LMFDB - Embedded newform 405.4.a.k.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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 $ x^6 - 2x^5 - 38x^4 + 42x^3 + 393x^2 - 72*x - 432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.23336$ 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)

$f(q)$	$=$	$q-5.23336 q^{2} +19.3881 q^{4} -5.00000 q^{5} +33.0180 q^{7} -59.5981 q^{8} +O(q^{10})$	\(q-5.23336 q^{2} +19.3881 q^{4} -5.00000 q^{5} +33.0180 q^{7} -59.5981 q^{8} +26.1668 q^{10} -6.08311 q^{11} -64.4507 q^{13} -172.795 q^{14} +156.794 q^{16} +76.9845 q^{17} -118.716 q^{19} -96.9405 q^{20} +31.8351 q^{22} -77.6004 q^{23} +25.0000 q^{25} +337.294 q^{26} +640.157 q^{28} -62.8149 q^{29} -106.914 q^{31} -343.774 q^{32} -402.888 q^{34} -165.090 q^{35} +108.268 q^{37} +621.285 q^{38} +297.990 q^{40} -142.766 q^{41} +339.568 q^{43} -117.940 q^{44} +406.111 q^{46} +598.346 q^{47} +747.191 q^{49} -130.834 q^{50} -1249.58 q^{52} -488.041 q^{53} +30.4155 q^{55} -1967.81 q^{56} +328.733 q^{58} -242.826 q^{59} -499.339 q^{61} +559.519 q^{62} +544.744 q^{64} +322.254 q^{65} -921.635 q^{67} +1492.58 q^{68} +863.977 q^{70} +60.6882 q^{71} -338.439 q^{73} -566.606 q^{74} -2301.68 q^{76} -200.852 q^{77} -556.311 q^{79} -783.969 q^{80} +747.147 q^{82} +64.8420 q^{83} -384.923 q^{85} -1777.08 q^{86} +362.542 q^{88} -941.159 q^{89} -2128.04 q^{91} -1504.52 q^{92} -3131.36 q^{94} +593.581 q^{95} -1042.17 q^{97} -3910.32 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8}+O(q^{10}) $ 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8	$ 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + 20 q^{10} - 88 q^{11} + 20 q^{13} - 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} - 170 q^{20} - 74 q^{22} - 210 q^{23} + 150 q^{25} - 4 q^{26} + 352 q^{28} - 296 q^{29} - 104 q^{31} - 722 q^{32} - 428 q^{34} - 200 q^{35} - 204 q^{37} + 20 q^{38} + 330 q^{40} - 344 q^{41} + 512 q^{43} - 716 q^{44} - 186 q^{46} - 238 q^{47} + 68 q^{49} - 100 q^{50} - 468 q^{52} - 850 q^{53} + 440 q^{55} - 2316 q^{56} + 890 q^{58} - 1840 q^{59} - 364 q^{61} - 1038 q^{62} - 990 q^{64} - 100 q^{65} + 88 q^{67} - 236 q^{68} + 900 q^{70} - 1364 q^{71} + 836 q^{73} - 1316 q^{74} - 2106 q^{76} - 840 q^{77} - 680 q^{79} - 290 q^{80} + 1742 q^{82} - 2148 q^{83} + 620 q^{85} - 2872 q^{86} + 1296 q^{88} - 3000 q^{89} - 3058 q^{91} - 1002 q^{92} - 3662 q^{94} + 230 q^{95} - 612 q^{97} - 1982 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8 + 20 * q^10 - 88 * q^11 + 20 * q^13 - 180 * q^14 + 58 * q^16 - 124 * q^17 - 46 * q^19 - 170 * q^20 - 74 * q^22 - 210 * q^23 + 150 * q^25 - 4 * q^26 + 352 * q^28 - 296 * q^29 - 104 * q^31 - 722 * q^32 - 428 * q^34 - 200 * q^35 - 204 * q^37 + 20 * q^38 + 330 * q^40 - 344 * q^41 + 512 * q^43 - 716 * q^44 - 186 * q^46 - 238 * q^47 + 68 * q^49 - 100 * q^50 - 468 * q^52 - 850 * q^53 + 440 * q^55 - 2316 * q^56 + 890 * q^58 - 1840 * q^59 - 364 * q^61 - 1038 * q^62 - 990 * q^64 - 100 * q^65 + 88 * q^67 - 236 * q^68 + 900 * q^70 - 1364 * q^71 + 836 * q^73 - 1316 * q^74 - 2106 * q^76 - 840 * q^77 - 680 * q^79 - 290 * q^80 + 1742 * q^82 - 2148 * q^83 + 620 * q^85 - 2872 * q^86 + 1296 * q^88 - 3000 * q^89 - 3058 * q^91 - 1002 * q^92 - 3662 * q^94 + 230 * q^95 - 612 * q^97 - 1982 * 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 $

$2$	−5.23336	−1.85027	−0.925137	−	0.379634i	$-0.876050\pi$
$2$	−5.23336	−1.85027	−0.925137	+	0.379634i	$0.876050\pi$
$3$	0	0
$3$	0	0
$4$	19.3881	2.42351
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	33.0180	1.78281	0.891403	−	0.453211i	$-0.149722\pi$
$7$	33.0180	1.78281	0.891403	+	0.453211i	$0.149722\pi$
$8$	−59.5981	−2.63389
$9$	0	0
$10$	26.1668	0.827468
$11$	−6.08311	−0.166739	−0.0833694	−	0.996519i	$-0.526568\pi$
$11$	−6.08311	−0.166739	−0.0833694	+	0.996519i	$0.526568\pi$
$12$	0	0
$13$	−64.4507	−1.37503	−0.687516	−	0.726169i	$-0.741299\pi$
$13$	−64.4507	−1.37503	−0.687516	+	0.726169i	$0.741299\pi$
$14$	−172.795	−3.29868
$15$	0	0
$16$	156.794	2.44990
$17$	76.9845	1.09832	0.549162	−	0.835716i	$-0.314947\pi$
$17$	76.9845	1.09832	0.549162	+	0.835716i	$0.314947\pi$
$18$	0	0
$19$	−118.716	−1.43344	−0.716720	−	0.697361i	$-0.754357\pi$
$19$	−118.716	−1.43344	−0.716720	+	0.697361i	$0.754357\pi$
$20$	−96.9405	−1.08383
$21$	0	0
$22$	31.8351	0.308512
$23$	−77.6004	−0.703513	−0.351757	−	0.936092i	$-0.614416\pi$
$23$	−77.6004	−0.703513	−0.351757	+	0.936092i	$0.614416\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	337.294	2.54419
$27$	0	0
$28$	640.157	4.32065
$29$	−62.8149	−0.402222	−0.201111	−	0.979568i	$-0.564455\pi$
$29$	−62.8149	−0.402222	−0.201111	+	0.979568i	$0.564455\pi$
$30$	0	0
$31$	−106.914	−0.619429	−0.309714	−	0.950830i	$-0.600233\pi$
$31$	−106.914	−0.619429	−0.309714	+	0.950830i	$0.600233\pi$
$32$	−343.774	−1.89910
$33$	0	0
$34$	−402.888	−2.03220
$35$	−165.090	−0.797295
$36$	0	0
$37$	108.268	0.481058	0.240529	−	0.970642i	$-0.422679\pi$
$37$	108.268	0.481058	0.240529	+	0.970642i	$0.422679\pi$
$38$	621.285	2.65226
$39$	0	0
$40$	297.990	1.17791
$41$	−142.766	−0.543813	−0.271906	−	0.962324i	$-0.587654\pi$
$41$	−142.766	−0.543813	−0.271906	+	0.962324i	$0.587654\pi$
$42$	0	0
$43$	339.568	1.20427	0.602136	−	0.798394i	$-0.294317\pi$
$43$	339.568	1.20427	0.602136	+	0.798394i	$0.294317\pi$
$44$	−117.940	−0.404093
$45$	0	0
$46$	406.111	1.30169
$47$	598.346	1.85697	0.928486	−	0.371368i	$-0.121111\pi$
$47$	598.346	1.85697	0.928486	+	0.371368i	$0.121111\pi$
$48$	0	0
$49$	747.191	2.17840
$50$	−130.834	−0.370055
$51$	0	0
$52$	−1249.58	−3.33241
$53$	−488.041	−1.26486	−0.632431	−	0.774617i	$-0.717943\pi$
$53$	−488.041	−1.26486	−0.632431	+	0.774617i	$0.717943\pi$
$54$	0	0
$55$	30.4155	0.0745678
$56$	−1967.81	−4.69571
$57$	0	0
$58$	328.733	0.744221
$59$	−242.826	−0.535819	−0.267909	−	0.963444i	$-0.586333\pi$
$59$	−242.826	−0.535819	−0.267909	+	0.963444i	$0.586333\pi$
$60$	0	0
$61$	−499.339	−1.04809	−0.524047	−	0.851689i	$-0.675578\pi$
$61$	−499.339	−1.04809	−0.524047	+	0.851689i	$0.675578\pi$
$62$	559.519	1.14611
$63$	0	0
$64$	544.744	1.06395
$65$	322.254	0.614933
$66$	0	0
$67$	−921.635	−1.68053	−0.840266	−	0.542175i	$-0.817601\pi$
$67$	−921.635	−1.68053	−0.840266	+	0.542175i	$0.817601\pi$
$68$	1492.58	2.66180
$69$	0	0
$70$	863.977	1.47521
$71$	60.6882	0.101442	0.0507209	−	0.998713i	$-0.483848\pi$
$71$	60.6882	0.101442	0.0507209	+	0.998713i	$0.483848\pi$
$72$	0	0
$73$	−338.439	−0.542621	−0.271311	−	0.962492i	$-0.587457\pi$
$73$	−338.439	−0.542621	−0.271311	+	0.962492i	$0.587457\pi$
$74$	−566.606	−0.890090
$75$	0	0
$76$	−2301.68	−3.47396
$77$	−200.852	−0.297263
$78$	0	0
$79$	−556.311	−0.792276	−0.396138	−	0.918191i	$-0.629650\pi$
$79$	−556.311	−0.792276	−0.396138	+	0.918191i	$0.629650\pi$
$80$	−783.969	−1.09563
$81$	0	0
$82$	747.147	1.00620
$83$	64.8420	0.0857510	0.0428755	−	0.999080i	$-0.486348\pi$
$83$	64.8420	0.0857510	0.0428755	+	0.999080i	$0.486348\pi$
$84$	0	0
$85$	−384.923	−0.491185
$86$	−1777.08	−2.22823
$87$	0	0
$88$	362.542	0.439171
$89$	−941.159	−1.12093	−0.560465	−	0.828178i	$-0.689377\pi$
$89$	−941.159	−1.12093	−0.560465	+	0.828178i	$0.689377\pi$
$90$	0	0
$91$	−2128.04	−2.45142
$92$	−1504.52	−1.70497
$93$	0	0
$94$	−3131.36	−3.43591
$95$	593.581	0.641054
$96$	0	0
$97$	−1042.17	−1.09089	−0.545445	−	0.838147i	$-0.683639\pi$
$97$	−1042.17	−1.09089	−0.545445	+	0.838147i	$0.683639\pi$
$98$	−3910.32	−4.03064
$99$	0	0
$100$	484.703	0.484703
$101$	−682.243	−0.672135	−0.336068	−	0.941838i	$-0.609097\pi$
$101$	−682.243	−0.672135	−0.336068	+	0.941838i	$0.609097\pi$
$102$	0	0
$103$	1213.66	1.16103	0.580514	−	0.814250i	$-0.302852\pi$
$103$	1213.66	1.16103	0.580514	+	0.814250i	$0.302852\pi$
$104$	3841.14	3.62168
$105$	0	0
$106$	2554.10	2.34034
$107$	1311.93	1.18532	0.592658	−	0.805454i	$-0.298079\pi$
$107$	1311.93	1.18532	0.592658	+	0.805454i	$0.298079\pi$
$108$	0	0
$109$	−294.780	−0.259035	−0.129518	−	0.991577i	$-0.541343\pi$
$109$	−294.780	−0.259035	−0.129518	+	0.991577i	$0.541343\pi$
$110$	−159.176	−0.137971
$111$	0	0
$112$	5177.02	4.36770
$113$	−1585.12	−1.31961	−0.659805	−	0.751437i	$-0.729361\pi$
$113$	−1585.12	−1.31961	−0.659805	+	0.751437i	$0.729361\pi$
$114$	0	0
$115$	388.002	0.314621
$116$	−1217.86	−0.974790
$117$	0	0
$118$	1270.80	0.991412
$119$	2541.88	1.95810
$120$	0	0
$121$	−1294.00	−0.972198
$122$	2613.22	1.93926
$123$	0	0
$124$	−2072.86	−1.50119
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−149.176	−0.104230	−0.0521149	−	0.998641i	$-0.516596\pi$
$127$	−149.176	−0.104230	−0.0521149	+	0.998641i	$0.516596\pi$
$128$	−100.655	−0.0695057
$129$	0	0
$130$	−1686.47	−1.13779
$131$	−1171.77	−0.781513	−0.390757	−	0.920494i	$-0.627787\pi$
$131$	−1171.77	−0.781513	−0.390757	+	0.920494i	$0.627787\pi$
$132$	0	0
$133$	−3919.77	−2.55555
$134$	4823.25	3.10944
$135$	0	0
$136$	−4588.13	−2.89286
$137$	1158.63	0.722541	0.361270	−	0.932461i	$-0.382343\pi$
$137$	1158.63	0.722541	0.361270	+	0.932461i	$0.382343\pi$
$138$	0	0
$139$	1586.52	0.968110	0.484055	−	0.875038i	$-0.339163\pi$
$139$	1586.52	0.968110	0.484055	+	0.875038i	$0.339163\pi$
$140$	−3200.79	−1.93226
$141$	0	0
$142$	−317.603	−0.187695
$143$	392.061	0.229271
$144$	0	0
$145$	314.075	0.179879
$146$	1771.18	1.00400
$147$	0	0
$148$	2099.11	1.16585
$149$	−517.109	−0.284317	−0.142158	−	0.989844i	$-0.545404\pi$
$149$	−517.109	−0.284317	−0.142158	+	0.989844i	$0.545404\pi$
$150$	0	0
$151$	555.586	0.299423	0.149712	−	0.988730i	$-0.452165\pi$
$151$	555.586	0.299423	0.149712	+	0.988730i	$0.452165\pi$
$152$	7075.25	3.77552
$153$	0	0
$154$	1051.13	0.550018
$155$	534.569	0.277017
$156$	0	0
$157$	1054.03	0.535802	0.267901	−	0.963446i	$-0.413670\pi$
$157$	1054.03	0.535802	0.267901	+	0.963446i	$0.413670\pi$
$158$	2911.38	1.46593
$159$	0	0
$160$	1718.87	0.849304
$161$	−2562.21	−1.25423
$162$	0	0
$163$	−635.916	−0.305575	−0.152788	−	0.988259i	$-0.548825\pi$
$163$	−635.916	−0.305575	−0.152788	+	0.988259i	$0.548825\pi$
$164$	−2767.96	−1.31794
$165$	0	0
$166$	−339.342	−0.158663
$167$	−1365.11	−0.632549	−0.316274	−	0.948668i	$-0.602432\pi$
$167$	−1365.11	−0.632549	−0.316274	+	0.948668i	$0.602432\pi$
$168$	0	0
$169$	1956.90	0.890713
$170$	2014.44	0.908827
$171$	0	0
$172$	6583.59	2.91857
$173$	−1134.93	−0.498770	−0.249385	−	0.968404i	$-0.580228\pi$
$173$	−1134.93	−0.498770	−0.249385	+	0.968404i	$0.580228\pi$
$174$	0	0
$175$	825.451	0.356561
$176$	−953.793	−0.408494
$177$	0	0
$178$	4925.43	2.07403
$179$	−1813.33	−0.757179	−0.378589	−	0.925565i	$-0.623591\pi$
$179$	−1813.33	−0.757179	−0.378589	+	0.925565i	$0.623591\pi$
$180$	0	0
$181$	2334.37	0.958634	0.479317	−	0.877642i	$-0.340884\pi$
$181$	2334.37	0.958634	0.479317	+	0.877642i	$0.340884\pi$
$182$	11136.8	4.53579
$183$	0	0
$184$	4624.84	1.85298
$185$	−541.340	−0.215136
$186$	0	0
$187$	−468.305	−0.183133
$188$	11600.8	4.50039
$189$	0	0
$190$	−3106.42	−1.18612
$191$	−442.644	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$191$	−442.644	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$192$	0	0
$193$	−3702.54	−1.38090	−0.690452	−	0.723378i	$-0.742589\pi$
$193$	−3702.54	−1.38090	−0.690452	+	0.723378i	$0.742589\pi$
$194$	5454.05	2.01844
$195$	0	0
$196$	14486.6	5.27938
$197$	−4491.92	−1.62455	−0.812274	−	0.583276i	$-0.801771\pi$
$197$	−4491.92	−1.62455	−0.812274	+	0.583276i	$0.801771\pi$
$198$	0	0
$199$	−2934.08	−1.04518	−0.522591	−	0.852584i	$-0.675034\pi$
$199$	−2934.08	−1.04518	−0.522591	+	0.852584i	$0.675034\pi$
$200$	−1489.95	−0.526778
$201$	0	0
$202$	3570.42	1.24363
$203$	−2074.03	−0.717084
$204$	0	0
$205$	713.830	0.243200
$206$	−6351.54	−2.14822
$207$	0	0
$208$	−10105.5	−3.36869
$209$	722.163	0.239010
$210$	0	0
$211$	−673.940	−0.219886	−0.109943	−	0.993938i	$-0.535067\pi$
$211$	−673.940	−0.219886	−0.109943	+	0.993938i	$0.535067\pi$
$212$	−9462.20	−3.06541
$213$	0	0
$214$	−6865.79	−2.19316
$215$	−1697.84	−0.538567
$216$	0	0
$217$	−3530.08	−1.10432
$218$	1542.69	0.479286
$219$	0	0
$220$	589.700	0.180716
$221$	−4961.71	−1.51023
$222$	0	0
$223$	178.516	0.0536068	0.0268034	−	0.999641i	$-0.491467\pi$
$223$	178.516	0.0536068	0.0268034	+	0.999641i	$0.491467\pi$
$224$	−11350.7	−3.38573
$225$	0	0
$226$	8295.53	2.44164
$227$	−5052.11	−1.47718	−0.738590	−	0.674154i	$-0.764508\pi$
$227$	−5052.11	−1.47718	−0.738590	+	0.674154i	$0.764508\pi$
$228$	0	0
$229$	1308.97	0.377725	0.188863	−	0.982004i	$-0.439520\pi$
$229$	1308.97	0.377725	0.188863	+	0.982004i	$0.439520\pi$
$230$	−2030.56	−0.582134
$231$	0	0
$232$	3743.65	1.05941
$233$	6591.30	1.85326	0.926632	−	0.375970i	$-0.122691\pi$
$233$	6591.30	1.85326	0.926632	+	0.375970i	$0.122691\pi$
$234$	0	0
$235$	−2991.73	−0.830463
$236$	−4707.95	−1.29856
$237$	0	0
$238$	−13302.6	−3.62302
$239$	−1584.50	−0.428841	−0.214421	−	0.976741i	$-0.568786\pi$
$239$	−1584.50	−0.428841	−0.214421	+	0.976741i	$0.568786\pi$
$240$	0	0
$241$	−1674.47	−0.447560	−0.223780	−	0.974640i	$-0.571840\pi$
$241$	−1674.47	−0.447560	−0.223780	+	0.974640i	$0.571840\pi$
$242$	6771.95	1.79883
$243$	0	0
$244$	−9681.23	−2.54007
$245$	−3735.95	−0.974210
$246$	0	0
$247$	7651.34	1.97102
$248$	6371.86	1.63151
$249$	0	0
$250$	654.171	0.165494
$251$	1200.75	0.301955	0.150978	−	0.988537i	$-0.451758\pi$
$251$	1200.75	0.301955	0.150978	+	0.988537i	$0.451758\pi$
$252$	0	0
$253$	472.052	0.117303
$254$	780.690	0.192854
$255$	0	0
$256$	−3831.19	−0.935350
$257$	5206.69	1.26375	0.631876	−	0.775070i	$-0.282285\pi$
$257$	5206.69	1.26375	0.631876	+	0.775070i	$0.282285\pi$
$258$	0	0
$259$	3574.80	0.857634
$260$	6247.89	1.49030
$261$	0	0
$262$	6132.31	1.44601
$263$	−7683.56	−1.80148	−0.900739	−	0.434362i	$-0.856974\pi$
$263$	−7683.56	−1.80148	−0.900739	+	0.434362i	$0.856974\pi$
$264$	0	0
$265$	2440.21	0.565663
$266$	20513.6	4.72846
$267$	0	0
$268$	−17868.7	−4.07279
$269$	624.277	0.141497	0.0707487	−	0.997494i	$-0.477461\pi$
$269$	624.277	0.141497	0.0707487	+	0.997494i	$0.477461\pi$
$270$	0	0
$271$	−1462.51	−0.327828	−0.163914	−	0.986475i	$-0.552412\pi$
$271$	−1462.51	−0.327828	−0.163914	+	0.986475i	$0.552412\pi$
$272$	12070.7	2.69078
$273$	0	0
$274$	−6063.51	−1.33690
$275$	−152.078	−0.0333477
$276$	0	0
$277$	−5898.04	−1.27935	−0.639673	−	0.768647i	$-0.720930\pi$
$277$	−5898.04	−1.27935	−0.639673	+	0.768647i	$0.720930\pi$
$278$	−8302.86	−1.79127
$279$	0	0
$280$	9839.06	2.09999
$281$	5272.91	1.11941	0.559707	−	0.828691i	$-0.310914\pi$
$281$	5272.91	1.11941	0.559707	+	0.828691i	$0.310914\pi$
$282$	0	0
$283$	−1119.26	−0.235100	−0.117550	−	0.993067i	$-0.537504\pi$
$283$	−1119.26	−0.235100	−0.117550	+	0.993067i	$0.537504\pi$
$284$	1176.63	0.245845
$285$	0	0
$286$	−2051.80	−0.424214
$287$	−4713.86	−0.969513
$288$	0	0
$289$	1013.62	0.206313
$290$	−1643.67	−0.332826
$291$	0	0
$292$	−6561.70	−1.31505
$293$	940.740	0.187572	0.0937860	−	0.995592i	$-0.470103\pi$
$293$	940.740	0.187572	0.0937860	+	0.995592i	$0.470103\pi$
$294$	0	0
$295$	1214.13	0.239626
$296$	−6452.57	−1.26705
$297$	0	0
$298$	2706.22	0.526064
$299$	5001.40	0.967353
$300$	0	0
$301$	11211.9	2.14698
$302$	−2907.58	−0.554015
$303$	0	0
$304$	−18613.9	−3.51179
$305$	2496.69	0.468722
$306$	0	0
$307$	1931.88	0.359148	0.179574	−	0.983744i	$-0.442528\pi$
$307$	1931.88	0.359148	0.179574	+	0.983744i	$0.442528\pi$
$308$	−3894.15	−0.720421
$309$	0	0
$310$	−2797.59	−0.512557
$311$	6298.16	1.14835	0.574174	−	0.818734i	$-0.305323\pi$
$311$	6298.16	1.14835	0.574174	+	0.818734i	$0.305323\pi$
$312$	0	0
$313$	−1818.81	−0.328450	−0.164225	−	0.986423i	$-0.552512\pi$
$313$	−1818.81	−0.328450	−0.164225	+	0.986423i	$0.552512\pi$
$314$	−5516.13	−0.991380
$315$	0	0
$316$	−10785.8	−1.92009
$317$	−3158.64	−0.559643	−0.279821	−	0.960052i	$-0.590275\pi$
$317$	−3158.64	−0.559643	−0.279821	+	0.960052i	$0.590275\pi$
$318$	0	0
$319$	382.110	0.0670660
$320$	−2723.72	−0.475815
$321$	0	0
$322$	13409.0	2.32066
$323$	−9139.31	−1.57438
$324$	0	0
$325$	−1611.27	−0.275006
$326$	3327.98	0.565398
$327$	0	0
$328$	8508.59	1.43234
$329$	19756.2	3.31062
$330$	0	0
$331$	−953.090	−0.158268	−0.0791338	−	0.996864i	$-0.525215\pi$
$331$	−953.090	−0.158268	−0.0791338	+	0.996864i	$0.525215\pi$
$332$	1257.16	0.207819
$333$	0	0
$334$	7144.14	1.17039
$335$	4608.17	0.751556
$336$	0	0
$337$	−568.893	−0.0919572	−0.0459786	−	0.998942i	$-0.514641\pi$
$337$	−568.893	−0.0919572	−0.0459786	+	0.998942i	$0.514641\pi$
$338$	−10241.1	−1.64806
$339$	0	0
$340$	−7462.92	−1.19039
$341$	650.368	0.103283
$342$	0	0
$343$	13345.6	2.10086
$344$	−20237.6	−3.17192
$345$	0	0
$346$	5939.50	0.922860
$347$	−8426.55	−1.30363	−0.651817	−	0.758376i	$-0.725993\pi$
$347$	−8426.55	−1.30363	−0.651817	+	0.758376i	$0.725993\pi$
$348$	0	0
$349$	7143.96	1.09572	0.547861	−	0.836569i	$-0.315442\pi$
$349$	7143.96	1.09572	0.547861	+	0.836569i	$0.315442\pi$
$350$	−4319.89	−0.659736
$351$	0	0
$352$	2091.21	0.316654
$353$	−5300.69	−0.799228	−0.399614	−	0.916684i	$-0.630856\pi$
$353$	−5300.69	−0.799228	−0.399614	+	0.916684i	$0.630856\pi$
$354$	0	0
$355$	−303.441	−0.0453661
$356$	−18247.3	−2.71659
$357$	0	0
$358$	9489.84	1.40099
$359$	9536.35	1.40198	0.700988	−	0.713173i	$-0.252743\pi$
$359$	9536.35	1.40198	0.700988	+	0.713173i	$0.252743\pi$
$360$	0	0
$361$	7234.52	1.05475
$362$	−12216.6	−1.77373
$363$	0	0
$364$	−41258.6	−5.94104
$365$	1692.20	0.242668
$366$	0	0
$367$	−730.208	−0.103860	−0.0519299	−	0.998651i	$-0.516537\pi$
$367$	−730.208	−0.103860	−0.0519299	+	0.998651i	$0.516537\pi$
$368$	−12167.3	−1.72354
$369$	0	0
$370$	2833.03	0.398060
$371$	−16114.2	−2.25500
$372$	0	0
$373$	12868.9	1.78640	0.893201	−	0.449657i	$-0.148454\pi$
$373$	12868.9	1.78640	0.893201	+	0.449657i	$0.148454\pi$
$374$	2450.81	0.338846
$375$	0	0
$376$	−35660.3	−4.89106
$377$	4048.47	0.553068
$378$	0	0
$379$	−4000.88	−0.542246	−0.271123	−	0.962545i	$-0.587395\pi$
$379$	−4000.88	−0.542246	−0.271123	+	0.962545i	$0.587395\pi$
$380$	11508.4	1.55360
$381$	0	0
$382$	2316.52	0.310270
$383$	65.3199	0.00871459	0.00435730	−	0.999991i	$-0.498613\pi$
$383$	65.3199	0.00871459	0.00435730	+	0.999991i	$0.498613\pi$
$384$	0	0
$385$	1004.26	0.132940
$386$	19376.7	2.55505
$387$	0	0
$388$	−20205.7	−2.64379
$389$	−5919.77	−0.771579	−0.385790	−	0.922587i	$-0.626071\pi$
$389$	−5919.77	−0.771579	−0.385790	+	0.922587i	$0.626071\pi$
$390$	0	0
$391$	−5974.03	−0.772685
$392$	−44531.2	−5.73766
$393$	0	0
$394$	23507.8	3.00586
$395$	2781.55	0.354317
$396$	0	0
$397$	−10228.7	−1.29310	−0.646551	−	0.762871i	$-0.723789\pi$
$397$	−10228.7	−1.29310	−0.646551	+	0.762871i	$0.723789\pi$
$398$	15355.1	1.93387
$399$	0	0
$400$	3919.84	0.489980
$401$	−12245.1	−1.52492	−0.762460	−	0.647036i	$-0.776008\pi$
$401$	−12245.1	−1.52492	−0.762460	+	0.647036i	$0.776008\pi$
$402$	0	0
$403$	6890.67	0.851734
$404$	−13227.4	−1.62893
$405$	0	0
$406$	10854.1	1.32680
$407$	−658.606	−0.0802110
$408$	0	0
$409$	9832.95	1.18877	0.594386	−	0.804180i	$-0.297395\pi$
$409$	9832.95	1.18877	0.594386	+	0.804180i	$0.297395\pi$
$410$	−3735.73	−0.449987
$411$	0	0
$412$	23530.6	2.81377
$413$	−8017.65	−0.955262
$414$	0	0
$415$	−324.210	−0.0383490
$416$	22156.5	2.61132
$417$	0	0
$418$	−3779.34	−0.442234
$419$	−12315.1	−1.43588	−0.717941	−	0.696104i	$-0.754915\pi$
$419$	−12315.1	−1.43588	−0.717941	+	0.696104i	$0.754915\pi$
$420$	0	0
$421$	−4318.42	−0.499921	−0.249961	−	0.968256i	$-0.580418\pi$
$421$	−4318.42	−0.499921	−0.249961	+	0.968256i	$0.580418\pi$
$422$	3526.97	0.406849
$423$	0	0
$424$	29086.3	3.33150
$425$	1924.61	0.219665
$426$	0	0
$427$	−16487.2	−1.86855
$428$	25435.8	2.87263
$429$	0	0
$430$	8885.42	0.996496
$431$	−5265.39	−0.588457	−0.294228	−	0.955735i	$-0.595063\pi$
$431$	−5265.39	−0.588457	−0.294228	+	0.955735i	$0.595063\pi$
$432$	0	0
$433$	5855.07	0.649831	0.324916	−	0.945743i	$-0.394664\pi$
$433$	5855.07	0.649831	0.324916	+	0.945743i	$0.394664\pi$
$434$	18474.2	2.04330
$435$	0	0
$436$	−5715.23	−0.627775
$437$	9212.42	1.00844
$438$	0	0
$439$	4022.69	0.437341	0.218671	−	0.975799i	$-0.429828\pi$
$439$	4022.69	0.437341	0.218671	+	0.975799i	$0.429828\pi$
$440$	−1812.71	−0.196403
$441$	0	0
$442$	25966.4	2.79434
$443$	−7502.23	−0.804608	−0.402304	−	0.915506i	$-0.631791\pi$
$443$	−7502.23	−0.804608	−0.402304	+	0.915506i	$0.631791\pi$
$444$	0	0
$445$	4705.80	0.501295
$446$	−934.240	−0.0991873
$447$	0	0
$448$	17986.4	1.89682
$449$	18574.2	1.95227	0.976135	−	0.217166i	$-0.0696813\pi$
$449$	18574.2	1.95227	0.976135	+	0.217166i	$0.0696813\pi$
$450$	0	0
$451$	868.461	0.0906746
$452$	−30732.5	−3.19809
$453$	0	0
$454$	26439.5	2.73319
$455$	10640.2	1.09631
$456$	0	0
$457$	14145.6	1.44792	0.723962	−	0.689840i	$-0.242319\pi$
$457$	14145.6	1.44792	0.723962	+	0.689840i	$0.242319\pi$
$458$	−6850.31	−0.698895
$459$	0	0
$460$	7522.62	0.762487
$461$	4910.75	0.496131	0.248066	−	0.968743i	$-0.420205\pi$
$461$	4910.75	0.496131	0.248066	+	0.968743i	$0.420205\pi$
$462$	0	0
$463$	9255.45	0.929022	0.464511	−	0.885567i	$-0.346230\pi$
$463$	9255.45	0.929022	0.464511	+	0.885567i	$0.346230\pi$
$464$	−9848.98	−0.985404
$465$	0	0
$466$	−34494.7	−3.42904
$467$	2825.10	0.279936	0.139968	−	0.990156i	$-0.455300\pi$
$467$	2825.10	0.279936	0.139968	+	0.990156i	$0.455300\pi$
$468$	0	0
$469$	−30430.6	−2.99606
$470$	15656.8	1.53658
$471$	0	0
$472$	14472.0	1.41129
$473$	−2065.63	−0.200799
$474$	0	0
$475$	−2967.90	−0.286688
$476$	49282.2	4.74547
$477$	0	0
$478$	8292.29	0.793474
$479$	−9173.48	−0.875046	−0.437523	−	0.899207i	$-0.644144\pi$
$479$	−9173.48	−0.875046	−0.437523	+	0.899207i	$0.644144\pi$
$480$	0	0
$481$	−6977.95	−0.661470
$482$	8763.10	0.828109
$483$	0	0
$484$	−25088.1	−2.35613
$485$	5210.85	0.487861
$486$	0	0
$487$	13663.7	1.27138	0.635690	−	0.771945i	$-0.280716\pi$
$487$	13663.7	1.27138	0.635690	+	0.771945i	$0.280716\pi$
$488$	29759.6	2.76056
$489$	0	0
$490$	19551.6	1.80255
$491$	5850.31	0.537721	0.268860	−	0.963179i	$-0.413353\pi$
$491$	5850.31	0.537721	0.268860	+	0.963179i	$0.413353\pi$
$492$	0	0
$493$	−4835.78	−0.441770
$494$	−40042.2	−3.64694
$495$	0	0
$496$	−16763.4	−1.51754
$497$	2003.81	0.180851
$498$	0	0
$499$	3842.97	0.344760	0.172380	−	0.985031i	$-0.444854\pi$
$499$	3842.97	0.344760	0.172380	+	0.985031i	$0.444854\pi$
$500$	−2423.51	−0.216766
$501$	0	0
$502$	−6283.97	−0.558700
$503$	−4242.88	−0.376104	−0.188052	−	0.982159i	$-0.560217\pi$
$503$	−4242.88	−0.376104	−0.188052	+	0.982159i	$0.560217\pi$
$504$	0	0
$505$	3411.21	0.300588
$506$	−2470.42	−0.217042
$507$	0	0
$508$	−2892.23	−0.252602
$509$	−7365.60	−0.641404	−0.320702	−	0.947180i	$-0.603919\pi$
$509$	−7365.60	−0.641404	−0.320702	+	0.947180i	$0.603919\pi$
$510$	0	0
$511$	−11174.6	−0.967388
$512$	20855.3	1.80016
$513$	0	0
$514$	−27248.5	−2.33829
$515$	−6068.32	−0.519227
$516$	0	0
$517$	−3639.80	−0.309629
$518$	−18708.2	−1.58686
$519$	0	0
$520$	−19205.7	−1.61966
$521$	8717.87	0.733084	0.366542	−	0.930401i	$-0.380542\pi$
$521$	8717.87	0.733084	0.366542	+	0.930401i	$0.380542\pi$
$522$	0	0
$523$	23515.4	1.96607	0.983037	−	0.183407i	$-0.0587127\pi$
$523$	23515.4	1.96607	0.983037	+	0.183407i	$0.0587127\pi$
$524$	−22718.4	−1.89401
$525$	0	0
$526$	40210.9	3.33323
$527$	−8230.71	−0.680333
$528$	0	0
$529$	−6145.18	−0.505069
$530$	−12770.5	−1.04663
$531$	0	0
$532$	−75997.0	−6.19340
$533$	9201.38	0.747760
$534$	0	0
$535$	−6559.63	−0.530089
$536$	54927.7	4.42633
$537$	0	0
$538$	−3267.07	−0.261809
$539$	−4545.24	−0.363224
$540$	0	0
$541$	3803.94	0.302300	0.151150	−	0.988511i	$-0.451702\pi$
$541$	3803.94	0.302300	0.151150	+	0.988511i	$0.451702\pi$
$542$	7653.87	0.606572
$543$	0	0
$544$	−26465.3	−2.08583
$545$	1473.90	0.115844
$546$	0	0
$547$	17335.9	1.35508	0.677540	−	0.735486i	$-0.263046\pi$
$547$	17335.9	1.35508	0.677540	+	0.735486i	$0.263046\pi$
$548$	22463.6	1.75109
$549$	0	0
$550$	795.878	0.0617025
$551$	7457.14	0.576561
$552$	0	0
$553$	−18368.3	−1.41248
$554$	30866.6	2.36714
$555$	0	0
$556$	30759.7	2.34623
$557$	−7797.62	−0.593170	−0.296585	−	0.955006i	$-0.595848\pi$
$557$	−7797.62	−0.593170	−0.296585	+	0.955006i	$0.595848\pi$
$558$	0	0
$559$	−21885.4	−1.65591
$560$	−25885.1	−1.95330
$561$	0	0
$562$	−27595.1	−2.07122
$563$	11535.6	0.863532	0.431766	−	0.901986i	$-0.357891\pi$
$563$	11535.6	0.863532	0.431766	+	0.901986i	$0.357891\pi$
$564$	0	0
$565$	7925.62	0.590147
$566$	5857.52	0.435000
$567$	0	0
$568$	−3616.90	−0.267186
$569$	10077.7	0.742496	0.371248	−	0.928534i	$-0.378930\pi$
$569$	10077.7	0.742496	0.371248	+	0.928534i	$0.378930\pi$
$570$	0	0
$571$	−16339.3	−1.19751	−0.598756	−	0.800932i	$-0.704338\pi$
$571$	−16339.3	−1.19751	−0.598756	+	0.800932i	$0.704338\pi$
$572$	7601.31	0.555641
$573$	0	0
$574$	24669.3	1.79386
$575$	−1940.01	−0.140703
$576$	0	0
$577$	18648.6	1.34550	0.672748	−	0.739871i	$-0.265114\pi$
$577$	18648.6	1.34550	0.672748	+	0.739871i	$0.265114\pi$
$578$	−5304.63	−0.381736
$579$	0	0
$580$	6089.31	0.435939
$581$	2140.95	0.152877
$582$	0	0
$583$	2968.81	0.210901
$584$	20170.3	1.42920
$585$	0	0
$586$	−4923.24	−0.347060
$587$	18097.1	1.27248	0.636241	−	0.771491i	$-0.280488\pi$
$587$	18097.1	1.27248	0.636241	+	0.771491i	$0.280488\pi$
$588$	0	0
$589$	12692.4	0.887913
$590$	−6354.00	−0.443373
$591$	0	0
$592$	16975.7	1.17855
$593$	−27154.1	−1.88041	−0.940207	−	0.340602i	$-0.889369\pi$
$593$	−27154.1	−1.88041	−0.940207	+	0.340602i	$0.889369\pi$
$594$	0	0
$595$	−12709.4	−0.875688
$596$	−10025.8	−0.689046
$597$	0	0
$598$	−26174.2	−1.78987
$599$	−3457.46	−0.235840	−0.117920	−	0.993023i	$-0.537623\pi$
$599$	−3457.46	−0.235840	−0.117920	+	0.993023i	$0.537623\pi$
$600$	0	0
$601$	−2938.43	−0.199436	−0.0997179	−	0.995016i	$-0.531794\pi$
$601$	−2938.43	−0.199436	−0.0997179	+	0.995016i	$0.531794\pi$
$602$	−58675.9	−3.97251
$603$	0	0
$604$	10771.8	0.725656
$605$	6469.98	0.434780
$606$	0	0
$607$	−7785.94	−0.520628	−0.260314	−	0.965524i	$-0.583826\pi$
$607$	−7785.94	−0.520628	−0.260314	+	0.965524i	$0.583826\pi$
$608$	40811.5	2.72225
$609$	0	0
$610$	−13066.1	−0.867264
$611$	−38563.8	−2.55339
$612$	0	0
$613$	23893.4	1.57430	0.787151	−	0.616760i	$-0.211555\pi$
$613$	23893.4	1.57430	0.787151	+	0.616760i	$0.211555\pi$
$614$	−10110.3	−0.664522
$615$	0	0
$616$	11970.4	0.782957
$617$	4880.46	0.318444	0.159222	−	0.987243i	$-0.449101\pi$
$617$	4880.46	0.318444	0.159222	+	0.987243i	$0.449101\pi$
$618$	0	0
$619$	15883.5	1.03136	0.515681	−	0.856780i	$-0.327539\pi$
$619$	15883.5	1.03136	0.515681	+	0.856780i	$0.327539\pi$
$620$	10364.3	0.671354
$621$	0	0
$622$	−32960.6	−2.12476
$623$	−31075.2	−1.99840
$624$	0	0
$625$	625.000	0.0400000
$626$	9518.48	0.607723
$627$	0	0
$628$	20435.7	1.29852
$629$	8334.96	0.528357
$630$	0	0
$631$	−12862.4	−0.811479	−0.405740	−	0.913989i	$-0.632986\pi$
$631$	−12862.4	−0.811479	−0.405740	+	0.913989i	$0.632986\pi$
$632$	33155.0	2.08677
$633$	0	0
$634$	16530.3	1.03549
$635$	745.878	0.0466130
$636$	0	0
$637$	−48157.0	−2.99537
$638$	−1999.72	−0.124090
$639$	0	0
$640$	503.275	0.0310839
$641$	29485.7	1.81687	0.908435	−	0.418025i	$-0.137278\pi$
$641$	29485.7	1.81687	0.908435	+	0.418025i	$0.137278\pi$
$642$	0	0
$643$	−14114.6	−0.865673	−0.432836	−	0.901473i	$-0.642487\pi$
$643$	−14114.6	−0.865673	−0.432836	+	0.901473i	$0.642487\pi$
$644$	−49676.5	−3.03964
$645$	0	0
$646$	47829.3	2.91303
$647$	13313.3	0.808962	0.404481	−	0.914546i	$-0.367452\pi$
$647$	13313.3	0.808962	0.404481	+	0.914546i	$0.367452\pi$
$648$	0	0
$649$	1477.14	0.0893418
$650$	8432.35	0.508837
$651$	0	0
$652$	−12329.2	−0.740566
$653$	−11506.2	−0.689541	−0.344771	−	0.938687i	$-0.612043\pi$
$653$	−11506.2	−0.689541	−0.344771	+	0.938687i	$0.612043\pi$
$654$	0	0
$655$	5858.86	0.349503
$656$	−22384.8	−1.33229
$657$	0	0
$658$	−103391.	−6.12556
$659$	25771.6	1.52340	0.761699	−	0.647931i	$-0.224365\pi$
$659$	25771.6	1.52340	0.761699	+	0.647931i	$0.224365\pi$
$660$	0	0
$661$	7882.88	0.463855	0.231928	−	0.972733i	$-0.425497\pi$
$661$	7882.88	0.463855	0.231928	+	0.972733i	$0.425497\pi$
$662$	4987.87	0.292838
$663$	0	0
$664$	−3864.46	−0.225859
$665$	19598.9	1.14287
$666$	0	0
$667$	4874.46	0.282968
$668$	−26467.0	−1.53299
$669$	0	0
$670$	−24116.2	−1.39058
$671$	3037.53	0.174758
$672$	0	0
$673$	−19631.3	−1.12441	−0.562207	−	0.826997i	$-0.690048\pi$
$673$	−19631.3	−1.12441	−0.562207	+	0.826997i	$0.690048\pi$
$674$	2977.23	0.170146
$675$	0	0
$676$	37940.5	2.15865
$677$	−598.326	−0.0339668	−0.0169834	−	0.999856i	$-0.505406\pi$
$677$	−598.326	−0.0339668	−0.0169834	+	0.999856i	$0.505406\pi$
$678$	0	0
$679$	−34410.4	−1.94485
$680$	22940.7	1.29373
$681$	0	0
$682$	−3403.61	−0.191101
$683$	19041.9	1.06679	0.533394	−	0.845867i	$-0.320916\pi$
$683$	19041.9	1.06679	0.533394	+	0.845867i	$0.320916\pi$
$684$	0	0
$685$	−5793.13	−0.323130
$686$	−69842.4	−3.88716
$687$	0	0
$688$	53242.2	2.95035
$689$	31454.6	1.73922
$690$	0	0
$691$	−27170.8	−1.49584	−0.747921	−	0.663787i	$-0.768948\pi$
$691$	−27170.8	−1.49584	−0.747921	+	0.663787i	$0.768948\pi$
$692$	−22004.1	−1.20877
$693$	0	0
$694$	44099.2	2.41208
$695$	−7932.62	−0.432952
$696$	0	0
$697$	−10990.8	−0.597282
$698$	−37386.9	−2.02739
$699$	0	0
$700$	16003.9	0.864131
$701$	27091.0	1.45965	0.729824	−	0.683635i	$-0.239602\pi$
$701$	27091.0	1.45965	0.729824	+	0.683635i	$0.239602\pi$
$702$	0	0
$703$	−12853.2	−0.689568
$704$	−3313.74	−0.177402
$705$	0	0
$706$	27740.5	1.47879
$707$	−22526.3	−1.19829
$708$	0	0
$709$	−4954.60	−0.262445	−0.131223	−	0.991353i	$-0.541890\pi$
$709$	−4954.60	−0.262445	−0.131223	+	0.991353i	$0.541890\pi$
$710$	1588.02	0.0839398
$711$	0	0
$712$	56091.3	2.95240
$713$	8296.56	0.435776
$714$	0	0
$715$	−1960.30	−0.102533
$716$	−35157.1	−1.83503
$717$	0	0
$718$	−49907.2	−2.59404
$719$	23414.3	1.21447	0.607235	−	0.794522i	$-0.292279\pi$
$719$	23414.3	1.21447	0.607235	+	0.794522i	$0.292279\pi$
$720$	0	0
$721$	40072.8	2.06989
$722$	−37860.9	−1.95157
$723$	0	0
$724$	45259.1	2.32326
$725$	−1570.37	−0.0804444
$726$	0	0
$727$	30955.7	1.57921	0.789603	−	0.613618i	$-0.210286\pi$
$727$	30955.7	1.57921	0.789603	+	0.613618i	$0.210286\pi$
$728$	126827.	6.45676
$729$	0	0
$730$	−8855.88	−0.449001
$731$	26141.5	1.32268
$732$	0	0
$733$	15572.7	0.784709	0.392354	−	0.919814i	$-0.371661\pi$
$733$	15572.7	0.784709	0.392354	+	0.919814i	$0.371661\pi$
$734$	3821.45	0.192169
$735$	0	0
$736$	26677.0	1.33604
$737$	5606.40	0.280210
$738$	0	0
$739$	−30909.9	−1.53862	−0.769310	−	0.638876i	$-0.779400\pi$
$739$	−30909.9	−1.53862	−0.769310	+	0.638876i	$0.779400\pi$
$740$	−10495.6	−0.521384
$741$	0	0
$742$	84331.3	4.17237
$743$	−35023.2	−1.72931	−0.864656	−	0.502365i	$-0.832463\pi$
$743$	−35023.2	−1.72931	−0.864656	+	0.502365i	$0.832463\pi$
$744$	0	0
$745$	2585.55	0.127150
$746$	−67347.8	−3.30533
$747$	0	0
$748$	−9079.55	−0.443825
$749$	43317.3	2.11319
$750$	0	0
$751$	−3610.78	−0.175445	−0.0877226	−	0.996145i	$-0.527959\pi$
$751$	−3610.78	−0.175445	−0.0877226	+	0.996145i	$0.527959\pi$
$752$	93816.8	4.54940
$753$	0	0
$754$	−21187.1	−1.02333
$755$	−2777.93	−0.133906
$756$	0	0
$757$	−19982.2	−0.959399	−0.479700	−	0.877433i	$-0.659254\pi$
$757$	−19982.2	−0.959399	−0.479700	+	0.877433i	$0.659254\pi$
$758$	20938.1	1.00330
$759$	0	0
$760$	−35376.3	−1.68846
$761$	−35157.3	−1.67471	−0.837354	−	0.546662i	$-0.815898\pi$
$761$	−35157.3	−1.67471	−0.837354	+	0.546662i	$0.815898\pi$
$762$	0	0
$763$	−9733.07	−0.461810
$764$	−8582.02	−0.406396
$765$	0	0
$766$	−341.843	−0.0161244
$767$	15650.3	0.736768
$768$	0	0
$769$	−39041.2	−1.83077	−0.915384	−	0.402583i	$-0.868113\pi$
$769$	−39041.2	−1.83077	−0.915384	+	0.402583i	$0.868113\pi$
$770$	−5255.67	−0.245975
$771$	0	0
$772$	−71785.2	−3.34664
$773$	18244.4	0.848907	0.424453	−	0.905450i	$-0.360466\pi$
$773$	18244.4	0.848907	0.424453	+	0.905450i	$0.360466\pi$
$774$	0	0
$775$	−2672.85	−0.123886
$776$	62111.3	2.87328
$777$	0	0
$778$	30980.3	1.42763
$779$	16948.6	0.779522
$780$	0	0
$781$	−369.173	−0.0169143
$782$	31264.3	1.42968
$783$	0	0
$784$	117155.	5.33686
$785$	−5270.16	−0.239618
$786$	0	0
$787$	42888.2	1.94256	0.971282	−	0.237930i	$-0.0764690\pi$
$787$	42888.2	1.94256	0.971282	+	0.237930i	$0.0764690\pi$
$788$	−87089.8	−3.93711
$789$	0	0
$790$	−14556.9	−0.655583
$791$	−52337.7	−2.35261
$792$	0	0
$793$	32182.7	1.44116
$794$	53530.3	2.39259
$795$	0	0
$796$	−56886.2	−2.53301
$797$	11473.1	0.509911	0.254956	−	0.966953i	$-0.417939\pi$
$797$	11473.1	0.509911	0.254956	+	0.966953i	$0.417939\pi$
$798$	0	0
$799$	46063.3	2.03955
$800$	−8594.35	−0.379820
$801$	0	0
$802$	64083.2	2.82152
$803$	2058.76	0.0904759
$804$	0	0
$805$	12811.1	0.560908
$806$	−36061.4	−1.57594
$807$	0	0
$808$	40660.4	1.77033
$809$	−6454.45	−0.280502	−0.140251	−	0.990116i	$-0.544791\pi$
$809$	−6454.45	−0.280502	−0.140251	+	0.990116i	$0.544791\pi$
$810$	0	0
$811$	−11250.0	−0.487102	−0.243551	−	0.969888i	$-0.578312\pi$
$811$	−11250.0	−0.487102	−0.243551	+	0.969888i	$0.578312\pi$
$812$	−40211.4	−1.73786
$813$	0	0
$814$	3446.73	0.148412
$815$	3179.58	0.136658
$816$	0	0
$817$	−40312.2	−1.72625
$818$	−51459.4	−2.19955
$819$	0	0
$820$	13839.8	0.589399
$821$	−532.438	−0.0226336	−0.0113168	−	0.999936i	$-0.503602\pi$
$821$	−532.438	−0.0226336	−0.0113168	+	0.999936i	$0.503602\pi$
$822$	0	0
$823$	−33016.7	−1.39841	−0.699204	−	0.714922i	$-0.746462\pi$
$823$	−33016.7	−1.39841	−0.699204	+	0.714922i	$0.746462\pi$
$824$	−72332.0	−3.05802
$825$	0	0
$826$	41959.3	1.76750
$827$	−36231.3	−1.52344	−0.761720	−	0.647906i	$-0.775645\pi$
$827$	−36231.3	−1.52344	−0.761720	+	0.647906i	$0.775645\pi$
$828$	0	0
$829$	36588.3	1.53289	0.766445	−	0.642310i	$-0.222024\pi$
$829$	36588.3	1.53289	0.766445	+	0.642310i	$0.222024\pi$
$830$	1696.71	0.0709562
$831$	0	0
$832$	−35109.2	−1.46297
$833$	57522.1	2.39259
$834$	0	0
$835$	6825.57	0.282885
$836$	14001.4	0.579243
$837$	0	0
$838$	64449.7	2.65677
$839$	15326.8	0.630677	0.315339	−	0.948979i	$-0.397882\pi$
$839$	15326.8	0.630677	0.315339	+	0.948979i	$0.397882\pi$
$840$	0	0
$841$	−20443.3	−0.838218
$842$	22599.9	0.924992
$843$	0	0
$844$	−13066.4	−0.532896
$845$	−9784.48	−0.398339
$846$	0	0
$847$	−42725.2	−1.73324
$848$	−76521.8	−3.09879
$849$	0	0
$850$	−10072.2	−0.406440
$851$	−8401.64	−0.338431
$852$	0	0
$853$	4573.30	0.183572	0.0917859	−	0.995779i	$-0.470742\pi$
$853$	4573.30	0.183572	0.0917859	+	0.995779i	$0.470742\pi$
$854$	86283.4	3.45733
$855$	0	0
$856$	−78188.3	−3.12199
$857$	19569.2	0.780011	0.390006	−	0.920813i	$-0.372473\pi$
$857$	19569.2	0.780011	0.390006	+	0.920813i	$0.372473\pi$
$858$	0	0
$859$	21897.0	0.869750	0.434875	−	0.900491i	$-0.356792\pi$
$859$	21897.0	0.869750	0.434875	+	0.900491i	$0.356792\pi$
$860$	−32917.9	−1.30522
$861$	0	0
$862$	27555.7	1.08881
$863$	13335.5	0.526008	0.263004	−	0.964795i	$-0.415287\pi$
$863$	13335.5	0.526008	0.263004	+	0.964795i	$0.415287\pi$
$864$	0	0
$865$	5674.65	0.223057
$866$	−30641.7	−1.20237
$867$	0	0
$868$	−68441.7	−2.67634
$869$	3384.10	0.132103
$870$	0	0
$871$	59400.0	2.31078
$872$	17568.4	0.682270
$873$	0	0
$874$	−48211.9	−1.86590
$875$	−4127.26	−0.159459
$876$	0	0
$877$	−21811.8	−0.839830	−0.419915	−	0.907563i	$-0.637940\pi$
$877$	−21811.8	−0.839830	−0.419915	+	0.907563i	$0.637940\pi$
$878$	−21052.2	−0.809201
$879$	0	0
$880$	4768.97	0.182684
$881$	3646.54	0.139450	0.0697248	−	0.997566i	$-0.477788\pi$
$881$	3646.54	0.139450	0.0697248	+	0.997566i	$0.477788\pi$
$882$	0	0
$883$	9019.80	0.343760	0.171880	−	0.985118i	$-0.445016\pi$
$883$	9019.80	0.343760	0.171880	+	0.985118i	$0.445016\pi$
$884$	−96198.1	−3.66006
$885$	0	0
$886$	39261.9	1.48875
$887$	−19731.9	−0.746938	−0.373469	−	0.927643i	$-0.621832\pi$
$887$	−19731.9	−0.746938	−0.373469	+	0.927643i	$0.621832\pi$
$888$	0	0
$889$	−4925.49	−0.185822
$890$	−24627.2	−0.927533
$891$	0	0
$892$	3461.09	0.129917
$893$	−71033.3	−2.66186
$894$	0	0
$895$	9066.67	0.338621
$896$	−3323.43	−0.123915
$897$	0	0
$898$	−97205.4	−3.61223
$899$	6715.78	0.249148
$900$	0	0
$901$	−37571.6	−1.38923
$902$	−4544.98	−0.167773
$903$	0	0
$904$	94470.4	3.47571
$905$	−11671.9	−0.428714
$906$	0	0
$907$	7706.14	0.282115	0.141057	−	0.990001i	$-0.454950\pi$
$907$	7706.14	0.282115	0.141057	+	0.990001i	$0.454950\pi$
$908$	−97950.7	−3.57997
$909$	0	0
$910$	−55684.0	−2.02847
$911$	−24765.1	−0.900665	−0.450332	−	0.892861i	$-0.648694\pi$
$911$	−24765.1	−0.900665	−0.450332	+	0.892861i	$0.648694\pi$
$912$	0	0
$913$	−394.441	−0.0142980
$914$	−74028.8	−2.67905
$915$	0	0
$916$	25378.4	0.915422
$917$	−38689.6	−1.39329
$918$	0	0
$919$	−4899.44	−0.175862	−0.0879312	−	0.996127i	$-0.528026\pi$
$919$	−4899.44	−0.175862	−0.0879312	+	0.996127i	$0.528026\pi$
$920$	−23124.2	−0.828676
$921$	0	0
$922$	−25699.7	−0.917978
$923$	−3911.40	−0.139486
$924$	0	0
$925$	2706.70	0.0962117
$926$	−48437.1	−1.71894
$927$	0	0
$928$	21594.1	0.763860
$929$	−13356.2	−0.471692	−0.235846	−	0.971790i	$-0.575786\pi$
$929$	−13356.2	−0.471692	−0.235846	+	0.971790i	$0.575786\pi$
$930$	0	0
$931$	−88703.6	−3.12260
$932$	127793.	4.49141
$933$	0	0
$934$	−14784.8	−0.517958
$935$	2341.53	0.0818996
$936$	0	0
$937$	4960.13	0.172935	0.0864677	−	0.996255i	$-0.472442\pi$
$937$	4960.13	0.172935	0.0864677	+	0.996255i	$0.472442\pi$
$938$	159254.	5.54353
$939$	0	0
$940$	−58003.9	−2.01264
$941$	−55574.3	−1.92526	−0.962631	−	0.270817i	$-0.912706\pi$
$941$	−55574.3	−1.92526	−0.962631	+	0.270817i	$0.912706\pi$
$942$	0	0
$943$	11078.7	0.382579
$944$	−38073.7	−1.31270
$945$	0	0
$946$	10810.2	0.371533
$947$	20945.5	0.718732	0.359366	−	0.933197i	$-0.382993\pi$
$947$	20945.5	0.718732	0.359366	+	0.933197i	$0.382993\pi$
$948$	0	0
$949$	21812.7	0.746121
$950$	15532.1	0.530451
$951$	0	0
$952$	−151491.	−5.15741
$953$	26967.8	0.916656	0.458328	−	0.888783i	$-0.348449\pi$
$953$	26967.8	0.916656	0.458328	+	0.888783i	$0.348449\pi$
$954$	0	0
$955$	2213.22	0.0749927
$956$	−30720.5	−1.03930
$957$	0	0
$958$	48008.2	1.61907
$959$	38255.6	1.28815
$960$	0	0
$961$	−18360.4	−0.616308
$962$	36518.2	1.22390
$963$	0	0
$964$	−32464.8	−1.08467
$965$	18512.7	0.617559
$966$	0	0
$967$	−18506.5	−0.615439	−0.307719	−	0.951477i	$-0.599566\pi$
$967$	−18506.5	−0.615439	−0.307719	+	0.951477i	$0.599566\pi$
$968$	77119.7	2.56066
$969$	0	0
$970$	−27270.3	−0.902676
$971$	56274.4	1.85987	0.929934	−	0.367726i	$-0.119864\pi$
$971$	56274.4	1.85987	0.929934	+	0.367726i	$0.119864\pi$
$972$	0	0
$973$	52383.9	1.72595
$974$	−71507.2	−2.35240
$975$	0	0
$976$	−78293.2	−2.56773
$977$	−23165.7	−0.758583	−0.379292	−	0.925277i	$-0.623832\pi$
$977$	−23165.7	−0.758583	−0.379292	+	0.925277i	$0.623832\pi$
$978$	0	0
$979$	5725.17	0.186902
$980$	−72433.1	−2.36101
$981$	0	0
$982$	−30616.8	−0.994931
$983$	−3286.86	−0.106647	−0.0533237	−	0.998577i	$-0.516982\pi$
$983$	−3286.86	−0.106647	−0.0533237	+	0.998577i	$0.516982\pi$
$984$	0	0
$985$	22459.6	0.726520
$986$	25307.4	0.817395
$987$	0	0
$988$	148345.	4.77680
$989$	−26350.6	−0.847221
$990$	0	0
$991$	35155.7	1.12690	0.563450	−	0.826150i	$-0.309474\pi$
$991$	35155.7	1.12690	0.563450	+	0.826150i	$0.309474\pi$
$992$	36754.2	1.17636
$993$	0	0
$994$	−10486.6	−0.334624
$995$	14670.4	0.467419
$996$	0	0
$997$	−13641.5	−0.433329	−0.216665	−	0.976246i	$-0.569518\pi$
$997$	−13641.5	−0.433329	−0.216665	+	0.976246i	$0.569518\pi$
$998$	−20111.7	−0.637900
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.k.1.1	✓	6
3.2	odd	2		405.4.a.l.1.6	yes	6
5.4	even	2		2025.4.a.z.1.6		6
9.2	odd	6		405.4.e.w.271.1		12
9.4	even	3		405.4.e.x.136.6		12
9.5	odd	6		405.4.e.w.136.1		12
9.7	even	3		405.4.e.x.271.6		12
15.14	odd	2		2025.4.a.y.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.1	✓	6	1.1	even	1	trivial
405.4.a.l.1.6	yes	6	3.2	odd	2
405.4.e.w.136.1		12	9.5	odd	6
405.4.e.w.271.1		12	9.2	odd	6
405.4.e.x.136.6		12	9.4	even	3
405.4.e.x.271.6		12	9.7	even	3
2025.4.a.y.1.1		6	15.14	odd	2
2025.4.a.z.1.6		6	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)

\(f(q)\)	\(=\)	\(q-5.23336 q^{2} +19.3881 q^{4} -5.00000 q^{5} +33.0180 q^{7} -59.5981 q^{8} +O(q^{10})\)	\(q-5.23336 q^{2} +19.3881 q^{4} -5.00000 q^{5} +33.0180 q^{7} -59.5981 q^{8} +26.1668 q^{10} -6.08311 q^{11} -64.4507 q^{13} -172.795 q^{14} +156.794 q^{16} +76.9845 q^{17} -118.716 q^{19} -96.9405 q^{20} +31.8351 q^{22} -77.6004 q^{23} +25.0000 q^{25} +337.294 q^{26} +640.157 q^{28} -62.8149 q^{29} -106.914 q^{31} -343.774 q^{32} -402.888 q^{34} -165.090 q^{35} +108.268 q^{37} +621.285 q^{38} +297.990 q^{40} -142.766 q^{41} +339.568 q^{43} -117.940 q^{44} +406.111 q^{46} +598.346 q^{47} +747.191 q^{49} -130.834 q^{50} -1249.58 q^{52} -488.041 q^{53} +30.4155 q^{55} -1967.81 q^{56} +328.733 q^{58} -242.826 q^{59} -499.339 q^{61} +559.519 q^{62} +544.744 q^{64} +322.254 q^{65} -921.635 q^{67} +1492.58 q^{68} +863.977 q^{70} +60.6882 q^{71} -338.439 q^{73} -566.606 q^{74} -2301.68 q^{76} -200.852 q^{77} -556.311 q^{79} -783.969 q^{80} +747.147 q^{82} +64.8420 q^{83} -384.923 q^{85} -1777.08 q^{86} +362.542 q^{88} -941.159 q^{89} -2128.04 q^{91} -1504.52 q^{92} -3131.36 q^{94} +593.581 q^{95} -1042.17 q^{97} -3910.32 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8}+O(q^{10}) \) 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8	\( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + 20 q^{10} - 88 q^{11} + 20 q^{13} - 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} - 170 q^{20} - 74 q^{22} - 210 q^{23} + 150 q^{25} - 4 q^{26} + 352 q^{28} - 296 q^{29} - 104 q^{31} - 722 q^{32} - 428 q^{34} - 200 q^{35} - 204 q^{37} + 20 q^{38} + 330 q^{40} - 344 q^{41} + 512 q^{43} - 716 q^{44} - 186 q^{46} - 238 q^{47} + 68 q^{49} - 100 q^{50} - 468 q^{52} - 850 q^{53} + 440 q^{55} - 2316 q^{56} + 890 q^{58} - 1840 q^{59} - 364 q^{61} - 1038 q^{62} - 990 q^{64} - 100 q^{65} + 88 q^{67} - 236 q^{68} + 900 q^{70} - 1364 q^{71} + 836 q^{73} - 1316 q^{74} - 2106 q^{76} - 840 q^{77} - 680 q^{79} - 290 q^{80} + 1742 q^{82} - 2148 q^{83} + 620 q^{85} - 2872 q^{86} + 1296 q^{88} - 3000 q^{89} - 3058 q^{91} - 1002 q^{92} - 3662 q^{94} + 230 q^{95} - 612 q^{97} - 1982 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8 + 20 * q^10 - 88 * q^11 + 20 * q^13 - 180 * q^14 + 58 * q^16 - 124 * q^17 - 46 * q^19 - 170 * q^20 - 74 * q^22 - 210 * q^23 + 150 * q^25 - 4 * q^26 + 352 * q^28 - 296 * q^29 - 104 * q^31 - 722 * q^32 - 428 * q^34 - 200 * q^35 - 204 * q^37 + 20 * q^38 + 330 * q^40 - 344 * q^41 + 512 * q^43 - 716 * q^44 - 186 * q^46 - 238 * q^47 + 68 * q^49 - 100 * q^50 - 468 * q^52 - 850 * q^53 + 440 * q^55 - 2316 * q^56 + 890 * q^58 - 1840 * q^59 - 364 * q^61 - 1038 * q^62 - 990 * q^64 - 100 * q^65 + 88 * q^67 - 236 * q^68 + 900 * q^70 - 1364 * q^71 + 836 * q^73 - 1316 * q^74 - 2106 * q^76 - 840 * q^77 - 680 * q^79 - 290 * q^80 + 1742 * q^82 - 2148 * q^83 + 620 * q^85 - 2872 * q^86 + 1296 * q^88 - 3000 * q^89 - 3058 * q^91 - 1002 * q^92 - 3662 * q^94 + 230 * q^95 - 612 * q^97 - 1982 * q^98

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