LMFDB - Embedded newform 405.4.a.g.1.3

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:	$3$
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			1.3
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	$-0.261702\pi$
$2$	3.85028	1.36128	0.680641	+	0.732617i	$0.261702\pi$
$3$	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	$-0.477821\pi$
$11$	5.07995	0.139242	0.0696210	+	0.997574i	$0.477821\pi$
$12$	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	$-0.622426\pi$
$17$	−52.5976	−0.750399	−0.375199	+	0.926944i	$0.622426\pi$
$18$	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	$-0.353047\pi$
$23$	98.2680	0.890883	0.445441	+	0.895311i	$0.353047\pi$
$24$	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	$-0.319304\pi$
$29$	167.936	1.07534	0.537671	+	0.843155i	$0.319304\pi$
$30$	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	$-0.568246\pi$
$41$	−111.713	−0.425527	−0.212763	+	0.977104i	$0.568246\pi$
$42$	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	$-0.382681\pi$
$47$	232.176	0.720560	0.360280	+	0.932844i	$0.382681\pi$
$48$	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	$-0.678553\pi$
$53$	−410.528	−1.06397	−0.531984	+	0.846754i	$0.678553\pi$
$54$	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	$-0.446361\pi$
$59$	152.013	0.335431	0.167716	+	0.985835i	$0.446361\pi$
$60$	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	$-0.612445\pi$
$71$	−413.938	−0.691908	−0.345954	+	0.938252i	$0.612445\pi$
$72$	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	$-0.892770\pi$
$83$	−1427.33	−1.88758	−0.943792	+	0.330539i	$0.892770\pi$
$84$	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	$-0.413628\pi$
$89$	450.084	0.536054	0.268027	+	0.963411i	$0.413628\pi$
$90$	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	$-0.863792\pi$
$101$	−1847.03	−1.81967	−0.909835	+	0.414969i	$0.863792\pi$
$102$	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	$-0.812194\pi$
$107$	−1839.39	−1.66187	−0.830936	+	0.556368i	$0.812194\pi$
$108$	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	$-0.0563721\pi$
$113$	2364.84	1.96872	0.984359	+	0.176174i	$0.0563721\pi$
$114$	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	$-0.683534\pi$
$131$	−1634.81	−1.09033	−0.545167	+	0.838327i	$0.683534\pi$
$132$	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	$-0.211246\pi$
$137$	2526.38	1.57550	0.787749	+	0.615996i	$0.211246\pi$
$138$	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	$-0.524430\pi$
$149$	−278.901	−0.153345	−0.0766727	+	0.997056i	$0.524430\pi$
$150$	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	$-0.332705\pi$
$167$	2165.49	1.00341	0.501707	+	0.865037i	$0.332705\pi$
$168$	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	$-0.403129\pi$
$173$	1363.69	0.599305	0.299653	+	0.954048i	$0.403129\pi$
$174$	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	$-0.580489\pi$
$179$	−1198.29	−0.500358	−0.250179	+	0.968200i	$0.580489\pi$
$180$	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	$-0.576091\pi$
$191$	−1250.03	−0.473554	−0.236777	+	0.971564i	$0.576091\pi$
$192$	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	$-0.458779\pi$
$197$	714.141	0.258276	0.129138	+	0.991627i	$0.458779\pi$
$198$	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	$-0.557320\pi$
$227$	−1225.11	−0.358209	−0.179104	+	0.983830i	$0.557320\pi$
$228$	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	$-0.476393\pi$
$233$	527.061	0.148193	0.0740963	+	0.997251i	$0.476393\pi$
$234$	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	$-0.495850\pi$
$239$	96.3386	0.0260738	0.0130369	+	0.999915i	$0.495850\pi$
$240$	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	$-0.674030\pi$
$251$	−4134.83	−1.03979	−0.519897	+	0.854229i	$0.674030\pi$
$252$	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	$-0.430409\pi$
$257$	1787.18	0.433778	0.216889	+	0.976196i	$0.430409\pi$
$258$	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	$-0.715701\pi$
$263$	−5348.15	−1.25392	−0.626960	+	0.779052i	$0.715701\pi$
$264$	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	$-0.113731\pi$
$269$	8266.59	1.87369	0.936846	+	0.349743i	$0.113731\pi$
$270$	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	$-0.689985\pi$
$281$	−5294.93	−1.12409	−0.562045	+	0.827107i	$0.689985\pi$
$282$	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	$-0.627632\pi$
$293$	−3915.07	−0.780617	−0.390309	+	0.920684i	$0.627632\pi$
$294$	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	$-0.565380\pi$
$311$	−2237.22	−0.407913	−0.203957	+	0.978980i	$0.565380\pi$
$312$	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	$-0.473449\pi$
$317$	940.460	0.166629	0.0833147	+	0.996523i	$0.473449\pi$
$318$	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	$-0.811316\pi$
$347$	−10722.3	−1.65879	−0.829397	+	0.558660i	$0.811316\pi$
$348$	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	$-0.784100\pi$
$353$	−10328.6	−1.55732	−0.778659	+	0.627447i	$0.784100\pi$
$354$	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	$-0.473710\pi$
$359$	1122.33	0.164998	0.0824992	+	0.996591i	$0.473710\pi$
$360$	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	$-0.646862\pi$
$383$	−6673.74	−0.890371	−0.445185	+	0.895438i	$0.646862\pi$
$384$	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	$-0.526230\pi$
$389$	−1263.00	−0.164619	−0.0823094	+	0.996607i	$0.526230\pi$
$390$	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	$-0.779936\pi$
$401$	−12372.4	−1.54077	−0.770385	+	0.637580i	$0.779936\pi$
$402$	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	$-0.187852\pi$
$419$	14252.0	1.66171	0.830855	+	0.556490i	$0.187852\pi$
$420$	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	$-0.445151\pi$
$431$	3068.41	0.342923	0.171462	+	0.985191i	$0.445151\pi$
$432$	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	$-0.499388\pi$
$443$	35.8248	0.00384218	0.00192109	+	0.999998i	$0.499388\pi$
$444$	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	$-0.456334\pi$
$449$	2602.16	0.273505	0.136752	+	0.990605i	$0.456334\pi$
$450$	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	$-0.389289\pi$
$461$	6747.31	0.681678	0.340839	+	0.940122i	$0.389289\pi$
$462$	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	$-0.617008\pi$
$467$	−7253.46	−0.718737	−0.359369	+	0.933196i	$0.617008\pi$
$468$	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	$-0.335635\pi$
$479$	10351.8	0.987447	0.493724	+	0.869619i	$0.335635\pi$
$480$	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	$-0.463929\pi$
$491$	2460.52	0.226154	0.113077	+	0.993586i	$0.463929\pi$
$492$	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	$-0.0859697\pi$
$503$	21744.3	1.92750	0.963749	+	0.266810i	$0.0859697\pi$
$504$	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	$-0.360538\pi$
$509$	9743.82	0.848501	0.424251	+	0.905545i	$0.360538\pi$
$510$	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	$-0.229012\pi$
$521$	17889.5	1.50432	0.752161	+	0.658980i	$0.229012\pi$
$522$	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	$-0.642610\pi$
$557$	−11389.0	−0.866371	−0.433185	+	0.901305i	$0.642610\pi$
$558$	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	$-0.664778\pi$
$563$	−13221.1	−0.989704	−0.494852	+	0.868977i	$0.664778\pi$
$564$	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	$-0.400575\pi$
$569$	8341.79	0.614597	0.307299	+	0.951613i	$0.400575\pi$
$570$	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	$-0.818094\pi$
$587$	−23924.2	−1.68221	−0.841105	+	0.540872i	$0.818094\pi$
$588$	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	$-0.635034\pi$
$593$	−11887.8	−0.823226	−0.411613	+	0.911359i	$0.635034\pi$
$594$	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	$-0.446472\pi$
$599$	4907.42	0.334744	0.167372	+	0.985894i	$0.446472\pi$
$600$	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	$-0.520972\pi$
$617$	−2018.05	−0.131675	−0.0658375	+	0.997830i	$0.520972\pi$
$618$	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	$-0.604385\pi$
$641$	−10454.2	−0.644177	−0.322089	+	0.946709i	$0.604385\pi$
$642$	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	$-0.364123\pi$
$647$	13627.3	0.828045	0.414022	+	0.910267i	$0.364123\pi$
$648$	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	$-0.525347\pi$
$653$	−2654.67	−0.159089	−0.0795446	+	0.996831i	$0.525347\pi$
$654$	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	$-0.681422\pi$
$659$	−18256.8	−1.07919	−0.539593	+	0.841926i	$0.681422\pi$
$660$	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	$-0.235778\pi$
$677$	25999.2	1.47597	0.737984	+	0.674818i	$0.235778\pi$
$678$	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	$-0.624593\pi$
$683$	−13619.4	−0.763004	−0.381502	+	0.924368i	$0.624593\pi$
$684$	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	$-0.596694\pi$
$701$	−11103.4	−0.598244	−0.299122	+	0.954215i	$0.596694\pi$
$702$	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	$-0.804414\pi$
$719$	−31506.0	−1.63418	−0.817089	+	0.576511i	$0.804414\pi$
$720$	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	$-0.871437\pi$
$743$	−37246.3	−1.83908	−0.919538	+	0.393001i	$0.871437\pi$
$744$	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	$-0.394336\pi$
$761$	13682.9	0.651778	0.325889	+	0.945408i	$0.394336\pi$
$762$	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	$-0.270459\pi$
$773$	28378.8	1.32046	0.660230	+	0.751064i	$0.270459\pi$
$774$	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	$-0.282470\pi$
$797$	28414.4	1.26285	0.631425	+	0.775437i	$0.282470\pi$
$798$	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	$-0.552880\pi$
$809$	−7610.15	−0.330728	−0.165364	+	0.986233i	$0.552880\pi$
$810$	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	$-0.648283\pi$
$821$	−21133.1	−0.898355	−0.449178	+	0.893442i	$0.648283\pi$
$822$	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	$-0.496700\pi$
$827$	493.085	0.0207331	0.0103665	+	0.999946i	$0.496700\pi$
$828$	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	$-0.432391\pi$
$839$	10246.1	0.421615	0.210807	+	0.977528i	$0.432391\pi$
$840$	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	$-0.329621\pi$
$857$	25593.3	1.02013	0.510065	+	0.860136i	$0.329621\pi$
$858$	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	$-0.584315\pi$
$863$	−13274.3	−0.523596	−0.261798	+	0.965123i	$0.584315\pi$
$864$	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	$-0.580088\pi$
$881$	−13020.2	−0.497913	−0.248956	+	0.968515i	$0.580088\pi$
$882$	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	$-0.408371\pi$
$887$	14999.8	0.567806	0.283903	+	0.958853i	$0.408371\pi$
$888$	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	$-0.624214\pi$
$911$	−20919.3	−0.760799	−0.380400	+	0.924822i	$0.624214\pi$
$912$	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	$-0.441562\pi$
$929$	10338.5	0.365119	0.182559	+	0.983195i	$0.441562\pi$
$930$	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	$-0.539947\pi$
$941$	−7226.11	−0.250334	−0.125167	+	0.992136i	$0.539947\pi$
$942$	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	$-0.804863\pi$
$947$	−47671.3	−1.63581	−0.817903	+	0.575356i	$0.804863\pi$
$948$	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	$-0.171556\pi$
$953$	50498.6	1.71649	0.858243	+	0.513243i	$0.171556\pi$
$954$	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	$-0.663820\pi$
$971$	−29787.4	−0.984472	−0.492236	+	0.870462i	$0.663820\pi$
$972$	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	$-0.562722\pi$
$977$	−11957.2	−0.391551	−0.195776	+	0.980649i	$0.562722\pi$
$978$	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	$-0.583245\pi$
$983$	−15937.0	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$984$	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.g.1.3	✓	3
3.2	odd	2		405.4.a.i.1.1	yes	3
5.4	even	2		2025.4.a.r.1.1		3
9.2	odd	6		405.4.e.s.271.3		6
9.4	even	3		405.4.e.u.136.1		6
9.5	odd	6		405.4.e.s.136.3		6
9.7	even	3		405.4.e.u.271.1		6
15.14	odd	2		2025.4.a.p.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.g.1.3	✓	3	1.1	even	1	trivial
405.4.a.i.1.1	yes	3	3.2	odd	2
405.4.e.s.136.3		6	9.5	odd	6
405.4.e.s.271.3		6	9.2	odd	6
405.4.e.u.136.1		6	9.4	even	3
405.4.e.u.271.1		6	9.7	even	3
2025.4.a.p.1.3		3	15.14	odd	2
2025.4.a.r.1.1		3	5.4	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
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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