753 q^{19} +12.4566 q^{20} -157.080 q^{22} +27.1763 q^{23} -53.2495 q^{25} +53.6304 q^{26} -5.38401 q^{28} +145.246 q^{29} +72.0652 q^{31} +65.6493 q^{32} -43.4397 q^{34} -31.0122 q^{35} +371.278 q^{37} -262.561 q^{38} +204.987 q^{40} -348.178 q^{41} -246.686 q^{43} +90.3998 q^{44} +69.4428 q^{46} +269.231 q^{47} -329.596 q^{49} -136.067 q^{50} -30.8645 q^{52} +349.771 q^{53} +520.709 q^{55} -88.6001 q^{56} +371.144 q^{58} -78.9826 q^{59} -410.299 q^{61} +184.147 q^{62} +568.335 q^{64} -177.781 q^{65} -493.393 q^{67} +24.9997 q^{68} -79.2448 q^{70} -480.053 q^{71} +524.867 q^{73} +948.718 q^{74} +151.105 q^{76} -225.062 q^{77} -189.939 q^{79} +424.146 q^{80} -889.692 q^{82} +1044.00 q^{83} +144.000 q^{85} -630.350 q^{86} +1487.63 q^{88} -725.237 q^{89} +76.8411 q^{91} -39.9645 q^{92} +687.960 q^{94} +870.373 q^{95} -1752.23 q^{97} -842.208 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8}+O(q^{10}) $ 3 * q - 5 * q^2 + 13 * q^4 - 8 * q^5 - 8 * q^7 - 33 * q^8	$ 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8} - 38 q^{10} - 34 q^{11} + 36 q^{13} + 104 q^{14} - 79 q^{16} - 51 q^{17} - 142 q^{19} + 126 q^{20} - 248 q^{22} - 110 q^{23} - 193 q^{25} - 154 q^{26} - 472 q^{28} - 90 q^{29} - 148 q^{31} + 151 q^{32} + 85 q^{34} - 416 q^{35} + 110 q^{37} - 80 q^{38} - 202 q^{40} - 720 q^{41} - 146 q^{43} + 192 q^{44} + 748 q^{46} - 500 q^{47} + 379 q^{49} + 385 q^{50} + 1218 q^{52} - 610 q^{53} + 430 q^{55} + 1368 q^{56} + 1006 q^{58} + 216 q^{59} - 18 q^{61} + 904 q^{62} + 377 q^{64} + 966 q^{65} - 1404 q^{67} - 221 q^{68} + 1472 q^{70} + 960 q^{71} - 794 q^{73} + 1874 q^{74} - 392 q^{76} - 48 q^{77} - 276 q^{79} + 1130 q^{80} + 382 q^{82} + 1552 q^{83} + 136 q^{85} - 16 q^{86} + 1724 q^{88} - 1394 q^{89} - 3968 q^{91} - 2244 q^{92} + 3960 q^{94} + 602 q^{95} + 402 q^{97} - 4109 q^{98}+O(q^{100}) $ 3 * q - 5 * q^2 + 13 * q^4 - 8 * q^5 - 8 * q^7 - 33 * q^8 - 38 * q^10 - 34 * q^11 + 36 * q^13 + 104 * q^14 - 79 * q^16 - 51 * q^17 - 142 * q^19 + 126 * q^20 - 248 * q^22 - 110 * q^23 - 193 * q^25 - 154 * q^26 - 472 * q^28 - 90 * q^29 - 148 * q^31 + 151 * q^32 + 85 * q^34 - 416 * q^35 + 110 * q^37 - 80 * q^38 - 202 * q^40 - 720 * q^41 - 146 * q^43 + 192 * q^44 + 748 * q^46 - 500 * q^47 + 379 * q^49 + 385 * q^50 + 1218 * q^52 - 610 * q^53 + 430 * q^55 + 1368 * q^56 + 1006 * q^58 + 216 * q^59 - 18 * q^61 + 904 * q^62 + 377 * q^64 + 966 * q^65 - 1404 * q^67 - 221 * q^68 + 1472 * q^70 + 960 * q^71 - 794 * q^73 + 1874 * q^74 - 392 * q^76 - 48 * q^77 - 276 * q^79 + 1130 * q^80 + 382 * q^82 + 1552 * q^83 + 136 * q^85 - 16 * q^86 + 1724 * q^88 - 1394 * q^89 - 3968 * q^91 - 2244 * q^92 + 3960 * q^94 + 602 * q^95 + 402 * q^97 - 4109 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.55528	0.903426	0.451713	−	0.892163i	$-0.350813\pi$
$2$	2.55528	0.903426	0.451713	+	0.892163i	$0.350813\pi$
$3$	0	0
$3$	0	0
$4$	−1.47057	−0.183821
$5$	−8.47057	−0.757631	−0.378815	−	0.925472i	$-0.623668\pi$
$5$	−8.47057	−0.757631	−0.378815	+	0.925472i	$0.623668\pi$
$6$	0	0
$7$	3.66117	0.197685	0.0988424	−	0.995103i	$-0.468486\pi$
$7$	3.66117	0.197685	0.0988424	+	0.995103i	$0.468486\pi$
$8$	−24.1999	−1.06949
$9$	0	0
$10$	−21.6446	−0.684463
$11$	−61.4727	−1.68497	−0.842487	−	0.538717i	$-0.818909\pi$
$11$	−61.4727	−1.68497	−0.842487	+	0.538717i	$0.818909\pi$
$12$	0	0
$13$	20.9881	0.447773	0.223887	−	0.974615i	$-0.428125\pi$
$13$	20.9881	0.447773	0.223887	+	0.974615i	$0.428125\pi$
$14$	9.35531	0.178594
$15$	0	0
$16$	−50.0729	−0.782389
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−102.753	−1.24069	−0.620344	−	0.784330i	$-0.713007\pi$
$19$	−102.753	−1.24069	−0.620344	+	0.784330i	$0.713007\pi$
$20$	12.4566	0.139268
$21$	0	0
$22$	−157.080	−1.52225
$23$	27.1763	0.246376	0.123188	−	0.992383i	$-0.460688\pi$
$23$	27.1763	0.246376	0.123188	+	0.992383i	$0.460688\pi$
$24$	0	0
$25$	−53.2495	−0.425996
$26$	53.6304	0.404530
$27$	0	0
$28$	−5.38401	−0.0363386
$29$	145.246	0.930053	0.465026	−	0.885297i	$-0.346045\pi$
$29$	145.246	0.930053	0.465026	+	0.885297i	$0.346045\pi$
$30$	0	0
$31$	72.0652	0.417526	0.208763	−	0.977966i	$-0.433056\pi$
$31$	72.0652	0.417526	0.208763	+	0.977966i	$0.433056\pi$
$32$	65.6493	0.362664
$33$	0	0
$34$	−43.4397	−0.219113
$35$	−31.0122	−0.149772
$36$	0	0
$37$	371.278	1.64967	0.824835	−	0.565374i	$-0.191268\pi$
$37$	371.278	1.64967	0.824835	+	0.565374i	$0.191268\pi$
$38$	−262.561	−1.12087
$39$	0	0
$40$	204.987	0.810282
$41$	−348.178	−1.32625	−0.663126	−	0.748508i	$-0.730771\pi$
$41$	−348.178	−1.32625	−0.663126	+	0.748508i	$0.730771\pi$
$42$	0	0
$43$	−246.686	−0.874865	−0.437433	−	0.899251i	$-0.644112\pi$
$43$	−246.686	−0.874865	−0.437433	+	0.899251i	$0.644112\pi$
$44$	90.3998	0.309734
$45$	0	0
$46$	69.4428	0.222582
$47$	269.231	0.835561	0.417781	−	0.908548i	$-0.362808\pi$
$47$	269.231	0.835561	0.417781	+	0.908548i	$0.362808\pi$
$48$	0	0
$49$	−329.596	−0.960921
$50$	−136.067	−0.384856
$51$	0	0
$52$	−30.8645	−0.0823102
$53$	349.771	0.906506	0.453253	−	0.891382i	$-0.350263\pi$
$53$	349.771	0.906506	0.453253	+	0.891382i	$0.350263\pi$
$54$	0	0
$55$	520.709	1.27659
$56$	−88.6001	−0.211423
$57$	0	0
$58$	371.144	0.840234
$59$	−78.9826	−0.174282	−0.0871412	−	0.996196i	$-0.527773\pi$
$59$	−78.9826	−0.174282	−0.0871412	+	0.996196i	$0.527773\pi$
$60$	0	0
$61$	−410.299	−0.861204	−0.430602	−	0.902542i	$-0.641699\pi$
$61$	−410.299	−0.861204	−0.430602	+	0.902542i	$0.641699\pi$
$62$	184.147	0.377204
$63$	0	0
$64$	568.335	1.11003
$65$	−177.781	−0.339247
$66$	0	0
$67$	−493.393	−0.899664	−0.449832	−	0.893113i	$-0.648516\pi$
$67$	−493.393	−0.899664	−0.449832	+	0.893113i	$0.648516\pi$
$68$	24.9997	0.0445832
$69$	0	0
$70$	−79.2448	−0.135308
$71$	−480.053	−0.802420	−0.401210	−	0.915986i	$-0.631410\pi$
$71$	−480.053	−0.802420	−0.401210	+	0.915986i	$0.631410\pi$
$72$	0	0
$73$	524.867	0.841521	0.420760	−	0.907172i	$-0.361763\pi$
$73$	524.867	0.841521	0.420760	+	0.907172i	$0.361763\pi$
$74$	948.718	1.49036
$75$	0	0
$76$	151.105	0.228065
$77$	−225.062	−0.333094
$78$	0	0
$79$	−189.939	−0.270505	−0.135252	−	0.990811i	$-0.543184\pi$
$79$	−189.939	−0.270505	−0.135252	+	0.990811i	$0.543184\pi$
$80$	424.146	0.592762
$81$	0	0
$82$	−889.692	−1.19817
$83$	1044.00	1.38065	0.690323	−	0.723501i	$-0.257468\pi$
$83$	1044.00	1.38065	0.690323	+	0.723501i	$0.257468\pi$
$84$	0	0
$85$	144.000	0.183752
$86$	−630.350	−0.790376
$87$	0	0
$88$	1487.63	1.80207
$89$	−725.237	−0.863764	−0.431882	−	0.901930i	$-0.642150\pi$
$89$	−725.237	−0.863764	−0.431882	+	0.901930i	$0.642150\pi$
$90$	0	0
$91$	76.8411	0.0885180
$92$	−39.9645	−0.0452890
$93$	0	0
$94$	687.960	0.754868
$95$	870.373	0.939984
$96$	0	0
$97$	−1752.23	−1.83415	−0.917073	−	0.398720i	$-0.869455\pi$
$97$	−1752.23	−1.83415	−0.917073	+	0.398720i	$0.869455\pi$
$98$	−842.208	−0.868121
$99$	0	0
$100$	78.3070	0.0783070
$101$	−1052.52	−1.03693	−0.518463	−	0.855100i	$-0.673496\pi$
$101$	−1052.52	−1.03693	−0.518463	+	0.855100i	$0.673496\pi$
$102$	0	0
$103$	−862.054	−0.824667	−0.412333	−	0.911033i	$-0.635286\pi$
$103$	−862.054	−0.824667	−0.412333	+	0.911033i	$0.635286\pi$
$104$	−507.910	−0.478891
$105$	0	0
$106$	893.762	0.818961
$107$	1503.32	1.35824	0.679120	−	0.734027i	$-0.262361\pi$
$107$	1503.32	1.35824	0.679120	+	0.734027i	$0.262361\pi$
$108$	0	0
$109$	1164.76	1.02352	0.511762	−	0.859127i	$-0.328993\pi$
$109$	1164.76	1.02352	0.511762	+	0.859127i	$0.328993\pi$
$110$	1330.55	1.15330
$111$	0	0
$112$	−183.325	−0.154666
$113$	−940.942	−0.783331	−0.391665	−	0.920108i	$-0.628101\pi$
$113$	−940.942	−0.783331	−0.391665	+	0.920108i	$0.628101\pi$
$114$	0	0
$115$	−230.198	−0.186662
$116$	−213.594	−0.170963
$117$	0	0
$118$	−201.822	−0.157451
$119$	−62.2399	−0.0479456
$120$	0	0
$121$	2447.89	1.83914
$122$	−1048.43	−0.778034
$123$	0	0
$124$	−105.977	−0.0767500
$125$	1509.87	1.08038
$126$	0	0
$127$	−2554.20	−1.78464	−0.892319	−	0.451405i	$-0.850923\pi$
$127$	−2554.20	−1.78464	−0.892319	+	0.451405i	$0.850923\pi$
$128$	927.058	0.640165
$129$	0	0
$130$	−454.280	−0.306485
$131$	−100.915	−0.0673055	−0.0336527	−	0.999434i	$-0.510714\pi$
$131$	−100.915	−0.0673055	−0.0336527	+	0.999434i	$0.510714\pi$
$132$	0	0
$133$	−376.195	−0.245265
$134$	−1260.75	−0.812780
$135$	0	0
$136$	411.398	0.259391
$137$	−1745.14	−1.08830	−0.544150	−	0.838988i	$-0.683148\pi$
$137$	−1745.14	−1.08830	−0.544150	+	0.838988i	$0.683148\pi$
$138$	0	0
$139$	−2022.87	−1.23437	−0.617186	−	0.786817i	$-0.711727\pi$
$139$	−2022.87	−1.23437	−0.617186	+	0.786817i	$0.711727\pi$
$140$	45.6056	0.0275313
$141$	0	0
$142$	−1226.67	−0.724927
$143$	−1290.20	−0.754487
$144$	0	0
$145$	−1230.32	−0.704636
$146$	1341.18	0.760252
$147$	0	0
$148$	−545.990	−0.303244
$149$	−2319.95	−1.27555	−0.637777	−	0.770221i	$-0.720146\pi$
$149$	−2319.95	−1.27555	−0.637777	+	0.770221i	$0.720146\pi$
$150$	0	0
$151$	−3332.68	−1.79609	−0.898044	−	0.439905i	$-0.855012\pi$
$151$	−3332.68	−1.79609	−0.898044	+	0.439905i	$0.855012\pi$
$152$	2486.61	1.32691
$153$	0	0
$154$	−575.096	−0.300926
$155$	−610.434	−0.316330
$156$	0	0
$157$	1472.78	0.748664	0.374332	−	0.927295i	$-0.377872\pi$
$157$	1472.78	0.748664	0.374332	+	0.927295i	$0.377872\pi$
$158$	−485.348	−0.244381
$159$	0	0
$160$	−556.087	−0.274766
$161$	99.4970	0.0487047
$162$	0	0
$163$	761.196	0.365776	0.182888	−	0.983134i	$-0.441455\pi$
$163$	761.196	0.365776	0.182888	+	0.983134i	$0.441455\pi$
$164$	512.020	0.243793
$165$	0	0
$166$	2667.70	1.24731
$167$	−1382.94	−0.640811	−0.320406	−	0.947280i	$-0.603819\pi$
$167$	−1382.94	−0.640811	−0.320406	+	0.947280i	$0.603819\pi$
$168$	0	0
$169$	−1756.50	−0.799499
$170$	367.959	0.166007
$171$	0	0
$172$	362.768	0.160819
$173$	3064.45	1.34674	0.673370	−	0.739305i	$-0.264846\pi$
$173$	3064.45	1.34674	0.673370	+	0.739305i	$0.264846\pi$
$174$	0	0
$175$	−194.956	−0.0842129
$176$	3078.11	1.31830
$177$	0	0
$178$	−1853.18	−0.780347
$179$	−3002.33	−1.25366	−0.626830	−	0.779156i	$-0.715648\pi$
$179$	−3002.33	−1.25366	−0.626830	+	0.779156i	$0.715648\pi$
$180$	0	0
$181$	2711.15	1.11336	0.556680	−	0.830727i	$-0.312075\pi$
$181$	2711.15	1.11336	0.556680	+	0.830727i	$0.312075\pi$
$182$	196.350	0.0799695
$183$	0	0
$184$	−657.663	−0.263498
$185$	−3144.94	−1.24984
$186$	0	0
$187$	1045.04	0.408666
$188$	−395.923	−0.153594
$189$	0	0
$190$	2224.04	0.849206
$191$	−1310.96	−0.496637	−0.248318	−	0.968678i	$-0.579878\pi$
$191$	−1310.96	−0.496637	−0.248318	+	0.968678i	$0.579878\pi$
$192$	0	0
$193$	4404.35	1.64265	0.821327	−	0.570458i	$-0.193234\pi$
$193$	4404.35	1.64265	0.821327	+	0.570458i	$0.193234\pi$
$194$	−4477.43	−1.65702
$195$	0	0
$196$	484.693	0.176637
$197$	495.157	0.179078	0.0895392	−	0.995983i	$-0.471461\pi$
$197$	495.157	0.179078	0.0895392	+	0.995983i	$0.471461\pi$
$198$	0	0
$199$	4019.17	1.43171	0.715857	−	0.698246i	$-0.246036\pi$
$199$	4019.17	1.43171	0.715857	+	0.698246i	$0.246036\pi$
$200$	1288.63	0.455600
$201$	0	0
$202$	−2689.47	−0.936786
$203$	531.771	0.183857
$204$	0	0
$205$	2949.27	1.00481
$206$	−2202.78	−0.745026
$207$	0	0
$208$	−1050.94	−0.350333
$209$	6316.48	2.09053
$210$	0	0
$211$	−3727.51	−1.21617	−0.608086	−	0.793871i	$-0.708062\pi$
$211$	−3727.51	−1.21617	−0.608086	+	0.793871i	$0.708062\pi$
$212$	−514.363	−0.166635
$213$	0	0
$214$	3841.40	1.22707
$215$	2089.57	0.662825
$216$	0	0
$217$	263.843	0.0825385
$218$	2976.29	0.924679
$219$	0	0
$220$	−765.738	−0.234664
$221$	−356.798	−0.108601
$222$	0	0
$223$	5492.27	1.64928	0.824641	−	0.565657i	$-0.191377\pi$
$223$	5492.27	1.64928	0.824641	+	0.565657i	$0.191377\pi$
$224$	240.353	0.0716933
$225$	0	0
$226$	−2404.37	−0.707682
$227$	3392.42	0.991907	0.495954	−	0.868349i	$-0.334819\pi$
$227$	3392.42	0.991907	0.495954	+	0.868349i	$0.334819\pi$
$228$	0	0
$229$	2592.52	0.748116	0.374058	−	0.927405i	$-0.377966\pi$
$229$	2592.52	0.748116	0.374058	+	0.927405i	$0.377966\pi$
$230$	−588.220	−0.168635
$231$	0	0
$232$	−3514.94	−0.994687
$233$	5964.88	1.67713	0.838567	−	0.544798i	$-0.183394\pi$
$233$	5964.88	1.67713	0.838567	+	0.544798i	$0.183394\pi$
$234$	0	0
$235$	−2280.54	−0.633047
$236$	116.149	0.0320368
$237$	0	0
$238$	−159.040	−0.0433153
$239$	−4061.05	−1.09911	−0.549556	−	0.835457i	$-0.685203\pi$
$239$	−4061.05	−1.09911	−0.549556	+	0.835457i	$0.685203\pi$
$240$	0	0
$241$	2719.55	0.726894	0.363447	−	0.931615i	$-0.381600\pi$
$241$	2719.55	0.726894	0.363447	+	0.931615i	$0.381600\pi$
$242$	6255.04	1.66153
$243$	0	0
$244$	603.373	0.158307
$245$	2791.86	0.728023
$246$	0	0
$247$	−2156.58	−0.555547
$248$	−1743.97	−0.446542
$249$	0	0
$250$	3858.14	0.976042
$251$	1647.74	0.414360	0.207180	−	0.978303i	$-0.433571\pi$
$251$	1647.74	0.414360	0.207180	+	0.978303i	$0.433571\pi$
$252$	0	0
$253$	−1670.60	−0.415137
$254$	−6526.70	−1.61229
$255$	0	0
$256$	−2177.79	−0.531687
$257$	1695.48	0.411523	0.205761	−	0.978602i	$-0.434033\pi$
$257$	1695.48	0.411523	0.205761	+	0.978602i	$0.434033\pi$
$258$	0	0
$259$	1359.31	0.326115
$260$	261.440	0.0623607
$261$	0	0
$262$	−257.867	−0.0608055
$263$	−5598.45	−1.31261	−0.656303	−	0.754498i	$-0.727881\pi$
$263$	−5598.45	−1.31261	−0.656303	+	0.754498i	$0.727881\pi$
$264$	0	0
$265$	−2962.76	−0.686797
$266$	−961.283	−0.221579
$267$	0	0
$268$	725.568	0.165377
$269$	−4878.07	−1.10565	−0.552827	−	0.833296i	$-0.686451\pi$
$269$	−4878.07	−1.10565	−0.552827	+	0.833296i	$0.686451\pi$
$270$	0	0
$271$	−1242.71	−0.278559	−0.139279	−	0.990253i	$-0.544479\pi$
$271$	−1242.71	−0.278559	−0.139279	+	0.990253i	$0.544479\pi$
$272$	851.239	0.189757
$273$	0	0
$274$	−4459.30	−0.983198
$275$	3273.39	0.717792
$276$	0	0
$277$	2767.97	0.600402	0.300201	−	0.953876i	$-0.402946\pi$
$277$	2767.97	0.600402	0.300201	+	0.953876i	$0.402946\pi$
$278$	−5168.99	−1.11516
$279$	0	0
$280$	750.493	0.160180
$281$	7470.07	1.58586	0.792931	−	0.609311i	$-0.208554\pi$
$281$	7470.07	1.58586	0.792931	+	0.609311i	$0.208554\pi$
$282$	0	0
$283$	3599.57	0.756087	0.378043	−	0.925788i	$-0.376597\pi$
$283$	3599.57	0.756087	0.378043	+	0.925788i	$0.376597\pi$
$284$	705.951	0.147502
$285$	0	0
$286$	−3296.81	−0.681623
$287$	−1274.74	−0.262180
$288$	0	0
$289$	289.000	0.0588235
$290$	−3143.80	−0.636587
$291$	0	0
$292$	−771.853	−0.154689
$293$	5659.34	1.12840	0.564201	−	0.825637i	$-0.309184\pi$
$293$	5659.34	1.12840	0.564201	+	0.825637i	$0.309184\pi$
$294$	0	0
$295$	669.028	0.132042
$296$	−8984.90	−1.76431
$297$	0	0
$298$	−5928.11	−1.15237
$299$	570.378	0.110320
$300$	0	0
$301$	−903.159	−0.172948
$302$	−8515.91	−1.62263
$303$	0	0
$304$	5145.12	0.970701
$305$	3475.47	0.652474
$306$	0	0
$307$	−2164.87	−0.402461	−0.201231	−	0.979544i	$-0.564494\pi$
$307$	−2164.87	−0.402461	−0.201231	+	0.979544i	$0.564494\pi$
$308$	330.969	0.0612297
$309$	0	0
$310$	−1559.83	−0.285781
$311$	−6424.23	−1.17133	−0.585666	−	0.810552i	$-0.699167\pi$
$311$	−6424.23	−1.17133	−0.585666	+	0.810552i	$0.699167\pi$
$312$	0	0
$313$	−6872.79	−1.24113	−0.620564	−	0.784156i	$-0.713096\pi$
$313$	−6872.79	−1.24113	−0.620564	+	0.784156i	$0.713096\pi$
$314$	3763.35	0.676363
$315$	0	0
$316$	279.319	0.0497244
$317$	−9341.77	−1.65516	−0.827581	−	0.561346i	$-0.810284\pi$
$317$	−9341.77	−1.65516	−0.827581	+	0.561346i	$0.810284\pi$
$318$	0	0
$319$	−8928.67	−1.56711
$320$	−4814.12	−0.840992
$321$	0	0
$322$	254.242	0.0440011
$323$	1746.80	0.300911
$324$	0	0
$325$	−1117.61	−0.190750
$326$	1945.07	0.330452
$327$	0	0
$328$	8425.89	1.41842
$329$	985.702	0.165178
$330$	0	0
$331$	4546.85	0.755038	0.377519	−	0.926002i	$-0.376777\pi$
$331$	4546.85	0.755038	0.377519	+	0.926002i	$0.376777\pi$
$332$	−1535.27	−0.253792
$333$	0	0
$334$	−3533.80	−0.578926
$335$	4179.32	0.681613
$336$	0	0
$337$	−2205.83	−0.356555	−0.178277	−	0.983980i	$-0.557052\pi$
$337$	−2205.83	−0.356555	−0.178277	+	0.983980i	$0.557052\pi$
$338$	−4488.34	−0.722288
$339$	0	0
$340$	−211.761	−0.0337776
$341$	−4430.05	−0.703520
$342$	0	0
$343$	−2462.49	−0.387644
$344$	5969.77	0.935664
$345$	0	0
$346$	7830.52	1.21668
$347$	−1772.21	−0.274170	−0.137085	−	0.990559i	$-0.543773\pi$
$347$	−1772.21	−0.274170	−0.137085	+	0.990559i	$0.543773\pi$
$348$	0	0
$349$	1757.53	0.269565	0.134783	−	0.990875i	$-0.456966\pi$
$349$	1757.53	0.269565	0.134783	+	0.990875i	$0.456966\pi$
$350$	−498.165	−0.0760801
$351$	0	0
$352$	−4035.64	−0.611080
$353$	1252.22	0.188808	0.0944039	−	0.995534i	$-0.469905\pi$
$353$	1252.22	0.188808	0.0944039	+	0.995534i	$0.469905\pi$
$354$	0	0
$355$	4066.32	0.607938
$356$	1066.51	0.158778
$357$	0	0
$358$	−7671.79	−1.13259
$359$	7383.80	1.08552	0.542761	−	0.839887i	$-0.317379\pi$
$359$	7383.80	1.08552	0.542761	+	0.839887i	$0.317379\pi$
$360$	0	0
$361$	3699.11	0.539307
$362$	6927.74	1.00584
$363$	0	0
$364$	−113.000	−0.0162715
$365$	−4445.92	−0.637562
$366$	0	0
$367$	−4809.91	−0.684128	−0.342064	−	0.939677i	$-0.611126\pi$
$367$	−4809.91	−0.684128	−0.342064	+	0.939677i	$0.611126\pi$
$368$	−1360.79	−0.192762
$369$	0	0
$370$	−8036.18	−1.12914
$371$	1280.57	0.179202
$372$	0	0
$373$	10359.7	1.43808	0.719042	−	0.694967i	$-0.244581\pi$
$373$	10359.7	1.43808	0.719042	+	0.694967i	$0.244581\pi$
$374$	2670.35	0.369200
$375$	0	0
$376$	−6515.37	−0.893629
$377$	3048.44	0.416453
$378$	0	0
$379$	1333.12	0.180680	0.0903401	−	0.995911i	$-0.471205\pi$
$379$	1333.12	0.180680	0.0903401	+	0.995911i	$0.471205\pi$
$380$	−1279.94	−0.172789
$381$	0	0
$382$	−3349.86	−0.448675
$383$	8185.00	1.09199	0.545997	−	0.837787i	$-0.316151\pi$
$383$	8185.00	1.09199	0.545997	+	0.837787i	$0.316151\pi$
$384$	0	0
$385$	1906.40	0.252362
$386$	11254.3	1.48402
$387$	0	0
$388$	2576.78	0.337155
$389$	−929.154	−0.121105	−0.0605527	−	0.998165i	$-0.519286\pi$
$389$	−929.154	−0.121105	−0.0605527	+	0.998165i	$0.519286\pi$
$390$	0	0
$391$	−461.996	−0.0597549
$392$	7976.19	1.02770
$393$	0	0
$394$	1265.26	0.161784
$395$	1608.90	0.204943
$396$	0	0
$397$	7497.21	0.947794	0.473897	−	0.880580i	$-0.342847\pi$
$397$	7497.21	0.947794	0.473897	+	0.880580i	$0.342847\pi$
$398$	10270.1	1.29345
$399$	0	0
$400$	2666.35	0.333294
$401$	3645.49	0.453982	0.226991	−	0.973897i	$-0.427111\pi$
$401$	3645.49	0.453982	0.226991	+	0.973897i	$0.427111\pi$
$402$	0	0
$403$	1512.51	0.186957
$404$	1547.80	0.190609
$405$	0	0
$406$	1358.82	0.166101
$407$	−22823.5	−2.77965
$408$	0	0
$409$	−13585.3	−1.64242	−0.821210	−	0.570625i	$-0.806701\pi$
$409$	−13585.3	−1.64242	−0.821210	+	0.570625i	$0.806701\pi$
$410$	7536.19	0.907771
$411$	0	0
$412$	1267.71	0.151591
$413$	−289.169	−0.0344530
$414$	0	0
$415$	−8843.25	−1.04602
$416$	1377.85	0.162392
$417$	0	0
$418$	16140.4	1.88864
$419$	−12991.8	−1.51477	−0.757386	−	0.652967i	$-0.773524\pi$
$419$	−12991.8	−1.51477	−0.757386	+	0.652967i	$0.773524\pi$
$420$	0	0
$421$	3036.69	0.351542	0.175771	−	0.984431i	$-0.443758\pi$
$421$	3036.69	0.351542	0.175771	+	0.984431i	$0.443758\pi$
$422$	−9524.81	−1.09872
$423$	0	0
$424$	−8464.44	−0.969503
$425$	905.241	0.103319
$426$	0	0
$427$	−1502.18	−0.170247
$428$	−2210.74	−0.249673
$429$	0	0
$430$	5339.42	0.598813
$431$	−1911.68	−0.213648	−0.106824	−	0.994278i	$-0.534068\pi$
$431$	−1911.68	−0.213648	−0.106824	+	0.994278i	$0.534068\pi$
$432$	0	0
$433$	6112.37	0.678387	0.339194	−	0.940717i	$-0.389846\pi$
$433$	6112.37	0.678387	0.339194	+	0.940717i	$0.389846\pi$
$434$	674.192	0.0745675
$435$	0	0
$436$	−1712.87	−0.188145
$437$	−2792.43	−0.305675
$438$	0	0
$439$	13062.6	1.42014	0.710072	−	0.704129i	$-0.248662\pi$
$439$	13062.6	1.42014	0.710072	+	0.704129i	$0.248662\pi$
$440$	−12601.1	−1.36530
$441$	0	0
$442$	−911.717	−0.0981130
$443$	−10276.7	−1.10216	−0.551082	−	0.834451i	$-0.685785\pi$
$443$	−10276.7	−1.10216	−0.551082	+	0.834451i	$0.685785\pi$
$444$	0	0
$445$	6143.17	0.654414
$446$	14034.3	1.49000
$447$	0	0
$448$	2080.77	0.219436
$449$	6486.32	0.681756	0.340878	−	0.940108i	$-0.389276\pi$
$449$	6486.32	0.681756	0.340878	+	0.940108i	$0.389276\pi$
$450$	0	0
$451$	21403.5	2.23470
$452$	1383.72	0.143993
$453$	0	0
$454$	8668.57	0.896115
$455$	−650.888	−0.0670640
$456$	0	0
$457$	−9213.29	−0.943063	−0.471531	−	0.881849i	$-0.656299\pi$
$457$	−9213.29	−0.943063	−0.471531	+	0.881849i	$0.656299\pi$
$458$	6624.61	0.675868
$459$	0	0
$460$	338.522	0.0343124
$461$	−13304.1	−1.34410	−0.672052	−	0.740504i	$-0.734587\pi$
$461$	−13304.1	−1.34410	−0.672052	+	0.740504i	$0.734587\pi$
$462$	0	0
$463$	−17035.1	−1.70991	−0.854955	−	0.518703i	$-0.826415\pi$
$463$	−17035.1	−1.70991	−0.854955	+	0.518703i	$0.826415\pi$
$464$	−7272.89	−0.727663
$465$	0	0
$466$	15241.9	1.51517
$467$	−16651.7	−1.65000	−0.824999	−	0.565134i	$-0.808824\pi$
$467$	−16651.7	−1.65000	−0.824999	+	0.565134i	$0.808824\pi$
$468$	0	0
$469$	−1806.40	−0.177850
$470$	−5827.41	−0.571911
$471$	0	0
$472$	1911.37	0.186394
$473$	15164.4	1.47413
$474$	0	0
$475$	5471.52	0.528528
$476$	91.5281	0.00881341
$477$	0	0
$478$	−10377.1	−0.992966
$479$	8278.34	0.789660	0.394830	−	0.918754i	$-0.370804\pi$
$479$	8278.34	0.789660	0.394830	+	0.918754i	$0.370804\pi$
$480$	0	0
$481$	7792.43	0.738678
$482$	6949.19	0.656695
$483$	0	0
$484$	−3599.79	−0.338072
$485$	14842.4	1.38960
$486$	0	0
$487$	−5597.10	−0.520799	−0.260399	−	0.965501i	$-0.583854\pi$
$487$	−5597.10	−0.520799	−0.260399	+	0.965501i	$0.583854\pi$
$488$	9929.20	0.921053
$489$	0	0
$490$	7133.98	0.657715
$491$	4563.12	0.419410	0.209705	−	0.977765i	$-0.432750\pi$
$491$	4563.12	0.419410	0.209705	+	0.977765i	$0.432750\pi$
$492$	0	0
$493$	−2469.18	−0.225571
$494$	−5510.67	−0.501896
$495$	0	0
$496$	−3608.51	−0.326668
$497$	−1757.56	−0.158626
$498$	0	0
$499$	−15257.5	−1.36878	−0.684388	−	0.729118i	$-0.739930\pi$
$499$	−15257.5	−1.36878	−0.684388	+	0.729118i	$0.739930\pi$
$500$	−2220.37	−0.198596
$501$	0	0
$502$	4210.43	0.374344
$503$	−4407.84	−0.390727	−0.195364	−	0.980731i	$-0.562589\pi$
$503$	−4407.84	−0.390727	−0.195364	+	0.980731i	$0.562589\pi$
$504$	0	0
$505$	8915.43	0.785607
$506$	−4268.84	−0.375045
$507$	0	0
$508$	3756.13	0.328054
$509$	7762.09	0.675931	0.337965	−	0.941159i	$-0.390261\pi$
$509$	7762.09	0.675931	0.337965	+	0.941159i	$0.390261\pi$
$510$	0	0
$511$	1921.63	0.166356
$512$	−12981.3	−1.12051
$513$	0	0
$514$	4332.42	0.371780
$515$	7302.08	0.624793
$516$	0	0
$517$	−16550.4	−1.40790
$518$	3473.42	0.294621
$519$	0	0
$520$	4302.29	0.362823
$521$	4435.23	0.372957	0.186479	−	0.982459i	$-0.440292\pi$
$521$	4435.23	0.372957	0.186479	+	0.982459i	$0.440292\pi$
$522$	0	0
$523$	−8007.36	−0.669479	−0.334739	−	0.942311i	$-0.608648\pi$
$523$	−8007.36	−0.669479	−0.334739	+	0.942311i	$0.608648\pi$
$524$	148.403	0.0123722
$525$	0	0
$526$	−14305.6	−1.18584
$527$	−1225.11	−0.101265
$528$	0	0
$529$	−11428.5	−0.939299
$530$	−7570.68	−0.620470
$531$	0	0
$532$	553.221	0.0450849
$533$	−7307.61	−0.593860
$534$	0	0
$535$	−12734.0	−1.02904
$536$	11940.1	0.962186
$537$	0	0
$538$	−12464.8	−0.998878
$539$	20261.1	1.61913
$540$	0	0
$541$	4747.43	0.377279	0.188639	−	0.982046i	$-0.439592\pi$
$541$	4747.43	0.377279	0.188639	+	0.982046i	$0.439592\pi$
$542$	−3175.47	−0.251657
$543$	0	0
$544$	−1116.04	−0.0879591
$545$	−9866.21	−0.775454
$546$	0	0
$547$	−19924.5	−1.55742	−0.778712	−	0.627381i	$-0.784127\pi$
$547$	−19924.5	−1.55742	−0.778712	+	0.627381i	$0.784127\pi$
$548$	2566.34	0.200052
$549$	0	0
$550$	8364.41	0.648472
$551$	−14924.4	−1.15391
$552$	0	0
$553$	−695.401	−0.0534746
$554$	7072.93	0.542419
$555$	0	0
$556$	2974.77	0.226904
$557$	−502.841	−0.0382515	−0.0191257	−	0.999817i	$-0.506088\pi$
$557$	−502.841	−0.0382515	−0.0191257	+	0.999817i	$0.506088\pi$
$558$	0	0
$559$	−5177.47	−0.391741
$560$	1552.87	0.117180
$561$	0	0
$562$	19088.1	1.43271
$563$	−3139.90	−0.235046	−0.117523	−	0.993070i	$-0.537495\pi$
$563$	−3139.90	−0.235046	−0.117523	+	0.993070i	$0.537495\pi$
$564$	0	0
$565$	7970.31	0.593475
$566$	9197.90	0.683068
$567$	0	0
$568$	11617.2	0.858184
$569$	−1954.85	−0.144027	−0.0720135	−	0.997404i	$-0.522942\pi$
$569$	−1954.85	−0.144027	−0.0720135	+	0.997404i	$0.522942\pi$
$570$	0	0
$571$	5443.14	0.398928	0.199464	−	0.979905i	$-0.436080\pi$
$571$	5443.14	0.398928	0.199464	+	0.979905i	$0.436080\pi$
$572$	1897.32	0.138691
$573$	0	0
$574$	−3257.32	−0.236860
$575$	−1447.12	−0.104955
$576$	0	0
$577$	17676.5	1.27536	0.637679	−	0.770302i	$-0.279895\pi$
$577$	17676.5	1.27536	0.637679	+	0.770302i	$0.279895\pi$
$578$	738.475	0.0531427
$579$	0	0
$580$	1809.27	0.129527
$581$	3822.26	0.272933
$582$	0	0
$583$	−21501.4	−1.52744
$584$	−12701.7	−0.900002
$585$	0	0
$586$	14461.2	1.01943
$587$	−4972.44	−0.349633	−0.174816	−	0.984601i	$-0.555933\pi$
$587$	−4972.44	−0.349633	−0.174816	+	0.984601i	$0.555933\pi$
$588$	0	0
$589$	−7404.90	−0.518019
$590$	1709.55	0.119290
$591$	0	0
$592$	−18591.0	−1.29068
$593$	−5167.46	−0.357845	−0.178922	−	0.983863i	$-0.557261\pi$
$593$	−5167.46	−0.357845	−0.178922	+	0.983863i	$0.557261\pi$
$594$	0	0
$595$	527.208	0.0363251
$596$	3411.64	0.234474
$597$	0	0
$598$	1457.47	0.0996664
$599$	19327.9	1.31839	0.659196	−	0.751972i	$-0.270897\pi$
$599$	19327.9	1.31839	0.659196	+	0.751972i	$0.270897\pi$
$600$	0	0
$601$	−21546.2	−1.46238	−0.731189	−	0.682175i	$-0.761034\pi$
$601$	−21546.2	−1.46238	−0.731189	+	0.682175i	$0.761034\pi$
$602$	−2307.82	−0.156245
$603$	0	0
$604$	4900.93	0.330159
$605$	−20735.0	−1.39339
$606$	0	0
$607$	18276.7	1.22212	0.611061	−	0.791583i	$-0.290743\pi$
$607$	18276.7	1.22212	0.611061	+	0.791583i	$0.290743\pi$
$608$	−6745.64	−0.449954
$609$	0	0
$610$	8880.77	0.589462
$611$	5650.65	0.374142
$612$	0	0
$613$	1261.91	0.0831456	0.0415728	−	0.999135i	$-0.486763\pi$
$613$	1261.91	0.0831456	0.0415728	+	0.999135i	$0.486763\pi$
$614$	−5531.84	−0.363594
$615$	0	0
$616$	5446.49	0.356242
$617$	7208.71	0.470359	0.235180	−	0.971952i	$-0.424432\pi$
$617$	7208.71	0.470359	0.235180	+	0.971952i	$0.424432\pi$
$618$	0	0
$619$	2789.48	0.181129	0.0905643	−	0.995891i	$-0.471133\pi$
$619$	2789.48	0.181129	0.0905643	+	0.995891i	$0.471133\pi$
$620$	897.685	0.0581482
$621$	0	0
$622$	−16415.7	−1.05821
$623$	−2655.22	−0.170753
$624$	0	0
$625$	−6133.31	−0.392532
$626$	−17561.9	−1.12127
$627$	0	0
$628$	−2165.82	−0.137620
$629$	−6311.73	−0.400104
$630$	0	0
$631$	22237.3	1.40294	0.701469	−	0.712700i	$-0.252528\pi$
$631$	22237.3	1.40294	0.701469	+	0.712700i	$0.252528\pi$
$632$	4596.52	0.289303
$633$	0	0
$634$	−23870.8	−1.49532
$635$	21635.6	1.35210
$636$	0	0
$637$	−6917.59	−0.430275
$638$	−22815.2	−1.41577
$639$	0	0
$640$	−7852.71	−0.485009
$641$	−17554.3	−1.08168	−0.540838	−	0.841127i	$-0.681893\pi$
$641$	−17554.3	−1.08168	−0.540838	+	0.841127i	$0.681893\pi$
$642$	0	0
$643$	−8076.29	−0.495331	−0.247666	−	0.968846i	$-0.579663\pi$
$643$	−8076.29	−0.495331	−0.247666	+	0.968846i	$0.579663\pi$
$644$	−146.317	−0.00895295
$645$	0	0
$646$	4463.54	0.271851
$647$	−17091.8	−1.03856	−0.519280	−	0.854604i	$-0.673800\pi$
$647$	−17091.8	−1.03856	−0.519280	+	0.854604i	$0.673800\pi$
$648$	0	0
$649$	4855.27	0.293661
$650$	−2855.79	−0.172328
$651$	0	0
$652$	−1119.39	−0.0672373
$653$	1032.07	0.0618501	0.0309250	−	0.999522i	$-0.490155\pi$
$653$	1032.07	0.0618501	0.0309250	+	0.999522i	$0.490155\pi$
$654$	0	0
$655$	854.811	0.0509927
$656$	17434.3	1.03764
$657$	0	0
$658$	2518.74	0.149226
$659$	−20405.8	−1.20622	−0.603109	−	0.797659i	$-0.706071\pi$
$659$	−20405.8	−1.20622	−0.603109	+	0.797659i	$0.706071\pi$
$660$	0	0
$661$	15338.8	0.902588	0.451294	−	0.892375i	$-0.350963\pi$
$661$	15338.8	0.902588	0.451294	+	0.892375i	$0.350963\pi$
$662$	11618.5	0.682122
$663$	0	0
$664$	−25264.7	−1.47659
$665$	3186.59	0.185820
$666$	0	0
$667$	3947.24	0.229142
$668$	2033.71	0.117795
$669$	0	0
$670$	10679.3	0.615787
$671$	25222.2	1.45111
$672$	0	0
$673$	−19367.8	−1.10932	−0.554662	−	0.832076i	$-0.687153\pi$
$673$	−19367.8	−1.10932	−0.554662	+	0.832076i	$0.687153\pi$
$674$	−5636.49	−0.322121
$675$	0	0
$676$	2583.05	0.146965
$677$	16111.2	0.914629	0.457314	−	0.889305i	$-0.348811\pi$
$677$	16111.2	0.914629	0.457314	+	0.889305i	$0.348811\pi$
$678$	0	0
$679$	−6415.22	−0.362583
$680$	−3484.78	−0.196522
$681$	0	0
$682$	−11320.0	−0.635579
$683$	41.7084	0.00233664	0.00116832	−	0.999999i	$-0.499628\pi$
$683$	41.7084	0.00233664	0.00116832	+	0.999999i	$0.499628\pi$
$684$	0	0
$685$	14782.3	0.824529
$686$	−6292.34	−0.350208
$687$	0	0
$688$	12352.3	0.684485
$689$	7341.04	0.405909
$690$	0	0
$691$	5712.73	0.314504	0.157252	−	0.987559i	$-0.449736\pi$
$691$	5712.73	0.314504	0.157252	+	0.987559i	$0.449736\pi$
$692$	−4506.49	−0.247559
$693$	0	0
$694$	−4528.48	−0.247692
$695$	17134.9	0.935198
$696$	0	0
$697$	5919.03	0.321663
$698$	4490.96	0.243532
$699$	0	0
$700$	286.695	0.0154801
$701$	−18596.0	−1.00194	−0.500971	−	0.865464i	$-0.667024\pi$
$701$	−18596.0	−1.00194	−0.500971	+	0.865464i	$0.667024\pi$
$702$	0	0
$703$	−38149.8	−2.04673
$704$	−34937.1	−1.87037
$705$	0	0
$706$	3199.78	0.170574
$707$	−3853.45	−0.204984
$708$	0	0
$709$	13428.3	0.711297	0.355649	−	0.934620i	$-0.384260\pi$
$709$	13428.3	0.711297	0.355649	+	0.934620i	$0.384260\pi$
$710$	10390.6	0.549227
$711$	0	0
$712$	17550.7	0.923791
$713$	1958.46	0.102868
$714$	0	0
$715$	10928.7	0.571622
$716$	4415.14	0.230449
$717$	0	0
$718$	18867.6	0.980689
$719$	5148.06	0.267024	0.133512	−	0.991047i	$-0.457375\pi$
$719$	5148.06	0.267024	0.133512	+	0.991047i	$0.457375\pi$
$720$	0	0
$721$	−3156.13	−0.163024
$722$	9452.24	0.487224
$723$	0	0
$724$	−3986.93	−0.204659
$725$	−7734.28	−0.396198
$726$	0	0
$727$	4944.39	0.252238	0.126119	−	0.992015i	$-0.459748\pi$
$727$	4944.39	0.252238	0.126119	+	0.992015i	$0.459748\pi$
$728$	−1859.55	−0.0946696
$729$	0	0
$730$	−11360.6	−0.575990
$731$	4193.66	0.212186
$732$	0	0
$733$	9650.69	0.486298	0.243149	−	0.969989i	$-0.421820\pi$
$733$	9650.69	0.486298	0.243149	+	0.969989i	$0.421820\pi$
$734$	−12290.6	−0.618059
$735$	0	0
$736$	1784.10	0.0893517
$737$	30330.2	1.51591
$738$	0	0
$739$	−11809.7	−0.587856	−0.293928	−	0.955828i	$-0.594963\pi$
$739$	−11809.7	−0.587856	−0.293928	+	0.955828i	$0.594963\pi$
$740$	4624.85	0.229747
$741$	0	0
$742$	3272.22	0.161896
$743$	−5901.27	−0.291381	−0.145691	−	0.989330i	$-0.546540\pi$
$743$	−5901.27	−0.291381	−0.145691	+	0.989330i	$0.546540\pi$
$744$	0	0
$745$	19651.3	0.966399
$746$	26471.9	1.29920
$747$	0	0
$748$	−1536.80	−0.0751215
$749$	5503.93	0.268504
$750$	0	0
$751$	−518.294	−0.0251835	−0.0125918	−	0.999921i	$-0.504008\pi$
$751$	−518.294	−0.0251835	−0.0125918	+	0.999921i	$0.504008\pi$
$752$	−13481.2	−0.653734
$753$	0	0
$754$	7789.61	0.376234
$755$	28229.7	1.36077
$756$	0	0
$757$	−16294.6	−0.782349	−0.391175	−	0.920316i	$-0.627931\pi$
$757$	−16294.6	−0.782349	−0.391175	+	0.920316i	$0.627931\pi$
$758$	3406.49	0.163231
$759$	0	0
$760$	−21063.0	−1.00531
$761$	−15850.1	−0.755015	−0.377507	−	0.926007i	$-0.623219\pi$
$761$	−15850.1	−0.755015	−0.377507	+	0.926007i	$0.623219\pi$
$762$	0	0
$763$	4264.40	0.202335
$764$	1927.85	0.0912923
$765$	0	0
$766$	20914.9	0.986537
$767$	−1657.70	−0.0780390
$768$	0	0
$769$	39962.5	1.87397	0.936987	−	0.349364i	$-0.113602\pi$
$769$	39962.5	1.87397	0.936987	+	0.349364i	$0.113602\pi$
$770$	4871.39	0.227990
$771$	0	0
$772$	−6476.90	−0.301954
$773$	7681.94	0.357439	0.178719	−	0.983900i	$-0.442805\pi$
$773$	7681.94	0.357439	0.178719	+	0.983900i	$0.442805\pi$
$774$	0	0
$775$	−3837.44	−0.177864
$776$	42403.8	1.96161
$777$	0	0
$778$	−2374.25	−0.109410
$779$	35776.3	1.64547
$780$	0	0
$781$	29510.1	1.35206
$782$	−1180.53	−0.0539841
$783$	0	0
$784$	16503.8	0.751814
$785$	−12475.2	−0.567211
$786$	0	0
$787$	16191.1	0.733357	0.366679	−	0.930348i	$-0.380495\pi$
$787$	16191.1	0.733357	0.366679	+	0.930348i	$0.380495\pi$
$788$	−728.162	−0.0329184
$789$	0	0
$790$	4111.17	0.185150
$791$	−3444.95	−0.154853
$792$	0	0
$793$	−8611.40	−0.385624
$794$	19157.4	0.856262
$795$	0	0
$796$	−5910.46	−0.263179
$797$	4632.12	0.205870	0.102935	−	0.994688i	$-0.467177\pi$
$797$	4632.12	0.205870	0.102935	+	0.994688i	$0.467177\pi$
$798$	0	0
$799$	−4576.93	−0.202653
$800$	−3495.79	−0.154494
$801$	0	0
$802$	9315.22	0.410140
$803$	−32265.0	−1.41794
$804$	0	0
$805$	−842.796	−0.0369002
$806$	3864.89	0.168902
$807$	0	0
$808$	25470.8	1.10899
$809$	−10914.2	−0.474319	−0.237159	−	0.971471i	$-0.576216\pi$
$809$	−10914.2	−0.474319	−0.237159	+	0.971471i	$0.576216\pi$
$810$	0	0
$811$	2928.60	0.126803	0.0634014	−	0.997988i	$-0.479805\pi$
$811$	2928.60	0.126803	0.0634014	+	0.997988i	$0.479805\pi$
$812$	−782.006	−0.0337968
$813$	0	0
$814$	−58320.3	−2.51121
$815$	−6447.76	−0.277123
$816$	0	0
$817$	25347.6	1.08543
$818$	−34714.2	−1.48381
$819$	0	0
$820$	−4337.10	−0.184705
$821$	−10429.7	−0.443361	−0.221681	−	0.975119i	$-0.571154\pi$
$821$	−10429.7	−0.443361	−0.221681	+	0.975119i	$0.571154\pi$
$822$	0	0
$823$	25715.8	1.08918	0.544591	−	0.838702i	$-0.316685\pi$
$823$	25715.8	1.08918	0.544591	+	0.838702i	$0.316685\pi$
$824$	20861.6	0.881977
$825$	0	0
$826$	−738.907	−0.0311257
$827$	−31929.2	−1.34255	−0.671274	−	0.741209i	$-0.734253\pi$
$827$	−31929.2	−1.34255	−0.671274	+	0.741209i	$0.734253\pi$
$828$	0	0
$829$	31374.4	1.31445	0.657224	−	0.753695i	$-0.271730\pi$
$829$	31374.4	1.31445	0.657224	+	0.753695i	$0.271730\pi$
$830$	−22596.9	−0.945002
$831$	0	0
$832$	11928.3	0.497042
$833$	5603.13	0.233058
$834$	0	0
$835$	11714.3	0.485498
$836$	−9288.82	−0.384283
$837$	0	0
$838$	−33197.6	−1.36849
$839$	−38253.1	−1.57407	−0.787034	−	0.616910i	$-0.788384\pi$
$839$	−38253.1	−1.57407	−0.787034	+	0.616910i	$0.788384\pi$
$840$	0	0
$841$	−3292.57	−0.135002
$842$	7759.58	0.317593
$843$	0	0
$844$	5481.56	0.223558
$845$	14878.5	0.605725
$846$	0	0
$847$	8962.16	0.363570
$848$	−17514.1	−0.709240
$849$	0	0
$850$	2313.14	0.0933412
$851$	10090.0	0.406439
$852$	0	0
$853$	1728.49	0.0693813	0.0346906	−	0.999398i	$-0.488955\pi$
$853$	1728.49	0.0693813	0.0346906	+	0.999398i	$0.488955\pi$
$854$	−3838.47	−0.153805
$855$	0	0
$856$	−36380.3	−1.45263
$857$	−17426.4	−0.694603	−0.347301	−	0.937754i	$-0.612902\pi$
$857$	−17426.4	−0.694603	−0.347301	+	0.937754i	$0.612902\pi$
$858$	0	0
$859$	−10053.4	−0.399323	−0.199661	−	0.979865i	$-0.563984\pi$
$859$	−10053.4	−0.399323	−0.199661	+	0.979865i	$0.563984\pi$
$860$	−3072.85	−0.121841
$861$	0	0
$862$	−4884.86	−0.193015
$863$	35498.0	1.40019	0.700096	−	0.714049i	$-0.253140\pi$
$863$	35498.0	1.40019	0.700096	+	0.714049i	$0.253140\pi$
$864$	0	0
$865$	−25957.7	−1.02033
$866$	15618.8	0.612873
$867$	0	0
$868$	−388.000	−0.0151723
$869$	11676.1	0.455793
$870$	0	0
$871$	−10355.4	−0.402846
$872$	−28187.2	−1.09465
$873$	0	0
$874$	−7135.43	−0.276155
$875$	5527.91	0.213574
$876$	0	0
$877$	−30475.1	−1.17340	−0.586699	−	0.809805i	$-0.699573\pi$
$877$	−30475.1	−1.17340	−0.586699	+	0.809805i	$0.699573\pi$
$878$	33378.5	1.28299
$879$	0	0
$880$	−26073.4	−0.998788
$881$	−2396.70	−0.0916537	−0.0458268	−	0.998949i	$-0.514592\pi$
$881$	−2396.70	−0.0916537	−0.0458268	+	0.998949i	$0.514592\pi$
$882$	0	0
$883$	11248.6	0.428705	0.214352	−	0.976756i	$-0.431236\pi$
$883$	11248.6	0.428705	0.214352	+	0.976756i	$0.431236\pi$
$884$	524.696	0.0199632
$885$	0	0
$886$	−26259.7	−0.995725
$887$	−19513.0	−0.738648	−0.369324	−	0.929301i	$-0.620411\pi$
$887$	−19513.0	−0.738648	−0.369324	+	0.929301i	$0.620411\pi$
$888$	0	0
$889$	−9351.39	−0.352796
$890$	15697.5	0.591215
$891$	0	0
$892$	−8076.76	−0.303173
$893$	−27664.2	−1.03667
$894$	0	0
$895$	25431.5	0.949811
$896$	3394.12	0.126551
$897$	0	0
$898$	16574.3	0.615916
$899$	10467.2	0.388321
$900$	0	0
$901$	−5946.12	−0.219860
$902$	54691.7	2.01889
$903$	0	0
$904$	22770.7	0.837768
$905$	−22965.0	−0.843516
$906$	0	0
$907$	−1024.73	−0.0375146	−0.0187573	−	0.999824i	$-0.505971\pi$
$907$	−1024.73	−0.0375146	−0.0187573	+	0.999824i	$0.505971\pi$
$908$	−4988.79	−0.182333
$909$	0	0
$910$	−1663.20	−0.0605873
$911$	46205.4	1.68041	0.840205	−	0.542269i	$-0.182435\pi$
$911$	46205.4	1.68041	0.840205	+	0.542269i	$0.182435\pi$
$912$	0	0
$913$	−64177.4	−2.32635
$914$	−23542.5	−0.851988
$915$	0	0
$916$	−3812.48	−0.137520
$917$	−369.469	−0.0133053
$918$	0	0
$919$	−44026.2	−1.58030	−0.790148	−	0.612916i	$-0.789996\pi$
$919$	−44026.2	−1.58030	−0.790148	+	0.612916i	$0.789996\pi$
$920$	5570.78	0.199634
$921$	0	0
$922$	−33995.5	−1.21430
$923$	−10075.4	−0.359302
$924$	0	0
$925$	−19770.4	−0.702752
$926$	−43529.3	−1.54478
$927$	0	0
$928$	9535.30	0.337297
$929$	19511.4	0.689073	0.344537	−	0.938773i	$-0.388036\pi$
$929$	19511.4	0.689073	0.344537	+	0.938773i	$0.388036\pi$
$930$	0	0
$931$	33866.8	1.19220
$932$	−8771.77	−0.308293
$933$	0	0
$934$	−42549.7	−1.49065
$935$	−8852.05	−0.309618
$936$	0	0
$937$	−34871.5	−1.21580	−0.607900	−	0.794014i	$-0.707988\pi$
$937$	−34871.5	−1.21580	−0.607900	+	0.794014i	$0.707988\pi$
$938$	−4615.84	−0.160674
$939$	0	0
$940$	3353.69	0.116367
$941$	16423.1	0.568945	0.284473	−	0.958684i	$-0.408182\pi$
$941$	16423.1	0.568945	0.284473	+	0.958684i	$0.408182\pi$
$942$	0	0
$943$	−9462.18	−0.326756
$944$	3954.89	0.136357
$945$	0	0
$946$	38749.3	1.33176
$947$	−37007.6	−1.26989	−0.634944	−	0.772558i	$-0.718977\pi$
$947$	−37007.6	−1.26989	−0.634944	+	0.772558i	$0.718977\pi$
$948$	0	0
$949$	11016.0	0.376811
$950$	13981.3	0.477486
$951$	0	0
$952$	1506.20	0.0512776
$953$	−2540.43	−0.0863510	−0.0431755	−	0.999068i	$-0.513747\pi$
$953$	−2540.43	−0.0863510	−0.0431755	+	0.999068i	$0.513747\pi$
$954$	0	0
$955$	11104.6	0.376267
$956$	5972.05	0.202040
$957$	0	0
$958$	21153.4	0.713399
$959$	−6389.25	−0.215140
$960$	0	0
$961$	−24597.6	−0.825672
$962$	19911.8	0.667341
$963$	0	0
$964$	−3999.28	−0.133618
$965$	−37307.4	−1.24452
$966$	0	0
$967$	−29865.4	−0.993183	−0.496591	−	0.867984i	$-0.665415\pi$
$967$	−29865.4	−0.993183	−0.496591	+	0.867984i	$0.665415\pi$
$968$	−59238.8	−1.96695
$969$	0	0
$970$	37926.4	1.25541
$971$	40980.9	1.35442	0.677209	−	0.735790i	$-0.263189\pi$
$971$	40980.9	1.35442	0.677209	+	0.735790i	$0.263189\pi$
$972$	0	0
$973$	−7406.08	−0.244017
$974$	−14302.1	−0.470503
$975$	0	0
$976$	20544.9	0.673796
$977$	−29118.0	−0.953497	−0.476748	−	0.879040i	$-0.658185\pi$
$977$	−29118.0	−0.953497	−0.476748	+	0.879040i	$0.658185\pi$
$978$	0	0
$979$	44582.3	1.45542
$980$	−4105.63	−0.133826
$981$	0	0
$982$	11660.0	0.378906
$983$	35854.0	1.16334	0.581671	−	0.813424i	$-0.302399\pi$
$983$	35854.0	1.16334	0.581671	+	0.813424i	$0.302399\pi$
$984$	0	0
$985$	−4194.26	−0.135675
$986$	−6309.44	−0.203787
$987$	0	0
$988$	3171.41	0.102121
$989$	−6703.99	−0.215545
$990$	0	0
$991$	22194.9	0.711448	0.355724	−	0.934591i	$-0.384234\pi$
$991$	22194.9	0.711448	0.355724	+	0.934591i	$0.384234\pi$
$992$	4731.03	0.151422
$993$	0	0
$994$	−4491.04	−0.143307
$995$	−34044.6	−1.08471
$996$	0	0
$997$	−7611.65	−0.241789	−0.120894	−	0.992665i	$-0.538576\pi$
$997$	−7611.65	−0.241789	−0.120894	+	0.992665i	$0.538576\pi$
$998$	−38987.1	−1.23659
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.f.1.3		3
3.2	odd	2		51.4.a.e.1.1	✓	3
4.3	odd	2		2448.4.a.bd.1.1		3
12.11	even	2		816.4.a.s.1.3		3
15.14	odd	2		1275.4.a.q.1.3		3
21.20	even	2		2499.4.a.n.1.1		3
51.50	odd	2		867.4.a.k.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.4.a.e.1.1	✓	3	3.2	odd	2
153.4.a.f.1.3		3	1.1	even	1	trivial
816.4.a.s.1.3		3	12.11	even	2
867.4.a.k.1.1		3	51.50	odd	2
1275.4.a.q.1.3		3	15.14	odd	2
2448.4.a.bd.1.1		3	4.3	odd	2
2499.4.a.n.1.1		3	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.a (trivial)

\(f(q)\)	\(=\)	\(q+2.55528 q^{2} -1.47057 q^{4} -8.47057 q^{5} +3.66117 q^{7} -24.1999 q^{8} +O(q^{10})\)	\(q+2.55528 q^{2} -1.47057 q^{4} -8.47057 q^{5} +3.66117 q^{7} -24.1999 q^{8} -21.6446 q^{10} -61.4727 q^{11} +20.9881 q^{13} +9.35531 q^{14} -50.0729 q^{16} -17.0000 q^{17} -102.753 q^{19} +12.4566 q^{20} -157.080 q^{22} +27.1763 q^{23} -53.2495 q^{25} +53.6304 q^{26} -5.38401 q^{28} +145.246 q^{29} +72.0652 q^{31} +65.6493 q^{32} -43.4397 q^{34} -31.0122 q^{35} +371.278 q^{37} -262.561 q^{38} +204.987 q^{40} -348.178 q^{41} -246.686 q^{43} +90.3998 q^{44} +69.4428 q^{46} +269.231 q^{47} -329.596 q^{49} -136.067 q^{50} -30.8645 q^{52} +349.771 q^{53} +520.709 q^{55} -88.6001 q^{56} +371.144 q^{58} -78.9826 q^{59} -410.299 q^{61} +184.147 q^{62} +568.335 q^{64} -177.781 q^{65} -493.393 q^{67} +24.9997 q^{68} -79.2448 q^{70} -480.053 q^{71} +524.867 q^{73} +948.718 q^{74} +151.105 q^{76} -225.062 q^{77} -189.939 q^{79} +424.146 q^{80} -889.692 q^{82} +1044.00 q^{83} +144.000 q^{85} -630.350 q^{86} +1487.63 q^{88} -725.237 q^{89} +76.8411 q^{91} -39.9645 q^{92} +687.960 q^{94} +870.373 q^{95} -1752.23 q^{97} -842.208 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8}+O(q^{10}) \) 3 * q - 5 * q^2 + 13 * q^4 - 8 * q^5 - 8 * q^7 - 33 * q^8	\( 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8} - 38 q^{10} - 34 q^{11} + 36 q^{13} + 104 q^{14} - 79 q^{16} - 51 q^{17} - 142 q^{19} + 126 q^{20} - 248 q^{22} - 110 q^{23} - 193 q^{25} - 154 q^{26} - 472 q^{28} - 90 q^{29} - 148 q^{31} + 151 q^{32} + 85 q^{34} - 416 q^{35} + 110 q^{37} - 80 q^{38} - 202 q^{40} - 720 q^{41} - 146 q^{43} + 192 q^{44} + 748 q^{46} - 500 q^{47} + 379 q^{49} + 385 q^{50} + 1218 q^{52} - 610 q^{53} + 430 q^{55} + 1368 q^{56} + 1006 q^{58} + 216 q^{59} - 18 q^{61} + 904 q^{62} + 377 q^{64} + 966 q^{65} - 1404 q^{67} - 221 q^{68} + 1472 q^{70} + 960 q^{71} - 794 q^{73} + 1874 q^{74} - 392 q^{76} - 48 q^{77} - 276 q^{79} + 1130 q^{80} + 382 q^{82} + 1552 q^{83} + 136 q^{85} - 16 q^{86} + 1724 q^{88} - 1394 q^{89} - 3968 q^{91} - 2244 q^{92} + 3960 q^{94} + 602 q^{95} + 402 q^{97} - 4109 q^{98}+O(q^{100}) \) 3 * q - 5 * q^2 + 13 * q^4 - 8 * q^5 - 8 * q^7 - 33 * q^8 - 38 * q^10 - 34 * q^11 + 36 * q^13 + 104 * q^14 - 79 * q^16 - 51 * q^17 - 142 * q^19 + 126 * q^20 - 248 * q^22 - 110 * q^23 - 193 * q^25 - 154 * q^26 - 472 * q^28 - 90 * q^29 - 148 * q^31 + 151 * q^32 + 85 * q^34 - 416 * q^35 + 110 * q^37 - 80 * q^38 - 202 * q^40 - 720 * q^41 - 146 * q^43 + 192 * q^44 + 748 * q^46 - 500 * q^47 + 379 * q^49 + 385 * q^50 + 1218 * q^52 - 610 * q^53 + 430 * q^55 + 1368 * q^56 + 1006 * q^58 + 216 * q^59 - 18 * q^61 + 904 * q^62 + 377 * q^64 + 966 * q^65 - 1404 * q^67 - 221 * q^68 + 1472 * q^70 + 960 * q^71 - 794 * q^73 + 1874 * q^74 - 392 * q^76 - 48 * q^77 - 276 * q^79 + 1130 * q^80 + 382 * q^82 + 1552 * q^83 + 136 * q^85 - 16 * q^86 + 1724 * q^88 - 1394 * q^89 - 3968 * q^91 - 2244 * q^92 + 3960 * q^94 + 602 * q^95 + 402 * q^97 - 4109 * q^98

Introduction

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