LMFDB - Embedded newform 405.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.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:	$23.8957735523$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37228 q^{2} +3.37228 q^{4} -5.00000 q^{5} +1.62772 q^{7} -15.6060 q^{8} +O(q^{10})$	\(q+3.37228 q^{2} +3.37228 q^{4} -5.00000 q^{5} +1.62772 q^{7} -15.6060 q^{8} -16.8614 q^{10} +32.8614 q^{11} -33.0217 q^{13} +5.48913 q^{14} -79.6060 q^{16} -110.307 q^{17} -54.3070 q^{19} -16.8614 q^{20} +110.818 q^{22} +67.4375 q^{23} +25.0000 q^{25} -111.359 q^{26} +5.48913 q^{28} -274.519 q^{29} -6.00000 q^{31} -143.606 q^{32} -371.986 q^{34} -8.13859 q^{35} -347.723 q^{37} -183.139 q^{38} +78.0298 q^{40} -291.337 q^{41} +201.388 q^{43} +110.818 q^{44} +227.418 q^{46} +482.095 q^{47} -340.351 q^{49} +84.3070 q^{50} -111.359 q^{52} +175.228 q^{53} -164.307 q^{55} -25.4021 q^{56} -925.755 q^{58} +183.242 q^{59} +436.595 q^{61} -20.2337 q^{62} +152.568 q^{64} +165.109 q^{65} -831.500 q^{67} -371.986 q^{68} -27.4456 q^{70} +118.951 q^{71} +183.318 q^{73} -1172.62 q^{74} -183.139 q^{76} +53.4891 q^{77} -638.293 q^{79} +398.030 q^{80} -982.470 q^{82} +1494.64 q^{83} +551.535 q^{85} +679.139 q^{86} -512.834 q^{88} +1437.27 q^{89} -53.7501 q^{91} +227.418 q^{92} +1625.76 q^{94} +271.535 q^{95} -891.769 q^{97} -1147.76 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + q^{4} - 10 q^{5} + 9 q^{7} + 9 q^{8}+O(q^{10}) $ 2 * q + q^2 + q^4 - 10 * q^5 + 9 * q^7 + 9 * q^8	$ 2 q + q^{2} + q^{4} - 10 q^{5} + 9 q^{7} + 9 q^{8} - 5 q^{10} + 37 q^{11} - 112 q^{13} - 12 q^{14} - 119 q^{16} - 77 q^{17} + 35 q^{19} - 5 q^{20} + 101 q^{22} + 267 q^{23} + 50 q^{25} + 76 q^{26} - 12 q^{28} - 325 q^{29} - 12 q^{31} - 247 q^{32} - 451 q^{34} - 45 q^{35} - 638 q^{37} - 395 q^{38} - 45 q^{40} - 238 q^{41} - 97 q^{43} + 101 q^{44} - 246 q^{46} + 901 q^{47} - 629 q^{49} + 25 q^{50} + 76 q^{52} - 224 q^{53} - 185 q^{55} + 156 q^{56} - 806 q^{58} + 85 q^{59} - 247 q^{61} - 6 q^{62} + 713 q^{64} + 560 q^{65} - 606 q^{67} - 451 q^{68} + 60 q^{70} - 394 q^{71} - 811 q^{73} - 484 q^{74} - 395 q^{76} + 84 q^{77} - 840 q^{79} + 595 q^{80} - 1109 q^{82} + 387 q^{83} + 385 q^{85} + 1387 q^{86} - 411 q^{88} + 1065 q^{89} - 636 q^{91} - 246 q^{92} + 632 q^{94} - 175 q^{95} - 1031 q^{97} - 463 q^{98}+O(q^{100}) $ 2 * q + q^2 + q^4 - 10 * q^5 + 9 * q^7 + 9 * q^8 - 5 * q^10 + 37 * q^11 - 112 * q^13 - 12 * q^14 - 119 * q^16 - 77 * q^17 + 35 * q^19 - 5 * q^20 + 101 * q^22 + 267 * q^23 + 50 * q^25 + 76 * q^26 - 12 * q^28 - 325 * q^29 - 12 * q^31 - 247 * q^32 - 451 * q^34 - 45 * q^35 - 638 * q^37 - 395 * q^38 - 45 * q^40 - 238 * q^41 - 97 * q^43 + 101 * q^44 - 246 * q^46 + 901 * q^47 - 629 * q^49 + 25 * q^50 + 76 * q^52 - 224 * q^53 - 185 * q^55 + 156 * q^56 - 806 * q^58 + 85 * q^59 - 247 * q^61 - 6 * q^62 + 713 * q^64 + 560 * q^65 - 606 * q^67 - 451 * q^68 + 60 * q^70 - 394 * q^71 - 811 * q^73 - 484 * q^74 - 395 * q^76 + 84 * q^77 - 840 * q^79 + 595 * q^80 - 1109 * q^82 + 387 * q^83 + 385 * q^85 + 1387 * q^86 - 411 * q^88 + 1065 * q^89 - 636 * q^91 - 246 * q^92 + 632 * q^94 - 175 * q^95 - 1031 * q^97 - 463 * 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 $

$2$	3.37228	1.19228	0.596141	−	0.802880i	$-0.296700\pi$
$2$	3.37228	1.19228	0.596141	+	0.802880i	$0.296700\pi$
$3$	0	0
$3$	0	0
$4$	3.37228	0.421535
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	1.62772	0.0878885	0.0439443	−	0.999034i	$-0.486008\pi$
$7$	1.62772	0.0878885	0.0439443	+	0.999034i	$0.486008\pi$
$8$	−15.6060	−0.689693
$9$	0	0
$10$	−16.8614	−0.533204
$11$	32.8614	0.900735	0.450368	−	0.892843i	$-0.351293\pi$
$11$	32.8614	0.900735	0.450368	+	0.892843i	$0.351293\pi$
$12$	0	0
$13$	−33.0217	−0.704507	−0.352253	−	0.935905i	$-0.614584\pi$
$13$	−33.0217	−0.704507	−0.352253	+	0.935905i	$0.614584\pi$
$14$	5.48913	0.104788
$15$	0	0
$16$	−79.6060	−1.24384
$17$	−110.307	−1.57373	−0.786864	−	0.617126i	$-0.788297\pi$
$17$	−110.307	−1.57373	−0.786864	+	0.617126i	$0.788297\pi$
$18$	0	0
$19$	−54.3070	−0.655731	−0.327865	−	0.944724i	$-0.606329\pi$
$19$	−54.3070	−0.655731	−0.327865	+	0.944724i	$0.606329\pi$
$20$	−16.8614	−0.188516
$21$	0	0
$22$	110.818	1.07393
$23$	67.4375	0.611378	0.305689	−	0.952131i	$-0.401113\pi$
$23$	67.4375	0.611378	0.305689	+	0.952131i	$0.401113\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−111.359	−0.839970
$27$	0	0
$28$	5.48913	0.0370481
$29$	−274.519	−1.75782	−0.878912	−	0.476984i	$-0.841730\pi$
$29$	−274.519	−1.75782	−0.878912	+	0.476984i	$0.841730\pi$
$30$	0	0
$31$	−6.00000	−0.0347623	−0.0173812	−	0.999849i	$-0.505533\pi$
$31$	−6.00000	−0.0347623	−0.0173812	+	0.999849i	$0.505533\pi$
$32$	−143.606	−0.793318
$33$	0	0
$34$	−371.986	−1.87633
$35$	−8.13859	−0.0393050
$36$	0	0
$37$	−347.723	−1.54501	−0.772504	−	0.635010i	$-0.780996\pi$
$37$	−347.723	−1.54501	−0.772504	+	0.635010i	$0.780996\pi$
$38$	−183.139	−0.781816
$39$	0	0
$40$	78.0298	0.308440
$41$	−291.337	−1.10974	−0.554868	−	0.831938i	$-0.687231\pi$
$41$	−291.337	−1.10974	−0.554868	+	0.831938i	$0.687231\pi$
$42$	0	0
$43$	201.388	0.714220	0.357110	−	0.934062i	$-0.383762\pi$
$43$	201.388	0.714220	0.357110	+	0.934062i	$0.383762\pi$
$44$	110.818	0.379692
$45$	0	0
$46$	227.418	0.728935
$47$	482.095	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$47$	482.095	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$48$	0	0
$49$	−340.351	−0.992276
$50$	84.3070	0.238456
$51$	0	0
$52$	−111.359	−0.296974
$53$	175.228	0.454140	0.227070	−	0.973878i	$-0.427085\pi$
$53$	175.228	0.454140	0.227070	+	0.973878i	$0.427085\pi$
$54$	0	0
$55$	−164.307	−0.402821
$56$	−25.4021	−0.0606161
$57$	0	0
$58$	−925.755	−2.09582
$59$	183.242	0.404340	0.202170	−	0.979350i	$-0.435201\pi$
$59$	183.242	0.404340	0.202170	+	0.979350i	$0.435201\pi$
$60$	0	0
$61$	436.595	0.916397	0.458199	−	0.888850i	$-0.348495\pi$
$61$	436.595	0.916397	0.458199	+	0.888850i	$0.348495\pi$
$62$	−20.2337	−0.0414465
$63$	0	0
$64$	152.568	0.297984
$65$	165.109	0.315065
$66$	0	0
$67$	−831.500	−1.51618	−0.758089	−	0.652152i	$-0.773867\pi$
$67$	−831.500	−1.51618	−0.758089	+	0.652152i	$0.773867\pi$
$68$	−371.986	−0.663382
$69$	0	0
$70$	−27.4456	−0.0468626
$71$	118.951	0.198829	0.0994146	−	0.995046i	$-0.468303\pi$
$71$	118.951	0.198829	0.0994146	+	0.995046i	$0.468303\pi$
$72$	0	0
$73$	183.318	0.293914	0.146957	−	0.989143i	$-0.453052\pi$
$73$	183.318	0.293914	0.146957	+	0.989143i	$0.453052\pi$
$74$	−1172.62	−1.84208
$75$	0	0
$76$	−183.139	−0.276414
$77$	53.4891	0.0791643
$78$	0	0
$79$	−638.293	−0.909033	−0.454517	−	0.890738i	$-0.650188\pi$
$79$	−638.293	−0.909033	−0.454517	+	0.890738i	$0.650188\pi$
$80$	398.030	0.556264
$81$	0	0
$82$	−982.470	−1.32312
$83$	1494.64	1.97661	0.988304	−	0.152498i	$-0.0487317\pi$
$83$	1494.64	1.97661	0.988304	+	0.152498i	$0.0487317\pi$
$84$	0	0
$85$	551.535	0.703793
$86$	679.139	0.851551
$87$	0	0
$88$	−512.834	−0.621231
$89$	1437.27	1.71180	0.855900	−	0.517142i	$-0.173004\pi$
$89$	1437.27	1.71180	0.855900	+	0.517142i	$0.173004\pi$
$90$	0	0
$91$	−53.7501	−0.0619181
$92$	227.418	0.257717
$93$	0	0
$94$	1625.76	1.78388
$95$	271.535	0.293252
$96$	0	0
$97$	−891.769	−0.933458	−0.466729	−	0.884400i	$-0.654568\pi$
$97$	−891.769	−0.933458	−0.466729	+	0.884400i	$0.654568\pi$
$98$	−1147.76	−1.18307
$99$	0	0
$100$	84.3070	0.0843070
$101$	153.744	0.151466	0.0757332	−	0.997128i	$-0.475870\pi$
$101$	153.744	0.151466	0.0757332	+	0.997128i	$0.475870\pi$
$102$	0	0
$103$	135.294	0.129426	0.0647131	−	0.997904i	$-0.479387\pi$
$103$	135.294	0.129426	0.0647131	+	0.997904i	$0.479387\pi$
$104$	515.336	0.485893
$105$	0	0
$106$	590.919	0.541463
$107$	718.783	0.649414	0.324707	−	0.945815i	$-0.394734\pi$
$107$	718.783	0.649414	0.324707	+	0.945815i	$0.394734\pi$
$108$	0	0
$109$	−2010.56	−1.76676	−0.883378	−	0.468661i	$-0.844736\pi$
$109$	−2010.56	−1.76676	−0.883378	+	0.468661i	$0.844736\pi$
$110$	−554.090	−0.480276
$111$	0	0
$112$	−129.576	−0.109320
$113$	226.745	0.188764	0.0943820	−	0.995536i	$-0.469912\pi$
$113$	226.745	0.188764	0.0943820	+	0.995536i	$0.469912\pi$
$114$	0	0
$115$	−337.188	−0.273417
$116$	−925.755	−0.740985
$117$	0	0
$118$	617.943	0.482087
$119$	−179.549	−0.138313
$120$	0	0
$121$	−251.128	−0.188676
$122$	1472.32	1.09260
$123$	0	0
$124$	−20.2337	−0.0146535
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1132.24	0.791100	0.395550	−	0.918444i	$-0.370554\pi$
$127$	1132.24	0.791100	0.395550	+	0.918444i	$0.370554\pi$
$128$	1663.35	1.14860
$129$	0	0
$130$	556.793	0.375646
$131$	−777.505	−0.518557	−0.259278	−	0.965803i	$-0.583485\pi$
$131$	−777.505	−0.518557	−0.259278	+	0.965803i	$0.583485\pi$
$132$	0	0
$133$	−88.3966	−0.0576312
$134$	−2804.05	−1.80771
$135$	0	0
$136$	1721.45	1.08539
$137$	−623.638	−0.388913	−0.194456	−	0.980911i	$-0.562294\pi$
$137$	−623.638	−0.388913	−0.194456	+	0.980911i	$0.562294\pi$
$138$	0	0
$139$	1282.68	0.782702	0.391351	−	0.920241i	$-0.372008\pi$
$139$	1282.68	0.782702	0.391351	+	0.920241i	$0.372008\pi$
$140$	−27.4456	−0.0165684
$141$	0	0
$142$	401.136	0.237060
$143$	−1085.14	−0.634574
$144$	0	0
$145$	1372.59	0.786123
$146$	618.199	0.350428
$147$	0	0
$148$	−1172.62	−0.651275
$149$	−1524.31	−0.838098	−0.419049	−	0.907964i	$-0.637636\pi$
$149$	−1524.31	−0.838098	−0.419049	+	0.907964i	$0.637636\pi$
$150$	0	0
$151$	3162.56	1.70441	0.852203	−	0.523211i	$-0.175266\pi$
$151$	3162.56	1.70441	0.852203	+	0.523211i	$0.175266\pi$
$152$	847.514	0.452253
$153$	0	0
$154$	180.380	0.0943861
$155$	30.0000	0.0155462
$156$	0	0
$157$	−2388.70	−1.21426	−0.607131	−	0.794602i	$-0.707680\pi$
$157$	−2388.70	−1.21426	−0.607131	+	0.794602i	$0.707680\pi$
$158$	−2152.50	−1.08382
$159$	0	0
$160$	718.030	0.354783
$161$	109.769	0.0537331
$162$	0	0
$163$	−2544.79	−1.22284	−0.611422	−	0.791305i	$-0.709402\pi$
$163$	−2544.79	−1.22284	−0.611422	+	0.791305i	$0.709402\pi$
$164$	−982.470	−0.467793
$165$	0	0
$166$	5040.36	2.35667
$167$	1374.56	0.636925	0.318462	−	0.947935i	$-0.396834\pi$
$167$	1374.56	0.636925	0.318462	+	0.947935i	$0.396834\pi$
$168$	0	0
$169$	−1106.56	−0.503670
$170$	1859.93	0.839119
$171$	0	0
$172$	679.139	0.301069
$173$	−2360.45	−1.03735	−0.518675	−	0.854971i	$-0.673575\pi$
$173$	−2360.45	−1.03735	−0.518675	+	0.854971i	$0.673575\pi$
$174$	0	0
$175$	40.6930	0.0175777
$176$	−2615.96	−1.12037
$177$	0	0
$178$	4846.87	2.04095
$179$	−1305.11	−0.544963	−0.272482	−	0.962161i	$-0.587844\pi$
$179$	−1305.11	−0.544963	−0.272482	+	0.962161i	$0.587844\pi$
$180$	0	0
$181$	3099.43	1.27281	0.636406	−	0.771355i	$-0.280420\pi$
$181$	3099.43	1.27281	0.636406	+	0.771355i	$0.280420\pi$
$182$	−181.261	−0.0738238
$183$	0	0
$184$	−1052.43	−0.421663
$185$	1738.61	0.690948
$186$	0	0
$187$	−3624.84	−1.41751
$188$	1625.76	0.630696
$189$	0	0
$190$	915.693	0.349639
$191$	−380.712	−0.144227	−0.0721135	−	0.997396i	$-0.522974\pi$
$191$	−380.712	−0.144227	−0.0721135	+	0.997396i	$0.522974\pi$
$192$	0	0
$193$	1546.04	0.576613	0.288307	−	0.957538i	$-0.406908\pi$
$193$	1546.04	0.576613	0.288307	+	0.957538i	$0.406908\pi$
$194$	−3007.30	−1.11294
$195$	0	0
$196$	−1147.76	−0.418279
$197$	4284.60	1.54957	0.774784	−	0.632226i	$-0.217859\pi$
$197$	4284.60	1.54957	0.774784	+	0.632226i	$0.217859\pi$
$198$	0	0
$199$	−1402.85	−0.499727	−0.249863	−	0.968281i	$-0.580386\pi$
$199$	−1402.85	−0.499727	−0.249863	+	0.968281i	$0.580386\pi$
$200$	−390.149	−0.137939
$201$	0	0
$202$	518.468	0.180591
$203$	−446.840	−0.154493
$204$	0	0
$205$	1456.68	0.496289
$206$	456.249	0.154312
$207$	0	0
$208$	2628.73	0.876296
$209$	−1784.61	−0.590640
$210$	0	0
$211$	−2150.46	−0.701628	−0.350814	−	0.936445i	$-0.614095\pi$
$211$	−2150.46	−0.701628	−0.350814	+	0.936445i	$0.614095\pi$
$212$	590.919	0.191436
$213$	0	0
$214$	2423.94	0.774285
$215$	−1006.94	−0.319409
$216$	0	0
$217$	−9.76631	−0.00305521
$218$	−6780.16	−2.10647
$219$	0	0
$220$	−554.090	−0.169803
$221$	3642.53	1.10870
$222$	0	0
$223$	−2558.15	−0.768190	−0.384095	−	0.923294i	$-0.625486\pi$
$223$	−2558.15	−0.768190	−0.384095	+	0.923294i	$0.625486\pi$
$224$	−233.750	−0.0697236
$225$	0	0
$226$	764.646	0.225060
$227$	4084.54	1.19428	0.597138	−	0.802139i	$-0.296305\pi$
$227$	4084.54	1.19428	0.597138	+	0.802139i	$0.296305\pi$
$228$	0	0
$229$	−2741.91	−0.791225	−0.395612	−	0.918418i	$-0.629468\pi$
$229$	−2741.91	−0.791225	−0.395612	+	0.918418i	$0.629468\pi$
$230$	−1137.09	−0.325990
$231$	0	0
$232$	4284.13	1.21236
$233$	−5084.70	−1.42966	−0.714828	−	0.699300i	$-0.753495\pi$
$233$	−5084.70	−1.42966	−0.714828	+	0.699300i	$0.753495\pi$
$234$	0	0
$235$	−2410.48	−0.669115
$236$	617.943	0.170443
$237$	0	0
$238$	−605.489	−0.164908
$239$	−1476.47	−0.399603	−0.199801	−	0.979836i	$-0.564030\pi$
$239$	−1476.47	−0.399603	−0.199801	+	0.979836i	$0.564030\pi$
$240$	0	0
$241$	1763.46	0.471345	0.235673	−	0.971833i	$-0.424271\pi$
$241$	1763.46	0.471345	0.235673	+	0.971833i	$0.424271\pi$
$242$	−846.874	−0.224955
$243$	0	0
$244$	1472.32	0.386294
$245$	1701.75	0.443759
$246$	0	0
$247$	1793.31	0.461967
$248$	93.6358	0.0239753
$249$	0	0
$250$	−421.535	−0.106641
$251$	−1705.16	−0.428801	−0.214400	−	0.976746i	$-0.568780\pi$
$251$	−1705.16	−0.428801	−0.214400	+	0.976746i	$0.568780\pi$
$252$	0	0
$253$	2216.09	0.550690
$254$	3818.22	0.943214
$255$	0	0
$256$	4388.74	1.07147
$257$	−228.596	−0.0554842	−0.0277421	−	0.999615i	$-0.508832\pi$
$257$	−228.596	−0.0554842	−0.0277421	+	0.999615i	$0.508832\pi$
$258$	0	0
$259$	−565.995	−0.135788
$260$	556.793	0.132811
$261$	0	0
$262$	−2621.97	−0.618266
$263$	1288.76	0.302161	0.151081	−	0.988521i	$-0.451725\pi$
$263$	1288.76	0.302161	0.151081	+	0.988521i	$0.451725\pi$
$264$	0	0
$265$	−876.141	−0.203098
$266$	−298.098	−0.0687127
$267$	0	0
$268$	−2804.05	−0.639122
$269$	−973.981	−0.220761	−0.110380	−	0.993889i	$-0.535207\pi$
$269$	−973.981	−0.220761	−0.110380	+	0.993889i	$0.535207\pi$
$270$	0	0
$271$	−4021.83	−0.901508	−0.450754	−	0.892648i	$-0.648845\pi$
$271$	−4021.83	−0.901508	−0.450754	+	0.892648i	$0.648845\pi$
$272$	8781.10	1.95747
$273$	0	0
$274$	−2103.08	−0.463693
$275$	821.535	0.180147
$276$	0	0
$277$	−3379.17	−0.732977	−0.366489	−	0.930422i	$-0.619440\pi$
$277$	−3379.17	−0.732977	−0.366489	+	0.930422i	$0.619440\pi$
$278$	4325.56	0.933201
$279$	0	0
$280$	127.011	0.0271083
$281$	−733.307	−0.155678	−0.0778388	−	0.996966i	$-0.524802\pi$
$281$	−733.307	−0.155678	−0.0778388	+	0.996966i	$0.524802\pi$
$282$	0	0
$283$	6909.03	1.45123	0.725617	−	0.688099i	$-0.241555\pi$
$283$	6909.03	1.45123	0.725617	+	0.688099i	$0.241555\pi$
$284$	401.136	0.0838135
$285$	0	0
$286$	−3659.40	−0.756591
$287$	−474.214	−0.0975331
$288$	0	0
$289$	7254.64	1.47662
$290$	4628.78	0.937280
$291$	0	0
$292$	618.199	0.123895
$293$	4649.49	0.927051	0.463525	−	0.886084i	$-0.346584\pi$
$293$	4649.49	0.927051	0.463525	+	0.886084i	$0.346584\pi$
$294$	0	0
$295$	−916.209	−0.180826
$296$	5426.55	1.06558
$297$	0	0
$298$	−5140.41	−0.999248
$299$	−2226.91	−0.430720
$300$	0	0
$301$	327.804	0.0627717
$302$	10665.0	2.03213
$303$	0	0
$304$	4323.16	0.815627
$305$	−2182.97	−0.409825
$306$	0	0
$307$	−7361.42	−1.36853	−0.684264	−	0.729234i	$-0.739877\pi$
$307$	−7361.42	−1.36853	−0.684264	+	0.729234i	$0.739877\pi$
$308$	180.380	0.0333705
$309$	0	0
$310$	101.168	0.0185354
$311$	2709.21	0.493971	0.246986	−	0.969019i	$-0.420560\pi$
$311$	2709.21	0.493971	0.246986	+	0.969019i	$0.420560\pi$
$312$	0	0
$313$	−6243.72	−1.12753	−0.563763	−	0.825936i	$-0.690647\pi$
$313$	−6243.72	−1.12753	−0.563763	+	0.825936i	$0.690647\pi$
$314$	−8055.37	−1.44774
$315$	0	0
$316$	−2152.50	−0.383189
$317$	−2084.03	−0.369245	−0.184623	−	0.982809i	$-0.559106\pi$
$317$	−2084.03	−0.369245	−0.184623	+	0.982809i	$0.559106\pi$
$318$	0	0
$319$	−9021.08	−1.58333
$320$	−762.840	−0.133263
$321$	0	0
$322$	370.173	0.0640650
$323$	5990.45	1.03194
$324$	0	0
$325$	−825.544	−0.140901
$326$	−8581.76	−1.45797
$327$	0	0
$328$	4546.59	0.765377
$329$	784.715	0.131498
$330$	0	0
$331$	−227.876	−0.0378406	−0.0189203	−	0.999821i	$-0.506023\pi$
$331$	−227.876	−0.0378406	−0.0189203	+	0.999821i	$0.506023\pi$
$332$	5040.36	0.833210
$333$	0	0
$334$	4635.39	0.759394
$335$	4157.50	0.678055
$336$	0	0
$337$	3417.48	0.552409	0.276205	−	0.961099i	$-0.410923\pi$
$337$	3417.48	0.552409	0.276205	+	0.961099i	$0.410923\pi$
$338$	−3731.65	−0.600517
$339$	0	0
$340$	1859.93	0.296673
$341$	−197.168	−0.0313116
$342$	0	0
$343$	−1112.30	−0.175098
$344$	−3142.86	−0.492592
$345$	0	0
$346$	−7960.10	−1.23681
$347$	−3609.60	−0.558425	−0.279212	−	0.960229i	$-0.590073\pi$
$347$	−3609.60	−0.558425	−0.279212	+	0.960229i	$0.590073\pi$
$348$	0	0
$349$	−8800.83	−1.34985	−0.674925	−	0.737886i	$-0.735824\pi$
$349$	−8800.83	−1.34985	−0.674925	+	0.737886i	$0.735824\pi$
$350$	137.228	0.0209576
$351$	0	0
$352$	−4719.09	−0.714570
$353$	−8510.49	−1.28319	−0.641597	−	0.767042i	$-0.721728\pi$
$353$	−8510.49	−1.28319	−0.641597	+	0.767042i	$0.721728\pi$
$354$	0	0
$355$	−594.755	−0.0889192
$356$	4846.87	0.721584
$357$	0	0
$358$	−4401.19	−0.649750
$359$	−4320.35	−0.635152	−0.317576	−	0.948233i	$-0.602869\pi$
$359$	−4320.35	−0.635152	−0.317576	+	0.948233i	$0.602869\pi$
$360$	0	0
$361$	−3909.75	−0.570017
$362$	10452.2	1.51755
$363$	0	0
$364$	−181.261	−0.0261006
$365$	−916.588	−0.131442
$366$	0	0
$367$	6600.39	0.938794	0.469397	−	0.882987i	$-0.344471\pi$
$367$	6600.39	0.938794	0.469397	+	0.882987i	$0.344471\pi$
$368$	−5368.43	−0.760459
$369$	0	0
$370$	5863.10	0.823805
$371$	285.222	0.0399137
$372$	0	0
$373$	−8608.45	−1.19498	−0.597492	−	0.801875i	$-0.703836\pi$
$373$	−8608.45	−1.19498	−0.597492	+	0.801875i	$0.703836\pi$
$374$	−12224.0	−1.69007
$375$	0	0
$376$	−7523.56	−1.03191
$377$	9065.10	1.23840
$378$	0	0
$379$	−7129.80	−0.966314	−0.483157	−	0.875534i	$-0.660510\pi$
$379$	−7129.80	−0.966314	−0.483157	+	0.875534i	$0.660510\pi$
$380$	915.693	0.123616
$381$	0	0
$382$	−1283.87	−0.171959
$383$	−5927.46	−0.790807	−0.395404	−	0.918507i	$-0.629395\pi$
$383$	−5927.46	−0.790807	−0.395404	+	0.918507i	$0.629395\pi$
$384$	0	0
$385$	−267.446	−0.0354034
$386$	5213.68	0.687486
$387$	0	0
$388$	−3007.30	−0.393485
$389$	−1038.86	−0.135405	−0.0677024	−	0.997706i	$-0.521567\pi$
$389$	−1038.86	−0.135405	−0.0677024	+	0.997706i	$0.521567\pi$
$390$	0	0
$391$	−7438.83	−0.962143
$392$	5311.50	0.684365
$393$	0	0
$394$	14448.9	1.84752
$395$	3191.47	0.406532
$396$	0	0
$397$	13441.4	1.69926	0.849628	−	0.527382i	$-0.176826\pi$
$397$	13441.4	1.69926	0.849628	+	0.527382i	$0.176826\pi$
$398$	−4730.81	−0.595815
$399$	0	0
$400$	−1990.15	−0.248769
$401$	−13075.4	−1.62831	−0.814157	−	0.580644i	$-0.802801\pi$
$401$	−13075.4	−1.62831	−0.814157	+	0.580644i	$0.802801\pi$
$402$	0	0
$403$	198.130	0.0244903
$404$	518.468	0.0638484
$405$	0	0
$406$	−1506.87	−0.184199
$407$	−11426.7	−1.39164
$408$	0	0
$409$	−5273.19	−0.637512	−0.318756	−	0.947837i	$-0.603265\pi$
$409$	−5273.19	−0.637512	−0.318756	+	0.947837i	$0.603265\pi$
$410$	4912.35	0.591716
$411$	0	0
$412$	456.249	0.0545577
$413$	298.266	0.0355368
$414$	0	0
$415$	−7473.22	−0.883966
$416$	4742.12	0.558898
$417$	0	0
$418$	−6018.19	−0.704209
$419$	−13846.1	−1.61438	−0.807192	−	0.590289i	$-0.799014\pi$
$419$	−13846.1	−1.61438	−0.807192	+	0.590289i	$0.799014\pi$
$420$	0	0
$421$	1097.37	0.127037	0.0635184	−	0.997981i	$-0.479768\pi$
$421$	1097.37	0.127037	0.0635184	+	0.997981i	$0.479768\pi$
$422$	−7251.94	−0.836538
$423$	0	0
$424$	−2734.60	−0.313217
$425$	−2757.68	−0.314746
$426$	0	0
$427$	710.654	0.0805408
$428$	2423.94	0.273751
$429$	0	0
$430$	−3395.69	−0.380825
$431$	−15912.8	−1.77841	−0.889205	−	0.457509i	$-0.848742\pi$
$431$	−15912.8	−1.77841	−0.889205	+	0.457509i	$0.848742\pi$
$432$	0	0
$433$	3566.31	0.395810	0.197905	−	0.980221i	$-0.436586\pi$
$433$	3566.31	0.395810	0.197905	+	0.980221i	$0.436586\pi$
$434$	−32.9348	−0.00364267
$435$	0	0
$436$	−6780.16	−0.744750
$437$	−3662.33	−0.400900
$438$	0	0
$439$	−580.822	−0.0631461	−0.0315730	−	0.999501i	$-0.510052\pi$
$439$	−580.822	−0.0631461	−0.0315730	+	0.999501i	$0.510052\pi$
$440$	2564.17	0.277823
$441$	0	0
$442$	12283.6	1.32188
$443$	10792.8	1.15752	0.578759	−	0.815499i	$-0.303537\pi$
$443$	10792.8	1.15752	0.578759	+	0.815499i	$0.303537\pi$
$444$	0	0
$445$	−7186.34	−0.765540
$446$	−8626.80	−0.915898
$447$	0	0
$448$	248.338	0.0261894
$449$	−2894.01	−0.304180	−0.152090	−	0.988367i	$-0.548600\pi$
$449$	−2894.01	−0.304180	−0.152090	+	0.988367i	$0.548600\pi$
$450$	0	0
$451$	−9573.74	−0.999578
$452$	764.646	0.0795707
$453$	0	0
$454$	13774.2	1.42391
$455$	268.751	0.0276906
$456$	0	0
$457$	3199.92	0.327540	0.163770	−	0.986499i	$-0.447635\pi$
$457$	3199.92	0.327540	0.163770	+	0.986499i	$0.447635\pi$
$458$	−9246.49	−0.943363
$459$	0	0
$460$	−1137.09	−0.115255
$461$	−7316.38	−0.739171	−0.369585	−	0.929197i	$-0.620500\pi$
$461$	−7316.38	−0.739171	−0.369585	+	0.929197i	$0.620500\pi$
$462$	0	0
$463$	7011.98	0.703832	0.351916	−	0.936032i	$-0.385530\pi$
$463$	7011.98	0.703832	0.351916	+	0.936032i	$0.385530\pi$
$464$	21853.3	2.18646
$465$	0	0
$466$	−17147.1	−1.70455
$467$	8002.63	0.792971	0.396485	−	0.918041i	$-0.370230\pi$
$467$	8002.63	0.792971	0.396485	+	0.918041i	$0.370230\pi$
$468$	0	0
$469$	−1353.45	−0.133255
$470$	−8128.80	−0.797774
$471$	0	0
$472$	−2859.67	−0.278870
$473$	6617.91	0.643323
$474$	0	0
$475$	−1357.68	−0.131146
$476$	−605.489	−0.0583037
$477$	0	0
$478$	−4979.08	−0.476439
$479$	3178.84	0.303225	0.151613	−	0.988440i	$-0.451553\pi$
$479$	3178.84	0.303225	0.151613	+	0.988440i	$0.451553\pi$
$480$	0	0
$481$	11482.4	1.08847
$482$	5946.87	0.561976
$483$	0	0
$484$	−846.874	−0.0795336
$485$	4458.84	0.417455
$486$	0	0
$487$	13060.3	1.21523	0.607615	−	0.794232i	$-0.292126\pi$
$487$	13060.3	1.21523	0.607615	+	0.794232i	$0.292126\pi$
$488$	−6813.49	−0.632033
$489$	0	0
$490$	5738.79	0.529086
$491$	4401.61	0.404566	0.202283	−	0.979327i	$-0.435164\pi$
$491$	4401.61	0.404566	0.202283	+	0.979327i	$0.435164\pi$
$492$	0	0
$493$	30281.4	2.76634
$494$	6047.56	0.550794
$495$	0	0
$496$	477.636	0.0432389
$497$	193.619	0.0174748
$498$	0	0
$499$	18725.2	1.67987	0.839935	−	0.542687i	$-0.182593\pi$
$499$	18725.2	1.67987	0.839935	+	0.542687i	$0.182593\pi$
$500$	−421.535	−0.0377033
$501$	0	0
$502$	−5750.29	−0.511251
$503$	3811.68	0.337882	0.168941	−	0.985626i	$-0.445965\pi$
$503$	3811.68	0.337882	0.168941	+	0.985626i	$0.445965\pi$
$504$	0	0
$505$	−768.720	−0.0677378
$506$	7473.29	0.656577
$507$	0	0
$508$	3818.22	0.333477
$509$	−4495.48	−0.391471	−0.195735	−	0.980657i	$-0.562709\pi$
$509$	−4495.48	−0.391471	−0.195735	+	0.980657i	$0.562709\pi$
$510$	0	0
$511$	298.390	0.0258317
$512$	1493.27	0.128894
$513$	0	0
$514$	−770.891	−0.0661528
$515$	−676.469	−0.0578812
$516$	0	0
$517$	15842.3	1.34767
$518$	−1908.69	−0.161898
$519$	0	0
$520$	−2576.68	−0.217298
$521$	−12095.0	−1.01706	−0.508531	−	0.861043i	$-0.669811\pi$
$521$	−12095.0	−1.01706	−0.508531	+	0.861043i	$0.669811\pi$
$522$	0	0
$523$	7385.38	0.617476	0.308738	−	0.951147i	$-0.400093\pi$
$523$	7385.38	0.617476	0.308738	+	0.951147i	$0.400093\pi$
$524$	−2621.97	−0.218590
$525$	0	0
$526$	4346.06	0.360261
$527$	661.842	0.0547064
$528$	0	0
$529$	−7619.18	−0.626217
$530$	−2954.59	−0.242150
$531$	0	0
$532$	−298.098	−0.0242936
$533$	9620.45	0.781816
$534$	0	0
$535$	−3593.91	−0.290427
$536$	12976.4	1.04570
$537$	0	0
$538$	−3284.54	−0.263209
$539$	−11184.4	−0.893778
$540$	0	0
$541$	−5935.19	−0.471670	−0.235835	−	0.971793i	$-0.575783\pi$
$541$	−5935.19	−0.471670	−0.235835	+	0.971793i	$0.575783\pi$
$542$	−13562.7	−1.07485
$543$	0	0
$544$	15840.7	1.24847
$545$	10052.8	0.790117
$546$	0	0
$547$	−10157.2	−0.793951	−0.396976	−	0.917829i	$-0.629940\pi$
$547$	−10157.2	−0.793951	−0.396976	+	0.917829i	$0.629940\pi$
$548$	−2103.08	−0.163940
$549$	0	0
$550$	2770.45	0.214786
$551$	14908.3	1.15266
$552$	0	0
$553$	−1038.96	−0.0798936
$554$	−11395.5	−0.873915
$555$	0	0
$556$	4325.56	0.329937
$557$	−5709.62	−0.434334	−0.217167	−	0.976134i	$-0.569682\pi$
$557$	−5709.62	−0.434334	−0.217167	+	0.976134i	$0.569682\pi$
$558$	0	0
$559$	−6650.20	−0.503173
$560$	647.881	0.0488892
$561$	0	0
$562$	−2472.92	−0.185612
$563$	−12939.1	−0.968595	−0.484297	−	0.874903i	$-0.660925\pi$
$563$	−12939.1	−0.968595	−0.484297	+	0.874903i	$0.660925\pi$
$564$	0	0
$565$	−1133.72	−0.0844178
$566$	23299.2	1.73028
$567$	0	0
$568$	−1856.34	−0.137131
$569$	6125.42	0.451302	0.225651	−	0.974208i	$-0.427549\pi$
$569$	6125.42	0.451302	0.225651	+	0.974208i	$0.427549\pi$
$570$	0	0
$571$	−19282.4	−1.41321	−0.706605	−	0.707608i	$-0.749774\pi$
$571$	−19282.4	−1.41321	−0.706605	+	0.707608i	$0.749774\pi$
$572$	−3659.40	−0.267495
$573$	0	0
$574$	−1599.18	−0.116287
$575$	1685.94	0.122276
$576$	0	0
$577$	−4988.14	−0.359894	−0.179947	−	0.983676i	$-0.557593\pi$
$577$	−4988.14	−0.359894	−0.179947	+	0.983676i	$0.557593\pi$
$578$	24464.7	1.76055
$579$	0	0
$580$	4628.78	0.331378
$581$	2432.86	0.173721
$582$	0	0
$583$	5758.24	0.409060
$584$	−2860.85	−0.202710
$585$	0	0
$586$	15679.4	1.10531
$587$	15294.0	1.07539	0.537693	−	0.843141i	$-0.319296\pi$
$587$	15294.0	1.07539	0.537693	+	0.843141i	$0.319296\pi$
$588$	0	0
$589$	325.842	0.0227947
$590$	−3089.71	−0.215596
$591$	0	0
$592$	27680.8	1.92175
$593$	11090.9	0.768040	0.384020	−	0.923325i	$-0.374539\pi$
$593$	11090.9	0.768040	0.384020	+	0.923325i	$0.374539\pi$
$594$	0	0
$595$	897.744	0.0618553
$596$	−5140.41	−0.353288
$597$	0	0
$598$	−7509.75	−0.513539
$599$	28201.6	1.92368	0.961840	−	0.273612i	$-0.0882184\pi$
$599$	28201.6	1.92368	0.961840	+	0.273612i	$0.0882184\pi$
$600$	0	0
$601$	21144.0	1.43508	0.717539	−	0.696518i	$-0.245268\pi$
$601$	21144.0	1.43508	0.717539	+	0.696518i	$0.245268\pi$
$602$	1105.45	0.0748416
$603$	0	0
$604$	10665.0	0.718467
$605$	1255.64	0.0843785
$606$	0	0
$607$	−8798.98	−0.588369	−0.294184	−	0.955749i	$-0.595048\pi$
$607$	−8798.98	−0.588369	−0.294184	+	0.955749i	$0.595048\pi$
$608$	7798.81	0.520203
$609$	0	0
$610$	−7361.60	−0.488627
$611$	−15919.6	−1.05407
$612$	0	0
$613$	19539.8	1.28745	0.643724	−	0.765257i	$-0.277388\pi$
$613$	19539.8	1.28745	0.643724	+	0.765257i	$0.277388\pi$
$614$	−24824.8	−1.63167
$615$	0	0
$616$	−834.750	−0.0545991
$617$	3273.87	0.213616	0.106808	−	0.994280i	$-0.465937\pi$
$617$	3273.87	0.213616	0.106808	+	0.994280i	$0.465937\pi$
$618$	0	0
$619$	9386.47	0.609490	0.304745	−	0.952434i	$-0.401429\pi$
$619$	9386.47	0.609490	0.304745	+	0.952434i	$0.401429\pi$
$620$	101.168	0.00655326
$621$	0	0
$622$	9136.21	0.588953
$623$	2339.47	0.150448
$624$	0	0
$625$	625.000	0.0400000
$626$	−21055.6	−1.34433
$627$	0	0
$628$	−8055.37	−0.511854
$629$	38356.3	2.43142
$630$	0	0
$631$	9647.08	0.608628	0.304314	−	0.952572i	$-0.401573\pi$
$631$	9647.08	0.608628	0.304314	+	0.952572i	$0.401573\pi$
$632$	9961.19	0.626954
$633$	0	0
$634$	−7027.94	−0.440245
$635$	−5661.18	−0.353791
$636$	0	0
$637$	11239.0	0.699065
$638$	−30421.6	−1.88778
$639$	0	0
$640$	−8316.75	−0.513669
$641$	7003.77	0.431563	0.215782	−	0.976442i	$-0.430770\pi$
$641$	7003.77	0.431563	0.215782	+	0.976442i	$0.430770\pi$
$642$	0	0
$643$	−6288.89	−0.385707	−0.192853	−	0.981228i	$-0.561774\pi$
$643$	−6288.89	−0.385707	−0.192853	+	0.981228i	$0.561774\pi$
$644$	370.173	0.0226504
$645$	0	0
$646$	20201.5	1.23037
$647$	5900.85	0.358557	0.179279	−	0.983798i	$-0.442624\pi$
$647$	5900.85	0.358557	0.179279	+	0.983798i	$0.442624\pi$
$648$	0	0
$649$	6021.58	0.364203
$650$	−2783.97	−0.167994
$651$	0	0
$652$	−8581.76	−0.515472
$653$	7885.15	0.472542	0.236271	−	0.971687i	$-0.424075\pi$
$653$	7885.15	0.472542	0.236271	+	0.971687i	$0.424075\pi$
$654$	0	0
$655$	3887.53	0.231906
$656$	23192.2	1.38034
$657$	0	0
$658$	2646.28	0.156782
$659$	−28757.8	−1.69992	−0.849959	−	0.526848i	$-0.823374\pi$
$659$	−28757.8	−1.69992	−0.849959	+	0.526848i	$0.823374\pi$
$660$	0	0
$661$	8260.23	0.486060	0.243030	−	0.970019i	$-0.421859\pi$
$661$	8260.23	0.486060	0.243030	+	0.970019i	$0.421859\pi$
$662$	−768.464	−0.0451166
$663$	0	0
$664$	−23325.4	−1.36325
$665$	441.983	0.0257735
$666$	0	0
$667$	−18512.9	−1.07470
$668$	4635.39	0.268486
$669$	0	0
$670$	14020.3	0.808432
$671$	14347.1	0.825431
$672$	0	0
$673$	804.962	0.0461055	0.0230528	−	0.999734i	$-0.492661\pi$
$673$	804.962	0.0461055	0.0230528	+	0.999734i	$0.492661\pi$
$674$	11524.7	0.658627
$675$	0	0
$676$	−3731.65	−0.212315
$677$	−1363.27	−0.0773926	−0.0386963	−	0.999251i	$-0.512320\pi$
$677$	−1363.27	−0.0773926	−0.0386963	+	0.999251i	$0.512320\pi$
$678$	0	0
$679$	−1451.55	−0.0820403
$680$	−8607.24	−0.485401
$681$	0	0
$682$	−664.907	−0.0373323
$683$	11434.6	0.640604	0.320302	−	0.947315i	$-0.396216\pi$
$683$	11434.6	0.640604	0.320302	+	0.947315i	$0.396216\pi$
$684$	0	0
$685$	3118.19	0.173927
$686$	−3751.00	−0.208766
$687$	0	0
$688$	−16031.7	−0.888378
$689$	−5786.34	−0.319945
$690$	0	0
$691$	−19530.6	−1.07522	−0.537611	−	0.843193i	$-0.680673\pi$
$691$	−19530.6	−1.07522	−0.537611	+	0.843193i	$0.680673\pi$
$692$	−7960.10	−0.437280
$693$	0	0
$694$	−12172.6	−0.665799
$695$	−6413.41	−0.350035
$696$	0	0
$697$	32136.5	1.74642
$698$	−29678.9	−1.60940
$699$	0	0
$700$	137.228	0.00740962
$701$	11041.4	0.594903	0.297452	−	0.954737i	$-0.403863\pi$
$701$	11041.4	0.594903	0.297452	+	0.954737i	$0.403863\pi$
$702$	0	0
$703$	18883.8	1.01311
$704$	5013.60	0.268405
$705$	0	0
$706$	−28699.8	−1.52993
$707$	250.252	0.0133122
$708$	0	0
$709$	3939.08	0.208653	0.104327	−	0.994543i	$-0.466731\pi$
$709$	3939.08	0.208653	0.104327	+	0.994543i	$0.466731\pi$
$710$	−2005.68	−0.106017
$711$	0	0
$712$	−22430.0	−1.18062
$713$	−404.625	−0.0212529
$714$	0	0
$715$	5425.71	0.283790
$716$	−4401.19	−0.229721
$717$	0	0
$718$	−14569.4	−0.757280
$719$	23298.2	1.20845	0.604225	−	0.796814i	$-0.293483\pi$
$719$	23298.2	1.20845	0.604225	+	0.796814i	$0.293483\pi$
$720$	0	0
$721$	220.220	0.0113751
$722$	−13184.8	−0.679621
$723$	0	0
$724$	10452.2	0.536535
$725$	−6862.97	−0.351565
$726$	0	0
$727$	−9504.48	−0.484872	−0.242436	−	0.970167i	$-0.577946\pi$
$727$	−9504.48	−0.484872	−0.242436	+	0.970167i	$0.577946\pi$
$728$	838.823	0.0427044
$729$	0	0
$730$	−3090.99	−0.156716
$731$	−22214.6	−1.12399
$732$	0	0
$733$	7320.81	0.368895	0.184448	−	0.982842i	$-0.440950\pi$
$733$	7320.81	0.368895	0.184448	+	0.982842i	$0.440950\pi$
$734$	22258.4	1.11931
$735$	0	0
$736$	−9684.43	−0.485018
$737$	−27324.3	−1.36567
$738$	0	0
$739$	−1274.52	−0.0634424	−0.0317212	−	0.999497i	$-0.510099\pi$
$739$	−1274.52	−0.0634424	−0.0317212	+	0.999497i	$0.510099\pi$
$740$	5863.10	0.291259
$741$	0	0
$742$	961.849	0.0475884
$743$	−1941.15	−0.0958465	−0.0479232	−	0.998851i	$-0.515260\pi$
$743$	−1941.15	−0.0958465	−0.0479232	+	0.998851i	$0.515260\pi$
$744$	0	0
$745$	7621.56	0.374809
$746$	−29030.1	−1.42476
$747$	0	0
$748$	−12224.0	−0.597531
$749$	1169.98	0.0570761
$750$	0	0
$751$	16540.0	0.803667	0.401834	−	0.915713i	$-0.368373\pi$
$751$	16540.0	0.803667	0.401834	+	0.915713i	$0.368373\pi$
$752$	−38377.6	−1.86102
$753$	0	0
$754$	30570.1	1.47652
$755$	−15812.8	−0.762234
$756$	0	0
$757$	21145.7	1.01526	0.507631	−	0.861575i	$-0.330521\pi$
$757$	21145.7	1.01526	0.507631	+	0.861575i	$0.330521\pi$
$758$	−24043.7	−1.15212
$759$	0	0
$760$	−4237.57	−0.202254
$761$	20134.2	0.959087	0.479543	−	0.877518i	$-0.340802\pi$
$761$	20134.2	0.959087	0.479543	+	0.877518i	$0.340802\pi$
$762$	0	0
$763$	−3272.62	−0.155278
$764$	−1283.87	−0.0607968
$765$	0	0
$766$	−19989.1	−0.942865
$767$	−6050.96	−0.284860
$768$	0	0
$769$	2194.72	0.102917	0.0514587	−	0.998675i	$-0.483613\pi$
$769$	2194.72	0.102917	0.0514587	+	0.998675i	$0.483613\pi$
$770$	−901.902	−0.0422108
$771$	0	0
$772$	5213.68	0.243063
$773$	8327.70	0.387486	0.193743	−	0.981052i	$-0.437937\pi$
$773$	8327.70	0.387486	0.193743	+	0.981052i	$0.437937\pi$
$774$	0	0
$775$	−150.000	−0.00695246
$776$	13916.9	0.643799
$777$	0	0
$778$	−3503.34	−0.161441
$779$	15821.6	0.727688
$780$	0	0
$781$	3908.90	0.179093
$782$	−25085.8	−1.14715
$783$	0	0
$784$	27093.9	1.23424
$785$	11943.5	0.543034
$786$	0	0
$787$	−18963.3	−0.858918	−0.429459	−	0.903086i	$-0.641296\pi$
$787$	−18963.3	−0.858918	−0.429459	+	0.903086i	$0.641296\pi$
$788$	14448.9	0.653197
$789$	0	0
$790$	10762.5	0.484701
$791$	369.076	0.0165902
$792$	0	0
$793$	−14417.1	−0.645608
$794$	45328.2	2.02599
$795$	0	0
$796$	−4730.81	−0.210652
$797$	−34265.1	−1.52288	−0.761439	−	0.648237i	$-0.775507\pi$
$797$	−34265.1	−1.52288	−0.761439	+	0.648237i	$0.775507\pi$
$798$	0	0
$799$	−53178.5	−2.35459
$800$	−3590.15	−0.158664
$801$	0	0
$802$	−44093.9	−1.94141
$803$	6024.08	0.264739
$804$	0	0
$805$	−548.847	−0.0240302
$806$	668.152	0.0291993
$807$	0	0
$808$	−2399.33	−0.104465
$809$	−36425.5	−1.58300	−0.791502	−	0.611166i	$-0.790701\pi$
$809$	−36425.5	−1.58300	−0.791502	+	0.611166i	$0.790701\pi$
$810$	0	0
$811$	45174.0	1.95595	0.977973	−	0.208732i	$-0.0669335\pi$
$811$	45174.0	1.95595	0.977973	+	0.208732i	$0.0669335\pi$
$812$	−1506.87	−0.0651241
$813$	0	0
$814$	−38533.9	−1.65923
$815$	12724.0	0.546872
$816$	0	0
$817$	−10936.8	−0.468336
$818$	−17782.7	−0.760093
$819$	0	0
$820$	4912.35	0.209203
$821$	6919.30	0.294136	0.147068	−	0.989126i	$-0.453016\pi$
$821$	6919.30	0.294136	0.147068	+	0.989126i	$0.453016\pi$
$822$	0	0
$823$	−11398.5	−0.482780	−0.241390	−	0.970428i	$-0.577603\pi$
$823$	−11398.5	−0.482780	−0.241390	+	0.970428i	$0.577603\pi$
$824$	−2111.39	−0.0892643
$825$	0	0
$826$	1005.84	0.0423699
$827$	−34712.0	−1.45956	−0.729779	−	0.683683i	$-0.760377\pi$
$827$	−34712.0	−1.45956	−0.729779	+	0.683683i	$0.760377\pi$
$828$	0	0
$829$	2732.97	0.114500	0.0572498	−	0.998360i	$-0.481767\pi$
$829$	2732.97	0.114500	0.0572498	+	0.998360i	$0.481767\pi$
$830$	−25201.8	−1.05394
$831$	0	0
$832$	−5038.06	−0.209932
$833$	37543.1	1.56157
$834$	0	0
$835$	−6872.79	−0.284841
$836$	−6018.19	−0.248975
$837$	0	0
$838$	−46693.0	−1.92480
$839$	9575.79	0.394032	0.197016	−	0.980400i	$-0.436875\pi$
$839$	9575.79	0.394032	0.197016	+	0.980400i	$0.436875\pi$
$840$	0	0
$841$	50971.7	2.08994
$842$	3700.64	0.151464
$843$	0	0
$844$	−7251.94	−0.295761
$845$	5532.82	0.225248
$846$	0	0
$847$	−408.766	−0.0165825
$848$	−13949.2	−0.564879
$849$	0	0
$850$	−9299.66	−0.375265
$851$	−23449.6	−0.944584
$852$	0	0
$853$	9041.89	0.362941	0.181470	−	0.983396i	$-0.441914\pi$
$853$	9041.89	0.362941	0.181470	+	0.983396i	$0.441914\pi$
$854$	2396.52	0.0960274
$855$	0	0
$856$	−11217.3	−0.447896
$857$	2315.28	0.0922851	0.0461426	−	0.998935i	$-0.485307\pi$
$857$	2315.28	0.0922851	0.0461426	+	0.998935i	$0.485307\pi$
$858$	0	0
$859$	−31403.1	−1.24733	−0.623667	−	0.781690i	$-0.714358\pi$
$859$	−31403.1	−1.24733	−0.623667	+	0.781690i	$0.714358\pi$
$860$	−3395.69	−0.134642
$861$	0	0
$862$	−53662.6	−2.12036
$863$	40883.5	1.61262	0.806310	−	0.591493i	$-0.201461\pi$
$863$	40883.5	1.61262	0.806310	+	0.591493i	$0.201461\pi$
$864$	0	0
$865$	11802.3	0.463917
$866$	12026.6	0.471917
$867$	0	0
$868$	−32.9348	−0.00128788
$869$	−20975.2	−0.818798
$870$	0	0
$871$	27457.6	1.06816
$872$	31376.7	1.21852
$873$	0	0
$874$	−12350.4	−0.477985
$875$	−203.465	−0.00786099
$876$	0	0
$877$	−28314.7	−1.09021	−0.545107	−	0.838366i	$-0.683511\pi$
$877$	−28314.7	−1.09021	−0.545107	+	0.838366i	$0.683511\pi$
$878$	−1958.70	−0.0752879
$879$	0	0
$880$	13079.8	0.501046
$881$	−6479.51	−0.247787	−0.123893	−	0.992296i	$-0.539538\pi$
$881$	−6479.51	−0.247787	−0.123893	+	0.992296i	$0.539538\pi$
$882$	0	0
$883$	−17769.0	−0.677208	−0.338604	−	0.940929i	$-0.609955\pi$
$883$	−17769.0	−0.677208	−0.338604	+	0.940929i	$0.609955\pi$
$884$	12283.6	0.467357
$885$	0	0
$886$	36396.3	1.38009
$887$	25888.1	0.979976	0.489988	−	0.871729i	$-0.337001\pi$
$887$	25888.1	0.979976	0.489988	+	0.871729i	$0.337001\pi$
$888$	0	0
$889$	1842.96	0.0695287
$890$	−24234.4	−0.912739
$891$	0	0
$892$	−8626.80	−0.323819
$893$	−26181.2	−0.981096
$894$	0	0
$895$	6525.54	0.243715
$896$	2707.47	0.100949
$897$	0	0
$898$	−9759.41	−0.362668
$899$	1647.11	0.0611060
$900$	0	0
$901$	−19328.9	−0.714694
$902$	−32285.3	−1.19178
$903$	0	0
$904$	−3538.57	−0.130189
$905$	−15497.2	−0.569219
$906$	0	0
$907$	−45188.2	−1.65430	−0.827150	−	0.561981i	$-0.810040\pi$
$907$	−45188.2	−1.65430	−0.827150	+	0.561981i	$0.810040\pi$
$908$	13774.2	0.503429
$909$	0	0
$910$	906.303	0.0330150
$911$	−13174.2	−0.479121	−0.239560	−	0.970881i	$-0.577003\pi$
$911$	−13174.2	−0.479121	−0.239560	+	0.970881i	$0.577003\pi$
$912$	0	0
$913$	49116.1	1.78040
$914$	10791.0	0.390520
$915$	0	0
$916$	−9246.49	−0.333529
$917$	−1265.56	−0.0455752
$918$	0	0
$919$	−54078.5	−1.94112	−0.970558	−	0.240867i	$-0.922568\pi$
$919$	−54078.5	−1.94112	−0.970558	+	0.240867i	$0.922568\pi$
$920$	5262.14	0.188574
$921$	0	0
$922$	−24672.9	−0.881299
$923$	−3927.97	−0.140077
$924$	0	0
$925$	−8693.07	−0.309002
$926$	23646.4	0.839166
$927$	0	0
$928$	39422.6	1.39451
$929$	45717.8	1.61459	0.807294	−	0.590150i	$-0.200931\pi$
$929$	45717.8	1.61459	0.807294	+	0.590150i	$0.200931\pi$
$930$	0	0
$931$	18483.4	0.650666
$932$	−17147.1	−0.602651
$933$	0	0
$934$	26987.1	0.945445
$935$	18124.2	0.633931
$936$	0	0
$937$	−5055.54	−0.176262	−0.0881309	−	0.996109i	$-0.528089\pi$
$937$	−5055.54	−0.176262	−0.0881309	+	0.996109i	$0.528089\pi$
$938$	−4564.21	−0.158877
$939$	0	0
$940$	−8128.80	−0.282056
$941$	−28112.0	−0.973883	−0.486942	−	0.873435i	$-0.661888\pi$
$941$	−28112.0	−0.973883	−0.486942	+	0.873435i	$0.661888\pi$
$942$	0	0
$943$	−19647.0	−0.678468
$944$	−14587.1	−0.502935
$945$	0	0
$946$	22317.4	0.767022
$947$	−30120.4	−1.03356	−0.516780	−	0.856118i	$-0.672870\pi$
$947$	−30120.4	−1.03356	−0.516780	+	0.856118i	$0.672870\pi$
$948$	0	0
$949$	−6053.47	−0.207064
$950$	−4578.46	−0.156363
$951$	0	0
$952$	2802.03	0.0953933
$953$	−19362.7	−0.658154	−0.329077	−	0.944303i	$-0.606738\pi$
$953$	−19362.7	−0.658154	−0.329077	+	0.944303i	$0.606738\pi$
$954$	0	0
$955$	1903.56	0.0645003
$956$	−4979.08	−0.168447
$957$	0	0
$958$	10719.9	0.361530
$959$	−1015.11	−0.0341810
$960$	0	0
$961$	−29755.0	−0.998792
$962$	38721.9	1.29776
$963$	0	0
$964$	5946.87	0.198689
$965$	−7730.20	−0.257869
$966$	0	0
$967$	−30724.5	−1.02175	−0.510876	−	0.859654i	$-0.670679\pi$
$967$	−30724.5	−1.02175	−0.510876	+	0.859654i	$0.670679\pi$
$968$	3919.10	0.130129
$969$	0	0
$970$	15036.5	0.497724
$971$	49791.0	1.64559	0.822796	−	0.568336i	$-0.192413\pi$
$971$	49791.0	1.64559	0.822796	+	0.568336i	$0.192413\pi$
$972$	0	0
$973$	2087.84	0.0687906
$974$	44042.9	1.44890
$975$	0	0
$976$	−34755.6	−1.13985
$977$	22391.9	0.733244	0.366622	−	0.930370i	$-0.380514\pi$
$977$	22391.9	0.733244	0.366622	+	0.930370i	$0.380514\pi$
$978$	0	0
$979$	47230.7	1.54188
$980$	5738.79	0.187060
$981$	0	0
$982$	14843.5	0.482357
$983$	24619.3	0.798815	0.399408	−	0.916773i	$-0.369216\pi$
$983$	24619.3	0.798815	0.399408	+	0.916773i	$0.369216\pi$
$984$	0	0
$985$	−21423.0	−0.692988
$986$	102117.	3.29825
$987$	0	0
$988$	6047.56	0.194735
$989$	13581.1	0.436658
$990$	0	0
$991$	24540.3	0.786629	0.393314	−	0.919404i	$-0.371328\pi$
$991$	24540.3	0.786629	0.393314	+	0.919404i	$0.371328\pi$
$992$	861.636	0.0275776
$993$	0	0
$994$	652.937	0.0208349
$995$	7014.26	0.223485
$996$	0	0
$997$	45956.1	1.45982	0.729912	−	0.683542i	$-0.239561\pi$
$997$	45956.1	1.45982	0.729912	+	0.683542i	$0.239561\pi$
$998$	63146.7	2.00288
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.e.1.2		2
3.2	odd	2		405.4.a.d.1.1		2
5.4	even	2		2025.4.a.j.1.1		2
9.2	odd	6		45.4.e.a.31.2	yes	4
9.4	even	3		135.4.e.a.46.1		4
9.5	odd	6		45.4.e.a.16.2	✓	4
9.7	even	3		135.4.e.a.91.1		4
15.14	odd	2		2025.4.a.l.1.2		2
45.2	even	12		225.4.k.a.49.1		8
45.14	odd	6		225.4.e.a.151.1		4
45.23	even	12		225.4.k.a.124.1		8
45.29	odd	6		225.4.e.a.76.1		4
45.32	even	12		225.4.k.a.124.4		8
45.38	even	12		225.4.k.a.49.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.a.16.2	✓	4	9.5	odd	6
45.4.e.a.31.2	yes	4	9.2	odd	6
135.4.e.a.46.1		4	9.4	even	3
135.4.e.a.91.1		4	9.7	even	3
225.4.e.a.76.1		4	45.29	odd	6
225.4.e.a.151.1		4	45.14	odd	6
225.4.k.a.49.1		8	45.2	even	12
225.4.k.a.49.4		8	45.38	even	12
225.4.k.a.124.1		8	45.23	even	12
225.4.k.a.124.4		8	45.32	even	12
405.4.a.d.1.1		2	3.2	odd	2
405.4.a.e.1.2		2	1.1	even	1	trivial
2025.4.a.j.1.1		2	5.4	even	2
2025.4.a.l.1.2		2	15.14	odd	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+3.37228 q^{2} +3.37228 q^{4} -5.00000 q^{5} +1.62772 q^{7} -15.6060 q^{8} +O(q^{10})\)	\(q+3.37228 q^{2} +3.37228 q^{4} -5.00000 q^{5} +1.62772 q^{7} -15.6060 q^{8} -16.8614 q^{10} +32.8614 q^{11} -33.0217 q^{13} +5.48913 q^{14} -79.6060 q^{16} -110.307 q^{17} -54.3070 q^{19} -16.8614 q^{20} +110.818 q^{22} +67.4375 q^{23} +25.0000 q^{25} -111.359 q^{26} +5.48913 q^{28} -274.519 q^{29} -6.00000 q^{31} -143.606 q^{32} -371.986 q^{34} -8.13859 q^{35} -347.723 q^{37} -183.139 q^{38} +78.0298 q^{40} -291.337 q^{41} +201.388 q^{43} +110.818 q^{44} +227.418 q^{46} +482.095 q^{47} -340.351 q^{49} +84.3070 q^{50} -111.359 q^{52} +175.228 q^{53} -164.307 q^{55} -25.4021 q^{56} -925.755 q^{58} +183.242 q^{59} +436.595 q^{61} -20.2337 q^{62} +152.568 q^{64} +165.109 q^{65} -831.500 q^{67} -371.986 q^{68} -27.4456 q^{70} +118.951 q^{71} +183.318 q^{73} -1172.62 q^{74} -183.139 q^{76} +53.4891 q^{77} -638.293 q^{79} +398.030 q^{80} -982.470 q^{82} +1494.64 q^{83} +551.535 q^{85} +679.139 q^{86} -512.834 q^{88} +1437.27 q^{89} -53.7501 q^{91} +227.418 q^{92} +1625.76 q^{94} +271.535 q^{95} -891.769 q^{97} -1147.76 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{4} - 10 q^{5} + 9 q^{7} + 9 q^{8}+O(q^{10}) \) 2 * q + q^2 + q^4 - 10 * q^5 + 9 * q^7 + 9 * q^8	\( 2 q + q^{2} + q^{4} - 10 q^{5} + 9 q^{7} + 9 q^{8} - 5 q^{10} + 37 q^{11} - 112 q^{13} - 12 q^{14} - 119 q^{16} - 77 q^{17} + 35 q^{19} - 5 q^{20} + 101 q^{22} + 267 q^{23} + 50 q^{25} + 76 q^{26} - 12 q^{28} - 325 q^{29} - 12 q^{31} - 247 q^{32} - 451 q^{34} - 45 q^{35} - 638 q^{37} - 395 q^{38} - 45 q^{40} - 238 q^{41} - 97 q^{43} + 101 q^{44} - 246 q^{46} + 901 q^{47} - 629 q^{49} + 25 q^{50} + 76 q^{52} - 224 q^{53} - 185 q^{55} + 156 q^{56} - 806 q^{58} + 85 q^{59} - 247 q^{61} - 6 q^{62} + 713 q^{64} + 560 q^{65} - 606 q^{67} - 451 q^{68} + 60 q^{70} - 394 q^{71} - 811 q^{73} - 484 q^{74} - 395 q^{76} + 84 q^{77} - 840 q^{79} + 595 q^{80} - 1109 q^{82} + 387 q^{83} + 385 q^{85} + 1387 q^{86} - 411 q^{88} + 1065 q^{89} - 636 q^{91} - 246 q^{92} + 632 q^{94} - 175 q^{95} - 1031 q^{97} - 463 q^{98}+O(q^{100}) \) 2 * q + q^2 + q^4 - 10 * q^5 + 9 * q^7 + 9 * q^8 - 5 * q^10 + 37 * q^11 - 112 * q^13 - 12 * q^14 - 119 * q^16 - 77 * q^17 + 35 * q^19 - 5 * q^20 + 101 * q^22 + 267 * q^23 + 50 * q^25 + 76 * q^26 - 12 * q^28 - 325 * q^29 - 12 * q^31 - 247 * q^32 - 451 * q^34 - 45 * q^35 - 638 * q^37 - 395 * q^38 - 45 * q^40 - 238 * q^41 - 97 * q^43 + 101 * q^44 - 246 * q^46 + 901 * q^47 - 629 * q^49 + 25 * q^50 + 76 * q^52 - 224 * q^53 - 185 * q^55 + 156 * q^56 - 806 * q^58 + 85 * q^59 - 247 * q^61 - 6 * q^62 + 713 * q^64 + 560 * q^65 - 606 * q^67 - 451 * q^68 + 60 * q^70 - 394 * q^71 - 811 * q^73 - 484 * q^74 - 395 * q^76 + 84 * q^77 - 840 * q^79 + 595 * q^80 - 1109 * q^82 + 387 * q^83 + 385 * q^85 + 1387 * q^86 - 411 * q^88 + 1065 * q^89 - 636 * q^91 - 246 * q^92 + 632 * q^94 - 175 * q^95 - 1031 * q^97 - 463 * q^98

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