LMFDB - Embedded newform 261.4.a.f.1.1

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			1.1
Root			$-3.68360$ 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)

$f(q)$	$=$	$q-4.47236 q^{2} +12.0020 q^{4} +6.52855 q^{5} +5.22706 q^{7} -17.8986 q^{8} +O(q^{10})$	\(q-4.47236 q^{2} +12.0020 q^{4} +6.52855 q^{5} +5.22706 q^{7} -17.8986 q^{8} -29.1981 q^{10} +21.1299 q^{11} +83.4615 q^{13} -23.3773 q^{14} -15.9674 q^{16} -11.3273 q^{17} -7.68096 q^{19} +78.3559 q^{20} -94.5005 q^{22} -153.169 q^{23} -82.3780 q^{25} -373.270 q^{26} +62.7354 q^{28} +29.0000 q^{29} +270.530 q^{31} +214.601 q^{32} +50.6597 q^{34} +34.1251 q^{35} -298.404 q^{37} +34.3520 q^{38} -116.852 q^{40} +184.710 q^{41} +208.337 q^{43} +253.602 q^{44} +685.028 q^{46} +553.098 q^{47} -315.678 q^{49} +368.424 q^{50} +1001.71 q^{52} +321.465 q^{53} +137.948 q^{55} -93.5569 q^{56} -129.699 q^{58} -104.930 q^{59} +464.230 q^{61} -1209.91 q^{62} -832.032 q^{64} +544.883 q^{65} +745.813 q^{67} -135.950 q^{68} -152.620 q^{70} +509.252 q^{71} -0.374979 q^{73} +1334.57 q^{74} -92.1871 q^{76} +110.447 q^{77} +610.912 q^{79} -104.244 q^{80} -826.092 q^{82} -791.431 q^{83} -73.9507 q^{85} -931.757 q^{86} -378.194 q^{88} +342.011 q^{89} +436.258 q^{91} -1838.34 q^{92} -2473.65 q^{94} -50.1455 q^{95} +601.476 q^{97} +1411.83 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.47236	−1.58122	−0.790610	−	0.612321i	$-0.790236\pi$
$2$	−4.47236	−1.58122	−0.790610	+	0.612321i	$0.790236\pi$
$3$	0	0
$3$	0	0
$4$	12.0020	1.50025
$5$	6.52855	0.583931	0.291966	−	0.956429i	$-0.405691\pi$
$5$	6.52855	0.583931	0.291966	+	0.956429i	$0.405691\pi$
$6$	0	0
$7$	5.22706	0.282235	0.141117	−	0.989993i	$-0.454930\pi$
$7$	5.22706	0.282235	0.141117	+	0.989993i	$0.454930\pi$
$8$	−17.8986	−0.791012
$9$	0	0
$10$	−29.1981	−0.923324
$11$	21.1299	0.579173	0.289586	−	0.957152i	$-0.406482\pi$
$11$	21.1299	0.579173	0.289586	+	0.957152i	$0.406482\pi$
$12$	0	0
$13$	83.4615	1.78062	0.890310	−	0.455355i	$-0.150488\pi$
$13$	83.4615	1.78062	0.890310	+	0.455355i	$0.150488\pi$
$14$	−23.3773	−0.446275
$15$	0	0
$16$	−15.9674	−0.249491
$17$	−11.3273	−0.161604	−0.0808020	−	0.996730i	$-0.525748\pi$
$17$	−11.3273	−0.161604	−0.0808020	+	0.996730i	$0.525748\pi$
$18$	0	0
$19$	−7.68096	−0.0927438	−0.0463719	−	0.998924i	$-0.514766\pi$
$19$	−7.68096	−0.0927438	−0.0463719	+	0.998924i	$0.514766\pi$
$20$	78.3559	0.876046
$21$	0	0
$22$	−94.5005	−0.915799
$23$	−153.169	−1.38861	−0.694303	−	0.719683i	$-0.744287\pi$
$23$	−153.169	−1.38861	−0.694303	+	0.719683i	$0.744287\pi$
$24$	0	0
$25$	−82.3780	−0.659024
$26$	−373.270	−2.81555
$27$	0	0
$28$	62.7354	0.423424
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	270.530	1.56737	0.783687	−	0.621156i	$-0.213337\pi$
$31$	270.530	1.56737	0.783687	+	0.621156i	$0.213337\pi$
$32$	214.601	1.18551
$33$	0	0
$34$	50.6597	0.255531
$35$	34.1251	0.164806
$36$	0	0
$37$	−298.404	−1.32587	−0.662936	−	0.748676i	$-0.730690\pi$
$37$	−298.404	−1.32587	−0.662936	+	0.748676i	$0.730690\pi$
$38$	34.3520	0.146648
$39$	0	0
$40$	−116.852	−0.461897
$41$	184.710	0.703584	0.351792	−	0.936078i	$-0.385573\pi$
$41$	184.710	0.703584	0.351792	+	0.936078i	$0.385573\pi$
$42$	0	0
$43$	208.337	0.738861	0.369430	−	0.929258i	$-0.379553\pi$
$43$	208.337	0.738861	0.369430	+	0.929258i	$0.379553\pi$
$44$	253.602	0.868906
$45$	0	0
$46$	685.028	2.19569
$47$	553.098	1.71654	0.858272	−	0.513194i	$-0.171538\pi$
$47$	553.098	1.71654	0.858272	+	0.513194i	$0.171538\pi$
$48$	0	0
$49$	−315.678	−0.920344
$50$	368.424	1.04206
$51$	0	0
$52$	1001.71	2.67138
$53$	321.465	0.833144	0.416572	−	0.909103i	$-0.363231\pi$
$53$	321.465	0.833144	0.416572	+	0.909103i	$0.363231\pi$
$54$	0	0
$55$	137.948	0.338197
$56$	−93.5569	−0.223251
$57$	0	0
$58$	−129.699	−0.293625
$59$	−104.930	−0.231538	−0.115769	−	0.993276i	$-0.536933\pi$
$59$	−104.930	−0.231538	−0.115769	+	0.993276i	$0.536933\pi$
$60$	0	0
$61$	464.230	0.974403	0.487201	−	0.873290i	$-0.338018\pi$
$61$	464.230	0.974403	0.487201	+	0.873290i	$0.338018\pi$
$62$	−1209.91	−2.47836
$63$	0	0
$64$	−832.032	−1.62506
$65$	544.883	1.03976
$66$	0	0
$67$	745.813	1.35993	0.679967	−	0.733243i	$-0.261994\pi$
$67$	745.813	1.35993	0.679967	+	0.733243i	$0.261994\pi$
$68$	−135.950	−0.242447
$69$	0	0
$70$	−152.620	−0.260594
$71$	509.252	0.851227	0.425613	−	0.904905i	$-0.360058\pi$
$71$	509.252	0.851227	0.425613	+	0.904905i	$0.360058\pi$
$72$	0	0
$73$	−0.374979	−0.000601205 0	−0.000300603	−	1.00000i	$-0.500096\pi$
$73$	−0.374979	−0.000601205 0	−0.000300603		1.00000i	$0.500096\pi$
$74$	1334.57	2.09649
$75$	0	0
$76$	−92.1871	−0.139139
$77$	110.447	0.163463
$78$	0	0
$79$	610.912	0.870037	0.435018	−	0.900422i	$-0.356742\pi$
$79$	610.912	0.870037	0.435018	+	0.900422i	$0.356742\pi$
$80$	−104.244	−0.145686
$81$	0	0
$82$	−826.092	−1.11252
$83$	−791.431	−1.04664	−0.523319	−	0.852137i	$-0.675306\pi$
$83$	−791.431	−1.04664	−0.523319	+	0.852137i	$0.675306\pi$
$84$	0	0
$85$	−73.9507	−0.0943656
$86$	−931.757	−1.16830
$87$	0	0
$88$	−378.194	−0.458132
$89$	342.011	0.407338	0.203669	−	0.979040i	$-0.434713\pi$
$89$	342.011	0.407338	0.203669	+	0.979040i	$0.434713\pi$
$90$	0	0
$91$	436.258	0.502553
$92$	−1838.34	−2.08326
$93$	0	0
$94$	−2473.65	−2.71423
$95$	−50.1455	−0.0541560
$96$	0	0
$97$	601.476	0.629594	0.314797	−	0.949159i	$-0.398063\pi$
$97$	601.476	0.629594	0.314797	+	0.949159i	$0.398063\pi$
$98$	1411.83	1.45526
$99$	0	0
$100$	−988.704	−0.988704
$101$	−402.327	−0.396367	−0.198183	−	0.980165i	$-0.563504\pi$
$101$	−402.327	−0.396367	−0.198183	+	0.980165i	$0.563504\pi$
$102$	0	0
$103$	−1338.38	−1.28033	−0.640166	−	0.768236i	$-0.721134\pi$
$103$	−1338.38	−1.28033	−0.640166	+	0.768236i	$0.721134\pi$
$104$	−1493.84	−1.40849
$105$	0	0
$106$	−1437.71	−1.31738
$107$	−500.501	−0.452199	−0.226099	−	0.974104i	$-0.572597\pi$
$107$	−500.501	−0.452199	−0.226099	+	0.974104i	$0.572597\pi$
$108$	0	0
$109$	1274.80	1.12022	0.560108	−	0.828420i	$-0.310760\pi$
$109$	1274.80	1.12022	0.560108	+	0.828420i	$0.310760\pi$
$110$	−616.951	−0.534764
$111$	0	0
$112$	−83.4628	−0.0704151
$113$	−335.278	−0.279118	−0.139559	−	0.990214i	$-0.544568\pi$
$113$	−335.278	−0.279118	−0.139559	+	0.990214i	$0.544568\pi$
$114$	0	0
$115$	−999.972	−0.810851
$116$	348.059	0.278590
$117$	0	0
$118$	469.287	0.366113
$119$	−59.2083	−0.0456103
$120$	0	0
$121$	−884.528	−0.664559
$122$	−2076.21	−1.54074
$123$	0	0
$124$	3246.91	2.35146
$125$	−1353.88	−0.968756
$126$	0	0
$127$	755.312	0.527741	0.263870	−	0.964558i	$-0.415001\pi$
$127$	755.312	0.527741	0.263870	+	0.964558i	$0.415001\pi$
$128$	2004.35	1.38407
$129$	0	0
$130$	−2436.91	−1.64409
$131$	253.351	0.168973	0.0844863	−	0.996425i	$-0.473075\pi$
$131$	253.351	0.168973	0.0844863	+	0.996425i	$0.473075\pi$
$132$	0	0
$133$	−40.1488	−0.0261755
$134$	−3335.55	−2.15035
$135$	0	0
$136$	202.742	0.127831
$137$	2477.49	1.54501	0.772504	−	0.635010i	$-0.219004\pi$
$137$	2477.49	1.54501	0.772504	+	0.635010i	$0.219004\pi$
$138$	0	0
$139$	423.114	0.258187	0.129094	−	0.991632i	$-0.458793\pi$
$139$	423.114	0.258187	0.129094	+	0.991632i	$0.458793\pi$
$140$	409.571	0.247251
$141$	0	0
$142$	−2277.56	−1.34598
$143$	1763.53	1.03129
$144$	0	0
$145$	189.328	0.108433
$146$	1.67704	0.000950637 0
$147$	0	0
$148$	−3581.45	−1.98914
$149$	1263.82	0.694873	0.347437	−	0.937703i	$-0.387052\pi$
$149$	1263.82	0.694873	0.347437	+	0.937703i	$0.387052\pi$
$150$	0	0
$151$	3369.67	1.81603	0.908013	−	0.418943i	$-0.137599\pi$
$151$	3369.67	1.81603	0.908013	+	0.418943i	$0.137599\pi$
$152$	137.478	0.0733615
$153$	0	0
$154$	−493.960	−0.258470
$155$	1766.17	0.915238
$156$	0	0
$157$	−3688.61	−1.87505	−0.937527	−	0.347914i	$-0.886890\pi$
$157$	−3688.61	−1.87505	−0.937527	+	0.347914i	$0.886890\pi$
$158$	−2732.22	−1.37572
$159$	0	0
$160$	1401.03	0.692258
$161$	−800.624	−0.391913
$162$	0	0
$163$	−1975.81	−0.949432	−0.474716	−	0.880139i	$-0.657449\pi$
$163$	−1975.81	−0.949432	−0.474716	+	0.880139i	$0.657449\pi$
$164$	2216.90	1.05555
$165$	0	0
$166$	3539.57	1.65496
$167$	1608.44	0.745297	0.372649	−	0.927973i	$-0.378450\pi$
$167$	1608.44	0.745297	0.372649	+	0.927973i	$0.378450\pi$
$168$	0	0
$169$	4768.82	2.17061
$170$	330.734	0.149213
$171$	0	0
$172$	2500.46	1.10848
$173$	−4445.71	−1.95376	−0.976881	−	0.213784i	$-0.931421\pi$
$173$	−4445.71	−1.95376	−0.976881	+	0.213784i	$0.931421\pi$
$174$	0	0
$175$	−430.595	−0.186000
$176$	−337.390	−0.144498
$177$	0	0
$178$	−1529.60	−0.644091
$179$	−1461.35	−0.610203	−0.305101	−	0.952320i	$-0.598690\pi$
$179$	−1461.35	−0.610203	−0.305101	+	0.952320i	$0.598690\pi$
$180$	0	0
$181$	3789.62	1.55624	0.778122	−	0.628113i	$-0.216172\pi$
$181$	3789.62	1.55624	0.778122	+	0.628113i	$0.216172\pi$
$182$	−1951.11	−0.794646
$183$	0	0
$184$	2741.50	1.09840
$185$	−1948.14	−0.774218
$186$	0	0
$187$	−239.344	−0.0935966
$188$	6638.30	2.57525
$189$	0	0
$190$	224.269	0.0856326
$191$	4782.10	1.81163	0.905813	−	0.423678i	$-0.139261\pi$
$191$	4782.10	1.81163	0.905813	+	0.423678i	$0.139261\pi$
$192$	0	0
$193$	−3557.27	−1.32673	−0.663363	−	0.748298i	$-0.730871\pi$
$193$	−3557.27	−1.32673	−0.663363	+	0.748298i	$0.730871\pi$
$194$	−2690.02	−0.995527
$195$	0	0
$196$	−3788.78	−1.38075
$197$	2290.24	0.828290	0.414145	−	0.910211i	$-0.364081\pi$
$197$	2290.24	0.828290	0.414145	+	0.910211i	$0.364081\pi$
$198$	0	0
$199$	−2788.00	−0.993146	−0.496573	−	0.867995i	$-0.665409\pi$
$199$	−2788.00	−0.993146	−0.496573	+	0.867995i	$0.665409\pi$
$200$	1474.45	0.521296
$201$	0	0
$202$	1799.35	0.626742
$203$	151.585	0.0524097
$204$	0	0
$205$	1205.89	0.410845
$206$	5985.71	2.02449
$207$	0	0
$208$	−1332.67	−0.444249
$209$	−162.298	−0.0537147
$210$	0	0
$211$	628.449	0.205044	0.102522	−	0.994731i	$-0.467309\pi$
$211$	628.449	0.205044	0.102522	+	0.994731i	$0.467309\pi$
$212$	3858.24	1.24993
$213$	0	0
$214$	2238.42	0.715025
$215$	1360.14	0.431444
$216$	0	0
$217$	1414.08	0.442367
$218$	−5701.36	−1.77131
$219$	0	0
$220$	1655.65	0.507382
$221$	−945.391	−0.287755
$222$	0	0
$223$	136.439	0.0409714	0.0204857	−	0.999790i	$-0.493479\pi$
$223$	136.439	0.0409714	0.0204857	+	0.999790i	$0.493479\pi$
$224$	1121.73	0.334593
$225$	0	0
$226$	1499.49	0.441347
$227$	−4180.45	−1.22232	−0.611159	−	0.791508i	$-0.709296\pi$
$227$	−4180.45	−1.22232	−0.611159	+	0.791508i	$0.709296\pi$
$228$	0	0
$229$	−1352.95	−0.390417	−0.195209	−	0.980762i	$-0.562538\pi$
$229$	−1352.95	−0.390417	−0.195209	+	0.980762i	$0.562538\pi$
$230$	4472.24	1.28213
$231$	0	0
$232$	−519.058	−0.146887
$233$	−2175.34	−0.611635	−0.305818	−	0.952090i	$-0.598930\pi$
$233$	−2175.34	−0.611635	−0.305818	+	0.952090i	$0.598930\pi$
$234$	0	0
$235$	3610.93	1.00234
$236$	−1259.38	−0.347367
$237$	0	0
$238$	264.801	0.0721198
$239$	4512.18	1.22121	0.610604	−	0.791936i	$-0.290927\pi$
$239$	4512.18	1.22121	0.610604	+	0.791936i	$0.290927\pi$
$240$	0	0
$241$	1950.53	0.521346	0.260673	−	0.965427i	$-0.416056\pi$
$241$	1950.53	0.521346	0.260673	+	0.965427i	$0.416056\pi$
$242$	3955.93	1.05081
$243$	0	0
$244$	5571.70	1.46185
$245$	−2060.92	−0.537417
$246$	0	0
$247$	−641.064	−0.165141
$248$	−4842.09	−1.23981
$249$	0	0
$250$	6055.03	1.53182
$251$	−27.4143	−0.00689393	−0.00344696	−	0.999994i	$-0.501097\pi$
$251$	−27.4143	−0.00689393	−0.00344696	+	0.999994i	$0.501097\pi$
$252$	0	0
$253$	−3236.44	−0.804243
$254$	−3378.03	−0.834474
$255$	0	0
$256$	−2307.91	−0.563454
$257$	−4458.31	−1.08211	−0.541053	−	0.840988i	$-0.681974\pi$
$257$	−4458.31	−1.08211	−0.541053	+	0.840988i	$0.681974\pi$
$258$	0	0
$259$	−1559.77	−0.374207
$260$	6539.70	1.55990
$261$	0	0
$262$	−1133.08	−0.267183
$263$	−4641.08	−1.08814	−0.544071	−	0.839039i	$-0.683118\pi$
$263$	−4641.08	−1.08814	−0.544071	+	0.839039i	$0.683118\pi$
$264$	0	0
$265$	2098.70	0.486499
$266$	179.560	0.0413893
$267$	0	0
$268$	8951.28	2.04025
$269$	235.021	0.0532694	0.0266347	−	0.999645i	$-0.491521\pi$
$269$	235.021	0.0532694	0.0266347	+	0.999645i	$0.491521\pi$
$270$	0	0
$271$	3816.09	0.855392	0.427696	−	0.903923i	$-0.359325\pi$
$271$	3816.09	0.855392	0.427696	+	0.903923i	$0.359325\pi$
$272$	180.867	0.0403188
$273$	0	0
$274$	−11080.2	−2.44300
$275$	−1740.64	−0.381689
$276$	0	0
$277$	−2024.79	−0.439198	−0.219599	−	0.975590i	$-0.570475\pi$
$277$	−2024.79	−0.439198	−0.219599	+	0.975590i	$0.570475\pi$
$278$	−1892.32	−0.408251
$279$	0	0
$280$	−610.791	−0.130363
$281$	−5451.81	−1.15739	−0.578697	−	0.815543i	$-0.696439\pi$
$281$	−5451.81	−1.15739	−0.578697	+	0.815543i	$0.696439\pi$
$282$	0	0
$283$	5026.80	1.05587	0.527937	−	0.849284i	$-0.322966\pi$
$283$	5026.80	1.05587	0.527937	+	0.849284i	$0.322966\pi$
$284$	6112.06	1.27706
$285$	0	0
$286$	−7887.16	−1.63069
$287$	965.493	0.198576
$288$	0	0
$289$	−4784.69	−0.973884
$290$	−846.744	−0.171457
$291$	0	0
$292$	−4.50051	−0.000901961 0
$293$	−6160.34	−1.22830	−0.614148	−	0.789191i	$-0.710500\pi$
$293$	−6160.34	−1.22830	−0.614148	+	0.789191i	$0.710500\pi$
$294$	0	0
$295$	−685.043	−0.135203
$296$	5340.99	1.04878
$297$	0	0
$298$	−5652.26	−1.09875
$299$	−12783.7	−2.47258
$300$	0	0
$301$	1088.99	0.208532
$302$	−15070.4	−2.87153
$303$	0	0
$304$	122.645	0.0231388
$305$	3030.75	0.568984
$306$	0	0
$307$	−4329.54	−0.804885	−0.402443	−	0.915445i	$-0.631839\pi$
$307$	−4329.54	−0.804885	−0.402443	+	0.915445i	$0.631839\pi$
$308$	1325.59	0.245236
$309$	0	0
$310$	−7898.94	−1.44719
$311$	6411.83	1.16907	0.584536	−	0.811368i	$-0.301276\pi$
$311$	6411.83	1.16907	0.584536	+	0.811368i	$0.301276\pi$
$312$	0	0
$313$	−19.5263	−0.00352617	−0.00176309	−	0.999998i	$-0.500561\pi$
$313$	−19.5263	−0.00352617	−0.00176309	+	0.999998i	$0.500561\pi$
$314$	16496.8	2.96487
$315$	0	0
$316$	7332.18	1.30528
$317$	8198.11	1.45253	0.726265	−	0.687415i	$-0.241255\pi$
$317$	8198.11	1.45253	0.726265	+	0.687415i	$0.241255\pi$
$318$	0	0
$319$	612.767	0.107550
$320$	−5431.97	−0.948926
$321$	0	0
$322$	3580.68	0.619701
$323$	87.0043	0.0149878
$324$	0	0
$325$	−6875.39	−1.17347
$326$	8836.54	1.50126
$327$	0	0
$328$	−3306.05	−0.556543
$329$	2891.08	0.484469
$330$	0	0
$331$	2374.46	0.394297	0.197148	−	0.980374i	$-0.436832\pi$
$331$	2374.46	0.394297	0.197148	+	0.980374i	$0.436832\pi$
$332$	−9498.79	−1.57022
$333$	0	0
$334$	−7193.52	−1.17848
$335$	4869.08	0.794108
$336$	0	0
$337$	−4985.34	−0.805842	−0.402921	−	0.915235i	$-0.632005\pi$
$337$	−4985.34	−0.805842	−0.402921	+	0.915235i	$0.632005\pi$
$338$	−21327.9	−3.43221
$339$	0	0
$340$	−887.559	−0.141572
$341$	5716.26	0.907780
$342$	0	0
$343$	−3442.95	−0.541988
$344$	−3728.92	−0.584448
$345$	0	0
$346$	19882.8	3.08933
$347$	2023.98	0.313121	0.156560	−	0.987668i	$-0.449959\pi$
$347$	2023.98	0.313121	0.156560	+	0.987668i	$0.449959\pi$
$348$	0	0
$349$	−2651.99	−0.406755	−0.203378	−	0.979100i	$-0.565192\pi$
$349$	−2651.99	−0.406755	−0.203378	+	0.979100i	$0.565192\pi$
$350$	1925.78	0.294106
$351$	0	0
$352$	4534.49	0.686616
$353$	1729.22	0.260729	0.130364	−	0.991466i	$-0.458385\pi$
$353$	1729.22	0.260729	0.130364	+	0.991466i	$0.458385\pi$
$354$	0	0
$355$	3324.68	0.497058
$356$	4104.83	0.611111
$357$	0	0
$358$	6535.68	0.964864
$359$	−6875.38	−1.01078	−0.505388	−	0.862892i	$-0.668651\pi$
$359$	−6875.38	−1.01078	−0.505388	+	0.862892i	$0.668651\pi$
$360$	0	0
$361$	−6800.00	−0.991399
$362$	−16948.6	−2.46076
$363$	0	0
$364$	5235.99	0.753957
$365$	−2.44807	−0.000351063 0
$366$	0	0
$367$	−7435.15	−1.05753	−0.528763	−	0.848770i	$-0.677344\pi$
$367$	−7435.15	−1.05753	−0.528763	+	0.848770i	$0.677344\pi$
$368$	2445.72	0.346445
$369$	0	0
$370$	8712.80	1.22421
$371$	1680.32	0.235142
$372$	0	0
$373$	1397.48	0.193991	0.0969954	−	0.995285i	$-0.469077\pi$
$373$	1397.48	0.193991	0.0969954	+	0.995285i	$0.469077\pi$
$374$	1070.43	0.147997
$375$	0	0
$376$	−9899.65	−1.35781
$377$	2420.38	0.330653
$378$	0	0
$379$	7009.57	0.950020	0.475010	−	0.879980i	$-0.342444\pi$
$379$	7009.57	0.950020	0.475010	+	0.879980i	$0.342444\pi$
$380$	−601.848	−0.0812478
$381$	0	0
$382$	−21387.3	−2.86458
$383$	11728.7	1.56477	0.782387	−	0.622792i	$-0.214002\pi$
$383$	11728.7	1.56477	0.782387	+	0.622792i	$0.214002\pi$
$384$	0	0
$385$	721.060	0.0954510
$386$	15909.4	2.09784
$387$	0	0
$388$	7218.94	0.944552
$389$	4367.15	0.569212	0.284606	−	0.958645i	$-0.408137\pi$
$389$	4367.15	0.569212	0.284606	+	0.958645i	$0.408137\pi$
$390$	0	0
$391$	1734.99	0.224404
$392$	5650.18	0.728003
$393$	0	0
$394$	−10242.8	−1.30971
$395$	3988.37	0.508042
$396$	0	0
$397$	1632.74	0.206410	0.103205	−	0.994660i	$-0.467090\pi$
$397$	1632.74	0.206410	0.103205	+	0.994660i	$0.467090\pi$
$398$	12469.0	1.57038
$399$	0	0
$400$	1315.37	0.164421
$401$	−5028.65	−0.626232	−0.313116	−	0.949715i	$-0.601373\pi$
$401$	−5028.65	−0.626232	−0.313116	+	0.949715i	$0.601373\pi$
$402$	0	0
$403$	22578.8	2.79090
$404$	−4828.74	−0.594651
$405$	0	0
$406$	−677.942	−0.0828712
$407$	−6305.23	−0.767908
$408$	0	0
$409$	−2497.21	−0.301904	−0.150952	−	0.988541i	$-0.548234\pi$
$409$	−2497.21	−0.301904	−0.150952	+	0.988541i	$0.548234\pi$
$410$	−5393.19	−0.649635
$411$	0	0
$412$	−16063.3	−1.92082
$413$	−548.477	−0.0653482
$414$	0	0
$415$	−5166.90	−0.611164
$416$	17910.9	2.11095
$417$	0	0
$418$	725.854	0.0849347
$419$	−12909.4	−1.50517	−0.752585	−	0.658496i	$-0.771193\pi$
$419$	−12909.4	−1.50517	−0.752585	+	0.658496i	$0.771193\pi$
$420$	0	0
$421$	−4019.30	−0.465293	−0.232647	−	0.972561i	$-0.574739\pi$
$421$	−4019.30	−0.465293	−0.232647	+	0.972561i	$0.574739\pi$
$422$	−2810.65	−0.324219
$423$	0	0
$424$	−5753.76	−0.659027
$425$	933.118	0.106501
$426$	0	0
$427$	2426.56	0.275010
$428$	−6007.03	−0.678413
$429$	0	0
$430$	−6083.02	−0.682208
$431$	−8768.94	−0.980012	−0.490006	−	0.871719i	$-0.663005\pi$
$431$	−8768.94	−0.980012	−0.490006	+	0.871719i	$0.663005\pi$
$432$	0	0
$433$	4496.26	0.499021	0.249511	−	0.968372i	$-0.419730\pi$
$433$	4496.26	0.499021	0.249511	+	0.968372i	$0.419730\pi$
$434$	−6324.26	−0.699480
$435$	0	0
$436$	15300.2	1.68061
$437$	1176.48	0.128785
$438$	0	0
$439$	−7316.62	−0.795451	−0.397725	−	0.917504i	$-0.630200\pi$
$439$	−7316.62	−0.795451	−0.397725	+	0.917504i	$0.630200\pi$
$440$	−2469.06	−0.267518
$441$	0	0
$442$	4228.13	0.455004
$443$	12801.1	1.37291	0.686454	−	0.727173i	$-0.259166\pi$
$443$	12801.1	1.37291	0.686454	+	0.727173i	$0.259166\pi$
$444$	0	0
$445$	2232.84	0.237858
$446$	−610.204	−0.0647847
$447$	0	0
$448$	−4349.08	−0.458649
$449$	−6412.22	−0.673968	−0.336984	−	0.941510i	$-0.609407\pi$
$449$	−6412.22	−0.673968	−0.336984	+	0.941510i	$0.609407\pi$
$450$	0	0
$451$	3902.91	0.407496
$452$	−4024.02	−0.418748
$453$	0	0
$454$	18696.5	1.93275
$455$	2848.14	0.293456
$456$	0	0
$457$	−2153.32	−0.220412	−0.110206	−	0.993909i	$-0.535151\pi$
$457$	−2153.32	−0.220412	−0.110206	+	0.993909i	$0.535151\pi$
$458$	6050.89	0.617336
$459$	0	0
$460$	−12001.7	−1.21648
$461$	1850.55	0.186960	0.0934800	−	0.995621i	$-0.470201\pi$
$461$	1850.55	0.186960	0.0934800	+	0.995621i	$0.470201\pi$
$462$	0	0
$463$	1892.04	0.189914	0.0949572	−	0.995481i	$-0.469729\pi$
$463$	1892.04	0.189914	0.0949572	+	0.995481i	$0.469729\pi$
$464$	−463.056	−0.0463293
$465$	0	0
$466$	9728.89	0.967129
$467$	3739.40	0.370533	0.185267	−	0.982688i	$-0.440685\pi$
$467$	3739.40	0.370533	0.185267	+	0.982688i	$0.440685\pi$
$468$	0	0
$469$	3898.41	0.383821
$470$	−16149.4	−1.58493
$471$	0	0
$472$	1878.10	0.183150
$473$	4402.13	0.427928
$474$	0	0
$475$	632.742	0.0611204
$476$	−710.621	−0.0684270
$477$	0	0
$478$	−20180.1	−1.93100
$479$	7260.30	0.692550	0.346275	−	0.938133i	$-0.387446\pi$
$479$	7260.30	0.692550	0.346275	+	0.938133i	$0.387446\pi$
$480$	0	0
$481$	−24905.2	−2.36087
$482$	−8723.46	−0.824363
$483$	0	0
$484$	−10616.1	−0.997008
$485$	3926.77	0.367640
$486$	0	0
$487$	4756.40	0.442573	0.221287	−	0.975209i	$-0.428974\pi$
$487$	4756.40	0.442573	0.221287	+	0.975209i	$0.428974\pi$
$488$	−8309.05	−0.770764
$489$	0	0
$490$	9217.18	0.849775
$491$	2007.83	0.184546	0.0922730	−	0.995734i	$-0.470587\pi$
$491$	2007.83	0.184546	0.0922730	+	0.995734i	$0.470587\pi$
$492$	0	0
$493$	−328.491	−0.0300091
$494$	2867.07	0.261125
$495$	0	0
$496$	−4319.66	−0.391046
$497$	2661.89	0.240246
$498$	0	0
$499$	−8952.55	−0.803149	−0.401574	−	0.915826i	$-0.631537\pi$
$499$	−8952.55	−0.803149	−0.401574	+	0.915826i	$0.631537\pi$
$500$	−16249.3	−1.45338
$501$	0	0
$502$	122.607	0.0109008
$503$	−20564.9	−1.82295	−0.911477	−	0.411351i	$-0.865057\pi$
$503$	−20564.9	−1.82295	−0.911477	+	0.411351i	$0.865057\pi$
$504$	0	0
$505$	−2626.61	−0.231451
$506$	14474.6	1.27168
$507$	0	0
$508$	9065.28	0.791746
$509$	−13321.9	−1.16008	−0.580041	−	0.814587i	$-0.696964\pi$
$509$	−13321.9	−1.16008	−0.580041	+	0.814587i	$0.696964\pi$
$510$	0	0
$511$	−1.96004	−0.000169681 0
$512$	−5712.97	−0.493125
$513$	0	0
$514$	19939.2	1.71105
$515$	−8737.66	−0.747626
$516$	0	0
$517$	11686.9	0.994176
$518$	6975.87	0.591703
$519$	0	0
$520$	−9752.62	−0.822462
$521$	13047.0	1.09712	0.548558	−	0.836113i	$-0.315177\pi$
$521$	13047.0	1.09712	0.548558	+	0.836113i	$0.315177\pi$
$522$	0	0
$523$	−20207.4	−1.68950	−0.844750	−	0.535161i	$-0.820251\pi$
$523$	−20207.4	−1.68950	−0.844750	+	0.535161i	$0.820251\pi$
$524$	3040.73	0.253502
$525$	0	0
$526$	20756.6	1.72059
$527$	−3064.36	−0.253294
$528$	0	0
$529$	11293.8	0.928228
$530$	−9386.16	−0.769262
$531$	0	0
$532$	−481.868	−0.0392700
$533$	15416.2	1.25281
$534$	0	0
$535$	−3267.55	−0.264053
$536$	−13349.0	−1.07572
$537$	0	0
$538$	−1051.10	−0.0842306
$539$	−6670.23	−0.533038
$540$	0	0
$541$	−17866.0	−1.41981	−0.709906	−	0.704297i	$-0.751263\pi$
$541$	−17866.0	−1.41981	−0.709906	+	0.704297i	$0.751263\pi$
$542$	−17067.0	−1.35256
$543$	0	0
$544$	−2430.84	−0.191583
$545$	8322.59	0.654129
$546$	0	0
$547$	−8027.79	−0.627502	−0.313751	−	0.949505i	$-0.601586\pi$
$547$	−8027.79	−0.627502	−0.313751	+	0.949505i	$0.601586\pi$
$548$	29734.9	2.31790
$549$	0	0
$550$	7784.76	0.603534
$551$	−222.748	−0.0172221
$552$	0	0
$553$	3193.27	0.245555
$554$	9055.59	0.694468
$555$	0	0
$556$	5078.23	0.387347
$557$	−17357.6	−1.32040	−0.660201	−	0.751089i	$-0.729529\pi$
$557$	−17357.6	−1.32040	−0.660201	+	0.751089i	$0.729529\pi$
$558$	0	0
$559$	17388.1	1.31563
$560$	−544.891	−0.0411176
$561$	0	0
$562$	24382.5	1.83009
$563$	6073.90	0.454679	0.227340	−	0.973816i	$-0.426997\pi$
$563$	6073.90	0.454679	0.227340	+	0.973816i	$0.426997\pi$
$564$	0	0
$565$	−2188.88	−0.162986
$566$	−22481.7	−1.66957
$567$	0	0
$568$	−9114.87	−0.673330
$569$	−5084.04	−0.374577	−0.187288	−	0.982305i	$-0.559970\pi$
$569$	−5084.04	−0.374577	−0.187288	+	0.982305i	$0.559970\pi$
$570$	0	0
$571$	15552.3	1.13983	0.569914	−	0.821704i	$-0.306976\pi$
$571$	15552.3	1.13983	0.569914	+	0.821704i	$0.306976\pi$
$572$	21166.0	1.54719
$573$	0	0
$574$	−4318.03	−0.313992
$575$	12617.8	0.915125
$576$	0	0
$577$	−20714.3	−1.49454	−0.747269	−	0.664521i	$-0.768635\pi$
$577$	−20714.3	−1.49454	−0.747269	+	0.664521i	$0.768635\pi$
$578$	21398.9	1.53992
$579$	0	0
$580$	2272.32	0.162678
$581$	−4136.86	−0.295397
$582$	0	0
$583$	6792.52	0.482534
$584$	6.71159	0.000475561 0
$585$	0	0
$586$	27551.3	1.94221
$587$	1015.03	0.0713711	0.0356856	−	0.999363i	$-0.488639\pi$
$587$	1015.03	0.0713711	0.0356856	+	0.999363i	$0.488639\pi$
$588$	0	0
$589$	−2077.93	−0.145364
$590$	3063.76	0.213785
$591$	0	0
$592$	4764.74	0.330793
$593$	−3831.39	−0.265323	−0.132661	−	0.991161i	$-0.542352\pi$
$593$	−3831.39	−0.265323	−0.132661	+	0.991161i	$0.542352\pi$
$594$	0	0
$595$	−386.545	−0.0266333
$596$	15168.4	1.04249
$597$	0	0
$598$	57173.4	3.90969
$599$	16703.5	1.13938	0.569688	−	0.821861i	$-0.307064\pi$
$599$	16703.5	1.13938	0.569688	+	0.821861i	$0.307064\pi$
$600$	0	0
$601$	17781.6	1.20686	0.603432	−	0.797414i	$-0.293799\pi$
$601$	17781.6	1.20686	0.603432	+	0.797414i	$0.293799\pi$
$602$	−4870.35	−0.329735
$603$	0	0
$604$	40442.9	2.72450
$605$	−5774.69	−0.388057
$606$	0	0
$607$	7610.08	0.508869	0.254435	−	0.967090i	$-0.418111\pi$
$607$	7610.08	0.508869	0.254435	+	0.967090i	$0.418111\pi$
$608$	−1648.34	−0.109949
$609$	0	0
$610$	−13554.6	−0.899689
$611$	46162.4	3.05651
$612$	0	0
$613$	4207.29	0.277212	0.138606	−	0.990348i	$-0.455738\pi$
$613$	4207.29	0.277212	0.138606	+	0.990348i	$0.455738\pi$
$614$	19363.3	1.27270
$615$	0	0
$616$	−1976.85	−0.129301
$617$	−28539.4	−1.86216	−0.931082	−	0.364810i	$-0.881134\pi$
$617$	−28539.4	−1.86216	−0.931082	+	0.364810i	$0.881134\pi$
$618$	0	0
$619$	16799.9	1.09086	0.545432	−	0.838155i	$-0.316366\pi$
$619$	16799.9	1.09086	0.545432	+	0.838155i	$0.316366\pi$
$620$	21197.6	1.37309
$621$	0	0
$622$	−28676.0	−1.84856
$623$	1787.71	0.114965
$624$	0	0
$625$	1458.39	0.0933369
$626$	87.3288	0.00557565
$627$	0	0
$628$	−44270.9	−2.81306
$629$	3380.10	0.214266
$630$	0	0
$631$	−16207.1	−1.02249	−0.511247	−	0.859434i	$-0.670816\pi$
$631$	−16207.1	−1.02249	−0.511247	+	0.859434i	$0.670816\pi$
$632$	−10934.4	−0.688210
$633$	0	0
$634$	−36664.9	−2.29677
$635$	4931.09	0.308164
$636$	0	0
$637$	−26346.9	−1.63878
$638$	−2740.51	−0.170060
$639$	0	0
$640$	13085.5	0.808202
$641$	−10697.0	−0.659133	−0.329567	−	0.944132i	$-0.606903\pi$
$641$	−10697.0	−0.659133	−0.329567	+	0.944132i	$0.606903\pi$
$642$	0	0
$643$	7901.34	0.484601	0.242300	−	0.970201i	$-0.422098\pi$
$643$	7901.34	0.484601	0.242300	+	0.970201i	$0.422098\pi$
$644$	−9609.12	−0.587969
$645$	0	0
$646$	−389.115	−0.0236989
$647$	312.370	0.0189807	0.00949036	−	0.999955i	$-0.496979\pi$
$647$	312.370	0.0189807	0.00949036	+	0.999955i	$0.496979\pi$
$648$	0	0
$649$	−2217.17	−0.134101
$650$	30749.3	1.85552
$651$	0	0
$652$	−23713.7	−1.42439
$653$	−13732.3	−0.822950	−0.411475	−	0.911421i	$-0.634986\pi$
$653$	−13732.3	−0.822950	−0.411475	+	0.911421i	$0.634986\pi$
$654$	0	0
$655$	1654.02	0.0986684
$656$	−2949.35	−0.175538
$657$	0	0
$658$	−12929.9	−0.766051
$659$	−19869.0	−1.17449	−0.587244	−	0.809410i	$-0.699787\pi$
$659$	−19869.0	−1.17449	−0.587244	+	0.809410i	$0.699787\pi$
$660$	0	0
$661$	−11047.2	−0.650057	−0.325029	−	0.945704i	$-0.605374\pi$
$661$	−11047.2	−0.650057	−0.325029	+	0.945704i	$0.605374\pi$
$662$	−10619.5	−0.623470
$663$	0	0
$664$	14165.5	0.827902
$665$	−262.114	−0.0152847
$666$	0	0
$667$	−4441.90	−0.257858
$668$	19304.5	1.11814
$669$	0	0
$670$	−21776.3	−1.25566
$671$	9809.12	0.564347
$672$	0	0
$673$	−25673.8	−1.47051	−0.735253	−	0.677793i	$-0.762937\pi$
$673$	−25673.8	−1.47051	−0.735253	+	0.677793i	$0.762937\pi$
$674$	22296.2	1.27421
$675$	0	0
$676$	57235.6	3.25646
$677$	−4268.79	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$677$	−4268.79	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$678$	0	0
$679$	3143.95	0.177693
$680$	1323.61	0.0746443
$681$	0	0
$682$	−25565.2	−1.43540
$683$	−7370.85	−0.412940	−0.206470	−	0.978453i	$-0.566198\pi$
$683$	−7370.85	−0.412940	−0.206470	+	0.978453i	$0.566198\pi$
$684$	0	0
$685$	16174.4	0.902178
$686$	15398.1	0.857001
$687$	0	0
$688$	−3326.60	−0.184339
$689$	26830.0	1.48351
$690$	0	0
$691$	2112.00	0.116272	0.0581361	−	0.998309i	$-0.481484\pi$
$691$	2112.00	0.116272	0.0581361	+	0.998309i	$0.481484\pi$
$692$	−53357.5	−2.93114
$693$	0	0
$694$	−9051.97	−0.495113
$695$	2762.32	0.150764
$696$	0	0
$697$	−2092.27	−0.113702
$698$	11860.6	0.643169
$699$	0	0
$700$	−5168.02	−0.279047
$701$	−20945.5	−1.12853	−0.564266	−	0.825593i	$-0.690841\pi$
$701$	−20945.5	−1.12853	−0.564266	+	0.825593i	$0.690841\pi$
$702$	0	0
$703$	2292.02	0.122966
$704$	−17580.7	−0.941192
$705$	0	0
$706$	−7733.71	−0.412269
$707$	−2102.99	−0.111868
$708$	0	0
$709$	7826.11	0.414550	0.207275	−	0.978283i	$-0.433541\pi$
$709$	7826.11	0.414550	0.207275	+	0.978283i	$0.433541\pi$
$710$	−14869.2	−0.785958
$711$	0	0
$712$	−6121.51	−0.322210
$713$	−41436.8	−2.17646
$714$	0	0
$715$	11513.3	0.602200
$716$	−17539.1	−0.915459
$717$	0	0
$718$	30749.2	1.59826
$719$	−23373.7	−1.21237	−0.606183	−	0.795325i	$-0.707300\pi$
$719$	−23373.7	−1.21237	−0.606183	+	0.795325i	$0.707300\pi$
$720$	0	0
$721$	−6995.78	−0.361354
$722$	30412.1	1.56762
$723$	0	0
$724$	45483.2	2.33476
$725$	−2388.96	−0.122378
$726$	0	0
$727$	31240.8	1.59375	0.796875	−	0.604145i	$-0.206485\pi$
$727$	31240.8	1.59375	0.796875	+	0.604145i	$0.206485\pi$
$728$	−7808.40	−0.397525
$729$	0	0
$730$	10.9487	0.000555107 0
$731$	−2359.88	−0.119403
$732$	0	0
$733$	−4426.59	−0.223056	−0.111528	−	0.993761i	$-0.535574\pi$
$733$	−4426.59	−0.223056	−0.111528	+	0.993761i	$0.535574\pi$
$734$	33252.7	1.67218
$735$	0	0
$736$	−32870.2	−1.64621
$737$	15758.9	0.787636
$738$	0	0
$739$	−32119.5	−1.59883	−0.799414	−	0.600781i	$-0.794856\pi$
$739$	−32119.5	−1.59883	−0.799414	+	0.600781i	$0.794856\pi$
$740$	−23381.7	−1.16152
$741$	0	0
$742$	−7514.99	−0.371811
$743$	19215.0	0.948763	0.474382	−	0.880319i	$-0.342672\pi$
$743$	19215.0	0.948763	0.474382	+	0.880319i	$0.342672\pi$
$744$	0	0
$745$	8250.91	0.405758
$746$	−6250.02	−0.306742
$747$	0	0
$748$	−2872.61	−0.140419
$749$	−2616.15	−0.127626
$750$	0	0
$751$	3593.57	0.174609	0.0873045	−	0.996182i	$-0.472175\pi$
$751$	3593.57	0.174609	0.0873045	+	0.996182i	$0.472175\pi$
$752$	−8831.55	−0.428263
$753$	0	0
$754$	−10824.8	−0.522835
$755$	21999.1	1.06043
$756$	0	0
$757$	−32956.7	−1.58234	−0.791169	−	0.611598i	$-0.790527\pi$
$757$	−32956.7	−1.58234	−0.791169	+	0.611598i	$0.790527\pi$
$758$	−31349.4	−1.50219
$759$	0	0
$760$	897.532	0.0428381
$761$	−15689.7	−0.747373	−0.373687	−	0.927555i	$-0.621906\pi$
$761$	−15689.7	−0.747373	−0.373687	+	0.927555i	$0.621906\pi$
$762$	0	0
$763$	6663.45	0.316164
$764$	57394.9	2.71790
$765$	0	0
$766$	−52455.0	−2.47425
$767$	−8757.64	−0.412282
$768$	0	0
$769$	2134.77	0.100106	0.0500532	−	0.998747i	$-0.484061\pi$
$769$	2134.77	0.100106	0.0500532	+	0.998747i	$0.484061\pi$
$770$	−3224.84	−0.150929
$771$	0	0
$772$	−42694.5	−1.99043
$773$	−14780.2	−0.687721	−0.343861	−	0.939021i	$-0.611735\pi$
$773$	−14780.2	−0.687721	−0.343861	+	0.939021i	$0.611735\pi$
$774$	0	0
$775$	−22285.7	−1.03294
$776$	−10765.6	−0.498017
$777$	0	0
$778$	−19531.5	−0.900049
$779$	−1418.75	−0.0652530
$780$	0	0
$781$	10760.4	0.493007
$782$	−7759.49	−0.354832
$783$	0	0
$784$	5040.56	0.229618
$785$	−24081.3	−1.09490
$786$	0	0
$787$	35525.7	1.60909	0.804545	−	0.593892i	$-0.202409\pi$
$787$	35525.7	1.60909	0.804545	+	0.593892i	$0.202409\pi$
$788$	27487.6	1.24265
$789$	0	0
$790$	−17837.4	−0.803326
$791$	−1752.52	−0.0787768
$792$	0	0
$793$	38745.3	1.73504
$794$	−7302.21	−0.326380
$795$	0	0
$796$	−33461.7	−1.48997
$797$	−2190.00	−0.0973321	−0.0486660	−	0.998815i	$-0.515497\pi$
$797$	−2190.00	−0.0973321	−0.0486660	+	0.998815i	$0.515497\pi$
$798$	0	0
$799$	−6265.09	−0.277400
$800$	−17678.4	−0.781281
$801$	0	0
$802$	22490.0	0.990210
$803$	−7.92326	−0.000348202 0
$804$	0	0
$805$	−5226.91	−0.228850
$806$	−100981.	−4.41302
$807$	0	0
$808$	7201.07	0.313531
$809$	−21464.6	−0.932824	−0.466412	−	0.884568i	$-0.654454\pi$
$809$	−21464.6	−0.932824	−0.466412	+	0.884568i	$0.654454\pi$
$810$	0	0
$811$	−32288.8	−1.39804	−0.699021	−	0.715101i	$-0.746380\pi$
$811$	−32288.8	−1.39804	−0.699021	+	0.715101i	$0.746380\pi$
$812$	1819.33	0.0786279
$813$	0	0
$814$	28199.3	1.21423
$815$	−12899.2	−0.554403
$816$	0	0
$817$	−1600.22	−0.0685248
$818$	11168.4	0.477377
$819$	0	0
$820$	14473.2	0.616371
$821$	40334.4	1.71459	0.857296	−	0.514825i	$-0.172143\pi$
$821$	40334.4	1.71459	0.857296	+	0.514825i	$0.172143\pi$
$822$	0	0
$823$	8188.53	0.346822	0.173411	−	0.984850i	$-0.444521\pi$
$823$	8188.53	0.346822	0.173411	+	0.984850i	$0.444521\pi$
$824$	23955.0	1.01276
$825$	0	0
$826$	2452.99	0.103330
$827$	7630.74	0.320855	0.160427	−	0.987048i	$-0.448713\pi$
$827$	7630.74	0.320855	0.160427	+	0.987048i	$0.448713\pi$
$828$	0	0
$829$	12637.7	0.529462	0.264731	−	0.964322i	$-0.414717\pi$
$829$	12637.7	0.529462	0.264731	+	0.964322i	$0.414717\pi$
$830$	23108.3	0.966385
$831$	0	0
$832$	−69442.7	−2.89362
$833$	3575.77	0.148731
$834$	0	0
$835$	10500.8	0.435203
$836$	−1947.90	−0.0805857
$837$	0	0
$838$	57735.6	2.38000
$839$	−24059.1	−0.990005	−0.495002	−	0.868892i	$-0.664833\pi$
$839$	−24059.1	−0.990005	−0.495002	+	0.868892i	$0.664833\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	17975.8	0.735731
$843$	0	0
$844$	7542.67	0.307618
$845$	31133.5	1.26749
$846$	0	0
$847$	−4623.48	−0.187562
$848$	−5132.97	−0.207862
$849$	0	0
$850$	−4173.24	−0.168401
$851$	45706.2	1.84111
$852$	0	0
$853$	−18588.4	−0.746138	−0.373069	−	0.927804i	$-0.621695\pi$
$853$	−18588.4	−0.746138	−0.373069	+	0.927804i	$0.621695\pi$
$854$	−10852.5	−0.434852
$855$	0	0
$856$	8958.25	0.357695
$857$	−28616.2	−1.14062	−0.570310	−	0.821430i	$-0.693177\pi$
$857$	−28616.2	−1.14062	−0.570310	+	0.821430i	$0.693177\pi$
$858$	0	0
$859$	−31968.9	−1.26981	−0.634904	−	0.772591i	$-0.718960\pi$
$859$	−31968.9	−1.26981	−0.634904	+	0.772591i	$0.718960\pi$
$860$	16324.4	0.647276
$861$	0	0
$862$	39217.9	1.54961
$863$	−15417.8	−0.608145	−0.304072	−	0.952649i	$-0.598346\pi$
$863$	−15417.8	−0.608145	−0.304072	+	0.952649i	$0.598346\pi$
$864$	0	0
$865$	−29024.0	−1.14086
$866$	−20108.9	−0.789062
$867$	0	0
$868$	16971.8	0.663663
$869$	12908.5	0.503902
$870$	0	0
$871$	62246.7	2.42153
$872$	−22817.0	−0.886104
$873$	0	0
$874$	−5261.67	−0.203637
$875$	−7076.80	−0.273417
$876$	0	0
$877$	−41961.0	−1.61565	−0.807824	−	0.589424i	$-0.799355\pi$
$877$	−41961.0	−1.61565	−0.807824	+	0.589424i	$0.799355\pi$
$878$	32722.6	1.25778
$879$	0	0
$880$	−2202.67	−0.0843772
$881$	22884.6	0.875145	0.437573	−	0.899183i	$-0.355838\pi$
$881$	22884.6	0.875145	0.437573	+	0.899183i	$0.355838\pi$
$882$	0	0
$883$	29611.7	1.12855	0.564277	−	0.825585i	$-0.309155\pi$
$883$	29611.7	1.12855	0.564277	+	0.825585i	$0.309155\pi$
$884$	−11346.6	−0.431706
$885$	0	0
$886$	−57251.1	−2.17087
$887$	−27117.7	−1.02652	−0.513259	−	0.858234i	$-0.671562\pi$
$887$	−27117.7	−1.02652	−0.513259	+	0.858234i	$0.671562\pi$
$888$	0	0
$889$	3948.06	0.148947
$890$	−9986.06	−0.376105
$891$	0	0
$892$	1637.54	0.0614675
$893$	−4248.32	−0.159199
$894$	0	0
$895$	−9540.48	−0.356316
$896$	10476.8	0.390633
$897$	0	0
$898$	28677.8	1.06569
$899$	7845.36	0.291054
$900$	0	0
$901$	−3641.32	−0.134639
$902$	−17455.2	−0.644341
$903$	0	0
$904$	6001.00	0.220786
$905$	24740.7	0.908740
$906$	0	0
$907$	10010.9	0.366491	0.183245	−	0.983067i	$-0.441340\pi$
$907$	10010.9	0.366491	0.183245	+	0.983067i	$0.441340\pi$
$908$	−50173.9	−1.83379
$909$	0	0
$910$	−12737.9	−0.464019
$911$	−40394.1	−1.46906	−0.734532	−	0.678575i	$-0.762598\pi$
$911$	−40394.1	−1.46906	−0.734532	+	0.678575i	$0.762598\pi$
$912$	0	0
$913$	−16722.8	−0.606183
$914$	9630.44	0.348519
$915$	0	0
$916$	−16238.2	−0.585725
$917$	1324.28	0.0476899
$918$	0	0
$919$	2392.98	0.0858945	0.0429472	−	0.999077i	$-0.486325\pi$
$919$	2392.98	0.0858945	0.0429472	+	0.999077i	$0.486325\pi$
$920$	17898.1	0.641393
$921$	0	0
$922$	−8276.32	−0.295625
$923$	42502.9	1.51571
$924$	0	0
$925$	24581.9	0.873781
$926$	−8461.87	−0.300296
$927$	0	0
$928$	6223.42	0.220144
$929$	21920.3	0.774148	0.387074	−	0.922049i	$-0.373486\pi$
$929$	21920.3	0.774148	0.387074	+	0.922049i	$0.373486\pi$
$930$	0	0
$931$	2424.71	0.0853562
$932$	−26108.5	−0.917608
$933$	0	0
$934$	−16724.0	−0.585894
$935$	−1562.57	−0.0546540
$936$	0	0
$937$	4891.90	0.170556	0.0852782	−	0.996357i	$-0.472822\pi$
$937$	4891.90	0.170556	0.0852782	+	0.996357i	$0.472822\pi$
$938$	−17435.1	−0.606905
$939$	0	0
$940$	43338.5	1.50377
$941$	40152.3	1.39100	0.695499	−	0.718527i	$-0.255183\pi$
$941$	40152.3	1.39100	0.695499	+	0.718527i	$0.255183\pi$
$942$	0	0
$943$	−28291.9	−0.977001
$944$	1675.47	0.0577668
$945$	0	0
$946$	−19687.9	−0.676648
$947$	16057.5	0.551002	0.275501	−	0.961301i	$-0.411156\pi$
$947$	16057.5	0.551002	0.275501	+	0.961301i	$0.411156\pi$
$948$	0	0
$949$	−31.2963	−0.00107052
$950$	−2829.85	−0.0966448
$951$	0	0
$952$	1059.74	0.0360783
$953$	21912.3	0.744815	0.372408	−	0.928069i	$-0.378532\pi$
$953$	21912.3	0.744815	0.372408	+	0.928069i	$0.378532\pi$
$954$	0	0
$955$	31220.2	1.05787
$956$	54155.4	1.83212
$957$	0	0
$958$	−32470.7	−1.09507
$959$	12950.0	0.436055
$960$	0	0
$961$	43395.3	1.45666
$962$	111385.	3.73306
$963$	0	0
$964$	23410.3	0.782152
$965$	−23223.8	−0.774717
$966$	0	0
$967$	17618.1	0.585895	0.292948	−	0.956129i	$-0.405364\pi$
$967$	17618.1	0.585895	0.292948	+	0.956129i	$0.405364\pi$
$968$	15831.8	0.525674
$969$	0	0
$970$	−17561.9	−0.581319
$971$	−24152.9	−0.798252	−0.399126	−	0.916896i	$-0.630686\pi$
$971$	−24152.9	−0.798252	−0.399126	+	0.916896i	$0.630686\pi$
$972$	0	0
$973$	2211.64	0.0728694
$974$	−21272.4	−0.699805
$975$	0	0
$976$	−7412.56	−0.243105
$977$	43754.7	1.43279	0.716395	−	0.697695i	$-0.245791\pi$
$977$	43754.7	1.43279	0.716395	+	0.697695i	$0.245791\pi$
$978$	0	0
$979$	7226.66	0.235919
$980$	−24735.2	−0.806263
$981$	0	0
$982$	−8979.74	−0.291808
$983$	51651.4	1.67591	0.837957	−	0.545736i	$-0.183750\pi$
$983$	51651.4	1.67591	0.837957	+	0.545736i	$0.183750\pi$
$984$	0	0
$985$	14952.0	0.483665
$986$	1469.13	0.0474510
$987$	0	0
$988$	−7694.08	−0.247754
$989$	−31910.7	−1.02599
$990$	0	0
$991$	23894.0	0.765911	0.382956	−	0.923767i	$-0.374906\pi$
$991$	23894.0	0.765911	0.382956	+	0.923767i	$0.374906\pi$
$992$	58055.8	1.85814
$993$	0	0
$994$	−11904.9	−0.379881
$995$	−18201.6	−0.579929
$996$	0	0
$997$	5951.47	0.189052	0.0945261	−	0.995522i	$-0.469866\pi$
$997$	5951.47	0.189052	0.0945261	+	0.995522i	$0.469866\pi$
$998$	40039.1	1.26995
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.4.a.f.1.1		5
3.2	odd	2		29.4.a.b.1.5	✓	5
12.11	even	2		464.4.a.l.1.4		5
15.14	odd	2		725.4.a.c.1.1		5
21.20	even	2		1421.4.a.e.1.5		5
24.5	odd	2		1856.4.a.y.1.4		5
24.11	even	2		1856.4.a.bb.1.2		5
87.86	odd	2		841.4.a.b.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.b.1.5	✓	5	3.2	odd	2
261.4.a.f.1.1		5	1.1	even	1	trivial
464.4.a.l.1.4		5	12.11	even	2
725.4.a.c.1.1		5	15.14	odd	2
841.4.a.b.1.1		5	87.86	odd	2
1421.4.a.e.1.5		5	21.20	even	2
1856.4.a.y.1.4		5	24.5	odd	2
1856.4.a.bb.1.2		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)

\(f(q)\)	\(=\)	\(q-4.47236 q^{2} +12.0020 q^{4} +6.52855 q^{5} +5.22706 q^{7} -17.8986 q^{8} +O(q^{10})\)	\(q-4.47236 q^{2} +12.0020 q^{4} +6.52855 q^{5} +5.22706 q^{7} -17.8986 q^{8} -29.1981 q^{10} +21.1299 q^{11} +83.4615 q^{13} -23.3773 q^{14} -15.9674 q^{16} -11.3273 q^{17} -7.68096 q^{19} +78.3559 q^{20} -94.5005 q^{22} -153.169 q^{23} -82.3780 q^{25} -373.270 q^{26} +62.7354 q^{28} +29.0000 q^{29} +270.530 q^{31} +214.601 q^{32} +50.6597 q^{34} +34.1251 q^{35} -298.404 q^{37} +34.3520 q^{38} -116.852 q^{40} +184.710 q^{41} +208.337 q^{43} +253.602 q^{44} +685.028 q^{46} +553.098 q^{47} -315.678 q^{49} +368.424 q^{50} +1001.71 q^{52} +321.465 q^{53} +137.948 q^{55} -93.5569 q^{56} -129.699 q^{58} -104.930 q^{59} +464.230 q^{61} -1209.91 q^{62} -832.032 q^{64} +544.883 q^{65} +745.813 q^{67} -135.950 q^{68} -152.620 q^{70} +509.252 q^{71} -0.374979 q^{73} +1334.57 q^{74} -92.1871 q^{76} +110.447 q^{77} +610.912 q^{79} -104.244 q^{80} -826.092 q^{82} -791.431 q^{83} -73.9507 q^{85} -931.757 q^{86} -378.194 q^{88} +342.011 q^{89} +436.258 q^{91} -1838.34 q^{92} -2473.65 q^{94} -50.1455 q^{95} +601.476 q^{97} +1411.83 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 26 q^{4} - 10 q^{5} + 40 q^{7} + 84 q^{8}+O(q^{10}) \) 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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