LMFDB - Embedded newform 261.4.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,4,Mod(1,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, 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("261.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	261.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:	$15.3994985115$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13458092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 $ x^5 - x^4 - 14x^3 + 18x^2 + 20*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.957567$ of defining polynomial
Character	$\chi$	$=$	261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.84972 q^{2} +0.120922 q^{4} -12.8729 q^{5} +26.0540 q^{7} -22.4532 q^{8} +O(q^{10})$	\(q+2.84972 q^{2} +0.120922 q^{4} -12.8729 q^{5} +26.0540 q^{7} -22.4532 q^{8} -36.6841 q^{10} +62.8274 q^{11} +22.3936 q^{13} +74.2465 q^{14} -64.9528 q^{16} +57.9808 q^{17} +71.3143 q^{19} -1.55661 q^{20} +179.041 q^{22} +49.5307 q^{23} +40.7104 q^{25} +63.8155 q^{26} +3.15048 q^{28} +29.0000 q^{29} +62.9198 q^{31} -5.47182 q^{32} +165.229 q^{34} -335.389 q^{35} +119.123 q^{37} +203.226 q^{38} +289.037 q^{40} +414.916 q^{41} -348.009 q^{43} +7.59719 q^{44} +141.149 q^{46} -553.259 q^{47} +335.808 q^{49} +116.013 q^{50} +2.70787 q^{52} +107.308 q^{53} -808.768 q^{55} -584.994 q^{56} +82.6420 q^{58} -136.881 q^{59} -579.408 q^{61} +179.304 q^{62} +504.029 q^{64} -288.269 q^{65} +919.959 q^{67} +7.01113 q^{68} -955.765 q^{70} -781.802 q^{71} -133.237 q^{73} +339.467 q^{74} +8.62344 q^{76} +1636.90 q^{77} +868.196 q^{79} +836.127 q^{80} +1182.39 q^{82} +83.3560 q^{83} -746.379 q^{85} -991.730 q^{86} -1410.68 q^{88} +357.919 q^{89} +583.442 q^{91} +5.98933 q^{92} -1576.64 q^{94} -918.019 q^{95} -187.105 q^{97} +956.961 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8}+O(q^{10}) $ 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8	$ 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8} - 64 q^{10} - 12 q^{11} + 14 q^{13} + 192 q^{14} + 146 q^{16} - 66 q^{17} + 214 q^{19} - 6 q^{20} - 98 q^{22} - 164 q^{23} + 207 q^{25} - 56 q^{26} + 540 q^{28} + 145 q^{29} + 420 q^{31} + 652 q^{32} + 204 q^{34} + 52 q^{35} + 378 q^{37} + 496 q^{38} - 80 q^{40} + 1158 q^{41} - 204 q^{43} - 784 q^{44} + 580 q^{46} - 248 q^{47} - 283 q^{49} - 908 q^{50} + 1482 q^{52} + 554 q^{53} + 546 q^{55} + 608 q^{56} - 440 q^{59} + 618 q^{61} - 1250 q^{62} + 2594 q^{64} + 1656 q^{65} + 1164 q^{67} - 356 q^{68} - 2184 q^{70} + 692 q^{71} - 1950 q^{73} + 1832 q^{74} + 1376 q^{76} + 1616 q^{77} + 272 q^{79} + 890 q^{80} + 92 q^{82} - 512 q^{83} - 1628 q^{85} - 2446 q^{86} - 6954 q^{88} - 866 q^{89} + 2580 q^{91} - 3468 q^{92} - 5942 q^{94} - 2244 q^{95} + 1562 q^{97} + 3408 q^{98}+O(q^{100}) $ 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8 - 64 * q^10 - 12 * q^11 + 14 * q^13 + 192 * q^14 + 146 * q^16 - 66 * q^17 + 214 * q^19 - 6 * q^20 - 98 * q^22 - 164 * q^23 + 207 * q^25 - 56 * q^26 + 540 * q^28 + 145 * q^29 + 420 * q^31 + 652 * q^32 + 204 * q^34 + 52 * q^35 + 378 * q^37 + 496 * q^38 - 80 * q^40 + 1158 * q^41 - 204 * q^43 - 784 * q^44 + 580 * q^46 - 248 * q^47 - 283 * q^49 - 908 * q^50 + 1482 * q^52 + 554 * q^53 + 546 * q^55 + 608 * q^56 - 440 * q^59 + 618 * q^61 - 1250 * q^62 + 2594 * q^64 + 1656 * q^65 + 1164 * q^67 - 356 * q^68 - 2184 * q^70 + 692 * q^71 - 1950 * q^73 + 1832 * q^74 + 1376 * q^76 + 1616 * q^77 + 272 * q^79 + 890 * q^80 + 92 * q^82 - 512 * q^83 - 1628 * q^85 - 2446 * q^86 - 6954 * q^88 - 866 * q^89 + 2580 * q^91 - 3468 * q^92 - 5942 * q^94 - 2244 * q^95 + 1562 * q^97 + 3408 * 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 $

$2$	2.84972	1.00753	0.503765	−	0.863841i	$-0.331948\pi$
$2$	2.84972	1.00753	0.503765	+	0.863841i	$0.331948\pi$
$3$	0	0
$3$	0	0
$4$	0.120922	0.0151152
$5$	−12.8729	−1.15138	−0.575692	−	0.817667i	$-0.695267\pi$
$5$	−12.8729	−1.15138	−0.575692	+	0.817667i	$0.695267\pi$
$6$	0	0
$7$	26.0540	1.40678	0.703391	−	0.710804i	$-0.251669\pi$
$7$	26.0540	1.40678	0.703391	+	0.710804i	$0.251669\pi$
$8$	−22.4532	−0.992300
$9$	0	0
$10$	−36.6841	−1.16005
$11$	62.8274	1.72211	0.861053	−	0.508515i	$-0.169805\pi$
$11$	62.8274	1.72211	0.861053	+	0.508515i	$0.169805\pi$
$12$	0	0
$13$	22.3936	0.477759	0.238879	−	0.971049i	$-0.423220\pi$
$13$	22.3936	0.477759	0.238879	+	0.971049i	$0.423220\pi$
$14$	74.2465	1.41737
$15$	0	0
$16$	−64.9528	−1.01489
$17$	57.9808	0.827201	0.413601	−	0.910458i	$-0.364271\pi$
$17$	57.9808	0.827201	0.413601	+	0.910458i	$0.364271\pi$
$18$	0	0
$19$	71.3143	0.861086	0.430543	−	0.902570i	$-0.358322\pi$
$19$	71.3143	0.861086	0.430543	+	0.902570i	$0.358322\pi$
$20$	−1.55661	−0.0174034
$21$	0	0
$22$	179.041	1.73507
$23$	49.5307	0.449037	0.224519	−	0.974470i	$-0.427919\pi$
$23$	49.5307	0.449037	0.224519	+	0.974470i	$0.427919\pi$
$24$	0	0
$25$	40.7104	0.325683
$26$	63.8155	0.481356
$27$	0	0
$28$	3.15048	0.0212638
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	62.9198	0.364540	0.182270	−	0.983249i	$-0.441656\pi$
$31$	62.9198	0.364540	0.182270	+	0.983249i	$0.441656\pi$
$32$	−5.47182	−0.0302278
$33$	0	0
$34$	165.229	0.833429
$35$	−335.389	−1.61974
$36$	0	0
$37$	119.123	0.529288	0.264644	−	0.964346i	$-0.414746\pi$
$37$	119.123	0.529288	0.264644	+	0.964346i	$0.414746\pi$
$38$	203.226	0.867569
$39$	0	0
$40$	289.037	1.14252
$41$	414.916	1.58046	0.790231	−	0.612810i	$-0.209961\pi$
$41$	414.916	1.58046	0.790231	+	0.612810i	$0.209961\pi$
$42$	0	0
$43$	−348.009	−1.23421	−0.617103	−	0.786882i	$-0.711694\pi$
$43$	−348.009	−1.23421	−0.617103	+	0.786882i	$0.711694\pi$
$44$	7.59719	0.0260300
$45$	0	0
$46$	141.149	0.452418
$47$	−553.259	−1.71705	−0.858523	−	0.512775i	$-0.828618\pi$
$47$	−553.259	−1.71705	−0.858523	+	0.512775i	$0.828618\pi$
$48$	0	0
$49$	335.808	0.979033
$50$	116.013	0.328136
$51$	0	0
$52$	2.70787	0.00722142
$53$	107.308	0.278111	0.139055	−	0.990285i	$-0.455593\pi$
$53$	107.308	0.278111	0.139055	+	0.990285i	$0.455593\pi$
$54$	0	0
$55$	−808.768	−1.98280
$56$	−584.994	−1.39595
$57$	0	0
$58$	82.6420	0.187093
$59$	−136.881	−0.302041	−0.151020	−	0.988531i	$-0.548256\pi$
$59$	−136.881	−0.302041	−0.151020	+	0.988531i	$0.548256\pi$
$60$	0	0
$61$	−579.408	−1.21616	−0.608078	−	0.793877i	$-0.708059\pi$
$61$	−579.408	−1.21616	−0.608078	+	0.793877i	$0.708059\pi$
$62$	179.304	0.367285
$63$	0	0
$64$	504.029	0.984431
$65$	−288.269	−0.550084
$66$	0	0
$67$	919.959	1.67748	0.838738	−	0.544535i	$-0.183294\pi$
$67$	919.959	1.67748	0.838738	+	0.544535i	$0.183294\pi$
$68$	7.01113	0.0125033
$69$	0	0
$70$	−955.765	−1.63194
$71$	−781.802	−1.30680	−0.653400	−	0.757013i	$-0.726658\pi$
$71$	−781.802	−1.30680	−0.653400	+	0.757013i	$0.726658\pi$
$72$	0	0
$73$	−133.237	−0.213619	−0.106810	−	0.994279i	$-0.534064\pi$
$73$	−133.237	−0.213619	−0.106810	+	0.994279i	$0.534064\pi$
$74$	339.467	0.533273
$75$	0	0
$76$	8.62344	0.0130155
$77$	1636.90	2.42263
$78$	0	0
$79$	868.196	1.23645	0.618225	−	0.786001i	$-0.287852\pi$
$79$	868.196	1.23645	0.618225	+	0.786001i	$0.287852\pi$
$80$	836.127	1.16852
$81$	0	0
$82$	1182.39	1.59236
$83$	83.3560	0.110235	0.0551175	−	0.998480i	$-0.482447\pi$
$83$	83.3560	0.110235	0.0551175	+	0.998480i	$0.482447\pi$
$84$	0	0
$85$	−746.379	−0.952425
$86$	−991.730	−1.24350
$87$	0	0
$88$	−1410.68	−1.70885
$89$	357.919	0.426284	0.213142	−	0.977021i	$-0.431630\pi$
$89$	357.919	0.426284	0.213142	+	0.977021i	$0.431630\pi$
$90$	0	0
$91$	583.442	0.672102
$92$	5.98933	0.00678729
$93$	0	0
$94$	−1576.64	−1.72997
$95$	−918.019	−0.991440
$96$	0	0
$97$	−187.105	−0.195852	−0.0979260	−	0.995194i	$-0.531221\pi$
$97$	−187.105	−0.195852	−0.0979260	+	0.995194i	$0.531221\pi$
$98$	956.961	0.986404
$99$	0	0
$100$	4.92277	0.00492277
$101$	959.423	0.945209	0.472605	−	0.881275i	$-0.343314\pi$
$101$	959.423	0.945209	0.472605	+	0.881275i	$0.343314\pi$
$102$	0	0
$103$	78.8738	0.0754531	0.0377265	−	0.999288i	$-0.487988\pi$
$103$	78.8738	0.0754531	0.0377265	+	0.999288i	$0.487988\pi$
$104$	−502.808	−0.474080
$105$	0	0
$106$	305.797	0.280205
$107$	−713.851	−0.644959	−0.322479	−	0.946577i	$-0.604516\pi$
$107$	−713.851	−0.644959	−0.322479	+	0.946577i	$0.604516\pi$
$108$	0	0
$109$	536.561	0.471497	0.235749	−	0.971814i	$-0.424246\pi$
$109$	536.561	0.471497	0.235749	+	0.971814i	$0.424246\pi$
$110$	−2304.76	−1.99773
$111$	0	0
$112$	−1692.28	−1.42772
$113$	−1946.36	−1.62034	−0.810170	−	0.586195i	$-0.800625\pi$
$113$	−1946.36	−1.62034	−0.810170	+	0.586195i	$0.800625\pi$
$114$	0	0
$115$	−637.601	−0.517014
$116$	3.50673	0.00280682
$117$	0	0
$118$	−390.073	−0.304315
$119$	1510.63	1.16369
$120$	0	0
$121$	2616.28	1.96565
$122$	−1651.15	−1.22531
$123$	0	0
$124$	7.60837	0.00551009
$125$	1085.05	0.776397
$126$	0	0
$127$	−1995.14	−1.39402	−0.697009	−	0.717062i	$-0.745486\pi$
$127$	−1995.14	−1.39402	−0.697009	+	0.717062i	$0.745486\pi$
$128$	1480.12	1.02207
$129$	0	0
$130$	−821.488	−0.554225
$131$	−1544.84	−1.03033	−0.515164	−	0.857092i	$-0.672269\pi$
$131$	−1544.84	−1.03033	−0.515164	+	0.857092i	$0.672269\pi$
$132$	0	0
$133$	1858.02	1.21136
$134$	2621.63	1.69011
$135$	0	0
$136$	−1301.85	−0.820832
$137$	1294.93	0.807543	0.403771	−	0.914860i	$-0.367699\pi$
$137$	1294.93	0.807543	0.403771	+	0.914860i	$0.367699\pi$
$138$	0	0
$139$	1999.66	1.22021	0.610105	−	0.792320i	$-0.291127\pi$
$139$	1999.66	1.22021	0.610105	+	0.792320i	$0.291127\pi$
$140$	−40.5557	−0.0244828
$141$	0	0
$142$	−2227.92	−1.31664
$143$	1406.93	0.822752
$144$	0	0
$145$	−373.313	−0.213807
$146$	−379.688	−0.215228
$147$	0	0
$148$	14.4045	0.00800029
$149$	−1187.63	−0.652984	−0.326492	−	0.945200i	$-0.605867\pi$
$149$	−1187.63	−0.652984	−0.326492	+	0.945200i	$0.605867\pi$
$150$	0	0
$151$	−2257.61	−1.21670	−0.608350	−	0.793669i	$-0.708168\pi$
$151$	−2257.61	−1.21670	−0.608350	+	0.793669i	$0.708168\pi$
$152$	−1601.23	−0.854456
$153$	0	0
$154$	4664.72	2.44087
$155$	−809.958	−0.419725
$156$	0	0
$157$	1188.18	0.603995	0.301997	−	0.953309i	$-0.402347\pi$
$157$	1188.18	0.603995	0.301997	+	0.953309i	$0.402347\pi$
$158$	2474.12	1.24576
$159$	0	0
$160$	70.4379	0.0348038
$161$	1290.47	0.631697
$162$	0	0
$163$	−2452.33	−1.17841	−0.589207	−	0.807982i	$-0.700560\pi$
$163$	−2452.33	−1.17841	−0.589207	+	0.807982i	$0.700560\pi$
$164$	50.1722	0.0238890
$165$	0	0
$166$	237.541	0.111065
$167$	2020.14	0.936067	0.468034	−	0.883711i	$-0.344963\pi$
$167$	2020.14	0.936067	0.468034	+	0.883711i	$0.344963\pi$
$168$	0	0
$169$	−1695.53	−0.771746
$170$	−2126.97	−0.959597
$171$	0	0
$172$	−42.0818	−0.0186553
$173$	−2862.12	−1.25782	−0.628910	−	0.777478i	$-0.716499\pi$
$173$	−2862.12	−1.25782	−0.628910	+	0.777478i	$0.716499\pi$
$174$	0	0
$175$	1060.67	0.458165
$176$	−4080.81	−1.74774
$177$	0	0
$178$	1019.97	0.429494
$179$	232.651	0.0971460	0.0485730	−	0.998820i	$-0.484533\pi$
$179$	232.651	0.0971460	0.0485730	+	0.998820i	$0.484533\pi$
$180$	0	0
$181$	−2607.67	−1.07086	−0.535432	−	0.844578i	$-0.679851\pi$
$181$	−2607.67	−1.07086	−0.535432	+	0.844578i	$0.679851\pi$
$182$	1662.65	0.677163
$183$	0	0
$184$	−1112.12	−0.445580
$185$	−1533.45	−0.609413
$186$	0	0
$187$	3642.78	1.42453
$188$	−66.9010	−0.0259535
$189$	0	0
$190$	−2616.10	−0.998905
$191$	−1528.90	−0.579202	−0.289601	−	0.957147i	$-0.593523\pi$
$191$	−1528.90	−0.579202	−0.289601	+	0.957147i	$0.593523\pi$
$192$	0	0
$193$	1017.58	0.379518	0.189759	−	0.981831i	$-0.439229\pi$
$193$	1017.58	0.379518	0.189759	+	0.981831i	$0.439229\pi$
$194$	−533.197	−0.197327
$195$	0	0
$196$	40.6065	0.0147983
$197$	−3290.20	−1.18994	−0.594968	−	0.803749i	$-0.702835\pi$
$197$	−3290.20	−1.18994	−0.594968	+	0.803749i	$0.702835\pi$
$198$	0	0
$199$	−29.9190	−0.0106578	−0.00532891	−	0.999986i	$-0.501696\pi$
$199$	−29.9190	−0.0106578	−0.00532891	+	0.999986i	$0.501696\pi$
$200$	−914.079	−0.323176
$201$	0	0
$202$	2734.09	0.952326
$203$	755.565	0.261233
$204$	0	0
$205$	−5341.15	−1.81972
$206$	224.769	0.0760212
$207$	0	0
$208$	−1454.53	−0.484871
$209$	4480.49	1.48288
$210$	0	0
$211$	2267.20	0.739717	0.369859	−	0.929088i	$-0.379406\pi$
$211$	2267.20	0.739717	0.369859	+	0.929088i	$0.379406\pi$
$212$	12.9758	0.00420370
$213$	0	0
$214$	−2034.28	−0.649815
$215$	4479.87	1.42105
$216$	0	0
$217$	1639.31	0.512828
$218$	1529.05	0.475047
$219$	0	0
$220$	−97.7975	−0.0299705
$221$	1298.40	0.395203
$222$	0	0
$223$	4945.03	1.48495	0.742474	−	0.669875i	$-0.233652\pi$
$223$	4945.03	1.48495	0.742474	+	0.669875i	$0.233652\pi$
$224$	−142.562	−0.0425239
$225$	0	0
$226$	−5546.59	−1.63254
$227$	3559.46	1.04075	0.520374	−	0.853938i	$-0.325792\pi$
$227$	3559.46	1.04075	0.520374	+	0.853938i	$0.325792\pi$
$228$	0	0
$229$	6143.40	1.77278	0.886391	−	0.462937i	$-0.153204\pi$
$229$	6143.40	1.77278	0.886391	+	0.462937i	$0.153204\pi$
$230$	−1816.99	−0.520907
$231$	0	0
$232$	−651.143	−0.184266
$233$	1087.06	0.305648	0.152824	−	0.988253i	$-0.451163\pi$
$233$	1087.06	0.305648	0.152824	+	0.988253i	$0.451163\pi$
$234$	0	0
$235$	7122.03	1.97698
$236$	−16.5519	−0.00456540
$237$	0	0
$238$	4304.88	1.17245
$239$	−1079.00	−0.292027	−0.146013	−	0.989283i	$-0.546644\pi$
$239$	−1079.00	−0.292027	−0.146013	+	0.989283i	$0.546644\pi$
$240$	0	0
$241$	−989.224	−0.264405	−0.132202	−	0.991223i	$-0.542205\pi$
$241$	−989.224	−0.264405	−0.132202	+	0.991223i	$0.542205\pi$
$242$	7455.68	1.98045
$243$	0	0
$244$	−70.0629	−0.0183824
$245$	−4322.81	−1.12724
$246$	0	0
$247$	1596.98	0.411391
$248$	−1412.75	−0.361733
$249$	0	0
$250$	3092.09	0.782243
$251$	−900.246	−0.226386	−0.113193	−	0.993573i	$-0.536108\pi$
$251$	−900.246	−0.226386	−0.113193	+	0.993573i	$0.536108\pi$
$252$	0	0
$253$	3111.88	0.773290
$254$	−5685.61	−1.40451
$255$	0	0
$256$	185.693	0.0453353
$257$	3125.18	0.758534	0.379267	−	0.925287i	$-0.376176\pi$
$257$	3125.18	0.758534	0.379267	+	0.925287i	$0.376176\pi$
$258$	0	0
$259$	3103.62	0.744592
$260$	−34.8580	−0.00831462
$261$	0	0
$262$	−4402.35	−1.03808
$263$	−2814.89	−0.659976	−0.329988	−	0.943985i	$-0.607045\pi$
$263$	−2814.89	−0.659976	−0.329988	+	0.943985i	$0.607045\pi$
$264$	0	0
$265$	−1381.36	−0.320212
$266$	5294.84	1.22048
$267$	0	0
$268$	111.243	0.0253554
$269$	−4409.28	−0.999400	−0.499700	−	0.866199i	$-0.666556\pi$
$269$	−4409.28	−0.999400	−0.499700	+	0.866199i	$0.666556\pi$
$270$	0	0
$271$	−4417.97	−0.990304	−0.495152	−	0.868806i	$-0.664888\pi$
$271$	−4417.97	−0.990304	−0.495152	+	0.868806i	$0.664888\pi$
$272$	−3766.01	−0.839515
$273$	0	0
$274$	3690.19	0.813623
$275$	2557.73	0.560862
$276$	0	0
$277$	−887.577	−0.192525	−0.0962624	−	0.995356i	$-0.530689\pi$
$277$	−887.577	−0.192525	−0.0962624	+	0.995356i	$0.530689\pi$
$278$	5698.49	1.22940
$279$	0	0
$280$	7530.55	1.60727
$281$	4286.58	0.910021	0.455010	−	0.890486i	$-0.349636\pi$
$281$	4286.58	0.910021	0.455010	+	0.890486i	$0.349636\pi$
$282$	0	0
$283$	−1709.59	−0.359097	−0.179549	−	0.983749i	$-0.557464\pi$
$283$	−1709.59	−0.359097	−0.179549	+	0.983749i	$0.557464\pi$
$284$	−94.5367	−0.0197525
$285$	0	0
$286$	4009.36	0.828946
$287$	10810.2	2.22336
$288$	0	0
$289$	−1551.22	−0.315738
$290$	−1063.84	−0.215416
$291$	0	0
$292$	−16.1112	−0.00322890
$293$	−1145.01	−0.228301	−0.114151	−	0.993463i	$-0.536415\pi$
$293$	−1145.01	−0.228301	−0.114151	+	0.993463i	$0.536415\pi$
$294$	0	0
$295$	1762.05	0.347765
$296$	−2674.68	−0.525212
$297$	0	0
$298$	−3384.42	−0.657900
$299$	1109.17	0.214532
$300$	0	0
$301$	−9067.01	−1.73626
$302$	−6433.56	−1.22586
$303$	0	0
$304$	−4632.06	−0.873905
$305$	7458.63	1.40026
$306$	0	0
$307$	1079.39	0.200665	0.100333	−	0.994954i	$-0.468009\pi$
$307$	1079.39	0.200665	0.100333	+	0.994954i	$0.468009\pi$
$308$	197.937	0.0366185
$309$	0	0
$310$	−2308.16	−0.422885
$311$	−2313.90	−0.421895	−0.210947	−	0.977497i	$-0.567655\pi$
$311$	−2313.90	−0.421895	−0.210947	+	0.977497i	$0.567655\pi$
$312$	0	0
$313$	−7653.19	−1.38206	−0.691029	−	0.722827i	$-0.742842\pi$
$313$	−7653.19	−1.38206	−0.691029	+	0.722827i	$0.742842\pi$
$314$	3385.99	0.608542
$315$	0	0
$316$	104.984	0.0186892
$317$	3657.23	0.647982	0.323991	−	0.946060i	$-0.394975\pi$
$317$	3657.23	0.647982	0.323991	+	0.946060i	$0.394975\pi$
$318$	0	0
$319$	1821.99	0.319787
$320$	−6488.29	−1.13346
$321$	0	0
$322$	3677.48	0.636453
$323$	4134.86	0.712291
$324$	0	0
$325$	911.653	0.155598
$326$	−6988.47	−1.18729
$327$	0	0
$328$	−9316.18	−1.56829
$329$	−14414.6	−2.41551
$330$	0	0
$331$	3237.92	0.537681	0.268841	−	0.963185i	$-0.413360\pi$
$331$	3237.92	0.537681	0.268841	+	0.963185i	$0.413360\pi$
$332$	10.0795	0.00166622
$333$	0	0
$334$	5756.84	0.943115
$335$	−11842.5	−1.93142
$336$	0	0
$337$	−7976.89	−1.28940	−0.644702	−	0.764434i	$-0.723018\pi$
$337$	−7976.89	−1.28940	−0.644702	+	0.764434i	$0.723018\pi$
$338$	−4831.78	−0.777557
$339$	0	0
$340$	−90.2533	−0.0143961
$341$	3953.09	0.627777
$342$	0	0
$343$	−187.371	−0.0294959
$344$	7813.92	1.22470
$345$	0	0
$346$	−8156.25	−1.26729
$347$	8355.93	1.29271	0.646354	−	0.763037i	$-0.276293\pi$
$347$	8355.93	1.29271	0.646354	+	0.763037i	$0.276293\pi$
$348$	0	0
$349$	5544.56	0.850411	0.425205	−	0.905097i	$-0.360202\pi$
$349$	5544.56	0.850411	0.425205	+	0.905097i	$0.360202\pi$
$350$	3022.61	0.461615
$351$	0	0
$352$	−343.780	−0.0520555
$353$	1682.79	0.253727	0.126864	−	0.991920i	$-0.459509\pi$
$353$	1682.79	0.253727	0.126864	+	0.991920i	$0.459509\pi$
$354$	0	0
$355$	10064.0	1.50463
$356$	43.2801	0.00644337
$357$	0	0
$358$	662.990	0.0978774
$359$	−7143.13	−1.05014	−0.525069	−	0.851059i	$-0.675961\pi$
$359$	−7143.13	−1.05014	−0.525069	+	0.851059i	$0.675961\pi$
$360$	0	0
$361$	−1773.27	−0.258531
$362$	−7431.13	−1.07893
$363$	0	0
$364$	70.5507	0.0101590
$365$	1715.14	0.245958
$366$	0	0
$367$	−4456.16	−0.633814	−0.316907	−	0.948457i	$-0.602644\pi$
$367$	−4456.16	−0.633814	−0.316907	+	0.948457i	$0.602644\pi$
$368$	−3217.15	−0.455722
$369$	0	0
$370$	−4369.90	−0.614001
$371$	2795.79	0.391241
$372$	0	0
$373$	2508.28	0.348187	0.174094	−	0.984729i	$-0.444300\pi$
$373$	2508.28	0.348187	0.174094	+	0.984729i	$0.444300\pi$
$374$	10380.9	1.43525
$375$	0	0
$376$	12422.4	1.70383
$377$	649.414	0.0887176
$378$	0	0
$379$	−12733.9	−1.72585	−0.862926	−	0.505331i	$-0.831371\pi$
$379$	−12733.9	−1.72585	−0.862926	+	0.505331i	$0.831371\pi$
$380$	−111.008	−0.0149858
$381$	0	0
$382$	−4356.95	−0.583563
$383$	1027.19	0.137042	0.0685209	−	0.997650i	$-0.478172\pi$
$383$	1027.19	0.137042	0.0685209	+	0.997650i	$0.478172\pi$
$384$	0	0
$385$	−21071.6	−2.78937
$386$	2899.82	0.382376
$387$	0	0
$388$	−22.6250	−0.00296034
$389$	−5153.35	−0.671684	−0.335842	−	0.941918i	$-0.609021\pi$
$389$	−5153.35	−0.671684	−0.335842	+	0.941918i	$0.609021\pi$
$390$	0	0
$391$	2871.83	0.371444
$392$	−7539.97	−0.971495
$393$	0	0
$394$	−9376.17	−1.19890
$395$	−11176.2	−1.42363
$396$	0	0
$397$	6250.95	0.790242	0.395121	−	0.918629i	$-0.370703\pi$
$397$	6250.95	0.790242	0.395121	+	0.918629i	$0.370703\pi$
$398$	−85.2610	−0.0107381
$399$	0	0
$400$	−2644.25	−0.330532
$401$	−11083.8	−1.38029	−0.690145	−	0.723671i	$-0.742453\pi$
$401$	−11083.8	−1.38029	−0.690145	+	0.723671i	$0.742453\pi$
$402$	0	0
$403$	1409.00	0.174162
$404$	116.015	0.0142870
$405$	0	0
$406$	2153.15	0.263200
$407$	7484.17	0.911490
$408$	0	0
$409$	−3375.43	−0.408079	−0.204039	−	0.978963i	$-0.565407\pi$
$409$	−3375.43	−0.408079	−0.204039	+	0.978963i	$0.565407\pi$
$410$	−15220.8	−1.83342
$411$	0	0
$412$	9.53754	0.00114049
$413$	−3566.29	−0.424905
$414$	0	0
$415$	−1073.03	−0.126923
$416$	−122.534	−0.0144416
$417$	0	0
$418$	12768.2	1.49405
$419$	−6486.50	−0.756292	−0.378146	−	0.925746i	$-0.623438\pi$
$419$	−6486.50	−0.756292	−0.378146	+	0.925746i	$0.623438\pi$
$420$	0	0
$421$	−10938.6	−1.26631	−0.633153	−	0.774026i	$-0.718240\pi$
$421$	−10938.6	−1.26631	−0.633153	+	0.774026i	$0.718240\pi$
$422$	6460.89	0.745287
$423$	0	0
$424$	−2409.40	−0.275969
$425$	2360.42	0.269406
$426$	0	0
$427$	−15095.9	−1.71087
$428$	−86.3200	−0.00974867
$429$	0	0
$430$	12766.4	1.43174
$431$	−4124.34	−0.460934	−0.230467	−	0.973080i	$-0.574025\pi$
$431$	−4124.34	−0.460934	−0.230467	+	0.973080i	$0.574025\pi$
$432$	0	0
$433$	−1561.41	−0.173295	−0.0866473	−	0.996239i	$-0.527615\pi$
$433$	−1561.41	−0.173295	−0.0866473	+	0.996239i	$0.527615\pi$
$434$	4671.58	0.516689
$435$	0	0
$436$	64.8818	0.00712677
$437$	3532.25	0.386660
$438$	0	0
$439$	−15712.7	−1.70826	−0.854130	−	0.520060i	$-0.825909\pi$
$439$	−15712.7	−1.70826	−0.854130	+	0.520060i	$0.825909\pi$
$440$	18159.4	1.96754
$441$	0	0
$442$	3700.08	0.398178
$443$	−12763.5	−1.36888	−0.684439	−	0.729070i	$-0.739953\pi$
$443$	−12763.5	−1.36888	−0.684439	+	0.729070i	$0.739953\pi$
$444$	0	0
$445$	−4607.44	−0.490817
$446$	14092.0	1.49613
$447$	0	0
$448$	13131.9	1.38488
$449$	−3117.88	−0.327710	−0.163855	−	0.986484i	$-0.552393\pi$
$449$	−3117.88	−0.327710	−0.163855	+	0.986484i	$0.552393\pi$
$450$	0	0
$451$	26068.1	2.72172
$452$	−235.357	−0.0244918
$453$	0	0
$454$	10143.5	1.04858
$455$	−7510.56	−0.773847
$456$	0	0
$457$	−8479.41	−0.867943	−0.433972	−	0.900927i	$-0.642888\pi$
$457$	−8479.41	−0.867943	−0.433972	+	0.900927i	$0.642888\pi$
$458$	17507.0	1.78613
$459$	0	0
$460$	−77.0997	−0.00781477
$461$	−9253.32	−0.934859	−0.467430	−	0.884030i	$-0.654820\pi$
$461$	−9253.32	−0.934859	−0.467430	+	0.884030i	$0.654820\pi$
$462$	0	0
$463$	521.395	0.0523354	0.0261677	−	0.999658i	$-0.491670\pi$
$463$	521.395	0.0523354	0.0261677	+	0.999658i	$0.491670\pi$
$464$	−1883.63	−0.188460
$465$	0	0
$466$	3097.83	0.307949
$467$	−4337.35	−0.429783	−0.214891	−	0.976638i	$-0.568940\pi$
$467$	−4337.35	−0.429783	−0.214891	+	0.976638i	$0.568940\pi$
$468$	0	0
$469$	23968.6	2.35984
$470$	20295.8	1.99186
$471$	0	0
$472$	3073.42	0.299715
$473$	−21864.5	−2.12544
$474$	0	0
$475$	2903.24	0.280441
$476$	182.668	0.0175894
$477$	0	0
$478$	−3074.84	−0.294226
$479$	11258.1	1.07390	0.536949	−	0.843615i	$-0.319577\pi$
$479$	11258.1	1.07390	0.536949	+	0.843615i	$0.319577\pi$
$480$	0	0
$481$	2667.58	0.252872
$482$	−2819.02	−0.266395
$483$	0	0
$484$	316.365	0.0297112
$485$	2408.58	0.225501
$486$	0	0
$487$	4353.54	0.405088	0.202544	−	0.979273i	$-0.435079\pi$
$487$	4353.54	0.405088	0.202544	+	0.979273i	$0.435079\pi$
$488$	13009.6	1.20679
$489$	0	0
$490$	−12318.8	−1.13573
$491$	8458.77	0.777472	0.388736	−	0.921349i	$-0.372912\pi$
$491$	8458.77	0.777472	0.388736	+	0.921349i	$0.372912\pi$
$492$	0	0
$493$	1681.44	0.153607
$494$	4550.96	0.414489
$495$	0	0
$496$	−4086.82	−0.369967
$497$	−20369.0	−1.83838
$498$	0	0
$499$	14850.9	1.33230	0.666149	−	0.745819i	$-0.267942\pi$
$499$	14850.9	1.33230	0.666149	+	0.745819i	$0.267942\pi$
$500$	131.206	0.0117354
$501$	0	0
$502$	−2565.45	−0.228091
$503$	1686.45	0.149493	0.0747464	−	0.997203i	$-0.476185\pi$
$503$	1686.45	0.149493	0.0747464	+	0.997203i	$0.476185\pi$
$504$	0	0
$505$	−12350.5	−1.08830
$506$	8868.00	0.779112
$507$	0	0
$508$	−241.256	−0.0210709
$509$	11113.5	0.967773	0.483887	−	0.875131i	$-0.339225\pi$
$509$	11113.5	0.967773	0.483887	+	0.875131i	$0.339225\pi$
$510$	0	0
$511$	−3471.35	−0.300516
$512$	−11311.8	−0.976395
$513$	0	0
$514$	8905.90	0.764246
$515$	−1015.33	−0.0868754
$516$	0	0
$517$	−34759.8	−2.95694
$518$	8844.45	0.750198
$519$	0	0
$520$	6472.57	0.545848
$521$	15931.1	1.33964	0.669820	−	0.742523i	$-0.266371\pi$
$521$	15931.1	1.33964	0.669820	+	0.742523i	$0.266371\pi$
$522$	0	0
$523$	−7960.43	−0.665555	−0.332778	−	0.943005i	$-0.607986\pi$
$523$	−7960.43	−0.665555	−0.332778	+	0.943005i	$0.607986\pi$
$524$	−186.804	−0.0155736
$525$	0	0
$526$	−8021.67	−0.664945
$527$	3648.14	0.301548
$528$	0	0
$529$	−9713.71	−0.798366
$530$	−3936.49	−0.322623
$531$	0	0
$532$	224.675	0.0183099
$533$	9291.45	0.755079
$534$	0	0
$535$	9189.30	0.742594
$536$	−20656.0	−1.66456
$537$	0	0
$538$	−12565.2	−1.00692
$539$	21098.0	1.68600
$540$	0	0
$541$	13818.9	1.09819	0.549096	−	0.835760i	$-0.314972\pi$
$541$	13818.9	1.09819	0.549096	+	0.835760i	$0.314972\pi$
$542$	−12590.0	−0.997760
$543$	0	0
$544$	−317.261	−0.0250045
$545$	−6907.07	−0.542874
$546$	0	0
$547$	22093.3	1.72695	0.863474	−	0.504393i	$-0.168284\pi$
$547$	22093.3	1.72695	0.863474	+	0.504393i	$0.168284\pi$
$548$	156.585	0.0122062
$549$	0	0
$550$	7288.82	0.565084
$551$	2068.12	0.159900
$552$	0	0
$553$	22619.9	1.73942
$554$	−2529.35	−0.193974
$555$	0	0
$556$	241.802	0.0184437
$557$	3110.34	0.236606	0.118303	−	0.992978i	$-0.462255\pi$
$557$	3110.34	0.236606	0.118303	+	0.992978i	$0.462255\pi$
$558$	0	0
$559$	−7793.17	−0.589653
$560$	21784.4	1.64386
$561$	0	0
$562$	12215.6	0.916873
$563$	13284.6	0.994455	0.497227	−	0.867620i	$-0.334351\pi$
$563$	13284.6	0.994455	0.497227	+	0.867620i	$0.334351\pi$
$564$	0	0
$565$	25055.2	1.86563
$566$	−4871.86	−0.361801
$567$	0	0
$568$	17553.9	1.29674
$569$	6809.18	0.501680	0.250840	−	0.968029i	$-0.419293\pi$
$569$	6809.18	0.501680	0.250840	+	0.968029i	$0.419293\pi$
$570$	0	0
$571$	−13471.8	−0.987352	−0.493676	−	0.869646i	$-0.664347\pi$
$571$	−13471.8	−0.987352	−0.493676	+	0.869646i	$0.664347\pi$
$572$	170.128	0.0124361
$573$	0	0
$574$	30806.0	2.24010
$575$	2016.41	0.146244
$576$	0	0
$577$	5331.06	0.384636	0.192318	−	0.981333i	$-0.438399\pi$
$577$	5331.06	0.384636	0.192318	+	0.981333i	$0.438399\pi$
$578$	−4420.56	−0.318116
$579$	0	0
$580$	−45.1416	−0.00323173
$581$	2171.75	0.155077
$582$	0	0
$583$	6741.87	0.478936
$584$	2991.59	0.211974
$585$	0	0
$586$	−3262.96	−0.230020
$587$	−3333.96	−0.234425	−0.117212	−	0.993107i	$-0.537396\pi$
$587$	−3333.96	−0.234425	−0.117212	+	0.993107i	$0.537396\pi$
$588$	0	0
$589$	4487.09	0.313900
$590$	5021.36	0.350383
$591$	0	0
$592$	−7737.34	−0.537167
$593$	23405.3	1.62081	0.810404	−	0.585872i	$-0.199248\pi$
$593$	23405.3	1.62081	0.810404	+	0.585872i	$0.199248\pi$
$594$	0	0
$595$	−19446.1	−1.33985
$596$	−143.610	−0.00986998
$597$	0	0
$598$	3160.83	0.216147
$599$	10352.1	0.706133	0.353067	−	0.935598i	$-0.385139\pi$
$599$	10352.1	0.706133	0.353067	+	0.935598i	$0.385139\pi$
$600$	0	0
$601$	15171.1	1.02969	0.514843	−	0.857285i	$-0.327850\pi$
$601$	15171.1	1.02969	0.514843	+	0.857285i	$0.327850\pi$
$602$	−25838.5	−1.74933
$603$	0	0
$604$	−272.994	−0.0183907
$605$	−33679.0	−2.26322
$606$	0	0
$607$	−20823.0	−1.39239	−0.696193	−	0.717854i	$-0.745124\pi$
$607$	−20823.0	−1.39239	−0.696193	+	0.717854i	$0.745124\pi$
$608$	−390.219	−0.0260287
$609$	0	0
$610$	21255.0	1.41081
$611$	−12389.5	−0.820334
$612$	0	0
$613$	−19071.6	−1.25660	−0.628299	−	0.777972i	$-0.716249\pi$
$613$	−19071.6	−1.25660	−0.628299	+	0.777972i	$0.716249\pi$
$614$	3075.97	0.202176
$615$	0	0
$616$	−36753.7	−2.40397
$617$	−15200.4	−0.991806	−0.495903	−	0.868378i	$-0.665163\pi$
$617$	−15200.4	−0.991806	−0.495903	+	0.868378i	$0.665163\pi$
$618$	0	0
$619$	4358.76	0.283026	0.141513	−	0.989936i	$-0.454803\pi$
$619$	4358.76	0.283026	0.141513	+	0.989936i	$0.454803\pi$
$620$	−97.9414	−0.00634423
$621$	0	0
$622$	−6593.98	−0.425071
$623$	9325.20	0.599689
$624$	0	0
$625$	−19056.5	−1.21961
$626$	−21809.5	−1.39246
$627$	0	0
$628$	143.677	0.00912950
$629$	6906.83	0.437827
$630$	0	0
$631$	−2580.66	−0.162812	−0.0814062	−	0.996681i	$-0.525941\pi$
$631$	−2580.66	−0.162812	−0.0814062	+	0.996681i	$0.525941\pi$
$632$	−19493.8	−1.22693
$633$	0	0
$634$	10422.1	0.652861
$635$	25683.2	1.60505
$636$	0	0
$637$	7519.96	0.467742
$638$	5192.18	0.322195
$639$	0	0
$640$	−19053.3	−1.17680
$641$	−19858.9	−1.22368	−0.611840	−	0.790982i	$-0.709570\pi$
$641$	−19858.9	−1.22368	−0.611840	+	0.790982i	$0.709570\pi$
$642$	0	0
$643$	17371.9	1.06545	0.532723	−	0.846290i	$-0.321169\pi$
$643$	17371.9	1.06545	0.532723	+	0.846290i	$0.321169\pi$
$644$	156.046	0.00954823
$645$	0	0
$646$	11783.2	0.717654
$647$	−3275.07	−0.199005	−0.0995024	−	0.995037i	$-0.531725\pi$
$647$	−3275.07	−0.199005	−0.0995024	+	0.995037i	$0.531725\pi$
$648$	0	0
$649$	−8599.88	−0.520146
$650$	2597.96	0.156770
$651$	0	0
$652$	−296.540	−0.0178120
$653$	−20726.5	−1.24210	−0.621049	−	0.783772i	$-0.713293\pi$
$653$	−20726.5	−1.24210	−0.621049	+	0.783772i	$0.713293\pi$
$654$	0	0
$655$	19886.4	1.18630
$656$	−26949.9	−1.60399
$657$	0	0
$658$	−41077.6	−2.43370
$659$	18404.6	1.08792	0.543960	−	0.839111i	$-0.316924\pi$
$659$	18404.6	1.08792	0.543960	+	0.839111i	$0.316924\pi$
$660$	0	0
$661$	−7146.10	−0.420501	−0.210250	−	0.977648i	$-0.567428\pi$
$661$	−7146.10	−0.420501	−0.210250	+	0.977648i	$0.567428\pi$
$662$	9227.19	0.541729
$663$	0	0
$664$	−1871.61	−0.109386
$665$	−23918.0	−1.39474
$666$	0	0
$667$	1436.39	0.0833841
$668$	244.279	0.0141488
$669$	0	0
$670$	−33747.9	−1.94596
$671$	−36402.7	−2.09435
$672$	0	0
$673$	−26819.0	−1.53610	−0.768051	−	0.640389i	$-0.778773\pi$
$673$	−26819.0	−1.53610	−0.768051	+	0.640389i	$0.778773\pi$
$674$	−22731.9	−1.29911
$675$	0	0
$676$	−205.026	−0.0116651
$677$	20093.1	1.14068	0.570340	−	0.821408i	$-0.306811\pi$
$677$	20093.1	1.14068	0.570340	+	0.821408i	$0.306811\pi$
$678$	0	0
$679$	−4874.82	−0.275521
$680$	16758.6	0.945092
$681$	0	0
$682$	11265.2	0.632503
$683$	6876.22	0.385229	0.192614	−	0.981275i	$-0.438303\pi$
$683$	6876.22	0.385229	0.192614	+	0.981275i	$0.438303\pi$
$684$	0	0
$685$	−16669.5	−0.929791
$686$	−533.956	−0.0297180
$687$	0	0
$688$	22604.1	1.25258
$689$	2403.01	0.132870
$690$	0	0
$691$	17332.7	0.954224	0.477112	−	0.878842i	$-0.341684\pi$
$691$	17332.7	0.954224	0.477112	+	0.878842i	$0.341684\pi$
$692$	−346.092	−0.0190122
$693$	0	0
$694$	23812.1	1.30244
$695$	−25741.4	−1.40493
$696$	0	0
$697$	24057.1	1.30736
$698$	15800.5	0.856814
$699$	0	0
$700$	128.258	0.00692526
$701$	−11127.5	−0.599545	−0.299772	−	0.954011i	$-0.596911\pi$
$701$	−11127.5	−0.599545	−0.299772	+	0.954011i	$0.596911\pi$
$702$	0	0
$703$	8495.15	0.455762
$704$	31666.8	1.69530
$705$	0	0
$706$	4795.48	0.255638
$707$	24996.8	1.32970
$708$	0	0
$709$	17432.0	0.923374	0.461687	−	0.887043i	$-0.347244\pi$
$709$	17432.0	0.923374	0.461687	+	0.887043i	$0.347244\pi$
$710$	28679.7	1.51596
$711$	0	0
$712$	−8036.42	−0.423002
$713$	3116.46	0.163692
$714$	0	0
$715$	−18111.2	−0.947302
$716$	28.1325	0.00146838
$717$	0	0
$718$	−20355.9	−1.05805
$719$	−21082.7	−1.09353	−0.546767	−	0.837285i	$-0.684142\pi$
$719$	−21082.7	−1.09353	−0.546767	+	0.837285i	$0.684142\pi$
$720$	0	0
$721$	2054.97	0.106146
$722$	−5053.32	−0.260478
$723$	0	0
$724$	−315.323	−0.0161863
$725$	1180.60	0.0604779
$726$	0	0
$727$	−25839.1	−1.31818	−0.659091	−	0.752063i	$-0.729059\pi$
$727$	−25839.1	−1.31818	−0.659091	+	0.752063i	$0.729059\pi$
$728$	−13100.1	−0.666927
$729$	0	0
$730$	4887.67	0.247810
$731$	−20177.9	−1.02094
$732$	0	0
$733$	1278.54	0.0644256	0.0322128	−	0.999481i	$-0.489745\pi$
$733$	1278.54	0.0644256	0.0322128	+	0.999481i	$0.489745\pi$
$734$	−12698.8	−0.638586
$735$	0	0
$736$	−271.023	−0.0135734
$737$	57798.6	2.88879
$738$	0	0
$739$	4224.54	0.210287	0.105144	−	0.994457i	$-0.466470\pi$
$739$	4224.54	0.210287	0.105144	+	0.994457i	$0.466470\pi$
$740$	−185.427	−0.00921140
$741$	0	0
$742$	7967.23	0.394186
$743$	17992.3	0.888390	0.444195	−	0.895930i	$-0.353490\pi$
$743$	17992.3	0.888390	0.444195	+	0.895930i	$0.353490\pi$
$744$	0	0
$745$	15288.2	0.751835
$746$	7147.91	0.350809
$747$	0	0
$748$	440.491	0.0215320
$749$	−18598.6	−0.907316
$750$	0	0
$751$	−10082.4	−0.489895	−0.244948	−	0.969536i	$-0.578771\pi$
$751$	−10082.4	−0.489895	−0.244948	+	0.969536i	$0.578771\pi$
$752$	35935.7	1.74261
$753$	0	0
$754$	1850.65	0.0893856
$755$	29061.9	1.40089
$756$	0	0
$757$	10806.5	0.518851	0.259425	−	0.965763i	$-0.416467\pi$
$757$	10806.5	0.518851	0.259425	+	0.965763i	$0.416467\pi$
$758$	−36288.2	−1.73885
$759$	0	0
$760$	20612.5	0.983806
$761$	30710.4	1.46288	0.731439	−	0.681907i	$-0.238849\pi$
$761$	30710.4	1.46288	0.731439	+	0.681907i	$0.238849\pi$
$762$	0	0
$763$	13979.5	0.663293
$764$	−184.877	−0.00875475
$765$	0	0
$766$	2927.21	0.138074
$767$	−3065.26	−0.144303
$768$	0	0
$769$	10757.1	0.504436	0.252218	−	0.967670i	$-0.418840\pi$
$769$	10757.1	0.504436	0.252218	+	0.967670i	$0.418840\pi$
$770$	−60048.2	−2.81037
$771$	0	0
$772$	123.047	0.00573649
$773$	18077.2	0.841125	0.420563	−	0.907263i	$-0.361833\pi$
$773$	18077.2	0.841125	0.420563	+	0.907263i	$0.361833\pi$
$774$	0	0
$775$	2561.49	0.118725
$776$	4201.10	0.194344
$777$	0	0
$778$	−14685.6	−0.676742
$779$	29589.4	1.36091
$780$	0	0
$781$	−49118.6	−2.25045
$782$	8183.92	0.374241
$783$	0	0
$784$	−21811.7	−0.993608
$785$	−15295.3	−0.695430
$786$	0	0
$787$	−31543.7	−1.42873	−0.714366	−	0.699773i	$-0.753285\pi$
$787$	−31543.7	−1.42873	−0.714366	+	0.699773i	$0.753285\pi$
$788$	−397.857	−0.0179861
$789$	0	0
$790$	−31848.9	−1.43435
$791$	−50710.4	−2.27946
$792$	0	0
$793$	−12975.0	−0.581030
$794$	17813.5	0.796192
$795$	0	0
$796$	−3.61786	−0.000161095 0
$797$	−18280.9	−0.812477	−0.406239	−	0.913767i	$-0.633160\pi$
$797$	−18280.9	−0.812477	−0.406239	+	0.913767i	$0.633160\pi$
$798$	0	0
$799$	−32078.4	−1.42034
$800$	−222.760	−0.00984470
$801$	0	0
$802$	−31585.6	−1.39068
$803$	−8370.93	−0.367875
$804$	0	0
$805$	−16612.0	−0.727326
$806$	4015.26	0.175473
$807$	0	0
$808$	−21542.1	−0.937932
$809$	−21776.3	−0.946372	−0.473186	−	0.880963i	$-0.656896\pi$
$809$	−21776.3	−0.946372	−0.473186	+	0.880963i	$0.656896\pi$
$810$	0	0
$811$	17035.7	0.737612	0.368806	−	0.929506i	$-0.379767\pi$
$811$	17035.7	0.737612	0.368806	+	0.929506i	$0.379767\pi$
$812$	91.3640	0.00394858
$813$	0	0
$814$	21327.8	0.918353
$815$	31568.5	1.35681
$816$	0	0
$817$	−24818.0	−1.06276
$818$	−9619.03	−0.411151
$819$	0	0
$820$	−645.860	−0.0275054
$821$	45120.7	1.91805	0.959027	−	0.283314i	$-0.0914339\pi$
$821$	45120.7	1.91805	0.959027	+	0.283314i	$0.0914339\pi$
$822$	0	0
$823$	22261.7	0.942883	0.471442	−	0.881897i	$-0.343734\pi$
$823$	22261.7	0.942883	0.471442	+	0.881897i	$0.343734\pi$
$824$	−1770.97	−0.0748721
$825$	0	0
$826$	−10162.9	−0.428104
$827$	−10280.4	−0.432266	−0.216133	−	0.976364i	$-0.569345\pi$
$827$	−10280.4	−0.432266	−0.216133	+	0.976364i	$0.569345\pi$
$828$	0	0
$829$	28509.2	1.19441	0.597206	−	0.802088i	$-0.296278\pi$
$829$	28509.2	1.19441	0.597206	+	0.802088i	$0.296278\pi$
$830$	−3057.84	−0.127878
$831$	0	0
$832$	11287.0	0.470321
$833$	19470.4	0.809857
$834$	0	0
$835$	−26005.0	−1.07777
$836$	541.788	0.0224140
$837$	0	0
$838$	−18484.7	−0.761986
$839$	−4746.97	−0.195332	−0.0976661	−	0.995219i	$-0.531138\pi$
$839$	−4746.97	−0.195332	−0.0976661	+	0.995219i	$0.531138\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	−31172.0	−1.27584
$843$	0	0
$844$	274.153	0.0111810
$845$	21826.3	0.888576
$846$	0	0
$847$	68164.5	2.76524
$848$	−6969.94	−0.282251
$849$	0	0
$850$	6726.56	0.271434
$851$	5900.22	0.237670
$852$	0	0
$853$	35313.3	1.41747	0.708736	−	0.705473i	$-0.249266\pi$
$853$	35313.3	1.41747	0.708736	+	0.705473i	$0.249266\pi$
$854$	−43019.0	−1.72375
$855$	0	0
$856$	16028.2	0.639993
$857$	−32142.8	−1.28119	−0.640594	−	0.767880i	$-0.721312\pi$
$857$	−32142.8	−1.28119	−0.640594	+	0.767880i	$0.721312\pi$
$858$	0	0
$859$	37568.2	1.49221	0.746107	−	0.665826i	$-0.231921\pi$
$859$	37568.2	1.49221	0.746107	+	0.665826i	$0.231921\pi$
$860$	541.713	0.0214794
$861$	0	0
$862$	−11753.2	−0.464404
$863$	−3416.95	−0.134779	−0.0673896	−	0.997727i	$-0.521467\pi$
$863$	−3416.95	−0.134779	−0.0673896	+	0.997727i	$0.521467\pi$
$864$	0	0
$865$	36843.6	1.44823
$866$	−4449.59	−0.174599
$867$	0	0
$868$	198.228	0.00775149
$869$	54546.5	2.12930
$870$	0	0
$871$	20601.2	0.801429
$872$	−12047.5	−0.467867
$873$	0	0
$874$	10065.9	0.389571
$875$	28269.8	1.09222
$876$	0	0
$877$	−11891.0	−0.457847	−0.228923	−	0.973444i	$-0.573521\pi$
$877$	−11891.0	−0.457847	−0.228923	+	0.973444i	$0.573521\pi$
$878$	−44776.8	−1.72112
$879$	0	0
$880$	52531.7	2.01232
$881$	−20042.7	−0.766464	−0.383232	−	0.923652i	$-0.625189\pi$
$881$	−20042.7	−0.766464	−0.383232	+	0.923652i	$0.625189\pi$
$882$	0	0
$883$	−18042.8	−0.687641	−0.343821	−	0.939035i	$-0.611721\pi$
$883$	−18042.8	−0.687641	−0.343821	+	0.939035i	$0.611721\pi$
$884$	157.004	0.00597356
$885$	0	0
$886$	−36372.5	−1.37919
$887$	−2247.91	−0.0850929	−0.0425464	−	0.999094i	$-0.513547\pi$
$887$	−2247.91	−0.0850929	−0.0425464	+	0.999094i	$0.513547\pi$
$888$	0	0
$889$	−51981.4	−1.96108
$890$	−13129.9	−0.494512
$891$	0	0
$892$	597.960	0.0224453
$893$	−39455.3	−1.47852
$894$	0	0
$895$	−2994.88	−0.111852
$896$	38562.9	1.43783
$897$	0	0
$898$	−8885.09	−0.330177
$899$	1824.68	0.0676934
$900$	0	0
$901$	6221.80	0.230053
$902$	74286.7	2.74222
$903$	0	0
$904$	43702.1	1.60786
$905$	33568.1	1.23298
$906$	0	0
$907$	−1798.34	−0.0658354	−0.0329177	−	0.999458i	$-0.510480\pi$
$907$	−1798.34	−0.0658354	−0.0329177	+	0.999458i	$0.510480\pi$
$908$	430.416	0.0157311
$909$	0	0
$910$	−21403.0	−0.779674
$911$	−4602.02	−0.167367	−0.0836837	−	0.996492i	$-0.526669\pi$
$911$	−4602.02	−0.167367	−0.0836837	+	0.996492i	$0.526669\pi$
$912$	0	0
$913$	5237.04	0.189836
$914$	−24164.0	−0.874478
$915$	0	0
$916$	742.869	0.0267960
$917$	−40249.1	−1.44944
$918$	0	0
$919$	20622.0	0.740215	0.370107	−	0.928989i	$-0.379321\pi$
$919$	20622.0	0.740215	0.370107	+	0.928989i	$0.379321\pi$
$920$	14316.2	0.513033
$921$	0	0
$922$	−26369.4	−0.941898
$923$	−17507.4	−0.624335
$924$	0	0
$925$	4849.53	0.172380
$926$	1485.83	0.0527294
$927$	0	0
$928$	−158.683	−0.00561316
$929$	24061.8	0.849777	0.424889	−	0.905246i	$-0.360313\pi$
$929$	24061.8	0.849777	0.424889	+	0.905246i	$0.360313\pi$
$930$	0	0
$931$	23947.9	0.843032
$932$	131.449	0.00461992
$933$	0	0
$934$	−12360.2	−0.433019
$935$	−46893.0	−1.64018
$936$	0	0
$937$	−6581.96	−0.229481	−0.114740	−	0.993396i	$-0.536604\pi$
$937$	−6581.96	−0.229481	−0.114740	+	0.993396i	$0.536604\pi$
$938$	68303.8	2.37761
$939$	0	0
$940$	861.207	0.0298824
$941$	−23579.7	−0.816873	−0.408436	−	0.912787i	$-0.633926\pi$
$941$	−23579.7	−0.816873	−0.408436	+	0.912787i	$0.633926\pi$
$942$	0	0
$943$	20551.0	0.709686
$944$	8890.80	0.306537
$945$	0	0
$946$	−62307.8	−2.14144
$947$	36368.6	1.24796	0.623982	−	0.781439i	$-0.285514\pi$
$947$	36368.6	1.24796	0.623982	+	0.781439i	$0.285514\pi$
$948$	0	0
$949$	−2983.65	−0.102058
$950$	8273.42	0.282553
$951$	0	0
$952$	−33918.5	−1.15473
$953$	−44404.3	−1.50933	−0.754667	−	0.656108i	$-0.772202\pi$
$953$	−44404.3	−1.50933	−0.754667	+	0.656108i	$0.772202\pi$
$954$	0	0
$955$	19681.4	0.666884
$956$	−130.474	−0.00441404
$957$	0	0
$958$	32082.5	1.08198
$959$	33738.1	1.13604
$960$	0	0
$961$	−25832.1	−0.867111
$962$	7601.88	0.254776
$963$	0	0
$964$	−119.619	−0.00399653
$965$	−13099.2	−0.436971
$966$	0	0
$967$	21928.3	0.729230	0.364615	−	0.931158i	$-0.381201\pi$
$967$	21928.3	0.729230	0.364615	+	0.931158i	$0.381201\pi$
$968$	−58743.9	−1.95052
$969$	0	0
$970$	6863.77	0.227198
$971$	6352.95	0.209965	0.104982	−	0.994474i	$-0.466521\pi$
$971$	6352.95	0.209965	0.104982	+	0.994474i	$0.466521\pi$
$972$	0	0
$973$	52099.1	1.71657
$974$	12406.4	0.408138
$975$	0	0
$976$	37634.1	1.23426
$977$	−12792.8	−0.418913	−0.209456	−	0.977818i	$-0.567169\pi$
$977$	−12792.8	−0.418913	−0.209456	+	0.977818i	$0.567169\pi$
$978$	0	0
$979$	22487.1	0.734107
$980$	−522.721	−0.0170385
$981$	0	0
$982$	24105.1	0.783326
$983$	−7536.73	−0.244542	−0.122271	−	0.992497i	$-0.539018\pi$
$983$	−7536.73	−0.244542	−0.122271	+	0.992497i	$0.539018\pi$
$984$	0	0
$985$	42354.3	1.37007
$986$	4791.65	0.154764
$987$	0	0
$988$	193.110	0.00621826
$989$	−17237.1	−0.554205
$990$	0	0
$991$	−28088.3	−0.900357	−0.450179	−	0.892938i	$-0.648640\pi$
$991$	−28088.3	−0.900357	−0.450179	+	0.892938i	$0.648640\pi$
$992$	−344.286	−0.0110192
$993$	0	0
$994$	−58046.1	−1.85222
$995$	385.144	0.0122712
$996$	0	0
$997$	−44951.9	−1.42793	−0.713963	−	0.700184i	$-0.753101\pi$
$997$	−44951.9	−1.42793	−0.713963	+	0.700184i	$0.753101\pi$
$998$	42320.9	1.34233
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.4.a.f.1.4		5
3.2	odd	2		29.4.a.b.1.2	✓	5
12.11	even	2		464.4.a.l.1.2		5
15.14	odd	2		725.4.a.c.1.4		5
21.20	even	2		1421.4.a.e.1.2		5
24.5	odd	2		1856.4.a.y.1.2		5
24.11	even	2		1856.4.a.bb.1.4		5
87.86	odd	2		841.4.a.b.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.b.1.2	✓	5	3.2	odd	2
261.4.a.f.1.4		5	1.1	even	1	trivial
464.4.a.l.1.2		5	12.11	even	2
725.4.a.c.1.4		5	15.14	odd	2
841.4.a.b.1.4		5	87.86	odd	2
1421.4.a.e.1.2		5	21.20	even	2
1856.4.a.y.1.2		5	24.5	odd	2
1856.4.a.bb.1.4		5	24.11	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:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)

\(f(q)\)	\(=\)	\(q+2.84972 q^{2} +0.120922 q^{4} -12.8729 q^{5} +26.0540 q^{7} -22.4532 q^{8} +O(q^{10})\)	\(q+2.84972 q^{2} +0.120922 q^{4} -12.8729 q^{5} +26.0540 q^{7} -22.4532 q^{8} -36.6841 q^{10} +62.8274 q^{11} +22.3936 q^{13} +74.2465 q^{14} -64.9528 q^{16} +57.9808 q^{17} +71.3143 q^{19} -1.55661 q^{20} +179.041 q^{22} +49.5307 q^{23} +40.7104 q^{25} +63.8155 q^{26} +3.15048 q^{28} +29.0000 q^{29} +62.9198 q^{31} -5.47182 q^{32} +165.229 q^{34} -335.389 q^{35} +119.123 q^{37} +203.226 q^{38} +289.037 q^{40} +414.916 q^{41} -348.009 q^{43} +7.59719 q^{44} +141.149 q^{46} -553.259 q^{47} +335.808 q^{49} +116.013 q^{50} +2.70787 q^{52} +107.308 q^{53} -808.768 q^{55} -584.994 q^{56} +82.6420 q^{58} -136.881 q^{59} -579.408 q^{61} +179.304 q^{62} +504.029 q^{64} -288.269 q^{65} +919.959 q^{67} +7.01113 q^{68} -955.765 q^{70} -781.802 q^{71} -133.237 q^{73} +339.467 q^{74} +8.62344 q^{76} +1636.90 q^{77} +868.196 q^{79} +836.127 q^{80} +1182.39 q^{82} +83.3560 q^{83} -746.379 q^{85} -991.730 q^{86} -1410.68 q^{88} +357.919 q^{89} +583.442 q^{91} +5.98933 q^{92} -1576.64 q^{94} -918.019 q^{95} -187.105 q^{97} +956.961 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8}+O(q^{10}) \) 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8	\( 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8} - 64 q^{10} - 12 q^{11} + 14 q^{13} + 192 q^{14} + 146 q^{16} - 66 q^{17} + 214 q^{19} - 6 q^{20} - 98 q^{22} - 164 q^{23} + 207 q^{25} - 56 q^{26} + 540 q^{28} + 145 q^{29} + 420 q^{31} + 652 q^{32} + 204 q^{34} + 52 q^{35} + 378 q^{37} + 496 q^{38} - 80 q^{40} + 1158 q^{41} - 204 q^{43} - 784 q^{44} + 580 q^{46} - 248 q^{47} - 283 q^{49} - 908 q^{50} + 1482 q^{52} + 554 q^{53} + 546 q^{55} + 608 q^{56} - 440 q^{59} + 618 q^{61} - 1250 q^{62} + 2594 q^{64} + 1656 q^{65} + 1164 q^{67} - 356 q^{68} - 2184 q^{70} + 692 q^{71} - 1950 q^{73} + 1832 q^{74} + 1376 q^{76} + 1616 q^{77} + 272 q^{79} + 890 q^{80} + 92 q^{82} - 512 q^{83} - 1628 q^{85} - 2446 q^{86} - 6954 q^{88} - 866 q^{89} + 2580 q^{91} - 3468 q^{92} - 5942 q^{94} - 2244 q^{95} + 1562 q^{97} + 3408 q^{98}+O(q^{100}) \) 5 * q + 26 * q^4 - 10 * q^5 + 40 * q^7 + 84 * q^8 - 64 * q^10 - 12 * q^11 + 14 * q^13 + 192 * q^14 + 146 * q^16 - 66 * q^17 + 214 * q^19 - 6 * q^20 - 98 * q^22 - 164 * q^23 + 207 * q^25 - 56 * q^26 + 540 * q^28 + 145 * q^29 + 420 * q^31 + 652 * q^32 + 204 * q^34 + 52 * q^35 + 378 * q^37 + 496 * q^38 - 80 * q^40 + 1158 * q^41 - 204 * q^43 - 784 * q^44 + 580 * q^46 - 248 * q^47 - 283 * q^49 - 908 * q^50 + 1482 * q^52 + 554 * q^53 + 546 * q^55 + 608 * q^56 - 440 * q^59 + 618 * q^61 - 1250 * q^62 + 2594 * q^64 + 1656 * q^65 + 1164 * q^67 - 356 * q^68 - 2184 * q^70 + 692 * q^71 - 1950 * q^73 + 1832 * q^74 + 1376 * q^76 + 1616 * q^77 + 272 * q^79 + 890 * q^80 + 92 * q^82 - 512 * q^83 - 1628 * q^85 - 2446 * q^86 - 6954 * q^88 - 866 * q^89 + 2580 * q^91 - 3468 * q^92 - 5942 * q^94 - 2244 * q^95 + 1562 * q^97 + 3408 * q^98

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