LMFDB - Embedded newform 2070.4.a.bj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,4,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2070.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2070.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:	$122.133953712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 68x^{2} - 111x + 342 $ x^4 - 68x^2 - 111x + 342
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$8.73081$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +23.5622 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +23.5622 q^{7} +8.00000 q^{8} +10.0000 q^{10} +32.1352 q^{11} +40.0811 q^{13} +47.1245 q^{14} +16.0000 q^{16} -126.165 q^{17} +0.232742 q^{19} +20.0000 q^{20} +64.2704 q^{22} +23.0000 q^{23} +25.0000 q^{25} +80.1621 q^{26} +94.2490 q^{28} +137.226 q^{29} +112.866 q^{31} +32.0000 q^{32} -252.331 q^{34} +117.811 q^{35} +45.7057 q^{37} +0.465483 q^{38} +40.0000 q^{40} +135.385 q^{41} +543.528 q^{43} +128.541 q^{44} +46.0000 q^{46} -26.4344 q^{47} +212.180 q^{49} +50.0000 q^{50} +160.324 q^{52} -43.6958 q^{53} +160.676 q^{55} +188.498 q^{56} +274.451 q^{58} -202.248 q^{59} +150.279 q^{61} +225.732 q^{62} +64.0000 q^{64} +200.405 q^{65} -420.722 q^{67} -504.661 q^{68} +235.622 q^{70} -667.381 q^{71} +602.960 q^{73} +91.4115 q^{74} +0.930966 q^{76} +757.178 q^{77} -1378.88 q^{79} +80.0000 q^{80} +270.769 q^{82} +485.178 q^{83} -630.826 q^{85} +1087.06 q^{86} +257.082 q^{88} +1127.71 q^{89} +944.400 q^{91} +92.0000 q^{92} -52.8688 q^{94} +1.16371 q^{95} -1486.24 q^{97} +424.359 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8}+O(q^{10}) $ 4 * q + 8 * q^2 + 16 * q^4 + 20 * q^5 - q^7 + 32 * q^8	$ 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 20 q^{13} - 2 q^{14} + 64 q^{16} + 23 q^{17} + 53 q^{19} + 80 q^{20} + 78 q^{22} + 92 q^{23} + 100 q^{25} - 40 q^{26} - 4 q^{28} - 161 q^{29} + 388 q^{31} + 128 q^{32} + 46 q^{34} - 5 q^{35} + 466 q^{37} + 106 q^{38} + 160 q^{40} - 484 q^{41} + 894 q^{43} + 156 q^{44} + 184 q^{46} + 265 q^{47} + 1643 q^{49} + 200 q^{50} - 80 q^{52} - 576 q^{53} + 195 q^{55} - 8 q^{56} - 322 q^{58} + 94 q^{59} + 1153 q^{61} + 776 q^{62} + 256 q^{64} - 100 q^{65} - 1472 q^{67} + 92 q^{68} - 10 q^{70} - 200 q^{71} + 1147 q^{73} + 932 q^{74} + 212 q^{76} + 2176 q^{77} - 908 q^{79} + 320 q^{80} - 968 q^{82} + 1048 q^{83} + 115 q^{85} + 1788 q^{86} + 312 q^{88} + 1784 q^{89} + 2329 q^{91} + 368 q^{92} + 530 q^{94} + 265 q^{95} - 2047 q^{97} + 3286 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 + 20 * q^5 - q^7 + 32 * q^8 + 40 * q^10 + 39 * q^11 - 20 * q^13 - 2 * q^14 + 64 * q^16 + 23 * q^17 + 53 * q^19 + 80 * q^20 + 78 * q^22 + 92 * q^23 + 100 * q^25 - 40 * q^26 - 4 * q^28 - 161 * q^29 + 388 * q^31 + 128 * q^32 + 46 * q^34 - 5 * q^35 + 466 * q^37 + 106 * q^38 + 160 * q^40 - 484 * q^41 + 894 * q^43 + 156 * q^44 + 184 * q^46 + 265 * q^47 + 1643 * q^49 + 200 * q^50 - 80 * q^52 - 576 * q^53 + 195 * q^55 - 8 * q^56 - 322 * q^58 + 94 * q^59 + 1153 * q^61 + 776 * q^62 + 256 * q^64 - 100 * q^65 - 1472 * q^67 + 92 * q^68 - 10 * q^70 - 200 * q^71 + 1147 * q^73 + 932 * q^74 + 212 * q^76 + 2176 * q^77 - 908 * q^79 + 320 * q^80 - 968 * q^82 + 1048 * q^83 + 115 * q^85 + 1788 * q^86 + 312 * q^88 + 1784 * q^89 + 2329 * q^91 + 368 * q^92 + 530 * q^94 + 265 * q^95 - 2047 * q^97 + 3286 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	23.5622	1.27224	0.636121	−	0.771589i	$-0.280538\pi$
$7$	23.5622	1.27224	0.636121	+	0.771589i	$0.280538\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	32.1352	0.880830	0.440415	−	0.897794i	$-0.354831\pi$
$11$	32.1352	0.880830	0.440415	+	0.897794i	$0.354831\pi$
$12$	0	0
$13$	40.0811	0.855115	0.427557	−	0.903988i	$-0.359374\pi$
$13$	40.0811	0.855115	0.427557	+	0.903988i	$0.359374\pi$
$14$	47.1245	0.899611
$15$	0	0
$16$	16.0000	0.250000
$17$	−126.165	−1.79997	−0.899987	−	0.435916i	$-0.856424\pi$
$17$	−126.165	−1.79997	−0.899987	+	0.435916i	$0.856424\pi$
$18$	0	0
$19$	0.232742	0.00281024	0.00140512	−	0.999999i	$-0.499553\pi$
$19$	0.232742	0.00281024	0.00140512	+	0.999999i	$0.499553\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	64.2704	0.622841
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	80.1621	0.604657
$27$	0	0
$28$	94.2490	0.636121
$29$	137.226	0.878695	0.439347	−	0.898317i	$-0.355210\pi$
$29$	137.226	0.878695	0.439347	+	0.898317i	$0.355210\pi$
$30$	0	0
$31$	112.866	0.653914	0.326957	−	0.945039i	$-0.393977\pi$
$31$	112.866	0.653914	0.326957	+	0.945039i	$0.393977\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−252.331	−1.27277
$35$	117.811	0.568964
$36$	0	0
$37$	45.7057	0.203080	0.101540	−	0.994831i	$-0.467623\pi$
$37$	45.7057	0.203080	0.101540	+	0.994831i	$0.467623\pi$
$38$	0.465483	0.00198714
$39$	0	0
$40$	40.0000	0.158114
$41$	135.385	0.515696	0.257848	−	0.966186i	$-0.416987\pi$
$41$	135.385	0.515696	0.257848	+	0.966186i	$0.416987\pi$
$42$	0	0
$43$	543.528	1.92761	0.963805	−	0.266608i	$-0.0859030\pi$
$43$	543.528	1.92761	0.963805	+	0.266608i	$0.0859030\pi$
$44$	128.541	0.440415
$45$	0	0
$46$	46.0000	0.147442
$47$	−26.4344	−0.0820394	−0.0410197	−	0.999158i	$-0.513061\pi$
$47$	−26.4344	−0.0820394	−0.0410197	+	0.999158i	$0.513061\pi$
$48$	0	0
$49$	212.180	0.618599
$50$	50.0000	0.141421
$51$	0	0
$52$	160.324	0.427557
$53$	−43.6958	−0.113247	−0.0566235	−	0.998396i	$-0.518033\pi$
$53$	−43.6958	−0.113247	−0.0566235	+	0.998396i	$0.518033\pi$
$54$	0	0
$55$	160.676	0.393919
$56$	188.498	0.449805
$57$	0	0
$58$	274.451	0.621331
$59$	−202.248	−0.446278	−0.223139	−	0.974787i	$-0.571630\pi$
$59$	−202.248	−0.446278	−0.223139	+	0.974787i	$0.571630\pi$
$60$	0	0
$61$	150.279	0.315430	0.157715	−	0.987485i	$-0.449587\pi$
$61$	150.279	0.315430	0.157715	+	0.987485i	$0.449587\pi$
$62$	225.732	0.462387
$63$	0	0
$64$	64.0000	0.125000
$65$	200.405	0.382419
$66$	0	0
$67$	−420.722	−0.767155	−0.383577	−	0.923509i	$-0.625308\pi$
$67$	−420.722	−0.767155	−0.383577	+	0.923509i	$0.625308\pi$
$68$	−504.661	−0.899987
$69$	0	0
$70$	235.622	0.402318
$71$	−667.381	−1.11554	−0.557771	−	0.829995i	$-0.688343\pi$
$71$	−667.381	−1.11554	−0.557771	+	0.829995i	$0.688343\pi$
$72$	0	0
$73$	602.960	0.966728	0.483364	−	0.875420i	$-0.339415\pi$
$73$	602.960	0.966728	0.483364	+	0.875420i	$0.339415\pi$
$74$	91.4115	0.143600
$75$	0	0
$76$	0.930966	0.00140512
$77$	757.178	1.12063
$78$	0	0
$79$	−1378.88	−1.96374	−0.981872	−	0.189545i	$-0.939299\pi$
$79$	−1378.88	−1.96374	−0.981872	+	0.189545i	$0.939299\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	270.769	0.364652
$83$	485.178	0.641629	0.320815	−	0.947142i	$-0.396043\pi$
$83$	485.178	0.641629	0.320815	+	0.947142i	$0.396043\pi$
$84$	0	0
$85$	−630.826	−0.804973
$86$	1087.06	1.36303
$87$	0	0
$88$	257.082	0.311420
$89$	1127.71	1.34311	0.671556	−	0.740954i	$-0.265626\pi$
$89$	1127.71	1.34311	0.671556	+	0.740954i	$0.265626\pi$
$90$	0	0
$91$	944.400	1.08791
$92$	92.0000	0.104257
$93$	0	0
$94$	−52.8688	−0.0580106
$95$	1.16371	0.00125678
$96$	0	0
$97$	−1486.24	−1.55572	−0.777858	−	0.628441i	$-0.783694\pi$
$97$	−1486.24	−1.55572	−0.777858	+	0.628441i	$0.783694\pi$
$98$	424.359	0.437416
$99$	0	0
$100$	100.000	0.100000
$101$	−1888.86	−1.86087	−0.930437	−	0.366451i	$-0.880573\pi$
$101$	−1888.86	−1.86087	−0.930437	+	0.366451i	$0.880573\pi$
$102$	0	0
$103$	−1497.17	−1.43224	−0.716118	−	0.697979i	$-0.754083\pi$
$103$	−1497.17	−1.43224	−0.716118	+	0.697979i	$0.754083\pi$
$104$	320.649	0.302329
$105$	0	0
$106$	−87.3917	−0.0800777
$107$	585.150	0.528678	0.264339	−	0.964430i	$-0.414846\pi$
$107$	585.150	0.528678	0.264339	+	0.964430i	$0.414846\pi$
$108$	0	0
$109$	−139.166	−0.122290	−0.0611452	−	0.998129i	$-0.519475\pi$
$109$	−139.166	−0.122290	−0.0611452	+	0.998129i	$0.519475\pi$
$110$	321.352	0.278543
$111$	0	0
$112$	376.996	0.318060
$113$	1293.18	1.07656	0.538282	−	0.842765i	$-0.319073\pi$
$113$	1293.18	1.07656	0.538282	+	0.842765i	$0.319073\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	548.902	0.439347
$117$	0	0
$118$	−404.495	−0.315566
$119$	−2972.74	−2.29000
$120$	0	0
$121$	−298.328	−0.224138
$122$	300.557	0.223043
$123$	0	0
$124$	451.464	0.326957
$125$	125.000	0.0894427
$126$	0	0
$127$	−1298.43	−0.907221	−0.453610	−	0.891200i	$-0.649864\pi$
$127$	−1298.43	−0.907221	−0.453610	+	0.891200i	$0.649864\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	400.811	0.270411
$131$	525.247	0.350313	0.175157	−	0.984541i	$-0.443957\pi$
$131$	525.247	0.350313	0.175157	+	0.984541i	$0.443957\pi$
$132$	0	0
$133$	5.48392	0.00357531
$134$	−841.444	−0.542460
$135$	0	0
$136$	−1009.32	−0.636387
$137$	2428.12	1.51422	0.757109	−	0.653288i	$-0.226611\pi$
$137$	2428.12	1.51422	0.757109	+	0.653288i	$0.226611\pi$
$138$	0	0
$139$	2456.46	1.49895	0.749476	−	0.662031i	$-0.230305\pi$
$139$	2456.46	1.49895	0.749476	+	0.662031i	$0.230305\pi$
$140$	471.245	0.284482
$141$	0	0
$142$	−1334.76	−0.788808
$143$	1288.01	0.753211
$144$	0	0
$145$	686.128	0.392964
$146$	1205.92	0.683580
$147$	0	0
$148$	182.823	0.101540
$149$	2405.39	1.32253	0.661266	−	0.750151i	$-0.270019\pi$
$149$	2405.39	1.32253	0.661266	+	0.750151i	$0.270019\pi$
$150$	0	0
$151$	−649.276	−0.349916	−0.174958	−	0.984576i	$-0.555979\pi$
$151$	−649.276	−0.349916	−0.174958	+	0.984576i	$0.555979\pi$
$152$	1.86193	0.000993570 0
$153$	0	0
$154$	1514.36	0.792404
$155$	564.330	0.292439
$156$	0	0
$157$	3665.04	1.86307	0.931536	−	0.363650i	$-0.118470\pi$
$157$	3665.04	1.86307	0.931536	+	0.363650i	$0.118470\pi$
$158$	−2757.75	−1.38858
$159$	0	0
$160$	160.000	0.0790569
$161$	541.932	0.265281
$162$	0	0
$163$	−1055.27	−0.507088	−0.253544	−	0.967324i	$-0.581596\pi$
$163$	−1055.27	−0.507088	−0.253544	+	0.967324i	$0.581596\pi$
$164$	541.539	0.257848
$165$	0	0
$166$	970.356	0.453700
$167$	731.805	0.339095	0.169547	−	0.985522i	$-0.445769\pi$
$167$	731.805	0.339095	0.169547	+	0.985522i	$0.445769\pi$
$168$	0	0
$169$	−590.507	−0.268779
$170$	−1261.65	−0.569202
$171$	0	0
$172$	2174.11	0.963805
$173$	521.773	0.229304	0.114652	−	0.993406i	$-0.463425\pi$
$173$	521.773	0.229304	0.114652	+	0.993406i	$0.463425\pi$
$174$	0	0
$175$	589.056	0.254448
$176$	514.163	0.220208
$177$	0	0
$178$	2255.42	0.949723
$179$	1183.07	0.494005	0.247002	−	0.969015i	$-0.420554\pi$
$179$	1183.07	0.494005	0.247002	+	0.969015i	$0.420554\pi$
$180$	0	0
$181$	1723.92	0.707946	0.353973	−	0.935256i	$-0.384830\pi$
$181$	1723.92	0.707946	0.353973	+	0.935256i	$0.384830\pi$
$182$	1888.80	0.769270
$183$	0	0
$184$	184.000	0.0737210
$185$	228.529	0.0908204
$186$	0	0
$187$	−4054.35	−1.58547
$188$	−105.738	−0.0410197
$189$	0	0
$190$	2.32742	0.000888676 0
$191$	2798.12	1.06002	0.530012	−	0.847990i	$-0.322187\pi$
$191$	2798.12	1.06002	0.530012	+	0.847990i	$0.322187\pi$
$192$	0	0
$193$	−497.686	−0.185618	−0.0928089	−	0.995684i	$-0.529585\pi$
$193$	−497.686	−0.185618	−0.0928089	+	0.995684i	$0.529585\pi$
$194$	−2972.47	−1.10006
$195$	0	0
$196$	848.718	0.309300
$197$	−296.489	−0.107228	−0.0536141	−	0.998562i	$-0.517074\pi$
$197$	−296.489	−0.107228	−0.0536141	+	0.998562i	$0.517074\pi$
$198$	0	0
$199$	2347.43	0.836207	0.418103	−	0.908399i	$-0.362695\pi$
$199$	2347.43	0.836207	0.418103	+	0.908399i	$0.362695\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	−3777.71	−1.31584
$203$	3233.34	1.11791
$204$	0	0
$205$	676.923	0.230626
$206$	−2994.34	−1.01274
$207$	0	0
$208$	641.297	0.213779
$209$	7.47920	0.00247535
$210$	0	0
$211$	2838.61	0.926153	0.463076	−	0.886318i	$-0.346746\pi$
$211$	2838.61	0.926153	0.463076	+	0.886318i	$0.346746\pi$
$212$	−174.783	−0.0566235
$213$	0	0
$214$	1170.30	0.373832
$215$	2717.64	0.862053
$216$	0	0
$217$	2659.38	0.831937
$218$	−278.331	−0.0864723
$219$	0	0
$220$	642.704	0.196960
$221$	−5056.84	−1.53918
$222$	0	0
$223$	−2124.68	−0.638024	−0.319012	−	0.947751i	$-0.603351\pi$
$223$	−2124.68	−0.638024	−0.319012	+	0.947751i	$0.603351\pi$
$224$	753.992	0.224903
$225$	0	0
$226$	2586.35	0.761246
$227$	−1551.97	−0.453780	−0.226890	−	0.973920i	$-0.572856\pi$
$227$	−1551.97	−0.453780	−0.226890	+	0.973920i	$0.572856\pi$
$228$	0	0
$229$	−158.531	−0.0457469	−0.0228735	−	0.999738i	$-0.507281\pi$
$229$	−158.531	−0.0457469	−0.0228735	+	0.999738i	$0.507281\pi$
$230$	230.000	0.0659380
$231$	0	0
$232$	1097.80	0.310666
$233$	728.999	0.204971	0.102486	−	0.994734i	$-0.467320\pi$
$233$	728.999	0.204971	0.102486	+	0.994734i	$0.467320\pi$
$234$	0	0
$235$	−132.172	−0.0366891
$236$	−808.991	−0.223139
$237$	0	0
$238$	−5945.47	−1.61928
$239$	6201.60	1.67844	0.839222	−	0.543789i	$-0.183011\pi$
$239$	6201.60	1.67844	0.839222	+	0.543789i	$0.183011\pi$
$240$	0	0
$241$	−7223.04	−1.93061	−0.965305	−	0.261126i	$-0.915906\pi$
$241$	−7223.04	−1.93061	−0.965305	+	0.261126i	$0.915906\pi$
$242$	−596.656	−0.158490
$243$	0	0
$244$	601.115	0.157715
$245$	1060.90	0.276646
$246$	0	0
$247$	9.32853	0.00240308
$248$	902.928	0.231193
$249$	0	0
$250$	250.000	0.0632456
$251$	−5042.78	−1.26812	−0.634059	−	0.773285i	$-0.718612\pi$
$251$	−5042.78	−1.26812	−0.634059	+	0.773285i	$0.718612\pi$
$252$	0	0
$253$	739.110	0.183666
$254$	−2596.86	−0.641502
$255$	0	0
$256$	256.000	0.0625000
$257$	−2981.15	−0.723577	−0.361788	−	0.932260i	$-0.617834\pi$
$257$	−2981.15	−0.723577	−0.361788	+	0.932260i	$0.617834\pi$
$258$	0	0
$259$	1076.93	0.258367
$260$	801.621	0.191209
$261$	0	0
$262$	1050.49	0.247709
$263$	−7242.86	−1.69815	−0.849076	−	0.528271i	$-0.822841\pi$
$263$	−7242.86	−1.69815	−0.849076	+	0.528271i	$0.822841\pi$
$264$	0	0
$265$	−218.479	−0.0506456
$266$	10.9678	0.00252812
$267$	0	0
$268$	−1682.89	−0.383577
$269$	2567.58	0.581962	0.290981	−	0.956729i	$-0.406018\pi$
$269$	2567.58	0.581962	0.290981	+	0.956729i	$0.406018\pi$
$270$	0	0
$271$	8066.27	1.80808	0.904042	−	0.427443i	$-0.140586\pi$
$271$	8066.27	1.80808	0.904042	+	0.427443i	$0.140586\pi$
$272$	−2018.64	−0.449994
$273$	0	0
$274$	4856.23	1.07071
$275$	803.380	0.176166
$276$	0	0
$277$	8991.54	1.95036	0.975179	−	0.221418i	$-0.0710684\pi$
$277$	8991.54	1.95036	0.975179	+	0.221418i	$0.0710684\pi$
$278$	4912.92	1.05992
$279$	0	0
$280$	942.490	0.201159
$281$	−968.130	−0.205529	−0.102765	−	0.994706i	$-0.532769\pi$
$281$	−968.130	−0.205529	−0.102765	+	0.994706i	$0.532769\pi$
$282$	0	0
$283$	−2252.65	−0.473167	−0.236583	−	0.971611i	$-0.576028\pi$
$283$	−2252.65	−0.473167	−0.236583	+	0.971611i	$0.576028\pi$
$284$	−2669.52	−0.557771
$285$	0	0
$286$	2576.03	0.532600
$287$	3189.97	0.656090
$288$	0	0
$289$	11004.7	2.23991
$290$	1372.26	0.277868
$291$	0	0
$292$	2411.84	0.483364
$293$	−2734.35	−0.545196	−0.272598	−	0.962128i	$-0.587883\pi$
$293$	−2734.35	−0.545196	−0.272598	+	0.962128i	$0.587883\pi$
$294$	0	0
$295$	−1011.24	−0.199582
$296$	365.646	0.0717998
$297$	0	0
$298$	4810.78	0.935172
$299$	921.865	0.178304
$300$	0	0
$301$	12806.7	2.45239
$302$	−1298.55	−0.247428
$303$	0	0
$304$	3.72387	0.000702560 0
$305$	751.394	0.141065
$306$	0	0
$307$	−7977.09	−1.48299	−0.741493	−	0.670961i	$-0.765882\pi$
$307$	−7977.09	−1.48299	−0.741493	+	0.670961i	$0.765882\pi$
$308$	3028.71	0.560314
$309$	0	0
$310$	1128.66	0.206786
$311$	−7729.43	−1.40931	−0.704655	−	0.709550i	$-0.748898\pi$
$311$	−7729.43	−1.40931	−0.704655	+	0.709550i	$0.748898\pi$
$312$	0	0
$313$	5506.25	0.994350	0.497175	−	0.867650i	$-0.334371\pi$
$313$	5506.25	0.994350	0.497175	+	0.867650i	$0.334371\pi$
$314$	7330.08	1.31739
$315$	0	0
$316$	−5515.51	−0.981872
$317$	−5231.37	−0.926887	−0.463443	−	0.886126i	$-0.653386\pi$
$317$	−5231.37	−0.926887	−0.463443	+	0.886126i	$0.653386\pi$
$318$	0	0
$319$	4409.77	0.773981
$320$	320.000	0.0559017
$321$	0	0
$322$	1083.86	0.187582
$323$	−29.3639	−0.00505836
$324$	0	0
$325$	1002.03	0.171023
$326$	−2110.55	−0.358566
$327$	0	0
$328$	1083.08	0.182326
$329$	−622.854	−0.104374
$330$	0	0
$331$	−3551.61	−0.589771	−0.294885	−	0.955533i	$-0.595281\pi$
$331$	−3551.61	−0.589771	−0.294885	+	0.955533i	$0.595281\pi$
$332$	1940.71	0.320815
$333$	0	0
$334$	1463.61	0.239776
$335$	−2103.61	−0.343082
$336$	0	0
$337$	−7002.99	−1.13198	−0.565990	−	0.824412i	$-0.691506\pi$
$337$	−7002.99	−1.13198	−0.565990	+	0.824412i	$0.691506\pi$
$338$	−1181.01	−0.190055
$339$	0	0
$340$	−2523.31	−0.402487
$341$	3626.97	0.575987
$342$	0	0
$343$	−3082.42	−0.485234
$344$	4348.22	0.681513
$345$	0	0
$346$	1043.55	0.162143
$347$	10268.2	1.58854	0.794272	−	0.607562i	$-0.207852\pi$
$347$	10268.2	1.58854	0.794272	+	0.607562i	$0.207852\pi$
$348$	0	0
$349$	4515.58	0.692588	0.346294	−	0.938126i	$-0.387440\pi$
$349$	4515.58	0.692588	0.346294	+	0.938126i	$0.387440\pi$
$350$	1178.11	0.179922
$351$	0	0
$352$	1028.33	0.155710
$353$	−7838.63	−1.18189	−0.590947	−	0.806711i	$-0.701246\pi$
$353$	−7838.63	−1.18189	−0.590947	+	0.806711i	$0.701246\pi$
$354$	0	0
$355$	−3336.90	−0.498886
$356$	4510.84	0.671556
$357$	0	0
$358$	2366.14	0.349314
$359$	12464.8	1.83250	0.916250	−	0.400606i	$-0.131201\pi$
$359$	12464.8	1.83250	0.916250	+	0.400606i	$0.131201\pi$
$360$	0	0
$361$	−6858.95	−0.999992
$362$	3447.85	0.500594
$363$	0	0
$364$	3777.60	0.543956
$365$	3014.80	0.432334
$366$	0	0
$367$	3936.14	0.559850	0.279925	−	0.960022i	$-0.409690\pi$
$367$	3936.14	0.559850	0.279925	+	0.960022i	$0.409690\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	457.057	0.0642197
$371$	−1029.57	−0.144077
$372$	0	0
$373$	1920.72	0.266625	0.133313	−	0.991074i	$-0.457439\pi$
$373$	1920.72	0.266625	0.133313	+	0.991074i	$0.457439\pi$
$374$	−8108.69	−1.12110
$375$	0	0
$376$	−211.475	−0.0290053
$377$	5500.15	0.751385
$378$	0	0
$379$	−13074.5	−1.77201	−0.886004	−	0.463678i	$-0.846530\pi$
$379$	−13074.5	−1.77201	−0.886004	+	0.463678i	$0.846530\pi$
$380$	4.65483	0.000628389 0
$381$	0	0
$382$	5596.23	0.749550
$383$	−8838.22	−1.17914	−0.589571	−	0.807716i	$-0.700703\pi$
$383$	−8838.22	−1.17914	−0.589571	+	0.807716i	$0.700703\pi$
$384$	0	0
$385$	3785.89	0.501160
$386$	−995.372	−0.131252
$387$	0	0
$388$	−5944.94	−0.777858
$389$	12687.3	1.65365	0.826827	−	0.562456i	$-0.190144\pi$
$389$	12687.3	1.65365	0.826827	+	0.562456i	$0.190144\pi$
$390$	0	0
$391$	−2901.80	−0.375321
$392$	1697.44	0.218708
$393$	0	0
$394$	−592.977	−0.0758218
$395$	−6894.38	−0.878213
$396$	0	0
$397$	8060.49	1.01900	0.509501	−	0.860470i	$-0.329830\pi$
$397$	8060.49	1.01900	0.509501	+	0.860470i	$0.329830\pi$
$398$	4694.87	0.591287
$399$	0	0
$400$	400.000	0.0500000
$401$	−12325.1	−1.53488	−0.767440	−	0.641121i	$-0.778470\pi$
$401$	−12325.1	−1.53488	−0.767440	+	0.641121i	$0.778470\pi$
$402$	0	0
$403$	4523.79	0.559171
$404$	−7555.43	−0.930437
$405$	0	0
$406$	6466.69	0.790483
$407$	1468.76	0.178879
$408$	0	0
$409$	−10806.7	−1.30650	−0.653250	−	0.757143i	$-0.726595\pi$
$409$	−10806.7	−1.30650	−0.653250	+	0.757143i	$0.726595\pi$
$410$	1353.85	0.163077
$411$	0	0
$412$	−5988.67	−0.716118
$413$	−4765.41	−0.567774
$414$	0	0
$415$	2425.89	0.286945
$416$	1282.59	0.151164
$417$	0	0
$418$	14.9584	0.00175033
$419$	1823.74	0.212639	0.106319	−	0.994332i	$-0.466093\pi$
$419$	1823.74	0.212639	0.106319	+	0.994332i	$0.466093\pi$
$420$	0	0
$421$	5995.43	0.694060	0.347030	−	0.937854i	$-0.387190\pi$
$421$	5995.43	0.694060	0.347030	+	0.937854i	$0.387190\pi$
$422$	5677.23	0.654889
$423$	0	0
$424$	−349.567	−0.0400388
$425$	−3154.13	−0.359995
$426$	0	0
$427$	3540.91	0.401303
$428$	2340.60	0.264339
$429$	0	0
$430$	5435.28	0.609564
$431$	−3887.80	−0.434499	−0.217249	−	0.976116i	$-0.569708\pi$
$431$	−3887.80	−0.434499	−0.217249	+	0.976116i	$0.569708\pi$
$432$	0	0
$433$	−4422.93	−0.490884	−0.245442	−	0.969411i	$-0.578933\pi$
$433$	−4422.93	−0.490884	−0.245442	+	0.969411i	$0.578933\pi$
$434$	5318.75	0.588268
$435$	0	0
$436$	−556.662	−0.0611452
$437$	5.35306	0.000585976 0
$438$	0	0
$439$	−1748.08	−0.190048	−0.0950242	−	0.995475i	$-0.530293\pi$
$439$	−1748.08	−0.190048	−0.0950242	+	0.995475i	$0.530293\pi$
$440$	1285.41	0.139271
$441$	0	0
$442$	−10113.7	−1.08837
$443$	−3371.31	−0.361571	−0.180785	−	0.983523i	$-0.557864\pi$
$443$	−3371.31	−0.361571	−0.180785	+	0.983523i	$0.557864\pi$
$444$	0	0
$445$	5638.55	0.600658
$446$	−4249.37	−0.451151
$447$	0	0
$448$	1507.98	0.159030
$449$	−13314.7	−1.39947	−0.699734	−	0.714404i	$-0.746698\pi$
$449$	−13314.7	−1.39947	−0.699734	+	0.714404i	$0.746698\pi$
$450$	0	0
$451$	4350.61	0.454240
$452$	5172.70	0.538282
$453$	0	0
$454$	−3103.94	−0.320871
$455$	4722.00	0.486529
$456$	0	0
$457$	3767.20	0.385607	0.192803	−	0.981237i	$-0.438242\pi$
$457$	3767.20	0.385607	0.192803	+	0.981237i	$0.438242\pi$
$458$	−317.063	−0.0323480
$459$	0	0
$460$	460.000	0.0466252
$461$	−9674.06	−0.977366	−0.488683	−	0.872461i	$-0.662523\pi$
$461$	−9674.06	−0.977366	−0.488683	+	0.872461i	$0.662523\pi$
$462$	0	0
$463$	2977.73	0.298892	0.149446	−	0.988770i	$-0.452251\pi$
$463$	2977.73	0.298892	0.149446	+	0.988770i	$0.452251\pi$
$464$	2195.61	0.219674
$465$	0	0
$466$	1458.00	0.144937
$467$	10701.0	1.06035	0.530176	−	0.847888i	$-0.322126\pi$
$467$	10701.0	1.06035	0.530176	+	0.847888i	$0.322126\pi$
$468$	0	0
$469$	−9913.16	−0.976006
$470$	−264.344	−0.0259431
$471$	0	0
$472$	−1617.98	−0.157783
$473$	17466.4	1.69790
$474$	0	0
$475$	5.81854	0.000562048 0
$476$	−11890.9	−1.14500
$477$	0	0
$478$	12403.2	1.18684
$479$	−2337.15	−0.222937	−0.111469	−	0.993768i	$-0.535555\pi$
$479$	−2337.15	−0.222937	−0.111469	+	0.993768i	$0.535555\pi$
$480$	0	0
$481$	1831.94	0.173657
$482$	−14446.1	−1.36515
$483$	0	0
$484$	−1193.31	−0.112069
$485$	−7431.18	−0.695737
$486$	0	0
$487$	−7183.85	−0.668442	−0.334221	−	0.942495i	$-0.608473\pi$
$487$	−7183.85	−0.668442	−0.334221	+	0.942495i	$0.608473\pi$
$488$	1202.23	0.111521
$489$	0	0
$490$	2121.80	0.195618
$491$	12084.2	1.11070	0.555350	−	0.831617i	$-0.312584\pi$
$491$	12084.2	1.11070	0.555350	+	0.831617i	$0.312584\pi$
$492$	0	0
$493$	−17313.1	−1.58163
$494$	18.6571	0.00169923
$495$	0	0
$496$	1805.86	0.163478
$497$	−15725.0	−1.41924
$498$	0	0
$499$	7145.78	0.641060	0.320530	−	0.947238i	$-0.396139\pi$
$499$	7145.78	0.641060	0.320530	+	0.947238i	$0.396139\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	−10085.6	−0.896695
$503$	−20436.2	−1.81154	−0.905770	−	0.423771i	$-0.860706\pi$
$503$	−20436.2	−1.81154	−0.905770	+	0.423771i	$0.860706\pi$
$504$	0	0
$505$	−9444.29	−0.832208
$506$	1478.22	0.129871
$507$	0	0
$508$	−5193.72	−0.453610
$509$	−19721.7	−1.71738	−0.858690	−	0.512495i	$-0.828721\pi$
$509$	−19721.7	−1.71738	−0.858690	+	0.512495i	$0.828721\pi$
$510$	0	0
$511$	14207.1	1.22991
$512$	512.000	0.0441942
$513$	0	0
$514$	−5962.30	−0.511646
$515$	−7485.84	−0.640516
$516$	0	0
$517$	−849.475	−0.0722628
$518$	2153.86	0.182693
$519$	0	0
$520$	1603.24	0.135205
$521$	−3483.23	−0.292904	−0.146452	−	0.989218i	$-0.546785\pi$
$521$	−3483.23	−0.292904	−0.146452	+	0.989218i	$0.546785\pi$
$522$	0	0
$523$	15689.0	1.31172	0.655862	−	0.754881i	$-0.272305\pi$
$523$	15689.0	1.31172	0.655862	+	0.754881i	$0.272305\pi$
$524$	2100.99	0.175157
$525$	0	0
$526$	−14485.7	−1.20077
$527$	−14239.8	−1.17703
$528$	0	0
$529$	529.000	0.0434783
$530$	−436.958	−0.0358118
$531$	0	0
$532$	21.9357	0.00178765
$533$	5426.36	0.440979
$534$	0	0
$535$	2925.75	0.236432
$536$	−3365.78	−0.271230
$537$	0	0
$538$	5135.15	0.411510
$539$	6818.43	0.544881
$540$	0	0
$541$	7620.73	0.605621	0.302810	−	0.953051i	$-0.402075\pi$
$541$	7620.73	0.605621	0.302810	+	0.953051i	$0.402075\pi$
$542$	16132.5	1.27851
$543$	0	0
$544$	−4037.29	−0.318194
$545$	−695.828	−0.0546899
$546$	0	0
$547$	5925.62	0.463183	0.231592	−	0.972813i	$-0.425607\pi$
$547$	5925.62	0.463183	0.231592	+	0.972813i	$0.425607\pi$
$548$	9712.47	0.757109
$549$	0	0
$550$	1606.76	0.124568
$551$	31.9381	0.00246934
$552$	0	0
$553$	−32489.4	−2.49836
$554$	17983.1	1.37911
$555$	0	0
$556$	9825.85	0.749476
$557$	5456.16	0.415054	0.207527	−	0.978229i	$-0.433459\pi$
$557$	5456.16	0.415054	0.207527	+	0.978229i	$0.433459\pi$
$558$	0	0
$559$	21785.2	1.64833
$560$	1884.98	0.142241
$561$	0	0
$562$	−1936.26	−0.145331
$563$	9194.29	0.688265	0.344132	−	0.938921i	$-0.388173\pi$
$563$	9194.29	0.688265	0.344132	+	0.938921i	$0.388173\pi$
$564$	0	0
$565$	6465.88	0.481454
$566$	−4505.30	−0.334580
$567$	0	0
$568$	−5339.05	−0.394404
$569$	−338.831	−0.0249640	−0.0124820	−	0.999922i	$-0.503973\pi$
$569$	−338.831	−0.0249640	−0.0124820	+	0.999922i	$0.503973\pi$
$570$	0	0
$571$	−1725.34	−0.126451	−0.0632254	−	0.997999i	$-0.520139\pi$
$571$	−1725.34	−0.126451	−0.0632254	+	0.997999i	$0.520139\pi$
$572$	5152.06	0.376605
$573$	0	0
$574$	6379.93	0.463926
$575$	575.000	0.0417029
$576$	0	0
$577$	23300.0	1.68109	0.840547	−	0.541738i	$-0.182234\pi$
$577$	23300.0	1.68109	0.840547	+	0.541738i	$0.182234\pi$
$578$	22009.3	1.58385
$579$	0	0
$580$	2744.51	0.196482
$581$	11431.9	0.816307
$582$	0	0
$583$	−1404.18	−0.0997513
$584$	4823.68	0.341790
$585$	0	0
$586$	−5468.70	−0.385512
$587$	−15653.3	−1.10065	−0.550325	−	0.834950i	$-0.685496\pi$
$587$	−15653.3	−1.10065	−0.550325	+	0.834950i	$0.685496\pi$
$588$	0	0
$589$	26.2686	0.00183766
$590$	−2022.48	−0.141126
$591$	0	0
$592$	731.292	0.0507701
$593$	2658.08	0.184072	0.0920358	−	0.995756i	$-0.470663\pi$
$593$	2658.08	0.184072	0.0920358	+	0.995756i	$0.470663\pi$
$594$	0	0
$595$	−14863.7	−1.02412
$596$	9621.57	0.661266
$597$	0	0
$598$	1843.73	0.126080
$599$	−19417.6	−1.32451	−0.662256	−	0.749278i	$-0.730401\pi$
$599$	−19417.6	−1.32451	−0.662256	+	0.749278i	$0.730401\pi$
$600$	0	0
$601$	−18469.0	−1.25352	−0.626760	−	0.779213i	$-0.715619\pi$
$601$	−18469.0	−1.25352	−0.626760	+	0.779213i	$0.715619\pi$
$602$	25613.5	1.73410
$603$	0	0
$604$	−2597.10	−0.174958
$605$	−1491.64	−0.100238
$606$	0	0
$607$	3968.56	0.265369	0.132684	−	0.991158i	$-0.457640\pi$
$607$	3968.56	0.265369	0.132684	+	0.991158i	$0.457640\pi$
$608$	7.44773	0.000496785 0
$609$	0	0
$610$	1502.79	0.0997477
$611$	−1059.52	−0.0701531
$612$	0	0
$613$	−11478.7	−0.756311	−0.378156	−	0.925742i	$-0.623442\pi$
$613$	−11478.7	−0.756311	−0.378156	+	0.925742i	$0.623442\pi$
$614$	−15954.2	−1.04863
$615$	0	0
$616$	6057.42	0.396202
$617$	15691.1	1.02382	0.511911	−	0.859039i	$-0.328938\pi$
$617$	15691.1	1.02382	0.511911	+	0.859039i	$0.328938\pi$
$618$	0	0
$619$	−18249.4	−1.18499	−0.592494	−	0.805575i	$-0.701856\pi$
$619$	−18249.4	−1.18499	−0.592494	+	0.805575i	$0.701856\pi$
$620$	2257.32	0.146220
$621$	0	0
$622$	−15458.9	−0.996533
$623$	26571.4	1.70876
$624$	0	0
$625$	625.000	0.0400000
$626$	11012.5	0.703111
$627$	0	0
$628$	14660.2	0.931536
$629$	−5766.48	−0.365540
$630$	0	0
$631$	−23528.8	−1.48442	−0.742208	−	0.670169i	$-0.766222\pi$
$631$	−23528.8	−1.48442	−0.742208	+	0.670169i	$0.766222\pi$
$632$	−11031.0	−0.694288
$633$	0	0
$634$	−10462.7	−0.655408
$635$	−6492.15	−0.405722
$636$	0	0
$637$	8504.38	0.528973
$638$	8819.55	0.547287
$639$	0	0
$640$	640.000	0.0395285
$641$	−3748.79	−0.230996	−0.115498	−	0.993308i	$-0.536846\pi$
$641$	−3748.79	−0.230996	−0.115498	+	0.993308i	$0.536846\pi$
$642$	0	0
$643$	15924.3	0.976660	0.488330	−	0.872659i	$-0.337606\pi$
$643$	15924.3	0.976660	0.488330	+	0.872659i	$0.337606\pi$
$644$	2167.73	0.132640
$645$	0	0
$646$	−58.7278	−0.00357680
$647$	2854.11	0.173426	0.0867129	−	0.996233i	$-0.472364\pi$
$647$	2854.11	0.173426	0.0867129	+	0.996233i	$0.472364\pi$
$648$	0	0
$649$	−6499.27	−0.393095
$650$	2004.05	0.120931
$651$	0	0
$652$	−4221.09	−0.253544
$653$	−7925.97	−0.474988	−0.237494	−	0.971389i	$-0.576326\pi$
$653$	−7925.97	−0.474988	−0.237494	+	0.971389i	$0.576326\pi$
$654$	0	0
$655$	2626.24	0.156665
$656$	2166.15	0.128924
$657$	0	0
$658$	−1245.71	−0.0738035
$659$	5799.43	0.342813	0.171406	−	0.985200i	$-0.445169\pi$
$659$	5799.43	0.342813	0.171406	+	0.985200i	$0.445169\pi$
$660$	0	0
$661$	22324.8	1.31367	0.656833	−	0.754036i	$-0.271896\pi$
$661$	22324.8	1.31367	0.656833	+	0.754036i	$0.271896\pi$
$662$	−7103.22	−0.417031
$663$	0	0
$664$	3881.42	0.226850
$665$	27.4196	0.00159893
$666$	0	0
$667$	3156.19	0.183221
$668$	2927.22	0.169547
$669$	0	0
$670$	−4207.22	−0.242596
$671$	4829.24	0.277840
$672$	0	0
$673$	−23253.5	−1.33188	−0.665940	−	0.746005i	$-0.731969\pi$
$673$	−23253.5	−1.33188	−0.665940	+	0.746005i	$0.731969\pi$
$674$	−14006.0	−0.800430
$675$	0	0
$676$	−2362.03	−0.134389
$677$	19003.7	1.07884	0.539419	−	0.842038i	$-0.318644\pi$
$677$	19003.7	1.07884	0.539419	+	0.842038i	$0.318644\pi$
$678$	0	0
$679$	−35019.1	−1.97925
$680$	−5046.61	−0.284601
$681$	0	0
$682$	7253.94	0.407284
$683$	11222.5	0.628722	0.314361	−	0.949304i	$-0.398210\pi$
$683$	11222.5	0.628722	0.314361	+	0.949304i	$0.398210\pi$
$684$	0	0
$685$	12140.6	0.677179
$686$	−6164.85	−0.343112
$687$	0	0
$688$	8696.45	0.481902
$689$	−1751.38	−0.0968391
$690$	0	0
$691$	4297.68	0.236601	0.118301	−	0.992978i	$-0.462255\pi$
$691$	4297.68	0.236601	0.118301	+	0.992978i	$0.462255\pi$
$692$	2087.09	0.114652
$693$	0	0
$694$	20536.4	1.12327
$695$	12282.3	0.670352
$696$	0	0
$697$	−17080.8	−0.928240
$698$	9031.15	0.489734
$699$	0	0
$700$	2356.22	0.127224
$701$	−25769.6	−1.38845	−0.694225	−	0.719758i	$-0.744253\pi$
$701$	−25769.6	−1.38845	−0.694225	+	0.719758i	$0.744253\pi$
$702$	0	0
$703$	10.6376	0.000570705 0
$704$	2056.65	0.110104
$705$	0	0
$706$	−15677.3	−0.835725
$707$	−44505.7	−2.36748
$708$	0	0
$709$	−4900.01	−0.259554	−0.129777	−	0.991543i	$-0.541426\pi$
$709$	−4900.01	−0.259554	−0.129777	+	0.991543i	$0.541426\pi$
$710$	−6673.81	−0.352766
$711$	0	0
$712$	9021.67	0.474862
$713$	2595.92	0.136350
$714$	0	0
$715$	6440.07	0.336846
$716$	4732.28	0.247002
$717$	0	0
$718$	24929.6	1.29577
$719$	−7631.85	−0.395855	−0.197928	−	0.980217i	$-0.563421\pi$
$719$	−7631.85	−0.395855	−0.197928	+	0.980217i	$0.563421\pi$
$720$	0	0
$721$	−35276.6	−1.82215
$722$	−13717.9	−0.707101
$723$	0	0
$724$	6895.70	0.353973
$725$	3430.64	0.175739
$726$	0	0
$727$	−36985.6	−1.88682	−0.943410	−	0.331628i	$-0.892402\pi$
$727$	−36985.6	−1.88682	−0.943410	+	0.331628i	$0.892402\pi$
$728$	7555.20	0.384635
$729$	0	0
$730$	6029.60	0.305706
$731$	−68574.3	−3.46965
$732$	0	0
$733$	−22227.3	−1.12003	−0.560015	−	0.828482i	$-0.689205\pi$
$733$	−22227.3	−1.12003	−0.560015	+	0.828482i	$0.689205\pi$
$734$	7872.29	0.395874
$735$	0	0
$736$	736.000	0.0368605
$737$	−13520.0	−0.675733
$738$	0	0
$739$	−17899.1	−0.890974	−0.445487	−	0.895288i	$-0.646970\pi$
$739$	−17899.1	−0.890974	−0.445487	+	0.895288i	$0.646970\pi$
$740$	914.115	0.0454102
$741$	0	0
$742$	−2059.14	−0.101878
$743$	29843.4	1.47355	0.736776	−	0.676137i	$-0.236347\pi$
$743$	29843.4	1.47355	0.736776	+	0.676137i	$0.236347\pi$
$744$	0	0
$745$	12027.0	0.591455
$746$	3841.44	0.188532
$747$	0	0
$748$	−16217.4	−0.792736
$749$	13787.4	0.672606
$750$	0	0
$751$	−9399.65	−0.456722	−0.228361	−	0.973577i	$-0.573337\pi$
$751$	−9399.65	−0.456722	−0.228361	+	0.973577i	$0.573337\pi$
$752$	−422.950	−0.0205098
$753$	0	0
$754$	11000.3	0.531309
$755$	−3246.38	−0.156487
$756$	0	0
$757$	29871.6	1.43422	0.717109	−	0.696961i	$-0.245465\pi$
$757$	29871.6	1.43422	0.717109	+	0.696961i	$0.245465\pi$
$758$	−26149.0	−1.25300
$759$	0	0
$760$	9.30966	0.000444338 0
$761$	273.955	0.0130497	0.00652487	−	0.999979i	$-0.497923\pi$
$761$	273.955	0.0130497	0.00652487	+	0.999979i	$0.497923\pi$
$762$	0	0
$763$	−3279.05	−0.155583
$764$	11192.5	0.530012
$765$	0	0
$766$	−17676.4	−0.833780
$767$	−8106.31	−0.381619
$768$	0	0
$769$	−13273.8	−0.622450	−0.311225	−	0.950336i	$-0.600739\pi$
$769$	−13273.8	−0.622450	−0.311225	+	0.950336i	$0.600739\pi$
$770$	7571.78	0.354374
$771$	0	0
$772$	−1990.74	−0.0928089
$773$	−34161.1	−1.58951	−0.794753	−	0.606933i	$-0.792400\pi$
$773$	−34161.1	−1.58951	−0.794753	+	0.606933i	$0.792400\pi$
$774$	0	0
$775$	2821.65	0.130783
$776$	−11889.9	−0.550028
$777$	0	0
$778$	25374.6	1.16931
$779$	31.5096	0.00144923
$780$	0	0
$781$	−21446.4	−0.982603
$782$	−5803.60	−0.265392
$783$	0	0
$784$	3394.87	0.154650
$785$	18325.2	0.833191
$786$	0	0
$787$	−38489.6	−1.74334	−0.871669	−	0.490095i	$-0.836962\pi$
$787$	−38489.6	−1.74334	−0.871669	+	0.490095i	$0.836962\pi$
$788$	−1185.95	−0.0536141
$789$	0	0
$790$	−13788.8	−0.620990
$791$	30470.1	1.36965
$792$	0	0
$793$	6023.33	0.269729
$794$	16121.0	0.720544
$795$	0	0
$796$	9389.73	0.418103
$797$	−17176.8	−0.763405	−0.381702	−	0.924285i	$-0.624662\pi$
$797$	−17176.8	−0.763405	−0.381702	+	0.924285i	$0.624662\pi$
$798$	0	0
$799$	3335.10	0.147669
$800$	800.000	0.0353553
$801$	0	0
$802$	−24650.2	−1.08532
$803$	19376.2	0.851523
$804$	0	0
$805$	2709.66	0.118637
$806$	9047.58	0.395394
$807$	0	0
$808$	−15110.9	−0.657918
$809$	−8226.61	−0.357518	−0.178759	−	0.983893i	$-0.557208\pi$
$809$	−8226.61	−0.357518	−0.178759	+	0.983893i	$0.557208\pi$
$810$	0	0
$811$	27867.9	1.20662	0.603312	−	0.797505i	$-0.293847\pi$
$811$	27867.9	1.20662	0.603312	+	0.797505i	$0.293847\pi$
$812$	12933.4	0.558956
$813$	0	0
$814$	2937.53	0.126487
$815$	−5276.37	−0.226777
$816$	0	0
$817$	126.502	0.00541705
$818$	−21613.5	−0.923835
$819$	0	0
$820$	2707.69	0.115313
$821$	17637.5	0.749761	0.374880	−	0.927073i	$-0.377684\pi$
$821$	17637.5	0.749761	0.374880	+	0.927073i	$0.377684\pi$
$822$	0	0
$823$	28048.6	1.18799	0.593993	−	0.804470i	$-0.297551\pi$
$823$	28048.6	1.18799	0.593993	+	0.804470i	$0.297551\pi$
$824$	−11977.3	−0.506372
$825$	0	0
$826$	−9530.82	−0.401477
$827$	32592.9	1.37046	0.685228	−	0.728329i	$-0.259703\pi$
$827$	32592.9	1.37046	0.685228	+	0.728329i	$0.259703\pi$
$828$	0	0
$829$	18815.9	0.788303	0.394152	−	0.919045i	$-0.371038\pi$
$829$	18815.9	0.788303	0.394152	+	0.919045i	$0.371038\pi$
$830$	4851.78	0.202901
$831$	0	0
$832$	2565.19	0.106889
$833$	−26769.7	−1.11346
$834$	0	0
$835$	3659.03	0.151648
$836$	29.9168	0.00123767
$837$	0	0
$838$	3647.48	0.150358
$839$	−7612.72	−0.313254	−0.156627	−	0.987658i	$-0.550062\pi$
$839$	−7612.72	−0.313254	−0.156627	+	0.987658i	$0.550062\pi$
$840$	0	0
$841$	−5558.14	−0.227896
$842$	11990.9	0.490775
$843$	0	0
$844$	11354.5	0.463076
$845$	−2952.54	−0.120202
$846$	0	0
$847$	−7029.28	−0.285158
$848$	−699.134	−0.0283117
$849$	0	0
$850$	−6308.26	−0.254555
$851$	1051.23	0.0423452
$852$	0	0
$853$	−31421.4	−1.26125	−0.630627	−	0.776086i	$-0.717202\pi$
$853$	−31421.4	−1.26125	−0.630627	+	0.776086i	$0.717202\pi$
$854$	7081.81	0.283764
$855$	0	0
$856$	4681.20	0.186916
$857$	−20909.5	−0.833435	−0.416718	−	0.909036i	$-0.636820\pi$
$857$	−20909.5	−0.833435	−0.416718	+	0.909036i	$0.636820\pi$
$858$	0	0
$859$	23304.5	0.925655	0.462828	−	0.886448i	$-0.346835\pi$
$859$	23304.5	0.925655	0.462828	+	0.886448i	$0.346835\pi$
$860$	10870.6	0.431027
$861$	0	0
$862$	−7775.61	−0.307237
$863$	19146.3	0.755211	0.377606	−	0.925966i	$-0.376747\pi$
$863$	19146.3	0.755211	0.377606	+	0.925966i	$0.376747\pi$
$864$	0	0
$865$	2608.86	0.102548
$866$	−8845.87	−0.347107
$867$	0	0
$868$	10637.5	0.415968
$869$	−44310.5	−1.72972
$870$	0	0
$871$	−16863.0	−0.656005
$872$	−1113.32	−0.0432362
$873$	0	0
$874$	10.7061	0.000414347 0
$875$	2945.28	0.113793
$876$	0	0
$877$	31389.0	1.20859	0.604293	−	0.796762i	$-0.293456\pi$
$877$	31389.0	1.20859	0.604293	+	0.796762i	$0.293456\pi$
$878$	−3496.16	−0.134384
$879$	0	0
$880$	2570.82	0.0984798
$881$	−44496.3	−1.70161	−0.850806	−	0.525480i	$-0.823886\pi$
$881$	−44496.3	−1.70161	−0.850806	+	0.525480i	$0.823886\pi$
$882$	0	0
$883$	8906.65	0.339448	0.169724	−	0.985492i	$-0.445712\pi$
$883$	8906.65	0.339448	0.169724	+	0.985492i	$0.445712\pi$
$884$	−20227.4	−0.769592
$885$	0	0
$886$	−6742.62	−0.255669
$887$	36584.7	1.38489	0.692444	−	0.721472i	$-0.256534\pi$
$887$	36584.7	1.38489	0.692444	+	0.721472i	$0.256534\pi$
$888$	0	0
$889$	−30593.9	−1.15420
$890$	11277.1	0.424729
$891$	0	0
$892$	−8498.74	−0.319012
$893$	−6.15238	−0.000230551 0
$894$	0	0
$895$	5915.35	0.220926
$896$	3015.97	0.112451
$897$	0	0
$898$	−26629.5	−0.989573
$899$	15488.1	0.574591
$900$	0	0
$901$	5512.90	0.203842
$902$	8701.23	0.321197
$903$	0	0
$904$	10345.4	0.380623
$905$	8619.62	0.316603
$906$	0	0
$907$	−21346.1	−0.781463	−0.390731	−	0.920505i	$-0.627778\pi$
$907$	−21346.1	−0.781463	−0.390731	+	0.920505i	$0.627778\pi$
$908$	−6207.89	−0.226890
$909$	0	0
$910$	9444.00	0.344028
$911$	10208.0	0.371246	0.185623	−	0.982621i	$-0.440570\pi$
$911$	10208.0	0.371246	0.185623	+	0.982621i	$0.440570\pi$
$912$	0	0
$913$	15591.3	0.565166
$914$	7534.40	0.272665
$915$	0	0
$916$	−634.125	−0.0228735
$917$	12376.0	0.445683
$918$	0	0
$919$	1758.00	0.0631025	0.0315513	−	0.999502i	$-0.489955\pi$
$919$	1758.00	0.0631025	0.0315513	+	0.999502i	$0.489955\pi$
$920$	920.000	0.0329690
$921$	0	0
$922$	−19348.1	−0.691102
$923$	−26749.3	−0.953917
$924$	0	0
$925$	1142.64	0.0406161
$926$	5955.46	0.211348
$927$	0	0
$928$	4391.22	0.155333
$929$	845.008	0.0298426	0.0149213	−	0.999889i	$-0.495250\pi$
$929$	845.008	0.0298426	0.0149213	+	0.999889i	$0.495250\pi$
$930$	0	0
$931$	49.3830	0.00173841
$932$	2915.99	0.102486
$933$	0	0
$934$	21402.0	0.749782
$935$	−20271.7	−0.709045
$936$	0	0
$937$	3851.61	0.134287	0.0671433	−	0.997743i	$-0.478612\pi$
$937$	3851.61	0.134287	0.0671433	+	0.997743i	$0.478612\pi$
$938$	−19826.3	−0.690141
$939$	0	0
$940$	−528.688	−0.0183446
$941$	4379.36	0.151714	0.0758571	−	0.997119i	$-0.475831\pi$
$941$	4379.36	0.151714	0.0758571	+	0.997119i	$0.475831\pi$
$942$	0	0
$943$	3113.85	0.107530
$944$	−3235.96	−0.111570
$945$	0	0
$946$	34932.8	1.20059
$947$	−29746.9	−1.02074	−0.510371	−	0.859954i	$-0.670492\pi$
$947$	−29746.9	−1.02074	−0.510371	+	0.859954i	$0.670492\pi$
$948$	0	0
$949$	24167.3	0.826663
$950$	11.6371	0.000397428 0
$951$	0	0
$952$	−23781.9	−0.809638
$953$	−4306.61	−0.146385	−0.0731924	−	0.997318i	$-0.523319\pi$
$953$	−4306.61	−0.146385	−0.0731924	+	0.997318i	$0.523319\pi$
$954$	0	0
$955$	13990.6	0.474057
$956$	24806.4	0.839222
$957$	0	0
$958$	−4674.29	−0.157640
$959$	57211.9	1.92645
$960$	0	0
$961$	−17052.3	−0.572397
$962$	3663.87	0.122794
$963$	0	0
$964$	−28892.2	−0.965305
$965$	−2488.43	−0.0830108
$966$	0	0
$967$	−12233.4	−0.406824	−0.203412	−	0.979093i	$-0.565203\pi$
$967$	−12233.4	−0.406824	−0.203412	+	0.979093i	$0.565203\pi$
$968$	−2386.63	−0.0792449
$969$	0	0
$970$	−14862.4	−0.491960
$971$	−48207.7	−1.59326	−0.796631	−	0.604465i	$-0.793387\pi$
$971$	−48207.7	−1.59326	−0.796631	+	0.604465i	$0.793387\pi$
$972$	0	0
$973$	57879.8	1.90703
$974$	−14367.7	−0.472660
$975$	0	0
$976$	2404.46	0.0788575
$977$	−28662.9	−0.938595	−0.469298	−	0.883040i	$-0.655493\pi$
$977$	−28662.9	−0.938595	−0.469298	+	0.883040i	$0.655493\pi$
$978$	0	0
$979$	36239.2	1.18305
$980$	4243.59	0.138323
$981$	0	0
$982$	24168.4	0.785383
$983$	20944.1	0.679567	0.339783	−	0.940504i	$-0.389646\pi$
$983$	20944.1	0.679567	0.339783	+	0.940504i	$0.389646\pi$
$984$	0	0
$985$	−1482.44	−0.0479539
$986$	−34626.2	−1.11838
$987$	0	0
$988$	37.3141	0.00120154
$989$	12501.1	0.401934
$990$	0	0
$991$	−36440.2	−1.16808	−0.584038	−	0.811727i	$-0.698528\pi$
$991$	−36440.2	−1.16808	−0.584038	+	0.811727i	$0.698528\pi$
$992$	3611.71	0.115597
$993$	0	0
$994$	−31450.0	−1.00355
$995$	11737.2	0.373963
$996$	0	0
$997$	−11945.0	−0.379439	−0.189720	−	0.981838i	$-0.560758\pi$
$997$	−11945.0	−0.379439	−0.189720	+	0.981838i	$0.560758\pi$
$998$	14291.6	0.453298
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.bj.1.3		4
3.2	odd	2		230.4.a.h.1.4	✓	4
12.11	even	2		1840.4.a.m.1.1		4
15.2	even	4		1150.4.b.n.599.1		8
15.8	even	4		1150.4.b.n.599.8		8
15.14	odd	2		1150.4.a.p.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.h.1.4	✓	4	3.2	odd	2
1150.4.a.p.1.1		4	15.14	odd	2
1150.4.b.n.599.1		8	15.2	even	4
1150.4.b.n.599.8		8	15.8	even	4
1840.4.a.m.1.1		4	12.11	even	2
2070.4.a.bj.1.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +23.5622 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +23.5622 q^{7} +8.00000 q^{8} +10.0000 q^{10} +32.1352 q^{11} +40.0811 q^{13} +47.1245 q^{14} +16.0000 q^{16} -126.165 q^{17} +0.232742 q^{19} +20.0000 q^{20} +64.2704 q^{22} +23.0000 q^{23} +25.0000 q^{25} +80.1621 q^{26} +94.2490 q^{28} +137.226 q^{29} +112.866 q^{31} +32.0000 q^{32} -252.331 q^{34} +117.811 q^{35} +45.7057 q^{37} +0.465483 q^{38} +40.0000 q^{40} +135.385 q^{41} +543.528 q^{43} +128.541 q^{44} +46.0000 q^{46} -26.4344 q^{47} +212.180 q^{49} +50.0000 q^{50} +160.324 q^{52} -43.6958 q^{53} +160.676 q^{55} +188.498 q^{56} +274.451 q^{58} -202.248 q^{59} +150.279 q^{61} +225.732 q^{62} +64.0000 q^{64} +200.405 q^{65} -420.722 q^{67} -504.661 q^{68} +235.622 q^{70} -667.381 q^{71} +602.960 q^{73} +91.4115 q^{74} +0.930966 q^{76} +757.178 q^{77} -1378.88 q^{79} +80.0000 q^{80} +270.769 q^{82} +485.178 q^{83} -630.826 q^{85} +1087.06 q^{86} +257.082 q^{88} +1127.71 q^{89} +944.400 q^{91} +92.0000 q^{92} -52.8688 q^{94} +1.16371 q^{95} -1486.24 q^{97} +424.359 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8}+O(q^{10}) \) 4 * q + 8 * q^2 + 16 * q^4 + 20 * q^5 - q^7 + 32 * q^8	\( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} - 20 q^{13} - 2 q^{14} + 64 q^{16} + 23 q^{17} + 53 q^{19} + 80 q^{20} + 78 q^{22} + 92 q^{23} + 100 q^{25} - 40 q^{26} - 4 q^{28} - 161 q^{29} + 388 q^{31} + 128 q^{32} + 46 q^{34} - 5 q^{35} + 466 q^{37} + 106 q^{38} + 160 q^{40} - 484 q^{41} + 894 q^{43} + 156 q^{44} + 184 q^{46} + 265 q^{47} + 1643 q^{49} + 200 q^{50} - 80 q^{52} - 576 q^{53} + 195 q^{55} - 8 q^{56} - 322 q^{58} + 94 q^{59} + 1153 q^{61} + 776 q^{62} + 256 q^{64} - 100 q^{65} - 1472 q^{67} + 92 q^{68} - 10 q^{70} - 200 q^{71} + 1147 q^{73} + 932 q^{74} + 212 q^{76} + 2176 q^{77} - 908 q^{79} + 320 q^{80} - 968 q^{82} + 1048 q^{83} + 115 q^{85} + 1788 q^{86} + 312 q^{88} + 1784 q^{89} + 2329 q^{91} + 368 q^{92} + 530 q^{94} + 265 q^{95} - 2047 q^{97} + 3286 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 + 20 * q^5 - q^7 + 32 * q^8 + 40 * q^10 + 39 * q^11 - 20 * q^13 - 2 * q^14 + 64 * q^16 + 23 * q^17 + 53 * q^19 + 80 * q^20 + 78 * q^22 + 92 * q^23 + 100 * q^25 - 40 * q^26 - 4 * q^28 - 161 * q^29 + 388 * q^31 + 128 * q^32 + 46 * q^34 - 5 * q^35 + 466 * q^37 + 106 * q^38 + 160 * q^40 - 484 * q^41 + 894 * q^43 + 156 * q^44 + 184 * q^46 + 265 * q^47 + 1643 * q^49 + 200 * q^50 - 80 * q^52 - 576 * q^53 + 195 * q^55 - 8 * q^56 - 322 * q^58 + 94 * q^59 + 1153 * q^61 + 776 * q^62 + 256 * q^64 - 100 * q^65 - 1472 * q^67 + 92 * q^68 - 10 * q^70 - 200 * q^71 + 1147 * q^73 + 932 * q^74 + 212 * q^76 + 2176 * q^77 - 908 * q^79 + 320 * q^80 - 968 * q^82 + 1048 * q^83 + 115 * q^85 + 1788 * q^86 + 312 * q^88 + 1784 * q^89 + 2329 * q^91 + 368 * q^92 + 530 * q^94 + 265 * q^95 - 2047 * q^97 + 3286 * q^98

Introduction

L-functions

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