LMFDB - Embedded newform 405.4.a.i.1.1

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.7032.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 14x + 18 $ x^3 - x^2 - 14*x + 18
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.85028$ 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)

$f(q)$	$=$	$q-3.85028 q^{2} +6.82469 q^{4} -5.00000 q^{5} -26.3282 q^{7} +4.52526 q^{8} +O(q^{10})$	\(q-3.85028 q^{2} +6.82469 q^{4} -5.00000 q^{5} -26.3282 q^{7} +4.52526 q^{8} +19.2514 q^{10} -5.07995 q^{11} -82.7210 q^{13} +101.371 q^{14} -72.0211 q^{16} +52.5976 q^{17} -29.8611 q^{19} -34.1235 q^{20} +19.5593 q^{22} -98.2680 q^{23} +25.0000 q^{25} +318.499 q^{26} -179.682 q^{28} -167.936 q^{29} +190.774 q^{31} +241.100 q^{32} -202.516 q^{34} +131.641 q^{35} -365.864 q^{37} +114.974 q^{38} -22.6263 q^{40} +111.713 q^{41} -403.603 q^{43} -34.6691 q^{44} +378.360 q^{46} -232.176 q^{47} +350.174 q^{49} -96.2571 q^{50} -564.546 q^{52} +410.528 q^{53} +25.3997 q^{55} -119.142 q^{56} +646.601 q^{58} -152.013 q^{59} +532.250 q^{61} -734.535 q^{62} -352.134 q^{64} +413.605 q^{65} +613.405 q^{67} +358.962 q^{68} -506.855 q^{70} +413.938 q^{71} -114.484 q^{73} +1408.68 q^{74} -203.793 q^{76} +133.746 q^{77} +79.1053 q^{79} +360.106 q^{80} -430.126 q^{82} +1427.33 q^{83} -262.988 q^{85} +1553.99 q^{86} -22.9881 q^{88} -450.084 q^{89} +2177.90 q^{91} -670.649 q^{92} +893.943 q^{94} +149.306 q^{95} +1436.88 q^{97} -1348.27 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8}+O(q^{10}) $ 3 * q + q^2 + 5 * q^4 - 15 * q^5 - 25 * q^7 - 27 * q^8	$ 3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8} - 5 q^{10} + 58 q^{11} - 47 q^{13} + 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 25 q^{20} + 260 q^{22} - 51 q^{23} + 75 q^{25} + 253 q^{26} + 83 q^{28} + 350 q^{29} + 638 q^{31} + 245 q^{32} - 154 q^{34} + 125 q^{35} - 414 q^{37} + 397 q^{38} + 135 q^{40} + 179 q^{41} - 836 q^{43} + 332 q^{44} + 261 q^{46} + 235 q^{47} + 892 q^{49} + 25 q^{50} - 1335 q^{52} + 505 q^{53} - 290 q^{55} + 15 q^{56} + 1876 q^{58} + 535 q^{59} - 104 q^{61} + 348 q^{62} - 303 q^{64} + 235 q^{65} - 40 q^{67} + 830 q^{68} - 795 q^{70} + 452 q^{71} - 710 q^{73} + 1394 q^{74} + 849 q^{76} + 2148 q^{77} - 634 q^{79} + 635 q^{80} + 613 q^{82} + 1734 q^{83} - 170 q^{85} + 460 q^{86} - 768 q^{88} - 852 q^{89} - 1229 q^{91} - 1839 q^{92} + 1751 q^{94} + 25 q^{95} + 38 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 - 15 * q^5 - 25 * q^7 - 27 * q^8 - 5 * q^10 + 58 * q^11 - 47 * q^13 + 159 * q^14 - 127 * q^16 + 34 * q^17 - 5 * q^19 - 25 * q^20 + 260 * q^22 - 51 * q^23 + 75 * q^25 + 253 * q^26 + 83 * q^28 + 350 * q^29 + 638 * q^31 + 245 * q^32 - 154 * q^34 + 125 * q^35 - 414 * q^37 + 397 * q^38 + 135 * q^40 + 179 * q^41 - 836 * q^43 + 332 * q^44 + 261 * q^46 + 235 * q^47 + 892 * q^49 + 25 * q^50 - 1335 * q^52 + 505 * q^53 - 290 * q^55 + 15 * q^56 + 1876 * q^58 + 535 * q^59 - 104 * q^61 + 348 * q^62 - 303 * q^64 + 235 * q^65 - 40 * q^67 + 830 * q^68 - 795 * q^70 + 452 * q^71 - 710 * q^73 + 1394 * q^74 + 849 * q^76 + 2148 * q^77 - 634 * q^79 + 635 * q^80 + 613 * q^82 + 1734 * q^83 - 170 * q^85 + 460 * q^86 - 768 * q^88 - 852 * q^89 - 1229 * q^91 - 1839 * q^92 + 1751 * q^94 + 25 * q^95 + 38 * 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 $

$2$	−3.85028	−1.36128	−0.680641	−	0.732617i	$-0.738298\pi$
$2$	−3.85028	−1.36128	−0.680641	+	0.732617i	$0.738298\pi$
$3$	0	0
$3$	0	0
$4$	6.82469	0.853087
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−26.3282	−1.42159	−0.710795	−	0.703400i	$-0.751664\pi$
$7$	−26.3282	−1.42159	−0.710795	+	0.703400i	$0.751664\pi$
$8$	4.52526	0.199990
$9$	0	0
$10$	19.2514	0.608783
$11$	−5.07995	−0.139242	−0.0696210	−	0.997574i	$-0.522179\pi$
$11$	−5.07995	−0.139242	−0.0696210	+	0.997574i	$0.522179\pi$
$12$	0	0
$13$	−82.7210	−1.76482	−0.882411	−	0.470480i	$-0.844081\pi$
$13$	−82.7210	−1.76482	−0.882411	+	0.470480i	$0.844081\pi$
$14$	101.371	1.93518
$15$	0	0
$16$	−72.0211	−1.12533
$17$	52.5976	0.750399	0.375199	−	0.926944i	$-0.377574\pi$
$17$	52.5976	0.750399	0.375199	+	0.926944i	$0.377574\pi$
$18$	0	0
$19$	−29.8611	−0.360559	−0.180279	−	0.983615i	$-0.557700\pi$
$19$	−29.8611	−0.360559	−0.180279	+	0.983615i	$0.557700\pi$
$20$	−34.1235	−0.381512
$21$	0	0
$22$	19.5593	0.189548
$23$	−98.2680	−0.890883	−0.445441	−	0.895311i	$-0.646953\pi$
$23$	−98.2680	−0.890883	−0.445441	+	0.895311i	$0.646953\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	318.499	2.40242
$27$	0	0
$28$	−179.682	−1.21274
$29$	−167.936	−1.07534	−0.537671	−	0.843155i	$-0.680696\pi$
$29$	−167.936	−1.07534	−0.537671	+	0.843155i	$0.680696\pi$
$30$	0	0
$31$	190.774	1.10529	0.552646	−	0.833416i	$-0.313618\pi$
$31$	190.774	1.10529	0.552646	+	0.833416i	$0.313618\pi$
$32$	241.100	1.33190
$33$	0	0
$34$	−202.516	−1.02150
$35$	131.641	0.635754
$36$	0	0
$37$	−365.864	−1.62561	−0.812807	−	0.582533i	$-0.802062\pi$
$37$	−365.864	−1.62561	−0.812807	+	0.582533i	$0.802062\pi$
$38$	114.974	0.490822
$39$	0	0
$40$	−22.6263	−0.0894384
$41$	111.713	0.425527	0.212763	−	0.977104i	$-0.431754\pi$
$41$	111.713	0.425527	0.212763	+	0.977104i	$0.431754\pi$
$42$	0	0
$43$	−403.603	−1.43137	−0.715685	−	0.698423i	$-0.753885\pi$
$43$	−403.603	−1.43137	−0.715685	+	0.698423i	$0.753885\pi$
$44$	−34.6691	−0.118786
$45$	0	0
$46$	378.360	1.21274
$47$	−232.176	−0.720560	−0.360280	−	0.932844i	$-0.617319\pi$
$47$	−232.176	−0.720560	−0.360280	+	0.932844i	$0.617319\pi$
$48$	0	0
$49$	350.174	1.02092
$50$	−96.2571	−0.272256
$51$	0	0
$52$	−564.546	−1.50555
$53$	410.528	1.06397	0.531984	−	0.846754i	$-0.321447\pi$
$53$	410.528	1.06397	0.531984	+	0.846754i	$0.321447\pi$
$54$	0	0
$55$	25.3997	0.0622709
$56$	−119.142	−0.284304
$57$	0	0
$58$	646.601	1.46384
$59$	−152.013	−0.335431	−0.167716	−	0.985835i	$-0.553639\pi$
$59$	−152.013	−0.335431	−0.167716	+	0.985835i	$0.553639\pi$
$60$	0	0
$61$	532.250	1.11718	0.558588	−	0.829446i	$-0.311343\pi$
$61$	532.250	1.11718	0.558588	+	0.829446i	$0.311343\pi$
$62$	−734.535	−1.50461
$63$	0	0
$64$	−352.134	−0.687761
$65$	413.605	0.789252
$66$	0	0
$67$	613.405	1.11850	0.559248	−	0.829000i	$-0.311090\pi$
$67$	613.405	1.11850	0.559248	+	0.829000i	$0.311090\pi$
$68$	358.962	0.640155
$69$	0	0
$70$	−506.855	−0.865440
$71$	413.938	0.691908	0.345954	−	0.938252i	$-0.387555\pi$
$71$	413.938	0.691908	0.345954	+	0.938252i	$0.387555\pi$
$72$	0	0
$73$	−114.484	−0.183552	−0.0917761	−	0.995780i	$-0.529254\pi$
$73$	−114.484	−0.183552	−0.0917761	+	0.995780i	$0.529254\pi$
$74$	1408.68	2.21292
$75$	0	0
$76$	−203.793	−0.307588
$77$	133.746	0.197945
$78$	0	0
$79$	79.1053	0.112659	0.0563294	−	0.998412i	$-0.482060\pi$
$79$	79.1053	0.112659	0.0563294	+	0.998412i	$0.482060\pi$
$80$	360.106	0.503263
$81$	0	0
$82$	−430.126	−0.579262
$83$	1427.33	1.88758	0.943792	−	0.330539i	$-0.107230\pi$
$83$	1427.33	1.88758	0.943792	+	0.330539i	$0.107230\pi$
$84$	0	0
$85$	−262.988	−0.335589
$86$	1553.99	1.94850
$87$	0	0
$88$	−22.9881	−0.0278471
$89$	−450.084	−0.536054	−0.268027	−	0.963411i	$-0.586372\pi$
$89$	−450.084	−0.536054	−0.268027	+	0.963411i	$0.586372\pi$
$90$	0	0
$91$	2177.90	2.50885
$92$	−670.649	−0.760000
$93$	0	0
$94$	893.943	0.980885
$95$	149.306	0.161247
$96$	0	0
$97$	1436.88	1.50405	0.752024	−	0.659135i	$-0.229078\pi$
$97$	1436.88	1.50405	0.752024	+	0.659135i	$0.229078\pi$
$98$	−1348.27	−1.38975
$99$	0	0
$100$	170.617	0.170617
$101$	1847.03	1.81967	0.909835	−	0.414969i	$-0.136208\pi$
$101$	1847.03	1.81967	0.909835	+	0.414969i	$0.136208\pi$
$102$	0	0
$103$	47.7531	0.0456821	0.0228410	−	0.999739i	$-0.492729\pi$
$103$	47.7531	0.0456821	0.0228410	+	0.999739i	$0.492729\pi$
$104$	−374.334	−0.352947
$105$	0	0
$106$	−1580.65	−1.44836
$107$	1839.39	1.66187	0.830936	−	0.556368i	$-0.187806\pi$
$107$	1839.39	1.66187	0.830936	+	0.556368i	$0.187806\pi$
$108$	0	0
$109$	559.158	0.491354	0.245677	−	0.969352i	$-0.420990\pi$
$109$	559.158	0.491354	0.245677	+	0.969352i	$0.420990\pi$
$110$	−97.7963	−0.0847683
$111$	0	0
$112$	1896.19	1.59976
$113$	−2364.84	−1.96872	−0.984359	−	0.176174i	$-0.943628\pi$
$113$	−2364.84	−1.96872	−0.984359	+	0.176174i	$0.943628\pi$
$114$	0	0
$115$	491.340	0.398415
$116$	−1146.11	−0.917360
$117$	0	0
$118$	585.295	0.456616
$119$	−1384.80	−1.06676
$120$	0	0
$121$	−1305.19	−0.980612
$122$	−2049.32	−1.52079
$123$	0	0
$124$	1301.98	0.942910
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2403.43	−1.67929	−0.839646	−	0.543134i	$-0.817238\pi$
$127$	−2403.43	−1.67929	−0.839646	+	0.543134i	$0.817238\pi$
$128$	−572.983	−0.395664
$129$	0	0
$130$	−1592.50	−1.07439
$131$	1634.81	1.09033	0.545167	−	0.838327i	$-0.316466\pi$
$131$	1634.81	1.09033	0.545167	+	0.838327i	$0.316466\pi$
$132$	0	0
$133$	786.190	0.512566
$134$	−2361.78	−1.52259
$135$	0	0
$136$	238.018	0.150072
$137$	−2526.38	−1.57550	−0.787749	−	0.615996i	$-0.788754\pi$
$137$	−2526.38	−1.57550	−0.787749	+	0.615996i	$0.788754\pi$
$138$	0	0
$139$	1890.84	1.15381	0.576904	−	0.816812i	$-0.304261\pi$
$139$	1890.84	1.15381	0.576904	+	0.816812i	$0.304261\pi$
$140$	898.409	0.542353
$141$	0	0
$142$	−1593.78	−0.941881
$143$	420.219	0.245737
$144$	0	0
$145$	839.679	0.480907
$146$	440.795	0.249866
$147$	0	0
$148$	−2496.91	−1.38679
$149$	278.901	0.153345	0.0766727	−	0.997056i	$-0.475570\pi$
$149$	278.901	0.153345	0.0766727	+	0.997056i	$0.475570\pi$
$150$	0	0
$151$	−431.952	−0.232793	−0.116396	−	0.993203i	$-0.537134\pi$
$151$	−431.952	−0.232793	−0.116396	+	0.993203i	$0.537134\pi$
$152$	−135.129	−0.0721082
$153$	0	0
$154$	−514.960	−0.269459
$155$	−953.871	−0.494302
$156$	0	0
$157$	−445.107	−0.226264	−0.113132	−	0.993580i	$-0.536088\pi$
$157$	−445.107	−0.226264	−0.113132	+	0.993580i	$0.536088\pi$
$158$	−304.578	−0.153360
$159$	0	0
$160$	−1205.50	−0.595644
$161$	2587.22	1.26647
$162$	0	0
$163$	−2533.56	−1.21745	−0.608723	−	0.793382i	$-0.708318\pi$
$163$	−2533.56	−1.21745	−0.608723	+	0.793382i	$0.708318\pi$
$164$	762.406	0.363011
$165$	0	0
$166$	−5495.62	−2.56953
$167$	−2165.49	−1.00341	−0.501707	−	0.865037i	$-0.667295\pi$
$167$	−2165.49	−1.00341	−0.501707	+	0.865037i	$0.667295\pi$
$168$	0	0
$169$	4645.77	2.11460
$170$	1012.58	0.456830
$171$	0	0
$172$	−2754.47	−1.22108
$173$	−1363.69	−0.599305	−0.299653	−	0.954048i	$-0.596871\pi$
$173$	−1363.69	−0.599305	−0.299653	+	0.954048i	$0.596871\pi$
$174$	0	0
$175$	−658.205	−0.284318
$176$	365.864	0.156693
$177$	0	0
$178$	1732.95	0.729721
$179$	1198.29	0.500358	0.250179	−	0.968200i	$-0.419511\pi$
$179$	1198.29	0.500358	0.250179	+	0.968200i	$0.419511\pi$
$180$	0	0
$181$	1098.13	0.450959	0.225480	−	0.974248i	$-0.427605\pi$
$181$	1098.13	0.450959	0.225480	+	0.974248i	$0.427605\pi$
$182$	−8385.52	−3.41525
$183$	0	0
$184$	−444.689	−0.178168
$185$	1829.32	0.726997
$186$	0	0
$187$	−267.193	−0.104487
$188$	−1584.53	−0.614700
$189$	0	0
$190$	−574.869	−0.219502
$191$	1250.03	0.473554	0.236777	−	0.971564i	$-0.423909\pi$
$191$	1250.03	0.473554	0.236777	+	0.971564i	$0.423909\pi$
$192$	0	0
$193$	3241.51	1.20896	0.604480	−	0.796620i	$-0.293381\pi$
$193$	3241.51	1.20896	0.604480	+	0.796620i	$0.293381\pi$
$194$	−5532.39	−2.04743
$195$	0	0
$196$	2389.83	0.870929
$197$	−714.141	−0.258276	−0.129138	−	0.991627i	$-0.541221\pi$
$197$	−714.141	−0.258276	−0.129138	+	0.991627i	$0.541221\pi$
$198$	0	0
$199$	−666.927	−0.237574	−0.118787	−	0.992920i	$-0.537901\pi$
$199$	−666.927	−0.237574	−0.118787	+	0.992920i	$0.537901\pi$
$200$	113.132	0.0399981
$201$	0	0
$202$	−7111.61	−2.47708
$203$	4421.45	1.52869
$204$	0	0
$205$	−558.564	−0.190301
$206$	−183.863	−0.0621862
$207$	0	0
$208$	5957.66	1.98601
$209$	151.693	0.0502049
$210$	0	0
$211$	−4959.21	−1.61804	−0.809019	−	0.587782i	$-0.800001\pi$
$211$	−4959.21	−1.61804	−0.809019	+	0.587782i	$0.800001\pi$
$212$	2801.73	0.907657
$213$	0	0
$214$	−7082.16	−2.26227
$215$	2018.02	0.640128
$216$	0	0
$217$	−5022.74	−1.57127
$218$	−2152.92	−0.668871
$219$	0	0
$220$	173.346	0.0531225
$221$	−4350.92	−1.32432
$222$	0	0
$223$	1123.77	0.337458	0.168729	−	0.985662i	$-0.446034\pi$
$223$	1123.77	0.337458	0.168729	+	0.985662i	$0.446034\pi$
$224$	−6347.72	−1.89341
$225$	0	0
$226$	9105.30	2.67998
$227$	1225.11	0.358209	0.179104	−	0.983830i	$-0.442680\pi$
$227$	1225.11	0.358209	0.179104	+	0.983830i	$0.442680\pi$
$228$	0	0
$229$	−3578.18	−1.03254	−0.516272	−	0.856425i	$-0.672681\pi$
$229$	−3578.18	−1.03254	−0.516272	+	0.856425i	$0.672681\pi$
$230$	−1891.80	−0.542355
$231$	0	0
$232$	−759.954	−0.215058
$233$	−527.061	−0.148193	−0.0740963	−	0.997251i	$-0.523607\pi$
$233$	−527.061	−0.148193	−0.0740963	+	0.997251i	$0.523607\pi$
$234$	0	0
$235$	1160.88	0.322244
$236$	−1037.44	−0.286152
$237$	0	0
$238$	5331.87	1.45216
$239$	−96.3386	−0.0260738	−0.0130369	−	0.999915i	$-0.504150\pi$
$239$	−96.3386	−0.0260738	−0.0130369	+	0.999915i	$0.504150\pi$
$240$	0	0
$241$	923.946	0.246957	0.123478	−	0.992347i	$-0.460595\pi$
$241$	923.946	0.246957	0.123478	+	0.992347i	$0.460595\pi$
$242$	5025.37	1.33489
$243$	0	0
$244$	3632.45	0.953047
$245$	−1750.87	−0.456567
$246$	0	0
$247$	2470.14	0.636322
$248$	863.303	0.221048
$249$	0	0
$250$	481.286	0.121757
$251$	4134.83	1.03979	0.519897	−	0.854229i	$-0.325970\pi$
$251$	4134.83	1.03979	0.519897	+	0.854229i	$0.325970\pi$
$252$	0	0
$253$	499.197	0.124048
$254$	9253.90	2.28599
$255$	0	0
$256$	5023.22	1.22637
$257$	−1787.18	−0.433778	−0.216889	−	0.976196i	$-0.569591\pi$
$257$	−1787.18	−0.433778	−0.216889	+	0.976196i	$0.569591\pi$
$258$	0	0
$259$	9632.55	2.31095
$260$	2822.73	0.673301
$261$	0	0
$262$	−6294.47	−1.48425
$263$	5348.15	1.25392	0.626960	−	0.779052i	$-0.284299\pi$
$263$	5348.15	1.25392	0.626960	+	0.779052i	$0.284299\pi$
$264$	0	0
$265$	−2052.64	−0.475821
$266$	−3027.05	−0.697747
$267$	0	0
$268$	4186.30	0.954175
$269$	−8266.59	−1.87369	−0.936846	−	0.349743i	$-0.886269\pi$
$269$	−8266.59	−1.87369	−0.936846	+	0.349743i	$0.886269\pi$
$270$	0	0
$271$	−7114.58	−1.59476	−0.797380	−	0.603478i	$-0.793781\pi$
$271$	−7114.58	−1.59476	−0.797380	+	0.603478i	$0.793781\pi$
$272$	−3788.13	−0.844446
$273$	0	0
$274$	9727.28	2.14470
$275$	−126.999	−0.0278484
$276$	0	0
$277$	−3897.48	−0.845404	−0.422702	−	0.906269i	$-0.638918\pi$
$277$	−3897.48	−0.845404	−0.422702	+	0.906269i	$0.638918\pi$
$278$	−7280.29	−1.57066
$279$	0	0
$280$	595.710	0.127145
$281$	5294.93	1.12409	0.562045	−	0.827107i	$-0.310015\pi$
$281$	5294.93	1.12409	0.562045	+	0.827107i	$0.310015\pi$
$282$	0	0
$283$	−4642.99	−0.975255	−0.487628	−	0.873052i	$-0.662138\pi$
$283$	−4642.99	−0.975255	−0.487628	+	0.873052i	$0.662138\pi$
$284$	2825.00	0.590257
$285$	0	0
$286$	−1617.96	−0.334518
$287$	−2941.20	−0.604924
$288$	0	0
$289$	−2146.50	−0.436902
$290$	−3233.00	−0.654650
$291$	0	0
$292$	−781.317	−0.156586
$293$	3915.07	0.780617	0.390309	−	0.920684i	$-0.372368\pi$
$293$	3915.07	0.780617	0.390309	+	0.920684i	$0.372368\pi$
$294$	0	0
$295$	760.067	0.150009
$296$	−1655.63	−0.325107
$297$	0	0
$298$	−1073.85	−0.208746
$299$	8128.83	1.57225
$300$	0	0
$301$	10626.1	2.03482
$302$	1663.14	0.316897
$303$	0	0
$304$	2150.63	0.405747
$305$	−2661.25	−0.499616
$306$	0	0
$307$	6637.24	1.23390	0.616950	−	0.787002i	$-0.288368\pi$
$307$	6637.24	1.23390	0.616950	+	0.787002i	$0.288368\pi$
$308$	912.775	0.168864
$309$	0	0
$310$	3672.67	0.672884
$311$	2237.22	0.407913	0.203957	−	0.978980i	$-0.434620\pi$
$311$	2237.22	0.407913	0.203957	+	0.978980i	$0.434620\pi$
$312$	0	0
$313$	1314.19	0.237324	0.118662	−	0.992935i	$-0.462139\pi$
$313$	1314.19	0.237324	0.118662	+	0.992935i	$0.462139\pi$
$314$	1713.79	0.308009
$315$	0	0
$316$	539.869	0.0961077
$317$	−940.460	−0.166629	−0.0833147	−	0.996523i	$-0.526551\pi$
$317$	−940.460	−0.166629	−0.0833147	+	0.996523i	$0.526551\pi$
$318$	0	0
$319$	853.106	0.149733
$320$	1760.67	0.307576
$321$	0	0
$322$	−9961.54	−1.72402
$323$	−1570.62	−0.270563
$324$	0	0
$325$	−2068.03	−0.352964
$326$	9754.93	1.65729
$327$	0	0
$328$	505.530	0.0851013
$329$	6112.77	1.02434
$330$	0	0
$331$	−2242.46	−0.372376	−0.186188	−	0.982514i	$-0.559613\pi$
$331$	−2242.46	−0.372376	−0.186188	+	0.982514i	$0.559613\pi$
$332$	9741.07	1.61027
$333$	0	0
$334$	8337.73	1.36593
$335$	−3067.02	−0.500207
$336$	0	0
$337$	−3123.65	−0.504915	−0.252457	−	0.967608i	$-0.581239\pi$
$337$	−3123.65	−0.504915	−0.252457	+	0.967608i	$0.581239\pi$
$338$	−17887.5	−2.87856
$339$	0	0
$340$	−1794.81	−0.286286
$341$	−969.123	−0.153903
$342$	0	0
$343$	−188.878	−0.0297332
$344$	−1826.41	−0.286260
$345$	0	0
$346$	5250.61	0.815823
$347$	10722.3	1.65879	0.829397	−	0.558660i	$-0.188684\pi$
$347$	10722.3	1.65879	0.829397	+	0.558660i	$0.188684\pi$
$348$	0	0
$349$	−6101.65	−0.935856	−0.467928	−	0.883767i	$-0.654999\pi$
$349$	−6101.65	−0.935856	−0.467928	+	0.883767i	$0.654999\pi$
$350$	2534.28	0.387037
$351$	0	0
$352$	−1224.77	−0.185456
$353$	10328.6	1.55732	0.778659	−	0.627447i	$-0.215900\pi$
$353$	10328.6	1.55732	0.778659	+	0.627447i	$0.215900\pi$
$354$	0	0
$355$	−2069.69	−0.309430
$356$	−3071.69	−0.457301
$357$	0	0
$358$	−4613.74	−0.681128
$359$	−1122.33	−0.164998	−0.0824992	−	0.996591i	$-0.526290\pi$
$359$	−1122.33	−0.164998	−0.0824992	+	0.996591i	$0.526290\pi$
$360$	0	0
$361$	−5967.31	−0.869998
$362$	−4228.13	−0.613883
$363$	0	0
$364$	14863.5	2.14027
$365$	572.419	0.0820871
$366$	0	0
$367$	−3642.88	−0.518138	−0.259069	−	0.965859i	$-0.583416\pi$
$367$	−3642.88	−0.518138	−0.259069	+	0.965859i	$0.583416\pi$
$368$	7077.37	1.00254
$369$	0	0
$370$	−7043.41	−0.989647
$371$	−10808.5	−1.51253
$372$	0	0
$373$	9534.33	1.32351	0.661755	−	0.749721i	$-0.269812\pi$
$373$	9534.33	1.32351	0.661755	+	0.749721i	$0.269812\pi$
$374$	1028.77	0.142236
$375$	0	0
$376$	−1050.66	−0.144105
$377$	13891.8	1.89779
$378$	0	0
$379$	10826.9	1.46738	0.733691	−	0.679483i	$-0.237796\pi$
$379$	10826.9	1.46738	0.733691	+	0.679483i	$0.237796\pi$
$380$	1018.97	0.137557
$381$	0	0
$382$	−4812.96	−0.644640
$383$	6673.74	0.890371	0.445185	−	0.895438i	$-0.353138\pi$
$383$	6673.74	0.890371	0.445185	+	0.895438i	$0.353138\pi$
$384$	0	0
$385$	−668.730	−0.0885237
$386$	−12480.7	−1.64573
$387$	0	0
$388$	9806.24	1.28308
$389$	1263.00	0.164619	0.0823094	−	0.996607i	$-0.473770\pi$
$389$	1263.00	0.164619	0.0823094	+	0.996607i	$0.473770\pi$
$390$	0	0
$391$	−5168.66	−0.668517
$392$	1584.63	0.204173
$393$	0	0
$394$	2749.65	0.351587
$395$	−395.526	−0.0503825
$396$	0	0
$397$	−888.172	−0.112282	−0.0561411	−	0.998423i	$-0.517880\pi$
$397$	−888.172	−0.112282	−0.0561411	+	0.998423i	$0.517880\pi$
$398$	2567.86	0.323405
$399$	0	0
$400$	−1800.53	−0.225066
$401$	12372.4	1.54077	0.770385	−	0.637580i	$-0.220064\pi$
$401$	12372.4	1.54077	0.770385	+	0.637580i	$0.220064\pi$
$402$	0	0
$403$	−15781.0	−1.95064
$404$	12605.4	1.55234
$405$	0	0
$406$	−17023.8	−2.08098
$407$	1858.57	0.226354
$408$	0	0
$409$	−12647.3	−1.52902	−0.764508	−	0.644615i	$-0.777018\pi$
$409$	−12647.3	−1.52902	−0.764508	+	0.644615i	$0.777018\pi$
$410$	2150.63	0.259054
$411$	0	0
$412$	325.900	0.0389708
$413$	4002.24	0.476846
$414$	0	0
$415$	−7136.64	−0.844154
$416$	−19944.0	−2.35057
$417$	0	0
$418$	−584.061	−0.0683430
$419$	−14252.0	−1.66171	−0.830855	−	0.556490i	$-0.812148\pi$
$419$	−14252.0	−1.66171	−0.830855	+	0.556490i	$0.812148\pi$
$420$	0	0
$421$	2545.69	0.294701	0.147351	−	0.989084i	$-0.452925\pi$
$421$	2545.69	0.294701	0.147351	+	0.989084i	$0.452925\pi$
$422$	19094.4	2.20261
$423$	0	0
$424$	1857.75	0.212783
$425$	1314.94	0.150080
$426$	0	0
$427$	−14013.2	−1.58816
$428$	12553.3	1.41772
$429$	0	0
$430$	−7769.94	−0.871395
$431$	−3068.41	−0.342923	−0.171462	−	0.985191i	$-0.554849\pi$
$431$	−3068.41	−0.342923	−0.171462	+	0.985191i	$0.554849\pi$
$432$	0	0
$433$	−13982.0	−1.55180	−0.775900	−	0.630856i	$-0.782704\pi$
$433$	−13982.0	−1.55180	−0.775900	+	0.630856i	$0.782704\pi$
$434$	19339.0	2.13894
$435$	0	0
$436$	3816.08	0.419168
$437$	2934.39	0.321215
$438$	0	0
$439$	7692.44	0.836310	0.418155	−	0.908376i	$-0.362677\pi$
$439$	7692.44	0.836310	0.418155	+	0.908376i	$0.362677\pi$
$440$	114.941	0.0124536
$441$	0	0
$442$	16752.3	1.80277
$443$	−35.8248	−0.00384218	−0.00192109	−	0.999998i	$-0.500612\pi$
$443$	−35.8248	−0.00384218	−0.00192109	+	0.999998i	$0.500612\pi$
$444$	0	0
$445$	2250.42	0.239731
$446$	−4326.84	−0.459376
$447$	0	0
$448$	9271.04	0.977713
$449$	−2602.16	−0.273505	−0.136752	−	0.990605i	$-0.543666\pi$
$449$	−2602.16	−0.273505	−0.136752	+	0.990605i	$0.543666\pi$
$450$	0	0
$451$	−567.495	−0.0592512
$452$	−16139.3	−1.67949
$453$	0	0
$454$	−4717.02	−0.487623
$455$	−10889.5	−1.12199
$456$	0	0
$457$	8151.66	0.834395	0.417198	−	0.908816i	$-0.363012\pi$
$457$	8151.66	0.834395	0.417198	+	0.908816i	$0.363012\pi$
$458$	13777.0	1.40558
$459$	0	0
$460$	3353.25	0.339882
$461$	−6747.31	−0.681678	−0.340839	−	0.940122i	$-0.610711\pi$
$461$	−6747.31	−0.681678	−0.340839	+	0.940122i	$0.610711\pi$
$462$	0	0
$463$	838.556	0.0841706	0.0420853	−	0.999114i	$-0.486600\pi$
$463$	838.556	0.0841706	0.0420853	+	0.999114i	$0.486600\pi$
$464$	12094.9	1.21011
$465$	0	0
$466$	2029.33	0.201732
$467$	7253.46	0.718737	0.359369	−	0.933196i	$-0.382992\pi$
$467$	7253.46	0.718737	0.359369	+	0.933196i	$0.382992\pi$
$468$	0	0
$469$	−16149.8	−1.59004
$470$	−4469.71	−0.438665
$471$	0	0
$472$	−687.900	−0.0670830
$473$	2050.28	0.199307
$474$	0	0
$475$	−746.528	−0.0721117
$476$	−9450.83	−0.910038
$477$	0	0
$478$	370.931	0.0354937
$479$	−10351.8	−0.987447	−0.493724	−	0.869619i	$-0.664365\pi$
$479$	−10351.8	−0.987447	−0.493724	+	0.869619i	$0.664365\pi$
$480$	0	0
$481$	30264.7	2.86892
$482$	−3557.45	−0.336178
$483$	0	0
$484$	−8907.55	−0.836547
$485$	−7184.38	−0.672631
$486$	0	0
$487$	5202.70	0.484100	0.242050	−	0.970264i	$-0.422180\pi$
$487$	5202.70	0.484100	0.242050	+	0.970264i	$0.422180\pi$
$488$	2408.57	0.223424
$489$	0	0
$490$	6741.35	0.621516
$491$	−2460.52	−0.226154	−0.113077	−	0.993586i	$-0.536071\pi$
$491$	−2460.52	−0.226154	−0.113077	+	0.993586i	$0.536071\pi$
$492$	0	0
$493$	−8833.01	−0.806935
$494$	−9510.75	−0.866213
$495$	0	0
$496$	−13739.8	−1.24382
$497$	−10898.2	−0.983608
$498$	0	0
$499$	−12856.3	−1.15336	−0.576681	−	0.816969i	$-0.695653\pi$
$499$	−12856.3	−1.15336	−0.576681	+	0.816969i	$0.695653\pi$
$500$	−853.087	−0.0763024
$501$	0	0
$502$	−15920.3	−1.41545
$503$	−21744.3	−1.92750	−0.963749	−	0.266810i	$-0.914030\pi$
$503$	−21744.3	−1.92750	−0.963749	+	0.266810i	$0.914030\pi$
$504$	0	0
$505$	−9235.17	−0.813782
$506$	−1922.05	−0.168865
$507$	0	0
$508$	−16402.7	−1.43258
$509$	−9743.82	−0.848501	−0.424251	−	0.905545i	$-0.639462\pi$
$509$	−9743.82	−0.848501	−0.424251	+	0.905545i	$0.639462\pi$
$510$	0	0
$511$	3014.15	0.260936
$512$	−14756.9	−1.27377
$513$	0	0
$514$	6881.13	0.590494
$515$	−238.766	−0.0204296
$516$	0	0
$517$	1179.44	0.100332
$518$	−37088.0	−3.14586
$519$	0	0
$520$	1871.67	0.157843
$521$	−17889.5	−1.50432	−0.752161	−	0.658980i	$-0.770988\pi$
$521$	−17889.5	−1.50432	−0.752161	+	0.658980i	$0.770988\pi$
$522$	0	0
$523$	5012.48	0.419083	0.209542	−	0.977800i	$-0.432803\pi$
$523$	5012.48	0.419083	0.209542	+	0.977800i	$0.432803\pi$
$524$	11157.1	0.930150
$525$	0	0
$526$	−20591.9	−1.70694
$527$	10034.3	0.829410
$528$	0	0
$529$	−2510.39	−0.206328
$530$	7903.24	0.647726
$531$	0	0
$532$	5365.50	0.437263
$533$	−9241.00	−0.750979
$534$	0	0
$535$	−9196.94	−0.743212
$536$	2775.82	0.223689
$537$	0	0
$538$	31828.7	2.55062
$539$	−1778.87	−0.142154
$540$	0	0
$541$	12629.5	1.00367	0.501834	−	0.864964i	$-0.332659\pi$
$541$	12629.5	1.00367	0.501834	+	0.864964i	$0.332659\pi$
$542$	27393.2	2.17092
$543$	0	0
$544$	12681.3	0.999456
$545$	−2795.79	−0.219740
$546$	0	0
$547$	7331.85	0.573103	0.286552	−	0.958065i	$-0.407491\pi$
$547$	7331.85	0.573103	0.286552	+	0.958065i	$0.407491\pi$
$548$	−17241.8	−1.34404
$549$	0	0
$550$	488.981	0.0379095
$551$	5014.75	0.387724
$552$	0	0
$553$	−2082.70	−0.160154
$554$	15006.4	1.15083
$555$	0	0
$556$	12904.4	0.984299
$557$	11389.0	0.866371	0.433185	−	0.901305i	$-0.357390\pi$
$557$	11389.0	0.866371	0.433185	+	0.901305i	$0.357390\pi$
$558$	0	0
$559$	33386.5	2.52611
$560$	−9480.93	−0.715433
$561$	0	0
$562$	−20387.0	−1.53020
$563$	13221.1	0.989704	0.494852	−	0.868977i	$-0.335222\pi$
$563$	13221.1	0.989704	0.494852	+	0.868977i	$0.335222\pi$
$564$	0	0
$565$	11824.2	0.880438
$566$	17876.8	1.32760
$567$	0	0
$568$	1873.18	0.138375
$569$	−8341.79	−0.614597	−0.307299	−	0.951613i	$-0.599425\pi$
$569$	−8341.79	−0.614597	−0.307299	+	0.951613i	$0.599425\pi$
$570$	0	0
$571$	21602.5	1.58325	0.791625	−	0.611007i	$-0.209235\pi$
$571$	21602.5	1.58325	0.791625	+	0.611007i	$0.209235\pi$
$572$	2867.86	0.209635
$573$	0	0
$574$	11324.4	0.823472
$575$	−2456.70	−0.178177
$576$	0	0
$577$	−696.389	−0.0502444	−0.0251222	−	0.999684i	$-0.507997\pi$
$577$	−696.389	−0.0502444	−0.0251222	+	0.999684i	$0.507997\pi$
$578$	8264.63	0.594746
$579$	0	0
$580$	5730.55	0.410256
$581$	−37579.0	−2.68337
$582$	0	0
$583$	−2085.46	−0.148149
$584$	−518.069	−0.0367087
$585$	0	0
$586$	−15074.1	−1.06264
$587$	23924.2	1.68221	0.841105	−	0.540872i	$-0.181906\pi$
$587$	23924.2	1.68221	0.841105	+	0.540872i	$0.181906\pi$
$588$	0	0
$589$	−5696.73	−0.398522
$590$	−2926.47	−0.204205
$591$	0	0
$592$	26349.9	1.82935
$593$	11887.8	0.823226	0.411613	−	0.911359i	$-0.364966\pi$
$593$	11887.8	0.823226	0.411613	+	0.911359i	$0.364966\pi$
$594$	0	0
$595$	6923.99	0.477069
$596$	1903.41	0.130817
$597$	0	0
$598$	−31298.3	−2.14027
$599$	−4907.42	−0.334744	−0.167372	−	0.985894i	$-0.553528\pi$
$599$	−4907.42	−0.334744	−0.167372	+	0.985894i	$0.553528\pi$
$600$	0	0
$601$	11264.7	0.764552	0.382276	−	0.924048i	$-0.375140\pi$
$601$	11264.7	0.764552	0.382276	+	0.924048i	$0.375140\pi$
$602$	−40913.7	−2.76996
$603$	0	0
$604$	−2947.94	−0.198592
$605$	6525.97	0.438543
$606$	0	0
$607$	5022.66	0.335854	0.167927	−	0.985799i	$-0.446293\pi$
$607$	5022.66	0.335854	0.167927	+	0.985799i	$0.446293\pi$
$608$	−7199.51	−0.480228
$609$	0	0
$610$	10246.6	0.680118
$611$	19205.8	1.27166
$612$	0	0
$613$	19450.0	1.28153	0.640766	−	0.767736i	$-0.278617\pi$
$613$	19450.0	1.28153	0.640766	+	0.767736i	$0.278617\pi$
$614$	−25555.3	−1.67969
$615$	0	0
$616$	605.236	0.0395871
$617$	2018.05	0.131675	0.0658375	−	0.997830i	$-0.479028\pi$
$617$	2018.05	0.131675	0.0658375	+	0.997830i	$0.479028\pi$
$618$	0	0
$619$	9097.52	0.590728	0.295364	−	0.955385i	$-0.404559\pi$
$619$	9097.52	0.590728	0.295364	+	0.955385i	$0.404559\pi$
$620$	−6509.88	−0.421682
$621$	0	0
$622$	−8613.93	−0.555285
$623$	11849.9	0.762049
$624$	0	0
$625$	625.000	0.0400000
$626$	−5060.01	−0.323065
$627$	0	0
$628$	−3037.72	−0.193023
$629$	−19243.6	−1.21986
$630$	0	0
$631$	3118.35	0.196735	0.0983674	−	0.995150i	$-0.468638\pi$
$631$	3118.35	0.196735	0.0983674	+	0.995150i	$0.468638\pi$
$632$	357.972	0.0225307
$633$	0	0
$634$	3621.04	0.226829
$635$	12017.2	0.751002
$636$	0	0
$637$	−28966.8	−1.80173
$638$	−3284.70	−0.203828
$639$	0	0
$640$	2864.91	0.176946
$641$	10454.2	0.644177	0.322089	−	0.946709i	$-0.395615\pi$
$641$	10454.2	0.644177	0.322089	+	0.946709i	$0.395615\pi$
$642$	0	0
$643$	16125.6	0.989008	0.494504	−	0.869175i	$-0.335350\pi$
$643$	16125.6	0.989008	0.494504	+	0.869175i	$0.335350\pi$
$644$	17657.0	1.08041
$645$	0	0
$646$	6047.34	0.368312
$647$	−13627.3	−0.828045	−0.414022	−	0.910267i	$-0.635877\pi$
$647$	−13627.3	−0.828045	−0.414022	+	0.910267i	$0.635877\pi$
$648$	0	0
$649$	772.220	0.0467061
$650$	7962.49	0.480484
$651$	0	0
$652$	−17290.8	−1.03859
$653$	2654.67	0.159089	0.0795446	−	0.996831i	$-0.474653\pi$
$653$	2654.67	0.159089	0.0795446	+	0.996831i	$0.474653\pi$
$654$	0	0
$655$	−8174.04	−0.487612
$656$	−8045.68	−0.478858
$657$	0	0
$658$	−23535.9	−1.39441
$659$	18256.8	1.07919	0.539593	−	0.841926i	$-0.318578\pi$
$659$	18256.8	1.07919	0.539593	+	0.841926i	$0.318578\pi$
$660$	0	0
$661$	−7038.34	−0.414160	−0.207080	−	0.978324i	$-0.566396\pi$
$661$	−7038.34	−0.414160	−0.207080	+	0.978324i	$0.566396\pi$
$662$	8634.10	0.506909
$663$	0	0
$664$	6459.03	0.377499
$665$	−3930.95	−0.229226
$666$	0	0
$667$	16502.7	0.958003
$668$	−14778.8	−0.856000
$669$	0	0
$670$	11808.9	0.680922
$671$	−2703.81	−0.155558
$672$	0	0
$673$	−10043.1	−0.575233	−0.287617	−	0.957746i	$-0.592863\pi$
$673$	−10043.1	−0.575233	−0.287617	+	0.957746i	$0.592863\pi$
$674$	12027.0	0.687331
$675$	0	0
$676$	31705.9	1.80393
$677$	−25999.2	−1.47597	−0.737984	−	0.674818i	$-0.764222\pi$
$677$	−25999.2	−1.47597	−0.737984	+	0.674818i	$0.764222\pi$
$678$	0	0
$679$	−37830.4	−2.13814
$680$	−1190.09	−0.0671145
$681$	0	0
$682$	3731.40	0.209505
$683$	13619.4	0.763004	0.381502	−	0.924368i	$-0.375407\pi$
$683$	13619.4	0.763004	0.381502	+	0.924368i	$0.375407\pi$
$684$	0	0
$685$	12631.9	0.704584
$686$	727.236	0.0404752
$687$	0	0
$688$	29068.0	1.61076
$689$	−33959.3	−1.87771
$690$	0	0
$691$	−13722.9	−0.755492	−0.377746	−	0.925909i	$-0.623301\pi$
$691$	−13722.9	−0.755492	−0.377746	+	0.925909i	$0.623301\pi$
$692$	−9306.80	−0.511259
$693$	0	0
$694$	−41283.8	−2.25808
$695$	−9454.22	−0.515999
$696$	0	0
$697$	5875.82	0.319315
$698$	23493.1	1.27396
$699$	0	0
$700$	−4492.05	−0.242548
$701$	11103.4	0.598244	0.299122	−	0.954215i	$-0.403306\pi$
$701$	11103.4	0.598244	0.299122	+	0.954215i	$0.403306\pi$
$702$	0	0
$703$	10925.1	0.586129
$704$	1788.82	0.0957652
$705$	0	0
$706$	−39767.9	−2.11995
$707$	−48629.1	−2.58682
$708$	0	0
$709$	−3063.60	−0.162279	−0.0811395	−	0.996703i	$-0.525856\pi$
$709$	−3063.60	−0.162279	−0.0811395	+	0.996703i	$0.525856\pi$
$710$	7968.90	0.421222
$711$	0	0
$712$	−2036.75	−0.107206
$713$	−18747.0	−0.984686
$714$	0	0
$715$	−2101.09	−0.109897
$716$	8177.93	0.426849
$717$	0	0
$718$	4321.30	0.224609
$719$	31506.0	1.63418	0.817089	−	0.576511i	$-0.195586\pi$
$719$	31506.0	1.63418	0.817089	+	0.576511i	$0.195586\pi$
$720$	0	0
$721$	−1257.25	−0.0649411
$722$	22975.9	1.18431
$723$	0	0
$724$	7494.43	0.384707
$725$	−4198.40	−0.215068
$726$	0	0
$727$	15056.5	0.768107	0.384054	−	0.923311i	$-0.374528\pi$
$727$	15056.5	0.768107	0.384054	+	0.923311i	$0.374528\pi$
$728$	9855.55	0.501746
$729$	0	0
$730$	−2203.98	−0.111744
$731$	−21228.5	−1.07410
$732$	0	0
$733$	20267.4	1.02127	0.510637	−	0.859797i	$-0.329410\pi$
$733$	20267.4	1.02127	0.510637	+	0.859797i	$0.329410\pi$
$734$	14026.1	0.705331
$735$	0	0
$736$	−23692.4	−1.18657
$737$	−3116.06	−0.155742
$738$	0	0
$739$	−22434.4	−1.11673	−0.558365	−	0.829595i	$-0.688571\pi$
$739$	−22434.4	−1.11673	−0.558365	+	0.829595i	$0.688571\pi$
$740$	12484.6	0.620191
$741$	0	0
$742$	41615.6	2.05897
$743$	37246.3	1.83908	0.919538	−	0.393001i	$-0.128563\pi$
$743$	37246.3	1.83908	0.919538	+	0.393001i	$0.128563\pi$
$744$	0	0
$745$	−1394.51	−0.0685781
$746$	−36709.9	−1.80167
$747$	0	0
$748$	−1823.51	−0.0891365
$749$	−48427.7	−2.36250
$750$	0	0
$751$	−22231.0	−1.08019	−0.540093	−	0.841606i	$-0.681611\pi$
$751$	−22231.0	−1.08019	−0.540093	+	0.841606i	$0.681611\pi$
$752$	16721.6	0.810868
$753$	0	0
$754$	−53487.5	−2.58342
$755$	2159.76	0.104108
$756$	0	0
$757$	−11212.1	−0.538324	−0.269162	−	0.963095i	$-0.586747\pi$
$757$	−11212.1	−0.538324	−0.269162	+	0.963095i	$0.586747\pi$
$758$	−41686.5	−1.99752
$759$	0	0
$760$	675.647	0.0322478
$761$	−13682.9	−0.651778	−0.325889	−	0.945408i	$-0.605664\pi$
$761$	−13682.9	−0.651778	−0.325889	+	0.945408i	$0.605664\pi$
$762$	0	0
$763$	−14721.6	−0.698504
$764$	8531.05	0.403982
$765$	0	0
$766$	−25695.8	−1.21205
$767$	12574.7	0.591977
$768$	0	0
$769$	13674.3	0.641231	0.320616	−	0.947209i	$-0.396110\pi$
$769$	13674.3	0.641231	0.320616	+	0.947209i	$0.396110\pi$
$770$	2574.80	0.120506
$771$	0	0
$772$	22122.3	1.03135
$773$	−28378.8	−1.32046	−0.660230	−	0.751064i	$-0.729541\pi$
$773$	−28378.8	−1.32046	−0.660230	+	0.751064i	$0.729541\pi$
$774$	0	0
$775$	4769.35	0.221058
$776$	6502.25	0.300795
$777$	0	0
$778$	−4862.92	−0.224093
$779$	−3335.87	−0.153427
$780$	0	0
$781$	−2102.79	−0.0963426
$782$	19900.8	0.910040
$783$	0	0
$784$	−25219.9	−1.14887
$785$	2225.54	0.101188
$786$	0	0
$787$	28443.0	1.28829	0.644144	−	0.764904i	$-0.277214\pi$
$787$	28443.0	1.28829	0.644144	+	0.764904i	$0.277214\pi$
$788$	−4873.79	−0.220332
$789$	0	0
$790$	1522.89	0.0685848
$791$	62261.9	2.79871
$792$	0	0
$793$	−44028.3	−1.97162
$794$	3419.71	0.152848
$795$	0	0
$796$	−4551.58	−0.202671
$797$	−28414.4	−1.26285	−0.631425	−	0.775437i	$-0.717530\pi$
$797$	−28414.4	−1.26285	−0.631425	+	0.775437i	$0.717530\pi$
$798$	0	0
$799$	−12211.9	−0.540707
$800$	6027.49	0.266380
$801$	0	0
$802$	−47637.3	−2.09742
$803$	581.572	0.0255582
$804$	0	0
$805$	−12936.1	−0.566382
$806$	60761.5	2.65537
$807$	0	0
$808$	8358.32	0.363916
$809$	7610.15	0.330728	0.165364	−	0.986233i	$-0.447120\pi$
$809$	7610.15	0.330728	0.165364	+	0.986233i	$0.447120\pi$
$810$	0	0
$811$	25751.6	1.11499	0.557497	−	0.830179i	$-0.311762\pi$
$811$	25751.6	1.11499	0.557497	+	0.830179i	$0.311762\pi$
$812$	30175.0	1.30411
$813$	0	0
$814$	−7156.03	−0.308131
$815$	12667.8	0.544459
$816$	0	0
$817$	12052.0	0.516093
$818$	48695.6	2.08142
$819$	0	0
$820$	−3812.03	−0.162344
$821$	21133.1	0.898355	0.449178	−	0.893442i	$-0.351717\pi$
$821$	21133.1	0.898355	0.449178	+	0.893442i	$0.351717\pi$
$822$	0	0
$823$	−3127.25	−0.132453	−0.0662266	−	0.997805i	$-0.521096\pi$
$823$	−3127.25	−0.132453	−0.0662266	+	0.997805i	$0.521096\pi$
$824$	216.095	0.00913597
$825$	0	0
$826$	−15409.8	−0.649121
$827$	−493.085	−0.0207331	−0.0103665	−	0.999946i	$-0.503300\pi$
$827$	−493.085	−0.0207331	−0.0103665	+	0.999946i	$0.503300\pi$
$828$	0	0
$829$	2832.98	0.118689	0.0593447	−	0.998238i	$-0.481099\pi$
$829$	2832.98	0.118689	0.0593447	+	0.998238i	$0.481099\pi$
$830$	27478.1	1.14913
$831$	0	0
$832$	29128.8	1.21378
$833$	18418.3	0.766094
$834$	0	0
$835$	10827.4	0.448741
$836$	1035.26	0.0428291
$837$	0	0
$838$	54874.3	2.26205
$839$	−10246.1	−0.421615	−0.210807	−	0.977528i	$-0.567609\pi$
$839$	−10246.1	−0.421615	−0.210807	+	0.977528i	$0.567609\pi$
$840$	0	0
$841$	3813.44	0.156359
$842$	−9801.62	−0.401171
$843$	0	0
$844$	−33845.1	−1.38033
$845$	−23228.8	−0.945676
$846$	0	0
$847$	34363.4	1.39403
$848$	−29566.7	−1.19732
$849$	0	0
$850$	−5062.89	−0.204301
$851$	35952.8	1.44823
$852$	0	0
$853$	−24332.4	−0.976699	−0.488350	−	0.872648i	$-0.662401\pi$
$853$	−24332.4	−0.976699	−0.488350	+	0.872648i	$0.662401\pi$
$854$	53954.8	2.16194
$855$	0	0
$856$	8323.71	0.332358
$857$	−25593.3	−1.02013	−0.510065	−	0.860136i	$-0.670379\pi$
$857$	−25593.3	−1.02013	−0.510065	+	0.860136i	$0.670379\pi$
$858$	0	0
$859$	31050.0	1.23331	0.616654	−	0.787234i	$-0.288488\pi$
$859$	31050.0	1.23331	0.616654	+	0.787234i	$0.288488\pi$
$860$	13772.3	0.546085
$861$	0	0
$862$	11814.2	0.466815
$863$	13274.3	0.523596	0.261798	−	0.965123i	$-0.415685\pi$
$863$	13274.3	0.523596	0.261798	+	0.965123i	$0.415685\pi$
$864$	0	0
$865$	6818.47	0.268017
$866$	53834.5	2.11244
$867$	0	0
$868$	−34278.7	−1.34043
$869$	−401.851	−0.0156868
$870$	0	0
$871$	−50741.4	−1.97395
$872$	2530.34	0.0982661
$873$	0	0
$874$	−11298.3	−0.437264
$875$	3291.02	0.127151
$876$	0	0
$877$	51094.2	1.96731	0.983653	−	0.180074i	$-0.0576336\pi$
$877$	51094.2	1.96731	0.983653	+	0.180074i	$0.0576336\pi$
$878$	−29618.1	−1.13845
$879$	0	0
$880$	−1829.32	−0.0700753
$881$	13020.2	0.497913	0.248956	−	0.968515i	$-0.419912\pi$
$881$	13020.2	0.497913	0.248956	+	0.968515i	$0.419912\pi$
$882$	0	0
$883$	−4577.13	−0.174443	−0.0872213	−	0.996189i	$-0.527799\pi$
$883$	−4577.13	−0.174443	−0.0872213	+	0.996189i	$0.527799\pi$
$884$	−29693.7	−1.12976
$885$	0	0
$886$	137.936	0.00523029
$887$	−14999.8	−0.567806	−0.283903	−	0.958853i	$-0.591629\pi$
$887$	−14999.8	−0.567806	−0.283903	+	0.958853i	$0.591629\pi$
$888$	0	0
$889$	63278.0	2.38726
$890$	−8664.76	−0.326341
$891$	0	0
$892$	7669.39	0.287881
$893$	6933.03	0.259804
$894$	0	0
$895$	−5991.43	−0.223767
$896$	15085.6	0.562472
$897$	0	0
$898$	10019.1	0.372317
$899$	−32037.8	−1.18857
$900$	0	0
$901$	21592.8	0.798400
$902$	2185.02	0.0806576
$903$	0	0
$904$	−10701.5	−0.393725
$905$	−5490.67	−0.201675
$906$	0	0
$907$	−37611.0	−1.37691	−0.688453	−	0.725281i	$-0.741710\pi$
$907$	−37611.0	−1.37691	−0.688453	+	0.725281i	$0.741710\pi$
$908$	8361.00	0.305583
$909$	0	0
$910$	41927.6	1.52735
$911$	20919.3	0.760799	0.380400	−	0.924822i	$-0.375786\pi$
$911$	20919.3	0.760799	0.380400	+	0.924822i	$0.375786\pi$
$912$	0	0
$913$	−7250.75	−0.262831
$914$	−31386.2	−1.13585
$915$	0	0
$916$	−24420.0	−0.880849
$917$	−43041.5	−1.55001
$918$	0	0
$919$	18211.5	0.653689	0.326845	−	0.945078i	$-0.394015\pi$
$919$	18211.5	0.653689	0.326845	+	0.945078i	$0.394015\pi$
$920$	2223.44	0.0796791
$921$	0	0
$922$	25979.1	0.927956
$923$	−34241.4	−1.22109
$924$	0	0
$925$	−9146.61	−0.325123
$926$	−3228.68	−0.114580
$927$	0	0
$928$	−40489.3	−1.43225
$929$	−10338.5	−0.365119	−0.182559	−	0.983195i	$-0.558438\pi$
$929$	−10338.5	−0.365119	−0.182559	+	0.983195i	$0.558438\pi$
$930$	0	0
$931$	−10456.6	−0.368100
$932$	−3597.03	−0.126421
$933$	0	0
$934$	−27927.9	−0.978403
$935$	1335.96	0.0467280
$936$	0	0
$937$	−4046.02	−0.141065	−0.0705324	−	0.997509i	$-0.522470\pi$
$937$	−4046.02	−0.141065	−0.0705324	+	0.997509i	$0.522470\pi$
$938$	62181.5	2.16450
$939$	0	0
$940$	7922.64	0.274902
$941$	7226.11	0.250334	0.125167	−	0.992136i	$-0.460053\pi$
$941$	7226.11	0.250334	0.125167	+	0.992136i	$0.460053\pi$
$942$	0	0
$943$	−10977.8	−0.379095
$944$	10948.2	0.377471
$945$	0	0
$946$	−7894.18	−0.271313
$947$	47671.3	1.63581	0.817903	−	0.575356i	$-0.195137\pi$
$947$	47671.3	1.63581	0.817903	+	0.575356i	$0.195137\pi$
$948$	0	0
$949$	9470.22	0.323937
$950$	2874.35	0.0981643
$951$	0	0
$952$	−6266.58	−0.213341
$953$	−50498.6	−1.71649	−0.858243	−	0.513243i	$-0.828444\pi$
$953$	−50498.6	−1.71649	−0.858243	+	0.513243i	$0.828444\pi$
$954$	0	0
$955$	−6250.13	−0.211780
$956$	−657.482	−0.0222432
$957$	0	0
$958$	39857.5	1.34419
$959$	66515.0	2.23971
$960$	0	0
$961$	6603.78	0.221670
$962$	−116528.	−3.90540
$963$	0	0
$964$	6305.65	0.210676
$965$	−16207.6	−0.540663
$966$	0	0
$967$	23665.3	0.786996	0.393498	−	0.919325i	$-0.371265\pi$
$967$	23665.3	0.786996	0.393498	+	0.919325i	$0.371265\pi$
$968$	−5906.35	−0.196113
$969$	0	0
$970$	27661.9	0.915640
$971$	29787.4	0.984472	0.492236	−	0.870462i	$-0.336180\pi$
$971$	29787.4	0.984472	0.492236	+	0.870462i	$0.336180\pi$
$972$	0	0
$973$	−49782.5	−1.64024
$974$	−20031.9	−0.658997
$975$	0	0
$976$	−38333.3	−1.25719
$977$	11957.2	0.391551	0.195776	−	0.980649i	$-0.437278\pi$
$977$	11957.2	0.391551	0.195776	+	0.980649i	$0.437278\pi$
$978$	0	0
$979$	2286.41	0.0746413
$980$	−11949.2	−0.389491
$981$	0	0
$982$	9473.69	0.307859
$983$	15937.0	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$983$	15937.0	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$984$	0	0
$985$	3570.70	0.115505
$986$	34009.6	1.09847
$987$	0	0
$988$	16858.0	0.542837
$989$	39661.3	1.27518
$990$	0	0
$991$	42686.5	1.36830	0.684148	−	0.729343i	$-0.260174\pi$
$991$	42686.5	1.36830	0.684148	+	0.729343i	$0.260174\pi$
$992$	45995.6	1.47214
$993$	0	0
$994$	41961.4	1.33897
$995$	3334.64	0.106246
$996$	0	0
$997$	41369.5	1.31413	0.657063	−	0.753836i	$-0.271798\pi$
$997$	41369.5	1.31413	0.657063	+	0.753836i	$0.271798\pi$
$998$	49500.5	1.57005
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.i.1.1	yes	3
3.2	odd	2		405.4.a.g.1.3	✓	3
5.4	even	2		2025.4.a.p.1.3		3
9.2	odd	6		405.4.e.u.271.1		6
9.4	even	3		405.4.e.s.136.3		6
9.5	odd	6		405.4.e.u.136.1		6
9.7	even	3		405.4.e.s.271.3		6
15.14	odd	2		2025.4.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.g.1.3	✓	3	3.2	odd	2
405.4.a.i.1.1	yes	3	1.1	even	1	trivial
405.4.e.s.136.3		6	9.4	even	3
405.4.e.s.271.3		6	9.7	even	3
405.4.e.u.136.1		6	9.5	odd	6
405.4.e.u.271.1		6	9.2	odd	6
2025.4.a.p.1.3		3	5.4	even	2
2025.4.a.r.1.1		3	15.14	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-3.85028 q^{2} +6.82469 q^{4} -5.00000 q^{5} -26.3282 q^{7} +4.52526 q^{8} +O(q^{10})\)	\(q-3.85028 q^{2} +6.82469 q^{4} -5.00000 q^{5} -26.3282 q^{7} +4.52526 q^{8} +19.2514 q^{10} -5.07995 q^{11} -82.7210 q^{13} +101.371 q^{14} -72.0211 q^{16} +52.5976 q^{17} -29.8611 q^{19} -34.1235 q^{20} +19.5593 q^{22} -98.2680 q^{23} +25.0000 q^{25} +318.499 q^{26} -179.682 q^{28} -167.936 q^{29} +190.774 q^{31} +241.100 q^{32} -202.516 q^{34} +131.641 q^{35} -365.864 q^{37} +114.974 q^{38} -22.6263 q^{40} +111.713 q^{41} -403.603 q^{43} -34.6691 q^{44} +378.360 q^{46} -232.176 q^{47} +350.174 q^{49} -96.2571 q^{50} -564.546 q^{52} +410.528 q^{53} +25.3997 q^{55} -119.142 q^{56} +646.601 q^{58} -152.013 q^{59} +532.250 q^{61} -734.535 q^{62} -352.134 q^{64} +413.605 q^{65} +613.405 q^{67} +358.962 q^{68} -506.855 q^{70} +413.938 q^{71} -114.484 q^{73} +1408.68 q^{74} -203.793 q^{76} +133.746 q^{77} +79.1053 q^{79} +360.106 q^{80} -430.126 q^{82} +1427.33 q^{83} -262.988 q^{85} +1553.99 q^{86} -22.9881 q^{88} -450.084 q^{89} +2177.90 q^{91} -670.649 q^{92} +893.943 q^{94} +149.306 q^{95} +1436.88 q^{97} -1348.27 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8}+O(q^{10}) \) 3 * q + q^2 + 5 * q^4 - 15 * q^5 - 25 * q^7 - 27 * q^8	\( 3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8} - 5 q^{10} + 58 q^{11} - 47 q^{13} + 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 25 q^{20} + 260 q^{22} - 51 q^{23} + 75 q^{25} + 253 q^{26} + 83 q^{28} + 350 q^{29} + 638 q^{31} + 245 q^{32} - 154 q^{34} + 125 q^{35} - 414 q^{37} + 397 q^{38} + 135 q^{40} + 179 q^{41} - 836 q^{43} + 332 q^{44} + 261 q^{46} + 235 q^{47} + 892 q^{49} + 25 q^{50} - 1335 q^{52} + 505 q^{53} - 290 q^{55} + 15 q^{56} + 1876 q^{58} + 535 q^{59} - 104 q^{61} + 348 q^{62} - 303 q^{64} + 235 q^{65} - 40 q^{67} + 830 q^{68} - 795 q^{70} + 452 q^{71} - 710 q^{73} + 1394 q^{74} + 849 q^{76} + 2148 q^{77} - 634 q^{79} + 635 q^{80} + 613 q^{82} + 1734 q^{83} - 170 q^{85} + 460 q^{86} - 768 q^{88} - 852 q^{89} - 1229 q^{91} - 1839 q^{92} + 1751 q^{94} + 25 q^{95} + 38 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 - 15 * q^5 - 25 * q^7 - 27 * q^8 - 5 * q^10 + 58 * q^11 - 47 * q^13 + 159 * q^14 - 127 * q^16 + 34 * q^17 - 5 * q^19 - 25 * q^20 + 260 * q^22 - 51 * q^23 + 75 * q^25 + 253 * q^26 + 83 * q^28 + 350 * q^29 + 638 * q^31 + 245 * q^32 - 154 * q^34 + 125 * q^35 - 414 * q^37 + 397 * q^38 + 135 * q^40 + 179 * q^41 - 836 * q^43 + 332 * q^44 + 261 * q^46 + 235 * q^47 + 892 * q^49 + 25 * q^50 - 1335 * q^52 + 505 * q^53 - 290 * q^55 + 15 * q^56 + 1876 * q^58 + 535 * q^59 - 104 * q^61 + 348 * q^62 - 303 * q^64 + 235 * q^65 - 40 * q^67 + 830 * q^68 - 795 * q^70 + 452 * q^71 - 710 * q^73 + 1394 * q^74 + 849 * q^76 + 2148 * q^77 - 634 * q^79 + 635 * q^80 + 613 * q^82 + 1734 * q^83 - 170 * q^85 + 460 * q^86 - 768 * q^88 - 852 * q^89 - 1229 * q^91 - 1839 * q^92 + 1751 * q^94 + 25 * q^95 + 38 * q^98

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