LMFDB - Embedded newform 2070.4.a.bj.1.4

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			1.4
Root			$-3.74869$ 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)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +29.3684 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +29.3684 q^{7} +8.00000 q^{8} +10.0000 q^{10} +38.1645 q^{11} -22.5396 q^{13} +58.7368 q^{14} +16.0000 q^{16} +104.049 q^{17} +142.000 q^{19} +20.0000 q^{20} +76.3291 q^{22} +23.0000 q^{23} +25.0000 q^{25} -45.0792 q^{26} +117.474 q^{28} -241.429 q^{29} +99.2706 q^{31} +32.0000 q^{32} +208.097 q^{34} +146.842 q^{35} +59.9452 q^{37} +284.000 q^{38} +40.0000 q^{40} -249.248 q^{41} -163.863 q^{43} +152.658 q^{44} +46.0000 q^{46} +205.591 q^{47} +519.502 q^{49} +50.0000 q^{50} -90.1584 q^{52} -491.274 q^{53} +190.823 q^{55} +234.947 q^{56} -482.858 q^{58} -433.734 q^{59} +660.902 q^{61} +198.541 q^{62} +64.0000 q^{64} -112.698 q^{65} -323.564 q^{67} +416.195 q^{68} +293.684 q^{70} -893.243 q^{71} +196.273 q^{73} +119.890 q^{74} +567.999 q^{76} +1120.83 q^{77} -500.211 q^{79} +80.0000 q^{80} -498.496 q^{82} -800.944 q^{83} +520.243 q^{85} -327.726 q^{86} +305.316 q^{88} +729.016 q^{89} -661.952 q^{91} +92.0000 q^{92} +411.181 q^{94} +709.999 q^{95} +1139.96 q^{97} +1039.00 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	29.3684	1.58574	0.792872	−	0.609388i	$-0.208585\pi$
$7$	29.3684	1.58574	0.792872	+	0.609388i	$0.208585\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	38.1645	1.04609	0.523047	−	0.852304i	$-0.324795\pi$
$11$	38.1645	1.04609	0.523047	+	0.852304i	$0.324795\pi$
$12$	0	0
$13$	−22.5396	−0.480874	−0.240437	−	0.970665i	$-0.577291\pi$
$13$	−22.5396	−0.480874	−0.240437	+	0.970665i	$0.577291\pi$
$14$	58.7368	1.12129
$15$	0	0
$16$	16.0000	0.250000
$17$	104.049	1.48444	0.742221	−	0.670155i	$-0.233772\pi$
$17$	104.049	1.48444	0.742221	+	0.670155i	$0.233772\pi$
$18$	0	0
$19$	142.000	1.71458	0.857289	−	0.514835i	$-0.172147\pi$
$19$	142.000	1.71458	0.857289	+	0.514835i	$0.172147\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	76.3291	0.739701
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	−45.0792	−0.340029
$27$	0	0
$28$	117.474	0.792872
$29$	−241.429	−1.54594	−0.772969	−	0.634444i	$-0.781229\pi$
$29$	−241.429	−1.54594	−0.772969	+	0.634444i	$0.781229\pi$
$30$	0	0
$31$	99.2706	0.575146	0.287573	−	0.957759i	$-0.407152\pi$
$31$	99.2706	0.575146	0.287573	+	0.957759i	$0.407152\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	208.097	1.04966
$35$	146.842	0.709166
$36$	0	0
$37$	59.9452	0.266349	0.133175	−	0.991093i	$-0.457483\pi$
$37$	59.9452	0.266349	0.133175	+	0.991093i	$0.457483\pi$
$38$	284.000	1.21239
$39$	0	0
$40$	40.0000	0.158114
$41$	−249.248	−0.949414	−0.474707	−	0.880144i	$-0.657446\pi$
$41$	−249.248	−0.949414	−0.474707	+	0.880144i	$0.657446\pi$
$42$	0	0
$43$	−163.863	−0.581136	−0.290568	−	0.956854i	$-0.593844\pi$
$43$	−163.863	−0.581136	−0.290568	+	0.956854i	$0.593844\pi$
$44$	152.658	0.523047
$45$	0	0
$46$	46.0000	0.147442
$47$	205.591	0.638052	0.319026	−	0.947746i	$-0.396644\pi$
$47$	205.591	0.638052	0.319026	+	0.947746i	$0.396644\pi$
$48$	0	0
$49$	519.502	1.51458
$50$	50.0000	0.141421
$51$	0	0
$52$	−90.1584	−0.240437
$53$	−491.274	−1.27324	−0.636620	−	0.771178i	$-0.719668\pi$
$53$	−491.274	−1.27324	−0.636620	+	0.771178i	$0.719668\pi$
$54$	0	0
$55$	190.823	0.467828
$56$	234.947	0.560645
$57$	0	0
$58$	−482.858	−1.09314
$59$	−433.734	−0.957074	−0.478537	−	0.878067i	$-0.658833\pi$
$59$	−433.734	−0.957074	−0.478537	+	0.878067i	$0.658833\pi$
$60$	0	0
$61$	660.902	1.38721	0.693605	−	0.720355i	$-0.256021\pi$
$61$	660.902	1.38721	0.693605	+	0.720355i	$0.256021\pi$
$62$	198.541	0.406690
$63$	0	0
$64$	64.0000	0.125000
$65$	−112.698	−0.215053
$66$	0	0
$67$	−323.564	−0.589995	−0.294997	−	0.955498i	$-0.595319\pi$
$67$	−323.564	−0.589995	−0.294997	+	0.955498i	$0.595319\pi$
$68$	416.195	0.742221
$69$	0	0
$70$	293.684	0.501456
$71$	−893.243	−1.49308	−0.746538	−	0.665342i	$-0.768286\pi$
$71$	−893.243	−1.49308	−0.746538	+	0.665342i	$0.768286\pi$
$72$	0	0
$73$	196.273	0.314685	0.157343	−	0.987544i	$-0.449707\pi$
$73$	196.273	0.314685	0.157343	+	0.987544i	$0.449707\pi$
$74$	119.890	0.188337
$75$	0	0
$76$	567.999	0.857289
$77$	1120.83	1.65884
$78$	0	0
$79$	−500.211	−0.712381	−0.356191	−	0.934413i	$-0.615925\pi$
$79$	−500.211	−0.712381	−0.356191	+	0.934413i	$0.615925\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	−498.496	−0.671337
$83$	−800.944	−1.05922	−0.529609	−	0.848242i	$-0.677661\pi$
$83$	−800.944	−1.05922	−0.529609	+	0.848242i	$0.677661\pi$
$84$	0	0
$85$	520.243	0.663863
$86$	−327.726	−0.410925
$87$	0	0
$88$	305.316	0.369850
$89$	729.016	0.868264	0.434132	−	0.900849i	$-0.357055\pi$
$89$	729.016	0.868264	0.434132	+	0.900849i	$0.357055\pi$
$90$	0	0
$91$	−661.952	−0.762543
$92$	92.0000	0.104257
$93$	0	0
$94$	411.181	0.451171
$95$	709.999	0.766783
$96$	0	0
$97$	1139.96	1.19325	0.596626	−	0.802520i	$-0.296508\pi$
$97$	1139.96	1.19325	0.596626	+	0.802520i	$0.296508\pi$
$98$	1039.00	1.07097
$99$	0	0
$100$	100.000	0.100000
$101$	1669.38	1.64465	0.822326	−	0.569017i	$-0.192676\pi$
$101$	1669.38	1.64465	0.822326	+	0.569017i	$0.192676\pi$
$102$	0	0
$103$	−1110.63	−1.06246	−0.531229	−	0.847228i	$-0.678270\pi$
$103$	−1110.63	−1.06246	−0.531229	+	0.847228i	$0.678270\pi$
$104$	−180.317	−0.170015
$105$	0	0
$106$	−982.549	−0.900317
$107$	−39.1892	−0.0354071	−0.0177036	−	0.999843i	$-0.505636\pi$
$107$	−39.1892	−0.0354071	−0.0177036	+	0.999843i	$0.505636\pi$
$108$	0	0
$109$	807.545	0.709622	0.354811	−	0.934938i	$-0.384545\pi$
$109$	807.545	0.709622	0.354811	+	0.934938i	$0.384545\pi$
$110$	381.645	0.330804
$111$	0	0
$112$	469.894	0.396436
$113$	−1066.58	−0.887923	−0.443961	−	0.896046i	$-0.646427\pi$
$113$	−1066.58	−0.887923	−0.443961	+	0.896046i	$0.646427\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	−965.715	−0.772969
$117$	0	0
$118$	−867.468	−0.676753
$119$	3055.74	2.35395
$120$	0	0
$121$	125.532	0.0943139
$122$	1321.80	0.980906
$123$	0	0
$124$	397.082	0.287573
$125$	125.000	0.0894427
$126$	0	0
$127$	−641.707	−0.448364	−0.224182	−	0.974547i	$-0.571971\pi$
$127$	−641.707	−0.448364	−0.224182	+	0.974547i	$0.571971\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−225.396	−0.152066
$131$	−2389.92	−1.59396	−0.796979	−	0.604007i	$-0.793570\pi$
$131$	−2389.92	−1.59396	−0.796979	+	0.604007i	$0.793570\pi$
$132$	0	0
$133$	4170.31	2.71888
$134$	−647.128	−0.417189
$135$	0	0
$136$	832.390	0.524830
$137$	−899.291	−0.560814	−0.280407	−	0.959881i	$-0.590470\pi$
$137$	−899.291	−0.560814	−0.280407	+	0.959881i	$0.590470\pi$
$138$	0	0
$139$	309.341	0.188762	0.0943811	−	0.995536i	$-0.469913\pi$
$139$	309.341	0.188762	0.0943811	+	0.995536i	$0.469913\pi$
$140$	587.368	0.354583
$141$	0	0
$142$	−1786.49	−1.05576
$143$	−860.214	−0.503040
$144$	0	0
$145$	−1207.14	−0.691364
$146$	392.546	0.222516
$147$	0	0
$148$	239.781	0.133175
$149$	−2560.87	−1.40802	−0.704010	−	0.710190i	$-0.748609\pi$
$149$	−2560.87	−1.40802	−0.704010	+	0.710190i	$0.748609\pi$
$150$	0	0
$151$	−2463.15	−1.32747	−0.663737	−	0.747966i	$-0.731031\pi$
$151$	−2463.15	−1.32747	−0.663737	+	0.747966i	$0.731031\pi$
$152$	1136.00	0.606195
$153$	0	0
$154$	2241.66	1.17298
$155$	496.353	0.257213
$156$	0	0
$157$	−566.720	−0.288084	−0.144042	−	0.989572i	$-0.546010\pi$
$157$	−566.720	−0.288084	−0.144042	+	0.989572i	$0.546010\pi$
$158$	−1000.42	−0.503730
$159$	0	0
$160$	160.000	0.0790569
$161$	675.473	0.330650
$162$	0	0
$163$	960.902	0.461740	0.230870	−	0.972985i	$-0.425843\pi$
$163$	960.902	0.461740	0.230870	+	0.972985i	$0.425843\pi$
$164$	−996.992	−0.474707
$165$	0	0
$166$	−1601.89	−0.748980
$167$	4224.70	1.95759	0.978794	−	0.204847i	$-0.0656696\pi$
$167$	4224.70	1.95759	0.978794	+	0.204847i	$0.0656696\pi$
$168$	0	0
$169$	−1688.97	−0.768760
$170$	1040.49	0.469422
$171$	0	0
$172$	−655.451	−0.290568
$173$	−3448.98	−1.51573	−0.757865	−	0.652412i	$-0.773757\pi$
$173$	−3448.98	−1.51573	−0.757865	+	0.652412i	$0.773757\pi$
$174$	0	0
$175$	734.210	0.317149
$176$	610.633	0.261524
$177$	0	0
$178$	1458.03	0.613955
$179$	−1457.82	−0.608728	−0.304364	−	0.952556i	$-0.598444\pi$
$179$	−1457.82	−0.608728	−0.304364	+	0.952556i	$0.598444\pi$
$180$	0	0
$181$	−3634.97	−1.49274	−0.746368	−	0.665534i	$-0.768204\pi$
$181$	−3634.97	−1.49274	−0.746368	+	0.665534i	$0.768204\pi$
$182$	−1323.90	−0.539199
$183$	0	0
$184$	184.000	0.0737210
$185$	299.726	0.119115
$186$	0	0
$187$	3970.97	1.55287
$188$	822.362	0.319026
$189$	0	0
$190$	1420.00	0.542197
$191$	−448.811	−0.170025	−0.0850127	−	0.996380i	$-0.527093\pi$
$191$	−448.811	−0.170025	−0.0850127	+	0.996380i	$0.527093\pi$
$192$	0	0
$193$	4259.35	1.58857	0.794286	−	0.607544i	$-0.207845\pi$
$193$	4259.35	1.58857	0.794286	+	0.607544i	$0.207845\pi$
$194$	2279.92	0.843756
$195$	0	0
$196$	2078.01	0.757292
$197$	1767.37	0.639189	0.319594	−	0.947554i	$-0.396453\pi$
$197$	1767.37	0.639189	0.319594	+	0.947554i	$0.396453\pi$
$198$	0	0
$199$	1004.29	0.357750	0.178875	−	0.983872i	$-0.442754\pi$
$199$	1004.29	0.357750	0.178875	+	0.983872i	$0.442754\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	3338.76	1.16294
$203$	−7090.37	−2.45146
$204$	0	0
$205$	−1246.24	−0.424591
$206$	−2221.25	−0.751272
$207$	0	0
$208$	−360.634	−0.120218
$209$	5419.36	1.79361
$210$	0	0
$211$	5395.17	1.76028	0.880140	−	0.474714i	$-0.157449\pi$
$211$	5395.17	1.76028	0.880140	+	0.474714i	$0.157449\pi$
$212$	−1965.10	−0.636620
$213$	0	0
$214$	−78.3783	−0.0250366
$215$	−819.314	−0.259892
$216$	0	0
$217$	2915.42	0.912034
$218$	1615.09	0.501779
$219$	0	0
$220$	763.291	0.233914
$221$	−2345.22	−0.713830
$222$	0	0
$223$	−1504.79	−0.451876	−0.225938	−	0.974142i	$-0.572545\pi$
$223$	−1504.79	−0.451876	−0.225938	+	0.974142i	$0.572545\pi$
$224$	939.788	0.280323
$225$	0	0
$226$	−2133.16	−0.627856
$227$	1779.44	0.520288	0.260144	−	0.965570i	$-0.416230\pi$
$227$	1779.44	0.520288	0.260144	+	0.965570i	$0.416230\pi$
$228$	0	0
$229$	3976.16	1.14739	0.573694	−	0.819070i	$-0.305510\pi$
$229$	3976.16	1.14739	0.573694	+	0.819070i	$0.305510\pi$
$230$	230.000	0.0659380
$231$	0	0
$232$	−1931.43	−0.546572
$233$	2311.12	0.649815	0.324907	−	0.945746i	$-0.394667\pi$
$233$	2311.12	0.649815	0.324907	+	0.945746i	$0.394667\pi$
$234$	0	0
$235$	1027.95	0.285346
$236$	−1734.94	−0.478537
$237$	0	0
$238$	6111.49	1.66449
$239$	−824.267	−0.223085	−0.111543	−	0.993760i	$-0.535579\pi$
$239$	−824.267	−0.223085	−0.111543	+	0.993760i	$0.535579\pi$
$240$	0	0
$241$	3641.83	0.973406	0.486703	−	0.873567i	$-0.338199\pi$
$241$	3641.83	0.973406	0.486703	+	0.873567i	$0.338199\pi$
$242$	251.064	0.0666900
$243$	0	0
$244$	2643.61	0.693605
$245$	2597.51	0.677343
$246$	0	0
$247$	−3200.62	−0.824496
$248$	794.165	0.203345
$249$	0	0
$250$	250.000	0.0632456
$251$	7767.51	1.95331	0.976655	−	0.214813i	$-0.0689143\pi$
$251$	7767.51	1.95331	0.976655	+	0.214813i	$0.0689143\pi$
$252$	0	0
$253$	877.784	0.218126
$254$	−1283.41	−0.317042
$255$	0	0
$256$	256.000	0.0625000
$257$	1501.17	0.364359	0.182179	−	0.983265i	$-0.441685\pi$
$257$	1501.17	0.364359	0.182179	+	0.983265i	$0.441685\pi$
$258$	0	0
$259$	1760.49	0.422362
$260$	−450.792	−0.107527
$261$	0	0
$262$	−4779.84	−1.12710
$263$	5430.98	1.27334	0.636671	−	0.771136i	$-0.280311\pi$
$263$	5430.98	1.27334	0.636671	+	0.771136i	$0.280311\pi$
$264$	0	0
$265$	−2456.37	−0.569410
$266$	8340.61	1.92254
$267$	0	0
$268$	−1294.26	−0.294997
$269$	5014.20	1.13651	0.568255	−	0.822852i	$-0.307619\pi$
$269$	5014.20	1.13651	0.568255	+	0.822852i	$0.307619\pi$
$270$	0	0
$271$	7215.54	1.61739	0.808695	−	0.588228i	$-0.200174\pi$
$271$	7215.54	1.61739	0.808695	+	0.588228i	$0.200174\pi$
$272$	1664.78	0.371111
$273$	0	0
$274$	−1798.58	−0.396556
$275$	954.113	0.209219
$276$	0	0
$277$	−7439.60	−1.61373	−0.806863	−	0.590738i	$-0.798837\pi$
$277$	−7439.60	−1.61373	−0.806863	+	0.590738i	$0.798837\pi$
$278$	618.681	0.133475
$279$	0	0
$280$	1174.74	0.250728
$281$	2605.59	0.553155	0.276578	−	0.960992i	$-0.410800\pi$
$281$	2605.59	0.553155	0.276578	+	0.960992i	$0.410800\pi$
$282$	0	0
$283$	−7162.57	−1.50449	−0.752245	−	0.658884i	$-0.771029\pi$
$283$	−7162.57	−1.50449	−0.752245	+	0.658884i	$0.771029\pi$
$284$	−3572.97	−0.746538
$285$	0	0
$286$	−1720.43	−0.355703
$287$	−7320.01	−1.50553
$288$	0	0
$289$	5913.13	1.20357
$290$	−2414.29	−0.488868
$291$	0	0
$292$	785.093	0.157343
$293$	−9310.55	−1.85641	−0.928205	−	0.372069i	$-0.878649\pi$
$293$	−9310.55	−1.85641	−0.928205	+	0.372069i	$0.878649\pi$
$294$	0	0
$295$	−2168.67	−0.428016
$296$	479.562	0.0941687
$297$	0	0
$298$	−5121.74	−0.995620
$299$	−518.411	−0.100269
$300$	0	0
$301$	−4812.39	−0.921533
$302$	−4926.31	−0.938666
$303$	0	0
$304$	2272.00	0.428645
$305$	3304.51	0.620380
$306$	0	0
$307$	3359.07	0.624470	0.312235	−	0.950005i	$-0.398922\pi$
$307$	3359.07	0.624470	0.312235	+	0.950005i	$0.398922\pi$
$308$	4483.32	0.829419
$309$	0	0
$310$	992.706	0.181877
$311$	5509.20	1.00450	0.502248	−	0.864724i	$-0.332507\pi$
$311$	5509.20	1.00450	0.502248	+	0.864724i	$0.332507\pi$
$312$	0	0
$313$	2401.42	0.433661	0.216831	−	0.976209i	$-0.430428\pi$
$313$	2401.42	0.433661	0.216831	+	0.976209i	$0.430428\pi$
$314$	−1133.44	−0.203706
$315$	0	0
$316$	−2000.84	−0.356191
$317$	1057.36	0.187341	0.0936704	−	0.995603i	$-0.470140\pi$
$317$	1057.36	0.187341	0.0936704	+	0.995603i	$0.470140\pi$
$318$	0	0
$319$	−9214.02	−1.61720
$320$	320.000	0.0559017
$321$	0	0
$322$	1350.95	0.233805
$323$	14774.9	2.54519
$324$	0	0
$325$	−563.490	−0.0961748
$326$	1921.80	0.326500
$327$	0	0
$328$	−1993.98	−0.335669
$329$	6037.86	1.01179
$330$	0	0
$331$	−9680.21	−1.60747	−0.803735	−	0.594987i	$-0.797157\pi$
$331$	−9680.21	−1.60747	−0.803735	+	0.594987i	$0.797157\pi$
$332$	−3203.78	−0.529609
$333$	0	0
$334$	8449.40	1.38422
$335$	−1617.82	−0.263854
$336$	0	0
$337$	467.694	0.0755991	0.0377996	−	0.999285i	$-0.487965\pi$
$337$	467.694	0.0755991	0.0377996	+	0.999285i	$0.487965\pi$
$338$	−3377.93	−0.543596
$339$	0	0
$340$	2080.97	0.331931
$341$	3788.62	0.601657
$342$	0	0
$343$	5183.59	0.815998
$344$	−1310.90	−0.205463
$345$	0	0
$346$	−6897.96	−1.07178
$347$	−3371.55	−0.521597	−0.260798	−	0.965393i	$-0.583986\pi$
$347$	−3371.55	−0.521597	−0.260798	+	0.965393i	$0.583986\pi$
$348$	0	0
$349$	10958.7	1.68082	0.840412	−	0.541947i	$-0.182313\pi$
$349$	10958.7	1.68082	0.840412	+	0.541947i	$0.182313\pi$
$350$	1468.42	0.224258
$351$	0	0
$352$	1221.27	0.184925
$353$	1584.76	0.238947	0.119474	−	0.992837i	$-0.461879\pi$
$353$	1584.76	0.238947	0.119474	+	0.992837i	$0.461879\pi$
$354$	0	0
$355$	−4466.21	−0.667724
$356$	2916.06	0.434132
$357$	0	0
$358$	−2915.63	−0.430436
$359$	−5130.67	−0.754280	−0.377140	−	0.926156i	$-0.623092\pi$
$359$	−5130.67	−0.754280	−0.377140	+	0.926156i	$0.623092\pi$
$360$	0	0
$361$	13304.9	1.93978
$362$	−7269.94	−1.05552
$363$	0	0
$364$	−2647.81	−0.381271
$365$	981.366	0.140732
$366$	0	0
$367$	2811.27	0.399856	0.199928	−	0.979811i	$-0.435929\pi$
$367$	2811.27	0.399856	0.199928	+	0.979811i	$0.435929\pi$
$368$	368.000	0.0521286
$369$	0	0
$370$	599.452	0.0842271
$371$	−14427.9	−2.01903
$372$	0	0
$373$	−1952.42	−0.271026	−0.135513	−	0.990776i	$-0.543268\pi$
$373$	−1952.42	−0.271026	−0.135513	+	0.990776i	$0.543268\pi$
$374$	7941.94	1.09804
$375$	0	0
$376$	1644.72	0.225586
$377$	5441.71	0.743401
$378$	0	0
$379$	9609.27	1.30236	0.651181	−	0.758923i	$-0.274274\pi$
$379$	9609.27	1.30236	0.651181	+	0.758923i	$0.274274\pi$
$380$	2840.00	0.383391
$381$	0	0
$382$	−897.623	−0.120226
$383$	−5027.44	−0.670732	−0.335366	−	0.942088i	$-0.608860\pi$
$383$	−5027.44	−0.670732	−0.335366	+	0.942088i	$0.608860\pi$
$384$	0	0
$385$	5604.15	0.741855
$386$	8518.70	1.12329
$387$	0	0
$388$	4559.84	0.596626
$389$	−5892.29	−0.767997	−0.383999	−	0.923334i	$-0.625453\pi$
$389$	−5892.29	−0.767997	−0.383999	+	0.923334i	$0.625453\pi$
$390$	0	0
$391$	2393.12	0.309528
$392$	4156.02	0.535486
$393$	0	0
$394$	3534.75	0.451975
$395$	−2501.05	−0.318587
$396$	0	0
$397$	−2454.41	−0.310285	−0.155143	−	0.987892i	$-0.549584\pi$
$397$	−2454.41	−0.310285	−0.155143	+	0.987892i	$0.549584\pi$
$398$	2008.58	0.252967
$399$	0	0
$400$	400.000	0.0500000
$401$	−9458.39	−1.17788	−0.588939	−	0.808177i	$-0.700454\pi$
$401$	−9458.39	−1.17788	−0.588939	+	0.808177i	$0.700454\pi$
$402$	0	0
$403$	−2237.52	−0.276573
$404$	6677.53	0.822326
$405$	0	0
$406$	−14180.7	−1.73345
$407$	2287.78	0.278627
$408$	0	0
$409$	−5857.25	−0.708123	−0.354062	−	0.935222i	$-0.615200\pi$
$409$	−5857.25	−0.708123	−0.354062	+	0.935222i	$0.615200\pi$
$410$	−2492.48	−0.300231
$411$	0	0
$412$	−4442.51	−0.531229
$413$	−12738.1	−1.51767
$414$	0	0
$415$	−4004.72	−0.473696
$416$	−721.267	−0.0850073
$417$	0	0
$418$	10838.7	1.26827
$419$	5252.61	0.612426	0.306213	−	0.951963i	$-0.400938\pi$
$419$	5252.61	0.612426	0.306213	+	0.951963i	$0.400938\pi$
$420$	0	0
$421$	−203.517	−0.0235602	−0.0117801	−	0.999931i	$-0.503750\pi$
$421$	−203.517	−0.0235602	−0.0117801	+	0.999931i	$0.503750\pi$
$422$	10790.3	1.24471
$423$	0	0
$424$	−3930.20	−0.450158
$425$	2601.22	0.296888
$426$	0	0
$427$	19409.6	2.19976
$428$	−156.757	−0.0177036
$429$	0	0
$430$	−1638.63	−0.183771
$431$	−3107.72	−0.347317	−0.173659	−	0.984806i	$-0.555559\pi$
$431$	−3107.72	−0.347317	−0.173659	+	0.984806i	$0.555559\pi$
$432$	0	0
$433$	−5713.46	−0.634114	−0.317057	−	0.948406i	$-0.602695\pi$
$433$	−5713.46	−0.634114	−0.317057	+	0.948406i	$0.602695\pi$
$434$	5830.83	0.644905
$435$	0	0
$436$	3230.18	0.354811
$437$	3266.00	0.357514
$438$	0	0
$439$	−4587.14	−0.498706	−0.249353	−	0.968413i	$-0.580218\pi$
$439$	−4587.14	−0.498706	−0.249353	+	0.968413i	$0.580218\pi$
$440$	1526.58	0.165402
$441$	0	0
$442$	−4690.43	−0.504754
$443$	−5268.01	−0.564990	−0.282495	−	0.959269i	$-0.591162\pi$
$443$	−5268.01	−0.564990	−0.282495	+	0.959269i	$0.591162\pi$
$444$	0	0
$445$	3645.08	0.388299
$446$	−3009.58	−0.319525
$447$	0	0
$448$	1879.58	0.198218
$449$	−16866.8	−1.77282	−0.886409	−	0.462902i	$-0.846808\pi$
$449$	−16866.8	−1.77282	−0.886409	+	0.462902i	$0.846808\pi$
$450$	0	0
$451$	−9512.43	−0.993177
$452$	−4266.31	−0.443961
$453$	0	0
$454$	3558.87	0.367899
$455$	−3309.76	−0.341020
$456$	0	0
$457$	9124.25	0.933948	0.466974	−	0.884271i	$-0.345344\pi$
$457$	9124.25	0.933948	0.466974	+	0.884271i	$0.345344\pi$
$458$	7952.31	0.811326
$459$	0	0
$460$	460.000	0.0466252
$461$	−6011.92	−0.607382	−0.303691	−	0.952771i	$-0.598219\pi$
$461$	−6011.92	−0.607382	−0.303691	+	0.952771i	$0.598219\pi$
$462$	0	0
$463$	−8584.09	−0.861634	−0.430817	−	0.902439i	$-0.641775\pi$
$463$	−8584.09	−0.861634	−0.430817	+	0.902439i	$0.641775\pi$
$464$	−3862.86	−0.386484
$465$	0	0
$466$	4622.25	0.459488
$467$	−3954.09	−0.391806	−0.195903	−	0.980623i	$-0.562764\pi$
$467$	−3954.09	−0.391806	−0.195903	+	0.980623i	$0.562764\pi$
$468$	0	0
$469$	−9502.56	−0.935581
$470$	2055.91	0.201770
$471$	0	0
$472$	−3469.87	−0.338377
$473$	−6253.75	−0.607923
$474$	0	0
$475$	3549.99	0.342916
$476$	12223.0	1.17697
$477$	0	0
$478$	−1648.53	−0.157745
$479$	95.3377	0.00909413	0.00454707	−	0.999990i	$-0.498553\pi$
$479$	95.3377	0.00909413	0.00454707	+	0.999990i	$0.498553\pi$
$480$	0	0
$481$	−1351.14	−0.128080
$482$	7283.66	0.688302
$483$	0	0
$484$	502.127	0.0471570
$485$	5699.80	0.533638
$486$	0	0
$487$	13915.2	1.29478	0.647391	−	0.762158i	$-0.275860\pi$
$487$	13915.2	1.29478	0.647391	+	0.762158i	$0.275860\pi$
$488$	5287.22	0.490453
$489$	0	0
$490$	5195.02	0.478954
$491$	−6291.33	−0.578256	−0.289128	−	0.957290i	$-0.593365\pi$
$491$	−6291.33	−0.578256	−0.289128	+	0.957290i	$0.593365\pi$
$492$	0	0
$493$	−25120.3	−2.29486
$494$	−6401.24	−0.583007
$495$	0	0
$496$	1588.33	0.143786
$497$	−26233.1	−2.36764
$498$	0	0
$499$	−638.332	−0.0572658	−0.0286329	−	0.999590i	$-0.509115\pi$
$499$	−638.332	−0.0572658	−0.0286329	+	0.999590i	$0.509115\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	15535.0	1.38120
$503$	−15063.4	−1.33527	−0.667637	−	0.744487i	$-0.732694\pi$
$503$	−15063.4	−1.33527	−0.667637	+	0.744487i	$0.732694\pi$
$504$	0	0
$505$	8346.91	0.735510
$506$	1755.57	0.154238
$507$	0	0
$508$	−2566.83	−0.224182
$509$	13623.1	1.18631	0.593157	−	0.805087i	$-0.297881\pi$
$509$	13623.1	1.18631	0.593157	+	0.805087i	$0.297881\pi$
$510$	0	0
$511$	5764.23	0.499010
$512$	512.000	0.0441942
$513$	0	0
$514$	3002.33	0.257640
$515$	−5553.13	−0.475146
$516$	0	0
$517$	7846.27	0.667463
$518$	3520.99	0.298655
$519$	0	0
$520$	−901.584	−0.0760328
$521$	21659.4	1.82133	0.910666	−	0.413143i	$-0.135569\pi$
$521$	21659.4	1.82133	0.910666	+	0.413143i	$0.135569\pi$
$522$	0	0
$523$	−3813.85	−0.318868	−0.159434	−	0.987209i	$-0.550967\pi$
$523$	−3813.85	−0.318868	−0.159434	+	0.987209i	$0.550967\pi$
$524$	−9559.69	−0.796979
$525$	0	0
$526$	10862.0	0.900388
$527$	10329.0	0.853771
$528$	0	0
$529$	529.000	0.0434783
$530$	−4912.74	−0.402634
$531$	0	0
$532$	16681.2	1.35944
$533$	5617.95	0.456549
$534$	0	0
$535$	−195.946	−0.0158345
$536$	−2588.51	−0.208595
$537$	0	0
$538$	10028.4	0.803634
$539$	19826.6	1.58440
$540$	0	0
$541$	9727.32	0.773031	0.386516	−	0.922283i	$-0.373679\pi$
$541$	9727.32	0.773031	0.386516	+	0.922283i	$0.373679\pi$
$542$	14431.1	1.14367
$543$	0	0
$544$	3329.56	0.262415
$545$	4037.73	0.317353
$546$	0	0
$547$	782.647	0.0611766	0.0305883	−	0.999532i	$-0.490262\pi$
$547$	782.647	0.0611766	0.0305883	+	0.999532i	$0.490262\pi$
$548$	−3597.16	−0.280407
$549$	0	0
$550$	1908.23	0.147940
$551$	−34282.8	−2.65063
$552$	0	0
$553$	−14690.4	−1.12965
$554$	−14879.2	−1.14108
$555$	0	0
$556$	1237.36	0.0943811
$557$	−3472.87	−0.264184	−0.132092	−	0.991237i	$-0.542169\pi$
$557$	−3472.87	−0.264184	−0.132092	+	0.991237i	$0.542169\pi$
$558$	0	0
$559$	3693.40	0.279453
$560$	2349.47	0.177292
$561$	0	0
$562$	5211.18	0.391140
$563$	−7544.75	−0.564784	−0.282392	−	0.959299i	$-0.591128\pi$
$563$	−7544.75	−0.564784	−0.282392	+	0.959299i	$0.591128\pi$
$564$	0	0
$565$	−5332.89	−0.397091
$566$	−14325.1	−1.06383
$567$	0	0
$568$	−7145.94	−0.527882
$569$	−13222.5	−0.974193	−0.487096	−	0.873348i	$-0.661944\pi$
$569$	−13222.5	−0.974193	−0.487096	+	0.873348i	$0.661944\pi$
$570$	0	0
$571$	4069.49	0.298254	0.149127	−	0.988818i	$-0.452354\pi$
$571$	4069.49	0.298254	0.149127	+	0.988818i	$0.452354\pi$
$572$	−3440.85	−0.251520
$573$	0	0
$574$	−14640.0	−1.06457
$575$	575.000	0.0417029
$576$	0	0
$577$	846.627	0.0610841	0.0305421	−	0.999533i	$-0.490277\pi$
$577$	846.627	0.0610841	0.0305421	+	0.999533i	$0.490277\pi$
$578$	11826.3	0.851051
$579$	0	0
$580$	−4828.58	−0.345682
$581$	−23522.4	−1.67965
$582$	0	0
$583$	−18749.3	−1.33193
$584$	1570.19	0.111258
$585$	0	0
$586$	−18621.1	−1.31268
$587$	−4702.77	−0.330672	−0.165336	−	0.986237i	$-0.552871\pi$
$587$	−4702.77	−0.330672	−0.165336	+	0.986237i	$0.552871\pi$
$588$	0	0
$589$	14096.4	0.986133
$590$	−4337.34	−0.302653
$591$	0	0
$592$	959.123	0.0665874
$593$	14016.5	0.970640	0.485320	−	0.874337i	$-0.338703\pi$
$593$	14016.5	0.970640	0.485320	+	0.874337i	$0.338703\pi$
$594$	0	0
$595$	15278.7	1.05272
$596$	−10243.5	−0.704010
$597$	0	0
$598$	−1036.82	−0.0709010
$599$	−12885.0	−0.878910	−0.439455	−	0.898265i	$-0.644828\pi$
$599$	−12885.0	−0.878910	−0.439455	+	0.898265i	$0.644828\pi$
$600$	0	0
$601$	−4753.41	−0.322622	−0.161311	−	0.986904i	$-0.551572\pi$
$601$	−4753.41	−0.322622	−0.161311	+	0.986904i	$0.551572\pi$
$602$	−9624.77	−0.651622
$603$	0	0
$604$	−9852.61	−0.663737
$605$	627.659	0.0421785
$606$	0	0
$607$	−6800.06	−0.454705	−0.227352	−	0.973813i	$-0.573007\pi$
$607$	−6800.06	−0.454705	−0.227352	+	0.973813i	$0.573007\pi$
$608$	4543.99	0.303097
$609$	0	0
$610$	6609.02	0.438675
$611$	−4633.93	−0.306823
$612$	0	0
$613$	20842.3	1.37327	0.686634	−	0.727003i	$-0.259087\pi$
$613$	20842.3	1.37327	0.686634	+	0.727003i	$0.259087\pi$
$614$	6718.14	0.441567
$615$	0	0
$616$	8966.65	0.586488
$617$	−18917.7	−1.23436	−0.617179	−	0.786823i	$-0.711724\pi$
$617$	−18917.7	−1.23436	−0.617179	+	0.786823i	$0.711724\pi$
$618$	0	0
$619$	−28044.8	−1.82103	−0.910513	−	0.413480i	$-0.864313\pi$
$619$	−28044.8	−1.82103	−0.910513	+	0.413480i	$0.864313\pi$
$620$	1985.41	0.128607
$621$	0	0
$622$	11018.4	0.710285
$623$	21410.0	1.37684
$624$	0	0
$625$	625.000	0.0400000
$626$	4802.83	0.306645
$627$	0	0
$628$	−2266.88	−0.144042
$629$	6237.22	0.395380
$630$	0	0
$631$	−25796.2	−1.62746	−0.813732	−	0.581241i	$-0.802567\pi$
$631$	−25796.2	−1.62746	−0.813732	+	0.581241i	$0.802567\pi$
$632$	−4001.69	−0.251865
$633$	0	0
$634$	2114.71	0.132470
$635$	−3208.54	−0.200515
$636$	0	0
$637$	−11709.4	−0.728324
$638$	−18428.0	−1.14353
$639$	0	0
$640$	640.000	0.0395285
$641$	26482.3	1.63181	0.815903	−	0.578188i	$-0.196240\pi$
$641$	26482.3	1.63181	0.815903	+	0.578188i	$0.196240\pi$
$642$	0	0
$643$	30458.1	1.86804	0.934020	−	0.357219i	$-0.116275\pi$
$643$	30458.1	1.86804	0.934020	+	0.357219i	$0.116275\pi$
$644$	2701.89	0.165325
$645$	0	0
$646$	29549.8	1.79972
$647$	6746.24	0.409926	0.204963	−	0.978770i	$-0.434293\pi$
$647$	6746.24	0.409926	0.204963	+	0.978770i	$0.434293\pi$
$648$	0	0
$649$	−16553.3	−1.00119
$650$	−1126.98	−0.0680058
$651$	0	0
$652$	3843.61	0.230870
$653$	−12976.2	−0.777640	−0.388820	−	0.921314i	$-0.627117\pi$
$653$	−12976.2	−0.777640	−0.388820	+	0.921314i	$0.627117\pi$
$654$	0	0
$655$	−11949.6	−0.712840
$656$	−3987.97	−0.237354
$657$	0	0
$658$	12075.7	0.715442
$659$	6538.56	0.386504	0.193252	−	0.981149i	$-0.438097\pi$
$659$	6538.56	0.386504	0.193252	+	0.981149i	$0.438097\pi$
$660$	0	0
$661$	−25177.1	−1.48151	−0.740753	−	0.671777i	$-0.765531\pi$
$661$	−25177.1	−1.48151	−0.740753	+	0.671777i	$0.765531\pi$
$662$	−19360.4	−1.13665
$663$	0	0
$664$	−6407.55	−0.374490
$665$	20851.5	1.21592
$666$	0	0
$667$	−5552.86	−0.322350
$668$	16898.8	0.978794
$669$	0	0
$670$	−3235.64	−0.186573
$671$	25223.0	1.45115
$672$	0	0
$673$	18339.3	1.05041	0.525207	−	0.850975i	$-0.323988\pi$
$673$	18339.3	1.05041	0.525207	+	0.850975i	$0.323988\pi$
$674$	935.388	0.0534567
$675$	0	0
$676$	−6755.86	−0.384380
$677$	−31876.9	−1.80965	−0.904823	−	0.425789i	$-0.859997\pi$
$677$	−31876.9	−1.80965	−0.904823	+	0.425789i	$0.859997\pi$
$678$	0	0
$679$	33478.8	1.89219
$680$	4161.95	0.234711
$681$	0	0
$682$	7577.23	0.425436
$683$	28843.7	1.61592	0.807959	−	0.589239i	$-0.200572\pi$
$683$	28843.7	1.61592	0.807959	+	0.589239i	$0.200572\pi$
$684$	0	0
$685$	−4496.45	−0.250804
$686$	10367.2	0.576998
$687$	0	0
$688$	−2621.80	−0.145284
$689$	11073.1	0.612268
$690$	0	0
$691$	−18327.7	−1.00900	−0.504499	−	0.863413i	$-0.668323\pi$
$691$	−18327.7	−1.00900	−0.504499	+	0.863413i	$0.668323\pi$
$692$	−13795.9	−0.757865
$693$	0	0
$694$	−6743.09	−0.368825
$695$	1546.70	0.0844170
$696$	0	0
$697$	−25933.9	−1.40935
$698$	21917.5	1.18852
$699$	0	0
$700$	2936.84	0.158574
$701$	8329.48	0.448788	0.224394	−	0.974499i	$-0.427960\pi$
$701$	8329.48	0.448788	0.224394	+	0.974499i	$0.427960\pi$
$702$	0	0
$703$	8512.20	0.456677
$704$	2442.53	0.130762
$705$	0	0
$706$	3169.52	0.168961
$707$	49027.1	2.60800
$708$	0	0
$709$	−6169.85	−0.326818	−0.163409	−	0.986558i	$-0.552249\pi$
$709$	−6169.85	−0.326818	−0.163409	+	0.986558i	$0.552249\pi$
$710$	−8932.43	−0.472152
$711$	0	0
$712$	5832.12	0.306978
$713$	2283.22	0.119926
$714$	0	0
$715$	−4301.07	−0.224966
$716$	−5831.27	−0.304364
$717$	0	0
$718$	−10261.3	−0.533356
$719$	−28740.8	−1.49075	−0.745377	−	0.666644i	$-0.767730\pi$
$719$	−28740.8	−1.49075	−0.745377	+	0.666644i	$0.767730\pi$
$720$	0	0
$721$	−32617.3	−1.68479
$722$	26609.9	1.37163
$723$	0	0
$724$	−14539.9	−0.746368
$725$	−6035.72	−0.309188
$726$	0	0
$727$	11174.0	0.570041	0.285021	−	0.958521i	$-0.408000\pi$
$727$	11174.0	0.570041	0.285021	+	0.958521i	$0.408000\pi$
$728$	−5295.62	−0.269600
$729$	0	0
$730$	1962.73	0.0995123
$731$	−17049.7	−0.862663
$732$	0	0
$733$	−23560.7	−1.18722	−0.593612	−	0.804751i	$-0.702299\pi$
$733$	−23560.7	−1.18722	−0.593612	+	0.804751i	$0.702299\pi$
$734$	5622.54	0.282741
$735$	0	0
$736$	736.000	0.0368605
$737$	−12348.7	−0.617191
$738$	0	0
$739$	22713.6	1.13063	0.565313	−	0.824877i	$-0.308755\pi$
$739$	22713.6	1.13063	0.565313	+	0.824877i	$0.308755\pi$
$740$	1198.90	0.0595575
$741$	0	0
$742$	−28855.9	−1.42767
$743$	15882.7	0.784224	0.392112	−	0.919917i	$-0.371745\pi$
$743$	15882.7	0.784224	0.392112	+	0.919917i	$0.371745\pi$
$744$	0	0
$745$	−12804.4	−0.629685
$746$	−3904.84	−0.191644
$747$	0	0
$748$	15883.9	0.776433
$749$	−1150.92	−0.0561466
$750$	0	0
$751$	−30250.6	−1.46985	−0.734927	−	0.678146i	$-0.762784\pi$
$751$	−30250.6	−1.46985	−0.734927	+	0.678146i	$0.762784\pi$
$752$	3289.45	0.159513
$753$	0	0
$754$	10883.4	0.525664
$755$	−12315.8	−0.593664
$756$	0	0
$757$	−9611.05	−0.461453	−0.230726	−	0.973019i	$-0.574110\pi$
$757$	−9611.05	−0.461453	−0.230726	+	0.973019i	$0.574110\pi$
$758$	19218.5	0.920908
$759$	0	0
$760$	5679.99	0.271099
$761$	−5892.64	−0.280694	−0.140347	−	0.990102i	$-0.544822\pi$
$761$	−5892.64	−0.280694	−0.140347	+	0.990102i	$0.544822\pi$
$762$	0	0
$763$	23716.3	1.12528
$764$	−1795.25	−0.0850127
$765$	0	0
$766$	−10054.9	−0.474279
$767$	9776.19	0.460232
$768$	0	0
$769$	1683.72	0.0789553	0.0394777	−	0.999220i	$-0.487431\pi$
$769$	1683.72	0.0789553	0.0394777	+	0.999220i	$0.487431\pi$
$770$	11208.3	0.524571
$771$	0	0
$772$	17037.4	0.794286
$773$	−19880.8	−0.925048	−0.462524	−	0.886607i	$-0.653056\pi$
$773$	−19880.8	−0.925048	−0.462524	+	0.886607i	$0.653056\pi$
$774$	0	0
$775$	2481.76	0.115029
$776$	9119.68	0.421878
$777$	0	0
$778$	−11784.6	−0.543056
$779$	−35393.2	−1.62785
$780$	0	0
$781$	−34090.2	−1.56190
$782$	4786.24	0.218869
$783$	0	0
$784$	8312.04	0.378646
$785$	−2833.60	−0.128835
$786$	0	0
$787$	−7622.64	−0.345258	−0.172629	−	0.984987i	$-0.555226\pi$
$787$	−7622.64	−0.345258	−0.172629	+	0.984987i	$0.555226\pi$
$788$	7069.50	0.319594
$789$	0	0
$790$	−5002.11	−0.225275
$791$	−31323.7	−1.40802
$792$	0	0
$793$	−14896.5	−0.667074
$794$	−4908.82	−0.219405
$795$	0	0
$796$	4017.16	0.178875
$797$	22380.9	0.994693	0.497347	−	0.867552i	$-0.334308\pi$
$797$	22380.9	0.994693	0.497347	+	0.867552i	$0.334308\pi$
$798$	0	0
$799$	21391.4	0.947152
$800$	800.000	0.0353553
$801$	0	0
$802$	−18916.8	−0.832886
$803$	7490.67	0.329191
$804$	0	0
$805$	3377.36	0.147871
$806$	−4475.04	−0.195566
$807$	0	0
$808$	13355.1	0.581472
$809$	−8117.18	−0.352762	−0.176381	−	0.984322i	$-0.556439\pi$
$809$	−8117.18	−0.352762	−0.176381	+	0.984322i	$0.556439\pi$
$810$	0	0
$811$	7221.05	0.312658	0.156329	−	0.987705i	$-0.450034\pi$
$811$	7221.05	0.312658	0.156329	+	0.987705i	$0.450034\pi$
$812$	−28361.5	−1.22573
$813$	0	0
$814$	4575.56	0.197019
$815$	4804.51	0.206497
$816$	0	0
$817$	−23268.5	−0.996403
$818$	−11714.5	−0.500719
$819$	0	0
$820$	−4984.96	−0.212296
$821$	−34803.3	−1.47947	−0.739735	−	0.672899i	$-0.765049\pi$
$821$	−34803.3	−1.47947	−0.739735	+	0.672899i	$0.765049\pi$
$822$	0	0
$823$	490.883	0.0207911	0.0103956	−	0.999946i	$-0.496691\pi$
$823$	490.883	0.0207911	0.0103956	+	0.999946i	$0.496691\pi$
$824$	−8885.01	−0.375636
$825$	0	0
$826$	−25476.1	−1.07316
$827$	−30952.0	−1.30146	−0.650729	−	0.759310i	$-0.725537\pi$
$827$	−30952.0	−1.30146	−0.650729	+	0.759310i	$0.725537\pi$
$828$	0	0
$829$	8587.63	0.359784	0.179892	−	0.983686i	$-0.442425\pi$
$829$	8587.63	0.359784	0.179892	+	0.983686i	$0.442425\pi$
$830$	−8009.44	−0.334954
$831$	0	0
$832$	−1442.53	−0.0601092
$833$	54053.5	2.24831
$834$	0	0
$835$	21123.5	0.875460
$836$	21677.4	0.896806
$837$	0	0
$838$	10505.2	0.433051
$839$	−3077.94	−0.126653	−0.0633267	−	0.997993i	$-0.520171\pi$
$839$	−3077.94	−0.126653	−0.0633267	+	0.997993i	$0.520171\pi$
$840$	0	0
$841$	33898.9	1.38992
$842$	−407.035	−0.0166596
$843$	0	0
$844$	21580.7	0.880140
$845$	−8444.83	−0.343800
$846$	0	0
$847$	3686.67	0.149558
$848$	−7860.39	−0.318310
$849$	0	0
$850$	5202.43	0.209932
$851$	1378.74	0.0555377
$852$	0	0
$853$	−9193.14	−0.369012	−0.184506	−	0.982831i	$-0.559068\pi$
$853$	−9193.14	−0.369012	−0.184506	+	0.982831i	$0.559068\pi$
$854$	38819.3	1.55547
$855$	0	0
$856$	−313.513	−0.0125183
$857$	−21445.8	−0.854813	−0.427407	−	0.904059i	$-0.640573\pi$
$857$	−21445.8	−0.854813	−0.427407	+	0.904059i	$0.640573\pi$
$858$	0	0
$859$	−40276.7	−1.59979	−0.799897	−	0.600137i	$-0.795113\pi$
$859$	−40276.7	−1.59979	−0.799897	+	0.600137i	$0.795113\pi$
$860$	−3277.26	−0.129946
$861$	0	0
$862$	−6215.44	−0.245590
$863$	−43713.4	−1.72424	−0.862120	−	0.506703i	$-0.830864\pi$
$863$	−43713.4	−1.72424	−0.862120	+	0.506703i	$0.830864\pi$
$864$	0	0
$865$	−17244.9	−0.677855
$866$	−11426.9	−0.448386
$867$	0	0
$868$	11661.7	0.456017
$869$	−19090.3	−0.745218
$870$	0	0
$871$	7293.01	0.283713
$872$	6460.36	0.250889
$873$	0	0
$874$	6531.99	0.252801
$875$	3671.05	0.141833
$876$	0	0
$877$	24740.5	0.952596	0.476298	−	0.879284i	$-0.341978\pi$
$877$	24740.5	0.952596	0.476298	+	0.879284i	$0.341978\pi$
$878$	−9174.27	−0.352639
$879$	0	0
$880$	3053.16	0.116957
$881$	44027.2	1.68367	0.841835	−	0.539735i	$-0.181476\pi$
$881$	44027.2	1.68367	0.841835	+	0.539735i	$0.181476\pi$
$882$	0	0
$883$	30245.0	1.15269	0.576344	−	0.817207i	$-0.304479\pi$
$883$	30245.0	1.15269	0.576344	+	0.817207i	$0.304479\pi$
$884$	−9380.87	−0.356915
$885$	0	0
$886$	−10536.0	−0.399509
$887$	7317.82	0.277010	0.138505	−	0.990362i	$-0.455770\pi$
$887$	7317.82	0.277010	0.138505	+	0.990362i	$0.455770\pi$
$888$	0	0
$889$	−18845.9	−0.710991
$890$	7290.16	0.274569
$891$	0	0
$892$	−6019.17	−0.225938
$893$	29193.8	1.09399
$894$	0	0
$895$	−7289.08	−0.272232
$896$	3759.15	0.140161
$897$	0	0
$898$	−33733.7	−1.25357
$899$	−23966.8	−0.889140
$900$	0	0
$901$	−51116.5	−1.89005
$902$	−19024.9	−0.702282
$903$	0	0
$904$	−8532.63	−0.313928
$905$	−18174.8	−0.667572
$906$	0	0
$907$	−35436.0	−1.29728	−0.648640	−	0.761096i	$-0.724662\pi$
$907$	−35436.0	−1.29728	−0.648640	+	0.761096i	$0.724662\pi$
$908$	7117.75	0.260144
$909$	0	0
$910$	−6619.52	−0.241137
$911$	11359.1	0.413112	0.206556	−	0.978435i	$-0.433775\pi$
$911$	11359.1	0.413112	0.206556	+	0.978435i	$0.433775\pi$
$912$	0	0
$913$	−30567.7	−1.10804
$914$	18248.5	0.660401
$915$	0	0
$916$	15904.6	0.573694
$917$	−70188.2	−2.52761
$918$	0	0
$919$	40517.5	1.45435	0.727175	−	0.686452i	$-0.240833\pi$
$919$	40517.5	1.45435	0.727175	+	0.686452i	$0.240833\pi$
$920$	920.000	0.0329690
$921$	0	0
$922$	−12023.8	−0.429484
$923$	20133.3	0.717982
$924$	0	0
$925$	1498.63	0.0532699
$926$	−17168.2	−0.609267
$927$	0	0
$928$	−7725.72	−0.273286
$929$	−16242.2	−0.573614	−0.286807	−	0.957988i	$-0.592594\pi$
$929$	−16242.2	−0.573614	−0.286807	+	0.957988i	$0.592594\pi$
$930$	0	0
$931$	73769.2	2.59687
$932$	9244.50	0.324907
$933$	0	0
$934$	−7908.19	−0.277049
$935$	19854.9	0.694463
$936$	0	0
$937$	−47445.0	−1.65417	−0.827087	−	0.562074i	$-0.810004\pi$
$937$	−47445.0	−1.65417	−0.827087	+	0.562074i	$0.810004\pi$
$938$	−19005.1	−0.661556
$939$	0	0
$940$	4111.81	0.142673
$941$	48063.5	1.66506	0.832532	−	0.553976i	$-0.186890\pi$
$941$	48063.5	1.66506	0.832532	+	0.553976i	$0.186890\pi$
$942$	0	0
$943$	−5732.70	−0.197967
$944$	−6939.74	−0.239268
$945$	0	0
$946$	−12507.5	−0.429867
$947$	−13745.4	−0.471663	−0.235831	−	0.971794i	$-0.575781\pi$
$947$	−13745.4	−0.471663	−0.235831	+	0.971794i	$0.575781\pi$
$948$	0	0
$949$	−4423.92	−0.151324
$950$	7099.99	0.242478
$951$	0	0
$952$	24445.9	0.832245
$953$	13999.1	0.475839	0.237920	−	0.971285i	$-0.423535\pi$
$953$	13999.1	0.475839	0.237920	+	0.971285i	$0.423535\pi$
$954$	0	0
$955$	−2244.06	−0.0760377
$956$	−3297.07	−0.111543
$957$	0	0
$958$	190.675	0.00643052
$959$	−26410.7	−0.889308
$960$	0	0
$961$	−19936.4	−0.669207
$962$	−2702.28	−0.0905666
$963$	0	0
$964$	14567.3	0.486703
$965$	21296.7	0.710431
$966$	0	0
$967$	−800.187	−0.0266104	−0.0133052	−	0.999911i	$-0.504235\pi$
$967$	−800.187	−0.0266104	−0.0133052	+	0.999911i	$0.504235\pi$
$968$	1004.25	0.0333450
$969$	0	0
$970$	11399.6	0.377339
$971$	11363.5	0.375563	0.187782	−	0.982211i	$-0.439870\pi$
$971$	11363.5	0.375563	0.187782	+	0.982211i	$0.439870\pi$
$972$	0	0
$973$	9084.84	0.299328
$974$	27830.4	0.915549
$975$	0	0
$976$	10574.4	0.346803
$977$	17444.0	0.571220	0.285610	−	0.958346i	$-0.407804\pi$
$977$	17444.0	0.571220	0.285610	+	0.958346i	$0.407804\pi$
$978$	0	0
$979$	27822.5	0.908286
$980$	10390.0	0.338671
$981$	0	0
$982$	−12582.7	−0.408889
$983$	54032.7	1.75318	0.876590	−	0.481237i	$-0.159812\pi$
$983$	54032.7	1.75318	0.876590	+	0.481237i	$0.159812\pi$
$984$	0	0
$985$	8836.87	0.285854
$986$	−50240.7	−1.62271
$987$	0	0
$988$	−12802.5	−0.412248
$989$	−3768.84	−0.121175
$990$	0	0
$991$	28869.3	0.925392	0.462696	−	0.886517i	$-0.346882\pi$
$991$	28869.3	0.925392	0.462696	+	0.886517i	$0.346882\pi$
$992$	3176.66	0.101672
$993$	0	0
$994$	−52466.2	−1.67417
$995$	5021.45	0.159991
$996$	0	0
$997$	3358.78	0.106694	0.0533469	−	0.998576i	$-0.483011\pi$
$997$	3358.78	0.106694	0.0533469	+	0.998576i	$0.483011\pi$
$998$	−1276.66	−0.0404931
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.4.a.bj.1.4		4
3.2	odd	2		230.4.a.h.1.2	✓	4
12.11	even	2		1840.4.a.m.1.3		4
15.2	even	4		1150.4.b.n.599.3		8
15.8	even	4		1150.4.b.n.599.6		8
15.14	odd	2		1150.4.a.p.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.h.1.2	✓	4	3.2	odd	2
1150.4.a.p.1.3		4	15.14	odd	2
1150.4.b.n.599.3		8	15.2	even	4
1150.4.b.n.599.6		8	15.8	even	4
1840.4.a.m.1.3		4	12.11	even	2
2070.4.a.bj.1.4		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)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +29.3684 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} +29.3684 q^{7} +8.00000 q^{8} +10.0000 q^{10} +38.1645 q^{11} -22.5396 q^{13} +58.7368 q^{14} +16.0000 q^{16} +104.049 q^{17} +142.000 q^{19} +20.0000 q^{20} +76.3291 q^{22} +23.0000 q^{23} +25.0000 q^{25} -45.0792 q^{26} +117.474 q^{28} -241.429 q^{29} +99.2706 q^{31} +32.0000 q^{32} +208.097 q^{34} +146.842 q^{35} +59.9452 q^{37} +284.000 q^{38} +40.0000 q^{40} -249.248 q^{41} -163.863 q^{43} +152.658 q^{44} +46.0000 q^{46} +205.591 q^{47} +519.502 q^{49} +50.0000 q^{50} -90.1584 q^{52} -491.274 q^{53} +190.823 q^{55} +234.947 q^{56} -482.858 q^{58} -433.734 q^{59} +660.902 q^{61} +198.541 q^{62} +64.0000 q^{64} -112.698 q^{65} -323.564 q^{67} +416.195 q^{68} +293.684 q^{70} -893.243 q^{71} +196.273 q^{73} +119.890 q^{74} +567.999 q^{76} +1120.83 q^{77} -500.211 q^{79} +80.0000 q^{80} -498.496 q^{82} -800.944 q^{83} +520.243 q^{85} -327.726 q^{86} +305.316 q^{88} +729.016 q^{89} -661.952 q^{91} +92.0000 q^{92} +411.181 q^{94} +709.999 q^{95} +1139.96 q^{97} +1039.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} - q^{7} + 32 q^{8}+O(q^{10}) \) 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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